THEACCURACYOFREDUCEDCOMPONENT MODELS

G.J .T. J. Claessens

WFW-report: 95.159

Professor: Prof. dr. ir. D.H. van Campen, TUE Coaches: Dr. ir. A. de Kraker, TUE

Ir. J.J. Wijker, TUD/Fokker Space & Systems

Eindhoven, October 1995

Eindhoven University of Technology Department of Mechanical Engineering Division of Fundamental Mechanics

Summary

Notation

Introduction

System modeling and reduction 2.1 2.2 2.3 2.4 2.5

Introduction: Component Mode Synthesis . . . . . . . . . . . . . The reduced component model . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . Method of Guyan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Craig-Bampton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Rubin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Components without rigid body modes . . . . . . . . . . . . . . . . . . . . 2.5.2 Components with rigid body modes . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Coupling-method of Martinez . . . . . . . . . . . . . . . . . . . . . . . . . .

Base excitations and effective masses

4

9 9

10 11 12 13 13 16 18

28

4.1 4.2 4.3

4.4 4.5 4.6

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Base excitation mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Effective masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Accuracy of the reduced component model 25 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Mode shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.1 Modal Assurance Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.2 Cross-orthogonality check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Rigid body motion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Frequency- and impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Effective masses and reaction forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Application: reduction of a multi-dof discrete system 28 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Conclusions 38

Bibliography 39

1

A Derivation sum effective masses

B Multi-dof discrete system

C Correlation matrices for cross-orthogonality

40

42

43

2

N

For the analysis of the dynamic behaviour of complex systems, the Finite Element Method is often used. To yield good approximations of the system behaviour, a large number of finite elements and degrees of freedom is necessary. Consequently, large system matrices will result and much CPU-time is needed t o solve the system equations.

In linear dynamics, Component Mode Synthesis methods, which are based on the principle of superposition, can be used t o reduce the system model. The displacement field is approximated by a linear combination of well chosen modes, whose coefficients act as the dof of the reduced system. With CMS methods, the linear system is subdivided in a number of components. Each component is reduced and after that all reduced components are coupled t o obtain a compact system model.

The purpose of this investigation is to obtain a reduced component model which satisfies some essential criteria, such as eigenfrequencies, mode shapes, responses, rigid body motion energy, effective masses and reaction forces. Therefore, three CMS methods are used to obtain a reduced component model and to see which method is best.

The Guyan reduction only uses static modes for the approximation of the displacement field. The Craig-Bampton reduction is based on fixed-interface eigenmodes and constraint modes. At last, the Rubin reduction uses free-interface eigenmodes and correcting flexibility modes t o get a reduced model.

The Craig-Bampton and Rubin methods both yield good results, because the reduced component model satisfies all essential criteria. The method of Rubin is preferred if the component is excited in the form of force excitations. If the component is excited by a base excitation, the Craig-Bampton method should be used.

It is found that the Guyan reduction is the worst reduction method.

3

Scalars

Viscous damping constant Elastic energy Kinetic energy Frequency Cut-off frequency Imaginary unity Linear spring-stiffness, integer Mass k-th modal mass Number of dof of the component Number of boundary dof of the component Number of deleted (elastic) free-interface eigenmodes of the component Number of external dof of the component Number of elastic free-interface eigenmodes of the component, number of effective masses of the component Number of effective masses of the reduced component Number of internal dof of the component Number of kept free-/fixed-interface eigenmodes of the component Number of local dof of the component Number of kept elastic free-interface eigenmodes of the component Number of dof of the reduced component Number of rigid body modes of the component Number of internal dof minus number of rigid body dof of the component Time Rigid body motion energy dimensionless modal viscous damping k-th dimensionless modal viscous damping k-th eigenvalue k-th reduced eigenvalue Angular frequency Cut-off angular frequency k-th angular eigenfrequency corresponding t o elastic eigenmodes k-th angular eigenfrequency k-th reduced angular eigenfrequency Undamped angular eigenfrequency

4

Column matrices

Loads acting on the component Column with loads acting on t>he boundary dof of the Component Reduced column of loads Reaction loads exercised at the base of the component Dof of the reduced component Perturbation of p Dof of the reduced component with original boundary dof Boundary dof of the reduced component Internal dof of the reduced component Rigid body dof of the reduced component Dof of the component, displacement Perturbation of q Boundary dof of the component External dof of the component Elastic dof of the component Internal dof of the component Local dof of the component Rigid body dof of the component Internal dof different from rigid body dof of the component Dynamic displacement Elastic displacement Rigid body displacement Static displacement i-th component mode Column with a base displacement Relative displacement as a result of a base excitation Virtual work of generalized loads Natural coordinates Natural coordinates corresponding to elastic eigenmodes Natural coordinates of fixed component First n k natural coordinates Natural coordinates corresponding t o rigid body modes k-th eigenmode of the component k-th eigenmode of the reduced component k-th eigenmode of the reduced component with data of physical coordinates

5

Matrices

Damping matrix Reduced damping matrix Correlation matrix for cross-orthogonality Matrix with unit loads on boundary dof Flexibility matrix Elastic flexibility matrix Residual flexibility matrix Impulse response matrix Frequency response matrix Stiffness matrix Reduced stiffness matrix Modal participation matrix Participation matrix of k-th mode Modal mass matrix Dynamic mass matrix Effective mass matrix Mass matrix of the component as rigid body Reduced mass matrix Equivalent mass matrix at the base Modal Assurance Criterion matrix Total transformation matrix Transformation matrix with use of residual flexibility modes Transformation matrix with use of flexibility modes Transformation matrix which supplies the boundary dof again Matrix with eigenvalues corresponding to free-interface eigenmodes Matrix with eigenvalues corresponding t o deleted (elastic) free-interface eigenmodes Matrix with eigenvalues corresponding t o elastic free-interface eigenmodes Matrix with eigenvalues corresponding t o fixed-interface eigenmodes Matrix with eigenvalues corresponding t o kept free-interface eigenmodes Matrix with eigenvalues corresponding t o kept elastic free-interface eigenmodes Matrix with free-interface eigenmodes Matrix with deleted (elastic) free-interface eigenmodes Matrix with elastic free-interface eigenmodes Matrix with flexibility modes Matrix with residual flexibility modes Matrix with fixed-interface eigenmodes Matrix with kept fixed-interface eigenmodes Matrix with kept free-interface eigenmodes Matrix with kept elastic free-interface eigenmodes Matrix with rigid body modes Matrix with reduced eigenmodes Matrix with reduced eigenmodes with data of physical coordinates Matrix with eigenvalues corresponding t o elastic eigenmodes Matrix with angular eigenfrequencies

6

Chapter P

Introduction

For the calculation of the dynamic behaviour of complex elastic structures in general the Finite Element Method (FEM) is used. In order t o yield good approximations of the system behaviour such as displacements, strains and stresses, the physical model is divided in a large number of finite elements. Consequently, an excessive number of degrees of freedom (dof) appear in the idealization of a complicated structure, but because of the used computer storage capacity i t is not always useful, sometimes even impossible, t o deal with the resulting large matrices.

In linear dynamics, this problem can easily be circumvented through applying the Ritz reduction method, which is based on the principle of superposition. The displacement field is approximated by a linear combination of a limited number of well chosen displacement functions or modes, whose coefficients act as the dof of the reduced system. Usually a number of eigenmodes with eigenfrequencies lying in a frequency range of interest is chosen. The frequency range of interest is determined by the frequency spectrum of the external loads. In practice the significant part of this spectrum often ranges from zero to some cut-oflfrequency f c . The amount of CPU-time, which is saved by solving the reduced equations, can be very large. The extra CPU-time needed t o calculate static modes, eigenfrequencies and corresponding eigenmodes is quickly regained. In most analyses one wants t o know the eigenfrequencies and eigenmodes of the system anyway.

Component Mode Synthesis (CMS) methods are based on the Ritz reduction technique. With CMS methods, the linear system is first subdivided in a number of subsystems, also called compo- nents or substructures. A simple example of a system divided in two components (or subsystems) is given in Figure 1.1. A subdivision in components can be advantageous in many situations. These situations will be discussed later on. Subsequently the Ritz reduction method is carried out at component level. After reduction, component models are coupled t o obtain a compact system model.

The main contractor, who is responsible for the dynamic behaviour of the total system, asks his subcontractors to deliver a reduced component model that is as good as possible in a way that is as cheap as possible. The purpose of this investigation is to obtain such reduced component model. To check the quality of the reduced component model, the model must meet certain important requirements.

Suppose each of the components of Figure 1.1 needs t o be reduced. In the frequency range of interest, eigenfrequencies and eigenmodes of the reduced component have t o correspond t o those of the original component. Some possible ways to compare eigenmodes is t o make use of the Modal Assurance Criterion (MAC) and orthogonality checks. It is obvious that also response characteristics of the component must be produced by the reduced component model as well. Therefore, frequency- and impulse response functions are compared t o ensure that the reduced

7

component model is good. Component 2 of Figure 1.1 can make a motion as a rigid body. To guarantee that after reduction this rigid body motion is still present, the rigid body motion energyis a good measure. The concept of eflective masses plays an important part in the dynamic behaviour of constructions subjected to base excitations. This concept will be introduced in chapter 3. For reduced components, the effective masses and acting reaction forces at the base must have a certain accuracy in comparison with the original component.

CMS methods distinguish themselves by the use of different types of component modes. In the investigation three CMS methods are used, namely the method of Guyan, the method of Craig- Bampton and at last the method of Rubin. These methods are discussed in chapter 2. In chapter 4, the necessary accuracy of the reduced component model in comparison with the original component of all essential criteria will be discussed. An application of the several reduction methods to a multi-dof discrete system is described in chapter 5. Therefore, comparisons will be done between the reduction methods for all criteria. All results will be discussed. Finally, in chapter 6, a number of conclusions are drawn.

+ + + + + + +

I I

I I component 2 component 1

Figure 1.1: Example of a system subdivided in two components

8

Chapter 2

System modeling and reduction

2.1 Introduction: Component Mode Synthesis

Component Mode Synthesis (CMS) methods are used for modeling and analysis of large linear dynamic systems, which are undamped, proportionally damped or slightly damped. Their two major features are:

1. The subdivision of the system in a number of components, which can be advantageous in various situations:

o The components originate from different subcontractors. Each of them is responsible for the dynamic performance of his component. The main contractor, who is responsible for the total system, has t o check if requirements with regard t o system dynamics are satisfied or not.

o Some components can be modeled theoretically. Other components have to be identified experimentally, for example in case of unknown damping characteristics.

o A system contains some or many identical components. Only one of these components needs t o be modeled.

o Local modifications of the design only requires a new analysis of one or more components.

In general a component will have one or more interfaces. An interface will be part of two or more components and/or local nonlinearities [4], [9].

2. The number of dof of every component is reduced using the Ritz method. The displacement field q of the component is approximated by a reduced number of component modes t;:

n r e d

q = tip; = Tp

with:

q : dof of the component (n * 1);

T : transformation matrix (n * n,,d);

p : dof of the reduced component (n,,d * 1)

9

2.2 The reduced component model

The total system is subdivided in components. These components have in total n dof. The nodes which couple a component at the other components are the so-called boundary or interface nodes. The number of boundary dof is nb. The other nodes are internal nodes. So, n = nb + n;, with ni the number of internal dof. It is assumed that the equations of motion of the components are linear. The equations of motion of a component are:

M q + B q + K q = f (2.2)

with:

M : mass matrix (n JF n);

B : damping matrix (n * n);

K : stiffness matrix (n * n);

f : loads acting on the component (n * 1)

The displacement field of the component is approximated by writing q as a linear combination of nred generalized coordinates p :

4 = TP (2.3)

The transformation matrix T consists of a limited number of well chosen vectors. For the kinetic energy holds:

(2.4)

(2.5)

(2.6)

E k i n = 1/2qTMq = 1/2p T T T MTI) = 1/2pTMredI)

For the elastic energy holds: Eel = 1/2qTKq = 1/2p T T T K T p = I/2pTKredp

The virtual work of generalized loads is: SW = 6 q T ( f - Bq) = 6 p T T T (f - BTI)) 1 SpT(fred - Bred$)

The equations of motion of the reduced component now become:

Mredp f Bred@ f Kredp f r e d

with: T Mred = T M T : reduced mass matrix (nred * n r e d ) ;

I<red = TTKT : reduced stiffness matrix (nred * n r e d ) ; (2.10)

f r e d = T T f : reduced column of loads (?.%Ted * 1) (2.11)

So the component with n dof has been reduced t o a component with n r e d dof. In the investigation, only undamped components are considered ( B = 0 ) .

10

2.3 Method of Guyan

In the method of Guyan, the total set of dof is subdivided in ne external dof and n1 local dof:

(2.12)

It is important that the set qe contains all boundary doff&, because these dof have t o be available after reduction. The system can now be partitioned as:

(2.13)

Consider the second part of the equations of motion:

Mleie + Mllil + Kleqe + Kllql = 0 (2.14)

It is now supposed that the total inertia forces Mieqe + Milqi are negligible with regard t o each of the stiffness terms Kleqe and K l l q l . This will result in:

K i e q e + Kllql = 0 (2.15)

By a good choice of qe, Kil will be regular and so from (2.15) it follows:

41 = - K i l K l e q e (2.16)

So, the dof q can be written as a linear combination of the external dof qe:

4 = T P (2.17)

with:

The system with n dof is thus be reduced to a system with n,,d = ne dof. The transformation matrix T consists of so-called constraint modes or static modes. Therefore, this reduction method is often called the static reduction method. A static mode is defined as the static displacement field which results as a unit displacement or -rotation is imposed on one of the external dof, whereas the other external dof are suppressed.

If for a system with n dof, only the first n k eigenfrequencies lie in the frequency range of interest, then roughly ( 3 ù 4) c n k dof have t o be considered as external dof [2]. The selection of the external dof must be in such a way that the system behaviour of the original component is described by the reduced component as good as possible. Roughly, this means that if mass and stiffness are divided regularly, also al1 external dof (different from boundary dof) have to be divided regularly along the system. If mass and stiffness are not divided regularly, then the diagonal terms of the mass- and stiffness matrix give some mathematical support in the selection of the external dof, namely; keep those dof for which holds that [3]:

(2.18)

where w, is the cut-off angular frequency.

11

2.4 Method of Craig-Bampton

The total set of dof is subdivided in nb boundary dof and n; internal dof, so:

The system can be partitioned as follows:

(2.19)

(2.20)

The total displacement q can be split up in a static deformation part qs and a "dynamic" part q d , which is the contribution of the displacements owing t o the inertia forces, which act on the internal nodes:

q = qs + qd (2.21)

The static displacement qs is a linear combination of constraint modes. These are assumed to be completely determined by the boundary dof qb, in the way which also underlies the Guyan reduction:

(2.22)

with:

The "dynamic" contribution qd is a linear combination of so-called fixed-interfuce eigenmodes. These are eigenmodes of the free-vibrating component for which all boundary dof have been suppressed:

M..". 2 2 4 2 + K i q i = o (2.24)

Fixed-interface eigenmodes are calculated by solving the associated eigenvalue problem:

The eigenvalues XI, = wk and the corresponding fixed-interface eigenmodes q5h are stored in the matrices A;i and @ii respectively:

12

(2.26)

(2.27)

The fixed-interface modes are normalized on the mass matrix A&:

a+M;;Qjii = I (2.28)

@I(;;@;; zz = A;; (2.29)

The total displacement q can now be given by:

(2.30)

With this, the system has not been reduced yet. If only the first nk fixed-interface modes, which lie in the frequency range of interest are taken, the system will be reduced. This means tha t the "dynamic" part qd will be approximated as follows:

So, the total displacement field q can be approximated by writing:

4 = T P

(2.31)

(2.32)

with:

The system with n dof has now been reduced t o a system with n,,d = nb + nk dof. If k = O, it is a true Guyan reduction and if k = i, there is no reduction a t all.

2.5 Method of Rubin

When the method of Rubin is used, a distinction between two kinds of components has t o be made: completely fixed components and free components. Completely fixed components are components which are still kinematically determined after subdividing the construction. The stiffness matrix of these components is regular. If the system has free components, then the component can move without deforming. These components will have rigid body modes. The stiffness matrix of free components will be singular. Components with rigid body modes have t o be reduced otherwise than components without rigid body modes.

2.5.1

The response of a component t o loads acting on the boundary nodes can be described with the help of the frequency response function (FRF). The reduced component must produce this frequency response function in the frequency range of interest as well. Consider the equations of motion of the undamped, free-vibrating component:

Components without rigid body modes

M q + I ( q = o (2.33)

13

The associated eigenvalue problem with this is:

( - 4 M + K)$k = o (2.34)

In contrast with the method of Craig-Bampton, the so-called free-interface eigenmodes are con- sidered. corresponding fi-ee-interface eigenmodes c$k are stored iïì the matïices R and @ ïespecti-dy:

This means that no boundary dof are suppressed. The eigenvalues XI, = w i and the

(2.35)

The free-interface modes are normalized on the mass matrix M :

QTM@ = I

@'Ka = A

The frequency response function can now be written as follows:

(2.37)

(2.38)

(2.39)

For low excitation frequencies, if w2 < A h , the contribution of the higher eigenmodes will be more or less statically constant. So this gives:

(2.40)

This approximation for H ( w ) holds for frequencies which are considerably lower than the eigenfre- quency which belongs t o the nk-th eigenmode.

so the flexibility matrix G can be expressed as:

(2.42)

14

In this, @k is the matrix of kept free-interface eigenmodes and @d the matrix of deleted free-interfuce eigenmodes. h k k and A d d are diagonal matrices with corresponding eigenvalues. From (2.40) and (2.42) it follows that:

(2.43)

The approximation of the frequency response function can thus be written by the first n k eigenmodes and the inverse of the stiffness matrix. The last part of (2.43) is called the residualflexibility matrix, so:

G,,, = @ d h d , @ z = K-l - @kh,@T (2.44)

In general, it holds for the response of a component:

4 = W4.f (2.45)

Since dynamic loads act on the boundary dof, the generalized load vector will be as follows:

f b = [ ] with f b b ( n b * 1) and f b ( n * 1).

For the response of a component it follows with (2.43):

(2.46)

(2.47)

The columns which describe the behaviour of Gres f b are the so-called residual flexibility modes. They can be determined by setting a unit load on all boundary dof separately, so:

f b b Ibb (2.48)

In here, I b b is the ( n b * nb) identity matrix. So this gives for the residual flexibility modes:

@G = GresPb = (I(-1 - @ k h i L @ T ) F b (2.49)

with:

F b = [ oi, ] (2.50)

In case of a (quasi-) static load, the residual flexibility modes represent that part of the (quasi- ) static response, which originates from the deleted free-interface modes @d. The response of a c o m p c ~ e n t can thiis be written by the Sept free-interface modes @k and the resiclza! flexibility modes @G. The displacement field q can now be approximated by:

q = TlP (2.51)

15

with:

The system with n dof has now been reduced t o a system with n,,d = nb + n k dof.

In the calculatiori_ of the residual flexibility modes QG, the kept free-interface modes @k are used. But these kept free-interface modes are also used in the calculation of the transformation matrix Ti. The dof q are written as a linear combination of columns of Ti. So, in the calculation of @G, these kept free-interface modes can thus be omitted just as well. Then the transformation matrix becomes:

= [Q'F @kl

with:

(2.52)

(2.53)

@F is called the matrix of flexibility modes.

Also from a numerical point of view, use of flexibility modes is preferred instead of residual flexi- bility modes. If the number of kept free-interface modes is very large, the residual flexibility modes will become very small. This can cause ill-conditioned matrices. This will not occur when flexibility modes are used.

2.5.2

In the section before, the inverse of the stiffness matrix was used. This can only be calculated if this stiffness matrix is regular. If I< is not regular, then the component has rigid body modes. The number of rigid body modes n R will be six at most in the three-dimensional space. This implies for the rank of the stiffness matrix:

Components with rigid body modes

rank(I<) = n - n R (2.54)

Rigid body modes @R are those free-interface eigenmodes of a, for which the eigenfrequency is equal t o zero. From now on, free-interface eigenmodes exclusive rigid body modes are called elastic free-interface eigenmodes @ E , so:

@ = [ @ R @E] ; n=nR+nE (2.55)

Rigid body modes are defined by:

The matrix @R can be calculated from:

(2.56)

(2.57)

16

Within the set of internal dof qi, some dof are suppressed. The number of suppressed dof must be equal t o the number of rigid body modes, so:

(2.58)

Every unit, load Fb will result in a displacement, which will be the sum of a displacement as a rigid body Q' and an elastic displacement Q" ( Q is used instead of q because the total set Fb is considered) :

(2.59)

Q' = @RPR (2.60)

Because rigid body modes have been normalized on the mass matrix, substitution of (2.60) in (2.2), followed by premultiplication with and use of (PRM@>R = I , yields:

& = @sFb (2.61)

Substitution of (2.50) and (2.59) in (2.2), using (2.60) and (2.61), gives:

hdQe + I<Qe = RFb (2.62)

with:

R = I - M@R<PR (2.63)

In general (quasi-) static elastic displacements (Qe = ( j e = O) can only be calculated if these are considered relative t o Q', because I< is singular:

(2.64)

G is called the flexibility matrix. The matrix of flexibility modes @F is found by requiring these elastic displacements t o be orthogonal to @R with respect t o the mass matrix by premultiplication of (2.64) with RT:

@F = GEFb (2.65)

GE is the elastic flexibility matrix:

GE = R ~ G R (2.66)

Ar, alternative formulation fer GE fo!lows from premiiltiplication of (2.37) with @-Ti followed by postmultiplication with ( P T :

M@(PT = r (2.67)

17

Substitution of this equation in (2.63), using (2.55) and (2.34) gives:

R = M@P - ~zI@R<PR = M@& = I<@EA&@E (2.68)

The matrix @E consists of elastic free-interface modes and AEE is a diagonal matrix with corre- sponding eigenvalues.

Finally, substitution of (2.68) in (2.66) results in:

(2.69)

For the elastic flexibility matrix holds:

Just as in section 2.5.1, @d is the matrix of deleted elastic free-interface eigenmodes and @, is the matrix of kept elastic free-interface eigenmodes. A,, and Add are diagonal matrices with corre- sponding eigenvalues again.

For the residual flexibility modes holds:

(2.71)

The residual flexibility modes can thus be determined by G, the rigid body modes @R, the kept elastic free-interface modes @, and the mass matrix M . The transformation matrix now becomes:

1 T @G = G,,,Fb = (GE - @,AL.@:)Fb = (RTGR - @,A;, @,)Fb

Just like in the method without rigid body modes, it is advisable t o use flexibility modes instead of residual flexibility modes t o prevent numerical problems. So Ti becomes:

When the method of Rubin is used, i t is trivial that the first n k eigenfrequencies and eigenmodes of the reduced component are equal t o those of the original component. So take those npk eigenmodes which lie in the frequency range of interest in the transformation matrix. In general, the n b

highest "eigenfrequencies" will be inaccurate, because their corresponding eigenmodes will be linear combinations of residual flexibility modes. Therefore, these eigenfrequencies, referred t o as artificial eigenfrequencies, will be greater than or equal t o the lowest deleted genuine eigenfrequency. In linear dynamics, the artificial eigenfrequencies will not have a negative influence on the accuracy of the solution, because the external load signal does not contain these high frequencies.

2.5.3 Coupling-method of Martinez

After a component has been reduced, this will have a new set of generalized coordinates. To couple all components, a new transformation has t o take place in such a way that all boundary dof are available again. It is known that:

4 = Tip (2.74)

18

or

From the first equation of (2.75) follows:

Pb = @ i - q b - @ r j @ b k P k

This yields:

or

P = T2P'

With this, the total transformation matrix T becomes:

4 1 TiT2p' = Tp'

with:

(2.75)

(2.76)

(2.77)

(2.78)

(2.79)

In this way, it is easy to couple all reduced components again.

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Base excitations and effective masses

3.1 Introduction

A dynamic system can be excited in two ways. An excitation can take place in the form of force excitations, i.e. a dynamic force acting on the construction itself. A construction can also be subjected t o a dynamic base excitation, i.e. the construction is excited at the base by an appointed displacement, velocity or acceleration. Base excitations are often met in dynamic studies for space traveling, f.i. satellites and solar arrays.

In this chapter base excitations are considered. The concept of eSfectiwe masses plays an impor- tant part in the dynamic behaviour of constructions subjected to base excitations. The magnitude of an effective mass corresponding to a certain mode shape is a measure of the importance of that mode.

3.2 Base excitation mass-spring system

The mass-spring system with mass m, stiffness k and damper b is excited at the base by a displace- ment u(t) . The response of the mass is the displacement &). This is illustrated in Figure 3.1.

1 T

base

Figure 3.1: Base excitation mass-spring system

20

The equation of motion is:

mi$) + b (4(t) - .;l(t)) + i (q ( t ) - u( t ) ) = 0

The relative displacement of m is:

= d t ) -

This gives for the equation of m o t h :

mqt) + b&t) + k q t ) = -rnG(t>

The reaction force, exercised at the base by the mass-spring system is:

FR(t) = -kS(t) - b&) = mi$) (3-4)

Suppose a harmonic base excitation? which will result into a harmonic response of the mass-spring system, so:

u( t ) = u(w)ejwt (3-5)

d ( t ) = b(w)ejwt (3.6)

(-u2 + 2jCwow + w,2)b(w) = 0 2 U ( w ) (3.7)

Substitution of (3.5) and (3.6) in (3.3) with the introduction of wo2 = k / m and C = b/2& gives:

With the introduction of the frequency response function

The absolute displacement i ( w ) = &(u) + U(w) now becomes:

The absolute acceleration 4" then becomes ;(o) = -w24(o) , so:

&) = -o2 [ 1 + (E) 2 ~ ( w ) ] U(w)

The reaction force p'(u), exercised at the base as a result of U(w) is k ~ ( w ) = m$(w) , so:

(3.10)

(3.11)

(3.12)

The latter equation is of great importance for the concept of effective masses.

21

3.3 Effective masses

Consider the equations of motion of a n-dimensional, linear, undamped system:

M q + K 4 = f (3.13)

For the existence of effective masses, i t is necessary that the system cadn make a, motion a s a rigid body. The system can thus be partitioned in a R-set and a E-set, so:

(3.14)

The solution of the equations of motion can be obtained with the help of modal superposition. The displacement 4 can be written as follows:

4=@rl (3.15)

In this, @ is a (n * n) matrix with mode shapes of the system, is a set of n natural coordinates. In case of base excitations, the displacement can be split up in an imposed motion of the R-set as a rigid body and an elastic motion with regard zero. The modal base @ can thus be partitioned

to the R-set, i.e. the dof of the R-set are set t o as follows:

The rigid body modes @R (n * n ~ ) can be calculated just like in chapter 2, namely:

The elastic modes @E (n * n ~ ) can be determined by suppressing all dof of the R-set, so:

with

(3.16)

(3.17)

(3.18)

@EE ( n ~ * n ~ ) a matrix, which can be found by solving the eigenvalue problem:

MEE + K E E ) ~ ~ = 0 (3.19)

Substitution of (3.16) in (3.13), followed by premultiplication with @', yields:

If no external loads are set on the dof of the E s e t ( f E = O) and use of the orthogonality properties, the equations of motion become:

(3.21)

with:

22

Mo : mass matrix of the component as rigid body with regard t o the R-set (TLR * n ~ ) ;

m E : diagonal matrix with modal masses (TLE * n ~ ) ;

WE : diagonal matrix with eigenvalues of the E-set dof ( n ~ * n ~ ) ;

FR : column of reaction loads as a result of an imposed displacement in the R-set ( n ~ * 1)

Now, two equations are obtained:

M o i j ~ + L T e ~ = FR (3.22)

Equation (3.23) represents a set of uncoupled differential equations of relative motions of the system with regard t o the base, subjected to generalized inertia forces. Equation (3.22) permits to evaluate the reactions at the base as function of the imposed displacement and also elastic displacements. For lightly damped constructions, a modal viscous damping is introduced, denoted with ”k” for each mode shape, so (3.23) becomes:

(3.24) 2 mEkeEk f 2<krnEkwEk+Ek f W ~ ~ m E ~ r l E k = -L~c+R

In the frequency domain, the solution for the natural coordinate ?Ek ( t ) = rjEk ( u ) e j w t is:

or expressed in the transfer function:

with:

Lk = @E,hf@R

Writing (3.22) in the frequency domain:

(3.25)

(3.26)

(3.27)

(3.28)

23

and substitution of (3.26) in (3.28), yields:

where: 7

is the so-called dynamic muss matrix ( n ~ * n ~ ) and where

(3.29)

(3.30)

(3.31)

is the eflectiwe muss matrix ( n ~ * TZR) of the k-th mode. It can be seen that the effective masses are a measure of the modal reaction force as a result of a unit base displacement. So the calculation of effective masses permits t o characterize in a simple way, the contribution of different modes t o the dynamic mass and consequently also to the reaction forces in case of base excitations.

Assume all elastic modes @E have been calculated, then it can be proved (see appendix A) that:

(3.32) k=l

with:

MB = MRR - M R E M Ë ~ M E R (3.33)

MB is called the equivalent mass at the base. This means that every mode with corresponding effective mass represents a portion of the total mass MO - MB.

Equation (3.29) can also be written as:

FR(W) IZ a ( W ) $ R ( W ) (3.34)

and hence i t can be seen that &f(w) can be considered as the transfer matrix between accelerations at the base and corresponding reactions.

24

Chapter 4

Accuracy of the reduced component model

4.1 Introduction

After reduction, the reduced component model must have a certain accuracy in comparison with the original component. Therefore, to ensure that the reduced component is good, a comparison between the reduced component and original component is necessary. The comparison must be done for some essential criteria. In this chapter, the way of comparing and the necessary accuracy of the reduced component model of all these criteria will be discussed.

4.2 Eigenfrequencies

The eigenfrequencies and eigenmodes of an undamped component can be calculated by solving the eigenvalue problem:

(-ui&! + K ) (bk = o ( 4 4

For the original component, this yields n eigenvalues X I , = and n eigenmodes &. For the reduced component, this will give n r e d eigenvalues AhTed = wiTed and lZred eigenmodes &red. In the frequency range of interest, the eigenfrequencies of the reduced component must approximate those of the original component, f.i. with an accuracy of say, less than 3 %. Eigenfrequencies can easily be calculated by f = 4 1 2 ~ .

4.3 Mode shapes

The comparison of mode shapes of the reduced component with those of the original component is more difficult, because the reduced component has no longer physical dof. However, a comparison can still be done by rescaling the reduced eigenmodes with the help of the following retransforma- tion:

&red = T(bk, ,d (4.2)

In this way, i t is possible t o retransform the matrix Qred (n,,d * nr,d) with da ta of the reduced coordinates t o the matrix (n * nred) with da ta of the physical coordinates. So, a comparison

25

between the mutual data of quantifying the comparison between mode shapes and the check for orthogonality.

and @ can be done. Several techniques have been developed for

4.3.1 Modal Assurance Criterion

One of these techniques is referred t o as the Modal Assurance Criterion (MAC) and this provides a rriezsiire ~f the !east squares deïi2tiori of the poirits from the straight line correlation. This is defined by:

and is a real scalar, even if the mode shape data are complex and is lying between O and i. If q5z = a& , this will result in:

MAC(z , y) = 1

When eigenmodes of the original and reduced component must be compared, one must take the mutual modes in ip and @Ted. I t is difficult to say which value the MAC should take in order t o guarantee good results, because this depends on the size of the modes. However, i t can be said that a value in excess of 0.9 should be attained t o get a good approximation of an eigenmode [i].

4.3.2 Cross-orthogonality check

If one wants to check the orthogonality of reduced eigenmodes with regard t o the original eigen- modes, the correlation matrix for cross-orthogonality can be used. This is defined by:

c c = QTMQTed ( n r e d * n r e d ) (4.4) In this, @ is the (n t matrix with the first n r e d eigenmodes because only these eigenmodes are of importance. The goal is for the diagonal terms of the correlation matrix for cross-orthogonality t o be greater than 0.9 and the off-diagonal terms to be less than 0.2 [7].

4.4 Rigid body motion energy

If a system has rigid body modes, then the system has eigenmodes for which the eigenfrequency is equal t o zero. So, a rigid body mode is defined by:

K$R, = o (4.5)

Premultiplication of (4.5) with $4gk, gives the elastic energy V of the component subjected to a motion as rigid body, also called the rigid body motion energy :

1 vr, =

In theory, the rigid body motion energy must be equal to zero (Vr, = O). However, during matrix operations in the computer, rounding errors will occur. So, the rigid body motion energy will not exactly be zero. Therefore, it is required that:

V,=S<&

For example, E = is required. The value of E depends on the size and the complexity of the model. For reduced components, E may have a higher value because after reduction of the mass- and stiffness matrix, larger rounding errors are allowed. So, f.i. E = is required.

26

4.5 Frequency- and impulse responses

The quality of the reduced component can be assayed by looking a t its response characteristics. If a system is excited by a sine-waving signal, the frequency response function can be used t o analyse system responses. For an undamped system, this can be written as follows:

4k4Z (Xk - Lù2)

H ( w ) = (-w2M + = k=i

(4.7)

where #k is normalized on the mass matrix. In the frequency range of interest, the frequency responses of the component must be produced by the reduced component as well.

For systems excited by a transient signal such as an impulse, it is usual to consider the time domain for analysing responses. In the investigation, the impulse response function is used. For undamped systems, this is defined by:

h(t) = Q> Sin R-'Q>*

with:

Sin =

sin w1 t sin w2t

sin wnt

and where Q> is normalized on the mass matrix. Also here, it is clear that impulse responses must be produced by the reduced component as well.

Because the reduced model will have nb artificial eigenfrequencies (> f,), it can happen that the reduced impulse response still shows these high frequencies, where the original impulse response does not. This would mean that the reduced model does not satisfy. However, because in practice always some damping is present, the reduced model will neither produce these high frequencies in the impulse response.

4.6 Effective masses and reaction forces

The concept of effective masses has been completely explained in chapter 3 . It was found that the effective masses has its objective t o determine the most important modes for a system subjected to base excitations. For the original component, this yielded n E = n - n R effective masses. After reduction, there will be nEred = n,,d - nR reduced effective masses left. It is obvious that the most important effective masses of the system are still be supplied with a certain accuracy, f.i. say, less than 10 %.

In equation (3.29), the magnitude of the reaction force as a result of a base excitation was derived. In the frequency range of interest, the reaction forces of the reduced component must approximate those of the original component.

27

Chapter 5

Application: reduction of a multi-dof discrete system

5.1 Introduction

In this chapter, an application of the several reduction methods t o a multi-dof discrete system will be discussed. Therefore, the mass-spring model of appendix B will be considered. This model can be considered as one component with one boundary node (or interface node). The dof q are the vertical displacements of the masses. The linear, undamped component has in total 100 dof (n = 100) from where one is a boundary dof ( n b = i). Because the component can move as a rigid body, the component model has one rigid body mode (nR = 1). In case of base excitations, mass ml will be considered as base. This means that the imposed displacement at the base is 41 = qR.

The frequency range of interest is chosen from zero to 100 Hz. For the C B (Craig-Bampton) reduction, a cut-off frequency of 100 Hz results in a reduced model of 14 dof (13 fixed-interface eigenmodes + 1 constraint mode) and for Rubin, this also yields a reduced model of 14 dof (12 elastic free-interface eigenmodes + 1 rigid body mode + 1 flexibility mode). So, for this example, both reduction techniques will give the same number of dof for the reduced component model. Also the Guyan reduction is executed. To compare the Guyan method with the other methods, also 14 external dof are chosen. To show the dependence of the choice of the external dof, two Guyan reductions will be done; namely one with external dof divided regularly along the system and the other one with more external dof according to the system becomes more flexible. In future, these reductions will be denoted as Guyanl and Guyan2 respectively.

In the next section, comparisons between the reduced component model and the original com- ponent model will be done for all items discussed in chapter 4.

5.2 Results

Eigenfrequencies

Table 5.1 shows the eigenfrequencies of the reduced component model and the corresponding eigen- frequencies of the original component modei. This is done for all reduction techniques. In Table 5.2 the relative errors are given. From Table 5.2 it can be seen that for C B and Rubin the relative errors are very small up to the cut-off frequency.

28

no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Guyanl 0.000 10.884 18.988 26.245 38.311 47.585 56.410 68.748 78.470 82.019 112.187 125.453 148.922 188.088

Original 0.000 10.738 18.618 25.157 36.360 45.794 49.118 61.235 66.128 73.048 73.852 85.390 97.324 101.649 110.334

Guyan2 0.000 10.786 18.993 25.676 37.429 50.261 52.524 65.435 68.748 80.531 86.910 98.184 117.897 140.154

no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

CB

Guyanl 0.000 1.363 1.989 4.327 5.365 3.911 14.847 12.268 18.664 12.280 51.908 46.918 53.017 85.037

0.000 10.738 18.624 25.161 36.364 45.953 49.137 61.236 66.128 73.167 74.749 85.416 97.721 115.631

Rubin 0.000 10.738 18.618 25.15? 36.360 45.794 49.118 61.235 66.128 73.048 73.852 85.390 97.324 148.407

Table 5.1: Eigenfrequencies [Hz] of the original and reduced component model

Guyan2 0.000 0.452 2.013 2.064 2.940 9.754 6.935 6.858 3.962 10.243 17.682 14.983 21.140 371881

CB 0.000 0.005 0.034 0.016 0.010 0.346 0.040 0.001 0.000 0.162 1.215 0.030 0.408 13.755

Rubin 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

46.000

Table 5.2: Relative errors [%] in the eigenfrequencies of the reduced model

29

It is trivial that for Rubin these relative errors are exactly equal t o zero (see section 2.5). The Guyan reductions both give bad results if an accuracy of, say 3 9% is desired, although the second reduction gives somewhat better results than the first one. Note that all relative errors are posi- tive, which means that all eigenfrequencies of the reduced model are upper-bounds for the exact eigenfrequency; this fact is inherent in the use of the Ritz method.

no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode shapes

Guyanl 1.000 0.999 0.994 0.984 0.990 0.958 0.779 0.000 0.000 0.219 0.000 0.048 0.002 0.001

In Table 5.3 the MAC-values are given for the comparison of the i-th eigenmode of the reduced component model with the i-th eigenmode of the original model. I t can be seen that up t o the

Guyan2 1.000 1.000 0.999 0.998 0.992 0.550 0.558 0.971 0.996 0.808 0.695 0.823 0.481 0.016

CB 1.000 1.000 1.000 1 .o00 1.000 0.999 1.000 1.000 1.000 0.914 0.902 0.999 0.925 0.725

Rubin 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.455

Table 5.3: MAC-values eigenmodes original and reduced component model

cut-off frequency, both the CB and Rubin reduction give a good value of the MAC (> 0.9). Again, Rubin gives the best results (MAC = i), because all eigenmodes have been calculated exactly. Above the cut-off frequency the MAC gives bad results. Looking at the Guyan values, bad results are found. The second Guyan reduction gives good results up to the 9-th mode except the 6-th and 7-th mode, while for the first Guyan reduction even worse results are found. This reduction gives reasonable results up to the 6-th mode, above it it is really bad. Especially the MAC-values of the 8-th and 9-th mode are strange. These values (0.000) seem t o indicate a comparison of different eigenmodes.

Therefore, the correlation matrix for cross-orthogonality can be used t o see if the right eigen- modes are compared. Because large matrices ( 1 4 ~ ~ 14) will result, these are represented in appendix C for each reduction method. These matrices show good correlations (> 0.9) for the diagonal terms up to the cut-off frequency and bad correlations (< 0.2) for the off-diagonal terms for Rubin and CB. The relative high value of Cc(ll, 10) and Cc(iO, 11) for CB does not mean that different modes have been compared. But the eigenfrequencies corresponding t o both modes are lying so close together (see Table 5.11, tha t these eigenmodes also resemble each other. From the correlation mztrix of the first Guyan method it can be concluded that the 8-th eigenmode of the reduced model corresponds t o the 9-th eigenmode of the original model because Cc(9, 8) = 1.00. For higher modes it is difficult t o find out which eigenmodes correspond. The second Guyan method gives somewhat better results for correlating modes.

30

When comparing eigenmodes of the reduced model with those of the original component model, difficulties can appear if a Guyan reduction is used. This strongly depends on the choice of the external dof. As far as that goes, the second Guyan reduction is better than the first one. However, in the frequency range of interest only the Rubin and CB reductions satisfy.

Original Guyanl Guyan2 C B O. 186e- 1 O O. 255e- 1 O O. 742e- 1 O O. 14 7e-22

Rigid body motion energy

Rubin O. 327e-08

To check the quality of the reduced component model when this moves as a rigid body, the rigid body motion energy is calculated (see Table 5.4). All reduction methods give a sufficient low value

Table 5.4: Rigid body motion energy original and reduced model

of the rigid body motion energy ( S 5 ). The extremely low value for CB can be made clear because the constraint mode used in the transformation matrix T, is equal t o the rigid body mode of the model. Rubin gives a somewhat higher value. This is due t o rounding errors which appear after transformation Tz, which supplies the boundary dof again.

Frequency- and impulse responses

To look at response characteristics, frequency- and impulse responses can be used. In this ap- plication the response QSO as a result of an excitation in qb = q1 is considered. In Figure 5.1 - 5.4 the FRF of the reduced component model is compared with that of the original model for Guyanl, Guyana, C B and Rubin respectively. In here, the solid line represents the original model and the dashed line represents the reduced model. In Figure 5.5 - 5.8 this is done for the impulse response function. From the original FRF, i t strikes that for mode 9 no resonance can be seen. This is due to the fact that mso does not move in that mode.

Figure 5.1 and 5.2 show that the Guyan reductions give a reasonable result up t o about 40 Hz. Up there, the frequency responses are bad. Especially the first Guyan reduction shows a deviant characteristic with regard t o the original characteristic. This is due to the exchange of eigenmodes after reduction, what is not t o foresee. As can be seen in Figure 5.3 and 5.4 the CB and Rubin reductions yield very good results up to the cut-off frequency. For CB very small deviations appear in the high-frequency domain, because eigenmodes are not approximated so well any longer here.

The impulse response plots of Figure 5.5 and 5.6 show that both Guyan reductions are bad. Looking at the C B and Rubin reduction plots in Figure 5.7 and 5.8, it can be concluded that these reductions are good; amplitudes and frequencies correspond t o each other.

31

SO 1 0 0 1 2 0 140 160 1 8 0 200 f [Hzl

Figure 5.1: Frequency response plot H(80,l) original and Guyanl reduced model

Figure 5.2: Frequency response plot H(80,l) original and Guyana reduced model

Figure 5.3: Frequency response plot H(80,l) original and CB reduced model

32

f [Hzl

Figure 5.4: Frequency response plot H(80,l) original and Rubin reduced model

-1

- E

-- o s.3

I -

I : ,

O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -1

t [SI

, - . . . . . . . . . . . . . . . .:. . . . . . . . . . . . . . - ! I - l . ,

i\;

Figure 5.5: Impulse response plot h(80,l) original and Guyanl reduced model

. . . . . ~~ . . . . . . . . . .

. . . . . . . . . . z n s : : . . . . . . . . . . . . . . . . . . .

i ! ' ! . . . . . . . . . . . . . . . . . . . . ? ! . / ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ? ! a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.6: Impulse response plot h(80,l) original and Guyan2 reduced model

33

.5 x 1 o- - - E

o ï

I

m

O

-0.5

O 0.05 o. 1 O 1 5 0.2 O -1

t [SI

I !5 0.3 0.35 0.4

Figure 5.7: Impulse response plot h(80,l) original and CB reduced model

1

O I

E I 1 -- 8 F

-0

1 0.15 1

p-.4 0.25

3

Figure 5.8: Impulse response plot h(80,l) original and Rubin reduced model

Effective masses

The effective mass matrix of the original and reduced component model for each mode has been calculated. Because the system can only move in one direction, the effective mass matrix consists of one number. In Table 5.5 these values are reported. The table also shows the sum of the ef- fective masses. Because only the first effective masses are of importance, not all effective masses ( n ~ = 99) are given here. In Table 5.6 the relative errors are given. From the first two columns of Table 5.5, it can be found that the first mode shape is the most important mode shape in case of base excitations. Then successively, mode 2, mode 4, mode 7 and mode 3 are the most important modes. For other (higher) modes, the effective masses becorrie smaller, so these modes have no or little contribution to the total reaction force exercised a t the base. Also, i t can be seen that the sum of the effective masses is equal t o the total mass as rigid body minus the equivalent mass at the base (Mo - MB = 100 - 1 = 99).

34

no. 1 2 3

5 6 7 8 9 10 11 12 13 14 15 16 17

-

A T

no. 1 2 3 4 5 6 7 8 9 10 11 12 13

sum __

Guyanl 0.389 1.621 2.651 2.067

28.371 36.850 7.544

100.000 > 100 > 100 69.861 67.836 > 100

Original 76.399 9.125 1.532 5.622 0.815 0.038 1.930 0.228 0.000 0.002 0.090 0.887 0.037 0.087 0.357 0.116 0.109 99.000

Guyanl 76.697 8.977 1.573 5 .?38 0.584 0.052 2.075 0.000 0.012 0.959 0.152 0.285 0.279

97.385

Guyan2 76.291 9.068 1.345 4.287 1.571 0.024 0.000 0.000 0.071 0.400 0.005 0.001 0.000

93.065

CB 76.399 9.125 1.532 5.622 0.815 0.038 1.930 0.228 0.000 0.002 0.090 0.887 0.037

96.706

Rubin 76.399 9.125 1.532 5.623 0.817 0.038 1.885 0.284 0.000 0.002 0.050 1.101 0.749

97.605

Table 5.5: Effective masses [kg] of the original and reduced component model

Guyan2 0.142 0.623 12.154 23.750 92.705 36.926 99.979 100.000 > 100 > 100 94.497 99.939 99.964

CB 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Rubin 0.000 0.000 0.005 0.006 0.151 0.170 2.335

24.422 99.875 0.350

43.771 24.140 > 100

r- - - lable 5.6: Absolute value relative errors [%o] in the effective masses of the reduced model

35

It is obvious that the most important effective masses must be reduced most accurately. Cup- pose an accuracy of 10 % for the five most important effective masses (mode 1, 2, 3, 4 and 7) is required. Then the first Guyan reduction, the CB reduction and the Rubin reduction all satisfy (see Table 5.6). The second Guyan reduction only gives good results for the first two mode shapes. The C B reduction gives exact reduced values for all modes (relative error = 0.000). This is because the CB method uses fixed-interface eigenmodes in the calculation of reduced models, while in the calculation of effective masses also fixed-interface eigenmodes are needed. If higher modes are also of importance (f.i. mode 12) or if a higher accuracy is desired (f.i. a%), then this reduction method is preferred. However, the relative error will be zero for CB if and only if the component model has one boundary dof. If more boundary dof are present, then the above does not hold any longer. This means that the CB reduction is not the best reduction as a matter of course.

Note that the sum of the reduced effective masses does not say anything about the quality of the reduced model. If this was so, then the first Guyan reduction would be better than the CB reduction and this is absolutely not true!

Reaction forces

After effective masses of the component model are known, also reaction forces exercised a t the base can be calculated in the frequency domain. In Figure 5.9 - 5.12 the reaction forces of the original and reduced component model have been plotted. In the frequency range of interest only the CB and Rubin reductions satisfy, where the CB reduction is the best of both. The second Guyan reduction only satisfies up t o about 40 Hz. The first Guyan reduction is somewhat better (up to 60 Hz).

1

1

z r. - Y

1

1

Figure 5.9: Reaction force plot original and Guyanl reduced model

36

1

1

1

1

Figure 5.10: Reaction force plot original and Guyan2 reduced model

Figure 5.11: Reaction force plot original and CB reduced model

Figure 5.12: Reaction force plot original and Rubin reduced model

37

Chapter 6

Conclusions

In this last chapter, some major conclusions are drawn. In the investigation, three CMS methods have been considered to obtain a reduced component model, namely the method of Guyan, the method of Craig-Bampton and the method of Rubin. To check the quality of the reduced component model, the model has been assayed on the basis of some essential criteria.

It can be concluded that the Guyan reduction is the worst reduction method of all. In the frequency range of interest, both investigated Guyan reductions do not satisfy ail criteria discussed. Of course, the Guyan results can be improved by taking more external dof. But this would mean that the number of dof of the reduced model will be larger and so no optimum reduction will be achieved.

The reduced component model satisfies all essential criteria. So one of these reduction methods must be used. The Craig-Bampton and Rubin methods yield results which do not differ much from each other. A choice can still be made if one knows in what form excitations take place. In case of force excitations, the Rubin method is preferred because frequency and impulse response functions are reduced somewhat more accurately than when the method of Craig-Bampton is used. In case of base excitations, effective masses are used t o characterize the structure dynamics. In that case and if the component only has one boundary node, the Craig-Bampton method should be used because then effective masses are reduced exactly.

It should be noted that in the investigation, only one component has been considered. The main contractor, who is responsible for the dynamic behaviour of the total system, must couple all reduced component models which are delivered by different subcontractors, t o obtain a reduced system model. The accuracy of the total reduced system is not examined in this investigation. Therefore, one is referred t o other literature [i], [4], [lo].

The Craig-Bampton and Rubin reduction methods yield much better results.

38

Bibliography

[i] Kraker, A. de, Numeriek-experimentele analyse van dynamische systemen. Collegediktaat 4.668, Eindhoven, 1992.

[Z] Campen, D.H. v., Kraker, A. de, Het dynamisch gedrag van constructies. Collegediktaat 4.552, Eindhoven, 1984.

[3] Wijker, J.J., Ruimtevaartconstructies. Collegediktaat Lucht- en Ruimtevaarttechniek, Vakgroep C, TU Delft, Delft, 1993.

[4] Fey, R.H.B., Steady-state behaviour of reduced dynamic systems with local nonlinearities. Proef- schrift TUE, Eindhoven, 1991.

[5] Géradin, M., Rixen, D., Mechanical vibrations: theory and application t o structural dynamics. Wiley, Chichester, 1994.

[6] Pestel, E.C., Leckie, F.A., Matrix methods in elastomechanics. McGraw-Hill, London, 1963.

[7] Ricks, E.G., Guidelines for loads analyses and dynamic model verification of shuttle cargo ele- ments. Marshall Space Flight Center, Alabama, 1991.

[SI Imbert, J.F., Mamode, A., Effective mass concept, helpful t o characterize structure dynamics with base excitation. CNES, Toulouse, 1978.

[9] Wijckmans, M.W.J.E., Dynamic analyses of idealised solar panels with a bilinear snubber. WFW-rapport 95.071, Eindhoven, 1995.

[lo] Vorst, E.L.B. v.d., Component Mode Synthese. TNO-rapport BI-91-146, Rijswijk, 1991.

39

Appendix ,4

Derivation sum effective masses

From (3.21) it is known that:

L = ~ D E M Q R

with:

s = -K&KER

For the sum of effective masses holds: n E n E

L k L h - L T ~ - I L xMEk=x--- E h= 1 k= 1 m E k

so:

= [IRR ST] [ MRR MRE ] [ ORR ORE -1 T ] [ MRR M R E ] [ I F ] MER MEE OER Q E E ~ E @EE MER MEE

Substitution of

in (A.3) yields:

From (3.21) it is also known that:

Mo = @ R b f @ ~

40

From (A.5) and (A.6), it can easily be derived that:

k=i

41

Appendix

Multi-do€ discrete system

I T

> I I I I I

I I I I

I

I I I

n = 100 T

9 = [9, .929---i41001

qb= 41

mi= 1 kg ki= (80-i) *10 N/m i = 1, ..., 79

ki= (170-i) *10 N/m i = 90, ..., 99 k=5*1O3N/m

i = 1, ..., 100 5

5

5 ki= (160-i) * I O N/m i = 80, ..., 89

base

42

Appendix C

Correlation matrices for cross-ort hogonality

Guyanl reduction:

c, =

1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.02 0.01 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 1.00 0.05 0.01 0.00 0.02 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.06 0.99 0.00 0.01 0.04 0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 1.00 0.03 0.00 0.00 0.01 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.02 0.98 0.16 0.00 0.02 0.03 0.01 0.00 0.00 0.00 0.00 0.01 0.03 0.07 0.02 0.19 0.88 0.00 0.04 0.15 0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.07 0.04 0.02 0.40 0.00 0.12 0.46 0.06 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.04 0.01 0.00 0.01 0.00 0.67 0.47 0.00 0.01 0.02 0.00 0.00 0.01 0.01 0.02 0.01 0.01 0.01 0.00 0.72 0.57 0.01 0.03 0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.02 0.06 0.00 0.07 0.40 0.20 0.22 0.05 0.01 0.00 0.00 0.00 0.00 0.03 0.01 0.11 0.00 0.02 0.08 0.64 0.24 0.04 0.02 0.00 0.00 0.00 0.00 0.02 0.02 0.04 0.00 0.01 0.02 0.69 0.49 0.07 0.03

Guyan2 reduction:

c, =

1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 1.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 1.00 0.02 0.00 0.01 0.00 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.02 1.00 0.06 0.01 0.01 0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.01 0.01 0.06 0.74 0.64 0.03 0.00 0.01 0.07 0.02 0.01 0.00 0.00 0.00 0.00 0.01 0.03 0.65 0.75 0.08 0.00 0.01 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.07 0.04 0.99 0.00 0.07 0.03 0.01 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.05 0.04 0.05 0.00 0.90 0.34 0.00 0.02 0.02 0.00 0.00 0.01 0.00 0.02 0.06 0.11 0.05 0.00 0.36 0.83 0.09 0.01 0.01

0.00 0.00 0.01 0.01 0.01 0.02 0.07 0.05 0.00 0.03 0.21 0.02 0.69 0.09 0.00 0.00 0.01 0.02 0.00 0.06 0.08 0.02 0.00 0.08 0.29 0.21 0.36 0.13

û.00 0.00 0.00 0.01 0.02 0.01 0.03 0.01 0.00 0.02 0.13 0.91 0.0: 0.04

43

Craig-Bampton reduction:

c, =

1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 i . û û û.û2 U.ÛÛ û.ûû Û.ûû û.ûi û.ûû û.Oû Û.û% 0.00 0.00 0.00 0.00 0.00 0.02 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.96 0.29 0.01 0.01 0.04 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.00 0.29 0.95 0.01 0.01 0.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 1.00 0.01 0.02 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.01 0.05 0.01 0.96 0.23 0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.00 0.00 0.02 0.11 0.03 0.27 0.85

Rubin reduction:

c, =

1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.67

44