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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 MODALANALYSIS OF FINITE ELEMENT MODELSByMalcolm J. SmithB.A.Sc. (Engineering Physics), University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993© Malcolm J. Smith, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of ^Mechanical Engineering The University of British ColumbiaVancouver, CanadaDate  April 27. 1993DE-6 (2/88)AbstractComponent mode synthesis (CMS) is a condensation method for vibration analysis whichpreserves the low frequency vibrational characteristics of a structure. In this method, thestructure is treated as an assemblage of components whose displacements are describedin 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 compo-nent mode sets used by other researchers are reviewed with a view to establishing whichis most suitable for large-order finite element models. Two component mode sets areidentified as ideally satisfying the basic requirements for inter-component compatibility,high convergence rate, linear independence and completeness. Fixed-interface and free-interface CMS formulations in the form of matrix eigenvalue equations are derived fromthese mode sets and describe approximately the low-frequency free vibration modes ofthe structure. They are improvements over previous formulations in that they can be sys-tematically and efficiently applied to linear, undamped, discrete systems of an arbitrarilycomplex geometry. The free-interface formulation is derived both with and without anapproximation of the high-frequency component inertia, and this results in two differentstructural mass matrices. Two new developments of the free-interface formulation arepresented: (1) a method for calculating upper and lower bounds to the exact naturalfrequencies is given, providing a measure of the absolute accuracy of the structural fre-quencies; (2) the convergence and interlacing properties of the free-interface method areexplored through the analysis of a two-component vibrating rod.Both the fixed- and free-interface methods have been implemented in the general-purpose finite element program VAST. Three finite element models are analyzed and aiicomprehensive comparison of the frequency and mode shape results obtained with CMS,direct finite element analysis, and Guyan reduction is presented. The complexity of thetest 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 thelow frequency modes, and that by using a frequency cutoff criterion to select dynamicmodes, the natural frequencies converge in a fairly uniform manner. It is also shown thatin terms of computational cost and order-reduction, the relative advantages of using CMSincrease 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 overallperformance because of its high convergence rate and superior condensation in complextwo and three dimensional models.Application of CMS to structural dynamic modification and inverse modification isalso studied. These techniques use a baseline modal analysis as a reference point for themodified system dynamics. A generalized CMS formulation for the baseline system isused to derive a linear-equivalent perturbation equation from which modified modes canbe efficiently determined without recalculating the component modes. Also, two newmethods are presented for predicting design changes which satisfy prescribed frequencyconstraints. An iterative scheme is proposed in which the energy-balance perturbationequations are solved with a full account of the nonlinear coupling terms; and a Newton'smethod algorithm using inverse iteration eigenvector updating is applied to the linear-equivalent equation. Numerical results using a finite element model are presented whichshow that for large structural changes, the two new methods give similar or better resultsthan an established method.iiiTable of ContentsAbstract^ iiList of Tables viiiList of FiguresList of Symbols^ xiiAcknowledgement xix1 Background and Objectives^ 11.1 Thesis Objectives and Overview  ^42 Component Mode Representations^ 72.1 Introduction  ^72.2 Requirements for Approximating Functions of Structural Components^72.2.1 Component Modes for Discrete Models ^  102.3 Free-Free Modes^  122.4 Free-Free Modes with Interface Loading^  152.5 Static Approximation of Higher Modes  182.6 Inertial Approximation of Lower Modes^  242.7 Constraint Modes ^  252.8 Attachment Modes  302.9 Polynomial Functions ^  33iv2.10 Comparison of Component Mode Representations ^  352.11 Mode Selection ^  392.12 Summary  403 Substructure Synthesis^ 413.1 Introduction ^  413.2 Inter-Component Equilibrium and Compatibility^  423.3 Lagrangian Formulation of the System Equations  443.4 Modal Force Method ^  483.5 Free-Interface Formulation  503.6 Fixed-Interface Formulation ^  563.7 Condensation in the Fixed- and Free-Interface Formulations ^ 603.8 Component Mode Substitution ^  613.9 Error Estimation for Natural Frequencies ^  643.10 Modal Properties of Combined Systems; the Inclusion Principle ^ 683.11 Summary ^  774 Modal Analysis of Three Finite Element Models^ 794.1 Introduction ^  794.2 Analysis of a Container Ship ^  804.2.1 Natural Frequency Results  824.2.2 Mode Shape Results ^  884.2.3 Performance ^  914.3 Analysis of a Telescope Focus Unit ^  924.3.1 Natural Frequency Results  954.3.2 Mode Shape Results ^  1004.3.3 Performance ^  104v4.4 Analysis of a Telescope Model ^  1064.4.1 Natural Frequency Results  1084.4.2 Mode Shape Results and Performance ^  1144.5 Natural Frequency Error Estimation: Ship Model  1164.6 Reanalysis Following a Design Modification ^  1184.6.1 Reanalysis of the Container Ship  1194.6.2 Reanalysis of the Telescope Focus Unit ^  1224.7 Discussion and Summary of Numerical Results  1245 Structural Dynamic Modification^ 1265.1 Introduction ^  1265.2 Structural Changes  1275.3 Structural Dynamic Modification with CMS ^  1315.3.1 Application to Substructured Models  1335.3.2 Numerical Results ^  1375.4 Perturbation Methods for Inverse Modification ^  1425.4.1 Background ^  1425.4.2 Application to CMS ^  1445.5 Sensitivity Analysis; Newton's Method for Inverse Modification ^ 1505.5.1 Background ^  1505.5.2 Newton's Method for Substructured Problems ^ 1515.6 Numerical Results ^  1545.7 Summary  1626 Summary and Conclusions^ 164Bibliography^ 168viA Convergence Characteristics of the Modulation Matrix II^175B Eigenvalue Sensitivity to Residual Flexibility Changes^177C Implementation of Component Mode Synthesis in VASTO6^179C.1 The Substructure/Superelement Option ^  179C.2 Implementation of the Fixed-Interface Method ^  182C.3 Implementation of the Free-Interface Method  186C.4 Including Fluid Added-Mass in a CMS Analysis ^  193C.5 User's Guide to VASTO6 CMS (Pre-release Version)  196C.5.1 Format of input file PREFX.CMS ^  197C.5.2 Format of input file PREFX.GLM  200viiList of Tables4.1 Description of the container ship model ^  814.2 Natural frequency results for the container ship, free-interface method . ^ 834.3 Natural frequency results for the container ship, fixed-interface method ^ 854.4 Component frequencies of the container ship ^  864.5 Mode shape results for the container ship, free-interface method ^ 894.6 Mode shape results for the container ship, fixed-interface method . .^904.7 CPU times for modal analysis of the container ship ^ 914.8 Description of the TFU model ^  934.9 Natural frequency results for the TFU, free-interface method ^ 964.10 Natural frequency results for the TFU, fixed-interface method ^ 974.11 Component frequencies of the TFU ^  984.12 Mode shape results for TFU, free-interface method ^ 1014.13 Mode shape results for TFU, fixed-interface method  1024.14 CPU times for modal analysis of the TFU ^  1054.15 Natural frequency results for telescope model, free-interface method . .^1094.16 Natural frequency results for telescope model, fixed-interface method .^1104.17 Natural frequencies of spider components ^  1124.18 Mode shape results for telescope model, fixed-interface method ^ 1154.19 CPU times for modal analysis of the telescope model ^ 1154.20 Lower and upper bounds calculated with dynamic residual flexibility . . ^ 1174.21 Lower and upper bound estimates of four CMS frequencies, with predictedabsolute errors and exact frequencies ^  117viii4.22 Reanalysis of the container ship model, free-interface method ^ 1214.23 Reanalysis of the container ship model, fixed-interface method ^ 1214.24 Reanalysis of the TFU, free-interface method^  1234.25 Reanalysis of the TFU, fixed-interface method  123C.1 Description of VASTO6 modules ^  180C.2 T-file locations of VAST information relevant to CMS ^ 182C.3 Additional storage files created by the fixed-interface method. ^ 186C.4 Special storage files created by the fixed-interface program^ 186C.5 Additional storage files created by free-interface program  192C.6 Special storage files created by the free-interface program. ^ 192ixList of Figures2.1 A general distributed model for a structural component ^ 82.2 Two component system^ 163.1 Two-component continuous bar ^ 693.2 Graphical solution for continuous bar, two mode approximation ^ 733.3 Graphical solution for continuous bar, three mode approximation^. 744.1 Two-dimensional container ship model: a) stern; b) aft-body; c) deck-house; d) mid-body; e) fore-body ^ 814.2 Combined natural frequency results for the container ship^ 874.3 Components of the TFU model: a) inner tube with screw assembly at-tached to top, and chopping mechanism attached to bottom; b) outertube; c) support tube with external constraints. ^ 944.4 Combined natural frequency results for the TFU model ^ 994.5 Complete telescope model, with focus unit and spider 1074.6 Combined natural frequency results for the telescope model^ 1114.7 Mode 9 of the container ship, f = 5.91Hz. ^ 1194.8 Six groups of bar stiffeners used for modifications ^ 1205.1 Reanalysis results for modifications to hull-bottom stiffeners ^ 1385.2 Reanalysis results for modifications to bearing stiffnesses 1395.3 Reanalysis results for lumped-mass additions to the chopping mechanism 1415.4 Frequency modification of hull-bottom stiffeners for five modal approxi-mations ^ 156x5.5 Comparison of three methods for frequency modification of hull-bottomstiffeners ^ 1575.6 Three-parameter frequency modification of hull-bottom stiffeners ^ 1585.7 Frequency modification with mid-body stiffeners ^ 160C.1 Flow chart for Guyan reduction in VASTO6 181C.2 Flow chart for the fixed-interface CMS method in VASTO6 ^ 183C.3 Reanalysis option of the fixed-interface CMS method in VAST06 ^ 185C.4 Flow chart for the free-interface CMS method in VASTO6^ 187C.5 Reanalysis option for the free-interface CMS method in VASTO6 ^ 187C.6 Flow chart for the module CMS. ^ 189xiList of SymbolsA^equilibrium/compatibility connectivity matrixB^inertial complement to residual flexibility matrixzi^linear differential operator of maximum order 2/ — 1admixture coefficient vector for the ith modified modeindicates participation of the ja baseline mode in the ith modified modeC^admixture coefficient matrixD general diagonal matrixrank-deficient square matrix formed by constraint matrix Rapplied load vector for a componentindependent set of interface loadsith natural frequency (Hz)F*^penalty function for unconstrained optimizationgi^jth inequality constraint in penalty function F*static flexibility matrix6^residual flexibility matrixGC^constrained flexibility matrixH dynamic response matrixI^identity matrixk reduced-order component stiffness matrixelement stiffness matrix(k..);^element stiffness matrix, linearized about the jth propertyK component or structural stiffness matrixxiiI?^condensed system stiffness matrix given by a CMS formulationKCAL^interface stiffness coupling matrix generated from residual flexibilities(Kr )i^global coordinate accumulation of (ker )j over all elements1^indicates order of differential equations; or, the number of frequencyconstraintsL^Lagrangian of a dynamic systemlinear self-adjoint operator of order 21m^reduced-order component mass matrix; or, the number of structural designvariablesme^element mass matrix(mer )i^element mass matrix, linearized about the j th propertyM^component or structural mass matrixNI^condensed system mass matrix given by a CMS formulationMA^fluid added-mass matrix(Mr);^global coordinate accumulation of (m..)i over all elementsmass densityn^number of degrees of freedom in the discrete model of a structure orcomponentnumber of inequality constraints in penalty function F*ith continuous shape function(bar) over a vector indicates the uncoupled linear collocation of all suchcomponent vectors, over a matrix indicates the uncoupled diagonalcollocation of all such component matricesvector of free-free modal coordinatesvector of constraint modal coordinatespn^vector of fixed-interface normal modal coordinatesPr^vector of rigid body modal coordinatesvector of interface-loaded modal coordinatesP^projection matrixq number of baseline modes retainedr^property variableR^set of static determinate interface coordinates; general inter-componentconstraint matrix; the residual vector of equality constraints in penaltyfunction FtRayleigh quotient3^number of structural componentsset of redundant interface coordinateslocal to global coordinate transformation for element et^timeT^transformation between physical coordinates and generalized systemcoordinates in a CMS formulationTA^equilibrium/connectivity matrixTc^Guyan reduction transformation matrixtransformation between component generalized coordinates and systemcoordinatesT^kinetic energyto/^eigensolution tolerance parameteru^displacement vector for a componentuB^independent set of interface displacementsun n-mode approximation to uxivU general upper-triangular matrix^ potential or strain energyw(x), W(x) continuous displacement functions for component, structurewn^approximation to w based on n shape functionsx^spatial coordinatex^vector of spatial coordinatesX^matrix of unsubstructured mode shapesX$^ith unsubstructured mode shapeXr^ith mode shape for a wetted structureY^matrix of the zero-eigenvalue eigenvectors of DZ^matrix of condensed system eigenvectorsfractional change to a property, or a vector of fractional changes13^^transformation matrix between interface and full component coordinatesystemsr^condensed system stiffness matrix given by free-interface CMSf^matrix of residual acceleration modesb f^virtual force vectorSu^virtual displacement vectorA f %^percentage error in frequencyAke^change to element stiffness matrixAme^change to element mass matrixAK^change to stiffness matrixAM^change to mass matrixAu%^percentage error in mode shapeAX^change to mode shape matrixxvAce^change to system-mode eigenvalue matrix77^vector of component generalized coordinatesith eigenvector of condensed CMS equationseigenvector of i th wetted modeinertial complement to residual attachment modesA^eigenvalueAw^eigenvalue of a wetted structureA^diagonal matrix of component free-free eigenvaluesweighting coefficient used in penalty function F*II(w)^modulation matrix95;^ith free-interface or free-free mode(A^ith interface-loaded free vibration modematrix of free-free modes4fin^matrix of fixed-interface normal modes4.L^matrix of fixed-interface normal modes (interior partition)(br^matrix of rigid body modesmatrix of attachment modesqic^matrix of static constraint modesmatrix of static constraint modes (interior partition)f^inertia-relief modesmatrix of residual attachment modesw^frequencyw,^cutoff frequencywi^ith natural frequency (3 -1 )n2 diagonal matrix of system eigenvaluesxviSubscriptsconstraint modesd^dependent coordinatese^element numberI inertia-relief modesindependent, or generalized, coordinatesh^high frequency modes1 low-frequency modesle^low-frequency elastic modesm^medium-range frequency modesn fixed-interface normal modesr^rigid body modesSuperscriptspartition of interface coordinatesBB^interface coordinate partition from a symmetric matrixBI, IB^partition of interface and interior coupling termsBN, NB^partition of interface and modal coupling termsE partition of extra coordinatesI^partition of interior coordinatesII^interior coordinate partition from a symmetric matrix(k)^kth iteration; kth structural component-NN^partition of terms associated with fixed-interface vibration modesO partition of original coordinatesR^partition of determinate interface coordinatesS partition of redundant interface coordinatesxviiT^transpose of a vector or matrixxviiiAcknowledgementThe author would like to thank Professor Stan Hutton for the guidance and insighthe provided throughout the researching and writing of this thesis. The author wouldalso like to acknowledge the generous financial support provided by Defence ResearchEstablishment Atlantic and the Natural Sciences and Engineering Research Council ofCanada.xixChapter 1Background and ObjectivesA basic problem in structural dynamic analysis is the evaluation of natural modes ofvibration. Accurate knowledge of at least some of the modes is of considerable importancefor determining the dynamic response of a structure to applied loads. Mathematicalidealization 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 ofthese equations for complex structural systems has been studied extensively and remainsa primary concern of analytical modal analysis.Linear vibrations of a distributed, elastic, undamped structure are described by apartial 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 massdensity of the structure, A is a parameter, x is a vector of spatial variables and W(x) thedisplacement function for the structure [1). The displacement is further constrained bythe following boundary conditions which are defined at all points on the boundary:13;W(x) = 0 i = 1,2, ... ,1 (1.2)where Bi is a. linear partial differential operator of maximum order 2/ — 1. Solution ofthe eigenvalue problem consists of determining the eigenvalue, eigenfunction pairs AT.,Wr which represent the natural modes of the structure. While all elastic, undamped1Chapter 1. Background and Objectives^ 2structures can be described with equations (1.1) and (1.2), closed form solutions are rareand the analyst is left to seek an approximate solution.The most popular and well known approximate methods for solving these equationsare the finite element and the Rayleigh-Ritz method. Both seek to replace the continu-ous variable W(x) with a collection of discrete variables and to replace the differentialeigenvalue problem with an algebraic eigenvalue problem. Algebraic eigenvalue problemscan in general be solved, whereas differential eigenvalue problems cannot.In the finite element method a structure is divided into a number of sub-domains orfinite elements. The displacement of each element is described by a linear combinationof element shape functions, each of which is in turn defined by a unit displacement ofan element coordinate. Thus, the continuous displacement function of the structure isreplaced by a vector describing the displacement of the element nodes. The advantagesof the method are that a structure of arbitrary structural geometrical complexity canbe effectively modelled with a mesh of relatively simple elements and that if the correctelements are used, the results can be made to converge to exact solutions as the elementmesh is refined. One of the disadvantages, however, is that accurate results for complexstructures require a large number of nodal coordinates. The subsequent analysis maythen require lengthy computations.The Rayleigh-Ritz method, by contrast, uses a set of approximating shape functionsdefined over the whole domain of the structure. A solution in the Rayleigh-Ritz sensecorresponds to a configuration that is a stationary point in the Rayleigh quotient,R(W) = A = [W, 14/^(1.3)where [W, W} is the energy inner product defined by I W.CWdx and D denotes thespatial domain occupied by the structure. The continuous variable W in R is replacedby a number of discrete, generalized coordinates representing the relative participationh MW2 dxChapter 1. Background and Objectives^ 3of the approximating functions. This substitution transforms (1.3) into an algebraiceigenvalue problem. The advantage of the Rayleigh-Ritz method is that accurate resultscan 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 suitableapproximating functions is too difficult and the method is discarded in favor of finiteelements.A third method for modal analysis of a linear structure is component mode synthesiswhich combines features from the Rayleigh-Ritz method and finite elements. Structuresmay, in general, be treated as an assemblage of components or substructures; indeed, insome 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 componentthan for the entire structure. For a component, such approximating functions mightinclude static deflections or rigid and elastic mode shapes; they might be measuredexperimentally or they might be calculated from a detailed finite element model of thecomponent. The object is to replace the detailed model of a structural component witha simplified one based on a set of shape functions that provide a good approximation ina particular frequency range.In the Rayleigh-Ritz method, approximating functions must be defined over the entiredomain of the structure. These global approximating functions can be formed implicitlyfrom the component shape functions by maintaining displacement compatibility at thecomponent interfaces [2]. This same process is apparent in the finite element methodwhen compatible element types are used. Indeed, if a model is substructured so thateach component corresponds to a single finite element of the original model, a CMSanalysis will be same as a finite element analysis. On the other hand if no substructuringis performed, i.e., the entire model is treated as a single substructure, the CMS methodis no different than the Rayleigh-Ritz method. The proper course for using the CMSChapter 1. Background and Objectives^ 4method lies between these two extremes. The number of structural components is usuallyconsiderably less than the number of finite elements required in the modelling, whileat the same time, each component represents a much simpler model than the entirestructure. Having established a set of component shape functions for each component,the equations of motion of the structure are derived by enforcing displacement and slopecompatibility at the inter-component boundaries. Sometimes it is convenient to satisfyforce and moment equilibrium here as well. What results is a global stiffness and massmatrix in terms of generalized coordinates, which are themselves related to the componentshape functions.One of the important advantages of component mode synthesis is that equations ofmotion of the structure are of smaller order than, for instance, those that are obtainedwith the finite element method. For large-order models, reduction in the number ofdegrees 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 compo-nent modes are chosen so that the static and low frequency motion of the componentsis well represented. Any inaccuracies which arise from the reduction are confined to thehigh-frequency modes.1.1 Thesis Objectives and OverviewThis thesis provides an in-depth assessment of the effectiveness of component mode syn-thesis 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 andto identify those suitable for application to general, large-order structures;2. To present general formulations of the free vibration structural equations that canbe applied to complex discrete structural models, and directly implemented in aChapter 1. Background and Objectives^ 5general-purpose finite element program;3. To develop the general formulations further in order to gain a better understandingof convergence properties and to investigate the effects of approximations at thecomponent level;4. To present comprehensive frequency and mode shape results for realistic finite ele-ment models which illustrate the key performance characteristics of the CMS meth-ods 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 vibrationequations.To simplify the analysis energy dissipation is not considered, although the analysis isalso 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 compo-nent mode set is discussed, as well as the topics of convergence, mode selection andinter-component compatibility. In Chapter 3, attention is focussed on the synthesis ofstructural components—the process by which the free vibration equations of the systemare formulated. In this area, the treatment differs somewhat from previous work in thatthe emphasis is on deriving equations specific to a particular component mode set butwhich are applicable to systems with an arbitrary number of components and in the mostgeneral geometric configuration. Two advantages of this approach are that the synthesisof the equations can be done more efficiently and that predictions can be made aboutthe performance of the various component mode representations based on the form ofthe free vibration equations.Chapter 1. Background and Objectives^ 6Two component mode synthesis methods are selected for further study: the fixed-interface and free-interface method. In Chapter 4, modal analysis results for three finiteelement models are presented. The purpose is to compare the performance of the fixed-and free-interface methods with direct finite element analysis and Guyan reduction. Theinfluence 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 bemade, each of which requires its own modal analysis. Structural dynamic modificationtechniques have been developed by various researchers to improve the efficiency of multi-ple analyses. In Chapter 5, it is shown how structural dynamic modification techniquescan be applied when the baseline, or unmodified, structure is represented by a general-ized CMS formulation. The condensation inherent to the CMS formulations is helpfulin reducing the cost of the reanalysis, and it is demonstrated that in many instancesaccurate results for the modified structural modes can be obtained. If it is required thatthe modified structure have certain prescribed frequencies, the efficiency of the redesignprocess can be improved using inverse modification techniques. In these techniques,which in the past have been based on sensitivity or perturbation analysis, design changesare calculated which satisfy free vibration equations of the modified structure subjectto prescribed modal constraints. In Chapter 5, two methods—one based on Newton'smethod, and the other on an iterative solution of the perturbation equations—are used tosolve freqency modification problems. The equations of motion of the modified structureare based on a generalized CMS formulation and the results are compared with thoseobtained with an established method.Chapter 2Component Mode Representations2.1 IntroductionComponent mode synthesis is an analysis method for determining the natural modes ofvibration of a structure. As described in the introductory chapter, it combines featuresof the Rayleigh-Ritz method and the finite element method in an attempt to render anaccurate, reduced-order model of a physical structure. The method can be viewed ashaving two stages; the first is the subdivision of a structure into components and thedescription of each component in terms of approximating functions; the second consistsof reassembling the structure with the aid of the approximating functions and therebyestablishing coupled equations of motion for the entire structure.In the present chapter, attention is focussed on the first stage of the analysis—theselection of a set of approximating functions for a component, which will collectivelybe referred to as a component mode representation. First, a statement of the generalrequirements for component shape functions is given. Following that, a survey of theexisting component mode representations is presented along with some discussion oftheir strengths and weaknesses.2.2 Requirements for Approximating Functions of Structural ComponentsTo begin, consider the general structural component depicted in Figure 2.1. Assumingthat it is a linear, distributed elastic component, its free vibration response as part of a7V1Vrp^ ,,,,,,,, ,, ,,,,,,,,,,,,,,,,,,,,,,,,,Chapter 2. Component Mode Representations^ 8rAdjacent ComponentsIIFigure 2.1: A general distributed model for a structural componentlarger 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 sameas for (1.1). The displacement is subject to the boundary conditions of the typeBiw(x) = 0^i = 1, 2, . . . , 1^ (2.2)The boundary of the component in Figure 2.1 may be divided into five regions, eachof which possesses characteristic boundary conditions. The two fundamental types ofboundary conditions are geometric boundary conditions, where the order of the operatorBi 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 conditionsapply. There are no geometric boundary conditions along the free boundary. Second isChapter 2. Component Mode Representations^ 9the region where external constraints are applied. An external constraint is some phys-ical property of the component which manifests itself in either the geometric or naturalboundary 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 at-tached (V). This region is called the component interface or the inter-component bound-ary. Here, adjacent components exert forces and moments which quantitatively remainunknown until the dynamics of the entire structure are determined. For the purposesof the component analysis, the interfaces are regarded as having only non-homogeneousnatural boundary conditions.As in the analysis of the whole structure, closed form solutions to the componentproblem are rare because of the complexity of the governing equation and also becausethe boundary conditions on the component interfaces are unknown. Instead, a solutionis sought with a sequence of approximating functions, as in the Rayleigh-Ritz method.The displacement w may be approximated by a linear combination of functionswn(x) = E Ni(x)pii=i (2.3)where pi is the participation of the ith function in the displacement function wn. Forwn to be an approximation in the Rayleigh-Ritz sense, the functions Ni must be linearlyindependent, they must be 1 times differentiable, and they must satisfy the geometricboundary conditions. They do not have to satisfy either the differential equation or thenatural boundary conditions [4].Such a sequence of approximating functions should also be complete so that as n ismade arbitrarily large, the approximate displacement wn may be made to approach theexact displacement w to within an arbitrarily small difference in the sense of the energynorm. 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^ 10When selecting approximating functions, the objective is to closely approximate thecomponent displacement (and by association, its kinetic energy and strain energy) as itis undergoing free vibration motion as part of the larger structure. If a wide spectrumof natural modes is to be determined, then a large number of approximating functionswill be necessary to correctly model the energy in all the modes. Typically though, it isusually only a small portion of the structural modes that are of interest, especially in acomplex structure, and these are usually the lowest frequency modes. This simplifies thetask of selecting approximating functions since the kinetic and strain energies only needto be accurately modelled at low frequency.The approximating functions, which from now on will be referred to as componentmodes, should conform with the actual boundary conditions on the component as much aspossible. On the free boundary for instance, homogeneous natural boundary conditionsshould be used when calculating the component modes. On the component interfacesthe boundary conditions are unknown and so the conditions that ought to be applied tothe component modes are left to the judgement of the analyst.2.2.1 Component Modes for Discrete ModelsRelatively complex components are difficult to analyze with a differential equation andboundary conditions of the form (2.1) and (2.2). Instead it is often more practical to builda 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 equationof 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 dis-cretization procedure, u is the displacement vector and f is the vector of applied loads.Chapter 2. Component Mode Representations^ 11In 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 conditionson the component are handled in the following way: external constraints, whether ge-ometric or natural, are incorporated in K, M and u; natural boundary conditions atthe 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 boundaryconditions (2.2) with a system of ordinary differential equations.A sequence of component modes which may be used to approximate the displacementvector u is given by= E Xi1ji^ (2.5)i=iwhere the vector Xi is the ith component mode and where ni is the participation of the ithmode in the vector un. As un is only an approximation to u, the number of componentmodes n will be less than the number of coordinates in u. Applying the Rayleigh-Ritzmethod to the discrete representation gives the following elements of the generalizedstiffness 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 belinearly independent. If the component modes accurately represent the static and low-frequency motion of the component, k and m will give an accurate description of thecomponent's stiffness and mass distribution at low frequency.The remaining sections of this chapter describe a variety of component mode repre-sentations for a discrete component model. Reviews of component mode representationshave been given by Hurty [5], Kubomura [6], and Craig [7].Chapter 2. Component Mode Representations^ 122.3 Free-Free ModesThe simplest component mode representation describes the displacement as a linear com-bination of its unconstrained or free-free vibration modes, i.e.,21(t) = E Oipi = (bp(t)where the modal matrix (1) contains the free-free mode shapes as columns:= {(ai ... (an}and p(t) is the column vector of component modal coordinates. This type of componentmode representation has been used by Goldman [8], Hou [9], Dowell [10] and Yee andTsuei [11].Free-free component modes are calculated from the component equation with theapplied 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 — (41i110i = 0^ (2.11)From this equation are computed the natural frequencies of the component wi and theircorresponding free-free mode shapes fit. A useful convention is to incorporate any externalconstraints into K and M. The resulting free-free modes will therefore satisfy the externalconstraints applied to the component, guaranteeing the admissibility of the modes. Themode set will also include any rigid body modes in the component. The number of rigidbody modes is usually between zero and six, but more may exist if the component isarticulated.(2.8)(2.9)Chapter 2. Component Mode Representations^ 13The general equation of motion of a component is obtained by including a non-zeroforce 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 giveMs: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 nor-malized. 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 w2I]p(t) = (I)Tf(t) (2.14)where A is the diagonal matrix with the component eigenvalues A i = (4 on the diagonaland (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 themass 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 thenumber coordinates in the original model. It is the truncation at the component levelthat primarily distinguishes component mode synthesis from other dynamic condensationmethods [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 frequencycomponent modes are omitted as they contribute little to the low frequency dynamicresponse of the structure.Chapter 2. Component Mode Representations^ 14While coordinate reduction is important, a sufficient number of modes must be in-cluded 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 co-ordinates in all components must exceed the total number of compatibility equations ifthese are to be fully satisfied. The analyst will usually want to use enough modes toensure inter-component compatibility, plus additional modes to improve the accuracy ofthe results.The compatibility requirement becomes troublesome in structures whose componentinterfaces consist of meshed curves or surfaces. As finite element meshes are further re-fined, the number of degrees of freedom on the interfaces increases. At the same time, thenumber of component modes that can be reasonably calculated diminishes because of theincreased order of the governing matrices. As a result,"it may become very difficult to ob-tain enough component modes to enable the compatibility equations to be fully satisified;instead, only partial or approximate satisfaction of compatibility may be possible. Thisis not necessarily detrimental to the accuracy. Using a truncated mode set constrainsa component's motion and tends to produce a model that is too stiff. Conversely, theunder-constraint which stems from improperly satisfied constraint equations relaxes someof the constraints on the component, thereby reducing its stiffness. Obtaining a balancebetween these two operations could offset the ill-effects of each. However the amount bywhich the frequencies of the structure are raised and lowered by model truncation and byrelaxing interface constraints cannot be quantified. Furthermore, it is very easy to devisea substructured model in which the number of interface coordinates exceeds by an orderof magnitude the number of component modes that can be realistically computed. Inthese 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 shapeChapter 2. Component Mode Representations^ 15functions with homogeneous natural boundary conditions show poor convergence char-acteristics. Equation (2.8) is an example of this type of representation. The slow con-vergence stems from the inability of this type of mode, whether taken individually or insmall numbers, to satisfy non-zero natural boundary conditions. Even though a smallnumber of free-free modes may describe the displacement of a component very well in itsinterior, there are inherent inaccuracies at its boundary which do not disappear unlessvery large numbers of modes are used. This conclusion is supported by the investiga-tions of Hou [9] which show that the structural frequencies converge very slowly with thecomponent mode representation in (2.8).2.4 Free-Free Modes with Interface LoadingBenfield and Hruda [15] introduced a variation to the classical method by includingstiffness and inertial interface loading in the eigenvalue equation of a component. Thiswas an attempt to at least partially account for the presence of adjoining componentsin the dynamics and thereby render component modes closer to the structural modes infrequency and energy distribution. For this reason it is expected that these modes willconverge faster than classical free-free modes.Static condensation is used to calculate the interface loading. This method, whichis often referred to as Guyan reduction [16], will be discussed in another context inSection 2.7. For the present discussion it will be sufficient to quote the formula. Tocalculate the stiffness and inertia loadings on a particular component the stiffness andmass characteristics of the adjoining components are condensed on to the interface. Forsimplicity, consider the two component system shown in Figure 2.2. The stiffness andmass matrices of each component are partitioned into interface (B) and interior (I)Chapter 2. Component Mode Representations^ 16Interfacea^ bFigure 2.2: Two component systemcoordinates:KBB KBI^MBB mBIK=Kra KII^M= MIB mll^(2.15)The interface loadings on component a are determined through the condensation of thestiffness and mass properties of component b on to its interface:Ice il = Ter KbTeBB riICT ibr ITCMb = -Lb 4L‘b"b(2.16)(2.17)where 4313 and mr are the condensed stiffness and mass of component b and the matrixTbc is defined as,uBur lb =[ I1.,B^ri,C.,B(Job = J. b u,b_reII-1 KI, B'Lb(2.18)which is the general result from Guyan reduction [16].The free vibration modes of component a are now determined with the interfaceloadings applied. The appropriate eigenvalue equation is,+ iclb3B^KaBI sp 2 iti-aBB + meB maBI '',B[KBB=($) (2.19)K B IT^K II it mBIT^mlI p.Chapter 2. Component Mode Representations^ 17The component displacement is subsequently described by the truncated sequence ofmodes:na .. ,.., E 0,5,i.i(2.20)where na is the number of modes calculated for component a.It should be emphasized that the interface loadings are used only to determine a moreconvergent mode set; once the mode set is established the stiffness and mass propertiesof the components revert to their original forms and the synthesis of the components canproceed in the usual manner.The modal representation (2.20) is an improvement over the classical representationwith respect to its convergence properties but it is does not assist in the satisfaction ofcompatibility conditions at the component interfaces. A large number of dynamic modesare required to properly satisfy compatibility along complex interfaces while the compu-tational effort necessary to calculate these modes is greatly increased by the inclusionof interface loading. It is also a disadvantage to require knowledge of the stiffness andmass characteristics of adjacent components when choosing component modes. It maybe that structural components are being designed independently and that one designgroup may not have detailed information about the work of other design groups. Also, toreanalyze the structure after extensive modifications to one component means not onlythat the modes of that component have to be reanalyzed but that the interface loadingsof all components attached to it have to be adjusted as well. It is clear then, there aredistinct advantages to using component modes which allow components to be analyzedindependently.Chapter 2. Component Mode Representations^ 182.5 Static Approximation of Higher ModesThe dilemna of how to maintain compatibilty between components without having tocompute large numbers of dynamic modes is resolved by using a static approximationfor the higher component modes. Suppose that the mode shapes of a component arepartitioned into two groups such thatn(t)u(t) = [ 4)1 (kh J 1 ph(t) (2.21)The modal matrix is partitioned so that 4)1 contains rigid body modes 40,. and low fre-quency elastic component modes 4)/ e :4)1 = { Cr tie ]^(2.22)Modes that are considered low frequency have frequencies that are comparable to atarget frequency range. Modes in 41, have frequencies that are significantly higher thanthe target frequency range. With this partitioning, the component equations become{ At — w2 /^0^p,(t) 1^[ 4)T. ^f (t)^(2.23)0^Ah — w2/^ph(t)^4,TWhile 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 fre-quency. Define2We »W 2 (2.24)where we is the cutoff frequency, or the lowest frequency in the th set. The followingapproximation may then be used:ph(t) = Al; 1 4V(t)^ (2.25)Chapter 2. Component Mode Representations^ 19The 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 MacNeal-Rubin mode set in recognition of its originators [17, 18]. The second term of u(t) isa static approximation of the displacement contributed by the higher modes, G beingthe static flexibility in the higher modes. Urgueira and Ewins [19] used (2.27) in theirderivations. MacNeal [17], Rubin [18] and Craig and Chang [20] used instead the residualflexibility of the component:= G — (1) 1.4.14:1q. (2.28)where G is the full static flexibility of the component and (1)/. is the matrix of lowerelastic modes of the component. Note that (2.27) and (2.28) are equivalent when the setof higher modes is complete. However, by using (2.28) the higher modes no longer needto be computed.In the free vibration of the structure, f(t) has non-zero values at the interface locationsonly. As a result, the only columns of 6 that will contribute to the displacement willbe those associated with interface locations and each of these columns represents theresidual 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 staticattachment mode. DefiningAt) = Or (i )^(2.29)where fB(t) contains forces at the interface coordinates only and ,a is the coordinatetransformation from the interface to the full coordinate system of the component, theChapter 2. Component Mode Representations^ 20displacement of a component is then given by,u(t) = (1)p(t) 'tfB (t)^ (2.30)wherexi/ = Op (2.31)and where subscript 1 has been dropped. The matrix if contains the residual attachmentmodes as columns. By their definition, these modes are mass-orthogonal and stiffness-orthogonal to the lower normal modes.Because the interface forces appear on the right-hand side of (2.30), they may betreated as modal, or generalized, coordinates. A sufficient number are available to ensurethat compatibility and equilibrium can always be satisfied, but this requires that theresidual attachment modes be linearly independent.As each free-free mode is added, the rank of the residual flexibility matrix is reducedby one. Thus, with a full set of modes, the residual flexibility disappears and the modeset reduces to the classical representation. By this means, linear independence andcompleteness requirements are satisfied. But to maintain the linear independence ofthe residual attachment modes, the rank cannot be less than the number of interfaceforces, thus limiting the number of free-free modes that can be used. In most realisticsituations, however, accurate results can be obtained with a relatively small number offree-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 isvalid provided (2.24) is maintained, but this may not be possible in components havinghigh modal density in the target frequency range, as a very large number of modeswould have to be calculated to satisfy (2.24). To improve the accuracy, a second-orderapproximation may be employed in which[Ahh w 2 —1 = I cd 2 A h— 1 I —1 Ah 1 Ah 1 co 2 A h— 2^(2.32)Chapter 2. Component Mode Representations 21The expression for ph and the component displacement thus become,Ph =^(412 42) 41:f (2.33)u = (lop + Of + 2:6 fwhere ds is as before and,B = .1)hA.172 (1q:(2.34)(2.35)The symmetric matrix /3 can be calculated without knowledge of the neglected modesby using (2.28) and by observing that (2.36)(2.37)Applying (2.29),u = (pp + (.1 + w2611/1) afifBOr,U = Ifop + Off + ca2t) fB^ (2.38)where= 13,3 = Omit^ (2.39)It should be emphasized that the columns of E do not represent a new set of modes, butrather are complementary to the existing residual attachment modes. It will be shownin Section 3.5 that retaining this second-order contribution in the substructure synthesiscreates a supplementary global mass matrix which partially accounts for the inertia ofthe neglected modes.The MacLaurin series expansion of [Ah — ca 2 /1 -1 can be continued further so that,[Ah w211 —1 = Ah-1 w2Ah-2 w4Ah 3 w6Ah-4^(2.40)Chapter 2. Component Mode Representations^ 22This expansion is valid as long as GJ2 is smaller than all diagonal elements of Ah. Since thediagonals are eigenvalues of the neglected modes and are therefore always larger than 4,the expansion is valid for the frequency range 0 < w < we . The resulting displacementcontribution from the higher modes is,4/sPh 4//1 (Ah 1 W242 + W4 A173^.) 4:01:f^(2.41)By (2.27), (2.35), and (2.36), it can be stated that for integer n > 1,thAh-t*: n-1= ( 4PhAh - T^430;111:11:= (GM)n-1 us(2.42)(2.43)In other words, every term in the expansion (2.41) dependent on the higher modes canbe replaced with an equivalent expression in terms of 6 and M; i.e.,1/isPh = [a + C4/ 26Ma Cs.14 (GM) 2 G +...] f^(2.44)= a(w)f^(2.45)where G(w) is the dynamic residual flexibility of the component. By factoring out 6, theseries converges to the expression,6(w) = 6 (.1 — w 2m6) -1 (.1 — w 26m) -1 G^(2.46)provided liw 2 dMIlp < 1, for the general class of p-norms [21]. In Appendix A it is shownthat there exists a low frequency range 0 < w2 < 4/a, a > 1 in which this conditionholds. Furthermore, it can be shown through a similarity transformation that I — c‘, 26Mis positive definite and invertible when w < w e . Define A asA = 4) -1 — cv 26M)^ (2.47)Chapter 2. Component Mode Representations^ 23where (I) is a square matrix containing all free-free modes of the component. Matrix fl• isinvertible, and from mass-orthogonality,^= (DT M. Therefore,A =^(1)T^- W 2 e11161114)^(2.48)I — W2 41,T m ,f)h^m 4f, (2.49)1 0 2 0^0 - w (2.50)0^I 0^AllIt is inferred from (2.50) that the eigenvalues of A (i.e., its diagonals) are positive whenthe diagonal elements of c,./ 244 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—w26M is positive definite and invertible in the frequency range 0 < w < wc.The modulation matrix •II (w) = (/ — c4;26M) -1 (2.51)tunes the static residual flexibility 6 to frequency w. The evaluation of 6 at w andits substitution in (2.45) gives an exact account of the higher modes' contribution tothe component displacement. However, using d(w) in the CMS formulations is difficultbecause w is unknown and appears in a highly nonlinear form. Indeed, including termshigher than w 2 in (2.44) creates a nonlinear eigenvalue equation for the structure. How-ever, once estimates of the system natural frequencies CA have been obtained with eitherthe first- or second-order approximations, calculating d(cDi) is useful for estimating themagnitude 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 de-grees of freedom to satisfy compatibility constraints at interface locations on discrete com-ponents. By adding a linearly independent set of residual attachment modes, the modeset is expanded relatively inexpensively so as to ensure satisfaction of inter-componentChapter 2. Component Mode Representations^ 24compatibility, and a more accurate component mode representation is produced by in-cluding the static flexibility of the neglected free-free modes.2.6 Inertial Approximation of Lower ModesThe concept of using an approximation of the higher modes was carried further by Kubo-mura [6] and Kuang and Tsuei [22] who used approximations for both the high and thelow frequency modes. Suppose the component mode set in (2.8) is partitioned into threegroups such thatu(t )4PhilPPmPh(2.52)The set L contains component modes that are in the same frequency range of the systemmodes of interest. Modes with much lower frequencies are contained in (I)i and modeswith much higher frequencies are in (I)h. The equation of motion of the component isthen,At — co 2I^0^0 (DT0^A„, — cy 2/^0 Pm = 4,InT 1(0^(2.53)0^0^Ah - Q.,21 PhIf upper and lower frequency cutoffs are such that wc2, < w2 and c4h^wa, then=^1^At)^ (2.54)Ph = 41,101: f(t) (2.55)and the displacement of the component is given by1^TU =^fB +^+^h A h i (eh fB^(2.56)Chapter 2. Component Mode Representations^ 25Oru = (1)„,pm fB (2.57)As before, the approximation for the high frequency modes can be thought of as residualattachment modes as designated by xis. The approximation of the low frequency modesis purely inertial. Each column of t is the acceleration of the component induced by aunit force at one interface location with zero force elsewhere. Consequently, each of thesecolumns 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 stillyet to be determined. Thus a new set of inertial coordinates are defined: pc„ = 113 /w2 .Observe that the residual acceleration modes are linear combinations of the rigid bodyand low frequency elastic component modes. Approximating the low frequency modesdoes 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 struc-tures which are of interest in the present work. In these structures, it is the very lowestsystem 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 ModesHurty [23] proposed a component mode representation in which the displacement of acomponent is described by a combination of rigid body modes (r) (if the componentis unrestrained or partially restrained), static constraint modes (c), and cantileverednormal modes (n). Partitioning into determinate interface (R), redundant interface (5),I 0 0<Dr' p(t) I pc (t) 0 pra(t)^(2.58)uRuus, = ck4Chapter 2. Component Mode Representations^ 26and interior (I) coordinates, the component displacement is given by,The determinate interface coordinates are interface coordinates which, if constrained,make the component statically determinate. If a component is fully restrained with ex-ternal 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 zerodisplacement at other R coordinates. A static constraint mode is defined by a unit dis-placement 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 interfacelocations rigidly held. Craig and Bampton [24] recognized that a complete set of con-straint 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 needsto be made between determinate and redundant interface coordinates. The componentdisplacement is therefore given by the following, where B denotes the complete set ofinterface coordinates:or, in unpartitioned form,uB^I 0Pc + ^p.(1);,U = XlicPc (DnPn(2.59)(2.60)The component mode representation as expressed by (2.59) is commonly referred to asthe Craig-Bampton mode set and is statically equivalent to (2.58). This representationwas also adopted by Hurty et al. [25] and SandstrOm [26]. Note that each static modehas a unit displacement at one interface coordinate and that each normal mode has zeroChapter 2. Component Mode Representations^ 27displacement at all interface coordinates. By the top equation in (2.59), uB = pc and soI^B AIU = -rnPn (2.61)The matrices xli! and cl,„I are calculated in the following manner: the equation ofmotion of a component in partitioned form isKBB KBI mBI uB fBH uB}w2[MBB(2.62)KIB MIB^mII 111 0Setting w = 0 for a static analysis, the second equation in (2.62) becomesul =^KIBuB = TIuB (2.63)Holding the interface rigidly by setting uB = 0, the second equation in (2.62) becomes[KII 402mII1 ul = 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 componentequation in modal coordinates iskBB^0 uB mBN f B0^kNN pn[mBBM^MNB^NNlliiB0(2.65)wherekBB KBB KBIT cIkNNmBB MBB mBIC + TcITmIB^T cITAIIICMBN = MBI +emIl(DI = mNBTmN = OnChapter 2. Component Mode Representations^ 28Note that the modal stiffness matrix is block diagonal and that the static constraintmodes are orthogonal (with respect to the stiffness matrix) to the cantilevered normalmodes.There are as many static constraint modes as there are interface coordinates in acomponent. A complete set of these modes must be included in the analysis to ensurethat the rigid body motion and static behaviour of the component are preserved, as theseare important characteristics of the lower structural modes.The normal modes are included in the representation to account for some of theinertial properties of the component. To give a complete account of the componentinertia, a full set of normal modes has to be included. Generally this is not practicalbecause great computational effort is required to calculate a large number of normalmodes and also because the normal mode set has to be truncated if any coordinatereduction is to be achieved. The normal modes which are generally kept and included in4:1)n/ are the modes of lowest frequency. It is the lower component modes which make themost 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 theremaining set of constraint modes is statically complete, it tends to produce poor resultsin a free vibration analysis because of an inadequate account of the kinetic energy. Thiscan be improved within the confines of static condensation by including static modes fordegrees 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 asubset of the interior degrees of freedom such that the motion of the component is bestdescribed with a minimum of constraint modes. Various procedures for selecting mastershave been proposed [27, 28]. While masters chosen from the interior coordinates play nopart in the inter-component compatibility, they do foster a more accurate representation[KBB KBI 1[0^[mBB MB! 1[Vs(2.67)KIB KII^r^mIB mIIChapter 2. Component Mode Representations^ 29of the inertial properties at non-interface locations. However, this can be better achievedby including dynamic modes by means of a CMS procedure rather than by adding morestatic modes.Hintz [29] proposed a constraint mode method in which the constraint modes of theCraig-Bampton mode set are augmented with inertia-relief modes.0 uBuBl . [I(2.66)AF^41; piThe inertia-relief modes W if are defined as the static response of a component (withinterfaces held fixed) to rigid body inertia forces. They are calculated by the followingformula, in which a unit modal acceleration is applied to each rigid body mode, and inwhich RB is a vector of reaction forces necessary to maintain zero displacement at theinterface: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 con-straint modes. By (2.68), the number of inertia-relief modes is equal to the number ofrigid body modes. Thus, for a fully restrained component, the constraint mode method ofHintz [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 enhancethe component displacement field at low frequency by including rigid body inertial effects.However, these modes are not strictly necessary for static completeness, as the constraintChapter 2. Component Mode Representations^ 30modes defined in (2.63) are capable of exactly describing the static displacement to aninterface load.To improve the component representation in dynamics problems, Hintz added fixed-interface free vibration modes to (2.66), giving the following component mode represen-tation:uBP ^(2.69)PnNote 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 thenumber 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 inplace of dynamic modes in a CMS analysis, and thereby reduce the number of dynamicmodes 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 bestaccuracy.2.8 Attachment ModesHintz [29] proposed a static mode set consisting of rigid body and attachment modeswhich would be statically equivalent to the constraint mode representation (2.66). At-tachment modes are defined by placing a unit load at an interface location, with zero loadat all other locations. To show how attachment modes are calculated, statically restrainedcomponents are distinguished from those that are unrestrained, or partially restrained.For a restrained component, attachment modes IP are calculated by substituting w = 01 :1[ KBB KBI 1 1 TB 1 1 1=KIB ICH^WI 0(2.70)Chapter 2. Component Mode Representations^ 31and 1B = I into (2.62) and solving the following equations for':or, in unpartitioned form,KT = /3^ (2.71)Note thatTI = _KII-1 KIB TB = gicI AFB1 TB 1 [ I=^TB1 11 1^t 11 1(2.72)(2.73)Thus, attachment modes are linear combinations of constraint modes.For an unrestrained component, the applied forces must be equilibrated by the resul-tant inertia forces if the deformed displacement is to be isolated. This involves solvingthe statics problem,KT = (/ — API),.4:1:0,1 /3 = P/3 (2.74)where P is a projection matrix having the property P = P2 and where PP definesequilibrated 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 modifiedequationif/ = Gcps (2.75)is used where GC is the component flexibility after a set of statically determinate con-straints is applied. If the attachment modes are to be mass-orthogonal to rigid bodymodes, 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^ 32where(2.77)Assuming mass-normalized rigid body modes, the result of substituting (2.76) in (2.77)isPr = 4),T11141^(2.78)andtp . (i _ 4, rT 4, r m) if = pril = pTGepp^(2.79)Thus, premultiplying a force vector by P transforms it into an equilibrated, or inertia-relief, force vector; and premultiplying a displacement vector by PT makes it mass-orthogonal to the rigid body modes [30]. The attachment modes resulting from (2.79) arereferred to as inertia-relief attachment modes. This equation can be used as the generaldefinition of attachment modes by noting that for a restrained component, P = I andGc = 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 set41 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 setof component free-vibration modes. Hintz [29] suggested using either fixed-interface orfree-interface dynamic modes and showed examples derived with each. However, thefree vibration modes have to be selected carefully to avoid linear dependence with theattachment modes. This is cited as a common problem when attachment modes are usedwith complex components [31]. Only when fixed-interface free vibration modes are usedwith a fully restrained component is linear independence between the attachment andChapter 2. Component Mode Representations^ 33free vibration modes guaranteed. In this case, the attachment mode set is equivalent tothe Craig-Bampton mode set.The main advantage of attachment modes is that, in general, they are easier to obtainexperimentally than are constraint modes because the necessary boundary constraintsare easier to impose on a structural component [31]. Computationally, they are moreexpensive as they involve the inversion of the full stiffness matrix, whereas constraintmodes require only the inversion of If - II. Moreover, the Craig-Bampton mode set doesnot require explicit calculation of the rigid body modes, giving it an added computationaladvantage.2.9 Polynomial FunctionsMeirovitch and Hale [2, 3] observed that the basic requirements for component modesof distributed models—completeness, linear independence, and differentiability—are sat-isfied by a much larger class of admissible functions which may include, for instance,low-order polynomials. Independent polynomial functions are generally easier to de-rive than component modes because they can be established without knowledge of thestiffness and mass distribution in the component: they depend only on its physical di-mensions. For these functions to be admissible, it is also necessary that they satisfyexternal geometric constraints. This may be a difficult requirement to satisfy if the con-straints are distributed in a complex manner throughout the component. However, thisrequirement is not strictly necessary since the enforcement of the external constraints canbe postponed until later in the analysis when inter-component compatibility constraintsare satisfied. Therefore, it is permissible to generate component polynomial functionswithout taking into account external constraints.Admissible vectors are generated for discrete models by sampling polynomials definedChapter 2. Component Mode Representations^ 34over the spatial domain of the component. Translational elements of admissible displace-ment vectors are sampled directly from the polynomials; rotational elements are sampledfrom the spatial derivatives of the polynomials. Generally, an infinite number of indepen-dent polynomials are available from which only a finite number of linearly independentadmissible vectors are chosen. Therefore, sampling of the polynomials must be done withcare to ensure that the resulting admissible vectors are linearly independent.Admissible vectors defined in this way depend only on the spatial extent of the com-ponent and its node locations. They do not depend on the distribution of strain orkinetic energy in the vibrating component. If the material properties of the componentsare uniformly distributed, low-order polynomials can be used effectively to predict thelow frequency modes. This has been demonstrated by Meirovitch and Hale [2, 3], for asystem 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 materialproperties are distributed nonuniformly, there may be small-scale vibration effects thatcannot be represented with low-order polynomials. Higher-order polynomials will haveto be added, and although this in itself is not difficult, it has the effect of increasingthe order of the system equations and necessitates prolonged computation at the systemlevel.As an example, consider the segments of a ship hull as structural components. Alow-frequency mode of the hull may consist exclusively of bulkhead vibration. If this isthe case, it is likely that the bulkhead vibration will appear in the low frequency compo-nent modes as well. As a result, this local vibration mode can be adequately representedwith a small number of component modes. By contrast, none of the polynomial-derivedadmissible vectors are likely to be similar to the bulkhead mode. These vectors are deflec-tion shapes defined over the spatial dimensions of the component under the assumptionof a uniform distribution of mass and stiffness within the component. However, localChapter 2. Component Mode Representations^ 35vibration modes only appear as the result of a non-uniform distribution of mass andstiffness. Therefore, such modes can only be represented by taking large numbers ofpolynomial functions in linear combination, which, as was mentioned above, is harmfulto the computational efficiency of the analysis.Faced with this difficulty, a different approach might be to substructure the modelfurther so that all components are reduced to simple, uniformly distributed, plates andbeams. However, this severely restricts the way in which components can be designedseparately and then reassembled. Consequently, using polynomial-derived admissiblefunctions may be useful in certain types of structures, but in general application tofinite-element models convergence problems and insufficent order-reduction will likely beencountered.2.10 Comparison of Component Mode RepresentationsSome general conclusions can be drawn from the foregoing discussion on the subjects ofcompatibility, linear independence, completeness, convergence rate, and computationalaspects of the component mode representations.CompatibilityIn the most general applications, a component mode set should contain a mixture ofstatic and dynamic modes. Static modes can be calculated relatively cheaply and solarge enough numbers of them can always be assembled to satisfy the interface compat-ibility requirements. A dynamic complement is necessary to give a good account of thelow frequency component inertia. Four component mode representations described inthis chapter—the MacNeal-Rubin mode set, the Craig-Bampton mode set, and the con-straint and attachment mode sets of Hintz—are of this type. Mode sets using dynamicChapter 2. Component Mode Representations^ 36modes only, such as classical free-free modes or the interface-loaded modes of Benfieldand Hruda, have a more limited applicability because they are more difficult to calculatein large numbers. They would mainly be useful in structures composed of beam-like com-ponents, such as a series of robotic links, where the number of compatibility constraintsremains small.Linear Independence and CompletenessThe linear independence and completeness of a mode set are both necessary conditionsfor convergence of the system modes. Both the Craig-Bampton and the MacNeal-Rubinmode set satisfy these requirements, the latter by degenerating to the classical free-freemode set when a full complement of modes is used. The constraint and attachmentmode sets of Hintz can generally be made to satisfy the linear independence requirementprovided the truncted set of free vibration modes is chosen carefully. However, as thenumber of free vibration modes is increased, the linear independence of these two modesets will eventually be lost.Convergence Rate: the Relation to Mode AccelerationWhen comparing component mode representations, an important consideration is the rateat which the system modes converge as a result of increasing the number of componentmodes. It is known from the Rayleigh-Ritz method that a solution obtained with a set ofapproximating functions will converge to the exact solution as the number of functionsis increased, provided that they satisfy the requirements of linear independence andcompleteness. In this respect, the component mode representations presented in thischapter 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 thesame improvement in convergence over the classical method that the mode accelerationChapter 2. Component Mode Representations^ 37method enjoys over the mode displacement method. The mode acceleration method wasfirst suggested by Williams [33] as an alternative method for determining stresses inducedby transient loads on aircraft. In comparison to the more conventional mode displacementmethod 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 examiningdisplacements rather than stresses. In the mode displacement method, the displacementis approximated by a set of n dynamic modes,U = Ei=i(2.81)The equations of motion for an undamped structure can then be written in terms of themodal coordinates pi :pi + wfpi Orf^= 1,2,...,n^(2.82)This system of equations can be solved for pi using Duhamel's integral and the summationin (2.81) provides the dynamic response. Supposing (2.81)—(2.82) describe the motionof a structural component acting under periodic interface loads, the mode displacementmethod is then equivalent to the classical free-free mode representation.In the mode acceleration method, (2.82) is transposed:Pi = a (4)TfWiand the dynamic response becomes,mm^U = E^- E^i=1T f i=1(2.83)(2.84)If the summation were over all modes, the first term would reduce to simply the staticresponse to the applied load; it is the second term that accounts for the inertial effects.Chapter 2. Component Mode Representations^ 38Consequently, 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 isno longer dependent on the number of modes retained; it is determined directly throughthe 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 MacNeal-Rubin mode set will converge faster than the classical mode set as a consequence of thecomplete account of the static response of a component to interface loading.For the same reason, it is expected that the other statically complete mode setsdiscussed in this chapter—the Craig-Bampton mode set, and the constraint mode andattachment mode sets of Hintz—will also experience a higher convergence rate than theclassical method.Computational ConsiderationsIt has already been noted that constraint modes are computationally less expensive thanattachment modes because they are derived from the inversion of the submatrixrather than that of the full matrix K. A further point to note is that fixed-interfacefree vibration modes can be calculated more cheaply than free-free modes, owing tothe smaller order of its associated eigenvalue equation. It has also been noted thatthe Craig-Bampton mode set does not require the rigid body modes to be calculatedexplicitly. These considerations indicate that the Craig-Basnpton mode set is the mostChapter 2. Component Mode Representations^ 39advantageous computationally. But it will be shown in the next chapter that the extracomputation needed at the component level gives the MacNeal-Rubin mode set importantadvantages in the formulation of the system equations.2.11 Mode SelectionHaving established the requirements for component mode representations and havingdiscussed some aspects of their respective convergence rates, it is now necessary to con-sider the problem of component mode selection: namely, deciding how many componentmodes are required to get results of a certain accuracy. Unfortunately, this is a questionfor which no definitive theoretical answer can be given. It is known that if a full set ofcomponent modes are used, exact results are obtained; but this fact is of no practicaluse since CMS offers no advantages in this case. CMS is most attractive when modaltruncation, especially severe modal truncation, is possible.All analytical methods use a mathematical model to predict the behaviour of a phys-ical system. But the accuracy of one method is not known until its results are comparedto those of another. For example, the accuracy of a finite element analysis is largelydependent on the degree of mesh refinement; but this accura cy cannot be known with-out a comparison with experimental results or with results obtained by some methodknown to be accurate. Nevertheless, experienced analysts can choose an appropriatemesh refinement without foreknowledge of the correct results.The same is true of CMS with regard to mode selection. While the absolute accuracyof a particular choice of modes is unpredictible, general rules of thumb can be developedto guide the analyst in choosing modes for a particular problem. The appropriate choiceof 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 theirChapter 2. Component Mode Representations^ 40flexibilities, and their relative participation in the modes of interest. If it is the lowestfrequency modes that are chiefly of interest, a cutoff frequency criterion can be used. Acutoff frequency c' is chosen based on the analyst's intuition or experience: all componentmodes below this frequency are included in the analysis, all above are excluded. Thiscriterion can be adjusted as the accuracy requirements change. If good accuracy isrequired only in particular modes, component modes can be weighted according to acomponent's participation in the targeted modes.2.12 SummarySeveral component mode representations satisfying the basic requirements of linear de-pendence and completeness were presented in this chapter. The representations give areduced-order description of a structural component which preserves its rigid body, staticand low frequency elastic response. It was found that representations combining staticand dynamic modes are more successful in general applications for two reasons: first,the number of modes is always large enough to satisfy inter-component compatibilityconstraints; secondly, a higher convergence rate is expected because a complete staticresponse to interface loading is automatically included.Chapter 3Substructure Synthesis3.1 IntroductionIn the previous chapter several component mode sets were discussed and it was shownthat with them, reduced-order discrete representations of the component matrices couldbe generated. In the present chapter, attention is focussed on the general problem oflinking the reduced-order component representations together, a process which givesequations of motion for the entire structure. The coupling of the components is achievedby satisfying the compatibility and equilibrium conditions at the component interfaces.Specifically, interface displacement and rotation coordinates must match, and interfaceforces and moments must cancel.The object of the present chapter is to show how these conditions can be appliedin their most general form to the various component representations described in thelast chapter. A critical assessment of the various forms of system equations will begiven with regard to their applicability to substructured problems of large size and ofgeneral geometric complexity. In particular, two criteria by which they will be judgedare 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, asubstantial degree of coordinate reduction is desirable to minimize the size of the systemequations and thereby reduce the computational time.The second criterion is important because it is often the case that when large, complex41Chapter 3. Substructure Synthesis^ 42structures are analyzed with finite elements, the majority of the computational time isspent on the eigensolution. Any reduction in the size of this equation is beneficial if itdoes not sacrifice accuracy.3.2 Inter-Component Equilibrium and CompatibilityMeirovitch and Hale [2, 3] examined the problem of satisfying inter-component compati-bility in distributed models. A basic difficulty encountered with interfaces that are curvesor surfaces is that with a finite number of modes, it is impossible to satisfy the infinitenumber of constraints at the interface. Instead an intermediate structure is introduced.This is a structure made up of the original structural components but where interfacecompatibility is only partially satisfied with a finite number of weighting functions. Inthis respect, the intermediate structure lies between the system of uncoupled componentsand the actual, fully coupled, structure. One particular intermediate structure of impor-tance to the present work is where compatibility is exactly satisfied at a finite number ofdiscrete locations on the interface.In substructured finite element models with conforming elements, maintaining com-patibility at interface nodes guarantees its maintenance between the nodes. Therefore,no distinction between the intermediate and actual structures is needed. If the modelcontains non-conforming elements, the intermediate structure satisfying compatibility atthe interface nodes is the nearest configuration to the actual structure possible. And soin either case, this particular intermediate structure represents the optimal configurationfor the assembled finite element model.Of course, the option exists for satisfying compatibility at only a subset of the interfacenodes. One consequence of this is that any upper bound provided by the Rayleigh-Ritzmethod is immediately lost. Moreover, this approximated compatibility is unnecessaryChapter 3. Substructure Synthesis^ 43since 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. Inthe remainder of this section, general expressions for compatibility and equilibrium arepresented for discrete models.Equilibrium conditions describing force and moment balance at the inter-componentboundaries can be expressed in the following general form:= Aff^(3.1)-B iwhere f s a vector containing the interface forces and moments of each component insequence:e7B =^ (3.2)fBand J.]: is an independent set of global interface forces and moments partitioned from-Bf . Assuming that the relative positions and orientations of the components are time-invariant, and that the interfaces themselves possess no flexibility or inertia, the trans-formation matrix A contains constant coefficients that depend only on the structuralgeometry.In a similar manner, 77113 may be defined: UB =- uB1•••}(3.3)By the principle of virtual work,u!'_Br^-BT= u A 51gB = 0^ (3.4)Chapter 3. Substructure Synthesis^ 44where 6f: is a set of virtual interface loads. Therefore, the requirement for inter-component compatibility isATTu-B 0A different expression for the compatibility is,-B TAUSU = 1 AU(3.5)(3.6)which can be derived from (3.5) by selecting an independent set of coordinates u9 fromrsB . By combining (3.5) and (3.6) and by noting that u9 may assume arbitrary values,ATTA = 0^ (3.7)By the principle of virtual work,- B - Brf Br 81-4 = f^BTA6u = 0g (3.8)where Su: is a set of virtual interface displacements. Therefore, the requirement forinter-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 compo-nents and of an arbitrary geometrical configuration. Based on information detailing theconnectivity of the structural components, the matrices A and TA can be easily andsystematically constructed.3.3 Lagrangian Formulation of the System EquationsRegardless of what type of component representation is used, the total potential andkinetic energy of s uncoupled components may be written as,(3.10)Chapter 3. Substructure Synthesis^ 45T = —2 77i mt^(3.11)The vector 3 contains the component modal coordinates of each component written insequence:71ri= ^ (3.12)77.where 77,772,...,77, are vectors of modal coordinates corresponding to each of the s com-ponents. Depending on the component mode representation, these might include normalmodal coordinates, interface displacements, or interface loads. The quantities WI and iare block-diagonal matrices of the form,m1 0 .^.^. 0 k1 0^.. 00 m2 0 0 k2 0m = = (3.13)•^•0 0 .^.^. 771, 0 0^... lc.where ki and ms are the condensed stiffness and mass for the ith component. For a par-ticular mode set, these are determined from (2.6) and (2.7). The equations of constraintwhich 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 com-ponents, or any other constraints on the system. The overbar notation will be usedthroughout to indicate square or rectangular matrices of the form (3.13), or vectors ofthe form (3.12).The equations of motion of the system are obtained from the Euler-Lagrange equa-tions. Two methods are available for incorporating the constraints (3.14) [36]. TheChapter 3. Substructure Synthesis^ 46method of Lagrange multipliers can be used, in which the Euler-Lagrange equations areapplied 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 thecharacterstic equation of coupled systems from the classical free-free mode set. Thismethod is also used in Section 3.10 to illustrate solutions obtained from the MacNeal-Rubin mode set.The more usual method is to eliminate some of the coordinates in 7-7 by means of acoordinate transformation. Many researchers [9, 15, 23, 24, 29] have derived transforma-tion matrices by a direct partitioning of R. The vector r7 is partitioned into dependent(d) and independent (g) coordinates giving,Rd Rnd = 0 (3.16)g ri where Tld is selected so that Rd is invertible. The following transformation may now bederived:?id(3.17)Applying this transformation to V and T eliminates the dependent coordinates nd fromthe analysis and couples together the system components.Another elimination method has been given by Kuang and Tsuei [22]. Multiplying(3.14) by RT givesRTR =^= 0^ (3.18)where D is a rank-deficient square matrix. The eigenvectors of 2) corresponding to itszero eigenvalues are calculated from the equationDYi = 0^ (3.19)Chapter 3. Substructure Synthesis^ 47Each of these eigenvectors satisfies all of the constraint equations contained in R. Thelinear combination of all these eigenvectors therefore gives a coordinate transformationbetween the uncoupled and coupled system:= = [Yi • • • Yq] (3.20)Elimination schemes such as these create an overall transformation between the un-coupled component coordinates and the generalized coordinates of the system,= (3.21)Substituting this transformation into the energy expressions and applying the Euler-Lagrange equations gives the free vibration equations of motion for the coupled systemin 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 com-ponent mode representations given in the previous chapter and in general it is not nec-essary to use the same representation with each component. The order of (3.22) is equalto the difference between the number of columns and the number of rows in R, or inother words, the excess in the number of component modal coordinates over the numberof constraint equations. Equations (3.22) are legitimate provided that R contains allthe 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 cumbersomeand inefficient to use in complex problems. In the remainder of this chapter, a differentChapter 3. Substructure Synthesis^ 48approach is taken in which it is assumed that all structural components are expressedin terms of the same type of modal representation. The general equilibrium and com-patibility relations given in Section 3.2 are used to synthesize the system equations fordifferent component mode sets. This leads to a more efficient and economical handlingof the inter-component constraints which can nevertheless be applied to components ofan arbitrary geometrical complexity.3.4 Modal Force MethodThe classical free-free mode representation (2.8) has been used by Yee et al. [11, 37, 38] ina synthesis procedure called the modal force method. Rather than identify a stiffness andmass matrix, this method derives a single dynamic response matrix H(w) by combining(2.8) and (2.14):u(t) = 4:0{A w2I] At) = H(w)f(t) (3.24)Since a free vibration state of the structure is being considered, the vector At) containsinterface loads only. Equation (3.24) may be partitioned according to whether the co-ordinates of u(t) are located on the component interface or at an interior point. For asingle component,1UB =[HBB HatlIfnur^RIB H"^fi }(3.25)Since we are considering the free vibration of a whole structure, fl = 0. For a system ofs components, {uB24Hp 0 .^00 HBB^02^• • •Af2 7-1B B (w) .7.B (3.26)0^0^. . . HBB flChapter 3. Substructure Synthesis^ 49Satisfying compatibility using (3.5) givesATTI-B = ATTIBBt = 0^ (3.27)Satisfying equilibrium between the interface loads by applying (3.1) givesATTIBBAfgB = 0^ (3.28)The expression (3.28) is the most general form of the structural equations that canbe 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-connectedstructure is one in which only two components are joined at any single point on theinter-component boundaries. The size of (3.28) is equal to the number of compatibilityequations which, for simply-connected structures, is equivalent to the number of inde-pendent interface coordinates. For structures whose components have more complexinterconnections, the number of compatibility equations is somewhat larger and is in factequal to the number of independent interface loads.As the terms containing w in (3.28) are not in simple polynomial form, it cannot beput into the form of an algebraic eigenvalue problem, as is obtained when the Lagrangianformulation is used. The natural frequencies of the structure may nevertheless be foundby computing the zeros of the determinant of ATTI BBA. For each natural frequency withere is a corresponding non-trivial vector fgBi determined from (3.28). The structuralmode shape can be recovered from f9 with the following transformations:1.71? =^= TIBB (ciji)AfgBi^ (3.29)ui = FIB (uji)7B = RIB (4 ji)AfgBi (3.30)with HBB defined in (3.26) and with H1B similar in form to HBB . The vector Til3 givesthe interface portion of the mode shape and Fit gives the non-interface portion of themode shape.Chapter 3. Substructure Synthesis^ 50The accuracy of the system modes determined from (3.28) depends on the numberof component modes included in (2.8). It was established in Section 2.3 that there is aminimum number of normal modes that must be used in order to maintain compatibilitybetween components. Additional modes beyond this minimum will improve the accuracyof the final solution. Unlike some other CMS methods, the number of component modesdoes not change the size of the system equations (3.28). Their size is determined solelyby the number of compatibility equations that are defined for the structure.The chief drawback of the modal force method is the nonlinear character of thegoverning equation (3.28). This equation is both difficult to formulate and difficult tosolve unless the number of compatibility constraints is very small. As a result, the modalforce method is only of practical use in special types of structures where the componentinterfaces are simple, and where sufficient accuracy can be obtained with free-free modesalone.3.5 Free-Interface FormulationIn this section, a general synthesis procedure is presented for the MacNeal-Rubin modeset described in Section 2.5. This mode set has been used in various forms by a numberof researchers [17, 18, 19, 20]. MacNeal [17] derived equations based in the first-orderexpression (2.26). His assembled equations took the form of a stiffness matrix defined interms of physical displacements and modal coordinates. Rubin [18] derived similar equa-tions using the second-order expression (2.38). Chang [39] synthesized the equations ofmotion using the Lagrangian method with direct partitioning of the constraint equations.Irretier and Sinapius [40] developed the system equations from the general connectivitymatrix TA defined by (3.6) and (3.9). This derivation is relatively complicated and fur-ther approximations are used to simplify the system equations. A simpler form of theChapter 3. Substructure Synthesis^ 51system 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-orderapproximation (2.38), occupy the remainder of this section.In Section 2.5, the component displacement using residual attachment modes wasgiven asu(t) = 4>p(t)^fB (t)^ (3.31)where=From (3.31) the uncoupled interface displacements can be writteng^'.13—= 41' p^fB(3.32)(3.33)it.17^0^.^00^c ^.^00^0^4)Bswhere(3.34)77Band ill is similar. Likewise, the uncoupled component equations of motion may bewritten in the following compact form:- w 2/1 = -VT? (3.35)The compatibility equations are obtained by premultiplying (3.33) by AT , as in (3.5):ATV + ATiBr =0^ (3.36)In addition to compatibility, equilibrium is also satisfied by invoking (3.1),AT ^+ ATTF BAfr: =0^ (3.37)Chapter 3. Substructure Synthesis^ 52This allows the component interface loads to be expressed in terms of the modal coordi-nates:B 1 -1fB = —A [ATT A AT (I)B(3.38)Observe that (3.38) along with (2.25) defines a direct relationship between the higherand lower modal coordinates.Applying (3.38) to (3.35), the system equations take the form of a symmetric, positivedefinite eigenvalue problem in standard form:(3.39)whereT [ -B 1 -1 „, B-^T -r=A+T A A T A Al (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 free-free component modes. Unlike a mode shape, which describes how much each physicaldegree of freedom is participating in a natural mode of the system, the eigenvectordescribes how much each free-free component mode is participating in a natural modeof the system. Component modes which make a large contribution will have a largeamplitude in the eigenvector, those that do not will have a small amplitude. Structuralmode shapes can be recovered from the eigenvectors of (3.39) by writing the componentdisplacement in a similar manner to (3.33):-B2-1.4p+xlif (3.41)Applying (3.38) to (3.41), the i th structural mode shape is expressed in terms of the i theigenvector:-,7111 ]- 1us = apt —^[Al T A AT 4) pi (3.42)Chapter 3. Substructure Synthesis^ 53To 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. Thestiffness matrices of the uncoupled components can be represented using the overbarnotation as,X^(3.43)The original component stiffness matrices K can be partitioned and rearranged so thatB [,,,BB re,BA =I^K-IBNote that by (3.40) and (3.44), the matrix I' can be expressed as(3.44) T -BB^]-1K + A [AT tif A AT -KB1[- --B-IB Kzz r —st'r :B (3.45)-Since KBB is block-diagonal and A { B lANIArAT is in general full, the latter can beinterpreted, as was noted by Urgueira and Ewins [19], as the stiffness matrix of anintermediate, or coupling, spring system which links the interface degrees of freedom ofthe uncoupled system. Note that the inter-component links of the actual structure arerigid. The finite stiffness given to the links by virtue of the intermediate system is asoftening effect introduced by the residual approximation of the neglected modes. Thissoftening effect partially compensates for the overstiffness resulting from the truncatedfree-free mode sets.The order of the global matrix in (3.39) is equal to the total number of free vibrationmodes used for all components. This is quite different from the system equations derivedin (3.28) where the size is equal to the total number of compatibility equations. If thisnumber is less than the sum of all the component modes, then it may be advantageous touse (3.28). If instead, the number of compatibility equations is greater than the numberChapter 3. Substructure Synthesis^ 54of component modes, as is usually the case in complex structures with many degrees offreedom, then it would be better to use equations in (3.39). Not only is compatibilityguaranteed 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 uncoupledinterface displacements are expressed, from (2.38), as7, B^ 7,—13 ^B —u = (I) p + (41 + u.) 2E. ) t^ (3.46)Applying compatibility and equilibrium gives,B CBf = — A [AT 111 + co 2 F.,-7.7B) A] - 1 AT (i) B 15^(3.47)Since w 2 is unknown until the final solution is found, the inversion in the preceding--Bequation cannot be performed exactly. Defining KcPL = [411T if 14.1gives^—^ ]-1fB = —A {I + w 2 KcpLAT E A KcpLATTBpA linearized approximation for —fB is given by,^—B^ --,.Bf —^' —A [KCPL — w 2 &MAT F Axcpd ATTBT)- iand rearranging(3.48)(3.49)where the matrix inverse in (3.48) has been approximated by the first two terms of itsMacLaurin 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^ (3.50)k = r . A- + TBTAKcpL ATTB (3.51)IC/ = I + TBTAKcpLATrAKcpLATTB^( 3 . 5 2 )Chapter 3. Substructure Synthesis^ 55Thus, the net effect of including the second-order term in the approximation of theneglected component modes is the creation of a non-identity, supplementary global massmatrix. The order, symmetry and positive definiteness of global matrices are not affectedby the inclusion of the second-order term. Craig and Chang [42J derived expressionsequivalent 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 Lagrangianmethod. Using the first-order approximation to the neglected modes, potential andkinetic energies of the uncoupled components are given by (3.10) and (3.11), where [7^0^_ [A o- -0 XII M 111(3.53)andpfB (3.54)-= B^T -Note that^= G =^KW. The coupling of the components is achieved with (3.38)which suggests the transformation,IB^—AKcpLATTB 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-orderapproximation if the Lagrangian formulation is used, or of the second-order approxima--,-.13tion if the direct formulation is used. The form of E in (3.56) clearly indicates that thesupplementary mass matrix represents the inertia contributed by the residual attachmentChapter 3. Substructure Synthesis^ 56modes, a term which is significant if the criterion (2.24) is not strictly adhered to in theselection of component modes. Note that in developing the transformation (3.55), therelationship (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 constraintstake the form,AT r ATiB^p16i =^ = 0^(3.57)0^TT^.1Bwhere (3.9) and (3.36) have been used. The direct partitioning of R requires the inversionof the submatrix Rd defined in (3.14). In this case, the square matrix Rd has order equalto the number of compatibility plus the number of equilibrium constraints. Typically thisis 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 globalmatrices and the predicted natural modes, both approaches should produce identicalresults. The greater efficiency of the synthesis method presented above stems from themore economical handling of the constraint equations.3.6 Fixed -Interface FormulationIn Section 2.7, a component mode representation was described in which static constraintmodes were augmented with fixed-interface dynamic modes. This led to a componentequation of the form (2.65). Based on this equation, the uncoupled equation of the systemmay be written as,,B0 kNN0 11,1 B +[?-.TIBB r-FtBN 1{,BH}fin^Frt-NB^ 0(3.58)Chapter 3. Substructure Synthesis^ 57wherekBB=kBB 0^00 kBB .^0(3.59)0^0 kBB— 7-7-1BB —mBNand kNN ,^and r--1/NN are similar.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: } [ Tim-,BBTA-I-NN^7-Ti NB TA0^k^PnTir-TinN iign . Tir(3.60)--NNM -13n0The term TA r on the right-hand side of (3.60) is zero by (3.9). The coupled equationsof motion of the system at frequency w therefore take the form of an algebraic eigenvalueproblem:where—w2k = o (3.61)uB=n}(3.62)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 eigenvectordescribes the participation of the interface displacement coordinates and the participationof the fixed-interface dynamic modes in a natural mode of the system. The modaldisplacement at the interior degrees of freedom can be recovered from the eigenvector byarranging the interior displacement of all components in sequence. This can be expressedin the following compact form:I 1T/ 1--B^ gu = c u npn = w c TAug (Pnpn (3.63)Chapter 3. Substructure Synthesis^ 58The modal displacement of the interior degrees of freedom for the i th system mode cantherefore be obtained from the ith eigenvector:I = [ xli Iui^c TA^(Dni. (3.64)The vector uf, taken together with 4, defines the i th structural mode shape. Thecomponent displacement in the i th mode can be written in unpartitioned form asz7i = LTA „ Si (3.65)The order of (3.61) is equal to the number of independent interface coordinates inthe structure plus the total number of fixed-interface dynamics modes used. For a largestructure with many interface coordinates, this number can be be very large and the timerequired to extract the natural modes can be considerable. This is the chief drawback ofthe 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 incomparison to other types of component modes. Also, the synthesis procedure is simpleand requires little computation. The global matrices are derived from the componentmatrices through simple transformations which, when examined closely, are similar toassembling global matrices from element matrices, as done in the finite element method.This assembly process eliminates the need for multiplication of large matrices and matrixinversions at the system level. Therefore, this is a more efficient means of synthesizingthe system equations than is provided by the more general elimination methods describedin Section 3.3.SandstrOm [26] carries the fixed-interface method one step further by applying acondensation procedure to the system equations. A similar idea was introduced by Kuharand Stahle [43]. Assuming a periodic solution with frequency w, the first equation inChapter 3. Substructure Synthesis^ 59(3.60) can be written as,Aft'  m ^(TT BB ^ITT —BN-A ft' -L AU - L02^M -1- Ally + -L Am Pit) = 0Likewise, the second equation in (3.60) can be written as,(3.66)(kIN wzieN)._^2NB T B 0pn — —m AU g ^ (3.67)This establishes a relationship between the modal coordinates p„ and the interface coor-dinates uB:pn = w2^- w2—NN) -1 M NB ry, BA119Substituting (3.68) into (3.66) gives,(3.68)rrrATIBBTA w2 (TITfiBB TA + ce2TIBN [INN1- - W2MNNi-1—mNBTA)] ugB = 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 2and a nonlinear eigenvalue problem results. Consequently, the solution of (3.69) is bothsimplified, by a reduction in order, and complicated, by the introduction of nonlinearitiesin w2 . The advantages of this condensation may outweigh its difficulties if the coordinatereduction is significant, i.e., if the number of modal coordinates eliminated is comparableto 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 modalcoordinates and so this condensation procedure would have little value. As with themodal force method, it would be of more use in beam-type structures where the com-ponent interfaces are restricted to a small number of point locations. If the number ofinterface coordinates could be thus limited, the nonlinear equation (3.69) is potentiallyvery compact.Chapter 3. Substructure Synthesis^ 60Corresponding 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 interiorcoordinates, is constructed with the aid of (3.6), (3.63) and (3.68):-1ui = [471,TA^(TCNN - Wi2-M,NN) -MNB TA} UBgi (3.70)With this formula, the mode shapes can be constructed without having to explicitlycalculate the the modal coordinates To n .3.7 Condensation in the Fixed- and Free-Interface FormulationsThe Craig-Bampton and MacNeal-Rubin mode sets give fundamentally different formu-lations of the system equations. The formulation (3.60) produced by the Craig-Bamptonmode set will be referred to as the fixed-interface method. These equations are in termsof the interface displacements and the fixed-interface modal coordinates. Two separateformulations have been derived with the MacNeal-Rubin mode set. The first-order massformulation (3.39)—(3.40) and the second-order mass formulation (3.50)—(3.52) are ex-pressed in terms of the free-free modal coordinates only; all physical displacements andloads have been eliminated. Collectively these two formulations will be referred to as thefree-interface method.The distinctions between these two methods are of prime importance to the systemcondensation. Consider a system of complex components with meshed curve or meshedsurface interfaces. As the mesh is further refined, the u: set expands, increasing theorder of (3.60). But modal sets Ton and IT) do not experience a corresponding expansion,for the number of dynamic component modes is only weakly dependent on the modelcomplexity; in fact, it is more strongly influenced by the accuracy demands and thedegree of substructuring. Thus, as the complexity of a substructured finite elementChapter 3. Substructure Synthesis^ 61model 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 equi-librium are explicitly enforced, while in the fixed-interface method only compatibilityis enforced. Consequently, twice as many constraints are applied in the free-interfacemethod, since in most cases the number of equilibrium and compatibility constraints isapproximately the same. It was established earlier that the order of the coupled sys-tem equations is equal to the total number of component modes minus the number ofconstraints. Thus, as more constraints are applied, more coordinates are eliminatedfrom the governing equations. This provides an additional explanation for the improvedcondensation of the free-interface method.However, the benefits of the free-interface formulation come at a price. It was notedat the end of Chapter 2 that more extensive component level calculations are requiredfor 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-interfacemethod 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 thesystem 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 SubstitutionBenfield and Hruda [15] proposed a hybrid substructure synthesis method in which a clas-sical free-free mode representation is used for some components, and the Craig-Bamptonmode set for others. This approach attempts to resolve one of the basic difficulties ofthe fixed-interface method—the retention of interface degrees of freedom in the systemChapter 3. Substructure Synthesis^ 62equations.Consider a two-component discrete structure shown with components a and b. Com-ponent a is designated the main body and its displacement is described with free-freemodes:ua = (pp^ (3.71)or, in partitioned form,a{UBu[4113(3.72)Component b is designated a branch component and its displacement is describedwith the Craig-Bampton mode set:lutiJu l ^( Nuts"^xijcz 4)1,^pnFor the purposes of this example, compatibility equation (3.5) can be written(3.73)[ AT ATuBa =Ub(3.74)Because it is a two-component system, the number of interface coordinates in a andb is the same, and is exactly equal to the number of constraint equations in (3.74).Therefore, Aa and Ab are square matrices and, in general, the compatibility equationscan be arranged so that one of the two is an identity matrix. Letting Ab = I, and using(3.71) and (3.74),uB _AaTuaB = _AaTep^(3.75)Substituting (3.75) in (3.73) givesuf,'^—A„TV 0 ^p^p IPAbub^--tiPIAI(DB 4)4^Pn^Pn }(3.76)Chapter 3. Substructure Synthesis^ 63With this transformation, expressions for the kinetic and potential energies of thecoupled system can be derived. Applying Lagrange's equations givesii/-i + k^= owhereiff = Tc7,1', M Tab"k = TLY Tab(3.77)(3.78)(3.79)4. =1 P }(3.80)Pnand Tab is the general two-component transformation matrix,4)B^04)I^0Tab= (3. 81 )—AaT(DB^0—TIA/V3^cl.,Note that the static constraint modes defined for component b allow full satisfactionof 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 fromthe coupled equations. This keeps the order of the coupled equations small, regardless ofhow complex the interface is.The method is applied to multi-component models by executing the above steps suc-cessively. For instance, to synthesize a third component c, which may be connected toeither a or b, the procedure is to repeat (3.71)—(3.77) with the coupled system a-b definedas the main body and the new component defined as the branch. The total synthesis ofa multi-component system therefore requires intermediate eigensolutions which providefree-free modes for the main body in the next level. The number of intermediate eigen-solutions varies according to how the components are connected, but at most it is .s — 2,Chapter 3. Substructure Synthesis^ 64where s is the number of components. Although it is undesirable to have to perform ad-ditional eigensolutions, they are typically of small order, thus limiting the computationalexpense.The chief drawback of component mode substitution is the inconsistency in the treat-ment of the components. The representation for the branch component is statically com-plete in that the static response to interface loading is exact, whereas that of the mainbody is not. Moreover, the static incompleteness of the main body is carried througheach 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 inter-face loading, as described in Section 2.4. Although this innovation improves convergenceby providing more realistic modes for the main body, it also means that componentscan no longer be analyzed independently and it significantly increases the computationalburden. (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 modesof the branch components.)A further point to note is that if the structure is to be reanalyzed following a designmodification, many of the intermediate results may have to recalculated, in addition tothe modes of the modified components.3.9 Error Estimation for Natural FrequenciesThe task of selecting dynamic component modes for a CMS analysis provides certainchallenges. 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 selectedusing a cutoff frequency criterion, but some judgement is required in choosing a cutofffrequency suitable for the target frequency range. It is therefore of considerable interestChapter 3. Substructure Synthesis^ 65to be able to predict the accuracy of a particular choice of component modes, if not inadvance of the analysis, then immediately afterwards. It is also of interest to make thesepredictions without having to compare the results to those of other methods.Hurty [5] provided a convergence criterion for the fixed-interface method based onestimating the perturbation to a natural frequency resulting from the addition of com-ponent modes. It can be readily noted that introducing more fixed-interface dynamicmodes 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 andextras (E) representing the newly added component modes:A-4.00 A;40E= (3.82) These extra rows and columns can be condensed into the original equations througha process similar to that used in (3.66)—(3.69). In this way, the extra modes form afrequency-dependent addition to the system mass matrix of the form,6,A4- (03) = co2k0E rEE^2 -1 -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 ican be obtained with,AC44 = aAk(Wigi wi 1 14 (3.84)The formula is accurate for small frequency changes. If the set of extra modes is expandedto 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 erroris 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 thereChapter 3. Substructure Synthesis^ 66is a significant error in the mode. In this way, inaccurate natural frequencies can beidentified in the results. But this method has two drawbacks: the error estimate dependson the selection of extra modes; and, if they are not unused modes left over from theoriginal analysis, the extra modes have to be calculated anew.In the present section, an error estimation scheme is developed for the free-interfaceformulation in which a complete account of the neglected free-free modes is taken. Thisis possible because the residual flexibility can be tuned to a particular frequency w by re-placing G with d(w), as was shown in Section 2.5. The resulting component displacementgiven byu (I)p + d(w)f (3.85)is an exact representation at frequency w when 0 < w < wc . Using the synthesis procedurefor the first-order mass formulation in Section 3.5, the resulting free vibration equationof the system is,{r(w) — ca 2 1} P = 0 (3.86)wherer(w) = X + TBTA [AT-1i7B(w)A1 -1 ATTB (3.87)tr(w) = -6-,-BB(w) (3.88)Solving the characteristic equationdet [r(w) — w 2 .1] = 0 (3.89)gives the exact natural frequencies of the coupled system in the range 0 < w < coc .Although (3.86) provides a condensed representation of the system, a complete accountof the static and inertial effects of the neglected modes has been included by virtue ofthe dynamic residual flexibility.Chapter 3. Substructure Synthesis^ 67Now 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 < wc . It is of interest to investigatehow the eigenvalues of (3.90), a i , vary as a function of ci). It is shown in Appendix B thatthe Rayleigh quotientPi Piis a continously non-increasing function of ar in the range 0 < (2) < wc . If 65 is equal to anexact natural frequency of the system (C) = wi), the Rayleigh quotient is stationary atai = If ai is moved to a value larger than wi , the Rayleigh quotient will stay the sameor decrease; if (7) is moved to a value smaller than wi, it will stay the same or increase. Asit does not matter which mode is being considered, the same result can be applied to allmodes of the system. This leads to the following general conclusion: in a system wherethe modes are ordered such that w 1 < w2 < < < ..., and where Ca is situatedbetween two exact modes of the system such that wi < iw << 4^j^1,2,...,i^ (3.92)•^4^j=i+1,i+ 2... (3.93)In (3.92) lower bounds to the exact natural frequencies are obtained; in (3.93) upperbounds are obtained.The free-interface formulation with the first-order mass matrix is obtained by setting= 0 in (3.90). In this case,>32.^j = 1,2, ...^ (3.94)(3.91)and the frequencies obtained are upper bounds to the exact values. Defining f/ i , (22, • •to be frequencies calculated from the free-interface formulation, (3.90) is now solved withChapter 3. Substructure Synthesis^ 68=^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 similarto (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 targetfrequency range when no other means of comparison are available. By using the dynamicresidual flexibilities of the components, the cumulative effect of all neglected modes canbe evaluated at a particular frequency without computing additional free-free modes.3.10 Modal Properties of Combined Systems; the Inclusion PrincipleTo conclude this chapter, the convergence of CMS-derived frequencies is investigated.Particular attention is given to the improvement in convergence offered by the free-interface method.Rayleigh [44] showed that if a constraint is applied to a dynamical system, the modi-fied frequencies interlace the original frequencies in such a way that the former are greaterthan, or in exceptional cases equal to, the latter. In a discrete system, adding a constraintis often equivalent to removing a degree of freedom, and vice versa [1]. This leads to theinclusion principle, whereby the frequencies of a discrete system monotonically decreasetowards the actual frequencies as the number of degrees of freedom increases. This resultis important in situations where modal truncation is a factor. Meirovitch and Kwak [45]have investigated the applicability of the inclusion principle to substructured synthesisformulations. They prove the monotonic convergence of the frequencies provided that aChapter 3. Substructure Synthesis^ 69 a bx = 21Figure 3.1: Two-component continuous barsupplementary 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 systemmatrices resulting from classical free-free mode sets, and by examining (3.60) it can beverified that this property holds for the fixed-interface method. However, it does not holdfor the free-interface formulations. While each additional free-free mode contributes asingle row and column to F, the simultaneous reduction in the residual flexibility affectsmatrix elements throughout r. Consequently, the inclusion principle cannot be appliedin its canonical form. It is therefore important to the discussion of residual flexibilityformulations to investigate their convergence characteristics.Dowell [46] showed that if two substructures are joined at a single point, the fre-quencies of the combined system interlace the component frequencies. Thus, the generalinterlacing principle for a dynamic system also applies to a system of uncoupled compo-nents. Dowell used a classical free-free representation of the system components, a casefor which the inclusion principle is known to hold.To investigate the influence of residual flexibility on the modal properties, considerthe axial vibration of the two-component distributed system in Figure 3.1. Componenta is a fixed-free bar with a frequency spectrum wai = = 7r/2, 37r/2, 57r/2, ,Chapter 3. Substructure Synthesis^ 70while component b is a free-free bar with spectrum wbi = abj = 0, r, 21-, 3r, ....Two constraints exist between the components: one geometric, and one natural. In aclassical free-interface component mode representation, only the geometric constraint isused explicitly, but if residual flexibility is included both must be used. The displacementfunctions of the two components arewa = E 0.i(x)ai 0.(x)f^ (3.97)1=1JWb = E obi (x)bj — ti,b(x)f^ (3.98)j=1where Oa i (X) and Obi (X) are the fixed-free eigenfunctions of component a and the free-freeeigenfunctions of component b respectively; and where I and J are the number of modesselected for components a and b respectively. The functions Ti,a (x) and 1/;b(x) are theresidual attachment modes associated with f:0.i(x)Oai(1)L-s•^1=1+1^ata (x )^ Oat(x)95.(1) " ^(71^at(3.99)(3.100)(3.101)j=J+1bb(x)^E- obi(xA)boi bi(1) = bb(x) — Ej (1)17i(X)fkbi(1)^(3.102)j=1^Abjwhere ika (x) and/Pb(x) are the inertia-relief attachment modes associated with f . Notethat 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 bydtP 2^21 dti„ 2(3.103)+ [f^dx^(--=) dx1 f 22 o dx dx1 1^'1V =^Aai4 — E2 i=i 13-1Chapter 3. Substructure Synthesis^ 711 I^J=^a2^i) + [ /Ap2 dx + 1 21^P^(3.104)2 4.4 i^2 L's^2 ^a^ti=i J=1where mass normalization of the eigenfunctions has been assumed. The geometric con-straint is satisfied with the equationI^ JR =^Oai (1)a i + l'ska (l)f —^Obj(1)bj 114(1)f = 0^(3.105)i=1 j=1The method of Lagrangian multipliers is used to derive the equations of motion. Withthe Lagrangian,L=T —V —pR^ (3.106)the following three equations are derived in conjunction with (3.105):a i A a ia i — pOai(/) = 0^ (3.107)Abibi pcbbi (1) = 0 (3.108)Ei+df— ft (1'4a(1) + 1414 1)) = 0^ (3.109)whereE = 101 ,tp^!dx + f 21 1'4 dx=^1^00^2 7 \t° (1)2a,\i1) + E c'b, i2( ‘ )i=i+i^al^j=J+1^biG = fi ichdxpa \ 2^/21 ( dijIb 2JO^) dx + 1^dx^dxt (gii( 1 ) + t qi( 1 ) =1=1+1 1\az^j=J+1 A b3Assuming sinusoidal solutions of frequency cv for ai, ki and f, (3.107)-(3.109) are substi-tuted into (3.105) giving the result02 . (/) \--■ ^7)30)^1^ (3.110)A .a-1 co 2 + 47g b •0 — 4) 2 = k— W 2 A/1-i^at 3Chapter 3. Substructure Synthesis^ 72where= =G2 (3.111)The right-hand side of (3.110) is the residual contribution from the neglected modes givenby the second-order formulation. Setting this term to zero gives the equation resultingfrom the classical free-free component mode representation. Replacing the right-handside 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 areshown in Figures 3.2 and 3.3. In these figures, the left- and right-hand sides of (3.110) areplotted independently. The three curves corresponding to the three right-hand sides arelabelled according to the residual approximation used. In Figure 3.2, the fundamentalfixed-free mode of component a and the rigid body mode of component b are used. Thefundamental 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 indicatedby 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 theassembled 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 betweenthe poles of the left-hand side. The poles of the left-hand side occur at the componentfrequencies, which are the same as the frequencies of the uncoupled system. Thus, fre-quencies of the combined system always interlace the frequencies of the uncoupled systemsuch thatal <^< A2 <^< (3.112)where A i , A2, ... are the squares of the uncoupled system frequencies, ordered from lowestcobl 0)2w11, .^1^.^.^.^I/Classical\1st OrderChapter 3. Substructure Synthesis^ 732.00,oiea 1.0w0coco0 0.0Cas'..C'-1.02usco—1 -2.00.00^0.25^0.50^0.75^1.00^1.25^1.50w (-:. 1r slFigure 3.2: Graphical solution for continuous bar, two mode approximationWb1^co,^Wei^02^03 b2^0)3Classicali 1 14 order0.50^0.75^1.00^1.25^1.50^1.75^2.00co (+ ir S-1 )6 2.0,—ridul 1.0"6coa):0° 0.0lccos:16.2 -1 .o2asèco-2.0Chapter 3. Substructure Synthesis^ 74Figure 3.3: Graphical solution for continuous bar, three mode approximationChapter 3. Substructure Synthesis^ 75to highest, and Cpl. , 6.)2, . are the frequencies of the combined system. This provides anupper bound for the combined system frequencies. Secondly, including residual effectsgives one more mode than is obtained with the classical method. The additional modeappears in such a way that the classically-derived frequencies interlace the frequenciesderived with residual effects:(2)^- (1)^- (0)^- (2)^- (1)^- (0)W1 < W1 < W1^W2 < U-72^W2^• • • (3.113)- (0) - (1) - (2)where wi , wi , wi are frequencies calculated with the classical, first-order and second-order residual methods respectively. Note that the basic interlacing principle (3.112) isunaffected by the vertical position of the right-hand side curves.Thirdly, from the inclusion principle it is known that solutions obtained with theclassical CMS representation converge to the exact values as the number of componenteigenfunctions 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 curvesdegenerate to the classical case. To show this, the right-hand side of (3.110) can bewritten,1K -w2f1^1 — w2 Bco^( 4.02 )^co Oti (i) ( (02 \b^63 )4/2-132^i=l+1 Aai^Aai -^=J+102ai(1)^j^(gi(1)^(3.115)i=■+1^at^j= j+iIf c4.) 2 < Aai, i = I + 1, I + 2, ... and w 2 < Abj, j = J + 1, J + 2, ..., w 2 E is always lessthan 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 frequencyrange.(3.114)wherei^1 +1,1 +2,...2w11111^= 0/-000 )taiCV 211111^= 0J-400 Abs j= J +1,J +2,...Therefore, by (3.115)w2BJim^— 0/ ,J--000 GChapter 3. Substructure Synthesis^ 76As more free-interface modes are used in the analysis, the lowest numbered termsin the summation are removed; and since all terms are positive, G and w2B must bothdecrease in magnitude. Furthermore, for a fixed frequency w,and because Ern G = 0,Ern^G —^ (3.119)1 — W2BAs a result, the right-hand side of (3.110) vanishes as more component modes are con-tributed to the modal representations. Evidence of the diminishing residual terms canbe found by comparing the the ls t order and 2nd order curves in Figures 3.2 and 3.3.In this limiting process, the frequencies obtained with residual methods approachthose of the classical case. The combination of the two limits gives the required conver-gence properties:lim J.,C2) = lim Co•( 1) = lim w.°) = lim (3.120)I,J.00 2  1,J-400More importantly, the residual solutions are seen to converge to the exact results ahead ofthe classical solutions. The residual terms adjust themselves as more component modesare 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 tosupply the variables with numbers. The general form of (3.110) is in fact applicable to anytwo-component system subjected to a single constraint. By adding further summationsChapter 3. Substructure Synthesis^ 77to the left side, the equations can be used for any number of components. But whenanother constraint is imposed on the system, (3.112) is no longer applicable and theupper bound provided by this relation is lost. Systems with multiple constraints requirea more sophisticated treatment than the one presented above.3.11 SummaryA number of specific CMS formulations which are suitable for substructured systems ofan arbitrary geometrical complexity have been derived based on the component moderepresentations detailed in Chapter 2. Inter-component compatibility and equilibriumare expressed with the general relations stated in Section 3.2. The resulting formulationsare computationally more efficient than the more general elimination methods of Section3.3 and are therefore better suited for a finite element program.Of particular interest are the free- and fixed-interface formulations derived from theMacNeal-Rubin and Craig-Bampton mode sets. Two free-interface formulations werederived, in which first- and second-order mass matrices are employed. Both formulationsgive matrix equations in terms of the free-free modal coordinates only. On the otherhand, the fixed-interface equations are in terms of interface displacements and modalcoordinates. As a result, the free-interface formulations can be expected to providegreater condensation when applied to arbitrarily complex systems.New contributions made in this chapter are the following: the generalized formula-tions for the two free-interface methods and the modal force method using the generalconnectivity matrix; the method for calculating lower bounds to frequencies derived fromthe free-interface method using the dynamic residual flexibility concept; and the demon-stration that frequencies calculated with residual flexibility will converge to the exactChapter 3. Substructure Synthesis^ 78results as component modes are added (the inclusion principle), and that residual flexi-bility provides an accelerated convergence for the system modes.Chapter 4Modal Analysis of Three Finite Element Models4.1 IntroductionIn 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 first-and second-order mass terms;— direct finite element analysis;— Guyan reduction.• The computational time required for each method as a fuitction of the eigensolutiontolerance.• The effect of modal truncation on the CMS methods by presenting results obtainedwith various cutoff frequencies.79Chapter 4. Modal Analysis of Three Finite Element Models^ 80The accuracy of CMS-derived frequencies and mode shapes is determined by com-paring 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 finiteelement mesh, a CMS analysis can do no better than reproduce the results of a directanalysis. The CMS frequency and mode shape results can thus be presented as a per-centage difference from the direct results. The percentage difference in the ith naturalfrequency is,-%Af, = f.s x 100%fiA percentage difference in the ith mode shape is given by,%Au, =11%112 112 x 100%where 11%11 2 is the Euclidian norm of the vector u1.The natural frequency and mode shape results were computed using the Vibrationand Strength Analysis Program (VAST), Version 06 [47]. Originally, this program onlyhad the capability for direct analysis of finite element models and Guyan reduction ofsubstructured models. Modifications were made to the program to allow CMS analysisusing the fixed- or free-interface method. Details of the CMS implementation in VASTcan be found in Appendix C.4.2 Analysis of a Container ShipA two-dimensional model of a container ship is depicted in Figure 4.1. This type of modelis useful for predicting the vertical modes of vibration of the actual ship. Wave motionand propeller-induced pressure forces can excite resonances in the ship's structure andso accurate knowledge of the natural modes is important in the design stage. The modelis composed of 8-node membrane elements, and 3-node bar elements: the membranes(4. 1)(4.2)Chapter 4. Modal Analysis of Three Finite Element Models^ 81(c)(a)^(b)Figure 4.1: Two-dimensional container ship model: a) stern; b) aft-body; c) deckhouse;d) mid-body; e) fore-bodysimulate the sides of the hull and interior walls; the bar stiffeners simulate the hullbottom, decks and bulkheads. The substructuring scheme shown in Figure 4.1 is used forthe CMS analyses. Relevant data regarding the model and its constituent componentscan be found in Table 4.1.Components # nodes # d.o.f. # membranes # barsStern (a) 54 162 13 10Aft-body (b) 65 195 16 16Deckhouse (c) 121 363 32 32Mid-body (d) 121 363 32 16Fore-body (e) 121 363 32 24Complete Model 447 1341 125 106Table 4.1: Description of the container ship model(d)^(e)Chapter 4. Modal Analysis of Three Finite Element Models^ 824.2.1 Natural Frequency ResultsIn Table 4.2 are found the free-interface CMS natural frequency results for the shipmodel. Six cases are analyzed: first- and second-order approximations are employedwith cutoff frequencies on the free-free modes of 25, 30 and 35Hz. The CMS frequenciesare compared with the results of the direct analysis for the first 37 elastic modes. Itshould be noted that the model also has three rigid body modes that are omitted fromthe table. At the bottom of the table are the total number of degrees of freedom in theequations of motion resulting from each formulation.For each of the three frequency cutoffs, the second-order mass offers a distinct im-provement over the first-order mass. This does not come as a surprise since in Chapter2 it was demonstrated that the MacNeal-Rubin mode set employing the first-order ap-proximation is only accurate when the square of the target frequency is negligible incomparison 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 isnot met. In view of the cutoff frequencies that have been used here, this condition clearlyhas not been met for the target frequency range 0-30Hz. Satisfying this condition re-quires calculating a much larger number of component modes, something which shouldbe avoided. Including the second-order mass terms improves the inertial representationof the neglected modes, which partially compensates for low cutoff frequencies.Another important phenomenon is that as the cutoff frequency increases, the CMSfrequencies converge to the direct-analysis results and that, in general, the low frequenciesconverge before the high. However, there are some exceptions to this latter rule. Forexample, mode 20 converges faster than either mode 15 or 16. But it should be observedthat each mode has a different distribution of strain and kinetic energy and so theycannot be expected to converge at the same rate.Chapter 4. Modal Analysis of Three Finite Element Models^ 83Direct FEM Analysis % Error, Free-Interface CMS AnalysisMode Frequency(Hz)25 Hz. Cut-off 30 Hz. Cut-off 35 Hz. Cut-off1st 2nd 1st 2nd 1st 2nd4 0.881 0.00 0.00 0.00 0.00 0.00 0.005 2.14 0.02 0.00 0.02 0.00 0.01 0.006 3.52 0.07 0.01 0.05 0.01 0.04 0.017 4.55 0.18 0.06 0.09 0.02 0.07 0.028 5.58 0.19 0.02 0.12 0.01 0.11 0.019 5.91 0.57 0.33 0.28 0.10 0.23 0.0810 8.01 0.26 0.04 0.18 0.02 0.11 0.0211 8.93 0.39 0.09 0.30 0.07 0.23 0.0512 10.2 0.51 0.07 0.38 0.04 0.25 0.0313 12.3 0.72 0.14 0.44 0.08 0.35 0.0714 13.1 2.07 0.47 1.17 0.25 0.75 0.1715 13.7 2.61 1.01 1.71 0.90 1.07 0.4916 14.2 6.52 2.49 2.82 0.79 1.71 0.6617 15.0 2.22 0.38 1.19 0.24 0.81 0.1418 16.4 3:35 0.54 1.79 0.22 0.76 0.1019 17.3 2.58 1.00 1.89 0.84 1.13 0.3920 18.9 1.64 0.48 0.77 0.14 0.38 0.0921 19.2 3.85 2.16 2.56 1.65 1.19 0.5922 20.0 2.92 0.57 1.63 0.23 1.08 0.1023 20.2 5.98 3.21 1.86 0.56 0.91 0.4824 21.4 3.57 1.44 1.10 0.42 0.85 0.2425 21.7 6.82 3.26 2.00 0.35 1.45 0.2526 22.9 9.45 1.81 1.35 0.41 0.82 0.1327 23.7 6.06 1.73 1.88 0.59 1.36 0.3628 24.0 15.5 5.20 2.77 0.87 1.24 0.2829 24.9 20.0 4.48 3.43 1.21 1.89 0.5230 25.3 25.4 7.37 3.57 0.93 2.45 0.5231 25.7 26.6 8.77 2.98 1.28 1.01 0.3232 26.1 32.2 14.6 3.57 0.36 1.11 0.2033 26.9 35.9 14.8 5.42 1.54 2.36 0.5234 27.8 47.4 20.9 2.94 1.55 1.37 0.4935 28.1 74.3 40.8 2.56 0.89 1.18 0.1936 28.6 - - 4.94 1.53 2.17 0.5837 29.3 - - 8.16 1.67 1.14 0.4238 29.7 - - 14.0 4.00 1.44 0.3039 30.0 - - 15.2 4.26 2.59 0.6640 30.6 - - 17.7 4.60 2.40 0.92# d.o.f.^1341 36 36 50 50 62 62Table 4.2: Natural frequency results for the container ship, free-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 84Table 4.2 demonstrates the efficiency of the free-interface method's condensation,particularly when the second-order approximation is used. In the last column (35Hzcutoff, 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 theship have been reduced in order by a factor of 20 over those of the direct analysis, whileleaving the low-frequency spectrum nearly perfectly intact.The results of the fixed-interface method (Table 4.3) are similar in that convergenceincreases with the cutoff frequency and the low frequencies tend to converge before thehigh. The cutoff frequencies here refer to fixed-interface component frequencies, whichare always higher than their free-free counterparts. Thus, the 35Hz cutoff in the fixed-interface method allocates 45 dynamic modes to the analysis while in the free-interfacemethod 62 are needed. Table 4.4 shows the frequencies of the component modes used inboth cases. Note that the free-free modes include the three rigid body modes for eachcomponent. The mid-body and the fore-body contribute the majority of the modes, asthese 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-interfacemethods and displays their convergence characteristics as a function of the number of dy-namic component modes. This is in contrast with Tables 4.2 and 4.3 where the accuracyof the frequencies is recorded as a function of the cutoff frequency. For the curves cor-responding to the free-interface formulations, the number of dynamic component modesrefers to non-rigid body modes only. Rigid body and static modes are essential for goodaccuracy in all CMS analyses, while elastic dynamic modes are non-essential because theycan be added in varying numbers as the accuracy requires. The graph therefore showsthe effect of different types of dynamic modes on the overall accuracy and convergencerate of the system frequencies. In terms of overall accuracy, free-free modes are moreeffective provided that their inertia contribution is included; if not, fixed-interface modesChapter 4. Modal Analysis of Three Finite Element Models^ 85Direct FEM Analysis % Error, Fixed-,Interface CMS AnalysisMode Frequency(Hz)GuyanReduction20 Hz.Cut-off25 Hz.Cut-off30 Hz.Cut-off35 Hz.Cut-off40 Hz.Cut-off4 0.881 14.4 0.00 0.00 0.00 0.00 0.005 2.14 36.5 0.02 0.02 0.01 0.01 0.016 3.52 48.8 0.03 0.02 0.01 0.01 0.017 4.55 106 0.18 0.08 0.07 0.06 0.068 5.58 114 0.10 0.03 0.02 0.00 -0.019 5.91 210 0.15 0.07 0.06 0.05 0.0510 8.01 132 0.08 0.02 0.02 0.01 0.0011 8.93 166 0.92 0.34 0.26 0.20 0.1712 10.2 192 0.43 0.19 0.16 0.08 0.0513 12.3 194 0.74 0.55 0.39 0.34 0.2114 13.1 178 1.86 0.42 0.35 0.21 0.1515 13.7 181 1.26 0.84 0.71 0.67 0.5516 14.2 195 1.64 1.10 0.96 0.79 0.5417 15.0 218 0.92 0.51 0.29 0.19 0.1118 16.4 215 5.20 0.62 0.50 0.23 0.1619 17.3 202 .5.18 0.85 0.73 0.49 0.3820 18.9 193 7.36 0.60 0.43 0.23 0.2021 19.2 198 10.3 3.96 2.18 1.29 1.0422 20.0 209 14.5 1.53 0.87 0.65 0.5523 20.2 230 15.3 2.56 1.31 0.81 0.6824 21.4 232 17.8 3.04 1.60 0.72 0.4025 21.7 238 20.7 7.35 1.23 0.73 0.4726 22.9 229 42.9 3.90 2.28 1.09 0.6327 23.7 219 43.8 4.73 0.77 0.39 0.1928 24.0 247 47.1 9.68 2.71 1.25 0.9429 24.9 241 53.7 13.9 1.24 0.81 0.3730 25.3 246 52.8 14.3 2.53 1.49 0.4731 25.7 243 54.7 19.2 3.44 1.35 0.8532 26.1 244 58.2 26.6 4.71 2.15 0.5833 26.9 257 57.7 26.3 5.68 1.46 0.9034 27.8 257 88.0 23.9 2.81 1.10 0.5835 28.1 276 92.6 28.7 10.4 1.43 0.7636 28.6 288 102 35.2 13.2 1.85 0.9537 29.3 295 99.8 37.5 16.6 2.25 0.8938 29.7 290 109 43.5 15.8 2.24 1.8639 30.0 293 112 42.3 19.4 1.66 0.9240 30.6 297 114 43.6 22.3 1.15 0.42# d.o.f. 1341 102 120 127 133 147 160Table 4.3: Natural frequency results for the container ship, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 86Component Frequencies (Hz)Mode Stern Aft-body Deckhouse Mid-body Fore-bodyFree Fixed Free Fixed Free Fized Free Fized Free Fixed123456789101112131415161718192021222300018.626.528.732.86.4118.218.933.738.800022.625.028.930.324.437.700019.921.027.429.433.26.4816.116.925.426.932.635.739.10004.579.8811.415.519:020.623.424.027.227.728.128.829.331.933.633.84.118.3112.012.818.321.523.024.928.128.831.131.631.833.033.534.435.538.238.239.60004.139.2410.614.618.519.322.522.725.726.427.729.531.431.832.833.134.335.00.933.835.928.8013.416.218.821.324.124.927.228.430.031.231.632.434.334.335.235.836.838.439.2Table 4.4: Component frequencies of the container ship35^^^ Free, l st order mass^ Free, 2nd order mass30^O^ Fixed0 25Vvi 20E75 15z1010^20^30^40^50No. of dynamic component modesFigure 4.2: Combined natural frequency results for the container shipChapter 4. Modal Analysis of Three Finite Element Models^ 87Chapter 4. Modal Analysis of Three Finite Element Models^ 88are more effective. The same conclusion can be drawn for the convergence rate, which isindicated by the slope of the curves.Independent of accuracy and convergence rate considerations, the fixed-interface for-mulation is not as effective at condensing the equations of motion as the free-interfacemethod—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 be-cause in the formulation, more independent constraint equations can be specified withthe generalized coordinates.4.2.2 Mode Shape ResultsThe accuracy of the CMS mode shapes is not only a function of the cutoff frequencybut it also shows a strong dependency on the eigensolution tolerance. Eigensolutions aredetermined by an iterative procedure in which a tolerance factor is required to terminatethe solution. The VAST eigensolution routine EIGEN1 uses an algorithm based on theinverse power method with shifting. The condition,A (k) _  < to/ (4.3)Vieis used to terminate in the kth iteration. While variations in the tolerance may have onlya minor effect on the frequencies, a much more pronounced effect is visible in the modeshapes. This observation is verified by noting that first-order variations in the eigen-vectors 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 theorder of tot} in the mode shape.This is doubly true in a CMS analysis where accurate reconstruction of the systemmode shapes relies on accurate eigenvectors at both the component and system level. Themode shape results for the ship model (Tables 4.5 and 4.6) were calculated with tol = 10-6Chapter 4. Modal Analysis of Three Finite Element Models^ 89Direct FEM Analysis % Error, Free-Interface CMS AnalysisMode Frequency(Hz)25 Hz. Cut-off 30 Hz. Cut-off 35 Hz. Cut-off1st 2nd 1st 2nd 1st 2nd4 0.881 0.02 0.02 0.02 0.02 0.02 0.025 2.14 0.07 0.04 0.06 0.04 0.05 0.046 3.52 0.20 0.11 0.16 0.11 0.14 0.117 4.55 0.62 0.52 0.24 0.28 0.32 0.248 5.58 1.58 2.45 1.18 0.54 1.24 1.639 5.91 1.78 2.30 1.13 0.54 1.18 1.4210 8.01 1.23 0.58 1.16 0.31 0.81 0.2711 8.93 1.99 1.06 1.87 0.93 1.33 0.7412 10.2 2.74 1.04 2.37 0.80 1.15 0.6013 12.3 5.68 3.62 3.34 2.24 2.35 2.1314 13.1 12.3 7.70 15.7 6.88 5.40 3.8415 13.7 29.9 8.60 22.7 7.67 7.50 5.8416 14.2 23.1 7.90 21.6 3.77 5.39 4.6517 15.0 24.6 8.65 19.4 6.73 9.78 5.0018 16.4 24.3 8.21 16.7 5.94 5.89 2.8819 17.3 24.7 15.3 17.4 13.2 11.3 6.0820 18.9 37.1 20.6 22.0 8.01 14.9 4.32# d.o.f. 1341 36 36 50 50 62 62Table 4.5: Mode shape results for the container ship, free-interface methodand are compared with direct-analysis mode shapes of the same tolerance so as to bestisolate the effect of modal truncation. Although acceptable results for frequencies canbe 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 modeshapes than for frequencies and that larger absolute error can be expected in modeshapes than in frequencies at a particular cutoff value. Indeed, if a e% error is found inthe frequency, the error in the mode shape might be expected to be,2^1AU%  ( 1 + TZ) J^_ 1^1.0% (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 typicalof the differences between frequency and mode shape errors found in the tables.Chapter 4. Modal Analysis of Three Finite Element Models^ 90Direct FEM Analysis % Error, Fixed-Interface CMS AnalysisMode Frequency(Hz)GuyanReduction20 Hz.Cut-off25 Hz.Cut-off30 Hz.Cut-off35 Hz.Cut-off40 Hz.Cut-off4 0.881 15.5 0.03 0.02 0.02 0.02 0.025 2.14 42.4 0.14 0.10 0.07 0.06 0.066 3.52 103 0.37 0.24 0.18 0.17 0.167 4.55 108 0.98 0.51 0.49 0.45 0.438 5.58 68.0 1.03 0.94 0.88 1.00 0.999 5.91 95.2 1.02 0.75 0.69 0.80 0.8010 8.01 122 1.45 0.75 0.62 0.63 0.4011 8.93 114 5.58 2.29 1.88 1.53 1.2512 10.2 124 4.18 2.53 2.19 1.58 1.1313 12.3 101 12.4 7.27 6.05 4.74 3.7214 13.1 112 23.6 6.35 6.91 5.07 5.5615 13.7 113 13.8 12.3 8.25 7.76 6.0716 14.2 103 9.92 11.8 7.90 7.04 4.5117 15.0 130 11.2 6.20 4.82 3.82 3.5318 16.4 122 40.4 8.66 6.95 5.42 5.3219 17.3 114 49.1 11.2 9.01 8.24 6.8920 18.9 134 59.4 18.1 11.0 4.31 3.75# d.o.f. 1341 102 120 127 133 147 160Table 4.6: Mode shape results for the container ship, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 91CPU Times (s) for 40 ModesTolerances 10-2 10' 10 -6Direct Analysis 664 2448 5298Guyan Reduction 317 401 484Fixed-Interface CMS 793 1356 2379Free-Interface CMS 298 462 1028Table 4.7: CPU times for modal analysis of the container shipTables 4.5 and 4.6 show that comparable results are obtained with the fixed-interfaceand the free-interface methods, in spite of the fact that the latter uses significantly fewerindependent variables in the cases presented. This reiterates the free-interface method'ssuperiority in condensing the equations of motion whilst preserving the low-frequencyspectrum.4.2.3 PerformanceTable 4.7 shows the CPU time expenditure for the four methods at three different tol-erances. For the CMS analyses, the tolerance factor applies to both the componentand system analyses. Two general observations can be made. First, as the toleranceis reduced, the relative efficiency of CMS increases. Secondly, the free-interface methodmaintains a significant performance edge over the fixed-interface method for all threetolerances. For this latter comparison, it is only fair to use CMS runs which give a sim-ilar degree of accuracy. The two used in Table 4.7 are, for the free- and fixed-interfacemethods respectively, the 35Hz cutoff with second-order mass (see the last column ofTable 4.2) and the 40Hz cutoff (see last column of Table 4.3). The advantage enjoyed bythe free-interface method is partly explained by its superior condensation and partly bydifferences in the program implementation. Because the two methods had to be fit intoChapter 4. Modal Analysis of Three Finite Element Models^ 92an existing finite element program, certain redundancies in the algorithm were unavoid-able 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 poorcompared 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 analysisand the fixed-interface method somewhat slower. But the difference either way is notvery significant and unless a substructured treatment is needed for some other reason, nospecial advantage is gained from using CMS at this level of accuracy. Indeed, this rep-resents the lower limit of CMS's performance advantage. Models significantly smaller inorder 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 morethan 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 fre-quencies are to be resolved, a tolerance of about 10-6 is usually needed. At this level ofaccuracy, CMS has a clear performance advantage.4.3 Analysis of a Telescope Focus UnitThe telescope focus unit (TFU) is composed of the three concentric cylinders shown sepa-rately 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 andfour guide rails.Chapter 4. Modal Analysis of Three Finite Element Models^ 93Components # nodes # d.o.f. # plates # bricks # barsInner Tube (1) 76 456 20 0 8Outer Tube (2) 92 552 28 0 0Screw Assembly (3) 84 504 16 4 0Chopping Mechanism (. 4 ) 120 360 0 18 0Support Tube (5) 108 648 28 0 0Complete Model 428 2568 92 22 8Table 4.8: Description of the TFU modelScrew Assembly Allows the inner tube to be advanced or retracted for focussing.Chopping Mechanism Appended to the inner tube, this creates rapid oscillations inthe lens position necessary for superimposing infrared images of the object underobservation and its background.Support Tube Supports the outer tube along a flange with adjustable positioningscrews.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 inTable 4.8.Because vibration of the TFU inner components could degrade the telescope's imagequality, the dynamic response and the existence of resonances is of great importance tothe design. The source of excitation is the chopping mechanism, which exerts a square-wave periodic force and can therefore excite a wide range of vibrational modes. Inthis section, CMS is used to determine the low-frequency modes of the TFU with eightlocations on the support tube fixed. These eight locations are where the support strutswould normally be attached in the complete telescope (see Section 4.4).Chapter 4. Modal Analysis of Three Finite Element Models^ 94(a) (b) (c)Figure 4.3: Components of the TFU model: a) inner tube With screw assembly attachedto top, and chopping mechanism attached to bottom; b) outer tube; c) support tubewith external constraints.Chapter 4. Modal Analysis of Three Finite Element Models^ 954.3.1 Natural Frequency ResultsThe same general trends can be seen in Tables 4.9 and 4.10 as were found in the frequencyresults of the ship model. Clearly visible are the systematic convergence as cutofffrequency is increased (except in modes 1-3); the convergence of the low modes beforethe high; the greater condensation provided by the free-interface method over the fixed-interface method; and, in the former, the improved accuracy with the second-order massapproximation.However, one feature not evident in the ship results but more noticeable here is theapparent slow convergence of the fixed-interface method. Between Guyan reduction—which is essentially a zero-frequency cutoff—and 4000Hz cutoff, the improvement in theaccuracy of natural frequencies is noticeably less than what the free-interface methodachieves over a smaller range of cutoff frequencies. One might be tempted to conclude thatin this case the free-interface method enjoys a speedier convergence, but this is largely anillusion. Comparing the fixed- and free-interface modes of the telescope components (seeTable 4.11) reveals that the unconstrained components have either the same or a highermodal density than their constrained counterparts. Thus, if an incremental increase ismade to the cutoff frequency, more free-interface modes have to be added than fixed-interface modes. Because of this difference, the fixed-interface method may require ahigher cutoff frequency to get comparable results, but it may not mean that it requiresmore dynamic modes. Indeed, of the 63 free-free modes used in the last two columns ofTable 4.9, 24 are rigid-body modes which still leaves 39 elastic modes in comparison to the29 used by the fixed-interface method at 4000Hz cutoff. This point is reflected in Figure4.4 where the overall accuracy of the frequencies is plotted as a function of the numberof dynamic component modes. Here, as in the ship model, the fixed-interface curve fallsbetween the two free-interface curves. An interesting point is that the convergence ratesChapter 4. Modal Analysis of Three Finite Element Models^ 96Direct FEM Analysis % Error, Free-Interface CMS AnalysisMode Frequency(Hz)2000 Hz. 2500 Hz. 3000 Hz. 3500 Hz.1st 2nd 1st 2nd 1st 2nd 1st 2nd1 34.7 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.042 72.8 -0.05 -0.05 -0.05 -0.06 -0.06 -0.06 -0.06 -0.063 72.8 -0.02 -0.02 -0.02 -0.02 -0.03 -0.03 -0.03 -0.034 185.8 0.07 0.02 0.05 0.00 0.05 0.00 0.05 0.005 432 0.17 0.10 0.08 0.01 0.03 0.01 0.03 0.016 433 0.17 0.10 0.17 0.10 0.03 0.01 0.03 0.017 655 0.26 0.15 0.13 0.03 0.13 0.03 0.13 0.038 658 0.26 0.15 0.26 0.15 0.14 0.04 0.14 0.039 1002 0.34 0.22 0.34 0.22 0.34 0.22 0.34 0.2210 1059 1.03 0.21 0.92 0.11 0.23 0.10 0.23 0.1011 1067 1.08 0.22 1.08 0.22 0.23 0.09 0.23 0.0912 1231 1.55 1.40 0.27 0.13 0.27 0.14 0.28 0.1413 1232 1.54 1.39 1.54 1.39 0.28 0.13 0.26 0.1214 1273 3.48 0.23 3.40 0.19 3.27 0.17 0.68 0.0615 1347 0.04 0.00 0.04 0.00 0.02 0.00 0.02 0.0016 1444 0.16 0.01 0.16 0.01 0.14 0.01 0.05 0.0117 1447 2.73 0.20 2.73 0.21 2.73 0.20 0.16 0.0318 1486 5.86 0.00 5.86 0.00 5.85 0.00 0.00 0.0019 1576 2.13 0.00 2.12 0.00 2.11 0.00 0.00 0.0020 1693 13.0 12.1 1.88 1.20 1.88 1.20 1.79 1.1921 1913 1.31 0.07 0.79 0.06 0.42 0.02 0.30 0.0022 1955 0.34 0.31 0.34 0.31 0.34 0.31 0.12 0.1023 2210 26.4 6.55 0.02 0.00 0.02 0.00 0.01 0.0024 2289 26.5 4.53 15.7 1.88 3.48 0.56 3.48 0.5625 2321 57.4 48.2 17.5 3.10 3.56 0.60 3.54 0.6026 2562 69.8 40.6 11.8 5.81 0.03 0.02 0.03 0.0227 2563 72.7 51.0 13.0 6.58 0.69 0.48 0.24 0.0528 2596 75.9 57.8 40.7 32.5 0.60 0.12 0.31 0.0329 2699 69.5 53.8 36.8 33.5 1.64 1.16 1.64 1.1730 2702 81.7 64.5 61.0 34.9 1.65 1.14 1.64 1.1331 2911 99.1 52.7 52.4 26.5 1.08 1.04 1.08 1.0432 2941 175 62.8 55.2 31.7 7.80 6.61 2.77 1.4933 2979 179 80.9 64.8 37.5 13.8 5.64 1.82 1.8234 3032 206 97.5 91.8 37.0 16.6 13.5 7.77 3.4335 3118 203 97.0 93.0 42.6 17.2 12.5 5.03 4.7236 3264 214 101 148 65.1 13.1 10.4 3.01 2.6537 3315 239 107 150 69.4 31.4 9.97 7.23 3.7438 3323 247 115 179 70.8 33.5 16.3 9.94 6.5039 3359 280 145 181 78.3 45.6 20.9 9.30 5.7040 3533 261 152 191 85.5 54.6 17.6 5.71 1.99# d.o.f. 2568 42 42 47 47 56 56 63 63Table 4.9: Natural frequency results for the TFU, free-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 97Direct FEM Analysis % Error, Fixed-Interface CMS AnalysisMode Frequency(Hz)GuyanReduction2000 Hz.Cut-off3000 Hz.Cut-off4000 Hz.Cut-off1 34.7 0.04 -0.04 -0.04 -0.042 72.8 0.89 -0.06 -0.07 -0.073 72.8 0.91 -0.01 -0.01 -0.014 185.8 0.32 0.32 0.00 0.005 432 7.33 0.60 0.01 -0.036 433 7.42 2.00 0.00 -0.037 655 6.14 1.20 0.27 0.078 658 6.12 2.88 0.53 0.079 1002 10.9 0.41 0.41 0.4110 1059 5.71 4.10 0.62 0.2711 1067 5.88 4.18 3.52 0.2812 1231 5.75 1.93 0.27 0.2213 1232 25.9 5.51 1.10 0.2214 1273 53.9 5.97 2.08 2.0615 1347 60.3 7.23 0.09 0.0916 1444 49.8 2.81 0.07 0.0517 1447 69.6 7.05 2.62 0.1818 1486 67.6 6.14 4.21 -0.1119 1576 62.6 23.4 0.11 0.0020 1693 72.0 15.7 0.49 0.4921 1913 54.1 27.5 2.19 0.1022 1955 52.0 27.0 13.1 0.0123 2210 48.5 16.0 9.51 0.0124 2289 44.2 27.3 7.24 5.2425 2321 50.2 41.3 10.1 5.1826 2562 43.2 28.1 0.39 0.0427 2563 44.9 28.8 7.47 1.3228 2596 58.2 34.3 12.2 4.7929 2699 54.3 36.0 9.00 1.8730 2702 63.7 37.2 21.4 2.0231 2911 52.0 36.1 12.8 1.1332 2941 52.8 35.9 18.6 3.3433 2979 68.3 35.5 23.3 8.3434 3032 65.6 35.4 22.1 7.7335 3118 67.6 41.3 23.6 5.2936 3264 69.2 53.6 18.6 5.8137 3315 77.5 52.3 20.6 6.7238 3323 86.3 56.5 29.7 6.8239 3359 88.1 65.0 30.2 8.6940 3533 87.2 62.8 35.6 7.97# d.o.f. 2568 360 369 380 389Table 4.10: Natural frequency results for the TFU, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 98Component Frequencies (Hz)Mode Inner tube Outer tube Screw Assem. Chop Mech. Support tubeFree Fixed Free Fixed Free Fixed Free Fixed Free Fixed1 0 1233 0 2764 0 0 490 1338 14772 0 1233 0 3178 0 0 490 1428 15033 0 2093 0 3178 0 0 862 1482 15944 0 2411 0 3759 0 0 2074 1574 16005 0 2728 0 0 0 2955 2203 22306 0 2728 0 0 0 2955 2557 26037 968 3201 1399 1706 2560 29958 968 3245 1399 1706 2919 30619 1650 3839 1675 3031 307510 1859 1723 3246 348111 2439 181112 2439 181113 2562 183514 2654 194815 3035 236216 3343 245917 273118 285919 299520 299521 305122 308923 3298Table 4.11: Component frequencies of the TFU—o-- Free, l st order massA^ Free, 2nd order mass0^ Fixedcl 20Va..aCl)a)18 15E*60z105^10^15^20^25No. of dynamic component modesFigure 4.4: Combined natural frequency results for the TFU model25Chapter 4. Modal Analysis of Three Finite Element Models^ 99Chapter 4. Modal Analysis of Three Finite Element Models^ 100for the three cases are about the same. That is, regardless of the overall accuracies, theeffect of adding one additional dynamic mode is the same on average for each formulation.Another interesting feature is the appearance of negative frequency errors, particu-larly in modes 1-3. It has been argued in Chapter 2 that as long as inter-componentcompatibility is maintained, the CMS results should converge from above as more com-ponent modes are added. That is, modal truncation being the only source of error, itis 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 thetolerance 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 numbersof similar magnitude or when inverting a poorly condition matrix; and the influence ofnon-conforming elements in the structural model. Since both methods show the samenegative error in modes 1 and 2, it is unlikely that the tolerance factor or numerical round-off could be responsible. Instead, this shows the effect of using incompatible elementsin the mesh, in particular, the thick/thin shell element whose rotational coordinates areincompatible when two elements meet in perpendicular planes. With incompatibilities inthe component modes, the upper bound on the CMS frequencies is lost and the possibilityof a negative frequency error arises. Such negative errors are most likely to occur in thelowest frequency modes, as these are the first to converge, and in modes with significantdisplacement in the incompatible elements.4.3.2 Mode Shape ResultsThe convergence characteristics of the mode shapes (see Tables 4.12 and 4.13) are inmany ways similar to those of the ship. But now, because of the presence of repeatedfrequencies, there is an added complication: mode shapes cannot be defined uniquely forrepeated frequencies and so it makes little sense to compare them with Equation (4.2).Chapter 4. Modal Analysis of Three Finite Element Models^ 101Direct FEM Analysis % Error, Free-Interface CMS AnalysisMode Frequency(Hz)2000 Hz. 2500 Hz. 3000 Hz. 3500 Hz.1st 2nd 1st 2nd 1st 2nd 1st 2nd1 34.7 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.052 72.8 - - - - - - - -3 72.8 - - - - - - - -4 185.8 0.08 0.03 0.05 0.01 0.05 0.01 0.05 0.015 432 4.58 0.25 2.17 2.19 0.12 0.12 4.59 4.576 433 4.59 0.25 2.18 2.21 0.12 0.12 4.60 4.597 655 0.72 1.90 2.21 2.37 1.79 7.84 0.25 1.788 658 0.72 1.93 2.36 2.49 1.82 7.86 0.25 1.819 1002 1.14 0.80 1.16 0.79 1.19 1.16 1.19 0.7910 1059 2.49 3.15 2.29 1.05 2.26 2.23 2.25 0.4511 1067 2.39 3.32 2.51 2.70 2.17 2.12 2.18 0.4612 1231 5.80 7.88 14.6 13.4 13.9 13.8 18.5 22.713 1232 5.89 8.15 15.6 14.5 14.1 14.0 18.8 23.014 1273 5.81 3.52 6.22 2.33 6.01 1.84 1.85 1.5615 1347 1.57 1.27 2.66 0.33 2.49 0.31 1.06 0.5816 1444 14.6 7.47 14.5 3.76 13.6 7.33 10.9 16.817 1447 121 6.08 121 3.37 122 6.03 5.28 9.4518 1486 98.6 1.52 98.5 1.60 98.6 1.36 1.53 1.5219 1576 130 0.52 130 .0.55 130 0.59 0.48 0.4420 1693 15.8 15.6 5.01 4.36 5.00 4.36 4.62 4.35# d. o .f. 2568 42 42 47 47 56 56 63 63Table 4.12: Mode shape results for TFU, free-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 102Direct FEM Analysis % Error, Fixed-Interface CMS AnalysisMode Frequency(Hz)GuyanReduction2000 Hz.Cut-off3000 Hz.Cut-off4000 Hz.Cut-off1 34.7 0.02 0.05 0.05 0.052 72.8 - - - -3 72.8 - - - -4 185.8 .^0.38 0.38 0.03 0.035 432 11.1 52.2 8.63 4.606 433 11.0 53.0 8.71 4.617 655 11.4 57.0 57.1 0.338 658 11.6 59.2 57.5 0.349 1002 115 1.85 1.84 2.0010 1059 133 8.78 64.7 2.6011 1067 112 8.13 70.0 2.5512 1231 131 58.5 61.1 7.8713 1232 117 131 61.2 8.0414 1273 161 111 3.72 3.6115 1347 215 91.2 2.79 2.3316 1444 232 95.2 8.08 7.3617 1447 127 22.1 121 5.3018 1486 205 98.5 158 3.0619 1576 135 182 23.8 0.6520 1693 121 25.6 2.17 2.18# d.o.f. I 2568 360 369 380 389Table 4.13: Mode shape results for TFU, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 103For example, modes 2 and 3 have identical frequencies and so the mode shape error hasbeen omitted.Other modes have frequencies that are very close though not identical. This seems topresent difficulties for CMS in that the mode shapes can remain ill-defined long after thecorresponding frequencies have converged. In the free-interface results, this occurs mostnoticeably 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 errorsremain 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 showvery large mode shape errors until the 4000Hz cutoff where they all but disappear. Itcan 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 ishappening is a symptom of CMS condensation, of trying to specify modes with a severelyreduced quantity of independent variables. That CMS is able to accurately predict thefrequencies is a reflection of how a reduced set of generalized coordinates, correspondingto low frequency component modes, will render the Rayleigh quotient,ai = ttT -^ (4.5)ST MSistationary at the correct values of Ai. However, this does not necessarily mean that thecondensed eigenvector i corresponds to the true mode shape with the same degree ofaccuracy. If the size of the system has been reduced from 2568 to less than 400, as is inthe case of the focus unit, far fevier independent variables are available for determiningthe full extent of the mode shapes.The difficulty with the near-repeated modes is that they look very much like repeatedmodes; it is only a small asymmetry in the model which distinguishes them. At the lowercutoff frequencies, the condensation is too severe for the asymmetry to be detected; thatChapter 4. Modal Analysis of Three Finite Element Models^ 104is, in representing the components with a small number of component modes, the analysiscondenses out the asymmetric features in the model. But at a higher cutoff, the modesuddenly locks in—that is, enough coordinates have now been added for these closely-related modes to be distinguished. This sudden convergence can also be observed in thedistinct modes, but it is not delayed to the higher cutoffs as it is for the near-repeatedmodes. In general, mode shapes of the distinct modes are ultimately more accuratelypredicted 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 ofpredicting the near-repeated mode shapes, in spite of its greater condensation. Evidentlyfree-interface component modes do a better job of detecting the asymmetries than dofixed-interface modes.4.3.3 PerformanceTable 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-interfacemethod is unable to give results at the two larger tolerances. This is because of ill-conditioning in the residual flexibility matrix 6, which is calculated with the equation,= G — (1)11 -1 (DT (4.6)If too large a tolerance is used, errors occur in the free-free modes 4). These errorsare magnified by a loss of precision in the subtraction operation, and this can cause 6to lose its positive definiteness. In this particular example, the problem was caused byinaccurate calculation of modes 7 and 8 of the screw assembly. These are repeated modesof the component, but the eigenvalue solver only renders them so when the tolerance issufficiently small. In general, this is a problem in components with high modal density,Chapter 4. Modal Analysis of Three Finite Element Models^ 105CPU Time (s) for 40 ModesTolerances 10-2 10 -4 10 -6Directauyan52941524129992119533554836Free-Interface CMS2000 Hz. M1 i.c. i.c. 20452000 Hz. M2 i.c. i.c. 22992500 Hz. Mil i.c. i.c. 22292500 Hz. M2 i.c. i.c. 24833000 Hz. M1 i.c. i.c. 46213000 Hz. M2 i.c. i.c. 49363500 Hz. Ml i.c. i.c. 49913500 Hz. M2 i.c. i.c. 5094Fixed-Interface CMS2000 Hz. 4571 5742 63863000 Hz. 5010 5764 70134000 Hz. 4890 5711 8308Table 4.14: CPU times for modal analysis of the TFUChapter 4. Modal Analysis of Three Finite Element Models^ 106and a high tolerance is needed to obtain sufficiently accurate component modes. Thefixed-interface method is not sensitive to errors in the free vibration modes because itdoes not use a residual flexibility approximation, and at the two larger tolerances itperforms 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 timeexpended and therefore the improved accuracy is obtained cheaply. However, betweenthe 2500Hz and 3000Hz cutoffs, the time expenditure nearly doubles as a result of slowlyconverging component modes in this frequency range. As the density of the componentmodes increases, eigensolution routines require more iterations to resolve the modes.Therefore, in components with high modal density, the cost of the component analysisinevitably increases; but in proportion to a direct analysis, the cost of the total CMSanalysis declines. For the container ship model, the free-interface method was aboutfive times as fast as a direct analysis; for the TFU model, a similar level of accuracy isobtained at one-tenth the cost.4.4 Analysis of a Telescope ModelThe complete model of the telescope consists of the focus unit of Section 4.3, and asupporting framework suspending the focus unit above the centre of the main mirror (seeFigure 4.5) The supporting framework, or spider, consists of four separate and identicalcomponents, giving the telescope model a total of nine components. Each leg of thespider is composed of 38 2-node beam elements and one thick/thin shell element, fora total of 35 nodes and 210 degrees of freedom. The beam elements have rectangulartubular 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 vibrationChapter 4. Modal Analysis of Three Finite Element Models^ 107Figure 4.5: Complete telescope model, with focus unit and spiderChapter 4. Modal Analysis of Three Finite Element Models^ 108from the chopping mechanism and that this is potentially harmful to the telescope'simage quality. Particularly dangerous are resonances in the support structure causinglateral or skewing motion in the focus unit. It is with this in mind that the followingmodal analysis results are presented.4.4.1 Natural Frequency ResultsBoth CMS methods display very rapid and well-defined convergence for the natural fre-quencies (see Tables 4.15 and 4.16). Indeed, it is remarkable to note, especially in theresults for the fixed-interface method, how sharp the distinction is between regions ofhigh and low accuracy. This can be attributed to the dominance of spider vibration inthe natural modes of the telescope. In both of the CMS methods, the only dynamicmodes used were from the spider components; the free vibration modes of the TFU com-ponents were of such high frequency that they could be safely neglected. Another factorcontributing to the sharp convergence is the distinct and the well-separated frequencyspectrums of both the free-free and clamped modes of the spider (see Table 4.17). Bycontrast, the high modal density in the focus-unit components created a much slowerand less sharp convergence.Because of the high inertia of the TFU, many vibration modes of the telescope actlike individual modes of the four spider components under clamped boundary conditions.This is verified by noting that the spider component fixed-interface frequencies in Table4.17 closely correspond to many of the telescope frequencies. Thus, the fixed-interfacemodes are seen to give greater overall accuracy in terms of the number of dynamic modescontributed (see Figure 4.6). Indeed, it would be difficult to find shape functions moresuitable for describing the telescope modes. An equal number of free-free modes clearlycannot give the same level of accuracy, but as with the TFU, the convergence rate isabout the same in each case.Chapter 4. Modal Analysis of Three Finite Element Models^ 109Direct FEM Analysis % Error, Free-Interface CMS AnalysisMode Frequency(Hz)30 Hz. 90 Hz. 150 Hz.1st 2nd 1st 2nd 1st 2nd1 11.5 0.86 0.02 0.05 0.00 0.03 0.002 13.3 3.16 0.05 0.04 -0.01 0.01 -0.013 13.3 3.25 0.26 0.08 0.02 0.06 0.024 13.4 3.31 -0.03 0.01 -0.02 -0.03 -0.025 25.5 18.7 2.93 0.20 0.01 0.18 0.016 33.5 15.5 6.87 0.72 0.03 0.19 -0.047 34.0 25.8 6.55 0.54 -0.06 0.28 0.088 34.6 26.2 10.6 0.29 0.08 0.10 0.039 34.6 74.1 16.6 0.32 -0.01 0.08 -0.0810 39.1 55.3 7.08 1.96 0.03 0.48 0.0011 42.2 45.1 2.09 2.01 0.06 0.67 0.0312 45.3 46.5 4.88 0.02 -0.03 -0.06 0.0413 45.3 77.3 8.29 0.01 -0.02 0.04 0.0114 45.3 88.7 9.95 0.13 0.14 0.14 0.0315 45.6 152 9.33 0.38 -0.02 0.17 -0.0216 50.6 135 46.1 0.56 -0.03 0.26 -0.0617 61.0 119 30.1 1.03 0.07 0.28 -0.1018 61.0 123 . 42.5 1.11 0.07 0.29 -0.0619 61.1 191 72.1 1.12 0.07 0.26 -0.0120 71.0 159 54.7 4.10 0.89 0.96 0.0821 78.1 151 43.4 4.03 0.38 2.88 0.0522 83.7 168 67.7 0.89 0.18 0.16 0.0123 83.9 244 70.0 1.78 0.60 0.14 -0.1024 84.0 294 206 3.05 0.75 0.26 0.0425 85.6 341 225 5.18 1.38 0.22 0.0426 85.7 365 232 10.8 2.83 2.46 -0.0227 98.1 460 375 17.4 7.21 0.76 0.0228 99.0 469 391 62.3 9.57 1.10 0.2529 99.7 646 537 63.6 9.63 0.88 0.1930 104.0 615 578 92.8 8.10 0.45 0.0331 104.1 1039 670 106 12.0 1.22 0.2432 104.3 1108 943 116 12.2 3.99 0.3133 105.9 1091 1018 116 10.6 6.41 0.0734 117.3 1270 964 94.7 19.4 1.55 0.2235 117.5 1409 1000 107 19.4 1.60 0.2936 117.7 1525 1043 136 52.3 1.58 0.3837 120.1 1679 1072 141 49.7 2.65 0.4338 125.5 1604 1334 141 44.6 2.88 0.1039 127.4 2344 1388 160 78.5 4.03 0.9740 133.6 2267 1453 195 92.6 0.22 0.00# d.o.f. 3360 46 46 62 62 78 78Table 4.15: Natural frequency results for telescope model, free-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 110Direct FEM Analysis % Error, Fixed-Interface CMS AnalysisMode Frequency(Hz)GuyanReduction85 Hz.Cut-off100 Hz.Cut-off125 Hz.Cut-off1 11.5 58.8 0.01 0.00 0.002 13.3 171 0.00 0.00 0.003 13.3 175 0.00 0.00 0.004 13.4^. 200 0.00 0.00 0.005 25.5 207 0.15 0.08 0.056 33.5 157 0.11 0.05 0.047 34.0 209 0.02 0.10 0.018 34.6 207 0.00 0.06 0.099 34.6 208 0.04 -0.05 -0.0510 39.1 218 0.57 0.52 0.1411 42.2 289 0.62 0.57 0.1712 45.3 295 -0.06 -0.10 -0.1113 45.3 334 -0.05 -0.06 0.0114 45.3 451 0.15 0.18 0.1015 45.6 808 0.12 0.12 0.0816 50.6 728 1.00 0.34 0.1417 61.0 650 -0.03 -0.03 -0.0318 61.0 706 0.07 6.07 0.0219 61.1 722 0.02 0.01 0.0320 71.0 740 5.35 1.11 0.4421 78.1 841 1.80 0.84 0.6722 83.7 785 4.30 0.14 -0.0423 83.9 803 26.8 -0.03 0.1524 84.0 880 67.4 0.06 0.0325 85.6 944 68.2 0.31 0.2126 85.7 1090 192 0.90 0.6427 98.1 984 179 -0.02 -0.0228 99.0 1026 178 0.71 0.4329 99.7 1033 244 0.17 -0.0530 104.0 1013 251 2.35 0.4531 104.1 1097 382 2.96 0.4032 104.3 1224 383 36.1 0.3233 105.9 1419 376 36.6 0.3834 117.3 1341 345 113 0.1435 117.5 1418 434 157 -0.0436 117.7 1428 494 159 0.0837 120.1 1489 521 214 0.2638 125.5 1465 496 223 -0.0239 127.4 1543 701 295 1.6040 133.6 1481 735 278 80.2# d.o.f. 3360 408 424 432 440Table 4.16: Natural frequency results for telescope model, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 111—D---- Free, l it order mass—6,-- Free, rd order mass—0— Fixed10^15^20^25^30^35^40^45No. of dynamic component modesFigure 4.6: Combined natural frequency results for the telescope modelChapter 4. Modal Analysis of Three Finite Element Models^ 112Component Frequencies (Hz)Mode Free Fixed1 1.55 13.42 10.1 34.63 20.0 45.54 28.7 61.15 39.5 84.16 55.4 98.67 64.2 117.38 90.0 124.99 95.910 107.311 120.312 135.8Table 4.17: Natural frequencies of spider componentsOne impressive feature of CMS is how greatly it has simplified the analysis of thetelescope model. Whereas a direct finite element analysis of the complete telescoperepresents 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 wasthe focus unit, and in fact in some ways has made it much simpler. The simplificationstems from recognizing that with the spider attached, the focus unit acts by and largeas a rigid body whose deformations can be accounted for by static approximation. Thiscould 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 frequenciesare an order of magnitude larger than the spider frequencies. Consequently, in the free-interface 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 mightbe tempted to introduce a similar simplification in the direct analysis by replacing thedetailed model of the focus unit with a cruder equivalent. Such an attempt may beChapter 4. Modal Analysis of Three Finite Element Models^ 113partially successful but it should be noted that among the many repeated modes of thetelescope are a few that are entirely distinct. These modes are typified by simultaneousvibration of the spider and the inner components of the focus unit. Therefore, the detailin the modelling of the focus unit components needs to be maintained in order to predictthese interactive modes. Moreover, it is these interactive modes which are of the mostcritical interest in the dynamic response because they are most likely to affect the imagequality. It should be noted that CMS does not detract from the complexity of the originalmodel. Instead, it allows the analyst to discard information from the original model thatis of no importance to the targeted modes.In view of the CMS results, a cutoff frequency criterion can be confidently proposed forthe telescope model. To establish such a criterion, first a suitable definition of accuracymust be agreed upon. One reasonable definition might be that structural modes belowa target frequency of f* are accurate when at least 90% of these modes have less than1% error in the natural frequency. If this definition is satisfactory, the fixed-interfacemethod 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 witha 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 4.3that these cutoff frequencies would not be high enough to give the same accuracy. Inestablishing such rules of thumb, such factors as the modal density and the complexityof the component connectivity have to be accounted for. However, it is obviously notpractical to routinely engage in a detailed numerical study involving comparison to adirect finite element analysis in order to determine an appropriate cutoff frequency. Hencethe need for intuition or numerical experience on the part of the analyst. Of course, ifthere is doubt about a suitable value for the cutoff, it is safer to add too many modesthan too few.Chapter 4. Modal Analysis of Three Finite Element Models^ 114Although the two CMS methods seem capable of giving results of equal accuracy, thefree-interface method does so with more condensed equations and thus fewer independentvariables. In Table 4.15 the first 40 modes are resolved to within 1% error with only78 degrees of freedom retained; to achieve comparable results with the fixed-interfacemethod, more than 440 are necessary.4.4.2 Mode Shape Results and PerformanceThe mode shape results for the fixed-interface results are shown in Table 4.18. Theresults for the free-interface method are similar but have been omitted for the sake ofbrevity. As was the case for the focus unit, comparing mode shapes only makes sense fordistinct modes. On the whole, the mode shapes display slow convergence and this can beattributed to the fact that to/ 10-4 was used instead of tol 10 -6 . The direct analysisof the complete telescope model proved prohibitively expensive with a tolerance smallerthan 10 -4 , and while this is quite sufficient for giving reliable frequency results, somediscrepancy can be expected in the mode shapes. Since the direct-analysis mode shapesare the standard of comparison for Table 4.18, the percentage errors cannot be expectedto converge smoothly. However, this does not prevent accurate mode shape calculationsbecause 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 thoughmore 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 fortol 10 -4 , the former is more than 10 times as fast as a direct analysis. With the resultsfor the two previous examples in mind, the performance advantage for CMS evidentlyincreases with the size of the model. Also, the advantage of the free-interface method overthe fixed-interface method appears to increase with the model size. No ill-conditioningChapter 4. Modal Analysis of Three Finite Element Models^ 115Direct FEM Analysis % Error, Fixed-Interface CMS AnalysisMode Frequency(Hz)GuyanReduction65 Hz.Cut-off100 Hz.Cut-off125 Hz.Cut-off1 11.5 62.0 2.38 2.37 2.372 13.3 - - - -3 13.3 - - - -4 13.4 165 0.32 0.32 0.325 25.5 153 1.15 0.53 0.276 33.5 135 6.27 15.5 4.807 34.0 115 33.1 23.5 26.38 34.6 - - - -9 34.6 - - - -10 39.1 102 3.65 2.56 2.2311 42.2 134 3.60 3.71 1.3212 45.3 - - - -13 45.3 - - - -14 45.3 - - - -15 45.6 119 27.2 57.7 35.816 50.6 105 8.75 7.47 5.3617 61.0 - - - -18 61.0 - - - -19 61.1 188 71.2 76.1 39.220 71.0 119 31.7 9.90 4.62# d.o.f. 1 3360 408 424 432 440Table 4.18: Mode shape results for telescope model, fixed-interface methodCPU Time (s) for 40 modesTolerances 10-2 10 -4Direct Analysis 6324 25121Guyan Reduction 2637 3123Free-Interface CMS 2161 2203Fixed-Interface CMS 6225 5744Table 4.19: CPU times for modal analysis of the telescope modelChapter 4. Modal Analysis of Three Finite Element Models^ 116problems were encountered with the full telescope model; the modal density of the spidercomponents is low enough for accurate computations with the larger tolerances.4.5 Natural Frequency Error Estimation: Ship ModelIn Section 3.9, a procedure was described for calculating lower bounds to the exactfrequencies based on a free-interface CMS analysis. This procedure provides a means ofestimating the absolute error in the CMS-derived frequencies without having to comparethem to results of another method, and without having to calculate additional free-freemodes. In the present section, the results of lower bound analyses of the container shipmodel are given and compared to direct-analysis results of the assembled model.Table 4.20 shows natural frequency results obtained using the dynamic residual flex-ibility evaluated at four different frequencies: U.; = 2.14, 5.94, 10.24, 19.26Hz . These fre-quencies were originally calculated as the 5 th , 9th, 12th, and 20th modes in the 25Hz,first-order mass results listed in Table 4.2. The numbers listed in each of the four casesin 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; apositive error indicates an upper bound. Lower bounds generally appear in all modes oflower frequency than Cv, whereas upper bounds appear in all modes above Cv. Note thatthe 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 thevalues used for (1) are upper bounds calculated from the free-interface method, the errorin this mode ought to be slightly negative, according to (3.96). However, in this examplethe results are so close that secondary sources of error such as numerical round-off createa slightly positive frequency error.To summarize, the original results in Table 4.2 and the lower bound estimates in TableChapter 4. Modal Analysis of Three Finite Element Models^ 117Mode Direct(Hz)% Frequency Error with G(m)cii = 2.14Hz ci,' = 5.94Hz c 7, = 10.24Hz ii, = 19.26Hz4 0.881 -0.01 -0.11 -0.35 -1.625 2.14 0.003 -0.15 -0.51 -2.436 3.52 0.04 -0.11 -0.47 -2.347 4.55 0.14 -0.11 -0.73 -4.378 5.58 0.16 -0.03 -0.48 -3.479 5.91 0.50 -0.002 -1.19 -6.3710 8.01 0.24 0.12 -0.16 -1.6411 8.97 0.37 0.24 -0.08 -1.9012 10.2 0.49 0.35 0.01 -1.7813 12.3 0.70 0.58 0.27 -1.6214 13.1 2.02^• 1.72 0.96 -3.5615 13.7 2.57 2.29 1.56 -5.4916 14.2 6.39 5.48 3.34 -3.9717 15.0 2.18 1.93 1.32 -1.8818 16.4 3.30 2.98 2.20 -1.5719 17.3 2.55 2.35 1.87 -0.5720 18.9 1.62 1.53 1.29 -0.08Table 4.20: Lower and upper bounds calculated with dynamic residual flexibility4.20 are compiled in 4.21 for four selected modes. It should be noted that the lower boundsare much closer to the exact frequencies than are the upper bounds. By this means, agood estimate of the error in the upper bounds is obtained. The dynamic residualflexibility therefore provides a means of accurately estimating the absolute error of naturalfrequencies calculated with CMS. This is valuable when direct-analysis results are noti CZu (Hz) rvi (Hz) il)i - 6.-4 (Hz) wi (Hz)5 2.1406 2.1411 -0.0005 2.14069 5.9071 5.9412 -0.0341 5.907212 10.1870 10.2375 -0.0505 10.186020 18.9314 19.2566 -0.3252 18.9461Table 4.21: Lower and upper bound estimates of four CMS frequencies, with predictedabsolute errors and exact frequenciesChapter 4. Modal Analysis of Three Finite Element Models^ 118available for comparison. Although this technique is only valid for natural modes belowthe 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 onthe characteristics of the model being analyzed. Assuming that the cutoff frequency isalways chosen to be high enough to include the entire target frequency range, the accuracyof structural natural frequencies falling in this range can be tested using the dynamicflexibility concept. The cost of the error estimates is dominated by the calculation ofthe component modulation matrices (2.51). The same free-free modes are used as in theoriginal CMS analysis, and therefore the same static residual flexibilities apply.4.6 Reanalysis Following a Design ModificationMany changes may be made to a structural model before the design is finalized, anda separate modal analysis may be required for each change. If a direct finite elementanalysis has been performed, two possibilites exist: to perform a direct analysis on themodified model which will be just as expensive as the original; or to seek an approximatesolution using structural dynamic modification techniques [49].If instead a substructuring approach was taken and the original model was analyzedwith CMS, more options are available to the analyst. If the design modification affectsevery structural component, reanalysis requires a CMS analysis like the original. How-ever, if the design modification affects only some of the components, the componentmodes need only be recalculated for the components that have been modified; compo-nent 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 componentand so computational savings can be realized by omitting the reanalysis of unmodifiedChapter 4. Modal Analysis of Three Finite Element Models^ 119Figure 4.7: Mode 9 of the container ship, f = 5.91Hz.components.4.6.1 Reanalysis of the Container ShipIn the results for the container ship presented in Section 4.2, mode 9 was found tohave frequency 5.91Hz and the mode shape depicted in Figure 4.7. This mode couldbe excited in a resonant or near-resonant condition by pressure forces on the hull. Forinstance, a four-bladed propeller rotating at 90RPM would produce a 6Hz periodic forcein the immediate vicinity. Excessive vibration in this mode could bring about fatigueproblems in the structural members and panels and it would also create uncomfortableliving conditions for the crew in the deckhouse.One solution to this problem is to raise the frequency to a safe region without causingany other frequencies to fall into the region around the forcing frequency. As this is alocal mode in the stern and deckhouse components, it is susceptible to a purely localmodification. In this example, the four horizontal bar stiffener elements representingthe top deck of the stern component (groups 1 and 2 in Figure 4.8) are modified, andfour stiffener elements representing a new bulkhead (group 3) are added. Inspection ofChapter 4. Modal Analysis of Three Finite Element Models^ 120h1-± ■E■MIME■1■■4 5 6Figure 4.8: Six groups of bar stiffeners used for modificationsFigure 4.7 indicates that element groups 2 and 3 will be deformed axially. The stiffnessof group 2 is increased by doubling the cross-sectional area A, whereas the new bulkheadis given the same properties as the other bulkheads in the vessel. By contrast, group 1experiences large deflection but little deformation in this mode. Therefore, the frequencyis best raised by reducing the mass of these elements and hence, the cross-sectional areaof group 1 was reduced by half.Tables 4.22 and 4.23 show the reanalysis results with the modifications to the sterncomponent. The beam-like hull bending modes that predominate at low frequencyare little influenced by the stiffness modification in the stern and so only mode 9 issignificantly affected out of the first fourteen modes. Only at higher frequencies dosignificant frequency shifts begin to appear again. Nevertheless, raising mode 9 to 6.24Hzmay be enough to produce a satisfactory dynamic response. All that remains is totranslate 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 andChapter 4. Modal Analysis of Three Finite Element Models^ 121Free-interface MethodMode Original Re-analysis % Change4 0.881 0.877 -0.405 2.14 2.14 0.106 3.52 3.52 0.057 4.55 4.54 -0.268 5.58 5.60 0.539 5.91 6.24 5.5510 8.01 8.01 -0.0711 8.94 8.90 -0.4612 10.19 10.17 -0.1813 12.28 12.16 -0.9414 13.08 13.01 -0.4615 13.79 13.58 -1.4016 14.27 14.63 2.8617 15.00 15.00 0.0818 16.45 16.41 -0.2319 17.39 17.41 0.2520 18.96 18.95 -0.02CPU time 1028 176 -82.9Table 4.22: Reanalysis of the container ship model, free-interface methodFixed-Interface MethodMode Original Re-analysis % Change4 0.881 0.877 -0.405 2.14 2.14 0.106 3.52 3.52 0.057 4.55 4.54 -0.268 5.58 5.60 0.549 5.91 6.24 5.5410 8.01 8.01 -0.0711 8.94 8.91 -0.4612 10.19 10.17 -0.1813 12.28 12.18 -0.9214 13.08 13.01 -0.4915 13.79 13.61 -1.3416 14.27 14.66 2.8517 15.00 15.01 0.1118 16.45 16.42 -0.2119 17.39 17.43 0.2320 18.96 18.99 0.02CPU time 1561 662 -57.6Table 4.23: Reanalysis of the container ship model, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 1224.23. The reanalysis times are for calculation of 40 modes, as was done in the origi-nal analysis. The greater proportional time reduction in the free-interface method canbe attributed to two factors. First, this method places more computational empha-sis on component analysis, creating more condensed system equations. Conversely, thefixed-interface method performs fewer computations at the component level, instead al-lowing larger system equations and therefore placing more computational emphasis onthe system level. In a reanalysis mode, the free-interface method stands to benefit morebecause component-level calculations are skipped. Secondly, the implementation of thefixed-interface method in VAST had to be made less efficient, particularly in a reanalysismode where it was difficult to avoid unnecessary computations. These two factors com-bine to give the free-interface method the advantage, but this advantage diminishes as alarger proportion of the components undergo modification.4.6.2 Reanalysis of the Telescope Focus UnitAs a further example of the advantage of using CMS for reanalysis, the stiffness of thebearings connecting the inner tube to the outer tube will be modified to determine theireffect on the focus unit frequencies. These bearings have been modelled as simple barelements and are included as part of the inner tube component (see Figure 4.3). Thedifficulties involved with modelling bearings accurately justifies some investigation intothe effects of different modelling options. Tables 4.24 and 4.25 show the results of areanalysis 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 tobe re-calculated. The time savings obtained are similar to those found for the containership, with the free-interface method showing the largest proportional reduction. But itis here that the benefit of having condensed system equations is most apparent. WithChapter 4. Modal Analysis of Three Finite Element Models^ 123Free-Interface MethodMode OriginalA = 36mm2Re-analysisA = 72mm2% Change1 34.7 34.7 0.002 72.8 78.7 8.163 72.8 78.7 8.184 185.8 185.8 -0.015 431.6 468.2 8.476 433.2 470.4 8.597 655.4 710.0 8.348 658.0 714.2 8.549 1004 1003 -0.0610 1060 1148 8.2611 1067 1152 7.9012 1233 1274 3.3913 1234 1317 6.7314 1273 1321 3.6915 1347 1349 0.10CPU time 5094 1114 -78.1Table 4.24: Reanalysis of the TFU, free-interface methodFixed-Interface MethodMode OriginalA = 36mm2Re-analysisA = 72mm2% Change1 34.7 34.7 0.002 72.7 78.7 8.163 72.7 78.7 8.174 185.8 185.8 -0.015 431.5 467.9 8.446 433.0 470.1 8.577 655.6 710.0 8.298 658.3 714.2 8.499 1006 1005 -0.0610 1062 1151 8.3811 1070 1156 8.0212 1234 1299 5.3413 1235 1316 6.5414 1273 1319 3.6615 1349 1350 0.09CPU time 8308 6768 -18.5Table 4.25: Reanalysis of the TFU, fixed-interface methodChapter 4. Modal Analysis of Three Finite Element Models^ 124the fixed-interface method, whether in the original analysis or the reanalysis, the timeexpended for the final eigensolution alone is about 30003, while the entire reanalysis withthe free-interface method expends only 1114s (Table 4.24). These results were obtainedwith a eigensolution tolerance of 10 -6 . If a larger tolerance is used, the performance gapis narrowed at the price of obtaining less accurate mode shapes.4.7 Discussion and Summary of Numerical ResultsThe examples presented in this chapter show a variety of typical situations: modelsof differing size and complexity in two and three dimensions. In all cases, CMS hasproven capable of accurately predicting the low frequency modes with a reduced-orderformulation of the free vibration equations. Moreover, CMS is more efficient than directfinite element analysis and its relative efficiency increases with the number of degreesof freedom in the model, with the number of low-frequency structural modes that arecalculated, and with stringency of the accuracy requirements.Of the CMS methods used, the free-interface method with second-order mass termsappears to be the most economical. Although the relative speed of the methods is partlya function the efficiency of the software implementation, the free-interface method doesbenefit from a highly condensed set of global equations. A close study of the examplespresented in this chapter will strengthen this point further. The finite element modelsused in the examples have respectively 1341, 2568 and 3360 degrees of freedom but theresults for the free-interface method show that comparable accuracy between the differentexamples is obtained with equations of the same order. That is, the size of the condensedsystem 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 bythe desired range of accuracy. The opposite is true for the fixed-interface method whereChapter 4. Modal Analysis of Three Finite Element Models^ 125the original size and complexity does ultimately determine the size of the condensedsystem. Comparing the results for the ship model and the focus unit, a large jump canbe seen in the order of the condensed system because of the increased complexity of thecomponent interfaces in the latter.This point is illustrated by imagining the following process. Suppose that the coarse-ness of the mesh in the container ship model is reduced by a factor of two: that is, eachbar element is replaced by two others one-half its size and each membrane is replaced byfour others one-quarter its size. The overall effect will be to approximately double thenumber of nodes situated on the component interfaces (and thereby double the numberof static modes). If the fixed-interface method is used on the refined model, a largerorder system is the result.However, as the mesh was already sufficiently fine, one would not expect either thelow-frequency system or component modes to change significantly when the additionalrefinement takes place. That is, results of essentially the same accuracy would be obtainedwith the same cutoff frequencies, entailing the same dynamic component modes. Withthe free-interface method this leads to equations of the same order. The same processif carried out with the focus unit would give the same result. More generally, it can beremarked that the number of dynamic modes needed in a CMS analysis is not a functionof the mesh refinement.When applied to relatively complex models, the basic differences between the twoformulations have a great impact on their performance. Throughout the examples, thefree-interface method is consistently quicker than the fixed-interface method (exceptingthe cases for which ill-conditioning occurs); and as the model complexity increases, sodoes the discrepancy in the solution times. Furthermore, the difference is magnified inreanalysis problems where a partial reanalysis of the structural components precedes acomplete eigensolution of the condensed system equations.Chapter 5Structural Dynamic Modification5.1 IntroductionThe previous chapters of this thesis have dwelt on the problem of analyzing the naturalmodes of a system. In structural design, accurate knowledge of these modes is requiredto determine whether the vibrational response will be acceptable or not. Closely relatedto the analysis problem are the reanalysis and re-design problems, which are concernedwith determining the effects of modifications on the natural modes of the system andwith 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 ofwhich requires a free-vibration analysis. The purpose of developing structural dynamicmodification techniques is to provide a capability for efficient design and re-design, andto foster a better understanding of structural dynamic behaviour.Dynamic modification techniques all use an unmodified or baseline state as a referencepoint for the modified structure. In this respect, there is great potential for using a CMSformulation of the baseline system. Because of the compact, reduced-order nature of theCMS formulations, numerical techniques used in dynamic modification can be appliedwith less computational effort. As of yet, little work has appeared in the literaturerelating CMS analysis to structural dynamic modification.Reanalysis is concerned with the efficient evaluation of the natural modes of a mod-ified structure. This is often referred to as structural dynamic modification or forward126Chapter 5. Structural Dynamic Modification^ 127modification. 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 localmodifications. In this chapter, forward modification is dealt with in a broader sense, theintention being to demonstrate how techniques successfully applied to unsubstructuredsystems can be adapted to substructured situations.Inverse modification is concerned with assessing the changes to structural propertiesnecessary so that prescribed modal constraints are satisfied. The modal constraints mayconsist of constraints on the frequencies, mode shape constraints or a combination ofboth. Generally, mode shape constraints will take the form of limits on the relativeamplitudes of specified locations in the mode shapes. Techniques for solving problems ofthis kind are valuable because they eliminate the element of guesswork from the re-designprocess.Inverse modification techniques applicable to general structural changes fall into twoclasses: perturbation and sensitivity techniques. The application of these two techniquesto substructured models analyzed with CMS is the subject of Sections 5.4 and 5.5. InSection 5.6, numerical results are presented and the performance of the two techniquesis compared.5.2 Structural ChangesIt should be made clear what is meant by a modification of a structural model. In thepresent context, a structural modification can be any combination of the three followingtypes 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 existingmesh.These restrictions bring about two important advantages. First, because topologicalchanges are excluded, re-meshing of the model during reanalysis is not a concern. Sec-ondly, a structural modification defined as above can be expressed as a perturbation tothe existing system; i.e., the stiffness and mass matrices of the modified structure areperturbed 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 (orstructural component) are assemblies of all constituent element matrices. Likewise, theperturbed 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 deal-ing with substructured models, the above formulae define perturbations to a structuralcomponent. The perturbations to all components are expressed collectively using theoverbar notation, where AK is the uncoupled collocation of the various component stiff-ness perturbations:OK = diag {AK ( ')} (5.3)AM = diag {OM ( `)} (5.4)where the superscript (i) indicates the perturbed matrix of the ith 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, elasticmodulus, cross-sectional area, and second moment of area. Changes to any of the lastthree are admissible, according to the definition above; changing the length is forbiddenChapter 5. Structural Dynamic Modification^ 129since this requires relocating the element nodes. If the second moment of area is the onlyproperty 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 portionof the element stiffness matrix proportional to the property variable r. The perturbedelement stiffness Ake is therefore,Ake = — = (1o) Ar = (lc!) ra (5.6)where a is the fractional change to the property r. The above equation reflects the linearrelationship between the second moment of area and the element stiffness. Consequently,property changes of this type are called linear property changes. Changes to elementmass matrices are usually of this type.It could be argued that since the cross-sectional area and its second moment aredifficult to change independently, a better choice of property variables would include theactual dimensions of the cross section. For instance, choosing the width of a rectangularcross section as the property variable gives a perturbed element stiffness of the form,Ake = kr, (r' — r) kc ((r')3 — r3)= (kr, + 3icecr3) a + (3kcr3) a2 (keCr3) a3 (5.7)where r' = (1 a)r and where Icec is defined as the portion of the element stiffness matrixproportional to r3 . The higher-order terms in a reflect the nonlinear relationship betweenthe width of the beam and the element stiffness. Property changes of this type are callednonlinear property changes.In forward modification problems, the property changes are prescribed at the outsetand so element perturbations can be calculated exactly, whether they are of the form (5.6)Chapter 5. Structural Dynamic Modification^ 130or (5.7). On the other hand, for inverse modification problems the property changesare unknown and have to be determined to satisfy some modal constraints. In thisapplication, a is referred to as a design variable. When a design variable appears in anonlinear form it cannot easily be determined, particularly if it is coupled to anotherproperty change in the same element. Moreover, the linear and cubic portions of thestiffness, and /cc , are not always easy to separate, adding further complications to theexact determination of a.As a simplification, a linearized version of the perturbed element matrix is obtainedwith the first term of a Taylor expansion about the baseline value:Ar 8ake a (Ake )a=0(5.8)-a asT=TOFor a linear property Ake = Ake ; for a nonlinear property of the form (5.7),Ake^Ake = (Icezfr^31cecr3) a (5.9)where the equality is justified for small a only. In practical computations, the derivativesin (5.8) can be evaluated numerically by calculating small perturbations around thebaseline.Supposing the total number of design variables is m, the linearized element matrixperturbations are,Aice = Eln (al-2—e ) r 2•a • = Ein^(k.,)i a;^(5.10)^j=1 ar; ^3 :7=1171 ,((ame^m^Ark = E^—^ria, = E^(m.,.)i a;^(5.11)^J1. uri j=1Usually one would be concerned with property changes affecting groups of elements ratherthan just individual elements. Indeed, a single design variable can be used to describe aproperty change of a whole range of elements. The following expressions for the overallChapter 5. Structural Dynamic Modification^ 131perturbations are obtained by applying (5.10) and (5.11) to (5.1) and (5.2):^m ^ mAK = E E^(lcer )i Sea; = E (Kr ); a;^(5.12)^E j 1 j=1AM = E E Si (m ). a • E (m. ) a •.e^Spa;ej^T3 j (5.13)e j=1 j=1Similar 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 arange of elements within a component. But the process can be taken one step further byconsidering changes extending over a range of components. The appropriate expressionsare,m6,17 E Cg,.) ajj=1^3= E (sir) i=1dwhere CL) = iag {(Kr ); } and (MT ) i is similar.(5.14)(5.15)5.3 Structural Dynamic Modification with CMSA comprehensive review of structural dynamic modification techniques has been providedby Baldwin and Hutton [49]. They divide the various techniques into three classes: tech-niques for local modifications, techniques for small modifications, and techniques basedon modal truncation. In Section 4.6, some attention was given to local modifications ofsubstructured models within the context of CMS. The present discussion is relevant tomodal 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^ 132where A:, X: are the ith modified eigenvalue and mode shape of the structure. A directsolution of this equation gives accurate values for A:, X: at a cost equal to that ofthe original solution. To reduce this cost, a standard practice in forward and inversemodification is to use the original (baseline) modes as a basis for X::x: E xici; Xcij.1(5.17)That the modified and baseline mode shapes can be represented in the same vector spaceis 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 bythe 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 usuallyincludes 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 reducedmodal subspace. The strengths and weaknesses of this approximation are the subject ofmuch discussion in the modal analysis literature [50, 51].Applying (5.17) to (5.16) and premultiplying by XT gives,XT (K AK)Xci — A:XT (M AM) Xci = 0^(5.18)Assuming mass normalized modes, this becomes,(c12 XTIKX) ci — (/ XTAMX) = 0^(5.19)This equation is, like the original equation (5.16), an eigenvalue equation but where nowthe eigenvector is ci. Determining ci defines the modified mode shape within the limits setby (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-frequencyinformation.Chapter 5. Structural Dynamic Modification^ 133It is interesting to note that the procedure described in Section C.4 for calculatingthe wetted modes of a structure from its dry modes is a special application of (5.19). Inthat procedure, the fluid added-mass matrix serves as AM and the stiffness modificationis zero.5.3.1 Application to Substructured ModelsThe free-vibration equation of motion formulated by a CMS method is, in its most generalform,- Ai /Cgs = 0^ (5.20)where, k = 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 -1Are;^= (5.21)In the fixed-interface method, it takes the form,T = [^cTA =I 93(5.22)PnReformulating the CMS equations after a modification to the components gives,^(k OK) — A: (if + AM);^= 0 (5.23)where the modified condensed system matrices are defined by,^ff 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 componentmodes in either (5.21) or (5.22). Because T' is dependent on the modification, a simple,Chapter 5. Structural Dynamic Modification^ 134linear perturbation in the component stiffness or mass appears as a complex, nonlinearchange in the condensed system matrices. As a result, it is difficult to predict theprecise 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 thecomponent modes. In this section, an approximation to (5.23) is derived which can beused without recalculating the component modes.The structural modification equation (5.19) can be directly applied to a substructuredmodel once the mode shapes have been reconstructed from the baseline CMS analysiswith the general relation,= n (5.26)A modal approximation appropriate for substructured systems is derived by using ui inplace of Xi in (5.17):= E = E T64; = TZcij=1^j=1(5.27)The kinetic and potential energies of the baseline system oscillating in the ith modecan be written as,1^1^---T = iAiXTMXi = MV = —1 XTKX. = —1 fiTKii.2%^a^2s^s(5.28)(5.29)These expressions make use of the fact that the energy in the system is equal to thesum of the energy in all the components. Likewise, the energies of the modified systemoscillating in the it" modified mode are,= —2 A: T^+ AM) re^(5.30)_IT= ut + Ax) (5.31)Chapter 5. Structural Dynamic Modification^ 135Applying (5.27) and Lagrange's equation to T' and V' gives the following equation of themodified structure:[02 + ZTTTA-ITTZ - of (./ ZTTTAMTZ)]ci = 0^(5.32)where now the necessary definition of mass normalization is,S2 2 = ZTTTICTZ I = ZTTT MTZ^(5.33)The number of modes available for the baseline set is at most equal to the size of thebaseline CMS equations. Assuming that such a complete set is available, define as,= Z c.i (5.34)The matrix Z is square and invertible; thus,= (5.35)Substituting (5.35) into (5.32) and premultiplying by (Z -1 )T gives,[Z-TS2 2 Z-1^TTAKT -.A (Z-TZ -1^TTAMT)] = 0 (5.36)With (5.33), this equation reduces to,[k^TTAKT - A: (if^TTAMT)] i = 0 (5.37)Equation (5.37) is the same as the modified CMS equation (5.23) except that thetransformation T' is replaced by its baseline counterpart T. As a result, linear pertur-bations in the structural matrices now only appear as linear changes in the global per-turbation matrices, Ak and OM. Thus (5.37) can be described as the linear-equivalentequation for the modified system. Since (5.37) was derived assuming a full complementof CMS modes, it represents the optimal case for structural dynamic modification of aChapter 5. Structural Dynamic Modification^ 136substructured 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 cal-culated; all that is required is the preservation of k, Si and the baseline transformationmatrix T.Some advantages to using (5.37) are immediately apparent. The order of (5.37) is thesame as the original CMS equation and so the reanalysis benefits from the same degreeof condensation as the baseline analysis. Also, the component modes do not need tobe recalculated to solve (5.37); instead only the matrix products TT LIKT and TT AMTneed to be evaluated.For special types of modifications, the linear-equivalent equation gives an exact de-scription of the modified structure. Obviously, if the modification is such that the systemmode shapes are left unchanged, the modal truncation in (5.27) introduces no error andthe results from (5.37) will be exact. However, a stronger statement than this can bemade. Exact results will be obtained when the modal approximation (5.27) is capable ofexactly representing the modified mode shape. Using (5.35), this condition is equivalentfinding a 4 such that= = T (5.38)Modifications in which the columns of T' are linear combinations of the columns of T willsatisfy (5.38) since, in that case, the columns of both matrices span the same space. Inthe free-interface method, this situation arises when a perturbed component mass matrixAM is proportional to the baseline component matrix M. Such a modification leaves thecomponent modes unchanged except for an adjustment to the mass normalization factorin the free-interface modes (1). Thus, the columns of T' are scaled differently from thosein T, but through (5.38), an exact description of the modified mode shape can still beobtained using T. An example is given in the next section which shows that in specialChapter 5. Structural Dynamic Modification^ 137cases exact results can also be obtained when adding lumped masses.5.3.2 Numerical ResultsIn this section, four examples are presented which compare the linear-equivalent and CMSreanalysis methods. While the linear-equivalent equation (5.37) is in a general form whichcan be applied to either the fixed- or free-interface CMS methods, the examples here useresults 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 shipwere determined after a modification to the stern stiffeners (groups 1 and 2 in Figure4.8). The same modification is now made, but the natural modes are recalculated usingthe 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 bythe nature of the approximation in the linear-equivalent equation. In the derivation, themodified mode shape is projected on to a modal subspace spanned only by CMS-derivedmodes; the large number of higher frequency modes that cannot be calculated with theseequations are ignored. This effectively puts constraints on the modified structure whichraise its natural frequencies. The baseline modes are not an effective basis for the modifiedmode because the latter is a local mode in the stern and superstructure, as was shown inFigure 4.7. On the other hand, the majority of the baseline modes are vertical bendingmodes in the hull which are not very useful for characterizing localized displacements.To show better the effectiveness of modal truncation, consider raising the elasticmodulus of the stiffeners located at the bottom of hull (groups 4, 5, and 6 in Figure4.8). These stiffeners extend from the bow to the stern and are used to partly simulatethe bending stiffness of the hull bottom. Changing these stiffeners requires modificationsto three components, and so the change is not nearly as localized as in the previousexample. The effect of these stiffness changes on the fundamental flexible mode (h) is0.975—0--- Linear-equivalentCMS reanalysisN3, 0.950-8 0.9250.900Chapter 5. Structural Dynamic Modification^ 1380.0^0.5^1.0^1.5^2.0^2.5^3.0^3.5Fractional change to elastic modulusFigure 5.1: Reanalysis results for modifications to hull-bottom stiffenersshown in Figure 5.1. An excellent match is obtained between the linear-equivalent andCMS reanalysis results. Because the fundamental mode is the 2-node bending mode of thehull, changes to this mode are well approximated as a linear combination of the baselinemodes. The cost of the linear-equivalent method compares very favourably with theCMS reanalysis results. The average time for reanalyzing 10 modes in a CMS reanalysisis 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 wasreanalyzed after a modification to the bearing stiffnesses. Figure 5.2 shows the variationin the second and third frequencies (which are identical) resulting from modifications tothe bearing stiffness. The fractional change a refers to the cross-sectional area of the bar82.5171x- ...... -;" 80.00a=e.=g— 77.5Ez75.0—o— Linear-equivalent (free CMS)—0— CMS reanalysisChapter 5. Structural Dynamic Modification^ 1390.0^0.5^1.0^1.5^2.0^2.5^3.0Fractional change to cross-sectional area of stiffenersFigure 5.2: Reanalysis results for modifications to bearing stiffnessesChapter 5. Structural Dynamic Modification^ 140elements used to model the bearings. The inaccuracy of the linear-equivalent methodis caused by the inability of the truncated baseline mode set to describe the modifiedmode. While the linear-equivalent method is more accurate for small changes, it shouldbe emphasized that it is not a "small modification" method in the same sense that thesensitivity method is; the method is valid for large and small changes alike, as long asthe 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 ofthese are swinging modes affecting the inner tube and chopping mechanism. Considernow the effect of adding mass to the chopping mechanism. Four equally-distributed andidentical 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 shoulddrop. This is verified in Figure 5.3, where it can also be seen that the linear-equivalentmethod and the CMS reanalysis predict identical results. This is explained by notingthat the chopping mechanism is modelled as a block of aluminum solid enough for itsdynamics to be represented by rigid-body modes and static flexibility. Because of thecomponent's rigidity, adding lumped masses has an insignificant affect on the staticmodes. The effect on the rigid body modes is only to adjust the mass normalizationfactors, and consequently the modified matrix T' can be represented asT' = TD (5.39)where D is a general diagonal matrix. In such situations, exact representations of themodified mode shapes are obtained using (5.38). Not only the second and third, but allof the modes should be predicted with equal accuracy. This underscores the second inter-pretation of the linear-equivalent equation: it provides an equivalent CMS formulationof the modified system under the restriction that linear combinations of the columns ofT provide an accurate description of the modified mode shapes.—o— Linear-equivalent—a-- CMS reanalysisChapter 5. Structural Dynamic Modification^ 14172.5it7  70.0• ...—.._- ;coa)—0ciS 67.5mIE65.0m62.50.0^1.0^2.0^3.0Lumped added mass (Kg)4.0^5.0Figure 5.3: Reanalysis results for lumped-mass additions to the chopping mechanismChapter 5. Structural Dynamic Modification^ 1425.4 Perturbation Methods for Inverse ModificationAttention is now turned to inverse modification, which is concerned with determining aset of design changes which satisfy some prescribed modal constraints. To perform suchan analysis, the design variables must first be selected from among the allowable propertychanges. As was mentioned in the introduction, the modal constraints may consist offrequency goals or mode shape constraints. In the present discussion, only frequencygoals are considered; extra difficulties arise with mode shape constraints which need notbe discussed in this treatment.5.4.1 BackgroundThe original work by Stetson [52, 53, 54] in this area used an equation of motion of themodified 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 incremen-tal terms were deleted; and the perturbed modal matrix AX was approximated as thelinear combination of the baseline modes,AX = XC^ (5.41)where C is a square matrix of admixture coefficients in which the diagonal elements0. Note the slight difference between (5.41) and (5.17). With these two approximations,Stetson derived the linear perturbation equations,XTAKXi — AiXTAMXi = MiAAi for i = j^(5.42)= Mlci7^ for— Ai) f r i j^(5.43)Chapter 5. Structural Dynamic Modification^ 143Equation (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 ofunknown design variables a, as was described in Section 5.2. The solution a represents afirst-order estimate of the design change satisfying the dynamic equation of the modifiedstructure.Sands&Om and Anderson [55] presented a similar formulation, but expressed theprescribed mode shape shift explicitly, rather than with admixture coefficients. Kim etal. [56] proposed a method in which the modified perturbation equations were solved bymathematical programming, while retaining all the nonlinear terms in (5.40). Minimumweight solutions were found with the aid of a starting vector. A general dynamic reductionmethod combining static condensation with subspace iteration was used to compress theperturbation equations for large-order models [57, 58].Hoff et al. [59, 60] proposed a two-stage predictor-corrector method for frequencymodification which is more suitable for large structural changes. The predictor phase es-timates the design changes and the perturbed mode shapes using the linear perturbationequation. In the corrector phase, the general perturbation equation (5.40), incorporat-ing the estimated mode shape perturbations, are solved in an attempt to improve onthe predictor phase results. Welch [61] reported that the predictor-corrector scheme isadequate for problems with linear property changes but is unable to predict nonlinearproperty changes accurately. To improve the performance for large modifications, Bernit-sas and Kang [62] used the predictor-coirector incrementally to calculate a sequence ofsmall steps which cumulatively result in the complete solution. Gans and Anderson[63] adapted the predictor-corrector method for systems with significant centrifugal andcoriolis forces. The predictor-corrector method has also been implemented in the finiteelement program INSTRUM [64].Chapter 5. Structural Dynamic Modification^ 144The approach taken by Smith and Hutton [65] is somewhat different. The free vi-bration equation of the modified system (5.40) is premultiplied by the baseline modesrather than the modified modes. This gives the following equation which bears a strongresemblance to the forward modification equation (5.18):XT (K + AK) XC = XT (M + AM) XC (Se Af22) (5.44)The advantage of using (5.44) is that the number of nonlinear incremental terms is lessthan in (5.40), and this facilitates the solution of inverse modification problems with-out further approximation of the equations. Smith and Hutton described an interativemethod for calculating design changes exactly satisfying (5.44) in the presence of pre-scribed frequencies. The only limiting approximation is the truncation of the baselinemode set in (5.17).In a parallel development of perturbation methods, Ram and Braun [66] determineoptimal perturbing stiffness and mass matrices for a particular modal subspace. Thesubsidiary problem is then to relate the optimal perturbation to physical changes in theactual model, but this was not investigated by the authors. The philosophy adopted inthe present discussion is that all possible design solutions should be chosen from a setof perturbing matrices that are defined by a set of design parameters, as described inSection 5.2. As the design parameters are chosen by the analyst, this approach requiresgreater engineering judgment but at the same time affords greater flexibility in satisfyingthe modal constraints.5.4.2 Application to CMSLittle research work has as yet made use of CMS formulations in inverse perturbationstudies. Linearized perturbation equations for the fixed-interface CMS method have beendeveloped which are suitable for uniform stiffness and mass changes to the componentsChapter 5. Structural Dynamic Modification^ 145[67]. The present discussion is concerned with ways more general structural changes canbe predicted, and is an adaptation of the method described in [65].The basic perturbation equation for substructured systems is given by (5.23). Tobegin the development, consider a frequency modification problem having one frequencyconstraint, .X = A:. The task is to find the design variables a and the modified eigenvector4: satisfying (5.23). While the characterization of AK and AM in terms of designvariables is straightforward, a complication arises here because the transformation T' isalso a function of the design variables. The following frequency modification equationresults:7111- (a) (K + AK) r(a) : = A:T1T (a) CH T'(a) (5.45)The equation is clearly nonlinear in a. Moreover, the functional dependence of T' ona is not exactly known; for the relationship between the dynamic modes in T' and acan only be expressed with the help of approximate sensitivity techniques; and the staticmodes in T', because they are calculated with an inverted stiffness matrix, will generallyhave a complicated relationship to the structural properties. Consequently, the frequencymodification equation is exceedingly difficult to formulate to its full extent.Instead, consider the practice used in perturbation methods of expressing the modifiedmode shape as a linear combination of a truncated mode set, as in (5.17). In Section 5.3.1it was shown that if the baseline mode set included the maximum number obtainablefrom a CMS analysis—that is, when the number of baseline modes equalled the order ofthe CMS equations—the linear-equivalent perturbation equation (5.37) results. Applyingthe frequency constraint as = A: to that equation gives,TT (-1–f AK) 71e = A; TT (M AM) Te^(5.46)Because the baseline transformation T is invariant with respect to a, the linear-equivalentfrequency modification equation (5.46) is greatly simplified from (5.45).Chapter 5. Structural Dynamic Modification^ 146The baseline mode set is truncated further by the substitution,q1;; =EZ;ci; = Zcij=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 ZTTT AM(a)TZ] ci^(5.48)This equation is exactly analogous to the equation derived for unsubstructured systemsin [65].Solution AlgorithmAn algorithm is now given for calculating design solutions satisfying (5.48). This algo-rithm is applicable with linear design variables, though by linearizing nonlinear variablesaccurate solutions can still be obtained for small fractional changes. Consider the ith rowof (5.48):ZTTT [A-17(a) — A:AM(a)1TZci + (A i — A:) = 0^(5.49)This is called the design equation because it is used for determining the design variablesa satisfying the frequency constraint A: = A:. The remaining q —1 equations in (5.48),ZTTT [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 solvedsimultaneously; unknowns a and ci; are coupled in both equations. A solution is foundwith the following iterative algorithm, in which t designates the current iteration. Thisalgorithm is exactly analogous to the one proposed for unsubstructured systems in [65].Chapter 5. Structural Dynamic Modification^ 1471. Let t = 0, cT ) = 1, dicl) = 0 for j = 1, 2, ... , q; j^i.2. Calculate the design variables satisfying the frequency modification equation (5.49);i.e., solveZTTT [AK(a(t-I-1)) A:AT/02(t+1) )] TZc! t) + (Xi —^= 0^(5.51)(ait+i), ari ,) Tfor a( t+ 1) =^a(„+1)) . Note that if m = 1 the solution of thisequation is unique. When m > 1, the solution is underdetermined and it is nec-essary to use optimization. A description of the optimization algorithm is givenbelow under the heading "Mathematical Programming".3. Calculate the eigenvector change satisfying (5.50) subject to the design modificationa(t+1); i.e., solveZTTT {rf(ce(t+1) ) — A:AM(a (t+1) ) ] TZe l) + (Ai —^= 0j = 1, 2, . , q; j^i^(5.52)forr_^1) = (c(it+1) c(2t-I-1)^cltq-1-1)1) T and normalize such that liclt+1) 11 2 = 1. In thiscistep, q elements of ci(t+i) are determined from q — 1 equations. Because cl t+1) canonly be specified to within a common multiplicative factor, one of its elements canarbitrarily be assigned unit value. With this substitution, (5.52) becomes a set ofq — 1 equations in q — 1 unknowns. To avoid numerical difficulties, it is best toassign 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 differentfrom the baseline. This is the same assumption used by Stetson in his derivation ofthe linear perturbation equation. However, in subsequent iterations the above algorithmChapter 5. Structural Dynamic Modification^ 148differs from approach taken by Stetson and others. In (5.51) and (5.52), all of thecoupling terms between a and ci are retained and the only supporting approximation isthe truncation of the baseline modes in (5.48).Mathematical ProgrammingWhen more than one design variable have been defined, the solution of Step 2 is underde-termined and an infinite number of solutions are possible. To resolve this, mathematicalprogramming is used to solve (5.51). In this study, a penalty function method employinga minimum change objective is used. This involves minimizing the functional,ngF* = aT + iCi RT R ItEl/g; (5.53)The first term is a measure of the overall change that is to be minimized. Minimumchange solutions are often preferable to minimum weight solutions in redesign problemsand are less liable to produce pathological solutions [63]. The second term is an externalpenalty function where R is the residual of the equality constraints. In this case R is justthe residual of the design equation:R = ZTTT [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 areinequality constraints limiting the feasible domain of the design variables. For instance,a; < —1 are physically impossible, and soai + 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 itto be successful, the starting point (a = 0) must be in the feasible region, otherwise theinterior penalty function cannot work.Chapter 5. Structural Dynamic Modification^ 149The minimum value of F* is calculated using a quasi-Newton algorithm with a weakline search and a BFGS update, a description of which is given in [69]. The factor it ischosen based on an initial estimate of the minimum of F*. In the course of the quasi-Newton algorithm, is decreased by a factor of 5 after each iteration, progressivelyde-emphasizing the inequality constraints in favour of the equality constraints.It should be emphasized that mathematical programming is only used in Step 2 andtherefore constitues just one part of the larger algorithm. By contrast, Kim [57] usesa mathematical program to solve the full set of dynamic equilibrium equations. Theunknowns include both the design variables and all the unspecified components of theperturbed mode shapes, thus making the number of unknowns very large for complexmodels. The procedure adopted in this study is simpler in that design variables aloneare determined through optimization; modified mode shapes are subsequently determinedfrom the remaining perturbation equations. Modal truncation allows the iterative methodto be executed inexpensively even for large-order complex systems.Multiple Frequency ConstraintsThe above algorithm is easily adaptable for simultaneous multiple frequency constraints.If 1 frequencies are prescribed, 1 sets of q equations are obtained by substituting theappropriate index i in (5.48). One design equation is selected from each set, giving 1equations with m unknowns in Step 2. A solution is possible when m > 1, providedthe 1 design equations are independent; a unique solution is possible when m = 1, andmathematical programming is used when m > 1. In Step 3, the mode shape perturbationis obtained by solving each set of the 1 sets of q — 1 equations separately. Generally, thenumber of frequency constraints is small compared to q. Thus, the number of additionalequations is not too large to jeopardize the numerical efficiency of the method.Chapter 5. Structural Dynamic Modification^ 1505.5 Sensitivity Analysis; Newton's Method for Inverse Modification5.5.1 BackgroundMany developments in dynamic optimization have been based upon sensitivity methods[68]. General discussions of sensitivity methods in structural dynamics are given byAdelman and Haftka [70] and Brandon [71]. The basic sensitivity equations were derivedoriginally by Lancaster [72] and Fox and Kapoor [73]. For a general, undamped, discretestructure with distinct eigenvalues, these equations take the formaA, _ tT  a k^kA aiar — 3`1 [ Or — s Or (5.56)3T T^511;11[k Ask]ar 4.3 LaT At a7.1 (5.57)where derivatives are taken with respect to a structural property r, and where the eigen-vectors are assumed to be mass normalized. Eigenvalue derivatives are readily obtainedwith (5.56). Fox and Kapoor [73] and Nelson [74] established methods for calculatingeigenvector derivatives using (5.57). Rudisill [75] investigated adding second-order termsto the sensitivity equations. Johnson and Jen [32] calculated sensivities for a multi-linkrobot analyzed with a CMS procedure based on monomial functions. A complete sen-sitivity analysis of the fixed-interface CMS formulation was given by Heo and Ehmann[76].Several researchers proposed solving inverse eigenvalue problems with a Newton'smethod iteration approach [77, 78, 79]. The equations studied were of the form441i = AM;^A = Ao E AkCk^ (5.58)kAn adaptation of Newton's method to discrete structures was shown to involve repeatedsolution of (5.56) combined with accurate updating of modal and structural parametersChapter 5. Structural Dynamic Modification^ 151[80]. To avoid the expense of repeated eigensolutions and the truncation errors inherentto the modal subspace approximation, inverse iteration is used to update the eigenvectors.5.5.2 Newton's Method for Substructured ProblemsA Newton's method application of the sensitivity equations is suitable for frequencymodification problems. Using the general CMS formulation for the baseline system (5.20),the following expressions are obtained:a k^T OK^aT (...,TyaT)T= T T + TT-K- Tr -I- TOr Orail^aM^p_aT)T= TT --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 ex-tending over a range of components, and it can be either linear or nonlinear. Derivativesof K and M can be evaluated by assembling derivatives of element matrices. Deriva-tives of T are more difficult to express as they involve rates of change of componenteigenvectors.In the derivation of the linear-equivalent equation in Section 5.3.1, it was shown thata useful simplification is to treat T as an invariant transformation. This is justifiedwhen components change in a uniform manner, and for more general changes it allowscomponent modes to change within the subspace provided by the baseline analysis. Acompletely accurate prediction of the component mode derivatives by methods proposedin [73, 74], is much more costly and therefore diminishes the value of the inverse modifi-cation procedure.The resulting linear-equivalent sensitivity equation is given byOA,^Lrai^T a mT^ T - T^Ti^(5.61)Or OrChapter 5. Structural Dynamic Modification^ 152When multiple properties are allowed to change simultaneously, the total perturbation tothe eigenvalue is estimated with a first-order Taylor expansion about the baseline values:aA,A: = Ai + E^rkak^ (5.62)k=1 arkwhere the derivatives are evaluated at the baseline property values. Substituting (5.61)in (5.62) for an appropriate rk givesA: — at =^[TT aK T — TTan7I I T14'irkak^(5.63)ark^arkThe frequency modification equation is obtained by substituting A: = A: in (5.63). Designvariables ak which solve (5.63) give a first-order estimate of the design changes satisfy-ing the linear-equivalent equation. For a single design variable the solution of (5.63) isunique; with more than one design variable, this equation can be solved using the math-ematical programming technique suggested in Section 5.4.2. In the more general case ofI 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 theeigenvector (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 dynamicequation; (3) using a single step of inverse iteration [80}. This requires solving thefollowing linear system for -yi:(5.64)mOK = E (xr) k akk=1EM m (R.) akk=1^ka-K=^ark rkakk=1m am= ^rk akark[k + Ak - (/Cf + AM. )] -yi =where the perturbation matrices are given by:(5.65)(5.66)Chapter 5. Structural Dynamic Modification^ 153The updated eigenvector is obtained by mass normalizing= [7r ( 1171^(5.67)The eigenvalue is updated with Rayleigh's quotient:(k k)= ^_^s^ (5.68)V. (M M)With nonlinear property changes, the derivatives in (5.63) must be recalculated aftereach iteration. Inverse iteration has also been used to update eigenvectors in a forwardmodification procedure called Rayleigh quotient iteration [81].Using inverse iteration to update is particularly advantageous with the compactCMS formulation. The order of (5.64) is small and it can be solved in much less time thanis needed for a CMS reanalysis of the eigenvectors. Another advantage of inverse iterationis 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 baselinemodes, 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 bysubstituting Ai = A: in the right-hand side of (5.63). Smith and Hutton [80] showed withan example of an unsubstructured model, that the zone of convergence is sufficient forengineering purposes.Solutions found with this procedure satisfy the linear-equivalent equation, not theactual equation of the modified structure. At this point the analyst should check theaccuracy of the predicted solution with a CMS reanalysis. Using the reanalysis as thenew baseline, further frequency modification calculations can be done as required.Newton's method is expected to give equal accuracy to the perturbation method whenthe latter is supplied with a full set of baseline modes. Because of the impracticality of thisChapter 5. Structural Dynamic Modification^ 154requirement, Newton's method generally gives somewhat better results. Although bothare iterative methods, the nature of the iterations is fundamentally different. Newton'smethod converges to a solution through a sequence of linear steps, the length of whichprogressively diminishes to zero. On the other hand, the perturbation method solvesthe same system of equations repeatedly, each time making adjustments to the couplingterms, until all the equations are satisfied simultaneously.5.6 Numerical ResultsIn this section, the results of three frequency modification problems are presented. Ineach example, a single frequency goal is varied over a range of values. Three inversemodification methods are compared: the Newton's method algorithm of Section 5.5, thenew 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 op-erate within a smaller subspace. This means that the perturbation methods are lesscapable of accurately predicting the modified mode shape.The two perturbation methods differ from each other in that the predictor-correctoruses a linearized perturbation equation to predict the modified mode shapes of the struc-ture. With small structural changes this is adequate; but with large changes, inaccuratemodified mode shapes lead to erroneous design changes. A further difference is that theiterative method calculates the design changes using (5.51) alone. There is one such equa-tion for every frequency constraint. On the other hand, the predictor-corrector methodattempts to satisfy one equation associated with the frequency constraint plus q —1 otherequations associated with mode shape constraints. The latter are necessary conditionsfor making the predicted mode shapes orthogonal with respect to the corrected struc-ture. A variation of this method is to satisfy both the necessary and sufficient conditionsChapter 5. Structural Dynamic Modification^ 155for orthogonality by using a total of q(q — 1) constraint equations [60, 62]. In eithercase, the mode shape constraints are somewhat artificial in that they are not prescribedin the original specification of the problem; the only genuine constraint to be satisifiedis the frequency constraint. Furthermore, the imposition of these extra constraints isapt to make a naturally underconstrained problem overconstrained, thus necessitating aminimum error rather than a minimum change solution [59]. In such cases, Welch [64]suggested relaxing the mode shape constraints, and using only equations associated withthe frequency constraint to determine the design variables. This is the approach used forthe predictor-corrector results in the following examples.Consider the second example presented in Section 5.3.2 in which modifications aremade to the elastic modulus of the hull-bottom stiffeners in the container ship. Tak-ing the elastic modulus as the design property, frequency constraints are applied to thefundamental elastic mode (14). The results using the new perturbation method for fivedifferent 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 ef-fect of additional modes in the approximation is the inclusion of correction terms in thefrequency constraint equation. These correction terms enable the prediction of progres-sively more accurate design variables and mode shape changes. As the number of modesapproaches the order of the CMS equations, the results should converge to the linear-equivalent curve. But this process converges slowly, and with 30 modes the optimal curvehas still not been obtained.Figure 5.5 shows the results of the new perturbation method and the predictor-corrector, both using a 30-mode approximation, in comparison with Newton's methodand CMS reanalysis results. The two perturbation methods show almost identical re-sults, and both match the CMS reanalysis reasonably well, although Newton's methodLinear-equivalent--.---6------ 1 mode.■....07.■•■—0—■,..m.■4:1■■■■5 modes10 modes20 modes30 modesChapter 5. Structural Dynamic Modification^ 1560=• 3.0cscoas0.9coa) 2.0cco.c0TC.2tEs 1.0LL0.00.900^0.925^0.950^0.975^1.000Prescribed frequency f: (Hz)Figure 5.4: Frequency modification of hull-bottom stiffeners for five modal approxima-tions0.0Chapter 5. Structural Dynamic Modification^ 1570.900^0.925^0.950^0.975^1.000Prescribed frequency f; (Hz)Figure 5.5: Comparison of three methods for frequency modification of hull-bottom stiff-enersChapter 5. Structural Dynamic Modification^ 1586.05.0Eo 4.0=;c0CDCD00 3.0c.c0Tac 2.0.9tu.1.00.00.900^0.925^0.950^0.975^1.000Prescribed frequency f; (Hz)Figure 5.6: Three-parameter frequency modification of hull-bottom stiffenersdoes slightly better. Clearly, the modified mode is fairly well approximated by the 30-mode subspace. Also, because the stiffeners are changed uniformly using only one designvariable, the mode shape change is small. Thus, the linear perturbation equation usedin the predictor-corrector method is sufficient for predicting the modified mode shape.Consider applying the same frequency constraints, but where now the stiffeners be-longing to each component can vary independently. In Figure 5.6 a 30-mode approxima-tion has been used for the perturbation results. Design variable a l is the fractional changeof the elastic modulus of the stiffeners in the after-body, a2 the fractional change in themid-body, and a3 the fractional change in the fore-body. The frequency constraint equa-tion is solved using optimization with a minimum change objective. A description of theChapter 5. Structural Dynamic Modification^ 159optimization algorithm is given above under the heading "Mathematical Programming".The analysis produces fairly accurate results for two of the methods. A reanalysisbased on the design prediction for g = 1.00 gives the following results:Newton : f: = 0.996HzNew perturbation : f4 = 0.995HzPredictor-corrector : fl = 0.931HzBecause the stiffeners are being changed in a nonuniform manner, the mode shape changesassociated with these modifications are more significant. Thus, the linearized perturba-tion equation is not adequate for predicting the perturbed mode shape when the designchange is very large. This accounts for the poor performance of the predictor-correctormethod in the region where large design changes are required.It is interesting to note that although both Newton's method and the new perturba-tion method predict changes giving similar frequencies, the distribution of the change issignificantly different. Of many possible solutions, both methods seek a minimum changesolution. The magnitude of the change is, for the f4 = 1.00 results,Newton : E al! = 38.4New perturbation : E cz2k = 26.3The perturbation method is more economical because in each step it determines a min-imum solution for the total modification. By contrast, Newton's method determines aminimum partial solution in each iteration. The accumulation of these partial solutionsdoes not necessarily lead to a minimum total change. In Figure 5.6, Newton's methodhas weighted the mid-body component too heavily at the expense of the fore-body, givinga modification that may be more difficult to realize in the actual structure.Predictor-correctorPerturbationNewton, baseline a-0Newton, baseline a-4.48CMS ReanalysisECoCu 7.52001 2.5LL0.012.5Chapter 5. Structural Dynamic Modification^ 1600.900^0.925^0.950^0.975^1.000Prescribed frequency f4. (Hz)Figure 5.7: Frequency modification with mid-body stiffenersNow consider changing only the stiffeners in the mid-body component. The predicteddesign changes for prescribed fundamental-mode frequencies are shown in Figure 5.7. Theeffect of two separate approximations is clearly visible. In the region f: > 0.925Hz, thepredictions based on the approximate methods begin to diverge from the CMS reanalysiscurve. This is the influence of modal truncation. Both perturbation methods use a35 mode approximation but since large design changes are being made along just onesegment of the hull, the mode shape change is severe and is not easily represented withinthe 35-mode subspace. Also in this region, the Newton's method curve begins to divergefrom the perturbation curves. The better accuracy obtained with Newton's methodresults from using equations of larger order (62 degrees of freedom versus 35 for theChapter 5. Structural Dynamic Modification^ 161perturbation equations). This enables the modified mode shape to be represented moreaccurately, although over the range of prescribed frequencies presented, the extra degreesof freedom make only a small improvement in the predictions.In the region f: > 0.95Hz, the new method begins to diverge from the predictor-corrector curve as a result of errors in the latter stemming from the linear perturbationequation. 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 predictany design changes above a = 7.60. The shape of the predictor-corrector curve is similarto those in Figure 5.6; the peak and decline in regions of high a and large mode shapechange 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 reanalysiscurve approaches infinity, which means that this frequency becomes insensitive to changesin the mid-body stiffeners. In this region, the mid-body stiffeners are completely rigid inthe fundamental mode, a design change which in practical terms would be impossible tomake.Because of the increasing insensitivity of the mode, predicting design changes in theregion just below f: 0.98 is very difficult, regardless of the number of baseline modesavailable. It may be useful in situations like this to perform an inverse modificationanalysis in two or more steps, with an accurate reanalysis between each step. Becauseonly the mid-body component is changed, a CMS reanalysis can be done fairly inexpen-sively and each reanalysis provides a new reference point, or baseline, for the subsequentperturbation analysis. In Figure 5.7, a second set of Newton's method results are shownusing a = 4.48 as the baseline. This was the result obtained for f4 = 0.975Hz in the firstperturbation 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 secondChapter 5. Structural Dynamic Modification^ 162baseline. 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 moreaccurate design changes can be calculated.5.7 SummaryStructural dynamic modification has been treated as two separate problems: forwardmodification, which is concerned with the modal analysis of a modified structure; andinverse modification, which is concerned with finding a set of design changes which sat-isfy prescribed modal constraints. In this chapter, perturbation methods for structuraldynamic modification have been applied to substructured systems, using a CMS formu-lation for the baseline analysis. Approximating the modified mode shapes by projectingthem on to a modal subspace, it was shown that if the size of the subspace equals theorder of the CMS equations, the linear-equivalent equation (5.37) results. This equationrepresents the optimal description of the modified structure that can be obtained with-out considering changes to the component modes. Forward modification problems canbe directly and efficiently solved using the linear-equivalent equation.A new method for frequency modification problems has been presented which uses theenergy-balance formulation of the perturbation equations. An iterative scheme is usedwhich converges to a solution of the full perturbation equations. Examples have beenpresented which show thit the new perturbation method compares favourably with thepredictor-corrector technique when large structural changes occur in conjunction withsignificant mode shape changes. Also, a Newton's method algorithm based on the linear-equivalent equation has been described which gives a sequence of converging solutionswhich do not depend on the number of baseline modes available. Examples given showChapter 5. Structural Dynamic Modification^ 163that Newton's method is generally more accurate than perturbation methods, as modifiedmode shapes are represented using a larger subspace.Chapter 6Summary and ConclusionsComponent mode synthesis (CMS) has been studied as a modal analysis technique forlinear, undamped, discrete models. Of principle interest is the application of this tech-nique to large-order finite element models of an arbitrary complexity where condensationis most beneficial.It is often natural and convenient to treat a finite element model as an assemblage ofstructural components. CMS generates reduced-order representations of the componentsby approximating their displacement with a truncated sequence of component modes.The Craig-Bampton and MacNeal-Rubin representations are found to be the most ap-plicable to arbitrarily complex models as they best satisfy the basic requirements forcomponent mode sets. By expressing the compatibility and equilibrium constraints interms of physical displacements and loads, free vibration equations of motion are de-rived for each mode set. These two formulations are referred to respectively as the fixed-and free-interface methods. Two variations of the free-interface method are presented inwhich first- and second-order mass approximations are used. It was shown that theseformulations can be applied to discrete models of an arbitrary geometrical complexity,and that they handle constraint equations more efficiently than the direct eliminationmethod.Important differences distinguish the fixed- and free-interface formulations. The fixed-interface equations are in terms of interface displacements and free vibration modal co-ordinates; the free-interface equations are in terms of free vibration modal coordinates164Chapter 6. Summary and Conclusions^ 165alone. This gives the latter a higher degree condensation in complex models, particu-larly when component interfaces are meshed curves or surfaces; because as the mesh isrefined, the order of the fixed-interface equations steadily increases, while that of thefree-interface equations stays the same.Consequently, free-interface CMS is preferable for the majority of large-order struc-tural models; only in cases where interface coordinates are limited to a small numberof discrete points is the fixed-interface method favourable. Also the second-order massformulation of free-interface CMS is generally more effective than the first-order formu-lation, as significantly better results are obtained with little additional cost. This isparticularly true for components with high modal density in the target frequency range.Another factor affecting the performance of CMS is the tolerance to which eigenval-ues are calculated. As the tolerance is reduced, the relative efficiency of CMS methodsincreases in comparison to a direct finite element analysis. Moreover, when componentshave high modal density, significant numerical loss of precision can result from not usinga sufficiently small tolerance in determining the component modes. In particular, this isa problem in the free-interface method when loss of precision causes component residualflexibilities 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 withshifting.CMS can also be used advantageously when multiple variations of the same model areanalyzed. For each variation, only components that have changed must be reanalyzed.Therefore, the efficiency of each analysis increases as the structural change becomes morelocalized. This is especially true for the free-interface formulation where, for large-ordermodels, a greater emphasis is placed on component-level computations. If only one or twoof the structural components are changed, a much larger proportion of the computationaltime is saved than would be with the fixed-interface method. Under these circumstances,Chapter 6. Summary and Conclusions^ 166it is advisable to use the free-interface method.The results obtained with CMS depend on the care with which component modes areselected. Truncation of the component mode sets is necessary for order-reduction, andgenerally it is the high frequency modes that are eliminated. To distinguish between lowfrequency and high frequency modes, a cutoff frequency criterion is used. It is shown inChapter 4 that fairly uniform convergence in the low frequency modes can be achievedin this way. Selection of the cutoff frequency requires some judgement, but generally itshould 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 thecomponent level (i.e., in the baseline CMS analysis), not the system level. This givesa more easily adaptable description of the modified structure, as subsequent versionsare derived by updating the component modes rather than the system modes. Thelinear-equivalent equations also provide a basis for developing the frequency modificationequations. Two methods are developed: a Newton's method algorithm originally used forunsubstructured models is applied to the linear equivalent equation; and a new iterativemethod is proposed for solving the energy-balance perturbation equations, in which allcoupling terms are accounted for. In the examples presented, Newton's method givesslightly more accurate results, even though it does not require a large baseline mode set.The iterative solution of the perturbation equations also exhibits superior performanceto the predictor-corrector method when large design and large mode shape changes occursimultaneously.The accuracy of the structural dynamic and inverse modification techniques varieswith the character of the structural change. Widely distributed modifications of limitedmagnitude can be accurately represented by the linear-equivalent approximation; but thecondensed subspace afforded by this approximation makes severe, localized changes moreChapter 6. Summary and Conclusions^ 167difficult to represent. In the latter case, more accurate results can be achieved by applyingNewton's method to the unsubstructured model, assuming that such a model exists. Thisrequires 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, theorder of the governing equations can be reduced by 10 to 20 times, while maintaining theintegrity of the low frequency spectrum. As a result, subsequent modification calculationsare inexpensive and many useful predictions can be made about the dynamic behaviourof a modified structure of significant complexity.Throughout this thesis, damping has not been considered because it is not of criticalimportance for determining the natural modes of lightly damped structures. However, in-cluding damping effects in the techniques presented may make the work more relevant tosituations involving experimentally derived modes and in systems where damping playsa more significant role. Other areas in which additional work could be done are the fol-lowing. 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J., "Structural Modification Using Rayleigh QuotientIteration," International Journal of the Mechanical Sciences, Vol. 32, No. 3, 1990,pp. 169-179.Appendix AConvergence Characteristics of the Modulation Matrix IIThe series^II . E. (402 dm) k^ (A.1)k=0converges to the expression^II = (/ - w2 dm) -1^(A.2)if .11w 2OMIlp < 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 as11II^AI, P^max 'Olixiilip(A.3)x^pwhere the p-norm of a n x 1 vector x of isn11x111,= (E xl) P^1 ,^ (A.4)k=1A general property held by all p-norms is that the norm of a product of two matrices, Aand B, obeys the inequality,^IIABIlp 5_ 11A llplIB Ilp^(A.5)Thus, for any particular p-normIIOMII = OhAV`DIMii^(A.6)PhilliAViiiiqMii (A.7)1752awaw2 (min(Ah)k) -1 < — < 1k ^,2—e,2wW2 < c—awhich holds whena > 1Appendix A. Convergence Characteristics of the Modulation Matrix II^176Because the higher modes 4h are mass normalized,11(13IMIIII(13hil = a _>11 ,1q:McIth il= j1/11 = 1^(A.8)Furthermore, the norm of a diagonal matrix is equal to the largest element on the diag-onal. Therefore,1141 II = max(Ah)i l = (mkin(Ah)kr i^(A.9)Combining (A.7), (A.8), and (A.9) gives the resultilaMii < a (qn(Ah )k)-1^(A. 10)It is guaranteed that Ilw 2 dMil < 1 whena w2 (min(Ah )k) -1 < 1^ (A.11)The term min(Ah)k is the smallest eigenvalue among the neglected modes and is thereforegreater that are equal to the square of the cutoff frequency w c . ThereforeIn conclusion, the infinite series (A.1) is guaranteed to converge in the low frequencyrange 0 < w 2 < w! /a.Appendix BEigenvalue Sensitivity to Residual Flexibility ChangesThe eigenvalues of the equation,[r (a). —^= 0^ (B.1)in which r (LD) is a symmetric matrix of the form,rp)^+ ciBTA [ATte()A1ATTB^(B.2)are stationary points of the Rayleigh quotientai 137r(65)Fli —^ (B.3)Pi PiDifferentiation of the eigenvalues with respect to C4.7 2 gives the first-order sensitivity equa-tion,where_T aA t•^armPi Pi^= -T  awl Ps-;-, BB^-11 y (65) A [ATT (w)A1 ATTarp)^crTA [ATi(P) A1-1 A Ta acD25,7,2(B.4)(B.5)From (3.38), (B.1), (B.4) and assuming that the eigenvectors pi are normalized suchthat gp-i = 1,BaAi7.B2, alp (65),B5,7,2 =^ac,2 (B.6)where f a isi ^the interface load distribution in the ith mode. This expression is equivalentto a summation of individual terms contributed by each component,aA, _^DT NIB (6) ) _ E f„. ^ f„,0652 — ac4,2 (B.7)177Appendix B. Eigenvalue Sensitivity to Residual Flexibility Changes^178where .9 is the number of components. Note that for any componentNIB^_ aT ad(ei,) aawe —^5652 P.-firà — 652 M^Ai^— 652 m 6) -1PT a(cD)mOPV= 44: (AV +6J 24 2 65443 + ...) 2 (11T=^(42 + 261243 + 36)444(B.8)(B.9)(B.10)(B.11)(B.12)The diagonals of Ah-2 , Ah-3 , ... are always positive and as a result, (B.12) is a positivesemi-definite matrix; i.e., for a non-trivial f1 3 ,BT aii/B(w) Bfi 086,2 (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 resultaAiaez,2 5- 0^ (B.14)Thus, the eigenvalue Ai is a continuously non-increasing function of 65.Appendix CImplementation of Component Mode Synthesis in VASTO6In Chapters 2 and 3 it was found that the fixed- and free-interface methods were themost promising CMS techniques for application to general finite element models. Bothof these methods have been implemented in the finite element program VAST06, with acapability for using either first- or second-order mass approximations in the free-interfaceimplementation. The present chapter gives a description of the VAST-CMS program andguides the user in beginning a CMS analysis.C.1 The Substructure/Superelement OptionThe VASTO6 program contains a substructuring/superelement option which permits thefollowing: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 ma-trices;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 longas they include all the interface nodes. In CMS applications, the same superelement179Appendix C. Implementation of Component Mode Synthesis in VASTO6^180ELEMS 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 UTDU form.REDSM^Performs Guyan reduction on stiffness and mass matrices, giving kBB andMBB in (2.65). An in-core algorithm REDSM1 and an out-of-core algorithmREDSM2 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 UTDUform.EIGEN1 Solves global eigenvalue problem.EIGNSE Reconstructs global mode shapes from Guyan eigenvectors.Table C.1: Description of VASTO6 modulesdefinition 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 thanby 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. Adescription of the function of each program module is found in Table C.1.Appendix C. Implementation of Component Mode Synthesis in VASTO6^181 ELEMS1ELEMS2 ASSEM2PARTSMI = 1, NSE )ASSEM1STEMMASSMDECOM1EIGEN1EIGNSEFigure C.1: Flow chart for Guyan reduction in VASTO6Appendix C. Implementation of Component Mode Synthesis in VASTO6^182T54^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 KIIT21^Submatrices KBB , KBI , MII ,I^I, mBB ) mBIT22^_Tr' = (KII--1 KLB) T (out-of-core solution only)T23^Submatrix K".Table C.2: T-file locations of VAST information relevant to CMSThe information regarding the substructure and superelement matrices is stored invarious T-files. Table C.2 shows the T-file locations of information relevant to a CMSanalysis.C.2 Implementation of the Fixed-Interface MethodThe flow chart for the VASTO6 implementation of the fixed-interface method is shown inFigure C.2. The only difference from Figure C.1 is the inclusion of module CMS_1 in theELEMS2 loop. This module is responsible for calculating the fixed-interface normal modesand 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 stageof the program and matrix M" is available on file T21.ASSEM2ELEMS2CMS 1ASSEM1I =1 NSE )ELEMS1PAR1SMSTIFM4MASSMDECOM14EIGEN14EIGNSEDECOM2REDSMAppendix C. Implementation of Component Mode Synthesis in VASTO6^183Figure C.2: Flow chart for the fixed-interface CMS method in VASTO6Appendix C. Implementation of Component Mode Synthesis in VASTO6^184The second task makes use of the fixed-interface normal modes to calculate the fol-lowing terms:kNN^(= ANN , for mass normalized modes)^(C.2)mBN = mBI + rM"mNBT^ (C.3)niNN = 4,/Tm114/ (= I, for mass normalized modes) (C.4)One difficulty which arises here is that the static constraint modes calculated in REDSMhave to be saved if MBN is to be calculated. There are two algorithms in REDSM: anin-core solution and an out-of-core solution. The in-core solution never explicitly formsthe constraint modes; only in the out-of-core solution are they calculated and saved onfile T22 (see Table C.2.) Therefore, if the in-core solution has been used in REDSM, theconstraint modes first have to be constructed before MBN can be calculated . Becauseof 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 subroutinein ASSEM1 for assembling the submatrices (C.2)—(C.4) into the global equations; modi-fications to EIGEN1 so that the extra modal coordinates in the system eigenvectors areaccounted for; and modifications to EIGNSE so that the fixed-interface component modesare used in the reconstruction of the system mode shapes.Figure C.3 shows the flow chart for the reanalysis option in the fixed-interface imple-mentation. The algorithm is interrupted after DECOM2 if reanalysis of the component isnot desired. The implementation could be improved by skipping the entire loop if thecomponent is not being reanalyzed, but it is necessary to store beforehand the decom-posed Ku matrices elsewhere so that they are not overwritten during reanalysis. Thesematrices 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 contentsAppendix C. Implementation of Component Mode Synthesis in VASTO6^185ELEMS1ELEMS2ASSEM1STIFMMASSMDECOM1EIGEN1EIGNSEASSEM2PARIBMDECOM2REDSMCMS 1( I = 1, NSE )AFigure C.3: Reanalysis option of the fixed-interface CMS method in VASTO6Appendix C. Implementation of Component Mode Synthesis in VASTO6^186C42 Reduced submatrices kNN , TnNN and mBN .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. Thisinformation is later moved to C42.S46 Submatrices kBB and TriBB for one component. This information is later movedto T56.S51 Fixed-interface component modes calculated for one component. These arelater 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 eachcomponent, which contain CMS data for a single component. These will appear witha prefix supplied by the user and unique to a particular component. These special filesonly need to be saved if a subsequent reanalysis is to be performed.C.3 Implementation of the Free-Interface MethodThe flow chart for the free-interface method based on the MacNeal -Rubin mode set isshown in Figure C.4. A flow chart for the reanalysis option of the same method is shown inFigure C.5. Whereas the fixed-interface method follows the Guyan reduction programclosely, the free-interface implementation differs greatly. The two modules added forthis method are CMS_2 which calculates the free-free and residual attachment componentAppendix C. Implementation of Component Mode Synthesis in VASTO6^187ELEMS1ASSEM2ELEMS2ASSEM4 CMS 2I - 1, NSEFigure C.4: Flow chart for the free-interface CMS method in VASTO6ELEMS1ASSEM2ELEMS2noI = 1, NSE )4^ASSEM4 CMS 2Figure C.5: Reanalysis option for the free-interface CMS method in VASTO6Appendix C. Implementation of Component Mode Synthesis in VASTO6^188modes, 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 differenceto note here is that the modules STIFM and MASSM, for stiffness and mass modifica-tion respectively, are located inside the component. loop. By contrast, in the Guyan-reduction/fixed-interface implementation they were located outside the component loop(see Figures C.1—C.3). There are two principal advantages to making these modificationsat the component level. First, the free-free component modes calculated in EIGEN4 willautomatically incorporate any external constraints. This saves the trouble of having toform special constraint equations later on in the analysis. Secondly, the spring and massadditions can be applied to any node in the structure, not just at the master nodes. In-deed, for the free-interface method master nodes should be defined only on the componentinterfaces, otherwise the complex assembly algorithm will produce spurious results. Onthe 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 checksfor and corrects certain types of linear dependencies in the stiffness matrix. It is necessaryto make these corrections to the component stiffness matrix to ensure that the componentmodes, both static and dynamic, can be computed correctly. One case in which a lineardependency correction is necessary is a two-dimensional model composed of membraneelements (such as the container ship model described in Chapter 4.) Flat membraneelements have no stiffness in the out-of-plane direction and therefore have degeneratestiffness matrices. If the membranes are oriented in one of the coordinate planes, saythe x-y plane, the linear dependency correction is simply to add large springs to the z-coordinate diagonals of the stiffness matrix. However, if the membranes are not orientedin one of the coordinate planes, the linear dependency will not be obvious at first glanceand a more subtle detection and correction algorithm is used.RESMASSV DECOMKINTRLFSOLVEGEIGEN4DECOM1STIFMMASSMFULL GRESFLXAppendix C. Implementation of Component Mode Synthesis in VASTO6^189Figure C.6: Flow chart for the module CMS.Appendix C. Implementation of Component Mode Synthesis in VASTO6^190The fixed-interface program only does linear dependency checking on the Ku por-tion 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 thefixed-interface normal modes. On the other hand, the free-interface program has to de-compose the whole component stiffness matrix, as is required for calculating the free-freemodes and the component flexibility. In this case, the linear dependency checking cantherefore affect both the interior and interface coordinates of the stiffness matrix. A dif-ficulty arises here if a correction is made at an interface coordinate and, after calculatingthe component modes, it is not removed from the flexibility, then the inter-componentcompatibility and equilibrium constraints may not be properly satisfied.Linear dependency corrections of the first type, where large stiffness are added tothe diagonal, can easily be detected as a zero diagonal in the flexibility. If this diagonalcorresponds to an interface coordinate, the constraint that should be applied here is zeroload, with non-zero displacement allowable. Unless the effect of the correction is removedby replacing this zero with a large flexibility value, the constraint that will actually beapplied is a zero displacement, with a non-zero load allowable. This may seem a strangedistinction to make in light of the membrane example, where it seems natural to apply azero-displacement constraint at all out-of-plane coordinates, interface or otherwise. Butin this case it makes no difference whether a zero-displacement or zero-load condition isimposed; the net effect will be the same.As an example of a situation where it does make a difference, consider one componentwith 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 canonly be applied in the axial direction. The linear dependency corrections will effectivelyeliminate non-axial displacement of the bars, as is needed to calculate the componentmodes. But when this modified bar element is connected to the adjacent brick elementAppendix C. Implementation of Component Mode Synthesis in VASTO6^191during the synthesis phase, these corrections will effectively put zero-displacement con-straints on the connecting node, constraints which in reality do not exist. If any constraintis applied here it should be zero load in the non-axial directions, for the bar elements aretwo force members and therefore should only support a load passing through the two endnodes. To re-establish the proper interface constraints, the linear dependency correctionhas to be removed from the interface nodes of the bar elements by making an appropriatecorrection to the flexibility matrix.Although the free-interface program can handle the first type of linear dependencycorrection, the second type is much more difficult to detect. These corrections involvemanipulations of the triads situated along the diagonal of the stiffness matrix, and wouldbe invoked, for instance, when bar elements are not co-directional with one of the coordi-nate axes. If these corrections cannot be detected, some method needs to be devised bywhich the sequence of manipulations can be reversed in the flexibility matrix. It must beemphasized that an error in the analysis will only occur if the second type of correction ismade to an interface coordinate, and if the connecting elements in the adjacent compo-nent do not have the same degeneracy. Whether or not this type of error will occur canbe discovered by considering what would happen if the model were not substructured. Inthe above example, the stiffness matrix would not be degenerate at the node connectingthe bar element to the solid element and so no correction would be made there in theunsubstructured case.The free-interface program provides two different algorithms for calculating the com-ponent flexibility (see Figure C.6). If NCON = 0, the flexibility can be obtained directlyfrom the stiffness matrix. If NCON > 0, the more complicated method described by(2.79) is necessary. Module RESFLX determines the residual flexibility matrix if, andRESMASS determines the residual mass matrix Lt.The additional storage files created by the free-interface program are listed in TableAppendix C. Implementation of Component Mode Synthesis in VASTO6^192T23 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. Theseare later stored in T23 and T24.S38 Information regarding the number of free-free modes calculated, the numberretained, and the number of rigid-body modes.S51 Free-free component modes calculated for one component. These are latermoved 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 arealso some special files which are stored under a component-specific prefix defined by theuser (see Table C.6). These files have to be saved if a subsequent reanalysis is to beperformed.The free-interface program has been designed for efficient manipulation of data onmodern computer systems. So as to avoid excessive I/O operations, as much of theintermediate data as possible is kept in the internal memory space allocated for theprogram. 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 largestAppendix C. Implementation of Component Mode Synthesis in VASTO6^193component that can be analyzed is Ns 1/2x KORE VVV^5(C.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:LEN2xKORE 3112xKORE^(C.6)).:%2 min (07(017E,LEN^5where LEN is the total number of independent forces and moments acting on the com-ponent 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 numberof degrees of freedom per node.C.4 Including Fluid Added-Mass in a CMS AnalysisSuppose 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 vibrationof 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 fluidadded-mass matrix MA in the equation of motion:Mii + Ku = -MAU^ (C.10)Appendix C. Implementation of Component Mode Synthesis in VASTO6^194There are various methods for computing the added-mass matrix, but once it is estab-lished, 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 substruc-tured model, an analogous approach is to include added-mass effects in the componentlevel analyses and synthesis the equations of motion of the structure on this basis. How-ever, unlike the structural matrices, the added-mass matrix is fully coupled and cannotbe substructured. A different approach is therefore required.An approximate but generally efficient way to solve this problem is to first calculatethe dry modes using (C.9) and then approximate the motion of the wetted structure asa linear combination of the dry modes:u.-Ex4=x4. (C.12)where X is a rectangular matrix containing the eigenvectors as columns, is the vectorof generalized coordinates and q is the number of modes shapes. This transformation isexact only if a complete set of modes is included in X. This is seldom the case thoughand 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 fromthe dry mode shapes.Applying (C.12) to (C.10) and premultiplying by XT gives the following equation:+ 'CIA) + o (C.13)where,= xTmxAppendix C. Implementation of Component Mode Synthesis in VASTO6^195MA = XT MAX= xTifxThe 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 correspondingeigenvalue 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 andan accompanying fluid-element model. If the full-sized and substructured models arebased on the same finite element mesh, there is a relationship between the coordinatesystems of the two models which enables the generalized added-mass MA = XT MAXto be calculated from CMS-derived mode shapes. When this is case, (0.16) can beestablished regardless of whether the dry modes were calculated with a substructured orunsubstructured analysis.The VASTO6 program already provides for calculating wetted modes of unsubstruc-tured models with this method. To apply it in conjunction with a CMS analysis, two extramodules had to be created. The program MATCH finds matching node numbers in a sub-structured 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 afterAppendix C. Implementation of Component Mode Synthesis in VASTO6^196the dry modes have been calculated with CMS. The module AMEIGN then assembles theCMS-derived mode shapes into equivalent modes in the full-model coordinate systemand stores them in file C52. In this form, the mode shapes are passed on to the moduleEIGNWM 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-interfaceanalysis, set IELEMS = 7. The remainder of the master control codes should be set as fora 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 thedry modes, set the master control code IEIGEN to 3.The structural components are defined in exactly the same way as substructures andsuperelements are defined in the existing version of VAST06. However, some restrictionsare 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 CMSanalysis.• 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 thecomponent interfaces and at all nodes which have prescribed displacements andlumped masses. For a free-interface analysis, master nodes are defined at the com-ponent interfaces only; prescribed displacements and lumped masses are handledat the component level.To specify the component modes for each component, a data file PREFX.CMS mustAppendix C. Implementation of Component Mode Synthesis in VASTO6^197be supplied with the other VAST data files. To calculate wetted modes of a structure, adata file PREFX.GLM must also be included.C.5.1 Format of input file PREFX.CMSCard 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 componentsCard 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 theprevious component. If the component is the same as the previous one exceptfor 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 filesOmit 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 VASTO6^198Card 4 (15, E10.3) NSK, SPRING• NSK = the number of displacement nodes to be assigned prescribed displace-ments.• SPRING = default value for added spring stiffness (10 20 is assumed if SPRINGis 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 ...IDC6 , SKi ...SK6 .• NI = component node number to be assigned a prescribed displacement• IDCi = codes for specifying degrees of freedom to be assigned prescribed dis-placment.• SK; = translational spring stiffnesses or rotational spring stiffnesses to be as-signed to the degrees of freedom indicated by IDC ; = 1. When spring stiffnessesare 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 ...AML6• 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 VASTO6^199Omit 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 fig-ures.• IPTC = 2, normalized eigenvectors are printed out with eight significant fig-ures.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 forthis 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 is0.001.Card 11 (15) NMKP• NMKP = the number of component modes desired for this component.Card 12 (1615) MODES 1 MODESNmKpAppendix C. Implementation of Component Mode Synthesis in VASTO6^200• MODES1 MODESNMKP = the component modes to be used, listed in orderof increasing frequency.Omit Cards 13 through 15 if IELEMS = 6.Card 13 (15) NCON• NCON = number of nodes to be assigned prescribed displacements in orderfor 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 ...IDC6• NI = node number to be assigned a prescribed displacement.• IDCi = codes for specifying degrees of freedom to be assigned prescribed dis-placements.Exactly N RIG degrees of freedom should be assigned prescribed displacements inCard 14.Card 15 (15) IMSS2• IMSS2 = 0, second order mass terms are not calculated.C.5.2 Format of input file PREFX.GLMThis file contains pairs of nodes, matching nodes in the substructure model to nodes inthe complete structure model used to compute the fluid added-mass matrix. This fileis necessary if wet modes are to be calculated from dry modes computed with a CMSanalysis. If the PREFX.T41 file exists for both the complete and substructured models,Appendix C. Implementation of Component Mode Synthesis in VASTO6^201and the PREFX.T54 file exists for the substructured model, the PREFX.GLM can begenerated 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 modelInclude 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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