APPROVAL SHEET Title of Thesis: Vibration-based Damage Detection of Structures Name of Candidate: Benjamin H. Emory Doctor of Philosophy, 2010 Thesis and Abstract Approved: Date Approved: Dr. Weidong Zhu Professor Department of Mechanical Engineering Curriculum Vitae Name: Benjamin H. Emory Degree and date to be conferred: Doctor of Philosophy, July 2010 Secondary Education: Leonardtown High School, Leonardtown, MD Collegiate institutions attended: University of Maryland, Baltimore County, B.S., Mechanical Engineering, 2003 University of Maryland, Baltimore County, M.S., Mechanical Engineering, 2005 University of Maryland, Baltimore County, Ph.D., Mechanical Engineering, 2010 Major: Mechanical Engineering Minor: N/A Professional publications: Emory, B.H., Zhu, W.D., and Kaczmarczyk, S. (2009), Modal Testing and Modeling of a Simplied Elevator System, Proceedings of the IMAC-XXVII: A Conference & Exposition on Structural Dynamics. Emory, B.H., and Zhu, W.D. (2008), Development of a Summer Educational Program in Dynamic Systems, Proceedings of the IMAC-XXVI: A Conference & Exposition on Structural Dynamics. Xu G.Y. , Zhu W.D., and Emory B.H., (2007), Experimental and Numerical Investigation of Structural Damage Detection Using Changes in Natural Frequencies, ASME Journal of Vibration and Acoustics, Vol. 129, pp. 686-700. Emory, B.H., and Zhu, W.D. (2006), Experimental Modal Analysis of Rectangular and Circular Beams. Journal of STEM Education: Innovation and Research, 3&4, http://www.auburn.edu/research/litee/ jstem/include/getdoc.php?id=829&article=259&mode=pdf, accessed 4/4/07. Zhu, W.D., and Emory, B.H. (2006), The Future Engineers in Dynamic Systems Academy. Symposium on Multi-/Inter- Disciplinary Engineering Education, East China University of Science and Technology, Shanghai, China, August 16-20. Received a Bayer Teaching Excellence Award. Emory, B. H. (2005). Experimental Modal Analysis of Rectangular and Circular Beams, Masters Thesis, University of Maryland, Baltimore County, Department of Mechanical Engineering. Zhu, W.D. and Emory, B.H. (2005), On a Simple Impact Test Method for Accurate Measurement of Material Properties. Journal of Sound and Vibration, 287(3), pp. 637-643. Zhu, W.D., Xu, G.Y., He, K., and Emory, B.H. (2005), Vibration-Based Structural Damage Detection: Theory and Experiments. NSF-sponsored US-China Workshop on Smart Structures and Smart Systems (WSSSS2005), Oct. 16-18. Emory, B. H., Zhu, W.D. (2003), Experimental Modal Analysis of Rectangular and Circular Beams. ASME International Mechanical Engineering Congress and RD&D Expo. Professional positions held: National Air and Space Administration, Goddard Space Flight Center, Mechanical Systems Analysis and Simulation Branch, Structural Dynamics Group, Aerospace Engineering Co-op Trainee, January 2009-Current The University of Northampton, School of Applied Sciences, Lift Technology and High Performance Engineering Group, Visiting Ph.D. Research Assistant, March-June 2008. ABSTRACT Title of Thesis: Vibration-based Damage Detection of Structures Benjamin H. Emory, Doctor of Philosophy, 2010 Thesis directed by: Dr. Weidong Zhu Professor Department of Mechanical Engineering Vibration-based damage detection methods are of great interest because of their global nature as compared to standard localized non-destructive methods which are time consuming for large structures. In this work, vibration-based damage detection procedures are developed around a robust iterative algorithm that has been expanded to include two new optimization procedures, an improved computer program, and procedures for feature selection on structures with or without tension. The previous implementation of the iterative algorithm was successfully applied to beams and lightning masts using a few lower natural frequencies. The new implementation was successful at detecting stiness changes in rectangular beams with and without tension, circular beams, and wire ropes modeled with beam elements. Detection of mass increase in a pipeline system using solid and shell elements is also shown. The new computer implementation allows for easy use of external models by computing the sensitivity matrices using the nite dierence method. Including mass modication, new methods to overcome symmetry using length modication and targeted investigation are presented. The importance of step length is shown when the method fails because full step length is used. The importance of choosing a detection parameter is demonstrated when some the natural frequencies of a circular bar and a tensioned beam system increase and some decrease. Currently, there are two main methods used to perform modal testing: impact testing or shaker testing. The random impact method was successfully used on lightning masts to overcome external noise due to wind and on heavily damped elevator ropes. The main drawback is that for large structures, a heavy hammer has to be swung for 2-6 minutes for each test which is tiring. To overcome problems associated with the random impact method, a mechanical system was designed, built, and tested to implement the method. In the current research funding process educational outreach for the benet of society is often required. Demonstrations of how this research and that of the Dynamic Systems and Vibrations Laboratory benets society the development, delivery, and evaluation of an educational summer camp for high school students over six summers is presented. VIBRATION-BASED DAMAGE DETECTION OF STRUCTURES by Benjamin H. Emory Thesis submitted to the Faculty of the Graduate School of the University of Maryland in partial fulllment of the requirements for the degree of Doctor of Philosophy 2010 UMI Number: 3422804 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3422804 Copyright 2010 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 Е Copyright Benjamin H. Emory, 2010 To God for my strength. My parents Martin and Barbara, and my siblings Michael and Carla for their love and support. ii ACKNOWLEDGMENTS I would rst like to thank my family, friends, and co-workers for supporting me during my studies. I would like to say a special thank you to my father for the use of his talent, tools, time, and donation of materials for building many of my experiments. I would like to thank Dr. Weidong Zhu for being my adviser and my committee members Drs. Spence, Nichols, Zupan, and Carmi for their advice and time. I would also like to thank Dr. Kazcmarczyk who advised me during my semester at the University of Northampton. During my studies at the Dynamic Systems and Vibrations Laboratory, I had the fortune of collaborating with my labmates (in the order I met them): Drs. Chun Nam Wong, Yan Chen, Guangyao Xu, and Negan Zheng; Pratik Desai, Homayoun Dahaghin, Kun He, Hui Ren, Yi Chen, Yongfeng Xu, and Chuang Xiao. Without support from the technicians John Gottschalk, Victor Fulda, and Victor Grubbs at UMBC and my Uncle Charles Manny Wallace from Hollywood, building much of the experimental apparatus would have been very dicult. I would like to acknowledge Edward Bob Diedrich and Theodore Ted Green from Baltimore Gas and Electric for helping me get access to BG&E's electrical substations for testing. I would nally like to thank those who helped me proofread my thesis; Jane Porter, Doug Howle, Kathie Mitchell, Kun He, Ryan Simmons, and Amber Reynolds. During my studies I was funded by BG&E, the Graduate Assistance in Areas of National Need (GAANN) fellowship from the Department of Education, and through my co-op position at NASA Goddard Space Flight Center. Below is a more complete listing of the funding much of this work was performed under as well as those who played a key part in the specied project. Damage Detection: This work was supported by the Maryland Technology Development Corporation, the Baltimore Gas and Electric Company, and the National iii Science Foundation through Award CMS-0600559. Elevator Testing: This work was supported by the National Science Foundation through Award CMS-0600559 and its International Research and Education in Engineering (IREE) supplement, U.S. Department of Education Graduate Assistance in Areas of National Need (GAANN) fellowship, Dr. Stefan Kazcmarczyk and the University of Northampton Lift Engineering Group, and ThyssenKrupp AG who donated the aramid rope. Hammer Device: This work has been supported by the Maryland Technology Development Corporation, the Baltimore Gas and Electric Company, and the National Science Foundation through Award CMS-0600559, and Pratt and Whitney. I would like to thank Barry Gleit from G&G Technical, Inc. for helping me with the Copley Controls device, as well as the mechanical engineering senior design students who I supervised during their various senior projects related to the device. Engineering Education: This work was supported by the CAREER Award CMS0348605 and Award CMS-0600559 from the National Science Foundation. I would like to thank the Shriver Center sta members, especially Michele Wol (Director of the Shriver Center), Miryn Alcantara, Lori Hardesty, and Kerry Kidwell-Slack for their support of the program. I would also like to thank Dr. Yan Chen, Dr. Guangyao Xu, Pratik Desai, Kun He, Samuel Bronson; undergraduate students including Doug Howle, Jonathan Pandolni, Eric Nakamura, Yan-Jin Zhu, Amy Mensch, John Wetzel, Alan Harris, Andrew Fenner, James McFall, Jordan Ragos, and Don Mella for their assistance with the program; and Dr. Anne Spence for the use of her laboratory at UMBC for some of the activities. List of Figures 2.6.1 Circular segment properties. . . . . . . . . . . . . . . . . . . . . . . 46 2.6.2 Circular segmental section properties. . . . . . . . . . . . . . . . . . 46 2.7.1 Damage detection process owchart. . . . . . . . . . . . . . . . . . . 56 3.1.1 Healthy beam setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.2 Damage beam with tip masses. 67 . . . . . . . . . . . . . . . . . . . . 3.1.3 Healthy damage detection using the elastic modulus without convexication using ve natural frequencies. . . . . . . . . . . . . . . . . 69 3.1.4 Healthy damage detection using the elastic modulus with convexication using ve natural frequencies. . . . . . . . . . . . . . . . . . 70 3.1.5 Healthy damage detection using the thickness without convexication using ve natural frequencies. . . . . . . . . . . . . . . . . . . . . . 71 3.1.6 Healthy damage detection using the thickness with convexication using ve natural frequencies. . . . . . . . . . . . . . . . . . . . . . 72 3.1.7 Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. . . . . . . 73 3.1.8 Scenario 2 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . . . . . . v 74 3.1.9 Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . . . . . . . 75 3.1.10 Scenario 2 damage detection results using three to ve natural frequencies and the thickness with convexication. . . . . . . . . . . . 76 3.1.11 Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. . . . . . . 77 3.1.12 Scenario 3 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . . . . . . 78 3.1.13 Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . . . . . . . 79 3.1.14 Scenario 3 damage detection results using three to ve natural frequencies and the thickness with convexication. . . . . . . . . . . . 80 3.1.15 Scenario 4 damage detection results using four to six natural frequencies per beam and the elastic modulus without convexication. . . . 81 3.1.16 Scenario 4 damage detection results using seven and eight natural frequencies total and the thickness without convexication. . . . . . 82 3.1.17 Scenario 4 damage detection results using nine natural frequencies, six singular values, the thickness without convexication last, and the last three elements not updated. . . . . . . . . . . . . . . . . . . 83 3.1.18 Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. . . . . . . 84 3.1.19 Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . . . . . . . 85 3.1.20 Scenario 5 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . . . . . . 86 3.1.21 Scenario 5 damage detection results using three to ve natural frequencies and the thickness with convexication. . . . . . . . . . . . 87 3.1.22 Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 91 3.1.23 Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . 92 3.1.24 Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . 93 3.1.25 Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the thickness with convexication. . . . . . 94 3.1.26 Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 95 3.1.27 Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . 96 3.1.28 Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . 97 3.1.29 Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the thickness with convexication. . . . . . 98 3.1.30 Simulation of Scenario 4 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 99 3.1.31 Simulation of Scenario 4 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . 100 3.1.32 Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 101 3.1.33 Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. . . 102 3.1.34 Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . 103 3.1.35 Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the thickness without convexication. . . . . 104 3.2.1 Test setup for the circular beam with a machined damage. . . . . . 106 3.2.2 FRF's of the bending and torsional excitation tests. . . . . . . . . . 107 3.2.3 Changes in circular section properties as a function of upper radius. 112 3.2.4 Fixed-free rod simulated damage detection using thickness without convexication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.2.5 Fixed-free rod simulated damage detection using thickness without convexication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.2.6 Fixed-free rod simulated damage detection using thickness without convexication and recomputed derivatives at each step. . . . . . . 115 3.2.7 Fixed-free rod simulated damage detection using thickness using the Levenberg-Marquardt method. . . . . . . . . . . . . . . . . . . . . . 116 3.2.8 Fixed-free rod simulated damage detection using thickness using the Levenberg-Marquardt method. . . . . . . . . . . . . . . . . . . . . . 117 3.2.9 Fixed-free rod experimental damage detection using thickness without convexication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2.10 Fixed-free rod experimental damage detection using thickness without convexication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2.11 Fixed-free rod experimental damage detection using the LevenbergMarquardt method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.2.12 Fixed-free rod experimental damage detection using the LevenbergMarquardt method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.1 Two section lightning mast. . . . . . . . . . . . . . . . . . . . . . . 131 4.1.2 50' lightning mast damage detection results showing the dimensionless stiness (elastic modulus) vs. the height. . . . . . . . . . . . . . 132 4.1.3 Scale model of the 50' lightning mast with damage. . . . . . . . . . 133 4.1.4 Damage detection results for the scale mast showing the dimensionless stiness (elastic modulus) vs. the height. . . . . . . . . . . . . . 134 4.2.1 Free-free pipe system setup with mass added. . . . . . . . . . . . . . 139 4.2.2 Experimental results using 4-7 natural frequencies showing the dimensionless density vs. the group number where Group 0 is the free end of the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.2.3 Simulation results using 4-7 natural frequencies showing the dimensionless density vs. the group number where Group 0 is the free end of the pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2.1 Dimensions of the ropeelevator car system, L1 and L2 are the inclined and vertical rope lengths respectively, M is the mass of the car, and Ms is the apparent mass of the sheave. . . . . . . . . . . . 149 5.2.2 Side and front views of the rope system. . . . . . . . . . . . . . . . 150 5.2.3 Inclined rope termination. . . . . . . . . . . . . . . . . . . . . . . . 151 5.2.4 Capacitance accelerometer attached to the sheave. . . . . . . . . . . 152 5.2.5 CAD model of the sheave. . . . . . . . . . . . . . . . . . . . . . . . 153 5.2.6 Rope termination with strain gage. . . . . . . . . . . . . . . . . . . 154 5.2.7 CAD model of car-mass shaker system. . . . . . . . . . . . . . . . . 155 5.3.1 FRF vs. input voltage. . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.3.2 Clamped vs. un-clamped FRF's 0-180 Hz. . . . . . . . . . . . . . . 158 5.3.3 Clamped vs. un-clamped FRF's 0-2000 Hz. . . . . . . . . . . . . . . 159 5.3.4 Eect of shaker armature stiness on the rst longitudinal mode. . . 160 5.3.5 FFT of pendulum mode excitation. . . . . . . . . . . . . . . . . . . 161 5.3.6 FFT of roll mode excitation. . . . . . . . . . . . . . . . . . . . . . . 162 5.3.7 FFT of torsional mode excitation. . . . . . . . . . . . . . . . . . . . 163 5.3.8 FFT of yaw mode excitation. . . . . . . . . . . . . . . . . . . . . . . 164 5.4.1 Longitudinal vibration model. . . . . . . . . . . . . . . . . . . . . . 170 5.4.2 Normalized frequency vs. car mass sensitivity . . . . . . . . . . . . . 171 5.4.3 Normalized frequency vs. EA. . . . . . . . . . . . . . . . . . . . . . 172 5.4.4 Normalized frequency vs. apparent eective sheave mass sensitivity . 173 5.4.5 Normalized frequency vs. inclined rope length. . . . . . . . . . . . . 174 5.4.6 Normalized frequency vs. vertical rope length. . . . . . . . . . . . . 175 5.5.1 Bending vibration sensitivity of the vertical rope. . . . . . . . . . . 185 5.5.2 Bending vibration sensitivity of the inclined rope. . . . . . . . . . . 186 5.5.3 Bending vibration sensitivity to EI. . . . . . . . . . . . . . . . . . . 187 5.5.4 Bending vibration sensitivity to car mass. . . . . . . . . . . . . . . . 188 5.5.5 First mode shape of the inclined rope. . . . . . . . . . . . . . . . . . 189 5.5.6 Second mode shape of the inclined rope. . . . . . . . . . . . . . . . 190 5.5.7 Third mode shape of the inclined rope. . . . . . . . . . . . . . . . . 191 5.5.8 First mode shape of the vertical rope. . . . . . . . . . . . . . . . . . 192 5.5.9 Second mode shape of the vertical rope. . . . . . . . . . . . . . . . . 193 5.5.10 Third mode shape of the vertical rope. . . . . . . . . . . . . . . . . 194 5.5.11 Undeformed nite element model. . . . . . . . . . . . . . . . . . . . 196 5.5.12 First modes of the inclined and vertical ropes respectively. . . . . . 197 5.5.13 Second and third modes of the inclined rope respectively. . . . . . . 198 5.5.14 Second and fourth modes of the vertical and inclined ropes respectively.199 5.5.15 Fifth and third modes of the inclined and vertical ropes respectively. 200 6.1.1 Simplied model of the scaled elevator system shown in Chapter 6. . 207 6.1.2 Condition number at each iteration with a step size of 1.0. . . . . . 208 6.1.3 Damage detection results showing the percent reduction in elastic modulus vs. length with a step size of 0.1. . . . . . . . . . . . . . . 209 6.1.4 Condition number at each iteration with a step size of 0.1. . . . . . 210 6.1.5 Damage detection results showing the percent reduction in elastic modulus vs. length with a step size of 0.1. . . . . . . . . . . . . . . 211 6.1.6 Comparison of step sizes 1 in orange and 0.1 in blue with the error in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.1.7 Damage detection using the elastic modulus for a stiness increase of 20%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.2.1 Experimental setup overview. . . . . . . . . . . . . . . . . . . . . . 219 6.2.2 Adjustable end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.2.3 Characteristic equation plot for the beam with full thickness and half thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2.4 Characteristic equation plot for the beam with full tension and reduced tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.2.5 Close up of the damage section of the beam. . . . . . . . . . . . . . 225 6.2.6 Healthy detection results showing the dimensionless thickness vs. group number using four natural frequencies for each beam. . . . . . 228 6.2.7 Damage detection showing the dimensionless thickness vs. group number using three natural frequencies for each beam. . . . . . . . . 229 6.2.8 Damage detection showing the dimensionless thickness vs. group number using four natural frequencies for each beam. . . . . . . . . 230 6.2.9 Simulated damage detection showing the dimensionless thickness vs. group number using three natural frequencies for each beam. . . . . 231 6.2.10 Simulated damage detection showing the dimensionless thickness vs. group number using four natural frequencies for each beam. . . . . . 232 6.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.3.2 FRF of the longitudinal vibration of the rope. . . . . . . . . . . . . 239 6.3.3 Rope with damaged bers. . . . . . . . . . . . . . . . . . . . . . . . 241 6.3.4 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (A). . . . . . . . . . . . . . . . . 243 6.3.5 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (B). . . . . . . . . . . . . . . . . 244 6.3.6 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (C). . . . . . . . . . . . . . . . . 245 6.3.7 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 3m. . . . . . . . . . . . . . . . . . . . 246 6.3.8 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m and 3m (A). . . . . . . . . . . . . 247 6.3.9 Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m and 3m (B). . . . . . . . . . . . . 248 6.3.10 Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 3m. . . . . . . . . . 249 6.3.11 Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 1m. . . . . . . . . . 250 6.3.12 Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 1m and 3m. . . . . . 251 6.3.13 Standard deviation and average of the damage detection results from gures 6.3.4 through 6.3.12. . . . . . . . . . . . . . . . . . . . . . . 252 6.3.14 Simulated damage detection for the rope with 50% damage at Group 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.2.1 Average expected force for dierent combinations of distributions. . 262 7.3.1 70' lightning mast. . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.3.2 Single coherence (top) and FRF (bottom) from tests on an aluminum plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.3.3 Single coherence (top) and FRF (bottom) from tests on an aluminum plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.3.4 The Modal Shop impact device. . . . . . . . . . . . . . . . . . . . . 270 7.4.1 Impact device positioned to impact the aluminum plate. . . . . . . . 275 7.4.2 Multiple impact series from engineer 1. . . . . . . . . . . . . . . . . 276 7.4.3 Test engineer's multiple impact test measurements. . . . . . . . . . 277 7.4.4 Average and standard deviations of test engineer's impulse shape. . 278 7.4.5 Distributions of seconds between impacts for several test engineers. 279 7.4.6 Distribution of impact forces for several test engineers. . . . . . . . 280 7.4.7 Power spectral density of impact forces for several test engineers. . . 281 7.4.8 Impacts per 1 second time interval by test engineer 1. . . . . . . . . 282 7.4.9 Number of groups with the same number of impacts per time interval by test engineer 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.4.10 Measured and expected probability mass functions for test engineer 1. 284 7.4.11 Impact signal generated by the device. . . . . . . . . . . . . . . . . 285 7.4.12 Average and standard deviations of device impulse shape. . . . . . . 286 7.4.13 Distribution of impact forces from the device. . . . . . . . . . . . . 287 7.4.14 Distribution of seconds between impacts from the device. . . . . . . 288 7.4.15 Power spectral density of the impact forces from the device. . . . . . 289 7.5.1 Pipe setup showing impact location and noise impact locations. . . 293 7.5.2 Single impact by test engineer 1 with no noise. . . . . . . . . . . . . 294 7.5.3 Single impact test by test engineer 1 with noise at location 1. . . . . 295 7.5.4 Single impact by test engineer 1 with noise at location 2. . . . . . . 296 7.5.5 Multiple impact test by test engineer 1 with no noise. . . . . . . . . 297 7.5.6 Multiple impact by test engineer 1 with noise at location 1. . . . . . 298 7.5.7 Multiple impact by test engineer 1 with noise at location 2. . . . . . 299 7.5.8 Device multiple impact with no noise input. . . . . . . . . . . . . . 300 7.5.9 Device multiple impact FRF and PSD with noise at location 1. . . . 301 7.5.10 Device multiple impact FRF and PSD with noise at location 2. . . . 302 7.5.11 Device burst impact test with no noise. . . . . . . . . . . . . . . . 303 7.5.12 Device burst impact test with noise at location 1. . . . . . . . . . . 304 7.5.13 Device burst impact test with noise at location 2. . . . . . . . . . . 305 7.6.1 Overview of device setup for lightning mast testing. . . . . . . . . 308 7.6.2 Device impact and accelerometer locations for lightning mast testing . 309 7.6.3 Manual single impact test FRF and coherence. . . . . . . . . . . . . 310 7.6.4 Manual multiple impact test FRF and coherence. . . . . . . . . . . 311 7.6.5 Device multiple impact test FRF and coherence. . . . . . . . . . . . 312 7.6.6 Device burst impact test FRF and coherence. . . . . . . . . . . . . 313 8.2.1 FEDS Academy Logo . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.3.1 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.3.2 Acceleration response (top) of the pendulum and its FFT (bottom). 8.3.3 Students testing the strings of a guitar in 2006. 335 . . . . . . . . . . . 336 8.3.4 Voice maps of four speakers . . . . . . . . . . . . . . . . . . . . . . 337 8.3.5 Students learning about the scaled elevator in the DSVL in 2005. . 339 8.3.6 Measured acceleration (a) and calculated velocity (b) and displacement (c) of the scaled elevator. . . . . . . . . . . . . . . . . . . . . . 340 8.3.7 Lego project and video of a Lego robot designed by a group of students in 2007. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 8.3.8 A student testing the beam with a uid damper. . . . . . . . . . . . 342 8.3.9 Damaged beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8.3.10 Damage detection GUI and output. . . . . . . . . . . . . . . . . . . 344 8.4.1 Students performing a random impact test on a light pole in 2005. . 350 8.4.2 A student testing a cymbal with a pen-sized hammer. . . . . . . . . 351 8.4.3 A mode shape of the cymbal from I-DEAS. . . . . . . . . . . . . . . 352 8.4.4 Mode shapes and their animation videos of the top plate of a guitar measured by a group of students in 2006. . . . . . . . . . . . . . . . 353 8.4.5 A student performing an impact test on a softball bat in 2005. . . 354 8.4.6 First measured bending mode of a wooden bat with the animation video. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.4.7 First computed mode shape of the wooden bat with the animation video. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 8.4.8 Mbira with a capacitance probe. . . . . . . . . . . . . . . . . . . . . 357 8.4.9 A measured mode shape of the top plate with its animation video. 358 8.4.10 First computed mode shape of the middle key. . . . . . . . . . . . . 359 8.4.11 Bamboo bridge with accelerometers attached. . . . . . . . . . . . . 360 8.4.12 Finite element model of the bridge in Figure 22. . . . . . . . . . . . 361 8.4.13 Rotordynamics test stand and the associated videos. . . . . . . . . . 362 8.4.14 String mode shape experiment and the associated videos. . . . . . . 363 8.4.15 A simple tower on a shake table. . . . . . . . . . . . . . . . . . . . . 364 8.6.1 Pre-survey results from 2005-2009 on the numbers of students who had taken (a), or would take in the upcoming year (b), various courses.389 8.6.2 Pre-survey results on students' interest in: engineering; mechanical engineering; and dynamics, vibration, and control (Dynamics). The four bars, from left to right, in each subject area correspond to 2005 to 2009. The number of students who answered the question on Dynamics is much fewer in 2008 than those in the previous years because the students who responded with a neutral choice were not counted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 8.6.3 Vibration measurement classroom (a) and laboratory (b) survey results in 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 8.6.4 Pendulum classroom (a) and laboratory (b) survey results. The three bars, from left to right, for each question correspond to 2006 to 2008. 392 8.6.5 Guitar classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2008. 393 8.6.6 Voice classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2009. . . 394 8.6.7 Elevator classroom (a) and laboratory (b) survey results. The three bars, from left to right, for each question correspond to 2005 to 2008. 395 8.6.8 Survey results for the system control classroom (a) and the beam control laboratory. The four bars, from left to right, for each question correspond to 2005 to 2008. . . . . . . . . . . . . . . . . . . . . . . 396 8.6.9 Damage detection classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 8.6.10 Survey results for the Lego project in 2008 and 2009. . . . . . . . . 398 8.6.11 Classroom length survey results. The four bars, from left to right, for each classroom activity correspond to 2005 to 2009. Number of results vary due to some students failing to respond. . . . . . . . . . 399 8.6.12 Overall classroom survey results. The four bars, from left to right, for each classroom activity correspond to 2005 to 2009. . . . . . . . 400 8.6.13 Overall laboratory survey results. The four bars, from left to right, for each laboratory activity correspond to 2005 to 2008 or 2009. . . 401 8.6.14 Post-survey results: (a) part 1, (b) part 2, and (c) part 3. The four bars, from left to right, for each question or topic correspond to 2005 to 2007 or 2009. The number of students who answered the question on Curriculum in (b) is much lower in 2008 than those in the previous years because the students who responded with a neutral choice were not counted. . . . . . . . . . . . . . . . . . . . . . . . . 403 List of Tables 3.1.1 Measured versus calculated natural frequencies of the undamaged cantilever beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.1 Torsional natural frequency comparison for 28% stiness reduction from 23.9 cm through 37.15 cm . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 Comparison of the natural frequencies for the rst ten modes of the two section mast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4.1 Analytical and experimental longitudinal natural frequencies. . . . . . 176 5.5.1 Inclined rope MAC matrix for original FEM . . . . . . . . . . . . . . 184 5.5.2 Inclined rope MAC matrix for updated FEM . . . . . . . . . . . . . . 184 5.5.3 Vertical rope MAC matrix for original FEM. . . . . . . . . . . . . . . 184 5.5.4 Vertical rope MAC matrix for updated FEM. . . . . . . . . . . . . . 184 5.5.5 Bending vibration frequency comparisons. . . . . . . . . . . . . . . . 195 6.2.1 Torsion natural frequencies comparisons for a healthy and damaged beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.2.2 Comparison of natural frequencies of a tension beam 0.0127m wide, 0.0031m thick, and 3m long with 3000N of tension. . . . . . . . . . . 222 xviii 6.2.3 Damage detection results using the measured natural frequencies of the healthy beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.2.4 Damage detection results using the measured natural frequencies of the damaged beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.3.1 Comparison of the measured and modeled natural frequencies of the healthy rope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6.3.2 Comparison of the measured and modeled natural frequencies of the damaged rope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.1.1 Comparison of excitation techniques used in modal testing. . . . . . . 257 8.2.1 FEDS Academy's Schedule in 2006 . . . . . . . . . . . . . . . . . . . 325 8.3.1 Formant frequencies in Hz and vowel sounds of four speakers. . . . . 338 8.5.1 Connections among the ABET criteria, the MTEVSC standards, and the FEDS Academy activities. . . . . . . . . . . . . . . . . . . . . . . 373 8.5.2 Connections between the Maryland Mathematics Standards and the FEDS Academy activities. . . . . . . . . . . . . . . . . . . . . . . . . 374 8.5.3 Connections between the AP Calculus AB/BC Standards and the FEDS Academy activities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.5.4 Connections between the Maryland and AP Physics C Standards and the FEDS Academy activities. . . . . . . . . . . . . . . . . . . . . . . 376 8.5.5 Bloom's Taxonomy domains and subdomains. The complexity of the subdomains increases from the top to the bottom. . . . . . . . . . . . 377 8.6.1 Average scores for the laboratory activities and the MATLAB session from 2005 to 2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 8.6.2 Ranked aspects of the nal projects. . . . . . . . . . . . . . . . . . . 404 Contents List of Figures v List of Tables xviii Chapter 1 Introduction 1.1 1 Methodologies in Structural Damage Detection . . . . . . . . . . . . . 2 1.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Data Collection and Selection . . . . . . . . . . . . . . . . . . 4 1.1.3 Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Results Classication and Prognosis . . . . . . . . . . . . . . . 11 1.2 Damage Detection of Beam Structures . . . . . . . . . . . . . . . . . 13 1.3 Damage Detection of Cables . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Random Impact Hammer . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Objectives of the Current Work . . . . . . . . . . . . . . . . . . . . . 17 Chapter 2 Vibration-Based Damage Detection Algorithm 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Eigenparameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 xx 2.5 2.6 2.7 2.4.1 Non-Linear Least Squares . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Convexication of the Cost Function . . . . . . . . . . . . . . 28 Minimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.1 Pseudoinverse and Regularization . . . . . . . . . . . . . . . . 34 2.5.2 Levenberg-Marquardt and Trust Region Methods . . . . . . . 37 2.5.3 Parameter Updating and Iteration Termination . . . . . . . . 40 Updating Parameter Selection . . . . . . . . . . . . . . . . . . . . . . 41 2.6.1 Torsional Vibration . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6.2 Bending Vibration . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6.3 Longitudinal Vibration . . . . . . . . . . . . . . . . . . . . . 50 2.6.4 Tensioned System Vibration . . . . . . . . . . . . . . . . . . . 51 Program Design and Implementation . . . . . . . . . . . . . . . . . . 54 Chapter 3 Beam Damage Detection 3.1 3.2 57 Fixed Free Rectangular Beam . . . . . . . . . . . . . . . . . . . . . . 57 3.1.1 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Experimental Results and Discussion . . . . . . . . . . . . . . 59 3.1.3 Simulation Results and Discussion . . . . . . . . . . . . . . . 88 3.1.4 Conclusions for the Rectangular Beam . . . . . . . . . . . . . 90 Circular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.2 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 118 3.2.4 Conclusions for the Circular Beam . . . . . . . . . . . . . . . 125 Chapter 4 Lightning Mast Damage Detection 126 4.1 4.2 Introduction and Previous Results . . . . . . . . . . . . . . . . . . . . 126 4.1.1 50 Foot Lightning Mast . . . . . . . . . . . . . . . . . . . . . 127 4.1.2 Scaled Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Pipe Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.2 Experimental Results and Discussion . . . . . . . . . . . . . . 136 4.2.3 Simulation Results and Discussion . . . . . . . . . . . . . . . . 137 4.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Chapter 5 Elevator Testing 142 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.2 Aramid Suspension Rope-Elevator Car System Overview . . . . . . . 145 5.3 General Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4 Longitudinal Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.5 5.6 5.4.1 Vibration Testing . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.2 Longitudinal Analytical Model 5.4.3 Longitudinal Vibration Results . . . . . . . . . . . . . . . . . 167 . . . . . . . . . . . . . . . . . 165 Bending Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.5.1 Bending Vibration Modal Testing . . . . . . . . . . . . . . . 177 5.5.2 Finite Element Modeling of Bending Vibrations . . . . . . . . 177 5.5.3 Analytical Decoupled Models for Bending 5.5.4 Bending Vibration Results . . . . . . . . . . . . . . . . . . . 180 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter 6 Tensioned Member Damage Detection 6.1 . . . . . . . . . . . 178 202 Elevator System Simulations . . . . . . . . . . . . . . . . . . . . . . . 203 6.2 6.3 Tensioned Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.2.1 Experimental Setup and Testing . . . . . . . . . . . . . . . . . 214 6.2.2 Model Creation and Parameter Selection . . . . . . . . . . . . 215 6.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 216 6.2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Rope System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.3.1 Experimental Setup and Testing . . . . . . . . . . . . . . . . . 233 6.3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 6.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Chapter 7 Multiple Impact Device 254 7.1 Introduction and Previous Results . . . . . . . . . . . . . . . . . . . . 254 7.2 Previous Theory and Discussion . . . . . . . . . . . . . . . . . . . . . 258 7.3 Device Research, Development, and Design 7.4 Square Aluminum Plate Test . . . . . . . . . . . . . . . . . . . . . . . 271 7.5 . . . . . . . . . . . . . . 263 7.4.1 Manual Multiple Impact Testing . . . . . . . . . . . . . . . . . 271 7.4.2 Device Impact Test . . . . . . . . . . . . . . . . . . . . . . . . 273 Pipeline Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.5.1 Manual Multiple Impact Testing . . . . . . . . . . . . . . . . . 290 7.5.2 Device Multiple Impact Testing . . . . . . . . . . . . . . . . . 291 7.6 Lightning Mast Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 306 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Chapter 8 Engineering Education 315 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 8.2 Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.3 Daily Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 8.4 8.5 8.3.1 Vibration Measurement Module . . . . . . . . . . . . . . . . . 326 8.3.2 Pendulum Module . . . . . . . . . . . . . . . . . . . . . . . . 326 8.3.3 Guitar Module . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.3.4 Voice Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 8.3.5 Elevator Module . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.3.6 System Control Module 8.3.7 Damage Detection Module . . . . . . . . . . . . . . . . . . . . 332 . . . . . . . . . . . . . . . . . . . . . 331 Final Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.4.1 Cymbal Project . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.4.2 Guitar Project 8.4.3 Baseball Bat Project . . . . . . . . . . . . . . . . . . . . . . . 346 8.4.4 Mbira Project . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.4.5 Bridge Project . . . . . . . . . . . . . . . . . . . . . . . . . . 347 8.4.6 Rotordynamics Project 8.4.7 Bike Wheel Project 8.4.8 Shoe Project 8.4.9 Earthquake Project . . . . . . . . . . . . . . . . . . . . . . . . . . 345 . . . . . . . . . . . . . . . . . . . . . 347 . . . . . . . . . . . . . . . . . . . . . . . 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 . . . . . . . . . . . . . . . . . . . . . . . 349 Connections of Program Activities to Various Criteria and Standards 8.5.1 Comparison of Dierent Programs 365 . . . . . . . . . . . . . . . 366 8.5.1.1 Cognitive Domain . . . . . . . . . . . . . . . . . . . 367 8.5.1.2 Aective Domain 8.5.1.3 Psychomotor Domain . . . . . . . . . . . . . . . . . 371 . . . . . . . . . . . . . . . . . . . 370 8.6 8.7 Survey Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.6.1 Pre-survey Results . . . . . . . . . . . . . . . . . . . . . . . . 378 8.6.2 Laboratory Survey Results and Observations 8.6.3 Post-survey Results 8.6.4 Longitudinal Survey Results . . . . . . . . . 379 . . . . . . . . . . . . . . . . . . . . . . . 384 . . . . . . . . . . . . . . . . . . 387 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Chapter 9 Conclusions 407 Bibliography 412 Appendix A Damage Detection Program 440 Appendix B Derivation of longitudinal vibration modes. 479 Appendix C Derivation of bending vibration modes. 482 Chapter 1 Introduction The research presented in this thesis covers two very important thematic areas in structural dynamics research, namely vibration based damage detection or structural health monitoring and the improvement of modal testing through the design of a device to implement multiple impact testing. Structural damage detection using changes in vibration characteristics has recently received much attention. The main idea behind vibration based damage detection is that damage in a structure, such as cracks, corrosion, loose joints, or deteriorated boundaries etc., will manifest itself as changes in vibration characteristics, such as natural frequencies, mode shapes and modal damping, of the structure. In the current research funding process, government agencies such as the National Science Foundation (NSF) and at government agencies such as the National Aeronautics and Space Administration (NASA), education outreach for the benet of society is required for all research grants or projects. At the end of this thesis research will be presented that chronicles the development of an educational summer camp for high school students featuring research conducted in this thesis along with other theses in the Dynamic Systems and Vibrations Laboratory. 1 1.1 Methodologies in Structural Damage Detection Damage in a structure can be dened as any change in the structure that reduces its functionality, which can result from deterioration of its components, connections, or boundaries during its service life. All structures continuously accumulate damage during their service life; early detection, assessment, and monitoring of this damage can avoid catastrophic failures. Because of the shortfalls in the existing nondestructive testing (NDT) methods such as visual inspection (VI), penetrant testing (PT), magnetic particle testing (MPT), electromagnetic testing (ET), radiographic testing (RT), and ultrasonic testing (UT), structural damage detection using changes in vibration characteristics has received much attention. There are many ways to classify and organize structural health monitoring using vibration data as shown in extensive literature reviews [13]. In most vibration-based damage detection studies the focus is the intermediate step of feature selection before using statistical modeling or as a means to generate a model to be used for prognosis and future life prediction [4]. Most classications of vibration-based damage detection methods are comprised of some combination of the following steps where some steps are combined or omitted: problem formulation, data collection and reduction, feature selection, modeling, damage detection, results classication, and prognosis [4]. This research will focus on the feature selection and modeling steps. An overview of the steps above will be discussed to help put this research in perspective. In this research there are not enough specimens to make statistical models to look at the damage detection results in the feature vector to make a nal decision as to the location, extent, and type or the remaining life of the structure. 2 1.1.1 Problem Formulation At the beginning of any structural damage detection investigation, a plan needs to be generated that takes into account anything that might have an adverse eect on the damage detection process. One example would be the need for some methods and structures like bridges that need to be in operation to have adequate excitation and response to measure modal parameters, such as natural frequency and operational mode shape or operational deection shape. A similar approach can be taken with lightning masts with wind excitation, pipe lines with ow induced vibration, and elevator cables shortly after the car has stopped. Another eect that might need to be taken into account would be weather conditions such as temperature for structures such as large civil structures [5, 6] or simple plate structures [7] because these eects can change eigenparameters of the structures. Also included in the problem formulation are limitations in acquiring data, such as prior knowledge of damage, limited access of sensor placement, sensor type, and environmental or operational status. Prior knowledge is important, for example, in the use of NDT methods, such as visual inspection (VI), penetrant testing (PT), magnetic particle testing (MPT), electromagnetic testing (ET), radiographic testing (RT), and ultrasonic testing (UT) because they are localized experimental methods, which only inspect a small region of the structure at a time [812]. Sensor limitations often limit the eectiveness of NDT methods such as VI and PT methods because they are only capable of nding surface aws [11, 12]. The MPT and ET methods have similar sensor limitations as they can only nd surface or shallow subsurface aws. Vibration-based methods allow for a wide range of structures to be investigated because they don't depend on the material having certain properties as do the PT, MPT, ET, and UT methods [812]. Even though most vibration-based methods are global, sensor positioning problems arise 3 for methods that use mode shape or modal curvature information because easy access to the positions of interest aren't always guaranteed. Even for frequency based damage detection, sensor location is important. In the case of a beam, the sensor needs to be perpendicular to the damage because modes parallel to the damage don't change. Another important limitation to consider is the amount and type of equipment that can be taken to the structure when testing in the eld. Problem formulation also needs to take into account the limits on model development as is the case with many old civil structures and was the case for the lightning masts investigations in this research because limited blueprints and documentation were available. This limitation is usually not as critical for NDT methods because most are not model based. The type of structure being investigated is also important because it needs to be able to be modeled or represented as a linear structure for most modeled-based and even many signal-based methods. 1.1.2 Data Collection and Selection The method presented can use natural frequencies and mode shapes measured using accelerometers, laser vibrometers, dynamic strain gages, displacement sensors, and modal hammers / force sensors using a spectral analyzer. Location is not very critical because the method is a global method, but if mode shapes will be used the best practices for modal testing should be followed. For some structures being examined sensor locations are limited to what can be reached or what the sensor can be axed to. An example of this is the measurement of the longitudinal vibration of cables where a good choice of sensor location and excitation is at the boundary conditions. Location can sometimes be inuenced by the amount the sensor can change the vibration of the structure as in the case of a light xed-free beam. For lightning mast 4 and cable type structures, economics is a driver on sensor use and placement due to the number of structures to be tested and their size. Data collection rate is a key factor in structural health monitoring, especially for structures that accumulate damage rapidly. For structures in service, the frequency would be based on the expected life of structure and expected rate of change. For systems like elevator cables, it could be done daily because the cable properties vary with the number of loading cycles and magnitude of loading [13, 14]. For steel structures in wet or corrosive locations, the inspection interval could be based on corrosion loss rates. In cases such as civil infrastructure, it would be based on severe events such as earthquakes, severe storms, or severe loadings. If the natural frequencies of the structure are continuously monitored, this approach can be used by waiting until a minimum change in natural frequency is detected to run the damage detection model. An example would be in elevator systems where the material properties of the rope change with the number of cycles. In this approach normalization is not dicult because there is no need for natural frequencies and the mode shapes can be normalized to a unit vector. If only the natural frequencies are being used and the structure is heavily damped, then the natural frequency shift due to damping would need to be accounted for in this method. The main form of data reduction in this method is curve tting the natural frequencies and mode shapes, if they are used. The main source of data reduction variability would be due to the investigator performing the modal analysis. In comparison with NDT methods, data collection and selection are straight forward because local methods are mostly limited to small aws, don't require physical or numerical models, and in some cases can give the crack geometry [12]. One method where data ltering and selection is dicult is the AET methods, which provide comprehensive information on the origination of a aw in a stressed component and 5 nformation pertaining to the development of the aw as the component is subjected i to continuous or repetitive stress [10]. The principal limitations of the acoustic emission test (AET) method are repeatability, attenuation, history dependence, and noise [10]. Acoustic emission is stress unique and each loading is dierent, so the method works best when the loading history of the structure is known because the acoustic emission can be subject to extraneous noise [10]. The limitations of AET methods often mean that a specic location should always be chosen for similar structures as well as the data reduction algorithm, which is very important because the structure is continuously monitored. The VI, PT, MPT, and ET methods require less operator training, are more time ecient, and more cost-eective if only a small area has to be examined [11, 12]. The main data drawback to these methods in terms of data collection is that it depends heavily on investigator training, repeatability, and data processing, all of which can vary greatly. 1.1.3 Feature Selection Feature selection is the process of nding the information about the structure that gives the best possibility of determining if a structure is damaged. Determining the best features to use to detect damage is sometimes an iterative process based largely on numerical experiments, past experience, and experimental testing of known damages, just to name a few. Other information that can be used in feature selection is based on what types of data can be measured and processed repeatedly and reliably. The geometry of the structure can also be important in the feature selection process. For example, the thickness of a rectangular bar could be important so the natural frequencies of bending modes in the suspected damage direction could be measured. The selection process is also based on sensitivity analysis of the structure 6 to its geometry or material properties. Many times feature selection is based on past experience and research results from other investigators, as well as previous failure of similar structures. Experience gained during model development and correlation with the healthy structure is also a valuable source of information. As with all damage detection methods, features that aren't sensitive to variations due to operating or environmental conditions are usually the best choices. Feature selection is also based on the ability to modify them in a known way to expand the information base used during damage detection [17]. The features are also based on what can be measured and modeled [15, 16]. The aspect of structural health monitoring that receives the most attention is feature selection [4]. The main idea behind vibration-based damage detection is that damage in a structure will manifest itself as changes in vibration characteristics, such as natural frequencies, mode shapes, and modal damping, of the structure. A summary of structural health monitoring using vibration by Doebling et al. [9] goes into great detail about the dierent feature selection methods which outside of Farrar's classication scheme [4] are normally known as damage detection methods. Damage detection performed by tting a mathematical model to the test data is an inverse minimization or optimization problem. The most studied vibration-based methods are direct methods [18], which directly modify stiness and mass matrices using test data in most cases a single step [16, 19]. The two most common direct methods are the optimal matrix update [2024] and control-based eigenstructure assignment [18, 25, 26] methods. Direct methods were rst studied because they are computationally inexpensive and reproduce test data with very little error. Direct methods operate on the stiness and/or mass matrices and thus connectivity is lost and the model is no longer physically realizable because the system matrices lose their sparsity. This 7 drawback can be overcome by applying the correct constraints [20]. Another drawback of direct methods is that mode shapes can't be measured completely or are expanded [16, 19] to the same number of degrees of freedom as the model. Indirect methods change physical or elemental properties in the model to reproduce test data [16, 19], which yields a more physically realizable model while using incomplete mode shapes or even just natural frequencies [16, 18, 19]. Two drawbacks of indirect methods is that they are computationally expensive because they are usually iterative and depending on the minimization procedure used, can have convergence issues [16, 18, 19]. Sensitivity-based indirect methods are also very common and have a broad range of implementation [18]. The work presented is based on an iterative sensitivity-based algorithm [27, 28], while a similar algorithm was presented in Ref [29]. The knowledge that natural frequencies can be measured with more ease and accuracy than mode shapes makes damage detection using the rst several natural frequencies of great interest [30, 31]. Two common assumptions made for damage detection using measured natural frequencies is that there is one unknown damage location [3235] and/or the type of damage is known [32, 3437]. Forward methods are suitable for detecting a single crack described by two parameters: the crack location and extent; they become much more complicated for multiple cracks [39]. The forward problems for beams or rods with multiple cracks were studied in [37, 39 41]. The forward problem for more complex structures has been studied by using transfer function parameter changes and a hypothesis-based damage detection scheme [42, 43]. With the increase in computer power, inverse methods are widely used in feature extraction [3639, 4447] for both continuous [33, 3638, 43, 44] and discrete models [33, 39, 45, 46, 48, 49]. Many inverse techniques often use a spring to model a crack, 8 but linear elastic fracture mechanics approaches have been formulated [40, 46]. When a nite element (FE) model is used, damage is usually averaged over the element(s) in question [45, 4749] or a cracked element can be used [39]. Many feature selection methods are structure specic and the models investigated are limited to a specic type of vibration or element. The parameter of choice in many sensitivity-based methods utilizing a FE model is the elastic moduli [26, 48, 49]. To narrow down the number of features and data to be used a small level of damage is assumed and the mode shapes are unchanged [34, 36, 37, 45, 48]. The methods that rst predict damage location and then its extent are usually only applicable to simple structures. When a ner mesh is used, many of the methods cannot group multiple elements together to reduce the number of unknowns to be identied. The underdetermined problem, where the number of measured natural frequencies is less than the number of unknowns [37, 48], has not been investigated as much as the determined or overdetermined cases. The simulation example in Ref. [37] used three known natural frequency changes to determine ve unknown stinesses using the generalized inverse method for a structure with very small levels of damage. In the simulations, if a parameter is healthy after an iteration, the parameter is removed to make the system equations more determined [37]. Few damage detection algorithms that use measured natural frequencies have been validated experimentally. Of those few algorithms, a single notch or saw cut was placed on the structure. Very few used multiple damage locations or damage over a larger area [36, 39, 40, 44, 45, 49]. Another class of feature selection methods using vibration are non-model-based. Non-model-based methods focus on using the measured time responses from sensors or the mode shapes to nd sensitive features. An overview of the most common techniques is provided here for comparison to the model-based method presented and 9 more detailed overviews are provided in the review papers [13]. One of the most popular methods using time series data to extract features from modal properties are based on the auto-regressive moving average model [13, 50]. An extended ARMA method using exogenous inputs (ARMAX) to account for external excitations has also received much attention. Signal processing methods using wavelets, the Hilbert Huang Transform, neural networks, and time-series analysis have also received attention as methods to nd sensitive features [13]. The drawbacks for many of these methods is that without previous data, simulations, or training in the case of neural networks the location and extent of damages cannot be determined [51]. In many cases to be able to locate the damage a large sensor network or multiple measurement locations are needed. Feature selection using mode shape changes mainly use the modal assurance criteria (MAC) number and coordinate (COMAC) to locate changes in the structure, though other methods have also been proposed [1, 3]. The major drawbacks of the mode-shape-based methods based on MAC/COMAC is that even with large damages the mode shapes of some structures do not change enough to see noticeable dierences and MAC/COMAC are relatively insensitive to some types of damage [1, 3]. Modal curvature based methods use the derivatives of the measured mode shapes and can be related to the strain mode shape because there is a direct relationship between curvature and bending strain for beams and shells [5254]. The exibility matrix is the inverse of the static stiness matrix and relates the static displacement prole with a unit load at a degree-of-freedom (DOF). The exibility matrix is often computed from the rst few mass-normalized modes [16]. There are a wide range of feature selections based on the dynamic exibility matrix [1, 3], though the most straight forward method is to compare the measured healthy or FEM exibility matrix with that of the suspect structure. A more elaborate method is to use the exibility 10 matrix or its pseudoinverse, the stiness matrix whose calculation is susceptible to signicant error [16]. Residual force vector methods compute the harmonic force excitation that would need to be applied to the structure at the damaged natural frequency to reproduce the damaged mode shape [55, 56]. The diculties associated with this method are that there either needs to be a large number of measurements for the mode shapes, the mode shapes need to be expanded which can add uncertainty, or the model needs to be reduced if a FE model is used instead of a measured model [1, 3, 55, 56]. 1.1.4 Results Classication and Prognosis After the features are extracted from the model, they need to be classied as to their location and extent if that is not already given in the feature selection [1]. The classication stage as outlined by Farrar et. al. [57] uses statistical methods to determine if the features detected are signicant enough to be considered damaged or an erroneous result. In view of Rytter's [58] damage scheme, the nal step would be to determine the remaining life of the structure using the statistically classied results. Even though damage location and extent are given in the feature selection stage, statistical analysis is needed to determine a more accurate result because a way to model damage needs to be assumed during the feature selection stage of this method. An example of this could be selecting a full radius reduction for a circular beam instead of a at type reduction from just one side of the beam. Statistical based methods can be separated into three main types: group classication, outlier analysis, and regression analysis. These three methods can be used in an unsupervised learning approach, where data is only available from healthy structure, and supervised when information from both the healthy and damaged structure are available. Group 11 classication is the sorting of data into damaged and undamaged categories and is often used in a supervised learning approach because what constitutes damage needs to be known. Outlier analysis is used when data from the damaged structure is absent to determine if what is measured can be explained with prior information and models. Regression analysis can be used in a supervised and unsupervised learning approach that correlates measured features with known types, locations, and extents of damage. Group classication is often performed on signal-based methods where tests of the undamaged and damaged structures are performed. The most common types of signal processing methods for group classication methods are the auto-regressive methods, though non-linear based; neural network based methods are also used. An example of using an auto-regressive method with the Fisher's discriminant is given by [57] for a concrete beam structure. Another group classication approach to damage detection using Bayesian classication is shown for a simple 8 DOF structure by [59]. Group classication can also be done by monitoring the amount of nonlinearity in a signal as shown in [61], the authors also demonstrate using the receiver operation curve [60]; to quantify the amount of false positive or negative detections. Outlier analysis also uses many of the same signal processing techniques as group classication but without prior knowledge of how the damaged structure will react. An outlierbased method using the statistical properties of nonlinear signals, which can identify global and local damage through the use of measurements distributed over a plate, is demonstrated by Trendalova and Manoach [62]. For the same concrete beam structure as [57], Fugate et. al. [63] use a statistical process control chart to monitor the mean and variance of the acceleration signal models computed using an autoregressive method. Lei et. al. [64] use an outlier analysis approach using autoregressive signal processing and condence intervals applied to the residual error of 12 the auto-regressive model to locate damage in a building structure. For the same building structure as Lei et. al. [64], Silva et. al. [65] demonstrate a regression analysis method using a similar signal processing approach as Lei et. al. but with a fuzzy c-means clustering approach to identify the location and extent of damages. Besides being used for group classication, Bayesian approaches are used for regression analysis to nd the locations and extents as unsupervised methods as shown by [66 69]. One of the major advantages of Bayesian probabilistic approaches is that they give a probability distribution associated with the parameter(s) being used for damage detection. Bayesian approaches have been applied to a wide range of structures such as concrete beams [67], metallic beams [69], building structures [68], and composite aircraft wings [66]. Besides being applicable to a wide range of structures, they are not limited to a specic data type and have been used with modal parameters [67, 68] and time responses [66, 69] just to name a few. 1.2 Damage Detection of Beam Structures Beam structures are widely used in civil, mechanical, and aerospace engineering and were some of the rst applications of vibration-based damage detection [32, 34]. Detecting concentrated cracks in beams has received much attention and their eects are often modeled as rotational [32, 3638, 44] or translational springs [35, 38, 44]. An example where beam damage detection would have been useful was in September 2000 when a lightning mast collapsed in an electrical substation in the Baltimore area. The mast collapsed prematurely, causing damage to a surrounding structure resulting in a power outage. The mast failed at the sleeve joint which was 22 ft above ground due to internal corrosion, which was not visible from the ground. Similar failures have 13 been reported in trac signal structures [70], with a large percentage of surveyed structures having some type of damage. Damage detection could also prove useful for inspection of light poles after strong wind or snow events [71]. Bridge decks [72] and radio or transmission towers are also modeled as beams in some cases. Concrete beams have been studied widely to examine the eects of cracks [73, 74] and corrosion of the steel reinforcement [75]. One reason that vibration-based damage detection is useful for beam structures is that it is a global method unlike most NDT methods. For example, acoustic emissions have been used for trac control structures but were applied in a more local manner and some initial destructive testing needed to be performed for pattern recognition and initialization of search criteria [70]. Damage detection and classication of wood utility poles is also an area of interest because of the large number of wood poles that are in service or replaced annually [76, 77]. Helicopter blades [78, 79] and wind turbine blades [80] are other beam structures in which damage detection and health monitoring are of major concern because these structures are designed to be as light as possible to improve eciency. The guided wave ultrasonic testing (GWUT) method could work for the beams studied in this thesis based on similar studies by Rose [81], Rose et al. [82], Chinn et al. [83], and Bartoli et al. [84]. The method has been shown to be able to detect defects in beam structures with joints and curves over distances of 55 feet [81, 83, 84]. The GWUT method can also have diculties with complex geometries due to the need for guided waves to have an uninterrupted path to the areas to be investigated. With a simple beam the GWUT and other NDT methods yield similar results as damage detection, but beams with joints such as lightning masts or trac poles run into diculties because the joint alters the transmission path. One drawback of the GWUT method is the need for a nely meshed nite element (FE) model to capture the spatial and 14 temporal physics of the waves. 1.3 Damage Detection of Cables Research on the eects of local damage on the dynamics of cables and vibrationbased detection is still relatively sparse [85]. Cable and belt damage detection is important in stationary structures such as bridges [86], guyed towers [87], and transmission lines along with dynamic applications such as elevators, cranes, and material handling [88]. In many of these applications cables are replaced at regular intervals due to vibration-induced fatigue from wind, rain, snow, periodic loading or other service related stresses. Some of the most widely used methods for cable inspection are ultrasonic, acoustic emission, and electromagnetic. The methods all have distinct advantages but they are more localized, and in some cases do not or cannot give an indication of remaining strength or tension, and are not easily or cannot be applied to nonmetallic cables. Currently, there are very few proven technologies for health monitoring and damage detection of non-metallic cables other than visual inspection, which is very time consuming, expensive, and subjective. One major problem with visual inspection of jacketed, non-metallic cables is that due to internal friction between strands, the inner section can turn to dust before there is visible damage to the Nylon jacket. There are technologies and techniques for monitoring steel cables and belts in systems such as bridges, transmission lines, and material handling systems. For the eddy current and magnetic inspection techniques the material has to be conductive; thus, inspection of aramid cables with these methods is not possible. Ultrasonic testing is not feasible due to the fact that the system is not homogeneous or isotropic and has voids between the bers. Vibration-based methods for cable inspection are new 15 but promising due to their ability to overcome many of the drawbacks of the other methods. 1.4 Random Impact Hammer Currently, there are two main methods used to perform modal testing. The rst method relies on an impact hammer generating a single impact and the second relies on an electromagnetic or hydraulic shaker. A shaker can be used to generate all types of input from an impulse to continuously random input. The main advantages of the impact hammer method is that it is easy to use, very portable, low cost, and introduces no mass loading. But the disadvantages are it has low controllability, repeatability, and energy input. Shaker testing using a continuous random excitation signal is mainly the opposite and can average out slight nonlinearities. A new technique developed in the lab known as random impact testing combines the advantages of both of these methods [89]. The method involves continuously impacting the structure with a series of random impacts with random amplitude and arrival time. Like sine testing with the shaker, the frequency of random impacts can be used to focus energy in a desired bandwidth [89]. Currently, there is no device or system specically designed to perform random impact testing. Several investigators and companies have developed impact devices to increase repeatability but have still been focusing on single impact testing. The main purpose of automated impact devices is to be able to improve the results of modal testing by increasing repeatability [9092]. Bissinger and Ye demonstrated how using an automated impact device with a scanning laser vibrometer can save time and improve results when testing violins [93]. 16 1.5 Objectives of the Current Work The main objective of this work will be to develop a damage detection system based on the iterative algorithm in Refs. [27, 28]. The system will include both a general use computer program and methodology for use of the program. The system will be demonstrated both experimentally and numerically on a wide range of systems using the rst few natural frequencies to determine the locations and extent of damage in slender structures. The algorithm is tested on both rectangular and circular cantilever beams with dierent damage scenarios to demonstrate the use of damage detection using a length parameter, the use of structural modication for information enrichment, and the use of two dierent types of modes simultaneously. The addition of a convexication scheme is also investigated. A pipeline test scenario is carried out to investigate the exibility of the proposed program to be applied to external models from other researchers and the ability to perform damage detection using the density. The pipeline scenario also demonstrates the experimental and simulated detection of a parameter increase instead of decrease. To show the importance of damage feature selection the method will be demonstrated on tensioned slender structures. These tensioned beam test case will show the use of length modication for information enrichment as well as the Levenberg-Marquardt method. The need for step size modication will also be demonstrated. A secondary objective of this work will be to develop a multiple impact device for modal testing. The device will be able to achieve results equal to or better than those achieved by an experienced modal test engineer. Results will be compared by testing three structures of increasing complexity, a simple plate, a bolted pipe, and a 50' lightning mast. Comparisons of the pulse shapes, frequency response functions, 17 coherences, distributions of impact amplitude, arrival times, and power spectral densities generated by an engineer and the multiple impact device for a variety of tests will be made to validate the device. Another secondary objective will be to demonstrate the use and eectiveness of the Dynamic Systems and Vibrations Laboratories research when used in a two week summer camp over the course of six summers. 18 Chapter 2 Vibration-Based Damage Detection Algorithm 2.1 Introduction The focus of this chapter will be to give a derivation of the computational aspects of the feature selection method as well as the design of the method. In Section 2, the derivation of the eigenvalue problem associated with the healthy and damaged systems using a rst order Taylor series expansion is presented. Section 3 will review the eigenparameter sensitivity as used in the method being presented as well as a nite dierence calculation of the sensitivity matrices. In Section 4 the least squares cost function will be shown with and without convexication. Section 5 will show two approaches to solve the minimization problem using both the Moore-Penrose pseudoinverse method with the singular value decomposition used for regularization and the Levenberg-Marquardt method. Section 7 will demonstrate the theoretical approach used to choose the updating parameters for beam type structures. The 19 nal section will give an overview of the program design as implements in MATLAB [94] using the SDTools [95] toolbox. 2.2 Eigenvalue Problem A linear structure can be spatially discretized as an m-element system with N DOF using the FE method. If only the changes in the stiness properties due to damage are considered the changes in the inertial properties are assumed to be negligible. The dimensionless model parameters of the undamaged structure are denoted by Gh,i (i = 1, 2, . . . , m), where the subscript h denotes the undamaged or healthy struc- ture, and damage is characterized by reduction in the stiness parameters. In most cases the dimensionless model parameters are chosen as the one which means detection begins from the original model parameters. The estimated stiness parameters of the damaged structure before each iteration are denoted by Gd,i (i = 1, 2, . . . , m), where the subscript d denotes the damaged structure, and the mass and stiness matrices, which depends linearly on G, are denoted by K =K (G) and M =M (G) respectively, where G = [G1 , G2 , ..., Gm ]T , in which the superscript T denotes matrix transpose. The eigenvalue problem of the structure can be expressed in terms of the stiness parameters G as K?k = ?k M?k (2.2.1) where ?k = ?k (G) and ?k = ?k (G)(k = 1, 2, ..., N ) are the k -th eigenvalue and mass-normalized eigenvector, respectively. It is noted that ?k = ?k2 , where ?k is the k -th natural frequency. The normalized eigenvectors of Eq. 2.2.1. satisfy the 20 orthonormality relations ?k T M?u = ?ku ?k T K?u = ?k ?ku (2.2.2) where 1 ? u ? N and ?ku is the Kronecker delta. Before the rst iteration, the (0) initial stiness parameters of the damaged structure are assumed to be Gi = ?i Gh,i (i = 1, 2, ...m), where 0 < ?i ? 1. Unless stated otherwise, we start the iteration from the stiness parameters of the undamaged structure and set all ?i = 1. The eigenvalue problem of the damaged structure is Kd ?kd = ?kd M?kd (2.2.3) where Kd = K (Gd ) and Md = M (Gd ) are the stiness and mass matrices respectively with Gd = [Gd,1 , Gd,2 , ...Gd,m ]T , and ?kd = ?k (Gd ) and ?kd = ?k (Gd ) are the k -th eigenvalue and mass-normalized eigenvector, respectively. The stiness matrix Kd is related to K through the Taylor expansion m ?K ?Gi Kd = K + ?Gi i=1 (2.2.4) where ?Gi = Gd,i ? Gi (i = 1, 2, ..., m) are the changes in the stiness parameters, and the higher-order derivatives of K with respect to Gi vanish because K is a linear function of Gi . The mass matrix Md is related to M through the Taylor expansion Md = M + m ?M i=1 ?Gi ?Gi (2.2.5) where ?Gi = Gd,i ? Gi (i = 1, 2, ..., m) are the changes in the stiness parameters. The 21 derivative of M with respect to Gi will vanish if we assume that the changes in the inertial properties due to damage are negligible. 2.3 Eigenparameter Sensitivity The change of a structure's stiness and/or mass, will cause changes in its natural frequencies and mode shapes. The sensitivity of the eigenvalues and eigenvectors were rst derived by Wittrick [96] and Fox and Kapoor [97] for structural design optimization, which is based on the relationships between the changes in structural response and structural parameters. The derivation that follows is closely based on that presented in Ref. [16] with additions needed for the damage detection algorithm presented. To calculate the sensitivity of the eigenvalues to a set of design parameters, the eigenvalue equation (Eq. 2.2.1) is dierentiated with respect to the design parameters Gi (i = 1, 2, . . . , m) and gives ??j ?M ?K ??i =? ? ?j ? M ?j [K ? ?j M] ?Gi ?Gi ?Gi ?Gi (2.3.1) where the subsist j denotes the j -th eigenparameter. Pre-multiplying Eq. 2.3.1 by the mass normalized eigenvector ?Tj , and noting that M and K are symmetric, yields the eigenvalue sensitivities ??j ?K ?M T ?j = ?j ? ?j ?Gi ?Gi ?Gi (2.3.2) In the cases where a geometry parameter such as thickness or length is used, Eq. 2.3.2 is utilized to account for the change in the mass matrix. If mass changes are 22 gnored Eq. 2.3.2 becomes i ??j ?K T ?j = ?j ?Gi ?Gi (2.3.3) In the cases where a mass parameter such as density or non-structural mass is used, Eq. 2.3.2 is utilized to account for the change in the mass matrix. If stiness changes are ignored Eq. 2.3.2 becomes ??j ?M T ?j = ?j ?? ?Gi ?Gi (2.3.4) Equation 2.3.2 is used when using a stiness parameter such as the elastic modulus to do damage detection. To obtain the eigenvector sensitivities equation the derivative in Eq. 2.3.2 is substituted into Eq. 2.3.1 and yields: [K ? ?j M] ??j = uj,i ?Gi (2.3.5) where ?K ?M ?M ?K T =? ? ?j ? ?j ? ?j ?j M ?j ?Gi ?Gi ?Gi ?Gi uj,i (2.3.6) The complete eigenvector derivative is ??j = v j,i + cj,i ?j ?Gi (2.3.7) The mass normalization equation ?Tj M?j = 1 is dierentiated with respect to Gi and combined with Eq. 2.3.7 to eliminate 23 (2.3.8) ??j ?Gi 1 ?M cj,i = ??Tj Mv j,i ? ?Tj ? 2 ?Gi j To solve for v (2.3.9) it is noted that rank [K ? ?j M] = n ? 1 which allows for the o-th term to be set to zero ? ? [K ? ?j M]11 0 [K ? ?j M]13 ? ? 0 1 0 ? ? [K ? ?j M]31 0 [K ? ?j M]33 where the oth (2.3.10) ?? ? ? v 1,i ?? ? ?? ? ? ? vo,i ?? ? ? ? v 3,i ? ? ? ? ? ? ? ? ? u1,i ? ? ? = 0 ? ? ? ? ? ? ? ? ? ? u3,i ? ? ? ? ? ? ? ? ? ? ? (2.3.11) term is chosen from max ?j o ?K (2.3.12) ?M When calculating ?G or ?G there are several approaches that can be used, each i i ?K ?M having its own advantages and disadvantages. It is noted that ?G and ?G in Eq. i i 2.3.1 for each i is a constant matrix if the nondimensional parameter is used with a linear parameter such as the elastic modulus or the the sensitivity matrices can be assumed to be linear based on the initial sensitivities around the starting parameters. For structures that depend linearly on a parameter, the simplest method to formulate ?K ?M and ?G is to set ?Gi i in K. Gi = 1 and the rest of the parameters to zero, i.e., Gj = 0 (j = i), By the denition of dierentiation, one has ?k(1),i = ??k ??k = lim ?Gi ?Gi ?0 ?Gi 24 (2.3.13) ?k(1),i = hence ?k(1),i and ?k(1),i ??k ??k = lim ?Gi ?Gi ?0 ?Gi (2.3.14) can also be calculated from Eq. 2.3.13 using the nite dierence ?k , ? k , method, especially when ?K ?M and/or ?G and ?G are not directly available in the i i FE program. However, the changes in the eigenparameters in Eqs. 2.3.13 and 2.3.14 associated with the change in each stiness parameter ?Gi (i = 1, 2, ...m) need to be calculated in each iteration, and much more computational time is needed to calculate the eigenparameters from the eigenvalue problem in Eq. 2.2.1. The third method is to use the central dierence method ?K K (Gi + ?) ? K (Gi ? ?) ? lim ?Gi ??0 2h (2.3.15) ?M M (Gi + ?) ? M (Gi ? ?) ? lim ?Gi ??0 2? (2.3.16) directly with the mass and stiness matrices when they are available. The advantage here is that the eigenparameters don't have to be computed, which is very costly compared to computing the mass and stiness matrices twice because even with the computation of the eigenparameters, the mass and stiness matrices have to be computed once for every Gi . Another advantage is that one does not need to have prior knowledge of the element formulation, which allows for the use of geometrybased parameters or materials that are isotropic. For elements that don't depend linearly on G ?K ?M there can be a need to form ?G and ?G for each iteration, which is computationally very expensive but is sometimes necessary or benecial. 25 2.4 Cost Functions 2.4.1 Non-Linear Least Squares In order to use the eigenvalue problem and it's sensitivity information to determine the location and extent of damage in the underdetermined minimization, a non-linear least-squares cost function 1 2 Q(G) = Ri (G) = ?RT ?R 2 i=1 n (2.4.1) is needed where ? ? ? ? R1 ? ? ?1 ? ? ? ? ? R2 ? ? ? 2 ? ? d 2 ? ? ? ?R = ? . ? = ? .. ? .. ? ? . ? ? ? ? ? ? Rn ?n ? ?dn ?d1 and ?i is the i-th current result, ?di ? ? ? ? ? ? ? ? ? (2.4.2) is the i-th damaged measurement, and n is the number of parameters being compared in the cost function. The rst derivative of the cost function ? Q n with respect to nondimensional scaling parameters ?Ri i=1 Ri ?G1 ? ? n ?Ri ? i=1 Ri ?G2 ? ?Q = ? .. ? . ? ? n ?Ri i=1 Ri ?Gm ? ? ? ? ? ? ? ? ? ?=? ? ? ? ? ?? ?R1 ?G1 иии .. . .. . ?R1 ?Gm ?Rn ?G1 иии 26 ?Rn ?Gm G is ? ? ? R1 ? ?? . ? ? ? .. ? = J T ?R ?? ? ?? ? Rn (2.4.3) where J is the Jacobian or sensitivity matrix in the eigenvalue problem. If Gc is a point close to the local minimizer G?, then Q (G) around G = Gc can be expressed as 1 (G ? Gc )T ?2 Q (Gc ) (G ? Gc ) 2 Q (G) = Q (Gc ) + ?QT (Gc ) (G ? Gc ) + (2.4.4) where ?2Q (G) is the second derivative of Q with respect to G evaluated at the current position Gc. At a local minimizer G?, the rst derivative of Q equals to zero: (2.4.5) ?Q (G) = ?Q (Gc ) + ?2 Q (Gc ) (G ? Gc ) = 0 To simplify Eq. 2.4.5, the second derivative of ?Q is computed from Eq. 2.4.3 as ?2 Q = J T ??R + ?J T ?R = J T J + ?J T ?R ?G (2.4.6) and expanding the derivative of J T in Eq. 2.4.3 gives ? 2 ? Q=J J+ T n i=1 ? ? Ri ? ? ? ? 2 Ri ?G21 ? 2 Ri ?G1 ?G2 иии ? 2 Rn ?Gm ?G1 ? 2 Rn ?Gm ?G2 иии ... ... ? 2 Ri ?G1 ?Gl ... ?Ri ?G2m ? ? ? ? ? ? (2.4.7) Using the denition of a Hessian matrix, Eq. 2.4.7 can be written as 2 ? Q=J J+ T n i=1 27 Ri H i (G) (2.4.8) where H i (G) is the Hessian matrix of 2.4.5 gives J T ?R + JT J + Ri . n Substituting Eqs. 2.4.3 and 2.4.8 into Ri H i (G) ?G = 0 (2.4.9) i=1 where ?G = G ? Gc. The new search direction in Newton's method is obtained by solving Eq. 2.4.9. Applying the assumptions for the Gauss-Newton method that for zero and/or a small residual least-squares problem, Ri is very small when G ? G? and the objective function is more like a linear function; hence ni=1 RiH i (G) T J J , and n Ri H i (G) can be neglected compared to J T J thus ?2 Q = J T J i=1 and Eq. 2.4.9 J T ?R + J T J ?G = 0 (2.4.10) Finally, the Gauss-Newton minimization problem is given as min Q (G) = J ?G ? ?R22 (2.4.11) 2.4.2 Convexication of the Cost Function To try to bound the optimization and speed convergence, the convexication approach as suggested by Lo [98] is applied to the objective function 1 ?Ri2 (G) e W (G) = 2 j=1 n (2.4.12) where ? is the convexication parameter. The rst derivative of the cost function W with respect to nondimensional scaling parameters G is 28 ? n ?Ri ?Ri2 ?R e i i=1 ?G1 ? ? n 2 ?Ri ?Ri ? ?R e i i=1 ? ?G2 ?W = ? . ? . . ? ? n ?Ri ?Ri2 ?R e i ?Gm i=1 ? ? ? ? ? ? ? ? ? ?=? ? ? ? ? ?? ?R1 ?G1 иии ?Rn ?G1 . . . . . . ?R1 ?Gm иии ?Rn ?Gm ? ?R12 ? ? e ?R1 ?? . ?? . . ?? ?? 2 e?Rn ?Rn ? ? ? = Jv T ?Rv ? ? (2.4.13) where J is the Jacobian or sensitivity matrix in the eigenvalue problem of the damage detection algorithm. around G = Gc If Gc is a point close to the local minimizer G? , then W (G) can be expressed as W (G) = W (Gc ) + ?W T (Gc ) (G ? Gc ) + 1 (G ? Gc )T ?2 W (Gc ) (G ? Gc ) 2 (2.4.14) where ?2 W (G) current position is the second derivative of Gc . If ?2 W (G? ) W with respect to is positive denite, G evaluated at the W (G) = W (G? ) when ?W (G) = 0. ?W (G) = ?W (Gc ) + ?2 W (Gc ) (G ? Gc ) = 0 To simplify Eq. 2.4.15 the second derivative of ?2 W = Jv T expanding the derivative of Jv T from ?Q is computed from Eq. 2.4.13 as ??R + Jv T ?Rv ?G Eq. 2.4.13 gives 29 (2.4.15) (2.4.16) ? ??Rv = Jv T Jv T ?G 2 ? e?R1 ?R1 ? ? ? ? ? ?G1 ? . . . 2 e?Rn ?Rn иии 2 ? e?Rn ?Rn ?G1 ?Gm иии ? . . . 2 e?Rn ?Rn ? ? ? ? ? ? (2.4.17) ?Gm after further expansion ? = Jv T 2 ?R 1 2?R12 ) ?e?R1 ?G 1 2 n 2?R12 ) ?e?R1 ?R ?G1 и и и (1 + ? (1 + ? . . ? . . . . ? ? 2 ?R 2 ?R n n (1 + 2?Rn2 ) ?e?Rn ?G и и и (1 + 2?Rn2 ) ?e?Rn ?G m m ? ? ? ? ? ? (2.4.18) Eq. 2.4.18 can be simplied by noting that the second matrix term can be decomposed ? to = Jv T ? (1 + ? ? ? ? ? 2 2?R12 ) ?e?R1 .. . 2 (1 + 2?Rn2 ) ?e?Rn ? ? ? Jv ? ? (2.4.19) and nally written as = J v T ?v J v (2.4.20) Using the denition of a Hessian matrix, Eq. 2.4.20 can be written as ??Rv ?Ri2 + e ?Ri Hv i (G) ?G i=1 n 2 ? W = Jv where H i (G) T is the Hessian matrix of Ri . 2.4.5 gives 30 Substituting Eqs. (2.4.21) 2.4.3 and 2.4.8 into Jv T ?R + J v T ?v J v + n 2 ?G = 0 e?Ri ?Ri Hv i (2.4.22) i=1 If the second term in the second order derivative, pared to J v T ?v J v n 2 i=1 e?Ri ?Ri Hv i , is small com- and the Hessian matrix of the residuals, the generalized Gauss- Newton method can be used to solve the convexied least-squares problem. If an element in by Hv i,j J v T ?v J v is denoted by then Jv i,j = Jv i,j n i=1 Hv i,j = and those in m n 2 ?Ri e?Ri 2 ?e?Ri i=1 holds for all 2 ?Ri ?Ri Ri ?G i ?Gj Ri then 2 (2.4.23) (2.4.24) n i=1 ?R2 i Ri ?Gi ?G j for the small residual problem. If Hv i,j are denoted ?Ri2 ?Gi ?Gj The Gauss-Newton method makes the assumption that n e?Rfi ?Ri Hv i 2 ?Ri ?Ri 1 + 2?Ri2 ?e?Ri ?Gi ?Gj i=1 compared to 2 i=1 2 (1 + 2?Ri2 ) ?e?Ri > can be neglected when compared to 2 (1 + 2?Ri2 ) ?e?Ri ? ?e?Ri = 2? 2 Ri2 e?Ri > 0 for all Ri . is negligible Neglecting n i=1 Jv i,j because 2 e?Ri ?Ri Hv i in Eq. 2.4.22 yields the generalized Gauss-Newton method with convexication Jv T ?Rv + Jv T ?v Jv ?G = 0 (2.4.25) Finally, the Gauss-Newton minimization problem with convexication is given as min W (G) = ?v Jv ?G ? ?Rv 22 G 31 (2.4.26) 2.5 Minimization Procedure The eigenvalue sensitivity equations 2.3.1-2.3.11 serve both the forward and inverse problems. The former determines the changes in the eigenvalues from the changes in the stiness parameters. Damage detection is an inverse problem, in which one identies the changes in the stiness parameters from a selected set of n? measured eigenvalues of the damaged structure, where 1 ? n? ? N . By neglecting the higherorder eigenvalue residuals, the resulting system of equations from Eq. 2.4.11 or 2.4.26, involve n? scalar equations with m unknowns and can be written in the form (2.5.1) A?G = F where A, the system coecient matrix, is computed as A?ji = ?Tj ?K ?Gi ?j (2.5.2) ?dj when only stiness updating is used with the natural frequencies alone, A?ji = ?m ?Tj ??j ?G ?j i (2.5.3) ?dj when only mass updating is used with the natural frequencies alone, A?ji = ?Tj ?K ?Gi ?M ? ?j ?G ?j i (2.5.4) ?dj when mass and stiness updating are used with the natural frequencies. puted as 32 A is com- ? ? ? ? A?ji A?ji ? ? ? ?=? ?T j ?K ?M ??j ?G ?Gi i ?dj v j,i + cj,i ?j ?j ? ? ? (2.5.5) and when mass and stiness updating are used with the natural frequencies mode shapes A?ji , in which j = 1, 2, ...n? and A?ji and i = 1, 2, ...m, ? d ? ?c ?d ? ?c ?d ? ? c F = 1 d 1 , 2 d 2 , ... n d n ?n ?1 ?2 T ? (2.5.6) is the normalized dierence of the measured and estimated eigenvalues, and ? ? ? ? ? ?d ? ?c 2 ? 2 F? = ? .. ? . ? ? ?dn ? ?cn ?d1 ?c1 ? ? ? ? ? ? ? ? ? (2.5.7) is the normalized dierence of the measured and estimated eigenvectors, which can be stacked as in Eq.(2.5.5) when used in conjunction with the natural frequencies, and ?G = [?G1 , ?G2 , ...?Gm ]T is the optimal change in the stiness parameters to be found. In the practical application of the method when natural frequencies are used, Fi is normalized as Fi = ?i ? ?di ?di (2.5.8) or the higher natural frequencies will dominate the minimization because the absolute size of the error is larger than that of the lower modes even when the percent error might be small. When the mode shapes are used there is no need for normalization if the shapes are normalized to unit magnitude. 33 2.5.1 Pseudoinverse and Regularization While the gradient and quasi-Newton methods can be used in each iteration to nd ?G in Eq. 2.5.1, where the superscript Z denotes the iteration number, by minimizing a proper objective function, the generalized inverse method is most ecient because it does not involve nested iterations [27]; and the changes in all the stiness parameters ?G are determined from Z Z i (2.5.9) ?GZ = A+ F where ?G = ?G , ?G , ..., ?G , and A is the generalized inverse of A. Using only the rst several measured eigenvalues to detect damage can result in severely underdetermined system equations, i.e., n m in which an innite number of solutions are possible in each iteration. Using the generalized inverse method gives a unique minimum-norm solution such that Z Z 1 Z 2 Z T m + ? (2.5.10) A?GZ ? F = 0 where ?G < ?G for any possible solution ?G = ?G and и is the L ?norm. Even though the mode shapes can be used to improve the conditioning of the system of equations for the slender structures investigated in this research, using the mode shapes doesn't improve the results enough to warrant their measurement or use. This is because there is very little change in the mode shapes due to damage in these structures. The mode shapes are best used for making sure the natural frequencies being compared are for the same modes. The minimum norm solution in Eq. 2.5.2 results in small changes in the stiness Z Z 34 2 parameters in each iteration for a well-posed problem. When the underdetermined system equations are ill-conditioned, which arises because some rows of A can be almost linearly dependent, at least a small but nonzero singular value of A will be obtained. The small singular value leads to a large condition number of the generalized inverse A+ , and therefore will result in a magnication of the small changes in F giving large changes in the stiness parameters. It is well known that the higher the condition number, the closer the problem is to being ill-conditioned [99]. Keeping the condition number small, the problem is kept from being ill-conditioned [99]. The ill-conditioning problem usually does not occur for overdetermined system equations because when the number of rows well exceeds the number of its columns, the almost linear dependency of some rows would not result in a small singular value of A. While overdetermined system equations can be ill-conditioned when some columns of A are almost linearly dependent, this is not addressed here because in all cases shown the system of equations is underdetermined. To ensure the robustness of the iterative algorithm, the truncated singular value decomposition (SVD) [100] is used to regularize the ill-conditioned system equations in the least squares problem [16, 101]. Other regularization methods to solve the linear discrete ill-posed problem could [101, 102] also be used. To use regularization in the least squares problem an estimate of the noise is needed or a regularization parameter needs to be chosen [16]. Using the truncated SVD is a form of regularization and can often give the same results [100]. The amount of regularization depends on the problem. Based on singular value decomposition, one has An? Оm = T Un? Оn? Sn? Оm VmОm = T Un? Оr SrОr VrОm = r i=1 35 si ui v T (2.5.11) and ?1 T A+ mОn? = VmОr SrОr UrОn? = r vi i=1 si uTi (2.5.12) where n? < m is the number of the system of equations, r is the rank of A, Un? Оn? = [u1 u2 ...ur ...un? ] and VmОm = [v1 v2 ...vr ...vm ] are the orthogonal matrices, in which ui (i = 1, 2, ..., n? ) and v i (i = 1, 2, ..., r) are the n? О 1 and m О 1 column vectors of U and V, respectively, and S has positive singular values si (i = 1, 2, ..., r) in decreasing order along the main diagonal and zero entries elsewhere [103]. It is noted from Eq. 2.5.12 that a small singular value will render A+ singular. Setting all the singular values of A less than a threshold value to zero, i.e., si = 0 for i = q + 1, q + 2, ..., r, one has ? A?A= q si,i ui v Ti (2.5.13) i=1 ? Note that A diers from A slightly, but due to modeling error, measurement noise, and truncation of the higher-order eigenvalue residuals, the system coecient matrix A is not exact in each iteration anyway. The regularized A+ is taken to be ?+ A = q vi i=1 si v Ti (2.5.14) ?+ and consequently ?GZ = A F. For each generalized inversion the integer q is determined separately. During the investigation of various damage detection problems it was often noticed that in problems using two models and/or convexication A was often ill-conditioned. It was often noticed that using the convexication method would speed up the problem to be ill-conditioned. 36 2.5.2 Levenberg-Marquardt and Trust Region Methods The Gauss-Newton method presented in Chapter 2.4.1 is for the unconstrained problem, and can have diculty in converging when the Jacobian is rank-decient. If the parameters are normalized and bounded between 0 and 1, the nonlinear least square problem is a constrained problem. The Levenberg-Marquardt method is an improvement on a damped version of the Gauss-Newton method that incorporates the steepest decent method [104]. The steepest decent method uses the gradient direction to move towards a minimum [104]. The Levenberg-Marquardt method is one of the rst trust region methods that aims to minimize the subproblem 1 min J ?G ? ?R22 , ?G 2 subject to ?G ? ?k (2.5.15) where the trust region radius ?Z > 0 at each iteration. This is similar to approximating the cost function in the neighborhood of G at each iteration by MZ (G) = 1 F Z 2 + (?G)T J Z T F Z + 1 (?G)T J Z T J Z ?G 2 2 (2.5.16) For ease of notation the superscript Z will be only shown when necessary during the proceeding derivation. As in the Gauss-Newton method, the magnitude of each step is restricted to ensure the approximation of Q (G) by the model function M (G) is acceptable. When the solution of the Gauss-Newton equation lies strictly inside the trust region, then the step also solves the Levenberg-Marquardt sub-problem, otherwise, there is a? ? 0 such that the solution satises 37 ? ? ? ??G =? (2.5.17) ? ? ? J T J + ?I ?G = ?J T R The damping parameter ? leads to three key observations about the Levenberg- Marquardt method. As long as and thus ?G ? > 0 then the matrix J T J + ?I is positive denite is in the decent direction [105]. When haves like the steepest decent method and when ? ? is large then the method be- is small the method behaves like the Gauss-Newton method [105]. These two key behaviors help the method when the current iterate is far from the solution and gives almost quadratic convergence near the solution [105]. Following the deviation of a nearly exact solution to the subproblem in Eq. 2.5.15 for the model function Eq. 2.5.16, ?G? is a global solution of Eq. 2.5.17 if the following conditions are met: J T J + ?I ?G? = ?J T R ? (? ? ?G? ) = 0 T is J J + ?I (2.5.18) (2.5.19) positive semidenite To nd a solution to the model function it can be seen that if 2.5.19, and 2.5.20 lead to ?G? ? ?. can be written as a function of ? If ? = 0 (2.5.20) ? = 0 then Eqs. then solution for ?G 2.5.18, in Eq. 2.5.19 as ?1 T ?G (?) = ? J T J + ?I J R 38 (2.5.21) such that ?>0 while making Eq. 2.5.20 positive denite while the solution remains in the trust region [105]. To solve for an optimal damping parameter the algorithms presented by Nocedal [105], More [106], and Hebden [107] are used to obtain consequently ?G [105]. The rst step is to solve Eq. ? and 2.5.21 using the Cholesky factorization as follows: J T J + ?I = LT L (2.5.22) LT L?G = ?J T R (2.5.23) LT q = ?G (2.5.24) The damping parameter can be updated as ! ? Z+1 =? + Z ?G q "2 ! ?G ? ? ? " (2.5.25) which is a result of applying Newton's root nding method to the linear approximation of the value of ? needed to make J T J + ?I positive denite as shown in More and Nocedal . The sequence in Eqs. 2.5.22- 2.5.25 is repeated until the dierence between consecutive values of ? < 1e ? 3, though other approaches have been shown by More [106], and Hebden [107]. By using the ratio of the actual reduction (numerator) to the predicted reduction (denominator) Q GZ ? Q GZ + ?GZ ? = M (0) ? M ?GZ Z (2.5.26) gives an indication of the agreement of the model function and the cost function from the previous step. The predicted reduction will always be nonnegative if the ratio Eq. 39 2.5.26 is negative, then the step being taken is not an improvement on the previous position. If the ratio is close to 1 then the step was good and the trust region can be expanded; if it is close to zero then the radius needs to be reduced, otherwise the radius is left the same for the next step. The algorithm used to update the trust region radius is as shown by various authors [104107]. If ?Z < 0.25 then ?Z+1 = 0.25 ?GZ , else if ?Z > 0.75 and ?GZ < ?Z then ?Z+1 = min 2?Z , ?max , else ?Z+1 = ?Z . If ?Z > 0 then update the parameter vector G using the rules described below, otherwise don't update G. 2.5.3 Parameter Updating and Iteration Termination The estimated stiness parameters of the damaged structure are updated by GZ+1 = GZ + ??GZ (2.5.27) where ? is the step size of the increment vector. Depending on the step length ? the objective function might not decrease at each iteration. Since ?GZ is a search direction unless Q(GZ ) is a stationary point then Q(GZ + ??GZ) ) < Q(GZ) ) for all suciently small ? > 0. If the minimization diverges, one option is to shorten the step length, ? of the increment vector, ?GZ in the updating formula. If the increment vector is too long, but it points in a minimizing direction, going just part of the way will decrease the objective function Q(G). With the updated damage detection parameters, the eigenparameters are re-calculated from the eigenvalue problem in Eq. 2.2.1 and the eigenvalue sensitivities are re-evaluated if needed. One subsequently < ? is nds ?GZ+1 . This process is continued until the termination criterion ?G? i i satised for all i = 1, 2, ...m, where ? is the last iteration number. Note that after 40 the Z -th (1 ? Z < ?) iteration, GZi is set to Gh,i if GZi > Gh,i , and to some small stiness value, because the stiness parameters cannot be greater than those of the undamaged structure or negative. The iteration procedure is needed to accurately detect a large level of damage. Another method that was used to terminate the routine is based on direction cosine ?GZ+1 и ?GZ cos (?) = Z+1 Z ?G ?G between the current ?GZ+1 and the previous ?GZ (2.5.28) being less than a specied tolerance because the angle between the search direction and the negative gradient are zero when a minimum has been reached [105]. Two simple termination criteria that were used were based on the maximum number of iterations and the change in the step length of ?G between consecutive iterations. 2.6 Updating Parameter Selection In much of this research beam elements are used for damage detection with either the longitudinal, torsional, or bending degrees of freedom separately or combined. A few notes are in order as to selecting the type of vibration for slender beam type structures with or without tension. 2.6.1 Torsional Vibration The equation of motion for torsional vibration of a uniform rod is 2 ? 2 ? (x, t) 2 ? ? (x, t) = c ?t2 ?x2 41 (2.6.1) where # ?? ?P c= (2.6.2) is the wave speed, ? is the shear modulus, t is time, ? is angular displacement, ? is the torsional constant, ? is the mass density, and P = Ix + Iy is the polar moment of inertia. From the equation for the wave speed it can be seen that for a circular cross section uniform reduction in the radius cannot be detected because the torsional constant and polar moment of inertia are equal for circular sections. The element mass and stiness matrices for a torsional bar element are given as [108] ? ke = ? ? ? ?? ? 1 ?1 ? ? ? L ?1 1 and me = ?P6 L ?? 2 1 ? ? 1 2 (2.6.3) For a rectangular section the area moments of inertia are given by Ix = ba3 ab3 and Iy = 12 12 (2.6.4) and the torsional constant is given by [109] ?= a ! b "3 16 2 2 b ? 3.36 3 a ! b4 1? 12a4 " (2.6.5) If mass is included in damage detection and b is the parameter being updated then the mass and stiness matrices as a function of Gb are ? me = ? ? (Ix (a, Gb) + Iy (a, Gb)) L ? 2 1 ? ? ? 6 1 2 42 (2.6.6) ? ke = ?? (a, Gb) ? 1 ?1 ? ? ? L ?1 1 because the torsional constant is a function of b ? a and b (2.6.7) the derivative with respect to is % ! "3 $ 16 ?? (a, Gi b) ? a Gi b Gi b (Gi b)4 = ? 3.36 1? ?Gi ?Gi 2 2 3 a 12a4 (2.6.8) and expanding the derivative yields % ! "3 $ 4 a 16 b b b) G (G ?? (a, Gi b) i i 3G2i = ? 3.36 1? ?Gi 2 2 3 a 12a4 a ! G b "3 i + 2 2 $ 16 b (Gi b)4 ? ? 3.36 1? 3 a 12a4 % Gi b (4G3i ) (b)4 ? 3.36 1? a 12a4 (2.6.9) (2.6.10) The same analysis follows for the element mass matrix ?Ix (a, Gi b) + Iy (a, Gi b) ? = ?Gi ?Gi ? ?Gi a (Gi b)3 + a3 Gi b 12 43 = a (Gi b)3 + a3 Gi b 12 3G2i ab3 + a3 b 12 (2.6.11) (2.6.12) For a circular segment area the area moments of inertia about the x and y axes are given as Ix and Iy , respectively [109] 1 Ix = r4 ? + 2sin3 (?) cos (?) ? sin (?) cos (?) 4 (2.6.13) 1 Iy = Iy = r4 3? + 2sin3 (?) cos (?) ? 3sin (?) cos (?) 4 (2.6.14) and area is given as A = r2 (? ? sin (?) cos (?)) (2.6.15) where y and x are the centroidal axes (Fig. 2.6.1 [109]). For a circular segment the torsion constant is given by ? = 2?r4 (2.6.16) for 0 ? hr ? 1.0 where h = r(1 ? cos (?)) and h ? = 0.7854 ? 0.0333 ? 2.6183 r ! "2 ! "3 ! "4 ! "5 h h h h + 4.1595 ? 3.0769 ? 0.9299 r r r r (2.6.17) where h and r are given in gure 2.6.2 [109] The main damage detection observation in using the torsional vibration is that for a circular shaft the method cannot locate the damage where the radius is decreased uniformly because the torsional constant and polar moment of inertia are equal and c = & ? ? . For damage that is not symmetric the torsional constant diers from the polar moment and damage can be located. In some cases the mass loss can be more signicant than the stiness change and some natural frequencies will increase, some will remain the same, and some will decrease depending on the extent and location of the damage(s). A benet of torsional vibration is that in most cases of 44 solid beams it doesn't depend on tension so it is an eective choice for tensioned beams. If it is possible that the damage is uniform around the circumference then bending vibration is the best choice because longitudinal vibration isn't eective and is sometimes very dicult to measure. Mass updating is important in the torsional case because depending on the level of damage some of the natural frequencies will increase because the change in mass is larger than the change in stiness and updating E in this case will never reproduce these results. 45 Figure 2.6.1: Circular segment properties. Figure 2.6.2: Circular segmental section properties. 46 2.6.2 Bending Vibration The bending vibration for a beam using Euler-Bernoulli beam theory is given by the partial dierential equation 4 ? 2 w (x, t) 2 ? w (x, t) + c =0 ?t2 ?x4 where (2.6.18) # EI ?A c= (2.6.19) is the transverse wave speed, w is the transverse displacement, and E is the elastic modulus. The element mass and stiness matrices for a Euler-Bernoulli beam are given as [108] ? ? ? EI ? ? ke = ? L ? ? ? 12 6L 4L2 symm ?12 ? ? 6L ? ? ?6L 2L2 ? ? ? 12 ?6L ? ? ? 2 4L ? ? ?AL ? and me = 420 ??? ? ? 156 ?22L 4L2 symm 54 13L ? ? ?13L ?3L2 ? ? ? 156 22L ? ? ? 2 4L (2.6.20) If rotary inertia and shear stress can't be ignored then Timoshenko's beam theory needs to be used as given by [110112] ? 4 w (x, t) m? ? 4 w (x, t) m ? 2 w (x, t) ? + =0 ?x4 AG ?x2 и ?t2 EI ?t2 (2.6.21) where ? is the shear correction factor and m is the mass per unit length. The element stiness matrix is given by [108] 47 ? ? ? ? EI ? ? ke = ? L ? ? ? 0 0 0 ? ? 0 ? 1 ? ? ? 1 0 ?1 ? ?EA ? ? ? + ? ? ? ? 2L(1 + ?) 0 0 ? ? ? ? symm 1 symm ?L 2 L2 4 ? ?1 L 2 ?L 2 L2 1 ?L 2 4 L2 4 ? ? ? ? ? ? ? ? (2.6.22) where the shear correction factor for a circular section is given by [111, 112] ?= 6(1 + ?)2 7 + 12? + 4? 2 (2.6.23) and the shear correction factor for a rectangular section is given by [111, 112] ?= 5(1 + ?) 6 + 5? (2.6.24) For a Timoshenko beam with tension the resulting partial dierential equation (PDE) is ? 4 w (x, t) m? ? 4 w (x, t) m ? 2 w (x, t) P ? 2 w (x, t) ? + + =0 ?x4 AG ?x2 и ?t2 EI ?t2 EI ?x2 and the stiness matrix is given by 48 (2.6.25) ? ke 0 ? ? 1 0 ?1 ? ? ? 0 0 ? ? ? symm 1 ? L ?1 1 2 ? ? L2 ?L ?EA ? ? 4 2 + ? 2L(1 + ?) ? 1 ? ? symm ? ? 0 0 0 0 ? ? ? ? ? 0 2 0 1 LP ? ? ? + ? ? 6 ? 0 0 ? ? ? ? ? symm 2 ? ? EI ? ? = ? L ? ? ? 0 ? 0 0 (2.6.26) ? L 2 L2 4 ?L 2 L2 4 ? ? ? ? ? ? ? ? The mass matrix for a Euler-Bernoulli or Timoshenko beam with or without tension is given by [108] ? ? ? ? ? me = ? ? ? ? 2A 0 ? 0 ? ? 0 2I 0 I ? ? ? 2A 0 ? ? ? symm 2I A (2.6.27) The analysis for the area moment of inertia used for torsional vibration can also be used in the Euler-Bernoulli bending case as well. A very important observation to note is that when performing damage detection using the thickness parameter of the beam, the reduction in thickness cannot be perpendicular to the bending direction 49 being measured or the reductions in I and A will be proportional and the natural frequencies will not change. This result is similar to that shown in the torsional vibration case when the radius of the beam is reduced uniformly. 2.6.3 Longitudinal Vibration The longitudinal vibration of a uniform beam is given by 2 ? 2 z (x, t) 2 ? z (x, t) = c ?t2 ?x2 where (2.6.28) # c= E ? (2.6.29) is the wave speed, E is the elastic modulus and ? is the mass density. The mass and stiness for a longitudinal bar element are given as [108] ? ke = ? ? ? EA ? 1 ?1 ? ?AL ? 2 1 ? ? ? and me = ? ? L 6 ?1 1 1 2 (2.6.30) When performing damage detection using longitudinal vibration for the lower modes the natural frequencies do not change because they do not depend on the area of the beam, only the elastic modulus and density. For a small machined damage or a small cut neither of these values change. This result was experimentally validated on both the rectangular aluminum beam and the circular steel bar. A system that this is useful for is stranded ropes or cables where changes in cross section change the stiness EA. The most eective method for a solid cross section with damage like this is ultrasonic testing due to the wave being reected o the damage. 50 2.6.4 Tensioned System Vibration The tension and EI can be estimated using Eq. 2.6.25 and an optimization procedure. If there are many local minima a good educated guess is needed for the initial values of tension and EI . Once the tension is found the initial stress in the system can be readily estimated for solid but not so easily for stranded ropes because EI changes with the tension. Torsion was selected for solid sectioned tension members because bending vibration is most sensitive to changes in tension, while torsion is only sensitive to initial twist if there is any. Longitudinal vibration can't be used because even though the cross section of the beam changes and this vibration is not dependent on tension the wave speed does not change. One case where this would happen is if two cuts on opposites sides of the beam were to cross the center line of the beam and make the longitudinal wave travel around the cracks. Even this is dicult to detect because longitudinal waves are often hard to excite and are at such a high frequency that they travel an extra length into the boundary. One reason for using A to do damage detection is that for torsional vibration, damage can increase some natural frequencies while decreasing others. This depends on the location of the damage and which anti-nodal point it is near. In this case damage detection using EI will still work, but poorly, and will give all the natural frequencies being lower than what has been measured. In this case the mass change is more signicant than the stiness change. Another reason torsion was chosen was because there are innite beam cross section and tensions that will yield the same natural frequencies and mode shapes. When trying to nd the damage, if the tension is updated based on the area the system the method diverges to the lowest area and tension possible set by the minimum thickness allowable. For a solid section the tension updating analysis was as follows: 51 Starting from the stress equations T A (2.6.31) ? = E (2.6.32) ?= the initial strain in the beam was estimated from the estimated tension T and elastic modulus E as Ti EAi i = (2.6.33) The initial length of the beam before tensioning is estimated as Lf 1 + i Li = (2.6.34) and the initial change in length ?L = Lf ? Li (2.6.35) The total change in length can also be expressed as the summation of the change in element lengths of the discretized beam ?L = n ?lij (2.6.36) j=1 The nal length can be expressed in a similar fashion Lf = n i=1 52 lf (2.6.37) The initial subelement elongation can be expressed as T li AE ?li = n n T f l fj Af j E j=1 ?lj = j=1 The new tension is given by ?LE Tf = lf j n (2.6.38) (2.6.39) (2.6.40) j=1 Afj where n is the number of elements the beam is discretized into, lj and Aj are the length and cross section area of the j th element, respectively. This is the methodology that is implemented in the damage detection program and was experimentally checked and validated. For a beam with thickness t and tension P the natural frequencies are the same as those for a beam with thickness 2t and tension P2 . To estimate the tension in a rope or other tension member with unknown EI an optimization method such as f minseach in MATLAB [94] was used with the formula for the tensioned beam. The cost function was given by ?c ? ?m min ?m 2 (2.6.41) where ?m and ?c are the vectors of the measured and calculated natural frequencies of the system. The error is normalized by ?m to give all frequencies equal weight in the optimization, otherwise the higher frequencies will receive more weight and the optimization will ignore the lower frequencies leading to large errors. 53 2.7 Program Design and Implementation A ow chart of the algorithm is shown in gure 2.7.1. The algorithm as implemented in MATLAB [94] and SDTools [95] code is given in Appendix A. The rst consideration when designing the code was to allow it to be as modular and expandable as possible. The code needed to allow for the user to easily add extra physical phenomenon to the analysis such as tension or temperature dependence etc. A brief overview of all the functions and variables used in the program are give below in their code header function. Driver File: can be used to run a single case as would be when doing experimental damage detection using a series of natural frequencies and/or mode shapes (experimentalDD.m) or multiple cases such as roving the damage location and extent over a structure to gain an understanding of the robustness of the method applied to a given structure (multistudy.m). Main Function (mainsdtdd.m): is the implementation of the optimization routine and is called by the driver le and calls the functions listed below. Input File (inputsdd.m): contains the measurement information such as the eigenparameters, tension, initial non-dimensional parameters, and links to the model les. Model File(s) (multibeam.m): these les contain the nite element models and related functions and data such as cross section updating function and derivatives for the sensitivity matrix if the nite dierence method is not used to calculate the sensitivity matrices. This le also synchronizes the parameters (stiness, cross sections, etc.) across multiple models. In this case a typical beam model is shown. Termination Function (TERMSDTDD.m): implements various termination routines. 54 Model Solver Function (LDDSDT.m): builds the mass and stiness matrices and solves the eigenvalue equations. MAC Function (MACSDTDD.m): sorts the modes and natural frequencies compared to the damage input values in cases where mode switching might occur. Sensitivity Matrix Function (BDKDGSDT2.m): builds the sensitivity matrices using an analytical of nite dierence derivative. System Matrix Equation Creation Function (BAFSDT.m): builds A and F using the old method or convexication with the natural frequencies and mode shapes if they are selected. Solver Function (InverSDT.m): solves the system of equations in 2.5.10 using the pseudoinverse, singular value decomposition, or Levenberg-Marquardt method. G Updating Function (GSDTDD.m): is used to update the damage parameters and check to make sure they are within the limits set by the user. 55 Chapter 3 Beam Damage Detection In this chapter the damage detection algorithm and feature selection criteria will be applied to an aluminum rectangular beam (Fig. 3.1.1) and steel circular beam (Fig. 3.2.1). For the rectangular beam, the rst goal will be to validate the re-written and improved program as outlined in Chapter 2.7 against previous results. The addition of damage detection using a thickness parameter will also be investigated. The use of convexication will also be tested on both experimental and simulated test cases. 3.1 Fixed Free Rectangular Beam The beams used are 6061-T651 aluminum with a cross section of 2.54 cm wide by 0.635 cm thick and a clamped length of 45 cm and an overall length of 50 cm and were cut from the same 2 m piece of stock to ensure that the material properties and dimensions were as uniform as possible. The results using the area, convexication, and singular value decomposition for case four are new while the results using the elastic modulus (denoted E in the gures) serve as a validation of the new code and 57 comparison for the new results. A full discussion of the old results can be found in [27, 28]. 3.1.1 Testing Procedure A short overview of the the testing procedure that was used in [28] is outlined here. To ensure repeatable results the following setup routine was followed. A torque wrench is used to alternately tighten the bolts and when a new beam is tested, the bolts are loosened only one turn to reduce positional changes due to reassembly. The length of the beam is adjusted to the proper length, it is centered in the clamp, and reference lines are drawn on the beam to be used for later reassembly. The bolts are turned roughly Й of a turn by hand and a wrench is then used to turn the bolts roughly 1/8 of a turn. The bolts are tightened in 3 Nm increments to 47 Nm. To reduce mass loading the accelerometer is axed 1 cm from the clamped end of the beam. A computer and signal analyzer are used to collect data from the impact hammer and accelerometer to calculate the frequency response functions. A laser vibrometer was also used for checking the mass loading of the accelerometer. The impact hammer tip was changed depending on the frequency range of interest. Only one accelerometer was used to keep the mass loading to a minimum. When the laser vibrometer is used, it measures a location closer to the center of the beam. During each test three impacts were averaged to ensure that repeatable results were obtained as based on the coherence. A boxcar window was applied to the response and input when testing at lower frequencies because the long measurement time allowed the amplitude of the response to decay suciently. For testing at frequencies above 100 Hz, an exponential window was applied to the response because the amplitude of the response does not decay suciently in the shorter measurement time. 58 3.1.2 Experimental Results and Discussion Dierent scenarios for damage identication were performed experimentally on four beam specimens (Fig. 3.1.1), using only the changes in the rst several natural frequencies. Damage is considered to occur in this paper when the estimated stiness is less than 90% of its undamaged value when using the elastic modulus or 96.55% when using the thickness perpendicular to the bending direction. Note that the stiness parameters are dened as the bending stiness in what follows, and the stiness reduction in this section mainly results from the reduction of the area moment of inertia. When the initial stiness parameters dier from the desired solution, errors in stiness estimation can occur in the experimental investigation, because there are modeling errors and measurement noise and the system equations are underdetermined. When the updated stiness parameters in each iteration are bounded by the corresponding undamaged stiness values, errors in stiness estimation are manifested as erroneous stiness reduction. In what follows, erroneous stiness reduction in excess of 10% when using the modulus or 3.55% when using the thickness of the undamaged parameter value is referred to as erroneous damage. Some erroneous parameter reduction of about 10% of E or 3.55% of the thickness or less is not discussed in what follows. When damage detection is done using the area, the thickness of the beam is selected as the damage parameter. This is the obvious choice because the transverse modes in the thickness direction are used and the modes in the width bending direction don't change because the ratio of mass to stiness loss is linear. In all cases presented below, the nite dierence method was used with a step size of 1E-6. The maximum number of iterations chosen was 200 and the step size was chosen to be 0.1. The minimum change used for the change in G was 1E-8 and the maximum was 1E-12. The value used for the size of the convexication parameter was 59 0.1. These parameters were chosen based on multiple simulations as will be shown in Chapter 3.1.3. The minimum step size used in the nite dierence method was chosen because of the calculation limits of the computer because a larger step size would give more error due to the maximum precision of the machine. This is because the change is cubic when the thickness is changed. The maximum number of integrations was chosen due to time limits. For some simulations, a reduced number of singular values is used due to the condition number of the A matrix being too large for MATLAB to work with because of numerical or experimental noise. At most the necessary reduction in singular values is two. This represents a 58.14% stiness reduction in terms of the area moment of inertia / elastic modulus or a 25.2% reduction in area and 0.08 cm deep was machined from both the top and bottom surfaces of a beam specimen, as shown in gure 3.1.2. (.635cm)3 ? (.635cm ? 2 О 0.08cm)3 (.635cm)3 О 100% = 58.14% .635cm ? (.635cm ? 2 О 0.08cm) О 100% = 25.2% .635cm Scenario 1: (3.1.1) (3.1.2) An undamaged beam was rst tested to verify that the boundary conditions and material properties were correct. The rst ve measured natural frequencies of the beam are given in Table 3.1.1, along with the natural frequencies calculated from the 45-element (90 DOF) FE model. The maximum dierence between the measured and calculated natural frequencies is 0.22%, which occurs for the fourth mode. The damage detection results in Fig. 3.1.3, using the rst ve measured natural frequencies, show that there is some erroneous stiness reduction (5%, 5%, and 3.5%) at about 11 cm, 23 cm, and 35 cm from the cantilevered end using E and 60 no convexication. The damage detection results in Fig. 3.1.4, using the rst ve measured natural frequencies, show that there is some erroneous stiness reduction (11%, 10%, and 47%) at about 11 cm, 22 cm, and 45 cm from the cantilevered end using E and convexication. The reason there is a much larger error at the end of the beam is due to the low sensitivity to changes in stiness. The damage detection results in Fig. 3.1.5, using the rst ve measured natural frequencies, show that there is some erroneous stiness reduction (2%, 2.5%, and 1.2%) at about 11 cm, 22 cm, and 36 cm from the cantilevered end using A and no convexication. The damage detection results in Fig. 3.1.6, using the rst ve measured natural frequencies, show that there is erroneous stiness reduction throughout the beam using A and convexication. Scenario 2: The second scenario demonstrates damage detection on a cantilever beam specimen with a machined section from 9.6 cm to 15 cm from the cantilevered end, with 0.08 cm removed from both the top and bottom. Damage estimation is performed using the 90-element (180 DOF) FE model with 45 groups of two elements and the rst three to ve natural frequencies. Elements are grouped together to reduce the number of unknowns while allowing for more accurate calculation of the higher natural frequencies. Detection results using the elastic modulus with and without convexication are presented in gures 3.1.9 and 3.1.7, respectively. The damage detection results using the elastic modulus without convexication are nearly the same as presented in [28] and were used to validate the new code. The detection results without convexication show an overall detection over groups 4-16, which is about 50% too wide, while the maximum damage at group 13 is 15% larger than it should be. The detected dimensionless modulus using convexication shows a 61 more accurate location result but the extent results are grossly overestimated. The same model is used to perform damage detection using the thickness of the beam by reducing the thickness in the bending direction as shown with (Fig. 3.1.10) and without convexication (Fig. 3.1.8). The location results using the thickness as the detection parameter without convexication are slightly better than the modulus results, while the error in maximum extent is 1.7% greater and there is erroneous damage near the free end of the beam. One reason that error is likely to be seen near the free end of the beam is that the natural frequencies aren't sensitive to change in this location and when reducing the thickness the mass is also reduced, so it is possible that the ratio of mass to stiness in the reduced case is the same as in the healthy case. For scenario three a cantilever beam with a machined section from 24.6 cm to 30 cm from the cantilevered end with the same level of damage as in scenario two is investigated. Damage estimation was performed using a 90-element (180 DOF) FE model with two elements per group and the rst three to ve measured natural frequencies. The experimental results using the elastic modulus without convexication for this case are very good, especially when the rst ve measured natural frequencies are used. The results using the modulus without convexication show that the estimated damage extends from 18 cm from the cantilevered end to 33 cm, with the estimated average stiness reduction from 25 cm to 30 cm being 53% (Fig. 3.1.11). Detection results using the elastic modulus with convexication are presented in gure 3.1.13 and show a more constrained region of damage but an overestimation of the stiness reduction. The same model is used to perform damage detection using the thickness of the beam by reducing the thickness in the bending direction as Scenario 3: 62 shown by the detection results with (Fig. 3.1.14) and without convexication (Fig. 3.1.12). The estimated location results using the thickness are more constrained than the modulus results by about 5 groups but the error in the maximum reduction is about 0.9% larger, while the average reduction over groups 25-30 is 0.8%. The results for thickness detection using convexication show the right location for the maximum damage except for at the free end but there is an overall oset error. The error at the free end is because the sensitivity Scenario 4: Scenario four features a cantilever beam specimen with a machined section from 14.8 cm to 20 cm from the cantilevered end. In this case using only the rst several measured natural frequencies and starting from dierent initial stiness values leads to multiple solutions. The damage estimation was performed using the 90-element (180 DOF) FE model in 45 groups of two elements. As was shown in [28], without information enrichment, the solution using the rst ve measured has erroneous damages at three locations all of the same magnitude. It was shown that this was a possible solution in the sense that the minimum norm condition is satised. Theoretically, with enough information the ill-conditioning problem can be overcome but using higher frequencies is prone to modeling and measurement error, as well as increased variability. One solution to this problem was to initialize the damage estimation with good estimates of the locations and extent of damage or group neighboring elements together for the increasingly determined system equations. To detect the damage within 35-50% of the length of the beam from the cantilevered end by using the rst several measured natural frequencies a known mass was added to the free end of the structure. This modication doesn't increase the number of the system equations but increases the amount of knowns because the same set of parameters is 63 shared between both models. In this example two 5 g masses were attached on both the top and bottom surfaces of the beam at its tip (Fig. 3.1.2). By using the rst four to ve measured natural frequencies of both the original and modied structures, the estimated stiness parameters are shown in gure 3.1.15. The maximum error is 28% at group 19 but the error in location is only two cm on each side. Damage detection using convexication wouldn't converge because the condition number of A was so large it resulted in a computational error and is not shown. When using the thickness for damage detection using seven and eight natural frequencies the location was estimated between 14 and 22 cm for both cases while the errors were -9.6% and 19.8%, respectively (Fig. 3.1.16). In the cases using seven and eight natural frequencies it can be seen that there is erroneous damage at the end of the beam. If 9 natural frequencies are used with six singular values with the last three elements assumed to be healthy, the main damage location is between groups 15 and 21 and the maximum reduction is 36%, which is a 14.4% error as shown in gure 3.1.17. It can also be seen that by considering the last three elements healthy there isn't error at the xed end as in the cases with seven and eight natural frequencies. When the added mass is small, the rst few eigenvalue sensitivities of the modied structure are similar to those of the original structure. This leads to the rst few rows of the system coecient matrix being almost linearly dependent, resulting in a small singular value of and the relatively large generalized inverse or condition number. The treatment of the ill-conditioning problem here follows that in Chapter 2.5.1, even though only six singular values were used at each iteration. Another approach that was also successful was to keep the singular values up to a certain ratio. Even though increasing the total tip mass can reduce the ill-conditioning, it was found that if too much mass was added the eects 64 of stress stiening due to the mass loading would need to be included. This makes the problem much more dicult because the static deection would need to be computed at each iteration and most likely the sensitivities as well. Scenario 5: For scenario ve the cantilever beam was machined from 20 cm to 25.2 cm from the cantilevered end. Damage estimation is performed using the 90-element (180 DOF) FE model, with neighboring elements grouped in pairs, and the rst three to ve measured natural frequencies. When the modulus is used without convexication the error at the location of maximum damage is -18.7% and the location is too wide by two cm on each side as seen in previous scenarios (Fig. 3.1.18). In both the previous thickness and modulus updating cases, the extent is over estimated but in the correct location for the maximum damage is correct (Figs. 3.1.19 and 3.1.21). In the convexication cases using both the modulus and thickness there is also erroneous damage at the free end. When the thickness is used without convexication the error is only 6.4% but is also too wide as was seen in the modulus updating case (Fig. 3.1.20). 65 Figure 3.1.1: Healthy beam setup 66 Figure 3.1.2: Damage beam with tip masses. 67 Measurement (Hz) Finite Element (Hz) Dierence % 25.1 25.08 0.08 157.22 157.21 0.01 439.83 439.13 0.16 858.85 860.76 -0.22 1420.6 1420.5 0.01 Table 3.1.1: Measured versus calculated natural frequencies of the undamaged cantilever beam. 68 Figure 3.1.3: Healthy damage detection using the elastic modulus without convexication using ve natural frequencies. 69 Figure 3.1.4: Healthy damage detection using the elastic modulus with convexication using ve natural frequencies. 70 Figure 3.1.5: Healthy damage detection using the thickness without convexication using ve natural frequencies. 71 Figure 3.1.6: Healthy damage detection using the thickness with convexication using ve natural frequencies. 72 Figure 3.1.7: Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 73 Figure 3.1.8: Scenario 2 damage detection results using three to ve natural frequencies and the thickness without convexication. 74 Figure 3.1.9: Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. 75 Figure 3.1.10: Scenario 2 damage detection results using three to ve natural frequencies and the thickness with convexication. 76 Figure 3.1.12: Scenario 3 damage detection results using three to ve natural frequencies and the thickness without convexication. 78 Figure 3.1.14: Scenario 3 damage detection results using three to ve natural frequencies and the thickness with convexication. 80 3.1.3 Simulation Results and Discussion To gain more insight into the convergence of the estimated stiness parameters and the regions of the beam within which the damage detection is dicult numerical simulations of the experimental scenarios were carried out. The numerical simulations are useful because they are more controllable and have no noise or error associated with modeling or measurement error or variability. From the simulations the ndings concerning the regions of the beam that are dicult to detect damage in between 3550% and 95-100% of the length of the beam from the cantilevered end are conrmed. The four simulations are related to Scenarios 2 through 5 in Chapter 3.1.2, and are hence referred to as Simulations of Scenarios 2 through 5, respectively. The only dierences with the experimental scenarios is that all the simulations use a 50% damage when testing modulus detection or 20.8% damage for the thickness, a 45element (90 DOF) FE model is used, and damage is uniformly distributed over an integer number of elements. Even though there is no modeling error or measurement noise, errors are still possible due to the underdetermined systems of equations and in cases where two models are used, ill-conditioning. The simulations were done by reducing the thickness of the beam instead of reducing the modulus because this is what is done in the experimental cases. In this simulation the damaged section extends from 10 cm to 15 cm from the cantilevered end. For the simulations the rst three to ve natural frequencies were used. When damage is performed using the modulus without convexication the error is 4% at the maximum damage and the damage is too wide by ve elements to the left and two to the right. This location error is very similar to that of Scenario 2 as is the error near the free end of the beam at element 36. When using Simulation of Scenario 2: 88 convexication for the modulus or thickness similar behaviors as were observed in the experimental case are present. When using the thickness without convexication the error at the maximum damage is only 0.25%, which is less than that of the modulus as would be expected because the eects of mass loss are accounted for. It should also be noted that the error at the free end of the beam is present but smaller than the experimental case. The location of damage is nearly the same as for the experimental case as well. Simulation of Scenario 3: For the simulation of Scenario 3 the damage is from 25 cm to 30 cm from the cantilevered end. The location results using ve natural frequencies are similar to those in simulation two in that there are ve erroneous elements to the left and two to the right of the expected damage but this is much smaller than the experimental results. The error at the maximum damage is much more compared to simulation two at 20%. As in previous results, the parameter values found using convexication match the expected results in terms of a smaller damage region and an overestimation of extent. The simulation using the thickness as the damage parameter is about 0.25%, which is similar to that found in the Simulation of Scenario 2. Simulation of Scenario 4: Simulation four is an investigation of the cantilever beam with damage from 15 cm to 20 cm from the cantilevered end. The rst three to ve natural frequencies were used with the addition of a tip mass of 5% of the cantilever beam's mass. The simulation results are much better than those of the experiments due to the lack of noise and modeling error. The simulation results give a good understanding of the eects of noise on the damage detection results. The levels of error in the simulations using the modulus and thickness are of a similar 89 magnitude as those of other simulations. Also, as would be expected, the error using the thickness (Fig. 3.1.31) is less than that using the modulus (Fig. 3.1.30). The nal simulation is of a cantilever beam damage from 20 cm to 25 cm from the cantilevered end. The error location when using the modulus as the parameter without convexication is similar to that of the experimental case but the extent is estimated more accurately (Fig. 3.1.32). The results using convexication are similar to those in the majority of experimental and simulation results (Figs. 3.1.33 and 3.1.35). The results using the thickness as the damage parameter show the same locations of error near the clamped and free ends of the beam as the experimental results (Fig. 3.1.34). This simulation also shows that the results using the thickness is more accurate than those using the modulus. Simulation of Scenario 5: 3.1.4 Conclusions for the Rectangular Beam For the rectangular beam it was found that using the thickness as the updating parameter resulted in more accurate predictions than those compared with just using the elastic modulus. It was shown that using the convexication approach outlined in Chapter 2.4.2 gives a tighter bound on the damage location when used with the elastic modulus but results in an overall reduction in thickness. Due to the lack of sensitivity near the free end of the beam it would be benecial to not use the last few elements for damage detection. 90 Figure 3.1.22: Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 91 Figure 3.1.23: Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. 92 Figure 3.1.24: Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the thickness without convexication. 93 Figure 3.1.25: Simulation of Scenario 2 damage detection results using three to ve natural frequencies and the thickness with convexication. 94 Figure 3.1.26: Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 95 Figure 3.1.27: Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. 96 Figure 3.1.28: Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the thickness without convexication. 97 Figure 3.1.29: Simulation of Scenario 3 damage detection results using three to ve natural frequencies and the thickness with convexication. 98 Figure 3.1.30: Simulation of Scenario 4 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 99 Figure 3.1.31: Simulation of Scenario 4 damage detection results using three to ve natural frequencies and the thickness without convexication. 100 Figure 3.1.32: Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus without convexication. 101 Figure 3.1.33: Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the elastic modulus with convexication. 102 Figure 3.1.34: Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the thickness without convexication. 103 Figure 3.1.35: Simulation of Scenario 5 damage detection results using three to ve natural frequencies and the thickness without convexication. 104 3.2 Circular Beam 3.2.1 Testing To examine another type of beam geometry a steel circular beam with a diameter of 0.0127 m and a length of 0.9144 m with xed-free boundary conditions was tested. The xed boundary was implemented by welding the beam to a at plate and bolting the plate to an optical table as shown in gure 3.2.1. The bending vibration of the circular beam was measured using a laser vibrometer. The bending vibration was measured perpendicular and parallel to the damage. Measurement of the torsional vibration was done by aligning the laser vibrometer o the centerline of the machine damage section and the beam was impacted on the opposite edge of the damage. Damage was introduced starting at 23.9 cm and extending to 37.15 cm from the clamped end of the beam by machining the thickness to 1.096 cm. For the model being used this represents 26% reduction in thickness over parameters seven through 12. The frequency ranges measured were 0-1000 Hz and 0-2000 Hz both with 8192 spectral lines. The natural frequencies were curve t with a polynomial curve t in ME'Scope. A comparison of the FRF's showing how the torsional natural frequencies can be identied is shown in gure 3.2.2. 105 Figure 3.2.1: Test setup for the circular beam with a machined damage. 106 Figure 3.2.2: FRF's of the bending and torsional excitation tests. 107 3.2.2 Simulations Results Simulations were performed before experiments to determine the number and type of natural frequencies needed to detect damage and if any location would need information enrichment similar to the rectangular beam. The location of the damage was randomly chosen and a slot type damage was chosen because if the beam's radius is uniformly decreased then the torsional vibration won't change because the J and ? reductions cancel out each other. A slot was also tried to simulate a more realistic damage. The thickness needed to be used because the torsional natural frequencies can both increase and decrease as shown in table 3.2.1. Both the models for the exural and torsional natural frequencies use 30 groups with four elements per group. For the cases where the torsional vibration is included, it is included as a second model to avoid mode switching. In some cases the last mode solved for can switch from bending to torsion and even by computing the MAC matrix to sort out the modes because they can change order. Using two models also allows for the numbers of bending and torsional modes to be specied with more ease. The eect of recomputing the sensitivity matrix with respect to thickness change was also investigated because the change in cross sectional properties was not linear with respect to the thickness (Fig. 3.2.3) like the reduction in modulus or density. The computation of the derivative at each step is very costly and makes the damage detection process very long, but if the changes in the sensitivity matrices are very nonlinear this is necessary. The rst set of simulations were done using the pseudoinverse method with a step size of 0.01 and a maximum of 300 iterations. A smaller step size was used because of the scale and sensitivity of the damage feature. The scale factor was determined by decreasing the step size from one until the condition number didn't have large oscillations (see Chapter 6.3). In these cases using only the bending modes (Fig. 3.2.4) 108 is more accurate (9.5% error for six natural frequencies) at determining the extent of damage compared to using both bending and torsional natural frequencies (Fig. 3.2.5). The damage estimate using both exural and torsional natural frequencies is often overestimated by about 25% but the location of damage is more conned. In all cases the natural frequencies are underestimated especially at the higher natural frequencies. The addition of a torsional natural frequency to six bending modes makes the condition number of A increase from 8.5 to 14.5, whereas the condition number with ten bending frequencies is approximately 12. The condition number increases with the addition of one torsional mode because the rst bending mode shape is similar to that of the rst torsional mode. In the cases where the sensitivity matrix is recomputed at each iteration there is error at parameter 18 or 19 in all cases but this error is almost within the allowable error of 3% when more than eight bending natural frequencies, and nine or ten bending natural frequencies and one torsional frequency were used (Fig. 3.2.6). Damage detection using only the torsional frequencies was investigated but didn't work as well. Even though the error in the natural frequencies was very small this could be due to some of the damage locations making some of the natural frequencies increase and also the type of damage assumed. Another reason for the results of the torsional damage detection is that the formula for the torsional constant is approximate. In the cases where damage detection was preformed with the Levenberg-Marquardt method a step size of two and a maximum of 300 iterations were used. The magnitude of damage is always underestimated and there is erroneous damage near the xed end of the beam when both the bending alone (Fig. 3.2.7) and bending with torsional (Fig. 3.2.8) natural frequencies are used. In the cases where both the bending and torsional natural frequencies are used the damage is more uniform but the location 109 is wider than in the cases when the bending natural frequencies alone are used. The extents for bending alone and bending and torsion are nearly the identical for the same number of bending modes used. The error where the damage is largest is approximately 11% for both bending alone and bending and torsion. As the number of natural frequencies used increase the error at the free end of the beam decreases as is expected because the higher natural frequencies have a higher sensitivity to change at the free end compared to the lower natural frequencies. 110 Undamaged (Hz) Damaged (Hz) 880.14 865.07 2640.4 2691.4 4400.8 4350.9 6161.4 6092.5 7922.1 7976.6 9683.1 9649 11444 11415 13206 13128 14968 14938 16731 16762 18494 18370 Table 3.2.1: Torsional natural frequency comparison for 28% stiness reduction from 23.9 cm through 37.15 cm 111 (a) Change in A vs. upper radius. (c) Change in Iy (b) Change in vs. upper radius. Ix vs. upper radius. (d) Change in torsional constant vs. upper radius. Figure 3.2.3: Changes in circular section properties as a function of upper radius. 112 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies. natural frequencies. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies. natural frequencies. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies. Figure 3.2.4: Fixed-free rod simulated damage detection using thickness without convexication. 113 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies and one torsional natural frequency. Figure 3.2.5: Fixed-free rod simulated damage detection using thickness without convexication. 114 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies . natural frequencies. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies. natural frequencies and one torsional natural frequency. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies and one torsional natural frequency. Figure 3.2.6: Fixed-free rod simulated damage detection using thickness without convexication and recomputed derivatives at each step. 115 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies. natural frequencies. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies. natural frequencies. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies. Figure 3.2.7: Fixed-free rod simulated damage detection using thickness using the Levenberg-Marquardt method. 116 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies and one torsional natural frequency. Figure 3.2.8: Fixed-free rod simulated damage detection using thickness using the Levenberg-Marquardt method. 117 3.2.3 Experimental Results Damage detection using the experimental results was performed using the same models and properties as in the simulations. The cases using only bending are more accurate at determining the extent of damage but often have erroneous damage at group six near the clamped end and in groups 26 through 30 at the free end. As has been discussed before some erroneous damage is expected at the free end due to low sensitivity. In the rst cases with bending modes alone the erroneous damage at the free end is absent in the cases using 8 or 9 natural frequencies and is still relatively small when using 10. For the torsional cases the damage is severely overestimated in most cases at one group while the other groups are usually about 13% overestimated. The locations in most cases is from about 7-16 for the bending and torsion case and 6-13 excluding for bending alone. The location near the clamped end is better in both cases than in the simulation but the extent is worse in the bending and torsion case. There is erroneous damage near the free end and the damage is not as uniform over the estimated damage location. Compared to the simulations the cases using only bending modes the damage location is more focused but in all the cases using six, seven, and ten frequencies there is damage near the free end. In the cases using eight, nine, and ten there is damage at group six but not at seven as in the simulation cases where the damage is more continuous. The case using ten natural frequencies estimates the extent of damage nearly exactly between 26.5% and 27.25% but the width of the damage is only two elements at the maximum extents which is usually about four groups. When the bending and torsional modes are used together the extent of damage is always overestimated but gives the correct location except for erroneous damage near the end of the beam. The erroneous damage shifts from group 29 to groups 24 or 25 as the number of natural frequencies is increased so this could 118 possibly be used as an indicator that this damage location is a false positive. This is similar to the case where only the bending modes are used. Another indication that the damage near the free end is erroneous is that as the number of natural frequencies increase the extent of damage decreases from 67% to 16%, whereas the extents at the true damage locations increases from 70% to 57%. When the Levenberg-Marquardt method is used with only the bending modes the results are very similar to those from the pseudoinverse method. This is expected because the Levenberg-Marquardt method becomes more like the Gauss-Newton method when the damping parameter becomes small. The extent of damage when only using the bending modes is close to the expected value and the location of the damage is smaller than the true damage. Just like the Gauss-Newton method the LevenbergMarquardt method has a disconnected damage too close to the clamped end. The Levenberg-Marquardt method also has an erroneous damage at the free end but unlike the Gauss-Newton case the damage always stays at group 30. The LevenbergMarquardt method also has less uctuation over the damage region than the GaussNewton method. Two good examples of this are the cases where seven or eight natural frequencies are used. When a torsional natural frequency is added the damage region becomes much wider than when using the bending modes as is also seen in the simulated cases. There is some erroneous damage at group 19 but this error is close to the 3% limit set for ignorable damage due to noise. In the simulated cases using the Levenberg-Marquardt method with ten bending natural frequencies and one torsional natural frequency the extents and location of damage is very similar except for the large uctuation at group six. In general, the experimental cases have larger uctuation in damage extent in the damage region than those using simulations. The experimental results using the Levenberg-Marquardt method and one torsional fre119 quency also mimic the simulations in that they too have erroneous damage near the free end. 120 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies. natural frequencies. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies. natural frequencies. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies. Figure 3.2.9: Fixed-free rod experimental damage detection using thickness without convexication. 121 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies and one torsional natural frequency. Figure 3.2.10: Fixed-free rod experimental damage detection using thickness without convexication. 122 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies. natural frequencies. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies. natural frequencies. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies. Figure 3.2.11: Fixed-free rod experimental damage detection using the LevenbergMarquardt method. 123 (a) Dimensionless length parameter vs. (b) Dimensionless length parameter vs. parameter number using six bending parameter number using seven bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (c) Dimensionless length parameter vs. (d) Dimensionless length parameter vs. parameter number using eight bending parameter number using nine bending natural frequencies and one torsional natural frequencies and one torsional natural frequency. natural frequency. (e) Dimensionless length parameter vs. parameter number using ten bending natural frequencies and one torsional natural frequency. Figure 3.2.12: Fixed-free rod experimental damage detection using the LevenbergMarquardt method. 124 3.2.4 Conclusions for the Circular Beam Using the forward problem it was shown that damage detection with the elastic modulus would not be able to be used if the torsional modes were used. In general, the results when using the bending natural frequencies alone gave a tighter range for the location of damage and better estimate of the extent. The results when adding the rst torsional natural frequency were consistently overestimated for the Gauss-Newton method and underestimated for the Levenberg-Marquardt method. For this system the recommended damage detection approach is to use the Levenberg-Marquardt method without the addition of the rst torsional natural frequency. In general it had a faster rate of convergence with only the bending natural frequencies because using the a torsional mode tends to make the results worse. 125 Chapter 4 Lightning Mast Damage Detection 4.1 Introduction and Previous Results During a two year testing project, approximately 160 masts were tested at 55 Baltimore Gas and Electric (BGE) substations, which includes 45 dierent types of masts in four main categories. The masts ranged from 30 ft to 160 ft tall with two and three section tubular, octagonal tapered, dodecagonal tapered, and tapered tubular cross sections. Models for all but three masts were created because some masts don't have drawings or details on their construction. A testing procedure was developed to produce consistent, uniform results and simplify data management (see Chapter 7.6). Initially damage detection was done by comparing the FRF's of similar masts with each other to see if there were any dierences. If dierences were found, simple model updating techniques were rst used to check if there was any loosening at the base because in many cases this was the main cause of the natural frequency reduction. 126 4.1.1 50 Foot Lightning Mast Shown in gure 4.1.1 is a lightning mast in an electric substation, which is 50 feet tall with two dierent cross-sections connected by a bolted ange joint and an oset spike connected by two bolted joints. The bottom pipe is 8 inch schedule 40 while the top pipe is 6 inch schedule 40. Each section is 22.5 feet long and the spike is 1.5 inch solid steel welded to two anges which are bolted to the top pipe. In this case being able to use only the natural frequencies is advantageous because the measurement of the mode shapes is dicult, time consuming, and prone to wind noise. Two accelerometers were usually used to measure the natural frequencies of the structure because it has closely-spaced modes due to the slight asymmetry caused by the spike. A multiple impact test was used to reduce the eect of noise caused by the wind (see Chapter 7.6). To check to ensure the right modes were being compared, the mode shapes were measured using a laser vibrometer after reective tape was placed on the structure at 3 foot intervals using a man lift. The mode shapes correlated well with the nite element model. Measurements from several similar structures were taken to perform model updating after the initial model was created from available blueprints. After the experimental data was collected a FE model was made in the MATLAB toolbox SDTools using 12 DOF beam elements. For the rst model, all the joints were considered to be innitely sti because the natural frequencies of the model were fairly close to those measured (Table 4.1.1). The rst several measured natural frequencies were used to detect damage and the results in gure 4.1.2 show that the stiness of the elements near the joints, at 8.2 m and 13.72 m from the base, have some damage. This is most likely due to the joints being modeled with innite stiness. The mast is considered to be healthy because most of the similar structures measured had the same natural frequencies. Further research conducted 127 n the Dynamic System and Vibrations Laboratory by He et. al. show that modeling i the mast with shell elements improves the detection results [114]. 4.1.2 Scaled Model To allow for more test scenarios, several scale models of the 50 ft lightning mast of approximately 19 th scale (Fig. 4.1.3) were designed and fabricated in the laboratory. Standard materials were used to allow for multiple structures to be made eciently and maximize interchangeability. In the scale model the spike was welded directly to the top pipe due to the small size of the oset. Only one accelerometer was used to test the scale mast due to concerns for mass loading. The laser vibrometer was also used to measure the eect of mass loading and ensure that is was negligible. The mode shapes were also measured to ensure that the correct frequencies and modes were being compared and that the scale structure was behaving in a similar fashion as the full scale structures. Shown in gure 4.1.3 is one of the scaled models with a uniform damage section between 85 cm and 90.1 cm from the base. To create the damage, 0.0254 cm of material was machined from the surface around the whole circumference of the upper pipe, which is a 40% reduction in the area moment of inertia. Damage detection is performed using a 90-element (40 elements for the lower pipe, 40 elements for the upper pipe, and 10 elements for the spike) FE model. In the model the joints are considered to be innitely sti. The damage detection was started using the parameters of the undamaged structure. The damage estimation results in gure 4.1.4 show that the stiness estimation closely indicates the damage location and extent levels. Some of the damage around 80 cm is thought to be due to the exible joint. During the course of modeling it was found that the welds at the joint needed to be modeled better to capture the dynamics of the system. For 128 the full scale system the welds don't seem to inuence the dynamics as much as the bolted joints. Further research conducted in the Dynamic System and Vibrations Laboratory by He et. al. show that modeling the mast with shell elements improves the detection results [114]. 129 Table 4.1.1: Comparison of the natural frequencies for the rst ten modes of the two section mast. Mode Measured Finite Element % Error 1 1.17 1.18 -0.70 2 5.09 5.23 -2.77 3 7.81 7.95 -1.76 4 16.56 16.57 -0.08 5 29.31 30.42 -3.77 6 45.22 48.07 -6.30 7 47.65 48.07 -0.87 8 53.98 50.91 5.68 9 67.29 68.84 -2.29 10 74.52 79.37 -6.51 130 Figure 4.1.1: Two section lightning mast. 131 Figure 4.1.2: 50' lightning mast damage detection results showing the dimensionless stiness (elastic modulus) vs. the height. 132 Figure 4.1.3: Scale model of the 50' lightning mast with damage. 133 Figure 4.1.4: Damage detection results for the scale mast showing the dimensionless stiness (elastic modulus) vs. the height. 134 4.2 Pipe Setup To be able to investigate lightning mast damage detection and joint modeling, a simplied pipe system with a joint similar to the lightning mast was made. The system consisted of two pieces of 8 ft, 8 inch schedule 40 steel pipe with two octagonal anges having the same bolt pattern and thickness were welded to the pipes to form the joint. The pipes were suspended on two ropes to simulate free boundary conditions as shown in gure 4.2.1 due to limited ceiling height and oor mass in the laboratory that would be needed to make a cantilevered system. This test scenario is carried out to investigate two things: the exibility of the proposed program to be applied to external models from other researchers, in this case the model presented in [113] and the ability to perform damage detection using the density. This scenario also demonstrates the experimental and simulated detection of a parameter increase (a simulation example of stiness increase is given in Chapter 6.1). The density was chosen because of the cost of the fabrication of the experimental system, its multiple uses, and the diculty in accurately imparting a known damage to it. 4.2.1 Testing The pipe system was excited as detailed in Chapter 7.5 using the multiple impact testing method. Frequency response functions were measured from 0-1000 Hz using 8192 frequency lines in two perpendicular directions at the open end of the pipe to determine if there was any asymmetry. Because the pipe system is shared between researchers for both damage detection studies and joint monitoring studies in the lab, several test cycles were conducted to adjust the tension in the bolts such that they were tight enough to match the model in [114]. 135 4.2.2 Experimental Results and Discussion The model used has approximately 10 thousand elements including shell, solid, and rigid elements, and 44 thousand DOF [113]. Shell elements were used for the pipe sections to capture the breathing modes and better simulate the joint based on similar work performed in the lab [113]. Damage was simulated by the addition of 2.27 kg to the pipe over a .075 m length which gave a local density increase of 33.7 %. To add mass, modeling clay that wouldn't dry out was used because it was easy to form around the pipe and had a very low stiness. The mass was added between 1.12 m to 1.195 m as measured from the open / free end of the pipe system. The length of the groups of elements with the same density in the model were 0.1055 m giving 20 groups over the length of the pipe. Each group forms a complete ring but if more accuracy is needed the groups could be partitioned into quadrants, etc. If the mass is assumed to be spread over the nite element group of length 0.1055 m, then the density increase is 24%. The assumption is made that only one side of the system is damaged and thus only one side is used for damage detection in the model. This assumption is made because the system is symmetric about the ange and allows for the method to be tested on a system where only local damage in one part of the system is being searched for. The standard methods for information enrichment could also be used if the whole structure needed to be inspected. For the damage detection process, the modes with the most change were used because some frequencies didn't change signicantly enough to determine if the change was due to the mass addition or just experimental noise. The maximum frequency used was approximately 660 Hz which represented the 32-33 modes counting six rigid body modes. It should be noted that with the exception of approximately four torsional modes and the six rigid body modes, the reset of the modes are orthogonal pairs (see results in Chapter 7.5 136 for FRF's). The rst two bending modes were not used because they are the global bending modes and mostly inuenced by the stiness of the joint. The sensitivity matrices were computed using the nite dierence method and stored for later reuse because of their computational cost. The Levenberg-Marquardt method was used to detect the density change while the stiness matrix held constant. The LevenbergMarquardt method was used because of the reduced number of iterations necessary to reach convergence, in most cases approximately 40 or fewer iterations. The damage detection results show that the amplitude of the mass change is estimated very well in the case where 7 natural frequencies are used but steadily drops o as fewer frequencies are used (Fig. 4.2.2). If the total mass addition is considered then the case using only ve natural frequencies does the best. The extent of damage should be between groups 10 and 11 and is conrmed when six or seven natural frequencies are used but is wider at the base by one group when only four or ve natural frequencies are used. There is erroneous damage centered at group four for all the cases and when fewer frequencies are used there is erroneous detection near the ange at element 20. The erroneous damage near the ange is to be expected because this location isn't very sensitive to the addition of mass. 4.2.3 Simulation Results and Discussion To gain insight into the experimental results a simulated density increase of 25% at Group 11 was done. The simulation gives a more consistent damage extent at group 11 but the error is about 56% at its minimum; if the overall added mass between groups 10 and 11 is considered, then the error is only 12%. The simulation result also has erroneous damage centered around group four and at the ange. The erroneous density increase at group four is much less though than in the experimental cases by 137 about half. 4.2.4 Conclusions The implemented program was successfully used with a model from another research without signicant modication to detect the increase in mass using the density as the damage parameter. The use of the Levenberg-Marquardt method helped to reduce the number of iterations to reach convergence, which was benecial because of the computational cost of solving the model at each iteration. The experimental results were similar to the simulation results in terms of the locations of erroneous damage especially in the cases using four or ve natural frequencies. At the expected damage location the amplitude of the density increase was best for the experimental cases using six or seven natural frequencies. The best overall mass estimation in the experimental cases was for four or ve natural frequencies, while all the simulations gave nearly the same mass increase. 138 Figure 4.2.1: Free-free pipe system setup with mass added. 139 Figure 4.2.2: Experimental results using 4-7 natural frequencies showing the dimensionless density vs. the group number where Group 0 is the free end of the pipe. 140 Chapter 5 Elevator Testing 5.1 Introduction Vertical transportation systems such as traction drive elevators and drum drive hoists employ tension elements of low exural rigidity as a means of suspension and compensation. A typical element used in an elevator suspension is a wire rope. A conventional steel wire rope consists of a number of small diameter strands of wire assembled together to form a composite element. New suspension rope technologies use synthetic ropes [115], such as aramid (Kevlar) ropes, which are constructed of light, high-strength bers [116]. They oer many advantages over steel ropes not only in elevator applications but also in marine and logging applications. In elevator systems the main advantages are the reduction of the ratio of the sheave diameter to the rope diameter, the power consumption, and the rope-borne noise; and longer life. There is also no need for compensation ropes for new technologies [115]. In marine applications synthetic ropes are safer in failure due to the slower energy release of the rope upon breaking, have much higher specic strength, and are the only feasible 142 choice for deep mooring applications where self weight is critical [117, 118]. In the logging industry synthetic ropes help reduce worker fatigue, fatigue related accidents, and injuries due to broken wires [119]. The inertial and elastic characteristics of suspension ropes depend on the rope construction. Despite recent improvements in material design and the introduction of the new technologies, steel and synthetic ropes suer from excessive vibration. The lengths of the suspension and compensation ropes vary when the installation is in operation, and the length variation results in the change of the mass, stiness, and damping characteristics of the entire system. The dynamic stability associated with the lateral vibration of translating media with variable length has been recently studied [120, 121], along with the methods for dissipating the lateral vibration [122]. A scaled elevator model was designed and constructed to simulate the lateral vibration of a suspension rope with variable length [123], and the theoretical predictions for the uncontrolled and controlled lateral vibrations of a suspension rope were validated [124]. Forced lateral vibrations of a suspension rope due to building sway, pulley eccentricities, and guide rail irregularities have also been studied [125, 127]. For both metallic and synthetic ropes, strength and material property testing is usually done using a tension testing machine that is often very expensive [126] and requires a very experienced researcher to ensure for accurate results [117, 126, 128]. Another problem is the correct boundary condition setup because the rope terminations need to be correctly constructed [126] for the test and these terminations often dier from those seen in service [129]. Indirect material property and tension measurements using vibration-based methods are simpler and less time-consuming than direct methods employing static load cells or hydraulic jacks [130]. Knowledge of the bending stiness and bending vibration is important because bending leads to 143 fatigue [130, 131] and abrasion [117, 132]. Damage modeling [117, 133, 134], detection [118, 135], and life expectancy prediction [117, 130, 133] is of growing importance [126] especially when synthetic ber ropes are replacing metallic ropes. In metallic wire ropes the rst signs of damage can be seen from outer strand breakage, which is an indicator of fatigue [135]. Several nondestructive testing (NDT) and nondestructive evaluation (NDE) techniques [136, 137] can be used to inspect metallic wire ropes. For synthetic ropes new techniques are needed to inspect their health because the existing NDT and NDE techniques [8 11, 138 ? ] such as visual inspection, ultrasonic testing, magnetic particle inspection, and electromagnetic testing, are not applicable due to the voids in cross-section of the ropes and the material being nonmetallic or ferrous. In addition, visual inspection is not eective for jacketed aramid ropes because the jackets can show no damage but the inner strands can be damaged due to abrasion [132]. Also, visual inspection is often not possible because it is dicult to have access to ropes either in deep water or high elevations as seen in elevator systems. Continuous monitoring is of great interest because it can be dicult to determine the right inspection intervals [131, 132, 137]. One option to perform continuous monitoring is with natural frequency measurements. It has recently been shown that one can use the changes in the rst several natural frequencies to detect damage in slender structures, the lengths of which are much longer than their cross-section dimensions [27, 28]. In this work, a simplied aramid suspension ropeelevator car system, which can represent elevators, cranes, mine hoists, and other systems where a rope passes over a sheave [116], is constructed. A similar test rig has been used in Ref. [129] to investigate the feasibility of estimating cable tensions in cable-stayed bridges. The methodology was used to nd the resonances of the systems, examine the couplings 144 between the two rope sections separated by a sheave, and determine the material properties of the ropes. It can also be used in the future to develop a model-based damage detection method for ropes using the iterative algorithm described in [27, 28]. 5.2 Aramid Suspension Rope-Elevator Car System Overview The simplied aramid suspension ropeelevator car system in gures 5.2.1 and 5.2.2 consists of ve main parts. The rst part is lower boundary which is attached to the inclined section of the rope. The lower boundary consists of an eye, an eye bolt attached to a steel plate on the frame, and three U-clamps (Fig. 5.2.3). The rope is the second part and is made of three inner Teon coated aramid strands, six outer aramid strands, and a nylon jacket. The cross-section diameter is 9.53 mm and the density is 1437.5 kg m3 . The third part of the system is the sheave which is mounted to the frame and spins freely (Figs.5.2.4, and 5.2.5). The wrap angle of the rope around the sheave is approximately 165 degrees. The mass of the sheave is 5.45 kg, the diameter is 20.8 cm, and the thickness is 7.6 cm. The fourth part of the system is the termination for the vertical section of the rope. The termination is an asymmetric wedge type termination as shown in gure5.2.6. The fth part of the system is the rigid base assembly representing the car to which the termination is bolted. The base assembly in all cases consisted of two C-channel rails, one thin and one thick top plate, and a thick bottom plate to which the inertial shaker could be mounted (Figs.5.2.2b, 5.2.7). Various combinations of steel blocks of mass 24.45 kg each could be added to change the mass of the basecar assembly and tension in the system. In this arrangement the boundary conditions are not simple, and because the cable is 145 so short, they have substantial inuence on the dynamics of the system. To measure the longitudinal natural frequencies, both multiple impact modal hammer and inertial shaker tests were conducted using a standard modal accelerometer along with a dynamic strain gage. A four channel BrЧel & KjТr spectrum analyzer was connected to a PC to record the measurements and control the inertial shaker. One accelerometer was attached to the top of the car or the termination. The strain gage was bonded to the side of the termination with epoxy (Fig. 5.2.6). The strain gage was mounted to the termination because the levels of strain uctuation were within the measurement limits of the device. The strain in the cable was too large for the dynamic strain gage to measure and the gage could not be mounted to the cable rigidly enough to measure strain. An accelerometer was attached to the top of the inertial shaker to measure the input into the system. Between each setup the rope was allowed to unwind and stretch to come to equilibrium. The rst set of tests were to check the linearity of the system to changes in excitation force. The system was excited using a random excitation from 0-800 Hz over a range of voltages from 0.25 V to 1.25 V and the response was measured over the excitation frequency range with a Hanning window applied. The system was rst tested at 0.25 V as measured on the amplier then the voltage was increased in increments of 0.25 V up to 1.25 V. In addition to checking the linearity of the responses measured by the accelerometers, similar measurements were also taken using the strain gage for comparison. The strain gage was able to measure the longitudinal and bending natural frequencies simultaneously because the longitudinal and bending strains were captured due to orientation of the sensor. A multiple impact hammer test was also carried out by impacting the top of the car parallel with the rope and as close to the termination as possible without striking it. For each test 10 averages 146 were taken over 6400 spectral lines. To check the eects of sheave rotation on the system two tests were performed. In both tests the system was excited using a random excitation over 0-800 Hz with the same voltage amplitude each time. The rst test was done on the system as initially congured. In the second test the sheave was clamped so there was little to no rotation. In both tests the frequency response function (FRF) was recorded and the results were overlaid for comparison. To check for slip between the rope and the sheave two capacitance accelerometers were used to check that the motion of the car was in phase with the rotation of the sheave. The acceleration signal from the accelerometer on the top of the sheave was integrated over several seconds to determine if the sheave rotated independent of the rope. The system was also tested at the longitudinal resonance frequencies using a sinusoidal input to determine if there were any frequencies for which the sheave would experience rotation independent of the rope and wouldn't return to its initial static position. The natural frequency of the linearized pendulum model 5.2.1 for the system was used as a check of the measured length. ' ?= g L (5.2.1) where L = L2 + ltermination + lcarCG . The length of the rope L2, is added to the length of the termination ltermination, and distance to the center of gravity of the car lcarCG , to nd the total length of the pendulum. The system was excited by rocking the car along the three dierent axes of the system and orienting the capacitance accelerometer accordingly. The fast Fourier transform (FFT) of the response motions were recorded to determine the natural frequencies of the system. The length of the 147 termination was measured directly and the length to the center of gravity of the car was found by using a computer aided design (CAD) model 5.2.7. 148 Figure 5.2.1: Dimensions of the ropeelevator car system, L1 and L2 are the inclined and vertical rope lengths respectively, M is the mass of the car, and Ms is the apparent mass of the sheave. 149 Figure 5.2.2: Side and front views of the rope system. 150 Figure 5.2.3: Inclined rope termination. 151 Figure 5.2.4: Capacitance accelerometer attached to the sheave. 152 Figure 5.2.5: CAD model of the sheave. 153 Figure 5.2.6: Rope termination with strain gage. 154 Figure 5.2.7: CAD model of car-mass shaker system. 155 5.3 General Testing Results From the FRF's of the system over several voltages it is evident that the system was linear over the parameters tested (Fig. 5.3.1). The natural frequencies obtained from the linearity test using the inertial shaker were also compared to the multiple impact modal hammer test and found to be identical. From the results of the clamped and free sheave test it was found little to no dierence in the natural frequencies of the system other than the rst mode and second modes (Figs. 5.3.2 and 5.3.3). After integrating the time responses from the sheave rotation test it was determined that the sheave didn't rotate relative to the rope and the assumption that the rope can be modeled as being xed to the sheave is valid under the conditions tested. After performing hammer tests it was found that the shaker needed to be removed because the natural frequency associated with the shaker was 10 Hz and this was close to the rst natural frequency of the system and changed the dynamics of the system (Fig. 5.3.4). With the shaker removed and mass added in its place the results from the two excitation methods compared very well. The one major drawback of the hammer tests is that it is dicult to excite frequencies over 1 kHz. Another drawback is that the car swings (Fig. 5.3.5), rolls (Fig. 5.3.6), twists (Fig. 5.3.7), and yaws (Fig. 5.3.8) when struck by the hammer because the impact is not parallel with the rope. The length of the rope found using the pendulum motion matched that of the measurement and longitudinal model for the rst two natural frequencies. The length also matched that found in the nite element (FE) bending vibration model. 156 157 Figure 5.3.1: FRF vs. input voltage. 158 Figure 5.3.2: Clamped vs. un-clamped FRF's 0-180 Hz. 159 Figure 5.3.3: Clamped vs. un-clamped FRF's 0-2000 Hz. 161 Figure 5.3.5: FFT of pendulum mode excitation. 162 Figure 5.3.6: FFT of roll mode excitation. 163 Figure 5.3.7: FFT of torsional mode excitation. 164 Figure 5.3.8: FFT of yaw mode excitation. 5.4 Longitudinal Vibration 5.4.1 Vibration Testing The longitudinal natural frequencies were measured using both the strain gage and accelerometer, both attached to the vertical rope termination at the car-mass. The system was excited using a random excitation with a bandwidth of 0-1600 Hz. The response was sampled over 0-2000 Hz using 6400 spectral lines of resolution. In the case when the system has many more bending modes than longitudinal modes, tests over smaller frequency ranges were conducted around the frequencies where the longitudinal modes were expected. An accelerometer was also placed perpendicular to the rope to measure only the bending vibration as another means of separating the natural frequencies into longitudinal or bending. 5.4.2 Longitudinal Analytical Model The length of the inclined rope was measured from the beginning of the eye termination to the point where the rope contacted the pulley and the vertical rope was measured from the pulley to the top of the termination. There was initially some uncertainty about the apparent mass and radius of gyration of the sheave. The sheave was disassembled and its parts weighed. A CAD model was used to calculate the inertial properties of the assembly after the part weights were updated. The mass of the car was measured using a digital scale and its parts were also weighed separately. The main assumption in the model is that the rope does not slip relative to the sheave. A free body diagram is shown in gure 5.4.1 where Ms = RI is the equivalent mass of the sheave. 2 165 The equations of motion for the longitudinal vibration of the system are 2 ? 2 U1 2 ? U1 ? c = 0 ?t2 ?x21 2 ? 2 U2 2 ? U2 = 0 ?c ?t2 ?x22 (5.4.1) and the boundary conditions are stated as where c2 = EA m Ms U1 (0, t) = 0 (5.4.2) U1 (L1 , t) = U2 (0, t) (5.4.3) ?U1 ?U2 ? 2 U1 (L1,t ) = ?EA (L1 , t) + EA (0, t) 2 ?t ?x1 ?x2 (5.4.4) ? 2 U2 ?U2 (L2 , t) = ?EA (L2 , t) 2 ?t ?x2 (5.4.5) M The free vibrations are expressed as U1 (x1 , t) = Y1 (x1 ) f (t) (5.4.6) U2 (x2 , t) = Y2 (x2 ) f (t) (5.4.7) where Y1and Y2 are the mode shapes (eigenfunctions) and f (t) = Acos (?t)+Bsin (?t) is a harmonic function of time. The frequency equation is given by ! ? sin (?L1 ) " ! " M 2 Ms ? cos (?L2 ) + ?sin (?L2 ) + ?sin (?L1 ) ? ?cos (?L1 ) (5.4.8) m m ! " M 2 О = 0 ? sin (?L2 ) ? ?sin (?L2 ) m A full derivation of the longitudinal vibration equations of motion can be found in 166 Appendix B. 5.4.3 Longitudinal Vibration Results The longitudinal natural frequencies were determined by comparing the FRF's of the system measured with the accelerometers that were perpendicular and parallel to the rope. This was done to identify and separate the bending and longitudinal modes of the system because there were many more bending modes than longitudinal modes. This process along with the zoomed frequency range tests proved to be an eective way of identifying the longitudinal natural frequencies. Once the longitudinal frequencies were identied they were curve t using a polynomial curve t in ME'Scope. When the initial model did not compare well with the measured results, a sensitivity analysis was performed to determine the most sensitive parameters of the model. The rst mode is dominated by the mass of the car (Fig. 5.4.2) and the longitudinal stiness of the rope as is shown for all the modes (Fig. 5.4.3). The second natural frequency is dominated by the apparent eective mass of the sheave Ms, (Fig. 5.4.4). The third, fth, and sixth modes are mainly related to the length of the inclined rope (Fig. 5.4.5). The fourth and seventh modes are related to the length of the vertical rope (Fig. 5.4.6). The rst and second modes are shown to be less sensitive to the length changes than the third through seventh modes. During the model updating process the mass of the car was measured and uniquely determined. The apparent mass of the pulley was unknown as well, but the detailed model helped establish a good rst estimate. With these two parameters considered known and the lengths of the rope between the boundaries measured the model could be solved. The exact value for EA was not known so this parameter was varied rst. The value for the apparent axial elastic stiness EA was rst estimated from the rst two natural frequencies 167 because the length could be measured directly to the outsides of the boundaries. It was determined that the model could not predict either of the rst two modes accurately when the higher modes were correct and vice versa. If the rst two modes were correct the higher modes were estimated to be higher than measured. With this known the boundaries were examined further. A hypothesis was made that the higher frequency modes with shorter wave lengths acted over a longer length of the rope than the rst two frequencies because the waves travel into the boundaries. With this in consideration a second model was made to solve for the higher modes which would be used to estimate the length the wave traveled into the boundary. The material properties used in the rst model were also used in this model. Next, the lengths for the higher modes were updated using the modes associated with each section while keeping EA xed. The updated lengths were then checked against the maximum length possible found by measuring the boundaries to check that what was estimated was indeed physically possible. The length of rope inside the termination boundary was about 9.0 cm and the updated length was 5.5 cm extra. The length of the clamped boundary was 20 cm and the updated length was 9.5 cm extra. For both boundaries this was found to be true so it was concluded that the higher frequency waves travel into the rope under the clamps. This phenomenon has also been seen when testing a thin steel band in the laboratory as well. Some of the length could also be due to the rope wrapping the pulley because the sheave motion is smaller at the higher frequency modes. After both models were updated the measured and analytical results matched within 0.63% (Table 5.4.1). The apparent axial stiness EA was found to be 1.29E6 N, which would give an estimation of E to be 1.8085E10 Pa if the rope is considered a solid section. For comparison, the values given for EA at 8896 N of tension for aramid rope of the same size is between 120E6 N and 147E6 168 N for 1 million and 3 million cycles, respectively [115]. 169 Figure 5.4.1: Longitudinal vibration model. 170 Figure 5.4.2: Normalized frequency vs. car mass sensitivity . 171 Figure 5.4.3: Normalized frequency vs. EA. 172 Figure 5.4.4: Normalized frequency vs. apparent eective sheave mass sensitivity . 173 Figure 5.4.5: Normalized frequency vs. inclined rope length. 174 Figure 5.4.6: Normalized frequency vs. vertical rope length. 175 Table 5.4.1: Analytical and experimental longitudinal natural frequencies. Models Experiment % Error 6.6 6.6 0 111 110.3 -0.63 560 560.2 0.04 949.3 950 0.07 1107.7 1108 0.03 1657.7 1658 0.02 1886 1887.5 0.08 176 5.5 Bending Vibration 5.5.1 Bending Vibration Modal Testing To measure the transverse mode shapes of the system a multiple impact test is used. The hammer was roved over the structure so that the accelerometers and strain gage would not have to be moved and this allowed for the eects of mass loading by the sensors to be minimized. For the transverse case the accelerometer on the inclined rope was mounted 1.2 m from the lower boundary and was axed using zip ties. The test had a bandwidth of 0-800 Hz and used 3200 spectral lines and ve averages with a Hanning window for both the input and response signals. One accelerometer was axed perpendicular to the inclined rope. The other accelerometer was axed perpendicular to the vertical rope and the strain gage was xed to the termination in the vertical direction parallel to the rope. 5.5.2 Finite Element Modeling of Bending Vibrations The inclined rope was measured to be 2.93 m from the top of the boundary where the rope is clamped. The vertical rope was measured to be 1.64 m in length. The car mass was 183.1 kg. The material properties of the initial model were assumed to be the same as those used for the longitudinal vibration. The exact value for EI was unknown so this would also need to be updated. The area and second moment of inertia were estimated by measuring the cross section of the rope with calipers. This is only an estimate because some of the rope's thickness is due to the nylon jacket, there are voids between the bers, and the geometry is not uniform. The model used tensioned beam elements to model the inclined and vertical ropes, rigid bar elements and a mass element with the inertial properties of the sheave to model 177 the sheave, and a mass element to model the car. The rope was modeled as being rigidly connected to the sheave. This was achieved by linking the degrees of freedom from the beam element that would be connected with the sheave to the rigid link that extended from the point mass the radius of the sheave. The tension in both spans was considered to be the same because the pulley is free to rotate when the system is being loaded. The vertical rope is modeled using 24 elements and the inclined rope is modeled using 45 elements. To check that the model had converged a model with 200 element for each rope was created. The sheave only had one rotational degree of freedom and the lower boundary of the inclined rope was assumed to be xed. To try to improve the rst nite element model several steps were taken. The rst was to modify the boundary conditions by decreasing the rotational stiness of the lower boundary where the rope has an eye termination. The second was to include the mass of the three clamps by adding lumped masses of 157 g. The third was to include the stiness of the additional rope and the clamps. The mass of the accelerometers (10.6 g) on the vertical and inclined ropes were included as lumped mass elements. In this model 200 elements were used for each section of rope to allow for more precise positioning of the masses and added stiness for the section of rope under the clamps. 5.5.3 Analytical Decoupled Models for Bending In the following analytical models the inclined and vertical sections of the rope are modeled as being clamped-clamped and clamped with an end mass, respectively. The equation of motion for bending vibrations of an axially loaded beam is [139] ? 4V P ? 2 V (x, t) ? ? 2 V (x, t) + + =0 ?x4 EI ?x2 EI ?t2 178 (5.5.1) where V is the displacement with respect to the position along the beam x and time t. The bending stiness is EI , the load P is considered positive for compressive loads, and ? is the mass per unit length of the beam. The frequency equation for the clamped-clamped beam is given by 2 2 2? (1 ? cosh (?1 ) cos (?2 )) ? k sinh (?1 ) sin (?2 ) = 0 (5.5.2) where and ? = ?l ,x = xl ,? = ?l, ? ?1 = ?? 2 # k + 2 and 4 P EI ?? 2 = EI k2 = (5.5.3) ?4 (5.5.4) k = kl are the nondimensionalized parameters and ?1/2 k 4 +? ? 4 ? and ?2 = ? # 2 k + 2 ?1/2 4 k 4 +? ? 4 (5.5.5) are the real roots of the characteristic equation [139]. The frequency equation for the clamped-end mass beam is given by [139] 4 4 4 2 2 ? 2? + ? k sinh (?1 ) sin (?2 ) ? 2? k cosh (?1 ) cos (?2 ) + 2 ?T ? ?12 + ?22 (?1 cosh (?1 ) sin (?2 ) ? ?2 sinh (?1 ) cos (?2 )) = 0 where ?T = beam MB = ?l. MT which is the ratio of the tip mass MB MT to the total mass of the ?n for ? in equa"1/2 4 + ?n and ?2n = The mode shapes are found by substituting tions 5.5.2 and 5.5.6 yielding ?n4 = 2 ??n EI ! , ?1n = 179 2 ? k2 + (5.5.6) & 4 k 4 ! 2 k 2 + & 4 k 4 + 4 ?n "1/2 then solving the matrix equation associated with the boundary conditions of the system gives ? (x) = cosh (?1n x) ? cosh (?2n x) (5.5.7) ! " cosh (?1n x) ? cos (?2n ) ?1n ? sinh (?1n x) ? sin (?2n x) ?2n sinh (?1n x) ? ?1n sin (?2n ) ?2n for the clamped-clamped beam and ? (x) = cosh (?1n x) ? cosh (?2n x) (5.5.8) ? ? 2 ? " ! cosh (?1n x) + ?2n 2 cos (?2n ) ?1n 1n ? ? sin (?2n x) sinh (?1n x) ? ? sinh (?1n x) + ??2n sin (?2n ) ?2n 1n for the clamped-mass beam [139]. To solve the transcendental characteristic equations (5.4.7, 5.5.2, and 5.5.6) for the analytical models, a script in MATLAB was programmed that combined a simple line search to determine an interval for the f zero command to nd the root in the selected interval [140]. A full derivation of the bending vibration equations of motion can be found in Appendix C. 5.5.4 Bending Vibration Results The modal parameters were extracted using a multi-degree of freedom polynomial using the curve tting software tools available within ME'Scope package. The damping for all the modes was less than 0.7% and the phases were very near 0 or 180 degrees. The mode shapes were converted from complex to real by assigning the magnitudes with phases near 0 to be positive and those near 180 to be negative. The mode shapes 180 were extracted separately for the inclined and vertical ropes because the modes associated with the opposite rope were not excited very well and were generally noisy with low coherence. The displacements from the nite element model for the inclined rope were rotated in the plane of vibration to obtain the magnitude so that they could be compared to the measurements. In both cases the mode shapes were normalized to obtain unit vectors. The mode shapes were compared using the modal assurance criterion (MAC) [19] and found to correlate very well with only two of the higher modes having MAC numbers below 90% (Tables 5.5.1, 5.5.2, 5.5.3, and 5.5.4). The initial bending model did not match well with the measured results. As a rst approach a sensitivity analysis was carried out using the nite element model to investigate the causes of these dierences. Optimization was not used because the boundary conditions were also studied by changing some from xed to pinned. When the length of the vertical rope was varied the modes that changed were 2, 5, and 8 (Fig. 5.5.2). The higher modes are more sensitive than the lower modes. When the length of the inclined rope was varied modes 1, 3, 4, 6, and 7 changed with the seventh mode being the most sensitive (Fig. 5.5.2). As with both the vertical and inclined ropes, when EI was varied the higher frequencies were found to be more sensitive (Fig. 5.5.3). When the mass of the car and tension due to the car were varied the lower modes were shown to be more sensitive (Fig. 5.5.4). From the sensitivity analysis it can be seen that the two sections of rope can be considered to be decoupled for small amplitude vibrations. There is little dierence between the natural frequencies found between the nite element model with the coupled ropes and the uncoupled ropes modeled separately (Fig. 5.5.5). The uncoupled analytical solutions match nearly exactly with the initial nite elements solutions and this gives support for the results found using the nite 181 element method. This also indicates that there is little to no coupling between the bending modes between the two spans. There is some dierence between the mode shapes of the initial FE and analytical models and the measurements for the inclined section due to the mass of the accelerometer and the stiness of the lower boundary condition (Figs. 5.5.8, 5.5.9, and 5.5.10). The correlation of results between the analytical and FE models for the vertical rope are better because the boundary condition at the car end was closer to a rigid connection than that of the inclined rope at the lower boundary (Figs. 5.5.5, 5.5.6, and 5.5.7). The same reasoning is also thought to apply to the natural frequency predictions. The mass associated with the accelerometer made a very large dierence especially for the third mode of the vertical rope. A similar result was found for the inclined rope for the rst mode. The updated frequency results are given in table 5.5.5. The average absolute natural frequency error decreased by 29% and now only one frequency has an error over two per cent. Another possible source of error for the natural frequency estimation is that the material properties and behavior of the rope are not exactly known especially for material and geometric behaviors at higher frequencies. The MAC matrices give positive evidence that the multiple impact hammer test worked well for exciting the system for mode shape measurement. The updated nite element model of the system (Fig. 5.5.11) and the rst three bending mode shapes of the vertical and rst ve mode shapes of the inclined ropes are shown in gures 5.5.12, 5.5.13, 5.5.14, and 5.5.15. Even though the mode shapes of the updated nite element model show some degradation compared to the initial model the updated model is considered to be better because the natural frequencies overall have less error and the model includes more of the physical properties of the true system. To further improve the test results more points could be measured in order to more accurately 182 capture the mode shapes. The accelerometer should also be placed closer to one of the boundaries. A stier boundary that didn't rotate or use clamps could also be used to reduce some of the unknown boundary eects. The bending stiness EI was found to be 9 N m2 which gives an estimate of E as 2.22e10 Pa if the rope is considered a solid section. A direct comparison should not be made between the estimated E for the axial stiness and the estimated E for the bending stiness due to the dierences in the geometry of the rope and because the tension in the two systems was dierent for the two test setups. 183 Table 5.5.1: Inclined rope MAC matrix for original FEM FEM / Measured 23.67 47.63 71.37 23.8 0.97 0.02 0.06 48.01 0 0.91 0 73.03 0.01 0.01 0.84 Table 5.5.2: Inclined rope MAC matrix for updated FEM FEM / Measured 23.67 47.63 71.37 23.24 0.96 0.02 0.07 47.83 0.01 0.9 0 72.79 0.01 0 0.81 Table 5.5.3: Vertical rope MAC matrix for original FEM. FEM / Measured 42.56 87.4 136.59 42.12 0.99 0.04 0 88.26 0 0.93 0 130.5 0 0 0.96 Table 5.5.4: Vertical rope MAC matrix for updated FEM. FEM / Measured 42.56 87.4 136.59 42.12 0.99 0.04 0 88.26 0 0.93 0 130.5 0 0 0.96 184 Figure 5.5.1: Bending vibration sensitivity of the vertical rope. 185 Figure 5.5.2: Bending vibration sensitivity of the inclined rope. 186 Figure 5.5.3: Bending vibration sensitivity to EI. 187 Figure 5.5.4: Bending vibration sensitivity to car mass. 188 Figure 5.5.5: First mode shape of the inclined rope. 189 Figure 5.5.6: Second mode shape of the inclined rope. 190 Figure 5.5.7: Third mode shape of the inclined rope. 191 Figure 5.5.8: First mode shape of the vertical rope. 192 Figure 5.5.9: Second mode shape of the vertical rope. 193 Figure 5.5.10: Third mode shape of the vertical rope. 194 195 Mode Type FEM Update Inclined 23.24 Vertical 42.56 Inclined 47.83 Inclined 72.79 Vertical 86.35 Inclined 97.05 Inclined 127.6 Vertical 132.7 FEM Experimental FEM New/Exp FEM / Exp Error Analytical Decoupled AD/Exp Error FEM/AD Dierence 23.8 23.67 -1.82 -0.56 23.8 -0.55 -0.01 42.56 42.12 1.04 -1.04 42.56 -1.04 0 48.01 47.63 0.42 -0.81 48 -0.78 -0.03 73.03 71.37 1.99 -2.33 73 -2.28 -0.05 87.4 88.26 -2.16 0.97 87.4 0.97 0 99.23 97.09 -0.04 -2.21 99.23 -2.2 0 126.95 127.92 -0.25 0.76 126.95 0.76 0 136.59 130.5 1.69 -4.67 136.61 -4.68 0.01 Table 5.5.5: Bending vibration frequency comparisons. Figure 5.5.11: Undeformed nite element model. 196 Figure 5.5.12: First modes of the inclined and vertical ropes respectively. 197 Figure 5.5.13: Second and third modes of the inclined rope respectively. 198 Figure 5.5.14: Second and fourth modes of the vertical and inclined ropes respectively. 199 Figure 5.5.15: Fifth and third modes of the inclined and vertical ropes respectively. 200 5.6 Conclusions Using the aramid roped test stand new testing methods, modal models, and a modal survey were completed successfully. The system was shown to be linear up to an excitation voltage of 1.25 V. The measurement of both the bending and longitudinal modes using the multiple impact hammer technique and dynamic strain gage were of the same quality as those obtained with the inertial shaker and accelerometers. The length estimate using pendulum modes were comparable with that of directly measuring between the boundaries. The longitudinal modes were experimentally shown to be a coupling of both sections of the rope. During the model updating process it was determined that two models were needed: one model for the rst two modes and another for the higher modes. The transverse modes of the rope were shown both experimentally and numerically to be decoupled. The transverse natural frequencies and mode shapes measured using the multiple impact testing technique compared well with both the FE and analytical modes. After model updating of both models the material properties of the aramid cables were identied and for the axial stiness to be reasonable when compared with those given in the literature. The results of this investigation could also be applied to the determination of the tension in elevator suspension ropes. With the FE model of the system complete research on performing damage detection on lift ropes can begin. 201 Chapter 6 Tensioned Member Damage Detection The goals of these simulations and experiments is to demonstrate damage detection on a tensioned beam and rope. In this research the rope will be modeled using the same principle used and validated in Chapter 5. The original goal was to do damage detection on the elevator system shown in Chapter 5 but there was not enough time during the author's visit to the University of Northampton. A set of simulation for the system is demonstrated for the system to give proof for the reduction of step size used throughout this thesis because this was the rst system this idea was tested on. From simulations, model updating and correlation, and sensitivity analysis it was found that the bending and longitudinal modes for the aluminum beam couldn't be used, so the second highlight is to perform damage detection using the torsional natural frequencies. As with the round bar in Chapter 3 for the beam, some of the torsional natural frequencies increase so the thickness needed to be used. Due to the diculty in exciting and measure the torsional natural frequencies for a rope and because good 202 results were obtained in Chapter 5, the longitudinal modes of the rope are used for damage detection. The last unique feature of this test is the additional data provided by shortening the beam by moving one of the boundary conditions. This last feature allows for the testing of the method using two models with some common elements updated simultaneously to detect the location and extent in a similar fashion as the cantilever beam test using the addition of a tip mass. 6.1 Elevator System Simulations In the following simulation an elevator system similar to that tested in Chapter 5 is studied. The longitudinal natural frequencies were chosen because they can be modeled and measured very accurately as previously shown. The stiness properties of the rope depend on the tension but for the system being examined, the tension remains constant due to the suspended mass and the tension eect on the stiness properties of the rope can be ignored. The rope can be eectively approximated by a rod element for the longitudinal natural frequencies. In the simulation due to the symmetry of the system for the higher modes (3-?) two systems need to be used for information enrichment because both sections of rope are essentially symmetric systems. The rst system consists of 100 bar elements each 0.0498m in length and the other system has 90 bar elements 0.0498m in length with a 1.94 kg sheave mass and a 183 kg car mass (Fig. 6.1.1). The system with 90 elements is made by shortening the rope below the sheave. For the simulation the third, fourth, and fth natural frequencies are used due to the rst two modes not being based as much on the rope stiness as the masses of the sheave and car. As shown in Chapter 5, the third and fth 203 frequencies are mainly due to the inclined section and pulley mass and the fourth is from the vertical rope and car mass. The masses and EA are kept constant in each simulation though they can be dierent if needed because the G parameter is dimensionless. During the investigation a damage scenario was found that would not converge if ? = 1 but would converge if ? = 0.3 or ? = 0.1 when there was 50% damage over the length 0.4m - 1.4m or elements 8-28. This fact was discovered by looking at the condition number of A in the equation ?G = AT F and the estimated damage at each iteration where it was noticed that the damage seemed to cycle between three or four dierent cases and then repeat. Even though this approach is very crude, it proved extremely useful in all the simulations and experiments presented in this research. In gure 6.1.2 it can be seen that the condition number oscillates and at some iterations becomes very large. A typical example of one of the damage plots is shown in gure 6.1.3. When the step size is decreased the condition number reaches a maximum of approximately 1.8E5 (Fig.6.1.4) compared to almost 7E6 for the full step size case. It can also be seen in gure 6.1.4 that the condition number does oscillate like with the full step and near iteration 30 the condition number decreases after the method begins to nd the correct location of the damage at iteration 20. The condition number graph shown in gure 6.1.4 is dierent than the typical behavior that was witnessed in just about all the simulations and experiments performed in this thesis where the condition number increases and then levels o when the correct location is found. In just about all cases the correct damage location is identied rst and then the extent of damage is determined. In gure 6.1.5 it can be seen that the expected solution is obtained with the reduced step size. Another investigation that was done with the step size was to verify that the same solution was obtained in both cases for a system where both step sizes would converge. 204 Damage was roamed over the system at a level of 40% over the ve elements noted in the gure 6.1.6. For a step size of one, the termination is the maximum change in G between iterations less than 0.001 with the maximum iterations of 200. For a step size of 0.1, the minimization is terminated when the maximum change in G between iterations is less than 0.0001 or 1000 iterations. To achieve the same results using a smaller step size the change in G between iterations the must be decreased by the same factor as the step size. If the max change in G between iterations remains the same, the result will not be the same because smaller steps are taken and more steps are needed. The largest dierence between a step size one and 0.1 in all the cases was in case 71-75 with about .8% dierence (Fig. 6.1.6). In general, the dierences are mainly around the areas where there are large changes in the dimensionless stiness from one element to the next. The case where the damage is in groups 71-75 needs more investigation because there is a large error near 4.5m, which is on the vertical rope just below the middle (4.1m). During the investigation shown in Chapter 5 when the aramid rope was cut the natural frequencies increased. This behavior is thought to be due to the strands of the rope becoming straighter, which increased AE . In this case the E part of AE increases more than the associated decrease in A. For the simulations the rst ve natural frequencies were used. The maximum stiness was set at 1.5 times the healthy value. The Gauss-Newton method was used with a step size of 0.1, the termination for the maximum change in G between iterations was 0.0001, and a maximum of 500 cycles were used. The damage used was a 20% stiness over ve elements because this gives results close to those measured. The label numbers give the rst stiened element from the top of the rope (element number zero) where it is clamped (Fig. 6.1.7). The results show the correct location but in most cases the stiness increase 205 is only about 8% compared to the true value of 20%. The maximum stiness location is correct in all cases but the width of damage is often 15 elements too wide. 206 Figure 6.1.1: Simplied model of the scaled elevator system shown in Chapter 6. 207 Figure 6.1.2: Condition number at each iteration with a step size of 1.0. 208 Figure 6.1.3: Damage detection results showing the percent reduction in elastic modulus vs. length with a step size of 0.1. 209 Figure 6.1.4: Condition number at each iteration with a step size of 0.1. 210 Figure 6.1.5: Damage detection results showing the percent reduction in elastic modulus vs. length with a step size of 0.1. 211 (a) Damage detection results for a system with 40% damage over elements 21-25. (b) Damage detection results for a system with 40% damage over elements 71-75. Figure 6.1.6: Comparison of step sizes 1 in orange and 0.1 in blue with the error in yellow. 212 (a) Damage detection results for a system with 20% stiness increase over elements 1-10. (b) Damage detection results for a system with 20% stiness increase over elements 51-59. Figure 6.1.7: Damage detection using the elastic modulus for a stiness increase of 20%. 213 6.2 Tensioned Beam 6.2.1 Experimental Setup and Testing The aluminum beam used in this section has a cross section of 0.0127 m wide by 0.0031 m thick with a maximum length of 3.27 m (Fig. 6.2.1). To apply tension to the beam the mechanism shown in gure 6.2.2 is used. After tension was applied, the top of the clamp was bolted down. During initial testing it was found that the band had unwanted motion at the boundary so a large mass was placed on top of the clamp. Once the beam was secured a dynamic strain gage or laser vibrometer was used to measure the bending, longitudinal, and torsional vibrations. The different vibration modes were used to estimate the material properties of the band using formulas presented in Chapter 2. For the bending vibration the tension was unknown, so a least squares curve t was used as a cost function to estimate the tension with the elastic modulus estimated from both the longitudinal and torsional vibration natural frequencies. Once the tension was estimated, one boundary was then moved to shorten the rope to obtain more information about the system when doing damage detection because the system was symmetric. For the shortened beam the measurement procedure was repeated using the single impact hammer test. The tension was then re-estimated using the optimization procedure with the previously estimated material properties. The longitudinal vibration was used to estimate the length accurately. If the tensions were the same, the boundary was moved to the initial position and the tension measurement procedure was repeated to ensure the tension remained constant when the boundaries were moved. If the tensions were dierent, the boundary was moved back to the initial position and the tension verication procedure was repeated. Once the model and test were correlated damage 214 detection could be carried out. In all tests the band was excited using the single impact hammer. The natural frequencies were curve t with a rational polynomial curve t. 6.2.2 Model Creation and Parameter Selection The beams were modeled using 282 or 327 beam elements with only the torsional degree of freedom retained. The number of elements was chosen such that the diering lengths could be broken evenly into elements of approximately the same length. The beam was made into 94 and 109 groups of three elements respectively to reduce the number of unknowns, the time, and computing power needed to analyze the problem. The model was updated by using the damage detection method to the measured healthy frequencies. Through forward modeling it was found that when the thickness of the beam is reduced some of the natural frequencies increase while some decrease depending on the location of the damage. These increases and decreases are due to the nodal and antinodal points of the mode shape and in some cases the reduction in mass has more eect on the vibration than the stiness reduction. These changes can be seen by comparing the torsional wave speeds of two cross sections one slightly thicker than the other but of the same width. Using a nite element model of the beam, this can be seen for the experimental damage case to be presented as well (Fig. 6.2.1). A more interesting example can be seen if damage detection is tried using the bending natural frequencies. For example, using a 3 m beam with the same cross section and material properties as the test setup. If the thickness is reduced by half and the tension reduced by half, the natural frequencies computed using the nite element method are the same (Fig. 6.2.2). When the analytical characteristic equations for 215 a tensioned rectangular beam with full thickness and tension and й thickness and й tension are plotted, there is very little dierence in the roots (Fig. 6.2.3), much less than can be accurately measured repeatably. If the beam has 10% damage over 3.6% of the length, the tension is reduced 4 N and the analytical characteristic equation changes very little (Fig. 6.2.4). In hindsight, these three facts provided insight into the parameter selection process. When damage detection was attempted for the tensioned beam using the thickness and tension updating based on the cross-sectional area of the elements, the nondimensional thickness was reduced to the minimum allowable thickness. Using the thickness without tension updating did not work because damage alone could not reproduce the frequency changes the way a change in tension does because the tension dominates the stiness of the system. 6.2.3 Experimental Results The experimental damage detection was tested by removing 10% of the thickness from the top of the beam from 1.20-1.35m (group numbers 40-45) from the xed boundary (Fig. 6.2.5). Damage detection was done using the Gauss-Newton method with only the elements shared by both models. Some of the erroneous damage can be attributed to moving the boundary condition to shorten the beam for information enrichment. To check that the boundary conditions and experimental setup worked correctly, damage detection was performed using the healthy natural frequencies (Figs. 6.2.3 and 6.2.6). The updated healthy frequencies show that the movement of the boundary conditions is not perfectly accounted for. The results in table 6.2.4 show that the natural frequencies change very little between the healthy and damaged cases. Comparing gures 6.2.7 and 6.2.8 it can be seen that adding a fourth natural frequency of each system improves the results greatly near both boundary conditions but much 216 more so near the boundary closest to group 109 as shown in (Fig. 6.2.7). Another important observation is that the damage found using four natural frequencies for each system results in a more focused damage region by ve groups but the overall extent isn't much more (0.9871 vs. 0.9857). The damage is greatly underestimated but there is very little error in the natural frequencies, which is likely due to the overestimated damage location. 6.2.4 Simulations The simulations are used to nd the best approach and possible bad locations that are hard to detect. They show that even though the boundary conditions are perfect there is still some erroneous damage along the beam (Figs. 6.2.9 and 6.2.10). The simulations still have an error of between three to ve percent in terms of extent of damage. Similar to the experimental results, the width of the damage is decreased by ve groups when adding the fourth natural frequencies. The change in the extent of damage is much larger though, when compared to the experimental results (0.0014 vs. 1.69). Another noticeable dierence is that the case using four natural frequencies for each model has one more erroneous damage location than the case using three natural frequencies for each system. A small part of this error is due to the sensitivity matrix only being calculated at the beginning iteration of the damage detection process because the sensitivity matrix changes with each change of the system. The sensitivity is not recomputed at each step due to the high computational cost. One very important point to note is that 94 equations are being solved with only six or eight knowns, so the system of equations is severely underdetermined. The systems are still well conditioned though, with a condition number about 7.5 for the system with six natural frequencies. The dierence between the systems due to the length 217 change are great enough so the system of equations is full row rank. 6.2.5 Conclusions The tensioned beam test case demonstrates an approach to parameter and vibration type selection due to lack of sensitivity to bending and axial stiness. This test case shows that damage less than 10% can be found but noise and uncertainty starts to make accurate damage detection dicult. Even though the natural frequencies had little error after damage detection for the experimental results the extent was severely underestimated while the location was overestimated. This nding supports the common knowledge that underdetermined systems have innite solutions. 218 Figure 6.2.1: Experimental setup overview. 219 Figure 6.2.2: Adjustable end. 220 Table 6.2.1: Torsion natural frequencies comparisons for a healthy and damaged beam. Torsion healthy 15-21 5% 229.74 229.36 459.49 459.37 689.27 690.34 919.09 922.07 1149 1154 1378.9 1385.3 1609 1615.3 1839.1 1843.7 2069.3 2070.9 2299.7 2298.2 2530.2 2527 2760.9 2758.1 2991.8 2991.3 221 Table 6.2.2: Comparison of natural frequencies of a tension beam 0.0127m wide, 0.0031m thick, and 3m long with 3000N of tension. Full Thickness and Tension (Hz) Half Thickness and Tension (Hz) 28.46 28.46 56.99 56.99 85.66 85.66 114.54 114.54 143.68 143.68 173.17 173.17 203.06 203.06 233.42 233.42 264.31 264.31 295.79 295.79 327.91 327.91 360.73 360.73 222 Figure 6.2.5: Close up of the damage section of the beam. 225 Table 6.2.3: Damage detection results using the measured natural frequencies of the healthy beam. Measured Healthy Damage Detection Results % Error % Error Short Long Short Long Short Long 213.75 248.13 213.74 248.11 -0.005 -0.006 427.5 496.25 427.5 496.23 0.000 -0.004 641.88 744.38 641.85 744.36 -0.004 -0.002 855.6 990.3 855.57 990.29 -0.004 -0.001 226 227 Damaged Measured Short Long 213.75 248.13 426.88 494.38 640 743.75 856.25 989.38 Damaged Found 3 % Error % Error Short Long Short Long 213.75 248.13 0.000 -0.002 426.25 493.13 0.146 0.252 640 743.75 0.000 0.000 854.72 989.46 0.179 -0.009 Damaged Found 4 % Error % Error Short Long Short Long 213.77 248.1 -0.009 0.010 426.3 493.1 0.135 0.258 640.01 743.73 -0.002 0.003 856.23 987.46 0.002 0.194 Table 6.2.4: Damage detection results using the measured natural frequencies of the damaged beam. Figure 6.2.7: Damage detection showing the dimensionless thickness vs. group number using three natural frequencies for each beam. 229 Figure 6.2.8: Damage detection showing the dimensionless thickness vs. group number using four natural frequencies for each beam. 230 Figure 6.2.9: Simulated damage detection showing the dimensionless thickness vs. group number using three natural frequencies for each beam. 231 . Figure 6.2.10: Simulated damage detection showing the dimensionless thickness vs. group number using four natural frequencies for each beam. 232 6.3 Rope System 6.3.1 Experimental Setup and Testing The rope used was 163 in 7X19 stranded steel. The boundaries were a 25.4 cm threaded steel pipe with cast iron caps anchored in large concrete blocks (Figs. 6.3.1a and 6.3.1b). The boundary conditions were poured as a single block of concrete approximately 0.45 m on a side. The rope ends were crimped with aluminum ferrules and then sanded smooth to attach the accelerometer (Fig. 6.3.1c). To excite the rope the ferrule was impacted next to the accelerometer to excite the longitudinal vibration of the rope (Fig. 6.3.1c). This worked very well for the second and higher modes as shown in the FRF in gure 6.3.2 for the healthy rope in the 7.5m long conguration. As expected, the second and higher natural frequencies are harmonics of the fundamental frequency. To ensure that they would not slip, extra mass was added in the forms of two solid steel cylinders and a 22 kg bucket of sand. Overall the boundary conditions had an estimated weight of about 150 kg each. The added mass was a clamp that had a mass of 30 grams and is shown in gure 6.3.1b. Table 6.3.1 shows that the comparison between the nite element model using bar element to model the longitudinal vibration and the measurements as in good agreement. 6.3.2 Experiments To investigate damage detection for the rope, a small cut was made using a small rotary tool with a cutting wheel at 2 m from the boundary cap as shown in gure 6.3.3. None of the 7 larger strands were cut all the way through but some bers from multiple strands were cut through. For damage detection no attempt was made to model the damage other than reducing the elastic modulus because to fully model the 233 rope is beyond the scope of this thesis. The rope was modeled using 75 groups with ve elements per group, one DOF per node, and 374 nodes (excluding the boundary nodes). The large number of nodes allows for a ner positioning of the lumped mass elements instead of increasing the group density because this changes the wave speed, which is not what really happens and leads to having to adjust the wave number on each iteration. Table 6.3.1 shows that the comparison between the nite element model and the measurements for the damaged rope and they are in good agreement giving condence that the bar elements can mimic the stiness reduction introduced. To test dierent methods of modifying the structure to enrich the information and avoid the symmetry problem, 34 separate approaches to damage detection were carried out using diering number of frequencies, dierent combinations of masses, and dierent methods of modeling the additional mass. The rst set of experiments was to perform modal testing on the rope with no mass to verify the boundary conditions and model. The second set of experiments was to perform modal testing with a mass applied at 1m. The nal set of experiments was done with a mass added at 3 m. During the testing regime after a mass was applied and removed the rope was tested again with no mass to determine if the rope changed due to applying the mass or handling the rope, i.e. minor twisting or pulling associated with tightening the screws on the clamp. In all damage detection simulations, the Levenberg-Marquardt method was used and in most cases the rst natural frequencies were not used because of the diculty of exciting them fully and curve tting repeatably, though a few trials are shown. For the rst damage detection investigations, the rope with no mass and mass at 1 m was tested using various combinations of frequencies in gures 6.3.4, 6.3.5, and 6.3.6 where the rst set of numbers gives the natural frequencies for the rope without 234 mass and the second set is for the rope with mass at 1m. In all cases shown the largest damage is near group 20, which corresponded to 2 m along the rope. In all cases there are erroneous damages. Even though the two cases in gure 6.3.6 have split damages around group 20, the two damages at groups 9, 10, and 47 are less than 10% which is considered acceptable for cases using the elastic modulus. Most cases show damage near group 44 but not all, so at this point it is dicult to have condence that there is damage anywhere other than group 20. The next damage detection test was using the rope without mass and the rope with mass attached at 3 m (Fig. 6.3.7). For all but one case the damage is located between group 57-59 and in one case the two damages are at groups 55 and 61. Since the rope is symmetric this is a second solution to the problem and in this situation the damage should be found at group 55, so these results are somewhat promising compared with those for the case of rope without mass and at 1m. The next attempt was with a mass on the rope at 1m and then with the mass moved to 3 m and retested. In these cases the tests with 1-3 frequencies do the best but all cases still show the strongest damage at around group 20 (Figs. 6.3.8 and 6.3.9). Spreading the concentrated mass at 3m over three nodes (not groups) does just about the same as with a single lumped mass (Fig. 6.3.10). Spreading the mass at 1m does just about the same as with a single mass as shown in gure 6.3.11. Only one case where the mass is distributed at 1 m and 3 m does very well and the strongest damage is at group 20 while the other damages aren't constant (Fig. 6.3.12). The majority of the experimental error is thought to be due to the positioning and application of the masses because handling the rope can change the damage if it is twisted. Another source of error is how the damage was spread over groups that were 5 cm long when the damage was only a thin cut, though it is possible that the thin cut changes the local properties of the rope over a longer length 235 than the cut. The tension in the rope was assumed not to change enough to aect its material properties. To make an overall judgment about the experiments, the average and standard deviation of the 34 approaches were computed and displayed in gure 6.3.13. If anything less than 10% is considered as error as originally proposed, then the most likely location of the damage is at group 20. If one standard deviation is added to the damage, then there are only a few locations where the damage would be more than 10% and the damage extent near group 20 would be closer to the 40% reduction found in many cases. 6.3.3 Simulations To examine the eectiveness of the experimental damage detection, three simulations were performed with 50% damage at group 20. For all simulations the additional mass is spread over three nodes at 1 m (Fig. 6.3.14a), 3 m (Fig. 6.3.14b), or both 1 m and 3 m (Fig. 6.3.14c) at the same time. For the simulations natural frequencies 2-4, 2-5, and 2-6 were used. In all cases the damage is located correctly and the extent is nearly perfect. There is a little erroneous damage near group 55, which is the symmetric location at the other end of the cable. The most important nding is that the damage detected when the mass is placed at 3 m is at group 20 instead of 55 as in the experimental case. 6.3.4 Conclusions The experimental boundary conditions were shown to be adequate in the comparison of the experimental and model results. In the experimental results it was shown that adding a clamped mass can overcome the symmetry problem but that the damage 236 can be at either symmetric location. The statistical results show that through a series of tests the correct damage location can be estimated. The experimental results also demonstrate that even though the measured and detected are very close there is still error in the detected location and extent. As expected, the simulation results show that the expected damage and location can be found with just a few natural frequencies. 237 (a) Left boundary condition. (b) Right boundary condition with the clamped mass at 1m. (c) Accelerometer attachment at the end of the rope. Figure 6.3.1: Boundary Conditions 238 Figure 6.3.2: FRF of the longitudinal vibration of the rope. 239 Table 6.3.1: Comparison of the measured and modeled natural frequencies of the healthy rope. Healthy Model Healthy Measured Percent Error 203.95 203.7 0.12 407.89 611.84 815.79 1019.7 1223.7 1427.7 1631.6 1835.6 410 611 813 1018 1217 1426 1628 1825 240 -0.51 0.14 0.34 0.17 0.55 0.12 0.22 0.58 Figure 6.3.3: Rope with damaged bers. 241 Table 6.3.2: Comparison of the measured and modeled natural frequencies of the damaged rope. Damaged Model Damage Measured Percent Error 203.1 203.7 0.29 407.85 409 0.28 608.1 608 -0.02 808.66 808 -0.08 1017.4 1014 -0.34 1222.7 1212 -0.88 1416.5 1414 -0.18 1619.5 1618 -0.09 1833.8 1814 -1.09 242 Figure 6.3.4: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (A). 243 Figure 6.3.5: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (B). 244 Figure 6.3.6: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m (C). 245 Figure 6.3.7: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 3m. 246 Figure 6.3.8: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m and 3m (A). 247 Figure 6.3.9: Damage detection of the rope showing the dimensionless stiness vs. group number using a mass at 1m and 3m (B). 248 Figure 6.3.10: Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 3m. 249 Figure 6.3.11: Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 1m. 250 Figure 6.3.12: Damage detection of the rope showing the dimensionless stiness vs. group number using a distributed mass around 1m and 3m. 251 Figure 6.3.13: Standard deviation and average of the damage detection results from gures 6.3.4 through 6.3.12. 252 (a) Simulated damage detection for the rope showing the dimensionless stiness vs. group number using a mass at 1m. (b) Simulated damage detection for the rope showing the dimensionless stiness vs. group num- ber using a mass at 3m. (c) Simulated damage detection for the rope showing the dimensionless stiness vs. group number using a mass at 1m and 3m. Figure 6.3.14: Simulated damage detection for the rope with 50% damage at Group 20. 253 Chapter 7 Multiple Impact Device 7.1 Introduction and Previous Results The single impact hammer test is a common modal testing method for exciting a structure to extract the modal parameters [19, 141]. It is convenient, inexpensive, and portable for on-site usage, but its low energy input and signal-to-noise ratio can make it unreliable and non-repeatable for large structures. The impact hammer can only be used on relatively small structures with low noise level; otherwise some modes might be missed. Another drawback of single impact testing is that the low randomization of the input force signal make it unsuitable for nonlinear structures [141]. Electromechanical, hydraulic, and inertial shakers testing is another common excitation method [19] because their persistent excitation provides large energy input and increased signal-to-noise ratio. The shaker can be programmed to provide random excitation to average out slight nonlinearities so linearized eigenparameters of the structure can be extracted. The drawbacks of the shaker test are that it is expen254 sive and inconvenient for on-site measurement. For example, a 50 lbf electrodynamic shaker weighs around 45 lbs alone, without considering its separate power amplier compared to 3 lb for a modal hammer or 15 lbs for an linear electric motor, tripod, and amplier / controller. The need to connect the shaker to the structure by using a stinger has several drawbacks, such as needing extra support structure for the shaker, which need to be sti enough not to move considerably compared to the test structure, mass loading and local stiening, and the inability to modify some structures to attach the stinger. Electromechanical shakers are also expensive in comparison to the random impact device to be presented. A typical 50 lbf shaker costs around $5000 and the necessary amplier is about the same. For hydraulic shakers the cost is much more because they are typically used to provide a larger input force and require more maintenance. Inertial shakers can use a rotating unbalance or a reverse electromechanical shaker. The rotating unbalance shaker is limited to sine dwell and sine sweep but a reversed electromechanical shaker has all the same abilities as a regular electromechanical shaker. Inertial shakers require a large inertial mass to be able to produce a large force which can require extra xturing. Some other methods are: non-contact electromagnetic or ultrasonic exciters and the static relaxation method but are not considered here because of their limited use and the diculty encountered in measuring their excitation signal. Table 7.1.1 gives an overview of the four most common excitation techniques used in modal testing. The random impact hammer test can combine the advantages of the above two excitation methods [141]. The input energy to the structure is increased, the signalto-noise ratio is improved, and the random nature of the force input can average out slight nonlinearities [142] that can exist in the structure. The random impact test method is easy to use in the eld and doesn't add mass loading as in the shaker 255 test. Huo and Zhang [143] demonstrate the extraction of the modal parameters of a diesel locomotive using the random impact test. In their derivation of the method they assumed the force pulses to be a half sine wave, which is usually not the case in practice, and calculated the spectrum of a sample force signal. While the amplitudes and arrival times of the force pulses were stated to be random variables in Ref. [143], the analysis was essentially deterministic in nature as the number of the pulses was treated as a constant and the stochastic averages associated with the model were not determined. In addition, the mean value of a sum involving the products of the pulse amplitudes was erroneously concluded to be zero. 256 Table 7.1.1: Comparison of excitation techniques used in modal testing. Mechanical features Excitation force & Energy output Excitation functions Performance Adaptability Overall Cost Multiple Impact Device Simple Compact Medium weight Good mobility Impact force Large force High energy output Transient Continuous random Burst random Hammer (Single Impact) Very simple Compact Light weight Excellent mobility Impact force Large force Low energy output Transient High to medium High eciency eciency High Low signal-to-noise signal-to-noise ratio ratio Low accuracy No mass loading No mass loading High accuracy Uncontrollable Uncontrollable bandwidth bandwidth No xture, Easy setup No xture, Easy setup Medium Low 257 Shaker Inertia Shaker Complicated Heavy Poor mobility Simple Heavy Poor mobility Contacting force Medium force High energy output Transient Sinusoid Swept Sine Continuous random Burst random Chirp Medium eciency High signal-to-noise ratio High accuracy Controllable bandwidth Contacting force Large force High energy output Sinusoid Swept sine Fixture required High Low eciency High signal-to-noise ratio Medium accuracy Fixture required High 7.2 Previous Theory and Discussion An overview of the theory that the multiple impact methods is based on is presented in [89]. A short overview will be presented to facilitate discussion that will arise from the experimental results. An impact series can be modeled as a sum of force pulses with the same shape because the pulse shape is determined by the material properties of the structure and hammer tip. In the following tests in Chapter 7.3 this will be shown. The input force can be modeled as x (t) = N ?i y (t ? ?i ) (7.2.1) i=1 where t is time, y (и) is the normalized shape function with a unit maximum amplitude, ?i is the amplitude of the i-th pulse, ?i is the arrival time of the i-th pulse, and N is the number of pulses. Taking the Fourier transform (FT) of the input force gives the input force spectrum N X (j?) = ?i e?j?? Y (j?) (7.2.2) i i=1 where Y (j?) is the FT of the shape function. Note that ?i and ?i impose the amplitude and phase weighting factors e?j?t, respectively, on Y (j?) and the amplitude of the phase weighting factors is one. The magnitude of the force spectrum, |X (j?)|, for a single pulse is not aected by the phase weighting factor or arrival time but for a series of multiple pulses it can be. The optimal energy input for a linear structure is when all the pulses have the maximum acceptable amplitude for the structure but having diering amplitudes is benecial when there is some amplitude dependent nonlinearity. If the pulse shape function is a Dirac delta ? function where ?i = 1 then X (j?) = 258 N i=1 e?j??i . When ?i are uniformly distributed at frequencies that are integer mul- tiples of the impact frequency, one another and peaks of ? = |X (j?)| 2? , (?i ??i?1 ) e?j??i accumulate instead of canceling appear at those frequencies. This observation is the key point of random impact testing and in the design and selection of the device because most devices / control systems are designed for deterministic impacts. The fundamental frequency of oscillations of |X (j?)| x (t) is dependent on the frequency of impacts while the are dependent on force spectrum is to randomize E |X (j?)|, ?i . several combinations of ?i . One way to achieve a more broadband To investigate the averaged ?i and ?i The impact arrival times are assumed to be |X (j?)|, denoted are investigated using 100 averages. ?i = i + ri , where ri are normally ran- dom variables with zero mean and 0.5 standard deviation or uniformly distributed with a unit mean and 0.5 standard deviation. The impact amplitudes ?i can be nor- mally distributed random variables with zero mean and 0.5 standard deviation, or uniformly distributed random variables with a unit mean and 0.5 standard deviation, or constant amplitudes. It can be seen that all combinations of distributions give very similar expected force distributions (Fig. 7.2.1), which allows for some exibility in the design of the device. A random impact series in modal testing can be modeled as a sum of force pulses with the same shape and random amplitudes x (t) = N (t) ?i y (t ? ?i ) , ?i ?i ? (0, t ] and arrival times and ?i t ? (0, ?) (7.2.3) i=1 where val N (t) (0, T ], is the random number of pulses that have arrived in the time inter- which was assumed to be Poisson counting process with stationary incre- ments in [89]. All the pulses are assumed to have width 259 ?? , and y (t ? ?i ) satises y (t ? ?i ) = 0 if t < ?i and t > ?i + ?? . The experimental results show that the pulse widths are all nearly the same. It should be noted that the length of a typical test and the frequency with which a test engineer can impact the structure make it dicult to produce a Poisson's distribution because even for a Poisson's parameter of ? = 3, about 18% of the time should have 5 or more impacts per second. The device is even slower with a Poisson's parameter of about 1.5. From the tests results the number of impacts per time interval is fairly constant but the arrival times are still random. A nite time interval is used in modal testing so only the pulses in the interval (0, T ] T of length where T ?? force signal at time t can be replaced by N (T ). between T and and y (0) and y (?? ) is independent of pulses after T + ?? t<T For pulses arriving between consider the time interval don't have to be zero. The the random process T ? ?? and T N (t) and ending t ? (0, T + ?? ] instead of t ? (0, T ]. N (T ) x (t) = ?i y (t ? ?i ) , ?i ? (0, T ] and t ? ( 0, T + ?? ] (7.2.4) i=1 During the measurement process no attempt is made to ensure that the last pulse is within the measurement window. During the measurement process even if the last impact is cut in half, the energy input into the system associated with the output at the end of the measurement record is still accounted for. During the measurement process allowing for some time before the rst impact is most important so it is fully captured if the measurement process is triggered by the increasing amplitude of the N (t) with stationary increments, the probability ??t of the event {N (t) = n}, where n is an integer, is P{N } (n, t) = e (?t)n /n!, where rst impact. For the Poisson process ? is the constant arrival rate of the pulses [144, 145]. 260 By replacing t with T, the probability of the event {N (t) = n} is P{N } (n, T ) = e??T (?T )n n! (7.2.5) All the arrival times ?i , where i = 1, 2, . . . , N (T ) , are a sequence of identically distributed, mutually independent random variables. Each ?i is a uniformly distributed random variable in the time domain (0, T ] with probability density function p?i (? ) = ? ? ? ?1, 0 < ? ? T T ? ? ?0, (7.2.6) elsewhere Similarly, ?i (i = 1, 2, . . . , N (T )) are a sequence of identically distributed, mutually independent random variables, which are independent of ?i . The distribution of ?i (i = 1, 2, . . . , N (T )) was rst assumed to satisfy the normal distribution p?i (?) = e ? (???)2 2? 2 ? 2?? , 0<?<? where ? = E [?i ] is the mean and ? 2 = E [?2i ] ? E 2 [?i ] is the variance of ?i . 261 (7.2.7) (a) Uniform random ? with ?=1 or (b) Normal random uniform random. normal random. (c) Normal random ? and uniform ran? vs. uniform random ? and normal random ?. (d) Uniform random dom dom ? ? with ? and normal ran- vs. normal random form random ?. ? = 1 ? or and uni- Figure 7.2.1: Average expected force for dierent combinations of distributions. 262 7.3 Device Research, Development, and Design The need for a multiple impact device became evident during lightning mast testing when the multiple impact test was found to give better results than the single impact test. The duration of the lightning mast test is between 1.5 to 6 minutes with about 4 tests per structure. This is tiring because a 2.4 lb mini-sledge hammer is used and ten to fteen masts are tested in a day. The results are for tests performed on a 65 ft structure with a 5 ft spike, which will be referred to as a 70 ft mast (Fig. 7.3.1). The 70 ft mast has two constant cross-section schedule 40 pipes of equal length as shown below. From the results shown in Figures 7.3.2 and 7.3.3 for the 70 ft lightning mast, it can be easily seen that the multiple impact test has much better coherence, is less noisy away from the resonances, and can pick up the modes that were missing in the single impact test. The multiple impact test is better at exciting the lower modes that can also be excited by the wind, which can be seen by comparing the frequency response functions (FRF) between 0 and 30 Hz. It can be seen from the FRF for the multiple impact tests that there are some modes that are missed at near 4 Hz and 7 Hz from the single impact test, and the mode at 10 Hz is much improved compared to the single impact test. An experimental test was conducted with the shaker to determine if it could be used to generate impacts similar to the modal hammer by attaching a hammer tip to the force sensor at the end of the shaker. It was found that the pulse shape is dicult to control because the tip contacts the structure for too long and the pulse shape couldn't be made similar to that of the hammer or The Modal Shop, Inc. Electric Impact Hammer, Model 086M92ES (Fig. 7.3.4) [146]. This length of impact causes bad coherence because the long contact changes the dynamics of the structure. To 263 achieve a shorter duration of contact the shaker is driven at a higher frequency. One of the main problems with using the shaker is that the armature is too sti to bounce back o the structure. The Modal Shop, Inc. Electric Impact Hammer was tested but the frequency is not adjustable and the force can't be varied easily. This device is made for single impact testing at a repeatable force and is often used for quality control testing on assembly lines. The maximum frequency of the device is roughly 2 Hz. During the design phase three types of systems were considered: a hammer swinging device, a pneumatic actuator, and a linear electric actuator. The hammer swinging device was eliminated due to the complexity of the design, the limited range of motion, test setup and positioning diculty, the complexity of the control design needed, and cost. The pneumatic actuator was not feasible due to the complexity of the system, lack of robustness, cost, and ease of controllability and interfacing with controls software. One of the most important considerations is that random time intervals needed to be generated, which is a feature not often considered in the design of many systems where any noise is undesired. The Copley Controls STA2510 linear actuator (Fig. 7.4.1) [147]with Xenus Plus controller [148] was chosen because the hardware and controls software systems were already developed and ready to use with LabVIEW [149] and only the random impact algorithm had to be implemented in LabVIEW. LabVIEW was benecial because MATLAB scripting can also be used within the M athScript block. The rst step in the development of the impact system was to build a mounting system to secure the STA2510 linear motor to a standard tripod and gimble head. This allowed for easy and quick positioning of the device. Two dierent couplers were machined to interface with two dierent force sensors to the piston of the STA2510. 264 Acceleration measurements were taken to help characterize the dynamics of the system and improve the design of the mounting xture. In the laboratory a pull test was used to estimate the coecient of friction between the feet of the tripod to the oor in order to determine how much mass needed to be added to the tripod to keep the device from pushing itself away from the structure after repeated impacts. The control program was implemented in LabVIEW using driver libraries supplied by Copley Controls. The motion prole selected was a triangular position wave with selectable pulse width and amplitude. The program was constructed such that dierent types of random number generators could be selected to generate the random time intervals. The program also allowed for burst random excitation as well as a single impact. Due to the physical limits of the device, a minimum time between pulses was needed because the maximum frequency of the device is dependent on the amplitude of the displacement. During initial research and development it was found that the stando distance is important as well as the stiness of the support. If the support is too sti, the impacter won't rebound fast enough and the pulse width becomes wider or the force sensor will saturate because of the device time constant, which leads to a response that looks like a double impact. This behavior is similar to a person locking their wrist once the hammer hits the structure so the hammer contacts the structure for a longer time. Also, if the pulse width is too long the device will not rebound fast enough. In all cases shown the random impacts were generated by using a uniform random number generator in LabVIEW to continuously generate numbers between zero and one. When the random number generated was greater than 0.9 the device would impact the structure. The number 0.9 was chosen because of limitations of the device to impact the structure with sucient amplitude. Due to the limited acceleration of the device, sucient distance needs 265 to be maintained between the device and the structure for the impacter to reach a high enough velocity. This will generate enough force to push the device back some so the impact tip rebounds of the structure quickly enough to keep the impact shape narrow. No eort was made to generate random amplitudes due to the complexity of the relationship between the stando distance and necessary velocity to achieve a suitable rebound. Also, no eort was made to generate a Poisson distribution of impacts due to the frequency limitations of the device, which for small amplitudes was about 5 Hz. Another reason a Poisson distribution wasn't attempted was because of the short test time, which makes it dicult to even gather enough information to reliably compute statistical parameters of the input. It should also be noted that the multiple impact test works when performed by a test engineer with no consideration as to the types of distributions originally used. Even though it would seem like many of these facts are design limitations when viewed in terms of the assumptions made in the theory presented in Chapter 7.2, it will be shown that the device performs just as well or better than an experienced test engineer. 266 Figure 7.3.1: 70' lightning mast. 267 Figure 7.3.2: Single coherence (top) and FRF (bottom) from tests on an aluminum plate. 268 Figure 7.3.3: Single coherence (top) and FRF (bottom) from tests on an aluminum plate. 269 Figure 7.3.4: The Modal Shop impact device. 270 7.4 Square Aluminum Plate Test Much of the initial testing for research and development was conducted using a small impact hammer or the device on an aluminum plate .91 m square and .00635 m thick clamped in a vice as shown in gure 7.4.1. A spectrum analyzer was congured to capture a continuous long time record of 25 blocks with 8192 spectral lines each over a bandwidth of 500 Hz. Each test took approximately 2 minutes and 40 seconds to complete. The negative force at the ends of the impulses was mostly due to the inertia of the hammer but can also be partially due to the discharge time constant of the force sensor as well. The main goals of this test were to determine the distribution properties of the impact amplitudes and arrival times, the variations in pulse shape and input power spectral density. The input power spectral density is the most important comparison of the device because if the power spectral densities are similar then the responses, i.e. FRF's, will be similar. 7.4.1 Manual Multiple Impact Testing To compare the random distribution generated by the device to the distribution generated by a human, ve dierent experimentalists were asked to perform a multiple impact test on the aluminum plate. Two of the engineers were experienced at modal testing while three had little to no experience. The time history of the results with the peaks numbered using the peak picking software tool by [150] are shown in gures 7.4.2 and 7.4.3 where gure 7.4.2b shows a close up view. The shape of the picked peaks were then normalized to a unit height and the average and standard deviations were calculated for the nine sample points, four points before and after the peak point (Figs. 7.4.4). It was found that there is little dierence between the shapes of 271 the peaks for dierent experimentalists. This was expected because the pulse shape depends mostly on the stiness of the hammer tip and the structure. Distributions of the impulse forces and time between consecutive impacts were made by separating the signal properties into 50 bins of equal width. It is easily seen that none of the impulse force distributions or time distributions correspond to a well known statistical distribution or that any of the experimentalist distributions are similar. Examining the power spectral densities of the force signals (Fig. 7.4.7) shows that the results from all the experimentalists are nearly the same. This would indicate that the type of random distribution of impact force amplitude and arrival times are not very important and the shape of the tip has the most inuence on the power spectral density (PSD) (see Chapter 7.5). To compare the distribution of impacts over the time record to a Poisson distribution the Pearson's Chi-squared test is used. The Pearson's Chi-square test is given as: ?2 = k Oi ? E?i E?i is the expected probability of the tribution being tested and test. To calculate E?i Oi (7.4.1) E?i i=1 where 2 i-th group as computed using the dis- is the observed probability of the i-th group from the for a Poisson distribution the Poisson's parameter is found by calculating the average number of impacts per time period 1 ?? = y = yi n i=1 n where and n yi is the number of impacts per time period in group (7.4.2) i where i = 0, 1, 2, 3, ...n is the maximum number of impacts in any of the time periods. The sample 272 size is chosen to be approximately 1 second which gives 160 samples or bins. The number of impacts per second are rst grouped into 160 bins (Fig. 7.4.8). In each of these samples the number of impacts are counted and grouped into thirteen bins (Fig. 7.4.9) because the Chi-Squared test needs bins with 5 or more to be useful and the number of bins should be roughly ? N where N is the number of events [151]. The expected probability for the Poisson's distribution is calculated as E?i = P? (Y = i) = ??i e??? i! (7.4.3) The observed probability is computed by dividing the number of impacts per time period in group i by the total number of samples. The probability mass functions for test engineer one and the expected results are shown in gure 7.4.10. Even without computing the Chi-Squared statistic it is easily seen that the measured probability mass function and the expected probability mass function are very dierent. None of the tests performed by humans showed a dominant distribution type for either the arrival times or amplitudes. 7.4.2 Device Impact Test The same measurement parameters as used for the human test were used for sampling data during the device test. Figure 7.4.1 gives the typical setup of the device when used to excite the aluminum plate. An example of the input history generated by the device is given in gure 7.4.11. The biggest dierence between the device input prole and manual impact proles is the average time between impulses is greater for the device. Even though the device wasn't programmed to have random amplitudes, the amplitudes are still random because of the exibility of the mounting system. 273 From the results of the engineers it is thought that the amplitudes of the peaks being random is not overly important and because the method is focused on inputting more energy into the test article, having the maximum amplitude at every impact would be advantageous. The peak shape measured from the device is very similar to that generated by the experimentalists, as would be expected. One interesting nding is that the way the random impacts were generated isn't as random as was expected, which could be due to the 0.1 seconds needed to execute an impact. From the PSD result this doesn't seem to be a problem though because the PSD is similar to that from the experimentalists. The PSD from the device concentrated more energy at the lower frequencies compared to the human testers because the mass of the rod is heavier than the mass of the modal hammer. The increased mass has more momentum and thus takes more time to change direction so the contact time is longer. As the contact time becomes longer the amount of high frequency energy is decreased. 274 Figure 7.4.1: Impact device positioned to impact the aluminum plate. 275 (a) Total impact signal measurement. (b) Detailed view of the impact signal. Figure 7.4.2: Multiple impact series from engineer 1. 276 (a) Multiple impact series from engi- (b) Multiple impact series from engineer 3. neer 2. (c) Multiple impact series from engineer (d) Multiple impact series from engineer 4. 5. Figure 7.4.3: Test engineer's multiple impact test measurements. 277 (a) Average and standard deviations of (b) Average and standard deviations of the impulses from engineer 1. the impulses from engineer 2. (c) Average and standard deviations of (d) Average and standard deviations of the impulses from engineer 3. the impulses from engineer 4. (e) Average and standard deviations of the impulses from engineer 5. Figure 7.4.4: Average and standard deviations of test engineer's impulse shape. 278 (a) Seconds between impacts by engineer 1. (b) Seconds between impacts by engineer 2. (c) Seconds between impacts by engineer 3. (d) Seconds between impacts by engineer 4. (e) Seconds between impacts by engineer 5. Figure 7.4.5: Distributions of seconds between impacts for several test engineers. 279 (a) Distribution of impact forces by engineer (b) Distribution of impact forces by engineer 1. 2. (c) Distribution of impact forces by engineer (d) Distribution of impact forces by engineer 3. 4. (e) Distribution of impact forces by engineer 5. Figure 7.4.6: Distribution of impact forces for several test engineers. 280 (a) Power spectral density of impact forces by (b) Power spectral density of impact forces by engineer 1. engineer 2. (c) Power spectral density of impact forces by (d) Power spectral density of impact forces by engineer 3. engineer 4. (e) Power spectral density of impact forces by engineer 5. Figure 7.4.7: Power spectral density of impact forces for several test engineers. 281 Figure 7.4.8: Impacts per 1 second time interval by test engineer 1. 282 Figure 7.4.10: Measured and expected probability mass functions for test engineer 1. 284 (a) Impact signal generated by the device. (b) Detailed view of the impact signal. Figure 7.4.11: Impact signal generated by the device. 285 Figure 7.4.12: Average and standard deviations of device impulse shape. 286 Figure 7.4.13: Distribution of impact forces from the device. 287 Figure 7.4.14: Distribution of seconds between impacts from the device. 288 Figure 7.4.15: Power spectral density of the impact forces from the device. 289 7.5 Pipeline Test To test the impact device's ability to excite larger structures and examine its eectiveness in the presence of noise, a pipe setup that is the same as the central section of the lightning mast in section 6 was used. The measurement parameters used were 4096 spectral lines over a bandwidth of 0-1024 Hz. For each test 10 averages were taken. Noise was generated by having a person randomly impact the structure at one of the two dierent locations as shown in gure 7.5.1. The accelerometers were placed at right angles to each other on the inside of the pipe section where the hammer and noise inputs were applied. The main focus of these tests are to exam in the FRF and coherence of the responses at two accelerometer locations in the presence of noise. The tests compared are a single impact, manual multi impact, device multi impact both continuous and burst random. The same uniform random method used for the aluminum plate was used to test the pipe. 7.5.1 Manual Multiple Impact Testing The manual multiple impact test results will be used to benchmark the device multiple impact tests. The results from the single impact with no noise show that even though the PSD of the input does drop o signicantly until about 900 Hz, the energy input into the system isn't large enough to excite the structure enough to produce good results as seen by the coherences and FRF's in gure 7.5.2. The structure is not excited very well below 100 Hz as is shown by the FRF's and coherences. As expected, when noise is added at location 1 (Fig. 7.5.3) or 2 (Fig. 7.5.4) the results are worse even though the noise is still about a tenth of the input. The advantages of the multiple impact method can clearly be seen in gure 7.5.5 when there is no 290 extra noise input into the system. The FRF's are very smooth and the coherences are nearly one except at the anti-resonances, both of which are indicators of a good test. In comparison with the single impact test it can be seen that the PSD drops o near 900 Hz but encloses the PSD of the single impact test as noticed in Chapter 7.4. When the multiple impact test is used and the noise input is at the same level, the results are still better than in the single input case (Fig. 7.5.6). In the second noise cases the input PSD is about 2 to 3 times greater on average and the FRF's and coherences are also better than in the rst case as would be expected (Fig. 7.5.7). 7.5.2 Device Multiple Impact Testing The multiple impact tests were repeated with the device using continuous random and burst random inputs. Low frequency energy and coherence are better because the device has better repeatability in terms of impact location compared to a human. The mass of the rod helps with the low frequency energy but the coherence suers some in the higher frequencies above 800 Hz. This is considered acceptable because the lower frequency information below 100 Hz is of importance in this case. In the two test cases with noise, it can be seen that once the power spectral density of the device input crosses that of the noise input, the FRF's and coherences become noisy. Even in noise case 2 the high frequency results are better than those of the manual test most likely because the error in input location aects the higher frequencies more than the lower frequencies. This is expected because the lower modes are more global in nature and have a higher modal mass. The burst random results with no noise are similar to those of the continuous random test but the low frequency results are not as good because the structure is allowed to freely vibrate and dissipate energy. This free vibration is detrimental to the noise cases even though the PSD results are 291 similar to those of the continuous random cases. The greatest advantage to using the device to test is seen in the case using the continuous random input due to the repeatability of the device. 292 Figure 7.5.1: Pipe setup showing impact location and noise impact locations. 293 (a) Power spectral density of the input and instrument noise. (b) FRF and coherence. Figure 7.5.2: Single impact by test engineer 1 with no noise. 294 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.3: Single impact test by test engineer 1 with noise at location 1. 295 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.4: Single impact by test engineer 1 with noise at location 2. 296 (a) Power spectral density of the input and instrument noise. (b) FRF and coherence. Figure 7.5.5: Multiple impact test by test engineer 1 with no noise. 297 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.6: Multiple impact by test engineer 1 with noise at location 1. 298 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.7: Multiple impact by test engineer 1 with noise at location 2. 299 (a) Power spectral density of the input and instrument noise. (b) FRF and coherence. Figure 7.5.8: Device multiple impact with no noise input. 300 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.9: Device multiple impact FRF and PSD with noise at location 1. 301 (a) Power spectral density of the input and noise. (b) FRF and coherence. Figure 7.5.10: Device multiple impact FRF and PSD with noise at location 2. 302 (a) Power spectral density of the input and instrument noise. (b) FRF and coherence. Figure 7.5.11: Device burst impact test with no noise. 303 (a) Power spectral density of the input and input noise. (b) FRF and coherence. Figure 7.5.12: Device burst impact test with noise at location 1. 304 (a) Power spectral density of the input and input noise. (b) FRF and coherence. Figure 7.5.13: Device burst impact test with noise at location 2. 305 7.6 Lightning Mast Testing The main objectives of this test are to compare the FRF and coherence results from several excitation techniques on a lightning mast structure to determine if there are any advantages to using the multiple impact device. An overview of the eld setup is shown in gure 7.6.1. The lightning mast was impacted at 1m above the ground and the acceleration was measured at 1.3 m above the ground in two directions, one parallel to the spike and one perpendicular to the spike (Fig. 7.6.2). The mast was excited once parallel to the spike and once perpendicular to the spike for each test because the modes are not the same in both directions due to the spike being oset from the center. Four dierent types of tests were conducted: manual single impact (Fig. 7.6.3), manual multiple impact (Fig. 7.6.4), device multiple impact (Fig. 7.6.5), and device burst random (Fig. 7.6.6). For the manual impact tests a 2.4 lb miniature sledge hammer was used to excite the structure. For the device tests a force sensor was mounted to the front of the armature of the impact device (Fig. 7.6.2). Ideally, the optimal location to excite this structure would be at the free end or as high o the ground as possible but being in an 115 kV electrical substation one is limited to what is reachable while standing on the ground. This makes excitation of the lowest modes dicult especially when only using a single impact because the mass of the structure is roughly 2 tons. The measurements comprised 10 averages over a bandwidth of 0-100 Hz at a sampling rate of 256 Hz, which gave a frequency resolution of 0.0625 Hz. Each test took approximately 160 seconds to complete. After just a few tests this is fairly tiring when using the sledge hammer. This is one denite advantage of having the automated device. The device multiple and burst impact tests have better coherence and less noise at the lower natural frequencies as shown in gures 7.6.5 and 306 7.6.6, respectively. Even though the coherence for the multiple impact tests are lower in some cases than the single impact test, most of these cases are at anti-resonances, which are not signicantly important. The coherence from the multiple impact test using the device (Fig. 7.6.5) is better than the manual impact test (Fig. 7.6.4) and this is mostly due to the ability of the device to consistently hit the same location each time. There was very little noise during these tests because there was almost no wind. In all tests the FRF's after about 15 Hz are nearly all the same. The main reasons for dierences would be due to positioning dierence in impacting the structure between tests and setups, i.e. going from manual testing to device testing. 307 Figure 7.6.1: Overview of device setup for lightning mast testing. 308 Figure 7.6.2: Device impact and accelerometer locations for lightning mast testing . 309 (a) FRF (b) Coherence Figure 7.6.3: Manual single impact test FRF and coherence. 310 (a) FRF (b) Coherence Figure 7.6.4: Manual multiple impact test FRF and coherence. 311 (a) FRF (b) Coherence Figure 7.6.5: Device multiple impact test FRF and coherence. 312 (a) FRF (b) Coherence Figure 7.6.6: Device burst impact test FRF and coherence. 313 7.7 Conclusions In this chapter a random impact device was designed, built, and successfully tested in the laboratory on an aluminum plate and steel pipe line, and in the eld on a lightning mast. During the rst series of tests on the aluminum plate it was shown that neither test engineers nor the device generate a particular distribution for the time interval between pulses or magnitude of impact force but they do generate the same pulse shape and power spectral density. By testing multiple types of distribution the average expected force was shown to vary little and thus give condence that the time and force distribution type matter very little as long as the impact times are not deterministic. When testing the device on the pipe line it was shown to overcome the noise a little better at the lower frequencies because the mass of the impacter is large. In the cases where there was no added noise the device did better as well because it was more consistent at impacting the same location each time. This increase in consistency was also seen when the lightning mast was tested. When the burst random test was used to excite the lightning mast the results were not as good as the continuous impact test because of the energy lost when the device temporally stopped impacting the mast. Overall the device was just as successful or more successful when compared to the multiple impact tests performed by a skilled engineer. 314 Chapter 8 Engineering Education 8.1 Introduction The United States produces roughly one third of the world's technology [152] with about two million engineers [153]. The average annual growth rate of nonacademic science and engineering jobs between 1980 and 2000 was 4.9% compared with just 1.1% for the entire labor force [152] and the number of engineering jobs is expected to grow by about 9.4% between 2000 and 2010 [152] . More than half of science and engineering degreed workers will be of retirement age in 10-15 years [152] . The number of bachelor's degrees in engineering has decreased about 6.5% from 1995-96 to 2000-01 [154] with the number in mechanical engineering decreasing about 13.4% [155]. While engineering salaries are in the top 5% of all professions for starting bachelor's and master's degree graduates [153], 80% of newly graduated engineering employees work in another eld after 5 years [156]. One way to increase the number of engineering majors is to introduce students to engineering in high schools. Currently only one state has implemented mandatory 315 standards for engineering education at the K-12 level. Despite the innate tendency among all children to explore how things work and to structure and restructure the items in their environment, this childish curiosity has not translated into young people willing to study engineering. A common feeling shared by many in the engineering community is that many young people are uninterested in engineering due to the stereotypes and images associated with engineers. It is believed that many of these negative feelings are due to young people's lack of knowledge and interaction with engineers. For many students, the idea of engineering often suggests nothing more than being assigned projects working on a product removed from the interests of daily life. Young people see engineers as locked into technology to the exclusion of creative design or interaction with people [153, 157]. Some of the disinterest seen by prospective students is due to the way engineering is portrayed in K-12 education. Engineering projects tend to focus on the building aspects and are graded on nal outcomes [158]. It is obvious that these projects don't mimic engineering as taught at the college level, especially in the way they are graded [158]. To make things worse there is little or no guidance on the design process [158]. The design process is important because just focusing on the building aspect alienates girls because of their style of play, which tends to focus more on design than construction [158]. For many, having a familiar role model introduce them to engineering can dispel most of the common misconceptions of engineers [157, 159]. There are many approaches being used by engineering educators to increase the number of new engineering undergraduates; summer camps, after-school programs, engineering education, career days, graduate and undergraduate teaching fellows in the classroom, competitions, teacher training, and innovative classroom materials are just a few. All of these approaches have their pros and cons and many educators 316 combine these approaches to improve their outreach. Developed as an educational component of a CAREER award from the National Science Foundation, the Future Engineers in Dynamic Systems (FEDS) Academy is a unique two-week, full-day summer educational program for junior and senior high school students in the Baltimore region. With the available funding, sta, and resources, a summer camp for high school students was the most feasible option for the authors. Even though summer camps only reach a small number of students, they still play an important role. They introduce students to the college experience, equipment, role models, and knowledgeable instructors, in order to help them understand dierent levels of concepts. The FEDS Academy focuses on dynamic systems including dynamics, vibration, acoustics, and system control in mechanical engineering. Participating students are engaged in thought-provoking experiments and lectures led by undergraduate and graduate teaching fellows currently working in the faculty author's laboratory, the Dynamic Systems and Vibrations Laboratory (DSVL), and/or enrolled in the Mechanical Engineering Department at the University of Maryland, Baltimore County (UMBC). The goal of exposing high school students to concepts and applications not typically found in their K-12 curriculum. Due to the level of content presented, the focus of the program is on high school students. Some educators may feel that this is a little late, but encouraging students to investigate a career in engineering throughout their K-12 education is important [160]. While there are a wide range of summer camps to introduce high school students to engineering [157, 160163], very few of the camps [160, 161] introduce students to current laboratory research. Many programs also charge students to apply and/or attend [160, 161]. While many programs cover a wide range of subject areas or multiple types of engineering [157, 160170], there are focused camps in such areas as 317 mechanical engineering [169172], mechatronics [173175], civil engineering [170, 176, 177], aerospace engineering [178181], chemical/biochemical engineering [182, 183], and simulation-based engineering [184]. Other than the aerospace engineering camps few discuss any topics in dynamic systems. There are three programs that are similar in design to the current program: one is focused on mechatronics at Ohio Northern University [173], one on Robotic Autonomy at Carnegie Mellon University [174], and the other on simulation-based engineering at the University of Wisconsin [184]. A unique feature of the FEDS Academy is that it is the only program that is focused on dynamic systems. A distinguishing feature of the program is that it integrates the current research in the DSVL into the program. Another dierentiating aspect of the program is that it combines elements of design/build/test (DBT) and laboratorybased programs [162]. The level of technology presented is much higher than most programs. The level of diculty is also higher because students are exposed to content that is presented to upper-level undergraduate and graduate students in mechanical engineering. The program's nal projects also stand out because they are chosen by students and are open-ended because students dene their own goals. The FEDS Academy, which started in the summer of 2004 and is free for all students, seamlessly integrates various aspects of Science, Technology, Engineering, and Mathematics (STEM) education. To address the image problem, students are taught by undergraduate and graduate teaching fellows who are closer in age and interests. The program links engineering to real world activities that students have experience with, such as music, speech, and sports. Through the use of real world examples students are introduced to new concepts to spark their interest in taking more challenging courses in school and are shown how the content they have learned is used in engineering. In the engineering discussions, students learn how engineers can help 318 people and society. The program employs various types of learning, including handson and active learning, lectures, self-paced reading, research, and exploration-based learning, which shows students that there are many ways to learn about engineering. Students are introduced to the engineering design process, which shows that engineering is not just about building things and that analyzing trade-os is important. The program strives to enhance students' understanding and awareness of engineering as a potential educational and career pursuit, and to expose students to the connection of engineering theories and applications to everyday concepts. 8.2 Program Overview The FEDS Academy is designed to achieve a variety of goals. A major goal is to encourage high school students, especially underrepresented students (i.e., low socioeconomic status, ethnic/racial minorities, and women), to consider a career in engineering. Another major goal is to introduce students to dynamic systems and to show how it relates to real life experiences and situations. A secondary goal is to provide undergraduate students opportunities to engage in research and gain teaching and laboratory experience. Another secondary goal is for graduate students to gain teaching experience at various levels such as peer to peer, undergraduate, and high school. With the help of the Shriver Center at UMBC, iers are sent to high schools in the Baltimore region, especially those with large underrepresented populations, to recruit rising juniors and seniors. To encourage underrepresented students to attend, minority and female students are featured on the program iers. The Shriver Center has over 25 years of experience in organizing and leading work- and service-based 319 learning programs that facilitate the connection of theory to practice and thought to action for undergraduate and graduate students. It also manages several nationally recognized programs that provide service and support to the local and regional community, its families, youth, and K-12 schools. The applications are also posted on the faculty author's prole on the departmental website [185], and some national summer camp directories [161, 186] have listed the program. A logo shown in Figure 8.2.1 was developed for advertisement and recruitment purposes and to help generate interest in the program. It shows some wave or vibratory motion inside a magnifying glass, because the program has an overall theme of investigation of motion. Student applicants are asked for their basic information along with an essay on why they are interested in the program. In addition to student recruitment, the program seeks undergraduate and graduate teaching fellows. Both undergraduate and graduate applicants address in writing their interest in being a FEDS Academy teaching fellow and how the program compliments their academic pursuits, professional aspirations, and personal goals. They are also asked to submit a transcript, a resume, and three letters of reference. Students in the program are chosen based on grade point average (GPA) and essay content with special weight given to seniors. Thus far, the program has mostly realized about ten to fteen student applications each year with selection criteria in place to recruit graduating seniors if student applications were to falter. Similarly, selection criteria are in place for applications from undergraduate and graduate students. In the years that the program has been in place, the number of students enrolled in the program has increased from three in 2004 to nine in 2005 and 2006, and twelve in 2007 through 2009. African Americans and Asian Americans have been represented in nearly equal percentages (30% and 26% respectively) and women have accounted for 320 14% of the participants. The participants have been highly capable, with an average GPA of 3.62; many students have a weighted GPA of about 4.0. During the two-week program students gain insight into engineering through three types of content: 1) Current research conducted in the DSVL, such as damage detection, elevator dynamics, beam control. 2) Classic dynamics topics, such as modal testing and model updating, and random impact testing. 3) Everyday life experiences, such as baseball bat dynamics, rotor dynamics, elevator dynamics, bridge vibration, guitar vibration and acoustics, vocal acoustics, and Lego control. Lectures on statics, kinematics and dynamics, single- and multiple-degree-of-freedom (DOF) system vibrations, ordinary dierential equations (ODEs), and open- and closed-loop system control. During the rst day students create name tags and introduce themselves to each other. They are given a brief overview of the current research projects taking place in the DSVL to give them a feeling for what they will be learning and how this information is used in the real world. They discuss engineering and college life, as well as major careers and academic elds in mechanical engineering. During the rst six days students attend classroom and laboratory sessions on various aspects of dynamic systems. They also begin to conduct preliminary research about an open-ended, nal project of their interest, and to work on a Lego design project that concludes with a competition. The program is exible in order to allow students to nish what they start if they need more time or want to explore a topic in more depth. Within the rst several days students take a tour of the campus and go to the library to do research on their nal projects. During the last four days, they work on their nal projects that are often related to one of the topics from the rst six days. On the last day, they give a presentation of what they have accomplished on their projects. Table 8.2.1 shows a typical program in 2006-2009. During most of the laboratories 321 and the Lego and nal project periods, students are divided into groups of four or ve. A certicate of completion is provided to the students who successfully complete the program. At the beginning of the program, students are surveyed about their educational background, feelings, and understanding of engineering in general and mechanical engineering in particular. All surveys are conducted on a voluntary basis, and mainly administrated by the Shriver Center sta members. The pre-survey results are used as a comparison to their responses at the end of the program to assess their attitude toward engineering, and more specically, mechanical engineering. Students are asked open-ended, yes or no, and Likert scaled questions. At the end of each day, students are surveyed about their feelings and understanding of the classroom and laboratory sessions. Most of the questions are Likert scaled with some free response questions. Students rate the instructors on preparedness, conveying material eectively, and responding to questions appropriately. The surveys are summarized at the end of the day and the feedback is given to the instructors the same day. In some cases the feedback is implemented during the next day to tailor the lessons to the students, so they gain as much as possible from the program. Such immediacy is crucial because the cohorts of students can vary widely in ability from year to year and the speed at which the program is delivered one year can be too slow or fast the next. The undergraduate and graduate teaching fellows are also asked to write down notes about their observations about the occurrences of the day for internal use. At the end of the program students are surveyed about the classroom and laboratory sessions from the rst six days, the nal projects, and their feelings towards engineering. The undergraduate and graduate teaching fellows are also asked to write 322 down notes about their observations on the nal projects and the program overall for internal use. 323 Figure 8.2.1: FEDS Academy Logo 324 325 Monday Introduction | Pre-survey Pendulum Lecture | Pre-lab Pendulum Lab Lunch Pendulum Lab Data Analysis Engineering Discussions Project Overview | Survey Monday Control Lecture Beam Control | Lego Project Beam Control | Data Analysis Lunch Lego Project | Data Analysis Project Project Time\Day 9:00-10:00 10:00-11:00 11:00-12:00 12:00-13:00 13:00-14:00 14:00-15:00 15:00-16:00 Time\Day 9:00-10:00 10:00-11:00 11:00-12:00 12:00-13:00 13:00-14:00 14:00-15:00 15:00-16:00 Tuesday Lego Project Project Project Lunch Lego Project Project Campus Tour Tuesday Guitar Introduction Guitar Lab | MATLAB MATLAB | Guitar Lab Lunch Project Project Project | Survey Week 1 Wednesday Elevator Classroom Elevator Lab Data Analysis Lunch Lego Project Project Project | Survey Week 2 Wednesday Lego Project Project Project Lunch Project Lego Project Project Thursday Project Project Lego Competition Lunch Project Project Project Thursday Beam Test | Finite Elements Finite Elements | Beam Test Damage Detection | Lego Project Lunch Damage Detection | Lego Project Project Project | Survey Table 8.2.1: FEDS Academy's Schedule in 2006 Friday Practice Presenting | Project Project |Practice Presenting Project Lunch Project Presentation Awards | Post-survey Friday Research on Hearing Presentation | Speech Physics Voice Map | Lego Project Lunch Lego Project | Voice Map Project Project | Survey 8.3 Daily Lessons The program modules during the rst six days are described below. Each module contains some classroom content and some laboratory content. 8.3.1 Vibration Measurement Module A lecture introduced the fundamental concepts in statics, such as vector notation and DOFs. The creation of free-body diagrams was reviewed. During the second half of the lecture, students solved a few problems on their own and at the end the solutions are reviewed. The laboratory with a clamped-free beam [187] in 2004 and 2005 served as an introduction to vibration. Students were introduced to the vibration testing equipment and their underlying physical principles. They were introduced to modal testing, and to mode shapes through the use of a strobe light. 8.3.2 Pendulum Module The pendulum module took place on the rst day of the program from 2006 to 2009; it replaced the vibration measurement module in 2004 and 2005. A lecture was presented to introduce students to the concepts of vibration through a single-DOF pendulum (Fig. 8.3.1). Students were shown how to formulate the equation of motion for the pendulum, and were introduced to the concept of linearization through the small angle approximation. They guessed the solution of the second-order ODE and veried the solution by showing that it satises the ODE. In the laboratory session students measured the acceleration of the pendulum in gure 8.3.1 with a capacitive accelerometer using Siglab (Fig. 8.3.2). They compared 326 the natural frequency calculated from the fast Fourier transform (FFT) (Fig. 8.3.2) with that from the period. 8.3.3 Guitar Module The classroom sessions included a lecture on the physics of a classic acoustic guitar. Students were rst introduced to music notation and the mathematical relationships between the tones on a scale. They were taught about the parts of the guitar and their physical functions, and that sound is pressure waves in the air. Students saw that a string by itself alone produces very little sound, but that when the string is attached to the top plate of the guitar, the body acts like a Helmholtz resonator. They were introduced to partial dierentiation and partial dierential equations (PDEs) through the wave equation for a vibrating string; PDEs are introduced as equations involving partial derivatives. Fourier analysis was also discussed along with its use in some subject areas presented throughout the program. Students were also taught about logarithms and decibels and their usefulness by looking at dierent plots of the same data. In another classroom session students used MATLAB, to perform basic arithmetic, work with arrays, plot, and interpret command syntax. They computed the FFT of dierent sounds and plotted the results. For students to see a wave composed of two frequencies, they recreated the sounds for key presses on a telephone. As a nal exercise students modied the sound of the human voice by re-sampling it in MATLAB. In the pre-laboratory session, students used calipers to measure the lengths and diameters of six pieces of guitar string to nd their densities. They were given the formula for the natural frequencies of a string and asked to nd the tension as a function 327 of the fundamental frequency. Next, students determined the note and corresponding fundamental frequency for each string under the standard tuning. They used a capacitance probe to measure the vibration of a string using the spectral acquisition notebook in LMS Test.Lab (Fig. 8.3.3). They collected data, such as the displacement, spectrum, and octaves. Once students were familiar with the equipment, they tuned the guitar and found the tension in each string by recording the fundamental frequency from the FFT. The harmonics of each string were shown using the spectrum and harmonic cursor. Students gained insight into how careful they have to be when performing testing with the non-contact capacitance probe because of its short stando distance. Students saw how the movement of the guitar when plucked at dierent amplitudes aects the time response but not the spectrum. They discovered that the probe only measures along one axis and the string should be plucked perpendicular, instead of parallel, to the probe face. Students also used a microphone with the LMS Test.Lab spectral acquisition. They were taught about the physical principles of the microphone and then measured the acoustic response of each of the strings. They then compared the microphone results with those from the capacitance probe and discussed why there were dierences in the measured results. Students were taught that the modes of the body of the guitar give its sound a distinct character when they notice more peaks in the spectrum of the pressure from the microphone versus that of the displacement from the capacitance probe. Students also measured the response of chords and found that all the amplitudes were not equal in the FFTs. 8.3.4 Voice Module The classroom session addressed the physics and physiology of speech and psychoacoustics. Students read a handout and websites [188190] about the human voice, 328 which cover topics such as formants, harmonics, and simple ways to approximate the vocal tract as a pipe. Formants are peaks in the acoustic frequency spectrum of speech, which result from the resonant frequencies of the acoustical system (vocal tract). They were introduced to the acoustics response of tubes including pressure waves, pressure nodes, and anti nodes. They were shown how the resonant frequencies of the vocal tract can amplify the laryngeal tone and its harmonics formants. Students were shown that the laryngeal tone is what separates the voice from noise. They were introduced to the use of psychoacoustics in engineering design. Students were also introduced to the physics and mechanics of hearing and the frequency content of dierent types of noise. They did research on the Internet to learn more about the physics of hearing. They created and gave short presentations using PowerPoint about what they learned and were given feedback about their presentations. They were given pointers on slide production, readability, technical content presentation, and public speaking. During the voice laboratory students gained experience in signal processing and analog-to-digital conversion. They are shown some of the diculties of computergenerated speech. The objective of this laboratory was for students to make voice maps by plotting the rst and second formant frequencies. Students rst used the Vowel Sound Essentials website [191] to hear and practice the vowel sounds. They then used COLEA [192] to measure and determine the formant frequencies. They plotted the rst formant as the abscissa, which corresponds to the amount of opening of the mouth, and the second formant as the ordinate, which corresponds to the position of the tongue (Fig. 8.3.4). Two male and two female speakers' voices were mapped for ten dierent vowel sounds (Table 8.3.1). During the laboratory sessions from 2004 to 2006, students were joined by two female sta members from the Shriver 329 Center. It is best if one male and one female are native English speakers and for two speakers English is a second language, so students can learn about vowel sounds used in other languages and sounds not commonly used in the English language. Students discussed the dierences between the male and female voices and between the native and non-native English speakers. Using the spectrograms of dierent words and sentences, students saw how the frequency content of speech changes with time and that consonants have very little sound associated with them. 8.3.5 Elevator Module Students started the day by reading about how elevators work [193] and answering questions about dierent elevator designs. During the second classroom session students were taught the basics of control systems, such as transfer functions, block diagrams, and Laplace transforms. They saw how scaling laws were used in the model elevator design in the DSVL [123, 124, 194] and in other elds. The laboratory session introduced students to the workings of the scaled elevator in the DSVL (Fig. 8.3.5). Students were introduced to some problems with current elevators, such as increased noise and vibratory energy while approaching the top of the shaft. Two dierent motion proles were recorded using a capacitive accelerometer with one example shown in gure 8.3.6a. Students calculated the velocity and displacement proles from the acceleration prole by performing numerical integration using Simpson's 3/8 rule in Excel (Figs. 8.3.6b,c). Students compared their data with previous data [194]. 330 8.3.6 System Control Module Lego Control In the classroom sessions from 2005 to 2007, students designed Lego robots to grip an object and move it over or around a wall (Fig. 8.3.7). In 2008 students designed Lego robots to identify and move three objects at random positions on a straight line to one side of the line based on their colors. Students were led through the use of the basic principles of the engineering design process. They brainstormed as a group and used the basic physical principles they were exposed to previously to draw free body diagrams of their designs. Students had to take cost and manufacturability into account. They were introduced to moments, torques, gear ratios, and open- and closed-loop control. A derivation of open- and closed-loop control block diagrams was presented for a simple single-DOF system. Students worked through a few iterations of a closed-loop control block diagram to see how error is reduced. They programmed using the Lego Robolab environment in 2005-2007 and the Lego NXT environment in 2008. Beginning in 2006 the Lego project was extended to allow students more time to work on their robots. The goals of the project were scored to give students a better understanding of trade-os that are made in the engineering design process. Students programmed open-loop motion proles. They used stop watches to time their robots and nd the average velocity of their robots to make programming more ecient. They tried to make their robots as light as possible while keeping the cost as low as possible. The pieces were priced according to their availability, size, and complexity. Students gained experience with stability, repeatability, and robustness because many times the robot would not do the exact same thing twice. 331 Beam Control Students were taught the dierence between active and passive control, and were shown how passive control can be used to reduce the vibration of one or more modes of a thin beam. The beam was tested with a damper at two locations with dierent levels of damping [195]. The damper used can be the air damper used for the scaled elevator in Section 3.4, a uid damper (Fig. 8.4.2), or a magnetic damper. Students examined the frequency response functions (FRFs) with ME'scope VES to determine how much extra damping was added when using the damper at each position as compared to the beam with no damper. 8.3.7 Damage Detection Module The nite element method was introduced through various classroom activities. Students saw the derivation of the mass and stiness matrices for the longitudinal vibration of a beam. They were introduced to pre-processing, processing, and postprocessing along with the relevant terminology used in nite element modeling. They did static and dynamic analysis of a cantilever beam using I-DEAS. From 2006 to 2008 students programmed the nite elements in MATLAB to calculate the static displacement of a truss made of bar elements. They were shown a research poster in the DSVL to see how damage detection is applied to dierent structures, such as a truss and lightning masts [196]. In the laboratory session from 2006 to 2008 students were shown how damage can be detected on a cantilever beam with dierent types of damage, such as cuts and machined sections (Fig. 8.3.9) [27, 28]. They saw how nite element modeling is performed using the OpenFEM [197] toolbox in MATLAB. A graphical user interface (GUI) was designed (Fig. 8.3.10a) so students could use the damage detection code developed in the DSVL by inputting the natural frequencies they measured. The 332 damage detection results are shown in Fig. 8.3.10b for the beam in Figure 10, which has 56% stiness reduction between 20 cm and 25 cm from the base. In the laboratory session in 2005 students went outside and tested light poles in a parking lot at UMBC (Fig. 8.4.1) and compared the measured FRFs to each other. This activity is similar to the research conducted in the DSVL on lightning masts. Students were taught how to perform a random impact test and its importance in obtaining better data when compared to the single impact test [89]. 333 Figure 8.3.1: Pendulum 334 Figure 8.3.2: Acceleration response (top) of the pendulum and its FFT (bottom). 335 Figure 8.3.3: Students testing the strings of a guitar in 2006. 336 Figure 8.3.4: Voice maps of four speakers 337 Table 8.3.1: Formant frequencies in Hz and vowel sounds of four speakers. Speaker1 Speaker 2 Speaker 3 Speaker 4 Sound Formant 1 Formant 1 Formant 1 Formant 1 short a 754 1895 1055 2046 926 2541 1335 3402 short e 603 1938 642 1938 517 2347 538 2110 short i 538 1981 452 2390 474 1981 495 2347 short o 1055 3101 904 1314 818 2304 947 1378 short u 517 3553 603 1206 624 2455 538 1400 long a 560 2283 603 2261 538 2347 517 2283 long e 323 2433 345 2907 301 2412 258 2972 long i 712 1421 991 1486 904 1335 947 1464 long o 517 2950 581 1270 560 904 517 1464 long u 301 1658 336 2218 336 2326 366 2562 338 Figure 8.3.5: Students learning about the scaled elevator in the DSVL in 2005. 339 Figure 8.3.6: Measured acceleration (a) and calculated velocity (b) and displacement (c) of the scaled elevator. 340 Figure 8.3.7: Lego project and video of a Lego robot designed by a group of students in 2007. 341 Figure 8.3.8: A student testing the beam with a uid damper. 342 Figure 8.3.9: Damaged beam. 343 (a) Graphical user interface (GUI). (b) Damage detection results. Figure 8.3.10: Damage detection GUI and output. 344 8.4 Final Projects Some nal projects undertaken by students are shown below. Each group has one undergraduate teaching fellow as the mentor. Students rst brainstormed ideas and then discussed them with their mentors. Some mentors had already completed the projects, and oered students suggestions and ideas. 8.4.1 Cymbal Project During the 2005 program a group of students investigated the vibration and acoustics of a cymbal. They used a pen-sized hammer to impact the cymbal (Fig. 8.4.2), and recorded the FFTs and FRFs using a microphone to examine the acoustics and a laser vibrometer to measure the vibration. They modeled the cymbal using I-DEAS and visualized its mode shapes (Fig. 8.4.3). 8.4.2 Guitar Project During the 2006 program a group of students investigated the vibration and acoustics of a guitar. They were rst given journal articles and websites to study how other researchers test guitars. They then did research to nd more information about guitars. Students rst repeated the guitar laboratory to make sure the guitar was in tune and to get more practice. They then measured the sound when a chord was played to see that all the notes were present in the spectrum. They noticed some extra resonant frequencies in the spectrum, which they found later to be related to the vibration of the top plate. To investigate this further they measured the rst few mode shapes of the top plate using a roving hammer test (Fig. 8.4.4) and compared them with those presented in [198]. They found that the dierent designs had similar 345 mode shapes but dierent natural frequencies. 8.4.3 Baseball Bat Project To learn more about the dynamics of baseball bats students were given the task of creating a better baseball bat. They rst tested a bat by holding it and using a tethered ball to impact the bat at dierent locations, and reported if there was any sting. They entered the geometries of various bats in the LMS Test.Lab or ME'Scope software, and used the roving hammer test to measure their mode shapes (Figs. 8.4.5 and 8.4.6). They compared the experimental and numerical (Fig. 8.4.10) results and discussed the reasons for dierences between the two. Finally, they used the nite element model to see if they could increase the sweet zone of a bat. 8.4.4 Mbira Project During the 2007 program a group of students studied the vibration and acoustics of a mbira (Fig. 8.4.8). They set up free boundary conditions using foam (Fig. 8.4.8), and measured its acoustic response using a microphone when each key was played. They also measured the vibration of each key with a noncontact capacitance probe (Fig. 8.4.8) and determined the natural frequencies of the key from the FFT. Next, they used a small modal hammer and the laser vibrometer to conduct a roving hammer test. They measured the mode shapes of the top plate of the mbira and animated them (Fig. 8.4.9). They used I-DEAS to create a nite element model of the longest key on the mbira (Fig. 8.4.10) and tried to correlate this to their measurements. 346 8.4.5 Bridge Project During the 2007 program a group of students decided to study bridges and made a model from bamboo (Fig. 8.4.11). Using LMS Test.Lab, they entered the geometry of the bridge, instrumented the structure with an array of accelerometers, and performed a roving hammer test to excite as many modes as possible. They used a random impact test because the single impact results were poor due to some loose joints in the structure and high damping. Using modal analysis software, they obtained the natural frequencies and mode shapes of the structure from the FRFs. They also constructed a beam element model of the bridge in I-DEAS (Fig. 8.4.12). 8.4.6 Rotordynamics Project During the 2007 program a group of students studied rotordynamics and rotating systems using a reference [199] that they found at the library. They rst studied the equations of a simple rotor system and used them to design a test stand (Fig. 8.4.13). They took static measurements with a laser displacement sensor to nd the center of mass. They then used the laser displacement sensor to measure the rotational frequency of the shaft and a strobe light to estimate the critical speed of the rotor system and observe the rst two mode shapes of the shaft. After all the measurements were completed, they updated their mathematical model to better match what they observed and discussed the possible reasons for discrepancies. Students also conducted an experiment described in [199], where they used a motor to excite the vibration modes of a long weighted string (Fig. 8.4.14). They rst learned about the frequency equation of the string. They then determined the maximum rotational frequency of the drill using the laser displacement sensor as a frequency counter. They wrote a 347 MATLAB program to determine the length of the string and the necessary tension in the string. By adjusting the frequency of the motor they were able to excite the rst ve modes of the string. 8.4.7 Bike Wheel Project During the 2008 program a group of students worked on a method to true a bike wheel by measuring the distance from the rim to a laser displacement sensor. They built an electromagnet, attached it to a spectrum analyzer, and monitored the voltage to determine the rotational frequency of the wheel. They also found that they could identify one rotation by attaching a thin raised marker to the rim as a reference, which would show up as an impulse in the displacement measured by the laser displacement sensor. They used the measured displacement to identify the locations of anomalies or where to true the wheel. They varied the tension in the spokes to create a plot of the spoke tension versus the distance out of true. They measured the spoke tension with a Wheelsmith spoke tension tool. 8.4.8 Shoe Project During the 2008 program a group of students looked at the testing of the eectiveness of shoe soles and the measurement and replication of the human foot fall force. They instrumented two dierent types of shoes, and measured the acceleration and force by placing an accelerometer and a force transducer in the heel of the shoes. They used a linear actuator developed in the DSVL to try to replicate the measured results by programming several motion proles. 348 8.4.9 Earthquake Project During the 2008 program a group of students looked at the resonance of a toy building and learned to make a tuned mass damper for the building. They rst used wooden blocks for the building and applied a vertical base excitation. The blocks were loosely connected, and to make a good building analog, they used Legos to make a simple tower (Fig. 8.4.15). They made a shake table by placing ball bearings in the holes of an optical table and putting an aluminum plate on top of the ball bearings. They then attached a shaker to the optical table and used a stinger to drive the plate. They attached an accelerometer to a block at the end of the stinger to measure the input acceleration and frequency, and measured the mode shapes and modal damping of the tower. They did research to learn about passive vibration dampers and decided to use a pendulum-type absorber. They then calculated the length of the pendulum that would have the same rst natural frequency as the tower. They attached the pendulum to the tower and re-measured the mode shapes and damping of the tower; they found the damping increased and the rst mode response amplitude decreased. 349 Figure 8.4.1: Students performing a random impact test on a light pole in 2005. 350 Figure 8.4.2: A student testing a cymbal with a pen-sized hammer. 351 Figure 8.4.3: A mode shape of the cymbal from I-DEAS. 352 Figure 8.4.4: Mode shapes and their animation videos of the top plate of a guitar measured by a group of students in 2006. 353 Figure 8.4.5: A student performing an impact test on a softball bat in 2005. 354 Figure 8.4.6: First measured bending mode of a wooden bat with the animation video. 355 Figure 8.4.7: First computed mode shape of the wooden bat with the animation video. 356 Figure 8.4.8: Mbira with a capacitance probe. 357 Figure 8.4.9: A measured mode shape of the top plate with its animation video. 358 Figure 8.4.10: First computed mode shape of the middle key. 359 Figure 8.4.11: Bamboo bridge with accelerometers attached. 360 361 Figure 8.4.12: Finite element model of the bridge in Figure 22. Figure 8.4.13: Rotordynamics test stand and the associated videos. 362 Figure 8.4.14: String mode shape experiment and the associated videos. 363 Figure 8.4.15: A simple tower on a shake table. 364 8.5 Connections of Program Activities to Various Criteria and Standards The FEDS Academy activities have close links to the Accreditation Board for Engineering and Technology (ABET) criteria [200]. The ABET criteria are similar in many ways to the Maryland Technology Education Voluntary State Curriculum (MTEVSC) standards [201]. Table 8.5.1 shows the connections between the ABET criteria and the MTEVSC standards and how the FEDS Academy activities are related to them. Note that there are many MTEVSC standards that will fulll/mirror the ABET criteria and only one or two standards are shown in Table 8.5.1, and the FEDS Academy activities listed are the ones with the strongest links to the ABET criteria and the corresponding MTEVSC standards. If the program were to be made longer, stronger connections could be made for technical communication. Currently the 15 to 20 minute nal presentation is the main activity in the FEDS Academy for technical communication. In addition to engineering and technology, the FEDS Academy activities have strong links to the two other subjects in STEM: mathematics in the contexts of algebra, geometry, and calculus, and science in the context of physics. The Maryland Mathematical Standards [202, 203] for Algebra One and Two are outlined in Table 8.5.2. Maryland does not have standards for calculus, but many students in the program indicated in the pre-survey that they have had or will have Advanced Placement (AP) Calculus; so the concepts discussed in AP Calculus AB/BC [204] are linked to the FEDS Academy activities (Table 8.5.3). The Maryland Physics Standards [205] are for algebra-based physics, while the AP Physics C standards [206] are for calculusbased physics, which a few students indicated that they had taken or planned to take. 365 In the current program calculus-based physics is mainly used with some algebra-based physics used for the more dicult problems. The connections between the two physics standards and the FEDS Academy activities are outlined in Table 8.5.4. Overall, the current program provides a wide range of educational opportunities in STEM. 8.5.1 Comparison of Dierent Programs Bloom's Taxonomy [207209] is used to compare the FEDS Academy format with those of some other summer programs. Bloom's Taxonomy comprises three types of learning domains: cognitive [207], aective [208], and psychomotor [209]. Each type is broken into subdomains, from the simplest at the top to the most complex at the bottom in Table 8.5.5. The cognitive domain deals with knowledge and intellectual skills [207]; the aective domain deals with emotions, values, and attitudes [208]; and the psychomotor domain deals with physical skills and interaction with the physical world [209]. This comparison will be mainly made in the cognitive domain because all the programs used in the comparison list their cognitive educational features. The features of the FEDS Academy in the aective and psychomotor domains are only briey reviewed because most programs cover these two domains fairly well. The key elements of a DBT program are problem denition, modeling, design method, prototyping, and project management [162]. A laboratory-based program often includes learning through a combination of classroom sessions/lectures, experiments, and analysis of experimental results (CEA) [184, 210]; or a combination of experiments and analysis of experimental results without classroom sessions [165]. The FEDS Academy and the Robotic Autonomy program [174] can be looked at as a mixture of DBT and CEA. 366 8.5.1.1 Cognitive Domain The FEDS Academy is unique in the cognitive domain among DBT and laboratorybased programs. In the cognitive domain, knowledge is the simplest subdomain and is addressed in all the programs. In the knowledge subdomain most DBT programs do not cover the same depth of material as the FEDS Academy due to the lack of formal lectures and experiments. In the FEDS Academy and laboratory-based programs, learning is rst done through lectures and experiments at the beginning of the program. This gives students a new knowledge base upon which to build. In the DBT programs the lecture and initial experiment parts are often skipped because students often comment that they don't like lectures and there exists the notion that the best way to learn engineering is to do it [162]. Another drawback of many of the DBT programs is that students don't learn how to use new equipment [169]. The ideology of the FEDS Academy is to give students an overall picture because many students have wrong preconceptions about why things happen in the world around them and how science and mathematics are applied to practical problems [211]. In many of the programs reviewed there is felt to be a lack in comprehension because the tasks students often do are the ones they have already done before [169]. In the FEDS Academy most students have never had any formal experience with the advanced mathematics and physics they are exposed to, so they need to learn how to state new complex problems in their own words to connect old knowledge to new knowledge. The laboratory-based programs reviewed, where students solve new problems with new knowledge, techniques, and equipment, were often felt to improve comprehension the most. In the nal projects of the FEDS Academy, comprehension is important because many of the projects are completely new to students. If the problem is simple or familiar, there is little eort needed in the problem denition 367 stage. In the FEDS Academy and many laboratory-based programs students apply what they learn in classroom sessions to laboratories and data analysis. In contrast, many DBT programs and some laboratory based-programs focus on hands-on problem solving or laboratories using familiar concepts. In cases like these there is little new knowledge to apply. In many of the laboratory-based programs reviewed there is no relationship between the experiments presented. In these cases there is little or no progression of the experiments and little application of what is learned in a previous experiment to the next. Another drawback to the nature of laboratory-based programs is that students do little or no focus on designing using the content presented. Even though the focus of DBT programs is design, in the programs reviewed students rarely picked their projects, or have multiple projects to pick from, which limits their possibility to see how engineering applies to something that interests them. This is especially important for female students who are often interested in problems not addressed in a co-educational camp. In a longitudinal survey a female student commented that the nal project, which was the analysis of the dierent materials for the bottom part of a shoe, was the activity she enjoyed the most. In contrast to the FEDS Academy and laboratory-based programs, there is usually no formal scientic or mathematical analysis mentioned in the DBT programs reviewed to verify observed outcomes. The types of analysis used in DBT programs are problem denition, design analysis, and design method. In the FEDS Academy there is some analysis in all the activities. In the Lego and nal projects, and MATLAB programming, students learn how to break a complex problem into smaller parts. This type of analysis is also shown in programs that focus on programming [174, 184]. For the nal projects, students are tasked with choosing their own projects and iden- 368 tifying the main aspects of the problems to explore using what they have experienced in the laboratories. At the end of the nal projects students do an analysis to verify their results, and correlate their results to those of others or theoretical predictions. In many of the DBT projects reviewed the level of synthesis is generally lower than that in the FEDS Academy because the projects are familiar to the students. Students in the FEDS Academy face more challenges in synthesizing ideas because they are exposed to more dicult and unfamiliar material. During the laboratories and projects they also learn how to troubleshoot and x equipment. In the FEDS Academy students design and build their own experiments and equipment in their nal projects by synthesizing ideas. An example of this is that in the rotordynamics project students performed calculations using MATLAB to determine how to size their experiments based on the equipment and parts available. They used the speed of the drill to determine the length of their string, so they could excite the rst ve modes. An analog to this level of synthesis is seen in the Robotic Autonomy program [174], where instead of designing new hardware they design software. Bloom classied evaluation as the most complex subdomain in the cognitive domain because it encompasses all the other cognitive subdomains and builds upon them [207]. The laboratory-based programs reviewed mainly use external quantitative evaluation, where students compare their experimental results with theory or references. The laboratories in the FEDS Academy include external quantitative and qualitative evaluations, where students compare their results with those from other researchers; for instance, they do so when they make voice maps in the voice laboratory. There is little external quantitative evaluation in DBT programs because there are often no scientic measurements made by students in the programs reviewed. Internal qualitative evaluation is mainly used in DBT programs because students apply 369 the relationships and rules learned in the program to evaluate the logic and accuracy of their results. Often, they evaluate their designs based on the constraints given to design within. In the FEDS Academy students evaluate their designs for both the Lego and nal projects based on the scientic methods and engineering design principles using both internal and external, quantitative and qualitative criteria. They use newly presented mathematical evaluation techniques to compare experimental and theoretical results. The nal projects, such as the baseball bat, guitar, mbira, and rotordynamics projects, include external evaluations, where students compare their results with those of others. 8.5.1.2 Aective Domain The level of receiving phenomena in the FEDS Academy is matched by only a few of the programs in that the material is so new to the students, and if they don't pay attention, they won't be able to complete any of the laboratories. Responding to phenomena is very important in the FEDS Academy because students are rst shown how to use the equipment and they then use the equipment to proceed in nishing the laboratories. Students usually need to ask many questions about the concepts they are learning to successfully complete the laboratories. This is in contrast to other programs, where students are led step by step through activities. The possibility of failure is an important aspect of the FEDS Academy, which reminds students that attention to detail is important and that engineering is not always as expected. Students value engineering more when they choose their own projects and experience engineering culture through discussions and interactions with the sta. They gain a sense of value by explaining what they do to their classmates through project presentations at the end of the program. They learn a great deal about eective orga- 370 nization through the voice, guitar, pendulum, and beam control laboratories, where teamwork is an important part of the laboratories. The group projects show students the importance of organization, cooperation in a group, and self-reliance, when work is divided for eciency. Programs that don't have lectures can give students the wrong impression that engineering is just building things and doing design, which is what the FEDS Academy students often think at the beginning of the program. The FEDS Academy gives students a more broad impression of being an engineer and the life-long learning that takes place through reading, researching, and attending lectures or seminars. It strives to foster a genuine appreciation of engineering by showing that engineers don't just do without having to learn before they do and that what they are learning is relevant to the world around them. While most students who come to the FEDS Academy are already interested in engineering, they don't exactly know what engineering entails. 8.5.1.3 Psychomotor Domain The FEDS Academy extends the level of students' perception when they have to rely on the response of an instrument instead of what they can directly perceive. The level of guided response in the FEDS Academy is more challenging than in most other programs because students are introduced to more complex skills. The mechanism subdomain is focused on how students increase their prociency with the instruments and tools. The use of most of the instruments and tools in all the laboratories increases their prociency with them. A more complex overt response is seen as students become more familiar with the instruments and software they are using. An example of this is when students using an impact hammer know whether or not they have correctly used it without being told or asking. During such nal 371 projects as the rotordynamics and earthquake projects, students adapt equipment for purposes other than their original use. In the origination stage, which is the most complex psychomotor subdomain, students set up their experiments using the skills and physical intuition they previously learned and acquired. 372 Table 8.5.1: Connections among the ABET criteria, the MTEVSC standards, and the FEDS Academy activities. ABET Criteria MTEVSC Standards A. Apply mathematics, science, and engineering B. Design and conduct experiments and analyze and interpret data C. Design a system, component, or process within realistic constraints D. Function in multi-disciplinary teams E. Identify, formulate, and solve engineering problems F. Understand professional and ethical responsibility G. Communicate eectively Identify and demonstrate how technological progress promotes the advancement of science and mathematics. Demonstrate how many technological problems require a multidisciplinary approach. FEDS Academy Activities All modules in Section 3, Lego and nal projects Conduct a structured research and development process as part of a design problem. All modules, Lego and nal projects Identify and describe that requirements of a design, such as criteria, constraints, and eciency, sometimes compete with each other. Lego and nal projects Explain that brainstorming is a group problem-solving design process in which each person in the group presents his or her ideas in an open forum. Demonstrate knowledge of and apply the engineering design and development process. Lego and nal projects, most modules Lego and nal projects H. Understand the broad educational impact I. Recognize the importance of engaging in life long learning Explain that ethical considerations are important in the development, Group discussion, mentor role models selection, and use of technologies. Evaluate nal solutions and communicate observation, processes, and results of the entire design process, using verbal, graphic, quantitative, virtual, and written means, in addition to three-dimensional models. Analyze the connections that exist both within the various elds of science and among science and other disciplines including mathematics, social studies, language arts, ne arts, and technology. Develop an understanding of the cultural, social, economic, and political impacts of engineering design and development. Identify and use resources and strategies for keeping abreast of advances in technologies. J. Have a knowledge Identify and use resources and strategies for keeping abreast of of contemporary advances in technologies. Design and conduct research related to the issues nature of technology. K. Use modern tools, skills, and techniques in engineering practice Use knowledge of the core technologies in the engineering design process. 373 Final presentation, elevator and voice modules Elevator Module, Damage Detection Module, Guitar Module, Voice Module Final projects, laboratory and campus tours, engineering discussion Elevator, beam control, and damage detection modules; laboratory and campus tours; engineering discussion All especially the use of MATLAB, LMS Test.Lab, Siglab, ME'Scope, I-DEAS, Excel, and PowerPoint Table 8.5.2: Connections between the Maryland Mathematics Standards and the FEDS Academy activities. Maryland Mathematics Standards for Algebra One and Two Algebra One goals: functions and algebra, patterns and functional relationships using language of math and appropriate technology, model and interpret real world situations Geometry, measurement, and reasoning: represent and analyze two and three dimensional gures using tools and technology, apply geometric properties and relationships to solve problems using tools and technology, concepts of measurement using tools and technology Data analysis: collect, organize, analyze, and present data Algebra Two goals: functional notation; domain and range rules; add, subtract, multiply, and divide functions; logarithmic expressions Presentation of results: reasoning and process for solutions, ascribe meaning to solutions, alternative representation of functions Linear versus nonlinear functions, graph functions, solve equations/inequalities, linear systems of equations FEDS Academy Activities Pendulum and guitar modules I-DEAS session, nal projects, pole testing laboratory Voice, beam control, and guitar modules; nal projects Pendulum damping measurement, elevator motion proles Final projects, elevator and voice modules MATLAB introduction , MATLAB statics exercise using the nite element method 374 Table 8.5.3: Connections between the AP Calculus AB/BC Standards and the FEDS Academy activities. AP Calculus AB/BC Standards Derivatives FEDS Academy Activities Pendulum lecture, lecture on kinematics and dynamics, elevator motion proles Integrals Pendulum laboratory, elevator motion proles Series Lecture on Fourier series and analysis Vectors MATLAB statics exercise, MATLAB introduction, vibration measurement and damage detection modules, I-DEAS session, introduction to mode shapes Introduce and solve ordinary dierential equations Pendulum lecture Dene partial dierential equations Guitar lecture on the string vibration 375 Table 8.5.4: Connections between the Maryland and AP Physics C Standards and the FEDS Academy activities. Maryland and AP Physics C Standards FEDS Academy Activities Mechanics of materials Damage detection module, MATLAB statics exercise Scalar and vector quantities MATLAB statics exercise, pendulum module Object motion: constant Pendulum motion, elevator motion velocity/acceleration, linear frame of proles reference, projectile motion Newton's laws: balanced and unbalanced Pendulum and elevator modules, forces, inertia, action/reaction MATLAB statics exercise, Lego project, nal projects, lecture on kinematics and dynamics Friction, gravity, work and power, impulse Lecture on dampers, elevator and and momentum pendulum modules, Lego project, discussion on fundamentals of the impact hammer Physical waves: longitudinal, transverse, Beam control, elevator, and damage torsional detection modules Wavelength, frequency, velocity, Pendulum, guitar, and voice modules; amplitude, Doppler eect introduction to the laser vibrometer 376 Table 8.5.5: Bloom's Taxonomy domains and subdomains. The complexity of the subdomains increases from the top to the bottom. COGNITIVE AFFECTIVE PSYCHOMOTOR Knowledge Receiving Phenomenon Perception Comprehension Responding to Phenomenon Set Application Valuing Guided Response Analysis Organization Mechanism Synthesis Internalizing Values Complex Overt Response Evaluation Adaptation Origination 377 8.6 Survey Results The survey results for the 2005-2008 programs, along with many of the survey questions, are shown below. The survey results for the 2004 program are not included because there were fewer students in 2004. Some survey questions were changed in 2008. In particular, a neutral choice was added to some questions in 2008; the neutral choice was not used in the calculations shown below. Some survey questions for the Lego and nal projects were also added in 2008. 8.6.1 Pre-survey Results As noted earlier, students come from a talented pool. From 2005-2009, nearly half of students had taken a physics class, and 40% a calculus AB or BC class (Figure 8.6.1a); the numbers of students who were enrolled in various courses in the upcoming year are shown in Figure 8.6.1b. Their interest level in engineering was high from 2005 to 2007, with 66% being very interested (Figure 8.6.2). In 2008 and 2009 the pre-survey question concerning an interest in engineering was changed to allow students more choices for dierent types of engineering, including mechanical engineering. Ten out of eleven students selected one or more engineering disciplines in 2008, and one selected architecture. In terms of their interest in mechanical engineering, 85% of students responded positively (Figure 8.6.2). Note that, in 2008 and 2009, the students who selected mechanical engineering from the list of engineering choices were counted as Very True and those who didn't select mechanical engineering were counted as Not True at All. When asked if they planned to attend college and major in engineering, 80% of students responded positively. 378 8.6.2 Laboratory Survey Results and Observations Classroom: Content was appropriate to my knowledge base (Knowledge Base) I learned new skills/gained new insight and understanding (Skills/Insight) ! "# # $ % & $ ' ( $ $ ) ( $ * $ + $ & , - $ . / +00000000000000000000000000000000000 Overall, this classroom activity was: % # 1 ' ) / * / +00000000000000000000000000000000000 Laboratory: " Activity/demonstration was appropriate to my knowledge base %23 (Knowledge Base) I learned new skills/gained new insight and understanding (Skills/Insight) ! " # " $ % Overall, this laboratory activity was: ' ( ! $ # ) % ) * ++++++++++++++++++++++++++++++++++ ,- % . - - - - - - - / !0-1-- . 2 , . '- 34. 3 435 4- 3.4 6 - 3 .4 7 7 - - - - . 8 , - . 9% The vibration measurement laboratory was fairly well received by students, with 90% of them responding that they gained new skills and insight, and 86% that the material presented was appropriate to their knowledge base. The pendulum lecture was a little too advanced for students and 73% of them felt the lecture was appropriate for their knowledge base. Ninety-three percent of students responded positively when asked if they gained new skills and insight. Forty-six percent of students rated the lecture as very good and 46% as average. The pendulum laboratory was more appropriate, with 83% of students responding that the laboratory was appropriate to their knowledge base, and 94% responding that they gained new skills and insight. Overall, the laboratory was considered to be very good by 66% of students, with 34% responding that the laboratory was average. A majority of students (81%) thought the MATLAB session was very good and 95% of the students thought it was appropriate to their knowledge base. All of the students responded that they gained new skills and insight. One of the students commented that the MATLAB was fun while another said It encouraged me to learn more and come early tomorrow. The acoustics laboratory was well received by students with all students responding that the activity was appropriate to their knowledge base and the activity engaged their interest. All of the students responded that they gained new skills, insight, and understanding. When asked for comments about the laboratory activity one student wrote Loved it. Perfect Day. Ninetyseven percent of students in the voice laboratory responded that the content was appropriate to their knowledge base and all responded they gained new skills and understanding. A student commented that the laboratory was Excellent overall, and another wrote Loved the material. It gave me an understanding of a number of concepts in a systematic process. Ninety-two percent of students responded that 381 the elevator demonstration was appropriate for their knowledge base, with 97% responding they had a better understanding. Overall, 75% of the students rated the demonstration as very good. Three of the students described the activity as cool. Eighty-two percent of students responded that the control lecture content was appropriate and that they gained new skills and understanding. The introduction to ODEs lecture given in 2005 was relatively appropriate with 67% positive responses and 67% of the students stating that they gained new understanding; the ODEs were introduced in the pendulum lecture from 2006 to 2009. Some students commented that they had seen ODEs before. The beam control laboratory was very well received with 96% of students responding that the content was appropriate and engaging, and 96% of students stating they gained new skills and understanding. The Lego activity was rated by all students in 2005 to be appropriate to their knowledge base; 89% of the students found the activity engaging, and 67% learned new skills. The Lego project in 2008 was rated by 91% of students to be appropriate to their knowledge base and engaging, and the same percent learned new skills. The I-DEAS session was well received by all students in 2005 in terms of being appropriate to the knowledge base, being engaging, and learning new skills. The pole testing conducted in 2005 was well received by all students as well, even though it was quite hot that day. The damage detection laboratory was rated as engaging by 92% of students and very good overall by 76% from 2006 to 2008, with 93% considering the laboratory appropriate to their knowledge base and 87% responding that they gained new skills and insight. The survey results for the Lego project in 2008 and 2009 are shown in Figure 36, where the survey questions were essentially the same as those for the laboratories, except that the Understanding statement in the laboratory survey was replaced by 382 the statement The activity improved my understanding of the connections between dierent subject areas in engineering (Connections) and the overall rating statement became Overall, this activity was very good. The classroom length survey results are shown in Figure 8.6.11. The overall classroom and laboratory ratings are shown in Figures 8.6.12 and 8.6.13, respectively. For the three main survey questions, the average scores for the vibration measurement laboratory, the pendulum laboratory, the MATLAB session in the guitar module, the guitar laboratory, the voice laboratory, the elevator laboratory, the beam control laboratory, the damage detection laboratory, and the pole testing laboratory from 2005 to 2009, are shown in Table 8.6.1; the average corresponding to each main question, over all the activities, is also shown. As would be expected from high school students, repetition of material proved problematic. They would prefer not to cover material learned during their regular school year more than once. However, because a range of abilities and attention spans were evident even in this talented pool, repetition of some lecture material was necessary. Similarly, students at this level want more experiential work than they can often handle. To accommodate some of their concerns, the lecture material is immediately and continually updated, and work on the projects has been moved up to the rst day of the program. Accommodating students' input has met with measured success, as is evident in their responses to the laboratory questions. In most instances at least two-thirds of students responded in a highly positive fashion (the activity/demonstration was appropriate to my knowledge base: Very True; I learned new skills/gained new insight and understandingVery True; and overall, this activity was: Outstanding). The vibration measurement, pendulum, and elevator laboratories can be further improved. Overall, the content seems well suited to the 383 ! "#$$ % &%'( "$# % &#'( ! "#) %( * + "# %( 8.6.3 Post-survey Results , ! - / / $ 0 "1 2( "#&&&'( 0 "3( "#&&&'( 0 "3 ( "#&&&'( 0 " ( "#&&&'( , "3$( , "3#( 4 "3%( 5 "3)( 5 "3( %') What is the likelihood that you will apply to the UMBC's engineering program for college? (Apply) (2005-08) Based on your experiences in this program, what is the likelihood that you will consider engineering as a college major? (Consider) (2005-08) Very High (C1) High (C2) Intermediate (C3) Low (C4) Very Low (C5) In comparison to your knowledge of what engineering is and what engineers do at the beginning of the program, how would you rate your present level of knowledge? (Knowledge) (2005-07) In comparison to your interest in engineering at the beginning of the week, how would you rate your present level of interest? (Interest) (2005-07) Much Higher (C1) Higher (C2) About the Same (C3) Lower (C4) Much Lower (C5) Part 2 The lectures were eective in introducing advanced college level calculus (Calculus) (2005-07) The lectures were eective in introducing physics (Physics) (2005-07) The lectures were eective in introducing engineering (Engineering) (2005-07) The introduction to advanced calculus, physics, and engineering will help make me consider studying engineering in college (Study) (2005-07) The laboratories were of appropriate diculty for my level of education (Labs) (2005-08) I found the curriculum to be challenging (Curriculum) (2005-08) 385 !" #$ % &' &' %! (! ) %! ) * + * ,' + - ! " ! .! / 0' 1 1 ' 2! ! ! 3" 45$%$ 3 667 668$ ! 569 1 .! !.. . " ! ! !"!!"$ 9 !. 3 667 66:$ 889 1 .! . % !.!.!" 1 !"! . !. 889 !"!!" / '!0!! ! 1 .! !;.! ! !"!!" !. 489 . % !.!.!" 1 !"!!" ! . <'$ 89 1 .! !;.! % ! !"! !. 889 . % !.!.!" 1 !"!!" =! 664 !. 66: .! . ! !! 1 ;! /$ 1 .> .> : $ !" !% ! ! % 45 ," .! 1 !.. 2! % !" !% 45 8.6.4 Longitudinal Survey Results ! ! "##$ %& '( ( ) ! ' ! * + - %& . " . -! /( . 0 1 2 ! 3 3 4. 5 1 ! 67+ # 8 "8 """98 ""9"98 "9":98 ":908 00"98 0"9 098 090:98 0:95. 9 1 3 3. ; %& ) ! 3. -! . : 7 %& < ( ! => ! ? ! ' '( * # @! ( 0$: one graduated with a bachelor's degree in mechanical engineering; he served as a teaching assistant for the undergraduate vibrations course and his senior design group received a design award for their project from the Baltimore section of the American Society of Mechanical Engineers. Thirteen students are studying engineering, science, or mathematics in college; one student is studying business in college. Three students are still in high school, but plan to study engineering. Two of the students who returned the survey commented that the program didn't aect their choices of major, two cited a better understanding of dierent aspects of engineering, two were swayed to choose mechanical engineering, and two were more interested in mechanics or physics. One found that engineering wasn't for him even though he had originally thought to be an engineer. The student did, however, mention that the program was a great experience and it helped him decide what he really wanted to do. For question seven all the students commented that the FEDS Academy was a good or great learning experience. Many students commented that the introduction to engineering was very helpful and that the program gave them experience that none of their classmates had. Students also commented that they had learned a lot more that summer than any other summer and had fun. They also mentioned that the sta were great and it was nice to get to know them. 388 (a) a (b) b Figure 8.6.1: Pre-survey results from 2005-2009 on the numbers of students who had taken (a), or would take in the upcoming year (b), various courses. 389 390 Figure 8.6.2: Pre-survey results on students' interest in: engineering; mechanical engineering; and dynamics, vibration, and control (Dynamics). The four bars, from left to right, in each subject area correspond to 2005 to 2009. The number of students who answered the question on Dynamics is much fewer in 2008 than those in the previous years because the students who responded with a neutral choice were not counted. 391 Figure 8.6.3: Vibration measurement classroom (a) and laboratory (b) survey results in 2005. 392 Figure 8.6.4: Pendulum classroom (a) and laboratory (b) survey results. The three bars, from left to right, for each question correspond to 2006 to 2008. 393 Figure 8.6.5: Guitar classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2008. 394 Figure 8.6.6: Voice classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2009. 395 Figure 8.6.7: Elevator classroom (a) and laboratory (b) survey results. The three bars, from left to right, for each question correspond to 2005 to 2008. 396 Figure 8.6.8: Survey results for the system control classroom (a) and the beam control laboratory. The four bars, from left to right, for each question correspond to 2005 to 2008. 397 Figure 8.6.9: Damage detection classroom (a) and laboratory (b) survey results. The four bars, from left to right, for each question correspond to 2005 to 2009. 398 Figure 8.6.10: Survey results for the Lego project in 2008 and 2009. 399 Figure 8.6.11: Classroom length survey results. The four bars, from left to right, for each classroom activity correspond to 2005 to 2009. Number of results vary due to some students failing to respond. 400 Figure 8.6.12: Overall classroom survey results. The four bars, from left to right, for each classroom activity correspond to 2005 to 2009. 401 Figure 8.6.13: Overall laboratory survey results. The four bars, from left to right, for each laboratory activity correspond to 2005 to 2008 or 2009. Table 8.6.1: Average scores for the laboratory activities and the MATLAB session from 2005 to 2009. Activity Question Knowledge Base Skills / Insight Overall Vibration 2.44 2.77 1.88 Pendulum 2.1 2.48 1.74 MATLAB 2.4 2.56 2.07 Guitar 2.44 2.48 2.22 Voice 2.63 2.49 2.26 Elevator 2.37 2.43 1.94 Control 2.46 2.43 2.01 Damage 2.46 2.39 2.13 Pole 2.88 2.66 2.22 Average 2.47 2.52 2.05 402 (a) (a) (b) (b) (c) (c) Figure 8.6.14: Post-survey results: (a) part 1, (b) part 2, and (c) part 3. The four bars, from left to right, for each question or topic correspond to 2005 to 2007 or 2009. The number of students who answered the question on Curriculum in (b) is much lower in 2008 than those in the previous years because the students who responded with a neutral choice were not counted. 403 Table 8.6.2: Ranked aspects of the nal projects. Project Aspect Average Ranking Creating Models 7.38 Theoretical Analysis 7.13 Designing the Experiment 6.75 Ideation/ Brainstorming 4.5 Presenting 4.38 Preparing Presentation 4.13 Picking a Project 3.5 Picking a Group 3.13 Working Within a Group 3.13 404 8.7 Concluding Remarks The FEDS Academy was successful at introducing junior and senior high school students to dynamic systems as practiced within mechanical engineering. The main unique features of the program are its focus on dynamic systems, the level at which it integrates DBT and laboratory-based programs, and the range and depth of learning skills it covers. Dynamic systems can integrate various aspects of STEM education and be eectively used in an outreach program for high school students. The FEDS Academy has been ethnically diverse and included 38% of African American and female students. The program activities have close links to ABET criteria and Maryland and AP standards. Students in the program favored more the laboratory content than the classroom content. because the program intends to give students a feel for a collegiate education, some lecture content is necessary. While bringing academic insight to high school students is dicult, students generally had more condence in their engineering skills, a better understanding of engineering in the real world, and a higher interest in engineering when they completed the program. The longitudinal survey results indicate that many of the FEDS Academy students remain in engineering, science, or mathematics, and the program helps students make informed decisions about their future studies. In addition to the FEDS Academy, the authors have provided similar laboratory demonstrations or experiences to high school students in the Worthwhile to Help High School Students (WORTHY) program administrated by Northrop Grumman and the Shriver Center, and in the UMBC Classic Upward Bound program. Similar demonstrations have also been provided to middle school students in the Teaching Enhancement Partnership Project (TEPP) supported by the National Science Foundation, and in the Junior Engineers in Maryland Schools 405 (JEMS) program supported by the Maryland Department of Education, at UMBC. 406 Chapter 9 Conclusions The computational aspects of the feature selection method using an iterative algorithm is presented. The method is derived from the sensitivity of the eigenvalue problem to changes in model parameters using a rst order Taylor series expansion. The method was tested using the exact sensitivity matrices and those calculated using the nite dierence method. The method was shown to have a slight improvement when the sensitivity matrices are recalculated at each iteration if the sensitivity matrices are not linear with respect to the damage parameter. A convexied least squares cost function was found not to be useful in the damage detection scenarios investigated most likely due to the small number of parameters in the model. Two approaches to solve the minimization problem are shown; the Moore-Penrose pseudoinverse method with or without the singular value decomposition and the LevenbergMarquardt method. It was found that the Levenberg-Marquardt method was more robust and converged in fewer iterations. Several demonstrations of theoretical approaches to choose the updating parameters for beam type structures are presented and used successfully. A program based on the algorithms was designed and imple407 mented that was modular and expandable. The modularity and exibility of program was tested on several slender structures. For the rectangular beam it was found that using the thickness as the updating parameter resulted in more accurate predictions than those compared with just using the elastic modulus. It was shown that using the convexication approach outlined in Chapter 2.4.2 gives a tighter bound on the damage location when used with the elastic modulus but results in an overall reduction in thickness. The use of the SVD was demonstrated when there is a large amount of noise due to multiple models being used. Due to the lack of sensitivity near the free end of xed free beams, it is benecial to not ignore the last few elements. For both beams using the forward problem to investigate the best parameters to use proved to be very useful. The use of bending and torsional modes to do damage detection was demonstrated and it was determined that using the bending modes alone was best. The damage extent when the rst torsional natural frequency was included was consistently overestimated for the Gauss-Newton method and underestimated for the Levenberg-Marquardt method. In general the Levenberg-Marquardt method converged in fewer iterations than the Gauss-Newton method. The implemented program successfully used another researcher's model without signicant modication to detect an increase in mass using the density. The experimental results were similar to the simulation results in terms of the locations of erroneous damage especially in the cases using four or ve natural frequencies. At the expected damage location the amplitude of the density increase was best for the experimental cases using six or seven natural frequencies. The best overall mass estimation in the experimental cases was for four or ve natural frequencies while all the simulations gave nearly the same mass increase. 408 Using the aramid roped test stand new testing methods, modal models, and a modal survey were completed successfully. The measurement of both the bending and longitudinal modes using the multiple impact hammer technique and dynamic strain gage were of the same quality as those obtained with the inertial shaker and accelerometers. The longitudinal modes were experimentally shown to be a coupling of both sections of the rope and that one model is needed for the rst two modes and another for the higher modes. The transverse modes of the rope were shown both experimentally and numerically to be decoupled measured eigenparameters compared well with both the FE and analytical modes. The material properties of the aramid cables were identied and for the axial stiness to be reasonable when compared with those given in the literature. The results of this investigation could also be applied to the determination of the tension in elevator suspension ropes as well as damage detection. The tensioned beam test case demonstrates an approach to parameter and vibration type selection due to lack of sensitivity to bending and axial stiness. Using a system similar to the elevator system in Chapter 5, the length of the step size was shown to be important when the system of equations could switch between wellconditioned and ill-conditioned. For the rectangular tensioned beam, damage less than 10% was found but noise and uncertainty made accurate damage detection dicult. This case as well as the elevator simulation shows the use of two dierent length models using both the torsional modes and longitudinal modes, respectively. Even though the natural frequencies of the tensioned beam had little error after damage detection for the experimental results, the extent was severely underestimated while the location was overestimated. For the steel rope experimental results showed that adding a clamped mass can overcome the symmetry problem but that the damage 409 can be at either symmetric location. The statistical results show that through a series of tests, the correct damage location can be estimated. The experimental results also demonstrate that even though the measured and detected natural frequencies are very close, there is still error in the detected location and extent. A random impact device was designed, built, and successfully tested in the laboratory and in the eld. It was found that neither test engineers or the device generate a particular distribution for the time interval between pulses or magnitude of impact force. They do, however, generate the same pulse shape and more importantly, the same power spectral density. Using several combinations of random distributions, the average expected force varied little giving condence that the time and force distribution type do not matter unless the impact frequency is deterministic. The pipe line testing showed that the device overcame the lower frequency noise a little better due to the larger impacter mass. The device did better in many cases because it was more consist ant at impacting the same location. The use of burst random testing to excite the lightning mast is not recommended because of the energy lost when the device temporarily stops impacting the mast. Overall the device was just as successful or more successful when compared to the multiple impact tests performed by a skilled engineer even though it didn't match many of the theoretical assumptions put forth in previous research. The FEDS Academy successfully introduced upper level high school students to dynamic systems as practiced in the Dynamic Systems and Vibrations Laboratory during the six summers it was conducted. The program was unique because of its focus on dynamic systems, the level it integrated DBT and laboratory-based programs, and the range and depth of learning skills covered. Using dynamic systems to bring together the subject encompassed by STEM education was eectively applied in the 410 outreach program. 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Washington, DC: The National Academic Press. 439 Appendix A Damage Detection Program ExperimentDD.m % % % % This is the damage detection study f i l e . For simple damage detection set STUDYOPTS = [ ] ; For parametric the STUDYOPTS structure needs to be adjusted using for or while loops of the users design . clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % OPTIONS % Probably don ' t need to edit t h i s but need to add DG s e l e c t o r OPTIONS = struct ( ' term ' , [1 e ? 6 ,300 ,1e ? 6] , 'DGP' , [ 1 ] , ' sens ' , [ 0 ] , . . . 'SPB ' , [ 0 ] , 'CVX' , [ 0 ] , ' a ' , [ 1 ] , 'bd ' , [ . 9 0 ] , 'b ' , [ 1 ] , 'NFQ' , [ 0 ] , . . . 'NSVDS' , [ 6 ] , 'GP' , [ 'A' ] , 'DKMeth ' , [ 2 ] , 'MS' , [ 0 ] , 'MSA' , [ 0 ] , 'TRM' , [ 2 ] , . . . 'INVMETH' , [ 0 ] , 'SVDMAG' , [ 6 ] , 'DH' ,[1 e ? 6] , 'DER' , [ 0 ] , ' n1 ' , [ 0 . 0 ] ) ; % term [DG change , runsmax , a parameter for convex ] % DPG multiplication factor for DG % s e n s i t i v i t y build i f 0 load i f 1 % SPB is special boundaries i . e . reload tension 440 % CVX = 0 no convexification 1 is on % a is the exponential parameter % b is the updating parameter for a % NFQ is the number of frequencies to be used for detection in each model % NSVDS is the number of frequencies to use in the SVDS for % psuedoinverse % In studyopts configure the damage , mass addition , length change etc % for your models . STUDYOPTS should be a structure i f multiple % variables need to be passed and can ' t be made into an array . STUDYOPTS = s t r u c t ( ' runtype ' , {1} , ' d l ' , { [ ] } , ' de ' , { [ ] } , . . . ' ml ' , { [ ] } , 'me ' , { [ ] } , ' m u l t i ' , {0} , ' f i r s t r u n ' , { 0 } ) ; % dl is damage location ( s ) in element numbers % de is damage extent ( s ) % ml is the location ( s ) of mass addition in element numbers % me is the % increase ( s ) in density % S t i l l need to work out length change i . e . how to pad in BAFSDT.m % Run type 1 used for simulation % Run type 2 used for damage detection % Multi == 0 is for the i n i t a l run INP = [ ] ; [ modelCON , defCON , INP ,SOLN] = MAINDDSDT(STUDYOPTS, OPTIONS, INP ) ; STUDYOPTS = s t r u c t ( ' runtype ' , {2} , ' d l ' , {1} , ' de ' , { 1 } , . . . ' ml ' , {1} , 'me ' , {1} , ' m u l t i ' , {1} , ' f i r s t r u n ' , { 1 } ) ; [ modelCON , defCON , INP ,SOLN] = MAINDDSDT(STUDYOPTS, OPTIONS, INP , modelCON ) ; 441 multistudy.m % % % % This is the damage detection study f i l e . For simple damage detection set STUDYOPTS = [ ] ; For parametric the STUDYOPTS structure needs to be adjusted using for or while loops of the users design . clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % OPTIONS OPTIONS = struct ( ' term ' , [1 e ? 6 ,200 ,1e ? 6] , 'DGP' , [ 1 ] , ' sens ' , [ 0 ] , . . . 'SPB ' , [ 0 ] , 'CVX' , [ 0 ] , ' a ' , [ 1 ] , 'bd ' , [ . 9 0 ] , 'b ' , [ 1 ] , 'NFQ' , [ 0 ] , . . . 'NSVDS' , [ 6 ] , 'GP' , [ 'M' ] , 'DKMeth ' , [ 2 ] , 'MS' , [ 0 ] , 'MSA' , [ 0 ] , 'TRM' , [ 2 ] , . . . 'INVMETH' , [ 0 ] , 'SVDMAG' , [ 6 ] , 'DH' , [ . 0 0 1 ] , 'DER' , [ 0 ] , ' n1 ' , [ 0 . 0 ] ) ; % term [DG change , runsmax , a parameter for convex ] % DPG multiplication factor for DG % s e n s i t i v i t y build i f 0 load i f 1 % SPB is special boundaries i . e . reload tension % CVX = 0 no convexification 1 is on % a is the exponential parameter % b is the updating parameter for a % NFQ is the number of frequencies to be used for detection in each model % NSVDS is the number of frequencies to use in the SVDS for % psuedoinverse % DKMeth 1 for a n a l y t i c a l 2 for f i n i t e difference central difference method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for NFQ = [ 9 ] %number of frequencies OPTIONS.NFQ = NFQ; OPTIONS.NMS = 0; %Number of mode shapes f p r i n t f ( 'Number of Frequencies %i \n ' ,NFQ) OPTIONS.NSVDS = NFQ; %7;%2+2 * (NFQ? 5); 442 for %:2:5% Extent ext = 1 fprintf i = 0; for h ( ' Extent %i \n ' , e x t ) %Counter Variable = 9%1:3:15% Location i = i +1; fprintf ( ' L o c a t i o n %i \n ' , i ) % In studyopts configure the damage , mass addition , length change etc % for your models . STUDYOPTS should be a structure i f multiple % variables need to be passed and can ' t be made into an array . STUDYOPTS = s t r u c t ( ' r u n t y p e ' , { 1 } , ' ml ' , { [ ] } , 'me ' , { [ ] } , ' dl ' , { [ ] } , ' multi ' , {0} , ' de ' , {[]} ,... ' firstrun ' , {0}); % dl is damage location ( s ) in element numbers % de is damage extent ( s ) % ml is the location ( s ) of mass addition in element numbers % me is the % increase ( s ) in density % S t i l l need to work out length change i . e . how to pad in BAFSDT.m % Run type 1 used for simulation % Run type 2 used for damage detection % Multi == 0 is for the i n i t a l run %STUDYOPTS. de = 1? ext * .28;% * .26;%.1757;%.44;%.1757; %or .44;% STUDYOPTS. de = 1 . 0 4 8 ; STUDYOPTS. d l = [ h ] ; %Groups STUDYOPTS. r u n t y p e = 1 ; INP = [ ] ; % First run for simulated damage i . e . use dl and de [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS, OPTIONS, INP ) ; %for % You also need to modify INPUTSDTDD.m % Collect def . data for each round for the study % noise can be added here i f need be 443 INP ( 1 , 1 ) . FreqH=(normrnd ( defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) , . . . 0)*2* pi ) . ^ 2 ; INP ( 1 , 1 ) . ModeH = defCON ( 1 , 1 ) . d e f . d e f ( : , INP ( 1 , 1 ) . ModeS ) ; % ModeH are unit normalized and p o s i t i v e averaged %INP(1 ,2). FreqH = (normrnd(defCON(1 ,2). def . data (INP(1 ,2).ModeN) , %0) * 2 * pi ).^2; %INP(1 ,2).ModeH = defCON(1 ,1). def . def (: ,INP(1 ,1).ModeS ); %for zz = 1: length (INP(1 ,1).ModeS) %INP(1 ,1).ModeH(: , zz ) = defCON(1 ,1). def . def (: , zz ) . / . . . %norm(defCON(1 ,1). def . def (: , zz ) ) ; %i f mean(INP(1 ,1).ModeH(: , zz ))<0 %INP(1 ,1).ModeH(: , zz ) = ?INP(1 ,1).ModeH(: , zz ); %Make a p o s i t i v e average %end %end %INP(1 ,3). FreqH = (defCON(1 ,3). def . data (INP(1 ,3).ModeN) * 2 * pi ); % Set the input frequencies to the simulated r e s u l t s . SFREQH1 ( : , i ) = defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) ; SMODEH1 ( : , : , i ) = defCON ( 1 , 1 ) . d e f . d e f ( : , INP ( 1 , 1 ) . ModeS ) ; % Save the input frequencies . %SFREQH2(: , i ) = defCON(1 ,2). def . data (INP(1 ,2).ModeN); %SMODEH2( : , : , i ) = defCON(1 ,2). def . def (: ,INP(1 ,2).ModeS ); % Save the input frequencies . %SFREQH3(: , i ) = defCON(1 ,3). def . data (INP(1 ,3).ModeN); % Save the input frequencies . STUDYOPTS = s t r u c t ( ' r u n t y p e ' , ' ml ' , if {1} , ' me ' , i == 11 & e x t == 1 {1} , {2} , ' multi ' , ' dl ' , {1} , ' de ' , {1} , ' f i r s t r u n ' , % Check location and extent STUDYOPTS . f i r s t r u n = 1 ; 444 {1} ,... {0}); end %OPTIONS.DGP = 1 ? .05 * ( i ? 1); % Set the runtype to damage detection and elements back to healthy . % Set multi == 1 so we don ' t clear simulated damage information . % Below is the main command for damage detection . [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS , . . . OPTIONS , INP , modelCON ) ; mds = INP ( 1 , 1 ) . ModeN ( : , 1 ) ; err = 0.0; for hh = 1 : err s i z e ( INP ( 1 , 1 ) . ModeN , 2 ) sum ( ( ( defCON ( 1 , hh ) . d e f . d a t a ( mds ) ? . . . = e r r+ INP ( 1 , hh ) . FreqH ( : ) ) ) . . . . / defCON ( 1 , hh ) . d e f . d a t a ( mds ) . ^ 2 ) ; end Serr ( i ) = err ; i f s i z e ( INP ( 1 , 1 ) . ModeN, 1 ) > 0 %f p r i n t f ( 'The noise in the eigenvalue : \n ') Snoise ( i ) 2 * pi ) . ^ 2 ? INP ( 1 = max(max( abs ( ( ( defCON ( 1 , 1 ) . d e f . d a t a ( mds ) , 1 ) . FreqH ( : ) ) . / ( defCON ( 1 , 1 ) . d e f . d a t a ( mds ) * 2 * pi ) . ^ 2 ) ) ) ; end %f p r i n t f ( ' Total Iteration Number : \n ') %SOLN. cycle Sfreq ( : , i ) = defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) ; %Sfreq2 (: , i ) = defCON(1 ,2). def . data (INP(1 ,2).ModeN); % Scond (: , i ) = [SOLN(1 ,1). condA , . . . %[ zeros (1 ,OPTIONS. term(2)+3 ? length (SOLN(1 ,1). condA ) ) ] ] ; Scycle ( i ) = SOLN ( 1 , 1 ) . c y c l e ; %Snoise ( i ) = noise ; %Serr ( i ) = err ; %Serr (: , i ) = [ err , [ zeros (1,1001 ? length (condA ) ) ] ] ; 445 *... SG ( : , i ) = modelCON ( 1 , 1 ) .G; SCOS ( : , i )=[SOLN( 1 , 1 ) . cos ' , . . . [ zeros ( 1 ,OPTIONS. term(2)+3 ? length (SOLN( 1 , 1 ) . cos ' ) ) ] ] ; Sdl ( : , i ) = STUDYOPTS. d l ; Sde ( : , i ) = STUDYOPTS. de ; end % % Store and plot cond , err , vs cycle and f i n a l G, % % write these to f i l e along with parameters used and model % Figure out the best way to save data for what I need . % Location savefile =... ( [ ' TRM_2model_roundcanti_1tor_ ' i n t 2 s t r ( ext ) ' _freq ' i n t 2 s t r (NFQ ) ] ) ; save ( s a v e f i l e ) end % Extent clear SFREQH1 S f r e q S c y c l e SG SCOS Sdl Sde S n o i s e SFREQH2 SMODEH1 end %number of frequencies 446 MAINDDSDT.m function [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS, OPTIONS , INP , . . . modelCON ) KMS = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % I n i t i a l i z e defCON structure defCON = struct ( ' def ' , {[]}); % I n i t i a l i z e Inputs You can s u b s t i t u t e your own function here for % INPUTSDTDD() but you s t i c k with STUDYOPTS to pass study options . if STUDYOPTS . m u l t i [ modelCON , INP ] == 0 = INPUTSDTDD(STUDYOPTS, OPTIONS ) ; else [ modelCON ] = INPUTSDTDD(STUDYOPTS, OPTIONS , modelCON ) ; % Let ' s r e s t a r t with a fresh model . This needs to be fixed to allow % for INPUTSDTDD to have d i f f e r e n t names end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % I n i t i a l i z e SOLN structure SOLN = s t r u c t ( ' dkdg ' , { [ ] } , 'A ' , { [ ] } , ' F ' , { [ ] } , 'DG ' , { [ ] } , . . . 'GG ' , ?modelCON ( 1 , 1 ) . G, ' c y c l e ' , 1 , ' e r r ' , { [ ] } , ' condA ' , { [ ] } , . . . ' F_old ' , { 0 } , ' d f ' , { 1 } ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Some house keeping and l o c a l variable for main clear FE * %Decide how many modes to solve for . maxfreq = 35; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % To get simulated data if STUDYOPTS . r u n t y p e == 1 447 [ modelCON , defCON ] = LDDSDT( modelCON , defCON , m a x f r e q , 1 , OPTIONS ) ; return end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (OPTIONS .TRM == 1 SOLN ( 1 , 1 ) . d e l t a if = OPTIONS .TRM == = SOLN ( 1 , 1 ) . d e l t a 0 SOLN ( 1 , 1 ) . t | 2) & STUDYOPTS . r u n t y p e ~= 1 1; = 5.0 * norm ( modelCON ( 1 , 1 ) .G ) ; 1; OPTIONS .TRM == 1 modelCON ( 1 , 1 ) . G1=o n e s ( s i z e ( modelCON ( 1 , 1 ) .G) ) *1; end [ modelCON , defCON ] if = LDDSDT( modelCON , defCON , m a x f r e q , 1 , OPTIONS ) ; OPTIONS . DER == 0 if KMS == 0 & OPTIONS .TRM ~= & STUDYOPTS . f i r s t r u n 1 == 1 SOLN = BDKDGSDT2( modelCON , SOLN , STUDYOPTS, OPTIONS ) ; save p i p e 8 d d 2 . mat SOLN STUDYOPTS . f i r s t r u n = 0 ; elseif KMS == 0 load p i p e 8 d d . mat SOLN end else SOLN = BDKDGSDT2( modelCON , SOLN , STUDYOPTS, OPTIONS ) ; end if OPTIONS . MSA == 1 [ SOLN ] = BAFSDTMODES( modelCON , defCON , INP , SOLN , OPTIONS ) ; [ SOLN ] = BAFSDT( modelCON , defCON , INP , SOLN , OPTIONS ) ; else end k i =1; cMN = length ( INP ) ; 448 kk = 1 :cMN for mds = INP ( 1 , kk ) . ModeN ( : , 1 ) ; rMN = length ( INP ( 1 , kk ) . ModeN ) ; SOLN( 1 , 1 ) . F ( [ k i : k i+rMN ? 1 ] , 1 ) = ?(INP ( 1 , kk ) . FreqH ( : ) . ^ . 5 ? . . . ( defCON ( 1 , kk ) . d e f . data ( mds ) * 2 * p i ) ) . / INP ( 1 , kk ) . FreqH ( : ) . ^ . 5 ; k i = k i+rMN; end end while if 1 OPTIONS .TRM == 1 | OPTIONS .TRM == 2 SOLN( 1 , 1 ) . F_old=SOLN( 1 , 1 ) . F ; SOLN( 1 , 1 ) . A_old = SOLN( 1 , 1 ) .A; SOLN( 1 , 1 ) . J_old = sum (SOLN( 1 , 1 ) . F_old . ^ 2 ) ; SOLN( 1 , 1 ) . M_old=SOLN( 1 , 1 ) . J_old ; SOLN( 1 , 1 ) . dg_old=SOLN( 1 , 1 ) . A_old ' * SOLN( 1 , 1 ) . F_old ; SOLN( 1 , 1 ) . ddg_old=SOLN( 1 , 1 ) . A_old ' * SOLN( 1 , 1 ) . A_old ; modelCON ( 1 , 1 ) . G_old=modelCON ( 1 , 1 ) .G; if OPTIONS .TRM == 1 modelCON ( 1 , 1 ) .G=OPTIONS . bd . / ( 1 + . . . exp(?OPTIONS . b * modelCON ( 1 , 1 ) . G1))+(1 ?OPTIONS . bd ) ; end if OPTIONS .GP == 'E ' JL = 1 : length (modelCON)% a p p l y t o b o t h models for modelCON ( 1 , JL ) . model . p l ( : , 3 ) = modelCON ( 1 , 1 ) .G . . . ( 1 : length (modelCON ( 1 , JL ) . model .EM) ) . * . . . modelCON ( 1 , JL ) . model .EM' ; end elseif OPTIONS .GP == 'M' modelCON ( 1 , 1 ) . model . p l ( [ 1 : length (modelCON ( 1 , 1 ) .G) ] , 5 ) = . . . 449 modelCON(1 ,1).G * 7800; else modelCON = modelCON(1 ,1). mdl(STUDYOPTS,modelCON,OPTIONS) ; end end INDEX_H = 1; % I don ' t want t o t r a n s f e r SOLN . d k d g u n l e s s I h a v e [SOLN(1 ,1). cycle ,SOLN(1 ,1). err ] = TERMSDTDD(SOLN(1 ,1).DG, . . . SOLN(1 ,1).GG,modelCON(1 ,1).G, . . . INDEX_H,OPTIONS. term ,SOLN(1 ,1). cycle ,SOLN(1 ,1).F,OPTIONS) ; clear SOLN(1 ,1).F % Check if to see if we had an to . error . SOLN(1 ,1). err == inf disp ( ' Stopped at TERMSDTDD' ) break end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Choose wh at to build and solve [ modelCON , defCON ] = % LDDSDT( modelCON , defCON , INP , s o l v e t y p e , OPTIONS ) ; if KMS == 0 | | OPTIONS.GP == 'A' % Regular initial solve full build [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,1 ,OPTIONS) ; e l s e i f KMS == 0 | | OPTIONS.GP == 'M' % j u s t mass b u i l d and s o l v e [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,2 ,OPTIONS) ; e l s e i f KMS == 1 % j u s t s t i f f n e s s b u i l d and s o l v e [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,2 ,OPTIONS) ; else disp ( ' Error at : switch what to build . ' ) break end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 450 % Call DKDG and/or DMDG OPTIONS.DER == 0 & OPTIONS.TRM ~= 1 KMS == 0 & STUDYOPTS. f i r s t r u n == 1 SOLN = BDKDGSDT2(modelCON,SOLN,STUDYOPTS,OPTIONS) ; pipe8dd2 . mat SOLN KMS == 0 pipe8dd . mat SOLN if if save elseif load end else SOLN = BDKDGSDT2(modelCON,SOLN,STUDYOPTS,OPTIONS) ; end KMS = 1; % Turn o f f M and DKDG b u i l d i f OPTIONS. sens == 1 % e x i t for s e n s i t i v i t y run ( s ) break ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mode Checking if OPTIONS.MS == 1 [ modelCON, defCON ,SOLN] = MACSDTDD(modelCON, defCON , INP ,SOLN,OPTIONS) ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Build A and F if OPTIONS.MSA == 1 [SOLN] = BAFSDTMODES(modelCON, defCON , INP ,SOLN,OPTIONS) ; else [SOLN] = BAFSDT(modelCON, defCON , INP ,SOLN,OPTIONS) ; end SOLN( 1 , 1 ) . condA(SOLN( 1 , 1 ) . cycle ?1) = 451 cond (SOLN( 1 , 1 ) .A) ; % The above can be turned on of o f f depending on the options %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SOLN( 1 , 1 ) . d f=abs (SOLN( 1 , 1 ) . F_old?max( abs (SOLN( 1 , 1 ) . F ) ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Solve for DG using the pseudoinverese so far [SOLN] = InverSDT (SOLN,OPTIONS ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Choose a way to update G with DG i f OPTIONS.CVX == 0 [SOLN, modelCON ,OPTIONS]=GSDTDD(SOLN, modelCON ,OPTIONS ) ; e l s e i f OPTIONS.CVX == 1 % I f using c o n v e x i f i c a t i o n . [SOLN, modelCON ,OPTIONS]=GSDTDD(SOLN, modelCON ,OPTIONS ) ; i f max ( abs (SOLN ( : , 1 ) . F)) <1e ?6 OPTIONS. b = 1 . 1 ; e l s e i f SOLN ( : , 1 ) . df <1e ?6 OPTIONS. b = 0 . 9 ; end end t r u n c = length (modelCON ( 1 , 1 ) .G) ; modelCON ( 1 , 1 ) .G( 1 : t ru n c ) = 1 . / ( 1 + . . . exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1 ( 1 : t ru n c ) ) ) ; % update G with the extra parameter G1 G i s g e t t i n g too small here % need to bound i t . modelCON ( 1 , 1 ) .G( find (modelCON ( 1 , 1 ) .G< . 3 ) ) = . 3 ; else break ; end i f OPTIONS.GP == 'E '& OPTIONS.TRM ~= 1 & OPTIONS.TRM ~= 2 for JL = 1 : length (modelCON)% apply to both models 452 modelCON(1 ,JL ) . model . pl ( : ,3 ) = modelCON(1 ,1).G. . . (1: (modelCON(1 ,JL ) . model .EM) ) . * . . . modelCON(1 ,JL ) . model .EM' ; length end OPTIONS.GP == 'A' & OPTIONS.TRM ~= 1 & OPTIONS.TRM ~= 2 modelCON = modelCON(1 ,1). mdl(STUDYOPTS,modelCON,OPTIONS) ; elseif end SOLN(1 ,1). delG ( : ,SOLN(1 ,1). cycle ?1)=SOLN(1 ,1).A' * SOLN(1 ,1).F; %SOLN( 1 , 1 ) . delG1 ( : ,SOLN( 1 , 1 ) . c y c l e ?1)=max( abs ( (SOLN( 1 , 1 ) .A) ' * SOLN( 1 , 1 ) . %F ' ) ) ; SOLN(1 ,1). (SOLN(1 ,1). cycle ?1,1)=SOLN(1 ,1).DG' * . . . SOLN(1 ,1). delG ( : ,SOLN(1 ,1). cycle ? 1)/( (SOLN(1 ,1).DG) * . . . (SOLN(1 ,1). delG ) ) ; cos norm norm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear SOLN.A %d i s p ( 1 ) if ( (SOLN(1 ,1).F))< OPTIONS. term (3) ( 'Minimum s t i f f n e s s change s a t i s f i e d 1 ' ) norm abs disp break end if abs (SOLN(1 ,1). (SOLN(1 ,1). cycle ? 1)) < .00000000808 ( 'Minimum s t i f f n e s s change s a t i s f i e d 2 ' ) cos disp break end %i f SOLN( 1 , 1 ) . delG1 (SOLN( 1 , 1 ) . c y c l e ?1) <1e ?6 % d i s p ( ' Minimum s t i f f n e s s change s a t i s f i e d 3 ' ) %b r e a k end if ( (SOLN(1 ,1).F))<1e ?6 ( 'Minimum s t i f f n e s s change s a t i s f i e d 4 ' ) max abs disp 453 break end figure ( 1 ) ; pause ( 0 . 1 ) plot ( modelCON ( 1 , 1 ) .G) end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fprintf ( ' Number of c y c l e s %i \n ' , SOLN . c y c l e ) 454 INPUTSDTDD.m function if [ modelCON , INP ] STUDYOPTS . m u l t i INP = = INPUTSDTDD(STUDYOPTS, OPTIONS , modelCON ) == 0 s t r u c t ( ' FreqH ' , { [ ] } , ' ModeH ' , { [ ] } , ' ModeN ' , { [ ] } , ' ModeS ' , { [ ] } , . . . ' dof ' , { [ ] } ) ; modelCON = s t r u c t ( ' m o d e l ' , { [ ] } , 'G ' , { [ ] } , 'GH ' , { [ ] } , INP . FreqH = [ ] ; INP . ModeH = [ ] ; ' mdl ' , {[]}); %Use t h i s for multi study %INP(1 ,1). FreqH = ( [ ] ' * 2 * pi ).^2; %INP(1 ,2). FreqH = ( [ ] ' * 2 * pi ).^2; INP. mode = [ ] ; %Rigid Body Modes % RB = 6; INP ( 1 , 1 ) . ModeN = [ 1 +RB : 1 : RB+OPTIONS . NFQ ] ' ; INP ( 1 , 1 ) . ModeS = [ 1 +RB : 1 : RB+OPTIONS . NFQ ] ' ; end modelCON ( 1 , 1 ) . mdl = @pipe8dd ; modelCON = modelCON ( 1 , 1 ) . mdl (STUDYOPTS, modelCON , OPTIONS ) ; % Probably don ' t need to e d i t t h i s unless you want to allow for greater G modelCON ( 1 , 1 ) . G = ones ( 1 , 2 0 ) ; % i . e . GH modelCON ( 1 , 1 ) .GH = modelCON ( 1 , 1 ) . G ; modelCON ( 1 , 1 ) . G1 = if [ ] ; OPTIONS . CVX ==1 modelCON ( 1 , 1 ) . G1 = modelCON ( 1 , 1 ) . G = zeros zeros (1 , (1 , size size ( modelCON ( 1 , 1 ) . m o d e l . p l , 1 ) ) ; %./(1+ exp( ? 1000 * modelCON(1 ,1).G1) ) ; % updating G end INP ( 1 , 1 ) . d o f % update G1 %. . . ( modelCON ( 1 , 1 ) . m o d e l . p l , 1 ) ) ; = modelCON ( 1 , 1 ) . m o d e l . a d o f ; 455 multibeam.m function [ modelCON ] = m u l t i b e a m (STUDYOPTS, modelCON , OPTIONS) femesh ; % p a s t e u s e r model h e r e % use STUDYOPTS t o c o n f i g u e t h e model f o r p a r a m e t r i c s t u d i e s % STUDYOPTS i s t o t a l l y u s e r c o n f i g u r a b l e and i s l o c a t e d i n DDSTUDY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % model numbering f o r STUDYOPTS % mdl = 1 ; e t c . % Put your s e c t i o n d e r i v a t i v e s h e r e if STUDYOPTS . r u n t y p e == 2 && OPTIONS . GP == modelCON ( 1 , 1 ) .AK = ones ( size %&& OPTIONS.DER == 1 'A ' ( modelCON ( 1 , 1 ) . G ) ) ; %3*modelCON ( 1 , 1 ) .G. ^ 2 ; % Enter your d e r i v i t i v e term f o r s t i f f n e s s % ( 1 / 1 2 ) * b *(G*h )^3 modelCON ( 1 , 1 ) .AM = a = ones ( = .5 *.25 * = (1/12) *a .*b.^3; Iy1 = (1/12) *b.* a .^3; a. .0254; *b; modelCON ( 1 , 1 ) . m o d e l . i l ( : , [ 3 : 6 ] ) = %* 1 0 0 ; *(1 ?(b . ^ 4 . / ( 1 2 . * a . ^ 4 ) ) ) ) ) ; Ix1 l i l % and mass b *(G*h ) *a .*((.5.* b).^3).*((16/3) ?(3.36.*... (b ./ a ). A1 = ( modelCON ( 1 , 1 ) . G ) ) ; * . 0 2 5 4 ; %* 1 0 0 ; 1 b = modelCON ( 1 , 1 ) . G' k1 size size = [ k1 Ix1 Iy1 A1 ] ; ( modelCON ( 1 , 2 ) . m o d e l . i l , 1 ) ; modelCON ( 1 , 2 ) . m o d e l . i l ( ( [ 1 : l i l ] ) , [ 3 : 6 ] ) [ k1 ( [ 1 : l i l ] ) Ix1 ( [ 1 : l i l ] ) =... Iy1 ( [ 1 : l i l ] ) A1 ( [ 1 : l i l ] ) ] ; return end %I n i t i a l i z e%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% modelCON ( 1 , 1 ) .AK = ones ( size ( modelCON ( 1 , 1 ) . G ) ) ; 456 %3*modelCON ( 1 , 1 ) .G. ^ 2 ; % Enter your d e r i v i t i v e term f o r s t i f f n e s s % (1/12) * b * (G* h)^3 modelCON(1 ,1).AM = ones ( s i z e (modelCON(1 ,1).G) ) ; % and mass b * (G* h ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L = .45; % With t h i s we are assured the model i s evenly d i v i s i b l e EPG = 2; NG = 45; elements = EPG* ones (1 ,NG) ; NEL = sum( elements ) ; a = ones (NG,1) * 1 * .0254; %* 100; b = ones (NG,1) * .25 * .0254;%* 100; i f OPTIONS.GP == 'A' b(STUDYOPTS. dl ) = STUDYOPTS. de . * b(STUDYOPTS. dl ) ; end k1 = .5 * a . * ( ( . 5 . * b).^3). * ((16/3) ? (3.36. * (b./ a). * (1 ? (b .^4./(12. * a . ^ 4 ) ) ) ) ) ; Ix1 = (1/12) * a . * b .^3; Iy1 = (1/12) * b. * a .^3; A1 = a . * b ; E=67.5e9 ; rho = 2739; sh = 5 * (1+.33)/(6+5 * .33); FEnode = [ ] ; LG = L/ length ( elements ) ; FEnode = [1 , 0 0 0 0 0 0 ] ; count = 2; nodespace = 0; f o r i j = 1: length ( elements ) f o r ik = 1: elements ( i j ) nodespace = nodespace + LG/elements ( i j ) ; 457 FEnode = [ FEnode ; count , 0 0 0 0 0 nodespace ] ; count = count +1; end end IR = s i z e ( FEnode ,1)+1; FEnode = [ FEnode ; IR , zeros ( 1 , 3 ) , 1 , 0 , 0 ] ; OPTIONS.SPB == 0 if LineBeam = [ Inf 98 101 97 109 49 0]; LineBeam = [ Inf 98 101 97 else 109 49 116]; end count = 1 ; i i = 1 : length ( elements ) for FEelt ( count , 1 : 7 ) = LineBeam ; count = count + 1 ; for i i i =1: elements ( i i ) FEelt ( count , 1 : 7 ) = [ ( count ? i i ) ( count ? i i +1) i i count = count + 1 ; end end clear LineBeam i 1 i 2 LENGTH i NEL = length ( elements ) ; E = E* ones (NG, 1 ) ; % if Introduce damage OPTIONS.GP == 'E ' E(STUDYOPTS. dl ) = STUDYOPTS. de . * E(STUDYOPTS. dl ) ; 458 ii IR 0 0]; end model=femesh ( ' model ' ) ; model . pl =[(1:NEL) ' ones (NG,1) E, .33 * ones (NG,1) rho * ones (NG, 1 ) ] ; model .EM = E; model . i l = [ ( 1 :NG) ' ones (NG,1) k1 Ix1 Iy1 A1 ] ; mdof = f e u t i l ( ' getdof ' , model ) ; model . adof = fe_c (mdof , [ . 0 1 .05] ' , ' dof ' ) ; i1 = femesh ( ' findnode z==0' ) ; model . adof = fe_c ( model . adof , i1 , ' dof ' ,2); model=fe_case (model , ' SetCase1 ' , . . . 'KeepDof ' , ' f i n a l DOF l i s t ' , model . adof ) ; [ model . Case , model .DOF]=fe_mknl( ' i n i t ' , model ) ; modelCON(1 ,1). model = model ; clear FE* 459 TERMSDTDD.m function if [ cycle , err ] = TERMSDTDD(DG, GG, G, INDEX_H, t e r m , c y c l e , F , OPTIONS) OPTIONS . CVX == 0 if INDEX_H > 0 DG = G ; end err1 = max ( abs (DG ) ) ; err2 = max ( abs (GG?G ) ) ; err if = min ( err < if err1 , err2 ) ; term ( 1 ) err == err1 fprintf ( ' The min increment in DG ' ) ( ' The min increment in G' ' err1 else fprintf ?G0 ' ) err2 end err elseif = inf ; c y c l e >t e r m ( 2 ) ; cycle ; fprintf err = ( ' The result may need further inf ; end else err1 = max ( abs (DG ) ) ; err2 = max ( abs (GG?G ) ) ; err if = err min ( < err1 , err2 ) ; term ( 1 ) 460 improvement \n ' ) if e r r == e r r 1 fprintf ( ' The min i n c r e m e n t i n DG ' ) err1 else fprintf ( ' The min i n c r e m e n t i n G ' ' ?G0 ' ) err2 end err = inf ; elseif c y c l e >term ( 2 ) ; cycle ; fprintf ( ' \n The r e s u l t may need f u r t h e r improvement \n ' ) err = inf ; end if max ( abs (F)) < term disp ( (3); ' \n Minimum s t i f f n e s s change s a t i s f i e d \n ' ) err = inf ; elseif c y c l e >term ( 2 ) ; cycle ; fprintf ( ' \n The r e s u l t may need f u r t h e r improvement \n ' ) err = inf ; end end c y c l e=c y c l e +1; 461 LDDSDT.m function [ modelCON , defCON ] = LDDSDT( modelCON , defCON , NF , s o l v e t y p e , . . . OPTIONS ) ; if solvetype for ii == 1 = 1: length ( modelCON ) [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] . . . =f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 1 ) ; end clear modelCON ( 1 , i i ) . m o d e l . k modelCON ( 1 , i i ) . m o d e l .m modelCON ( 1 , i i ) . m o d e l .m=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . m od el , . . . modelCON ( 1 , i i ) . m o d e l . C a s e , 2 ) ; modelCON ( 1 , i i ) . m o d e l . k=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . mo de l , . . . modelCON ( 1 , i i ) . m o d e l . C a s e , 1 ) ; defCON ( 1 , i i ) . d e f=f e _ e i g ( { modelCON ( 1 , i i ) . m o d e l . m , . . . modelCON ( 1 , i i ) . m o d e l . k , . . . modelCON ( 1 , i i ) . m o d e l . a d o f } , [ 5 clear FE NF 500 11 1e ?5]); * end elseif for solvetype ii = 1: == 2 % 2+ runs w i t h j u s t b u i l d i n g k length ( modelCON ) [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] . . . =f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 0 ) ; end clear modelCON ( 1 , i i ) . m o d e l . k modelCON ( 1 , i i ) . m o d e l . k=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . mo de l , . . . 462 modelCON(1 , i i ) . model . Case , 1 ) ; defCON(1 , i i ) . def=fe_eig ({modelCON(1 , i i ) . model .m, . . . modelCON(1 , i i ) . model . k , . . . modelCON(1 , i i ) . model . adof } ,[5 NF 0 0 1e ? 6]); clear FE* end solvetype == 3 for i i = 1: length (modelCON)%need t o r e b u i l d model t o add mass . [modelCON(1 , i i ) . model . Case ,modelCON(1 , i i ) . model .DOF] . . . =fe_mknl( ' i n i t ' ,modelCON(1 , i i ) . model ) ; i f OPTIONS.SPB == 1 [modelCON] = SPBFunctions (modelCON,OPTIONS, ii , 0 ) ; elseif end modelCON(1 , i i ) . model .m modelCON(1 , i i ) . model .m=fe_mknl( ' assemble ' ,modelCON(1 , i i ) . model , . . . modelCON(1 , i i ) . model . Case , 2 ) ; defCON(1 , i i ) . def=fe_eig ({modelCON(1 , i i ) . model .m, . . . modelCON(1 , i i ) . model . k , . . . modelCON(1 , i i ) . model . adof } ,[5 NF 500 11 1e ? 5]); clear FE* clear end else disp ( ' Error in : LDDSDT' ) end 463 MACSDTDD.m function [ modelCON , defCON , SOLN ] = MACSDTDD( modelCON , defCON , INP , . . . SOLN , OPTIONS) for k = size 1:1: ( modelCON , 2 ) [ modedof , modeind ] for r = for = f e _ c ( modelCON ( 1 , k ) . m o d e l . a d o f , INP ( 1 , k ) . d o f ) ; 1: length ( INP ( 1 c = 1: , k ) . ModeS ) length ( INP ( 1 SOLN ( 1 , k ) . mac ( r , c ) , k ) . ModeS ) = ( sum ( INP ( 1 , k ) . ModeH ( : , r ) . defCON ( 1 , k ) . d e f . d e f ( : , c ) ) ) ^ 2 / ( sum ( INP ( 1 , k ) . ModeH ( : , r ) . *sum ( defCON ( 1 , k ) . . . . * INP ( 1 , k ) . ModeH ( : def . def ( : , c ). *... * defCON ( 1 , k ) . , r ) ) . . . def . def ( : , c ) ) ) ; end end [ mmac , i n d ] for g = max (SOLN ( 1 = 1:1: , k ) . mac ) ; length ( mmac ) defCON ( 1 , k ) . d e f . d e f ( : , g )= norm ( defCON ( 1 if mean ( defCON ( 1 defCON ( 1 , k ) . d e f . d e f ( : , g ) . / . . . , k ) . def . def ( : , g ) ) ; , k ) . d e f . d e f ( : , g )) <0 defCON ( 1 , k ) . d e f . d e f ( : , g ) = ?defCON ( 1 , k ) . def . def ( : , g ) ; end end for g if = g 1:1: ~= length ( mmac ) ind ( g ) place = ind ( g ) ; t em pz = defCON ( 1 , k ) . d e f . d e f ( : , p l a c e ) ; defCON ( 1 , k ) . d e f . d e f ( : , p l a c e ) defCON ( 1 , k ) . d e f . d e f ( : , g ) tempf = = defCON ( 1 , k ) . d e f . d e f ( : , g ) ; t em p z ; = defCON ( 1 , k ) . d e f . d a t a ( p l a c e ) ; defCON ( 1 , k ) . d e f . d a t a ( p l a c e ) 464 = defCON ( 1 , k ) . d e f . d a t a ( g ) ; defCON ( 1 , k ) . d e f . data ( g ) = tempf ; tempmac = SOLN( 1 , k ) . mac ( : , p l a c e ) ; SOLN( 1 , k ) . mac ( : , p l a c e ) = SOLN( 1 , k ) . mac ( : , g ) ; SOLN( 1 , k ) . mac ( : , g ) = tempmac ; ind ( g)=g ; ind ( p l a c e ) = p l a c e ; else continue end end end 465 BDKDGSDT2.m function [ SOLN ] = BDKDGSDT2( modelCON , SOLN, STUDYOPTS, OPTIONS) s t d t m p = STUDYOPTS . r u n t y p e ; detmp = STUDYOPTS . d e ; for i i = 1 : s i z e ( modelCON , 2 ) dkdgSTR = if s t r u c t ( ' dkdg ' , { [ ] } ) ; OPTIONS . GP == dmdgSTR = 'A ' | OPTIONS . GP == 'M' s t r u c t ( ' dmdg ' , { [ ] } ) ; end rpl = l e n g t h ( modelCON ( 1 , i i ) . G ) ; E = modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) ; r h o = modelCON ( 1 , i i ) . m o d e l . p l ( : , 5 ) ; I L = modelCON ( 1 , i i ) . m o d e l . i l ; if OPTIONS . DKMeth == 1 for v = if 1: rpl OPTIONS . GP == 'E ' modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) = modelCON ( 1 , i i ) . m o d e l . p l ( v , 3 ) = E( v ) ; elseif OPTIONS . GP == zeros ( rpl , 1 ) ; 'M' modelCON ( 1 , i i ) . m o d e l . p l ( [ 1 : l e n g t h ( modelCON ( 1 , 1 ) . G ) ] , 5 ) .0000000001 * ones ( rpl ,1); modelCON ( 1 , i i ) . m o d e l . p l ( v , 5 ) elseif OPTIONS . GP == = 7800; 'A ' modelCON ( 1 , i i ) . m o d e l . i l ( : , [ 3 : 8 ] ) modelCON ( 1 , i i ) . m o d e l . i l ( v , : ) = zeros ( rpl , 6 ) ; = IL ( v , : ) ; else disp ( ' Error in : B u i l d DKDG and DMDG' ) end 466 =... [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] = . . . f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 0 ) ; end if OPTIONS . GP == 'A' | OPTIONS . GP == 'E ' dkdgSTR ( v ) . dkdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . modelCON ( 1 , i i ) . mo de l , modelCON ( 1 , i i ) . m o d e l . C a s e , 1 ) ) ; end if OPTIONS . GP == 'A' | OPTIONS . GP == 'M' dmdgSTR ( v ) . dmdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . modelCON ( 1 , i i ) . mo de l , modelCON ( 1 , i i ) . m o d e l . C a s e , 2 ) ) ; end modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) SOLN ( 1 , i i ) . dkdg if = E; = dkdgSTR ; OPTIONS . GP == 'A' modelCON ( 1 , i i ) . m o d e l . i l = IL ; SOLN ( 1 , i i ) . dmdg = dmdgSTR ; end end elseif OPTIONS . DKMeth == 2 STUDYOPTS . r u n t y p e for vv = = 1; 1: rpl STUDYOPTS . d l = [ 1 : rpl ] ' ; STUDYOPTS . d e = ones ( rpl , 1 ) STUDYOPTS . d e ( vv ) * 1 e ?10; = modelCON ( 1 , 1 ) . G( vv )+OPTIONS . DH ; modelCON = modelCON ( 1 , i i ) . mdl (STUDYOPTS, modelCON , OPTIONS ) ; if OPTIONS . GP == 'E ' | OPTIONS . GP == 'A' dkdgSTR ( vv ) . dkdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . 467 modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 1 ) ) ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' dmdgSTR(vv ) . dmdg=sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 2 ) ) ; end STUDYOPTS. de (vv) = modelCON(1 ,1).G(vv) ?OPTIONS.DH; modelCON = modelCON(1 , i i ) . mdl(STUDYOPTS,modelCON,OPTIONS) ; i f OPTIONS.GP == 'E' | OPTIONS.GP == 'A' dkdgSTR(vv ) . dkdg = (dkdgSTR(vv ) . dkdg ? ... sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 1 ) ) ) / . . . (2 * OPTIONS.DH) ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' dmdgSTR(vv ) . dmdg= (dmdgSTR(vv ) . dmdg ? ... sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 2 ) ) ) / . . . (2 * OPTIONS.DH) ; end end %replace E modelCON(1 , i i ) . model . pl ( : ,3 ) = E; % put the DKDGstr in the right place in SOLN SOLN(1 , i i ) . dkdg = dkdgSTR ; i f OPTIONS.GP == 'E' | OPTIONS.GP == 'A' modelCON(1 , i i ) . model . i l = IL ; %Replace i l SOLN(1 , i i ) . dkdg = dkdgSTR ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' 468 else end end modelCON(1 , i i ) . model . i l = IL ; %Replace i l SOLN(1 , i i ) . dmdg = dmdgSTR; disp ( ' Error in : Build DKDG and DMDG' ) return end STUDYOPTS. runtype = stdtmp ; STUDYOPT. de = detmp ; 469 BAFSDT.m function cMN = if [ SOLN ] = BAFSDT( modelCON , defCON , INP , SOLN , OPTIONS ) ; length ( INP ) ; OPTIONS . CVX == 0 & OPTIONS . GP == OPTIONS .TRM == 'E ' & (OPTIONS .TRM ~= 1 | . . . 2) %OLD METHOD j u s t K for ii = 1 : cMN %rMN = length (INP(1 , i i ) .ModeN) ; rMN = for length ( INP ( 1 , i i ) . ModeN ) ; d = 1 :rMN dd = INP ( 1 , i i ) . ModeN ( d ) ; for e = 1: length (SOLN ( 1 R1 ( e , d , i i ) , i i ) . dkdg ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . *SOLN ( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end end elseif OPTIONS . CVX == 1 OPTIONS .TRM ~= for & OPTIONS . GP == 'E ' & OPTIONS .TRM ~= 2 %number of models %rMN = length (INP(1 , i i ) .ModeN) ; length % number of modes ii = 1 : cMN rMN = for ( INP ( 1 , i i ) . ModeN ) ; d = 1 :rMN dd = INP ( 1 , i i ) . ModeN ( d ) ; for e = 1: length (SOLN ( 1 R1 ( e , d , i i ) , i i ) . dkdg ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . *SOLN ( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . 470 1 &... * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS. a * exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; end %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay %the same . end end e l s e i f OPTIONS.TRM == 1 & OPTIONS.GP == 'E ' %number of models %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; rMN = length (INP ( 1 , i i ) . ModeN ) ; % number of modes for i i = 1 :cMN for d=1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : length (SOLN( 1 , i i ) . dkdg ) R1( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . % *SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS. bd *OPTIONS. b * exp(?OPTIONS. b * . . . modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS. b * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end end e l s e i f OPTIONS.CVX == 0 & OPTIONS.GP == 'A ' & (OPTIONS.TRM ~= 1 | . . . OPTIONS.TRM == 2) %OLD METHOD j u s t K % Later allow for d i f f e r e n t s i z e INP .ModeN or INF end pad for i i = 1 :cMN %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; 471 rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; for d =1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : l e n g t h (SOLN( 1 , i i ) . dkdg ) R1 ( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . * ( modelCON ( 1 , 1 ) .AK( e ) * SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . ? ((defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * p i ) . ^ 2 ) . . . * modelCON ( 1 * defCON ( 1 , end , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay %the same . end end elseif OPTIONS .CVX == 1 & OPTIONS .GP == 'A ' &... OPTIONS .TRM ~= 1 & OPTIONS .TRM == 2 for %number of models %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; % number of modes i i = 1 :cMN for d =1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : l e n g t h (SOLN( 1 , i i ) . dkdg ) R1 ( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . * ( modelCON ( 1 , 1 ) .AK( e ) * SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . ? ((defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * p i ) . ^ 2 ) . . . * modelCON ( 1 * defCON ( 1 , , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS . a * exp(?OPTIONS . a * modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS . a * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay end end 472 %t h e same . end OPTIONS.TRM == 1 & OPTIONS.GP == 'A' for i i = 1:cMN %number o f m o d e l s elseif %rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; rMN = length (INP(1 , i i ) .ModeN) ; % number o f modes for d=1:rMN dd = INP(1 , i i ) .ModeN(d ) ; for e = 1: length (SOLN(1 , i i ) . dkdg ) R1( e , d , i i ) = defCON(1 , i i ) . def . def ( : , dd ) ' . . . * (modelCON( 1 , 1 ) .AK( e ) * SOLN(1 , i i ) . dkdg (1 , e ) . dkdg . . . ? ((defCON(1 , i i ) . def . data (dd) * 2 * pi ) . ^ 2 ) . . . * modelCON( 1 , 1 ) .AM( e ) * SOLN(1 , i i ) . dmdg(1 , e ) . dmdg ) . . . * defCON(1 , i i ) . def . def ( : , dd ) . . . *OPTIONS. bd *OPTIONS. b * exp(?OPTIONS. b * modelCON( 1 , 1 ) .G1( e ) ) . . . /(1+ exp(?OPTIONS. b *modelCON( 1 , 1 ) .G1( e )))^2; end end %t h e %Check % healthy to to see the damaged difference since most in the people mode shapes assume mode and R1 f r o m shapes same . end elseif OPTIONS.CVX == 0 & OPTIONS.GP == 'M' & . . . (OPTIONS.TRM ~= 1 |OPTIONS.TRM == 2) %OLD METHOD % Later for allow for different size INP . ModeN or INF end just M p ad i i = 1:cMN %rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; rMN = length (INP(1 , i i ) .ModeN) ; for d=1:rMN dd = INP(1 , i i ) .ModeN(d ) ; for e = 1: length (SOLN(1 , i i ) . dkdg ) R1( e , d , i i ) = defCON(1 , i i ) . def . def ( : , dd ) ' . . . 473 stay * ( ? ( ( defCON ( 1 , * modelCON ( 1 * defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * pi ) . ^ 2 ) . . . , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) ; end %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end else disp ( ' E r r o r i n BAFSDT OPTIONS .CVX unknown \n ' ) end %F i r s t assume ModeN i s the same for a l l models k i =1; k j =1; SOLN ( 1 , 1 ) . A = [ ] ; for kk = 1 :cMN % This needs to be based on ModeN mds = INP ( 1 , kk ) . ModeN ( : , 1 ) ; rMN = length ( INP ( 1 , kk ) . ModeN ) ; SOLN ( 1 , 1 ) . F ( [ k i : k i+rMN ? 1 ] , 1 ) = ?(INP ( 1 , kk ) . FreqH ( : ) ? ( defCON ( 1 , kk ) . d e f . d a t a ( mds ) * 2 * pi ) . ^ 2 ) . / INP ( 1 , kk ) . FreqH ( : ) ; k i = k i+rMN ; for g = 1 :rMN SOLN ( 1 , 1 ) . A( k j , : ) = R1 ( : , g , kk ) ' . / INP ( 1 , kk ) . FreqH ( g ) ; k j = k j +1; %count rows end end SOLN ( 1 , 1 ) . J = sum (SOLN ( 1 , 1 ) . F . ^ 2 ) ; 474 ... InverSDT.m function if [ SOLN ] = InverSDT (SOLN , OPTIONS ) ; OPTIONS . INVMETH == 0 SOLN ( 1 , 1 ) .DG = elseif | OPTIONS .TRM == 1 ?pinv (SOLN ( 1 , 1 ) . A) * SOLN ( 1 ,1).F; OPTIONS . INVMETH == 1 [ U , S , V]= s v d s (SOLN ( 1 , 1 ) . A , OPTIONS . NSVDS ) ; for i =s i z e ( S , 1 ) : ?1:1 i f max ( diag ( S ) / S ( i , i )) <OPTIONS .SVDMAG break ; end end *S([1: SOLN ( 1 , 1 ) .DG=(V ( : , [ 1 : i ] ) i ] , [ 1 : i ])^ ? 1 *U ( : ,[1: i ]) ') *... (SOLN ( 1 , 1 ) . F ) ; end if (OPTIONS .TRM == 1 if norm (SOLN ( 1 | OPTIONS .TRM == 2) & SOLN ( 1 , 1 ) . c y c l e >= 2 , 1 ) .DG)>SOLN ( 1 , 1 ) . d e l t a SOLN ( 1 , 1 ) . d t = 1; while abs (SOLN ( 1 , 1 ) . d t )>=1e SOLN ( 1 , 1 ) . t _ o l d ?3 = SOLN ( 1 , 1 ) . t ; SOLN ( 1 , 1 ) . g=SOLN ( 1 , 1 ) . A' * SOLN ( 1 SOLN ( 1 , 1 ) . H = SOLN ( 1 , 1 ) . A' ,1).F; * SOLN ( 1 , 1 ) .A; SOLN ( 1 , 1 ) . B = SOLN ( 1 , 1 ) . H+SOLN ( 1 , 1 ) . t * eye ( s i z e (SOLN ( 1 , 1 ) .H ) ) ; [ SOLN ( 1 , 1 ) . R , SOLN ( 1 , 1 ) . u ]= c h o l (SOLN ( 1 , 1 ) . B ) ; if SOLN ( 1 , 1 ) . u==0 SOLN ( 1 , 1 ) . p = SOLN ( 1 , 1 ) . R\ (SOLN ( 1 , 1 ) . R'\ ?SOLN ( 1 ,1). g ); SOLN ( 1 , 1 ) . q = SOLN ( 1 , 1 ) . R' \ SOLN ( 1 , 1 ) . p ; SOLN ( 1 , 1 ) . t norm (SOLN ( 1 , 1 ) . q ))^2 = SOLN ( 1 , 1 ) . t +(norm (SOLN ( 1 , 1 ) . p ) / . . . * ( norm (SOLN ( 1 SOLN ( 1 , 1 ) . d t , 1 ) . p) ?SOLN ( 1 = SOLN ( 1 , 1 ) . t 475 , 1 ) . d e l t a ) /SOLN ( 1 , 1 ) . d e l t a ; ?SOLN ( 1 , 1 ) . t_old ; else SOLN( 1 , 1 ) . Ei=eig (SOLN( 1 , 1 ) .B ) ; SOLN( 1 , 1 ) . t = ?max(SOLN( 1 , 1 ) . Ei ( find (SOLN( 1 , 1 ) . Ei < 0 ) ) ) ; end end SOLN( 1 , 1 ) .DG=?inv (SOLN( 1 , 1 ) .B) * SOLN( 1 , 1 ) . g ; end SOLN( 1 , 1 ) .M=SOLN( 1 , 1 ) . J_old+SOLN( 1 , 1 ) . dg_old ' * SOLN( 1 , 1 ) .DG+ . . . 0 . 5 * SOLN( 1 , 1 ) .DG' * SOLN( 1 , 1 ) . ddg_old *SOLN( 1 , 1 ) .DG; SOLN( 1 , 1 ) .dM=SOLN( 1 , 1 ) . M_old?SOLN( 1 , 1 ) .M; SOLN( 1 , 1 ) . rho=(SOLN( 1 , 1 ) . J_old ?SOLN( 1 , 1 ) . J )/SOLN( 1 , 1 ) .dM; end 476 GSTDD.m function trunc if = [ SOLN , modelCON , OPTIONS ] length (SOLN ( 1 OPTIONS . CVX == 0 = GSDTDD(SOLN , modelCON , OPTIONS) , 1 ) .DG ) ; & OPTIONS .TRM == 0 SOLN ( 1 , 1 ) .GG = modelCON ( 1 , 1 ) . G ; modelCON ( 1 , 1 ) . G ( 1 : t r u n c ) = modelCON ( 1 , 1 ) . G ( 1 : t r u n c ) +... * OPTIONS .DGP SOLN ( 1 , 1 ) .DG( 1 : t r u n c ) ' ; modelCON ( 1 , 1 ) . G = min ( modelCON ( 1 modelCON ( 1 , 1 ) . G0 = modelCON ( 1 , 1 ) . G = elseif .6 * modelCON ( 1 , 1 ) .GH; max ( modelCON ( 1 OPTIONS . CVX == 1 , 1 ) . GH, modelCON ( 1 , 1 ) . G ) ; , 1 ) . G+modelCON ( 1 , 1 ) . G0 ) ; & OPTIONS .TRM == 0 modelCON ( 1 , 1 ) . G1 ( 1 : t r u n c ) % G here i s r e a l l y G1 = modelCON ( 1 , 1 ) . G1 ( 1 : t r u n c ) +... * OPTIONS .DGP SOLN ( 1 , 1 ) .DG( 1 : t r u n c ) ' ; modelCON ( 1 , 1 ) . G1 ( f i n d ( modelCON ( 1 f i n d ( modelCON ( 1 modelCON ( 1 , 1 ) . G1 ( if ?1e 1 0 ) * 1 0 ^ 8 ; , 1 ) . G1< f i n d ( modelCON ( 1 f i n d ( modelCON ( 1 ?1e 1 0 ) ) = . . . , 1 ) . G1< , 1 ) . G1>1 e 1 0 ) , 1 ) . G1>1 e 1 0 ) ) = . . . * 1 0 e ?8; OPTIONS . a >10 OPTIONS . b = . 1 ; end OPTIONS . a=OPTIONS . a elseif if *OPTIONS . b ; OPTIONS .TRM == 1 | OPTIONS .TRM == 2 %Updata the t r u s t region radius SOLN ( 1 , 1 ) . r h o < . 2 5 SOLN ( 1 , 1 ) . d e l t a = . 2 5 else if *norm (SOLN ( 1 SOLN ( 1 , 1 ) . r h o > . 7 5 & , 1 ) .DG ) ; norm (SOLN ( 1 , 1 ) .DG)==SOLN ( 1 , 1 ) . d e l t a SOLN ( 1 , 1 ) . d e l t a=min (SOLN ( 1 , 1 ) . d e l t a 0 , 2 * SOLN ( 1 end end if OPTIONS .TRM == 1 & SOLN ( 1 , 1 ) . r h o> OPTIONS . n1 477 ,1). delta ); modelCON(1 ,1).G1(1: trunc ) = modelCON(1 ,1).G1(1: trunc ) +... OPTIONS.DGP*SOLN(1 ,1).DG(1: trunc ) ' ; modelCON(1 ,1).G1( f i n d (modelCON(1 ,1).G1<?1e10))= ?1e10 ; modelCON(1 ,1).G1( f i n d (modelCON(1 ,1).G1>1e10))=1e10 ; e l s e i f OPTIONS.TRM == 2 & SOLN(1 ,1). rho>= OPTIONS. n1 modelCON(1 ,1).G(1: trunc ) = modelCON(1 ,1).G(1: trunc ) +... OPTIONS.DGP*SOLN(1 ,1).DG(1: trunc ) ' ; modelCON(1 ,1).G( f i n d (modelCON(1 ,1).G<1e ?1))=1e ? 1; modelCON(1 ,1).G( f i n d (modelCON(1 ,1).G>1.3))=1.3; else disp ( 'GSTDD error ' ) return end else disp ( 'GSTDD error ' ) return end 478 Appendix B Derivation of longitudinal vibration modes. The equations of motion for the longitudinal vibration of the system are 2 ? 2 U1 2 ? U1 ? c = 0 ?t2 ?x21 2 ? 2 U2 2 ? U2 ? c = 0 ?t2 ?x22 (B.0.1) (B.0.2) and the boundary conditions are stated as where c2 = EA m Ms U1 (0, t) = 0 (B.0.3) U1 (L1 , t) = U2 (0, t) (B.0.4) ? 2 U1 ?U1 ?U2 (L ) = ?EA (L , t) + EA (0, t) 1,t 1 ?t2 ?x1 ?x2 479 (B.0.5) M where Ms = ? 2 U2 ?U2 (L2 , t) = ?EA (L2 , t) 2 ?t ?x2 (B.0.6) I is the apparent mass of the sheave. R2 The free vibrations are expressed as where U1 (x1 , t) = Y1 (x1 ) f (t) (B.0.7) U2 (x2 , t) = Y2 (x2 ) f (t) (B.0.8) Y1 and Y2 are the mode shapes (eigenfunctions) and f (t) = Acos (?t)+Bsin (?t)is a harmonic function of time. Inserting Eqs. B.0.7 and B.0.8 into Eqs. B.0.1 and B.0.2 yields Y1 (x1 ) + ? 2 Y1 (x1 ) = 0 Y2 (x2 ) + ? 2 Y2 (x2 ) = 0 where ?2 = ? 2 c From Eq. B.0.3: (B.0.9) (B.0.10) . It is evident that Y1 (x1 ) = A1 cos (?X1 ) + B1 sin (?x1 ) (B.0.11) Y2 (x2 ) = A2 cos (?X2 ) + B2 sin (?x2 ) (B.0.12) A1 = 0 so that Y1 (x1 ) = B1 sin (?x1 ) (B.0.13) B1 sin (?x1 ) ? A1 = 0 (B.0.14) From Eq. B.0.4: 480 The boundary condition Eq. B.0.5 yields ! " ? 2 Ms sin (?L1 ) ? cos (?L1 ) B1 + B2 = 0 EA? (B.0.15) Also, the boundary condition Eq. B.0.6 gives ! " ! 2 " ?2M ? M cos (?L2 ) + ?sin (?L2 ) A2 + sin (?L2 ) + ?cos (?L2 ) B2 EA EA (B.0.16) Thus, combining Eqs. B.0.14, B.0.15, and B.0.16 yields the following ? ?? ?1 s1 ? ? ? ? ? ? 2 M2 s EA? 1 ? c1 0 0 ?2 M c EA 1 0 ? + ?s2 ? ? B1 ? ?? ? ?? A ? = 0 1 ?? 2 ? ?? ? 2 ? M s ? ?c B 2 2 EA 2 (B.0.17) where s1 ? sin (?L1), s2 ? sin (?L2), c1 ? cos (?L1), and c2 ? cos (?L2). Thus, for a non-trivial solution the following must be satised ? ? ? ? det ? ? ? s1 ?2 M 2 EA? s1 ? c 1 0 ?1 0 0 1 ?2 M c EA 1 + ?s2 ?2 M s EA 2 ? ?c2 ? ? ?=0 ? ? (B.0.18) which yields the frequency equation ! ? sin (?L1 ) " ! " Ms M 2 ? cos (?L2 ) + ?sin (?L2 ) + ?sin (?L1 ) ? ?cos (?L1 ) (и B.0.19) m m ! " M 2 ? sin (?L2 ) ? ?sin (?L2 ) = 0 m 481 Appendix C Derivation of bending vibration modes. In the following analytical models the inclined and vertical sections of the rope are modeled as being clamped-clamped and clamped with an end mass, respectively. The equation of motion for bending vibrations of an axially loaded beam is [139] ? 4V P ? 2 V (x, t) ? ? 2 V (x, t) + + =0 ?x4 EI ?x2 EI ?t2 where V is the displacement with respect to the position along the beam t. The bending stiness is and ? EI , the load P (C.0.1) x and time is considered positive for compressive loads, is the mass per unit length of the beam. Assuming V to have the form [139] V (x, t) = ? (x) sin (?t) (C.0.2) 2 d4 ? (x) 2 d ? (x) +k ? ? 4 ? (x) = 0 4 2 dx dx (C.0.3) yields 482 where P EI ?? 2 = EI k2 = ?4 To simplify the analysis more the equation can be nondimensionalized using the length of the beam l which gives [139] ? where ? = ?l ,x = xl ,? = ?l , and 2 (C.0.4) 4 (x) + k ? ? ? ? (x) = 0 k = kl. Assuming the mode shapes to be of the same form as a beam without tension gives ? (x) = Acosh (?1 x) + Bsinh (?1 x) + Ccos (?2 x) + Dsin (?2 x) (C.0.5) where ? ?1 = ?? 2 k + 2 # 4 ?1/2 k 4 +? ? 4 ? and ?2 = ? 2 # k + 2 4 ?1/2 k 4 +? ? 4 (C.0.6) are the real roots of the characteristic equation [139]. The boundary conditions for a clamped end are ? = 0 and v = 0 Applying the boundary conditions at x = 0 and x = 1and 483 (C.0.7) rearranging the equations nto matrix form gives i ? ? ? ? ? ? ? ? ? ? A ? ?? ? ?? ?? B ? 0 ?1 0 ?2 ? ?? ?=0 ?? ? C ? cosh (?1 ) sinh (?1 ) cos (?2 ) sin (?2 ) ? ? ?? ? ?? D ?1 sinh (?1 ) ?1 cosh (?1 ) ??2 sin (?2 ) ?2 cos (?2 ) 1 0 1 0 ?? (C.0.8) to avoid the trivial solution we take the derivative of the coecient matrix which with simplication yields [139] 2 2 2? (1 ? cosh (?1 ) cos (?2 )) ? k sinh (?1 ) sin (?2 ) = 0 (C.0.9) For the clamped-mass system the free-mass end boundary condition is 2 4 (C.0.10) ? (1) + k ? + ?T ? ? (1) = 0 where ?T = MT MB which is the ratio of the tip mass MT to the total mass of the beam MB = ?l [139]. Applying the boundary conditions for the clamped-free mass beam at x = 0 and x = 1 and arranging the equations into matrix form gives [139] ? ? ? ? ? ? ? ? ? 1 0 1 0 0 ?1 0 ?12 ch1 ?12 sh1 ??22 c2 F sh1 + Hch1 F ch1 + Hsh1 Gs + Hc2 484 ?? ? ?? A ? ? ?? ?? B ? ?2 ? ?? ? = 0 (C.0.11) ?? ? ? ? 2 ??2 s2 ?? C ? ? ?? D ?Gc2 + Hch2 where F = G = ?13 + ?1 k 2 ?23 ? ?2 k 2 H = ?T ? 4 ch1 = cosh (?1 ) sh1 = sinh (?1 ) ch2 = cosh (?2 ) sh2 = sinh (?2 ) c2 = cos (?2 ) s2 = sin (?2 ) Following the same procedure as above with the clamped clamped beam give the frequency equation after simplication to be as shown by [139] 4 4 4 2 2 ? 2? + ? k sinh (?1 ) sin (?2 ) ? 2? k cosh (?1 ) cos (?2 ) + 2 ?T ? ?12 + ?22 (?1 cosh (?1 ) sin (?2 ) ? ?2 sinh (?1 ) cos (?2 )) = 0 (C.0.12) The mode shapes !are found by substituting ?n f or ? in equations 10 and 13 to yield "1/2 "1/2 ! & & 4 4 ?? k k k k ?n4 = EI , ?1n = ? 2 + + ?n and ?2n = 2 + + ?n then solv4 4 2 n 2 4 2 485 4 ng the matrix equation associated with the boundary conditions of the system gives i cosh (?1n x) ? cos (?2n ) (C.0.13) ? (x) = cosh (?1n x) ? cosh (?2n x) ? sinh (?1n x) ? ??2n sin (? ) 2n 1n " ! ?1n sin (?2n x) и sinh (?1n x) ? ?2n for the clamped-clamped beam and ? ? 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Washington, DC: The National Academic Press. 439 Appendix A Damage Detection Program ExperimentDD.m % % % % This is the damage detection study f i l e . For simple damage detection set STUDYOPTS = [ ] ; For parametric the STUDYOPTS structure needs to be adjusted using for or while loops of the users design . clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % OPTIONS % Probably don ' t need to edit t h i s but need to add DG s e l e c t o r OPTIONS = struct ( ' term ' , [1 e ? 6 ,300 ,1e ? 6] , 'DGP' , [ 1 ] , ' sens ' , [ 0 ] , . . . 'SPB ' , [ 0 ] , 'CVX' , [ 0 ] , ' a ' , [ 1 ] , 'bd ' , [ . 9 0 ] , 'b ' , [ 1 ] , 'NFQ' , [ 0 ] , . . . 'NSVDS' , [ 6 ] , 'GP' , [ 'A' ] , 'DKMeth ' , [ 2 ] , 'MS' , [ 0 ] , 'MSA' , [ 0 ] , 'TRM' , [ 2 ] , . . . 'INVMETH' , [ 0 ] , 'SVDMAG' , [ 6 ] , 'DH' ,[1 e ? 6] , 'DER' , [ 0 ] , ' n1 ' , [ 0 . 0 ] ) ; % term [DG change , runsmax , a parameter for convex ] % DPG multiplication factor for DG % s e n s i t i v i t y build i f 0 load i f 1 % SPB is special boundaries i . e . reload tension 440 % CVX = 0 no convexification 1 is on % a is the exponential parameter % b is the updating parameter for a % NFQ is the number of frequencies to be used for detection in each model % NSVDS is the number of frequencies to use in the SVDS for % psuedoinverse % In studyopts configure the damage , mass addition , length change etc % for your models . STUDYOPTS should be a structure i f multiple % variables need to be passed and can ' t be made into an array . STUDYOPTS = s t r u c t ( ' runtype ' , {1} , ' d l ' , { [ ] } , ' de ' , { [ ] } , . . . ' ml ' , { [ ] } , 'me ' , { [ ] } , ' m u l t i ' , {0} , ' f i r s t r u n ' , { 0 } ) ; % dl is damage location ( s ) in element numbers % de is damage extent ( s ) % ml is the location ( s ) of mass addition in element numbers % me is the % increase ( s ) in density % S t i l l need to work out length change i . e . how to pad in BAFSDT.m % Run type 1 used for simulation % Run type 2 used for damage detection % Multi == 0 is for the i n i t a l run INP = [ ] ; [ modelCON , defCON , INP ,SOLN] = MAINDDSDT(STUDYOPTS, OPTIONS, INP ) ; STUDYOPTS = s t r u c t ( ' runtype ' , {2} , ' d l ' , {1} , ' de ' , { 1 } , . . . ' ml ' , {1} , 'me ' , {1} , ' m u l t i ' , {1} , ' f i r s t r u n ' , { 1 } ) ; [ modelCON , defCON , INP ,SOLN] = MAINDDSDT(STUDYOPTS, OPTIONS, INP , modelCON ) ; 441 multistudy.m % % % % This is the damage detection study f i l e . For simple damage detection set STUDYOPTS = [ ] ; For parametric the STUDYOPTS structure needs to be adjusted using for or while loops of the users design . clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % OPTIONS OPTIONS = struct ( ' term ' , [1 e ? 6 ,200 ,1e ? 6] , 'DGP' , [ 1 ] , ' sens ' , [ 0 ] , . . . 'SPB ' , [ 0 ] , 'CVX' , [ 0 ] , ' a ' , [ 1 ] , 'bd ' , [ . 9 0 ] , 'b ' , [ 1 ] , 'NFQ' , [ 0 ] , . . . 'NSVDS' , [ 6 ] , 'GP' , [ 'M' ] , 'DKMeth ' , [ 2 ] , 'MS' , [ 0 ] , 'MSA' , [ 0 ] , 'TRM' , [ 2 ] , . . . 'INVMETH' , [ 0 ] , 'SVDMAG' , [ 6 ] , 'DH' , [ . 0 0 1 ] , 'DER' , [ 0 ] , ' n1 ' , [ 0 . 0 ] ) ; % term [DG change , runsmax , a parameter for convex ] % DPG multiplication factor for DG % s e n s i t i v i t y build i f 0 load i f 1 % SPB is special boundaries i . e . reload tension % CVX = 0 no convexification 1 is on % a is the exponential parameter % b is the updating parameter for a % NFQ is the number of frequencies to be used for detection in each model % NSVDS is the number of frequencies to use in the SVDS for % psuedoinverse % DKMeth 1 for a n a l y t i c a l 2 for f i n i t e difference central difference method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for NFQ = [ 9 ] %number of frequencies OPTIONS.NFQ = NFQ; OPTIONS.NMS = 0; %Number of mode shapes f p r i n t f ( 'Number of Frequencies %i \n ' ,NFQ) OPTIONS.NSVDS = NFQ; %7;%2+2 * (NFQ? 5); 442 for %:2:5% Extent ext = 1 fprintf i = 0; for h ( ' Extent %i \n ' , e x t ) %Counter Variable = 9%1:3:15% Location i = i +1; fprintf ( ' L o c a t i o n %i \n ' , i ) % In studyopts configure the damage , mass addition , length change etc % for your models . STUDYOPTS should be a structure i f multiple % variables need to be passed and can ' t be made into an array . STUDYOPTS = s t r u c t ( ' r u n t y p e ' , { 1 } , ' ml ' , { [ ] } , 'me ' , { [ ] } , ' dl ' , { [ ] } , ' multi ' , {0} , ' de ' , {[]} ,... ' firstrun ' , {0}); % dl is damage location ( s ) in element numbers % de is damage extent ( s ) % ml is the location ( s ) of mass addition in element numbers % me is the % increase ( s ) in density % S t i l l need to work out length change i . e . how to pad in BAFSDT.m % Run type 1 used for simulation % Run type 2 used for damage detection % Multi == 0 is for the i n i t a l run %STUDYOPTS. de = 1? ext * .28;% * .26;%.1757;%.44;%.1757; %or .44;% STUDYOPTS. de = 1 . 0 4 8 ; STUDYOPTS. d l = [ h ] ; %Groups STUDYOPTS. r u n t y p e = 1 ; INP = [ ] ; % First run for simulated damage i . e . use dl and de [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS, OPTIONS, INP ) ; %for % You also need to modify INPUTSDTDD.m % Collect def . data for each round for the study % noise can be added here i f need be 443 INP ( 1 , 1 ) . FreqH=(normrnd ( defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) , . . . 0)*2* pi ) . ^ 2 ; INP ( 1 , 1 ) . ModeH = defCON ( 1 , 1 ) . d e f . d e f ( : , INP ( 1 , 1 ) . ModeS ) ; % ModeH are unit normalized and p o s i t i v e averaged %INP(1 ,2). FreqH = (normrnd(defCON(1 ,2). def . data (INP(1 ,2).ModeN) , %0) * 2 * pi ).^2; %INP(1 ,2).ModeH = defCON(1 ,1). def . def (: ,INP(1 ,1).ModeS ); %for zz = 1: length (INP(1 ,1).ModeS) %INP(1 ,1).ModeH(: , zz ) = defCON(1 ,1). def . def (: , zz ) . / . . . %norm(defCON(1 ,1). def . def (: , zz ) ) ; %i f mean(INP(1 ,1).ModeH(: , zz ))<0 %INP(1 ,1).ModeH(: , zz ) = ?INP(1 ,1).ModeH(: , zz ); %Make a p o s i t i v e average %end %end %INP(1 ,3). FreqH = (defCON(1 ,3). def . data (INP(1 ,3).ModeN) * 2 * pi ); % Set the input frequencies to the simulated r e s u l t s . SFREQH1 ( : , i ) = defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) ; SMODEH1 ( : , : , i ) = defCON ( 1 , 1 ) . d e f . d e f ( : , INP ( 1 , 1 ) . ModeS ) ; % Save the input frequencies . %SFREQH2(: , i ) = defCON(1 ,2). def . data (INP(1 ,2).ModeN); %SMODEH2( : , : , i ) = defCON(1 ,2). def . def (: ,INP(1 ,2).ModeS ); % Save the input frequencies . %SFREQH3(: , i ) = defCON(1 ,3). def . data (INP(1 ,3).ModeN); % Save the input frequencies . STUDYOPTS = s t r u c t ( ' r u n t y p e ' , ' ml ' , if {1} , ' me ' , i == 11 & e x t == 1 {1} , {2} , ' multi ' , ' dl ' , {1} , ' de ' , {1} , ' f i r s t r u n ' , % Check location and extent STUDYOPTS . f i r s t r u n = 1 ; 444 {1} ,... {0}); end %OPTIONS.DGP = 1 ? .05 * ( i ? 1); % Set the runtype to damage detection and elements back to healthy . % Set multi == 1 so we don ' t clear simulated damage information . % Below is the main command for damage detection . [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS , . . . OPTIONS , INP , modelCON ) ; mds = INP ( 1 , 1 ) . ModeN ( : , 1 ) ; err = 0.0; for hh = 1 : err s i z e ( INP ( 1 , 1 ) . ModeN , 2 ) sum ( ( ( defCON ( 1 , hh ) . d e f . d a t a ( mds ) ? . . . = e r r+ INP ( 1 , hh ) . FreqH ( : ) ) ) . . . . / defCON ( 1 , hh ) . d e f . d a t a ( mds ) . ^ 2 ) ; end Serr ( i ) = err ; i f s i z e ( INP ( 1 , 1 ) . ModeN, 1 ) > 0 %f p r i n t f ( 'The noise in the eigenvalue : \n ') Snoise ( i ) 2 * pi ) . ^ 2 ? INP ( 1 = max(max( abs ( ( ( defCON ( 1 , 1 ) . d e f . d a t a ( mds ) , 1 ) . FreqH ( : ) ) . / ( defCON ( 1 , 1 ) . d e f . d a t a ( mds ) * 2 * pi ) . ^ 2 ) ) ) ; end %f p r i n t f ( ' Total Iteration Number : \n ') %SOLN. cycle Sfreq ( : , i ) = defCON ( 1 , 1 ) . d e f . d a t a ( INP ( 1 , 1 ) . ModeN ) ; %Sfreq2 (: , i ) = defCON(1 ,2). def . data (INP(1 ,2).ModeN); % Scond (: , i ) = [SOLN(1 ,1). condA , . . . %[ zeros (1 ,OPTIONS. term(2)+3 ? length (SOLN(1 ,1). condA ) ) ] ] ; Scycle ( i ) = SOLN ( 1 , 1 ) . c y c l e ; %Snoise ( i ) = noise ; %Serr ( i ) = err ; %Serr (: , i ) = [ err , [ zeros (1,1001 ? length (condA ) ) ] ] ; 445 *... SG ( : , i ) = modelCON ( 1 , 1 ) .G; SCOS ( : , i )=[SOLN( 1 , 1 ) . cos ' , . . . [ zeros ( 1 ,OPTIONS. term(2)+3 ? length (SOLN( 1 , 1 ) . cos ' ) ) ] ] ; Sdl ( : , i ) = STUDYOPTS. d l ; Sde ( : , i ) = STUDYOPTS. de ; end % % Store and plot cond , err , vs cycle and f i n a l G, % % write these to f i l e along with parameters used and model % Figure out the best way to save data for what I need . % Location savefile =... ( [ ' TRM_2model_roundcanti_1tor_ ' i n t 2 s t r ( ext ) ' _freq ' i n t 2 s t r (NFQ ) ] ) ; save ( s a v e f i l e ) end % Extent clear SFREQH1 S f r e q S c y c l e SG SCOS Sdl Sde S n o i s e SFREQH2 SMODEH1 end %number of frequencies 446 MAINDDSDT.m function [ modelCON , defCON , INP , SOLN ] = MAINDDSDT(STUDYOPTS, OPTIONS , INP , . . . modelCON ) KMS = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % I n i t i a l i z e defCON structure defCON = struct ( ' def ' , {[]}); % I n i t i a l i z e Inputs You can s u b s t i t u t e your own function here for % INPUTSDTDD() but you s t i c k with STUDYOPTS to pass study options . if STUDYOPTS . m u l t i [ modelCON , INP ] == 0 = INPUTSDTDD(STUDYOPTS, OPTIONS ) ; else [ modelCON ] = INPUTSDTDD(STUDYOPTS, OPTIONS , modelCON ) ; % Let ' s r e s t a r t with a fresh model . This needs to be fixed to allow % for INPUTSDTDD to have d i f f e r e n t names end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % I n i t i a l i z e SOLN structure SOLN = s t r u c t ( ' dkdg ' , { [ ] } , 'A ' , { [ ] } , ' F ' , { [ ] } , 'DG ' , { [ ] } , . . . 'GG ' , ?modelCON ( 1 , 1 ) . G, ' c y c l e ' , 1 , ' e r r ' , { [ ] } , ' condA ' , { [ ] } , . . . ' F_old ' , { 0 } , ' d f ' , { 1 } ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Some house keeping and l o c a l variable for main clear FE * %Decide how many modes to solve for . maxfreq = 35; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % To get simulated data if STUDYOPTS . r u n t y p e == 1 447 [ modelCON , defCON ] = LDDSDT( modelCON , defCON , m a x f r e q , 1 , OPTIONS ) ; return end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (OPTIONS .TRM == 1 SOLN ( 1 , 1 ) . d e l t a if = OPTIONS .TRM == = SOLN ( 1 , 1 ) . d e l t a 0 SOLN ( 1 , 1 ) . t | 2) & STUDYOPTS . r u n t y p e ~= 1 1; = 5.0 * norm ( modelCON ( 1 , 1 ) .G ) ; 1; OPTIONS .TRM == 1 modelCON ( 1 , 1 ) . G1=o n e s ( s i z e ( modelCON ( 1 , 1 ) .G) ) *1; end [ modelCON , defCON ] if = LDDSDT( modelCON , defCON , m a x f r e q , 1 , OPTIONS ) ; OPTIONS . DER == 0 if KMS == 0 & OPTIONS .TRM ~= & STUDYOPTS . f i r s t r u n 1 == 1 SOLN = BDKDGSDT2( modelCON , SOLN , STUDYOPTS, OPTIONS ) ; save p i p e 8 d d 2 . mat SOLN STUDYOPTS . f i r s t r u n = 0 ; elseif KMS == 0 load p i p e 8 d d . mat SOLN end else SOLN = BDKDGSDT2( modelCON , SOLN , STUDYOPTS, OPTIONS ) ; end if OPTIONS . MSA == 1 [ SOLN ] = BAFSDTMODES( modelCON , defCON , INP , SOLN , OPTIONS ) ; [ SOLN ] = BAFSDT( modelCON , defCON , INP , SOLN , OPTIONS ) ; else end k i =1; cMN = length ( INP ) ; 448 kk = 1 :cMN for mds = INP ( 1 , kk ) . ModeN ( : , 1 ) ; rMN = length ( INP ( 1 , kk ) . ModeN ) ; SOLN( 1 , 1 ) . F ( [ k i : k i+rMN ? 1 ] , 1 ) = ?(INP ( 1 , kk ) . FreqH ( : ) . ^ . 5 ? . . . ( defCON ( 1 , kk ) . d e f . data ( mds ) * 2 * p i ) ) . / INP ( 1 , kk ) . FreqH ( : ) . ^ . 5 ; k i = k i+rMN; end end while if 1 OPTIONS .TRM == 1 | OPTIONS .TRM == 2 SOLN( 1 , 1 ) . F_old=SOLN( 1 , 1 ) . F ; SOLN( 1 , 1 ) . A_old = SOLN( 1 , 1 ) .A; SOLN( 1 , 1 ) . J_old = sum (SOLN( 1 , 1 ) . F_old . ^ 2 ) ; SOLN( 1 , 1 ) . M_old=SOLN( 1 , 1 ) . J_old ; SOLN( 1 , 1 ) . dg_old=SOLN( 1 , 1 ) . A_old ' * SOLN( 1 , 1 ) . F_old ; SOLN( 1 , 1 ) . ddg_old=SOLN( 1 , 1 ) . A_old ' * SOLN( 1 , 1 ) . A_old ; modelCON ( 1 , 1 ) . G_old=modelCON ( 1 , 1 ) .G; if OPTIONS .TRM == 1 modelCON ( 1 , 1 ) .G=OPTIONS . bd . / ( 1 + . . . exp(?OPTIONS . b * modelCON ( 1 , 1 ) . G1))+(1 ?OPTIONS . bd ) ; end if OPTIONS .GP == 'E ' JL = 1 : length (modelCON)% a p p l y t o b o t h models for modelCON ( 1 , JL ) . model . p l ( : , 3 ) = modelCON ( 1 , 1 ) .G . . . ( 1 : length (modelCON ( 1 , JL ) . model .EM) ) . * . . . modelCON ( 1 , JL ) . model .EM' ; end elseif OPTIONS .GP == 'M' modelCON ( 1 , 1 ) . model . p l ( [ 1 : length (modelCON ( 1 , 1 ) .G) ] , 5 ) = . . . 449 modelCON(1 ,1).G * 7800; else modelCON = modelCON(1 ,1). mdl(STUDYOPTS,modelCON,OPTIONS) ; end end INDEX_H = 1; % I don ' t want t o t r a n s f e r SOLN . d k d g u n l e s s I h a v e [SOLN(1 ,1). cycle ,SOLN(1 ,1). err ] = TERMSDTDD(SOLN(1 ,1).DG, . . . SOLN(1 ,1).GG,modelCON(1 ,1).G, . . . INDEX_H,OPTIONS. term ,SOLN(1 ,1). cycle ,SOLN(1 ,1).F,OPTIONS) ; clear SOLN(1 ,1).F % Check if to see if we had an to . error . SOLN(1 ,1). err == inf disp ( ' Stopped at TERMSDTDD' ) break end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Choose wh at to build and solve [ modelCON , defCON ] = % LDDSDT( modelCON , defCON , INP , s o l v e t y p e , OPTIONS ) ; if KMS == 0 | | OPTIONS.GP == 'A' % Regular initial solve full build [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,1 ,OPTIONS) ; e l s e i f KMS == 0 | | OPTIONS.GP == 'M' % j u s t mass b u i l d and s o l v e [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,2 ,OPTIONS) ; e l s e i f KMS == 1 % j u s t s t i f f n e s s b u i l d and s o l v e [modelCON, defCON ] = LDDSDT(modelCON, defCON , maxfreq ,2 ,OPTIONS) ; else disp ( ' Error at : switch what to build . ' ) break end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 450 % Call DKDG and/or DMDG OPTIONS.DER == 0 & OPTIONS.TRM ~= 1 KMS == 0 & STUDYOPTS. f i r s t r u n == 1 SOLN = BDKDGSDT2(modelCON,SOLN,STUDYOPTS,OPTIONS) ; pipe8dd2 . mat SOLN KMS == 0 pipe8dd . mat SOLN if if save elseif load end else SOLN = BDKDGSDT2(modelCON,SOLN,STUDYOPTS,OPTIONS) ; end KMS = 1; % Turn o f f M and DKDG b u i l d i f OPTIONS. sens == 1 % e x i t for s e n s i t i v i t y run ( s ) break ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mode Checking if OPTIONS.MS == 1 [ modelCON, defCON ,SOLN] = MACSDTDD(modelCON, defCON , INP ,SOLN,OPTIONS) ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Build A and F if OPTIONS.MSA == 1 [SOLN] = BAFSDTMODES(modelCON, defCON , INP ,SOLN,OPTIONS) ; else [SOLN] = BAFSDT(modelCON, defCON , INP ,SOLN,OPTIONS) ; end SOLN( 1 , 1 ) . condA(SOLN( 1 , 1 ) . cycle ?1) = 451 cond (SOLN( 1 , 1 ) .A) ; % The above can be turned on of o f f depending on the options %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SOLN( 1 , 1 ) . d f=abs (SOLN( 1 , 1 ) . F_old?max( abs (SOLN( 1 , 1 ) . F ) ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Solve for DG using the pseudoinverese so far [SOLN] = InverSDT (SOLN,OPTIONS ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Choose a way to update G with DG i f OPTIONS.CVX == 0 [SOLN, modelCON ,OPTIONS]=GSDTDD(SOLN, modelCON ,OPTIONS ) ; e l s e i f OPTIONS.CVX == 1 % I f using c o n v e x i f i c a t i o n . [SOLN, modelCON ,OPTIONS]=GSDTDD(SOLN, modelCON ,OPTIONS ) ; i f max ( abs (SOLN ( : , 1 ) . F)) <1e ?6 OPTIONS. b = 1 . 1 ; e l s e i f SOLN ( : , 1 ) . df <1e ?6 OPTIONS. b = 0 . 9 ; end end t r u n c = length (modelCON ( 1 , 1 ) .G) ; modelCON ( 1 , 1 ) .G( 1 : t ru n c ) = 1 . / ( 1 + . . . exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1 ( 1 : t ru n c ) ) ) ; % update G with the extra parameter G1 G i s g e t t i n g too small here % need to bound i t . modelCON ( 1 , 1 ) .G( find (modelCON ( 1 , 1 ) .G< . 3 ) ) = . 3 ; else break ; end i f OPTIONS.GP == 'E '& OPTIONS.TRM ~= 1 & OPTIONS.TRM ~= 2 for JL = 1 : length (modelCON)% apply to both models 452 modelCON(1 ,JL ) . model . pl ( : ,3 ) = modelCON(1 ,1).G. . . (1: (modelCON(1 ,JL ) . model .EM) ) . * . . . modelCON(1 ,JL ) . model .EM' ; length end OPTIONS.GP == 'A' & OPTIONS.TRM ~= 1 & OPTIONS.TRM ~= 2 modelCON = modelCON(1 ,1). mdl(STUDYOPTS,modelCON,OPTIONS) ; elseif end SOLN(1 ,1). delG ( : ,SOLN(1 ,1). cycle ?1)=SOLN(1 ,1).A' * SOLN(1 ,1).F; %SOLN( 1 , 1 ) . delG1 ( : ,SOLN( 1 , 1 ) . c y c l e ?1)=max( abs ( (SOLN( 1 , 1 ) .A) ' * SOLN( 1 , 1 ) . %F ' ) ) ; SOLN(1 ,1). (SOLN(1 ,1). cycle ?1,1)=SOLN(1 ,1).DG' * . . . SOLN(1 ,1). delG ( : ,SOLN(1 ,1). cycle ? 1)/( (SOLN(1 ,1).DG) * . . . (SOLN(1 ,1). delG ) ) ; cos norm norm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear SOLN.A %d i s p ( 1 ) if ( (SOLN(1 ,1).F))< OPTIONS. term (3) ( 'Minimum s t i f f n e s s change s a t i s f i e d 1 ' ) norm abs disp break end if abs (SOLN(1 ,1). (SOLN(1 ,1). cycle ? 1)) < .00000000808 ( 'Minimum s t i f f n e s s change s a t i s f i e d 2 ' ) cos disp break end %i f SOLN( 1 , 1 ) . delG1 (SOLN( 1 , 1 ) . c y c l e ?1) <1e ?6 % d i s p ( ' Minimum s t i f f n e s s change s a t i s f i e d 3 ' ) %b r e a k end if ( (SOLN(1 ,1).F))<1e ?6 ( 'Minimum s t i f f n e s s change s a t i s f i e d 4 ' ) max abs disp 453 break end figure ( 1 ) ; pause ( 0 . 1 ) plot ( modelCON ( 1 , 1 ) .G) end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fprintf ( ' Number of c y c l e s %i \n ' , SOLN . c y c l e ) 454 INPUTSDTDD.m function if [ modelCON , INP ] STUDYOPTS . m u l t i INP = = INPUTSDTDD(STUDYOPTS, OPTIONS , modelCON ) == 0 s t r u c t ( ' FreqH ' , { [ ] } , ' ModeH ' , { [ ] } , ' ModeN ' , { [ ] } , ' ModeS ' , { [ ] } , . . . ' dof ' , { [ ] } ) ; modelCON = s t r u c t ( ' m o d e l ' , { [ ] } , 'G ' , { [ ] } , 'GH ' , { [ ] } , INP . FreqH = [ ] ; INP . ModeH = [ ] ; ' mdl ' , {[]}); %Use t h i s for multi study %INP(1 ,1). FreqH = ( [ ] ' * 2 * pi ).^2; %INP(1 ,2). FreqH = ( [ ] ' * 2 * pi ).^2; INP. mode = [ ] ; %Rigid Body Modes % RB = 6; INP ( 1 , 1 ) . ModeN = [ 1 +RB : 1 : RB+OPTIONS . NFQ ] ' ; INP ( 1 , 1 ) . ModeS = [ 1 +RB : 1 : RB+OPTIONS . NFQ ] ' ; end modelCON ( 1 , 1 ) . mdl = @pipe8dd ; modelCON = modelCON ( 1 , 1 ) . mdl (STUDYOPTS, modelCON , OPTIONS ) ; % Probably don ' t need to e d i t t h i s unless you want to allow for greater G modelCON ( 1 , 1 ) . G = ones ( 1 , 2 0 ) ; % i . e . GH modelCON ( 1 , 1 ) .GH = modelCON ( 1 , 1 ) . G ; modelCON ( 1 , 1 ) . G1 = if [ ] ; OPTIONS . CVX ==1 modelCON ( 1 , 1 ) . G1 = modelCON ( 1 , 1 ) . G = zeros zeros (1 , (1 , size size ( modelCON ( 1 , 1 ) . m o d e l . p l , 1 ) ) ; %./(1+ exp( ? 1000 * modelCON(1 ,1).G1) ) ; % updating G end INP ( 1 , 1 ) . d o f % update G1 %. . . ( modelCON ( 1 , 1 ) . m o d e l . p l , 1 ) ) ; = modelCON ( 1 , 1 ) . m o d e l . a d o f ; 455 multibeam.m function [ modelCON ] = m u l t i b e a m (STUDYOPTS, modelCON , OPTIONS) femesh ; % p a s t e u s e r model h e r e % use STUDYOPTS t o c o n f i g u e t h e model f o r p a r a m e t r i c s t u d i e s % STUDYOPTS i s t o t a l l y u s e r c o n f i g u r a b l e and i s l o c a t e d i n DDSTUDY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % model numbering f o r STUDYOPTS % mdl = 1 ; e t c . % Put your s e c t i o n d e r i v a t i v e s h e r e if STUDYOPTS . r u n t y p e == 2 && OPTIONS . GP == modelCON ( 1 , 1 ) .AK = ones ( size %&& OPTIONS.DER == 1 'A ' ( modelCON ( 1 , 1 ) . G ) ) ; %3*modelCON ( 1 , 1 ) .G. ^ 2 ; % Enter your d e r i v i t i v e term f o r s t i f f n e s s % ( 1 / 1 2 ) * b *(G*h )^3 modelCON ( 1 , 1 ) .AM = a = ones ( = .5 *.25 * = (1/12) *a .*b.^3; Iy1 = (1/12) *b.* a .^3; a. .0254; *b; modelCON ( 1 , 1 ) . m o d e l . i l ( : , [ 3 : 6 ] ) = %* 1 0 0 ; *(1 ?(b . ^ 4 . / ( 1 2 . * a . ^ 4 ) ) ) ) ) ; Ix1 l i l % and mass b *(G*h ) *a .*((.5.* b).^3).*((16/3) ?(3.36.*... (b ./ a ). A1 = ( modelCON ( 1 , 1 ) . G ) ) ; * . 0 2 5 4 ; %* 1 0 0 ; 1 b = modelCON ( 1 , 1 ) . G' k1 size size = [ k1 Ix1 Iy1 A1 ] ; ( modelCON ( 1 , 2 ) . m o d e l . i l , 1 ) ; modelCON ( 1 , 2 ) . m o d e l . i l ( ( [ 1 : l i l ] ) , [ 3 : 6 ] ) [ k1 ( [ 1 : l i l ] ) Ix1 ( [ 1 : l i l ] ) =... Iy1 ( [ 1 : l i l ] ) A1 ( [ 1 : l i l ] ) ] ; return end %I n i t i a l i z e%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% modelCON ( 1 , 1 ) .AK = ones ( size ( modelCON ( 1 , 1 ) . G ) ) ; 456 %3*modelCON ( 1 , 1 ) .G. ^ 2 ; % Enter your d e r i v i t i v e term f o r s t i f f n e s s % (1/12) * b * (G* h)^3 modelCON(1 ,1).AM = ones ( s i z e (modelCON(1 ,1).G) ) ; % and mass b * (G* h ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L = .45; % With t h i s we are assured the model i s evenly d i v i s i b l e EPG = 2; NG = 45; elements = EPG* ones (1 ,NG) ; NEL = sum( elements ) ; a = ones (NG,1) * 1 * .0254; %* 100; b = ones (NG,1) * .25 * .0254;%* 100; i f OPTIONS.GP == 'A' b(STUDYOPTS. dl ) = STUDYOPTS. de . * b(STUDYOPTS. dl ) ; end k1 = .5 * a . * ( ( . 5 . * b).^3). * ((16/3) ? (3.36. * (b./ a). * (1 ? (b .^4./(12. * a . ^ 4 ) ) ) ) ) ; Ix1 = (1/12) * a . * b .^3; Iy1 = (1/12) * b. * a .^3; A1 = a . * b ; E=67.5e9 ; rho = 2739; sh = 5 * (1+.33)/(6+5 * .33); FEnode = [ ] ; LG = L/ length ( elements ) ; FEnode = [1 , 0 0 0 0 0 0 ] ; count = 2; nodespace = 0; f o r i j = 1: length ( elements ) f o r ik = 1: elements ( i j ) nodespace = nodespace + LG/elements ( i j ) ; 457 FEnode = [ FEnode ; count , 0 0 0 0 0 nodespace ] ; count = count +1; end end IR = s i z e ( FEnode ,1)+1; FEnode = [ FEnode ; IR , zeros ( 1 , 3 ) , 1 , 0 , 0 ] ; OPTIONS.SPB == 0 if LineBeam = [ Inf 98 101 97 109 49 0]; LineBeam = [ Inf 98 101 97 else 109 49 116]; end count = 1 ; i i = 1 : length ( elements ) for FEelt ( count , 1 : 7 ) = LineBeam ; count = count + 1 ; for i i i =1: elements ( i i ) FEelt ( count , 1 : 7 ) = [ ( count ? i i ) ( count ? i i +1) i i count = count + 1 ; end end clear LineBeam i 1 i 2 LENGTH i NEL = length ( elements ) ; E = E* ones (NG, 1 ) ; % if Introduce damage OPTIONS.GP == 'E ' E(STUDYOPTS. dl ) = STUDYOPTS. de . * E(STUDYOPTS. dl ) ; 458 ii IR 0 0]; end model=femesh ( ' model ' ) ; model . pl =[(1:NEL) ' ones (NG,1) E, .33 * ones (NG,1) rho * ones (NG, 1 ) ] ; model .EM = E; model . i l = [ ( 1 :NG) ' ones (NG,1) k1 Ix1 Iy1 A1 ] ; mdof = f e u t i l ( ' getdof ' , model ) ; model . adof = fe_c (mdof , [ . 0 1 .05] ' , ' dof ' ) ; i1 = femesh ( ' findnode z==0' ) ; model . adof = fe_c ( model . adof , i1 , ' dof ' ,2); model=fe_case (model , ' SetCase1 ' , . . . 'KeepDof ' , ' f i n a l DOF l i s t ' , model . adof ) ; [ model . Case , model .DOF]=fe_mknl( ' i n i t ' , model ) ; modelCON(1 ,1). model = model ; clear FE* 459 TERMSDTDD.m function if [ cycle , err ] = TERMSDTDD(DG, GG, G, INDEX_H, t e r m , c y c l e , F , OPTIONS) OPTIONS . CVX == 0 if INDEX_H > 0 DG = G ; end err1 = max ( abs (DG ) ) ; err2 = max ( abs (GG?G ) ) ; err if = min ( err < if err1 , err2 ) ; term ( 1 ) err == err1 fprintf ( ' The min increment in DG ' ) ( ' The min increment in G' ' err1 else fprintf ?G0 ' ) err2 end err elseif = inf ; c y c l e >t e r m ( 2 ) ; cycle ; fprintf err = ( ' The result may need further inf ; end else err1 = max ( abs (DG ) ) ; err2 = max ( abs (GG?G ) ) ; err if = err min ( < err1 , err2 ) ; term ( 1 ) 460 improvement \n ' ) if e r r == e r r 1 fprintf ( ' The min i n c r e m e n t i n DG ' ) err1 else fprintf ( ' The min i n c r e m e n t i n G ' ' ?G0 ' ) err2 end err = inf ; elseif c y c l e >term ( 2 ) ; cycle ; fprintf ( ' \n The r e s u l t may need f u r t h e r improvement \n ' ) err = inf ; end if max ( abs (F)) < term disp ( (3); ' \n Minimum s t i f f n e s s change s a t i s f i e d \n ' ) err = inf ; elseif c y c l e >term ( 2 ) ; cycle ; fprintf ( ' \n The r e s u l t may need f u r t h e r improvement \n ' ) err = inf ; end end c y c l e=c y c l e +1; 461 LDDSDT.m function [ modelCON , defCON ] = LDDSDT( modelCON , defCON , NF , s o l v e t y p e , . . . OPTIONS ) ; if solvetype for ii == 1 = 1: length ( modelCON ) [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] . . . =f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 1 ) ; end clear modelCON ( 1 , i i ) . m o d e l . k modelCON ( 1 , i i ) . m o d e l .m modelCON ( 1 , i i ) . m o d e l .m=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . m od el , . . . modelCON ( 1 , i i ) . m o d e l . C a s e , 2 ) ; modelCON ( 1 , i i ) . m o d e l . k=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . mo de l , . . . modelCON ( 1 , i i ) . m o d e l . C a s e , 1 ) ; defCON ( 1 , i i ) . d e f=f e _ e i g ( { modelCON ( 1 , i i ) . m o d e l . m , . . . modelCON ( 1 , i i ) . m o d e l . k , . . . modelCON ( 1 , i i ) . m o d e l . a d o f } , [ 5 clear FE NF 500 11 1e ?5]); * end elseif for solvetype ii = 1: == 2 % 2+ runs w i t h j u s t b u i l d i n g k length ( modelCON ) [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] . . . =f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 0 ) ; end clear modelCON ( 1 , i i ) . m o d e l . k modelCON ( 1 , i i ) . m o d e l . k=f e _ m k n l ( ' a s s e m b l e ' , modelCON ( 1 , i i ) . mo de l , . . . 462 modelCON(1 , i i ) . model . Case , 1 ) ; defCON(1 , i i ) . def=fe_eig ({modelCON(1 , i i ) . model .m, . . . modelCON(1 , i i ) . model . k , . . . modelCON(1 , i i ) . model . adof } ,[5 NF 0 0 1e ? 6]); clear FE* end solvetype == 3 for i i = 1: length (modelCON)%need t o r e b u i l d model t o add mass . [modelCON(1 , i i ) . model . Case ,modelCON(1 , i i ) . model .DOF] . . . =fe_mknl( ' i n i t ' ,modelCON(1 , i i ) . model ) ; i f OPTIONS.SPB == 1 [modelCON] = SPBFunctions (modelCON,OPTIONS, ii , 0 ) ; elseif end modelCON(1 , i i ) . model .m modelCON(1 , i i ) . model .m=fe_mknl( ' assemble ' ,modelCON(1 , i i ) . model , . . . modelCON(1 , i i ) . model . Case , 2 ) ; defCON(1 , i i ) . def=fe_eig ({modelCON(1 , i i ) . model .m, . . . modelCON(1 , i i ) . model . k , . . . modelCON(1 , i i ) . model . adof } ,[5 NF 500 11 1e ? 5]); clear FE* clear end else disp ( ' Error in : LDDSDT' ) end 463 MACSDTDD.m function [ modelCON , defCON , SOLN ] = MACSDTDD( modelCON , defCON , INP , . . . SOLN , OPTIONS) for k = size 1:1: ( modelCON , 2 ) [ modedof , modeind ] for r = for = f e _ c ( modelCON ( 1 , k ) . m o d e l . a d o f , INP ( 1 , k ) . d o f ) ; 1: length ( INP ( 1 c = 1: , k ) . ModeS ) length ( INP ( 1 SOLN ( 1 , k ) . mac ( r , c ) , k ) . ModeS ) = ( sum ( INP ( 1 , k ) . ModeH ( : , r ) . defCON ( 1 , k ) . d e f . d e f ( : , c ) ) ) ^ 2 / ( sum ( INP ( 1 , k ) . ModeH ( : , r ) . *sum ( defCON ( 1 , k ) . . . . * INP ( 1 , k ) . ModeH ( : def . def ( : , c ). *... * defCON ( 1 , k ) . , r ) ) . . . def . def ( : , c ) ) ) ; end end [ mmac , i n d ] for g = max (SOLN ( 1 = 1:1: , k ) . mac ) ; length ( mmac ) defCON ( 1 , k ) . d e f . d e f ( : , g )= norm ( defCON ( 1 if mean ( defCON ( 1 defCON ( 1 , k ) . d e f . d e f ( : , g ) . / . . . , k ) . def . def ( : , g ) ) ; , k ) . d e f . d e f ( : , g )) <0 defCON ( 1 , k ) . d e f . d e f ( : , g ) = ?defCON ( 1 , k ) . def . def ( : , g ) ; end end for g if = g 1:1: ~= length ( mmac ) ind ( g ) place = ind ( g ) ; t em pz = defCON ( 1 , k ) . d e f . d e f ( : , p l a c e ) ; defCON ( 1 , k ) . d e f . d e f ( : , p l a c e ) defCON ( 1 , k ) . d e f . d e f ( : , g ) tempf = = defCON ( 1 , k ) . d e f . d e f ( : , g ) ; t em p z ; = defCON ( 1 , k ) . d e f . d a t a ( p l a c e ) ; defCON ( 1 , k ) . d e f . d a t a ( p l a c e ) 464 = defCON ( 1 , k ) . d e f . d a t a ( g ) ; defCON ( 1 , k ) . d e f . data ( g ) = tempf ; tempmac = SOLN( 1 , k ) . mac ( : , p l a c e ) ; SOLN( 1 , k ) . mac ( : , p l a c e ) = SOLN( 1 , k ) . mac ( : , g ) ; SOLN( 1 , k ) . mac ( : , g ) = tempmac ; ind ( g)=g ; ind ( p l a c e ) = p l a c e ; else continue end end end 465 BDKDGSDT2.m function [ SOLN ] = BDKDGSDT2( modelCON , SOLN, STUDYOPTS, OPTIONS) s t d t m p = STUDYOPTS . r u n t y p e ; detmp = STUDYOPTS . d e ; for i i = 1 : s i z e ( modelCON , 2 ) dkdgSTR = if s t r u c t ( ' dkdg ' , { [ ] } ) ; OPTIONS . GP == dmdgSTR = 'A ' | OPTIONS . GP == 'M' s t r u c t ( ' dmdg ' , { [ ] } ) ; end rpl = l e n g t h ( modelCON ( 1 , i i ) . G ) ; E = modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) ; r h o = modelCON ( 1 , i i ) . m o d e l . p l ( : , 5 ) ; I L = modelCON ( 1 , i i ) . m o d e l . i l ; if OPTIONS . DKMeth == 1 for v = if 1: rpl OPTIONS . GP == 'E ' modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) = modelCON ( 1 , i i ) . m o d e l . p l ( v , 3 ) = E( v ) ; elseif OPTIONS . GP == zeros ( rpl , 1 ) ; 'M' modelCON ( 1 , i i ) . m o d e l . p l ( [ 1 : l e n g t h ( modelCON ( 1 , 1 ) . G ) ] , 5 ) .0000000001 * ones ( rpl ,1); modelCON ( 1 , i i ) . m o d e l . p l ( v , 5 ) elseif OPTIONS . GP == = 7800; 'A ' modelCON ( 1 , i i ) . m o d e l . i l ( : , [ 3 : 8 ] ) modelCON ( 1 , i i ) . m o d e l . i l ( v , : ) = zeros ( rpl , 6 ) ; = IL ( v , : ) ; else disp ( ' Error in : B u i l d DKDG and DMDG' ) end 466 =... [ modelCON ( 1 , i i ) . m o d e l . C a s e , modelCON ( 1 , i i ) . m o d e l . DOF ] = . . . f e _ m k n l ( ' i n i t ' , modelCON ( 1 , i i ) . m o d e l ) ; if OPTIONS . SPB == 1 [ modelCON ] = S P B F u n c t i o n s ( modelCON , OPTIONS , i i , 0 ) ; end if OPTIONS . GP == 'A' | OPTIONS . GP == 'E ' dkdgSTR ( v ) . dkdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . modelCON ( 1 , i i ) . mo de l , modelCON ( 1 , i i ) . m o d e l . C a s e , 1 ) ) ; end if OPTIONS . GP == 'A' | OPTIONS . GP == 'M' dmdgSTR ( v ) . dmdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . modelCON ( 1 , i i ) . mo de l , modelCON ( 1 , i i ) . m o d e l . C a s e , 2 ) ) ; end modelCON ( 1 , i i ) . m o d e l . p l ( : , 3 ) SOLN ( 1 , i i ) . dkdg if = E; = dkdgSTR ; OPTIONS . GP == 'A' modelCON ( 1 , i i ) . m o d e l . i l = IL ; SOLN ( 1 , i i ) . dmdg = dmdgSTR ; end end elseif OPTIONS . DKMeth == 2 STUDYOPTS . r u n t y p e for vv = = 1; 1: rpl STUDYOPTS . d l = [ 1 : rpl ] ' ; STUDYOPTS . d e = ones ( rpl , 1 ) STUDYOPTS . d e ( vv ) * 1 e ?10; = modelCON ( 1 , 1 ) . G( vv )+OPTIONS . DH ; modelCON = modelCON ( 1 , i i ) . mdl (STUDYOPTS, modelCON , OPTIONS ) ; if OPTIONS . GP == 'E ' | OPTIONS . GP == 'A' dkdgSTR ( vv ) . dkdg=s p a r s e ( f e _ m k n l ( ' a s s e m b l e ' , . . . 467 modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 1 ) ) ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' dmdgSTR(vv ) . dmdg=sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 2 ) ) ; end STUDYOPTS. de (vv) = modelCON(1 ,1).G(vv) ?OPTIONS.DH; modelCON = modelCON(1 , i i ) . mdl(STUDYOPTS,modelCON,OPTIONS) ; i f OPTIONS.GP == 'E' | OPTIONS.GP == 'A' dkdgSTR(vv ) . dkdg = (dkdgSTR(vv ) . dkdg ? ... sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 1 ) ) ) / . . . (2 * OPTIONS.DH) ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' dmdgSTR(vv ) . dmdg= (dmdgSTR(vv ) . dmdg ? ... sparse (fe_mknl ( ' assemble ' , . . . modelCON(1 , i i ) . model ,modelCON(1 , i i ) . model . Case , 2 ) ) ) / . . . (2 * OPTIONS.DH) ; end end %replace E modelCON(1 , i i ) . model . pl ( : ,3 ) = E; % put the DKDGstr in the right place in SOLN SOLN(1 , i i ) . dkdg = dkdgSTR ; i f OPTIONS.GP == 'E' | OPTIONS.GP == 'A' modelCON(1 , i i ) . model . i l = IL ; %Replace i l SOLN(1 , i i ) . dkdg = dkdgSTR ; end i f OPTIONS.GP == 'A' | OPTIONS.GP == 'M' 468 else end end modelCON(1 , i i ) . model . i l = IL ; %Replace i l SOLN(1 , i i ) . dmdg = dmdgSTR; disp ( ' Error in : Build DKDG and DMDG' ) return end STUDYOPTS. runtype = stdtmp ; STUDYOPT. de = detmp ; 469 BAFSDT.m function cMN = if [ SOLN ] = BAFSDT( modelCON , defCON , INP , SOLN , OPTIONS ) ; length ( INP ) ; OPTIONS . CVX == 0 & OPTIONS . GP == OPTIONS .TRM == 'E ' & (OPTIONS .TRM ~= 1 | . . . 2) %OLD METHOD j u s t K for ii = 1 : cMN %rMN = length (INP(1 , i i ) .ModeN) ; rMN = for length ( INP ( 1 , i i ) . ModeN ) ; d = 1 :rMN dd = INP ( 1 , i i ) . ModeN ( d ) ; for e = 1: length (SOLN ( 1 R1 ( e , d , i i ) , i i ) . dkdg ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . *SOLN ( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end end elseif OPTIONS . CVX == 1 OPTIONS .TRM ~= for & OPTIONS . GP == 'E ' & OPTIONS .TRM ~= 2 %number of models %rMN = length (INP(1 , i i ) .ModeN) ; length % number of modes ii = 1 : cMN rMN = for ( INP ( 1 , i i ) . ModeN ) ; d = 1 :rMN dd = INP ( 1 , i i ) . ModeN ( d ) ; for e = 1: length (SOLN ( 1 R1 ( e , d , i i ) , i i ) . dkdg ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . *SOLN ( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . 470 1 &... * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS. a * exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS. a * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; end %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay %the same . end end e l s e i f OPTIONS.TRM == 1 & OPTIONS.GP == 'E ' %number of models %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; rMN = length (INP ( 1 , i i ) . ModeN ) ; % number of modes for i i = 1 :cMN for d=1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : length (SOLN( 1 , i i ) . dkdg ) R1( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . % *SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . * defCON ( 1 , i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS. bd *OPTIONS. b * exp(?OPTIONS. b * . . . modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS. b * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end end e l s e i f OPTIONS.CVX == 0 & OPTIONS.GP == 'A ' & (OPTIONS.TRM ~= 1 | . . . OPTIONS.TRM == 2) %OLD METHOD j u s t K % Later allow for d i f f e r e n t s i z e INP .ModeN or INF end pad for i i = 1 :cMN %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; 471 rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; for d =1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : l e n g t h (SOLN( 1 , i i ) . dkdg ) R1 ( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . * ( modelCON ( 1 , 1 ) .AK( e ) * SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . ? ((defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * p i ) . ^ 2 ) . . . * modelCON ( 1 * defCON ( 1 , end , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay %the same . end end elseif OPTIONS .CVX == 1 & OPTIONS .GP == 'A ' &... OPTIONS .TRM ~= 1 & OPTIONS .TRM == 2 for %number of models %rMN = l e n g t h (INP(1 , i i ) .ModeN) ; rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; % number of modes i i = 1 :cMN for d =1:rMN dd = INP ( 1 , i i ) . ModeN( d ) ; for e = 1 : l e n g t h (SOLN( 1 , i i ) . dkdg ) R1 ( e , d , i i ) = defCON ( 1 , i i ) . d e f . d e f ( : , dd ) ' . . . * ( modelCON ( 1 , 1 ) .AK( e ) * SOLN( 1 , i i ) . dkdg ( 1 , e ) . dkdg . . . ? ((defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * p i ) . ^ 2 ) . . . * modelCON ( 1 * defCON ( 1 , , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) . . . *OPTIONS . a * exp(?OPTIONS . a * modelCON ( 1 , 1 ) . G1( e ) ) . . . /(1+ exp(?OPTIONS . a * modelCON ( 1 , 1 ) . G1( e ) ) ) ^ 2 ; %Check to see the d i f f e r e n c e in the mode shapes and R1 from % healthy to damaged since most people assume mode shapes stay end end 472 %t h e same . end OPTIONS.TRM == 1 & OPTIONS.GP == 'A' for i i = 1:cMN %number o f m o d e l s elseif %rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; rMN = length (INP(1 , i i ) .ModeN) ; % number o f modes for d=1:rMN dd = INP(1 , i i ) .ModeN(d ) ; for e = 1: length (SOLN(1 , i i ) . dkdg ) R1( e , d , i i ) = defCON(1 , i i ) . def . def ( : , dd ) ' . . . * (modelCON( 1 , 1 ) .AK( e ) * SOLN(1 , i i ) . dkdg (1 , e ) . dkdg . . . ? ((defCON(1 , i i ) . def . data (dd) * 2 * pi ) . ^ 2 ) . . . * modelCON( 1 , 1 ) .AM( e ) * SOLN(1 , i i ) . dmdg(1 , e ) . dmdg ) . . . * defCON(1 , i i ) . def . def ( : , dd ) . . . *OPTIONS. bd *OPTIONS. b * exp(?OPTIONS. b * modelCON( 1 , 1 ) .G1( e ) ) . . . /(1+ exp(?OPTIONS. b *modelCON( 1 , 1 ) .G1( e )))^2; end end %t h e %Check % healthy to to see the damaged difference since most in the people mode shapes assume mode and R1 f r o m shapes same . end elseif OPTIONS.CVX == 0 & OPTIONS.GP == 'M' & . . . (OPTIONS.TRM ~= 1 |OPTIONS.TRM == 2) %OLD METHOD % Later for allow for different size INP . ModeN or INF end just M p ad i i = 1:cMN %rMN = l e n g t h ( INP ( 1 , i i ) . ModeN ) ; rMN = length (INP(1 , i i ) .ModeN) ; for d=1:rMN dd = INP(1 , i i ) .ModeN(d ) ; for e = 1: length (SOLN(1 , i i ) . dkdg ) R1( e , d , i i ) = defCON(1 , i i ) . def . def ( : , dd ) ' . . . 473 stay * ( ? ( ( defCON ( 1 , * modelCON ( 1 * defCON ( 1 , i i ) . d e f . d a t a ( dd ) * 2 * pi ) . ^ 2 ) . . . , 1 ) .AM( e ) * SOLN( 1 , i i ) . dmdg ( 1 , e ) . dmdg ) . . . i i ) . d e f . d e f ( : , dd ) ; end %Check to see the d i f f e r e n c e in the mode shapes and R1 from end % healthy to damaged since most people assume mode shapes stay %the same . end else disp ( ' E r r o r i n BAFSDT OPTIONS .CVX unknown \n ' ) end %F i r s t assume ModeN i s the same for a l l models k i =1; k j =1; SOLN ( 1 , 1 ) . A = [ ] ; for kk = 1 :cMN % This needs to be based on ModeN mds = INP ( 1 , kk ) . ModeN ( : , 1 ) ; rMN = length ( INP ( 1 , kk ) . ModeN ) ; SOLN ( 1 , 1 ) . F ( [ k i : k i+rMN ? 1 ] , 1 ) = ?(INP ( 1 , kk ) . FreqH ( : ) ? ( defCON ( 1 , kk ) . d e f . d a t a ( mds ) * 2 * pi ) . ^ 2 ) . / INP ( 1 , kk ) . FreqH ( : ) ; k i = k i+rMN ; for g = 1 :rMN SOLN ( 1 , 1 ) . A( k j , : ) = R1 ( : , g , kk ) ' . / INP ( 1 , kk ) . FreqH ( g ) ; k j = k j +1; %count rows end end SOLN ( 1 , 1 ) . J = sum (SOLN ( 1 , 1 ) . F . ^ 2 ) ; 474 ... InverSDT.m function if [ SOLN ] = InverSDT (SOLN , OPTIONS ) ; OPTIONS . INVMETH == 0 SOLN ( 1 , 1 ) .DG = elseif | OPTIONS .TRM == 1 ?pinv (SOLN ( 1 , 1 ) . A) * SOLN ( 1 ,1).F; OPTIONS . INVMETH == 1 [ U , S , V]= s v d s (SOLN ( 1 , 1 ) . A , OPTIONS . NSVDS ) ; for i =s i z e ( S , 1 ) : ?1:1 i f max ( diag ( S ) / S ( i , i )) <OPTIONS .SVDMAG break ; end end *S([1: SOLN ( 1 , 1 ) .DG=(V ( : , [ 1 : i ] ) i ] , [ 1 : i ])^ ? 1 *U ( : ,[1: i ]) ') *... (SOLN ( 1 , 1 ) . F ) ; end if (OPTIONS .TRM == 1 if norm (SOLN ( 1 | OPTIONS .TRM == 2) & SOLN ( 1 , 1 ) . c y c l e >= 2 , 1 ) .DG)>SOLN ( 1 , 1 ) . d e l t a SOLN ( 1 , 1 ) . d t = 1; while abs (SOLN ( 1 , 1 ) . d t )>=1e SOLN ( 1 , 1 ) . t _ o l d ?3 = SOLN ( 1 , 1 ) . t ; SOLN ( 1 , 1 ) . g=SOLN ( 1 , 1 ) . A' * SOLN ( 1 SOLN ( 1 , 1 ) . H = SOLN ( 1 , 1 ) . A' ,1).F; * SOLN ( 1 , 1 ) .A; SOLN ( 1 , 1 ) . B = SOLN ( 1 , 1 ) . H+SOLN ( 1 , 1 ) . t * eye ( s i z e (SOLN ( 1 , 1 ) .H ) ) ; [ SOLN ( 1 , 1 ) . R , SOLN ( 1 , 1 ) . u ]= c h o l (SOLN ( 1 , 1 ) . B ) ; if SOLN ( 1 , 1 ) . u==0 SOLN ( 1 , 1 ) . p = SOLN ( 1 , 1 ) . R\ (SOLN ( 1 , 1 ) . R'\ ?SOLN ( 1 ,1). g ); SOLN ( 1 , 1 ) . q = SOLN ( 1 , 1 ) . R' \ SOLN ( 1 , 1 ) . p ; SOLN ( 1 , 1 ) . t norm (SOLN ( 1 , 1 ) . q ))^2 = SOLN ( 1 , 1 ) . t +(norm (SOLN ( 1 , 1 ) . p ) / . . . * ( norm (SOLN ( 1 SOLN ( 1 , 1 ) . d t , 1 ) . p) ?SOLN ( 1 = SOLN ( 1 , 1 ) . t 475 , 1 ) . d e l t a ) /SOLN ( 1 , 1 ) . d e l t a ; ?SOLN ( 1 , 1 ) . t_old ; else SOLN( 1 , 1 ) . Ei=eig (SOLN( 1 , 1 ) .B ) ; SOLN( 1 , 1 ) . t = ?max(SOLN( 1 , 1 ) . Ei ( find (SOLN( 1 , 1 ) . Ei < 0 ) ) ) ; end end SOLN( 1 , 1 ) .DG=?inv (SOLN( 1 , 1 ) .B) * SOLN( 1 , 1 ) . g ; end SOLN( 1 , 1 ) .M=SOLN( 1 , 1 ) . J_old+SOLN( 1 , 1 ) . dg_old ' * SOLN( 1 , 1 ) .DG+ . . . 0 . 5 * SOLN( 1 , 1 ) .DG' * SOLN( 1 , 1 ) . ddg_old *SOLN( 1 , 1 ) .DG; SOLN( 1 , 1 ) .dM=SOLN( 1 , 1 ) . M_old?SOLN( 1 , 1 ) .M; SOLN( 1 , 1 ) . rho=(SOLN( 1 , 1 ) . J_old ?SOLN( 1 , 1 ) . J )/SOLN( 1 , 1 ) .dM; end 476 GSTDD.m function trunc if = [ SOLN , modelCON , OPTIONS ] length (SOLN ( 1 OPTIONS . CVX == 0 = GSDTDD(SOLN , modelCON , OPTIONS) , 1 ) .DG ) ; & OPTIONS .TRM == 0 SOLN ( 1 , 1 ) .GG = modelCON ( 1 , 1 ) . G ; modelCON ( 1 , 1 ) . G ( 1 : t r u n c ) = modelCON ( 1 , 1 ) . G ( 1 : t r u n c ) +... * OPTIONS .DGP SOLN ( 1 , 1 ) .DG( 1 : t r u n c ) ' ; modelCON ( 1 , 1 ) . G = min ( modelCON ( 1 modelCON ( 1 , 1 ) . G0 = modelCON ( 1 , 1 ) . G = elseif .6 * modelCON ( 1 , 1 ) .GH; max ( modelCON ( 1 OPTIONS . CVX == 1 , 1 ) . GH, modelCON ( 1 , 1 ) . G ) ; , 1 ) . G+modelCON ( 1 , 1 ) . G0 ) ; & OPTIONS .TRM == 0 modelCON ( 1 , 1 ) . G1 ( 1 : t r u n c ) % G here i s r e a l l y G1 = modelCON ( 1 , 1 ) . G1 ( 1 : t r u n c ) +... * OPTIONS .DGP SOLN ( 1 , 1 ) .DG( 1 : t r u n c ) ' ; modelCON ( 1 , 1 ) . G1 ( f i n d ( modelCON ( 1 f i n d ( modelCON ( 1 modelCON ( 1 , 1 ) . G1 ( if ?1e 1 0 ) * 1 0 ^ 8 ; , 1 ) . G1< f i n d ( modelCON ( 1 f i n d ( modelCON ( 1 ?1e 1 0 ) ) = . . . , 1 ) . G1< , 1 ) . G1>1 e 1 0 ) , 1 ) . G1>1 e 1 0 ) ) = . . . * 1 0 e ?8; OPTIONS . a >10 OPTIONS . b = . 1 ; end OPTIONS . a=OPTIONS . a elseif if *OPTIONS . b ; OPTIONS .TRM == 1 | OPTIONS .TRM == 2 %Updata the t r u s t region radius SOLN ( 1 , 1 ) . r h o < . 2 5 SOLN ( 1 , 1 ) . d e l t a = . 2 5 else if *norm (SOLN ( 1 SOLN ( 1 , 1 ) . r h o > . 7 5 & , 1 ) .DG ) ; norm (SOLN ( 1 , 1 ) .DG)==SOLN ( 1 , 1 ) . d e l t a SOLN ( 1 , 1 ) . d e l t a=min (SOLN ( 1 , 1 ) . d e l t a 0 , 2 * SOLN ( 1 end end if OPTIONS .TRM == 1 & SOLN ( 1 , 1 ) . r h o> OPTIONS . n1 477 ,1). delta ); modelCON(1 ,1).G1(1: trunc ) = modelCON(1 ,1).G1(1: trunc ) +... OPTIONS.DGP*SOLN(1 ,1).DG(1: trunc ) ' ; modelCON(1 ,1).G1( f i n d (modelCON(1 ,1).G1<?1e10))= ?1e10 ; modelCON(1 ,1).G1( f i n d (modelCON(1 ,1).G1>1e10))=1e10 ; e l s e i f OPTIONS.TRM == 2 & SOLN(1 ,1). rho>= OPTIONS. n1 modelCON(1 ,1).G(1: trunc ) = modelCON(1 ,1).G(1: trunc ) +... OPTIONS.DGP*SOLN(1 ,1).DG(1: trunc ) ' ; modelCON(1 ,1).G( f i n d (modelCON(1 ,1).G<1e ?1))=1e ? 1; modelCON(1 ,1).G( f i n d (modelCON(1 ,1).G>1.3))=1.3; else disp ( 'GSTDD error ' ) return end else disp ( 'GSTDD error ' ) return end 478 Appendix B Derivation of longitudinal vibration modes. The equations of motion for the longitudinal vibration of the system are 2 ? 2 U1 2 ? U1 ? c = 0 ?t2 ?x21 2 ? 2 U2 2 ? U2 ? c = 0 ?t2 ?x22 (B.0.1) (B.0.2) and the boundary conditions are stated as where c2 = EA m Ms U1 (0, t) = 0 (B.0.3) U1 (L1 , t) = U2 (0, t) (B.0.4) ? 2 U1 ?U1 ?U2 (L ) = ?EA (L , t) + EA (0, t) 1,t 1 ?t2 ?x1 ?x2 479 (B.0.5) M where Ms = ? 2 U2 ?U2 (L2 , t) = ?EA (L2 , t) 2 ?t ?x2 (B.0.6) I is the apparent mass of the sheave. R2 The free vibrations are expressed as where U1 (x1 , t) = Y1 (x1 ) f (t) (B.0.7) U2 (x2 , t) = Y2 (x2 ) f (t) (B.0.8) Y1 and Y2 are the mode shapes (eigenfunctions) and f (t) = Acos (?t)+Bsin (?t)is a harmonic function of time. Inserting Eqs. B.0.7 and B.0.8 into Eqs. B.0.1 and B.0.2 yields Y1 (x1 ) + ? 2 Y1 (x1 ) = 0 Y2 (x2 ) + ? 2 Y2 (x2 ) = 0 where ?2 = ? 2 c From Eq. B.0.3: (B.0.9) (B.0.10) . It is evident that Y1 (x1 ) = A1 cos (?X1 ) + B1 sin (?x1 ) (B.0.11) Y2 (x2 ) = A2 cos (?X2 ) + B2 sin (?x2 ) (B.0.12) A1 = 0 so that Y1 (x1 ) = B1 sin (?x1 ) (B.0.13) B1 sin (?x1 ) ? A1 = 0 (B.0.14) From Eq. B.0.4: 480 The boundary condition Eq. B.0.5 yields ! " ? 2 Ms sin (?L1 ) ? cos (?L1 ) B1 + B2 = 0 EA? (B.0.15) Also, the boundary condition Eq. B.0.6 gives ! " ! 2 " ?2M ? M cos (?L2 ) + ?sin (?L2 ) A2 + sin (?L2 ) + ?cos (?L2 ) B2 EA EA (B.0.16) Thus, combining Eqs. B.0.14, B.0.15, and B.0.16 yields the following ? ?? ?1 s1 ? ? ? ? ? ? 2 M2 s EA? 1 ? c1 0 0 ?2 M c EA 1 0 ? + ?s2 ? ? B1 ? ?? ? ?? A ? = 0 1 ?? 2 ? ?? ? 2 ? M s ? ?c B 2 2 EA 2 (B.0.17) where s1 ? sin (?L1), s2 ? sin (?L2), c1 ? cos (?L1), and c2 ? cos (?L2). Thus, for a non-trivial solution the following must be satised ? ? ? ? det ? ? ? s1 ?2 M 2 EA? s1 ? c 1 0 ?1 0 0 1 ?2 M c EA 1 + ?s2 ?2 M s EA 2 ? ?c2 ? ? ?=0 ? ? (B.0.18) which yields the frequency equation ! ? sin (?L1 ) " ! " Ms M 2 ? cos (?L2 ) + ?sin (?L2 ) + ?sin (?L1 ) ? ?cos (?L1 ) (и B.0.19) m m ! " M 2 ? sin (?L2 ) ? ?sin (?L2 ) = 0 m 481 Appendix C Derivation of bending vibration modes. In the following analytical models the inclined and vertical sections of the rope are modeled as being clamped-clamped and clamped with an end mass, respectively. The equation of motion for bending vibrations of an axially loaded beam is [139] ? 4V P ? 2 V (x, t) ? ? 2 V (x, t) + + =0 ?x4 EI ?x2 EI ?t2 where V is the displacement with respect to the position along the beam t. The bending stiness is and ? EI , the load P (C.0.1) x and time is considered positive for compressive loads, is the mass per unit length of the beam. Assuming V to have the form [139] V (x, t) = ? (x) sin (?t) (C.0.2) 2 d4 ? (x) 2 d ? (x) +k ? ? 4 ? (x) = 0 4 2 dx dx (C.0.3) yields 482 where P EI ?? 2 = EI k2 = ?4 To simplify the analysis more the equation can be nondimensionalized using the length of the beam l which gives [139] ? where ? = ?l ,x = xl ,? = ?l , and 2 (C.0.4) 4 (x) + k ? ? ? ? (x) = 0 k = kl. Assuming the mode shapes to be of the same form as a beam without tension gives ? (x) = Acosh (?1 x) + Bsinh (?1 x) + Ccos (?2 x) + Dsin (?2 x) (C.0.5) where ? ?1 = ?? 2 k + 2 # 4 ?1/2 k 4 +? ? 4 ? and ?2 = ? 2 # k + 2 4 ?1/2 k 4 +? ? 4 (C.0.6) are the real roots of the characteristic equation [139]. The boundary conditions for a clamped end are ? = 0 and v = 0 Applying the boundary conditions at x = 0 and x = 1and 483 (C.0.7) rearranging the equations nto matrix form gives i ? ? ? ? ? ? ? ? ? ? A ? ?? ? ?? ?? B ? 0 ?1 0 ?2 ? ?? ?=0 ?? ? C ? cosh (?1 ) sinh (?1 ) cos (?2 ) sin (?2 ) ? ? ?? ? ?? D ?1 sinh (?1 ) ?1 cosh (?1 ) ??2 sin (?2 ) ?2 cos (?2 ) 1 0 1 0 ?? (C.0.8) to avoid the trivial solution we take the derivative of the coecient matrix which with simplication yields [139] 2 2 2? (1 ? cosh (?1 ) cos (?2 )) ? k sinh (?1 ) sin (?2 ) = 0 (C.0.9) For the cl

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