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An evaluation of component mode synthesis for modal analysis of finite element models Smith, Malcolm J. 1993

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AN EVALUATION OF COMPONENT MODE SYNTHESIS FOR MODAL ANALYSIS OF FINITE ELEMENT MODELS By Malcolm J. Smith B.A.Sc. (Engineering Physics), University of British Columbia  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  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 1993 © Malcolm J. Smith, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of ^Mechanical Engineering The University of British Columbia Vancouver, Canada  Date April 27. 1993  DE-6 (2/88)  Abstract  Component mode synthesis (CMS) is a condensation method for vibration analysis which preserves the low frequency vibrational characteristics of a structure. In this method, the structure is treated as an assemblage of components whose displacements are described in terms of component modes. These modes may be some combination of static response, free vibration, or rigid body displacements of a component. In this thesis, the component mode sets used by other researchers are reviewed with a view to establishing which is most suitable for large-order finite element models. Two component mode sets are identified as ideally satisfying the basic requirements for inter-component compatibility, high convergence rate, linear independence and completeness. Fixed-interface and freeinterface CMS formulations in the form of matrix eigenvalue equations are derived from  these mode sets and describe approximately the low-frequency free vibration modes of the structure. They are improvements over previous formulations in that they can be systematically and efficiently applied to linear, undamped, discrete systems of an arbitrarily complex geometry. The free-interface formulation is derived both with and without an approximation of the high-frequency component inertia, and this results in two different structural mass matrices. Two new developments of the free-interface formulation are presented: (1) a method for calculating upper and lower bounds to the exact natural frequencies is given, providing a measure of the absolute accuracy of the structural frequencies; (2) the convergence and interlacing properties of the free-interface method are explored through the analysis of a two-component vibrating rod. Both the fixed- and free-interface methods have been implemented in the generalpurpose finite element program VAST. Three finite element models are analyzed and a ii  comprehensive comparison of the frequency and mode shape results obtained with CMS, direct finite element analysis, and Guyan reduction is presented. The complexity of the test cases is sufficient to infer general performance characteristics of the CMS methods. It is shown that with CMS, accuracy equal to a direct analysis is readily obtained in the low frequency modes, and that by using a frequency cutoff criterion to select dynamic modes, the natural frequencies converge in a fairly uniform manner. It is also shown that in terms of computational cost and order-reduction, the relative advantages of using CMS increase with the size of the model and with the stringency of the accuracy requirements. The free-interface method with second-order mass approximation gives the best overall performance because of its high convergence rate and superior condensation in complex two and three dimensional models. Application of CMS to structural dynamic modification and inverse modification is also studied. These techniques use a baseline modal analysis as a reference point for the modified system dynamics. A generalized CMS formulation for the baseline system is used to derive a linear-equivalent perturbation equation from which modified modes can be efficiently determined without recalculating the component modes. Also, two new methods are presented for predicting design changes which satisfy prescribed frequency constraints. An iterative scheme is proposed in which the energy-balance perturbation equations are solved with a full account of the nonlinear coupling terms; and a Newton's method algorithm using inverse iteration eigenvector updating is applied to the linearequivalent equation. Numerical results using a finite element model are presented which show that for large structural changes, the two new methods give similar or better results than an established method.  iii  Table of Contents  ii  Abstract^  viii  List of Tables^ List of Figures List of Symbols^  xii  Acknowledgement^  xix  1 Background and Objectives ^  1  ^4  1.1 Thesis Objectives and Overview 2 Component Mode Representations ^  7  ^7  2.1 Introduction  2.2 Requirements for Approximating Functions of Structural Components ^7 2.2.1 Component Modes for Discrete Models ^  10  2.3 Free-Free Modes ^  12  2.4 Free-Free Modes with Interface Loading ^  15  2.5 Static Approximation of Higher Modes ^  18  2.6 Inertial Approximation of Lower Modes ^  24  2.7 Constraint Modes ^  25  2.8 Attachment Modes ^  30  2.9 Polynomial Functions ^  33  iv  2.10 Comparison of Component Mode Representations ^  35  2.11 Mode Selection ^  39  2.12 Summary ^  40  3 Substructure Synthesis^  41  3.1 Introduction ^  41  3.2 Inter-Component Equilibrium and Compatibility ^  42  3.3 Lagrangian Formulation of the System Equations ^  44  3.4 Modal Force Method ^  48  3.5 Free-Interface Formulation ^  50  3.6 Fixed-Interface Formulation ^  56  3.7 Condensation in the Fixed- and Free-Interface Formulations ^ 60 3.8 Component Mode Substitution ^  61  3.9 Error Estimation for Natural Frequencies ^  64  3.10 Modal Properties of Combined Systems; the Inclusion Principle ^ 68 3.11 Summary ^  77  4 Modal Analysis of Three Finite Element Models^  79  4.1 Introduction ^  79  4.2 Analysis of a Container Ship ^  80  4.2.1 Natural Frequency Results ^  82  4.2.2 Mode Shape Results ^  88  4.2.3 Performance ^  91  4.3 Analysis of a Telescope Focus Unit ^  92  4.3.1 Natural Frequency Results ^  95  4.3.2 Mode Shape Results ^  100  4.3.3 Performance ^  104  v  4.4 Analysis of a Telescope Model ^  106  4.4.1 Natural Frequency Results ^  108  4.4.2 Mode Shape Results and Performance ^  114  4.5 Natural Frequency Error Estimation: Ship Model ^ 116 4.6 Reanalysis Following a Design Modification ^  118  4.6.1 Reanalysis of the Container Ship ^  119  4.6.2 Reanalysis of the Telescope Focus Unit ^  122  4.7 Discussion and Summary of Numerical Results ^  124  5 Structural Dynamic Modification^  126  5.1 Introduction ^  126  5.2 Structural Changes ^  127  5.3 Structural Dynamic Modification with CMS ^  131  5.3.1 Application to Substructured Models ^  133  5.3.2 Numerical Results ^  137  5.4 Perturbation Methods for Inverse Modification ^  142  5.4.1 Background ^  142  5.4.2 Application to CMS ^  144  5.5 Sensitivity Analysis; Newton's Method for Inverse Modification ^ 150 5.5.1 Background ^  150  5.5.2 Newton's Method for Substructured Problems ^ 151 5.6 Numerical Results ^  154  5.7 Summary ^  162  6 Summary and Conclusions^  164  Bibliography^  168  vi  A Convergence Characteristics of the Modulation Matrix II^175 B Eigenvalue Sensitivity to Residual Flexibility Changes ^177 C Implementation of Component Mode Synthesis in VASTO6^179 C.1 The Substructure/Superelement Option ^  179  C.2 Implementation of the Fixed-Interface Method ^  182  C.3 Implementation of the Free-Interface Method ^  186  C.4 Including Fluid Added-Mass in a CMS Analysis ^  193  C.5 User's Guide to VASTO6 CMS (Pre-release Version) ^ 196 C.5.1 Format of input file PREFX.CMS ^  197  C.5.2 Format of input file PREFX.GLM ^  200  vii  List of Tables  4.1 Description of the container ship model ^  81  4.2 Natural frequency results for the container ship, free-interface method . ^ 83 4.3 Natural frequency results for the container ship, fixed-interface method ^ 85 4.4 Component frequencies of the container ship ^  86  4.5 Mode shape results for the container ship, free-interface method ^ 89 4.6 Mode shape results for the container ship, fixed-interface method . .^90 4.7 CPU times for modal analysis of the container ship ^ 91 4.8 Description of the TFU model ^  93  4.9 Natural frequency results for the TFU, free-interface method ^ 96 4.10 Natural frequency results for the TFU, fixed-interface method ^ 97 4.11 Component frequencies of the TFU ^  98  4.12 Mode shape results for TFU, free-interface method ^ 101 4.13 Mode shape results for TFU, fixed-interface method ^ 102 4.14 CPU times for modal analysis of the TFU ^  105  4.15 Natural frequency results for telescope model, free-interface method . . ^109 4.16 Natural frequency results for telescope model, fixed-interface method . ^110 4.17 Natural frequencies of spider components ^  112  4.18 Mode shape results for telescope model, fixed-interface method ^ 115 4.19 CPU times for modal analysis of the telescope model ^ 115 4.20 Lower and upper bounds calculated with dynamic residual flexibility . . ^ 117 4.21 Lower and upper bound estimates of four CMS frequencies, with predicted absolute errors and exact frequencies ^ viii  117  4.22 Reanalysis of the container ship model, free-interface method ^ 121 4.23 Reanalysis of the container ship model, fixed-interface method ^ 121 4.24 Reanalysis of the TFU, free-interface method ^  123  4.25 Reanalysis of the TFU, fixed-interface method ^  123  C.1 Description of VASTO6 modules ^  180  C.2 T-file locations of VAST information relevant to CMS ^ 182 C.3 Additional storage files created by the fixed-interface method. ^ 186 C.4 Special storage files created by the fixed-interface program ^ 186 C.5 Additional storage files created by free-interface program ^ 192 C.6 Special storage files created by the free-interface program. ^ 192  ix  List of Figures  2.1  A general distributed model for a structural component ^  2.2  Two component system ^  16  3.1  Two-component continuous bar ^  69  3.2  Graphical solution for continuous bar, two mode approximation ^  73  3.3  Graphical solution for continuous bar, three mode approximation ^.  74  4.1  Two-dimensional container ship model: a) stern; b) aft-body; c) deck-  8  house; d) mid-body; e) fore-body ^  81  4.2  Combined natural frequency results for the container ship ^  87  4.3  Components of the TFU model: a) inner tube with screw assembly attached to top, and chopping mechanism attached to bottom; b) outer tube; c) support tube with external constraints. ^  94  4.4  Combined natural frequency results for the TFU model ^  99  4.5  Complete telescope model, with focus unit and spider ^  107  4.6  Combined natural frequency results for the telescope model ^  111  4.7  Mode 9 of the container ship, f  = 5.91Hz.  119  4.8  Six groups of bar stiffeners used for modifications ^  120  5.1  Reanalysis results for modifications to hull-bottom stiffeners  ^  138  5.2  Reanalysis results for modifications to bearing stiffnesses ^  139  5.3  Reanalysis results for lumped-mass additions to the chopping mechanism  141  5.4  Frequency modification of hull-bottom stiffeners for five modal approximations ^  ^  156 x  5.5  Comparison of three methods for frequency modification of hull-bottom stiffeners ^  157  Three-parameter frequency modification of hull-bottom stiffeners ^  158  5.7 Frequency modification with mid-body stiffeners ^  160  C.1 Flow chart for Guyan reduction in VASTO6 ^  181  C.2 Flow chart for the fixed-interface CMS method in VASTO6 ^  183  C.3 Reanalysis option of the fixed-interface CMS method in VAST06 ^  185  C.4 Flow chart for the free-interface CMS method in VASTO6 ^  187  C.5 Reanalysis option for the free-interface CMS method in VASTO6 ^  187  C.6 Flow chart for the module CMS. ^  189  5.6  xi  List of Symbols  A^equilibrium/compatibility connectivity matrix  B^inertial complement to residual flexibility matrix zi^linear differential operator of maximum order 2/ — 1 admixture coefficient vector for the ith modified mode indicates participation of the ja baseline mode in the ith modified mode C^admixture coefficient matrix  D  general diagonal matrix rank-deficient square matrix formed by constraint matrix R applied load vector for a component independent set of interface loads i th  natural frequency (Hz)  F*^penalty function for unconstrained optimization gi^jth  inequality constraint in penalty function F*  static flexibility matrix  6^residual flexibility matrix GC^constrained flexibility matrix  H  dynamic response matrix  I^identity matrix k^reduced-order component stiffness matrix element stiffness matrix (k..);^element stiffness matrix, linearized about the jth  K  component or structural stiffness matrix  xi i  property  I?^condensed system stiffness matrix given by a CMS formulation KCAL^interface stiffness coupling matrix generated from residual flexibilities (Kr ) i^global coordinate accumulation of (k er ) j over all elements 1^indicates order of differential equations; or, the number of frequency  constraints  L^Lagrangian of a dynamic system linear self-adjoint operator of order 21  m^reduced-order component mass matrix; or, the number of structural design variables me^element mass matrix (m er ) i^element mass matrix, linearized about the j th property  M^component or structural mass matrix  NI^condensed system mass matrix given by a CMS formulation MA^fluid added-mass matrix  (Mr) ;^global coordinate accumulation of (m..) i over all elements mass density  n^number of degrees of freedom in the discrete model of a structure or component number of inequality constraints in penalty function F* i th continuous shape function  (bar) over a vector indicates the uncoupled linear collocation of all such component vectors, over a matrix indicates the uncoupled diagonal collocation of all such component matrices vector of free-free modal coordinates vector of constraint modal coordinates  pn^vector of fixed-interface normal modal coordinates Pr^vector  of rigid body modal coordinates  vector of interface-loaded modal coordinates P^projection matrix q^number of baseline modes retained r^property variable R^set of static determinate interface coordinates; general inter-component  constraint matrix; the residual vector of equality constraints in penalty function Ft Rayleigh quotient 3^number of structural components  set of redundant interface coordinates local to global coordinate transformation for element e t^time T^transformation between physical coordinates and generalized system  coordinates in a CMS formulation TA^equilibrium/connectivity Tc^Guyan  matrix  reduction transformation matrix  transformation between component generalized coordinates and system coordinates  T^kinetic energy to/^eigensolution tolerance parameter  u^displacement vector for a component u B^independent set of interface displacements  u n^n-mode approximation to u  xiv  ^  U  general upper-triangular matrix  ^  potential or strain energy  w(x), W(x) continuous displacement functions for component, structure wn^approximation to w based on n shape functions x^spatial coordinate  x^vector of spatial coordinates X^matrix of unsubstructured mode shapes X$^ith  unsubstructured mode shape  Xr^ith  mode shape for a wetted structure  Y^matrix of the zero-eigenvalue eigenvectors of D Z^matrix of condensed system eigenvectors  fractional change to a property, or a vector of fractional changes  13^transformation matrix between interface and full component coordinate systems  r^condensed system stiffness matrix given by free-interface CMS f^matrix of residual acceleration modes b f^virtual force vector Su^virtual displacement vector  A f %^percentage error in frequency Ake^change to element stiffness matrix Am e^change to element mass matrix AK^change to stiffness matrix AM^change to mass matrix Au%^percentage error in mode shape AX^change to mode shape matrix  xv  Ace^change to system-mode eigenvalue matrix 77^vector  of component generalized coordinates  i th eigenvector of condensed CMS equations  eigenvector of i th wetted mode inertial complement to residual attachment modes A^eigenvalue Aw^eigenvalue of a wetted structure A^diagonal matrix of component free-free eigenvalues weighting coefficient used in penalty function F* II(w)^modulation matrix  95;^ith free-interface or free-free mode (A^ith  interface-loaded free vibration mode  matrix of free-free modes f n^matrix  4 i  of fixed-interface normal modes  4.L^matrix of fixed-interface normal modes (interior partition) br^matrix  (  of rigid body modes  matrix of attachment modes qic^matrix of static constraint modes matrix of static constraint modes (interior partition) f^  inertia-relief modes matrix of residual attachment modes  w^frequency  w,^cutoff frequency wi^ith natural frequency (3 -1 ) n2^diagonal matrix of system eigenvalues  xvi  Subscripts  constraint modes d e  ^  ^  I  ^  ^  h 1  ^  le  ^  dependent coordinates  element number inertia-relief modes independent, or generalized, coordinates  high frequency modes low-frequency modes  low-frequency elastic modes  m^medium-range frequency modes  ^ n fixed-interface normal modes r^rigid body modes Superscripts  partition of interface coordinates BB^interface coordinate partition from a symmetric matrix BI, IB^partition of interface and interior coupling terms BN, NB^partition of interface and modal coupling terms  E  partition of extra coordinates  I^partition of interior coordinates II^interior coordinate partition from a symmetric matrix (k)^kth iteration; kth structural componentNN^partition of terms associated with fixed-interface vibration modes  O  partition of original coordinates  R^partition of determinate interface coordinates S  partition of redundant interface coordinates  xv ii  T^transpose of a vector or matrix  xviii  Acknowledgement  The author would like to thank Professor Stan Hutton for the guidance and insight he provided throughout the researching and writing of this thesis. The author would also like to acknowledge the generous financial support provided by Defence Research Establishment Atlantic and the Natural Sciences and Engineering Research Council of Canada.  xix  Chapter 1  Background and Objectives  A basic problem in structural dynamic analysis is the evaluation of natural modes of vibration. Accurate knowledge of at least some of the modes is of considerable importance for determining the dynamic response of a structure to applied loads. Mathematical idealization generally results in a system of linear equations which, under zero loading, describe the free vibration state of a structure. The efficient formulation and solution of these equations for complex structural systems has been studied extensively and remains a primary concern of analytical modal analysis. Linear vibrations of a distributed, elastic, undamped structure are described by a partial differential eigenvalue equation of the form, ZW(x) = \M(x)W(x) (1.1) where .0 is a linear self-adjoint, partial differential operator of order 2/, M(x) is the mass density of the structure, A is a parameter, x is a vector of spatial variables and W(x) the displacement function for the structure [1). The displacement is further constrained by the following boundary conditions which are defined at all points on the boundary: 13;W(x) = 0 i = 1,2, ... ,1 (1.2) where B i is a. linear partial differential operator of maximum order 2/ — 1. Solution of the eigenvalue problem consists of determining the eigenvalue, eigenfunction pairs AT., Wr which represent the natural modes of the structure. While all elastic, undamped 1  Chapter 1. Background and Objectives ^  2  structures can be described with equations (1.1) and (1.2), closed form solutions are rare and the analyst is left to seek an approximate solution. The most popular and well known approximate methods for solving these equations are the finite element and the Rayleigh-Ritz method. Both seek to replace the continuous variable W(x) with a collection of discrete variables and to replace the differential eigenvalue problem with an algebraic eigenvalue problem. Algebraic eigenvalue problems can in general be solved, whereas differential eigenvalue problems cannot. In the finite element method a structure is divided into a number of sub-domains or finite elements. The displacement of each element is described by a linear combination  of element shape functions, each of which is in turn defined by a unit displacement of an element coordinate. Thus, the continuous displacement function of the structure is replaced by a vector describing the displacement of the element nodes. The advantages of the method are that a structure of arbitrary structural geometrical complexity can be effectively modelled with a mesh of relatively simple elements and that if the correct elements are used, the results can be made to converge to exact solutions as the element mesh is refined. One of the disadvantages, however, is that accurate results for complex structures require a large number of nodal coordinates. The subsequent analysis may then require lengthy computations. The Rayleigh-Ritz method, by contrast, uses a set of approximating shape functions defined over the whole domain of the structure. A solution in the Rayleigh-Ritz sense corresponds to a configuration that is a stationary point in the Rayleigh quotient, R(W) = A = [W, 14/^(1.3)  h  MW2 dx  where [W, W} is the energy inner product defined by I W.CWdx and D denotes the spatial domain occupied by the structure. The continuous variable W in R is replaced by a number of discrete, generalized coordinates representing the relative participation  Chapter 1. Background and Objectives^  3  of the approximating functions. This substitution transforms (1.3) into an algebraic eigenvalue problem. The advantage of the Rayleigh-Ritz method is that accurate results can be found with fewer discrete variables than is required with the finite element method. However, for structures of great geometrical complexity, the task of finding suitable approximating functions is too difficult and the method is discarded in favor of finite elements. A third method for modal analysis of a linear structure is component mode synthesis which combines features from the Rayleigh-Ritz method and finite elements. Structures may, in general, be treated as an assemblage of components or substructures; indeed, in some cases it may be both natural and convenient to describe a structure in this way. Moreover, approximating functions are more easily derived for a structural component than for the entire structure. For a component, such approximating functions might include static deflections or rigid and elastic mode shapes; they might be measured experimentally or they might be calculated from a detailed finite element model of the component. The object is to replace the detailed model of a structural component with a simplified one based on a set of shape functions that provide a good approximation in a particular frequency range. In the Rayleigh-Ritz method, approximating functions must be defined over the entire domain of the structure. These global approximating functions can be formed implicitly from the component shape functions by maintaining displacement compatibility at the component interfaces [2]. This same process is apparent in the finite element method when compatible element types are used. Indeed, if a model is substructured so that each component corresponds to a single finite element of the original model, a CMS analysis will be same as a finite element analysis. On the other hand if no substructuring is performed, i.e., the entire model is treated as a single substructure, the CMS method is no different than the Rayleigh-Ritz method. The proper course for using the CMS  Chapter 1. Background and Objectives^  4  method lies between these two extremes. The number of structural components is usually considerably less than the number of finite elements required in the modelling, while at the same time, each component represents a much simpler model than the entire structure. Having established a set of component shape functions for each component, the equations of motion of the structure are derived by enforcing displacement and slope compatibility at the inter-component boundaries. Sometimes it is convenient to satisfy force and moment equilibrium here as well. What results is a global stiffness and mass matrix in terms of generalized coordinates, which are themselves related to the component shape functions. One of the important advantages of component mode synthesis is that equations of motion of the structure are of smaller order than, for instance, those that are obtained with the finite element method. For large-order models, reduction in the number of degrees of freedom means computational savings, but with a possible loss of accuracy. However, the accuracy of the low-frequency structural modes can be preserved if component modes are chosen so that the static and low frequency motion of the components is well represented. Any inaccuracies which arise from the reduction are confined to the high-frequency modes. 1.1 Thesis Objectives and Overview This thesis provides an in-depth assessment of the effectiveness of component mode synthesis in finite element applications. The specific objectives of this thesis are the following: 1. To give a comprehensive review of existing component mode synthesis methods and to identify those suitable for application to general, large-order structures; 2. To present general formulations of the free vibration structural equations that can be applied to complex discrete structural models, and directly implemented in a  Chapter 1. Background and Objectives^  5  general-purpose finite element program; 3. To develop the general formulations further in order to gain a better understanding of convergence properties and to investigate the effects of approximations at the component level; 4. To present comprehensive frequency and mode shape results for realistic finite element models which illustrate the key performance characteristics of the CMS methods in large-order problems; 5. To develop efficient methods for structural dynamic modification analysis which, through the use of CMS formulations, exploit order-reduction in the free vibration equations. To simplify the analysis energy dissipation is not considered, although the analysis is also applicable to damped systems that possess normal modes. In Chapter 2, a detailed survey of the basic component mode representations is given. The degree to which a representation satisfies the the basic requirements for a component mode set is discussed, as well as the topics of convergence, mode selection and inter-component compatibility. In Chapter 3, attention is focussed on the synthesis of structural components—the process by which the free vibration equations of the system are formulated. In this area, the treatment differs somewhat from previous work in that the emphasis is on deriving equations specific to a particular component mode set but which are applicable to systems with an arbitrary number of components and in the most general geometric configuration. Two advantages of this approach are that the synthesis of the equations can be done more efficiently and that predictions can be made about the performance of the various component mode representations based on the form of the free vibration equations.  Chapter 1. Background and Objectives^  6  Two component mode synthesis methods are selected for further study: the fixedinterface and free-interface method. In Chapter 4, modal analysis results for three finite element models are presented. The purpose is to compare the performance of the fixedand free-interface methods with direct finite element analysis and Guyan reduction. The influence on the performance of these methods of model complexity, modal truncation, modal density, and accuracy demands is evaluated. In the course of developing a finite element model, numerous modifications may be made, each of which requires its own modal analysis. Structural dynamic modification techniques have been developed by various researchers to improve the efficiency of multiple analyses. In Chapter 5, it is shown how structural dynamic modification techniques can be applied when the baseline, or unmodified, structure is represented by a generalized CMS formulation. The condensation inherent to the CMS formulations is helpful in reducing the cost of the reanalysis, and it is demonstrated that in many instances accurate results for the modified structural modes can be obtained. If it is required that the modified structure have certain prescribed frequencies, the efficiency of the redesign process can be improved using inverse modification techniques. In these techniques, which in the past have been based on sensitivity or perturbation analysis, design changes are calculated which satisfy free vibration equations of the modified structure subject to prescribed modal constraints. In Chapter 5, two methods—one based on Newton's method, and the other on an iterative solution of the perturbation equations—are used to solve freqency modification problems. The equations of motion of the modified structure are based on a generalized CMS formulation and the results are compared with those obtained with an established method.  Chapter 2  Component Mode Representations  2.1 Introduction Component mode synthesis is an analysis method for determining the natural modes of vibration of a structure. As described in the introductory chapter, it combines features of the Rayleigh-Ritz method and the finite element method in an attempt to render an accurate, reduced-order model of a physical structure. The method can be viewed as having two stages; the first is the subdivision of a structure into components and the description of each component in terms of approximating functions; the second consists of reassembling the structure with the aid of the approximating functions and thereby establishing coupled equations of motion for the entire structure. In the present chapter, attention is focussed on the first stage of the analysis—the selection of a set of approximating functions for a component, which will collectively be referred to as a component mode representation. First, a statement of the general requirements for component shape functions is given. Following that, a survey of the existing component mode representations is presented along with some discussion of their strengths and weaknesses. 2.2 Requirements for Approximating Functions of Structural Components To begin, consider the general structural component depicted in Figure 2.1. Assuming that it is a linear, distributed elastic component, its free vibration response as part of a  7  Chapter 2. Component Mode Representations ^  8  r  Adjacent Components 1  V  V  rp^  ,,,,,, ,,,,, ,,,,,,,,,,,,,,,,,,,,,,,,,  II  Figure 2.1: A general distributed model for a structural component larger structure can be expressed as, Cw(x) = AM(x)w(x)  ^  (2.1)  where w is the displacement of the component and the rest of the notation is the same as for (1.1). The displacement is subject to the boundary conditions of the type Biw(x) = 0^i = 1, 2, . . . , 1^ (2.2) The boundary of the component in Figure 2.1 may be divided into five regions, each of which possesses characteristic boundary conditions. The two fundamental types of boundary conditions are geometric boundary conditions, where the order of the operator B i is 1— 1 or less, and natural boundary conditions, where the order of Bi is 1 to 21— 1. First is the free boundary (I) along which homogeneous natural boundary conditions apply. There are no geometric boundary conditions along the free boundary. Second is  Chapter 2. Component Mode Representations^  9  the region where external constraints are applied. An external constraint is some physical property of the component which manifests itself in either the geometric or natural boundary conditions. These may be geometric constraints (II), natural constraints (III) or some mixture of the two (IV). Last is the region where adjacent components are attached (V). This region is called the component interface or the inter-component boundary. Here, adjacent components exert forces and moments which quantitatively remain unknown until the dynamics of the entire structure are determined. For the purposes of the component analysis, the interfaces are regarded as having only non-homogeneous natural boundary conditions. As in the analysis of the whole structure, closed form solutions to the component problem are rare because of the complexity of the governing equation and also because the boundary conditions on the component interfaces are unknown. Instead, a solution is sought with a sequence of approximating functions, as in the Rayleigh-Ritz method. The displacement w may be approximated by a linear combination of functions  E  wn (x) = i=i Ni(x)pi  (2.3)  where pi is the participation of the i th function in the displacement function wn. For wn  to be an approximation in the Rayleigh-Ritz sense, the functions Ni must be linearly  independent, they must be 1 times differentiable, and they must satisfy the geometric boundary conditions. They do not have to satisfy either the differential equation or the natural boundary conditions [4]. Such a sequence of approximating functions should also be complete so that as n is made arbitrarily large, the approximate displacement wn may be made to approach the exact displacement w to within an arbitrarily small difference in the sense of the energy norm. A sequence N1 , N2 . . . is said to be complete in energy if the energy inner product [w — wn, w — wn] can be made less than any arbitrarily small postive number e [4].  Chapter 2. Component Mode Representations^  10  When selecting approximating functions, the objective is to closely approximate the component displacement (and by association, its kinetic energy and strain energy) as it is undergoing free vibration motion as part of the larger structure. If a wide spectrum of natural modes is to be determined, then a large number of approximating functions will be necessary to correctly model the energy in all the modes. Typically though, it is usually only a small portion of the structural modes that are of interest, especially in a complex structure, and these are usually the lowest frequency modes. This simplifies the task of selecting approximating functions since the kinetic and strain energies only need to be accurately modelled at low frequency. The approximating functions, which from now on will be referred to as component modes, should conform with the actual boundary conditions on the component as much as  possible. On the free boundary for instance, homogeneous natural boundary conditions should be used when calculating the component modes. On the component interfaces the boundary conditions are unknown and so the conditions that ought to be applied to the component modes are left to the judgement of the analyst. 2.2.1 Component Modes for Discrete Models  Relatively complex components are difficult to analyze with a differential equation and boundary conditions of the form (2.1) and (2.2). Instead it is often more practical to build a component model with the aid of a discretization procedure such as finite elements. Such a procedure enables the distributed representation to be replaced by an equation of the form, Mii(t)-F Ku(t) = f(t)^  (2.4)  where M and K are real, symmetric, mass and stiffness matrices derived by the discretization procedure, u is the displacement vector and f is the vector of applied loads.  Chapter 2. Component Mode Representations ^  11  In general, M is positive definite and K is either positive semidefinite or positive definite, depending on whether or not rigid body motion is possible. The boundary conditions on the component are handled in the following way: external constraints, whether geometric or natural, are incorporated in K, M and u; natural boundary conditions at the component interfaces are represented by force and moment terms in the load vector. The discretization thus replaces the partial differential equation (2.1) and its boundary conditions (2.2) with a system of ordinary differential equations. A sequence of component modes which may be used to approximate the displacement vector u is given by  = E Xi1ji^  (2.5) i=i where the vector Xi is the i th component mode and where ni is the participation of the i th mode in the vector un. As un is only an approximation to u, the number of component modes n will be less than the number of coordinates in u. Applying the Rayleigh-Ritz method to the discrete representation gives the following elements of the generalized stiffness and mass matrices: ki;^XTKXi^i,j =1,2,...,n^(2.6)  mi; = mii = XTMXi i,j =1,2,...,n (2.7) Note that Xi, i = 1, 2, ... , n do not need to be normal modes although they should be linearly independent. If the component modes accurately represent the static and lowfrequency motion of the component, k and m will give an accurate description of the component's stiffness and mass distribution at low frequency. The remaining sections of this chapter describe a variety of component mode representations for a discrete component model. Reviews of component mode representations have been given by Hurty [5], Kubomura [6], and Craig [7].  Chapter 2. Component Mode Representations ^  12  2.3 Free-Free Modes The simplest component mode representation describes the displacement as a linear combination of its unconstrained or free-free vibration modes, i.e., 21(t)  = E Oipi = (bp(t)  (2.8)  where the modal matrix (1) contains the free-free mode shapes as columns: (2.9)  = {(ai ... (an}  and p(t) is the column vector of component modal coordinates. This type of component mode representation has been used by Goldman [8], Hou [9], Dowell [10] and Yee and Tsuei [11]. Free-free component modes are calculated from the component equation with the applied loads set to zero: ^Mu(t)  -F  Ku(t)  =  0^  (2.10)  This is transformed to an eigenvalue problem by assuming a sinusoidal solution for u(t) at frequency w: K — (41i1 10i = 0^  ^[  (2.11)  From this equation are computed the natural frequencies of the component wi and their corresponding free-free mode shapes fit. A useful convention is to incorporate any external constraints into K and M. The resulting free-free modes will therefore satisfy the external constraints applied to the component, guaranteeing the admissibility of the modes. The mode set will also include any rigid body modes in the component. The number of rigid body modes is usually between zero and six, but more may exist if the component is articulated.  Chapter 2. Component Mode Representations ^  13  The general equation of motion of a component is obtained by including a non-zero force vector on the right-hand side of (2.10): Mi(t) + Ku(t) = f(t)^  (2.12)  The component mode representation (2.8) is substituted into (2.12) to give Ms:1) 7;W . + 1‘40p(t) = f(t) (2.13) It may be assumed without loss of generality that the mode shapes in I, are mass normalized. By premultiplying (2.13) by 4 T and by assuming a sinusoidal solution for p(t), the equation of motion of a component becomes [A 7 w 2 I]p(t) = (I) T f(t) (2.14) where A is the diagonal matrix with the component eigenvalues A i = (4 on the diagonal and (DT At) is the vector of modal forces acting on the component. Comparing (2.14) to (2.12), it is seen that in modal coordinates, the component stiffness matrix is A and the mass matrix is I. If coordinate reduction is to be achieved, the set of component modes included in (I) must be truncated; i.e., the number of component modes must be fewer than the number coordinates in the original model. It is the truncation at the component level that primarily distinguishes component mode synthesis from other dynamic condensation methods [12, 13]. Generally, since it is the lower structural modes that are of interest, rigid body and low frequency component modes are included in (1); the high frequency component modes are omitted as they contribute little to the low frequency dynamic response of the structure.  Chapter 2. Component Mode Representations ^  14  While coordinate reduction is important, a sufficient number of modes must be included in 4 to satisfy the compatibility equations between components. Each Oi con,  tributes one modal coordinate to a component. The total number of these modal coordinates in all components must exceed the total number of compatibility equations if these are to be fully satisfied. The analyst will usually want to use enough modes to ensure inter-component compatibility, plus additional modes to improve the accuracy of the results. The compatibility requirement becomes troublesome in structures whose component interfaces consist of meshed curves or surfaces. As finite element meshes are further refined, the number of degrees of freedom on the interfaces increases. At the same time, the number of component modes that can be reasonably calculated diminishes because of the increased order of the governing matrices. As a result,"it may become very difficult to obtain enough component modes to enable the compatibility equations to be fully satisified; instead, only partial or approximate satisfaction of compatibility may be possible. This is not necessarily detrimental to the accuracy. Using a truncated mode set constrains a component's motion and tends to produce a model that is too stiff. Conversely, the under-constraint which stems from improperly satisfied constraint equations relaxes some of the constraints on the component, thereby reducing its stiffness. Obtaining a balance between these two operations could offset the ill-effects of each. However the amount by which the frequencies of the structure are raised and lowered by model truncation and by relaxing interface constraints cannot be quantified. Furthermore, it is very easy to devise a substructured model in which the number of interface coordinates exceeds by an order of magnitude the number of component modes that can be realistically computed. In these situations the under-constraint will be severe and would lead to unreliable results. It has been shown by Meirovitch [4, 14] that mode sets containing admissible shape  Chapter 2. Component Mode Representations ^  15  functions with homogeneous natural boundary conditions show poor convergence characteristics. Equation (2.8) is an example of this type of representation. The slow convergence stems from the inability of this type of mode, whether taken individually or in small numbers, to satisfy non-zero natural boundary conditions. Even though a small number of free-free modes may describe the displacement of a component very well in its interior, there are inherent inaccuracies at its boundary which do not disappear unless very large numbers of modes are used. This conclusion is supported by the investigations of Hou [9] which show that the structural frequencies converge very slowly with the component mode representation in (2.8). 2.4 Free-Free Modes with Interface Loading Benfield and Hruda [15] introduced a variation to the classical method by including stiffness and inertial interface loading in the eigenvalue equation of a component. This was an attempt to at least partially account for the presence of adjoining components in the dynamics and thereby render component modes closer to the structural modes in frequency and energy distribution. For this reason it is expected that these modes will converge faster than classical free-free modes. Static condensation is used to calculate the interface loading. This method, which is often referred to as Guyan reduction [16], will be discussed in another context in Section 2.7. For the present discussion it will be sufficient to quote the formula. To calculate the stiffness and inertia loadings on a particular component the stiffness and mass characteristics of the adjoining components are condensed on to the interface. For simplicity, consider the two component system shown in Figure 2.2. The stiffness and mass matrices of each component are partitioned into interface (B) and interior (I)  16  Chapter 2. Component Mode Representations^  Interface  a  ^  b  Figure 2.2: Two component system coordinates: K=  KB B KBI^MBB mBI ^(2.15) ^M=  K ra  KII  MIB mll  The interface loadings on component a are determined through the condensation of the stiffness and mass properties of component b on to its interface:  Ter KbTe  (2.16)  BB riICT ibr ITC Mb = -Lb L‘b"b  (2.17)  Ice il =  4  where 4313 and mr are the condensed stiffness and mass of component b and the matrix Tbc is defined as, uB  ur lb = [  _reII I-1 KII, B 'Lb  1.,B^ri,C.,B (Job = J. b u,b  (2.18)  which is the general result from Guyan reduction [16]. The free vibration modes of component a are now determined with the interface loadings applied. The appropriate eigenvalue equation is, [KBB  + icbl3B^KaBI  K B I T^K  II  sp it  2  =($)  iti-aBB +  me  B maBI  '',B  mBIT^mlI  p.  (2.19)  Chapter 2. Component Mode Representations ^  17  The component displacement is subsequently described by the truncated sequence of modes:  na . . ,.., E 0,5, i.i  (2.20)  where n a is the number of modes calculated for component a. It should be emphasized that the interface loadings are used only to determine a more convergent mode set; once the mode set is established the stiffness and mass properties of the components revert to their original forms and the synthesis of the components can proceed in the usual manner. The modal representation (2.20) is an improvement over the classical representation with respect to its convergence properties but it is does not assist in the satisfaction of compatibility conditions at the component interfaces. A large number of dynamic modes are required to properly satisfy compatibility along complex interfaces while the computational effort necessary to calculate these modes is greatly increased by the inclusion of interface loading. It is also a disadvantage to require knowledge of the stiffness and mass characteristics of adjacent components when choosing component modes. It may be that structural components are being designed independently and that one design group may not have detailed information about the work of other design groups. Also, to reanalyze the structure after extensive modifications to one component means not only that the modes of that component have to be reanalyzed but that the interface loadings of all components attached to it have to be adjusted as well. It is clear then, there are distinct advantages to using component modes which allow components to be analyzed independently.  Chapter 2. Component Mode Representations ^  18  2.5 Static Approximation of Higher Modes The dilemna of how to maintain compatibilty between components without having to compute large numbers of dynamic modes is resolved by using a static approximation for the higher component modes. Suppose that the mode shapes of a component are partitioned into two groups such that  n(t) u(t) = [ 4) 1 (k h 1 ph(t)  (2.21)  J  The modal matrix is partitioned so that 4)1 contains rigid body modes 40,. and low frequency elastic component modes 4)/ e :  4) 1 = { C r t ie ]  ^  (2.22)  Modes that are considered low frequency have frequencies that are comparable to a target frequency range. Modes in  41, have frequencies that are significantly higher than  the target frequency range. With this partitioning, the component equations become { At — w 2 /^0^p,(t) 1^[ 4)T 0^Ah — w 2 /^ph(t)^4,T .  ^  f (t)^(2.23)  While undergoing free vibration in a structural mode, the component force vector At) contains the forces applied by adjacent components and w is a structural natural frequency. Define  2 »W 2 We where we is the cutoff frequency, or the lowest frequency in the  (2. 2 4) th  set. The following  approximation may then be used: ph(t) = Al; 1 4V(t)^  (2.25)  Chapter 2. Component Mode Representations^  19  The displacement of a component is now modified to the following expression using (2.25): u(t) = vi(t) + Gf(0^ (2.26) where  (2.27) The component mode representation (2.26) is commonly referred to as the MacNealRubin mode set in recognition of its originators [17, 18]. The second term of u(t) is a static approximation of the displacement contributed by the higher modes, G being the static flexibility in the higher modes. Urgueira and Ewins [19] used (2.27) in their derivations. MacNeal [17], Rubin [18] and Craig and Chang [20] used instead the residual flexibility of the component:  = G — (1) 1.4. 1 4:1q. (2.28) where G is the full static flexibility of the component and (1)/. is the matrix of lower elastic modes of the component. Note that (2.27) and (2.28) are equivalent when the set of higher modes is complete. However, by using (2.28) the higher modes no longer need to be computed. In the free vibration of the structure, f(t) has non-zero values at the interface locations only. As a result, the only columns of 6 that will contribute to the displacement will be those associated with interface locations and each of these columns represents the residual displacement caused by a unit force at that location and zero force elsewhere. It therefore follows that each of these columns can be thought of as a residual static attachment mode. Defining  At) = Or (i )  ^  (2.29)  where fB(t) contains forces at the interface coordinates only and ,a is the coordinate transformation from the interface to the full coordinate system of the component, the  Chapter 2. Component Mode Representations^  20  displacement of a component is then given by, u(t) = (1)p(t) 'tfB (t)^ (2.30) where  xi/ =  Op  (2.31)  and where subscript 1 has been dropped. The matrix if contains the residual attachment modes as columns. By their definition, these modes are mass-orthogonal and stiffnessorthogonal to the lower normal modes. Because the interface forces appear on the right-hand side of (2.30), they may be treated as modal, or generalized, coordinates. A sufficient number are available to ensure that compatibility and equilibrium can always be satisfied, but this requires that the residual attachment modes be linearly independent. As each free-free mode is added, the rank of the residual flexibility matrix is reduced by one. Thus, with a full set of modes, the residual flexibility disappears and the mode set reduces to the classical representation. By this means, linear independence and completeness requirements are satisfied. But to maintain the linear independence of the residual attachment modes, the rank cannot be less than the number of interface forces, thus limiting the number of free-free modes that can be used. In most realistic situations, however, accurate results can be obtained with a relatively small number of free-free modes and so this limitation does not play a significant role. The first-order approximation [Ah — w 2 /] -1 Ah- 1 is used in (2.25). This result is valid provided (2.24) is maintained, but this may not be possible in components having high modal density in the target frequency range, as a very large number of modes would have to be calculated to satisfy (2.24). To improve the accuracy, a second-order approximation may be employed in which h  [Ah  w  2  —1  =I  cd 2  A h— 1 I —1 Ah 1 Ah 1  co 2 A h— 2^(2.32)  21  Chapter 2. Component Mode Representations The expression for ph and the component displacement thus become, Ph  =^(412 42) 4 1: f  (2.33)  u = (lop + Of + 2:6 f  (2. 34)  B = .1) h A. 17 2 (1q:  (2.35)  where ds is as before and,  The symmetric matrix /3 can be calculated without knowledge of the neglected modes by using (2.28) and by observing that (2.36)  Applying (2.29),  u = (p p + (.1 + w 2 6 11/1) afifB  (2.37)  Or, U  = Ifop + Off + ca 2t) fB  where  = 13,3 = O mit  ^  ^  (2.38)  (2.39)  It should be emphasized that the columns of E do not represent a new set of modes, but rather are complementary to the existing residual attachment modes. It will be shown in Section 3.5 that retaining this second-order contribution in the substructure synthesis creates a supplementary global mass matrix which partially accounts for the inertia of the neglected modes. The MacLaurin series expansion of [Ah — ca 2 /1 -1 can be continued further so that, ^ [A h w 211 —1 = A h-1 w 2A h-2 w 4A h 3 w 6A h-4 (2.40)  Chapter 2. Component Mode Representations ^  22  This expansion is valid as long as GJ 2 is smaller than all diagonal elements of Ah. Since the diagonals are eigenvalues of the neglected modes and are therefore always larger than 4, the expansion is valid for the frequency range 0 < w < w e . The resulting displacement contribution from the higher modes is, 4 /sPh 4/ /1  (Ah 1 W 2 4 2 + W 4 A17 3^.) 4:0 1:f^(2.41)  By (2.27), (2.35), and (2.36), it can be stated that for integer n > 1,  *: = ( PhAh  thAh-t  4  -  T  = (G M)n-1  n-1  ^430;111:11:  u s  (2.42) (2.43)  In other words, every term in the expansion (2.41) dependent on the higher modes can be replaced with an equivalent expression in terms of 1/ isPh  6 and M; i.e.,  = [a + C4/ 2 6Ma Cs.14 (GM) 2  = a(w)f  G +...] f^(2.44)  ^  where G(w) is the dynamic residual flexibility of the component. By factoring out  (2.45)  6, the  series converges to the expression,  6(w) = 6 (.1 — w 2 m6) -1 (.1 — w 2 6m) -1 G^(2.46) provided liw 2 dMIl p < 1, for the general class of p-norms [21]. In Appendix A it is shown that there exists a low frequency range 0 < w 2 < 4/a, a > 1 in which this condition holds. Furthermore, it can be shown through a similarity transformation that I — c‘, 2 6M is positive definite and invertible when w < w e . Define A as A = 4) -1 — cv 2 6M)^  (2.47)  Chapter 2. Component Mode Representations ^  23  where (I) is a square matrix containing all free-free modes of the component. Matrix fl• is invertible, and from mass-orthogonality, ^= (D T M. Therefore,  A =^(1)T  ^-  W 2 e11161114)^(2.48)  I — W2 41,T m ,f)h^m 4f,^(2.49) 1 0 -  0^I  w  2  0^0  (2.50)  0^All  It is inferred from (2.50) that the eigenvalues of A (i.e., its diagonals) are positive when the diagonal elements of c,./ 2 44 are less than 1, a condition which holds for w < wc. Because of the similarity transformation, A and I — (26 M have identical eigenvalues. Therefore, I—w 2 6M is positive definite and invertible in the frequency range 0 < w < wc. The modulation matrix •  II (w) = (/ — c4; 2 6M) tunes the static residual flexibility  -1  (2.51)  6 to frequency w. The evaluation of 6  at w and  its substitution in (2.45) gives an exact account of the higher modes' contribution to the component displacement. However, using d(w) in the CMS formulations is difficult because w is unknown and appears in a highly nonlinear form. Indeed, including terms higher than w 2 in (2.44) creates a nonlinear eigenvalue equation for the structure. However, once estimates of the system natural frequencies CA have been obtained with either the first- or second-order approximations, calculating d(cDi) is useful for estimating the magnitude of the error in wi. This is explored further in Section 3.9. In summary, a free-free mode representation by itself will often not provide enough degrees of freedom to satisfy compatibility constraints at interface locations on discrete components. By adding a linearly independent set of residual attachment modes, the mode set is expanded relatively inexpensively so as to ensure satisfaction of inter-component  Chapter 2. Component Mode Representations ^  24  compatibility, and a more accurate component mode representation is produced by including the static flexibility of the neglected free-free modes. 2.6 Inertial Approximation of Lower Modes The concept of using an approximation of the higher modes was carried further by Kubomura [6] and Kuang and Tsuei [22] who used approximations for both the high and the low frequency modes. Suppose the component mode set in (2.8) is partitioned into three groups such that P  u( t )  (2.52)  Pm 4Phil Ph  The set L contains component modes that are in the same frequency range of the system modes of interest. Modes with much lower frequencies are contained in (I)i and modes with much higher frequencies are in (I)h. The equation of motion of the component is then,  (DT  At — co 2 I^0^0  0^A„, — cy 2 /^0  Pm  0^0^Ah - Q.,2 1  Ph  =  4, In T  1(0^(2.53)  If upper and lower frequency cutoffs are such that wc2, < w 2 and c4h^wa, then 1^  At)^  (2.54)  Ph = 4 1,101: f(t)^  (2.55)  =^  and the displacement of the component is given by U  1^T B =^ f +^+^h A h i (eh  f B^(2.56)  Chapter 2. Component Mode Representations ^  25  Or  u = (1)„,pm f B  (2.57)  As before, the approximation for the high frequency modes can be thought of as residual attachment modes as designated by xis. The approximation of the low frequency modes is purely inertial. Each column of  t is the acceleration of the component induced by a  unit force at one interface location with zero force elsewhere. Consequently, each of these columns can be called residual acceleration modes. It is convenient to keep the w 2 on the right-hand side of (2.57) with fB, as it is still yet to be determined. Thus a new set of inertial coordinates are defined: pc„ = 113 /w 2 . Observe that the residual acceleration modes are linear combinations of the rigid body and low frequency elastic component modes. Approximating the low frequency modes does not preclude the need for computing them, as it did with high frequency modes. The component mode representation in (2.57) will not be useful for the type of structures which are of interest in the present work. In these structures, it is the very lowest system modes that are of interest and so an inertial approximation is not appropriate. However, approximations of the higher modes will prove to be very useful. 2.7 Constraint Modes  Hurty [23] proposed a component mode representation in which the displacement of a component is described by a combination of rigid body modes (r) (if the component is unrestrained or partially restrained), static constraint modes (c), and cantilevered normal modes (n). Partitioning into determinate interface (R), redundant interface (5),  Chapter 2. Component Mode Representations ^  26  and interior (I) coordinates, the component displacement is given by, I  uR  <Dr'  uus,  0  0 p(t)  I  pc (t)  =  0  pra(t)^(2.58)  ck4  The determinate interface coordinates are interface coordinates which, if constrained, make the component statically determinate. If a component is fully restrained with external constraints, the R set is null and all interface coordinates appear in the S set. A rigid body mode is defined by a unit displacement at an R coordinate, with zero displacement at other R coordinates. A static constraint mode is defined by a unit displacement at an S coordinate with zero displacement at all other coordinates in R+S. The cantilevered normal modes are free vibration modes calculated with all interface locations rigidly held. Craig and Bampton [24] recognized that a complete set of constraint modes would automatically include any possible rigid body motion. As a result, the rigid body modes can be explicitly dropped from (2.58) and no distinction needs to be made between determinate and redundant interface coordinates. The component displacement is therefore given by the following, where B denotes the complete set of interface coordinates: u B^I  Pc +0^p.  (2.59)  (1);,  or, in unpartitioned form, U = Xli cPc (DnPn  (2.60)  The component mode representation as expressed by (2.59) is commonly referred to as the Craig-Bampton mode set and is statically equivalent to (2.58). This representation was also adopted by Hurty et al. [25] and SandstrOm [26]. Note that each static mode has a unit displacement at one interface coordinate and that each normal mode has zero  27  Chapter 2. Component Mode Representations^  displacement at all interface coordinates. By the top equation in (2.59), uB = pc and so I^B U =^AI -rnPn  (2.61)  The matrices xli! and cl, „I are calculated in the following manner: the equation of motion of a component in partitioned form is KBB KBI  H  uB}  w2  K IB  [MBB  mBI  uB  fB  MIB^mII  111  0  (2.62)  Setting w = 0 for a static analysis, the second equation in (2.62) becomes ul  =^KIBuB = TIuB  (2.63)  Holding the interface rigidly by setting uB = 0, the second equation in (2.62) becomes [KII 40 2m II1 u l = 0^  (2.64)  The eigenvector solutions to this equation define the cantilevered normal modes 4)n. Applying (2.59) to (2.62) and by premultiplying by the modal matrix, the component equation in modal coordinates is k BB^0  uB  0^kNN  pn  [mBB  mBN lliiB  NB^NN M^M  0  where kBB  KBB  KBIT cI  MBB  mBIC + TcITmIB^T cITAIIIC  MBI  +emIl(DI  kNN  m BB MBN mN  = =  On  = mNB  fB  T  (2.65)  Chapter 2. Component Mode Representations ^  28  Note that the modal stiffness matrix is block diagonal and that the static constraint modes are orthogonal (with respect to the stiffness matrix) to the cantilevered normal modes. There are as many static constraint modes as there are interface coordinates in a component. A complete set of these modes must be included in the analysis to ensure that the rigid body motion and static behaviour of the component are preserved, as these are important characteristics of the lower structural modes. The normal modes are included in the representation to account for some of the inertial properties of the component. To give a complete account of the component inertia, a full set of normal modes has to be included. Generally this is not practical because great computational effort is required to calculate a large number of normal modes and also because the normal mode set has to be truncated if any coordinate reduction is to be achieved. The normal modes which are generally kept and included in 4:1) n/ are the modes of lowest frequency. It is the lower component modes which make the most significant contribution to the lower structural modes. An important special case of the Craig-Bampton representation is static condensation, or Guyan reduction, in which all dynamic modes are deleted from (2.59). While the remaining set of constraint modes is statically complete, it tends to produce poor results in a free vibration analysis because of an inadequate account of the kinetic energy. This can be improved within the confines of static condensation by including static modes for degrees of freedom other than those on the interface. In the parlance of Guyan reduction, a set of master degrees of freedom are selected which include both the boundary and a subset of the interior degrees of freedom such that the motion of the component is best described with a minimum of constraint modes. Various procedures for selecting masters have been proposed [27, 28]. While masters chosen from the interior coordinates play no part in the inter-component compatibility, they do foster a more accurate representation  Chapter 2. Component Mode Representations ^  29  of the inertial properties at non-interface locations. However, this can be better achieved by including dynamic modes by means of a CMS procedure rather than by adding more static modes. Hintz [29] proposed a constraint mode method in which the constraint modes of the Craig-Bampton mode set are augmented with inertia-relief modes. uBl  .  [I  0  AF^41;  uB  pi  (2.66)  The inertia-relief modes W fi are defined as the static response of a component (with interfaces held fixed) to rigid body inertia forces. They are calculated by the following formula, in which a unit modal acceleration is applied to each rigid body mode, and in which RB is a vector of reaction forces necessary to maintain zero displacement at the interface: [KBB KBI  1[0^  [mBB MB!  1[Vs  KIB KII^r^mIB mII  (2.67)  The inertia-relief modes can therefore be expressed as,  vf^KH-1 (m/B,I,Br MI/(DT)  ^(2.68)  With respect to the stiffness matrix, the inertia-relief modes are orthogonal to the constraint modes. By (2.68), the number of inertia-relief modes is equal to the number of rigid body modes. Thus, for a fully restrained component, the constraint mode method of Hintz [29] reduces to the definition of constraint modes used by Craig and Bampton [24]. For an unrestrained or partially restrained component, the inertia-relief modes enhance the component displacement field at low frequency by including rigid body inertial effects. However, these modes are not strictly necessary for static completeness, as the constraint  Chapter 2. Component Mode Representations ^  30  modes defined in (2.63) are capable of exactly describing the static displacement to an interface load. To improve the component representation in dynamics problems, Hintz added fixedinterface free vibration modes to (2.66), giving the following component mode representation:  1 :1  uB P  ^  (2.69)  Pn  Note that for a fully restrained component, the inertia-relief modes disappear and (2.69) reduces to the Craig-Bampton mode set. One difficulty that may arise is that xli if and 43./ span the same vector space and therefore may not be linearly independent modes. If the number of fixed-interface modes is small, this problem can be generally avoided [7, 29]. In fact, the most significant benefit of inertia-relief modes is that they can be used in place of dynamic modes in a CMS analysis, and thereby reduce the number of dynamic modes that need to be calculated. However, results presented by Hintz [29] comparing (2.69) with the Craig-Bampton mode set were inconclusive as to which offered the best accuracy. 2.8 Attachment Modes Hintz [29] proposed a static mode set consisting of rigid body and attachment modes which would be statically equivalent to the constraint mode representation (2.66). Attachment modes are defined by placing a unit load at an interface location, with zero load at all other locations. To show how attachment modes are calculated, statically restrained components are distinguished from those that are unrestrained, or partially restrained. For a restrained component, attachment modes IP are calculated by substituting w = 0  Chapter 2. Component Mode Representations ^  1 1 1=1 1  31  and 1B = I into (2.62) and solving the following equations for': [ KBB KBI  TB  KIB ICH^WI  (2.70)  0  or, in unpartitioned form, KT = /3^ Note that  1 1  TI = _KII -1 KIB T B = gicI AFB TB  (2.72)  [ I =^TB  1 11 1^t  (2.71)  (2.73)  11 1  Thus, attachment modes are linear combinations of constraint modes. For an unrestrained component, the applied forces must be equilibrated by the resultant inertia forces if the deformed displacement is to be isolated. This involves solving the statics problem, KT = (/ — API),.4:1:0,1 /3 = P/3  (2.74)  where P is a projection matrix having the property P = P 2 and where PP defines equilibrated force vectors resulting from applying unit loads to the interface coordinates. Because K is singular, T cannot be determined directly from (2.74). Instead, the modified equation  if/ = Gcps  (2.75)  is used where GC is the component flexibility after a set of statically determinate constraints is applied. If the attachment modes are to be mass-orthogonal to rigid body modes, it is necessary to remove the rigid body component of if [7]; i.e., -  T =4  -  4'rPr  ^  (2.76)  Chapter 2. Component Mode Representations ^  32  where (2.77)  Assuming mass-normalized rigid body modes, the result of substituting (2.76) in (2.77) is Pr = 4) ,T11141  and tp  ^  (2.78)  . ( i _ 4, rT 4, r m ) if = pril = pTGepp^(2.79)  Thus, premultiplying a force vector by P transforms it into an equilibrated, or inertiarelief, force vector; and premultiplying a displacement vector by PT makes it massorthogonal to the rigid body modes [30]. The attachment modes resulting from (2.79) are referred to as inertia-relief attachment modes. This equation can be used as the general definition of attachment modes by noting that for a restrained component, P = I and Gc = If' and hence (2.79) is equivalent to solving (2.70). If a component is unrestrained, inertia-relief attachment modes are not linear combinations of the constraint modes. However, the static mode set  ^ 41 a = [ 4) r 4' ]  (2.80)  is equivalent to the expanded constraint mode set used in (2.66) [29]. For dynamic analysis, the static mode set (2.80) is augmented with a truncated set of component free-vibration modes. Hintz [29] suggested using either fixed-interface or free-interface dynamic modes and showed examples derived with each. However, the free vibration modes have to be selected carefully to avoid linear dependence with the attachment modes. This is cited as a common problem when attachment modes are used with complex components [31]. Only when fixed-interface free vibration modes are used with a fully restrained component is linear independence between the attachment and  Chapter 2. Component Mode Representations ^  33  free vibration modes guaranteed. In this case, the attachment mode set is equivalent to the Craig-Bampton mode set. The main advantage of attachment modes is that, in general, they are easier to obtain experimentally than are constraint modes because the necessary boundary constraints are easier to impose on a structural component [31]. Computationally, they are more expensive as they involve the inversion of the full stiffness matrix, whereas constraint modes require only the inversion of If II. Moreover, the Craig-Bampton mode set does -  not require explicit calculation of the rigid body modes, giving it an added computational advantage. 2.9 Polynomial Functions  Meirovitch and Hale [2, 3] observed that the basic requirements for component modes of distributed models—completeness, linear independence, and differentiability—are satisfied by a much larger class of admissible functions which may include, for instance, low-order polynomials. Independent polynomial functions are generally easier to derive than component modes because they can be established without knowledge of the stiffness and mass distribution in the component: they depend only on its physical dimensions. For these functions to be admissible, it is also necessary that they satisfy external geometric constraints. This may be a difficult requirement to satisfy if the constraints are distributed in a complex manner throughout the component. However, this requirement is not strictly necessary since the enforcement of the external constraints can be postponed until later in the analysis when inter-component compatibility constraints are satisfied. Therefore, it is permissible to generate component polynomial functions without taking into account external constraints. Admissible vectors are generated for discrete models by sampling polynomials defined  Chapter 2. Component Mode Representations ^  34  over the spatial domain of the component. Translational elements of admissible displacement vectors are sampled directly from the polynomials; rotational elements are sampled from the spatial derivatives of the polynomials. Generally, an infinite number of independent polynomials are available from which only a finite number of linearly independent admissible vectors are chosen. Therefore, sampling of the polynomials must be done with care to ensure that the resulting admissible vectors are linearly independent. Admissible vectors defined in this way depend only on the spatial extent of the component and its node locations. They do not depend on the distribution of strain or kinetic energy in the vibrating component. If the material properties of the components are uniformly distributed, low-order polynomials can be used effectively to predict the low frequency modes. This has been demonstrated by Meirovitch and Hale [2, 3], for a system composed of flat, rectangular plate components, and by Johnson and Jen [32], for beam components comprising the links of a flexible robot arm. But if the material properties are distributed nonuniformly, there may be small-scale vibration effects that cannot be represented with low-order polynomials. Higher-order polynomials will have to be added, and although this in itself is not difficult, it has the effect of increasing the order of the system equations and necessitates prolonged computation at the system level. As an example, consider the segments of a ship hull as structural components. A low-frequency mode of the hull may consist exclusively of bulkhead vibration. If this is the case, it is likely that the bulkhead vibration will appear in the low frequency component modes as well. As a result, this local vibration mode can be adequately represented with a small number of component modes. By contrast, none of the polynomial-derived admissible vectors are likely to be similar to the bulkhead mode. These vectors are deflection shapes defined over the spatial dimensions of the component under the assumption of a uniform distribution of mass and stiffness within the component. However, local  Chapter 2. Component Mode Representations ^  35  vibration modes only appear as the result of a non-uniform distribution of mass and stiffness. Therefore, such modes can only be represented by taking large numbers of polynomial functions in linear combination, which, as was mentioned above, is harmful to the computational efficiency of the analysis. Faced with this difficulty, a different approach might be to substructure the model further so that all components are reduced to simple, uniformly distributed, plates and beams. However, this severely restricts the way in which components can be designed separately and then reassembled. Consequently, using polynomial-derived admissible functions may be useful in certain types of structures, but in general application to finite-element models convergence problems and insufficent order-reduction will likely be encountered. 2.10 Comparison of Component Mode Representations Some general conclusions can be drawn from the foregoing discussion on the subjects of compatibility, linear independence, completeness, convergence rate, and computational aspects of the component mode representations. Compatibility In the most general applications, a component mode set should contain a mixture of static and dynamic modes. Static modes can be calculated relatively cheaply and so large enough numbers of them can always be assembled to satisfy the interface compatibility requirements. A dynamic complement is necessary to give a good account of the low frequency component inertia. Four component mode representations described in this chapter—the MacNeal-Rubin mode set, the Craig-Bampton mode set, and the constraint and attachment mode sets of Hintz—are of this type. Mode sets using dynamic  Chapter 2. Component Mode Representations ^  36  modes only, such as classical free-free modes or the interface-loaded modes of Benfield and Hruda, have a more limited applicability because they are more difficult to calculate in large numbers. They would mainly be useful in structures composed of beam-like components, such as a series of robotic links, where the number of compatibility constraints remains small. Linear Independence and Completeness  The linear independence and completeness of a mode set are both necessary conditions for convergence of the system modes. Both the Craig-Bampton and the MacNeal-Rubin mode set satisfy these requirements, the latter by degenerating to the classical free-free mode set when a full complement of modes is used. The constraint and attachment mode sets of Hintz can generally be made to satisfy the linear independence requirement provided the truncted set of free vibration modes is chosen carefully. However, as the number of free vibration modes is increased, the linear independence of these two mode sets will eventually be lost. Convergence Rate: the Relation to Mode Acceleration  When comparing component mode representations, an important consideration is the rate at which the system modes converge as a result of increasing the number of component modes. It is known from the Rayleigh-Ritz method that a solution obtained with a set of approximating functions will converge to the exact solution as the number of functions is increased, provided that they satisfy the requirements of linear independence and completeness. In this respect, the component mode representations presented in this chapter are generally convergent but not all will converge at the same rate. It was demonstrated by Rubin [18] that the MacNeal-Rubin mode set enjoys the same improvement in convergence over the classical method that the mode acceleration  ^  Chapter 2. Component Mode Representations^  37  method enjoys over the mode displacement method. The mode acceleration method was first suggested by Williams [33] as an alternative method for determining stresses induced by transient loads on aircraft. In comparison to the more conventional mode displacement method it is generally accepted that mode acceleration is a faster converging method; that is, results of equal accuracy can be obtained with fewer modes [34]. How this improved convergence is achieved can be just as easily analyzed by examining displacements rather than stresses. In the mode displacement method, the displacement is approximated by a set of n dynamic modes, U  =E  (2.81)  i=i  The equations of motion for an undamped structure can then be written in terms of the modal coordinates p i : pi  ^ = 1,2,...,n^(2.82) + wfpi Orf  This system of equations can be solved for pi using Duhamel's integral and the summation in (2.81) provides the dynamic response. Supposing (2.81)—(2.82) describe the motion of a structural component acting under periodic interface loads, the mode displacement method is then equivalent to the classical free-free mode representation. In the mode acceleration method, (2.82) is transposed: Pi = a  Wi  (4 Tf  (2.83)  )  and the dynamic response becomes, U  T mm  = E^ f- E  ^i=1^  (2.84)  i=1  If the summation were over all modes, the first term would reduce to simply the static response to the applied load; it is the second term that accounts for the inertial effects.  38  Chapter 2. Component Mode Representations ^  Consequently, the dynamic response for a truncated mode set can be written:  u = Gf  (2.85)  ^ —  The improved convergence is derived from fact that the static portion of the response is no longer dependent on the number of modes retained; it is determined directly through the static flexibility [34, 35]. The response given by the mode acceleration method is transformed by substituting (2.82) into (2.85) giving, u = Gi^071,-.24(01. — c4Pi) = stkp +  Of  ^  (2.86)  which is precisely the MacNeal-Rubin mode set. Thus it is expected that the MacNealRubin mode set will converge faster than the classical mode set as a consequence of the complete account of the static response of a component to interface loading. For the same reason, it is expected that the other statically complete mode sets discussed in this chapter—the Craig-Bampton mode set, and the constraint mode and attachment mode sets of Hintz—will also experience a higher convergence rate than the classical method.  Computational Considerations It has already been noted that constraint modes are computationally less expensive than attachment modes because they are derived from the inversion of the submatrix rather than that of the full matrix K. A further point to note is that fixed-interface free vibration modes can be calculated more cheaply than free-free modes, owing to the smaller order of its associated eigenvalue equation. It has also been noted that the Craig-Bampton mode set does not require the rigid body modes to be calculated explicitly. These considerations indicate that the Craig-Basnpton mode set is the most  Chapter 2. Component Mode Representations^  39  advantageous computationally. But it will be shown in the next chapter that the extra computation needed at the component level gives the MacNeal-Rubin mode set important advantages in the formulation of the system equations. 2.11 Mode Selection  Having established the requirements for component mode representations and having discussed some aspects of their respective convergence rates, it is now necessary to consider the problem of component mode selection: namely, deciding how many component modes are required to get results of a certain accuracy. Unfortunately, this is a question for which no definitive theoretical answer can be given. It is known that if a full set of component modes are used, exact results are obtained; but this fact is of no practical use since CMS offers no advantages in this case. CMS is most attractive when modal truncation, especially severe modal truncation, is possible. All analytical methods use a mathematical model to predict the behaviour of a physical system. But the accuracy of one method is not known until its results are compared to those of another. For example, the accuracy of a finite element analysis is largely dependent on the degree of mesh refinement; but this accura c y cannot be known without a comparison with experimental results or with results obtained by some method known to be accurate. Nevertheless, experienced analysts can choose an appropriate mesh refinement without foreknowledge of the correct results. The same is true of CMS with regard to mode selection. While the absolute accuracy of a particular choice of modes is unpredictible, general rules of thumb can be developed to guide the analyst in choosing modes for a particular problem. The appropriate choice of component modes is problem-dependent. It is a function of the number of components, their size with respect to the whole structure, their modal density, the differences in their  Chapter 2. Component Mode Representations ^  40  flexibilities, and their relative participation in the modes of interest. If it is the lowest frequency modes that are chiefly of interest, a cutoff frequency criterion can be used. A cutoff frequency c' is chosen based on the analyst's intuition or experience: all component modes below this frequency are included in the analysis, all above are excluded. This criterion can be adjusted as the accuracy requirements change. If good accuracy is required only in particular modes, component modes can be weighted according to a component's participation in the targeted modes. 2.12 Summary  Several component mode representations satisfying the basic requirements of linear dependence and completeness were presented in this chapter. The representations give a reduced-order description of a structural component which preserves its rigid body, static and low frequency elastic response. It was found that representations combining static and dynamic modes are more successful in general applications for two reasons: first, the number of modes is always large enough to satisfy inter-component compatibility constraints; secondly, a higher convergence rate is expected because a complete static response to interface loading is automatically included.  Chapter 3  Substructure Synthesis  3.1 Introduction In the previous chapter several component mode sets were discussed and it was shown that with them, reduced-order discrete representations of the component matrices could be generated. In the present chapter, attention is focussed on the general problem of linking the reduced-order component representations together, a process which gives equations of motion for the entire structure. The coupling of the components is achieved by satisfying the compatibility and equilibrium conditions at the component interfaces. Specifically, interface displacement and rotation coordinates must match, and interface forces and moments must cancel. The object of the present chapter is to show how these conditions can be applied in their most general form to the various component representations described in the last chapter. A critical assessment of the various forms of system equations will be given with regard to their applicability to substructured problems of large size and of general geometric complexity. In particular, two criteria by which they will be judged are the following: the system equations should be in a form that is convenient to solve, (an algebraic eigenvalue problem in standard form being the most preferable); and, a substantial degree of coordinate reduction is desirable to minimize the size of the system equations and thereby reduce the computational time. The second criterion is important because it is often the case that when large, complex  41  Chapter 3. Substructure Synthesis ^  42  structures are analyzed with finite elements, the majority of the computational time is spent on the eigensolution. Any reduction in the size of this equation is beneficial if it does not sacrifice accuracy. 3.2 Inter-Component Equilibrium and Compatibility Meirovitch and Hale [2, 3] examined the problem of satisfying inter-component compatibility in distributed models. A basic difficulty encountered with interfaces that are curves or surfaces is that with a finite number of modes, it is impossible to satisfy the infinite number of constraints at the interface. Instead an intermediate structure is introduced. This is a structure made up of the original structural components but where interface compatibility is only partially satisfied with a finite number of weighting functions. In this respect, the intermediate structure lies between the system of uncoupled components and the actual, fully coupled, structure. One particular intermediate structure of importance to the present work is where compatibility is exactly satisfied at a finite number of discrete locations on the interface. In substructured finite element models with conforming elements, maintaining compatibility at interface nodes guarantees its maintenance between the nodes. Therefore, no distinction between the intermediate and actual structures is needed. If the model contains non-conforming elements, the intermediate structure satisfying compatibility at the interface nodes is the nearest configuration to the actual structure possible. And so in either case, this particular intermediate structure represents the optimal configuration for the assembled finite element model. Of course, the option exists for satisfying compatibility at only a subset of the interface nodes. One consequence of this is that any upper bound provided by the Rayleigh-Ritz method is immediately lost. Moreover, this approximated compatibility is unnecessary  Chapter 3. Substructure Synthesis ^  43  since it was shown in Chapter 2 that by using a mixture of static and dynamic modes, enough can be readily obtained to allow all compatibility constraints to be satisfied. In the remainder of this section, general expressions for compatibility and equilibrium are presented for discrete models. Equilibrium conditions describing force and moment balance at the inter-component boundaries can be expressed in the following general form:  = Aff  ^  (3.1)  -B  where f i s a vector containing the interface forces and moments of each component in sequence:  7B =  e  ^  (3.2)  fB and  J.]: is an independent set of global interface forces and moments partitioned from  -B  f . Assuming that the relative positions and orientations of the components are time-  invariant, and that the interfaces themselves possess no flexibility or inertia, the transformation matrix A contains constant coefficients that depend only on the structural geometry. In a similar manner, 77113 may be defined: uB 1 UB =  • • •  (3.3) }  u!' By the principle of virtual work, _Br^-BT  = u A 51gB = 0^  (3.4)  Chapter 3. Substructure Synthesis ^  44  where 6f: is a set of virtual interface loads. Therefore, the requirement for intercomponent compatibility is ATTu-B  0  (3.5)  A different expression for the compatibility is, U 1 AU -B=TAUS  (3.6)  which can be derived from (3.5) by selecting an independent set of coordinates u9 from  rsB . By combining (3.5) and (3.6) and by noting that u9 may assume arbitrary values, A T TA = 0^  (3.7)  By the principle of virtual work, - Br B - Br  f  814 = f^B TA6u g = 0 -  (3.8)  where Su: is a set of virtual interface displacements. Therefore, the requirement for inter-component equilibrium is,  TT? = o  ^  (3.9)  The expressions for equilibrium, (3.1) and (3.9), and the expressions for compatibility, (3.5) and (3.6), can be applied to structural models of an arbitrary number of components and of an arbitrary geometrical configuration. Based on information detailing the connectivity of the structural components, the matrices A and TA can be easily and systematically constructed. 3.3 Lagrangian Formulation of the System Equations  Regardless of what type of component representation is used, the total potential and kinetic energy of s uncoupled components may be written as, (3.10)  45  Chapter 3. Substructure Synthesis^  T = —2 77i mt  ^  (3.11)  The vector 3 contains the component modal coordinates of each component written in sequence: 71  (3.12)  ri= ^ 77.  where 77,772,...,77, are vectors of modal coordinates corresponding to each of the s components. Depending on the component mode representation, these might include normal modal coordinates, interface displacements, or interface loads. The quantities  WI and  i  are block-diagonal matrices of the form,  m=  m1  0  0  m2  . ^.^.  0 0  =  k1  0^..  0  0  k2  0  0  0^...  lc.  (3.13)  •^•  0  0  .^.^.  771,  where ki and ms are the condensed stiffness and mass for the i th component. For a particular mode set, these are determined from (2.6) and (2.7). The equations of constraint which link the various components together may be written in the general matrix form, RI/ =  0  (3.14)  The rows of R may express compatibility or equilibrium relationships between the components, or any other constraints on the system. The overbar notation will be used throughout to indicate square or rectangular matrices of the form (3.13), or vectors of the form (3.12). The equations of motion of the system are obtained from the Euler-Lagrange equations. Two methods are available for incorporating the constraints (3.14) [36]. The  46  Chapter 3. Substructure Synthesis^  method of Lagrange multipliers can be used, in which the Euler-Lagrange equations are applied to the modified Lagrangian, L = T — V - Rff (3.15)  where p is a vector of Lagrange multipliers. Dowell [10] used this method to derive the characterstic equation of coupled systems from the classical free-free mode set. This method is also used in Section 3.10 to illustrate solutions obtained from the MacNealRubin mode set. The more usual method is to eliminate some of the coordinates in 77 by means of a -  coordinate transformation. Many researchers [9, 15, 23, 24, 29] have derived transformation matrices by a direct partitioning of R. The vector r7 is partitioned into dependent (d) and independent (g) coordinates giving, Rd  (3.16) Rnd = 0 ri g  where Tld is selected so that Rd is invertible. The following transformation may now be derived: ?id  (3.17)  Applying this transformation to V and T eliminates the dependent coordinates nd from the analysis and couples together the system components. Another elimination method has been given by Kuang and Tsuei [22]. Multiplying (3.14) by RT gives  RTR =^= 0^  (3.18)  where D is a rank-deficient square matrix. The eigenvectors of 2) corresponding to its zero eigenvalues are calculated from the equation DYi = 0^  (3.19)  47  Chapter 3. Substructure Synthesis ^  Each of these eigenvectors satisfies all of the constraint equations contained in R. The linear combination of all these eigenvectors therefore gives a coordinate transformation between the uncoupled and coupled system: = = [Yi • • • Yq] (3.20) Elimination schemes such as these create an overall transformation between the uncoupled component coordinates and the generalized coordinates of the system, =  (3.21)  Substituting this transformation into the energy expressions and applying the EulerLagrange equations gives the free vibration equations of motion for the coupled system in the following general form: .  +  =O  (3.22)  where  = Terlit = TeTin-Te  (3.23)  This system of algebraic differential equations can be synthesized from any of the component mode representations given in the previous chapter and in general it is not necessary to use the same representation with each component. The order of (3.22) is equal to the difference between the number of columns and the number of rows in R, or in other words, the excess in the number of component modal coordinates over the number of constraint equations. Equations (3.22) are legitimate provided that R contains all the inter-component compatibility constraints. Other constraints can be added through (3.14) as desired. Although it is not essential to include the equilibrium relations as well, it will be shown in the subsequent sections that it is sometimes advantageous to do so. The main drawback of elimination methods is that they are generally cumbersome and inefficient to use in complex problems. In the remainder of this chapter, a different  48  Chapter 3. Substructure Synthesis ^  approach is taken in which it is assumed that all structural components are expressed in terms of the same type of modal representation. The general equilibrium and compatibility relations given in Section 3.2 are used to synthesize the system equations for different component mode sets. This leads to a more efficient and economical handling of the inter-component constraints which can nevertheless be applied to components of an arbitrary geometrical complexity. 3.4 Modal Force Method  The classical free-free mode representation (2.8) has been used by Yee et al. [11, 37, 38] in a synthesis procedure called the modal force method. Rather than identify a stiffness and mass matrix, this method derives a single dynamic response matrix H(w) by combining (2.8) and (2.14): u(t) = 4:0{A w 2 I] At) = H(w)f(t) (3.24)  Since a free vibration state of the structure is being considered, the vector At) contains interface loads only. Equation (3.24) may be partitioned according to whether the coordinates of u(t) are located on the component interface or at an interior point. For a single component, 1UB =[HBB Ha tlIfn  u r^RIB H"^fi  (3.25) }  Since we are considering the free vibration of a whole structure, fl = 0. For a system of s components,  uB 2  4  Hp 0 .^0  A  0 HBB^ 0 2^• • •  f2 {  0^0^. . . H BB  fl  71-B B (w) 7. .B  (3.26)  Chapter 3. Substructure Synthesis^  49  Satisfying compatibility using (3.5) gives A TTI-B = A TTIBB t = 0^  (3.27)  Satisfying equilibrium between the interface loads by applying (3.1) gives  A TTI BBAfgB = 0^  (3.28)  The expression (3.28) is the most general form of the structural equations that can be obtained with the modal force method. Yee and Tsuei [11] have derived a form of (3.28) applicable to a simply-connected three-component structure. A simply-connected  structure is one in which only two components are joined at any single point on the inter-component boundaries. The size of (3.28) is equal to the number of compatibility equations which, for simply-connected structures, is equivalent to the number of independent interface coordinates. For structures whose components have more complex interconnections, the number of compatibility equations is somewhat larger and is in fact equal to the number of independent interface loads. As the terms containing w in (3.28) are not in simple polynomial form, it cannot be put into the form of an algebraic eigenvalue problem, as is obtained when the Lagrangian formulation is used. The natural frequencies of the structure may nevertheless be found by computing the zeros of the determinant of ATTI BBA. For each natural frequency wi there is a corresponding non-trivial vector fgBi determined from (3.28). The structural  f9 with the following transformations: .171? =^= TIBB (ciji)AfgBi^  (3.29)  ui = FIB (uji)7B = RIB  (3.30)  mode shape can be recovered from  (  4 ji )AfgBi^  with HBB defined in (3.26) and with H 1B similar in form to H BB . The vector Til3 gives the interface portion of the mode shape and Fit gives the non-interface portion of the mode shape.  Chapter 3. Substructure Synthesis^  50  The accuracy of the system modes determined from (3.28) depends on the number of component modes included in (2.8). It was established in Section 2.3 that there is a minimum number of normal modes that must be used in order to maintain compatibility between components. Additional modes beyond this minimum will improve the accuracy of the final solution. Unlike some other CMS methods, the number of component modes does not change the size of the system equations (3.28). Their size is determined solely by the number of compatibility equations that are defined for the structure. The chief drawback of the modal force method is the nonlinear character of the governing equation (3.28). This equation is both difficult to formulate and difficult to solve unless the number of compatibility constraints is very small. As a result, the modal force method is only of practical use in special types of structures where the component interfaces are simple, and where sufficient accuracy can be obtained with free-free modes alone.  3.5 Free-Interface Formulation In this section, a general synthesis procedure is presented for the MacNeal-Rubin mode set described in Section 2.5. This mode set has been used in various forms by a number of researchers [17, 18, 19, 20]. MacNeal [17] derived equations based in the first-order expression (2.26). His assembled equations took the form of a stiffness matrix defined in terms of physical displacements and modal coordinates. Rubin [18] derived similar equations using the second-order expression (2.38). Chang [39] synthesized the equations of motion using the Lagrangian method with direct partitioning of the constraint equations. Irretier and Sinapius [40] developed the system equations from the general connectivity matrix TA defined by (3.6) and (3.9). This derivation is relatively complicated and further approximations are used to simplify the system equations. A simpler form of the  Chapter 3. Substructure Synthesis ^  51  system equations is developed in this thesis using the connectivity matrix A, defined by (3.1) and (3.5) [41]. The details of this method and its extension to the second-order approximation (2.38), occupy the remainder of this section. In Section 2.5, the component displacement using residual attachment modes was given as u(t) = 4>p(t)^f B (t)^  (3.31)  where  =  (3.32)  From (3.31) the uncoupled interface displacements can be written g^' .13—B 41' p^f  =  (3.33)  where it.17^0^.^0  0^c ^.^0  (3.34)  0^0^4)Bs 77B  and ill is similar. Likewise, the uncoupled component equations of motion may be written in the following compact form:  - w 2 /1 =  V  -  T?  (3.35)  The compatibility equations are obtained by premultiplying (3.33) by A T , as in (3.5): ATV + ATiBr =0^  (3.36)  In addition to compatibility, equilibrium is also satisfied by invoking (3.1), AT  ^+ A T TF BAfr: =0^  (3.37)  52  Chapter 3. Substructure Synthesis^  This allows the component interface loads to be expressed in terms of the modal coordinates: fB =  B  1  -1  —A [A T T A A T (I)  B  (3.38)  Observe that (3.38) along with (2.25) defines a direct relationship between the higher and lower modal coordinates. Applying (3.38) to (3.35), the system equations take the form of a symmetric, positive definite eigenvalue problem in standard form: (3.39) where  - 1 B  -^T r=A+T A A T T  [  -1  „,  A Al  B  (3.40)  Corresponding to each natural frequency wi is an eigenvector p-i determined from (3.39). Each entry of this vector is a modal coordinate associated with one of the freefree component modes. Unlike a mode shape, which describes how much each physical degree of freedom is participating in a natural mode of the system, the eigenvector describes how much each free-free component mode is participating in a natural mode of the system. Component modes which make a large contribution will have a large amplitude in the eigenvector, those that do not will have a small amplitude. Structural mode shapes can be recovered from the eigenvectors of (3.39) by writing the component displacement in a similar manner to (3.33): -B  21.4p+xlif  (3.41)  -  Applying (3.38) to (3.41), the i th structural mode shape is expressed in terms of the i th eigenvector:  us  = ap  t  -,7111 -]  1  —^[Al T A A T 4) pi  (3.42)  53  Chapter 3. Substructure Synthesis ^  To interpret the physical meaning of matrix r consider a system of uncoupled com,  ponents each of which is represented by a truncated set of free-free modes only. The stiffness matrices of the uncoupled components can be represented using the overbar notation as,  X  ^  (3.43)  The original component stiffness matrices K can be partitioned and rearranged so that B [,,,BB  A=  re  ,B  (3.44)  I^-IB  K  Note that by (3.40) and (3.44), the matrix I' can be expressed as T  r :B Since  K  BB  -  -BB^  ]-1  KB1[-  K + A [AT tif A AT -IB  Kz z  --B  — st'  r  (3.45)  is block-diagonal and ANIArAT A { B lis in general full, the latter can be  interpreted, as was noted by Urgueira and Ewins [19], as the stiffness matrix of an intermediate, or coupling, spring system which links the interface degrees of freedom of the uncoupled system. Note that the inter-component links of the actual structure are rigid. The finite stiffness given to the links by virtue of the intermediate system is a softening effect introduced by the residual approximation of the neglected modes. This softening effect partially compensates for the overstiffness resulting from the truncated free-free mode sets. The order of the global matrix in (3.39) is equal to the total number of free vibration modes used for all components. This is quite different from the system equations derived in (3.28) where the size is equal to the total number of compatibility equations. If this number is less than the sum of all the component modes, then it may be advantageous to use (3.28). If instead, the number of compatibility equations is greater than the number  Chapter 3. Substructure Synthesis^  54  of component modes, as is usually the case in complex structures with many degrees of freedom, then it would be better to use equations in (3.39). Not only is compatibility guaranteed by (3.36), but the resulting equations are smaller and are easier to solve, because they are in the form of a symmetric, positive definite eigenvalue problem. If instead the second-order approximation to the higher modes is used, the uncoupled interface displacements are expressed, from (2.38), as —13 B— u =^ (I) p  7,  B^  7,  + (41 + u.) 2 E. ) t^ (3.46)  Applying compatibility and equilibrium gives, CB  B  f =  —A  -7.7B  [A T 111 + co 2 F., ) A] - 1 A T (i) B 15^(3.47)  Since w 2 is unknown until the final solution is found, the inversion in the preceding --B  equation cannot be performed exactly. Defining KcPL = [411 T if 14.1  - i  and rearranging  gives ^—^ B  ]-1  (3.48)  f = —A {I + w 2 KcpLA T E A KcpLA TTB p —B  A linearized approximation for f  is given by,  ^—B^  --,.B  f ^' —A [KCPL — w 2 &MA T F —  Axc pd ATTB T)  (3.49)  where the matrix inverse in (3.48) has been approximated by the first two terms of its MacLaurin series expansion. Note that this expression contains the original expression (3.38) in addition to the linearized contribution from the second-order term. Applying (3.49) to (3.35) gives the global matrix equation,  rk — w 2 i/j p = 0  ^  k = r . A- + TBTAKc p L ATTB  (3.50)  ^  IC/ = I + TBTAKc p L ATrAK c p L ATTB  ^  (3.51)  (3.52)  55  Chapter 3. Substructure Synthesis ^  Thus, the net effect of including the second-order term in the approximation of the neglected component modes is the creation of a non-identity, supplementary global mass matrix. The order, symmetry and positive definiteness of global matrices are not affected by the inclusion of the second-order term. Craig and Chang [42J derived expressions equivalent to (3.39)-(3.40) and (3.50)-(3.52) for a two component system. The same global stiffness and mass matrices can be generated via the Lagrangian method. Using the first-order approximation to the neglected modes, potential and kinetic energies of the uncoupled components are given by (3.10) and (3.11), where  [7^0^_ [A o -^ 0 XII M^  (3.53)  -  111  and p  (3.54)  fB -= B^T -  Note that^= G =^KW. The coupling of the components is achieved with (3.38) which suggests the transformation,  I B^  —AKcp LA TT  ^ B  13  (3.55)  Applying this transformation to (3.10) and (3.11) and by noting that, =-B^-7,-T  =^M  ^  (3.56)  global stiffness and mass matrices identical to (3.51) and (3.52) are obtained. Consequently, the supplementary global mass matrix is a product of the first-order approximation if the Lagrangian formulation is used, or of the second-order approxima-,-.13 tion if the direct formulation is used. The form of E in (3.56) clearly indicates that the supplementary mass matrix represents the inertia contributed by the residual attachment  Chapter 3. Substructure Synthesis^  56  modes, a term which is significant if the criterion (2.24) is not strictly adhered to in the selection of component modes. Note that in developing the transformation (3.55), the relationship (3.38) had to be used. If instead the direct partitioning procedure (3.16)(3.17) is used, the equations satisfying both compatibility and equilibrium constraints take the form,  AT r ATiB^p 16i =^ 0^TT^.1B  = 0^(3.57)  where (3.9) and (3.36) have been used. The direct partitioning of R requires the inversion of the submatrix Rd defined in (3.14). In this case, the square matrix Rd has order equal to the number of compatibility plus the number of equilibrium constraints. Typically this is about twice the dimension of KCPL and therefore the procedure leading to (3.51)-(3.52) is generally more efficient than direct partitioning. In terms of the order of the global matrices and the predicted natural modes, both approaches should produce identical results. The greater efficiency of the synthesis method presented above stems from the more economical handling of the constraint equations.  3.6 Fixed - Interface Formulation In Section 2.7, a component mode representation was described in which static constraint modes were augmented with fixed-interface dynamic modes. This led to a component equation of the form (2.65). Based on this equation, the uncoupled equation of the system may be written as,  0  NN 0 k  ,B 11,1B +[?-.TIBB -r BN 1{,BH} 1  fin^  Ft  Frt-NB^  0  (3.58)  Chapter 3. Substructure Synthesis^  57  where kBB  kBB  =  0^0  0 kB B .^0  (3.59)  kBB  0^0 —^ 7 m BN and r1/NN are similar. and k NN , -7-1BB — --  To enforce the displacement compatibility between components, (3.6) is applied to (3.58). If the top equation in (3.58) is then premultiplied by TT, the result is, [ TATIBB TA  0  1 u: } [  Ti m-,BBTA Ti-rTi nN  - I-  NN^-7Ti NB TA  0^k^Pn  --NN M  iign  .  Ti r 0  (3.60)  -13n  The term TA r on the right-hand side of (3.60) is zero by (3.9). The coupled equations of motion of the system at frequency w therefore take the form of an algebraic eigenvalue problem:  —w 2 k  =o  (3.61)  where uB  ( 3.62)  =  n  and  k  }  and /Cf are the coupled stiffness and mass matrices in (3.60).  Corresponding to each natural frequency w i is an eigenvector^This eigenvector describes the participation of the interface displacement coordinates and the participation of the fixed-interface dynamic modes in a natural mode of the system. The modal displacement at the interior degrees of freedom can be recovered from the eigenvector by arranging the interior displacement of all components in sequence. This can be expressed in the following compact form: I T u =1 / c1 --B^ u^npn = w c TAu g (P npn g  (3.63)  Chapter 3. Substructure Synthesis ^  58  The modal displacement of the interior degrees of freedom for the i th system mode can therefore be obtained from the ith eigenvector: I  ui^ = xli c TA^(D ni. I  The vector  [  uf, taken together with 4,  (3.64)  defines the i th structural mode shape. The  component displacement in the i th mode can be written in unpartitioned form as z7i =  L  TA  „  Si  (3.65)  The order of (3.61) is equal to the number of independent interface coordinates in the structure plus the total number of fixed-interface dynamics modes used. For a large structure with many interface coordinates, this number can be be very large and the time required to extract the natural modes can be considerable. This is the chief drawback of the fixed-interface method which, in other aspects, is a very useful and attractive method. The calculation of static constraint and fixed-interface dynamic modes is economical in comparison to other types of component modes. Also, the synthesis procedure is simple and requires little computation. The global matrices are derived from the component matrices through simple transformations which, when examined closely, are similar to assembling global matrices from element matrices, as done in the finite element method. This assembly process eliminates the need for multiplication of large matrices and matrix inversions at the system level. Therefore, this is a more efficient means of synthesizing the system equations than is provided by the more general elimination methods described in Section 3.3. SandstrOm [26] carries the fixed-interface method one step further by applying a condensation procedure to the system equations. A similar idea was introduced by Kuhar and Stahle [43]. Assuming a periodic solution with frequency w, the first equation in  Chapter 3. Substructure Synthesis^  59  (3.60) can be written as, t Af  m ^ ^ITT —BN(TT BB -1- Ally -L Am Pit) = 0 A ft' -L AU - L02^M  +  '  (3.6 6)  Likewise, the second equation in (3.60) can be written as, (kIN wzieN)._^2NB —  T g 0^ pn — m AUB  (3.67)  This establishes a relationship between the modal coordinates p„ and the interface coordinates uB: pn  = w2^  - w 2—N N)  -1 M NB ry, B  (3.68)  A119  Substituting (3.68) into (3.66) gives,  r  rr TIBB TA  A  w 2 (TITfiBB TA + ce 2TIBN [INN  1-  2  -W M  NNi-1—mNBTA)] u B g  = 0 (3.69)  The modal coordinates pr, have been condensed out of the equation of motion (3.69). Because of the condensation, the system mass matrix is a function of the parameter w  2  and a nonlinear eigenvalue problem results. Consequently, the solution of (3.69) is both simplified, by a reduction in order, and complicated, by the introduction of nonlinearities in w 2 . The advantages of this condensation may outweigh its difficulties if the coordinate reduction is significant, i.e., if the number of modal coordinates eliminated is comparable to the number of interface coordinates retained. In large complex structures in which the component interfaces are curves or surfaces, the number of interface coordinates is usually much larger than the number of modal coordinates and so this condensation procedure would have little value. As with the modal force method, it would be of more use in beam-type structures where the component interfaces are restricted to a small number of point locations. If the number of interface coordinates could be thus limited, the nonlinear equation (3.69) is potentially very compact.  60  Chapter 3. Substructure Synthesis^  Corresponding to each natural frequency wi is an eigenvector u9 determined from (3.69). The remainder of the mode shape, given by the displacement at the interior coordinates, is constructed with the aid of (3.6), (3.63) and (3.68): -1  M, NN ) ui = [471,TA^(TCNN - Wi2-  M NB  TA} Ugi B  (3.70)  With this formula, the mode shapes can be constructed without having to explicitly calculate the the modal coordinates To n .  3.7 Condensation in the Fixed- and Free-Interface Formulations The Craig-Bampton and MacNeal-Rubin mode sets give fundamentally different formulations of the system equations. The formulation (3.60) produced by the Craig-Bampton mode set will be referred to as the fixed-interface method. These equations are in terms of the interface displacements and the fixed-interface modal coordinates. Two separate formulations have been derived with the MacNeal-Rubin mode set. The first-order mass formulation (3.39)—(3.40) and the second-order mass formulation (3.50)—(3.52) are expressed in terms of the free-free modal coordinates only; all physical displacements and loads have been eliminated. Collectively these two formulations will be referred to as the free-interface method. The distinctions between these two methods are of prime importance to the system condensation. Consider a system of complex components with meshed curve or meshed surface interfaces. As the mesh is further refined, the u: set expands, increasing the order of (3.60). But modal sets To n and IT) do not experience a corresponding expansion, for the number of dynamic component modes is only weakly dependent on the model complexity; in fact, it is more strongly influenced by the accuracy demands and the degree of substructuring. Thus, as the complexity of a substructured finite element  Chapter 3. Substructure Synthesis^  61  model is increased, the free-interface method enjoys a greater degree of condensation. This point will be clearly demonstrated with the examples in Chapter 4. It should be noted that in the free-interface method, both compatibility and equilibrium are explicitly enforced, while in the fixed-interface method only compatibility is enforced. Consequently, twice as many constraints are applied in the free-interface method, since in most cases the number of equilibrium and compatibility constraints is approximately the same. It was established earlier that the order of the coupled system equations is equal to the total number of component modes minus the number of constraints. Thus, as more constraints are applied, more coordinates are eliminated from the governing equations. This provides an additional explanation for the improved condensation of the free-interface method. However, the benefits of the free-interface formulation come at a price. It was noted at the end of Chapter 2 that more extensive component level calculations are required for the MacNeal-Rubin mode set than for the Craig-Bampton mode set. Moreover, in the present chapter it was shown that the coupling procedure in the free-interface method is far more complicated than what is needed in the fixed-interface formulation. This illustrates an essential trade-off in the CMS method: to avoid computations at the system level, more are required at the component level and in the coupling algorithm. As model complexity increases, this trade-off works in the free-interface method's favour.  3.8 Component Mode Substitution Benfield and Hruda [15] proposed a hybrid substructure synthesis method in which a classical free-free mode representation is used for some components, and the Craig-Bampton mode set for others. This approach attempts to resolve one of the basic difficulties of the fixed-interface method—the retention of interface degrees of freedom in the system  Chapter 3. Substructure Synthesis ^  62  equations. Consider a two-component discrete structure shown with components a and b. Component a is designated the main body and its displacement is described with free-free modes: ua  = (pp^  (3.71)  or, in partitioned form, UB a  (3.72)  u  b is designated a branch component and its displacement is described {  Component  [4113  with the Craig-Bampton mode set:  ^  l  ( Nuts lut Ju i " ^xijcz 4 1 ^pn )  (3.73)  ,  For the purposes of this example, compatibility equation (3.5) can be written  [ AT  AT  uB a  =  (3.74)  Ub  Because it is a two-component system, the number of interface coordinates in a and  b is the same, and is exactly equal to the number of constraint equations in (3.74). Therefore, A a and Ab are square matrices and, in general, the compatibility equations can be arranged so that one of the two is an identity matrix. Letting Ab = I, and using  (3.71) and (3.74), uB  _ AaTuaB = _ AaT ep^(3.75)  Substituting (3.75) in (3.73) gives  uf,'^—A„TV 0  ^p^p Ab  IP  ub^--tiPIAI(DB 4)4^Pn^Pn  (3.76) }  Chapter 3. Substructure Synthesis^  63  With this transformation, expressions for the kinetic and potential energies of the coupled system can be derived. Applying Lagrange's equations gives  + k^= o  (3.77)  = Tc7, 1', M Tab  (3.78)  "k = TLY Tab  (3.79)  ii/i -  where  iff  4. =  1 P  }  (3.80)  Pn  and Tab is the general two-component transformation matrix, 4)B^0 Tab=  4)I^0  —A aT(DB^0  (3. 81 )  —TIA/V3^cl., Note that the static constraint modes defined for component b allow full satisfaction of the compatibility constraints at the discrete interface locations. On the other hand, the free-free modes of component a allow the interface coordinates to be eliminated from the coupled equations. This keeps the order of the coupled equations small, regardless of how complex the interface is. The method is applied to multi-component models by executing the above steps successively. For instance, to synthesize a third component c, which may be connected to either a or b, the procedure is to repeat (3.71)—(3.77) with the coupled system a-b defined as the main body and the new component defined as the branch. The total synthesis of a multi-component system therefore requires intermediate eigensolutions which provide free-free modes for the main body in the next level. The number of intermediate eigensolutions varies according to how the components are connected, but at most it is .s — 2,  Chapter 3. Substructure Synthesis^  64  where s is the number of components. Although it is undesirable to have to perform additional eigensolutions, they are typically of small order, thus limiting the computational expense. The chief drawback of component mode substitution is the inconsistency in the treatment of the components. The representation for the branch component is statically complete in that the static response to interface loading is exact, whereas that of the main body is not. Moreover, the static incompleteness of the main body is carried through each of the intermediate stages, resulting in poor convergence for the system equations. It is for this reason that Benfield and Hruda proposed using free-free modes with interface loading, as described in Section 2.4. Although this innovation improves convergence by providing more realistic modes for the main body, it also means that components can no longer be analyzed independently and it significantly increases the computational burden. (Note that the static condensation used to calculate the interface loadings is, for multi-component models, not the same as that given by the static constraint modes of the branch components.) A further point to note is that if the structure is to be reanalyzed following a design modification, many of the intermediate results may have to recalculated, in addition to the modes of the modified components.  3.9 Error Estimation for Natural Frequencies The task of selecting dynamic component modes for a CMS analysis provides certain challenges. There is a computational advantage in including a small number of modes, but using too few will compromise the accuracy of the results. Modes may be selected using a cutoff frequency criterion, but some judgement is required in choosing a cutoff frequency suitable for the target frequency range. It is therefore of considerable interest  Chapter 3. Substructure Synthesis ^  65  to be able to predict the accuracy of a particular choice of component modes, if not in advance of the analysis, then immediately afterwards. It is also of interest to make these predictions without having to compare the results to those of other methods. Hurty [5] provided a convergence criterion for the fixed-interface method based on estimating the perturbation to a natural frequency resulting from the addition of component modes. It can be readily noted that introducing more fixed-interface dynamic modes into the analysis adds rows and columns to the system equations (3.61). Thus, the matrices of the system can be partitioned into original (0) degrees of freedom and extras (E) representing the newly added component modes:  =  A4 0 0 - .  A;4 0E (3.82)  These extra rows and columns can be condensed into the original equations through a process similar to that used in (3.66)—(3.69). In this way, the extra modes form a frequency-dependent addition to the system mass matrix of the form, 2 -1 6,A4 (03) = co2k0E rEE^ A — w mEo -  (3.83)  where it is assumed that the fixed-interface modes are mass normalized. Hurty showed that a first-order estimate of the change to the natural frequency w i can be obtained with, AC44 = aAk(Wigi  w  i  1 14  (3.84)  The formula is accurate for small frequency changes. If the set of extra modes is expanded to include all neglected modes, Awi corresponds to the absolute error in the eigenvalue. Thus, if a frequency has sufficiently converged, a reasonable estimate of frequency error is obtained. If a frequency has not converged the frequency error will not be accurate. Nevertheless, the estimated frequency change will be large enough to indicate that there  Chapter 3. Substructure Synthesis^  66  is a significant error in the mode. In this way, inaccurate natural frequencies can be identified in the results. But this method has two drawbacks: the error estimate depends on the selection of extra modes; and, if they are not unused modes left over from the original analysis, the extra modes have to be calculated anew. In the present section, an error estimation scheme is developed for the free-interface formulation in which a complete account of the neglected free-free modes is taken. This is possible because the residual flexibility can be tuned to a particular frequency w by replacing G with d(w), as was shown in Section 2.5. The resulting component displacement given by u (I)p + d(w)f  (3.85)  is an exact representation at frequency w when 0 < w < w c . Using the synthesis procedure for the first-order mass formulation in Section 3.5, the resulting free vibration equation of the system is,  {r(w) — ca 2 1} P =  0  (3.86)  where  r(w) = X + TBTA [A T-1i7B(w)A1 -1 A TTB r t (w) = -6-,-BB(w)  (3.87) (3.88)  Solving the characteristic equation  det [r(w) — w 2 .1] = 0  (3.89)  gives the exact natural frequencies of the coupled system in the range 0 < w < co c . Although (3.86) provides a condensed representation of the system, a complete account of the static and inertial effects of the neglected modes has been included by virtue of the dynamic residual flexibility.  Chapter 3. Substructure Synthesis^  67  Now consider a second problem in which the following eigenvalue equation is solved:  [r (65) - A,i1 A  ^  (3.90)  where 65 is an arbitrary frequency such that 0 < w < w c . It is of interest to investigate how the eigenvalues of (3.90), a i vary as a function of ci). It is shown in Appendix B that ,  the Rayleigh quotient  (3.91)  Pi Pi  is a continously non-increasing function of ar in the range 0 < (2) < w c . If 65 is equal to an exact natural frequency of the system (C) = wi), the Rayleigh quotient is stationary at  ai = If ai is moved to a value larger than wi , the Rayleigh quotient will stay the same or decrease; if (7) is moved to a value smaller than wi, it will stay the same or increase. As it does not matter which mode is being considered, the same result can be applied to all modes of the system. This leads to the following general conclusion: in a system where the modes are ordered such that w 1 < w 2  <  <  < ..., and where Ca is situated  between two exact modes of the system such that wi < iw <  < 4^j^1,2,...,i^ •^4^j=i+1,i+ 2...^  (3.92) (3.93)  In (3.92) lower bounds to the exact natural frequencies are obtained; in (3.93) upper bounds are obtained. The free-interface formulation with the first-order mass matrix is obtained by setting = 0 in (3.90). In this case, >  2.^j = 1,2, ...^  3  (3.94)  and the frequencies obtained are upper bounds to the exact values. Defining f/ i , (2 2, • • to be frequencies calculated from the free-interface formulation, (3.90) is now solved with  68  Chapter 3. Substructure Synthesis^  =^for i > 1; i.e., the following eigenvalue equation is solved: [r(O 1 ) — 3tid P, = 0^  (3.95)  Evaluating the eigenvalues results in a series of lower bounds and upper bounds similar to (3.92) and (3.93). For the i th mode, in particular, At <^  (3.96)  Thus, an absolute measure of the accuracy of the i th mode is obtained. The evaluation of (3.95) is useful for determining the accuracy of modes in the target frequency range when no other means of comparison are available. By using the dynamic residual flexibilities of the components, the cumulative effect of all neglected modes can be evaluated at a particular frequency without computing additional free-free modes. 3.10 Modal Properties of Combined Systems; the Inclusion Principle  To conclude this chapter, the convergence of CMS-derived frequencies is investigated. Particular attention is given to the improvement in convergence offered by the freeinterface method. Rayleigh [44] showed that if a constraint is applied to a dynamical system, the modified frequencies interlace the original frequencies in such a way that the former are greater than, or in exceptional cases equal to, the latter. In a discrete system, adding a constraint is often equivalent to removing a degree of freedom, and vice versa [1]. This leads to the inclusion principle, whereby the frequencies of a discrete system monotonically decrease towards the actual frequencies as the number of degrees of freedom increases. This result is important in situations where modal truncation is a factor. Meirovitch and Kwak [45] have investigated the applicability of the inclusion principle to substructured synthesis formulations. They prove the monotonic convergence of the frequencies provided that a  Chapter 3. Substructure Synthesis^  a  b  69  x = 21  Figure 3.1: Two-component continuous bar supplementary degree of freedom adds a single row and column to the system matrices, leaving the original portion of the matrix the same. This is generally true for system matrices resulting from classical free-free mode sets, and by examining (3.60) it can be verified that this property holds for the fixed-interface method. However, it does not hold for the free-interface formulations. While each additional free-free mode contributes a  F, the simultaneous reduction in the residual flexibility affects matrix elements throughout r. Consequently, the inclusion principle cannot be applied single row and column to  in its canonical form. It is therefore important to the discussion of residual flexibility formulations to investigate their convergence characteristics. Dowell [46] showed that if two substructures are joined at a single point, the frequencies of the combined system interlace the component frequencies. Thus, the general interlacing principle for a dynamic system also applies to a system of uncoupled components. Dowell used a classical free-free representation of the system components, a case for which the inclusion principle is known to hold. To investigate the influence of residual flexibility on the modal properties, consider the axial vibration of the two-component distributed system in Figure 3.1. Component a is a fixed-free bar with a frequency spectrum w a i = = 7r/2, 37r/2, 57r/2, ,  Chapter 3. Substructure Synthesis^  70  while component b is a free-free bar with spectrum wbi = a bj = 0, r, 21-, 3r, .... Two constraints exist between the components: one geometric, and one natural. In a classical free-interface component mode representation, only the geometric constraint is used explicitly, but if residual flexibility is included both must be used. The displacement functions of the two components are  wa =  E 0.i(x)ai 0.(x)f  ^  (3.97)  1=1  J  Wb  = E obi (x)bj — ti, b (x)f^ j=1  (3.98)  where O a i (X) and Obi (X) are the fixed-free eigenfunctions of component a and the free-free eigenfunctions of component b respectively; and where I and J are the number of modes selected for components a and b respectively. The functions Ti, a (x) and 1/;b(x) are the residual attachment modes associated with f:  0.i(x)Oai(1) L-s• at ^1=1+1^  (3.99)  ) (x^Oat(x)95.(1)  ^a  ^"  (3.100)  ^(71^at  E bb(x)^ -  A bi obi(x ) bi(1)  o  (3.101)  j=J+1  = bb(x) — E(1)17i(X)fkbi(1)^(3.102) Abj j  j=1^  where ik a (x) and/Pb(x) are the inertia-relief attachment modes associated with f . Note that by using the same force f in (3.97) and (3.98) and 'by the sign inversion in (3.98), inter-component equilibrium is at once satisfied. The potential and kinetic energies of the two components is given by 1 V =^Aai4 2 i=i  E  1^'1 — 3-11  dtP 2^21 dti„ 2 o dx^dx  2  + [f^dx^(--=) dx1 f 2  (3.103)  71  Chapter 3. Substructure Synthesis^ J 1 I^ a2^i) =^  +[  /Ap2 dx + 1  21^  ^a^t  24 .4 i^2 L's^2 i=i^J=1  P^(3.104)  where mass normalization of the eigenfunctions has been assumed. The geometric constraint is satisfied with the equation J  I^  R =^O ai (1)a i + l'ks a (l)f —^Obj(1)bj 114(1)f = 0^(3.105) i=1^j=1  The method of Lagrangian multipliers is used to derive the equations of motion. With the Lagrangian,  L=T —V —pR^  (3.106)  the following three equations are derived in conjunction with (3.105):  a i A a ia i — pO a i(/) = 0^  (3.107)  Abibi pcb bi (1) = 0^  (3.108)  Ei+df— ft (1'4a(1) + 1414 1 )) = 0^ (3.109) where  E = 10 1 ,tp^ !dx + f 21 14' dx 1^ ^2 7 \ =^ t ° (1)2a,\i1) + E c',b i2( ‘ ) 00  i=i+i^al^j=J+1^bi  G = =  f i ichp a \ 2^/21 (  dx JO^  ) dx +  dijIb 2  1^dx^  dx  t (gii( 1 ) + t qi( 1 ) 1=1+1 1\az^j=J+1 A b3  Assuming sinusoidal solutions of frequency cv for ai, ki and  f,  (3.107)-(3.109) are substi-  tuted into (3.105) giving the result 02 . (/)  1^ \--■ ^7)30)^ 0 (3.110)  A . a1 co 2 + 47g b • i^at^ 3  — 4) 2 = k— W 2 A/1  -  Chapter 3. Substructure Synthesis^  72  where  = =  G2  (3.111)  The right-hand side of (3.110) is the residual contribution from the neglected modes given by the second-order formulation. Setting this term to zero gives the equation resulting from the classical free-free component mode representation. Replacing the right-hand side of (3.110) with  -d  gives the equation for the first-order free-interface formulation,  in which the kinetic energy of the residual mode is neglected. Graphical solutions for the classical, first-order and second-order approximations are shown in Figures 3.2 and 3.3. In these figures, the left- and right-hand sides of (3.110) are plotted independently. The three curves corresponding to the three right-hand sides are labelled according to the residual approximation used. In Figure 3.2, the fundamental fixed-free mode of component a and the rigid body mode of component b are used. The fundamental free-free mode of b is added in Figure 3.3. The poles of the left-hand side of (3.110) occur at the component frequencies (.0, 1 = 0, wai = 7r/2, wb2 = 7r, and are indicated by vertical dashed lines. The intersections of the solid curves define solutions to (3.110) for each of the three formulations. These can be compared to the exact spectrum of the assembled bar ch./i = 7r/4, 37r/4, 57r/4, ..., indicated by vertical dotted lines. Several observations can be made about the nature of the approximated frequencies. First, for each right-hand side curve there is one and only one intersection point between the poles of the left-hand side. The poles of the left-hand side occur at the component frequencies, which are the same as the frequencies of the uncoupled system. Thus, frequencies of the combined system always interlace the frequencies of the uncoupled system such that al <^< A2 <^<^  (3.112)  where A i , A2, ... are the squares of the uncoupled system frequencies, ordered from lowest  73  Chapter 3. Substructure Synthesis ^  co bl  0 , oi  w1  0)2  1  2.0  e  a 1.0 w 0 co co 0  /Classical  0.0  C as .'.C'  \ 1st Order  -1.0  2 us co  —1 -2.0  0.00^0.25^0.50  ,  .^1^.^.^.^I  ^ ^ 1.00 1.25^1.50 0.75 w (-:. 1r sl  ^  Figure 3.2: Graphical solution for continuous bar, two mode approximation  74  Chapter 3. Substructure Synthesis^  W b1  6  ^  co,^Wei^02^03 b2^0)3  2.0  ,— ri d  ul 1.0  "6  co a)  Classical  :0 ° 0.0 lc c os  i  1 14 order  :6 1  .2 -1 .o 2as è co  -2.0  0.50^0.75^1.00^1.25^1.50^1.75^2.00 co (+ ir S-1  )  Figure 3.3: Graphical solution for continuous bar, three mode approximation  75  Chapter 3. Substructure Synthesis^  to highest, and  Cpl. , 6.)2, .  are the frequencies of the combined system. This provides an  upper bound for the combined system frequencies. Secondly, including residual effects gives one more mode than is obtained with the classical method. The additional mode appears in such a way that the classically-derived frequencies interlace the frequencies derived with residual effects: (2)^- (1)^- (0)^- (2)^- (1)^- (0) W1 < W1 < W1^W2 < U-7 2^W2^• • •  (3.113)  - (0) - (1) - (2)  where w i , w i , w i are frequencies calculated with the classical, first-order and secondorder residual methods respectively. Note that the basic interlacing principle (3.112) is unaffected by the vertical position of the right-hand side curves. Thirdly, from the inclusion principle it is known that solutions obtained with the classical CMS representation converge to the exact values as the number of component eigenfunctions is increased. Also, it can be observed that as more functions are added, the magnitude of the residual terms diminishes and the first- and second-order curves degenerate to the classical case. To show this, the right-hand side of (3.110) can be written, 1 K -w2f1^1 where  —  (3.114)  w2 B  co^( 4.02 )^co  Oti (i) ( (0 2  \  =J+1 b^63 ) 2-132^i=l+1 Aai^Aai ^(3.115) 02ai(1)^j^(gi(1) i=■+1^at^j= j+i  4/  -^  If c4.) 2 < A a i, i = I + 1, I + 2, ... and w 2 < Abj, j = J + 1, J + 2, ..., w 2 E is always less than O. This is verified by comparing each term in the numerator and denominator of (3.115). Consequently, the denominator of (3.114) is always positive in this frequency range.  76  Chapter 3. Substructure Synthesis^  As more free-interface modes are used in the analysis, the lowest numbered terms in the summation are removed; and since all terms are positive, G and w2B must both decrease in magnitude. Furthermore, for a fixed frequency w, w2 11111^= /-000 ) tai  0  CV 2 11111^= 0 J -400 Ab s  Therefore, by (3.115)  i^1 +1,1 +2,... j  = J +1,J +2,...  w 2B Jim^— 0  / ,J--000  G  and because Ern G = 0, Ern^G —^ 1 — W 2B  (3.119)  As a result, the right-hand side of (3.110) vanishes as more component modes are contributed to the modal representations. Evidence of the diminishing residual terms can be found by comparing the the ls t order and 2n d order curves in Figures 3.2 and 3.3. In this limiting process, the frequencies obtained with residual methods approach those of the classical case. The combination of the two limits gives the required convergence properties: lim J.,C 2) = lim Co•( 1) = lim  I,J.00  2  1,J-400  w. °) = lim (3.120)  More importantly, the residual solutions are seen to converge to the exact results ahead of the classical solutions. The residual terms adjust themselves as more component modes are added, thus preventing the frequencies from falling below the exact solutions. In the derivation of (3.110), the two-component system in Figure 3.1 was used only to supply the variables with numbers. The general form of (3.110) is in fact applicable to any two-component system subjected to a single constraint. By adding further summations  Chapter 3. Substructure Synthesis ^  77  to the left side, the equations can be used for any number of components. But when another constraint is imposed on the system, (3.112) is no longer applicable and the upper bound provided by this relation is lost. Systems with multiple constraints require a more sophisticated treatment than the one presented above. 3.11 Summary A number of specific CMS formulations which are suitable for substructured systems of an arbitrary geometrical complexity have been derived based on the component mode representations detailed in Chapter 2. Inter-component compatibility and equilibrium are expressed with the general relations stated in Section 3.2. The resulting formulations are computationally more efficient than the more general elimination methods of Section 3.3 and are therefore better suited for a finite element program. Of particular interest are the free- and fixed-interface formulations derived from the MacNeal-Rubin and Craig-Bampton mode sets. Two free-interface formulations were derived, in which first- and second-order mass matrices are employed. Both formulations give matrix equations in terms of the free-free modal coordinates only. On the other hand, the fixed-interface equations are in terms of interface displacements and modal coordinates. As a result, the free-interface formulations can be expected to provide greater condensation when applied to arbitrarily complex systems. New contributions made in this chapter are the following: the generalized formulations for the two free-interface methods and the modal force method using the general connectivity matrix; the method for calculating lower bounds to frequencies derived from the free-interface method using the dynamic residual flexibility concept; and the demonstration that frequencies calculated with residual flexibility will converge to the exact  Chapter 3. Substructure Synthesis^  78  results as component modes are added (the inclusion principle), and that residual flexibility provides an accelerated convergence for the system modes.  Chapter 4  Modal Analysis of Three Finite Element Models  4.1 Introduction In this chapter, detailed results are presented for the following finite element models: • A two-dimensional model of a container ship; • A three dimensional model of a telescope focus unit (TFU); • A three dimensional model of a telescope focus unit and its support structure. The results are presented with a view to comparing the following: • The accuracy of the natural frequencies and mode shapes obtained with, — the fixed-interface method using the Craig-Bampton mode set; — the free-interface method using the MacNeal-Rubin mode set with both firstand second-order mass terms; — direct finite element analysis; — Guyan reduction. • The computational time required for each method as a fuitction of the eigensolution tolerance. • The effect of modal truncation on the CMS methods by presenting results obtained with various cutoff frequencies. 79  Chapter 4. Modal Analysis of Three Finite Element Models^  80  The accuracy of CMS-derived frequencies and mode shapes is determined by comparing these results with direct finite element analyses of equivalent full-size models. Because both the substructured and full-sized models are derived from the same finite element mesh, a CMS analysis can do no better than reproduce the results of a direct analysis. The CMS frequency and mode shape results can thus be presented as a percentage difference from the direct results. The percentage difference in the frequency is,  %Af, = f.s x 100% -  fi  ith  natural (4. 1)  A percentage difference in the it h mode shape is given by, %Au, =  11%112  112 x 100%  (4.2)  where 11%11 2 is the Euclidian norm of the vector u1. The natural frequency and mode shape results were computed using the Vibration and Strength Analysis Program (VAST), Version 06 [47]. Originally, this program only had the capability for direct analysis of finite element models and Guyan reduction of substructured models. Modifications were made to the program to allow CMS analysis using the fixed- or free-interface method. Details of the CMS implementation in VAST can be found in Appendix C. 4.2 Analysis of a Container Ship  A two-dimensional model of a container ship is depicted in Figure 4.1. This type of model is useful for predicting the vertical modes of vibration of the actual ship. Wave motion and propeller-induced pressure forces can excite resonances in the ship's structure and so accurate knowledge of the natural modes is important in the design stage. The model is composed of 8-node membrane elements, and 3-node bar elements: the membranes  Chapter 4. Modal Analysis of Three Finite Element Models^  (c  81  )  (a)^(b)  (d)  ^  (e)  Figure 4.1: Two-dimensional container ship model: a) stern; b) aft-body; c) deckhouse; d) mid-body; e) fore-body simulate the sides of the hull and interior walls; the bar stiffeners simulate the hull bottom, decks and bulkheads. The substructuring scheme shown in Figure 4.1 is used for the CMS analyses. Relevant data regarding the model and its constituent components can be found in Table 4.1.  Components Stern (a) Aft body (b) Deckhouse (c) Mid body (d) Fore body (e) Complete Model -  -  -  # nodes 54 65 121 121 121 447  # d.o.f. 162 195 363 363 363 1341  # membranes 13 16 32 32 32 125  # bars 10 16 32 16 24 106  Table 4.1: Description of the container ship model  Chapter 4. Modal Analysis of Three Finite Element Models ^  82  4.2.1 Natural Frequency Results  In Table 4.2 are found the free-interface CMS natural frequency results for the ship model. Six cases are analyzed: first- and second-order approximations are employed with cutoff frequencies on the free-free modes of 25, 30 and 35Hz. The CMS frequencies are compared with the results of the direct analysis for the first 37 elastic modes. It should be noted that the model also has three rigid body modes that are omitted from the table. At the bottom of the table are the total number of degrees of freedom in the equations of motion resulting from each formulation. For each of the three frequency cutoffs, the second-order mass offers a distinct improvement over the first-order mass. This does not come as a surprise since in Chapter 2 it was demonstrated that the MacNeal-Rubin mode set employing the first-order approximation is only accurate when the square of the target frequency is negligible in comparison to the square of the cutoff frequency (see Equation (2.24)). In other words, the second-order terms are expected to have a significant effect when this condition is not met. In view of the cutoff frequencies that have been used here, this condition clearly has not been met for the target frequency range 0-30Hz. Satisfying this condition requires calculating a much larger number of component modes, something which should be avoided. Including the second-order mass terms improves the inertial representation of the neglected modes, which partially compensates for low cutoff frequencies. Another important phenomenon is that as the cutoff frequency increases, the CMS frequencies converge to the direct-analysis results and that, in general, the low frequencies converge before the high. However, there are some exceptions to this latter rule. For example, mode 20 converges faster than either mode 15 or 16. But it should be observed that each mode has a different distribution of strain and kinetic energy and so they cannot be expected to converge at the same rate.  Chapter 4. Modal Analysis of Three Finite Element Models ^  Direct FEM Analysis Mode Frequency (Hz)  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.93 10.2 12.3 13.1 13.7 14.2 15.0 16.4 17.3 18.9 19.2 20.0 20.2 21.4 21.7 22.9 23.7 24.0 24.9 25.3 25.7 26.1 26.9 27.8 28.1 28.6 29.3 29.7 30.0 30.6 # d.o.f.^1341  % Error, Free-Interface CMS Analysis 25 Hz. Cut-off 30 Hz. Cut-off 35 Hz. Cut-off 2nd 1st 1st 2nd 1st 2nd  0.00 0.02 0.07 0.18 0.19 0.57 0.26 0.39 0.51 0.72 2.07 2.61 6.52 2.22 3:35 2.58 1.64 3.85 2.92 5.98 3.57 6.82 9.45 6.06 15.5 20.0 25.4 26.6 32.2 35.9 47.4 74.3 36  0.00 0.00 0.01 0.06 0.02 0.33 0.04 0.09 0.07 0.14 0.47 1.01 2.49 0.38 0.54 1.00 0.48 2.16 0.57 3.21 1.44 3.26 1.81 1.73 5.20 4.48 7.37 8.77 14.6 14.8 20.9 40.8 36  0.00 0.02 0.05 0.09 0.12 0.28 0.18 0.30 0.38 0.44 1.17 1.71 2.82 1.19 1.79 1.89 0.77 2.56 1.63 1.86 1.10 2.00 1.35 1.88 2.77 3.43 3.57 2.98 3.57 5.42 2.94 2.56 4.94 8.16 14.0 15.2 17.7 50  0.00 0.00 0.01 0.02 0.01 0.10 0.02 0.07 0.04 0.08 0.25 0.90 0.79 0.24 0.22 0.84 0.14 1.65 0.23 0.56 0.42 0.35 0.41 0.59 0.87 1.21 0.93 1.28 0.36 1.54 1.55 0.89 1.53 1.67 4.00 4.26 4.60 50  0.00 0.01 0.04 0.07 0.11 0.23 0.11 0.23 0.25 0.35 0.75 1.07 1.71 0.81 0.76 1.13 0.38 1.19 1.08 0.91 0.85 1.45 0.82 1.36 1.24 1.89 2.45 1.01 1.11 2.36 1.37 1.18 2.17 1.14 1.44 2.59 2.40 62  0.00 0.00 0.01 0.02 0.01 0.08 0.02 0.05 0.03 0.07 0.17 0.49 0.66 0.14 0.10 0.39 0.09 0.59 0.10 0.48 0.24 0.25 0.13 0.36 0.28 0.52 0.52 0.32 0.20 0.52 0.49 0.19 0.58 0.42 0.30 0.66 0.92 62  Table 4.2: Natural frequency results for the container ship, free-interface method  83  Chapter 4. Modal Analysis of Three Finite Element Models ^  84  Table 4.2 demonstrates the efficiency of the free-interface method's condensation, particularly when the second-order approximation is used. In the last column (35Hz cutoff, 2nd order mass), the first 37 non-zero frequencies are computed to within 1% error. The effectiveness of the condensation is admirable—the equations of motion of the ship have been reduced in order by a factor of 20 over those of the direct analysis, while leaving the low-frequency spectrum nearly perfectly intact. The results of the fixed-interface method (Table 4.3) are similar in that convergence increases with the cutoff frequency and the low frequencies tend to converge before the high. The cutoff frequencies here refer to fixed-interface component frequencies, which are always higher than their free-free counterparts. Thus, the 35Hz cutoff in the fixedinterface method allocates 45 dynamic modes to the analysis while in the free-interface method 62 are needed. Table 4.4 shows the frequencies of the component modes used in both cases. Note that the free-free modes include the three rigid body modes for each component. The mid-body and the fore-body contribute the majority of the modes, as these are geometrically the largest and therefore the most flexible of the components. Figure 4.2 combines the natural frequency results for the free- and fixed-interface methods and displays their convergence characteristics as a function of the number of dynamic component modes. This is in contrast with Tables 4.2 and 4.3 where the accuracy of the frequencies is recorded as a function of the cutoff frequency. For the curves corresponding to the free-interface formulations, the number of dynamic component modes refers to non-rigid body modes only. Rigid body and static modes are essential for good accuracy in all CMS analyses, while elastic dynamic modes are non-essential because they can be added in varying numbers as the accuracy requires. The graph therefore shows the effect of different types of dynamic modes on the overall accuracy and convergence rate of the system frequencies. In terms of overall accuracy, free-free modes are more effective provided that their inertia contribution is included; if not, fixed-interface modes  Chapter 4. Modal Analysis of Three Finite Element Models ^  Direct FEM Analysis Mode Frequency (Hz) 4 0.881 5 2.14 6 3.52 4.55 7 8 5.58 9 5.91 10 8.01 11 8.93 12 10.2 13 12.3 14 13.1 15 13.7 16 14.2 17 15.0 18 16.4 19 17.3 20 18.9 21 19.2 20.0 22 23 20.2 21.4 24 25 21.7 26 22.9 27 23.7 28 24.0 29 24.9 30 25.3 31 25.7 32 26.1 33 26.9 34 27.8 28.1 35 36 28.6 29.3 37 38 29.7 39 30.0 40 30.6 # d.o.f.  1341  % Guyan Reduction 14.4 36.5 48.8 106 114 210 132 166 192 194 178 181 195 218 215 202 193 198 209 230 232 238 229 219 247 241 246 243 244 257 257 276 288 295 290 293 297 102  Error, Fixed-,Interface CMS Analysis 20 Hz. 30 Hz. 25 Hz. 35 Hz. Cut-off Cut-off Cut-off Cut-off 0.00 0.00 0.00 0.00 0.02 0.02 0.01 0.01 0.03 0.02 0.01 0.01 0.18 0.08 0.07 0.06 0.10 0.03 0.02 0.00 0.15 0.07 0.06 0.05 0.08 0.02 0.02 0.01 0.92 0.34 0.26 0.20 0.43 0.19 0.16 0.08 0.74 0.55 0.39 0.34 1.86 0.42 0.35 0.21 1.26 0.84 0.71 0.67 1.64 1.10 0.96 0.79 0.92 0.51 0.29 0.19 5.20 0.62 0.50 0.23 .5.18 0.49 0.85 0.73 0.60 0.43 0.23 7.36 10.3 3.96 2.18 1.29 0.87 0.65 14.5 1.53 15.3 2.56 1.31 0.81 17.8 3.04 1.60 0.72 7.35 1.23 0.73 20.7 42.9 3.90 2.28 1.09 43.8 4.73 0.77 0.39 47.1 9.68 2.71 1.25 53.7 13.9 1.24 0.81 52.8 14.3 2.53 1.49 54.7 19.2 3.44 1.35 58.2 26.6 4.71 2.15 57.7 26.3 5.68 1.46 88.0 23.9 2.81 1.10 92.6 28.7 10.4 1.43 102 35.2 13.2 1.85 99.8 37.5 16.6 2.25 109 43.5 15.8 2.24 112 42.3 19.4 1.66 114 43.6 22.3 1.15 120  127  133  147  40 Hz. Cut-off 0.00 0.01 0.01 0.06 -0.01 0.05 0.00 0.17 0.05 0.21 0.15 0.55 0.54 0.11 0.16 0.38 0.20 1.04 0.55 0.68 0.40 0.47 0.63 0.19 0.94 0.37 0.47 0.85 0.58 0.90 0.58 0.76 0.95 0.89 1.86 0.92 0.42 160  Table 4.3: Natural frequency results for the container ship, fixed-interface method  85  Chapter 4. Modal Analysis of Three Finite Element Models ^  Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  Stern Free Fixed  Component Frequencies (Hz) Deckhouse Mid-body Aft-body Free Fixed Free Fized Free Fized  0 0 0 18.6 26.5 28.7 32.8  0 0 0 22.6 25.0 28.9 30.3  6.41 18.2 18.9 33.7 38.8  24.4 37.7  0 0 0 19.9 21.0 27.4 29.4 33.2  6.48 16.1 16.9 25.4 26.9 32.6 35.7 39.1  0 0 0 4.57 9.88 11.4 15.5 19:0 20.6 23.4 24.0 27.2 27.7 28.1 28.8 29.3 31.9 33.6 33.8  4.11 8.31 12.0 12.8 18.3 21.5 23.0 24.9 28.1 28.8 31.1 31.6 31.8 33.0 33.5 34.4 35.5 38.2 38.2 39.6  Fore-body Free Fixed 0 0 0 4.13 9.24 10.6 14.6 18.5 19.3 22.5 22.7 25.7 26.4 27.7 29.5 31.4 31.8 32.8 33.1 34.3 35.0  Table 4.4: Component frequencies of the container ship  0.93 3.83 5.92 8.80 13.4 16.2 18.8 21.3 24.1 24.9 27.2 28.4 30.0 31.2 31.6 32.4 34.3 34.3 35.2 35.8 36.8 38.4 39.2  86  Chapter 4. Modal Analysis of Three Finite Element Models ^  35^^^ Free, l st order mass  ^ Free, 2 nd order mass 30^O^ Fixed 0 25 V  vi 20  E 75 15  z 10  10^20^30^40^50 No. of dynamic component modes  Figure 4.2: Combined natural frequency results for the container ship  87  Chapter 4. Modal Analysis of Three Finite Element Models ^  88  are more effective. The same conclusion can be drawn for the convergence rate, which is indicated by the slope of the curves. Independent of accuracy and convergence rate considerations, the fixed-interface formulation is not as effective at condensing the equations of motion as the free-interface method—more degrees of freedom have to be retained to get results of a similar accuracy. As was shown in Chapter 3, the free-interface method yields more compact equations because in the formulation, more independent constraint equations can be specified with the generalized coordinates. 4.2.2 Mode Shape Results  The accuracy of the CMS mode shapes is not only a function of the cutoff frequency but it also shows a strong dependency on the eigensolution tolerance. Eigensolutions are determined by an iterative procedure in which a tolerance factor is required to terminate the solution. The VAST eigensolution routine EIGEN1 uses an algorithm based on the inverse power method with shifting. The condition, A (k) _  <  Vie  to/  (4.3)  is used to terminate in the kth iteration. While variations in the tolerance may have only a minor effect on the frequencies, a much more pronounced effect is visible in the mode shapes. This observation is verified by noting that first-order variations in the eigenvectors produce, through Rayleigh's quotient, second-order variations in the eigenvalues [48]. Thus an error equivalent to tol in the eigenvalue corresponds to an error on the order of tot} in the mode shape. This is doubly true in a CMS analysis where accurate reconstruction of the system mode shapes relies on accurate eigenvectors at both the component and system level. The mode shape results for the ship model (Tables 4.5 and 4.6) were calculated with tol = 10-6  Chapter 4. Modal Analysis of Three Finite Element Models ^  Direct FEM Analysis Mode Frequency (Hz)  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # d.o.f.  0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.93 10.2 12.3 13.1 13.7 14.2 15.0 16.4 17.3 18.9 1341  89  % Error, Free-Interface CMS Analysis 25 Hz. Cut-off 30 Hz. Cut-off 35 Hz. Cut-off 1st 1st 2nd 2nd 1st 2nd  0.02 0.07 0.20 0.62 1.58 1.78 1.23 1.99 2.74 5.68 12.3 29.9 23.1 24.6 24.3 24.7 37.1 36  0.02 0.04 0.11 0.52 2.45 2.30 0.58 1.06 1.04 3.62 7.70 8.60 7.90 8.65 8.21 15.3 20.6 36  0.02 0.06 0.16 0.24 1.18 1.13 1.16 1.87 2.37 3.34 15.7 22.7 21.6 19.4 16.7 17.4 22.0 50  0.02 0.04 0.11 0.28 0.54 0.54 0.31 0.93 0.80 2.24 6.88 7.67 3.77 6.73 5.94 13.2 8.01 50  0.02 0.05 0.14 0.32 1.24 1.18 0.81 1.33 1.15 2.35 5.40 7.50 5.39 9.78 5.89 11.3 14.9 62  0.02 0.04 0.11 0.24 1.63 1.42 0.27 0.74 0.60 2.13 3.84 5.84 4.65 5.00 2.88 6.08 4.32 62  Table 4.5: Mode shape results for the container ship, free-interface method and are compared with direct-analysis mode shapes of the same tolerance so as to best isolate the effect of modal truncation. Although acceptable results for frequencies can be obtained with a larger tolerance (requiring fewer iterations and therefore less time), the quality of the mode shapes declines. The results in Tables 4.5 and 4.6 show that convergence is much slower for mode shapes than for frequencies and that larger absolute error can be expected in mode shapes than in frequencies at a particular cutoff value. Indeed, if a e% error is found in the frequency, the error in the mode shape might be expected to be,  ^_  J  2^1  1^1.0% 1 + TZ) AU% (  (4.4)  By this formula a 0.10% error in frequency, whatever the source of the error may be, would correspond to a 4.5% error in the mode shape. This order of magnitude is typical of the differences between frequency and mode shape errors found in the tables.  Chapter 4. Modal Analysis of Three Finite Element Models ^  Direct FEM Analysis Mode Frequency (Hz)  % Error, Fixed-Interface CMS Analysis Guyan 20 Hz. 25 Hz. 30 Hz. 35 Hz. Reduction Cut-off Cut-off Cut-off Cut-off  40 Hz. Cut-off  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.93 10.2 12.3 13.1 13.7 14.2 15.0 16.4 17.3 18.9  15.5 42.4 103 108 68.0 95.2 122 114 124 101 112 113 103 130 122 114 134  0.03 0.14 0.37 0.98 1.03 1.02 1.45 5.58 4.18 12.4 23.6 13.8 9.92 11.2 40.4 49.1 59.4  0.02 0.10 0.24 0.51 0.94 0.75 0.75 2.29 2.53 7.27 6.35 12.3 11.8 6.20 8.66 11.2 18.1  0.02 0.07 0.18 0.49 0.88 0.69 0.62 1.88 2.19 6.05 6.91 8.25 7.90 4.82 6.95 9.01 11.0  0.02 0.06 0.17 0.45 1.00 0.80 0.63 1.53 1.58 4.74 5.07 7.76 7.04 3.82 5.42 8.24 4.31  0.02 0.06 0.16 0.43 0.99 0.80 0.40 1.25 1.13 3.72 5.56 6.07 4.51 3.53 5.32 6.89 3.75  # d.o.f.  1341  102  120  127  133  147  160  Table 4.6: Mode shape results for the container ship, fixed-interface method  90  Chapter 4. Modal Analysis of Three Finite Element Models^  CPU Times (s) for 40 Modes 10-2 10' Tolerances 664 2448 Direct Analysis 317 401 Guyan Reduction Fixed-Interface CMS 793 1356 Free-Interface CMS 298 462  91  10 -6 5298 484 2379 1028  Table 4.7: CPU times for modal analysis of the container ship Tables 4.5 and 4.6 show that comparable results are obtained with the fixed-interface and the free-interface methods, in spite of the fact that the latter uses significantly fewer independent variables in the cases presented. This reiterates the free-interface method's superiority in condensing the equations of motion whilst preserving the low-frequency spectrum. 4.2.3 Performance  Table 4.7 shows the CPU time expenditure for the four methods at three different tolerances. For the CMS analyses, the tolerance factor applies to both the component and system analyses. Two general observations can be made. First, as the tolerance is reduced, the relative efficiency of CMS increases. Secondly, the free-interface method maintains a significant performance edge over the fixed-interface method for all three tolerances. For this latter comparison, it is only fair to use CMS runs which give a similar degree of accuracy. The two used in Table 4.7 are, for the free- and fixed-interface methods respectively, the 35Hz cutoff with second-order mass (see the last column of Table 4.2) and the 40Hz cutoff (see last column of Table 4.3). The advantage enjoyed by the free-interface method is partly explained by its superior condensation and partly by differences in the program implementation. Because the two methods had to be fit into  Chapter 4. Modal Analysis of Three Finite Element Models ^  92  an existing finite element program, certain redundancies in the algorithm were unavoidable and tended to work against the fixed-interface method. Finally, of the four methods, Guyan reduction is the fastest but as was seen in Table 4.3, its accuracy is very poor compared to the other methods. If only natural frequency results are of interest, the larger tolerances are adequate. When to/ = 10 -2 , the free-interface method is somewhat faster than a direct analysis and the fixed-interface method somewhat slower. But the difference either way is not very significant and unless a substructured treatment is needed for some other reason, no special advantage is gained from using CMS at this level of accuracy. Indeed, this represents the lower limit of CMS's performance advantage. Models significantly smaller in order than the container ship would be more efficiently analyzed with the direct method; because for small-order models, the advantages gained by a CMS condensation are more than cancelled by the computational overhead required for two levels of analysis. On the other hand, if accurate mode shapes are wanted, or if closely grouped frequencies are to be resolved, a tolerance of about 10 -6 is usually needed. At this level of accuracy, CMS has a clear performance advantage. 4.3 Analysis of a Telescope Focus Unit  The telescope focus unit (TFU) is composed of the three concentric cylinders shown separately in Figure 4.3. The finite element modelling divides the TFU into five components, the function of each of which is described in the following: Inner Tube Supports a mirror which focusses light reflected by the main mirror. Outer Tube Surrounds and supports the inner tube by means of a screw assembly and  four guide rails.  Chapter 4. Modal Analysis of Three Finite Element Models^  Components Inner Tube (1) Outer Tube (2) Screw Assembly (3) Chopping Mechanism (. 4 ) Support Tube (5) Complete Model  # nodes  # d.o.f.  # plates  # bricks  # bars  76 92 84 120 108 428  456 552 504 360 648 2568  20 28 16 0 28 92  0 0 4 18 0 22  8 0 0 0 0 8  93  Table 4.8: Description of the TFU model Screw Assembly Allows the inner tube to be advanced or retracted for focussing. Chopping Mechanism Appended to the inner tube, this creates rapid oscillations in the lens position necessary for superimposing infrared images of the object under observation and its background. Support Tube Supports the outer tube along a flange with adjustable positioning screws. The model is composed of plate (8-node thick/thin shell) elements, brick (20-node solid) elements and 2-node bar elements. The exact composition of each component is listed in Table 4.8. Because vibration of the TFU inner components could degrade the telescope's image quality, the dynamic response and the existence of resonances is of great importance to the design. The source of excitation is the chopping mechanism, which exerts a squarewave periodic force and can therefore excite a wide range of vibrational modes. In this section, CMS is used to determine the low-frequency modes of the TFU with eight locations on the support tube fixed. These eight locations are where the support struts would normally be attached in the complete telescope (see Section 4.4).  Chapter 4. Modal Analysis of Three Finite Element Models^  (a)  (b)  94  (c)  Figure 4.3: Components of the TFU model: a) inner tube With screw assembly attached to top, and chopping mechanism attached to bottom; b) outer tube; c) support tube with external constraints.  Chapter 4. Modal Analysis of Three Finite Element Models ^  95  4.3.1 Natural Frequency Results  The same general trends can be seen in Tables 4.9 and 4.10 as were found in the frequency results of the ship model. Clearly visible are the systematic convergence as cutoff frequency is increased (except in modes 1-3); the convergence of the low modes before the high; the greater condensation provided by the free-interface method over the fixedinterface method; and, in the former, the improved accuracy with the second-order mass approximation. However, one feature not evident in the ship results but more noticeable here is the apparent slow convergence of the fixed-interface method. Between Guyan reduction— which is essentially a zero-frequency cutoff—and 4000Hz cutoff, the improvement in the accuracy of natural frequencies is noticeably less than what the free-interface method achieves over a smaller range of cutoff frequencies. One might be tempted to conclude that in this case the free-interface method enjoys a speedier convergence, but this is largely an illusion. Comparing the fixed- and free-interface modes of the telescope components (see Table 4.11) reveals that the unconstrained components have either the same or a higher modal density than their constrained counterparts. Thus, if an incremental increase is made to the cutoff frequency, more free-interface modes have to be added than fixedinterface modes. Because of this difference, the fixed-interface method may require a higher cutoff frequency to get comparable results, but it may not mean that it requires more dynamic modes. Indeed, of the 63 free-free modes used in the last two columns of Table 4.9, 24 are rigid-body modes which still leaves 39 elastic modes in comparison to the 29 used by the fixed-interface method at 4000Hz cutoff. This point is reflected in Figure 4.4 where the overall accuracy of the frequencies is plotted as a function of the number of dynamic component modes. Here, as in the ship model, the fixed-interface curve falls between the two free-interface curves. An interesting point is that the convergence rates  Chapter 4. Modal Analysis of Three Finite Element Models  Direct FEM Analysis Frequency Mode (Hz)  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 # d.o.f.  34.7 72.8 72.8 185.8 432 433 655 658 1002 1059 1067 1231 1232 1273 1347 1444 1447 1486 1576 1693 1913 1955 2210 2289 2321 2562 2563 2596 2699 2702 2911 2941 2979 3032 3118 3264 3315 3323 3359 3533 2568  ^  % Error, Free-Interface CMS Analysis 2500 Hz. 2000 Hz. 3000 Hz. 3500 Hz. 1st 1st 2nd 2nd 1st 2nd 1st 2nd  -0.04 -0.05 -0.02 0.07 0.17 0.17 0.26 0.26 0.34 1.03 1.08 1.55 1.54 3.48 0.04 0.16 2.73 5.86 2.13 13.0 1.31 0.34 26.4 26.5 57.4 69.8 72.7 75.9 69.5 81.7 99.1 175 179 206 203 214 239 247 280 261  -0.04 -0.05 -0.02 0.02 0.10 0.10 0.15 0.15 0.22 0.21 0.22 1.40 1.39 0.23 0.00 0.01 0.20 0.00 0.00 12.1 0.07 0.31 6.55 4.53 48.2 40.6 51.0 57.8 53.8 64.5 52.7 62.8 80.9 97.5 97.0 101 107 115 145 152  -0.04 -0.05 -0.02 0.05 0.08 0.17 0.13 0.26 0.34 0.92 1.08 0.27 1.54 3.40 0.04 0.16 2.73 5.86 2.12 1.88 0.79 0.34 0.02 15.7 17.5 11.8 13.0 40.7 36.8 61.0 52.4 55.2 64.8 91.8 93.0 148 150 179 181 191  -0.04 -0.06 -0.02 0.00 0.01 0.10 0.03 0.15 0.22 0.11 0.22 0.13 1.39 0.19 0.00 0.01 0.21 0.00 0.00 1.20 0.06 0.31 0.00 1.88 3.10 5.81 6.58 32.5 33.5 34.9 26.5 31.7 37.5 37.0 42.6 65.1 69.4 70.8 78.3 85.5  42  42  47  47  -0.04 -0.06 -0.03 0.05 0.03 0.03 0.13 0.14 0.34 0.23 0.23 0.27 0.28 3.27 0.02 0.14 2.73 5.85 2.11 1.88 0.42 0.34 0.02 3.48 3.56 0.03 0.69 0.60 1.64 1.65 1.08 7.80 13.8 16.6 17.2 13.1 31.4 33.5 45.6 54.6 56  -0.04 -0.06 -0.03 0.00 0.01 0.01 0.03 0.04 0.22 0.10 0.09 0.14 0.13 0.17 0.00 0.01 0.20 0.00 0.00 1.20 0.02 0.31 0.00 0.56 0.60 0.02 0.48 0.12 1.16 1.14 1.04 6.61 5.64 13.5 12.5 10.4 9.97 16.3 20.9 17.6 56  -0.04 -0.06 -0.03 0.05 0.03 0.03 0.13 0.14 0.34 0.23 0.23 0.28 0.26 0.68 0.02 0.05 0.16 0.00 0.00 1.79 0.30 0.12 0.01 3.48 3.54 0.03 0.24 0.31 1.64 1.64 1.08 2.77 1.82 7.77 5.03 3.01 7.23 9.94 9.30 5.71 63  Table 4.9: Natural frequency results for the TFU, free-interface method  -0.04 -0.06 -0.03 0.00 0.01 0.01 0.03 0.03 0.22 0.10 0.09 0.14 0.12 0.06 0.00 0.01 0.03 0.00 0.00 1.19 0.00 0.10 0.00 0.56 0.60 0.02 0.05 0.03 1.17 1.13 1.04 1.49 1.82 3.43 4.72 2.65 3.74 6.50 5.70 1.99 63  96  Chapter 4. Modal Analysis of Three Finite Element Models^  Direct FEM Analysis Mode Frequency (Hz)  % Error, Fixed-Interface CMS Analysis Guyan 2000 Hz. 3000 Hz. 4000 Hz. Reduction Cut-off Cut-off Cut-off  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  34.7 72.8 72.8 185.8 432 433 655 658 1002 1059 1067 1231 1232 1273 1347 1444 1447 1486 1576 1693 1913 1955 2210 2289 2321 2562 2563 2596 2699 2702 2911 2941 2979 3032 3118 3264 3315 3323 3359 3533  0.04 0.89 0.91 0.32 7.33 7.42 6.14 6.12 10.9 5.71 5.88 5.75 25.9 53.9 60.3 49.8 69.6 67.6 62.6 72.0 54.1 52.0 48.5 44.2 50.2 43.2 44.9 58.2 54.3 63.7 52.0 52.8 68.3 65.6 67.6 69.2 77.5 86.3 88.1 87.2  -0.04 -0.06 -0.01 0.32 0.60 2.00 1.20 2.88 0.41 4.10 4.18 1.93 5.51 5.97 7.23 2.81 7.05 6.14 23.4 15.7 27.5 27.0 16.0 27.3 41.3 28.1 28.8 34.3 36.0 37.2 36.1 35.9 35.5 35.4 41.3 53.6 52.3 56.5 65.0 62.8  -0.04 -0.07 -0.01 0.00 0.01 0.00 0.27 0.53 0.41 0.62 3.52 0.27 1.10 2.08 0.09 0.07 2.62 4.21 0.11 0.49 2.19 13.1 9.51 7.24 10.1 0.39 7.47 12.2 9.00 21.4 12.8 18.6 23.3 22.1 23.6 18.6 20.6 29.7 30.2 35.6  # d.o.f.  2568  360  369  380  -0.04 -0.07 -0.01 0.00 -0.03 -0.03 0.07 0.07 0.41 0.27 0.28 0.22 0.22 2.06 0.09 0.05 0.18 -0.11 0.00 0.49 0.10 0.01 0.01 5.24 5.18 0.04 1.32 4.79 1.87 2.02 1.13 3.34 8.34 7.73 5.29 5.81 6.72 6.82 8.69 7.97 389  Table 4.10: Natural frequency results for the TFU, fixed-interface method  97  Chapter 4. Modal Analysis of Three Finite Element Models^  Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  Inner tube Free Fixed 0 0 0 0 0 0 968 968 1650 1859 2439 2439 2562 2654 3035 3343  1233 1233 2093 2411 2728 2728 3201 3245 3839  Component Frequencies (Hz) Outer tube Screw Assem. Chop Mech. Free Fixed Free Fixed Free Fixed 0 0 0 0 0 0 1399 1399 1675 1723 1811 1811 1835 1948 2362 2459 2731 2859 2995 2995 3051 3089 3298  2764 3178 3178 3759  0 0 0 0 0 0 1706 1706  0 0 0 0 0 0  490 490 862 2074 2955 2955  Table 4.11: Component frequencies of the TFU  Support tube Free Fixed 1338 1428 1482 1574 2203 2557 2560 2919 3031 3246  1477 1503 1594 1600 2230 2603 2995 3061 3075 3481  98  Chapter 4. Modal Analysis of Three Finite Element Models^  25  —o-- Free, l st order mass A^ Free, 2 nd order mass 0^ Fixed  cl 20 V a.. a  a)Cl)  18 15  E  *6  0  z  10  5^10^15^20^25 No. of dynamic component modes  Figure 4.4: Combined natural frequency results for the TFU model  99  Chapter 4. Modal Analysis of Three Finite Element Models^ 100  for the three cases are about the same. That is, regardless of the overall accuracies, the effect of adding one additional dynamic mode is the same on average for each formulation. Another interesting feature is the appearance of negative frequency errors, particularly in modes 1-3. It has been argued in Chapter 2 that as long as inter-component compatibility is maintained, the CMS results should converge from above as more component modes are added. That is, modal truncation being the only source of error, it is theoretically impossible for CMS-derived frequencies to be less than the exact values. However, in practice three other possible sources of error exist: errors derived from the tolerance factor in the eigensolver, which have already been discussed in Section 4.2.2; numerical loss of precision, which often happens, for example, when subtracting numbers of similar magnitude or when inverting a poorly condition matrix; and the influence of non-conforming elements in the structural model. Since both methods show the same negative error in modes 1 and 2, it is unlikely that the tolerance factor or numerical roundoff could be responsible. Instead, this shows the effect of using incompatible elements in the mesh, in particular, the thick/thin shell element whose rotational coordinates are incompatible when two elements meet in perpendicular planes. With incompatibilities in the component modes, the upper bound on the CMS frequencies is lost and the possibility of a negative frequency error arises. Such negative errors are most likely to occur in the lowest frequency modes, as these are the first to converge, and in modes with significant displacement in the incompatible elements. 4.3.2 Mode Shape Results The convergence characteristics of the mode shapes (see Tables 4.12 and 4.13) are in many ways similar to those of the ship. But now, because of the presence of repeated frequencies, there is an added complication: mode shapes cannot be defined uniquely for repeated frequencies and so it makes little sense to compare them with Equation (4.2).  Chapter 4. Modal Analysis of Three Finite Element Models ^ 101  Direct FEM Analysis Frequency Mode (Hz)  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # d. o .f.  34.7 72.8 72.8 185.8 432 433 655 658 1002 1059 1067 1231 1232 1273 1347 1444 1447 1486 1576 1693 2568  % Error, Free-Interface CMS Analysis 2000 Hz. 2500 Hz. 3000 Hz. 3500 Hz. 1st 2nd 1st 2nd 1st 2nd 1st 2nd  0.05 0.08 4.58 4.59 0.72 0.72 1.14 2.49 2.39 5.80 5.89 5.81 1.57 14.6 121 98.6 130 15.8 42  0.05 0.03 0.25 0.25 1.90 1.93 0.80 3.15 3.32 7.88 8.15 3.52 1.27 7.47 6.08 1.52 0.52 15.6 42  0.05 0.05 0.05 0.01 2.17 2.19 2.18 2.21 2.21 2.37 2.36 2.49 1.16 0.79 2.29 1.05 2.51 2.70 14.6 13.4 15.6 14.5 6.22 2.33 2.66 0.33 14.5 3.76 121 3.37 98.5 1.60 130 .0.55 5.01 4.36 47 47  0.05 0.05 0.12 0.12 1.79 1.82 1.19 2.26 2.17 13.9 14.1 6.01 2.49 13.6 122 98.6 130 5.00 56  0.05 0.01 0.12 0.12 7.84 7.86 1.16 2.23 2.12 13.8 14.0 1.84 0.31 7.33 6.03 1.36 0.59 4.36 56  0.05 0.05 4.59 4.60 0.25 0.25 1.19 2.25 2.18 18.5 18.8 1.85 1.06 10.9 5.28 1.53 0.48 4.62 63  0.05 0.01 4.57 4.59 1.78 1.81 0.79 0.45 0.46 22.7 23.0 1.56 0.58 16.8 9.45 1.52 0.44 4.35 63  Table 4.12: Mode shape results for TFU, free-interface method  Chapter 4. Modal Analysis of Three Finite Element Models^ 102  Direct FEM Analysis Mode Frequency (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  # d.o.f. I  % Error, Fixed-Interface CMS Analysis Guyan 2000 Hz. 3000 Hz. 4000 Hz. Reduction Cut-off Cut-off Cut-off  34.7 72.8 72.8 185.8 432 433 655 658 1002 1059 1067 1231 1232 1273 1347 1444 1447 1486 1576 1693  0.02 .^0.38 11.1 11.0 11.4 11.6 115 133 112 131 117 161 215 232 127 205 135 121  2568  360  0.05 0.38 52.2 53.0 57.0 59.2 1.85 8.78 8.13 58.5 131 111 91.2 95.2 22.1 98.5 182 25.6  0.05 0.03 8.63 8.71 57.1 57.5 1.84 64.7 70.0 61.1 61.2 3.72 2.79 8.08 121 158 23.8 2.17  0.05 0.03 4.60 4.61 0.33 0.34 2.00 2.60 2.55 7.87 8.04 3.61 2.33 7.36 5.30 3.06 0.65 2.18  369  380  389  Table 4.13: Mode shape results for TFU, fixed-interface method  Chapter 4. Modal Analysis of Three Finite Element Models^ 103  For example, modes 2 and 3 have identical frequencies and so the mode shape error has been omitted. Other modes have frequencies that are very close though not identical. This seems to present difficulties for CMS in that the mode shapes can remain ill-defined long after the corresponding frequencies have converged. In the free-interface results, this occurs most noticeably in modes 12 and 13 whose frequencies are 1230.9Hz and 1232.0Hz respectively. Despite the accurate frequency predicitions provided by CMS, the mode shape errors remain large, and indeed get larger, as progressively higher cutoff frequencies are used. A similar trend is seen in the fixed-interface results. For example, modes 7 and 8 show very large mode shape errors until the 4000Hz cutoff where they all but disappear. It can be verified that these large errors are not caused by a reversal in the modes' order; if these modes are switched, the mode shape errors are just as large. Instead, what is happening is a symptom of CMS condensation, of trying to specify modes with a severely reduced quantity of independent variables. That CMS is able to accurately predict the frequencies is a reflection of how a reduced set of generalized coordinates, corresponding to low frequency component modes, will render the Rayleigh quotient, ai =  tt  T -^  (4.5)  ST MSi  stationary at the correct values of Ai. However, this does not necessarily mean that the condensed eigenvector i corresponds to the true mode shape with the same degree of accuracy. If the size of the system has been reduced from 2568 to less than 400, as is in the case of the focus unit, far fevier independent variables are available for determining the full extent of the mode shapes. The difficulty with the near-repeated modes is that they look very much like repeated modes; it is only a small asymmetry in the model which distinguishes them. At the lower cutoff frequencies, the condensation is too severe for the asymmetry to be detected; that  Chapter 4. Modal Analysis of Three Finite Element Models^ 104 is, in representing the components with a small number of component modes, the analysis condenses out the asymmetric features in the model. But at a higher cutoff, the mode suddenly locks in—that is, enough coordinates have now been added for these closelyrelated modes to be distinguished. This sudden convergence can also be observed in the distinct modes, but it is not delayed to the higher cutoffs as it is for the near-repeated modes. In general, mode shapes of the distinct modes are ultimately more accurately predicted with CMS than those of the near-repeated modes. The locking-in feature occurs throughout the fixed-interface results in Table 4.13. With the exception of modes 12 and 13, the free-interface method does a better job of predicting the near-repeated mode shapes, in spite of its greater condensation. Evidently free-interface component modes do a better job of detecting the asymmetries than do fixed-interface modes. 4.3.3 Performance  Table 4.14 shows the CPU times for the CMS methods with various cutoff frequencies, direct analysis and Guyan reduction. It is immediately noticeable that the free-interface method is unable to give results at the two larger tolerances. This is because of illconditioning in the residual flexibility matrix  6, which is calculated with the equation,  = G — (1)11 -1 (D T  (4.6)  If too large a tolerance is used, errors occur in the free-free modes 4). These errors are magnified by a loss of precision in the subtraction operation, and this can cause  6  to lose its positive definiteness. In this particular example, the problem was caused by inaccurate calculation of modes 7 and 8 of the screw assembly. These are repeated modes of the component, but the eigenvalue solver only renders them so when the tolerance is sufficiently small. In general, this is a problem in components with high modal density,  Chapter 4. Modal Analysis of Three Finite Element Models ^ 105  CPU Time (s) for 40 Modes Tolerances 10-2 10 -4 10 -6 Direct 5294 12999 53355 auyan 1524 2119 4836 Free-Interface CMS 2000 Hz. M1 i.c. i.c. 2045 2000 Hz. M2 i.c. i.c. 2299 2500 Hz. Mil i.c. i.c. 2229 2500 Hz. M2 i.c. i.c. 2483 3000 Hz. M1 i.c. i.c. 4621 3000 Hz. M2 i.c. i.c. 4936 3500 Hz. Ml i.c. i.c. 4991 3500 Hz. M2 i.c. i.c. 5094 Fixed-Interface CMS 4571 2000 Hz. 5742 6386 3000 Hz. 5010 5764 7013 Hz. 4890 5711 8308 4000 Table 4.14: CPU times for modal analysis of the TFU  Chapter 4. Modal Analysis of Three Finite Element Models^ 106  and a high tolerance is needed to obtain sufficiently accurate component modes. The fixed-interface method is not sensitive to errors in the free vibration modes because it does not use a residual flexibility approximation, and at the two larger tolerances it performs either comparable or favorable to direct analysis. At to/ 10 -6 , both CMS methods have a performance advantage over direct analysis. Using a second-order instead of a first-order mass makes little difference to the time expended and therefore the improved accuracy is obtained cheaply. However, between the 2500Hz and 3000Hz cutoffs, the time expenditure nearly doubles as a result of slowly converging component modes in this frequency range. As the density of the component modes increases, eigensolution routines require more iterations to resolve the modes. Therefore, in components with high modal density, the cost of the component analysis inevitably increases; but in proportion to a direct analysis, the cost of the total CMS analysis declines. For the container ship model, the free-interface method was about five times as fast as a direct analysis; for the TFU model, a similar level of accuracy is obtained at one-tenth the cost. 4.4 Analysis of a Telescope Model  The complete model of the telescope consists of the focus unit of Section 4.3, and a supporting framework suspending the focus unit above the centre of the main mirror (see Figure 4.5) The supporting framework, or spider, consists of four separate and identical components, giving the telescope model a total of nine components. Each leg of the spider is composed of 38 2-node beam elements and one thick/thin shell element, for a total of 35 nodes and 210 degrees of freedom. The beam elements have rectangular tubular sections of dimension 1 x 3 in. and a wall thickness of 8in. It is easy to appreciate that the supporting framework is susceptible to vibration  Chapter 4. Modal Analysis of Three Finite Element Models ^  Figure 4.5: Complete telescope model, with focus unit and spider  107  Chapter 4. Modal Analysis of Three Finite Element Models^ 108  from the chopping mechanism and that this is potentially harmful to the telescope's image quality. Particularly dangerous are resonances in the support structure causing lateral or skewing motion in the focus unit. It is with this in mind that the following modal analysis results are presented. 4.4.1 Natural Frequency Results  Both CMS methods display very rapid and well-defined convergence for the natural frequencies (see Tables 4.15 and 4.16). Indeed, it is remarkable to note, especially in the results for the fixed-interface method, how sharp the distinction is between regions of high and low accuracy. This can be attributed to the dominance of spider vibration in the natural modes of the telescope. In both of the CMS methods, the only dynamic modes used were from the spider components; the free vibration modes of the TFU components were of such high frequency that they could be safely neglected. Another factor contributing to the sharp convergence is the distinct and the well-separated frequency spectrums of both the free-free and clamped modes of the spider (see Table 4.17). By contrast, the high modal density in the focus-unit components created a much slower and less sharp convergence. Because of the high inertia of the TFU, many vibration modes of the telescope act like individual modes of the four spider components under clamped boundary conditions. This is verified by noting that the spider component fixed-interface frequencies in Table 4.17 closely correspond to many of the telescope frequencies. Thus, the fixed-interface modes are seen to give greater overall accuracy in terms of the number of dynamic modes contributed (see Figure 4.6). Indeed, it would be difficult to find shape functions more suitable for describing the telescope modes. An equal number of free-free modes clearly cannot give the same level of accuracy, but as with the TFU, the convergence rate is about the same in each case.  Chapter 4. Modal Analysis of Three Finite Element Models ^ 109  Direct FEM Analysis Mode Frequency (Hz)  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 # d.o.f.  11.5 13.3 13.3 13.4 25.5 33.5 34.0 34.6 34.6 39.1 42.2 45.3 45.3 45.3 45.6 50.6 61.0 61.0 61.1 71.0 78.1 83.7 83.9 84.0 85.6 85.7 98.1 99.0 99.7 104.0 104.1 104.3 105.9 117.3 117.5 117.7 120.1 125.5 127.4 133.6 3360  % Error, Free-Interface CMS Analysis 30 Hz. 90 Hz. 150 Hz. 1st 1st 2nd 1st 2nd 2nd  0.86 3.16 3.25 3.31 18.7 15.5 25.8 26.2 74.1 55.3 45.1 46.5 77.3 88.7 152 135 119 123 191 159 151 168 244 294 341 365 460 469 646 615 1039 1108 1091 1270 1409 1525 1679 1604 2344 2267 46  0.02 0.05 0.26 -0.03 2.93 6.87 6.55 10.6 16.6 7.08 2.09 4.88 8.29 9.95 9.33 46.1 30.1 . 42.5 72.1 54.7 43.4 67.7 70.0 206 225 232 375 391 537 578 670 943 1018 964 1000 1043 1072 1334 1388 1453 46  0.05 0.04 0.08 0.01 0.20 0.72 0.54 0.29 0.32 1.96 2.01 0.02 0.01 0.13 0.38 0.56 1.03 1.11 1.12 4.10 4.03 0.89 1.78 3.05 5.18 10.8 17.4 62.3 63.6 92.8 106 116 116 94.7 107 136 141 141 160 195 62  0.00 -0.01 0.02 -0.02 0.01 0.03 -0.06 0.08 -0.01 0.03 0.06 -0.03 -0.02 0.14 -0.02 -0.03 0.07 0.07 0.07 0.89 0.38 0.18 0.60 0.75 1.38 2.83 7.21 9.57 9.63 8.10 12.0 12.2 10.6 19.4 19.4 52.3 49.7 44.6 78.5 92.6 62  0.03 0.01 0.06 -0.03 0.18 0.19 0.28 0.10 0.08 0.48 0.67 -0.06 0.04 0.14 0.17 0.26 0.28 0.29 0.26 0.96 2.88 0.16 0.14 0.26 0.22 2.46 0.76 1.10 0.88 0.45 1.22 3.99 6.41 1.55 1.60 1.58 2.65 2.88 4.03 0.22 78  0.00 -0.01 0.02 -0.02 0.01 -0.04 0.08 0.03 -0.08 0.00 0.03 0.04 0.01 0.03 -0.02 -0.06 -0.10 -0.06 -0.01 0.08 0.05 0.01 -0.10 0.04 0.04 -0.02 0.02 0.25 0.19 0.03 0.24 0.31 0.07 0.22 0.29 0.38 0.43 0.10 0.97 0.00 78  Table 4.15: Natural frequency results for telescope model, free-interface method  Chapter 4. Modal Analysis of Three Finite Element Models^ 110  Direct FEM Analysis Frequency Mode (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  # d.o.f.  % Error, Fixed-Interface CMS Analysis Guyan 85 Hz. 100 Hz. 125 Hz. Cut-off Reduction Cut-off Cut-off  11.5 13.3 13.3 13.4^. 25.5 33.5 34.0 34.6 34.6 39.1 42.2 45.3 45.3 45.3 45.6 50.6 61.0 61.0 61.1 71.0 78.1 83.7 83.9 84.0 85.6 85.7 98.1 99.0 99.7 104.0 104.1 104.3 105.9 117.3 117.5 117.7 120.1 125.5 127.4 133.6  58.8 171 175 200 207 157 209 207 208 218 289 295 334 451 808 728 650 706 722 740 841 785 803 880 944 1090 984 1026 1033 1013 1097 1224 1419 1341 1418 1428 1489 1465 1543 1481  0.01 0.00 0.00 0.00 0.15 0.11 0.02 0.00 0.04 0.57 0.62 -0.06 -0.05 0.15 0.12 1.00 -0.03 0.07 0.02 5.35 1.80 4.30 26.8 67.4 68.2 192 179 178 244 251 382 383 376 345 434 494 521 496 701 735  3360  408  424  0.00 0.00 0.00 0.00 0.08 0.05 0.10 0.06 -0.05 0.52 0.57 -0.10 -0.06 0.18 0.12 0.34 -0.03 6.07 0.01 1.11 0.84 0.14 -0.03 0.06 0.31 0.90 -0.02 0.71 0.17 2.35 2.96 36.1 36.6 113 157 159 214 223 295 278  0.00 0.00 0.00 0.00 0.05 0.04 0.01 0.09 -0.05 0.14 0.17 -0.11 0.01 0.10 0.08 0.14 -0.03 0.02 0.03 0.44 0.67 -0.04 0.15 0.03 0.21 0.64 -0.02 0.43 -0.05 0.45 0.40 0.32 0.38 0.14 -0.04 0.08 0.26 -0.02 1.60 80.2  432  440  Table 4.16: Natural frequency results for telescope model, fixed-interface method  Chapter 4. Modal Analysis of Three Finite Element Models ^ 111  —D---- Free, l it order mass —6,-- Free, rd order mass —0— Fixed  10^15^20^25^30^35^40^45 No. of dynamic component modes  Figure 4.6: Combined natural frequency results for the telescope model  Chapter 4. Modal Analysis of Three Finite Element Models ^ 112  Component Frequencies (Hz) Fixed Mode Free  1 2 3 4 5 6 7 8 9 10 11 12  1.55 10.1 20.0 28.7 39.5 55.4 64.2 90.0 95.9 107.3 120.3 135.8  13.4 34.6 45.5 61.1 84.1 98.6 117.3 124.9  Table 4.17: Natural frequencies of spider components One impressive feature of CMS is how greatly it has simplified the analysis of the telescope model. Whereas a direct finite element analysis of the complete telescope represents a more difficult problem—by virtue of the increased number of coordinates— over that of the focus unit, using CMS has made this problem no more difficult than was the focus unit, and in fact in some ways has made it much simpler. The simplification stems from recognizing that with the spider attached, the focus unit acts by and large as a rigid body whose deformations can be accounted for by static approximation. This could be discovered in the course of an analysis by comparing the component frequencies (listed in Tables 4.11 and 4.17), and by noting that the focus unit component frequencies are an order of magnitude larger than the spider frequencies. Consequently, in the freeinterface method, only rigid body modes and static flexibility of the focus unit are needed, while in the fixed-interface method only static constraint modes are needed. One might be tempted to introduce a similar simplification in the direct analysis by replacing the detailed model of the focus unit with a cruder equivalent. Such an attempt may be  Chapter 4. Modal Analysis of Three Finite Element Models^ 113  partially successful but it should be noted that among the many repeated modes of the telescope are a few that are entirely distinct. These modes are typified by simultaneous vibration of the spider and the inner components of the focus unit. Therefore, the detail in the modelling of the focus unit components needs to be maintained in order to predict these interactive modes. Moreover, it is these interactive modes which are of the most critical interest in the dynamic response because they are most likely to affect the image quality. It should be noted that CMS does not detract from the complexity of the original model. Instead, it allows the analyst to discard information from the original model that is of no importance to the targeted modes. In view of the CMS results, a cutoff frequency criterion can be confidently proposed for the telescope model. To establish such a criterion, first a suitable definition of accuracy must be agreed upon. One reasonable definition might be that structural modes below a target frequency of f* are accurate when at least 90% of these modes have less than 1% error in the natural frequency. If this definition is satisfactory, the fixed-interface method ought to be used with a cutoff frequency fc = p. For the free-interface method,  L = p should be used with a second-order mass, while fc = 1.5f* should be used with a first-order mass. Of course, this criterion is only valid for the telescope model; if, for instance, it was applied to the focus unit, it is clear from the results of Section 4.3 that these cutoff frequencies would not be high enough to give the same accuracy. In establishing such rules of thumb, such factors as the modal density and the complexity of the component connectivity have to be accounted for. However, it is obviously not practical to routinely engage in a detailed numerical study involving comparison to a direct finite element analysis in order to determine an appropriate cutoff frequency. Hence the need for intuition or numerical experience on the part of the analyst. Of course, if there is doubt about a suitable value for the cutoff, it is safer to add too many modes than too few.  Chapter 4. Modal Analysis of Three Finite Element Models^ 114  Although the two CMS methods seem capable of giving results of equal accuracy, the free-interface method does so with more condensed equations and thus fewer independent variables. In Table 4.15 the first 40 modes are resolved to within 1% error with only 78 degrees of freedom retained; to achieve comparable results with the fixed-interface method, more than 440 are necessary. 4.4.2 Mode Shape Results and Performance  The mode shape results for the fixed-interface results are shown in Table 4.18. The results for the free-interface method are similar but have been omitted for the sake of brevity. As was the case for the focus unit, comparing mode shapes only makes sense for distinct modes. On the whole, the mode shapes display slow convergence and this can be attributed to the fact that to/ 10 -4 was used instead of tol 10 -6 . The direct analysis of the complete telescope model proved prohibitively expensive with a tolerance smaller than 10 -4 , and while this is quite sufficient for giving reliable frequency results, some discrepancy can be expected in the mode shapes. Since the direct-analysis mode shapes are the standard of comparison for Table 4.18, the percentage errors cannot be expected to converge smoothly. However, this does not prevent accurate mode shape calculations because with CMS stricter tolerances can be used without great expense. In Table 4.19, the CPU time expenditure for tolerances of 10 -2 and 10 -4 is shown. The performance of the CMS methods is little affected by the tolerance, even though more accurate mode shapes are obtained with the smaller of the two. In both cases, the free-interface method is 2.5 to 3 times faster than the fixed-interface method and for  tol 10 -4 , the former is more than 10 times as fast as a direct analysis. With the results for the two previous examples in mind, the performance advantage for CMS evidently increases with the size of the model. Also, the advantage of the free-interface method over the fixed-interface method appears to increase with the model size. No ill-conditioning  Chapter 4. Modal Analysis of Three Finite Element Models^ 115  Direct FEM Analysis Mode Frequency (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # d.o.f. 1  % Error, Fixed-Interface CMS Analysis 100 Hz. 125 Hz. Guyan 65 Hz. Reduction Cut-off Cut-off Cut-off  11.5 13.3 13.3 13.4 25.5 33.5 34.0 34.6 34.6 39.1 42.2 45.3 45.3 45.3 45.6 50.6 61.0 61.0 61.1 71.0  62.0 165 153 135 115 102 134 119 105 188 119  2.38 0.32 1.15 6.27 33.1 3.65 3.60 27.2 8.75 71.2 31.7  2.37 0.32 0.53 15.5 23.5 2.56 3.71 57.7 7.47 76.1 9.90  2.37 0.32 0.27 4.80 26.3 2.23 1.32 35.8 5.36 39.2 4.62  3360  408  424  432  440  Table 4.18: Mode shape results for telescope model, fixed-interface method  CPU Time (s) for 40 modes Tolerances 10-2 10 -4 Direct Analysis 6324 25121 Guyan Reduction 2637 3123 Free-Interface CMS 2161 2203 Fixed-Interface CMS 6225 5744 Table 4.19: CPU times for modal analysis of the telescope model  Chapter 4. Modal Analysis of Three Finite Element Models ^ 116  problems were encountered with the full telescope model; the modal density of the spider components is low enough for accurate computations with the larger tolerances. 4.5 Natural Frequency Error Estimation: Ship Model In Section 3.9, a procedure was described for calculating lower bounds to the exact frequencies based on a free-interface CMS analysis. This procedure provides a means of estimating the absolute error in the CMS-derived frequencies without having to compare them to results of another method, and without having to calculate additional free-free modes. In the present section, the results of lower bound analyses of the container ship model are given and compared to direct-analysis results of the assembled model. Table 4.20 shows natural frequency results obtained using the dynamic residual flexibility evaluated at four different frequencies: U.; = 2.14, 5.94, 10.24, 19.26Hz . These frequencies were originally calculated as the 5 th , 9 th, 12 th, and 20 th modes in the 25Hz, first-order mass results listed in Table 4.2. The numbers listed in each of the four cases in Table 4.20 are percentage errors calculated with respect to the direct analysis results. Thus, a negative percentage error indicates that a lower bound has been predicted; a positive error indicates an upper bound. Lower bounds generally appear in all modes of lower frequency than Cv, whereas upper bounds appear in all modes above Cv. Note that the smallest percentage errors occur in modes closest in frequency to ar. For example, when (.1"; = 2.14Hz, the smallest error is 0.003% which occurs in mode 5. Because the values used for (1) are upper bounds calculated from the free-interface method, the error in this mode ought to be slightly negative, according to (3.96). However, in this example the results are so close that secondary sources of error such as numerical round-off create a slightly positive frequency error. To summarize, the original results in Table 4.2 and the lower bound estimates in Table  Chapter 4. Modal Analysis of Three Finite Element Models ^  Mode 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  Direct (Hz) 0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.97 10.2 12.3 13.1 13.7 14.2 15.0 16.4 17.3 18.9  cii = 2.14Hz -0.01 0.003 0.04 0.14 0.16 0.50 0.24 0.37 0.49 0.70 2.02^• 2.57 6.39 2.18 3.30 2.55 1.62  117  % Frequency Error with G(m) ci,' = 5.94Hz c 7, = 10.24Hz ii, = 19.26Hz -1.62 -0.35 -0.11 -2.43 -0.15 -0.51 -2.34 -0.47 -0.11 -4.37 -0.73 -0.11 -3.47 -0.03 -0.48 -6.37 -0.002 -1.19 -1.64 0.12 -0.16 -1.90 -0.08 0.24 -1.78 0.01 0.35 0.58 -1.62 0.27 -3.56 1.72 0.96 -5.49 2.29 1.56 -3.97 5.48 3.34 1.32 -1.88 1.93 -1.57 2.98 2.20 2.35 1.87 -0.57 -0.08 1.53 1.29  Table 4.20: Lower and upper bounds calculated with dynamic residual flexibility 4.20 are compiled in 4.21 for four selected modes. It should be noted that the lower bounds are much closer to the exact frequencies than are the upper bounds. By this means, a good estimate of the error in the upper bounds is obtained. The dynamic residual flexibility therefore provides a means of accurately estimating the absolute error of natural frequencies calculated with CMS. This is valuable when direct-analysis results are not  i 5 9 12 20  CZu (Hz) 2.1406 5.9071 10.1870 18.9314  rvi (Hz) 2.1411 5.9412 10.2375 19.2566  il)i - 6.4 (Hz) -0.0005 -0.0341 -0.0505 -0.3252 -  wi (Hz) 2.1406 5.9072 10.1860 18.9461  Table 4.21: Lower and upper bound estimates of four CMS frequencies, with predicted absolute errors and exact frequencies  Chapter 4. Modal Analysis of Three Finite Element Models^ 118  available for comparison. Although this technique is only valid for natural modes below the cutoff frequency, it was shown in the earlier examples that the cutoff frequency must, in general, be above the target frequency range if accurate results are to be obtained. However, the accuracy for a particular cutoff frequency is uncertain, and depends on the characteristics of the model being analyzed. Assuming that the cutoff frequency is always chosen to be high enough to include the entire target frequency range, the accuracy of structural natural frequencies falling in this range can be tested using the dynamic flexibility concept. The cost of the error estimates is dominated by the calculation of the component modulation matrices (2.51). The same free-free modes are used as in the original CMS analysis, and therefore the same static residual flexibilities apply. 4.6 Reanalysis Following a Design Modification  Many changes may be made to a structural model before the design is finalized, and a separate modal analysis may be required for each change. If a direct finite element analysis has been performed, two possibilites exist: to perform a direct analysis on the modified model which will be just as expensive as the original; or to seek an approximate solution using structural dynamic modification techniques [49]. If instead a substructuring approach was taken and the original model was analyzed with CMS, more options are available to the analyst. If the design modification affects every structural component, reanalysis requires a CMS analysis like the original. However, if the design modification affects only some of the components, the component modes need only be recalculated for the components that have been modified; component modes from the previous analysis can be re-used for the unchanged components. For a complex model, design modifications may well be localized in a single component and so computational savings can be realized by omitting the reanalysis of unmodified  Chapter 4. Modal Analysis of Three Finite Element Models^ 119  Figure 4.7: Mode 9 of the container ship, f = 5.91Hz. components. 4.6.1 Reanalysis of the Container Ship In the results for the container ship presented in Section 4.2, mode 9 was found to have frequency 5.91Hz and the mode shape depicted in Figure 4.7. This mode could be excited in a resonant or near-resonant condition by pressure forces on the hull. For instance, a four-bladed propeller rotating at 90RPM would produce a 6Hz periodic force in the immediate vicinity. Excessive vibration in this mode could bring about fatigue problems in the structural members and panels and it would also create uncomfortable living conditions for the crew in the deckhouse. One solution to this problem is to raise the frequency to a safe region without causing any other frequencies to fall into the region around the forcing frequency. As this is a local mode in the stern and deckhouse components, it is susceptible to a purely local modification. In this example, the four horizontal bar stiffener elements representing the top deck of the stern component (groups 1 and 2 in Figure 4.8) are modified, and four stiffener elements representing a new bulkhead (group 3) are added. Inspection of  Chapter 4. Modal Analysis of Three Finite Element Models ^ 120  h1-±  ■E■ MIME ■1■■ 4  5  6  Figure 4.8: Six groups of bar stiffeners used for modifications Figure 4.7 indicates that element groups 2 and 3 will be deformed axially. The stiffness of group 2 is increased by doubling the cross-sectional area A, whereas the new bulkhead is given the same properties as the other bulkheads in the vessel. By contrast, group 1 experiences large deflection but little deformation in this mode. Therefore, the frequency is best raised by reducing the mass of these elements and hence, the cross-sectional area of group 1 was reduced by half. Tables 4.22 and 4.23 show the reanalysis results with the modifications to the stern component. The beam-like hull bending modes that predominate at low frequency are little influenced by the stiffness modification in the stern and so only mode 9 is significantly affected out of the first fourteen modes. Only at higher frequencies do significant frequency shifts begin to appear again. Nevertheless, raising mode 9 to 6.24Hz may be enough to produce a satisfactory dynamic response. All that remains is to translate this stiffness modification into an actual physical modification of the ship. The CPU time expended for the two CMS runs are also shown in Tables 4.22 and  Chapter 4. Modal Analysis of Three Finite Element Models ^  Mode  Free-interface Method Original Re-analysis  % Change  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.94 10.19 12.28 13.08 13.79 14.27 15.00 16.45 17.39 18.96  0.877 2.14 3.52 4.54 5.60 6.24 8.01 8.90 10.17 12.16 13.01 13.58 14.63 15.00 16.41 17.41 18.95  -0.40 0.10 0.05 -0.26 0.53 5.55 -0.07 -0.46 -0.18 -0.94 -0.46 -1.40 2.86 0.08 -0.23 0.25 -0.02  CPU time  1028  176  -82.9  Table 4.22: Reanalysis of the container ship model, free-interface method Mode  Fixed-Interface Method Original Re-analysis  % Change  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  0.881 2.14 3.52 4.55 5.58 5.91 8.01 8.94 10.19 12.28 13.08 13.79 14.27 15.00 16.45 17.39 18.96  0.877 2.14 3.52 4.54 5.60 6.24 8.01 8.91 10.17 12.18 13.01 13.61 14.66 15.01 16.42 17.43 18.99  -0.40 0.10 0.05 -0.26 0.54 5.54 -0.07 -0.46 -0.18 -0.92 -0.49 -1.34 2.85 0.11 -0.21 0.23 0.02  CPU time  1561  662  -57.6  Table 4.23: Reanalysis of the container ship model, fixed-interface method  121  Chapter 4. Modal Analysis of Three Finite Element Models^  122  4.23. The reanalysis times are for calculation of 40 modes, as was done in the original analysis. The greater proportional time reduction in the free-interface method can be attributed to two factors. First, this method places more computational emphasis on component analysis, creating more condensed system equations. Conversely, the fixed-interface method performs fewer computations at the component level, instead allowing larger system equations and therefore placing more computational emphasis on the system level. In a reanalysis mode, the free-interface method stands to benefit more because component-level calculations are skipped. Secondly, the implementation of the fixed-interface method in VAST had to be made less efficient, particularly in a reanalysis mode where it was difficult to avoid unnecessary computations. These two factors combine to give the free-interface method the advantage, but this advantage diminishes as a larger proportion of the components undergo modification. 4.6.2 Reanalysis of the Telescope Focus Unit  As a further example of the advantage of using CMS for reanalysis, the stiffness of the bearings connecting the inner tube to the outer tube will be modified to determine their effect on the focus unit frequencies. These bearings have been modelled as simple bar elements and are included as part of the inner tube component (see Figure 4.3). The difficulties involved with modelling bearings accurately justifies some investigation into the effects of different modelling options. Tables 4.24 and 4.25 show the results of a reanalysis of the focus unit with the cross-sectional area of the bar elements doubled. The results clearly indicate which modes are affected by the bearings and which are not. With a CMS reanalysis, only component modes associated with the inner tube need to be re-calculated. The time savings obtained are similar to those found for the container ship, with the free-interface method showing the largest proportional reduction. But it is here that the benefit of having condensed system equations is most apparent. With  Chapter 4. Modal Analysis of Three Finite Element Models ^ 123  Mode  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CPU time  Free-Interface Method Original Re-analysis A = 36mm2 A = 72mm2  34.7 72.8 72.8 185.8 431.6 433.2 655.4 658.0 1004 1060 1067 1233 1234 1273 1347 5094  34.7 78.7 78.7 185.8 468.2 470.4 710.0 714.2 1003 1148 1152 1274 1317 1321 1349 1114  % Change  0.00 8.16 8.18 -0.01 8.47 8.59 8.34 8.54 -0.06 8.26 7.90 3.39 6.73 3.69 0.10 -78.1  Table 4.24: Reanalysis of the TFU, free-interface method  Mode  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CPU time  Fixed-Interface Method Original Re-analysis A = 36mm2 A = 72mm 2  34.7 72.7 72.7 185.8 431.5 433.0 655.6 658.3 1006 1062 1070 1234 1235 1273 1349 8308  34.7 78.7 78.7 185.8 467.9 470.1 710.0 714.2 1005 1151 1156 1299 1316 1319 1350 6768  % Change  0.00 8.16 8.17 -0.01 8.44 8.57 8.29 8.49 -0.06 8.38 8.02 5.34 6.54 3.66 0.09 -18.5  Table 4.25: Reanalysis of the TFU, fixed-interface method  Chapter 4. Modal Analysis of Three Finite Element Models ^ 124  the fixed-interface method, whether in the original analysis or the reanalysis, the time expended for the final eigensolution alone is about 30003, while the entire reanalysis with the free-interface method expends only 1114s (Table 4.24). These results were obtained with a eigensolution tolerance of 10 -6 . If a larger tolerance is used, the performance gap is narrowed at the price of obtaining less accurate mode shapes. 4.7 Discussion and Summary of Numerical Results The examples presented in this chapter show a variety of typical situations: models of differing size and complexity in two and three dimensions. In all cases, CMS has proven capable of accurately predicting the low frequency modes with a reduced-order formulation of the free vibration equations. Moreover, CMS is more efficient than direct finite element analysis and its relative efficiency increases with the number of degrees of freedom in the model, with the number of low-frequency structural modes that are calculated, and with stringency of the accuracy requirements. Of the CMS methods used, the free-interface method with second-order mass terms appears to be the most economical. Although the relative speed of the methods is partly a function the efficiency of the software implementation, the free-interface method does benefit from a highly condensed set of global equations. A close study of the examples presented in this chapter will strengthen this point further. The finite element models used in the examples have respectively 1341, 2568 and 3360 degrees of freedom but the results for the free-interface method show that comparable accuracy between the different examples is obtained with equations of the same order. That is, the size of the condensed system equations is not determined by the size or complexity of the original model; instead, they are determined solely by the cutoff frequency, which itself is determined by the desired range of accuracy. The opposite is true for the fixed-interface method where  Chapter 4. Modal Analysis of Three Finite Element Models ^ 125  the original size and complexity does ultimately determine the size of the condensed system. Comparing the results for the ship model and the focus unit, a large jump can be seen in the order of the condensed system because of the increased complexity of the component interfaces in the latter. This point is illustrated by imagining the following process. Suppose that the coarseness of the mesh in the container ship model is reduced by a factor of two: that is, each bar element is replaced by two others one-half its size and each membrane is replaced by four others one-quarter its size. The overall effect will be to approximately double the number of nodes situated on the component interfaces (and thereby double the number of static modes). If the fixed-interface method is used on the refined model, a larger order system is the result. However, as the mesh was already sufficiently fine, one would not expect either the low-frequency system or component modes to change significantly when the additional refinement takes place. That is, results of essentially the same accuracy would be obtained with the same cutoff frequencies, entailing the same dynamic component modes. With the free-interface method this leads to equations of the same order. The same process if carried out with the focus unit would give the same result. More generally, it can be remarked that the number of dynamic modes needed in a CMS analysis is not a function of the mesh refinement. When applied to relatively complex models, the basic differences between the two formulations have a great impact on their performance. Throughout the examples, the free-interface method is consistently quicker than the fixed-interface method (excepting the cases for which ill-conditioning occurs); and as the model complexity increases, so does the discrepancy in the solution times. Furthermore, the difference is magnified in reanalysis problems where a partial reanalysis of the structural components precedes a complete eigensolution of the condensed system equations.  Chapter 5 Structural Dynamic Modification  5.1 Introduction The previous chapters of this thesis have dwelt on the problem of analyzing the natural modes of a system. In structural design, accurate knowledge of these modes is required to determine whether the vibrational response will be acceptable or not. Closely related to the analysis problem are the reanalysis and re-design problems, which are concerned with determining the effects of modifications on the natural modes of the system and with the ways in which a system can be modified in an efficient manner. The design cycle of a particular structure may entail numerous modifications, each of which requires a free-vibration analysis. The purpose of developing structural dynamic modification techniques is to provide a capability for efficient design and re-design, and to foster a better understanding of structural dynamic behaviour. Dynamic modification techniques all use an unmodified or baseline state as a reference point for the modified structure. In this respect, there is great potential for using a CMS formulation of the baseline system. Because of the compact, reduced-order nature of the CMS formulations, numerical techniques used in dynamic modification can be applied with less computational effort. As of yet, little work has appeared in the literature relating CMS analysis to structural dynamic modification. Reanalysis is concerned with the efficient evaluation of the natural modes of a modified structure. This is often referred to as structural dynamic modification or forward  126  Chapter 5. Structural Dynamic Modification^  127  modification. This subject was touched on in Section 4.6 where it was shown that the free-  interface CMS technique provides a means of accurate and efficient reanalysis for local modifications. In this chapter, forward modification is dealt with in a broader sense, the intention being to demonstrate how techniques successfully applied to unsubstructured systems can be adapted to substructured situations. Inverse modification is concerned with assessing the changes to structural properties  necessary so that prescribed modal constraints are satisfied. The modal constraints may consist of constraints on the frequencies, mode shape constraints or a combination of both. Generally, mode shape constraints will take the form of limits on the relative amplitudes of specified locations in the mode shapes. Techniques for solving problems of this kind are valuable because they eliminate the element of guesswork from the re-design process. Inverse modification techniques applicable to general structural changes fall into two classes: perturbation and sensitivity techniques. The application of these two techniques to substructured models analyzed with CMS is the subject of Sections 5.4 and 5.5. In Section 5.6, numerical results are presented and the performance of the two techniques is compared. 5.2  Structural Changes  It should be made clear what is meant by a modification of a structural model. In the present context, a structural modification can be any combination of the three following types of changes: • Changes to the material properties of the structure; • Modification of existing elements provided that node locations are left unchanged;  Chapter 5. Structural Dynamic Modification ^  128  • Adding new elements, externally connected springs or lumped masses to the existing mesh. These restrictions bring about two important advantages. First, because topological changes are excluded, re-meshing of the model during reanalysis is not a concern. Secondly, a structural modification defined as above can be expressed as a perturbation to the existing system; i.e., the stiffness and mass matrices of the modified structure are perturbed versions of the original matrices: K K + AK, and M --+ M + AM. When using finite element models, the stiffness and mass matrices of a structure (or structural component) are assemblies of all constituent element matrices. Likewise, the perturbed matrices are assemblies of perturbed element matrices: AK =E seT Akese  (5.1)  AM = E seTAmese (5.2) where S. is the appropriate transformation from local to global coordinates. When dealing with substructured models, the above formulae define perturbations to a structural component. The perturbations to all components are expressed collectively using the overbar notation, where AK is the uncoupled collocation of the various component stiffness perturbations: OK = diag {AK ' }  (5.3)  AM = diag {OM ` }  (5.4)  (  (  )  )  where the superscript (i) indicates the perturbed matrix of the i th component. The element matrices are functions of a number of independently varying properties. For example, the stiffness matrix of a beam element is a function of its length, elastic modulus, cross-sectional area, and second moment of area. Changes to any of the last three are admissible, according to the definition above; changing the length is forbidden  Chapter 5. Structural Dynamic Modification^  129  since this requires relocating the element nodes. If the second moment of area is the only property to be changed, ke is in the following form: = (1c!') r (5.5) where r denotes the current value of the property and where lc!' is defined as the portion of the element stiffness matrix proportional to the property variable r. The perturbed element stiffness Ak e is therefore, Ake = — = (1o) Ar = (lc!) ra (5.6) where a is the fractional change to the property r. The above equation reflects the linear relationship between the second moment of area and the element stiffness. Consequently, property changes of this type are called linear property changes. Changes to element mass matrices are usually of this type. It could be argued that since the cross-sectional area and its second moment are difficult to change independently, a better choice of property variables would include the actual dimensions of the cross section. For instance, choosing the width of a rectangular cross section as the property variable gives a perturbed element stiffness of the form,  Ake = kr, (r' — r ) kc ((r') 3 — r3) = ( kr, + 3icec r 3) a + (3kc r 3) a 2 (keCr3) a 3  (5.7)  where r' = (1 a)r and where Icec is defined as the portion of the element stiffness matrix proportional to r 3 . The higher-order terms in a reflect the nonlinear relationship between the width of the beam and the element stiffness. Property changes of this type are called nonlinear property changes. In forward modification problems, the property changes are prescribed at the outset and so element perturbations can be calculated exactly, whether they are of the form (5.6)  ^  Chapter 5. Structural Dynamic Modification^  130  or (5.7). On the other hand, for inverse modification problems the property changes are unknown and have to be determined to satisfy some modal constraints. In this application, a is referred to as a design variable. When a design variable appears in a nonlinear form it cannot easily be determined, particularly if it is coupled to another property change in the same element. Moreover, the linear and cubic portions of the stiffness, and /cc , are not always easy to separate, adding further complications to the exact determination of a. As a simplification, a linearized version of the perturbed element matrix is obtained with the first term of a Taylor expansion about the baseline value: Ar 8ake  -a  a (Ake )  as  T=TO  (5.8) a=0  For a linear property Ake = Ake ; for a nonlinear property of the form (5.7), Ake^Ake = (Icezfr^31cec r 3 ) a  (5.9)  where the equality is justified for small a only. In practical computations, the derivatives in (5.8) can be evaluated numerically by calculating small perturbations around the baseline. Supposing the total number of design variables is m, the linearized element matrix perturbations are,  E ( -alar2—;e ) r •a^ • = E in^(k.,)  Aice = ln  ^j=1  2  a E^ , 171 (  3  =  a;^(5.10)  m  = E^(m.,.)  ^me^ (  ^Ark  i  :7=1  i a ;^(5.11) J1. Usually one would be concerned with property changes affecting groups of elements rather  ri a, uri^j=1 —^  than just individual elements. Indeed, a single design variable can be used to describe a property change of a whole range of elements. The following expressions for the overall  Chapter 5. Structural Dynamic Modification^  131  perturbations are obtained by applying (5.10) and (5.11) to (5.1) and (5.2): ^m  ^AK ^E  ^  =  m  E E^(lcer ) i Se a; = E (Kr ) ; a;^(5.12)  j=1^ j=1  Si^S  .  a•; E (m. ) a •j e (m ). apej^T3 e j=1^ j=1  AM = E E  (5.13)  Similar expressions for substructured systems can be developed with the help of (5.3) and (5.4). In this case, (5.12) and (5.13) describe property changes extending over a range of elements within a component. But the process can be taken one step further by considering changes extending over a range of components. The appropriate expressions are,  6,17  m  E Cg,.) aj  (5.14)  j=1^3  = E (sir ) i =1  (5.15)  CL) =dwher iag {(Kr ) ; } and (M T ) i is similar. 5.3 Structural Dynamic Modification with CMS A comprehensive review of structural dynamic modification techniques has been provided by Baldwin and Hutton [49]. They divide the various techniques into three classes: techniques for local modifications, techniques for small modifications, and techniques based on modal truncation. In Section 4.6, some attention was given to local modifications of substructured models within the context of CMS. The present discussion is relevant to modal truncation techniques. First, a brief review of the fundamental equations is given; and in the subsequent section the method is applied to substructured systems. The free-vibration equation of motion for a modified structure is, (K + AK) — (M + AM) Xi = 0^(5.16)  Chapter 5. Structural Dynamic Modification^  132  where A:, X: are the ith modified eigenvalue and mode shape of the structure. A direct solution of this equation gives accurate values for A:, X: at a cost equal to that of the original solution. To reduce this cost, a standard practice in forward and inverse modification is to use the original (baseline) modes as a basis for X::  x:  E xi ci; Xci  j.1  (5.17)  That the modified and baseline mode shapes can be represented in the same vector space is a consequence of the definition of a structural modification stated in Section 5.2. In (5.17), the projection of the modified mode on to the baseline modal space is given by the vector ci. With a complete set of baseline modes, this representation of X: is exact. However, in realistic situations the baseline mode set is inevitably truncated and usually includes just the low frequency modes. This introduces an approximation whereby (5.17) is suitable only if X: can be adequately represented within the confines of the reduced modal subspace. The strengths and weaknesses of this approximation are the subject of much discussion in the modal analysis literature [50, 51]. Applying (5.17) to (5.16) and premultiplying by X T gives, X T (K AK)Xci — A:X T (M AM) Xci = 0^(5.18)  Assuming mass normalized modes, this becomes, (c1 2 X T IKX) ci — (/ X T AMX) = 0^(5.19) This equation is, like the original equation (5.16), an eigenvalue equation but where now the eigenvector is ci. Determining ci defines the modified mode shape within the limits set by (5.17). The order of (5.19) is reduced because of the truncation of the baseline modes; as with CMS, the equations of motion are condensed at the expense of high-frequency information.  Chapter 5. Structural Dynamic Modification^  It is interesting to note that the procedure described in Section C.4 for calculating the wetted modes of a structure from its dry modes is a special application of (5.19). In that procedure, the fluid added-mass matrix serves as AM and the stiffness modification is zero. 5.3.1 Application to Substructured Models The free-vibration equation of motion formulated by a CMS method is, in its most general form,  where, k  A i /Cgs = 0  ^  (5.20)  = TT—KT ,^TT MT, and T is the transformation from the physical coordi-  nate system of the structural components to the generalized coordinates of the system. In the free-interface method, T takes on the form, T= 71) —^A [grill A1 -1A re;^=  (5.21)  In the fixed-interface method, it takes the form, T = [^c TA  =  I 3 9  Pn  (5.22)  Reformulating the CMS equations after a modification to the components gives, ^(k  ^ff  133  OK) — A: (if + AM);^= 0^(5.23)  where the modified condensed system matrices are defined by, Ak  =  T IT (K + AK) T'^ (5.24)  ^+ OM = (m +  AM) T 1^(5.25)  and where T' is the new transformation formed by substituting updated component modes in either (5.21) or (5.22). Because T' is dependent on the modification, a simple,  Chapter 5. Structural Dynamic Modification^  134  linear perturbation in the component stiffness or mass appears as a complex, nonlinear change in the condensed system matrices. As a result, it is difficult to predict the precise form of the modified CMS equation unless the component modes are recalculated. The treatment of the reanalysis problem in Section 4.6 included recalculation of the component modes. In this section, an approximation to (5.23) is derived which can be used without recalculating the component modes. The structural modification equation (5.19) can be directly applied to a substructured model once the mode shapes have been reconstructed from the baseline CMS analysis with the general relation, =n  (5.26)  A modal approximation appropriate for substructured systems is derived by using ui in place of Xi in (5.17):  = E = E T6 4; = TZci j=1^j=1  (5.27)  The kinetic and potential energies of the baseline system oscillating in the i th mode can be written as, 1^1^---  i AiXTMXi =^M  (5.28)  V = —1 XTKX. = —1 fiTKii. 2%^a^2s^s  (5.29)  T=  These expressions make use of the fact that the energy in the system is equal to the sum of the energy in all the components. Likewise, the energies of the modified system oscillating in the i t" modified mode are, T  = — A: ^+ AM) re  2  _I T  = u + Ax) t  ^  (5.30) (5.31)  135  Chapter 5. Structural Dynamic Modification ^  Applying (5.27) and Lagrange's equation to T' and V' gives the following equation of the modified structure: [0 2 + Z T T T A-ITTZ - of (./ Z T T T AMTZ)]ci = 0^(5.32) where now the necessary definition of mass normalization is, S2 2 = Z T T TICTZ I = Z T T T MTZ  ^  (5.33)  The number of modes available for the baseline set is at most equal to the size of the baseline CMS equations. Assuming that such a complete set is available, define as, (5.34)  = Z c.i The matrix Z is square and invertible; thus, =  (5.35)  Substituting (5.35) into (5.32) and premultiplying by (Z -1 ) T gives, [Z -T S2 2 Z -1^T T AKT -.A (Z -T Z -1^T T AMT)]  =0  (5.36)  With (5.33), this equation reduces to, [k^T T AKT - A: (if^T T AMT)] i = 0  (5.37)  Equation (5.37) is the same as the modified CMS equation (5.23) except that the transformation T' is replaced by its baseline counterpart T. As a result, linear perturbations in the structural matrices now only appear as linear changes in the global perturbation matrices, Ak and OM. Thus (5.37) can be described as the linear-equivalent equation for the modified system. Since (5.37) was derived assuming a full complement of CMS modes, it represents the optimal case for structural dynamic modification of a  Chapter 5. Structural Dynamic Modification ^  136  substructured system when no recalculation of the component modes is performed. Yet (5.37) can be derived independently of the number of baseline modes that are actually calculated; all that is required is the preservation of  k, Si and the baseline transformation  matrix T. Some advantages to using (5.37) are immediately apparent. The order of (5.37) is the same as the original CMS equation and so the reanalysis benefits from the same degree of condensation as the baseline analysis. Also, the component modes do not need to be recalculated to solve (5.37); instead only the matrix products TT LIKT and T T AMT need to be evaluated. For special types of modifications, the linear-equivalent equation gives an exact description of the modified structure. Obviously, if the modification is such that the system mode shapes are left unchanged, the modal truncation in (5.27) introduces no error and the results from (5.37) will be exact. However, a stronger statement than this can be made. Exact results will be obtained when the modal approximation (5.27) is capable of exactly representing the modified mode shape. Using (5.35), this condition is equivalent finding a 4 such that =  =  T  (5.38)  Modifications in which the columns of T' are linear combinations of the columns of T will satisfy (5.38) since, in that case, the columns of both matrices span the same space. In the free-interface method, this situation arises when a perturbed component mass matrix AM is proportional to the baseline component matrix M. Such a modification leaves the component modes unchanged except for an adjustment to the mass normalization factor in the free-interface modes (1). Thus, the columns of T' are scaled differently from those in T, but through (5.38), an exact description of the modified mode shape can still be obtained using T. An example is given in the next section which shows that in special  137  Chapter 5. Structural Dynamic Modification^  cases exact results can also be obtained when adding lumped masses. 5.3.2 Numerical Results In this section, four examples are presented which compare the linear-equivalent and CMS reanalysis methods. While the linear-equivalent equation (5.37) is in a general form which can be applied to either the fixed- or free-interface CMS methods, the examples here use results from the free-interface method only, because of its superior condensation. Consider the example used in Section 4.6.1 where the modes of the container ship were determined after a modification to the stern stiffeners (groups 1 and 2 in Figure 4.8). The same modification is now made, but the natural modes are recalculated using the linear-equivalent equation. The ninth modified mode was found to be  A = 6.317Hz,  whereas in Table 4.22, it was predicted to be 6.235Hz. The descrepancy is explained by the nature of the approximation in the linear-equivalent equation. In the derivation, the modified mode shape is projected on to a modal subspace spanned only by CMS-derived modes; the large number of higher frequency modes that cannot be calculated with these equations are ignored. This effectively puts constraints on the modified structure which raise its natural frequencies. The baseline modes are not an effective basis for the modified mode because the latter is a local mode in the stern and superstructure, as was shown in Figure 4.7. On the other hand, the majority of the baseline modes are vertical bending modes in the hull which are not very useful for characterizing localized displacements. To show better the effectiveness of modal truncation, consider raising the elastic modulus of the stiffeners located at the bottom of hull (groups 4, 5, and 6 in Figure 4.8). These stiffeners extend from the bow to the stern and are used to partly simulate the bending stiffness of the hull bottom. Changing these stiffeners requires modifications to three components, and so the change is not nearly as localized as in the previous example. The effect of these stiffness changes on the fundamental flexible mode (h) is  138  Chapter 5. Structural Dynamic Modification ^  0.975 N 3, 0.950  8 0.925  -  —0--- Linear-equivalent CMS reanalysis  0.900  0.0^0.5^1.0^1.5^2.0^2.5  ^  3.0  ^  3.5  Fractional change to elastic modulus  Figure 5.1: Reanalysis results for modifications to hull-bottom stiffeners shown in Figure 5.1. An excellent match is obtained between the linear-equivalent and CMS reanalysis results. Because the fundamental mode is the 2-node bending mode of the hull, changes to this mode are well approximated as a linear combination of the baseline modes. The cost of the linear-equivalent method compares very favourably with the CMS reanalysis results. The average time for reanalyzing 10 modes in a CMS reanalysis is 1150 seconds; with the linear-equivalent method the average time is 43 seconds. Another example is furnished by Section 4.6.2 in which the telescope focus unit was reanalyzed after a modification to the bearing stiffnesses. Figure 5.2 shows the variation in the second and third frequencies (which are identical) resulting from modifications to the bearing stiffness. The fractional change a refers to the cross-sectional area of the bar  Chapter 5. Structural Dynamic Modification ^  139  82.5 171 x -..... .;" 80.0 0  a =e. =  g 77.5 —  E  —o— Linear-equivalent (free CMS)  z  75.0  —0— CMS reanalysis  0.0^0.5^1.0^1.5^2.0^2.5 Fractional change to cross-sectional area of stiffeners  ^  3.0  Figure 5.2: Reanalysis results for modifications to bearing stiffnesses  Chapter 5. Structural Dynamic Modification ^  140  elements used to model the bearings. The inaccuracy of the linear-equivalent method is caused by the inability of the truncated baseline mode set to describe the modified mode. While the linear-equivalent method is more accurate for small changes, it should be emphasized that it is not a "small modification" method in the same sense that the sensitivity method is; the method is valid for large and small changes alike, as long as the modal subspace is sufficiently large to describe the mode shape changes. As a third example, consider again the second and third modes of the TFU. Both of these are swinging modes affecting the inner tube and chopping mechanism. Consider now the effect of adding mass to the chopping mechanism. Four equally-distributed and identical lumped masses of varying magnitude are added to the bottom of the component. As this addition increases the kinetic energy in these two modes, the frequencies should drop. This is verified in Figure 5.3, where it can also be seen that the linear-equivalent method and the CMS reanalysis predict identical results. This is explained by noting that the chopping mechanism is modelled as a block of aluminum solid enough for its dynamics to be represented by rigid-body modes and static flexibility. Because of the component's rigidity, adding lumped masses has an insignificant affect on the static modes. The effect on the rigid body modes is only to adjust the mass normalization factors, and consequently the modified matrix T' can be represented as T'  =  TD  (5.39)  where D is a general diagonal matrix. In such situations, exact representations of the modified mode shapes are obtained using (5.38). Not only the second and third, but all of the modes should be predicted with equal accuracy. This underscores the second interpretation of the linear-equivalent equation: it provides an equivalent CMS formulation of the modified system under the restriction that linear combinations of the columns of T provide an accurate description of the modified mode shapes.  141  Chapter 5. Structural Dynamic Modification^  72.5  i t7 70.0  —o— Linear-equivalent a-- CMS reanalysis  —  •. ..— .._ -; co a) —  0  S  ci  m  67.5  I E  m 65.0  62.5 0.0^1.0^2.0^3.0  4.0  ^  5.0  Lumped added mass (Kg)  Figure 5.3: Reanalysis results for lumped-mass additions to the chopping mechanism  Chapter 5. Structural Dynamic Modification ^  142  5.4 Perturbation Methods for Inverse Modification  Attention is now turned to inverse modification, which is concerned with determining a set of design changes which satisfy some prescribed modal constraints. To perform such an analysis, the design variables must first be selected from among the allowable property changes. As was mentioned in the introduction, the modal constraints may consist of frequency goals or mode shape constraints. In the present discussion, only frequency goals are considered; extra difficulties arise with mode shape constraints which need not be discussed in this treatment. 5.4.1 Background  The original work by Stetson [52, 53, 54] in this area used an equation of motion of the modified structure in the following perturbed form: (X + OX ) T (K OK) (X AX) = (X + OX ) T (M OM) (X AX) (02 + An2)  (5.40) Two basic approximations were made: in the expansion of (5.40), all nonlinear incremental terms were deleted; and the perturbed modal matrix AX was approximated as the linear combination of the baseline modes, AX = XC^  (5.41)  where C is a square matrix of admixture coefficients in which the diagonal elements 0. Note the slight difference between (5.41) and (5.17). With these two approximations, Stetson derived the linear perturbation equations, XTAKXi — AiXTAMXi = Mi AAi for i = j^(5.42)  — A i ) for i j^(5.43) =^for Mlci7  Chapter 5. Structural Dynamic Modification^  143  Equation (5.42) is used for frequency modification, (5.43) for mode shape modification. A prescribed frequency shift is defined by AAi, a prescribed mode shape change by cii. The equations can be solved after expressing the AK, AM perturbations in terms of unknown design variables a, as was described in Section 5.2. The solution a represents a first-order estimate of the design change satisfying the dynamic equation of the modified structure. Sands&Om and Anderson [55] presented a similar formulation, but expressed the prescribed mode shape shift explicitly, rather than with admixture coefficients. Kim et al. [56] proposed a method in which the modified perturbation equations were solved by mathematical programming, while retaining all the nonlinear terms in (5.40). Minimum weight solutions were found with the aid of a starting vector. A general dynamic reduction method combining static condensation with subspace iteration was used to compress the perturbation equations for large-order models [57, 58]. Hoff et al. [59, 60] proposed a two-stage predictor-corrector method for frequency modification which is more suitable for large structural changes. The predictor phase estimates the design changes and the perturbed mode shapes using the linear perturbation equation. In the corrector phase, the general perturbation equation (5.40), incorporating the estimated mode shape perturbations, are solved in an attempt to improve on the predictor phase results. Welch [61] reported that the predictor-corrector scheme is adequate for problems with linear property changes but is unable to predict nonlinear property changes accurately. To improve the performance for large modifications, Bernitsas and Kang [62] used the predictor-coirector incrementally to calculate a sequence of small steps which cumulatively result in the complete solution. Gans and Anderson [63] adapted the predictor-corrector method for systems with significant centrifugal and coriolis forces. The predictor-corrector method has also been implemented in the finite element program INSTRUM [64].  Chapter 5. Structural Dynamic Modification ^  144  The approach taken by Smith and Hutton [65] is somewhat different. The free vibration equation of the modified system (5.40) is premultiplied by the baseline modes rather than the modified modes. This gives the following equation which bears a strong resemblance to the forward modification equation (5.18): X T (K + AK) XC = X T (M + AM) XC (Se Af2 2 ) (5.44)  The advantage of using (5.44) is that the number of nonlinear incremental terms is less than in (5.40), and this facilitates the solution of inverse modification problems without further approximation of the equations. Smith and Hutton described an interative method for calculating design changes exactly satisfying (5.44) in the presence of prescribed frequencies. The only limiting approximation is the truncation of the baseline mode set in (5.17). In a parallel development of perturbation methods, Ram and Braun [66] determine optimal perturbing stiffness and mass matrices for a particular modal subspace. The subsidiary problem is then to relate the optimal perturbation to physical changes in the actual model, but this was not investigated by the authors. The philosophy adopted in the present discussion is that all possible design solutions should be chosen from a set of perturbing matrices that are defined by a set of design parameters, as described in Section 5.2. As the design parameters are chosen by the analyst, this approach requires greater engineering judgment but at the same time affords greater flexibility in satisfying the modal constraints. 5.4.2 Application to CMS  Little research work has as yet made use of CMS formulations in inverse perturbation studies. Linearized perturbation equations for the fixed-interface CMS method have been developed which are suitable for uniform stiffness and mass changes to the components  Chapter 5. Structural Dynamic Modification^  145  [67]. The present discussion is concerned with ways more general structural changes can be predicted, and is an adaptation of the method described in [65]. The basic perturbation equation for substructured systems is given by (5.23). To begin the development, consider a frequency modification problem having one frequency constraint, .X = A:. The task is to find the design variables a and the modified eigenvector 4: satisfying (5.23). While the characterization of AK and AM in terms of design variables is straightforward, a complication arises here because the transformation T' is also a function of the design variables. The following frequency modification equation results: 7111- (a) (K + AK) r(a) : = A:T 1T (a) CH T'(a) (5.45) The equation is clearly nonlinear in a. Moreover, the functional dependence of T' on a is not exactly known; for the relationship between the dynamic modes in T' and a can only be expressed with the help of approximate sensitivity techniques; and the static modes in T', because they are calculated with an inverted stiffness matrix, will generally have a complicated relationship to the structural properties. Consequently, the frequency modification equation is exceedingly difficult to formulate to its full extent. Instead, consider the practice used in perturbation methods of expressing the modified mode shape as a linear combination of a truncated mode set, as in (5.17). In Section 5.3.1 it was shown that if the baseline mode set included the maximum number obtainable from a CMS analysis—that is, when the number of baseline modes equalled the order of the CMS equations—the linear-equivalent perturbation equation (5.37) results. Applying the frequency constraint as = A: to that equation gives, TT (-1f– AK) 71 e = A; TT (M AM) Te^(5.46) Because the baseline transformation T is invariant with respect to a, the linear-equivalent frequency modification equation (5.46) is greatly simplified from (5.45).  Chapter 5. Structural Dynamic Modification ^  146  The baseline mode set is truncated further by the substitution, q  1;; =EZ ci = Zci  j=i  ;  ;  (5.47)  where now q is less than the order of the baseline CMS equations. Premultiplying (5.46) by ZT gives, ZTTT AK(a)T^= A: [i Z T T T AM(a)TZ] c i^(5.48)  This equation is exactly analogous to the equation derived for unsubstructured systems in [65]. Solution Algorithm An algorithm is now given for calculating design solutions satisfying (5.48). This algorithm is applicable with linear design variables, though by linearizing nonlinear variables accurate solutions can still be obtained for small fractional changes. Consider the ith row of (5.48): ZTTT [A -17(a) — A:AM(a)1TZc i + (A i — A:) = 0^(5.49)  This is called the design equation because it is used for determining the design variables a satisfying the frequency constraint A: = A:. The remaining q —1 equations in (5.48), Z T T T [6:K(a) — A:AM(a)1T Zci (A ; —^= 0^j = 1,2,...,q; j i (5.50)  are the admixture coefficient equations. They are used to determine the parameters c i; defining the modified eigenvector Unfortunately (5.49) and (5.50) cannot be solved simultaneously; unknowns a and ci; are coupled in both equations. A solution is found with the following iterative algorithm, in which t designates the current iteration. This algorithm is exactly analogous to the one proposed for unsubstructured systems in [65].  Chapter 5. Structural Dynamic Modification^  147  1. Let t = 0, cT ) = 1, dilc ) = 0 for j = 1, 2, ... , q; j^i. 2. Calculate the design variables satisfying the frequency modification equation (5.49); i.e., solve ZTTT [AK(a(t-I-1)) A:AT/02(t+1) ) ] TZc! t) + (Xi —^= 0^(5.51)  (ait+i), ar i ) , T for a( t + 1 ) =^a(„+1)) . Note that if m = 1 the solution of this equation is unique. When m > 1, the solution is underdetermined and it is necessary to use optimization. A description of the optimization algorithm is given below under the heading "Mathematical Programming". 3. Calculate the eigenvector change satisfying (5.50) subject to the design modification a(t+1); i.e., solve ZTT T {rf(ce (t + 1) ) — A:AM(a (t + 1) ) ] TZe l) + (Ai —^= 0 j = 1, 2, . , q; j^i^(5.52)  for_^1) r ci  = ( c it+1) c( 2t-I-1)^cltq-1-1)1 T ( )  and normalize such that licl t+1) 11 2 = 1. In this  step, q elements of ci(t+i) are determined from q — 1 equations. Because cl t+1) can only be specified to within a common multiplicative factor, one of its elements can arbitrarily be assigned unit value. With this substitution, (5.52) becomes a set of q — 1 equations in q — 1 unknowns. To avoid numerical difficulties, it is best to  assign a unit value to the largest element in cl t+1) , which in most cases is 4 +1) . 4. Set t = t + 1 and repeat Steps 2, 3 and 4 until a(t+ 1 ) = a( t ). In Step 1, the initial assumption made is that the modified eigenvector is no different from the baseline. This is the same assumption used by Stetson in his derivation of the linear perturbation equation. However, in subsequent iterations the above algorithm  Chapter 5. Structural Dynamic Modification^  148  differs from approach taken by Stetson and others. In (5.51) and (5.52), all of the coupling terms between a and ci are retained and the only supporting approximation is the truncation of the baseline modes in (5.48). Mathematical Programming  When more than one design variable have been defined, the solution of Step 2 is underdetermined and an infinite number of solutions are possible. To resolve this, mathematical programming is used to solve (5.51). In this study, a penalty function method employing a minimum change objective is used. This involves minimizing the functional, ng F* = a T + iCi RT R ItEl/g; (5.53) The first term is a measure of the overall change that is to be minimized. Minimum change solutions are often preferable to minimum weight solutions in redesign problems and are less liable to produce pathological solutions [63]. The second term is an external penalty function where R is the residual of the equality constraints. In this case R is just the residual of the design equation: R = ZTT T [AK(d+ 1) ) — A:AM(a (t + 1) )] TZc! t) + (Ai — (5.54) The third term of F* is an interior penalty function in which g; > 0, j 1, 2, ... , ng are inequality constraints limiting the feasible domain of the design variables. For instance, a; < —1 are physically impossible, and so  ai + 1 > 0 j = 1,2,...,m (5.55) are necessary inequality constraints. Further constraints on a can be added as desired. The use of this type of penalty function is discussed by Haftka and Kamat [68]. For it to be successful, the starting point (a = 0) must be in the feasible region, otherwise the interior penalty function cannot work.  Chapter 5. Structural Dynamic Modification^  149  The minimum value of F* is calculated using a quasi-Newton algorithm with a weak line search and a BFGS update, a description of which is given in [69]. The factor it is chosen based on an initial estimate of the minimum of F*. In the course of the quasiNewton algorithm, is decreased by a factor of 5 after each iteration, progressively de-emphasizing the inequality constraints in favour of the equality constraints. It should be emphasized that mathematical programming is only used in Step 2 and therefore constitues just one part of the larger algorithm. By contrast, Kim [57] uses a mathematical program to solve the full set of dynamic equilibrium equations. The unknowns include both the design variables and all the unspecified components of the perturbed mode shapes, thus making the number of unknowns very large for complex models. The procedure adopted in this study is simpler in that design variables alone are determined through optimization; modified mode shapes are subsequently determined from the remaining perturbation equations. Modal truncation allows the iterative method to be executed inexpensively even for large-order complex systems. Multiple Frequency Constraints The above algorithm is easily adaptable for simultaneous multiple frequency constraints. If 1 frequencies are prescribed, 1 sets of q equations are obtained by substituting the appropriate index i in (5.48). One design equation is selected from each set, giving 1 equations with m unknowns in Step 2. A solution is possible when m > 1, provided the 1 design equations are independent; a unique solution is possible when m = 1, and mathematical programming is used when m > 1. In Step 3, the mode shape perturbation is obtained by solving each set of the 1 sets of q — 1 equations separately. Generally, the number of frequency constraints is small compared to q. Thus, the number of additional equations is not too large to jeopardize the numerical efficiency of the method.  Chapter 5. Structural Dynamic Modification^  150  5.5 Sensitivity Analysis; Newton's Method for Inverse Modification  5.5.1 Background Many developments in dynamic optimization have been based upon sensitivity methods [68]. General discussions of sensitivity methods in structural dynamics are given by Adelman and Haftka [70] and Brandon [71]. The basic sensitivity equations were derived originally by Lancaster [72] and Fox and Kapoor [73]. For a general, undamped, discrete structure with distinct eigenvalues, these equations take the form  aA, _ t T a k^ A ai k ar ` Or — s Or —  3T  [k As k]ar  (5.56)  31 [  T^511;11  4.3 LaT A a7.1  (5.57)  t  where derivatives are taken with respect to a structural property r, and where the eigenvectors are assumed to be mass normalized. Eigenvalue derivatives are readily obtained with (5.56). Fox and Kapoor [73] and Nelson [74] established methods for calculating eigenvector derivatives using (5.57). Rudisill [75] investigated adding second-order terms to the sensitivity equations. Johnson and Jen [32] calculated sensivities for a multi-link robot analyzed with a CMS procedure based on monomial functions. A complete sensitivity analysis of the fixed-interface CMS formulation was given by Heo and Ehmann [76]. Several researchers proposed solving inverse eigenvalue problems with a Newton's method iteration approach [77, 78, 79]. The equations studied were of the form 441i = AM;^A = Ao  E  AkCk^  (5.58)  k  An adaptation of Newton's method to discrete structures was shown to involve repeated solution of (5.56) combined with accurate updating of modal and structural parameters  Chapter 5. Structural Dynamic Modification^  151  [80]. To avoid the expense of repeated eigensolutions and the truncation errors inherent to the modal subspace approximation, inverse iteration is used to update the eigenvectors. 5.5.2 Newton's Method for Substructured Problems  A Newton's method application of the sensitivity equations is suitable for frequency modification problems. Using the general CMS formulation for the baseline system (5.20), the following expressions are obtained:  a k^ T OK^aT (..., Ty aT)T = T T + T T-K-I- T Or^ Tr  Or  ail^aM^p_aT)T ail = T T --T + T - M— + T - M— ar^ar^ar^ar  (5.59) (5.60)  The structural property r may refer to a property of a particular component or one extending over a range of components, and it can be either linear or nonlinear. Derivatives of K and M can be evaluated by assembling derivatives of element matrices. Derivatives of T are more difficult to express as they involve rates of change of component eigenvectors. In the derivation of the linear-equivalent equation in Section 5.3.1, it was shown that a useful simplification is to treat T as an invariant transformation. This is justified when components change in a uniform manner, and for more general changes it allows component modes to change within the subspace provided by the baseline analysis. A completely accurate prediction of the component mode derivatives by methods proposed in [73, 74], is much more costly and therefore diminishes the value of the inverse modification procedure. The resulting linear-equivalent sensitivity equation is given by OA,^Lrai^T a m T^ T - T^Ti^(5.61) Or^Or  Chapter 5. Structural Dynamic Modification ^  152  When multiple properties are allowed to change simultaneously, the total perturbation to the eigenvalue is estimated with a first-order Taylor expansion about the baseline values: A: = A i +  aA, rkak^ E^ k=1 ark  (5.62)  where the derivatives are evaluated at the baseline property values. Substituting (5.61) in (5.62) for an appropriate rk gives A: — a t =^[TT aK T —  T Tan7I I T14'irkak^(5.63)  ark^ark  The frequency modification equation is obtained by substituting A: = A: in (5.63). Design variables ak which solve (5.63) give a first-order estimate of the design changes satisfying the linear-equivalent equation. For a single design variable the solution of (5.63) is unique; with more than one design variable, this equation can be solved using the mathematical programming technique suggested in Section 5.4.2. In the more general case of I prescribed frequencies, 1 equations similar to (5.63) are solved simultaneously.  Succesively more accurate solutions can be obtained by solving (5.63) repeatedly, updating the quantities between iterations. Three methods are available for updating the eigenvector (1) re-analyzing the modified system with CMS; (2) using the expansion (5.47) and then calculating the admixture coefficients c i from the modified dynamic equation; (3) using a single step of inverse iteration [80}. This requires solving the following linear system for -yi:  [k + Ak - (/Cf + AM. )] -yi =  (5.64)  where the perturbation matrices are given by:  OK = EM  E (x m  r k  ak  (R.)  ak  )  k=1 m  k=1^k  a-K =^ k=1 ark rkak  (5.65)  am = m^ rk a k  (5.66)  ark  Chapter 5. Structural Dynamic Modification ^  153  The updated eigenvector is obtained by mass normalizing = [7r ( 1171^(5.67) The eigenvalue is updated with Rayleigh's quotient: (k k) = ^_^s^ V (M M)  (5.68)  .  With nonlinear property changes, the derivatives in (5.63) must be recalculated after each iteration. Inverse iteration has also been used to update eigenvectors in a forward modification procedure called Rayleigh quotient iteration [81]. Using inverse iteration to update is particularly advantageous with the compact CMS formulation. The order of (5.64) is small and it can be solved in much less time than is needed for a CMS reanalysis of the eigenvectors. Another advantage of inverse iteration is that the accuracy does not depend on the number of baseline modes available. Indeed, the Newton iterative cycle can be carried out without any knowledge of the baseline modes, excepting those with frequency constraints. The Newton procedure converges quadratically provided the starting point (baseline) is not too far removed from the solution. The convergence of the process is improved by substituting Ai = A: in the right-hand side of (5.63). Smith and Hutton [80] showed with an example of an unsubstructured model, that the zone of convergence is sufficient for engineering purposes. Solutions found with this procedure satisfy the linear-equivalent equation, not the actual equation of the modified structure. At this point the analyst should check the accuracy of the predicted solution with a CMS reanalysis. Using the reanalysis as the new baseline, further frequency modification calculations can be done as required. Newton's method is expected to give equal accuracy to the perturbation method when the latter is supplied with a full set of baseline modes. Because of the impracticality of this  Chapter 5. Structural Dynamic Modification^  154  requirement, Newton's method generally gives somewhat better results. Although both are iterative methods, the nature of the iterations is fundamentally different. Newton's method converges to a solution through a sequence of linear steps, the length of which progressively diminishes to zero. On the other hand, the perturbation method solves the same system of equations repeatedly, each time making adjustments to the coupling terms, until all the equations are satisfied simultaneously. 5.6 Numerical Results In this section, the results of three frequency modification problems are presented. In each example, a single frequency goal is varied over a range of values. Three inverse modification methods are compared: the Newton's method algorithm of Section 5.5, the new perturbation algorithm described in Section 5.4, and the predictor-corrector method. The two perturbation methods differ from Newton's method in that they generally operate within a smaller subspace. This means that the perturbation methods are less capable of accurately predicting the modified mode shape. The two perturbation methods differ from each other in that the predictor-corrector uses a linearized perturbation equation to predict the modified mode shapes of the structure. With small structural changes this is adequate; but with large changes, inaccurate modified mode shapes lead to erroneous design changes. A further difference is that the iterative method calculates the design changes using (5.51) alone. There is one such equation for every frequency constraint. On the other hand, the predictor-corrector method attempts to satisfy one equation associated with the frequency constraint plus q —1 other equations associated with mode shape constraints. The latter are necessary conditions for making the predicted mode shapes orthogonal with respect to the corrected structure. A variation of this method is to satisfy both the necessary and sufficient conditions  Chapter 5. Structural Dynamic Modification^  155  for orthogonality by using a total of q(q — 1) constraint equations [60, 62]. In either case, the mode shape constraints are somewhat artificial in that they are not prescribed in the original specification of the problem; the only genuine constraint to be satisified is the frequency constraint. Furthermore, the imposition of these extra constraints is apt to make a naturally underconstrained problem overconstrained, thus necessitating a minimum error rather than a minimum change solution [59]. In such cases, Welch [64] suggested relaxing the mode shape constraints, and using only equations associated with the frequency constraint to determine the design variables. This is the approach used for the predictor-corrector results in the following examples. Consider the second example presented in Section 5.3.2 in which modifications are made to the elastic modulus of the hull-bottom stiffeners in the container ship. Taking the elastic modulus as the design property, frequency constraints are applied to the fundamental elastic mode (14). The results using the new perturbation method for five different modal approximations are shown in Figure 5.4. The one-mode approximation, which does not allow for any mode shape correction, produces a linear curve. The effect of additional modes in the approximation is the inclusion of correction terms in the frequency constraint equation. These correction terms enable the prediction of progressively more accurate design variables and mode shape changes. As the number of modes approaches the order of the CMS equations, the results should converge to the linearequivalent curve. But this process converges slowly, and with 30 modes the optimal curve has still not been obtained. Figure 5.5 shows the results of the new perturbation method and the predictorcorrector, both using a 30-mode approximation, in comparison with Newton's method and CMS reanalysis results. The two perturbation methods show almost identical results, and both match the CMS reanalysis reasonably well, although Newton's method  156  Chapter 5. Structural Dynamic Modification ^  --.---6------  0 = •  .■....07.■•■  3.0  cs co as 0  0—  —  ■,..m.■4:1■■■■  Linear-equivalent 1 mode 5 modes 10 modes 20 modes 30 modes  .9  a) 2.0 co c co .c 0 T C .2 t Es 1.0 LL  0.0  0.900^0.925^0.950  ^  0.975  ^  1.000  Prescribed frequency f: (Hz)  Figure 5.4: Frequency modification of hull-bottom stiffeners for five modal approximations  157  Chapter 5. Structural Dynamic Modification^  0.0  0.900^0.925^0.950  ^  0.975  ^  1.000  Prescribed frequency f; (Hz)  Figure 5.5: Comparison of three methods for frequency modification of hull-bottom stiffeners  158  Chapter 5. Structural Dynamic Modification ^  6.0  5.0 E o 4.0 =; c0 CD CD 0  0 3.0 c .c 0  Ta c 2.0 .9 t  u.  1.0  0.0  0.900^0.925^0.950  ^  0.975  ^  1.000  Prescribed frequency f; (Hz)  Figure 5.6: Three-parameter frequency modification of hull-bottom stiffeners does slightly better. Clearly, the modified mode is fairly well approximated by the 30mode subspace. Also, because the stiffeners are changed uniformly using only one design variable, the mode shape change is small. Thus, the linear perturbation equation used in the predictor-corrector method is sufficient for predicting the modified mode shape. Consider applying the same frequency constraints, but where now the stiffeners belonging to each component can vary independently. In Figure 5.6 a 30-mode approximation has been used for the perturbation results. Design variable a l is the fractional change of the elastic modulus of the stiffeners in the after-body, a 2 the fractional change in the mid-body, and a3 the fractional change in the fore-body. The frequency constraint equation is solved using optimization with a minimum change objective. A description of the  Chapter 5. Structural Dynamic Modification ^  159  optimization algorithm is given above under the heading "Mathematical Programming". The analysis produces fairly accurate results for two of the methods. A reanalysis based on the design prediction for  g = 1.00 gives the following results: Newton : f: = 0.996Hz  New perturbation : f4  =  0.995Hz  Predictor-corrector : fl = 0.931Hz  Because the stiffeners are being changed in a nonuniform manner, the mode shape changes associated with these modifications are more significant. Thus, the linearized perturbation equation is not adequate for predicting the perturbed mode shape when the design change is very large. This accounts for the poor performance of the predictor-corrector method in the region where large design changes are required. It is interesting to note that although both Newton's method and the new perturbation method predict changes giving similar frequencies, the distribution of the change is significantly different. Of many possible solutions, both methods seek a minimum change solution. The magnitude of the change is, for the f4 = 1.00 results,  E a ! = 38.4 New perturbation : E cz = 26.3 Newton :  l  2 k  The perturbation method is more economical because in each step it determines a minimum solution for the total modification. By contrast, Newton's method determines a minimum partial solution in each iteration. The accumulation of these partial solutions does not necessarily lead to a minimum total change. In Figure 5.6, Newton's method has weighted the mid-body component too heavily at the expense of the fore-body, giving a modification that may be more difficult to realize in the actual structure.  160  Chapter 5. Structural Dynamic Modification^  12.5  Predictor-corrector Perturbation Newton, baseline a 0 Newton, baseline a 4.48 CMS Reanalysis -  -  E  Co  Cu 7.5  2 0 LL 01 2.5  0.0  0.900^0.925^0.950  ^  Prescribed frequency f4 (Hz)  0.975  ^  1.000  .  Figure 5.7: Frequency modification with mid-body stiffeners Now consider changing only the stiffeners in the mid-body component. The predicted design changes for prescribed fundamental-mode frequencies are shown in Figure 5.7. The effect of two separate approximations is clearly visible. In the region f: > 0.925Hz, the predictions based on the approximate methods begin to diverge from the CMS reanalysis curve. This is the influence of modal truncation. Both perturbation methods use a 35 mode approximation but since large design changes are being made along just one segment of the hull, the mode shape change is severe and is not easily represented within the 35-mode subspace. Also in this region, the Newton's method curve begins to diverge from the perturbation curves. The better accuracy obtained with Newton's method results from using equations of larger order (62 degrees of freedom versus 35 for the  Chapter 5. Structural Dynamic Modification^  161  perturbation equations). This enables the modified mode shape to be represented more accurately, although over the range of prescribed frequencies presented, the extra degrees of freedom make only a small improvement in the predictions. In the region f: > 0.95Hz, the new method begins to diverge from the predictorcorrector curve as a result of errors in the latter stemming from the linear perturbation equation. The new method is able to follow the general trend of the CMS reanalysis curve, if lagging behind it somewhat, whereas the predictor-corrector method fails to predict any design changes above a = 7.60. The shape of the predictor-corrector curve is similar to those in Figure 5.6; the peak and decline in regions of high a and large mode shape change signal the neglect of perturbation terms in the energy of the modified structure. It should be noted that in the region f: > 0.98Hz, the slope of the CMS reanalysis curve approaches infinity, which means that this frequency becomes insensitive to changes in the mid-body stiffeners. In this region, the mid-body stiffeners are completely rigid in the fundamental mode, a design change which in practical terms would be impossible to make. Because of the increasing insensitivity of the mode, predicting design changes in the region just below f: 0.98 is very difficult, regardless of the number of baseline modes available. It may be useful in situations like this to perform an inverse modification analysis in two or more steps, with an accurate reanalysis between each step. Because only the mid-body component is changed, a CMS reanalysis can be done fairly inexpensively and each reanalysis provides a new reference point, or baseline, for the subsequent perturbation analysis. In Figure 5.7, a second set of Newton's method results are shown using a = 4.48 as the baseline. This was the result obtained for f4 = 0.975Hz in the first perturbation analysis. Reanalysis for a = 4.48 gives  4  0.966Hz. From this baseline,  new design changes are predicted, where the fractional changes now refer to the second  Chapter 5. Structural Dynamic Modification ^  162  baseline. The overall change for multiple perturbation analyses is given by, oT ( 1 +^(1 + a (21)^( 1 + a(n)) — 1 ^(5.69)  where a(j is the change predicted from the j th analysis. In this way, progressively more )  accurate design changes can be calculated. 5.7 Summary Structural dynamic modification has been treated as two separate problems: forward modification, which is concerned with the modal analysis of a modified structure; and inverse modification, which is concerned with finding a set of design changes which satisfy prescribed modal constraints. In this chapter, perturbation methods for structural dynamic modification have been applied to substructured systems, using a CMS formulation for the baseline analysis. Approximating the modified mode shapes by projecting them on to a modal subspace, it was shown that if the size of the subspace equals the order of the CMS equations, the linear-equivalent equation (5.37) results. This equation represents the optimal description of the modified structure that can be obtained without considering changes to the component modes. Forward modification problems can be directly and efficiently solved using the linear-equivalent equation. A new method for frequency modification problems has been presented which uses the energy-balance formulation of the perturbation equations. An iterative scheme is used which converges to a solution of the full perturbation equations. Examples have been presented which show thit the new perturbation method compares favourably with the predictor-corrector technique when large structural changes occur in conjunction with significant mode shape changes. Also, a Newton's method algorithm based on the linearequivalent equation has been described which gives a sequence of converging solutions which do not depend on the number of baseline modes available. Examples given show  Chapter 5. Structural Dynamic Modification^  163  that Newton's method is generally more accurate than perturbation methods, as modified mode shapes are represented using a larger subspace.  Chapter 6 Summary and Conclusions  Component mode synthesis (CMS) has been studied as a modal analysis technique for linear, undamped, discrete models. Of principle interest is the application of this technique to large-order finite element models of an arbitrary complexity where condensation is most beneficial. It is often natural and convenient to treat a finite element model as an assemblage of structural components. CMS generates reduced-order representations of the components by approximating their displacement with a truncated sequence of component modes. The Craig-Bampton and MacNeal-Rubin representations are found to be the most applicable to arbitrarily complex models as they best satisfy the basic requirements for component mode sets. By expressing the compatibility and equilibrium constraints in terms of physical displacements and loads, free vibration equations of motion are derived for each mode set. These two formulations are referred to respectively as the fixedand free-interface methods. Two variations of the free-interface method are presented in which first- and second-order mass approximations are used. It was shown that these formulations can be applied to discrete models of an arbitrary geometrical complexity, and that they handle constraint equations more efficiently than the direct elimination method. Important differences distinguish the fixed- and free-interface formulations. The fixedinterface equations are in terms of interface displacements and free vibration modal coordinates; the free-interface equations are in terms of free vibration modal coordinates 164  165  Chapter 6. Summary and Conclusions^  alone. This gives the latter a higher degree condensation in complex models, particularly when component interfaces are meshed curves or surfaces; because as the mesh is refined, the order of the fixed-interface equations steadily increases, while that of the free-interface equations stays the same. Consequently, free-interface CMS is preferable for the majority of large-order structural models; only in cases where interface coordinates are limited to a small number of discrete points is the fixed-interface method favourable. Also the second-order mass formulation of free-interface CMS is generally more effective than the first-order formulation, as significantly better results are obtained with little additional cost. This is particularly true for components with high modal density in the target frequency range. Another factor affecting the performance of CMS is the tolerance to which eigenvalues are calculated. As the tolerance is reduced, the relative efficiency of CMS methods increases in comparison to a direct finite element analysis. Moreover, when components have high modal density, significant numerical loss of precision can result from not using a sufficiently small tolerance in determining the component modes. In particular, this is a problem in the free-interface method when loss of precision causes component residual flexibilities to become ill-conditioned. For this reason, a tolerance of about 10  -6  is rec-  ommended for the component-level eigensolutions using the inverse power method with shifting. CMS can also be used advantageously when multiple variations of the same model are analyzed. For each variation, only components that have changed must be reanalyzed. Therefore, the efficiency of each analysis increases as the structural change becomes more localized. This is especially true for the free-interface formulation where, for large-order models, a greater emphasis is placed on component-level computations. If only one or two of the structural components are changed, a much larger proportion of the computational time is saved than would be with the fixed-interface method. Under these circumstances,  Chapter 6. Summary and Conclusions^  166  it is advisable to use the free-interface method. The results obtained with CMS depend on the care with which component modes are selected. Truncation of the component mode sets is necessary for order-reduction, and generally it is the high frequency modes that are eliminated. To distinguish between low frequency and high frequency modes, a cutoff frequency criterion is used. It is shown in Chapter 4 that fairly uniform convergence in the low frequency modes can be achieved in this way. Selection of the cutoff frequency requires some judgement, but generally it should be 1-2 times higher than the upper limit of the target frequency range. The extension to reanalysis and re-design problems gives CMS a wide applicability. In the derivation of the linear-equivalent equation, modal truncation is performed at the component level (i.e., in the baseline CMS analysis), not the system level. This gives a more easily adaptable description of the modified structure, as subsequent versions are derived by updating the component modes rather than the system modes. The linear-equivalent equations also provide a basis for developing the frequency modification equations. Two methods are developed: a Newton's method algorithm originally used for unsubstructured models is applied to the linear equivalent equation; and a new iterative method is proposed for solving the energy-balance perturbation equations, in which all coupling terms are accounted for. In the examples presented, Newton's method gives slightly more accurate results, even though it does not require a large baseline mode set. The iterative solution of the perturbation equations also exhibits superior performance to the predictor-corrector method when large design and large mode shape changes occur simultaneously. The accuracy of the structural dynamic and inverse modification techniques varies with the character of the structural change. Widely distributed modifications of limited magnitude can be accurately represented by the linear-equivalent approximation; but the condensed subspace afforded by this approximation makes severe, localized changes more  Chapter 6. Summary and Conclusions^  167  difficult to represent. In the latter case, more accurate results can be achieved by applying Newton's method to the unsubstructured model, assuming that such a model exists. This requires solving a system of equations with potentially thousands of degrees of freedom, leading to great computational expense. It is shown that by using a CMS method, the order of the governing equations can be reduced by 10 to 20 times, while maintaining the integrity of the low frequency spectrum. As a result, subsequent modification calculations are inexpensive and many useful predictions can be made about the dynamic behaviour of a modified structure of significant complexity. Throughout this thesis, damping has not been considered because it is not of critical importance for determining the natural modes of lightly damped structures. However, including damping effects in the techniques presented may make the work more relevant to situations involving experimentally derived modes and in systems where damping plays a more significant role. Other areas in which additional work could be done are the following. (1) An error estimation technique for the free-interface method was described in Section 3.9. This technique was based on using the dynamic residual flexibility, evaluated at select frequencies, to determine the contribution of the neglected component modes. This idea may find further application in inverse modification problems with prescribed frequencies. (2) It was found that the eigenvalue tolerance has a significant effect on the accuracy of the mode shapes. Further work is recommended to determine more precisely the relationship between the accuracies of the component modes and the system modes. Finally, (3) the perturbation method that was presented for solving frequency modification problems could be adapted for situations involving prescribed mode shapes and nonlinear property changes.  Bibliography  [1] Meirovitch, L., Computational methods in structural dynamics. Sijhoff and Noordhoff, Alpen aan den Rijn, The Netherlands, 1980. [2] Hale, A.L., and Meirovitch, L. 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B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983, p. 325. [70] Adelman, H. M., and Haftka, R. T., "Sensitivity Analysis of Discrete Structural Systems," AIAA Journal, Vol. 24, No. 5, May 1986, pp. 823-832. [71] Brandon, J. A., Strategies for Structural Dynamic Modification, Research Studies Press, Somerset, U.K., 1990. [72] Lancaster, P., "On Eigenvalues of Matrices Dependent on a Parameter," Numerische Mathematik, Vol. 6, No. 5, December 1964, pp. 377-387. [73] Fox, R. L., and Kapoor, M. P., "Rates of Change of Eigenvalues and Eigenvectors," AIAA Journal, Vol. 6, No. 12, December 1968, pp. 2426-2429. [74] Nelson, R. B., "Simplified Calculation of Eigenvector Derivatives," AIAA Journal, Vol. 14, No. 9, September 1976, pp. 1201-1205. [75] Rudisill, C. S., "Derivatives of Eigenvalues and Eigenvectors for a General Matrix," AIAA Journal, Vol. 12, No. 5, May 1974, pp. 721-725. [76] Heo, J. H., and Ehmann, K. F., "A Method for Substructure Sensitivity Analysis," Journal of Vibration and Acoustics, Vol. 113, April 1991, pp. 201-208. [77] Bohte, Z., "Numerical Solution of the Inverse Algebraic Eigenvalue Problem," The Computer Journal, Vol. 10,1967-1968, pp. 385-388. [78] Biegler-KOnig, F. W., "A Newton Iteration Process for Inverse Eigenvalue Problems," Numerische Mathematik, Vol. 37,1981, pp. 349-354.  Bibliography^  174  [79] Friedland, S., Nocedal, J., and Overton, M. L., "The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems," SIAM Journal of Numerical Analysis, Vol. 24, No. 3, June 1987, pp. 634-667. [80] Smith, M. J., and Hutton, S. G., "Frequency Modification Using Newton's Method and Inverse Iteration Eigenvector Updating," AIAA Journal, Vol. 30, No. 7, July 1992, pp. 1886-1891. [81] To, W. M., and Ewins, D. J., "Structural Modification Using Rayleigh Quotient Iteration," International Journal of the Mechanical Sciences, Vol. 32, No. 3, 1990, pp. 169-179.  Appendix A  Convergence Characteristics of the Modulation Matrix II  The series  ^II  E .  .  (40 2  k=0  dm)  k^  (A.1)  converges to the expression  = (/ - w 2 dm) -1^(A.2)  ^II  if .11w 2 OMIl p < 1 for the general class of p-norms. This condition also guarantees that (A.2) will be invertible [21]. The p-norm of a matrix A is defined as ^AI,  ^max 'O 11II P  x  (A.3)  lixiilip ^p  where the p-norm of a n x 1 vector x of is  (E n  xl) P ,^ 11x111,= k=1  ^1  (A.4)  A general property held by all p-norms is that the norm of a product of two matrices, A and B, obeys the inequality,  ^IIABIlp 5_ 11A llplIB Ilp  ^(A.5)  Thus, for any particular p-norm IIOMII = OhAV`DIMii  ^  PhilliAViiiiqMii  175  ^  (A.6) (A.7)  Appendix A. Convergence Characteristics of the Modulation Matrix II^176  Because the higher modes 4h are mass normalized, 11(13IMIIII(13hil = a _>11 ,1q:McIt h il= j1/11 = 1^(A.8) Furthermore, the norm of a diagonal matrix is equal to the largest element on the diagonal. Therefore, 1141 II = max(Ah)i l = (min(Ah)kr i^(A.9) k  Combining (A.7), (A.8), and (A.9) gives the result ilaMii < a (qn(A h )k) -1  ^  (A. 10)  It is guaranteed that Ilw 2 dMil < 1 when a w 2 (min(A h )k) -1 < 1^  (A.11)  The term min(Ah)k is the smallest eigenvalue among the neglected modes and is therefore greater that are equal to the square of the cutoff frequency w c . Therefore aw 2  aw 2 (min(Ah)k) -1 <^— ,2 < 1 k  which holds when W  2  ,2 w <— c  a  —e  a>1  In conclusion, the infinite series (A.1) is guaranteed to converge in the low frequency range 0 < w 2 < w! /a.  Appendix B Eigenvalue Sensitivity to Residual Flexibility Changes  The eigenvalues of the equation,  [r (a). —^= 0^ in which  (B.1)  r (LD) is a symmetric matrix of the form, rp)^+ ciB TA [ATte()A1ATTB^(B.2)  are stationary points of the Rayleigh quotient 137r(65)Fli ai —^ Pi Pi  (B.3)  Differentiation of the eigenvalues with respect to C4.7 2 gives the first-order sensitivity equation,  _T aA  •^ -T arm awl Ps  (B.4)  Pi Pi^= t  where  arp)^crTA [ATi(P) 5,7,2  -;,B  1 -1 Ta y (65) A [A T B^-1 T (w  A1  A  acD2  )A1 ATT  From (3.38), (B.1), (B.4) and assuming that the eigenvectors that  gpi = 1,  (B.5)  pi are normalized such  -  B  aA i7.B2 alp (65),B 5,7,2 =^ac,2 ,  a  (B.6)  where f i is^the interface load distribution in the i th mode. This expression is equivalent to a summation of individual terms contributed by each component,  _^ _E  aA, DT NIB (6) ) f„. ^ f„, 065 2 —^ac4,2 177  (B.7)  Appendix B. Eigenvalue Sensitivity to Residual Flexibility Changes ^178  where .9 is the number of components. Note that for any component  ^_ aT ad(ei,) a aw e —^565 2 P.-  NIB  (B.8)  firà — 65 2 M^Ai^— 65 2  m 6) -1  (B.9)  P T a(cD)mOPV 2 = 44: (AV +6J 2 4 2 65 4 4 3 + ...) (11T  (B.10)  =^(42 + 261 2 4 3 + 36) 4 4 4  (B.12)  (B.11)  The diagonals of A h-2 , A h-3 , ... are always positive and as a result, (B.12) is a positive semi-definite matrix; i.e., for a non-trivial f1 3 , BT  fi  aii/B(w)  86,2  B  0  (B.13)  for all components. The above expression is equal to zero only when stbIfiB = 0; that is, when there is no interaction between the interface forces and the higher mode shapes. Applying (B.13) to (B.7) for each component gives the final result aA i aez,2 5- 0^  Thus, the eigenvalue Ai is a continuously non-increasing function of 65.  (B.14)  Appendix C Implementation of Component Mode Synthesis in VASTO6  In Chapters 2 and 3 it was found that the fixed- and free-interface methods were the most promising CMS techniques for application to general finite element models. Both of these methods have been implemented in the finite element program VAST06, with a capability for using either first- or second-order mass approximations in the free-interface implementation. The present chapter gives a description of the VAST-CMS program and guides the user in beginning a CMS analysis. C.1 The Substructure/Superelement Option The VASTO6 program contains a substructuring/superelement option which permits the following: 1. Defining a structure as a collection of components or substructures; 2. Selecting a set of master nodes for each substructure, thus defining a superelement; 3. A Guyan reduction of each superelement, giving condensed stiffness and mass matrices; 4. Assembling global stiffness and mass matrices from the superelement matrices, giving equations of motion of the structure. For Guyan reduction, any substructure nodes can be selected as master nodes as long as they include all the interface nodes. In CMS applications, the same superelement 179  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^180  ELEMS 1 Assembles element matrices. ELEMS2 Defines superelement master node numbers. ASSEM2 Assembles full-sized substructure matrices, K and M. PARTSM Partitions substructure matrices into interface and interior coordinates. DECOM2 Decomposes  Ku into UT DU form.  REDSM^Performs Guyan reduction on stiffness and mass matrices, giving k BB and MBB in (2.65). An in-core algorithm REDSM1 and an out-of-core algorithm REDSM2 are available. ASSEM1 Assembles global equations from reduced superelement matrices. STIFM^Modifies stiffness matrix to account for external constraints. MASSM^Modifies mass matrix to account for lumped masses or fluid added-mass. DECOM1 Decomposes global stiffness or stiffness/mass combination into U T DU form. EIGEN1 Solves global eigenvalue problem. EIGNSE Reconstructs global mode shapes from Guyan eigenvectors.  Table C.1: Description of VASTO6 modules definition is used, but the master nodes are generally restricted to interface nodes only. It is generally better to account for the interior nodes with normal modes, rather than by defining additional master nodes. The flow chart for a typical Guyan reduction analysis is shown in Figure C.1. Here, the number of superelements is NSE, with one superelement defined per substructure. A description of the function of each program module is found in Table C.1.  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^181  ELEMS1  ASSEM2  ELEMS2  PARTSM I = 1, NSE )  ASSEM1  STEM  MASSM  DECOM1  EIGEN1  EIGNSE  Figure C.1: Flow chart for Guyan reduction in VASTO6  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^182  T54^Boundary node numbers in local and global coordinate system. T55^Submatrix KB/. T56^Reduced matrices k" and mBB computed from Guyan reduction. T57/T82 UTDU decomposition of KII T21^Submatrices KBB  ,  KBI ^I, MIImBB mBI ,  I  ,  )  T22^_Tr' = (KII--1 KLB) T (out-of-core solution only) T23^Submatrix K". Table C.2: T-file locations of VAST information relevant to CMS The information regarding the substructure and superelement matrices is stored in various T-files. Table C.2 shows the T-file locations of information relevant to a CMS analysis. C.2 Implementation of the Fixed-Interface Method The flow chart for the VASTO6 implementation of the fixed-interface method is shown in Figure C.2. The only difference from Figure C.1 is the inclusion of module CMS_1 in the ELEMS2 loop. This module is responsible for calculating the fixed-interface normal modes and for calculating the extra submatrices that appear in the Craig-Bampton equation (2.65). The first task requires solving the eigenvalue equation, {KH. _ wwIl] q = 0^  (C.1)  A UTDU decomposed version of KII has already been calculated in DECOM2 at this stage of the program and matrix M" is available on file T21.  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^183  ELEMS1  ASSEM2  ELEMS2  PAR1SM I =1 NSE )  DECOM2  REDSM  ASSEM1  CMS 1  STIFM  4 MASSM  DECOM1  4 EIGEN1  4 EIGNSE  Figure C.2: Flow chart for the fixed-interface CMS method in VASTO6  Appendix C. Implementation of Component Mode Synthesis in VASTO6^184  The second task makes use of the fixed-interface normal modes to calculate the following terms: k NN^(= ANN , for mass normalized modes)^(C.2) m BN = mBI + rM "mNBT^ niNN  (C.3)  = 4,/Tm114/ (= I, for mass normalized modes) (C.4)  One difficulty which arises here is that the static constraint modes calculated in REDSM have to be saved if  MBN is  to be calculated. There are two algorithms in REDSM: an  in-core solution and an out-of-core solution. The in-core solution never explicitly forms the constraint modes; only in the out-of-core solution are they calculated and saved on file T22 (see Table C.2.) Therefore, if the in-core solution has been used in REDSM, the constraint modes first have to be constructed before M BN can be calculated . Because of this, it is usually more efficient to use the out-of-core solver with fixed-interface CMS. Other changes to the program modules include a call to an additional subroutine in ASSEM1 for assembling the submatrices (C.2)—(C.4) into the global equations; modifications to EIGEN1 so that the extra modal coordinates in the system eigenvectors are accounted for; and modifications to EIGNSE so that the fixed-interface component modes are used in the reconstruction of the system mode shapes. Figure C.3 shows the flow chart for the reanalysis option in the fixed-interface implementation. The algorithm is interrupted after DECOM2 if reanalysis of the component is not desired. The implementation could be improved by skipping the entire loop if the component is not being reanalyzed, but it is necessary to store beforehand the decomposed Ku matrices elsewhere so that they are not overwritten during reanalysis. These matrices are needed in EIGNSE to reconstruct the system mode shapes. Extra storage files created by the fixed-interface program are listed in Table C.3. These files will appear with the same prefix as the other VAST files. Their contents  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^185  ELEMS1  ASSEM2  ELEMS2  PARIBM ( I = 1, NSE ) A  DECOM2  REDSM  ASSEM1  CMS 1  STIFM  MASSM  DECOM1  EIGEN1  EIGNSE  Figure C.3: Reanalysis option of the fixed-interface CMS method in VASTO6  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^186  C42 Reduced submatrices k NN TnNN an d m BN . ,  C46 Global stiffness and mass matrices assembled from the reduced submatrices. C51 Fixed-interface component modes used in the analysis. Table C.3: Additional storage files created by the fixed-interface method.  S42 Reduced submatrices kNN, „INN, and mBN for a single component. This information is later moved to C42. S46 Submatrices kBB and TriB B for one component. This information is later moved to T56. S51 Fixed-interface component modes calculated for one component. These are later moved to C51. Table C.4: Special storage files created by the fixed-interface program. are described in Table C.4. There are also separate files with the suffix Sxx for each component, which contain CMS data for a single component. These will appear with a prefix supplied by the user and unique to a particular component. These special files only need to be saved if a subsequent reanalysis is to be performed. C.3 Implementation of the Free-Interface Method  The flow chart for the free - interface method based on the MacNeal - Rubin mode set is shown in Figure C.4. A flow chart for the reanalysis option of the same method is shown in  Figure C.5. Whereas the fixed-interface method follows the Guyan reduction program closely, the free-interface implementation differs greatly. The two modules added for this method are CMS_2 which calculates the free-free and residual attachment component  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^187  ELEMS1  ASSEM2  ELEMS2  CMS 2  I - 1, NSE  ASSEM4  Figure C.4: Flow chart for the free-interface CMS method in VASTO6  ELEMS1  ASSEM2  ELEMS2  no  ASSEM4  I = 1, NSE  )  4^  CMS 2  Figure C.5: Reanalysis option for the free-interface CMS method in VASTO6  Appendix C. Implementation of Component Mode Synthesis in VASTO6^188  modes, and ASSEM4 which assembles and solves the condensed system equations. The flow chart for module CMS_2 is shown in Figure C.6. An important difference to note here is that the modules STIFM and MASSM, for stiffness and mass modification respectively, are located inside the component. loop. By contrast, in the Guyanreduction/fixed-interface implementation they were located outside the component loop (see Figures C.1—C.3). There are two principal advantages to making these modifications at the component level. First, the free-free component modes calculated in EIGEN4 will automatically incorporate any external constraints. This saves the trouble of having to form special constraint equations later on in the analysis. Secondly, the spring and mass additions can be applied to any node in the structure, not just at the master nodes. Indeed, for the free-interface method master nodes should be defined only on the component interfaces, otherwise the complex assembly algorithm will produce spurious results. On the other hand, if spring and mass additions are required in the fixed-interface method, master nodes have to be defined specially for them. In addition to adding external constraints and springs, the module STIFM also checks for and corrects certain types of linear dependencies in the stiffness matrix. It is necessary to make these corrections to the component stiffness matrix to ensure that the component modes, both static and dynamic, can be computed correctly. One case in which a linear dependency correction is necessary is a two-dimensional model composed of membrane elements (such as the container ship model described in Chapter 4.) Flat membrane elements have no stiffness in the out-of-plane direction and therefore have degenerate stiffness matrices. If the membranes are oriented in one of the coordinate planes, say the x-y plane, the linear dependency correction is simply to add large springs to the zcoordinate diagonals of the stiffness matrix. However, if the membranes are not oriented in one of the coordinate planes, the linear dependency will not be obvious at first glance and a more subtle detection and correction algorithm is used.  Appendix C. Implementation of Component Mode Synthesis in VASTO6^189  STIFM  MASSM  DECOM1  EIGEN4  V  DECOMK  SOLVEG  INTRLF  FULL G  RESFLX  RESMASS  Figure C.6: Flow chart for the module CMS.  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^190  The fixed-interface program only does linear dependency checking on the Ku portion of stiffness matrix to ensure that a correct UT DU decomposition is performed. This decomposition is need for calculating both the static constraint modes and the fixed-interface normal modes. On the other hand, the free-interface program has to decompose the whole component stiffness matrix, as is required for calculating the free-free modes and the component flexibility. In this case, the linear dependency checking can therefore affect both the interior and interface coordinates of the stiffness matrix. A difficulty arises here if a correction is made at an interface coordinate and, after calculating the component modes, it is not removed from the flexibility, then the inter-component compatibility and equilibrium constraints may not be properly satisfied. Linear dependency corrections of the first type, where large stiffness are added to the diagonal, can easily be detected as a zero diagonal in the flexibility. If this diagonal corresponds to an interface coordinate, the constraint that should be applied here is zero load, with non-zero displacement allowable. Unless the effect of the correction is removed by replacing this zero with a large flexibility value, the constraint that will actually be applied is a zero displacement, with a non-zero load allowable. This may seem a strange distinction to make in light of the membrane example, where it seems natural to apply a zero-displacement constraint at all out-of-plane coordinates, interface or otherwise. But in this case it makes no difference whether a zero-displacement or zero-load condition is imposed; the net effect will be the same. As an example of a situation where it does make a difference, consider one component with bar elements on the interface, meshing with another composed of brick elements. Bar elements are degenerate in non-axial directions, which means that non-zero forces can only be applied in the axial direction. The linear dependency corrections will effectively eliminate non-axial displacement of the bars, as is needed to calculate the component modes. But when this modified bar element is connected to the adjacent brick element  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^191  during the synthesis phase, these corrections will effectively put zero-displacement constraints on the connecting node, constraints which in reality do not exist. If any constraint is applied here it should be zero load in the non-axial directions, for the bar elements are two force members and therefore should only support a load passing through the two end nodes. To re-establish the proper interface constraints, the linear dependency correction has to be removed from the interface nodes of the bar elements by making an appropriate correction to the flexibility matrix. Although the free-interface program can handle the first type of linear dependency correction, the second type is much more difficult to detect. These corrections involve manipulations of the triads situated along the diagonal of the stiffness matrix, and would be invoked, for instance, when bar elements are not co-directional with one of the coordinate axes. If these corrections cannot be detected, some method needs to be devised by which the sequence of manipulations can be reversed in the flexibility matrix. It must be emphasized that an error in the analysis will only occur if the second type of correction is made to an interface coordinate, and if the connecting elements in the adjacent component do not have the same degeneracy. Whether or not this type of error will occur can be discovered by considering what would happen if the model were not substructured. In the above example, the stiffness matrix would not be degenerate at the node connecting the bar element to the solid element and so no correction would be made there in the unsubstructured case. The free-interface program provides two different algorithms for calculating the component flexibility (see Figure C.6). If NCON = 0, the flexibility can be obtained directly from the stiffness matrix. If NCON > 0, the more complicated method described by (2.79) is necessary. Module RESFLX determines the residual flexibility matrix if, and RESMASS determines the residual mass matrix Lt.  The additional storage files created by the free-interface program are listed in Table  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^192  T23 Residual flexibility matrices Ail. T24 Residual mass matrices E. C42 Equilibrium/compatibility connectivity matrix A (see Eq. (3.1)). C51 Free-interface normal modes used for the analysis. Table C.5: Additional storage files created by free-interface program.  S24 Residual flexibility and residual mass matrices for a single component. These are later stored in T23 and T24. S38 Information regarding the number of free-free modes calculated, the number retained, and the number of rigid-body modes. S51 Free-free component modes calculated for one component. These are later moved to C51. Table C.6: Special storage files created by the free-interface program. C.5. These files are stored under the same prefix as the other VAST files. There are also some special files which are stored under a component-specific prefix defined by the user (see Table C.6). These files have to be saved if a subsequent reanalysis is to be performed. The free-interface program has been designed for efficient manipulation of data on modern computer systems. So as to avoid excessive I/O operations, as much of the intermediate data as possible is kept in the internal memory space allocated for the program. The maximum size of problem that can be solved is controlled by KORE, which is defined at the beginning of the program. In the module CMS, the largest  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^193  component that can be analyzed is  Ns  1/2x KORE  (C.5)  VVV^5  where NS is the number of degrees of freedom in the component's finite element model. A similar constraint governing ASSEM4 determines the total size of the system equations: LEN .:%2 min (07(017E,  2xKORE  3  112xKORE )  LEN^5  ^(C.6)  where LEN is the total number of independent forces and moments acting on the component interfaces. An estimate for LEN can be obtained with, LEN Ls_d NMN x NDF^  (C.7)  where NMN is the total number of master nodes defined in ELEMS2 and NDF is the number of degrees of freedom per node. C.4 Including Fluid Added-Mass in a CMS Analysis Suppose that the free vibrations of a structural model are described by the equation, Mil -I- Ku = 0 (C.8) where K and M are the structural stiffness and mass matrices. The modes of vibration of the structure in air are calculated by means of the associated eigenvalue problem: {---AiM In Xi = 0  (C.9)  If this equation refers to a ship hull or some other structure in a marine environment, the standard practice for taking into account the surrounding water is to include a fluid added-mass matrix MA in the equation of motion:  Mii + Ku = - MAU^  (C.10)  Appendix C. Implementation of Component Mode Synthesis in VASTO6^194  There are various methods for computing the added-mass matrix, but once it is established, the wet modes are computed by solving the eigenvalue problem associated with (C.10):  (m + MA ) + KJ = 0 (C.11) where Ay' and Xr define the wet natural frequencies and mode shapes. For a substructured model, an analogous approach is to include added-mass effects in the component level analyses and synthesis the equations of motion of the structure on this basis. However, unlike the structural matrices, the added-mass matrix is fully coupled and cannot be substructured. A different approach is therefore required. An approximate but generally efficient way to solve this problem is to first calculate the dry modes using (C.9) and then approximate the motion of the wetted structure as a linear combination of the dry modes:  u.-Ex4=x4.  (C.12)  where X is a rectangular matrix containing the eigenvectors as columns, is the vector of generalized coordinates and q is the number of modes shapes. This transformation is exact only if a complete set of modes is included in X. This is seldom the case though and q is usually small in comparison to the size of the original equations, making (C.12) an approximation which is valid if the wet mode shapes are not radically different from the dry mode shapes. Applying (C.12) to (C.10) and premultiplying by  XT gives the following equation:  + 'CIA) + o where,  = xTmx  (C.13)  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^195  MA = XT MA X  = xTifx The dry modes are usually normalized so that (C.14)  fc= diag {4} = si2^  (0.15)  Substituting these into (0.13), the wetted modes are computed from the corresponding eigenvalue equation:  {_Ar (1-+ AL) S12] ,=0^ (0.16) where Ar and^define the natural frequency and mode shape of the structure in water. The wetted mode shapes are reconstructed with the relation, = X67'^  (0.17)  The added-mass matrix is generally calculated using a full-sized structural model and an accompanying fluid-element model. If the full-sized and substructured models are based on the same finite element mesh, there is a relationship between the coordinate systems of the two models which enables the generalized added-mass MA = XT MAX to be calculated from CMS-derived mode shapes. When this is case, (0.16) can be established regardless of whether the dry modes were calculated with a substructured or unsubstructured analysis. The VASTO6 program already provides for calculating wetted modes of unsubstructured models with this method. To apply it in conjunction with a CMS analysis, two extra modules had to be created. The program MATCH finds matching node numbers in a substructured and full-sized model and lists the corresponding pairs in the file PREFX.GLM. This program should be run after the added-mass matrix has been calculated and after  Appendix C. Implementation of Component Mode Synthesis in VASTO6^196  the dry modes have been calculated with CMS. The module AMEIGN then assembles the CMS-derived mode shapes into equivalent modes in the full-model coordinate system and stores them in file C52. In this form, the mode shapes are passed on to the module EIGNWM which computes the wetted modes with the method described above. C.5 User's Guide to VASTO8 CMS (Pre-release Version)  The master control code IELEMS determines whether or not a CMS analysis is performed. To initiate a fixed-interface CMS analysis, set IELEMS = 6 while for a free-interface analysis, set IELEMS = 7. The remainder of the master control codes should be set as for a regular natural frequency analysis. Direct iteration must be used for eigenvalue analysis (IEIGEN = 1). To calculate the wet natural frequencies following a CMS analysis of the dry modes, set the master control code IEIGEN to 3. The structural components are defined in exactly the same way as substructures and superelements are defined in the existing version of VAST06. However, some restrictions are necessary on the parameters used in defining superelements (i.e. components): • N LEVEL = 1. Only first-level superelements may be used as components in a CMS analysis. • NSLN = 0. The number of slave nodes must always be zero. • For a fixed-interface analysis, master nodes must be defined at all nodes on the component interfaces and at all nodes which have prescribed displacements and lumped masses. For a free-interface analysis, master nodes are defined at the component interfaces only; prescribed displacements and lumped masses are handled at the component level. To specify the component modes for each component, a data file PREFX.CMS must  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^197  be supplied with the other VAST data files. To calculate wetted modes of a structure, a data file PREFX.GLM must also be included.  C.5.1 Format of input file PREFX.CMS Card 1 (215) NCOMP, IREAN • NCOMP = no. of components in the structure. • IREAN = 1 for analysis of all components. • IREAN = 2 for reanalysis of specified components only. Include Cards 2 through 15 for NCOMP components  Card 2 (215) IFLAG, NRIG • IFLAG > 1, calculate component modes for this component. • IFLAG = 2, use out-of-core solver if IELEMS = 6. • IFLAG = 0, use component mode from a previous analysis. • IFLAG = —1, the component and its component modes are identical to the previous component. If the component is the same as the previous one except for a change in orientation, this option cannot be used. Instead set IFLAG = 1. • NRIG = number of rigid-body modes for the component (if IELEMS = 6, NRIG = 0).  Card 3 (A5) CPREFX • CPREFX = prefix to be used for this component's files Omit Cards 4 through 15 if IREAN = 2 or IFLAG = —1. Omit Cards 4 through 7 if IELEMS = 6.  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^198  Card 4 (15, E10.3) NSK, SPRING • NSK = the number of displacement nodes to be assigned prescribed displacements. • SPRING = default value for added spring stiffness (10 20 is assumed if SPRING is not provided) If NSK = 0, omit Card 5. If NSK 0 0, provide Card 5 for NSK nodes. Card 5 (14, 612, 6E10.3) NI, IDC 1 ...IDC 6 , SK i ...SK 6 . • NI = component node number to be assigned a prescribed displacement • IDCi = codes for specifying degrees of freedom to be assigned prescribed displacment. • SK; = translational spring stiffnesses or rotational spring stiffnesses to be assigned to the degrees of freedom indicated by IDC ; = 1. When spring stiffnesses are not provided, default SPRING is used. Card 6 (15) NLM • NLM = number of nodes where lumped masses are to be assigned. If NLM = 0, omit Card 7. If NLM 0 0, provide Card 7 for NLM nodes. Card 7 (14, 6E10.3) NI, AML i ...AML 6 • NI = component node number to be assigned lumped masses • AM Li = lumped masses for each degree of freedom of node NI.  Appendix C. Implementation of Component Mode Synthesis in VASTO6^199  Omit Card 8 if IELEMS = 7. Card 8 (15) IGEN • IGEN = 1, for direct iteration. Card 9 (215) IOPT, IPTC • IOPT = 0, natural frequencies are computed. • IPTC = 0, printing of normalized eigenvectors to CPREFX.LPT is suppressed. • IPTC = 1, normalized eigenvectors are printed out with three significant figures. • IPTC = 2, normalized eigenvectors are printed out with eight significant figures. Card 10 (315, E10.3) NM1, NM2, MNIT, TOL • N M1 = 1, the first mode to be computed. • NM2 = the last mode to be computed. If no component modes are desired for this component set NM2 = 0. • MNIT = maximum number of iterations allowed. Default value is 20. • TOL = tolerance to which iterations are carried out. The default value is 0.001. Card 11 (15) NMKP • NMKP = the number of component modes desired for this component. Card 12 (1615) MODES 1 MODESNmKp  Appendix C. Implementation of Component Mode Synthesis in VASTO6^200  • MODES1 MODESNMKP = the component modes to be used, listed in order of increasing frequency. Omit Cards 13 through 15 if IELEMS = 6. Card 13 (15) NCON • NCON = number of nodes to be assigned prescribed displacements in order for the component to be statically determinate. If NCON = 0, omit Card 14. If NCON 0, provide Card 14 for NCON nodes. Card 14 (14, 612) NI, IDC 1 ...IDC 6 • NI = node number to be assigned a prescribed displacement. • IDCi = codes for specifying degrees of freedom to be assigned prescribed displacements. Exactly N RIG degrees of freedom should be assigned prescribed displacements in Card 14. Card 15 (15) IMSS2 • IMSS2 = 0, second order mass terms are not calculated. C.5.2 Format of input file PREFX.GLM This file contains pairs of nodes, matching nodes in the substructure model to nodes in the complete structure model used to compute the fluid added-mass matrix. This file is necessary if wet modes are to be calculated from dry modes computed with a CMS analysis. If the PREFX.T41 file exists for both the complete and substructured models,  Appendix C. Implementation of Component Mode Synthesis in VASTO6 ^201  and the PREFX.T54 file exists for the substructured model, the PREFX.GLM can be generated with the program MATCH. Card 1 (I5) NSUB, NDN • NSUB = the number of substructures in the CMS model. • NDN = the number of nodes in the complete model. Include cards 2 and 3 for each substructure. Card 2 (I5) NNODES • NNODES = the number of nodes in the substructure model Include Card 3 NNODES times. Card 3 (215) SSNN, FMNN • SSNN = substructure node number • FMNN = node number in the full model corresponding to SSNN.  

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