# STRUCTURAL MODIFICATIONS: HOW TO CIRCUMVENT THE

код для вставкиSTRUCTURAL MODIFICATIONS: TO CIRCUMVENT THE EFFECT MODAL TRUNCATION. Techion The paper outlines the basic theory, and then shows 2 examples demonstrating itвЂ™s implementation. An extended presentation is available in [6a]. HOW OF 2. Theory I.sucher, s BraIn 1srae1 Institute of Technology Haifa ISRAEL 2.1 Assigning Modes A common design goti during design and uoublcshooting stagesis to modify an existing vibrating structure in order to obtain desired dynamic properties. The supplied knowledge (model) about the vibrating structure, is limited due to practical reesons. This modification of the structurnl response may be motivated, for example, by a desire to minimi7x the response at specific locations. The description here is suitable for systems with very low damping or for systems having proportional damping (9;2]. In either case the natural vibrating frequencies and modeshapes are the solution for the following generalized eigenvalue problem. Abstract The prediction of the effect of structural modifications is problematic when the modal data is extracted via EMA, and hcnnce truncated. We show how to control a subset of the originally identified eigenfrequencies and eigenvectors, and assign exact values to them. Our method is based on choosing structural modifications, such that the new eigenvectors are spanned by the original modal base.Examplesare given for a discrete and continous (FE modelled) system. 1. Introduction (-o:M + K)x, = 0 r=l...n (1) Any structural or geometrical changes will yield appropriate mass and stiffness changes. The new mass and stiffness matriceswill be The modal information available to engineers comes usually from two possible sources: 2) An expeIimental test I) A FE model In both cases we only acquire a partial modal base, i.e. a limited number of modes up to a maximum frequency. This limitation is due to the desire to minimix the computational effort ( in FE analysis)or experimental limitations. The fact that only a subset of modal information is available limits the use of structural modification techniques. Many researchers have address the influence of such modal truncation on the possible prediction of the effect of structural modifications, when those are based on modal information only (31, 321. We address the design problem, whereas structural modifications are undertaken in order to achieve ao exa~f desired modal behavior.We show how to control a subset of the originally identified eigenfrequencies and eigenvectors, and assign exact values to them. The necessaryinformation for such an implementation is still the truncated modal base, as available from test? ( or a limited FE analysis1. Our method is based on extracting both right and left eigenvectors from the measured transfer functions (in FE analysis both can obviously be extracted from the model 1. The right eigenvectors are those extracted by classicalEMA, the left eigenvectors are extracted by techniques similar to those described by 1391. Our method is based on choosing structural modifications, such that the new eigenvectors are spanned by the original modal base ~?=M+AM li?=~+~~ (2) The modified structure will possesnew natural frequencies and mode shapes.These will be the solution of: (-rs:ti+qi, =o (3a) Substituting Eq. (2) in Eq. (3a) and rearranging we obtain (ti;M-K)i,=(-$AM+AK)i, (3b) Eq. (3b) is the modified structureвЂ™s eigenvalue problem. This expressions includes the original mass and stiffness matrices along with the new natural frequencies and mode-shapes. In particular when the rth mode is unchanged the left hand side of Eq. (3b) is zero. Any mass and stiffness changes maintaining Eq. (3~) bellow, do not alter this mode. i.e., iu.=w, 2,=x,, (-o:AM + AK)x, = 0 (3C) The mass and stiffness increments, in this case, cause inertia and potential forces that balance each other at the harmonic excitation frequency wr. 1188 2.2 Structural Changes Based on Truncated Data This section describes how to circumvent the incomplete knowledge due to truncation (incomplete number of modes due to finite measuremeвЂќt hand-width). The fix1 that the mass and stiffness matrices are not known can also be circumvented, using measured FRFs a~ will he shown. Solving Eq. (3h) for mass and stiffness modification requires knowledge of the original mass and stiffness matrices. This information, although it can be dctcrmined using finite element analysis,will be practically inaccurate due to modeling errors. In any case the number of extracted modes will be smaller than the number of degrees of freedom (i.e., a truncated model). This section describesthe constraints and a numerical process by which Eq. (3a) can he solved using strictly realistic experimental data. A proof for the validity of the above statementsis also given. Defme the rth Left Modal Vector (LMV), I,, [39] for an 2.3 Main theorems The method is based on 2 thalitimc ~ihichcan be found in 1W. Tlzeorem 1: Structural modil~icaiions maintaining (A,%+ whose steady state response vccto~.x(r)is a linear comhiaation of the truncated modal b&seI; ...вЂ�., modified structureвЂ™s modal vector, i, can he expressed exactly by the original modes, i.e., r вЂ™ c spm[ XI .q,, ] 2.4 Circumventing the knowledge of Mass and Stiffness matrices The above proofs allow us to use Eq.(3h). If we use the LMV definition and Eq.(4), we may write: nommlized modal vectors throughout the paper). We also define L=[l,...l,,]- a matrix whoвЂ™s columns are the (truncated modelвЂ™s) left modal vectors. In a recent paper 1391, it is shown that using the same information required for the extraction of the rth mode, x,, Mi, = MXc, = C,,@,[i] xвЂќ[, ,... I.,] (i.e., the frequency hand which includes this mode) the LMV, I,. can he extracted, as well. This is true for a generally C,EYP x,Em = Lc, Lip /... /вЂњ,I ml Similarly damped system, where the vibratory modes are complex. However, for the sake of simplicity we deal here only with proportional (or undamped) systems, which simplifies the presentation of the procedure. An improved procedure for extracting the LMV is given in [ 6a]. The given equations do not involve the extracted modal vectors and use only the measured FRFs. We choose that any assigned mode will he a linear combination of the original set of known modal vectors. We next show that any excitation wctor, spanned by the .sctof left modal vectors. yields a spatial motion which can he described strictly as a combination of the truncated modal set. An equivalent condition is given for AM, AK to yield similar results. Since we prescribe the modes, WCcan always describe them as the required linear combination. We write the new ti mode as: XE%- 2: If (-C,?m + AK)-, c Sl,un[l, .I,,,], then the Theorem undamped (or proportionally damped) system as /, LA%. The LMV is orthogonal to all vibratory modes, I,, hut one, i.e.. ljTX, = s,,, 6% is the kroneckcr delta (assuming mass i,=xc, c .SPUU[~ . ..l..]. yield a modified structure AKx) where W: is the rtb original natural frequency and c&j the ith element of cr Substituting (%I and (Sh) into the L.H.S. of (3b) (вЂњ:M- K):, = c;,(fi; - m,?)/,c,[i] = L.d;pg{W; -o;}.c, (6) Finally, we can write the eigcnpair assignment equation (~q. (3a)) in terms of the experimentally obtained values and the unknown massand stiffness modifications. (Ak5i:M4)xcr = LвЂ™rliog{Wj -c$}.c, (7) Here the modifications AM(h) and AK(h) are functions of a design vector, h, while cr is a vector вЂњ1вЂ� the linear combination tenвЂќ.% c,,i, (4) Eq. (7) has a solution if and only if there exists a non zero vector C, 01 equivalently 1189 The last equation gives the needed discrete mass modifications to obtain a new mode shape (%,fi,). The only required data for this calcolation is. as mentioned before: the truncated modal base (natural frequencies, right and left modes). equivalently: de+%) =0 [ Ee (Af-r3~AM)X-L~diog{+o~} i 1 m 4. Examples equations The importance of Eq. (7&b) is in those applications where only modal frequencies are to be assigned, still the resultant mode will be spanned by the original ones. Note that the new mode shape does not appear in this equation. The LMV c;m be extracted directly from measured FRFs.A description of the numerical procedurer can be found in [6x by means using the modification We proceed with some example. ranging from simple discrete spring mass systems towards discretized continuous (more realistic) systems. 391. 3. Assignment of modes mass modifications for 4.1 Example 1: Non-uniqueness of the solution Using the redundant d.o.1. for optimization of point We now present a solution imvolving point mass changes only. Such modifications, can be approximated with relative ease in practical situations. Another reason for our interest in such casesis that we are able to obtain closed form solutions. The following example casts the modilication problem into an optimization problem by introducing an appropriate cost function. The cost function can be chosen to obtain additional desired properties. Consider the spring-masssystem described bellow : 3k k Consider mass modification at the exact nodal points at which the mode shape is measured (EMA) or calculated (FE). In this case the mars modification matrix is diagonal. Fig.1 Three degrees of freedom vibrating system AM = diog@вЂњ+}, i=l...n. Under these terms, the L.H.S of Eq.(7) become: A? -iiifAAC, = -c$ ~dipg{Z,[il]~ вЂњ7 Assuming we have a truncated model consisting of the first i14 (8) Using the left modal vectors with Eq.(7) as before, we have a set of linear equations in the point massmodifications: = (L)d;pg{ri, hvo modes: I1 11 -oi}.c, (9) Solving for the massmodifications: We wish to assigna different first mode, вЂ�t, (from SpanXX il=[l=xcf)=ii 3~) and a matching natвЂќra, Again we вЂњse the modification Eq.(7). This can be written in the form Aq=B. A more detailed description of such a formulation will be given in the next example: 1190 Aq=B outlines the desired freedom at measured by a spetial locations is descibed procedure allowed mass Each mode in changes have the curve, the degrees to bc calculated. appendix. In our are to lie on the exact modification will have tl-anslatory cdse,the nodal mass of This points. and wary inertia. Consider next g ={~,,AIw~.A~+KA~~~} The general solution for Eq.(l3) Where AвЂ™ is the pseudo columns span inverse of A the Null-space [3], N is a matrix of A, /3 is a vector 0 0 solution function have only solve the next Q Fig.2 Clamped 0 can -I 0 still For changes standard (IS) be found to be minimized. positive The 1 while if we example adding wish third a if we wish tnass. with two FE elements desired 8 to to assign a vibrating mode specifying the demands: beam [Rad# introduce tbe least Linear-programing beam of We ! cost in the figure following A particular depicted @I coefficients. =(a beam (14) whose (AвЂ™B)вЂ™ cantilever is: q=AвЂ™B+NP arbitrary a the vibrating should while mode mass have having a naturitl the same shape. We (and inertia) apply frequency mode directly shape Eq.(ll) at to;=10 as the original to obtain the changes. We 6 problem: o-Desire&Mode , 4. Find q such that c~q-afnin 2. Subject to : where Aq=B cвЂ™=(l (If4 l l 0 0 O),A.B.q 0 are given in -4 Eq.(l3) 0 The solution 01-3 I - / 2 (oblaincd for exactly Fig.3 ) is: 10 First four modes describing Am, Any choice Eq.(46). of p. will If we realizable restrict say still all mainttdning F.q.(13). 4.2 2: Example without the mode the we have Since span desired without here one). is simple In mode the changes number a Natural demonstrates one. specifications ourselves an intinitc the the assigned original allowed Changing frequency third the rcquircd restrict changing example vibrating mode. the solution, modifications, This meet positive the ability to change affecting the from cast point Desired MC&S 20 30 a Shape the identical the the third where 0 il natural accompanied is to be exactly choose and The Mode frequency mode ( WC just Beam 50 of solutions, matching selection the general oblain to of a Cantilever the desired 40 / in to 30 4, A,,,, = Am, = 0 Ak, = I Akk, = 2 Akk, = 0 Ak, = 3 = 20 the Fig.4 to original mode as designer 1191 10 Desired and accepted Mod? (FE 40 c;IIalxi,wl so s Ori inal natural Modified natural Conclusions We have presented a theory enabling to assign specific eigenfrcqucncies and eigcnvcctors to existing StNctuTcS. The necessary information is extracted via classical modal analysis. Our results arc thus of interest both to analyst using PE models,and cxpcrimenttelists who use EMA results. One important contribution of our work lies in the possibility of obtaining exact results cvcn when the original unmodified model is incomplete. i.c., includes only a partial range of itвЂ™s eigcnsolutions. This is always the case when experimental test results arc used for the design of moditications~ fre uencies fRad/s]2 fre uencies [Rad/s12 m 4 Mass modifications A% AI, 2.65 0.098 0.0667 AI, 0.0181 Table 1 modification results The results show that the goal was obtained exactly. It is also obvious that the other natural frequency where decreased. The actual shape of the added masses can be chosen so as to satisfy Am1 and AI1 for each location. *Acknowledgement: This researchwt supportid by Ulc Bcrstcin Fund for the Promotion of Research REFERENCES 5. Discussion Example number 1 demonstrates the fact that multiple solutions exist, enables to add additional design requirement. In this example we then add the requirement of minimum addition of mass. Example 2 refers to continuous structures. A relatively simple requirement posed, namely a shifting of frequencies. The fact that one modal vector (the third one) is required to stay unchanged, automatically retains itвЂ™s position in the original modes span. It seems important to notice that this demonstrates the possibility of shifting a natural frequency without affecting the associated modal deflection. Ox additional aspect needs to be mentioned. This concerns the fact that the solution could include negative mass changes which might be impossible to implement (in some cases). A more complicated approach based on the same basic equations is required. This approach will seekan approximate solution still complying with the exact mcdclling constraints. Such an approach is not introduced in this paper. Another aspect not covered by us is that of sensitivity with respect to the structural changes. In our view it is possible to develop further the presented equations, in order to calculate the senstttvtttes with respect to mass and stiffness changes, using the same truncated modal data. Low senstttvthes could imply the possibility of relaxing some requirements. 1192 Appendix: Selecting spatial curw When points mode (discrete) least squares least sense) squwcs It is worth natural [2;25]. M, K - Mass and Stiffness AM. AK- Mass and Stiffness c, h vector of constants. vector of s!~ctural At-Pseudo inverse (i running matrices modification Determines index) matrices the modified and geometrical design of the mstix ilh mode. pxamertes A 1193 choice The to extracting the discrete corvc noting that would be the formulation and is described for for the in reference from closest above [6a]. to a (in lcast set of modes. The the disactc a Finite-element element the resort uses a set of interpolating spatial points at which one must by II given a These points case. fit some arc given. the grid In this spanned procedure entries. standard x.d procedure the continuous curve with are given. fomting the o coincide mode. coordinates is to be specified, the desired do not squares .., shape describing usually lo a mode modal basis problem functions, modal model, functions is fairly

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