ANALYSIS OF DAMPED LINEAR DYNAMIC SYSTEMS ANDAPPLICATION OF COMPONENT MODE SYNTHESISByAlexander MuravyovB. A. Sc. ,Chelyabinsk Polytechnic Institute, Russia, 1982A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© Alexander Muravyov, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of ths thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)__________________________________Department of Mehc ,i lea-f eetpThe University of British ColumbiaVancouver, CanadaDate i4p1i 2 £ / 994DE-6 (2/88)AbstractThe analysis of nongyroscopic damped (viscous) linear dynamic systems is presented.Discrete systems having symmetric mass, stiffness and damping matrices are considered.Discretization of the systems is accomplished by application of the finite element procedure.The general case of classically damped systems is considered, and the necessary andsufficient condition for classical damping is given. For a system of this type, it is possibleto specify the damping matrix that will result in each mode having either a prescribeddecay factor or a damped eigenfrequency. This may be accomplished with only a knowledge of the undamped eigenfrequencies. The equations required to accomplish this taskare presented. Graphical results are presented that illustrate the effect of damping formass-proportional, stiffness-proportional and for Rayleigh damping.Nonclassically damped systems are considered and the formulation of a componentmode synthesis (CMS) method for the solution of the free vibration problem is described.The component mode synthesis method is a procedure in which the exact solutionis approximated by one constructed from some basis vectors (e.g., mode shapes) of subsystems (components of the original system). This method allows significant reductionof the eigenvalue equation size due to the use of a limited number of basis vectors. Theapproximate solution found for the lower eigenvalues and eigenvectors is very close tothe exact one due to the proper selection of the basis vectors and the procedure followed(e.g., Galerkin’s method) that determines the best approximation. The use of CMSmethods is especially advantageous in the case of large systems, subjected to numerousmodifications.11In this work the formulation of the CMS method was developed for the general case ofnonclassically damped systems. It was tested for different cases of noriclassically dampedsystems and the excellent agreement with the exact results derived from nonsubdividedsystems was found. Also a new method to treat an unconstrained component for thepurpose of stiffness matrix inversion is presented. The selection procedure of componentmodes is generalized from the undamped system case to the damped one.Some examples of forced responses are considered, particularly, the case of sinusoidalexcitation and the influence of the damping factor is analyzed.The experimental part of this study consists of the designing and testing of a vibrationrig designed to simulate the behaviour of a rigid engine resting on isolators that in turn aresupported on flexible beams. Free and steady-state responses of the rig are experimentallydetermined.Comparison of analytical results with experimental ones show good agreement foreigenquantities and steady-state forced response.111Table of ContentsAbstract iiTable of Contents ivList of Tables viList of Figures viiList of Symbols ixAcknowledgement Xi1 Introduction 11.1 Objectives 11.2 Background 11.2.1 Undamped systems 11.2.2 Classically damped systems . 31.2.3 Nonclassically damped systems . . . 42 Analysis of damped systems 72.1 Classically damped systems 72.2 Nonclassically damped systems 152.3 Free response function computation 172.4 Steady-state response function computation 18iv3 Formulation of component mode synthesis method 253.1 Undamped systems 253.1.1 Case of an unconstrained component. Method of weak springs . 313.2 Nonclassically damped systems 343.2.1 Component mode selection procedure 403.3 Case of an arbitrary number of components 424 Numerical results 454.1 Comparison of CMSFR method with “VAST” program for undamped systems 454.2 Comparison of CMSFR method with “DREIGN” program for undampedand nonclassically damped systems 505 Experimental results 566 Summary 64Bibliography 66A User’s Manual for the CMSFR method 69B Parameters of the vibration rig 73VList of Tables2.1 Steady-state responses, examples 1,2 232.2 Steady-state responses, examples 3,4 232.3 Steady-state responses, examples 5,6 234.1 Four component beam element system: “a” 474.2 Three component beam-bar element system with one rigid-body mode: “b” 474.3 Three component beam-bar-membrane element system: “c” 484.4 Four component beam-brick element system.: “d” 484.5 Two component beam element system 514.6 System with dashpots and lumped masses 524.7 Comparison of eigenvalues for the undamped rig model 534.8 Comparison of the 1st eigenvectors for the undamped rig model 544.9 Comparison of eigenvalues for the damped rig model 554.10 Comparison of the 1st eigenvectors for the damped rig model 555.1 Stiffness and damping properties of the spring isolators (undamped rig) 585.2 Stiffness and damping properties of the spring isolators (damped rig) . 585.3 Experimental frequencies for two rig tests 595.4 Amplitudes, Phase angles for undamped rig 605.5 Amplitudes, Phase angles for undamped rig ... continued 625.6 Amplitudes, Phase angles for damped rig 625.7 Amplitudes, Phase angles for damped rig ... continued 63viList of FiguresEffect of mass-proportional dampingEffect of stiffness-proportional dampingEffect of Rayleigh dampingTwo degree of freedom systemEigenvalues of the system in Fig 2.4Model of the rig with selected nodesEffect of damping on transmissibilityTwo component systemLinear bar elementMode selection procedure131415161722242632414.24.34.44.54.64.74.85.1 Photo of the experimental rig5.2 Reference points on the test rig46494950525253542.12.22.32.42.52.62.73.13.23.34.1 Test examplesFirst mode shape of the system in Fig.4.1,aSecond mode shape of the system in Fig.4.1,aTwo component beam element systemSystem with dashpots and lumped masses . . .Finite element model of the rigFour component presentation of the system . . .Selected nodes for the eigenvector presentation .5758vii5.3 Spring isolator characteristic determination 595.4 8 - 24 Hz excitation sweep 61B.1 Vibration rig, side view 73B.2 Vibration rig, front view 74B.3 Vibration rig, top view of the box 75VI”List of SymbolsM mass matrix of systemK stiffness matrix of systemC damping matrix of systemsystem eigenvaluesystem eigenvectormatrix of mode shapesX displacement vector of systemp vector of modal coordinatesF external force vector, acting on systemF vector of forces acting on componentsm1 mass matrix of 1st componentIc1 stiffness matrix of 1st componentc1 damping matrix of 1st componentm2 mass matrix of 2nd componentIc2 stiffness matrix of 2nd componentc2 damping matrix of 2nd componentsubvector of system eigenvector for 1st componentq2 subvector of system eigenvector for 2nd componentfi force vector acting on 1st componentf2 force vector acting on 2nd componentmatrix of lower (retained) free-free modes of 1st componentmatrix of lower (retained) free-free modes of 2nd componentixmatrix of residual-attachment modes of 1st componentmatrix of residual-attachment modes of 2nd componentvector of free-free mode coordinates for 1st componentvector of free-free mode coordinates for 2nd componentp vector of residual-attachment mode coordinates for 1st componentvector of residual-attachment mode coordinates for 2nd componentAll the remaining notations are described as they occur in the text.xA ckn owledgernentThe author would like to offer sincere thanks to Professor Stanley G. Hutton for hisinsight and advices in the preparation of this thesis. The author would like also toacknowledge the financial support provided by Defense Research Establishment Atlantic.xiChapter 1Introduction1.1 ObjectivesThe objectives of the present research can be formulated as:1) investigating the conditions which lead to classically damped vibratory systems,2) analysis of nonclassically damped systems,3) development of a component mode synthesis method for analysing the free and steady-state responses of nonclassically damped systems,4) construction of an experimental model to investigate the effect of damping upon thesystem response; data acquisition, analysis of experimental results and comparison withthe analytical ones,5) investigating the effect on transmissibility of the damping characteristics of springisolators of the type used to isolate engine vibrations.1.2 Background1.2.1 Undamped systemsThe literature on applications of component mode synthesis techniques to discrete undamped systems contains description of many different methods such as fixed-interfacemethods [lj, [2], free-interface methods [3], [4], [5], [6], [7] and others [8], [9], [lO],[ii].Before discussing existing CMS techniques it is expedient to give a little introductionconcerned with the general idea of component mode synthesis.1Chapter 1. Introduction 2Consider the system of equations of free motion of a discrete undamped systemMX+KX=Owhich leads to the eigenvalue equation(—w2M + K)q 0where M = mass matrix, K = stiffness matrix of the system, w = system eigenfrequency,= system eigenvector.In CMS, the system is subdivided into a set of subsystems (components). Thus thesystem eigenvector (or displacement vector) defined on the entire domain occupied bythe system can be partitioned into parts which are defined on the subdomains. Within asubdomain occupied by kth component the corresponding part of the system eigenvector(denote it by qk) is approximated by a linear combination of some basis vectors 4’ (ingeneral different for each component):= Pkwhere p is a subset of the system generalized coordinates associated with kth component.CMS methods differ to some extent in the selection of these vectors. The basisvectors used can be classified into the following groups: 1) free-free vibration modes,2) fixed-interface vibration modes (by interface is meant a common boundary betweenadjacent components), 3) attachment modes, 4) constraint modes, 5) residual-attachmentmodes, 6) rigid-body modes. Free-free and fixed-interface modes are obtained by thesolution of corresponding eigenproblems for free, or fixed (clamped) interface components.Constraint and attachment modes are respectively vectors of static displacements causedby a unit displacement, or unit load applied at the interface. The definition of residualattachment modes will be given in chapter 3.Chapter 1. Introduction 3The combination of free-free modes (including rigid-body modes) with residual-attachmentones is usually viewed as a free-interface method. The fixed-interface method uses fixed-interface vibration modes with constraint (or attachment) modes, plus rigid-body modes.A review and comparison of fixed- and free-interface methods has been conducted by anumber of authors [2], [9], [12], [13].In [14], [15], the use of a set of admissible functions different to vibration modesis proposed, e.g., low order polynomial functions. This gives an opportunity to avoideigensolutions at the component level. For discrete systems these polynomial functionsare replaced by their values at the nodes of the system, i.e. by vectors. However the choiceof these functions seems to be problematic. For 3-ID systems with complex geometry,boundary conditions, nonuniform mass and stiffness distribution it is not clear how manysuch basis vectors would be required for an accurate approximation.1.2.2 Classically damped systemsDescription of this type of system can be found in many references, e.g., [16], [11], [17],[18], [19], [20]. Usually consideration was limited to systems having Raylegh dampingC = ciiI + /3K, where and /3 are constants. The necessary and sufficient conditionswhen a system has classical normal modes (the eigenvectors are the same for dampedand undamped systems, i.e. they are real vectors) was formulated in [21],[22] as M’Ccommutes with M’K, where C damping matrix of the system.The analysis of this type of system is considered in chapter 2. For a system withdistinct undamped natural frequencies the necessary and sufficient condition of proportionality for C will be given in section 2.1 in terms of matrices M, K.Note that application of CMS methods to this kind of system can be accomplished by:i) application to the associated undamped system (C = 0), ii) construction of undampedsystem eigenvector of interest q5, iii) computation of system eigenvalue A (complex) ofChapter 1. Introduction 4interest from the equation.;\2/5TMl + ,\cbTCcb + bTKq5 = 0When structural damping is considered (usually related with sinusoidal excitationof a given frequency) the equivalent viscous damping matrix can be determined. Inthis case it is taken as proportional to the stiffness matrix and inversely proportional tothe excitation frequency. Description of this type of damping can be found in [23] forexample.1.2.3 Nonclassically damped systemsThere have been some efforts to develop CMS methods for nonclassically damped systemsin the past, though there is very little literature on this topic.The system of equations of free motion of the system is expressed as following:MX+CX+KX=0 (1.1)where the damping matrix cannot be diagonalized simultaneously with M and K.One approach assumes that the system is lightly, or proportionally damped [1], [2],[11] and the system eigenvector is constructed from iindamped vibration modes (realvectors) of components. This representation does not take into account the dampingproperties of the components and cannot be accurate, especially in the case of significantdamping.To solve the eigenvalue problem for a nonclassically damped system it is expedient torewrite the equation of free motion (1.1) in the reduced, or so-called state-space form:AY+BY=0 (1.2)Chapter 1. Introduction 5CM KU Xwhere A , B = , YM 0 0 —M XEquation (1.2) leads to the eigenvalue equation(AA + B)q 0which yields complex eigenvalues A and complex state-space eigenvectors q.Hale [15] proposed to use vibration modes of undamped components and to improveaccuracy by iteratively generating component state-space vectors. Hasselman and Kaplan[241 used complex vibration modes, but it was not shown how they were determinedand the residual-attachment modes were not taken into account. In [25] a modifiedfixed-interface method is shown, where fixed-interface vibration modes of undampedcomponents are replaced by corresponding complex modes.Howsman and Craig [26] showed a state-space free-interface formulation that uses aset of free-interface vibration modes, but instead of residual-attachment modes a set ofattachment modes was used.Another application of a state-space free-interface CMS technique for nonclassicallydamped systems was shown in [27] for the case of a two component system, where alongwith free-interface vibration modes a set of residual-attachment modes was used. Thisformulation has similarity (in terms of coupling procedure) with the formulation developed in this thesis, however in [27] the selection of free-free vibration modes was notgeneralized from the undamped system case to the damped one and the authors useddifferent approach for determination of residual-attachment modes in the case of unconstrained components.In this thesis a formulation of state-space free-interface method (CMSFR) has beendeveloped for systems with nonclassical damping for the case when mass, stiffness anddamping matrices of components are symmetric. A system eigenvector is constructedChapter 1. Introduction 6from state-space free-free vibration modes of damped components and state-space residual-attachment modes. A complex eigensolver is used to determine the free-free vibrationmodes of each component. The formulation is demonstrated for an arbitrary number ofcomponents, which can be constrained or unconstrained. Also a new method is presentedto treat an unconstrained component, which is more economical in the computationalsense than the method shown in [27]. A selection procedure for the retained free-interfacevibration modes is developed for this nonclassically damped system case.Chapter 2Analysis of damped systems2.1 Classically damped systemsIn this chapter, unsubdivided systems are considered. The general case of classical damping is analysed. An important property of this kind of damped system is that it hasclassical mode shapes (eigenvectors), i.e, the same as the associated undamped system.The system of equations of free vibration of a discrete nongyroscopic damped systemcan be expressed in the formMX+CX+KX=O (2.1)where M, C, K represent mass, damping and stiffness matrices of order N. Under certainconditions this system of, in general, coupled differential equations can be transformedto a system of uncoupled equations by a congruence transformation. It is known thatthe mode shape matrix of the undamped system can make this transformation withrespect to the M and K matrices. In this study the necessary and sufficient conditionof diagonalization of C by this matrix of undamped mode shapes is considered. For ageneral system it has been formulated, e.g., in [21], [22] as M1C commutes with IVI’K.It should be noted that the necessary and sufficient condition for diagonalization ofmatrix C in terms of undamped system modes cP=orC = (T)_1[a]_17Chapter 2. Analysis of damped systems 8( [aj is a diagonal matrix) implies that N real coefficients a, is enough to describe allthe variety of possible damping matrices for the given system defined by I’I, K.Limiting consideration to systems that have distinct (no repeated) undamped naturalfrequencies the necessary and sufficient condition of proportionality imposed on matrixC will be formulated in terms of matrices M, K (M is positive definite and K is positivesemidefinite) in the theorem below.Theorem.For the given matrices M, K of order N in (2.1) the matrix C will be diagonalizedsimultaneously with M, K if, and only if, it is represented asC = Mg1(MK)’’ (2.2)where gj, 1 = 1, 2, ..., N are arbitrary coefficients.Proof.Necessity.There is such a matrix 4 consisting of undamped mode shapes thatTMpJ= [b}TC= [a2]where [b2] = b, [a2] = a are diagonal matrices and the matrix 4 is mass normalized (firstequation), so the elements of b will be the natural frequencies squared. Rewrite the aboveexpressions asC (T)_1a4_ (2.3)K = (T)_lb4_l (2.4)Chapter 2. Analysis of damped systems 9andM =(4T)1 (2.5)orM1 = (2.6)It is obvious that there is one to one mapping, namely, a —*--- C, b —‘÷— K. Expressmatrix a asa = g1I + g2b + ... + 9Nb (2.7)Equation (2.7) can be rewritten as a system of linear equations for coefficients 9k, Ic =gi a11 1. L2 N—1£ (J U1 . •- Ui92 a21 1. x2 lN—i‘ ‘-‘2 “2 ‘-‘2 (2.8)1 1 12 1N—iI “N ‘-‘N UN9N aNThe determinant of this system (the Vandermonde determinant) is non-zero because thereare no repeated values b1 (natural frequencies squared), so the solution is guaranteed andthis representation (2.7) holds. Rewrite (2.7) using (2.3),(2.4)TC gk(TK)klTherefore any matrix C can be represented asC = (T)1[9J+g2TK + + ...++gJ\TK4. TK]-iand using (2.6) one can obtainC =Chapter 2. Analysis of damped systems 10as required.Sufficiency.The damping matrix is given as in (2.2). Using (2.5),(2.6),(2.4) one can obtainC = (T)_1 gi(T(T)_lb_l)l_l == (T)_1(91+ g2b + ... + 9Nb)Thereforerepresents a diagonal matrix as a sum of diagonal matrices, which was required to prove.It may seen from (2.2) that if the first two terms are taken thenC=giM+g2K 1=1,2which is Raylegh damping. If more terms taken, e.g. three, thenC=gM+g2K+g3M’K 1=1,2,3and so on.Remark 1.Note that for systems with repeated undamped natural frequencies expression (2.2) willrepresent a sufficient condition, but not the necessary one. This means that there mightbe matrices C, which are not represented as in (2.2), but are diagonalizable.Remark 2.For a system with repeated undamped eigenvalues the representation (2.2) will hold asthe necessary condition if the vector of damping coefficients [ak] lies in the space ofeigenvectors of the system matrix (2.8). Then, for example, if there are m repeatedeigenvalues it means that eigenvectors of the Vandermonde matrix will represent theChapter 2. Analysis of damped systems 11vector subspace dimensioned by N— m + 1. The form of these eigenvectors is notconsidered here and an explicit expression of vector [ak] as a linear combination of theseeigenvectors is not shown. It is left to note that in this case there will be an infinitenumber of solutions (vectors [g.i]) which yield the given damping coefficient vector [ak].Consider now the general case of classical damping for a system with distinct un-.damped natural frequencies. Substituting (2.2) in (2.1) and using the transformation tomodal coordinates X p with premultiplication by T gives:+T [Mg(M_lK)k_l] + Kp = 0 (2.9)where 4 is the matrix of classical mode shapes. Assuming that 4 is mass normalizedthenwhere A is a diagonal matrix, whose elements are the undamped natural frequenciessquared , w. The matrix equation (2.9) can then be written as a system of uncoupledequations:i=1,2,...,N (2.10)Equations (2.10) yield the eigenvalues of the damped system, namely:= (—g’ + ), = —where D= (Z gkh)2 — 4i,. If D < 0 the roots are complex conjugate, otherwisethey are real.Considering the case of complex conjugate roots:Re()_—g1 (2.11)and_________________1== \J71i — (gk_1)2 (2.12)Chapter 2. Analysis of damped systems 12Prom which it may be seen that(Re())2+ (Im()j))2 (2.13)Therefore the absolute values of complex eigenvalues are independent of arbitrary proportional damping. It may also be noted from (2.12) that for classical damping the dampednatural frequencies are always less than the undamped frequencies.If it is required to modify the natural frequencies of a given system by modificationof the damping matrix, equations (2.11),(2.13) can be used to determine the coefficients9k in (2.2) that yield the required eigenvalues. For example, upon the prescription of theimaginary parts of the first (lowest) L eigenvalues (L = 1, or 2,. ..,N) equations (2.13)can be used to determine the corresponding real parts. Then the use of (2.11) yields thefollowing system of algebraic equations for the kept (non-zero) coefficients gk:1 ?7 ... 77L—1 ReAi)1 1 72 72 ... 72 g2 — Re(A2)21 7L ... gL Re(AL)The determinant of this system is the Vandermonde determinant, which is not zero if thereare no repeated . Therefore for a system with distinct undamped natural frequenciesthere will be a unique solution for 9k, k 1, 2, ..., L. Note that only a knowledge ofundamped natural frequencies is necessary to construct the required C.Three particular cases of proportional damping are now considered.i) Mass-proportional damping: the damping matrix is represented as C = aM, whichcorresponds to the case when all g, = 0, except g = a. The effect of mass-proportionaldamping upon the damped frequency (Im()’,)) and decay factor (Re(A)) is illustratedgraphically in Figure 2.1. The undamped natural frequencies Wmzn, W, and Wmax representrespectively the minimum, some intermediate and the maximum value for the system withChapter 2. Analysis of damped systems 13Re(A)2 2 Decay FactorFigure 2.1: Effect of mass-proportional dampingno damping. The equation (2.13) is shown by solid circles, while the dashed lines representequation (2.11). Intersection of the solid circles with the dashed lines corresponds to thelocation of eigenvalues. If no intersection occurs that mode is overdamped for that valueof damping. Thus if a < 2Wmj,-, all modes are oscillatory and if a > 2Wmax all modes areoverdamped. It may be observed that mass-proportional damping has a larger influenceon the lower mode frequencies than those of the higher modes.ii) Stiffness-proportional damping: the damping matrix is represented in this case asC = /3K, corresponding to the case when all g, = 0, except g2 = /3. To illustrate theeffect of stiffness-proportional damping introduce the following notations: x = Re),y = Im()). Equations (2.11),(2.12) lead toy2 + (x + 11/3)2 = 1//32which represents the equation of circles shown as dashed circles in Figure 2.2. Equations(2.13) are shown by solid circles and upon the given /3 the eigenvalue location is determined by the intersection of the corresponding dashed circle with the solid ones. In thiscase it may seen that the stiffness-proportional damping has a larger influence on theIm(A)Frequencymax(4)jC.) mmChapter 2. Analysis of damped systems 14Re(X)Decay FactorFigure 2.2: Effect of stiffness-proportional dampinghigher mode frequencies. From the figure it may be noted that if 3 > ---- all modes areoverdamped and if /3 < ---— all modes are oscillatory.iii) Rayleigh damping g = c, g = 3, i.e., C = cM + /3K.Using the same notations equations (2.11),(2.12) lead toThe graphs of these circles are shown as dashed lines in Figure 2.3. Upon the given c, /3the eigenvalue location is determined once again by the intersection of the correspondingdashed circle with the solid ones. Intersection of dashed circle with x axis gives two values:= ,x= ‘. The condition that all the modes are oscillatory is thesimultaneous satisfaction of the following inequalities: and >Wmax. If > Wmax, or < or a/3 > 1 all modes are overdamped.Note that at a/3 = 1 the radius of the dashed circle becomes zero.It may seen from the figure that it is possible to choose Rayleigh damping coefficientssuch that the lowest and higher modes are overdamped, but the intermediate modes areoscillatory.Im(X)Frequencymax(A)j“nunChapter 2. Analysis of damped systems 15Re(X)Decay FactorFigure 2.3: Effect of Rayleigh damping2.2 Nonclassically damped systemsIn the case of nonclassically damped systems the general damping matrix cannot beexpressed in the form defined by equation (2.2). The damping matrix of this type ofsystem cannot be diagonalized simultaneously with M and K. To solve the eigenvalueproblem it is expedient to rewrite equation (2.1) in the reduced or so-called state-spaceform:AY+BYO (2.14)whereCM K 0 XA , B= , Y= (2.15)M 0 0 —M XThe equation of free motion in a system mode Y(t) = e)tq5 with eigenvalue ) andstate-space eigenvector will have the following form:()iA + B)q = 0 (2.16)where A, B are real symmetric 2N x 2N matrices, which are not positive definite. Notea1Im(X)Frequency0 max(A) mmXL XRthat the eigenvectors will be orthogonal with respect to A, B. The eigenvectors canChapter 2. Analysis of damped systems 16be normalized by settingbAq=1 i=1,2,...,2Nor if 4 is defined as a matrix of A-normalized eigenvectors, then the following will hold:TB= —Awhere A is the diagonal matrix of eigenvalues.There are different numerical methods for the solution of such a kind of eigenproblems.The solution of this problem can be found, e.g., using the QR method [281, [29]. In generalthere will be 2m real eigenvalues and N — m pairs of complex conjugate ones.The influence of damping on the response is complicated as the eigenvectors themselves depend upon C. To illustrate the effect of nonclassical damping the behaviour ofthe simple system shown in Figure 2.4 is analyzed, where m1 m2 k1 = k2 = 1. Thelocation of eigenvalues for different values of c is shown in Figure 2.5.Figure 2.4: Two degree of freedom systemAs may be noted the behaviour is quite different to that of a classically dampedsystem. As the damping constant c is first increased the imaginary part (frequency)of the lowest eigenvalue increases (a phenomenon not observed in classically dampedsystems) and that of the higher mode decreases. The real parts both increase with c upto a certain point after which the real part of the second mode begin to decrease. At adamping value of 2 both roots coincide and at this point there is a discontinuity in theChapter 2. Analysis of damped systems 17—1—0.5 0 Re(A)Figure 2.5: Eigenvalues of the system in Fig 2.4monotonic behaviour of the eigenvalues. The imaginary part (frequency) of the secondeigenvalue then starts to increase monotonically to a frequency of 1 rad/s. This situationcorresponds to the physical condition when c is so high that the first mass does not moveand the system is responding essentially as a single degree of freedom system. The realpart of the first eigenvalue monotonically increases and at c = 2.5 this mode becomescritically damped.2.3 Free response function computationConsider a homogeneous equation of motion of a nonclassically damped system with Ndegrees of freedom:Im(X)1.618c=1.5Vc=10.8660.618c=2.5MX-i-CX+KX=OChapter 2. Analysis of damped systems 18To find a homogeneous solution the eigenproblem should be solved first. The solution ofeigenproblem is usually conducted by application of the state-space representation (2.14)-(2.16). Note that eigenvalues and eigenvectors of this problem are in general complex.The homogeneous solution will be presented in a complex formY AjeA3tj (2.17)where ,\,, are the eigenvalues and eigenvectors and A are complex coefficients.The real form solution can be extracted from (2.17). Assume in general there is 2mreal eigenvalues and N — m pairs of complex conjugate ones, then the real homogeneoussolution Xh can be presented as2m N—rn N--TnXhom(t) = ckektRe[cbk] + c2rn+jRe[et]+ CN+m+jIm[e3cI3jk1 jzz1 j1where are 2m real eigenvalues and are N — m representatives of complex conjugateeigenvalues, , 4 are the corresponding complex state-space eigenvectors. The realcoefficients ek, k = 1,2, .., 2N will be determined from the initial conditionXh(O) X(O) Xpar(0)which represents the system of 2N real linear equations, where X is the total solutionand Xpar a particular term (in the case of external force presence) of the solution.2.4 Steady-state response function computationThe forced response due to a sinusoidal excitation is considered in this section. In theexperimental part of this work this type of excitation will be considered. The equationof motion of a nonclassically damped system:MX-CX + KX F(t) (2.18)Chapter 2. Analysis of damped systems 19where the external force vectorF(t) F0sinwL (2.19)Although steady-state response X carl be found from (2.18) directly) the state-spacerepresentation will be used to show the relation between complex and real forms of thesolution:AY + BY — F (2.20)C M K 0 Fd(t)where A = , B= ,Y = , F5 =M 0 0 —M 0The seeking complex solution is(Y + z)et (2.21)as the response to a given complex forceFd (fi + if2)et = ficoswt f2sznwt + i(fisznwt +f2coswt) (2.22)The steady-state response in real form may be expressed asX = Re[}jj (2.23)orX = Im[YdjProceeding with (2.23)) the given force will beF(t) Re[Fdj = ficoswt — f.sinwtIn the case (2.19)f = 0 (2.24)f2 = —F0 (2.25)Chapter 2. Analysis of damped systems 20Substituting (2.21),(2.22) in (2.20) and cancelling e one can obtain(Y-I- iY)’iw Y + zY[c M]+ [ K o] = (f + if2) (2.26)—Cl’i + eI’2)w (}‘ + zY)iwRewriting equation (2.26) asYi(—Mw2+ K) - YCw + i(Y1Cw + Y(—Mw2+ K)) = L + if2one can obtain the system of linear equations for Y1,Y2 (taking into account (224),(2.25))Cw —Mw2+K Y —F0= (2.27)—Mw2+K —Cw 0The solution of (2.27) yields the steady-state responseX = Re{(Y-1- iY2)etj= Ycoswt — Ysznwtwhich can be presented in the following formX = D.si’n(wt + )whered2 ... dN]is the amplitude vector of the forced response andN]is the phase angle vector. In more detaild, \/‘‘ + }‘andY1zan’b, =2zChapter 2. Analysis of damped systems 21where Yh, Y2 are ith components of vectors Y,, Y respectively.Remark 1.Note that system matrix of (2.27) (denote it by S) can be inverted in the followingways: i) if damping matrix C 0, then0 (—Mw2 + K)-1(—Mw2+K)’ 0ii) if C is not a zero matrix and assumed invertible, then[ a11 a,2 ]a,2 —a11where a1, = G’(DG—1+ GD’)’, a12 = a,,GD’, D = Cw, G --Mw2 + K. Thisleads to economy of computer time because the inversion of a matrix 2N by 2N is replacedby inversion of a matrix N by N in case i) and by three inversions of a matrix N by Nin case ii).Below some numerical results are presented, which show the influence of dampingon the steady-state vibration response. As an illustration a finite element model of thevibration rig (Fig.2.6) was considered. Parameters of the rig are presented in AppendixB. A vertical concentrated load F = F0sinwt was applied at the centre of the box, whereF0 = 30 N and w = 32 Hz were the same for all examples. Of interest is the influence ofdamping properties in the spring elements a, b, c, d. The following values of a dampingconstant were assigned: 1) c 0 (undamped system), 2) c = 0.002k3, 3) c 0.005k3,4)c = 0.01k3, 5) c = 0.02k5, 6) c 0.05k3, where k5 = 12 N/mm was the vertical axialstiffness of the elements a, b, c, d and kept constant. The influence of c upon the responseof the system was analyzed. For the chosen nodes (Fig.2.6) the amplitudes and phaseangles were computed and the results for the vertical component of motion are shown inTables 2.1-2.3.Chapter 2. Analysis of damped systems 22YFigure 2.6: Model of the rig with selected nodesIt may be noted from Tables 2.1-2.3 that amplitudes of motion of points 5,6 aredecreasing with increase of c and phase angles change.The ratio A3/A0was calculated, where Ab0 is the amplitude of vertical motionat the centre of the box and A3 is the amplitude of vertical motion of the supportedbeams (the average amplitude of the points 5,6 was taken). This ratio characterizes thetransmissibility of motion from the box (points 1-4) to the supported beams. The changeof this transmissibility depending of c/k3 is presented in Fig. 2.7. It may be noted thatincrease of damping up to the c = 0.01k3 leads to decrease of amplitudes of vibration ofthe supported beams, in other words provides decrease of transmissibility of motion fromthe box to the supported beams. After c = 0.01k3 the influence of c upon transmissibilityis insignificant, though the phase angles continue to change.25xChapter 2. Analysis of damped systems 23Example 1,_c_= 0 Example 2, c 0.002k3Node Amplitude, mm Phase angle, ° Amplitude, rrirn Phase angle,1 0.0280 180. 0.0261 183.2 0.0306 180. 0.0242 188.3 0.0324 180. 0.0261 188.4 0.0264 180. 0.0241 184.5 0.224 0. 0.11.6 82.76 0.228 0. 0.118 82.6Table 2.1: Steady-state responses, examples 1,2Example 9, c = 0.005k5 Example , c 0.01k3Node Amplitude, mm Phase angle, ° Amplitude, mm Phase angle,1 0.0256 183. 0.0255 183.2 0.0224 184. 0.0221 181.3 0.0243 184. 0.0240 181.4 0.0235 183. 0.0234 184.5 0.0674 122. 0.0540 147.6 0.0687 122. 0.0550 147.Table 2.2: Steady-state responses, examples 3,4Example 5, c = 0.02k3 Example 6, c 0.05k5Node Amplitude, mm Phase angle, 0 Amplitude, mm Phase angle, 01 0.0253 186. 0.0243 193.2 0.0221 179. 0.0229 176.3 0.0241 178. 0.0249 175.4 0.0232 186. 0.0222 194.5 0.0499 162. 0.0489 173.6 0.0509 162. 0.0499 172.Table 2.3: Steady-state responses, examples 5,6Chapter 2. Analysis of damped systems 24D) II Print II rabo.plt II TransmissibilityI:0.00 0.01 0.02 0.03 0.04 0.05Damping constantFigure 2.7: Effect of damping on transmissibilityChapter 3Formulation of component mode synthesis method3.1 Undamped systemsIn this chapter, the formulation of CMS method developed in this study will be presentedfor the case of undamped systems (in this section) and nonclassically damped systems insection 3.2. This formulation will be shown on an example of a two component systemand the case of an arbitrary number of components will be considered in section 3.3.The component mode synthesis method is a procedure in which the exact solutionis approximated by one constructed from some basis vectors (e.g., mode shapes) of sub-systems (components of subdivided system). This method allows a significant reductionof the eigenvalue equation size due to the use of a limited number of basis vectors. Theapproximate solution for the lower eigenvalues and eigenvectors is very close to the exactone due to the proper selection of the basis vectors and the use of Calerkin’s methodthat determines the best approximation.Consider a system subdivided into two adjacent components (call them the 1st andthe 2nd) with interface S (Fig.3.1). The equation of free motion of the system in a naturalmode X(t) = etq with frequency w and eigenvector [& 2jT can be written in thefollowing form:2rn1 0 k1 0 fi+ I = (3.1)0 m2 0 k2 ) 2 f2where term et was cancelled, m1, m2 are the mass matrices of the components and25Chapter 3. Formulation of component mode synthesis method 26Figure 3.1: Two component systemk1, k2 the stiffness matrices. The force vectors f, f2 will contain only interface forces(interaction between components), which appear as external forces at the artificial subdivision of the system and all the remaining components of fi, f2 corresponding to thecomponent internal degrees of freedom will be zero, because of the absence of externalforces. As can be noted the system eigenvector was also subdivided into two subvectorsq, qS2 corresponding to the 1st and 2nd components. Rewrite (3.1) in the abbreviatedform:(w2M + K)= f (3.2)Subvectors q, q will be approximated in the following form:1’= [ ] [;;] (3.3)22 [ ] [:;] (3.4)where , 4 = matrices of lower (retained) free-free vibration modes (including rigidbody modes) for 1st and 2nd components respectively (determined as the result of eigenproblem solution for free-free component), , 4 = matrices of residual-attachmentmodes for 1st and 2nd components, p1, p vectors of free-free mode coordinates for 1stSYxChapter 3. Formulation of component mode synthesis method 27and 2nd components, p, p vectors of residual-attachment modes mode coordinates for1st and 2nd components.Consider the determination of residual-attachment modes. Take the 1st componentequation from (3.1)(—w2mi + k1) fiUse the transformation to modal coordinates q iPi ( complete set of free-freemass-normalized modes of the 1st component) and premultiplying by 4 one can obtain:1 10 w 0 Pu1w2 + I fi (3.5)0 I 0 w1 ) Phiwhere 4 was partitioned into two sets,(index I means lower (retained) modes,h higher modes of component), w1, w diagonal matrices with natural componentfrequencies squared.The assignment of the number of retained modes for each component will depend onthe range of system frequencies, which are supposed to be evaluated by an applicationof this method. It may seen from (3.5) that ifw2 <<min[w1j (3.6)then the approximate expression for the modal coordinates Phi follows [101, [30]:1 2 i—iTPhi LWhij ‘‘hiJ1Thus the contribution of the higher modes to the subvector q can be approximated asr 2i—icTrh1Phi ‘I’hi[Whll ‘hiJ1where the columns of matrix R1 = 4h1[wl]_1’l corresponding to the interface degreesof freedom are called residual-attachment modes. Due to (3.7) the interface forces in fiwill be identified with the residual-attachment mode coordinates p.Chapter 3. Formulation of component mode synthesis method 28The number of retained free-free component modes should be high enough, in orderthat condition (3.6) and consequently expression (3.7) are satisfied at the proper level,such that a good appoximation for the subvector q51 by means of the basis vectors cI andF (the selected columns of matrices , R1 respectively) will be obtained. The matrixR1 can be expressed in an advantageous form [10], [30], [4]:n i—i .c. 2i-1c.TfLi= — ‘‘11 Wj P11This means that computation of the lower modes only is required for each component,which leads to economy of computer time. In the case of an unconstrained component,inversion of the stiffness matrix will be considered below. An analogous determinationof residual-attachment modes is conducted for the second component.Thus combining (3.3), (3.4) the system eigenvector is approximated as1q•=4 0 0 p’0 0 4 4 paP2or in abbreviated formç=p (3.8)Imposing the condition of force continuity at the interface nodes gives:(3.9)Using (3.9) one can express the vector p through a vector q’, which will not contain p:p1 I 0 0P10 I 0=(3.10)0 0 IP2p 0 —I 0Chapter 3. Formulation of component mode synthesis method 29or in abbreviated formp —Subvectors 4 and c2 can be written in the following form:=[]=[: :][z]and2B ‘2B P2=jI a a‘P2 2i 2i Plwhere B = displacements at the interface, qY all the remaining displacements. Applythe equation of continuity of displacements at the interface:(3.11)thus it follows from the above expressions that4BPi + ‘1BP1 = 2BP2 2BP1Therefore one can express the interface forces in terms of free-free mode coordinates:p T1 T2P2where=+ 2B)(iB)=+ 2B)(2B)Thus the following relation will hold:P1T1 T2 (3.12)0 IChapter 3. Formulation of component mode synthesis method 30or in the abbreviated form:‘Iq =3 qTherefore the vector p of all generalized coordinates can be expressed in terms ofindependent generalized coordinates q = [p pjT:p =or(3.13)where 3= 13’13”.Substituting (3.13) in (3.8) the relation between the approximate system eigenvectorand vector q will be:(3.14)Substituting (3.14) in (3.2) one can obtain:(—w2M + K)/3q = f + ewhere the quantity e represents an approximation error. Galerkin’s method (premultiplying e by basis functions and setting it to zero) yields the following matrix equation+ k)/3q=(3.15)and the vector on the right side vanishes in the absence of external forces. This fact isdue to the equal displacements and opposite forces at the interface. To show it considerthe product of an arbitrary vector q and vector/3TTf, namely,qT/3Tf= [ ] [:]Chapter 3. Formulation of component mode synthesis method 31and if only interface forces are present, then the above expression is transformed to thefollowing one:iBiTi al i LBiTi aLY1i Pii i P2Taking into account the (3.11) and (3.9) the above expression becomes a zero vector.Therefore the product of vectors q and 13TTf is always zero for an arbitrary vectorq, which means that vector/3T4f must be zero and the right side of equation (3.15)vanishes.Thus the final condensed equation of motion for two coupled components (or thewhole system) in terms of generalized coordinates can be written in the following form:(—w2M4f K4)q = 0where j , K4 =/3TTk will be real symmetric matrices. Thisequation represents an eigenvalue problem. The eigenvalues w2 and eigenvectors q willbe all real quantities.3.1.1 Case of an unconstrained component. Method of weak springsA new approach to treat an unconstrained component is developed by imposing “weak”constraints (springs) on the system, which remove the singularity of the stiffness matrixand make it invertible. The stiffness of “weak” springs can be assumed to be 107kwhere k, are corresponding diagonal elements of the stiffness matrix, which are supposedto be modified. If a smaller value is chosen the modified stiffness matrix may not be invertible (the numerical aspect of inversion should be taken into account). This methodrequires less computational effort than the method described in [27]. The lowest (strictlyspeaking non-zero, but near zero) frequencies will correspond to the rigid-body modes.These rigid-body modes will be strictly speaking “flexible” modes due to the introducedChapter 3. Formulation of component mode synthesis method 32“weak” springs, but the stiffnesses of these “weak” springs should be negligible compared to the component and system stiffnesses, so these “flexible” modes will actuallycorrespond to the motions of the component as a rigid body.To illustrate this method consider the example of an unconstrained component suchas a linear bar element (Fig.3.2), which can move along the X axis. The bar element hastwo degrees of freedom x1, x2 and the corresponding stiffness matrix is1 —1L—1 1where B = modulus of elasticity, A = cross-section area, L = length of the element. Thedeterminant of k equals 0. The mass matrix has the following form:pAL 2 1m=—6 12where p density of material. The eigensolution will yield:0, = [a, a] (rigid — body mode)2, = [a, —a] (flexible mode)where ri = 6—s.Introduce a weak spring with stiffness = E, attached to one of the nodes (Fig.3.2),then the stiffness matrix becomes invertiblek’—-1+E —lL—1 1Xi__X2Figure 3.2: Linear bar elementChapter 3. Formulation of component mode synthesis method 33If = 0.00 1, then the eigensolution will yield:X1 0.00016, = [a, 1.0005a] (“rigid— body” mode)2.0005, q2 = [a, —0.99983a] (flexible mode)The approximation error for eigenvalues and eigenvectors is less than 0.05%. Thereforean unconstrained component can be modelled as constrained if is small enough.Chapter 3. Formulation of component mode synthesis method 343.2 Nonclassically damped systemsIn this section the formulation shown in section 3.1 is generalized to nonclassicallydamped systems, using state-space representation.Consider again a system subdivided into two adjacent components with interface S(Fig.3.1). The equation of free motion of the system subdivided into two componentswill have the following form:m 0 U1 c1 0 U1 k1 0 U1 f1(t)+ . + =0 m2 U2 0 C2 U2 0 k2 U2 f2(t)where U1, U2 are displacement vectors of the 1st and 2nd components respectively.The state-space representation reduces this equation to the following one:A+1Y=FC(t)whereA10 B10 Y1 F1A= ,B = , Y = , F =0 A2 0 B2 F2c1 m1 c2 m2 k1 0 k2 0A1 , A2 = , B1 , B2 =m1 0 m2 0 0 —m1 0 —m2U1 U2 fi 12, = , F1 = , FUi U2 0 0The equation of free motion of the system in a system mode Y e\tcb with systemeigenvalue ) and system state-space eigenvector= [ 2jT (subdivided also) will havethe following form:I A1 0 B1 0 F1+ I = (3.16)0 A2 0 B2 ) 2 F24Chapter 3. Formulation of component mode synthesis method 35or in the abbreviated form:(A 1 B) (317)where F is defined by the equation:The force vectors F2 will contain only interface forces (interaction between components) which appear as external forces at the artificial subdivision of the system and allthe remaining components of F2 corresponding to the component internal degreesof freedom will be zero, because of the absence of external forces. In this state-spacerepresentation \, q, F are assumed complex.The complex subvectors q and q2 are approximated in the following form:-_p11—a(3.18)P2= [ ] a (3.19)P2where free-free vibration modes ‘I, 4 are complex and a complex eigensolver is used tocompute them. Note the modal coordinates pt, pa are also complex in general.Consider the determination of residual-attachment modes 4’, . Take the 1st component equation from (3.16)(A1 + B1) = F1Use the transformation to modal coordinates & 4ii ( = complete set of free-freeA-normalized modes of the 1st component) and premultiplying by one can obtain:10 A11 0A-1- I = F1 (3.20)0 I 0 Ahl ) PhiChapter 3. Formulation of component mode synthesis method 36where 4 was partitioned into two sets,(index 1 means lower (retained) modes, hhigher modes of component), )khl diagonal matrices with component eigenvalues.The assignment of the number of retained modes for each component will depend onthe range of system eigenvalues, which are supposed to be evaluated by an applicationof this method. It may seen from (3.20) that ifI << rnin{ hi 1 (3.21)then the approximate expression for the modal coordinates Phi followsi i—iiT i—iPhi [Ahlj ‘hi”i*Thus the contribution of the higher modes to the subvector can be approximated ashi Phi 4’hi[)hi]hiFi* (3.22)where the columns of matrix R1 4, [)hih corresponding to the interface degreesof freedom will be called state-space residual-attachment modes, which will be complexin this case. Due to (3.22) the interface forces in F1 will be identified with the residual-attachment mode coordinates p.The number of retained component modes should be high enough, in order thatcondition (3.21) and consequently expression (3.22) are satisfied at the proper level, suchthat a good appoximation for the subvector 4 by means of basis vectors 4 (theselected columns of matrices 4, R1 respectively) will be obtained.The matrix R1 can be expressed in the following form:R1 B—This means that computation of the lower eigenvectors only is required for each component, which leads to economy of computer time. The inversion of matrix B1 is determinedChapter 3. Formulation of component mode synthesis method 37by independent inversion of matrices rn1,k1. In the case of an unconstrained componentthe inversion of the stiffness matrix was discussed before. An analogous determinationof residual-attachment modes is conducted for the second component.Thus combining (3.18), (3.19) the system eigenvector is approximated as= = 4 0 0 pZ0 0 F 4 paP2or in abbreviated form= 4p (3.23)Apply the equation of force continuity at the interface:p = —p (3.24)Equation (3.24) is used to eliminate the attachment mode coordinates p from the generalized coordinate vector p, i.e., using a matrix transformation (see analogous transformation (3.10)) one can obtain:p = (3.25)where vector q’ does not contain coordinates p.The subvectors,can be partitioned in the following way:1BandChapter 3. Formulation of component mode synthesis method 38where B = displacements at the interface, çi’ = all the remaining displacements. Applythe equation of displacement continuity at the interface:= (3.26)Expression (3.26) is used to eliminate residual-attachment mode coordinates fromthe vector q’ expressing them in terms of free-free mode coordinates, i.e, using a matrixtransformation (see analogous transformation (3.12)) one can obtain:q’= 3Uq (3.27)where the vectorPt’qP2Substituting (3.27) in (3.25) the vector p of all generalized coordinates is expressed interms of independent generalized coordinates q:p = /3’13”qorp = ,8q (3.28)where 3 =Substituting (3.28) in (3.23) the relation between the approximate system eigenvector4 and the vector of generalized coordinates q will be(3.29)Substitution of (3.29) in (3.17) gives:(A + B)/3q = F4 +Chapter 3. Formulation of component mode synthesis method 39where the quantity represents an approximation error. Galerkin’s method again yieldsthe following matrix equation:TT(4 + B)3q = /3T’F.and the vector on the right side vanishes (it was proved in section 3.1).Therefore the final condensed equation of motion in a system mode for two coupledcomponents (or the whole system) in terms of generalized coordinates can be written as:()4 + B4)q = 0where A4 = I3TTA13 , B4 = /3T4TB/3 will be complex symmetric matrices.The solution of this eigenvalue equation yields the complex conjugate eigenvalues )and eigenvectors q in general. There may be an even number of real eigerivalues, whichwill correspond to overdamped modes (depending upon the damping properties of thesystem). In the case when c1, c2 are zero matrices (undamped system) the eigenvalueswill be pure imaginary (zero real parts).Chapter 3. Formulation of component mode synthesis method 403.2.1 Component mode selection procedureSelection of lower (retained) modes is made on the basis of the absolute values of theeigenvalues which are complex numbers. Thus both the imaginary part (frequency) andthe real part of the eigenvalue are counted. This will be important when a componentis heavily damped. In the case of an undamped component, selection of retained modeswill be based on the lower frequencies (imaginary parts), because the real parts of theeigenvalues are zeros.The ratioI .‘comT =Ayields the boundary separating the lower (retained) and higher modes of a component,where A is the largest system eigenvalue of interest and Acorn is the boundary componenteigenvalue out of the retained eigenvalues. This procedure can be illustrated as follows inFig.3.3. ‘The solid circle indicates the level of system eigenvalue of interest and the dashedone corresponds to the level of retained eigenvalues (eigenvectors) for each component.All the eigenvectors with eigenvalues located in sector D (within of dashed circle) willbe retained modes, and all the eigenvalues beyond of this sector will correspond to thehigher modes, which are not used in the analysis.It has been established by conducting a series of numerical computations for differentsystems (including heavily damped systems) that a good accuracy for the given range0 j A5 can be achieved with r = 2. Note that the greater r the better accuracy, butthe larger the size of the condensed eigenproblem.Chapter 3. Formulation of component mode synthesis method 41Jm(A)ACOrn/4../ 0/_—07 /0-of®DRe(A)Figure 3.3: Mode selection procedureChapter 3. Formulation of component mode synthesis method 423.3 Case of an arbitrary number of componentsGeneralization of the developed method (sections 3.1,3.2) to a system with an arbitrarynumber of components is straightforward and shown here only in a brief form. This generalization is conducted simultaneously for the both cases: undamped and nonclassicallydamped system.Consider a system consisting of N components, which are joined by L interfaces.Consider the jth interface, which identifies two adjacent components (call them the 1stand 2nd one). Introduce the following notations:Ta a a a1— Pu P12 ... PJ •.. P1LandT-— a a a aP2—p21 p22 ... p3... P2Lwhere vectors and consist of interface force vectors of all interfaces. Apply theequation of force equilibrium at the jth interface:= —p (3.30)Using (3.30) one can express the vector of generalized coordinates p (see analogousmatrix transformation (3.10)) through the vector q’, which will not contain i3:I,Consider now the displacements at the jth interface and q. These subvectorsand can be expressed in the following form:iBPi + iBP1 2j 2BP2 + 2BP2 = 1,2, ..., Lwhere the vectors f, 75 combine all the interface force vectors, which belong to thecomponents 1 and 2 respectively:Pk]Chapter 3. Formulation of component; mode synthesis met hod 43andwhere i, ..., k = numbers of the interfaces, which belong to component 1 and 1, ..., n tocomponent 2.The compatibilty of displacements at the jth interface yields the following system ofequationsB B‘-)Yij — Y2j I —which can be used to express the interface forces p (j = 1,2, ..., L) in terms of free-freemode coordinates. Thus the vector q’ can be expressed (see analogous transformation(3.12)) by the following matrix transformation:q’ =whereq=[p.....and p is a set of free-free mode coordinates for the ith component. Therefore the vector ofall generalized coordinates p is expressed in terms of independent generalized coordinatesq in the following matrix transformation:p = /3’/311qorp = 13qwhere 3 =Thus the relation between the approximate system eigenvector qS and the vector ofindependent generalized coordinates q will be= 13qGhapter 3. Formulation of component mode synthesis method 44Analogous manipulations (see sections 3.1,3.2) with the equation of free motion of thesubdivided system will yield the final condensed eigenproblems in terms of generalizedcoordinates. For the case of an undamped system:(—w2M + K)q = 0where M = 3TTJ1J3 K 3TTk will be real symmetric matrices. For adamped system:(-4 + B)q 0where A I3TTA/3 , B = /3TTB4I3 will be complex symmetric matrices.Chapter 4Numerical results4.1 Comparison of CMSFR method with “VAST” program for undampedsystemsComparison of the results obtained by the program using CMSFR method with theresults obtained by the finite element program “VAST” [31] are shown in Tables 4.1-4,4.Note that the “VAST” analysis treated each system as a whole (without a componentsubdivision). The user’s manual for the CMSFR program is presented in Appendix A.Below numerical results are shown for some examples of undamped systems (Fig.4.1,a-d), which are shown in subdivided form.The geometric and physical parameters of the systems shown are as follows: in examples a) and b) the cross-section of beam and bar elements was 1 x 1 m, in examples c)and d) 0.02 x 0.02 m. The dimensions of the bearri and bar element components withineach example were the same and are shown in metres. The thickness of the membraneelement (2nd component) in example c) is 0.02 rn. In example d) the second componentis represented by a brick element with parameters of a cube. The material of all elementsis steel. For examples a),b),c) the motion is considered in the plane of the drawing. Forexample d) 3-D motion is considered.The total number of retained modes for example a) was 20, which determines thesize of the condensed eigenvalue problem (Table 4.1). This total number of the retainedmodes was composed of 4 + 6 + 6 + 4 respectively for each component.45Chapter 4. Numerical results 46©__ __©__2 2a) b)c_4 ‘‘Figure 4.1: Test examplesA rigid-body mode exists for the system in Figure 4.1,b (Table 4.2), which was detected by the appearence of a low (near zero) frequency (the method of weak springswas used to treat unconstrained components). This system has bar elements in the 2ndcomponent, that provides a rotational rigid-body degree of freedom at the joint of thebar elements. The current version of “VAST” does not detect such “hidden” rigid-bodymodes, because it has no algorithmic option to treat such systems.The results for examples c), d) are presented in rfables 4.3, 4.4. Comparison of the1st and 2nd mode shapes are shown in Fig.4.2,4.3 for the system in Fig.4.1,a. There isno difference between the modes calculated using CMSFR method and those calculatedby “VAST” using the unsubdivided model.Chapter 4. Numerical results 47Frequencies, Hz# CMSFR VAST Difference, %1 124.85 124.85 0.2 298.25 298.25 0.3 430.54 430.11 0.0994 516.62 516.23 0.0795 768.96 768.95 0.0016 878.51 876.93 0.18Size of ezgenvalue problemCMSFR VAST20x20 21x21Table 4.1: Four component beam element system: “a”Frequencies, Hz# CMSFR VAST Difference, %1 0.044 - -2 140.35 140.35 0.3 147.44 147.44 0.3 432.89 432.28 0.144 633.57 633.48 0.0145 643.38 643.30 0.0126 898.45 894.15 0.48Size of eigenval’ue problemCMSFR VAST11 x 11 14 x 14Table 4.2: Three component beam-bar element system with one rigid-body mode: “b”Chapter 4. Numerical results 48Frequencies, Hz# CMSFR VAST Difference, %1 1.4666 1.4651 0.102 96.6478 96.660 0.0123 126.485 126.46 0.0194 313.73 313.73 0.5 1979.3 1979.3 0.Size of eigenvalue problemCMSFR VAST12x12 9x9Table 4.3: Three component beam-bar-membrane element system: “c”Frequencies, Hz# CMSFR VAST Difference, %1 0.2633 0.2631 0.0762 0.3409 0.3409 0.3 0.3664 0.3663 0.0274 16.308 16.308 0.5 38.242 38.228 0.0366 38.255 38.241 0.036Size of eigenvalue problemCMSFR VAST12x12 33x33Table 4.4: Four component beam-brick element system: “d”Chapter 4. Numerical resultsFigure 4.2: First mode shape of the system in Fig.4.1,a4920)11 Print II Istmod.pIt ) 1st mode shape 1 SI -Crnsfr5.0 6.0 7.0NodeOLI)JLflI p nomoo.pIT II dna mooe snape isi .crnsIrG)D4-0E1.0 2.0 3.0 4.0 5.0 6.0Node7.0 8.0 9.0Figure 4.3: Second mode shape of the system in Fig.4.1,aChapter 4. Numerical results 504.2 Comparison of CMSFR method with “DREIGN” program for undarnpedand nonclassically damped systemsBelow some systems (without a component subdivision) are considered and their corresponding eigenvalue problems are solved by using a complex eigensolver “DREIGN”program [32], which uses the QR method. Then these results are compared with CMSFRmethod results for subdivided systems.A two component system consisting of beam elements (Fig.4.4) was considered. Theelement cross-section was 0.01 x 0.01 m. The damping matrices of the componentswere taken as stiffness-proportional ones on the component level, i.e., c1— 0.00002k1,c2 = 0.00001k2,which produces nonclassical damping on the level of the whole system.The motion of the system is considered in the plane of the drawing. Table 4.5 presentsa comparison of the results and shows excellent correspondence.u a a u -a——.0.6a a a0,4Figure 4.4: Two component beam element systemThen a three component system with lumped masses and dashpots was considered(Fig.4.5). The length of each component consisting of beam elements was 2 rn, the cross-section of elements was 0.1 x 0.1 rn, the lumped masses were 100 kg each, the dashpots10 N x s/rn each. The motion of the system is considered in the plane of the drawing.The results are shown in Table 4.6.Then two finite elements models of the vibration rig were considered (Fig.4.6). Parameters of the rig are shown in Appendix B. For application of the CMSFR methodthe finite element model of the rig was subdivided into four corriponents (Fig.4.7). InChapter 4. Numerical results 51EigenvaluesCMSFR DREIGN Difference, %# Real,1/s Imag, Hz Real, 1/s Imag, Hz Real Imag1 -0.8964 52.717 -0.8964 52.716 0. 0.2 -6.396 145.34 -6.395 145.32 0.015 0.0133 -26.303 285.06 -26.287 285.02 0.06 0.0144 -70.22 472.73 -69.847 471.74 0.534 0.2095 -157.15 707.65 -156.86 706.34 0.184 0.185Size of eigenvalue problemCMSFR DREIGN16 x 16 complex 54 x 54 realTable 4.5: Two component beam element systemthe 1st model there were no damping elements present. Some clamping properties wereassumed for the elements a, b, c, d (Fig.4.7) for the 2nd model, producing an example of anonclassically damped system. Comparison of the eigenvalues is presented in Tables 4.7,4.9. Note that complex eigenvalues are obtained in pairs of conjugate numbers, but justone representative of each pair is shown in Tables 4.5-4.7,4.9. Good agreement betweenthe results is obtained and a reduction of eigenvalue problem size is quite noticeable.Comparison of the 1st eigenvectors for the undamped rig model is shown in Table4.8. “Z” displacements of the 12 nodes (Fig. 4.8) were selected. The comparison of the1st eigenvectors for the damped rig model is shown in Table 4.10.Chapter 4. Numerical results 52Figure 4.5: System with dashpots arid lumped massesEigenvaluesCMSFR DREIGN Difference, %Real, 1/s Imag, Hz Real, 1/s Imag, liz Real Imag1 -0.004433 14.044 -0.004433 14.044 0. 0.2 -0.01609 33.699 -0.01609 33.698 0. 0.3 -0.02204 60.588 -0.02204 6(1.588 0. 0.4 -0.0140 101.94 -0.0140 1(11.89 0. 0.0495 -0.005491 208.61 -0.005442 207.55 0.9 0.516 -0.001634 313.93 -0.001617 313.18 1.0 0.23Size of eigenvlue problemCMSFR DREIGN18 x 18 complex 24 x 24 realTable 4.6: System with dashpots and lumped massesp‘‘ \p__Figure 4.6: Finite element model of the rigChapter 4. Numerical results 53EigenvaluesCMSFR DREIGN Difference# Real 1/s Irnag, Hz Real 1/s Imag, Hz %1 0. 4.791 0. 4.774 0.3562 0. 5.767 0. 5.757 0.1733 0. 6.121 0. 6.108 0.2124 0. 8.144 0. 8.130 0.1725 0. 8.357 0. 8.343 0.1676 0. 9.212 0. 9.192 0.2177 0. 31.46 0. 31.33 0.4148 0. 31.57 0. 31.42 0.477Size of eigenvalue problemCMSFR DREIGN60 x 60 complex 516 x 516 realFigure 4.7: Four component presentation of the systemTable 4.7: Comparison of eigenvalues for the undamped rig modelChapter 4. Numerical results 5411221053 6z114 78 xzYFigure 4.8: Selected nodes for the eigenvector presentationZ displacements in the 1st eigenvectorCMSFR DREIGNReal Imag Real Imag1 O.913E-06 -O.252E-11 O.105E-05 -O.443E-082 O.467E-O1 O.000E+OO O.481E-O1 -O.742E-043 O.134E+OO -O.378E-06 O.137E+OO -O.309E-034 O.229E+OO -O.882E-05 O234E+OO -O.286E-035 O.104E-05 -O.136E-09 O.105E-05 -O.443E-086 O.481E-O1 -O.189E-05 O.481E-O1 -0.742E-047 O.137E+OO O.668E-05 O.137E-fOO -O.309E-038 O.235E+OO -O.164E-04 O.234E+OO .-O.286E-039 -O.643E+OO O.277E-04 -O.625E+OO O.578E-0310 0.986E-i-00 -0.1O1E-04 0.100E+01 0.000E+0011 0.100E+01 0.000E+00 0.100E+01 0.000E+0012 -0.626E+00 -0.529E-04 -0.625E+00 0.578E-03 jTable 4.8: Comparison of the 1st eigenvectors for the undamped rig modelChapter 4. Numerical results 55Eig e nval’u esCMSFR DREIGN Difference# Real, 1/s Imag, Hz Real 1/s Imag, Hz Real, % Imag, %1 -0.1924 11.030 -0.1927 11.009 0.15 0.192 -1.0425 15.640 -1.0405 15.625 0.19 0.103 -3.4175 20.737 -3.4106 20.717 0.20 0.094 -1.6897 21.942 -1.6815 21.918 0.48 0.115 -10.086 33.729 -10.025 33.652 0.61 0.226 -11.680 35.268 -11.571 35.140 0.94 0.367 -29.472 65.501 -29.055 65.213 1.43 0.448 -34.059 68.942 -34.098 68.745 0.11 0.28Size of eigenvalue problemCMSFR DREIGN60 x 60 complex 516 x 516 realTable 4.9: Comparison of eigenvalues for the damped rig modelZ displacements in the 1st eigenvectorCMSFR DREIGN# Real Imag R,eal Imag1 0.648E-05 0,574E-07 0.653E-05 0.141E-072 0.965E-01 0.177E-03 0.968E-0i 0.1 43E-033 0.255E+00 0.333E-03 0.256Ei—00 0.209E-034 0.425E00 0.657E-03 0.426E+00 0.104E-025 0.657E-05 0.485E-07 0.652E-05 0.182E-076 0.956E-01 0.126E-03 0.954E-01 0.476E-047 0.252E+00 0.205E-03 0.251E-j-00 0.434E-038 0.420E+00 0.401E-03 0.419E+00 0.216E-039 0.102E+00 0.132E-03 0.102E+00 0.976E-0410 0.304E+00 -0.156E-02 0.304E+00 -0.1S1E-0211 0.305E+00 -0.165E-02 0.304E+00 -0.163E-0212 0.104E+00 0.531E-04 0.103E*00 0.694E-04Table 4.10: Comparison of the 1st eigenvectors for the damped rig modelChapter 5Experimental resultsThe experimental section of this work contains results obtained for the vibration rig(Fig.5.1), which was designed and constructed for the purpose of modelling the vibrationsof an engine mounted on a flexible support. Its geometric parameters (in inches) are givenin the Appendix B. The material used for all elements, except springs isolators, was steel.The rig contains the four spring isolators a, b, c, d (Fig.5.2) that can be adjusted to providedifferent damping and stiffness characteristics.For the experimental determination of stiffness and damping characteristics of thespring isolators the following procedure (Fig.5.3) was used. A vertical impact is appliedat the mass m3 and the natural (damped) frequency and logarithmic decrement of thefree vertical oscillations of mass m5 is measured.The natural frequency is obtained by using “Nicolet 660 A” spectrum analyzer and anaccelerometer attached to the iriass. The Fourier transform feature built into the analyzeris used to determine the natural damped frequency s. It is assumed that damping issmall (damped and undamnped frequencies are close), thus stiffness of the spring can bedetermined as= pm3The damping coefficient of the equivalent dashpot (Fig. 5.3) is determined asA,Cs = —m5pln—----—where A2,-4+i are the two consecutive amplitudes of oscillation separated by the period56Chapter 5. Experimental results 5727r/p.These experimentally determined characteristics of the spring isolators (Tables 5.1,5.2)were assigned to the elements a, b, c, d in the finite element model. The first set of damping properties were taken as zero, i.e. the 1st rig model was considered as an undampedsystem. The free oscillations of the undamped rig were considered at first. The natural frequencies (Table 5.3) were obtained using the “Nicolet” analyzer and piezoelectricaccelerometers attached to various points of the rig. It may seen from Table 4.7 that agood agreement with the analytical results was obtained.Figure 5.1: Photo of the experimental rigChapter 5. Experimental results 58Figure 5.2: Reference points on the test rigYProperties of the isolators# Stiffness, N/mm Damping coeff-t, N x s/mm1 12.2 0.2 14.1 0.3 12.5 0.4 11.3 0.Table 5.1: Stiffness and damping properties of the spring isolators (undamped rig)Properties of the 4 isolators# Stiffness, N/mm Damping coeff-t, N x s/mm1 257.9 0.1162 314.4 0.1803 281.9 0.1424 247.3 0.10225xTable 5.2: Stiffness and damping properties of the spring isolators (damped rig)Chapter 5. Experimental results 59L.ms. IksFtcs/Figure 5.3: Spring isolator characteristic determinationNatural frequencies) Hz# undamped rig damped rig (with different stiffness)1 4.65 11.12 5.75 15.753 6.1 22.54 8.20 23.255 8.35 35.756 9.50 36.757 30.25-8 30.75-Table 5.3: Experimental frequencies for two rig testsIn the second test damping was introduced into the spring isolators (due to the specific construction of the spring isolator the introduction of damping leads to increasingstiffness) and the natural frequencies were measured (Table 5.3). Note the good agreement with the analytical results in Table 49. It may seen that the higher frequencies areassociated with quite large real parts. The experimental determination of the higher frequencies (#7#8) was complicated due to the fast amplitude decay of the free vibrationresponse.Forced responses due to a sinusoidal vertical force applied to point “b” (Fig.5.2) wereChapter 5. Experimental results 60Excitation frequencies, Hzw=5.9 w=6.2Amplitude Phase,° Amplitude Phase,°# FIx. An. Ec An. FIx. An. Ex. An.1 0.365 0.397 0. 0. 0.565 0.525 180. 180.2 0.173 0.256 0. 0. 0.494 0.470 180. 180.3 0.515 0.505 0. 0. 0.403 0.448 180. 180.4 0.699 0.648 0. 0. 0.483 0.505 180. 180.5 0.192 0.232 0. 0. 0.112 0.109 180. 180.6 0.208 0.212 0. 0. 0.161 0.185 180. 180.Table 5.4: Amplitudes, Phase angles for undamped riganalyzed for the damped rig. At first an excitation frequency sweep with a constant forceamplitude (10 N) was conducted with a rate th- 0.2 liz/s. The mesuared amplitude ofvibration at the point 2 is shown in Fig.5.4. It may seen that the peaks of the responsecorrespond to the frequencies associated with the eigenvalues (Table 5.3).Next a sinusoidal vertical excitation at a fixed frequency and force magnitude wasapplied to the point “b”. Comparison of the analytical and experimental steady-stateresponses (amplitude vector and phase angle vector) for six points of the rig (Fig. 5.2)was conducted. The amplitudes (in dimensionless form) and phase angles are presentedin Tables 5.4,5.5 for the undamped rig model and in Tables 5.6,5.7 for the damped one.The experimental phase angles were determined with an accuracy of 5 It may seen agood agreement of the experimental and analytical results.Chapter 5. Experimental results 61Figure 5.4: 8 - 24 Hz excitation sweepChapter 5. Experimental results 62Excitation frequencies, Hzw16.0 w=33.0Amplitude Phase,° Amplitude Phase,°# Ex. An. Ex. An. Ex. An. Ex. An.1 0.077 0.089 180. 180. 0.026 0.022 0. 0.2 0.105 0.118 180. 180. 0.031 0.027 0. 0.3 0.014 0.022 0. 0. 0.002 0.004 180. 180.4 0.049 0.056 0. 0. 0.024 0.014 180. 180.5 0.989 0.986 0. 0. 0.998 0.999 180. 180.6 0.015 0.020 0. 0. 0.014 0.019 0. 0.Table 5.5: Amplitudes, Phase angles for undamped rig ... continuedExcitation frequencies, Hzw=i1.0 w=l6.0Amplitude Phase,° Amplitude Phase,°# Ex. An. Ex. An. Ex. An. Ex. An.1 0.071 0.081 240. 252. 0.248 0.303 242. 249.2 0.413 0.441 233. 250. 0.441 0.356 247. 245.3 0.474 0.441 236. 250. 0.300 0.329 77. 74.54 0.063 0.083 240. 250. 0.382 0.316 72. 69.65 0.530 0.545 248. 251. 0.534 0.541 242. 248.6 0.559 0.545 233. 250. 0.470 0.527 67. 73.5Table 5.6: Amplitudes, Phase angles for damped rigChapter 5. Experimental results 63Excitation frequencies, Hzw=24.OAmplitude Phase,°# Ex. An. Ex. An.1 0.373 0.303 253. 249.2 0.306 0.332 237. 250.3 0.324 0.328 67. 74.64 0.192 0.269 40. 74.75 0.621 0.560 243. 253.6 0.487 0.550 64. 74.7Table 5.7: Amplitudes, Phase angles for damped rig ... continuedChapter 6SummaryResults have been presented that allow the prediction of the effect of damping on thefree vibration response of classically damped discrete nongyroscopic systems. The resultspresented are general in the sense that they consider all possible damping conditions thatlead to classically damped systems, which have distinct undamped natural frequencies.For this class of systems it is possible to specify the damping matrix that will result ineach mode having prescribed decay factor or damped natural frequency and for that justthe knowledge of undamped natural frequencies is required. r lie equations required toaccomplish this task have been presented.For nonclassically damped systems the free response behaviour is more complex andno general rules concerning the influence of clamping on free response characteristicsare evident. Characteristics not observed in classically damped systems are shown anddiscussed. It has been demonstrated that increasing damping can lead to an increasein free response frequency. Modification of damping properties of a system may lead toobtaining of desirable system eigenquantities, free and •forced vibration responses.A component mode synthesis method that enables the determination of eigenquantities (for a given range of interest) has been developed. The use of this method isespecially advantageous in the case of large systems, subjected to numerous modifications. The numerical results presented confirm the validity of the method.It has been shown that the effect of damping properties of the spring isolators of thevibration rig model on transmissibility of motion from the box to the supported beams is64Chapter 6. Summary 65quite significant. Increase of damping constants in the isolators (at the constant stiffnesscharacteristics) leads to a decrease of amplitudes of vibration of the supported beams,in other words, it provides a decrease of transmissibility of motion from the box to thesupported beams.Comparison of analytical and experimental results is presented, which shows goodagreement for eigenvalues, and steady-state responses of the vibration rig.As the future work, the assignment of the optimum range of retained componenteigenvectors can be further investigated. Also the displacement compatibility betweencomponents can be generalized, i.e., for some types of interfaces the slippage betweencomponents can be introduced, which means the assumption of equality of normal relativedisplacements and discontinuity of relative tangential displacements at the interface.Bibliography[1] Hurty, W. C., “Dynamic Analysis of Structural Systems Using Component Modes,”AIAA Journal, Vol. 3, April 1965, pp. 678-685.[2] Hurty, W. C., “Introduction to Modal Synthesis rfech1iques Synthesis of VibratingSystems, ASME booklet, Nov. 1971, Pp. 1—13.[3j Craig, R. R., and Bampton, M. C. C., “Coupling of Substructures for DynamicAnalysis,” AJAA Journal, Vol. 6, July 1968, pp. 1313—1319.[4] Craig, R. R., and Chang, C., “Free-Interface Methods of Substructure Coupling forDynamic Analysis,” AIAA Journal, Vol. 14, No. 8, November 1976, pp. 1633—1635.[5] Hintz, R. M., “Analytical Methods in Corriponerit Modal Synthesis,” A MA Journal,Vol. 13, Aug. 1975, pp. 1007--1016.[61 Dowell, E.H., “Free Vibrations of an Arbitrary Structure in Terms of ComponentModes,” Journal of Applied Mechanics, Vol. 39, September 1972, pp. 727—732.[7] Goldman, R.L., “Vibration Analysis by Dynamic Partitioning,” AIAA Journal, Vol.7, No. 6, June, 1969, pp. 1152—1154.[8] Gladwell, G.M.L., “Branch Mode Analysis of Vibrating Systems”, Journal of Soundand Vibration) Vol. 1, 1964, pp. 41 59.[9] Hou, S., “Review of Modal Synthesis Techniques and a New Approach,” Shock andVibration Bulletin, Vol. 40, Part 4, Dec. 1969, pp. 25—39.[10] MacNeal, R. H., “A Hybrid Method of Component Mode Synthesis,” Computersand Structures, Vol. 1, Dec. 1971, pp. 581 601.[11] Meirovitch, L., “Computational Methods in Structural Dynamics”, Sijhoff and Noordhoff International Publishers, Alpen aan den Rijn, The Netherlands, 1980.[12] Craig, R. R., “A Review of Time-Domain and Frequency-Domain Component-ModeSynthesis Methods,” international Journal of Analytical and Experimental ModalAnalysis, Vol. 2, No. 2, April 1987, pp. 59- 72.[13] Smith M. J., “An evaluation of component mode synthesis for modal analysis offinite element models,” Ph.D thesis, UBC, Vancouver, B.C.,1993.66Bibliography 67[14] Meirovitch, L., and Hale A.L., “On the Substructure Synthesis Method,” AJAAJournal, Vol. 19, No. 7, July 1981, pp. 940-947.[15] Hale, A.L., and Meirovitch, L., “A General Substructure Synthesis Method for theDynamic Simulation of Complex Structures,” Journal of Sound and Vibration, Vol.69, No. 2, 1980, PP. 309—326.[16] Weaver, W., Young, D. H., Timosbenko, S. P., “Vibration Problems in Engineering,”John Wiley and Sons, 1990.[17] Meirovitch, L., “Elements of Vibration Analysis,” McGraw Full, 1975.[18] Thomson, W. T., “Theory of Vibrations with Applications,” Prentice Halt Inc.,1981.[19] Clough, R. W., Penzien, J., “Dynamics of Structures,” McGraw Hill, 1975.[20] Craig, R. R., “Structural Dynamics”, Macmillan, 1979.[21] Bellman R., “Introduction to Matrix Analysis,” McGraw-Hill Book Company, Inc.,New York, N.Y., 1960.[22] Caughey T. K., M. E. J. O’Kelly, “Classical Normal Modes in Damped LinearDynamic Systems,” Journal of Applied Mechanics, Vol. 12, 1965, pp. 583—588.[23] Bishop R. E. D., Gladwell C. M. L., “An investigation into the theory of resonancetesting,” Philosophical Transactions of the Royal Society of London, Vol. 255, SeriesA. 1963, pp. 241—280.[24] Hasselman, T. K. and Kaplan, A., “Dynamic Analysis of Large Systems by ComplexMode Synthesis,” Journal of Dynamic Systerns,Measurement,and Control, Vol. 96,Series G, 1974, pp. 327—333.[25] Beiveau, J., and Soucy, Y., “Damping Synthesis Using Complex Substructure Modes and a Hermitian System Representation,” Proceedings of theAIAA/ASME/ASHE/AHS, New York, 1985, pp. 581-586.[26] Howsman, T., and Craig, R.R., “A Substructure Coupling Procedure Applicable to General Linear Time-Invariant I)ynamic Systems,” Proceedings of theAIAA/ASME/ASHE/AHS, New York, 1984, pp. 164-171.[27] Craig, R. R., and Ni, Z., “Component Mode Synthesis for Model Order Reductionof Nonclassically Damped Systems,” J.Gumdance, Vol. 12, July-August. 1989, pp.577—584.Bibliography 68[28] Francis, J.G.F., “The QR Transformations, Parts I and Ii”, The Computer Journal,Vol. 4, 1961, PP. 265—271, 332—345.[29] Kublanovskaya, V.N., “On Some Algorithm fur the Solution of the Complete Eigenvalue Problem”, USSR Comput. Math. Math, Phys., Vol. 3, 1961, Pp. 637—657.[30] Rubin, S., “Improved Component-Mode Representation for Structural DynamicAnalysis,” AIAA Journal, Vol. 13, No. 8, August 1975, pp. 995—1006.[31] Martec Limited., “Vibration and Strength Analysis Program (VAST): User’s manual,” Halifax, N.S.,1990.[32] Nicol T., (editor), “UBC Matrix book (A Guide to Solving Matrix Problems),”Computing Centre, UBC, Vancouver, B.C., 1982.Appendix AUser’s Manual for the CMSFR methodThe main program in which the CMSFR method was implemented is named VASFIN. Itwas written by the author of this thesis. The computer language used was FORTRAN-77.Instructions to run VASFIN (gives the final results)The following files (* means prefix) are required to prepare to run VASFIN:1. *con2. *.ibn3 *rmk4 *dam5. *mod6. *1st7 *lms8. *ldp1. •c is ouput file of VASPRE2. *.ibn is ouput file of VASPRE3. *.rmk is ouput file of VASPRE4• *.dam is ouput file of VASDAM5. *.mod consists from one line, which contains the system eigenvalue range of interest69Appendix A. User’s Manual for the CMSFR method 70and adjustment coefficient i’ for the selection of component modes (it is created manually)6. *1st contains constraint and weak springs for specified nodes of each component (it iscreated manually)7. “.lms contains lumped masses for specified nodes of each cormiponent (it is createdmanually)8. *.ldp contains lumped dashpots for specified nodes of each component (it is createdmanually)The output files after execution of VASFIN are:1. contains eigenvalues, eigenvectors, the response function of the system.Instructions to run VASPRE******************************************************** ***The following files are required to prepare to run VASPRE:1. *.gogj ( = 4.gml, copy of *.gml)2. *sed3. *use1. *.gom is created from (basic) *.gom files of each component by running VASGENprogram2. *.sed is created from (basic) t.gom files of each component by running VASSEDprogram3. *.use. The same for all cases, only the first line (title of the problem) can be different.The output files after execution of VASPRE are:Appendix A. User’s Manual for the CMSFR method 711. (contains the quantity of components)2. * .ibn (component connection information)3. “.rmk (mass, stiffness matrices of components)Remark.For VASPRE, program VAST6O was taken as an initial program [31}, which was writtenin FORTRAN-77. Only subroutine “elems2” was slightly modified and instead of “cms2”,subroutine “compox” is used. These modifications were made by the author of this thesis.Instructions to run VASDAM********************** ****4***************************** ***The following files are required to prepare to run VASDAM:1. *.gom (= *.gm2, copy of *.gm2)2. *.sed3. *use1. *.gom is created from *.goIx. files of each component by running VASCEN program.Note that *.gom file will be as *.gm2, where damping properties of each componentelement are governed by two parameters: “modulus of elasticity” and “density”. Theyreflect two factors: i) the damping matrix of the element is proportional to the elementstiffness matrix, ii) the damping matrix of the element is proportional to the elementmass matrix. The value, e.g., 1O can be used for these two parameters in order toprescribe zero damping to the elements of each component2. *sed (the same as for VASPRE)3. *.use (the same as for VASPRE)Appendix A. User’s Manual for the CMSFR method 72The output files after execution of VASDAM are:1. *dam (damping matrices of components)Remark.For VASDAM, program VAST6O was taken as an initial program. Only subroutines“assem.2”, “elems2” were slightly modified. These modifications were made by the authorof this thesis.*0I—C I0CD144’iCD CDCDci:)aD-.2CD—.CD_4__L___UI—____C‘1II—.II8II—.UI 0 .p.0 ‘1Appendix B. Parameters of the vibration rig 74A— A± ±— 9/16+0_rH_10/FZF/////Z/7/77/////77///7777777/77//77Figure B.2: Vibration rig, front viewAppendix B. Parameters of the vibration rig 75o 0cv0 012Figure B.3: Vibration rig, top view of the box
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Analysis of damped linear dynamic systems and application of component mode synthesis Muravyov, Alexander 1994
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Title | Analysis of damped linear dynamic systems and application of component mode synthesis |
Creator |
Muravyov, Alexander |
Date Issued | 1994 |
Description | The analysis of nongyroscopic damped (viscous) linear dynamic systems is presented. Discrete systems having symmetric mass, stiffness and damping matrices are considered. Discretization of the systems is accomplished by application of the finite element proce dure. The general case of classically damped systems is considered, and the necessary and sufficient condition for classical damping is given. For a system of this type, it is possible to specify the damping matrix that will result in each mode having either a prescribed decay factor or a damped eigenfrequency. This may be accomplished with only a knowl edge of the undamped eigenfrequencies. The equations required to accomplish this task are presented. Graphical results are presented that illustrate the effect of damping for mass-proportional, stiffness-proportional and for Rayleigh damping. Nonclassically damped systems are considered and the formulation of a component mode synthesis (CMS) method for the solution of the free vibration problem is described. The component mode synthesis method is a procedure in which the exact solution is approximated by one constructed from some basis vectors (e.g., mode shapes) of sub systems (components of the original system). This method allows significant reduction of the eigenvalue equation size due to the use of a limited number of basis vectors. The approximate solution found for the lower eigenvalues and eigenvectors is very close to the exact one due to the proper selection of the basis vectors and the procedure followed (e.g., Galerkin’s method) that determines the best approximation. The use of CMS methods is especially advantageous in the case of large systems, subjected to numerous modifications. In this work the formulation of the CMS method was developed for the general case of nonclassically damped systems. It was tested for different cases of noriclassically damped systems and the excellent agreement with the exact results derived from nonsubdivided systems was found. Also a new method to treat an unconstrained component for the purpose of stiffness matrix inversion is presented. The selection procedure of component modes is generalized from the undamped system case to the damped one. Some examples of forced responses are considered, particularly, the case of sinusoidal excitation and the influence of the damping factor is analyzed. The experimental part of this study consists of the designing and testing of a vibration rig designed to simulate the behaviour of a rigid engine resting on isolators that in turn are supported on flexible beams. Free and steady-state responses of the rig are experimentally determined. Comparison of analytical results with experimental ones show good agreement for eigenquantities and steady-state forced response. |
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Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-02-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080875 |
URI | http://hdl.handle.net/2429/5087 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1994-05 |
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