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Analysis of damped linear dynamic systems and application of component mode synthesis Muravyov, Alexander 1994

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ANALYSIS OF DAMPED LINEAR DYNAMIC SYSTEMS AND APPLICATION OF COMPONENT MODE SYNTHESIS  By Alexander Muravyov B. A. Sc. ,Chelyabinsk Polytechnic Institute, Russia, 1982  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  April 1994  ©  Alexander Muravyov, 1994  In presenting this thesis in partial fulfilment of the requirements for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of ths thesis for scholarly purposes may be granted by the department  or  by  his  or  her  representatives.  It  is  understood  an advanced shall make it for extensive  head of my that copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of Mehc ,i lea-f eetp The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  p 1i 4 i  2 £ / 994  Abstract  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. 11  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.  111  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  vi  List of Figures  vii  List of Symbols  ix  Acknowledgement  Xi  1  2  Introduction  1  1.1  Objectives  1  1.2  Background  1  1.2.1  Undamped systems  1.2.2  Classically damped systems  1.2.3  Nonclassically damped systems  1 3  .  .  .  .  4  Analysis of damped systems  7  2.1  Classically damped systems  7  2.2  Nonclassically damped systems  15  2.3  Free response function computation  17  2.4  Steady-state response function computation  18  iv  3  Formulation of component mode synthesis method  25  3.1  25  Undamped systems 3.1.1  3.2  3.3 4  Case of an unconstrained component. Method of weak springs  .  Nonclassically damped systems  34  3.2.1  40  Component mode selection procedure  Case of an arbitrary number of components  Numerical results  4.1  42 45  Comparison of CMSFR method with “VAST” program for undamped sys tems  4.2  31  45  Comparison of CMSFR method with “DREIGN” program for undamped and nonclassically damped systems  50  5  Experimental results  56  6  Summary  64  Bibliography  66  A User’s Manual for the CMSFR method  69  B Parameters of the vibration rig  73  V  List of Tables  2.1  Steady-state responses, examples 1,2  23  2.2  Steady-state responses, examples 3,4  23  2.3  Steady-state responses, examples 5,6  23  4.1  Four component beam element system: “a”  47  4.2  Three component beam-bar element system with one rigid-body mode: “b” 47  4.3  Three component beam-bar-membrane element system: “c”  48  4.4  Four component beam-brick element system.: “d”  48  4.5  Two component beam element system  51  4.6  System with dashpots and lumped masses  52  4.7  Comparison of eigenvalues for the undamped rig model  53  4.8  Comparison of the 1st eigenvectors for the undamped rig model  54  4.9  Comparison of eigenvalues for the damped rig model  55  4.10 Comparison of the 1st eigenvectors for the damped rig model  55  5.1  Stiffness and damping properties of the spring isolators (undamped rig)  58  5.2  Stiffness and damping properties of the spring isolators (damped rig)  58  5.3  Experimental frequencies for two rig tests  59  5.4  Amplitudes, Phase angles for undamped rig  60  5.5  Amplitudes, Phase angles for undamped rig  5.6  Amplitudes, Phase angles for damped rig  5.7  Amplitudes, Phase angles for damped rig  vi  ...  continued  .  62 62  ...  continued  63  List of Figures  2.1  Effect of mass-proportional damping  2.2  Effect of stiffness-proportional damping  2.3  Effect of Rayleigh damping  2.4  Two degree of freedom system  2.5  Eigenvalues of the system in Fig 2.4  2.6  Model of the rig with selected nodes  2.7  Effect of damping on transmissibility  3.1  Two component system  3.2  Linear bar element  3.3  Mode selection procedure  4.1  Test examples  4.2  First mode shape of the system in Fig.4.1,a  49  4.3  Second mode shape of the system in Fig.4.1,a  49  4.4  Two component beam element system  50  4.5  System with dashpots and lumped masses  4.6  Finite element model of the rig  4.7  Four component presentation of the system  4.8  Selected nodes for the eigenvector presentation  5.1  Photo of the experimental rig  57  5.2  Reference points on the test rig  58  .  .  .  52 52  vii  .  .  .  .  53 54  5.3  Spring isolator characteristic determination  5.4  8  -  24 Hz excitation sweep  59 61  B.1 Vibration rig, side view  73  B.2 Vibration rig, front view  74  B.3 Vibration rig, top view of the box  75  VI”  List of Symbols  M  mass matrix of system  K  stiffness matrix of system  C  damping matrix of system system eigenvalue system eigenvector matrix of mode shapes  X  displacement vector of system  p  vector of modal coordinates  F  external force vector, acting on system  F  vector of forces acting on components  1 m  mass matrix of 1st component  1 Ic  stiffness matrix of 1st component  1 c  damping matrix of 1st component  2 m  mass matrix of 2nd component  2 Ic  stiffness matrix of 2nd component  2 c  damping matrix of 2nd component subvector of system eigenvector for 1st component  q2  subvector of system eigenvector for 2nd component  fi  force vector acting on 1st component  f2  force vector acting on 2nd component matrix of lower (retained) free-free modes of 1st component matrix of lower (retained) free-free modes of 2nd component ix  matrix of residual-attachment modes of 1st component matrix of residual-attachment modes of 2nd component vector of free-free mode coordinates for 1st component vector of free-free mode coordinates for 2nd component p  vector of residual-attachment mode coordinates for 1st component vector of residual-attachment mode coordinates for 2nd component  All the remaining notations are described as they occur in the text.  x  A ckn owledgernent  The author would like to offer sincere thanks to Professor Stanley G. Hutton for his insight and advices in the preparation of this thesis.  The author would like also to  acknowledge the financial support provided by Defense Research Establishment Atlantic.  xi  Chapter 1  Introduction  1.1  Objectives  The 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 steadystate responses of nonclassically damped systems, 4) construction of an experimental model to investigate the effect of damping upon the system response; data acquisition, analysis of experimental results and comparison with the analytical ones, 5) investigating the effect on transmissibility of the damping characteristics of spring isolators of the type used to isolate engine vibrations.  1.2  Background  1.2.1  Undamped systems  The literature on applications of component mode synthesis techniques to discrete un damped systems contains description of many different methods such as fixed-interface methods [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 introduction concerned with the general idea of component mode synthesis. 1  Chapter 1. Introduction  2  Consider the system of equations of free motion of a discrete undamped system MX+KX=O which leads to the eigenvalue equation M + K)q 2 (—w  where M =  =  mass matrix, K  =  0  stiffness matrix of the system, w  =  system eigenfrequency,  system eigenvector.  In CMS, the system is subdivided into a set of subsystems (components). Thus the system eigenvector (or displacement vector) defined on the entire domain occupied by the system can be partitioned into parts which are defined on the subdomains. Within a subdomain 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’ (in general different for each component):  =  Pk  where 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 basis  vectors 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 between adjacent components), 3) attachment modes, 4) constraint modes, 5) residual-attachment modes, 6) rigid-body modes. Free-free and fixed-interface modes are obtained by the solution of corresponding eigenproblems for free, or fixed (clamped) interface components. Constraint and attachment modes are respectively vectors of static displacements caused by a unit displacement, or unit load applied at the interface. The definition of residual attachment modes will be given in chapter 3.  Chapter 1. Introduction  3  The combination of free-free modes (including rigid-body modes) with residual-attachment ones is usually viewed as a free-interface method. The fixed-interface method uses fixedinterface 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 a number of authors [2], [9], [12], [13]. In [14], [15], the use of a set of admissible functions different to vibration modes is proposed, e.g., low order polynomial functions. This gives an opportunity to avoid eigensolutions at the component level. For discrete systems these polynomial functions are replaced by their values at the nodes of the system, i.e. by vectors. However the choice of 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 many such basis vectors would be required for an accurate approximation. 1.2.2  Classically damped systems  Description 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 damping  C  =  ciiI + /3K, where  and /3 are constants. The necessary and sufficient conditions  when a system has classical normal modes (the eigenvectors are the same for damped and undamped systems, i.e. they are real vectors) was formulated in [21],[22] as M’C commutes 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 with distinct undamped natural frequencies the necessary and sufficient condition of propor tionality 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 undamped  system eigenvector of interest q , iii) computation of system eigenvalue A (complex) of 5  Chapter 1. Introduction  4  interest from the equation /5TMl + ,\cbTCcb + bTKq5 2 .;\  =  0  When structural damping is considered (usually related with sinusoidal excitation of a given frequency) the equivalent viscous damping matrix can be determined. In this case it is taken as proportional to the stiffness matrix and inversely proportional to the excitation frequency. Description of this type of damping can be found in [23] for example.  1.2.3  Nonclassically damped systems  There have been some efforts to develop CMS methods for nonclassically damped systems in 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 (real vectors) of components. This representation does not take into account the damping properties of the components and cannot be accurate, especially in the case of significant damping. To solve the eigenvalue problem for a nonclassically damped system it is expedient to rewrite the equation of free motion (1.1) in the reduced, or so-called state-space form: AY+BY=0  (1.2)  Chapter 1. Introduction  where A  CM ,  M  B  5  KU =  0  ,  0  X  Y  —M  X  Equation (1.2) leads to the eigenvalue equation (AA + B)q  0  which yields complex eigenvalues A and complex state-space eigenvectors q. Hale [15] proposed to use vibration modes of undamped components and to improve accuracy by iteratively generating component state-space vectors. Hasselman and Kaplan  [241 used complex vibration modes, but it was not shown how they were determined and the residual-attachment modes were not taken into account.  In [25] a modified  fixed-interface method is shown, where fixed-interface vibration modes of undamped components are replaced by corresponding complex modes. Howsman and Craig [26] showed a state-space free-interface formulation that uses a set of free-interface vibration modes, but instead of residual-attachment modes a set of attachment modes was used. Another application of a state-space free-interface CMS technique for nonclassically damped systems was shown in [27] for the case of a two component system, where along with free-interface vibration modes a set of residual-attachment modes was used. This formulation has similarity (in terms of coupling procedure) with the formulation devel oped in this thesis, however in [27] the selection of free-free vibration modes was not generalized from the undamped system case to the damped one and the authors used different approach for determination of residual-attachment modes in the case of uncon strained components. In this thesis a formulation of state-space free-interface method (CMSFR) has been developed for systems with nonclassical damping for the case when mass, stiffness and damping matrices of components are symmetric. A system eigenvector is constructed  Chapter 1. Introduction  6  from state-space free-free vibration modes of damped components and state-space residualattachment modes. A complex eigensolver is used to determine the free-free vibration modes of each component. The formulation is demonstrated for an arbitrary number of components, which can be constrained or unconstrained. Also a new method is presented to treat an unconstrained component, which is more economical in the computational sense than the method shown in [27]. A selection procedure for the retained free-interface vibration modes is developed for this nonclassically damped system case.  Chapter 2  Analysis of damped systems  2.1  Classically damped systems  In this chapter, unsubdivided systems are considered. The general case of classical damp ing is analysed. An important property of this kind of damped system is that it has classical mode shapes (eigenvectors), i.e, the same as the associated undamped system. The system of equations of free vibration of a discrete nongyroscopic damped system can be expressed in the form  MX+CX+KX=O  (2.1)  where M, C, K represent mass, damping and stiffness matrices of order N. Under certain conditions this system of, in general, coupled differential equations can be transformed to a system of uncoupled equations by a congruence transformation. It is known that the mode shape matrix of the undamped system can make this transformation with respect to the M and K matrices. In this study the necessary and sufficient condition of diagonalization of C by this matrix of undamped mode shapes is considered. For a general system it has been formulated, e.g., in [21], [22] as M C commutes with IVI’K. 1 It should be noted that the necessary and sufficient condition for diagonalization of matrix C in terms of undamped system modes cP =  or  C  = (T)_1[a]_1  7  Chapter 2. Analysis of damped systems  (  8  [aj is a diagonal matrix) implies that N real coefficients a, is enough to describe all  the variety of possible damping matrices for the given system defined by I’I, K. Limiting consideration to systems that have distinct (no repeated) undamped natural frequencies the necessary and sufficient condition of proportionality imposed on matrix C will be formulated in terms of matrices M, K (M is positive definite and K is positive semidefinite) in the theorem below. Theorem. For the given matrices M, K of order N in (2.1) the matrix C will be diagonalized simultaneously with M, K if, and only if, it is represented as C where gj, 1  =  1, 2,  ...,  = 1 (M Mg K )’’  (2.2)  N are arbitrary coefficients.  Proof. Necessity. There is such a matrix 4 consisting of undamped mode shapes that TMpJ  TC  where [b ] 2  =  ] 2 b, [a  =  =  [b}  =  ] 2 [a  a are diagonal matrices and the matrix 4 is mass normalized (first  equation), so the elements of b will be the natural frequencies squared. Rewrite the above expressions as C K  =  (T)_1a _ 4 1  (2.3)  (T)_lb4_l  (2.4)  Chapter 2. Analysis of damped systems  9  and M  =  T)1 4 (  (2.5)  or 1 M  =  (2.6)  It is obvious that there is one to one mapping, namely, a  —*---  C, b  —‘÷—  K. Express  matrix a as a  =  g + ... + b I+2 1 g  (2.7)  9Nb  Equation (2.7) can be rewritten as a system of linear equations for coefficients  1  1.  L2  £  (J  1 U  1 ‘  ‘-‘2  1.  x2 “2  1  I  1  “N  . •-  N—1  gi  1 a  92  2 a  9k,  Ic  =  Ui  lN—i ‘-‘2  (2.8)  N—i 1  12  ‘-‘N  UN  9N  aN  The determinant of this system (the Vandermonde determinant) is non-zero because there are no repeated values b 1 (natural frequencies squared), so the solution is guaranteed and this representation (2.7) holds. Rewrite (2.7) using (2.3),(2.4) TC  gk(TK)kl  Therefore any matrix C can be represented as C  =  J 9 (T)1[  g T K + +2  +gJ\TK4.  and using (2.6) one can obtain C  =  TK]-i  + ...+  Chapter 2. Analysis of damped systems  10  as required. Sufficiency. The damping matrix is given as in (2.2). Using (2.5),(2.6),(2.4) one can obtain C  =  (T)_1 gi(T(T)_lb_l)l_l  = (T)_1( 91  g+ b +2  ...  +  =  9Nb)  Therefore  represents 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 then K 2 C=giM+g  1=1,2  which is Raylegh damping. If more terms taken, e.g. three, then K 2 C=gM+g K 3 M’K +g  1=1,2,3  and so on. Remark 1. Note that for systems with repeated undamped natural frequencies expression (2.2) will represent a sufficient condition, but not the necessary one. This means that there might be 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 as the necessary condition if the vector of damping coefficients [ak] lies in the space of eigenvectors of the system matrix (2.8). Then, for example, if there are m repeated eigenvalues it means that eigenvectors of the Vandermonde matrix will represent the  Chapter 2. Analysis of damped systems  vector subspace dimensioned by N  —  11  m + 1.  The form of these eigenvectors is not  considered here and an explicit expression of vector [ak] as a linear combination of these eigenvectors is not shown. It is left to note that in this case there will be an infinite number 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 to modal coordinates X  p with premultiplication by +  T  [Mg(M_lK)k_l  T  gives:  ] + Kp  =  0  (2.9)  where 4 is the matrix of classical mode shapes. Assuming that 4 is mass normalized then  where A is a diagonal matrix, whose elements are the undamped natural frequencies squared  w. The matrix equation (2.9) can then be written as a system of uncoupled  ,  equations: (2.10)  i=1,2,...,N  Equations (2.10) yield the eigenvalues of the damped system, namely: = (—g’  where D  =  2 (Z gkh)  —  +  ),  If D  4i,.  <  =  —  0 the roots are complex conjugate, otherwise  they are real. Considering the case of complex conjugate roots: Re()  1 _—g  (2.11)  and =  1 =  \J71i  —  (gk_1)2  (2.12)  Chapter 2. Analysis of damped systems  12  Prom which it may be seen that  2 + (Im()j)) (Re()) 2  (2.13)  Therefore the absolute values of complex eigenvalues are independent of arbitrary propor tional damping. It may also be noted from (2.12) that for classical damping the damped natural frequencies are always less than the undamped frequencies. If it is required to modify the natural frequencies of a given system by modification of the damping matrix, equations (2.11),(2.13) can be used to determine the coefficients 9k  in (2.2) that yield the required eigenvalues. For example, upon the prescription of the  imaginary 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 the following system of algebraic equations for the kept (non-zero) coefficients  1 2  1  ?7  1  72  1  7L  72  ...  L—1 77  ...  72  gk:  ReAi) g2  —  ) 2 Re(A  Re(AL)  ...  gL  The determinant of this system is the Vandermonde determinant, which is not zero if there are no repeated  .  Therefore for a system with distinct undamped natural frequencies  there will be a unique solution for  9k,  k  1, 2,  ...,  L. Note that only a knowledge of  undamped 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 corresponds to the case when all g,  =  0, except g  =  =  aM, which  a. The effect of mass-proportional  damping upon the damped frequency (Im()’,)) and decay factor (Re(A)) is illustrated graphically in Figure 2.1. The undamped natural frequencies  Wmzn, W,  and  Wmax  represent  respectively the minimum, some intermediate and the maximum value for the system with  Chapter 2. Analysis of damped systems  13  Im(A) Frequency max  (4)j  C.) mm  2  Re(A) Decay Factor  2  Figure 2.1: Effect of mass-proportional damping no damping. The equation (2.13) is shown by solid circles, while the dashed lines represent equation (2.11). Intersection of the solid circles with the dashed lines corresponds to the location of eigenvalues. If no intersection occurs that mode is overdamped for that value of damping. Thus if a  < 2 Wmj,-,  all modes are oscillatory and if a  > 2 Wmax  all modes are  overdamped. It may be observed that mass-proportional damping has a larger influence on the lower mode frequencies than those of the higher modes. ii) Stiffness-proportional damping: the damping matrix is represented in this case as C  =  /3K, corresponding to the case when all g,  =  0, except g2  =  /3. To illustrate the  effect of stiffness-proportional damping introduce the following notations: x y  =  =  Re),  Im()). Equations (2.11),(2.12) lead to 2 + (x + 11/3)2 y  =  1//32  which 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 deter  mined by the intersection of the corresponding dashed circle with the solid ones. In this case it may seen that the stiffness-proportional damping has a larger influence on the  Chapter 2. Analysis of damped systems  14  Im(X) Frequency  max  (A)j  “nun  Re(X) Decay Factor  Figure 2.2: Effect of stiffness-proportional damping higher mode frequencies. From the figure it may be noted that if 3 overdamped and if /3  <  ---—  iii) Rayleigh damping g  =  >  ----  all modes are  all modes are oscillatory.  c, g  =  3, i.e., C  =  cM + /3K.  Using the same notations equations (2.11),(2.12) lead to  The graphs of these circles are shown as dashed lines in Figure 2.3. Upon the given c, /3 the eigenvalue location is determined once again by the intersection of the corresponding dashed 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 the  simultaneous satisfaction of the following inequalities: Wmax.  If  Note that at a/3  > =  Wmax,  or  <  or a/3  and >  >  1 all modes are overdamped.  1 the radius of the dashed circle becomes zero.  It may seen from the figure that it is possible to choose Rayleigh damping coefficients such that the lowest and higher modes are overdamped, but the intermediate modes are oscillatory.  Chapter 2. Analysis of damped systems  15  Im(X) Frequency  1 a  0  (A)  XL  max  mm  Re(X) Decay Factor  XR  Figure 2.3: Effect of Rayleigh damping Nonclassically damped systems  2.2  In the case of nonclassically damped systems the general damping matrix cannot be expressed in the form defined by equation (2.2). The damping matrix of this type of system cannot be diagonalized simultaneously with M and K. To solve the eigenvalue problem it is expedient to rewrite equation (2.1) in the reduced or so-called state-space form: (2.14)  AY+BYO where A  CM ,  M  0  B=  K  0  0  —M  The equation of free motion in a system mode Y(t) state-space eigenvector  Y=  ,  X  (2.15)  X =  e)tq5 with eigenvalue ) and  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. Note that the eigenvectors will be orthogonal with respect to A, B. The eigenvectors  can  Chapter 2. Analysis of damped systems  16  be normalized by setting i=1,2,...,2N  bAq=1  or if 4 is defined as a matrix of A-normalized eigenvectors, then the following will hold: TB  =  —A  where 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 general there 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 them selves depend upon C. To illustrate the effect of nonclassical damping the behaviour of the simple system shown in Figure 2.4 is analyzed, where m 1  2 m  1 k  =  2 k  =  1. The  location of eigenvalues for different values of c is shown in Figure 2.5.  Figure 2.4: Two degree of freedom system As may be noted the behaviour is quite different to that of a classically damped system. As the damping constant c is first increased the imaginary part (frequency) of the lowest eigenvalue increases (a phenomenon not observed in classically damped systems) and that of the higher mode decreases. The real parts both increase with c up to a certain point after which the real part of the second mode begin to decrease. At a damping value of 2 both roots coincide and at this point there is a discontinuity in the  Chapter 2. Analysis of damped systems  17  Im(X) 1.618  V c=1.5 c=  1 0.866 0.618  c=2.5 —1  —0.5  0  Re(A)  Figure 2.5: Eigenvalues of the system in Fig 2.4 monotonic behaviour of the eigenvalues. The imaginary part (frequency) of the second eigenvalue then starts to increase monotonically to a frequency of 1 rad/s. This situation corresponds to the physical condition when c is so high that the first mass does not move and the system is responding essentially as a single degree of freedom system. The real part of the first eigenvalue monotonically increases and at c  =  2.5 this mode becomes  critically damped.  2.3  Free response function computation  Consider a homogeneous equation of motion of a nonclassically damped system with N degrees of freedom: MX-i-CX+KX=O  Chapter 2. Analysis of damped systems  18  To find a homogeneous solution the eigenproblem should be solved first. The solution of eigenproblem 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 form AjeA3tj  Y where ,\,,  (2.17)  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 2m real eigenvalues and N  —  m pairs of complex conjugate ones, then the real homogeneous  solution Xh can be presented as 2m  Xhom(t)  N—rn  ckektRe[cbk] +  =  t r 2 c ] n+jRe[e + jzz1  k1  where  are 2m real eigenvalues and  eigenvalues, , coefficients  ek,  k  4 =  N--Tn  j 3 CN+m+jIm[e3cI j1  are N  —  m representatives of complex conjugate  are the corresponding complex state-space eigenvectors. The real 1,2,  ..,  2N will be determined from the initial condition Xh(O)  X(O)  Xpar(0)  which represents the system of 2N real linear equations, where X is the total solution and Xpar a particular term (in the case of external force presence) of the solution.  2.4  Steady-state response function computation  The forced response due to a sinusoidal excitation is considered in this section. In the experimental part of this work this type of excitation will be considered. The equation of motion of a nonclassically damped system: MX  -  CX + KX  F(t)  (2.18)  Chapter 2. Analysis of damped systems  19  where the external force vector sinwL 0 F  F(t)  (2.19)  Although steady-state response X carl be found from (2.18) directly) the state-space representation will be used to show the relation between complex and real forms of the solution: AY + BY where A  C  M  =  ,  M  B  K  0  =  0  0  Y  ,  —  F  =  (2.20)  ,  5 F  Fd(t) =  —M  0  The seeking complex solution is t (Y + z)e  (2.21)  as the response to a given complex force Fd  )et 2 (fi + if  =  ficoswt  sznwt + i(fisznwt + f 2 f coswt) 2  (2.22)  The steady-state response in real form may be expressed as X  =  Re[}jj  X  =  Im[Ydj  (2.23)  or  Proceeding with (2.23)) the given force will be F(t)  Re[Fdj  =  ficoswt  f  =  —  f.sinwt  In the case (2.19)  f2 =  0  (2.24)  0 —F  (2.25)  Chapter 2. Analysis of damped systems  20  Substituting (2.21),(2.22) in (2.20) and cancelling e one can obtain  [c  M]  (Y -I- iY)’iw )w eI’ —Cl’i + 2  [  +  Y + zY  o]  K  =  (}‘ + zY)iw  (f  + if ) 2  L  2 + if  (2.26)  Rewriting equation (2.26) as 2 + K) Yi(—Mw  -  YCw + 1 i(Y C w+2 Y(—Mw + K))  =  one can obtain the system of linear equations for 2 ,Y (taking into account (224),(2.25)) 1 Y  Cw  Y  +K 2 —Mw  0 —F =  —Cw  +K 2 —Mw  0  The solution of (2.27) yields the steady-state response  X  =  Re{(Y -1- t )e 2 iY j  =  Ycoswt  which can be presented in the following form  X  =  D.si’n(wt +  )  where  2 d  ...  dN]  is the amplitude vector of the forced response and  N]  is the phase angle vector. In more detail  d,  \/‘‘  +  }‘  and  an’b,  =  z 1 Y 2z  —  Ysznwt  (2.27)  Chapter 2. Analysis of damped systems  21  where Yh, Y 2 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 following ways: i) if damping matrix C  0, then 0  1 + K)-  2 (—Mw  2 +K)’ (—Mw  0  ii) if C is not a zero matrix and assumed invertible, then  [ where a , 1  =  1 + GD’)’, G’(DG—  12 a  11 a  2 a,  2 a,  11 —a  =  ]  a,,GD’, D  =  Cw, G  2 + K. This --Mw  leads to economy of computer time because the inversion of a matrix 2N by 2N is replaced by inversion of a matrix N by N in case i) and by three inversions of a matrix N by N in case ii). Below some numerical results are presented, which show the influence of damping on the steady-state vibration response. As an illustration a finite element model of the vibration rig (Fig.2.6) was considered. Parameters of the rig are presented in Appendix B. A vertical concentrated load F  0 F  =  30 N and w  =  =  sinwt was applied at the centre of the box, where 0 F  32 Hz were the same for all examples. Of interest is the influence of  damping properties in the spring elements a, b, c, d. The following values of a damping constant were assigned: 1) c c  =  , 5) c 3 0.01k  =  , 6) c 5 0.02k  0 (undamped system), 2) c , where k 3 0.05k 5  =  =  , 3) c 3 0.002k  , 4) 3 0.005k  12 N/mm was the vertical axial  stiffness of the elements a, b, c, d and kept constant. The influence of c upon the response of the system was analyzed. For the chosen nodes (Fig.2.6) the amplitudes and phase angles were computed and the results for the vertical component of motion are shown in Tables 2.1-2.3.  Chapter 2. Analysis of damped systems  22  2 5  x  Y  Figure 2.6: Model of the rig with selected nodes It may be noted from Tables 2.1-2.3 that amplitudes of motion of points 5,6 are decreasing with increase of c and phase angles change. The ratio 0 /A was calculated, where Ab 3 A 0 is the amplitude of vertical motion at the centre of the box and A 3 is the amplitude of vertical motion of the supported beams (the average amplitude of the points 5,6 was taken). This ratio characterizes the transmissibility of motion from the box (points 1-4) to the supported beams. The change of this transmissibility depending of c/k 3 is presented in Fig. 2.7. It may be noted that increase of damping up to the c  =  3 leads to decrease of amplitudes of vibration of 0.01k  the supported beams, in other words provides decrease of transmissibility of motion from the box to the supported beams. After c  =  3 the influence of c upon transmissibility 0.01k  is insignificant, though the phase angles continue to change.  Chapter 2. Analysis of damped systems  Node 1 2 3 4 5 6  Example 1,_c_= 0 Amplitude, mm Phase angle, 0.0280 180. 0.0306 180. 0.0324 180. 0.0264 180. 0.224 0. 0.228 0.  23  °  Example 2, c 0.002k 3 Amplitude, rrirn Phase angle, 0.0261 183. 0.0242 188. 0.0261 188. 0.0241 184. 0.11.6 82.7 0.118 82.6  Table 2.1: Steady-state responses, examples 1,2  Node 1 2 3 4 5 6  Example 9, c = 0.005k 5 Amplitude, mm Phase angle, 0.0256 183. 0.0224 184. 0.0243 184. 0.0235 183. 0.0674 122. 0.0687 122.  °  Example , c 0.01k 3 Amplitude, mm Phase angle, 0.0255 183. 0.0221 181. 0.0240 181. 0.0234 184. 0.0540 147. 0.0550 147.  Table 2.2: Steady-state responses, examples 3,4  Node 1 2 3 4 5 6  Example 5, c = 0.02k 3 Amplitude, mm Phase angle, 0.0253 186. 0.0221 179. 0.0241 178. 0.0232 186. 0.0499 162. 0.0509 162.  0  Example 6, c 0.05k 5 Amplitude, mm Phase angle, 0.0243 193. 0.0229 176. 0.0249 175. 0.0222 194. 0.0489 173. 0.0499 172.  Table 2.3: Steady-state responses, examples 5,6  0  Chapter 2. Analysis of damped systems  D)  24  II Print II rabo.plt II Transmissibility  I: 0.00  0.01  0.02  0.03  0.04  0.05  Damping constant  Figure 2.7: Effect of damping on transmissibility  Chapter 3  Formulation of component mode synthesis method  3.1  Undamped systems  In this chapter, the formulation of CMS method developed in this study will be presented for the case of undamped systems (in this section) and nonclassically damped systems in section 3.2. This formulation will be shown on an example of a two component system and 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 solution is approximated by one constructed from some basis vectors (e.g., mode shapes) of subsystems (components of subdivided system). This method allows a significant reduction of the eigenvalue equation size due to the use of a limited number of basis vectors. The approximate solution for the lower eigenvalues and eigenvectors is very close to the exact one due to the proper selection of the basis vectors and the use of Calerkin’s method that determines the best approximation. Consider a system subdivided into two adjacent components (call them the 1st and the 2nd) with interface S (Fig.3.1). The equation of free motion of the system in a natural mode X(t)  =  etq  with frequency w and eigenvector  [&  jT 2  can be written in the  following form: 2  1 rn  1 k  0 +  0  2 m  0  0 2 k  fi  I  )  =  2  (3.1)  f2  where term et was cancelled, m , m 1 2 are the mass matrices of the components and 25  Chapter 3.  Formulation of component mode synthesis method  26  S Y  x Figure 3.1: Two component system , k 1 k 2 the stiffness matrices. The force vectors  f,  f2  will contain only interface forces  (interaction between components), which appear as external forces at the artificial sub division of the system and all the remaining components of fi,  f2  corresponding to the  component internal degrees of freedom will be zero, because of the absence of external forces. As can be noted the system eigenvector was also subdivided into two subvectors 2 corresponding to the 1st and 2nd components. Rewrite (3.1) in the abbreviated q, qS form: M + K) 2 (w  =  f  (3.2)  Subvectors q, q will be approximated in the following form:  1’=  22  where  ,  4  =  [ ] [;;] [ ] [:;]  (3.3)  (3.4)  matrices of lower (retained) free-free vibration modes (including rigid  body modes) for 1st and 2nd components respectively (determined as the result of eigen problem solution for free-free component),  ,  4  =  matrices of residual-attachment  modes for 1st and 2nd components, p , p vectors of free-free mode coordinates for 1st 1  Chapter 3. Formulation of component mode synthesis method  27  and 2nd components, p, p vectors of residual-attachment modes mode coordinates for 1st and 2nd components. Consider the determination of residual-attachment modes. Take the 1st component equation from (3.1) (—w m 2 i +k ) 1  fi  Use the transformation to modal coordinates q  iPi  (  complete set of free-free  mass-normalized modes of the 1st component) and premultiplying by 4 one can obtain:  1  2 1w  10  w  0  0  0  1 w  I  where 4 was partitioned into two sets h  )  fi  (3.5)  Phi  (index I means lower (retained) modes,  ,  higher modes of component), w , w 1  Pu  I  +  diagonal matrices with natural component  frequencies squared. The assignment of the number of retained modes for each component will depend on the range of system frequencies, which are supposed to be evaluated by an application of this method. It may seen from (3.5) that if 2 <<min[w w j 1  then the approximate expression for the modal coordinates Phi  1  (3.6) Phi  follows [101, [30]:  2 i—iT ‘‘hiJ1  LWhij  Thus the contribution of the higher modes to the subvector q can be approximated as h1Phi  where the columns of matrix R 1  r 2i—icTr ‘I’hi[Whll ‘hiJ1  = 4h1[wl]_1’l  corresponding to the interface degrees  of freedom are called residual-attachment modes. Due to (3.7) the interface forces in fi will be identified with the residual-attachment mode coordinates p.  Chapter 3.  Formulation of component mode synthesis method  28  The number of retained free-free component modes should be high enough, in order that condition (3.6) and consequently expression (3.7) are satisfied at the proper level, such that a good appoximation for the subvector q 1 by means of the basis vectors cI and 5 F (the selected columns of matrices  ,  1 respectively) will be obtained. The matrix R  1 can be expressed in an advantageous form [10], [30], [4]: R i—i n fLi =  —  .c. ‘‘11  2i-1c.T 11 P  Wj  This 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 determination of residual-attachment modes is conducted for the second component. Thus combining (3.3), (3.4) the system eigenvector is approximated as 1  q•= 0  4  0  0  p’  0  4  4  p a  P2 or in abbreviated form  (3.8)  ç=p  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: 1 p  I  0  0  0  I  0  0  0  I  1 P (3.10)  =  p  0  —I 0  P2  Chapter 3. Formulation of component mode synthesis method  29  or in abbreviated form p  Subvectors  4  and  c2  —  can be written in the following form:  =[]=[: :][z] and =  ‘P2  where  B  =  2B  ‘2B  jI 2i  a  2 P a  2i  Pl  displacements at the interface, qY all the remaining displacements. Apply  the equation of continuity of displacements at the interface: (3.11) thus it follows from the above expressions that +  BPi 4  ‘1BP1  =  2BP2  2BP1  Therefore one can express the interface forces in terms of free-free mode coordinates:  p  1 T  2 T 2 P  where =  =  +  2B)(iB)  +  2B)(2B)  Thus the following relation will hold: 1 P  1 T T 2 0  I  (3.12)  Chapter 3. Formulation of component mode synthesis method  30  or in the abbreviated form: ‘I  q =3 q Therefore the vector p of all generalized coordinates can be expressed in terms of independent 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 eigenvector and vector q will be: (3.14) Substituting (3.14) in (3.2) one can obtain:  M + K)/3q 2 (—w  f+  =  e  where the quantity e represents an approximation error. Galerkin’s method (premulti plying 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 is due to the equal displacements and opposite forces at the interface. To show it consider the product of an arbitrary vector q and vector TT / T q f 3 =  TTf, 3 /  []  namely,  [:]  Chapter 3. Formulation of component mode synthesis method  31  and if only interface forces are present, then the above expression is transformed to the following one: iBiTi al  LY1i  Pii  i  LBiTi a  i  P2  Taking into account the (3.11) and (3.9) the above expression becomes a zero vector. Therefore the product of vectors q and q, which means that vector  / T 4 T 3 f  TTf 13  is always zero for an arbitrary vector  must be zero and the right side of equation (3.15)  vanishes. Thus the final condensed equation of motion for two coupled components (or the whole system) in terms of generalized coordinates can be written in the following form:  4 fK M 2 (—w )q 4 where j  ,  4 K  =  TTk 3 /  =  0  will be real symmetric matrices.  This  equation represents an eigenvalue problem. The eigenvalues w 2 and eigenvectors q will be all real quantities.  3.1.1  Case of an unconstrained component. Method of weak springs  A new approach to treat an unconstrained component is developed by imposing “weak” constraints (springs) on the system, which remove the singularity of the stiffness matrix and make it invertible. The stiffness of “weak” springs can be assumed to be 10 k 7 where k, are corresponding diagonal elements of the stiffness matrix, which are supposed to be modified. If a smaller value is chosen the modified stiffness matrix may not be in vertible (the numerical aspect of inversion should be taken into account). This method requires less computational effort than the method described in [27]. The lowest (strictly speaking 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 introduced  Chapter 3. Formulation of component mode synthesis method  32  “weak” springs, but the stiffnesses of these “weak” springs should be negligible com pared to the component and system stiffnesses, so these “flexible” modes will actually correspond to the motions of the component as a rigid body. To illustrate this method consider the example of an unconstrained component such as a linear bar element (Fig.3.2), which can move along the X axis. The bar element has two degrees of freedom x , x 1 2 and the corresponding stiffness matrix is  L  where B  =  modulus of elasticity, A  =  1  —1  —1  1  cross-section area, L  =  length of the element. The  determinant of k equals 0. The mass matrix has the following form: pAL m=— 6  where p  1  12  density of material. The eigensolution will yield: 0,  =  2, where ri  2  =  (rigid  [a, a]  =  [a, —a]  —  body mode)  (flexible mode)  6—s.  Introduce a weak spring with stiffness  =  E,  attached to one of the nodes (Fig.3.2),  then the stiffness matrix becomes invertible k’—L  1+E —l —1  1  Xi__X2  Figure 3.2: Linear bar element  Chapter 3. Formulation of component mode synthesis method  If  =  33  0.00 1, then the eigensolution will yield: 1 X  0.00016,  2.0005,  =  2 q  [a, 1.0005a]  =  [a, —0.99983a]  (“rigid  —  body” mode)  (flexible mode)  The approximation error for eigenvalues and eigenvectors is less than 0.05%. Therefore an unconstrained component can be modelled as constrained if  is small enough.  Chapter 3. Formulation of component mode synthesis method  3.2  34  Nonclassically damped systems  In this section the formulation shown in section 3.1 is generalized to nonclassically damped 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 components will have the following form:  m  0  1 c  1 U  1 U  0  + 2 m  0  1 k  2 U  0  =  2 U  C2  (t) 1 f  1 U  0  +  .  2 k  0  2 U  (t) 2 f  where U , U 1 2 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) where  A  0 1 A =  ,  0  1 A  B  0 1 B =  ,  2 A  1 c  ,  1 m  2 A  2 c  ,  2 U ,  2 m  2 m  1 U =  Ui  ,  ,  F  1 F =  2 F  =  0  1 Y =  2 B  0  1 m  Y  1 F  1 k  1 B  ,  0  fi ,  2 B  2 k  0  0  2 —m  =  1 —m  0  =  2 U  0  12  F  0  0  The equation of free motion of the system in a system mode Y eigenvalue ) and system state-space eigenvector  =  [  jT 2  e\tcb with system  (subdivided also) will have  the following form:  I  1 A  1 B  0  0  +  0  2 A  0  2 B  1 F  I  )  =  2  24 F  (3.16)  Chapter 3. Formulation of component mode synthesis method  35  or in the abbreviated form: (A 1 B)  (317)  where F is defined by the equation:  The force vectors  2 will contain only interface forces (interaction between compo F  nents) which appear as external forces at the artificial subdivision of the system and all the remaining components of  2 corresponding to the component internal degrees F  of freedom will be zero, because of the absence of external forces. In this state-space representation \, q, F are assumed complex. The complex subvectors q and q 2 are approximated in the following form: p11  -  (3.18)  —  a  =  P2  []  (3.19)  a  2 P  where free-free vibration modes ‘I, 4 are complex and a complex eigensolver is used to compute them. Note the modal coordinates t p pa are also complex in general. , Consider the determination of residual-attachment modes 4’,  .  Take the 1st com  ponent equation from (3.16) 1 +B (A ) 1 Use the transformation to modal coordinates  &  =  1 F ii 4  (  =  complete set of free-free  A-normalized modes of the 1st component) and premultiplying by  A  10  11 A  0  0  Ahl  -10  I  I  )  =  Phi  one can obtain:  1 F  (3.20)  Chapter 3. Formulation of component mode synthesis method  where 4 was partitioned into two sets higher modes of component),  36  (index 1 means lower (retained) modes, h  ,  diagonal matrices with component eigenvalues.  )khl  The assignment of the number of retained modes for each component will depend on the range of system eigenvalues, which are supposed to be evaluated by an application of this method. It may seen from (3.20) that if  I  <<  rnin{  hi  1  then the approximate expression for the modal coordinates Phi  where the columns of matrix R 1  Phi  follows  i—iiT i—i i [Ahlj ‘hi”i*  Thus the contribution of the higher modes to the subvector hi Phi  (3.21)  can be approximated as  ’hi[)hi]hiFi* 4  (3.22)  4, [)hih corresponding to the interface degrees  of freedom will be called state-space residual-attachment modes, which will be complex in this case. Due to (3.22) the interface forces in F 1 will be identified with the residualattachment mode coordinates p. The number of retained component modes should be high enough, in order that condition (3.21) and consequently expression (3.22) are satisfied at the proper level, such that a good appoximation for the subvector  4  by means of basis vectors  4 (the  selected columns of matrices 4, R 1 respectively) will be obtained. The matrix R 1 can be expressed in the following form: 1 R  B  —  This means that computation of the lower eigenvectors only is required for each compo nent, which leads to economy of computer time. The inversion of matrix B 1 is determined  Chapter 3. Formulation of component mode synthesis method  37  by independent inversion of matrices rn ,k 1 . In the case of an unconstrained component 1 the inversion of the stiffness matrix was discussed before. An analogous determination of residual-attachment modes is conducted for the second component. Thus combining (3.18), (3.19) the system eigenvector is approximated as  =  =  0  4  0  0  pZ  0  F  4  p a  P2 or in abbreviated form =  p 4  (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 gen eralized coordinate vector p, i.e., using a matrix transformation (see analogous transfor mation (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: 1B  and  Chapter 3.  where  B  =  Formulation of component mode synthesis method  displacements at the interface, çi’  =  38  all the remaining displacements. Apply  the equation of displacement continuity at the interface: (3.26)  =  Expression (3.26) is used to eliminate residual-attachment mode coordinates from the vector q’ expressing them in terms of free-free mode coordinates, i.e, using a matrix transformation (see analogous transformation (3.12)) one can obtain: q’  =  3Uq  (3.27)  where the vector Pt’ q P2 Substituting (3.27) in (3.25) the vector p of all generalized coordinates is expressed in terms of independent generalized coordinates q: p  =  /3’13”q  or p where 3  =  ,8q  (3.28)  =  Substituting (3.28) in (3.23) the relation between the approximate system eigenvector  4  and the vector of generalized coordinates q will be (3.29)  Substitution of (3.29) in (3.17) gives: (A + B)/3q  =  4+ F  Chapter 3. Formulation of component mode synthesis method  where the quantity  39  represents an approximation error. Galerkin’s method again yields  the following matrix equation: TT(4  + B)3q  =  /3 F ’ T .  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 coupled components (or the whole system) in terms of generalized coordinates can be written as: ()4 + B )q 4  where A 4  =  I3TTA13  ,  4 B  =  /3T4TB/3  =  0  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, which will correspond to overdamped modes (depending upon the damping properties of the system). In the case when c , c 1 2 are zero matrices (undamped system) the eigenvalues will be pure imaginary (zero real parts).  Chapter 3. Formulation of component mode synthesis method  3.2.1  40  Component mode selection procedure  Selection of lower (retained) modes is made on the basis of the absolute values of the eigenvalues which are complex numbers. Thus both the imaginary part (frequency) and the real part of the eigenvalue are counted. This will be important when a component is heavily damped. In the case of an undamped component, selection of retained modes will be based on the lower frequencies (imaginary parts), because the real parts of the eigenvalues are zeros. The ratio T  =  I  .‘com  A  yields 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 component eigenvalue out of the retained eigenvalues. This procedure can be illustrated as follows in Fig.3.3. ‘The solid circle indicates the level of system eigenvalue of interest and the dashed one corresponds to the level of retained eigenvalues (eigenvectors) for each component. All the eigenvectors with eigenvalues located in sector D (within of dashed circle) will be retained modes, and all the eigenvalues beyond of this sector will correspond to the higher modes, which are not used in the analysis. It has been established by conducting a series of numerical computations for different systems (including heavily damped systems) that a good accuracy for the given range 5 0 j A  can be achieved with r  =  2. Note that the greater r the better accuracy, but  the larger the size of the condensed eigenproblem.  Chapter 3. Formulation of component mode synthesis method  Jm(A) ACOrn  /4.. /  0  / _—  07  /0  -of®  Re(A)  D  Figure 3.3: Mode selection procedure  41  Chapter 3. Formulation of component mode synthesis method  3.3  42  Case of an arbitrary number of components  Generalization of the developed method (sections 3.1,3.2) to a system with an arbitrary number of components is straightforward and shown here only in a brief form. This gen eralization is conducted simultaneously for the both cases: undamped and nonclassically damped 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 1st and 2nd one). Introduce the following notations: T 1  a  —  a  Pu  P12  a p 2 1  a p 2 2  a  ...  a  PJ  •..  P1L  and T  P2 where vectors  and  -—  —  ...  a p 3  a  ...  P2L  consist of interface force vectors of all interfaces. Apply the  equation of force equilibrium at the jth interface: =  —p  (3.30)  Using (3.30) one can express the vector of generalized coordinates p (see analogous matrix transformation (3.10)) through the vector q’, which will not contain i3: I,  Consider now the displacements at the jth interface and  and q. These subvectors  can be expressed in the following form: iBPi  +  iBP1  2j  2BP2  +  2BP2  =  1,2,  ...,  L  where the vectors f, 75 combine all the interface force vectors, which belong to the components 1 and 2 respectively: Pk]  Chapter 3. Formulation of component; mode synthesis met hod  43  and  where i,  ...,  k  =  numbers of the interfaces, which belong to component 1 and 1,  ...,  n to  component 2. The compatibilty of displacements at the jth interface yields the following system of equations B Yij  —  B Y2j  I  —  which can be used to express the interface forces p  ‘-)  (j  =  1,2,  ...,  L) in terms of free-free  mode coordinates. Thus the vector q’ can be expressed (see analogous transformation (3.12)) by the following matrix transformation:  q’ = where  q=[p  ..  ...  and p is a set of free-free mode coordinates for the ith component. Therefore the vector of all generalized coordinates p is expressed in terms of independent generalized coordinates q in the following matrix transformation:  p= or  where 3  =  q 11 /3’/3  p = 13q  Thus the relation between the approximate system eigenvector qS and the vector of independent generalized coordinates q will be  = 13q  Ghapter 3. Formulation of component mode synthesis method  44  Analogous manipulations (see sections 3.1,3.2) with the equation of free motion of the subdivided system will yield the final condensed eigenproblems in terms of generalized coordinates. For the case of an undamped system: M 2 (—w  where M  =  3TTJ J 1 3  K  + K)q  TTk 3  =  0  will be real symmetric matrices. For a  damped system: (-4 + B)q  where A  I3TTA/3  ,  B  =  /3TTB4I3  0  will be complex symmetric matrices.  Chapter 4  Numerical results  4.1  Comparison of CMSFR method with “VAST” program for undamped systems  Comparison of the results obtained by the program using CMSFR method with the results 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 component subdivision). 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 ex amples 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 within each example were the same and are shown in metres. The thickness of the membrane element (2nd component) in example c) is 0.02 rn. In example d) the second component is represented by a brick element with parameters of a cube. The material of all elements is steel. For examples a),b),c) the motion is considered in the plane of the drawing. For example d) 3-D motion is considered. The total number of retained modes for example a) was 20, which determines the size of the condensed eigenvalue problem (Table 4.1). This total number of the retained modes was composed of 4 + 6 + 6 + 4 respectively for each component.  45  Chapter 4. Numerical results  46  ©  © 2  2  a)  c  b)  _4  ‘‘  Figure 4.1: Test examples A rigid-body mode exists for the system in Figure 4.1,b (Table 4.2), which was de tected by the appearence of a low (near zero) frequency (the method of weak springs was used to treat unconstrained components). This system has bar elements in the 2nd component, that provides a rotational rigid-body degree of freedom at the joint of the bar elements. The current version of “VAST” does not detect such “hidden” rigid-body modes, 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 the 1st and 2nd mode shapes are shown in Fig.4.2,4.3 for the system in Fig.4.1,a. There is no difference between the modes calculated using CMSFR method and those calculated by “VAST” using the unsubdivided model.  Chapter 4. Numerical results  47  Frequencies, Hz CMSFR VAST Difference, % # 1 124.85 124.85 0. 2 298.25 298.25 0. 430.54 430.11 3 0.099 516.62 4 516.23 0.079 5 768.96 768.95 0.001 878.51 6 876.93 0.18 Size of ezgenvalue problem CMSFR VAST 20x20 21x21 Table 4.1: Four component beam element system: “a”  Frequencies, Hz CMSFR VAST Difference, # 0.044 1 2 140.35 140.35 0. 147.44 147.44 3 0. 432.89 432.28 3 0.14 4 633.57 633.48 0.014 643.38 643.30 5 0.012 898.45 894.15 6 0.48 Size of eigenval’ue problem CMSFR VAST 11 x 11 14 x 14 -  Table 4.2: Three component beam-bar element system  %  -  with one rigid-body mode: “b”  Chapter 4. Numerical results  Frequencies, Hz CMSFR VAST Difference, % # 1 1.4666 1.4651 0.10 2 96.6478 96.660 0.012 126.485 126.46 3 0.019 4 313.73 313.73 0. 1979.3 1979.3 5 0. Size of eigenvalue problem CMSFR VAST 12x12 9x9 Table 4.3: Three component beam-bar-membrane element system: “c”  Frequencies, Hz # CMSFR VAST Difference, % 0.2633 1 0.2631 0.076 0.3409 2 0.3409 0. 0.3664 0.027 0.3663 3 4 16.308 16.308 0. 38.242 38.228 0.036 5 38.255 38.241 6 0.036 Size of eigenvalue problem CMSFR VAST 12x12 33x33 Table 4.4: Four component beam-brick element system: “d”  48  Chapter 4. Numerical results  49  20)11 Print II Istmod.pIt ) 1st mode shape 1 SI -Crnsfr  5.0  6.0  7.0  Node  Figure 4.2: First mode shape of the system in Fig.4.1,a  OLI)JLflI  p nomoo.pIT II dna mooe snape  isi .crnsIr  G) D  4-  0  E  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  9.0  Node  Figure 4.3: Second mode shape of the system in Fig.4.1,a  Chapter 4. Numerical results  4.2  50  Comparison of CMSFR method with “DREIGN” program for undarnped and nonclassically damped systems  Below some systems (without a component subdivision) are considered and their cor responding eigenvalue problems are solved by using a complex eigensolver “DREIGN” program [32], which uses the QR method. Then these results are compared with CMSFR method results for subdivided systems. A two component system consisting of beam elements (Fig.4.4) was considered. The element cross-section was 0.01 x 0.01 m.  The damping matrices of the components  were taken as stiffness-proportional ones on the component level, i.e., c 1 2 c  =  —  , 1 0.00002k  , which produces nonclassical damping on the level of the whole system. 2 0.00001k  The motion of the system is considered in the plane of the drawing. Table 4.5 presents a comparison of the results and shows excellent correspondence.  u  a  a  0.6  u  -a——.  a  a  a  0,4  Figure 4.4: Two component beam element system Then 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 crosssection of elements was 0.1 x 0.1 rn, the lumped masses were 100 kg each, the dashpots 10 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). Pa rameters of the rig are shown in Appendix B. For application of the CMSFR method the finite element model of the rig was subdivided into four corriponents (Fig.4.7). In  Chapter 4. Numerical results  # 1 2 3 4 5  CMSFR Real,1/s Imag, Hz -0.8964 52.717 145.34 -6.396 285.06 -26.303 472.73 -70.22 -157.15 707.65 Size of CMSFR 16 x 16 complex  51  Eigenvalues DREIGN Difference, % Real, 1/s Imag, Hz Real Imag -0.8964 52.716 0. 0. -6.395 145.32 0.015 0.013 -26.287 285.02 0.014 0.06 -69.847 471.74 0.534 0.209 -156.86 706.34 0.184 0.185 eigenvalue problem DREIGN 54 x 54 real  Table 4.5: Two component beam element system the 1st model there were no damping elements present. Some clamping properties were assumed for the elements a, b, c, d (Fig.4.7) for the 2nd model, producing an example of a nonclassically 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 just one representative of each pair is shown in Tables 4.5-4.7,4.9. Good agreement between the 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 Table 4.8. “Z” displacements of the 12 nodes (Fig. 4.8) were selected. The comparison of the 1st eigenvectors for the damped rig model is shown in Table 4.10.  Chapter 4. Numerical results  52  p  ‘‘  p  \  Figure 4.5: System with dashpots arid lumped masses  CMSFR Real, 1/s Imag, Hz 1 2 3 4 5 6  -0.004433 -0.01609 -0.02204 -0.0140 -0.005491 -0.001634  14.044 33.699 60.588 101.94 208.61 313.93  Eigenvalues DREIGN Real, 1/s Imag, liz -0.004433 -0.01609 -0.02204 -0.0140  -0.005442 -0.001617  14.044 33.698 6(1.588 1(11.89 207.55 313.18  Difference, % Real Imag 0. 0. 0. 0. 0.9 1.0  0. 0. 0. 0.049 0.51  Size of eigenvlue problem CMSFR DREIGN 18 x 18 complex 24 x 24 real Table 4.6: System with dashpots and lumped masses  Figure  4.6:  Finite element model of the rig  0.23  Chapter 4. Numerical results  Figure 4.7: Four component presentation of the system  # 1 2 3 4 5 6 7 8  Eigenvalues CMSFR DREIGN Difference Real 1/s Irnag, Hz Real 1/s Imag, Hz % 4.791 0. 0. 4.774 0.356 0. 5.767 0. 5.757 0.173 6.121 6.108 0.212 0. 0. 0. 8.144 8.130 0.172 0. 8.343 0. 8.357 0. 0.167 9.212 0. 9.192 0.217 0. 31.46 31.33 0.414 0. 0. 0. 31.57 0. 31.42 0.477 Size of eigenvalue problem CMSFR DREIGN 516 x 516 real 60 x 60 complex  Table 4.7: Comparison of eigenvalues for the undamped rig model  53  Chapter 4. Numerical results  54  1 12  2  10  5 6  3  11  4  z  7 8  x  zY  Figure 4.8: Selected nodes for the eigenvector presentation  1 2 3 4 5 6 7 8 9 10 11 12  Z displacements in the 1st eigenvector CMSFR DREIGN Real Imag Imag Real -O.252E-11 O.913E-06 O.105E-05 -O.443E-08 O.467E-O1 O.481E-O1 -O.742E-04 O.000E+OO O.134E+OO -O.378E-06 O.137E+OO -O.309E-03 O.229E+OO -O.882E-05 O234E+OO -O.286E-03 -O.136E-09 O.104E-05 O.105E-05 -O.443E-08 O.481E-O1 -O.189E-05 O.481E-O1 -0.742E-04 O.137E O.137E+OO O.668E-05 -fOO -O.309E-03 O.235E+OO -O.164E-04 O.234E+OO .-O.286E-03 -O.643E+OO O.277E-04 -O.625E+OO O.578E-03 0.986E-i-00 -0.1O1E-04 0.100E+01 0.000E+00 0.100E+01 0.000E+00 0.100E+01 0.000E+00 -0.626E+00 -0.529E-04 -0.625E+00 0.578E-03  j  Table 4.8: Comparison of the 1st eigenvectors for the undamped rig model  Chapter 4. Numerical results  # 1 2 3 4 5 6 7 8  Eig e nval’u es DREIGN CMSFR Difference Real, 1/s Imag, Hz Real 1/s Imag, Hz Real, % Imag, % -0.1924 11.030 11.009 -0.1927 0.19 0.15 -1.0425 15.640 -1.0405 15.625 0.19 0.10 -3.4175 20.737 20.717 -3.4106 0.20 0.09 -1.6897 21.942 -1.6815 21.918 0.48 0.11 -10.086 33.729 -10.025 33.652 0.22 0.61 -11.680 35.268 -11.571 35.140 0.94 0.36 -29.472 65.501 65.213 -29.055 1.43 0.44 -34.059 68.942 -34.098 68.745 0.11 0.28 Size of eigenvalue problem CMSFR DREIGN 516 x 516 real 60 x 60 complex Table 4.9: Comparison of eigenvalues for the damped rig model  # 1 2 3 4 5 6 7 8 9 10 11 12  Z displacements in the 1st eigenvector CMSFR DREIGN Imag Real Imag R,eal 0,574E-07 0.648E-05 0.653E-05 0.141E-07 0.965E-01 0.177E-03 0.968E-0i 0.1 43E-03 0.255E+00 0.333E-03 0.256Ei—00 0.209E-03 0.425E00 0.657E-03 0.426E+00 0.104E-02 0.485E-07 0.182E-07 0.657E-05 0.652E-05 0.126E-03 0.956E-01 0.954E-01 0.476E-04 0.252E+00 0.205E-03 0.251E-j-00 0.434E-03 0.420E+00 0.401E-03 0.419E+00 0.216E-03 0.102E+00 0.132E-03 0.102E+00 0.976E-04 0.304E+00 -0.156E-02 0.304E+00 -0.1S1E-02 0.305E+00 -0.165E-02 0.304E+00 -0.163E-02 0.104E+00 0.531E-04 0.103E*00 0.694E-04  Table 4.10: Comparison of the 1st eigenvectors for the damped rig model  55  Chapter 5  Experimental results  The 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 vibrations of an engine mounted on a flexible support. Its geometric parameters (in inches) are given in 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 provide different damping and stiffness characteristics. For the experimental determination of stiffness and damping characteristics of the spring isolators the following procedure (Fig.5.3) was used. A vertical impact is applied at the mass m 3 and the natural (damped) frequency and logarithmic decrement of the free vertical oscillations of mass m 5 is measured. The natural frequency is obtained by using “Nicolet 660 A” spectrum analyzer and an accelerometer attached to the iriass. The Fourier transform feature built into the analyzer is used to determine the natural damped frequency  s.  It is assumed that damping is  small (damped and undamnped frequencies are close), thus stiffness of the spring can be determined as =  3 pm  The damping coefficient of the equivalent dashpot (Fig. 5.3) is determined as Cs =  where A , 2  -4+i  A, l—m p 5 n—----—  are the two consecutive amplitudes of oscillation separated by the period 56  Chapter 5. Experimental results  57  Figure 5.1: Photo of the experimental rig 27r/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 damp ing properties were taken as zero, i.e. the 1st rig model was considered as an undamped system. The free oscillations of the undamped rig were considered at first. The natu ral frequencies (Table 5.3) were obtained using the “Nicolet” analyzer and piezoelectric accelerometers attached to various points of the rig. It may seen from Table 4.7 that a good agreement with the analytical results was obtained.  Chapter 5. Experimental results  58  2 5  x  Y  Figure 5.2: Reference points on the test rig  # 1 2 3 4  Properties of the isolators Stiffness, N/mm Damping coeff-t, N x s/mm 12.2 0. 14.1 0. 12.5 0. 11.3 0.  Table 5.1: Stiffness and damping properties of the spring isolators (undamped rig)  # 1 2 3 4  Properties of the 4 isolators Stiffness, N/mm Damping coeff-t, N x s/mm 257.9 0.116 314.4 0.180 281.9 0.142 247.3 0.102  Table 5.2: Stiffness and damping properties of the spring isolators (damped rig)  Chapter 5. Experimental results  59  L.ms.  I  ksFtcs /  Figure 5.3: Spring isolator characteristic determination  # 1 2 3 4 5 6 7 8  Natural frequencies) Hz undamped rig damped rig (with different stiffness) 4.65 11.1 5.75 15.75 6.1 22.5 8.20 23.25 8.35 35.75 9.50 36.75 30.25 30.75 -  -  Table 5.3: Experimental frequencies for two rig tests In the second test damping was introduced into the spring isolators (due to the spe cific construction of the spring isolator the introduction of damping leads to increasing stiffness) and the natural frequencies were measured (Table 5.3). Note the good agree ment with the analytical results in Table 49. It may seen that the higher frequencies are associated with quite large real parts. The experimental determination of the higher fre quencies (#7#8) was complicated due to the fast amplitude decay of the free vibration response. Forced responses due to a sinusoidal vertical force applied to point “b” (Fig.5.2) were  Chapter 5. Experimental results  # 1 2 3 4 5 6  60  Excitation frequencies, Hz w=5.9 w=6.2 Amplitude Phase,° Amplitude Phase,° FIx. An. Ec An. FIx. An. Ex. An. 0.365 0.397 0. 0. 0.565 0.525 180. 180. 0.173 0.256 0. 0. 0.494 0.470 180. 180. 0.515 0.505 0. 0.403 0.448 180. 180. 0. 0.699 0.648 0.483 0.505 180. 180. 0. 0. 0.192 0.232 0. 0.112 0.109 180. 180. 0. 0.208 0.212 0.161 0.185 180. 180. 0. 0.  Table 5.4: Amplitudes, Phase angles for undamped rig analyzed for the damped rig. At first an excitation frequency sweep with a constant force amplitude (10 N) was conducted with a rate th  -  0.2 liz/s. The mesuared amplitude of  vibration at the point 2 is shown in Fig.5.4. It may seen that the peaks of the response correspond to the frequencies associated with the eigenvalues (Table 5.3). Next a sinusoidal vertical excitation at a fixed frequency and force magnitude was applied to the point “b”. Comparison of the analytical and experimental steady-state responses (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 presented in 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 good agreement of the experimental and analytical results.  It may seen a  Chapter 5. Experimental results  Figure 5.4: 8  61  -  24 Hz excitation sweep  Chapter 5. Experimental results  # 1 2 3 4 5 6  Excitation frequencies, Hz w16.0 w=33.0 Amplitude Phase,° Amplitude Phase,° Ex. Ex. An. An. Ex. An. Ex. An. 0.077 0.089 180. 180. 0.026 0.022 0. 0. 0.105 0.118 180. 180. 0.031 0.027 0. 0. 0.014 0.022 0. 0.002 0.004 180. 180. 0. 0.049 0.056 0. 0.024 0.014 180. 180. 0. 0.989 0.986 0. 0.998 0.999 180. 180. 0. 0.015 0.020 0. 0.014 0.019 0. 0. 0.  Table 5.5: Amplitudes, Phase angles for undamped rig  # 1 2 3 4 5 6  62  ...  continued  Excitation frequencies, Hz w=i1.0 w=l6.0 Amplitude Phase,° Amplitude Phase,° An. Ex. Ex. An. Ex. An. Ex. An. 0.071 0.081 240. 252. 0.248 0.303 242. 249. 0.413 0.441 233. 250. 0.441 0.356 247. 245. 0.474 0.441 236. 250. 0.300 0.329 77. 74.5 0.063 0.083 240. 250. 0.382 0.316 72. 69.6 0.530 0.545 248. 251. 0.534 0.541 242. 248. 0.559 0.545 233. 250. 0.470 0.527 67. 73.5 Table 5.6: Amplitudes, Phase angles for damped rig  Chapter 5. Experimental results  63  Excitation frequencies, Hz w=24.O Amplitude 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. 74.6 4 0.192 0.269 40. 74.7 5 0.621 0.560 243. 253. 6 0.487 0.550 64. 74.7 Table 5.7: Amplitudes, Phase angles for damped rig  ...  continued  Chapter 6  Summary  Results have been presented that allow the prediction of the effect of damping on the free vibration response of classically damped discrete nongyroscopic systems. The results presented are general in the sense that they consider all possible damping conditions that lead 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 in each mode having prescribed decay factor or damped natural frequency and for that just the knowledge of undamped natural frequencies is required.  r  lie equations required to  accomplish this task have been presented. For nonclassically damped systems the free response behaviour is more complex and no general rules concerning the influence of clamping on free response characteristics are evident. Characteristics not observed in classically damped systems are shown and discussed. It has been demonstrated that increasing damping can lead to an increase in free response frequency. Modification of damping properties of a system may lead to obtaining of desirable system eigenquantities, free and •forced vibration responses. A component mode synthesis method that enables the determination of eigenquan tities (for a given range of interest) has been developed. The use of this method is especially advantageous in the case of large systems, subjected to numerous modifica tions. 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 the vibration rig model on transmissibility of motion from the box to the supported beams is 64  Chapter 6. Summary  65  quite significant. Increase of damping constants in the isolators (at the constant stiffness characteristics) 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 the  supported beams. Comparison of analytical and experimental results is presented, which shows good agreement for eigenvalues, and steady-state responses of the vibration rig. As the future work, the assignment of the optimum  range  of  retained component  eigenvectors can be further investigated. 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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 Linear Dynamic 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 resonance testing,” Philosophical Transactions of the Royal Society of London, Vol. 255, Series A. 1963, pp. 241—280. [24] Hasselman, T. K. and Kaplan, A., “Dynamic Analysis of Large Systems by Complex Mode 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 Substruc ture Modes and a Hermitian System Representation,” Proceedings of the AIAA/ASME/ASHE/AHS, New York, 1985, pp. 581-586. [26] Howsman, T., and Craig, R.R., “A Substructure Coupling Procedure Appli cable to General Linear Time-Invariant I)ynamic Systems,” Proceedings of the AIAA/ASME/ASHE/AHS, New York, 1984, pp. 164-171. [27] Craig, R. R., and Ni, Z., “Component Mode Synthesis for Model Order Reduction of 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 Eigen value Problem”, USSR Comput. Math. Math, Phys., Vol. 3, 1961, Pp. 637—657. [30] Rubin, S., “Improved Component-Mode Representation for Structural Dynamic Analysis,” AIAA Journal, Vol. 13, No. 8, August 1975, pp. 995—1006. [31] Martec Limited., “Vibration and Strength Analysis Program (VAST): User’s man ual,” 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 A  User’s Manual for the CMSFR method  The main program in which the CMSFR method was implemented is named VASFIN. It was 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. *con 2. *.ibn 3 *rmk 4 *dam 5. *mod 6. *1st 7 *lms 8. *ldp  1. •c is ouput file of VASPRE 2. *.ibn is ouput file of VASPRE 3. *.rmk is ouput file of VASPRE 4• *.dam is ouput file of VASDAM 5. *.mod consists from one line, which contains the system eigenvalue range of interest  69  Appendix A. User’s Manual  and adjustment coefficient  i’  for  the CMSFR method  70  for the selection of component modes (it is created manually)  6. *1st contains constraint and weak springs for specified nodes of each component (it is created manually) 7. “.lms contains lumped masses for specified nodes of each cormiponent (it is created manually) 8. *.ldp contains lumped dashpots for specified nodes of each component (it is created manually)  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  (  =  .gml, copy of *.gml) 4  2. *sed 3. *use  1. *.gom is created from (basic) *.gom files of each component by running VASGEN program 2.  *.sed is created from (basic) t .gom files of each component by running VASSED  program 3. *.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  1. 2.  71  (contains the quantity of components) *  .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 written in 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. *.sed 3. *use  1. *.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 component element are governed by two parameters: “modulus of elasticity” and “density”. They reflect two factors: i) the damping matrix of the element is proportional to the element stiffness matrix, ii) the damping matrix of the element is proportional to the element mass matrix. The value, e.g., 1O can be used for these two parameters in order to prescribe zero damping to the elements of each component 2. *sed (the same as for VASPRE) 3. *.use (the same as for VASPRE)  Appendix A. User’s Manual for the CMSFR method  72  The 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 author of this thesis.  CD  CD  C  L  _4__  ci:) aD  I—  ‘1  0  .p.  0  UI  I—.  I—.  ‘1  UI  144’i  I  0  —  *0  8  C  I  I  I  I  -.2  —.  CD  CD  CD  CD  Appendix B. Parameters of the vibration rig  74  A— A  —  ±  ±  9/16  + 0  _rH_  10  /FZF/////Z/7/77/////77///7777777/77//77  Figure B.2: Vibration rig, front view  Appendix B. Parameters of the vibration rig  o  75  0  cv  0  0 12  Figure B.3: Vibration rig, top view of the box  

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