Simulation of Flexible Aircraft by Humayoon Abbasi A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto c 2010 by Humayoon Abbasi Copyright Library and Archives Canada Bibliothèque et Archives Canada Published Heritage Branch Direction du Patrimoine de l’édition 395 Wellington Street Ottawa ON K1A 0N4 Canada 395, rue Wellington Ottawa ON K1A 0N4 Canada Your file Votre référence ISBN: 978-0-494-72367-8 Our file Notre référence ISBN: 978-0-494-72367-8 NOTICE: AVIS: The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. . The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author’s permission. L’auteur a accordé une licence non exclusive permettant à la Bibliothèque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par télécommunication ou par l’Internet, prêter, distribuer et vendre des thèses partout dans le monde, à des fins commerciales ou autres, sur support microforme, papier, électronique et/ou autres formats. L’auteur conserve la propriété du droit d’auteur et des droits moraux qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation. In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis. Conformément à la loi canadienne sur la protection de la vie privée, quelques formulaires secondaires ont été enlevés de cette thèse. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Bien que ces formulaires aient inclus dans la pagination, il n’y aura aucun contenu manquant. Abstract Simulation of Flexible Aircraft Humayoon Abbasi Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto 2010 This study aims to improve flight simulation of flexible aircraft. More specifically, this thesis concentrates on comparing two flexible aircraft flight simulation models. Both modeling techniques considered use the same aircraft structural and aerodynamic data provided by the aircraft manufacturer. Simulation models were developed and tested using a number of control inputs in both longitudinal and lateral dimensions. Time history responses from the simulations were compared. The effect of increasing the flexibility of the aircraft model was also studied on both models. It was found that the two models produce very similar results for the original aircraft stiffness case. However, the lateral response of the two models diverges as the stiffness is lowered. A number of recommendations are made for further testing and research, based on the conclusions of the study. ii Acknowledgements First and foremost, I would like to acknowledge my thesis supervisor, Dr. Peter R. Grant, without whose assistance none of this work would have been possible. I am very grateful for his supervision throughout my research and study as well as for his patience and understanding. I would also like to express my thanks to Nestor X. Li for all his assistance in developing and troubleshooting the simulations as well as responding to my numerous requests in a timely fashion. I would also like to thank my friends M. Umer Ahmed, Terrence Fung, Andrew Sun, Amir Naseri, Steacie Liu, Bruce Haycock and others who made my time spent at UTIAS pleasant and enjoyable. Additionally, I would like to acknowledge the financial support of National Science and Engineering Research Council with their Canada Graduate Scholarship which made my graduate life much more comfortable than it would have been otherwise. Last but not least, I would like to thank my family for their continuous support and patience during my time of study at University of Toronto. As well, I am very grateful to my wife, Isra, for her contribution in prolonging my Masters. I dedicate this to you. Humayoon Abbasi iii Contents 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background 3 2.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Background Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Aerodynamic Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Natural Vibration Modes of the Aircraft . . . . . . . . . . . . . . 6 3 Methodology 9 4 Mean Axes Model 10 5 Fixed Axes Model 18 6 Results and Discussion 25 6.1 6.2 Comparison of Mean and Fixed Axes Models . . . . . . . . . . . . . . . . 25 6.1.1 Response to Elevator Impulse . . . . . . . . . . . . . . . . . . . . 25 6.1.2 Response to Elevator 2311 . . . . . . . . . . . . . . . . . . . . . . 30 6.1.3 Response to an Aileron Impulse . . . . . . . . . . . . . . . . . . . 34 6.1.4 Response to Aileron 2311 . . . . . . . . . . . . . . . . . . . . . . 39 . . . . . . . . . . . . . . . . . . . . . 42 6.2.1 Response to Elevator Impulse . . . . . . . . . . . . . . . . . . . . 42 6.2.2 Response to Elevator 2311 . . . . . . . . . . . . . . . . . . . . . . 53 6.2.3 Response to Aileron Impulse . . . . . . . . . . . . . . . . . . . . . 60 6.2.4 Response to Aileron 2311 70 Effect of Varying Aircraft Stiffness . . . . . . . . . . . . . . . . . . . . . . iv 7 Conclusions and Recommendations 78 References 80 v List of Figures 4.1 Inertial position of a mass element . . . . . . . . . . . . . . . . . . . . . 11 5.1 Component reference frames . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.1 Pitch rate time history of aircraft response to elevator impulse . . . . . . 26 6.2 A closer look at pitch rate time history just after the input . . . . . . . . 27 6.3 Pitch angle time history of aircraft response to elevator impulse . . . . . 27 6.4 X velocity time history of aircraft response to elevator impulse . . . . . . 28 6.5 Acceleration time history of aircraft response to elevator impulse . . . . . 28 6.6 Acceleration time history focusing on the response immediately after elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 29 Wingtip relative deflection time history of aircraft response to elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.8 Pitch rate time history of aircraft response to elevator 2311 . . . . . . . . 30 6.9 Pitch angle time history of aircraft response to elevator 2311 . . . . . . . 31 6.10 X velocity time history of aircraft response to elevator 2311 . . . . . . . . 32 6.11 Acceleration time history of aircraft response to elevator 2311 . . . . . . 32 6.12 Wingtip relative deflection time history of aircraft response to elevator 2311 33 6.13 Roll rate time history of aircraft response to aileron impulse . . . . . . . 34 6.14 A closer look at the roll rate time history immediately proceeding the input 35 6.15 Roll angle time history of aircraft response to aileron impulse . . . . . . . 36 6.16 Y velocity time history of aircraft response to aileron impulse . . . . . . 36 6.17 Acceleration time history of aircraft response to aileron impulse . . . . . 37 6.18 Focusing on acceleration time history response just after the input . . . . 37 6.19 Wingtip relative deflection time history of aircraft response to aileron impulse 38 6.20 Roll rate time history of aircraft response to aileron 2311 . . . . . . . . . 39 6.21 Roll angle time history of aircraft response to aileron 2311 . . . . . . . . 40 vi 6.22 Y velocity time history of aircraft response to aileron 2311 . . . . . . . . 40 6.23 Acceleration time history of aircraft response to aileron 2311 . . . . . . . 41 6.24 Wingtip relative deflection time history of aircraft response to aileron 2311 41 6.25 Pitch rate time history of mean axes model’s response to elevator impulse 43 6.26 Pitch rate time history of fixed axes model’s response to elevator impulse 44 6.27 A closer look at the pitch rate time history of mean axes model immediately after the elevator impulse . . . . . . . . . . . . . . . . . . . . . . . 45 6.28 A closer look at the pitch rate time history of fixed axes model immediately proceeding the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.29 Pitch angle time history of mean axes model’s response to elevator impulse 46 6.30 Pitch angle time history of fixed axes model’s response to elevator impulse 46 6.31 X velocity time history of mean axes model’s response to elevator impulse 47 6.32 X velocity time history of fixed axes model’s response to elevator impulse 47 6.33 X acceleration time history of mean axes model’s response to elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.34 X acceleration time history of fixed axes model’s response to elevator impulse 48 6.35 X acceleration time history of mean axes model: focusing on the response immediately proceeding the input . . . . . . . . . . . . . . . . . . . . . . 49 6.36 X acceleration time history of fixed axes model immediately proceeding the elevator impulse input . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.37 Z acceleration time history of mean axes model’s response to elevator impulse 50 6.38 Z acceleration time history of fixed axes model’s response to elevator impulse 50 6.39 Focusing on the Z acceleration time history of mean axes model immediately after the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.40 Focusing on the Z acceleration time history of fixed axes model immediately after the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.41 Wingtip relative deflection time history of mean axes model’s response to elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.42 Wingtip relative deflection time history of fixed axes model’s response to elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.43 Pitch rate time history of mean axes model’s response to elevator 2311 . 53 6.44 Pitch rate time history of fixed axes model’s response to elevator 2311 . . 54 6.45 Pitch angle time history of mean axes model’s response to elevator 2311 . 55 6.46 Pitch angle time history of fixed axes model’s response to elevator 2311 . 55 vii 6.47 X velocity time history of mean axes model’s response to elevator 2311 . 56 6.48 X velocity time history of fixed axes model’s response to elevator 2311 . . 56 6.49 X acceleration time history of mean axes model’s response to elevator 2311 57 6.50 X acceleration time history of fixed axes model’s response to elevator 2311 57 6.51 Z acceleration time history of mean axes model’s response to elevator 2311 58 6.52 Z acceleration time history of fixed axes model’s response to elevator 2311 58 6.53 Wingtip relative deflection time history of mean axes model’s response to elevator 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.54 Wingtip relative deflection time history of fixed axes model’s response to elevator 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.55 Roll rate time history of mean axes model’s response to aileron impulse . 60 6.56 Roll rate time history of fixed axes model’s response to aileron impulse . 61 6.57 Fousing on roll rate time history of mean axes model just after the aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.58 Focusing on roll rate time history of fixed axes model just after the aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.59 Roll angle time history of mean axes model’s response to aileron impulse 63 6.60 Roll angle time history of fixed axes model’s response to aileron impulse . 63 6.61 Y velocity time history of mean axes model’s response to aileron impulse 64 6.62 Y velocity time history of fixed axes model’s response to aileron impulse 64 6.63 X acceleration time history of mean axes model’s response to aileron impulse 65 6.64 X acceleration time history of fixed axes model’s response to aileron impulse 65 6.65 Focusing on X acceleration time history of mean axes model proceeding the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.66 Focusing on X acceleration time history of fixed axes model proceeding the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.67 Y acceleration time history of mean axes model’s response to aileron impulse 67 6.68 Y acceleration time history of fixed axes model’s response to aileron impulse 67 6.69 Focusing on Y acceleration time history of mean axes model immediately after aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.70 Focusing on Y acceleration time history of fixed axes model immediately after aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.71 Wingtip relative deflection time history of mean axes model’s response to aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 69 6.72 Wingtip relative deflection time history of fixed axes model’s response to aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.73 Roll rate time history of mean axes model’s response to aileron 2311 . . . 70 6.74 Roll rate time history of fixed axes model’s response to aileron 2311 . . . 71 6.75 Roll angle time history of mean axes model’s response to aileron 2311 . . 72 6.76 Roll angle time history of fixed axes model’s response to aileron 2311 . . 72 6.77 Y velocity time history of mean axes model’s response to aileron 2311 . . 73 6.78 Y velocity time history of fixed axes model’s response to aileron 2311 . . 73 6.79 X acceleration time history of mean axes model’s response to aileron 2311 74 6.80 X acceleration time history of fixed axes model’s response to aileron 2311 74 6.81 Y acceleration time history of mean axes model’s response to aileron 2311 75 6.82 Y acceleration time history of fixed axes model’s response to aileron 2311 75 6.83 Wingtip relative deflection time history of mean axes model’s response to aileron 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.84 Wingtip relative deflection time history of fixed axes model’s response to aileron 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 76 Nomenclature In order of appearance, grouped by section: c chord length CLα lift curve slope q dynamic pressure α angle of attack ρ air density Vx longitudinal body frame component of velocity Vz vertical body frame component of velocity CD0 drag coefficient corresponding to zero angle of attack k Oswald efficiency factor cs lateral chord length Csβ slope of the lateral force curve qs dynamic pressure of the lateral force β angle of sideslip Vy lateral body frame component of velocity x L the Lagrangian T kinetic energy of system U potential energy of system W external work done on system (•)(e) signifies application to an element ψ displacement of a finite element n vector of element shape functions d vector of the nodal displacements of the elements ρ density of element f body force vector m mass k stiffness B matrix consisting of shape function derivatives D constitutive matrix relating element stress to strain σ element stress ε element strain M global mass matrix K global stiffness matrix F(t) vector of external forces, possibly time varying ωi2 eigenvalues representing the natural frequencies φi eigenvectors representing the natural mode shapes xi ρdV mass element with volume dV R̄ inertial position of an element p̄ position of the element relative to a local reference system R̄0 position of the local reference system relative to the inertial frame ω̄ angular velocity of local reference frame relative to inertial reference frame Ug gravitational potential energy ḡ acceleration due to gravity s̄ (x, y, z) d¯(x, y, z, t) undeformed position of mass element Ue d¯ deformed position of a mass element strain energy displacement δ d¯ δt rate of displacement x̄cm position of the centre of mass relative to local reference frame M total mass of elastic body [I] inertia tensor of elastic body φi (x, y, z) mode shapes ηi (t) generalized coordinates Mi generalized mass of the i th mode ωi natural vibration frequency of the i th mode î, ĵ and k̂ unit vectors in each of the coordinate directions of the local reference frame φ, θ and ψ Euler angles commonly used in rigid body aircraft equations of motion Qi generalized force δW virtual work Qηi generalized forces for the elastic equations αv angle of attack of the aircraft φi b elastic displacement of a section due to bending mac aerodynamic moment per unit span about the line of aerodynamic centres αs local angle of attack of an aerodynamic section xii Cf transformation matrix Ef time rate of change of the transformation matrix V̄f velocity of a point on the fuselage Vf translational velocity vector for reference frame xf yf zf ωf angular velocity vector for reference frame xf yf zf rf position vector associated with the mass element dmf uf elastic displacement vector vf bending velocity αf torsion velocity r̃f skew-symmetric matrix corresponding to rf ũf skew-symmetric matrix corresponding to uf (•)i component type, (•)f for fuselage, (•)w for wing and (•)e for empennage rf i radius vector from xf yf zf to xi yi zi L̂i Lagrangian density for component i, not including the strain energy F̂ui Rayleigh’s dissipation function density for component i F̂ψi Rayleigh’s dissipation function density for component i Lui stiffness differential operator matrix for component i Lψi stiffness differential operator matrix for component i F net force vector acting on the aircraft in terms of fuselage body axes M net moment vector acting on the aircraft in terms of fuselage body axes Ûi net force density vector for component i Ψ̂i net moment density vector for component i S̃ matrix of first moments of inertia of the deformed aircraft J matrix of second moments of inertia of the deformed aircraft, simply referred to as the inertia matrix Cui and Cψi damping matrices p momentum vector for the entire aircraft Kui bending stiffness matrix Kψi torsional stiffness matrix fi net distributed forces over the component i δ spatial Dirac delta function rE position vector of engine FE δ (r − rE ) engine thrust xiii Chapter 1 Introduction Traditionally, aircraft simulation models assume the aircraft structure to be rigid and thus the aircraft is modeled as a rigid-body. This allows the aircraft equations of motion to be simplified greatly. Most of the flight simulation models make use of this assumption and the results are acceptable for a majority of aircraft present today. 1.1 Motivation With advances in material sciences, aircraft manufacturers are increasingly making use of lighter materials to save weight and as a result cut down on fuel costs and improve performance. One disadvantage of using lighter materials is an increase in the flexibility of the aircraft structures. The resulting aircraft have structures that are considerably more deformable even when encountering forces within their design envelopes and thus cannot be treated as rigid-bodies for flight simulation purposes. As a result of these developments, accurate modeling tools for modeling the dynamic response of flexible aircraft have become ever more important. Additionally, to allow pilot-in-the-loop simulations, an added constraint is placed on these techniques that they have to have sufficiently low computational complexity to enable real-time simulations using readily available hardware. 1.2 Objectives The primary goal of this effort is to identify a modeling technique for simulating dynamic flight response of a flexible aircraft. Additionally, the following sub-objectives are also 1 Chapter 1. Introduction 2 being pursued: 1. Studying the effect of varying the number of structural modes retained in the simulation model. For the model to be exact, an infinite number of structural dynamics modes is required. However, as this is impossible to achieve in practice, only a small number of modes are retained as dictated by the accuracy required, the frequency response for the flight task being flown, and the computing resources available. 2. Studying the effect of varying the flexibility of the aircraft structure. As the aircraft structure becomes increasingly flexible, the structural response becomes increasingly nonlinear and the overall system response departs farther away from that of the rigid model. Chapter 2 Background 2.1 Literature Review Traditionally, the fields of flight dynamics and aeroelasticity have been developed as separate disciplines. Even though the need for integrating the two fields was recognized long ago, not much work was done to fulfill this need in the early years due to a significant increase in the complexity of the resulting problem. Also, the lack of computing resources available at the time required the use of simplifying assumptions which allowed for analytical solutions to be found. One of the very first integrated analytical models for the dynamics of flexible aircraft is found in a two-part report by Milne [5]. In the first part, a linear flexible flight dynamics model was derived. The model consisted of linear equations about a steady state and assumed both rigid body motions as well as elastic deformations to be small. Mean axes (refer to Ch. 4 for a description of mean axes) constraints were specified, however, the formulation used body axes attached to the undeformed aircraft instead of mean axes which can change as the aircraft deforms. In the second part, longitudinal stability analysis was performed on the aircraft model. Taylor and Woodcock [8] simplified Milne’s [5] formulation into scalar form and used it to study selected special cases. Buttril, Zeiler and Arbuckle [1] developed a flight dynamics model for flexible aircraft by integrating nonlinear rigid body mechanics and linear structural dynamics using Lagrangian mechanics. The model made use of mean axes constraints and undamped natural vibration modes. The equations included nonlinear inertial coupling terms between angular and elastic momenta. The result of these coupling terms, however, was shown to be small. A steady-state, nonlinear aerodynamic model, augmented by an un3 Chapter 2. Background 4 steady aerodynamic model was used to calculate the aerodynamic loads. Waszak and Schmidt [10] used Langrange’s equations to derive the equations of motion for a flexible aircraft. The equations included nonlinear rigid-body motion equations, as well as linear equations for elastic degrees of freedom which were expressed in terms of natural vibrational modes of the aircraft. Inertial decoupling of the rigid-body motions and the elastic deformations was achieved by applying mean axes constraints. Strip theory was used to compute the aerodynamic loads which were then converted to generalized forces, ultimately coupling the rigid-body and elastic degrees of freedom. The resulting equations were very similar in appearance to the Euler equations for rigid-body motion, except for the augmented structural equations. Consequently, these equations have been very appealing for use in real-time flight simulation of flexible aircraft. Waszak, Buttril and Schmidt [9] presented a number of techniques to simplify flexible aircraft models, such as residualization and truncation of modes. Meirovitch and Tuzcu [4] have been critical of the use of mean axes in deriving the equations of motion for flexible aircraft. As these equations tend to be quite complex, earlier investigators looking to derive analytical models used the mean axes to help inertially decouple the system. However, Meirovitch and Tuzcu [4] argue that in a lot of these investigations, the additional constraints imposed by the use of mean axes were not fully satisfied. Additionally, if one were to satisfy these constraints, the resulting model would be more complicated than what could be derived without the use of mean axes. Meirovitch and Tuzcu [3] developed a unified formulation of the equations of motion in terms of quasi-coordinates. The model included hybrid (ordinary and partial) differential equations of motion which were derived without the use of mean axes. Strip theory was used to compute the aerodynamic forces; however, the need for a more accurate aerodynamic model was alluded to. The resulting state equations were functions of quasi-velocities, as well as generalized coordinates, velocities, and forces. A numerical example for a model of a flexible aircraft was presented and two flight dynamics problems were studied: the steady level cruise and the steady level turn manoeuvre. A controller consisting of a linear quadratic regulator was designed to alleviate gust loads and ensure flight stability. 5 Chapter 2. Background 2.2 2.2.1 Background Theory Aerodynamic Loads Aerodynamic forces constitute a major part of the total external forces that act on an aircraft. Therefore, in order to simulate the flight of an aircraft, it is required that the aerodynamic loading be computed at each time step of the simulation. There are a number of aerodynamic theories available which vary in terms of accuracy and computation costs. Usually the deciding factor in choosing one aerodynamic theory over another is the computation resources required for implementing that theory. Strip theory is an aerodynamic theory that provides a reasonable estimate of aerodynamic loading on an aircraft without being too resource-intensive. This is particularly important for simulation of flexible aircraft as the structure of a flexible aircraft can deform which can alter the aerodynamic loads exerted on the aircraft. Thus, it is important to have a quick way of estimating the aerodynamic loads on the deformed aircraft at each time step of the simulation. Strip theory regards all components of the aircraft as two-dimensional surfaces. For horizontal surfaces, the lift force per unit span is given by l = qcCLα α (2.1) where c is the chord length, CLα is the lift curve slope, q is the dynamic pressure and α is the angle of attack. The dynamic pressure is given by 2 2 q = 21 ρ(V x + V z ) (2.2) where ρ is the air density and V x and V z are the longitudinal and vertical components of the velocity in the component body frame, respectively. α = tan−1 (V z V x ) (2.3) is the expression for computing the angle of attack. In a similar fashion, the drag force per unit span is given by 2 d = qcCD = qc CD0 + kCL2 = qc CD0 + kCLα α2 (2.4) where CD0 is the drag coefficient corresponding to zero angle of attack and k is a constant dependent on efficiency. For vertical surfaces, the side force per unit span can be written as s = qs cs Csβ β (2.5) 6 Chapter 2. Background where cs is the lateral chord length, Csβ is the slope of the lateral force curve, qs is the dynamic pressure of the lateral force and β is the angle of sideslip. The lateral dynamic pressure can be expressed as 2 2 qs = 12 ρ(V x + V y ) (2.6) β = tan−1 (V y V x ) (2.7) and the sideslip angle as where V x and V y are the longitudinal and lateral components of velocity of the aircraft. The aerodynamic forces on the aircraft components can be written in the vector form as l sin α − d cos α (2.8) 0 fa = −l cos α − d sin α and the lateral aerodynamic force vector as s sin β fs = −s cos β 0 . (2.9) Typically, the lift, drag and lateral force per unit span are applied at the line of aerodynamic centers. Thus, the above mentioned force vectors may be integrated to obtain the net aerodynamic forces and moments, using the line of aerodynamic centers as the domain of integration. 2.2.2 Natural Vibration Modes of the Aircraft Developing the mean-axes flight dynamics model of a flexible aircraft requires knowledge of the aircraft’s structural modes. Silveira [7] performed modal analysis on the structural model of a small business jet aircraft in ANSYS, which will be used in this project. The following is a summary of the theory behind the modal analysis. The Extended Hamilton’s Principle can be used to express the motion of a system over time as Zt2 δ Zt2 Ldt + t1 Zt2 (T − U + W )dt = 0 δW dt = δ t1 (2.10) t1 where L = T − U is the Lagrangian and T , U and W are the kinetic energy, potential energy and external work done on the system. Representing the structure by an equivalent finite element model, Equation (2.10) can be applied to each element of the model, 7 Chapter 2. Background as Zt2 δ (T (e) − U (e) + W (e) )dt = 0 (2.11) t1 where the superscript (e) signifies application to an element. Assuming the displacement of each finite element, ψ, to be given by ψ = nT d (2.12) where n is a vector of element’s shape functions and d is a vector of the nodal displacements of the elements, then expressions can be written for the kinetic and potential energy and work, as follows: T (e) = 1 2 Z ρψ̇ T ψ̇dV = 21 ḋT mḋ (2.13) V U (e) = 12 dT kd (2.14) W (e) = dT f (2.15) where ρ is the element’s density, f is the body force vector and Z m = ρNT NdV (2.16) V Z k= BT DBdV (2.17) V are the mass and stiffness, respectively, wherein B is the matrix consisting of shape function derivatives and D is the constitutive matrix relating element stress, σ, to strain, ε, as follows: σ = Dε. (2.18) Inserting Eqs. (2.13) - (2.17) in the Hamiltonian equation, Eq. (2.11), the structural dynamic equation for an element can be derived as md̈ + kd = f . (2.19) Similarly, applying the Hamilton’s Principle to the whole finite element model yields Md̈ + Kd = F(t) (2.20) 8 Chapter 2. Background where M and K are the global mass and stiffness matrices, respectively, and F(t) is the vector of external forces, which may vary with time. The natural vibration modes of an elastic structure and their corresponding frequencies can be found by determining the homogeneous solution to Eqn. (2.20). Assuming a solution of the form d = φeiωt (2.21) −ω 2 Mφeiωt + Kφeiωt = 0 (2.22) we get which may be resolved by solving the generalized eigenvalue problem given below: K − ω 2 M φ = 0. (2.23) This can be solved by performing an eigenvalue analysis to obtain the eigenvalues, ωi2 , and the eigenvectors, φi , which represent the natural frequencies and mode shapes, respectively, of the vibrating structure. These mode shapes are used in the dynamic modeling of flexible aircraft. Chapter 3 Methodology In order to achieve the objectives for this project as laid out in Sec. 1.2, the following steps were performed: 1. Find data for a real aircraft, including structural, aerodynamic and geometric data. 2. Find the natural vibration modes of the aircraft. 3. Develop the rigid-body simulation model. 4. Develop the mean axes flexible simulation model of the aircraft. 5. Develop the fixed axes flexible aircraft simulation model. 6. Compare time histories of the response of the flexible models and the rigid-body model to various inputs. 7. Compare the effect of varying the elasticity of the aircraft structure, between the two flexible models. 8. Based on the above results, make recommendations for future work to be done in the field of flexible aircraft modeling and simulation. 9 Chapter 4 Mean Axes Model The simplified mean axes model for flight dynamics of flexible aircraft, described in this chapter, is based on the formulation developed by Waszak and Schmidt [10]. It derives nonlinear equations of motion for a deformable aircraft from first principles (Lagrange’s equation and Principle of Virtual Work). Inertial decoupling of the rigid-body and elastic momenta is achieved by choosing a particular form of local body reference axes, known as mean-axes, and with the use of the aircraft’s natural vibration modes. The resulting equations of motion are presented in scalar form, as a set of six nonlinear rigid-body equations of motion and n linear equations governing the elastic degrees of freedom, where n is the number of structural modes retained in the model. Consider the mass element, ρdV , of an elastic body shown in Fig. 4.1. The inertial position of this element is given by R̄ = R̄0 + p̄ (4.1) where p̄ is the position of the element relative to a local reference system oxyz and R̄0 is the position of the local reference system relative to the inertial frame OXY Z. The kinetic energy of the whole body is 1 T = 2 Z dR̄ dR̄ · ρdV dt dt (4.2) V where, treating each mass element as a point mass, the derivative terms are given by dR̄ dR̄0 δ p̄ = + + ω̄ × p̄ dt dt δt in which δ δt (4.3) denotes time derivative w.r.t. the local reference frame (attached to the body) and ω̄ is the angular velocity of the local reference frame relative to the inertial reference 10 11 Chapter 4. Mean Axes Model ρdV x p̄ y o R̄ z R̄0 O X Y Z Figure 4.1: Inertial position of a mass element frame. Inserting the above into the expression for the kinetic energy yields: Z 1 dR̄0 dR̄0 dR̄0 δ p̄ δ p̄ δ p̄ δ p̄ T = · +2 · + · + 2 · (ω̄ × p̄) + (ω̄ × p̄) · (ω̄ × p̄) 2 dt dt dt δt δt δt δt V dR̄0 +2 (ω̄ × p̄) · ρdV . (4.4) dt The potential energy of an elastic body is composed of the gravitational potential energy and the strain energy. The gravitational potential energy can be expressed as Z Ug = − ḡ · R̄0 + p̄ ρdV (4.5) V where ḡ is the acceleration due to gravity. To derive the expression for the strain energy, lets represent the position of a mass element in an elastic body as the sum of its undeformed position, s̄ (x, y, z), and its deformed position, d¯(x, y, z, t) (note that the 12 Chapter 4. Mean Axes Model deformed position is a function of time), as shown: p̄ = s̄ + d¯ (4.6) Then the strain energy can be written as (see Ref. [10]) Z 2¯ δ d ¯ 1 Ue = − · dρdV . 2 δt2 (4.7) V In order to simplify the equations of motion of an elastic body, a local reference system is used such that there is no inertial coupling between the elastic degrees of freedom and the rigid-body degrees of freedom. Such a reference system is referred to as a “mean axes” system. The mean axes are defined by the following condition: Z Z δ p̄ δ p̄ ρdV = p̄ × ρdV = 0 δt δt V (4.8) V which means that the linear and angular momenta due to elastic deformations are zero at every instant. These exact mean axes constraints are often difficult to apply, however, they can be used to derive the so-called “practical mean axes conditions” which are much easier to apply. Recalling Eq. (4.6) and the fact that only the deformed part of the position vector is time varying, the above equation can be reduced to Z Z ¯ δ d¯ δd s̄ + d¯ × ρdV = 0. ρdV = δt δt (4.9) V V ¯ and Assuming structural deformation to be small, the product of the displacement, d, displacement rate, δ d¯ , δt can be ignored. Also, assuming the mass density of each element to be constant, Eq. (4.9) simplifies to Z Z δ δ ¯ ¯ dρdV = s̄ × dρdV =0 δt δt V (4.10) V These are the practical mean axes constraints which will simplify the equations of motion for our problem. By applying the mean axes constraints derived above, the expression for kinetic energy of the system (Eq. (4.4)) can be simplified as follows. Applying Eq. (4.8) to the second term of the kinetic energy expression, we get Z Z dR̄0 δ p̄ dR̄0 δ p̄ · ρdV = · ρdV ∼ =0 dt δt dt δt V V (4.11) 13 Chapter 4. Mean Axes Model Similarly, the fourth term becomes Z Z δ p̄ δ p̄ · (ω̄ × p̄) ρdV = p̄ × ρdV · ω̄ ∼ = 0, δt δt (4.12) V V resulting in the following reduced expression for the kinetic energy of the system: Z 1 dR̄0 dR̄0 δ p̄ δ p̄ dR̄0 T = · + · + (ω̄ × p̄) · (ω̄ × p̄) + 2 (ω̄ × p̄) · ρdV . (4.13) 2 dt dt δt δt dt V Further simplification of the model can be done by selecting the origin of the local reference frame to lie at the instantaneous centre of mass of the elastic body, as expressed below: R p̄ρdV V x̄cm = R =0 ρdV (4.14) V where x̄cm is the position of the centre of mass relative to the local reference frame. This results in the last term in the kinetic energy expression, Eq. (4.4), to become zero, as follows: Z dR̄0 ρdV = ω̄ × (ω̄ × p̄) · dt Z p̄ρdV · dR̄0 = 0. dt (4.15) V V The kinetic energy expression now reduced to Z 1 dR̄0 dR̄0 δ p̄ δ p̄ T = · + · + (ω̄ × p̄) · (ω̄ × p̄) ρdV , 2 dt dt δt δt (4.16) V which can also be written as 1 dR̄0 dR̄0 1 T 1 T = M · + ω̄ [I] ω̄ + 2 dt dt 2 2 Z δ p̄ δ p̄ · ρdV δt δt (4.17) V where M is the total mass of the elastic body and [I] is the inertia tensor. Note, in this formulation, the inertia tensor is assumed to be constant since the elastic deformations that would generally cause variations in the inertia tensor, are assumed to be small. Additionally, using Eq. (4.14), the gravitational potential energy can be simplified as Z Z Z Ug = − ḡ · R̄0 + p̄ ρdV = −R̄0 · ḡ ρdV − p̄ρdV · ḡ = −R̄0 · ḡM. (4.18) V V V Using the natural vibration modes of the elastic body (see 2.2.2), any forced motion of the body can be described. The free vibration modes used in conjunction with the 14 Chapter 4. Mean Axes Model practical mean axes constraints can determine the origin and the orientation of the local reference frame which would decouple the kinetic energy expression. Once the natural vibration modes have been found, the relative displacements of the elastic body undergoing general elastic deformations can be described in terms of the mode shapes, φi (x, y, z), and the generalized coordinates, ηi (t), as follows: d¯ = ∞ X φ̄i (x, y, z) ηi (t). (4.19) i=1 Writing the practical mean axes constraints in terms of the above, we get Z Z ∞ ∞ X X dηi dηi φi ρdV = s̄ × φi ρdV = 0. dt dt i=1 i=1 V (4.20) V Expressing the last term in the kinetic energy expression, Eq. (4.17), in terms of the elastic displacements defined in Eq. (4.19), results in ) Z Z ¯ ¯ Z (X ∞ ∞ X δ p̄ δ p̄ δd δd dηi dηj φ̄i φ̄j · ρdV = · ρdV = · ρdV . δt δt δt δt dt j=1 dt i=1 V V (4.21) V By definition, the natural vibration modes are orthogonal to each other, which can be expressed as Z φ̄i · φ̄j ρdV ≡ 0, i 6= j. (4.22) V Using this property to further simplify Eq. (4.21) results in Z ∞ X δ p̄ δ p̄ · ρdV = Mi η̇i2 δt δt i=1 (4.23) V where Mi is the generalized mass of the i th mode, defined by Z Mi = φi · φi ρdV . (4.24) V The kinetic energy of the elastic body can now be expressed as ∞ 1X 1 dR̄0 dR̄0 1 T Mi η̇i2 . T = M · + ω̄ [I] ω̄ + 2 dt dt 2 2 i=1 (4.25) Expressing the strain energy, Eq. (4.7), in terms of the natural vibration modes yields ∞ Ue = 1X 2 2 ω i η i Mi 2 i=1 (4.26) 15 Chapter 4. Mean Axes Model where ωi is the natural vibration frequency of the i th mode. In order to apply the Lagrange’s equation, the motion of the elastic body needs to be described by generalized coordinates relative to an inertial reference frame. Defining the inertial position of the origin of the local, body-fixed, mean axes reference frame (the instantaneous centre of mass of the elastic body) as R̄0 = xî + y ĵ + z k̂ (4.27) where î, ĵ and k̂ are unit vectors in each of the coordinate directions of the local reference frame. The time rate of change of the inertial coordinates is given by dR̄0 δ R̄0 ∆ = + ω̄ × R̄0 = U î + V ĵ + W k̂. dt δt (4.28) Using the same set of Euler angles which are commonly used in rigid body aircraft equations of motion, φ, θ and ψ, the orientation of the local reference frame can be defined. Then we can define the angular velocity of the local reference frame relative to the inertial reference frame as ∆ ω̄ = pî + q ĵ + rk̂ (4.29) p = φ̇ − ψ̇ sin θ (4.30) q = ψ̇ cos θ sin φ + θ̇ cos φ (4.31) r = ψ̇ cos θ cos φ − θ̇ sin φ (4.32) where Then the inertial velocity of the elastic body in the local reference frame becomes U = ẋ + qz − ry (4.33) V = ẏ + rx − pz (4.34) W = ż + py − qx (4.35) Using the above expressions for inertial position and angular and translational velocities, the kinetic energy can be expressed as p ∞ i 1X 1h 1 2 2 2 Mi η̇i2 T = M (U + V + W ) + p q r [I] q + 2 2 2 i=1 r (4.36) 16 Chapter 4. Mean Axes Model and the potential energy expressions become Ug = −M g(−x sin θ + y sin φ cos θ + z cos φ cos θ) (4.37) ∞ Ue = 1X 2 2 ωi η i Mi . 2 i=1 The above energy expressions allow the application of Lagrange’s equation ∂T ∂U d ∂T − + = Qi . dt ∂ q̇i ∂qi ∂qi (4.38) (4.39) The resulting equations are the equations of motion of flexible aircraft and can be expressed in terms of the inertial velocities expressed in the local coordinate system (i.e. U , V , W , p, q and r). After some algebraic manipulations, these equations are h i M U̇ − rV + qW + g sin θ = QX h i M V̇ − pW + rU − g sin φ cos θ = QY h i M Ẇ − qU + pV − g cos φ cos θ = QZ Ixx ṗ − (Ixy q̇ + Ixz ṙ) + (Izz − Iyy ) qr + (Ixy r − Ixz q) p + r2 − q 2 Iyz = QφB Iyy q̇ − (Ixy ṗ + Iyz ṙ) + (Ixx − Izz ) pr + (Iyz p − Ixy r) q + p2 − r2 Ixz = QθB Izz ṙ − (Ixz ṗ + Iyz q̇) + (Iyy − Ixx ) pq + (Ixz q − Iyz p) r + q 2 − p2 Ixy = QψB Mi η̈i + ωi 2 ηi = Qηi i = 1, 2, . . . n. (4.40) (4.41) (4.42) (4.43) (4.44) (4.45) (4.46) Notice that the above equations have generalized force terms in them, represented by the Qi ’s. These generalized forces are found using the Principle of Virtual Work, expressed below: ∂ (δW ) (4.47) ∂qi where δW is the work associated with arbitrary virtual displacements of the generalized Qi = coordinates. If we define the external aerodynamic and propulsive forces, relative to the local reference frame, to be given by ∆ (4.48) ∆ (4.49) F̄ = X î + Y ĵ + Z k̂ and the moments by M = Lî + M ĵ + N k̂ 17 Chapter 4. Mean Axes Model then it can be shown that (Ref. [10]) QX = X, QY = Y, QZ = Z (4.50) QψB = N . (4.51) and QφB = L, QθB = M , Finally, the generalized forces for the elastic equations, Qηi , may be computed as follows: b Z2 −l cos αv φi Qηi = − b + (mac + le cos αs ) dφi b dx dy (4.52) b 2 where l is the aerodynamic force per unit span produced by each section of the body, αv is the angle of attack of the aircraft, φi b is the elastic displacement of each section due to bending, mac is the aerodynamic moment per unit span about the line of aerodynamic centres and αs is the local angle of attack of each section. For further details see Section 2.2.1 and Ref. [10]. Chapter 5 Fixed Axes Model The higher fidelity fixed axes flexible aircraft model is based on the integrated model developed by Meirovitch and Tuzcu [3]. It presents an alternative approach to modeling flight dynamics of elastic aircraft without the use of mean-axes. This derivation ultimately yields equations of motions in the form of ordinary differential equations in vector form, in terms of quasi-coordinates and generalized coordinates, velocities and forces. This chapter summarizes the derivation of this formulation for use in flexible aircraft simulation. In order to derive the equations of motion, a reference frame xf yf zf is attached to the undeformed fuselage, as well as reference frames xw yw zw and xe ye ze to the wing and empennage, respectively, as shown in Fig. 5.1. These local reference frames represent the respective components body axes. Consequently, the motion of the whole system can be described by six rigid body degrees of freedoms of the fuselage body axes and the elastic deformation of the flexible components relative to their respective undeformed body axes. Using the conventional set of Euler angles used in aircraft dynamics, φ, θ and ψ, the transformation matrix, Cf , that transforms the inertial axes XY Z into xf yf zf can be obtained as (assuming a 321-rotation) cψcθ sψcθ −sθ Cf = cψsθsφ sψcφ sψsθsφ cψcφ cθsφ − + cψsθcφ − sψsφ sψsθcφ − cψsφ cθcφ (5.1) where s = sin and c = cos. Similarly, the time rate of change of the transformation 18 19 Chapter 5. Fixed Axes Model xf F 0f xeV rfeR rfwR Z ywR xwR zwR yeR Rf xeR yf F zeR 0 zf F X Y Figure 5.1: Component reference frames matrix, Ef , is 1 0 −sθ . Ef = 0 cφ cθsφ 0 −sφ cθcφ (5.2) The velocity of a point on the fuselage, V̄f , is given by V̄f (rf , t) = Vf (t) + [r̃f + ũf (rf , t)]T [ωf (t) + αf (rf , t)] + vf (rf , t) ∼ = Vf + (r̃f + ũf )T ωf + r̃T αf + vf (5.3) f where Vf and ωf are translational and angular velocity vectors for reference frame xf yf zf , rf is the position vector associated with the mass element dmf , uf is the elastic displacement, vf and αf are the bending and torsion velocities, respectively and r̃f and ũf are the skew-symmetric matrices corresponding to rf and uf , respectively. The velocity of a point on the wing or empennage can be expressed as V̄i (ri , t) = Ci V̄f (rf i , t) + r̃iT Ci [Ωf (rf i , t) + αf (rf i , t)] + [r̃i + ũi (ri , t)]T [ωi (t) + αi (ri , t)] + vi (ri , t) ∼ = Ci Vf + [Ci (r̃f i + ũf i )T + (r̃i + ũi )T Ci ]ωf + r̃iT Ci (Ωf i + αf i ) + Ci (vf i + r̃fTi αf i ) + r̃iT αi + vi (5.4) i = w, e where i denotes the component type, f for fuselage, w for wing and e for empennage, Ci is the rotation matrix from the fuselage body axes xf yf zf to the component body axes 20 Chapter 5. Fixed Axes Model xi yi zi , rf i is the radius vector from xf yf zf to xi yi zi , Ωf i is the angular velocity of the fuselage at rf i due to bending, expressed as T Ωf i = [ 0 −∂ u̇f z /∂xf ∂ u̇f y /∂xf ] (5.5) rf i and αf i is the elastic velocity of the fuselage at rf i due to torsion, given by T αf i = [ αf i 0 0 ] . (5.6) rf i The hybrid equations of motion for flexible aircraft are given in Ref. [2]. These hybrid equations, derived from Lagrange’s equation, are a mix of ordinary and partial differential equations, in terms of quasi-coordinates, and are expressed as d ∂L ∂L ∂L + ω̃f ∂V − Cf ∂R =F dt ∂Vf f f −1 ∂L ∂L d ∂L ∂L + ω̃f ∂ω − Ef T =M + Ṽf ∂V dt ∂ωf ∂θf f f ∂ L̂i ∂ − ∂∂uL̂ii + ∂∂F̂u̇uii + Lui ui = Ûi ∂t ∂vi ∂ F̂ ∂ L̂i ∂ i = f, w, e + ∂ ψ̇ψi + Lψi ψi = Ψ̂i ∂t ∂αi (5.7) i where L is the Lagrangian for the entire aircraft, Ṽf and ω̃f are skew-symmetric matrices of Vf and ωf , respectively, Rf is the position vector of the origin Of of xf yf zf relative to XY Z, θf is the vector of Euler angles between xf yf zf and XY Z, ψi is the elastic angular displacement vector for component i, L̂i is the Lagrangian density for component i, not including the strain energy, F̂ui and F̂ψi are the Rayleigh’s dissipation function densities for component i, Lui and Lψi are stiffness differential operator matrices for component i, F and M are the net force and moment vectors acting on the aircraft in terms of fuselage body axes and Ûi and Ψ̂i are the net force and moment density vectors for component i, respectively. The hybrid equations of motion shown above include the Lagrangian L = T − U , in which T is the kinetic energy and U the potential energy, the Rayleigh dissipation function densities, F̂ui and F̂ψi , which contain structural damping information and the stiffness operators, Lui and Lψi , which are derived from the strain energy. In order to obtain an explicit set of expressions for the flexible aircraft equations of motion, it is necessary to derive each of these quantities in terms of the known rigid-body and elastic coordinates of the system. The kinetic energy for the entire aircraft can be expressed as T = Tf + Tw + Te (5.8) 21 Chapter 5. Fixed Axes Model where Ti are the kinetic energies of the aircraft components, given by Z 1 Ti = 2 V̄iT V̄i dmi , i = f, w, e (5.9) Therefore, the total kinetic energy becomes Z Z Z T T 1 1 1 T = 2 V̄f V̄f dmf + 2 V̄w V̄w dmw + 2 V̄eT V̄e dme . (5.10) The Rayleigh dissipation function densities can be expressed as F̂ui = 21 cui EIi ∂ 2 u̇Ti ∂ 2 u̇i , ∂xi 2 ∂xi 2 F̂ψi = 21 cψi GJi ∂ ψ̇iT ∂ ψ̇i , ∂xi ∂xi i = f, w, e (5.11) where cui and cψi are the bending and torsion damping functions, respectively, and EIi and GJi are, respectively, the flexural and torsional rigidities. As partial differential equations are difficult to work with, they are conventionally replaced by a set of ordinary differential equations in most practical applications, with the help of spatial discretization. In this case, the individual components of the aircraft are discretized separately, instead of using structural modes for the entire aircraft which can result in the loss of some geometric details of the aircraft structure. The following gives the discretization scheme being used for this formulation: ui (ri , t) = Φui (ri )qui (t), ψi (ri , t) = Φψi (ri )qψi (t), i = f, w, e (5.12) where Φui and Φψi are matrices of shape functions and qui and qψi are corresponding vectors of generalized coordinates. The time derivatives of these generalized coordinates are given by sui (t) = q̇ui (t), sψi (t) = q̇ψi (t), i = f, w, e (5.13) The velocity vectors can now be written using the discretized form of the elastic displacements, just shown, as V̄f (rf , t) = Vf + (r̃f + Φ̃uf quf )T ωf + Φuf suf + r̃f Φψf sψf V̄i (ri , t) = Ci Vf + [Ci (r̃f i + Φ̃uf i quf )T + (r̃i + Φ̃ui qui )T Ci ]ωf (5.14) +(r̃iT Ci ∆Φuf i + Ci Φuf i )suf + Φui sui +(r̃iT Ci Φψf i + Ci r̃fTi Φψf i )sψf + r̃iT Φψi sψi where 0 0 ∆= 0 0 0 ∂/∂xf 0 −∂/∂xf , 0 0 0 i = w, e T , Φuf = φ 0 uf y T 0 φuf z Φuf i = Φuf (rf i ), i = w, e (5.15) 22 Chapter 5. Fixed Axes Model Similar expressions can be written for Φψf and Φψf i . Inserting the above into the kinetic energy expression, Eq. (5.10), we get the following compact form for the total kinetic energy: T = 21 VT MV (5.16) where V is the discrete velocity vector for the entire system, expressed as iT h V = VfT ωfT sTuf sTuw sTue sTψf sTψw sTψe (5.17) and M = [Mmn ] is the complete system mass matrix. Refer to Ref. [4] for expressions of the submatrices that makeup the mass matrix. The matrix of first moments of inertia of the deformed aircraft, S̃, is given by S̃ = R R (r̃f + Φ̃uf quf )dmf + [(r̃f w + Φ̃uf w quf )CwT + CwT (r̃w + Φ̃uw quw )]Cw dmw R + [(r̃f e + Φ̃uf e quf )CeT + CeT (r̃e + Φ̃ue que )]Ce dme (5.18) and the matrix of second moments of inertia of the deformed aircraft, J, simply referred to as the inertia matrix can be computed as follows: Z J = (r̃f + Φ̃uf quf )T (r̃f + Φ̃uf quf )dmf Z + [Cw (r̃f w + Φ̃uf w quf )T + (r̃w + Φ̃uw quw )T Cw ]T × [Cw (r̃f w + Φ̃uf w quf )T + (r̃w + Φ̃uw quw )T Cw ]dmw Z + [Ce (r̃f e + Φ̃uf e quf )T + (r̃e + Φ̃ue que )T Ce ]T (5.19) × [Ce (r̃f e + Φ̃uf e quf )T + (r̃e + Φ̃ue que )T Ce ]dme . Similarly, substituting Eq. (5.12) into Eqs. (5.11) results in the following expressions for the generalized Rayleigh’s dissipation functions: T Fui = 21 q̇ui Cui q̇ui , T Fψi = 21 q̇ψi Cψi q̇ψi , i = f, w, e (5.20) where the damping matrices, Cui and Cψi are given by d2 ΦTui d2 Φui Cui = cui EIi dDi , dx2i dx2i Di Z Z Cψi = cψi GJi Di d2 ΦTψi d2 Φψi dDi , dx2i dx2i i = f, w, e (5.21) Expressing the momentum vector for the entire aircraft by h iT p = pTV f pTωf pTuf pTuw pTue pTψf pTψw pTψe , (5.22) 23 Chapter 5. Fixed Axes Model we can write p = ∂T /∂V = M V. (5.23) Having derived expressions for all of the required quantities, we can write the discretized state equations which describe the dynamics of a flexible aircraft as follows (from Equation (5.7)): Ṙf = CfT Vf , q̇ui = sui , θ̇f = Ef−1 wf q̇ψi = sψi , ṗV f = −ω̃f pV f + F, i = f, w, e ṗωf = −Ṽf pV f − ω̃f pωf + M ṗui = ∂T /∂qui − Kui qui − Cui sui + Qui , ṗψi = −Kψi qψi − Cψi sψi + Qψi , i = f, w, e i = f, w, e where, using Eqs. (5.10) and (5.14), we have R R R T T ∂T T T T T T T V̄e dme V̄ dm + Φ ω̃ C = Φ ω̃ V̄ dm + Φ ω̃ C w w f f e w uf e f uf f uf w f ∂quf R ∂T = ΦTuw C̃w ωf T V̄w dmw ∂quw R ∂T = ΦTue C̃e ωf T V̄e dme . ∂que Additionally, from strain energy, the bending stiffness matrix can be derived as Z ∂ 2 ΦTui ∂ 2 Φui Kui = [EI ] dDi , i = f, w, e, i 2 ∂x2i Di ∂xi and the torsional stiffness matrix as Z ∂ΦTψi ∂Φψi dDi , GJi Kψi = ∂xi ∂xi Di (5.24) i = f, w, e. (5.25) (5.26) (5.27) The only remaining quantities appearing in the state equations, Eq. (5.24) are the generalized force terms, F, M, Qui and Qψi . These forces are related to the actual loads applied to the aircraft and can be obtained from the principle of virtual work as R R R F = Df [ff + FE δ (r − rE )] dDf + CwT Dw fw dDw + CeT De fe dDe R M = Df r̃f + Φ̃uf quf [ff + FE δ (r − rE )] dDf i R h + Dw r̃f w + Φ̃uf w quf CwT + CwT r̃w + Φ̃uw quw fw dDw i R h T T + De r̃f e + Φ̃uf e quf Ce + Ce r̃e + Φ̃ue que fe dDe T R R Quf = Df ΦTuf [ff + FE δ (r − rE )] dDf + Dw r̃wT Cw ∆Φuf w + Cw Φuf w fw dDw T R + De r̃eT Ce ∆Φuf e + Ce Φuf e fe dDe T R R Qψf = Df ΦTψf r̃f [ff + FE δ (r − rE )] dDf + Dw r̃wT Cw Φψf w + Cw r̃fTw Φψf w fw dDw T R + De r̃eT Ce Φψf e + Ce r̃fTe Φψf e fe dDe R R Qui = Di ΦTui fi dDi , Qψi = Di ΦTψi r̃i fi dDi , i = w, e (5.28) Chapter 5. Fixed Axes Model 24 where fi are the net distributed forces over the component i, due to gravity, aerodynamics and control actions and FE δ (r − rE ) is the engine thrust, in which δ is the spatial Dirac delta function and rE is the engine’s position vector. This completes the derivation of all the quantities required by this flight dynamics model. Chapter 6 Results and Discussion The simulation results from the mean axes and fixed axes models were compared along with rigid body results for two control inputs for both longitudinal and lateral controls. 6.1 6.1.1 Comparison of Mean and Fixed Axes Models Response to Elevator Impulse The time history plots of the aircraft response to an elevator impulse are shown in Figures 6.1 to 6.7, below. From Figure 6.1 its clear that the two models have a very similar response in terms of the pitch rate of the aircraft. Additionally, the rigid aircraft response is also very similar to the flexible aircraft responses. In Figure 6.3, the pitch angle time histories are shown for the two flexible aircraft models as well as the rigid model response. Here, the response of the fixed axes model is offset from the other two models. This offset is caused by the fixed axes model trimming at a slightly different pitch angle initially, due to the fact that the structural component of the two models (the total modal mass) is not exactly the same. Figure 6.4 plots the longitudinal velocity response to the elevator impulse input. Again, the two flexible models and the rigid model are in close agreement with one another. The acceleration time histories, (Figure 6.5), are also very similar between the three models, except for some spikes seen in the vertical component of acceleration (Az ) of the mean axes model. The possible cause of these spikes is discussed later (see end of Sec. 25 Chapter 6. Results and Discussion 26 Figure 6.1: Pitch rate time history of aircraft response to elevator impulse 6.2.3). Figure 6.7 shows the vertical component of deflection of the right wingtip in response to the elevator impulse input to the models. As can be seen, the two flexible models are offset from one another, but have a similar trend. This offset is caused by different initial trim values. Overall, it can be concluded that the two models are very similar in response for the longitudinal elevator impulse input. Chapter 6. Results and Discussion Figure 6.2: A closer look at pitch rate time history just after the input Figure 6.3: Pitch angle time history of aircraft response to elevator impulse 27 Chapter 6. Results and Discussion Figure 6.4: X velocity time history of aircraft response to elevator impulse Figure 6.5: Acceleration time history of aircraft response to elevator impulse 28 Chapter 6. Results and Discussion 29 Figure 6.6: Acceleration time history focusing on the response immediately after elevator impulse Figure 6.7: Wingtip relative deflection time history of aircraft response to elevator impulse Chapter 6. Results and Discussion 6.1.2 30 Response to Elevator 2311 2311 inputs are designed to excite a wide range of frequencies in an aircraft’s response. 2311 inputs are alternating step inputs which satisfy the time duration ratio 2:3:1:1. Refer to [6] for details. The mean axes and fixed axes flexible aircraft models were both simulated with an elevator 2311 input. The resulting time histories are shown in Figures 6.8 to 6.12 below. For comparison, the response of the rigid aircraft model is also included in the plots. Figure 6.8: Pitch rate time history of aircraft response to elevator 2311 In the pitch rate time history, it can be seen that the response of the three models is very similar. Only slight differences in the response can be observed during the second peak and in the region after the control input has ended. However, the magnitude of the variation in response is negligible. From Figure 6.9, the pitch angle time histories of the two models can be compared for the case of an elevator 2311 input. The overall trend is the same for the two flexible models as well as the rigid model. However, unlike the elevator impulse case, the difference in the response of the two flexible models is more noticeable here. Overall though, it can be said that the models are in agreement. Once again, the longitudinal velocity time histories for the three models follow the Chapter 6. Results and Discussion 31 Figure 6.9: Pitch angle time history of aircraft response to elevator 2311 same trend (Figure 6.10). The difference in the three models becomes the most noticeable at the minimum, about halfway through the simulation. As expected, the time histories for the aircraft acceleration in response to the elevator 2311 input are similar for both flexible models. The rigid aircraft model acceleration is also in close agreement to the flexible models. Figure 6.12 gives the time history plots for the wing tip deflection of the aircraft relative to the wing root for both fixed and mean axes flexible aircraft models. Apart from a slight offset caused by a difference in the initial trim values, the two models are very similar in their response. Chapter 6. Results and Discussion Figure 6.10: X velocity time history of aircraft response to elevator 2311 Figure 6.11: Acceleration time history of aircraft response to elevator 2311 32 Chapter 6. Results and Discussion 33 Figure 6.12: Wingtip relative deflection time history of aircraft response to elevator 2311 Chapter 6. Results and Discussion 6.1.3 34 Response to an Aileron Impulse To compare the lateral response of the aircraft models, the models were simulated with an aileron impulse input same as the elevator impulse used for longitudinal response. The time history plots of the aircraft response to the aileron impulse are shown in Figures 6.13 to 6.19, below. Figure 6.13: Roll rate time history of aircraft response to aileron impulse The roll rate response, Figure 6.13, for the mean axes and fixed axes models is very similar, as well as the response of the rigid model. The roll angle time history response to the aileron impulse input is plotted in Figure 6.15. All three of the models start off at the same value and follow the same trend initially. However, after the control input ends, the mean axes model’s response drifts away from that of the other two models. Figure 6.16 shows the lateral velocity time history plots for the mean axes, fixed axes, as well as the rigid aircraft models. It can be seen that all three models have a similar response. The amplitude of the response is quite small though in all three cases. The acceleration time histories for the three models’ response to the aileron impulse are compared in Figure 6.17. The responses agree with one another for all three models, except for the mean axes model’s lateral acceleration Ay which has noticeable spikes and Chapter 6. Results and Discussion 35 Figure 6.14: A closer look at the roll rate time history immediately proceeding the input starts to drift away from the other two models. The relative wingtip deflection of the two models in response to the aileron impulse is plotted in Figure 6.19. The response is very similar for the two models with the exception of a small offset, likely caused by different trim points between the two models. Chapter 6. Results and Discussion Figure 6.15: Roll angle time history of aircraft response to aileron impulse Figure 6.16: Y velocity time history of aircraft response to aileron impulse 36 Chapter 6. Results and Discussion Figure 6.17: Acceleration time history of aircraft response to aileron impulse Figure 6.18: Focusing on acceleration time history response just after the input 37 Chapter 6. Results and Discussion 38 Figure 6.19: Wingtip relative deflection time history of aircraft response to aileron impulse Chapter 6. Results and Discussion 6.1.4 39 Response to Aileron 2311 Similar to the elevator 2311 input, an aileron 2311 input was used to simulate the response of the models. 2311 inputs excite a wider range of frequencies in the aircraft model output. Figures 6.20 to 6.24 show plots of the output of the flexible aircraft models along with the rigid model’s output. Figure 6.20: Roll rate time history of aircraft response to aileron 2311 Figure 6.20 shows a comparison of the roll rate time histories of all three models in response to an aileron 2311. The response from all three models is very similar. The roll angle time history of the simulation in response to an aileron 2311 is plotted in Figure 6.21. The response from all three models is about the same during the input, however, once the input ends the models diverge slightly. The lateral velocity time histories for the aileron 2311 response are compared in Figure 6.22. The velocities from all three of the models match well, for the whole time interval. The acceleration time history response to aileron 2311 is shown in Figure 6.23. All three models follow the same trend for all three acceleration components. Only the lateral acceleration, Ay , drifts slightly with time. Chapter 6. Results and Discussion Figure 6.21: Roll angle time history of aircraft response to aileron 2311 Figure 6.22: Y velocity time history of aircraft response to aileron 2311 40 Chapter 6. Results and Discussion 41 Figure 6.23: Acceleration time history of aircraft response to aileron 2311 Figure 6.24: Wingtip relative deflection time history of aircraft response to aileron 2311 Chapter 6. Results and Discussion 42 The time history for relative wingtip deflection in response to an aileron 2311 input for the mean axes and fixed axes models can be seen in Figure 6.24. Except for a small offset, the wingtip deflection is about the same for the two models. The offset can be attributed to the fact that the two models trim to slightly different initial values. By observing the response of the two flexible models given in this section, it can be concluded that the two models produce very similar output to a variety of inputs. As such, either of the two flexible models can be employed to simulate the flight dynamics of this particular aircraft. However, it can be noticed from the preceding results that the two flexible models’ responses do not differ much from the response of the rigid aircraft model. The reason being that the aircraft used in these simulations has a fairly rigid structure and thus it does not exhibit significant interactions between the structural dynamics and the flight dynamics. To study the response of the mean axes and fixed axes flexible flight simulation models, the aircraft structural data was modified to artificially make it more flexible. The effect of varying the stiffness of the aircraft structure on the response of the two flexible models is presented in the following section. 6.2 Effect of Varying Aircraft Stiffness In this section, the effect of varying the stiffness of the aircraft structure on the response of the flexible models is analysed. Each of the models was tested with one-half and onetenth the stiffness of the original aircraft and the results were compared for the different stiffness cases along with the rigid aircraft model. The same inputs were used as in the previous section for consistency. 6.2.1 Response to Elevator Impulse The time history plots of the aircraft response to an elevator impulse are shown in Figures 6.25 to 6.42, below. Figures 6.25 and 6.26 present the pitch rate response of the mean and fixed axes models, respectively. The original and half stiffness cases are very similar to the rigid model. The difference becomes significant for the tenth stiffness case, for both models. Comparing between the two models, the responses follow a similar trend with the exception of noticeable spikes in the mean axes model’s response at the beginning of the Chapter 6. Results and Discussion 43 Figure 6.25: Pitch rate time history of mean axes model’s response to elevator impulse input. The pitch angle time histories in response to elevator impulse for varying stiffnesses are given in Figures 6.29 and 6.30 for the two models. Once again, the original stiffness and half stiffness cases are similar to rigid response while the difference becomes noticeable in the case of one-tenth the original stiffness, for both models. The responses of the fixed axes model are offset due to different initial trim values, especially evident in the tenth stiffness case. The longitudinal component of velocity, Vx , simulation time histories are shown in Figures 6.31 and 6.32, for the mean axes and fixed axes models respectively. Again, the original and half stiffness cases match well with the rigid case, while the tenth stiffness cases differ noticeably. Figures 6.33 and 6.34 show the response of the longitudinal acceleration component, Ax . Similar to previous results, the response of the two flexible models is similar to one another with the exception of damped oscillations that start at the onset of the input in the mean axes model’s response. The vertical component of acceleration, Az , is compared for various aircraft stiffnesses, for the two flexible models in Figures 6.37 and 6.38. Similar to previous results, the two models produce similar responses except for the oscillations in the mean axes model and Chapter 6. Results and Discussion 44 Figure 6.26: Pitch rate time history of fixed axes model’s response to elevator impulse the effect of varying the structural stiffness of the model only becomes noticeable when the stiffness is decreased to one-tenth of the original stiffness. The effect of varying the stiffness of the structure of the aircraft model on the relative wingtip deflection in response to an elevator impulse input is demonstrated in Figures 6.41 and 6.42 for the mean axes and fixed axes models, respectively. As expected, the amount of deformation increases with decreasing stiffness. The response of the two models is similar for the original and half stiffness cases while there is a slightly greater offset for the tenth stiffness case. Chapter 6. Results and Discussion 45 Figure 6.27: A closer look at the pitch rate time history of mean axes model immediately after the elevator impulse Figure 6.28: A closer look at the pitch rate time history of fixed axes model immediately proceeding the input Chapter 6. Results and Discussion 46 Figure 6.29: Pitch angle time history of mean axes model’s response to elevator impulse Figure 6.30: Pitch angle time history of fixed axes model’s response to elevator impulse Chapter 6. Results and Discussion 47 Figure 6.31: X velocity time history of mean axes model’s response to elevator impulse Figure 6.32: X velocity time history of fixed axes model’s response to elevator impulse Chapter 6. Results and Discussion 48 Figure 6.33: X acceleration time history of mean axes model’s response to elevator impulse Figure 6.34: X acceleration time history of fixed axes model’s response to elevator impulse Chapter 6. Results and Discussion 49 Figure 6.35: X acceleration time history of mean axes model: focusing on the response immediately proceeding the input Figure 6.36: X acceleration time history of fixed axes model immediately proceeding the elevator impulse input Chapter 6. Results and Discussion 50 Figure 6.37: Z acceleration time history of mean axes model’s response to elevator impulse Figure 6.38: Z acceleration time history of fixed axes model’s response to elevator impulse Chapter 6. Results and Discussion 51 Figure 6.39: Focusing on the Z acceleration time history of mean axes model immediately after the input Figure 6.40: Focusing on the Z acceleration time history of fixed axes model immediately after the input Chapter 6. Results and Discussion 52 Figure 6.41: Wingtip relative deflection time history of mean axes model’s response to elevator impulse Figure 6.42: Wingtip relative deflection time history of fixed axes model’s response to elevator impulse Chapter 6. Results and Discussion 6.2.2 53 Response to Elevator 2311 An elevator 2311 input was used to further evaluate the effect of varying the stiffness of the aircraft structure on the response of flexible models. The results of these simulations are shown in Figures 6.43 to 6.54 below. Figure 6.43: Pitch rate time history of mean axes model’s response to elevator 2311 The pitch rate response of the two models (Figures 6.43 and 6.44) is in agreement with each other. Conforming with the previous results, both models have a very similar response for all stiffness cases. Once again, the effect of varying the stiffness only becomes evident when the stiffness is lowered to one-tenth of the original aircraft structure’s stiffness. Figures 6.43 and 6.44 show the pitch angle response to the elevator 2311 input for the mean axes and fixed axes models, respectively. As expected the results for both models are very similar for all stiffness cases. The longitudinal velocity component time histories for the elevator 2311 input response are given by Figures 6.47 and 6.48. All the responses for both models match well with each other with only slight differences in the time histories of the output. The difference in structural stiffnesses is most visible for the one-tenth stiffness cases, following the previous results. Chapter 6. Results and Discussion 54 Figure 6.44: Pitch rate time history of fixed axes model’s response to elevator 2311 Figures 6.49 to 6.52 show the longitudinal and vertical acceleration time histories in response to an elevator 2311 input for both flexible models with varying structural stiffnesses. Following the trend of the previous results there is a good match between the results of the two models for each of the aircraft stiffnesses modeled. The only small difference between the two models is the presence of oscillations in the mean axes model response, especially apparent in the tenth stiffness case. The reason behind these oscillations is discussed later in this chapter. The wingtip deflection in response to the elevator 2311 input, for the two models, is given in Figures 6.53 to 6.54. Apart from the results being slightly offset from one another caused by the models trimming to different values initially, the response of the two models is very comparable, for all three stiffness cases. Chapter 6. Results and Discussion 55 Figure 6.45: Pitch angle time history of mean axes model’s response to elevator 2311 Figure 6.46: Pitch angle time history of fixed axes model’s response to elevator 2311 Chapter 6. Results and Discussion 56 Figure 6.47: X velocity time history of mean axes model’s response to elevator 2311 Figure 6.48: X velocity time history of fixed axes model’s response to elevator 2311 Chapter 6. Results and Discussion 57 Figure 6.49: X acceleration time history of mean axes model’s response to elevator 2311 Figure 6.50: X acceleration time history of fixed axes model’s response to elevator 2311 Chapter 6. Results and Discussion 58 Figure 6.51: Z acceleration time history of mean axes model’s response to elevator 2311 Figure 6.52: Z acceleration time history of fixed axes model’s response to elevator 2311 Chapter 6. Results and Discussion 59 Figure 6.53: Wingtip relative deflection time history of mean axes model’s response to elevator 2311 Figure 6.54: Wingtip relative deflection time history of fixed axes model’s response to elevator 2311 Chapter 6. Results and Discussion 6.2.3 60 Response to Aileron Impulse To study the effect of varying the stiffness on the lateral response of the flexible aircraft models, simulations were run with inputs to the aileron. The first input used was an impulse, similar to the one used in Sec. 6.1.1 for the elevator. The time history plots of the response of the models to the aileron impulse for three stiffness cases are shown in the following Figures 6.55 to 6.72. Figure 6.55: Roll rate time history of mean axes model’s response to aileron impulse Figures 6.55 and 6.56 compare the time histories of the roll rate in response to an aileron impulse, respectively, for the mean and fixed axes flexible aircraft flight simulation models with varying structural stiffness. Both models compare favorably with one another for the original and half stiffness cases. Both models’ response shows damped oscillations in the tenth stiffness case. However, the mean axes model’s response for the tenth stiffness case starts to diverge from the other two responses about 15 seconds after the input. The roll angle simulation response to the aileron impulse is plotted in Figures 6.59 and 6.60. The response is similar between the two models for the original and half stiffness cases. However, the roll angle of the mean axes model diverges rapidly after the control input for the tenth stiffness case. The fixed axes model’s response does not diverge. Chapter 6. Results and Discussion 61 Figure 6.56: Roll rate time history of fixed axes model’s response to aileron impulse The response of the lateral component of velocity to an aileron impulse input for the mean axes and fixed axes models is shown in Figures 6.61 and 6.62. Again, the tenth stiffness case of the mean axes model diverges rapidly compared to the other cases which are similar to the response of the fixed axes model. The longitudinal and lateral acceleration response of the two flexible aircraft models is given by Figures 6.63 to 6.68. The time histories exhibit a good match between the two models for the original and half stiffness cases. However, the mean axes model’s response diverges for both acceleration components. Figures 6.71 and 6.72 give the relative wingtip deflection of the flexible aircraft models for the three stiffness cases in response to an aileron impulse. Both models produce similar results with a small offset caused by difference in the initial trim points of the two models. Chapter 6. Results and Discussion 62 Figure 6.57: Fousing on roll rate time history of mean axes model just after the aileron impulse Figure 6.58: Focusing on roll rate time history of fixed axes model just after the aileron impulse Chapter 6. Results and Discussion 63 Figure 6.59: Roll angle time history of mean axes model’s response to aileron impulse Figure 6.60: Roll angle time history of fixed axes model’s response to aileron impulse Chapter 6. Results and Discussion 64 Figure 6.61: Y velocity time history of mean axes model’s response to aileron impulse Figure 6.62: Y velocity time history of fixed axes model’s response to aileron impulse Chapter 6. Results and Discussion 65 Figure 6.63: X acceleration time history of mean axes model’s response to aileron impulse Figure 6.64: X acceleration time history of fixed axes model’s response to aileron impulse Chapter 6. Results and Discussion 66 Figure 6.65: Focusing on X acceleration time history of mean axes model proceeding the input Figure 6.66: Focusing on X acceleration time history of fixed axes model proceeding the input Chapter 6. Results and Discussion 67 Figure 6.67: Y acceleration time history of mean axes model’s response to aileron impulse Figure 6.68: Y acceleration time history of fixed axes model’s response to aileron impulse Chapter 6. Results and Discussion 68 Figure 6.69: Focusing on Y acceleration time history of mean axes model immediately after aileron impulse Figure 6.70: Focusing on Y acceleration time history of fixed axes model immediately after aileron impulse Chapter 6. Results and Discussion 69 Figure 6.71: Wingtip relative deflection time history of mean axes model’s response to aileron impulse Figure 6.72: Wingtip relative deflection time history of fixed axes model’s response to aileron impulse Chapter 6. Results and Discussion 6.2.4 70 Response to Aileron 2311 Similar to the elevator input, a 2311 aileron input was used to compare the outputs of the simulation models. The time history plots of the aircraft response to the aileron 2311 input are shown in Figures 6.73 to 6.84. Figure 6.73: Roll rate time history of mean axes model’s response to aileron 2311 The roll rate responses of the mean axes model for the original and half stiffness cases compare reasonably well with the fixed axes model (Figures 6.73 and 6.74). However, for the tenth stiffness case, the mean axes model’s response is out of phase and does not match the fixed axes model in magnitude either. The roll angle time histories of the flexible models for an aileron 2311 input are given in Figures 6.75 and 6.76. The two models’ response matches with one another for the original and half stiffness cases, while the tenth stiffness cases’ response diverges for the mean axes model. Figures 6.77 and 6.78 present the time histories of the lateral component of velocity of the two flexible models to the aileron 2311. The frequency of oscillations in the output of the mean axes model varies with stiffness, as can be noticed by observing the tenth stiffness case. However, no such trend is noticed in the response of the fixed axes model. Furthermore, the amplitude of the oscillations increases with decreasing stiffness in case Chapter 6. Results and Discussion 71 Figure 6.74: Roll rate time history of fixed axes model’s response to aileron 2311 of the mean axes model while the opposite is true for the fixed axes model. Figures 6.79 and 6.80 provide the longitudinal acceleration time histories of the two model’s response to an aileron 2311 input. The response of the mean axes model starts to diverge after the control input, something easily noticeable in the tenth stiffness case. Meanwhile, the response of the fixed axes model does not diverge. The lateral component of acceleration time histories in response to an aileron 2311 are shown in Figures 6.81 and 6.82 for varying aircraft stiffness. Once again, the response of the mean axes model diverges, while the fixed axes model’s response does not. Additionally, the mean axes model’s response exhibits small spikes which are less pronounced in the case of the fixed axes model. The wingtip deflection relative to the wing root in response to the aileron 2311 input for different aircraft structural stiffness ratios is given in Figures 6.83 and 6.84. The wingtip deflection time histories for the two models are very similar for each of the three aircraft stiffness ratios. Chapter 6. Results and Discussion Figure 6.75: Roll angle time history of mean axes model’s response to aileron 2311 Figure 6.76: Roll angle time history of fixed axes model’s response to aileron 2311 72 Chapter 6. Results and Discussion Figure 6.77: Y velocity time history of mean axes model’s response to aileron 2311 Figure 6.78: Y velocity time history of fixed axes model’s response to aileron 2311 73 Chapter 6. Results and Discussion 74 Figure 6.79: X acceleration time history of mean axes model’s response to aileron 2311 Figure 6.80: X acceleration time history of fixed axes model’s response to aileron 2311 Chapter 6. Results and Discussion 75 Figure 6.81: Y acceleration time history of mean axes model’s response to aileron 2311 Figure 6.82: Y acceleration time history of fixed axes model’s response to aileron 2311 Chapter 6. Results and Discussion 76 Figure 6.83: Wingtip relative deflection time history of mean axes model’s response to aileron 2311 Figure 6.84: Wingtip relative deflection time history of fixed axes model’s response to aileron 2311 Chapter 6. Results and Discussion 77 The mean axes and the fixed axes models’ response is comparable for longitudinal inputs for the original and half aircraft stiffness ratios. Even for the one-tenth stiffness ratio cases, the two models’ response is similar with the exception of some sharp oscillations found in the response of the mean axes model. However, these oscillations are damped and subside fairly quickly. These oscillations are likely caused by the particular choice of the natural vibrational modes that were used in the mean axes model for these simulations. More specifically, mode #4 which is primarily a horizontal empennage bending mode causes these oscillations. For the lateral cases, the response of the mean axes model does not match the fixed axes model as well as the longitudinal cases do. The response of the mean axes model diverges, especially for the tenth stiffness cases. The cause of this behaviour is not yet known, however it seems to be caused by the flexible part of the model as the rigid model does not show the same characteristics. Apart from that, the two models do correlate for the wing deflection output. Additionally, for the original and the half stiffness cases the diverging behaviour is not as pronounced and thus the response is comparable between the two flexible flight models. Chapter 7 Conclusions and Recommendations The objective of this study was to compare two available models for the simulation of aircraft with flexible structures. Using aircraft data available in the literature, a mean axes model and a fixed axes model were developed. Simulations were run with a couple of inputs to the models to obtain a comprehensive amount of data for performing the comparative analysis. The first input used was an impulse composed of modal frequencies obtained from Finite Element Analysis of the aircraft structure. The other input used was a 2311 input which was used to excite a wide range of frequencies in the aircraft dynamics. Each of these inputs was used in both longitudinal (elevator) and lateral (aileron) control inputs. Time history responses of the two models were compared with one another. It was found that the two models act in a very similar manner for all inputs tested in both lateral and longitudinal dimensions. However, comparing the response of the flexible models with a rigid aircraft model revealed the fact that the aircraft considered in this study is not very flexible by nature. To overcome this problem, the stiffness of the aircraft models was scaled down and the simulations were re-run. The effect of the increased flexibility on the response of the two models was analyzed by comparing simulation time histories of the half-stiffness and tenth-stiffness models. The longitudinal response of the two models was found to resemble each other for both decreased stiffness cases. However, in the lateral case, it was noticed that the tenth stiffness mean axes model diverges. The same behaviour was not noticed in the tenth stiffness fixed axes model’s response. The cause for this divergent behaviour was examined, however, no firm explanation has been found so far. As a result of this study, a number of recommendations are made for future work 78 Chapter 7. Conclusions and Recommendations 79 in the field of flexible aircraft flight simulation. First of all, further effort is needed to determine the cause of the divergent behaviour in the response of the mean axes model, most evident in the tenth-stiffness cases . Second, instead of just scaling the stiffness of the aircraft model, the whole structure of the aircraft needs to be modified to cause more oscillatory response to control inputs in the simulations. This will allow for a more dynamic analysis to be performed to study the performance of the two flexible flight models. Third, to improve the fidelity of the models, a more complete aerodynamic computation technique is needed to replace the simple aerodynamic strip theory used in this study. Such a technique will result in capturing non-linear aerodynamic effects which are currently ignored. Fourth, by optimizing the mode shapes included in the two flexible models, it may be possible to obtain similar oscillations from both models. Further effort is required in this area to help obtain a conclusive result. Last, a handling quality analysis is recommended to study the response of the two models in a motion flight simulator with human pilots. This will provide greater insight into the fidelity of the model in simulating the flight dynamics of the flexible aircraft being modeled. References [1] C. S. Buttril, T. A. Zeiler, and P. D. Arbuckle. Nonlinear simulation of a flexible aircraft in maneuvering flight. Number Paper 87-2501-CP, Monterey, CA, Aug. 17-19 1987. AIAA Flight Simulation Technologies Conference. [2] L. Meirovitch. A unified theory for the flight dynamics and aeroelasticity of whole aircraft. In Proceedings of the Eleventh Symposium on Structural Dynamics and Control, pages 461–468, Blacksburg, VA, May 12-14, 1997, pp. 1997. [3] L. Meirovitch and I. Tuzcu. Integrated approach to the dynamics and control of maneuvering flexible aircraft. CR 211748, NASA, June 2003. [4] L. Meirovitch and I. Tuzcu. The lure of mean axes. Journal of Applied Mechanics, 74:497–504, May 2007. [5] R. D. Milne. Dynamics of deformable aeroplane, parts i and ii. Technical Report Reports and Memoranda No. 3345, Her Majesty’s Stationery Office, London, 1962. [6] J. R. Raol, G. Girija, and J. Singh. Modelling and Parameter Estimation of Dynamic Systems. The Institution of Engineering and Technology, first edition, 2004. [7] S. C. S. Silveira. Modal analysis of a jet aircraft for use in flexible aircraft modeling. M. eng. research project, University of Toronto Institure for Aerospace Studies, Toronto, ON, April 2008. [8] A. S. Taylor and D. L. Woodcock. Mathematical approaches to the dynamics of deformable aircraft, parts i and ii. Technical Report Reports and Memoranda No. 3776, Her Majesty’s Stationery Office, London, 1971. [9] M. R. Waszak, C. S. Buttril, and D. K. Schmidt. Modeling and model simplification of aeroelastic vehicles: An overview. TM 107691, NASA, September 1992. 80 References 81 [10] M. R. Waszak and D. K. Schmidt. Flight dynamics of aeroelastic vehicles. Journal of Aircraft, 25(6):563–571, 1988.

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