37L

   
     

IHESIS

  

LlBRAflV
Michigan State
University

 

This is to certify that the

dissertation entitled

DIRECT GENERATION 0F EFFICIENT LOW-ORDER
MODELS FOR LINEAR MIXED-STRUCTURE SYSTEMS

presented by

William Frederick Resh

has been accepted towards fulfillment
of the requirements for

 

 

Ph.D. degree in Mechanical
Engineering

% flake/7
//

Major professor

 

Date {/[flL/Z/ 222/ /¢/%
fl

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

 

 

 

 

BEIURNING MATERIALS:
)VIESI.} Place in book drop to
LJBRARJES remove this checkout from
_3.___ your record. FINES will

 

 

be charged if book is
returned after the date
stamped below.

 

 

 

 

 

 

 

 

 

 

 

 

DIRECT GENERATION OF EFFICIENT LOW-ORDER

MODELS FOR LINEAR MIXED-STRUCTURE SYSTEMS

By

William Frederick Reah

A DISSERTATION
Submitted to
flichigan State University
in partial fulfillment of the require-ants
for the degree of

DOCTOR OF PHILOSOPHY

Department of lechanical Engineering

1984

ABSIRACT

DIRECT GENERATION OF EFFICIENT'LOI-ORDER
MODELS FOR LINEAR MIXED-STRUCTURE SYSTEMS

By

William Frederick Resh

Many physical systems consist of lumped-parameter subsystems
coupled by continuum elements. For example. in an automotive
drivetrain the engine. transmission. differential. and wheels could be
represented as lumped-parameter systems and the drive shaft and axles
as continuum elements. In modelling these mixed-structure systems.
the continuum elements give rise to partial differential equations and
the discrete subsystems give rise to ordinary differential equations.
Because of the coupling between the subsystems, analytical solutions

are unobtaiuable in general.

Typical modelling approaches for these problems discretize the
continuum elements. using large numbers of local variables in order to
get an accurate representation of the entire system's behavior.

Often, because of the model's size, it is very expensive to work with
directly, particularly in an iterative design context. One way to try
to achieve efficiency in the design process for linear systems is to

apply a model-order reduction procedure to the large-order model to

William Frederick Resh
gobtain a lower-order working model. In this work. a procedure is
develcped that permits forming an efficient low-order model directly.
bypassing the formulation of the large-order model and application of
a model-order reduction procedure. A procedure of this type is
especially attractive for use in preliminary design. where constraints
on time and computational resources may prohibit iterative analyses of

large systems models.

The particular type of linear mixed-structure systems to be
exmmined are those having a cascade structure. A procedure is
deve10ped based on parameter information. coupling effects. and
tabulated error results for some prototype cases. Applications of the

procedure are illustrated.

ACKNOWLEDGEMENTS

I would like to thank Ronald Rosenberg for the many timely and

insightful discussions we had, my wife for her support and patience.

and my parents for their support and interest through the years.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES Vi
LIST OF FIGURES viii
Chapter ' -
1. INTRODUCTION 1
1.1. Background 1
1.2. Research Objectives and Problem Restrictions A 1
1.3. Dissertation Organization 9
2. ELEMENTS AFFECTING DIRECT GENERATION 10
2.1. Discretizing the Continuum Elements and 10
Eigenvalue Convergence
2.2. The Prototype Linear Mixed-Structure System 13
2.2.1. Analytical Solutions 13
2.2.2. The Discretized Prototype Problem and 17
Evaluation Model Eigenvalues
2.3. Unit-Parameter Uniform Cascades 21
2.3.1. Clustering Behavior of Eigenvalues 21
2.3.2. Average Eigenvalues and Error Behavior 27
3. NON-UNIFORM CASCADE MODELLING PROCEDURE 31
3.1. Description of Procedure '" 31 '
3.2. Application Examples 32
3.3. Behavior of the Eigenvectors 44
4. MORE GENERAL SYSTEM MODELS 49
4.1. Relaxation of Restrictions 49

iv

4.2. Modifying the Procedure under Certain
Conditions
5. SUMMARY AND CONCLUSIONS

Appendices
A. SOLUTION T0 THE.FINITE DIFFERENCE PROBLEM

B. DIRECT SOLUTION TO THE BOUNDARY VALUE PROBLEM FOR
THE PROTOTYPE LINEAR MIXED‘STRUCTURE SYSTEM

C. ‘IHE ONEPSEGMENT ERROR TABLES

BIBLIOGRAPHY

Page

64
70

72

75

78

96

Table

2.1.

2.2.
2.3.
2.4.

3.1.
3.2.

3.3.

3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.

4.7.

. LIST OF TABLES

Comparison of Analytical Eigenvalue Solutions

Comparison of Curvefit Results and Analytical

Solutions

Eigenvalues for 3-Segment Unit-Parameter Uniform

Cascade

Average Eigenvalue Errors for Unit-Parameter
Uniform Cascades

Parameters
Eigenvalue

Eigenvalue

for Example 1
Data for Example

data for Example

Identical Discretizations

Parameters
Eigenvalue
Parameters
Eigenvalue
Parameters
Eigenvalue
Parameters
Eigenvalue
Parameters
Eigenvalue
Parameters
Eigenvalue

Parameters

for Example 2
Data for Example
for Example 3
Data for Example
for Example 4
Data for Example
for Example 5
Data for Example
for Example 6
Data for Example
for Example 7
Data for Example

for Example 8

vi

when Shafts have

Page

16

21..-
24
28

34
35

36

37
38
4o
41
43
44
53.
53
54
56
s7
58

58

Table Page

4.8. Eigenvalue Data for Example 8 60
4.9. Eigenvalue Data for Example 9 62
4.10. Parameters for AAB Example 66
4.11. Eigenvalue Data for AAB Example 66'
4.12. Evaluation Model Eigenvalues for Components of AAB 67
Cascade
4.13. Eigenvalue Data for AAB Example using Modified 68
Procedure
4.14. Eigenvalue Data for AAB Example with a Model 69
Quality Index Based on the First Ten Eigenvalues
C.1.1. First Eigenvalue 81
C.1.2. First Eigenvalue 82
C.1.3. First Eigenvalue 83
C.2.1. Second Eigenvalue 84
C.2.2. Second Eigenvalue 85
C.2.3. Second Eigenvalue 86
C.3.l. Third Eigenvalue _ _ 87
C.3.2. Third Eigenvalue 88
C.3.3. Third Eigenvalue 89
C.4.1. Fourth Eigenvalue . _90h
C.4.2. Fourth Eigenvalue 91
C.4.3. Fourth Eigenvalue 92
.c.s.1. Fifth Eigenvalue " ' 93 '
0.5.2. Fifth Eigenvalue 94
C.5.3. Fifth Eigenvalue 95

vii

Figure
1.1.
1.2.

1.3.

1.4.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
3.1.
3.2.
3.3.
3.4.
3.5.

3.6.

3.7.

LIST OF FIGURES

Typical Mixed-Structure System
Rotational Positioning Mechanism

Breakdown of LMS System into Mathematical
Components

Form of Discrete Subsystems

Uniform Continuum Element

Discretized Uniform Continuum Element

Prototype LMS System

One-Segment LMS System with Dependent Mass Element
Discretized Prototype LMS System

Least-Squares Curvefit to Eigenvalue Data
Cascade Systems of Increasing Size

Tho-Segment Cascade

Tho Identical One-Segments

Format of the One-Segment Error Tables

Form of Cascade System for Examples 1, 2, and 3
Low-Order Model for Example 1

Low-Order Model for Example 2

Form of Cascade System for Example 4

Low-Order Model for Example 4

Overlay of Low-Order and Evaluation Model
Eigenvectors for the BAA Model

Overlay of Low-Order and Evaluation Model
Eigenvectors for the ABA Model

viii

Page

11
11
14
14
18
22
23
25
25
30
33
33
.39“.,
42
42 '

46

47

Figure

3.8.

4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9.
4.10.
4.11.

4.12.

Overlay of Low-Order and Evaluation Model
Eigenvectors for the AAB Model

Mass-Spring Subsystem
Mass-Spring-Mass Subsystem
Spring-Mass-Spring Subsystem
Spring-Mass-Spring-Mass Subsystem
First-Oscillator-In Illustration
System of Example 5

System of Example 6

System of Example 7

System of Example 8

Discrete Subsystem from Example 8
Low-Order Model for Example 9

Physical Lumping Discretization Having Mass
Elements at Both Ends

ix

Page

48

so
so
so'
so
52
52
55
55
59
59
63

65

1. INTRODUCTION

1‘; 0 round

Many dynamical problems of practical interest are linear
mixed-structure systems. That is, they consist of linear subsystems
that are inherently discrete coupled by linear continuum elements.

- See Figure 1.1 for an abstract representation. The continua are shown
with each one coupling two discrete subsystems. which is the most
common case in engineering systems. Consider. for example, the
rotational positioning mechanism shown in Figure 1.2. In the process
of modelling the dynamics of such a system, the engineer is faced with
the problem of using finite dimensional models of the two shafts. The
questions emerge in this problem. The first of these is:. How should
one discretize each shaft in order to get a reasonable model of it?
Contained in this question is the matter of the type of discretization
to use (eg.. finite elements) and the number of elements or
degrees-of-freedom to use for each shaft. The second question is:

How do these discretized shaft models work in the combined system
model?

For the sake of example. assume that all the discrete-dynamic
e1ements-- inertias 31. 12, and 13. and springs k1 and k2 -- have unit
parameters. Assume that both shafts are uniform and identical except
that shaft 1 is twice the length of shaft 2. Also assume that modal
approximations are to be used to discretise the two shafts. Just
looking at the shafts. one might reasonably discretixe them using
twice as many modes for shaft 1 as for shaft 2 since they are .

identical except for shaft 1 being twice as long. Suppose one uses

 

 

Discrete V\\\\ Continuum (//’rDiscrete

 

 

 

 

 

Subsystem . Subsystem
Continuum Continuum ..
Discrete Discrete

 

Subsystem4//// \\\\¥Subsystem

Figure 1.1 Typical Mixed-Structure System

 

 

 

 

 

Ill:

JV

 

 

Figure 1.2 Rotational Positioning Mechanism

4
four modes for shaft 1 and two modes for shaft 2. How well will these

shaft models work in the combined system model? What if J3 and ‘prin‘
12 have a frequency near the third mode of shaft 2? How will using
only two modes for shaft 2 affect the resulting system model?

Assuming th‘t 33 and spring k2 have a frequency near the third mode of
‘h‘ft 2. 'h‘t if I3 is much less than or much greater than the inertia
of shaft 2? How will using only two modes for shaft 2 affect the
quality of the resulting system model? So even if the discretized
continuum representations are assumed to be good approximations of the
continuum elements. some thought must be given to how the 1 ‘
approximations will be interacting with the discrete dynamic elements
in the mixed-structure system.

Above it was stated that there is a need to discretise the '
continuum elements in a mixed-structure system. This need to
discretize the continuum elements results from the coupling that
exists between the partial differential equations (PDEs) representing
the continuum elements and the systems of ordinary differential
equations (ODEs) representing the inherently discrete subsystems. As
shown in Figure 1.3. the continuum elements in the linear
mixed-structure (LMS) system give rise to partial differential
equations (PDEs) and boundary conditions (BCs). The discrete
subsystems yield ordinary differential equations (ODEs). Because of
the physical coupling in the LMS system. variables from the PDE
representations appear in the ODEs and lumped parameter variables from
the ODEs appear in the boundary conditions. coupling the two types of
. representations. In a few degenerate cases, some of which are

discussed in Timoshenko [1]. the behavior of the discrete.subsystem,

 

   
     

 

 

 

 

PDEs
LMS Continuum Elements
and
J at;
Discrete

Subsystems Coupling

 

 

Figure 1.3 Breakdown of LMS System into Mathematical Components

 

kd mass

 

 

 

Figure 1.4 Form of Discrete Subsystems

6

can be described in terms of the coordinates of the continuum element.
permitting an analytical solution. But usually an analytical solution
is unobtainable. Moreover. the coupling in general has a nontrivial
effect on the subsystems and therefore cannot be neglected. Typically
what is done is to convert the PDE representations of the continuum
elements to ODE representations. so that all the dynamic equations are
ODEs.

Several methods are available to convert the PDE representations.
including finite element analysis. modal approximation. numerical
approximations to the Operators. and a physical lumping approach.
After a discretization technique has been applied. the discretised
. representations of the continuum elements and the inherently discrete
subsystems are then assembled into an ODE representation of the
complete system. To retain sufficient accuracy. large numbers of
variables are used to approximate the continuum elements. As a
result. the complete system model (hereafter referred to. as in
Skelton [2]. as the evaluation model) is often too large or expensive
to work with directly. In addition. one always has to address the
question of "How large is large enough?"

To save computation time in simulations and to facilitate control
system design. the evaluation model's size often is reduced in some
"optimal" fashion using a model-order reduction procedure. These
model order reduction techniques are of two basic types: aggregation
and singular perturbation. Both types require three basic elements.
namely. an evaluation model. a set of trial models. and a model
quality index. As described by Skelton. the model-order reduction

problem is fundamentally that of minimising some model quality index

7

for all models of a given order that are reduced-order models of a
given evaluation model.

Application of either type of model-order reduction procedure is
not a simple task. From Aoki [3.4] the aggregation method is seen to
be a generalization of the modal approximation procedure developed by
Davison [5.6] and others. As such. it typically requires knowledge of
the eigenvalues. and often also of the eigenvectors. of the evaluation
model. The singular perturbation method does not require the
evaluation model eigenvalues and eigenvectors. But it is a nontrivial
problem to formulate the system equations in the format required by
the theory.

Since the aim of the modelling procedure often is to obtain an
accurate system representation of relatively low order. it would be
useful in these cases if this representation could be formulated
directly. bypassing the formulation of the evaluation model and
application of a model-order reduction procedure. Consider. for.
example. the preliminary stages of the design process. Here the
systems engineer quite often has a large number of possible design
configurations to choose from. In addition. available computational
resources may be limited. prohibiting the formulation and reduction of
many evaluation models. Even if computational resources aren't
limited. the engineer's time is. so it is always an advantage to

reduce the number and size of evaluation models to be considered.

1...; 32mm! 9111.29.11”: an_d £_L_ro lam ___st_kestri ion_s

The goal of this work is to develop a set of guidelines that will

help the designer to directly generate efficient working-order models

8
for linear mixed-structure systems. With these guidelines the

designer:

1. specifies a model quality index.

2. breaks the mixed-structure system into segments.haviug one.
one continuum element in each segment.

3. distributes the eigenvalues among the segments.

4. chooses the number of spring-mass lumps to be used to
discretise the continuum elements by examining results for
some prototype cases. and

5. assembles the finite-dimensional model using the results
from step 4.

In this work a model quality index based on eigenvalues will be used.
To uphesize coupling effects and the coupling paths involved while
develOping the guidelines. linear mixed-structure systems examined
here will have cascade structure with discrete subsystems of the form
shown in Figure 1.4. To facilitate comparisons between the evaluation
model eigenvalues and the low-order model eigenvalues. work will be
limited to conservative systems where the continuum elements are
uniform and have both inertia and compliance effects.

So this work will pertain to the modelling of linear
mixed-structure systems having the following properties:

1. systems are cascade with one-dimensional dynamics.

2. systems are conservative.

3. continuum elements are uniform with dynamics described by

second-order equations of the form:

I'a’u/dt’ = d’u/ax’

9

4. discrete subsystems are of the form shown in Figure 1.4. and

5. the model quality index used is based on eigenvalue errors.
In what follows. the particular index used is that the first
I eigenvalues are each within Y percent of the corresponding

eigenvalues from the evaluation model.

143 Dissertation Organization

The remainder of the dissertation is organized as follows. In
chapter 2. section 2.1 discusses discretizing methods for.the
continuum elements and the convergence rate of the method used. In
section 2.2. the prototype linear mixed-structure system is examined.
A new method for solving the problem analytically is develOped. along
with a procedure for generating evaluation model eigenvalues from the
eigenvalues of low-order models. Unit-parameter uniform cascades and
the effect of coupling are discussed in section 2.3. The modelling
procedure for direct generation of low-order models is given in
chapter 3 along with some application examples. Also discussed in
chapter 3 is the eigenvector behavior of these models. In chapter 4 a
requirement on the form of the discrete subsystems used in develOpimg-
the modelling procedure is relaxed. and the usefulness of the

procedure in these cases is verified.

2. FACTORS AFFECTING DIRECT GENERATION OF LOW-ORDER MODELS

2‘; Discretizing‘thg Continuum Elements ggd Eigenvalug Convergence

Having set the evaluation framework. the modelling problem can
now be addressed. There are two factors affecting the direct
formulation of efficient low-order models. They are the strengths of
the couplings between subsystems and the discretization method used‘.
for the individual continuum elements.

There are many types of discretisation techniques available for:
representing continuum elements. Each of these techniques has its own
particular set of advantages and disadvantages with respect to items
of interest such as case of use. convergence behavior. and complexity
of the resulting finite dimensional model. The type of discretization
used for the continuum elements in this work will be a physical
lumping technique. If one considers the continuum element in Figure
2.1. the discretized model of Figure 2.2 results when physical lumping
techniques are applied. In determining the discretixed system's
parameters. the shaft's elastic modulus E. density 9. length h. and
cross-sectional area A are used to define a static stiffness. 1,-EA/h.
- and thO 'h‘ft" li“: ms=pAh. Then. in discretising. the shaft is .

broken into L spring-mass lumps. with each spring having stiffness -

k'kgL. and each mass having mass m=m,/L. The resulting eigenvalue

problem for the system in Figure 2.2 is:

’W'mlllx+k.[l]x=0 (2-1)

Here n is the system natural frequency.[I] is the LxL identity matrix.

10

11

 

V(t)—> E. A. o

 

 

 

 

 

Figure 2.1 Uniform Continuum Element

 

 

V(t)———>eJ\/V\,—IIJV\/\r-"--o..-I\N\,—

 

 

 

 

 

 

L spring-mass lumps

Figure 2.2 Discretized Uniform Continuum Element

 

 

 

 

12
x is the vector of displacements. and [I] is the banded stiffness

matrix given below.

'iL -L o -
-L 2L -L o
[r]: o -L 2L -L o ' ‘ (2-2)
-L 2L -L
L o -L Li

 

 

This system of equations is the same as the finite difference
formulation given below in equations (2-3) and (2-4). That is. if the
values of the index n are substituted in (2-3) and the boundary

conditions (2-4) applied. the system of equations in (2-1) results.
'EAL(°n+1—2en+en_1)/h = o’pAhOn/L (2—3)

eo=o (eLil-eL)/(h/L)=o (2-4)

Solving the finite difference equation for u (see Appendix A) and
using the known solution 5 of the continuum boundary value problem.
one can generate the following expression for the eigenvalue errors in

the discrete approximation.

((0-3)/3)j a (4L/(21-1)n)sin[(23-1)n/2(2L+1)l - 1 (2-5)

18132) e e e IL

Performing a Taylor series expansion of the sine term in equation

(2-5) above results in the error expression below.

(«n-FIE” - -1/(2L+1)- . . . (2-6)

’ So the convergence of the physical lumping discretization is of the

order (l/(2L+1)). or approximately order (1/2L) for large L.

2;; Th; Prototype Linear Miged-Structure System

2.2.1 Apalytigal Solutions

Define the system of Figure 2.3 to be the prototype linear

 

mixed-structure system. Analytical solutions for this case are more
difficult to obtain than for the system of Figure 2.4. whose solution
is discussed in Timoshenko. The difficulty in solving the prototype
linear mixed-structure system results from the appearance of the
independent coordinate y describing the position of the discrete mass.
Jacquot and Soedel [7] and Young [8] have found a solution to this
problem. In their approach. the discrete spring-mass subsystem is

replaced by a harmonic forcing function. Assuming that the
918°3V413°3 ”1 and eigenfunctions 01(x) of the shaft are known. the
forced solution can be obtained in terms of eigenfunction expansions.
Using the displacement impedance for the spring-mass system at the—H.
point of attachment. the forced solution can then be viewed as a
solution to the prototype linear mixed-structure system. .From this

one obtains the frequency relation:

14

 

 

 

 

 

 

\\\l\\\\
r

 

Figure 2.3 Prototype LMS System

 

 

 

 

 

/ %
////

L'. “(to t)

 

 

h

 

 

 

\\\l\ \\ \\

Figure 2.4 One-Segment LMS System with Dependent Mass Element

15
1’kd-aw'l(ta-nau"§§-1¢1'(h"PAuoin""i"“”'° (2’7)

Eere u is the natural frequency of the protdtype LMS system. ”i and

01(x) are the eigenvalues and eigenfunctions of the shaft. and kd gnd
'd are the stiffness and mass of the discrete subsystem.

In this work an alternative solution to the prototype linear
mixed-structure problem is develOped that treats the boundary value
problem directly.(See Appendix B). For the prototype linear

mixed-structure system. the boundary value problem is:

a'ulax’cule’n’e/at3 a’aa/p 1" (2-8)
may"(t)=-kd(y(t)-u(h.t)) (2-9)
u(0.t)=0 EA(3u/ax)'x=h=kd(y(t)-u(h.t)) (2—10)

At a natural frequency of the system. the motions are synchronous and
all amplitudes can be expressed in terms of the amplitude of one of
the points. So with the assumption that y(t)*cu(h.t). c constant. the
coordinate y can be removed from the problem. Solving using the
standard separation of variables technique results in the frequency

relation:

””(‘kd/EA)(u’fld/(kd-u'fld))tan(eh/a) (2-11)

Contrasting the two solutions one can see that the direct

solution to the boundary value problem has some advantages over the

16

solution from Jacquot and Soedel. In particular. one need not know
the eigenvalues and eigenfunctions of the shaft. Both equations (2-7)
and (2-11) require an iterative solution technique. But equation
(2-7) also requires that one truncate the infinite series before
beginning any computation for the natural frequencies. This is a
level of approximation not required in equation (2-11). The two
solutions were compared for a prototype system with elastic modulus
E8100. density 93100. cross-sectional area AF0.01. shaft length hpl.0.
and discrete subsystem parameters kd'l and .dal. For the .h.ft' the

eigenvalues and eigenfunctions are:

“i=in/2 01(x)=sin(inx) i=1.3.5. . .' . ‘ (2-12)

Keeping 10000 modes in equation (2-7). the results shown in Table 2.1

below were obtained.

Table 2.1 Comparison of Analytical Eigenvalue Solutions

Eigenvalue Solution from (2-7) Solution from (2-11)
1 0.6762 0.6762
2 2.117 2.117
3 4.921 4.921
4 7.980 7.980
5 11.086 11.086

One can see that the solution technique develOped here yields results
that agree with those derived from Jacquot and Soedel's technique.

The primary use of these analytical solutions in this work is as a

17

check for the procedure used to generate the evaluation model

eigenvalues for linear mixed-structure systems.

2.2.2 Th; Disgretized Prototype Problem ggd Evaluation Model

 

Elam

Now consider the discretized prototype linear mixed-structure
system in Figure 2.5. The elements labelled k and m are derived from

th' continuum. 'h119 kg and ma denote the discrete subsystem. The '

eigenvalue problem for this system is:

w: [u]x+lxlx=0 (2'13)
[M]=l; 0 (2-14)
0 m 0
0 m 0
0 m 0
0 md -

 

 

18

 

 

 

k k «————-
vm—SLMIM. m ....

Figure 2.5 Discretized Prototype LMS System

 

 

19

m- F21: -k 0
”E 2k 'k 0
O ’k 2k -k 0
-k (k+kd)
0 "kd

 

301° k‘kgL and m'ms/L. Premultiplying by [II]-1

expression:

(2-15)

 

gives the equivalent

-u’[I]x+[A]x=0 (2’15)
Where [A]=[M]’1[x]. (2-17)
[Al-1,,” ’21] -L’ ' (2—13)
-L' 2L’ -L'
‘13 (kd/k,+L)L wad/1:,
0 'kdmslk,md kdms’ks'dJ

 

 

One can see from (2-18) above that the parameters of interest in the

discretised problem are the stiffness ratio kd/ks- the mass :gtia

-‘/-d, and the number of spring-mass lumps. L.

20
If one lets L go to infinity. the eigenvalues of the discretised

problem will converge to the eigenvalues from the analytical solutions
discussed above. For a physical lumping discretizstion of the
continuum element. it was shown previously that the convergence was of
order (1/(2L*1)). Making explicit use of the rate of convergence. a
least-squares method was used to generate the evaluation model

eigenvalues without solving a very large eigenvalue problem. Let

”J‘Co+C./ (2L+1)+C./ (2L+1) ‘+C./ (2L+1) ' (2—19)

Bare C., c1, C3. and C. are constants to be determined. L is the

number of spring-mass lumps used to discretize the continuum. and 01.
is the corresponding eigenvalue for a given L. If a number of
relatively low-order models are run and a least-squares method is
applied using equation (2-19). the constants C., c1, ca, ‘nd c, can be
determined. Nb 0.thOt C. will correspond to the evaluation model
eigenvalue for a given sequence of eigenvalues {uj(L)} j-constant- To
check this procedure. a number of low-order problems were run for the
example above. and a least-squares curvefit was applied using equation
(2-19). Consider the second eigenvalue of the system. To apply the
curvefit procedure. 11 relatively low-order models were run. using
between 3 and 80 spring-mass lumps to approximate the shaft. Applying
_ a least-squares fit of equation (2-19) to the data for the second

eigenvalue gives:

C.-4.921 c,=-4.495 c,=-21.469 c,-2o.519

21
Figure 2.6 shows a plot of equation (2-19) using these parameters

through the eigenvalue data. Thble 2.2 below compares the results
obtained from the curvefit technique with the analytical solutions
from equation (2-7).The results from the least-squares curvefit
technique agree with the analytical solutions. supporting the idea
that the convergence for the linear mixed-structure system is the same
type as it is for the continuum element when physical lumping is used
‘ to discretize. So in future examples the evaluation model eigenvalues

will be generated using this curvefit technique.

Table 2.2 Comparison of Curvefit Results and Analytical Solutions

Eigenvalue Solution from Curvefit Solution from (2-7)
1 0.6762 0.6762
2 2.117 2.117
3 4.921 4.921.
4 7.980 7.980
5 11.086 11.086

2‘; Unit-ggrameter Uniform Cascade;

2.3,; Clustering Behavior 9; Eigenvalues

Raving dealt with the question of how to discretise the continuum
elements. the other main point to address is the coupling effect
between subsystems. In order to examine coupling effects. a study was
run using the sequence of unit-parameter uniform cascades shown in

Figure 2.7. For all the continuum elements. the parameters were k'31,

I,=l. And for all the discrete subsystems. they were kd-l. mdal.

22

euen oeueeuemmm cu amu0>ueo eeueeumlaeeea w.u enough

a enema neeatmuwumm we nemlez

Sm _ so * .3 SN 4.

 

 

 

om~e>mommu

 

 

 

 

 

 

 

23

 

 

 

Size

 

n8

 

 

 

n reasi

l.._.

 

 

 

 

l’“

 

 

 

 

 

 

J
a

 

 

 

 

 

 

Figure 2.7 Cascade Systems

 

 

 

 

ll

 

 

 

 

 

 

 

 

V(t) _)

V(t) ——)

 

 

 

 

V(t) —-)~

 

 

 

V(

24

Consider for example. the two-segment cascade shown in Figure
2.8. Since the two uncoupled segments in Figure 2.9 are identical.
they have identical spectra. Then when the segments are coupled. as
in Figure 2.8. one expects the coupling to perturb the eigenvalues
somewhat. But because the uncoupled segments have identical spectra.
one would intuitively expect the coupled system spectrum to have
clusters of eigenvalues. with two eigenvalues to a cluster.
Similarly. one would expect that as the number of segments in the
uniform cascade changes. the number of eigenvalues in a cluster A
changes such that the number of eigenvalues in a cluster is the same
as the number of segments in the cascade. As an example.'consider the
three-segment case. With L=15 for each shaft. the first 15

eigenvalues are those given in Table 2.3 below.

Table 2.3 Eigenvalues for the 3-Segment Unit-Parameter Uniform Cascade

0.2498 1.9863 4.7455 7.6415 10.503
0.7277 2.3269 4.9142 7.7489 10.579
1.1013 2.5446 4.9883 7.7797 10.594

Near the low end of the spectrum. some clustering is evident but the
width of the cluster is relatively large. As the frequency increases
through the spectrum the width of the clusters decreases. Using the
' following theorem from Crawford [9]. a bound can be obtained on the

perturbation introduced by the coupling.

1- T300133: L°t “n. In, F. and E be real symmetric nxn

matrices with I“ “a (nun?) positive definite. Let

25

 

 

 

 

 

 

 

 

 

 

 

 

 

V(t)—.-)[ I L :
Figure 2.8 Tho-Segment Cascade l”
_ - _—-_- _ l__-W’
l J“ l“N -
L____, _____,

 

 

 

 

Figure 2.9 Two Identical One-Segments

26

ijl-t(nu.su) tee {ijl-L(Mu+F.Ku+H)

with [Aj]. {:j} numbered in increasing order; Then

 

 

kuu+p)"

 

I‘j’le ‘ (llFlllljl +I'fllll i=1.....n (2—20)

For the cascade problem with physical lumping discretizations of the

continuum elements:

F=0 xj=ag>o ijfdj>o
Then from the theorem above,
< I
So the absolute perturbation in any eigenvalue is bounded by a

-i
ll“ l

relative perturbation,

9111

-'1
“u

 

I. u e-..)

 

 

 

 

 

constant

 

 

IIHII. More importantly, if one considers the

<

 

 

 

 

 

 

(mg—$3)/o3 uu“ l'nll/o3 -~ (2-22)
it is clear that as the frequency increases the effect of the coupling

decreases.

27
2,3,2 Avergge Eigenvalues gag Error Eehavior

In order to compare the systems of Figure 2.7 directly. average
eigenvalues were defined for each system. The average eigenvalues are
defined to be the averages of the eigenvalues in the clusters. With
this concept. the unit parameter uniform cascades can be compared
directly.

For each of the systems of Figure 2.7 a large numberdof
discretized models was evaluated. The average eigenvalues were
determined and the errors. compared to the average eigenvalues for the
evaluation model. were computed and tabulated. These results in Table
2.4 show that the errors in the average eigenvalues are nearly
stationary. At each location in the table. a set of four numbers
appears. From top down in a set. these four numbers are the
magnitudes of the errors in the average eigenvalues for the one. two.
three. and four-segment models respectively. Consider, for example.
the fourth average eigenvalue when the number of lumps per shaft L is
6. The errors for the one. two. three. and four-segment cases are
13.22. 13.28. 13.30. 13.31 percent respectively. Also. in examining
the table. it is clear that any increases in errors between the
one-segment case and the multi-segment cases are less than one
percent. From this it can be concluded that the cascade coupling has
little effect on the eigenvalue errors. Because the coupling has
little effect. large cascades can be viewed as assemblies of 1-segment
cascades. Discretisation can be performed on each segment separately
to meet the model quality index.

In order to apply the procedure suggested above. errors for the

l-segment case must be well documented. Recall that in section 2.2

28

"an.
e a see
«man

e
HHHF.

“Ch"
VHF!“ .Ffih ‘0'."

9000
0600

0
0

.00ee0uee0eeu

e0e0em nee-meetueeu see .eeumu .eeu .eee emu «cu eee0esee00e e0euese emu e0 .ueuue emu 0e ne‘euaemel emu
one chem-ea «ecu eeeAa .uee a £0 eeoo no» mean .eueemne cusp-em «ecu no use a .e0meu emu me noun-ee0 meee 04

00.0
00.0
00.0
00.00

00.9
99.9
00.9
90.9
00.0
00.0
00.0
00.0

00.0

00.0

00.00

00.00
00.00
00.00

00.00
00.00
00.00
00.00

00.9

00.9
00.9

00.0
00.0
00.0
00.0

00.0
00.0
00.0
09.0

00.00 00.90
00.00 00.90
00.00 00.90
00.00 00.90

09.00 09.00
00.00 00.00
00.00 00.00
99.00 99.00

00.0 00.00
00.0 00.00
00.0 00.00
90.0 00.00
00.0 00.9
00.0 00.9
00.0 00.9
00.0 00.9
00.0 09.0
00.0 09.0
00.0 00.0
00.0 00.0
00.0 09.0
00.0 09.0
00.0 00.0
09.0 00.0
09.0 09.0
90.0 00.0
00.0 00.0
90.0 09.0
00 0

00.00
00.00
00.00
00.00

00.00
00.00
00.00
00.00

00.00
00.00
00.00
00.00

00.0
00.0
00.0
00.0

00.0
00.0
00.0
99.0

00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0

00.00
00.00
00.00
90.00

00.00
00.00
00.00
00.00

00.00
00.00
00.00
00.00

90.00
00.00
00.00
00.00

09.9
09.9
09.9
90.9

00.0
00.0
.00.0
00.0

00.00
00.00
00.00
09.00

00.00
00.00
00.00
90.00

00.00
90.00
00.00
00.00

00.00
00.00
00.00
00.00

00.0
00.0
90.0
00.0

00.0
00.0
00.0
90.0
00.0
09.0
00.0
00.0

H «0.00 wen emu-0 no women:

90.00
00.00
00.00
00.00

00.00
00.00
00.00
09.00

00.00
90.00
00.00
09.00

00.00
00.00
90.00
00.00

00.0
00.0
00.0
90.0

00.0
00.0
00.0
00.0

new-ease muou0m0 none-enemluumb men eueuuu ee0e>me000 e0eue>< 0.0 e0meh

00.00
00.00
00.00
09.00

09.00
99.00
09.00
00.00

09.0
00.0
00.0

00.0
90.0
00.0
00.0

00.00
00.00
00.00
00.00

00.00
00.00
00.00
00.00

00.0
99.0
00.0
00.0

0

em0eeme000

28

09.0
09.0
09.0
00.0

0
F4

H?
0.0%
eee ee
“an“ hh'h

HCHI‘ 0

.0.
????

UMNN
???? 8..wa

”I”
??
e e
M"

00.0
00.0

09.0
09.0
09.0
00.0
00.0
00.0
00.0
90.0

00

one euemmee ueeu enema .ue. a a0 eee0 me» menu

00.0

00.9

00.0

00.0

0000 0000

use:

HHFF
“NM“
I

0
F.

.00eequeemeeu

e0e0el ace-meetueeu vie .eeumu .eeu .eee emu ecu eee0esee00e e0eueee emu l0 eueuue emu 0e ee0eu0e0el emu
.eueemme seems-e «ecu no use e .e0meu emu me ee0ueee0 nose «4

00.0
00.

00.0
00.0

00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0

00

00.00

00.00
00.00
00.00

00.00
00.00
00.00
00.00

00.9

00.9
00.9

00.0
00.0
00.0
00.0

00.0
00.0
00.0
09.0

00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0

00

00.0
00.0
00.0
09.0
09.0
90.0
00.0
90.0

00

00.0
09.0
00.0
00.0
09.0

00.0
00.

00.0
00.0
00.0

H «0e00 ue0 emu-0 0e we‘ll:

90.00
00.00
00.00
00.00

00.00
00.00
00.00
09.00

00.00
90.00
00.00
09.00

00.00
00.00
90.00
00.00

00.0
00.0
00.0
90.0

00.0
00.0
00.0
00.0

90.00
00.00
00.00
00.00

00.00

nee-coeu muou0m0 ueueleuemtu0e0 «on awoken ee0e>ee000 emeue>< 0.0 e0meh

00.00
00.00
00.00
00.00

00.00
00.00
00.00
00.00
00.0
99.0
00.0
00.0

0

0

ee0eeme000

29
the key parameters in the discretized l-segment (prototype) linear
mixed-structure system were determined to be kd/ks- 'sl'd' and L. The
l-segment case was studied for a number of variations of these
parameters and the error results were tabulated using tables of the
form shown in Figure 2.10.(See Appendix C). In examining these .
tables. keep in mind that L is a discrete variable. but the stiffness

ratio kdlk‘ and the mass ratio m‘lmd are continuous variables.

30

m /m
s

 

 

k Ik

L

Figure 2.10 Format of the One-Segment Error Tables

3. NON-UNIFORM.CASCADE MODELLING PROCEDURE.

:4; Dpscrippion p; Procedprp

Having examined the factors affecting direct generation of

efficient low-order models. a design procedure can now be given. The

following procedure is applicable to any linear cascade system having

the properties given in Chapter 1.

1.

Define a model quality index based on eigenvalue errors.
This index will be used for determining the accuracy of the
low-order model. In the examples that follow. the index
used is that the first X eigenvalues each have less than Y
percent error.

Account for any coupling effects that occur when joining
segments by reducing the acceptable eigenvalue error Y by
one percent. Y'=Y-l.

The X system eigenvalues of interest are now distributed
among the segments in the cascade. Say the cascade has M
segments. LOt "1 be the number of eigenvalues associated

with segment i. 1$iSM. For each segment define:
A1=(n/2hi Ei/pi (3’1)

DividO °fi°h A1 by the largest of the A1. call it Ak‘

Then for each eigenvalue associated with segment k. segment
i has (At/A1) eigenvalues associated with it. Solve

the f0110'inS for th° "1. If any of the Ni contain a

fractional part. round that Ni up to the next integer.

31

 

32
Ni-(Ak’Ai)Nk 181,2; e e e a

(3'2)
=N1+Nz+ . . . +Nu (3‘3)

4. Consider each segment as a separate problem where
the model quality index is that the first N (rounded up
to the nearest integer) eigenvalues each have less than Y'
percent error. Determine kd/k;. and ‘sl'd for egch
segment. Check the first N error tables in Appendix C to
find the number of lumps L needed for each segment to meet
the error criterion Y'.

5. Assemble the low-order system model.

'uwm

Several examples are given here to illustrate the application of
the procedure to a number of different cascade systems of increasing
generality. Ekamples 1.2. and 3 have the system configuration shown

in Figure 3.1.

Example One: Both shafts are identical.

The parameters for the two segments are given in Table 3.1 below.

 

 

 

 

 

 

33

0 o0maeum uou 0°00: Mechelle; 0.0 oum0um

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 van .0 .0 eo0naenm new aouuhm oueoeeu no such 0.0 onen0m

00

 

 

 

NU

0 unosmom

LC

 

 

Se

 

 

 

0 ueoamom

>|ll

 

00

 

 

34
Thble 3.1 Parameters for Example 1

Elastic Density Length Area k' "s
Modulus
segment 1 100 100 1.0 0.01 1.0 1.0
segment 2 100 100 1.0 0.01 1.0 1.0
kd lid kd/k' .8(.d
segment 1 16.0 1.0 16.0 1.0
segment 2 1.0 1.0 1.0 1.0

- Applying the Modelling Procedure:

1. Say that one wants the first 6 eigenvalues each with less than 10
percent error.

2. Account for coupling effects by reducing the acceptable error by 1
percent. Y'-10-1-9.

3. Distribute the eigenvalues among the segments.

Ai=(u/21:,L)«r‘7"ti1 p'. =(u/2Nioo7ioo'= u/2

A,”(fll2h3)i§3/pa =(fl/2) 00,100: “/2

N1‘(A3/A1)N2

N1=((n/2)/(u/2))N2-N2 - -

X“N1+N2

632N2 '9 N233 N133

35
4. Ni-Nz-S. so use the tables for the first three eigenvalues for

both segments. Looking in the l-segment error tables with the mass
and stiffness ratios from Table 3.1 above gives L1=4. L2-7. The
low-order system model with L1=4 and L287 was assembled(see Figure
3.2). and the eigenvalue simulation run. From the results in Table
3.2. it is clear that the requirements of the model quality index have

been met.

Table 3.2 Eigenvalue Data for Example 1

L1-4 Evaluation
L237 Model

Eigenvalue Percent Error

0.4476 0.4548 -1.58
1.0321 1.0419 -0.94
2.2337 2.3357 -4.36
3.1782 3.2644 -2.64
4.6767 5.0935 -8.18
5.5759 6.0834 -8.34

Note that if the discretization of the continuum elements was
based on the parameters of the continuum elements alone. the same
number of variables would be used on each shaft since they are
. identical. For comparison purposes. the simulation was also run using
the same number of lumps on each shaft. with L1=L2=6. The.results of

the simulation are as follows.

36

Thble 3.3 Eigenvalue Data for Example 1 when Shafts have
Identical Discretisations

Eigenvalue L1=6 Percent Error
L286
1 0.4484 -1.40
2 1.0416 -0.03
3 2.2264 -4.68
4 3.2237 -1.25
5 4.6156 -9.38
6 5.8481 -3.87

The results here also meet the model quality index although the
maximum error is higher than when the procedure is followed. But one
important thing to note here is the selection of the number of lumps
to use. Six lumps were used for L1 and L2 so that the total number of
lumps used would be nearly the same for the two simulations. But
choosing L1-L2 implies that the discrete subsystems do not affect the
eigenvalue errors. So the number of lumps to use would be determined
by equation (2-5). Using this relation shows that 7 lumps'should be
used for L1 and L2. approximately 30 percent more variables than the

procedure developed in this work requires.

Exemple Two: Shafts have different geometrical parameters.

In this example a 2-segment cascade is considered. .?h° shafts .
have the same stiffness and elastic modulus but have a difference in
the geometrical parameters. The second shaft is twice the length of

the first.

37
Thble 3.4 Parameters for Example 2

Elastic Density Length Area ks_ - m‘
Modulus
segment 1 100 100 1.0 0.01 1.0 1.0
segment 2 100 100 2.0 0.01 0.5 2.0
kd md kd/ks mglmd
segment 1 2.46 10.0 2.46 0.1
segment 2 1.23 2.0 2.46 1.0

Apply Modelling Procedure:

1. Say that one wants the first 6 eigenvalues each with less than 10
percent error.

2. Reduce the acceptable error by 1 percent to account for coupling
effects. Y'-10-1=9.

3. Distribute the eigenvalues among the segments.

A1'(tt/2h1)‘/El—lt-)I flunk/1307175 - n/2 A2014
N2'(A1/A2)N1=2N1

X=N1+N2

6===N1+2N1 -> N1==2 N2=4

4. Since N182 and N2-4. the one-segment tables for the first two

eigenvalues are used for segment 1 and the tables for the first four

38

eigenvalues are used for segment 2. Looking in appropriate error
tables gives L1=3. 12'8. Absembling the finite dimensional model for
g the system results in the configuration shown in Figure 3.3.

Simulation yields the results shown in Table 3.5 below.

Table 3.5 Eigenvalue Data for Example 2

Eigenvalue L1=3 Evaluation Percent Error

L288 Model

0.2122 0.2135 -0.61

0.4580 0.4617 -0.80

1.2898 1.3107 -1.59

2.2237 2.3923 -7.05

2.4649 2.6167 -5.80
4.0940 -9.06

6 3.7229

Again the procedure has resulted in a low order model that meets
the given model quality index. In contrast to the last example. note
here that the number of lumps used for the two continuum elements is
similar to what one would use if basing the decision on the continuum

elements alone.

Example Three: Shafts have different geometrical and constitutive
parameters.
This is another 2-segment example. But here the shafts have

different densities in addition to having different lengths.

39

 

 

0 o0naeum «on 0euox «ovuOIuoq 0.0 ounuam

0 «acumen

_

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 «moa0om

 

40
Thble 3.6 Parameters for Exmmple 3

Elastic Density Length Area k' m‘
Modulus
segment 1 100 100 1.0 0.01 1.0 1.0
segment 2 100 25 2.0 O 0.01 0.5 0.5
1:d "a kd/k. ‘s/‘d
segment 1 2.46 10.0 2.46 0.1
segment 2 1.23 0.5 2.46 1.0

Applying Modelling Procedure: ,. -

1. Assume that it is desired to have the first six eigenvalues with
less than 10 percent error in each.

2. Minus 1 percent for coupling effects. Y'-10-1=9.

3. Distribute the eigenvalues among the segments.
Ag=ul2 A2=n/2 9 N1=N2
X=N1+N2
6=2N1 9 N1=3 N2=3

4. N1‘N2'3' so use the tables for the first three eigenvalues for
both segments. looking in the 1-segment error tables with the mass
and stiffness ratios from Table 3.6 above gives L1=6. L286.
Assembling the low-order model and simulating gives the results shown

in Table 3.7 below.

41

Thble 3.7 Eigenvalue Data for Example 3

Eigenvalue L1=6 Evaluation Percent Error

L286 Model

0.2507 0.2513 -0.24
0.7924 0.8073 -1.87
2.3113 2.3921 -3.38
2.5275 2.5810 -2.07
4.7251 5.1658 -8.52
4.7805 5.2072

-8.19

Example Four: A General Cascade.
As a final example in illustrating the use of the modelling
procedure. consider the four-segment cascade of Figure 3.4 having the

parameters given in Table 3.8.

Applying Modelling Procedure:

1. Assume that one wants first 8 eigenvalues each with less than 12
percent error.

Y'=12-1-11.

2. Minus 1 percent for coupling effects.

3. Distribute the eigenvalues.

A1'8/4 A2=5u A3=nl6 A4=nl2

Nl-(Az/A1)N2-20N2 N3= (Ag/A3 ) N223 0N2 Ng" (Azti4)N2-10Nz _
X-N’1+N2+N3+N4

s-oinz -> N2=.13 -) N1=2.6 N3-3.9 N4=1.3

42

0 o0qaeum new 00008 augmented 0.0 one0wm

eee men 35 3e
-sea.sa-mmaana-a—ae.mé§|§

0 o0naeum new aouehm evacueu we each 0.0 oueu0m

v ucoamom 0 umofimom 0 ueoevom 0 umoemom

 

 

 

 

 

 

 

 

 

 

 

 

.(<. ./>\ .<<l T 30>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

43
4. So after rounding up. use N1-3. N2-1. N3-4. Ng-Z. Looking in the

. 1-segment error tables under the appropriate eigenvalues with the mass
and stiffness ratios from Table 3.8 above gives L134. chl. L386.

L485. Assembling the low-order model shown in Figure 3.5 and

simulating yields the results shown in Table 3.9 below.

Table 3.8 Parameters for Example 4

segment 1 segment 2 segment 3 segment 4

Elastic

Modulus 100 400 50 100

Density 100 4.0 50 25

length 2.0 1.0 3.0 2.0

Ares 0.01 0.01 0.01 0.01

k, 0.5 4.0 0.17 0.5

n, 2.0 0.04 1.5 0.5

kd 5.0 2.46 '1.65 44.41 , _
“a 2.0 0.4 1.5 0.05

kd/k, 10.0 0.62 9.87 22.21

-,/-d 1.0 0.1 1.0 10.0

44
Table 3.9 Eigenvalue Data for Example 4

Eigenvalue L1=4. L281 Evaluation Percent Error
L3=6. L4=5 Model

1 0.1845 0.1896 -2.69

2 0.4383 0.4470 -1.95

3 0.9628 0.9776 ' -1.51
1.451 1.507 -3.68
1.529 1.611 g -5.09
1.769 1.796 -l.45
2.332 2.343 —0.46
2.710

2.980 -9.04

Examining the results shows that the modelling procedure yields good

. results for the general case.

1‘; Eehavior p; pEp Eigenvectors

One other issue that bears mentioning here is the "correctness"
of the eigenvectors associated with the low-order models generated
from the modelling procedure. The modelling procedure is not
dependent on the order in which the 1-segments are coupled. so a
question naturally arises as to the "correctness" of the eigenvectors.

To examine this question. a 3-segment cascade was considered.
with two of the segments being identical and the third segment being
different. Let the identical segments be called 'A' segments. and the
different segment be called a 'B' segment. Three systems were
examined. an AAB. an ABA. and a BAA configuration. Low order models

and evaluation models were developed for each configuration. «

45

Eigenvectors were then examined to determine the "correctness" of the
low-order models' eigenvectors.

Examining Figures 3.6. 3.7. and 3.8. which are overlays of each
systems' low-order and evaluation model eigenvectors. it is clear that
the low-order models give good approximations to the eigenvectors of
their corresponding evaluation model. So the modelling procedure.
even though it does not account for how the cascade segments are
ordered. and has a model quality index based on eigenvalue information

only. yields useful information about the system eigenvectors.

46

0000: <40 ego uou enouoeouo000 000°: eo0uem0e>a 0mm «unwanted we he0ue>0 0.0 oue00m

neuoobee00m Auueom

 

 

Heuoo>mo000 cocoon

 

 

 

 

uouooemc0mm 0HOAH

N . s

 

é
_
.

_ _ in — Sat

_

E

uouooomou0m uauwm

 

 

 

 

47

aovoa <n< oAu now uuouoo>uouum dove: do«uuu~a>m can HovHOIDoA no hqauo>c h.m ouuuam

Hauoopdoudm Auuuom

 

 

 

 

¢ .m s
w _ T _ N-
\Jls
nouoo>noawm vacuum
w m s
n _ __ _ N-

 

 

acuoo>aonmm chunk

 

 

 

Houoo>nouam uaumm

 

 

48

none: n<< odu wow un0uoo>flouum Hove: acauannnvm an. HovHOIIQa «o ha~uo>o awn ounuum

 

 

 

 

 

 

 

nouoo>douam Auuuom Houoo>uoudm uumAH ~
._‘ m s . +_. .m s .
_ _ I _ _ n.
_ _ . m _ N
a Is
1 1|
Houoo>noumm vacuum Houoo>uouwm unuwm

 

 

 

 

 

 

4. MORE GENERAL CASCADE SYSTEMS

5;; Relaxation 2; Restrictions

In this section, some restrictions of Chapter 1 will be relaxed.
Some examples will be considered to demonstrate the usefulness of the
modelling procedure in situations where the restrictions of Chapter 1
are not met. The particular restriction to be relaxed is that the
discrete subsystems must be of the form shown in Figure 1.4. Now the
discrete subsystems will be permitted to have any one of the following
forms:

1. spring-mass

2. mass-spring

3. mass-spring-mass

4. spring-mass-spring

5. spring-mass-spring-mass
The first of these forms (spring-mass) is the form of the discrete
subsystem used in developing the modelling procedure and is shown in
Figure 1.4. The other permissible forms of the discrete subsystem are
shown in Figures 4.1, 4.2, 4.3. and 4.4.

The modelling procedure given in Chapter 3 requires that a
stiffness ratio and a mass ratio be defined for each segment. For the
discrete subsystem forms shown in Figures 4.2. 4.3. and 4.4. it is
possible to define more than one mass ratio or more than one stiffness
ratio for a segment. To use the modelling procedure unambiguously.
only a single mass and a single stiffness can be associated Vith.°!9hfl
discrete subsystem. Tb do this, a "first-oscillator-in" concept was

used. This notion is that for e segment, the discrete spring and
49 "

50

Figure 4.1 Mass-Spring Subsystem

Figure 4.2 Mass-Spring-lass Subsystem

kA k3
OJVVV

Figure 4.3 Spring-Mass-Spring Subsystem

 

 

 

 

Figure 4.4 Spring-Mess-Spring-Mass Subsystem

 

51
discrete mass nearest to the continuum element have a much greater

effect than the other springs and masses in the disorete,subsystem.c
Consider the segment shown in Figure 4.5. Using the
"first-oscillator-in" notion, the mass ratio and the stiffness ratio
to use with the modelling procedure are m‘/.d4 and de/k‘.

Using the "first-oscillator-in" concept. the modelling procedure
can be applied directly to systems having discrete subsystems with
more than one spring or more than one mass element. The following
examples demonstrate the usefulness of the modelling procedure for
cascade systems with more general discrete subsystems such as those

shown in Figures 4.1 through 4.4.

Example Five: Two segment cascade with a mass-spring discrete
subsystem.
Consider the system shown in Figure 4.6 with the parameters given

in Table 4.1 below. This example is the same as example two except

that the discrete subsystem in the first segment is reversed.

52

 

 

A B
B. A. }’VNJ\r "d 'd

Figure 4.5 First-Oscillator-In Illustration

 

 

 

V(t)‘4

 

 

 

 

AW”! W

Figure 4.6 System of Example 5

53
Thble 4.1 Parameters for Example 5

Elastic Density Length Area k‘ m,
lodulus

segment 1 100 100 1.0 .01 1.0 1.0

segment 2 100 100 2.0 .01 0.5 2.0
kd md kd/ks m‘lmd

segment 1 1.0 1.0 1.0 1.0

segment 2 1.0 1.0 1.0 1.0

Applying Modelling Procedure:

1. Say that one wants first 6 eigenvalues each with less than 10
percent error. ‘
Application of the modelling procedure follows exactly as in example
two. Tb meet the model quality index one should use Ll-3 and L2-8.

Assembling the low-order model and simulating gives the results shownv

in Table 4.2 below.

Thble 4.2 Eigenvalue Data for Example 5

Eigenvalue L1-3 Evaluation Percent Error
L2=8 Model
1 0.2377 0.2399 -0.92
2 0.4405 0.4450 -1.01
3 1.150 1.155 -0.43
4 2.212 2.271 -2.60
5 3.024 3.173 -4.70
6 3.449 3.621 -4.75

54
The results meet the specified model quality index. so the procedure

is seen to be useful for a cascade having this type of discrete

subsystem.

Example Six: Two-segment cascade having a mass-spring-mass discrete

subsystem.

Consider the system shown in Figure 4.7 with the parameters given

in Table 4.3 below.

Table 4.3 Parameters for Example 5

Elastic Density Length Area k8 m
s
Modulus
segment 1 100 25 1.0 .01 1.0 .25
segment 2 100 25 1.0 .01 1.0 .25
kd IdA kd/ks mslndA "43
segment 1 .6169 2.5 .6169 0.1 .25
segment 2 .6169 .25 .6169 1.0

Applying lodelling Procedure:

1. Say that one wants first 4 eigenvalues each with less than 12
percent error.

2. linus 1 percent for coupling effects. Y'=12-1=11.

3. Distributing the eigenvalues:

A13“ A381! '9 N1=N2

X‘Nl +N2

h oaqaeuu mo aeuehm w.¢ ouunmm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3;

 

N unciuem .n «aeauem
_ _
s___ r.</>.. u<<<.
v u u
M u
m «a
5
5
w cane—cum no 6393 E? 233m
N uncauom .n amen—mom
\ / \
e... as
v <
u

 

  

 

 

 

 

56
4-2N1 N132 "2-2

4. Nl-Nz-Z, so use the tables for the first two eigenvalues for both
segments. Looking in the appropriate 1-segment error tables with the
mass and stiffness ratios from the parameter table above gives L1-4.
L2-3. Assembling the low-order model and simulating yields the

results below.

Table 4.4 Eigenvalue Data for Example 6

Eigenvalue Ll=4 Evaluation Percent Error
L2-3 Iodel
1 0.5062 0.5093 -0.61
2 0.9321 0.9442 -1.28
3 1.916 1.869 2.51
4 4.188 4.634 -9.62

The errors are all within the 12 percent allowed by the specified
model quality index, so the procedure is seen to be useful for a

cascade system with a mass-spring-mass subsystem.

Example Seven: Two-segment cascade with a spring-mass-spring discrete
subsystem.
Consider the system shown in Figure 4.8 with the parameters given

in Table 4.5 below.

57
Table 4.5 Parameters for Example 7

Elastic Density Length Area k8 .‘
lodulus
segment 1 1.0 1.0 1.0 1.0 1.0 1.0
segment 2 1.0 1.0 1.0 1.0 1.0 1.0

‘4‘ ":1A de/ks ‘sI‘dA kdB
segment 1 1.0 1.0 1.0 1.0 1.0

segment 2 1.0 1.0 1.0 1.0

Applying Hodelling Procedure:

1. Say that one wants first 4 eigenvalues each with less than 10
percent error. O '
2. Iinus 1 percent for coupling effects. Y'=10-1=9.

3. Distributing the eigenvalues:

Aagfl/z Afr/2 9 N1=N2
13"]:sz
4-2N2 N122 N2=2

4. Ni-Nzt2, so use the tables for the first two eigenvalues for both
segments. Looking in the appropriate 1-segment error tables with the
mass and stiffness ratios from the parameter table above gives L1-3.
L2-3. Assembling the low-order model and simulating yields the

results below.

 

58
Table 4.6 Eigenvalue Data for Example 7

Eigenvalue L1-3 Evaluation Percent Error

L2-3 lodel
1 0.3280 0.3328 -1.42
2 0.9285 0.9377 -0.97
3 1.5685 1.596 -1.72
4 1.9713 2.143 -8.01

The results satisfy the specified model quality index. so the

procedure gives useful results for a cascade having a

spring-mass-spring discrete subsystem.

Example Eight:

Cascade with discrete subsystems having two

degrees-of-freedom in a spring-mass-spring-mass configuration.

Now consider the system of Figure 4.9 withlthe parameters given in

Table 4.7 below.

segment 1

segment 2

segment 1

segment 2

Table 4.7 Parameters for Example 8

Elastic Density Length Area k m

8 s

Modulus
1.0 1.0 1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0 1.0 1.0

kdA "'aA de/ks 'sl'dA kaB ”an
1.0 1.0 1.0 1.0 1.0 A. 1.0

1.0 1.0 1.0 1.0 1.0 1.0

Applying lodelling Procedure:

59

a canaenm flown leacheamm eueuouun oH.v cuunuh

 

 

”'6
fl

6%

d'fi
fl

\
\
\

 

 

 

 

 

m canaeum uo aeuehm m.v ouuumm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N udeaucm H uncancm
\. h _
, / \ . J
u u u
I a
an an a v
m < M u

 

 

 

 

 

 

60
1. Assume that it is desired to have the first 8 eigenvalues each

with less than 15 percent error.
2. Subtract 1 percent for coupling effects. Y'=15-1=14.

3. Distributing the eigenvalues:

Aian/z A.=n/2 N1=N2
X'N1+N2
8=2N2 ') N234 N134

4. Ni-N2=4. so use the tables for the first four eigenvalues for both
segments. Looking in the 1-segment error tables with the mass and
stiffness ratios from Table 4.7 above gives L126. L2=6. Simulation

gives the results shown in Table 4.8 below.

Table 4.8 Eigenvalue Data for Example 8

Eigenvalue L1=6 Evaluation Percent Error
L286 Model
1 0.2459 0.2468 -0.34
2 0.7056 0.7019 0.53
3 1.271 1.292 -1.61
4 1.567 1.566 0.09
5 2.057 2.144 -4.05
6 2.409 2.498 -3.54
7 4.458 4.921 -9.39
8 4.641 5.11 -9.17

61

The errors here are all well under the 15 percent allowed by the
choice of model quality index. So for cascade linear mixed-structure
systems where the discrete subsystems have a spring-mass-spring-mass
structure. the modelling procedure is still useful.

This example also serves to illustrate an interesting artifact
that occurs when identical segments are coupled together as they are
in example six. Note that some of the eigenvalues in the low-order
model have positive errors. In this example. the second and fourth
eigenvalues have positive errors. The interesting thing to note about
these eigenvalues is that their frequencies correspond closely with
those of the discrete subsystem. Consider the discrete subsystem

shown in Figure 4.10. The equations of motion for this system are:

Solving the system equations for the eigenvalues gives m-{0.62. 1.62}.
Comparing with the true spectrum for the linear system of example 6
shows that these eigenvalues correspond closely with the second
(0.7056) and the fourth (1.5663) eigenvalues.

In examples five through eight above. it has been shown that the
modelling procedure is useful for cascades more general than those
that were used in deve10ping the procedure. As a final illustration
of the usefulness of the procedure in general cascade systems.

consider example nine below.

Ekample Nine: Cascade with a force input.

 

62
The system in this example is identical to that in example two

(shown in Figure 3.1) with the exception that the velocity input in
example two is replaced with a force input here. Suppose that as in
example two. the model quality index is that the first six eigenvalues
each have less than 10 percent error. The modelling procedure does
not account for the type of input. so following through example two.
one finds that for the low-order model. L1=3 and L2-8. Assembling the
low-order model shown in Figure 4.11 and simulating yields the results

shown in Table 4.9 below.

Table 4.9 Eigenvalue Data for Example 9

Eigenvalue L1-3 Evaluation Percent Error
L288 Model
1 0.0000 0.0000 0.00
2 0.4319 0.4361 -0.96
3 1.262 1.179 7.04
4 1.343 1.319 1.82
5 2.462 2.615 -5.85
6 3.628 3.732 -2.79
7 3.723 4.094 -9.06

Even if the zero eigenvalue is assumed given. the results still meet

the model quality index. So the procedure is seen to be useful for a

cascade having a force input.

It is interesting to note here the type of physical lumping used
for the continuum element in segment one.

discretixations in this work. a springrmass physical lumping was used.

As in all other

m onmaenm now Home: enchanted HH.¢ ounuum

N udoauem H «don—new

63

 

 

 

 

 

 

 

 

 

 

 

 

E a a a T Auvm

 

 

 

 

 

 

 

 

 

 

 

64
But in segment one. the spring element next to the force input is

dependent. Since the continuum element is causally independent. one
might consider it "better” to use a physical lumping discretiration"
that has mass elements at both ends. such as the discretisation shown
in Figure 4.12. But comparing the errors in example nine.with those.
from example two. where the discretization matches the end conditions.
shows that they are quite similar although the correspondence in not
one-to-one. The effect on the errors of choosing the type of physical
lumping based on the shaft's end conditions is an area that has not

been adequately explored yet.

1‘; Modifying thg Procedure under Certgin Conditions

One last thing to be discussed is a caution concerning the use of
the modelling procedure. In modelling cascades where there are two
identical segments coupled together and also segments having higher
frequency spectra than the identical ones. if one is interested in
relatively few eigenvalues. then step 3 in the procedure.should be _
modified slightly. Before discussing this modification. the
difficulty is illustrated in the following example.

Consider an AAB configuration of a 3-segment cascade where the

'A' and '3' segments have the parameters given below.

65

lmllll
Figure 4.12 Physical Lumping Discretixation Having Mass
Elements at Both Ends

66
Table 4.10 Parameters for AAB Example

Elastic Density Length Area ks “s
Modulus

A segments 1.0 1.0 1.0 1.0 1.0 1.0

B segment 20.0 5.0 1.0 1.0 20.0 5.0
ha ma kd/k, .,/.d

A segments 1.0 1.0 1.0 1.0

B segment 200.0 0.5 10.0 10.0

Assume that one wants the first 5 eigenvalues with less than 10

percent error in each. Following through the modelling procedure

81'99 NAw2. NB-l. Checking the l-segment tables gives LAF4. 1885.
Assembling the low-order model and simulating yields the eigenvalues

below.

Table 4.11 Eigenvalue Data for AAB Example

Eigenvalue LAF4 Evaluation Percent Error
LB=5 Mode l
1 0.1853 0.1859 -0.32
2 0.7996 0.7916 1.01
3 1.860 2.029 -8.33
4 2.308 2.457 -6.06
5 4.174 4.896 -14.75

Examining the results shows that the fifth eigenvalue has too-large an
error. Tb explain this. examine the evaluation model eigenvalues for

the 'A’ segment. the '8' segment, and the 2-segment AA component given

67
in Table 4.12.

Table 4.12 Evaluation Model Eigenvalues for Components of AAB Cascade

Eigenvalue 'A' segment '3' segment 'AA' component
1 0.6762 2.85 0.3667
2 2.117 8.47 1.bos‘
3 4.921 13.85 2.042
4 7.98 19.04 2.495
5 11.086 24.3 4.91'

Comparing the eigenvalues with those from the AAB evaluation model
reveals the difficulty. The procedure indicated that one of the five
eigenvalues should come from the '8' segment (N331). But comparing
the eigenvalues shows that all five came from the 2-segment 'AA'
component. The solution to this problem is to modify step 3 in the

procedure to:
1. Round the N values up to the nearest integer.
2. For the identical coupled segments. increase their N value by

one .

Re-examining the problem using the modified procedure gives N‘ag.

Nh=1. and from the error tables one gets that LAP7. 1885." Assembling

the model and simulating gives the results below.

68
Table 4.13 Eigenvalue Data for AAB Example using Modified Procedure

Eigenvalue LAP7 Percent Error
lB=5
1 0.1855 -0.22
2 0.7962 0.58
3 1.932 -4.78
4 2.374 -3.38
5 4.521 -7.65

So the modification to the procedure corrected the difficulty. A note
on this is that the difficulty does not arise if one is interested in
a relatively large number of eigenvalues. Consider the same problem
but now assume that one wants the first 10 eigenvalues with less than
10 percent error in each. Using the unmodified procedure gives NAF4:
NB-2. and LAP9. lB=5. Simulating yields the following results. which

do meet the model quality index requirement.

69

Table 4.14 Eigenvalue Data for AAB Example with Model Quality
Index Based on the First Ten Eigenvalues

Eigenvalue LA¥9 Evaluation Percent Error
lB=5 Model
1 0.1856 0.1859 -0.16
2 0.7951 0.7916 0.44
3 1.9539 2.0299 -3.74
4 2.3935 2.4570 -2.58
5 4.6172 4.896 -5.69
6 4.755 4.901 -2.96
7 4.900 5.133 -4.53
8 7.343 7.999 -8.19
9 7.457 8.126 -8.22
10 9.438 9.872 -4.39

APPENDICES

5. SUMMARY AND CONCLUSIONS

The most important result of this work is the development of a
modelling procedure to directly generate efficient low-order models
for cascade linear mixed~structure systems. The procedure is a
five-step method in which the designer specifies a model quality
index. uncouples the cascade. distributes the eigenvalues of interest
among the segments. discretires the segments individually using the
error tables. and then assembles the low-order model. DevelOped using
some restrictive assumptions. this modelling procedure was shown in
Chapter 4 to still be useful when the assumption concerning the form
of the discrete subsystems is violated. This procedure gives the
systems modeller a rational approach to modelling linear .
mixed-structure systems. helping to remove the modelling of these
types of systems from the realm of a "black art".

In Chapter 2. two interesting techniques for generating the
eigenvalues of linear mixed-structure systems are introduced. The
first of these generates analytical solutions for linear
mixed-structure systems having only one continuum element. In
contrast to other solution techniques for these systems [7.8]. it
requires no a priori knowledge of the continuum element's eigenvalue
and eigenvector behavior.

The other technique generates evaluation model eigenvalues using
a sequence of relatively low-order discretized representations of the
system. It exploits the convergence behavior of the discretixing
technique in an absolute sense by applying a least squares technique

to a truncated convergence rate expression. in order to generate the

70

 

71

evaluation model eigenvalues. lhile applied here to linear
mixed-structure systems discretized using a physical lumping approach.
its usefulness does not appear to be limited to these cases.

Future efforts in modelling linear mixed-structure systems would
most naturally be directed toward examining application of the
modelling procedure to other types of system tapologies and
development of an analogous modelling procedure based on finite
element discretixations. With further development of the l-segment
error tables. it may be useful to implement the modelling procedure
within modelling and simulation programs such as ENFORT‘5-[10] and
MEDUSA [11].

One final note is that since the procedure is only a guideline
and cannot guarantee that the results will meet the model quality
index. it may be desireable in some cases to increase cne's confidence
in the results. One possible way to do this is to follow through the
procedure to find the number of lumps to use. But when assembling the
low-order model. replace the physical lumping discretizatibns‘with a
method having a faster convergence rate. such as a linear finite

element. using the number of lumps (elements) given by the procedure.

 

APTENDIX A
SGJITION TO THE FINITE DIFFERENCE PEWLEM
To solve the System in (A717.
-EAL/h) (en+l’20n+en-1)'”z (Mk/LNn
90.0 ‘°L+1’°L)"h’L"°
first use the central difference operator notation
A on‘en-i-l'zon'Hn-l
to rewrite the system equations (A-l) in the form (A-4).
-A en=((..’ph’)/(EL’))0n
Using the finite difference identity (A-5) from Goudreau [12].

sin(1n/L) sin(An/L)
A ]-—4.1n’ (1.121.)
cos(1n/L) cos(1n/L)

the general solution to the system equation is:

Gn-Asin(An/L)+Bcos(ln/L)

72

(A-l)

(A-2)

(A-3)

(A-4)

(ArS)

(A-6)

73

Th0 90-0 boundary condition gives B-O. Applying the other boundary

condition:
(em-0L) / (h/L)=0
sin(A(L+1)/h)-sink=0
Applying a trigonometric identity to equation (Ar7) gives:
cos(0.51(2L+l)IL)sin(0.51(1/L))80
So either:
A‘ZjflL j=1.2,3.. o 0
or
l=(L/(2L*1))(21'1)n j=1.2.3.. . .
Substituting (A-9) into the general solution (A76) gives

9n-Asin(2jnn)=0 . -

So x-ZjnL results in the trivial solution. Substituting (A-10)

the general solution yields:

entAsin[(Zj-1)nn/(ZL&1)] n=1.2.. e esL j=1,2,. 0 o

(A-2)

(A-7)

(Ar8)

(A-9)

(A-lO)

(A-ll)

into

(Ar12)

So the displacements at the points nP1.2.. . ..L are determined by a

set of sine functions. Comparing equations (A-4) and (A-5) results in

“QWA . mm

 

74

the following expression for the natural frequencies.
4sin’IA/2Llcu'ph‘laL' (A-13)

Using equation (Ar10) to substitute for 1 and solving (Ar13) for m

gives:
uj=(2L/h) E/p sin[(23-1)n/(4L+2)] j=1.2.. . ..L (A-14)
Dividing by the solution (A-15) to the continuum boundary value

problem and subtracting 1 gives the error expression shown in section

2.2 and reproduced below in (A-16).
Gj-((2j-1)n/2 Elp (Ar15)

l(0‘;)/;]'(4L/(21-1)n)sin[(Zj-l)n/(4L&2)]-1 j=1.2. . . . .L (A-16)

APPENDIX B

DIRECT SOLUTION TO THE BOUNDARY VALUE PROBLEM FOR THE

PROTOTYPE LINEAR MIXED-STRUCTURE SYSTEM.

For the prototype linear mixed-structure system of Figure 3.1a.

the boundary value problem is:

a‘u/ax’-1/a'a‘u(at‘ .‘-a/p (9-1)
md§(t)=-kd(y(t)-u(h.t)) (a-z)
u(0.t)=0 EA(au/3x) x..h-=kd(y(t)--u(h.t)) (B-3)
Assume that y(t)'cu(h.t). ' 7 (3'4)

substituting into (B-2) and (B-3) gives

mdca’u(h.t)/3t’=-kdn(h.t)(c-l) (n:s>‘

EA(0u/ax)x=hekdu(h.t)(c-l) " . (3-6)

Applying the standard separation of variables assumption

u(x.t)'P(x)Q(t) (3'7)

to the boundary value problem gives:

75

76
4’P"(x)/P(x)'0"(t)/Q(t)'-u'

P(0)=0

EAP'(h)'de(h)(c-1)

From equation (E-8). one obtains:

c"<t>+u‘c(t>-o

P"(x)+(m/a)'P(x)=0

, The solution to equation (B-12) is

P(x)-Dsin(uxla)+Fcos(ax/a)

Using the boundary condition in (B-9) gives F=0.

P(x)=Dsin(mx/a)

Substituting equation (B-14) into (E-10) gives

(o/a)cos(uh/a)-kd(c-1)lEA)sin(uh/a)

w-(akd/EA)(c-l)tan(mh/a)

So

This equation represents the frequency relation for the shaft.

(8-8)

(B-9)

(B-lO)

(B-ll)

(B-12)

(E-13)

(B—14)

(3-15)

(B-16)

77
Applying the same separation of variables to equation (B-5) yields

“ch(h)Q"(t)+kd(c-1)P(h)Q(t)=0 (ll-17)

Q"(t)+(kd(c-1)Indc)Q(t)=0 (B-18)
So the natural frequency of the discrete portion is

wa=(kd(c-l)lndc)1/a ' ‘ (3’19)
At a systo- natural frequency. Ufa. So Solving (B-19) for c and

substituting into equation (B-16) results in the following frequency ~

relation for the system.

w=(akd/EA)(u‘nd/(kd—u‘nd))t.n(uh/a) (n~20)

APPENDIX C
TEE ONE-SEGMENT ERROR TABLES

The results in the following tables are for one-segment cascade
systems as shown in Figure 2.2a. discretixed into the form in Figure
2.3. As discussed in section 2.2.2. there are three key parameters in
the discretised one-segment system. They are the stiffness ratio
kd/k'. the mass ratio ‘s/‘d' and the number of spring-mass lumps L
used to discretixe the continuum element.

The error tables are structured as follows. They are first
grouped according to the eigenvalue number. That is. there is a set
of tables for the first eigenvalue. a set for the second eigenvalue.
and so on. For each page in the tables. a mass ratio is specified.
Each row on a given table corresponds to a different L value and each
column on a given table to a different stiffness ratio. The entries
in the tables are the magnitudes of the errors (in percent) in the low

order models. The errors are calculated using the relation (C-l).

Error- (”-“truo/”true)100 (C-l)

To use the tables. one begins by obtaining the reduced acceptable
error Y'. the nu ratio "/Id. the stiffness ratio kd/k.. and the
number of eigenvalues N associated with the given 1-segment.‘ For the”
segment. the sets of error tables for the first N eigenvalues must be
examined. The mass ratio indicates which table in each set to use.

The stiffness ratio indicates a particular column in each table.

78

79
Beginning with the appropriate table and column for the first

eigenvalue. one merely reads down the column until an entry is found

that is less than Y'. Reading across the table gives the number of

lumps L(1) needed to meet the error criterion for the first
eigenvalue. Checking the apprOpriate table and column for the second
through the Nth eigenvalues gives values L(2) through L‘N). .Choosiug,
the largest of the values L‘l) through L(N) gives the number of lumps
L to use to discretize the segment.

To illustrate use of the tables. consider example two from
Chapter 3. Segment 1 in this example has 2 eigenvalues associated
with it. so the sets of tables C.1.1. C.1.2. C.1.3 and C.2.1. C.2.2.
C.2.3. corresponding to the first two eigenvalues. will be used. The
mass ratio is 0.1. This specifies that tables C.1.1 and C.2.1 are to
be used. The stiffness ratio is 2.46 which corresponds to the second
column in each table. So one follows down the second column in each
table until an entry is reached that has a value less than Y'.(Eere
Y'-9). Finding this entry and reading across the table shows that for
the first eigenvalue. L(1)=1. and for the second eigenvalue. L(2)-3.
Choosing the largest of these requires that 3 spring-mass lumps must
I be used to discretize segment 1.

Segment 2 has 4 eigenvalues associated with it. so the sets of.
tables: C.1.1. C.1.2. C.1.3; C.2.1. C.2.2. C.2.3: C.3.l. C.3.2.
C.3.3: and C.4.1. C.4.2. 0.4.3 will be used. The mass ratio is 1.0
for segment 2. therefore tables C.1.2. C.2.2. C.3.2. and C.4.2 will be
used. The stiffness ratio for this segment is 2.46. so one must
follow down the second column to find an entry less than Y'. Checking

this column shows that the first entry less than Y' occurs when

so
[JD-2. Lm-z. Lm-s. and L(4)-8. So a spring-mass lumps will be

used to discretize segment 2.

As another illustration of the use of the tables. consider
example one from Chapter 3. Three eigenvalues are associated with
each segment in the system. So the sets of tables corresponding to
the first three eigenvalues will be used. For both segments. the mass
ratio is one. so tables C.1.2. C.2.2. and C.3.2 will be used. For
segment 1. the stiffness ratio is 16. and for segment 2 it is 1.0.

Now just follow down the appropriate columns until the entry is less
than the reduced acceptable error Y'. In this problem. both stiffness
ratios are between columns on the table. So one must interpolate in
some fashion. One could either interpolate linearly. or more
conservatively. one could require the entries in both columns to be
less than Y'. Choosing the latter option. for segment one. kd/k‘alfi,
so one would search columns three and four. Following down these two
columns. at L(1)=2. L(2)=3. L(3)=4. the entries in both columns are
less than Y'=9. Choosing the largest of these values gives L1-4. .For
segment t'O: kd/kg-l. so examine columns one and two. When L(1)-2.
L(2)=4. and L(3)-7. the entries in both columns are less than Y'.
Again. choosing the largest of these gives L2=7. So in the low-order

model. one would use L1=4. L2=7.

81
Table C.1.1 First Eigenvalue

ms/md=O.1
Number
of Stiffness Ratio (kd/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

1 0.47 1.66 2.59 2.82 2.96 2.98 3.01 _3.03,
2 0.21 0.73 1.16 1.27 1.31 1.34 1.35 1.35
3 0.14 0.46 0.74 0.81 0.84 0.86 0.86 0.87
I 4 0.10 0.34 0.54 0.60 0.61 0.63 0.63 0.63
5 0. 08 0. 27 0. 43 0. 47 0. 48 0. 49 0. 50 0. 50
6 0.07 0.22 0.35 0.39 0.40 0.41 0.41 0.41
7 0.05 0.19 0.30 0.33 0.34 0.35 0.35 0.35
8 0.05 0.16 0.26 0.29 0.30 0.30 0.30 0.30
9 0. 04 0. 14 0. 23 0. 25 0. 26 0. 27 0. 27 0. 27
10 0.04 0.13 0.21 0.23 0.23 0.24 0.24 0.24
11 0.03 0.11 0.19 0.21 0.21 0.22 0.22 0.22

12 0.03 0.11 . 0.17 0.19 0.19 ‘0.20 0.20 0.20
13 0.03 0.09 0.16 0.17 0.18 0.18 0.18 0.18
14 0.03 0.09 0.15 0.16 0.17 0.17 0.17 0.17
15 0.02 0.08 0.14 0.15 0.15 0.16 0.16 0.16
16 0.02 0.08 0.13 0.14 0.14 0.15 0.15 0.15
17 0.02 0.07 0.12 0.13 0.14 0.14 0.14 0.14

18 0.02 0.07 0.11 0.12 0.13 0.13 0.13 0.13

82
Table 0.1.2 First Eigenvalue

ms/md31.0
Number
acmps Stiffness Ratio (id/ks)
0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90
1 5.57 13.37 16.63 17.27 17.51 17.62 17.67 17.72
2 2.39 6.40 8.39 8.77 8.91 8.98 9.02 9.04
3 1.51 4.17 5.59 5.85 5.95 6.00 6.03 6.05
4 1.10 3.09 4.19 4.38 4.46 4.50 4.52 4.54
5 0.86 2.46 3.35 3.50 3.57 3.60 3.61 3.63
6 0.71 2.04 2.79 2.92 2.97 3.00 3.01 3.02
7 0.60 1.74 2.40 2.50 2.55 2.57 2.58 2.59
8 0.52 1.52 2.10 2.18 2.23 2.25 2.26 2.27
9 0.46 1.34 1.87 1.94 1.98 2.00 2.00 2.02

10 0.42 1.21 1.69 1.75 1.78- 1.80 1.80 1.81
11 0.88 1.10 1.58 1.59 1.62 1.63 1.64 '1.65
12 0.34 1.00 1.41 1.45 1.48 1.50 1.50 1.51
18 0.82 0.98 1.80 1.84 1.87 1.88 1.88 1.39
14 0.29 0.86 1.21 1.25 1.27 1.28 1.29 1 29“
15 0.27 0.80 1.18 1.16 1.19 1.20 1.20 1.21
16 0.26 ' 0.75 1.06 1.09 1.11 1.12 1-12 _1.13.
17 0.24 0.70 1.00 1.02 1.05 1.06 1.06 1.07

18 0.23 0.67 0.95 0.97 0.99 1.00 1.00 1.01

Number

of
Lumps

10
11
12
18
14
15
16
17

18

31.19

17.41

12.01

9.15

7.38

6.19

5.33

4.67

4.16

3.75

3.42

3.14

2.90

2.69

2.52

2.36

2.22

32.83

18.84

13.17

10.11

8.20

6.90

5.96

5.24

4.67

4.22

3.53

3.27

3.04

2.84

2.66

83

Table C.1.3 First Eigenvalue
ms/md-10.0

Stiffness Ratio (Rd/ks)

33.17

19.11

13.38

10.28

8.35

7.03

6.07

5.34

4.76

4.30

3.92

3.60

3.33

3.10

2.90

2.72

2.56

2.42

33.20

19.15

13.41

10.31

8.36

7.05

6.09

5.35

4.78

4.32

3.93

3.61

3.34

3.11

2.91

2.73

2.57

2.43

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

33.24 33.24 33.24 33.24

19.17

13.43
10.32
8.38
7.06
6.09
5.36
4.78
4.82
3.94
8.62
3.35
8.11
2.91
2.78
2.57

2.43

19.17

13.43

10.33

8.39

7.06

6.10

5.36

4.79

4.32

3.94

3.62

3.35

3.11

2.91

2.73

2.57

2.43

19.18
18.44
10.88
8.89
7.06
6.10
5.87
4.79
4.32
3.94
3.62
8.85
8.12
2.91
2.78
2.58

2.43

19.18

413.44

10.33

8.39

7.06

6.10

5.37

4.79

4.32

3.94

3:62

3.35

3.12

2.91

2.73

2.57

2.43

84
Table C.2.1 Second Eigenvalue

ms/md=0.1
Number
of Stiffness Ratio (Rd/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

2 17.99 10.76 7.21 8.50 9.22 9.60 9.81 9.94
3 12.37 7.03 3.50 3.80 4.12 4.30 4.41 4.48
4 9.41 5.19 2.15 2.17 2.32 2.42 2.49 2.53
5 ' 7.58 4.09 1.50 1.41 1.49 1.55 1.60 1.62
6 6.35 3.38 1.12 0.99 1.04 1.08 1.11 1.13
7 5.46 2.87 0.89 0.74 0.76 0.79 0.81 0.83
8 4.79 2.50 0.73 0.57 0.59 0.61 0.62 0.63
9 4.26 2.21 0.61 0.46 0.46 0.48 0.49 0.50

10 -3.84 1.98 0.52 0.38 0.38 0.39 0.40 0.41
11 3.50 1.79 0.46 0.32 0.31 0.32 0.33 ” 0.33
12 3.21 1.64 0.41 0.27 0.26 0.27 0.28 0.28
13 2. 96 1. 51 0. 36 0. 23 O. 22 0. 23 0. 24 0. 24
14 2.75 1.40 0.33 0.20 0.19 0.20 0.20' 0.21‘
15 2.57 1.30 0.30 0.18 0.17 0.17 0.18 0.18
16 2.41 1.22 0.28 0.16 0.15 0.15 0.15 .0.16.
17 2.27 1.14 0.25 0.14 0.13 0.13 0.14 0.14

18 2.15 1.08 0.24 0.13 0.12 0.12 0.12 0.12

85
Table 0.2.2 Second Eigenvalue

ms/md-1.0
Number
of Stiffness Ratio (kd/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

2 16.18 6.86 9.07 11.58 12.51 12.96 18.22 18.86
8 11.12 4.86 8.92 5.82 5.96 6.25 6.44 6.58
4 8.46 8.16 2.18 8.06 8.52 8.75 8.86 8.92
5 6.81 2.47 1.89 2.00 2.34 2.50 2.60 2.65
6 5.70 2.02 0.97 1.42 1.69 1.81 1.90 1.98
7 4.89 1.71 0.71 1.06 1.29 1.38 1.46 1.49
8 4.28 1.48 0.55 0.88 1.02 1.10 1.16 1.19
9 8.81 1.30 0.43 0.67 0.83 0.90 0.96 0.97

10 3.43 1.16 0.35 0.55 0.70 0.75 0.80 0.82
11 3. 1 1 1. 05 0. 29 0. 46 0. 59 0. 64 0. 69 0. 70
12 2.85 0.96 0.25 '0.40 0.51 0.56 0.60 0.61
13 2.63 0.88 0.21 0.34 0.45 0.49 0.53 0.54
14 2.44 0.81 0.18 0.30 0.40 0.43 0.47 0.48
15 2.27 0.76 0.16 0.27 0.36 0.39 0.43 0.43
16 2. 13 0. 71 0. 14 0. 24 0. 333 0. 35 O. 39 0. 39
17 2.00 0.66 - 0.13 0.21 0.30 0.32 0.35 0.36

18 1.89 0.62 0.11 0.19 0.27 0.29 0.32 0.33

86
Table 0.2.3 Second Eigenvalue

ms/md-10.0
Number
0* Stiffness Ratio (Id/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

2 2.33 20.06 25.48 26.12 26.32 26.41 26.46 26.48
3 1.37 10.99 15.82 16.39 16.57 16.65 16.69 16.72
4 0.98 7.19 11.25 11.74 11.89 11.96 12.00 12.02
5 0. 77 5. 23 8. 65 9. 08 9. 21 9. 27 9. 30 9. 32
6 0.63 4.06 7.00 7.37 7.49 7.54 7.57 7.58
7 0.53 3.29 5.87 6.19 6.30 6.34 6.36 6.38
8 0.46 2.76 5.05 5.34 5.43 5.47 5.49 5.50
9 0.40 2.37 4.42 4.68 4.76 4.80 4.82 4.83

10 ' 0.36 2.07 8.98 4.17 4.24 4.28 4.29 4.30
11 0.88 1.88 3.54 8.76 8.82 8.85 8.87 ‘ 3.88

12 0.30 1.65 8.21 3.42 3.48 8.51 8.52 8.58
18 0.27 1.49 2.94 8.18 8.19 8.22 8.28 3.24
14 0.25 1.86 2.72 2.89 2.95 2.97 2.98 2.99
15 0.24 1.25 2.52 2.69 2.74 2.76 2.77 2.78
16 0.22 1.16 2.35 2 51 2.56 2.58 2.59 2.60
17 0.21 1.08 2.20 2.85 2.40 2.42 2.43 2.43

18 0.20 1.01 2.07 2.21 2.25 2.27 2.28 2.29

87

Table 6.3.1 Third Eigenvalue

ms/md-O.1
Number
of Stiffness Ratio (Rd/ts)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

8 20.82 19.58 13.83 15.01 15 95 16.45 16.74 16.91
4 15.15 18.80 8.65 8.58 9.06 9.87 9.56 9.68
5 11.80 10.54 6.10 5.61 5.84 6.08 6.16 6.24
6 9.68 8.46 4.62 8.99 4.09 4.21 4.29 4.35
7 8.10 7.04 8.66 8.00 8.02 8.10 8.16 8.20
8 6.99 6.01 8.01 2 85 2.88 2.88 2.42 2.45
9 6.18 5.28 2.58 1.90 1.85 1.88 1.91 1.94

10 5.46 4.63 2.18 1.58 1.51 1.53 1.55 1.57-

11 4.92 4.15 1.90 1.33 1.26 1.27 1.28 1.29

12 4.48 3.75 1.68 1.14 1.06 1.07 1.08 1.09

13 4.10 3.43 1.51 0.99 0.91 0.91 0.92 0.92

14 3.79 3.15 1.36 0.87 0.79 0.78 0.79 0.80

15 3.52 2.92 1.24 0.78 0.69 0.68 0.69 0.69

16 3.28 2.71 1.14 0.70 0.61 0.60 0.60 0.61

17 3.07 2.54 1.05 0.63 0.55 0.53 0.53 0.54

18 2.89 2.38 0.98 0.57 0.49 0.48 0.48 0.48

C

Table C.3.2 Third Eigenvalue

ms/md-1.0
Number
of Stiffness Ratio (Rd/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

3 20.82 18.80 13.68 15.93 17.15 17.62 17.99 18.12

4 15.16 13.29 8.04 8.94 9.79 10.12 10.42 10.51
5 11.82 10.14 5.47 5.72 6.32 6.53 6.77 "6.82
6 9.65 8.14 4.04 3.98 4.42 4.55 4.76 4.78
7 8.13 6.77 3.14 2.94 3.27 3.35 3.53 3.53
8 7.01 5.78 2.54 2.26 2.53 2.57 2.72 2.71"'
9 6.16 5.04 2.11 1.79 2.02 2.03 2.17 2.15

10 5.49 4.45 1.79 1.46 1.65 1.64 1.77 1.74
11 4.95 3.99 1.55 1.21 1.38 1.35 1.48 1.44
12 4.51 3.61 1.37 1.02 1.17 1.14 1.25 1.21
13 4.14 3.30 1.21 0.87 1.01 0.97 1.08 1.03
14 3.82 3.03 1.09 0.76 0.88 0.83 0.94 0.89
15 3.55 2.81 0.99 0.66 0.78 0.72 0.82 0.78
16 3.32 2.61 0.90 0.59 0.70 0.63 0.73 0.68
17 3.11 2.44 0.83 0.52 0.63 0.56 0.65 0.60

18 2.93 2.30 0.77 0.47 0.57 0.50 0.59 0.54

Number
of
Lumps

10
11
12
18
14
15
16
17

18

89

Table 0.3.3 Third Eigenvalue
ms/md810.0

Stiffness Ratio (id/ks)

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

20.12

14.63

11.39

9.

7.

28

81

.73

.91

.26

.74

.31

.95

.64

.38

.15

.96

.78

10.49 23.00 24.91 25.46 25.68 25.80 25.87

5.

4.

84

10

.15

.54

.12

.82

.59

.41

.27

.15

.06

.97

.91

85

.80

14.41 16.25 16.77 17.00 17.11 17.18
9.96 11.62 12.10 12.30 12.41 12.47
7.38 8.85 9.28 9.46 9.55 9.61
5. 74 7. 05 7. 43 p 7. 59 7. 68 7. 73
4.63 5.80 6.15 6.29 6.37 6.42
3. 84 4. 90 5. 21 5. 34 5. 42 5. 46
3. 25 4. 22 4. 51 4. 63 4. 69 4. 73
2.80 3.69 3.96 4.07 4.13 4.16
2.45 3.27 3.52 3.62 3.68 3.71
2. 17 2. 93 3. 16 3. 26 3. 31 3. 34
1.95 2.65 2.87 2.96 3.01 3.03
1.76 2.42 2.62 2.70 2.75 2.77
[.60 2.22 2.41 2.49 2.53 2.56
1.46 2.05 2.23 2.30 2.34 "2.37

1.34 1.90 2.07 2.14 2.18, 2.20

90

Table 0.4.1 Fourth Eigenvalue

ms/md=0.1
Number
of Stiffness Ratio (Rd/ts)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

4 22.25 22.49 18.61 19.11 20.05 20.58 20.89 21.08
5 16.80 16.79 18.19 12.48 12.98 18.85 18.58 18.74
6 18.82 13.19 9.97 8.89 9.10 9.34 9.51' 9.63
7 10.95 10.75 7.88 6.71 6.75 6.90 7.02 7.11
8 9.25 9.02 6.48 5.28 5.22 5.81 5.89 5.46
9 7.97 7.78 5.88 4.28 4.16 4.21 4.27 (4.32..

10 6.99 6.74 4.60 8.55 8.89 8.42 3.46 8.50
11 6.21 5.97 3.99 3.01 2.88 2.88 2.87 2.90
12 5.58 5.84 8.51 2.58 2.40 2.89 2.41 2.43
18 5.06 4.82 3.13 2.25 2.06 2.04 2.05 2.07
14 4.63 4.40 2.81 1.98 1.79 1.76 1.77 1.78
15 4.26 4.03 2.55 1.77 1.57 1.54 1.54 1.55
16 3.94 , 8.72 2.82 1.59 1.39 1.85 1.35 1.86
17 8.67 3.46 2.18 1.43 1.24 1.20 1.20 1.20

18 3.43 3.22 1.97 1.30 1.11 1.07 1.07 1.07

91

Table 0.4.2 Fourth Eigenvalue

mslmd-1.0
Number
of Sti9fness Ratio (Rd/k5)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

4 22.34 22.43 18.13 19.48 20.70 21.15 21.57 21.67
5 16.90 16.73 12.63 12.55 13.42 13.74 14.12 14.18
6 13.42 13.14 9.50 8.82 9.41 9.61 9.93 9.95.
7 11.06 10.71 7.48 6.57 6.97 7.08 7.36 7.35
8 9.36 8.99 6.10 5.11 5.39 5.43 5.68 5.64
9 8.09 7.70 5.10 4.10 4.30 4.29 4.51 4.46

10 7.10 6.72 4.36 3.37 3.52 3.47 3.68 3.62
11 6.33 5.95' 3.79 2.83 2.94 2.87 3.06 2.99
12 5.70 5.32 3.33 2.42 2.50 2.41 2.59 2.51
13 5.18 4.81 2.97 2.09 2.15 2.05 2.22 2.13
14 4.74 4.39 2.67 1.83 1.88 1.76 1.93 1.83
15 4.36 4.03 2.43 1.62 1.66 1.53 1.69 1.59
16 4.06 3.72 2.22 1.44 1.48 1.34 1.50 1.40
17 3.78 3.45 2.04 1.29 1.33 1.19 1.34 1.23

18 3.54 3.22 1.89 1.17 1.20 1.06 1.20 1.09

Number

of
Lumps

10
11
12
13
14
15
16
17

18

22.23
16.77
13.30
10.93
9.23
7.95
6.97
6.19
5.56
5.05
4.61
4.24
3.93

8.65

19.69

14.84

11.68

9.52

7.97

6.82

5.94

5.24

4.68

4.23

3.84

3.52

3.25

3.01

2.80

92

Table 6.4.3 Fourth Eigenvalue
me/md-10.0

Stiffness Ratio (Rd/ks)

21.69

13.85

9. 49

6.88

5.20

4.07

3.26

2.67

2.23

1.83

1.61

1.39

1.21

1.06

0.94

24.86

16.92

12.26

9.31

7.33

5.94

4.92

4.15

3.56

3.09

2.71

2.41

42.15

21.94

25.78

17.84

13.12

10.09

8.05

6.59

5.52

4.70

4.07

3.56

3.15

2.82

2.54

2.30

2.10

26.15

18.22

13.48

10.43

8.35

6.87

5.78

4.94

4.29

3.77

3.35

3.00

2.71

2.46

2.25

26. 84
18. 41
18. 66
10:59
8. 50
7. 01
5. 91
5. 06
4. 40
8. 88
8. 45
8. 09
2. 80
2. 55

2.. 88

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

26. 46
18. 52 "

13.76

'10.69

8.60
7.10
5.99
5.14
4.47
3.94
3.51
3.15
2.85
2.60

2.38

Number

of
Lumps

10
11

12
18
14
15
16
17

18

23.70

18.47

14.92

12.40

10.54

9.11

8.00

7.11

6.38

5.78

5.27

4.84

4.47

4.15

24.11

19.71

15.06

12.47

10.56

9.10

7.97

7.06

6.32

5.71

5.20

4.76

4.39

4.07

Table 0.5.1 F19th Eigenvalue

Stiffness Ratio (ho/he)

22.25

16.83

13.25

10.76

8.95

7.60

6.56

5.74

5.08

4. 54

4.09

3.71

3.39

3.11

93

ms/md=0.1

21.89

15.56

11.77

9.27

7.53

6.26

5.31

4.57

3.98

3.51

3.13

2.81

2.54

2.31

22.75

16.00

11.89

9.20

7.35

6.01

5.02

4.26

3.66

3.18

2.80

2.48

2.21

1. 99

23.26

16.38

12.14

9.36

7.43

6.05

5.02

4.23

3.62

3.13

2.73

2.41

2.14

1.91

28. 57
16. 68
12. 88
9. 50
7. 54
6. 12
5. 07
4. 27
8. 64
8. 14
2. 78
2. 40

2.13

1.90'

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

28.76
16.79
12. 46
9. 60
7. 61
6. 18
5. 11
4. 80
8. 66
8. 16
2. 75
2. 41
2. ‘14

1.90"

Number

of
Lumps

1o
11
12
18
14
15
16
17

18

23.85

18.63

15.09

12.58'

10.72

9.30

8.18

7.29

6.56

5.96

5.46

5.03

4.66

0.6169 2.4674

24. 11
18. 71
15. 05
12. 46
10. 55
9. 10
6. 96
7. 06
6. 82
5. 71
5. 20
4. 76
4. 89

4.07

Table C.5.2 Fifth Eigenvalue

Stiffness Ratio (id/Re)

21.93

16.63

13.14

10.72

8.96

7.65

6.63

5.83

5.19

4.66

4.23

3.86

3.55

3.28

94

ms/md-l.0

22.06 23.23

15.53
11.64
9.11
7.36
6.09
5.14
4.40
3.83
3.36
2.98
2.67
2.41

2.19

16.38

12.18

9.45

7.56

6.20

5.20

4. 48

3.83

3.35

2.96

2.65

2.38

9.8696 22.206 39.478 61.685 88.820

23.59 24.08

16.61

12.30

9.46

7.50

6.08

5.03

4.23

3.61

3.11

2.71

2.38

2.11

1.88

17. 09
12. 78
9. 85
7. 85
6. 41
5. 84
4. 52
8. 89
8. 88
2. 97
2. 68
2. 85

2.12

120.90

24.11
17.07

12.67

5.19
4.36
3.71
3.20
2.78

2. 44'

95
Table 6.5.3 Fifth Eigenvalue

ms/md-l0.0
Number
of Stiffness Ratio (Rd/ks)
Lumps

0.6169 2.4674 9.8696 22.206 39.478 61.685 88.820 120.90

5 23.70 23.85 21.93 25.17 26.36 26.86 27.11 27.26
6 18.47 18.46 14.95 17.89 19.10 19.61 19.86 20.01
7 14.92 14.83 10.89 13.29 14.43 14.92 15.17 15.31
8 12.40 12.26 8.37 10.23 11.28 11.74 11.98 12.11
9 10.53 10.37 6.68 8.10 9.06 9.49 9.71 9.83
10 9.11 8.94 5.49 6.56 7.44 7.83 8.04 8.16
11 8.00 7.81 4.61 5.42 6.22 6.59 6.78 6.89
12 7.10 6.92 3.95 4.55 5.28 5.62 5.80 5.90
13 6.37 6.19 3.43 3.87 4.55 4.86 5.03 5.12
14 5.77 5.59 3.01 3.33 3.95 4.25 4.40 4.49
15 5.26 5.08 2.67 2.89 3.47 3.76 3.90 3.98
16 4.83 4.66 2.39 2.53 3.07 3.33 3.47. 3.55.”
17 4.46 4.29 2.16 2.24 2.74 2.99 3.12 3.19
18 4.14 3.98 1.96 1.99 2.46 2.69

2.82

2.89

BIBLIOGRAPHY

10.

BIBLIOGRAPHY

Timoshenko. 8.. Young, D.B.. and Weaver. '.. Vibration Problems in
Engineering, John Wiley and Sons. Inc.. 1974.

Skelton, 3.8.. "Observability Measures and Performance Sensitivity
in the Model Reduction Problem". Int. J. Control. 29(1979).
pp541-556.

Aoki, M.. "Control of Large-Scale Dynamic Systems by Aggregation”.
IEEE Trans-AC. 13(1968). pp173-182.

Aoki, I.. "Some Approximation lethods for Estimation and Control
of Large Scale Systems". IEEE Trans-AC. 23(1978). pp173-182.

Davison. E.J.. ”A Bethod for Simplifying Linear Dynamic Systems”.
IEEE Trans-AC, 11(1966). pp93-101.

Davison. 8.1.. "A Nev Iethod for Simplifying Large Linear Dynamic
Systems". IEEE Trans-AC (Correspondence). 13(1968).pp214-215.

Jacquot. 3.6.. and Soedel, '.. "Vibrations of Elastic Surface
Systems Carrying Dynamic Elements". J. Acoust. Soc. Am.,
47(1970).pp1354-1358.

Young. D.. ”Vibration of a Beam with Concentrated lass, Spring.
and Dashpot”. Trans ASHE, J. Appl. lech.. 70(1948). 9965-72.

Crawford, C.R., ”A Stable Generalized Eigenvalue Problem". SIAI J.
Numer. Anal.. 6(1976). pp854-860.

Rosenberg. R.C.. "A Users Guide to ENPORI‘S". The Case Center for

.Computer Aided Design, Michigan State University. East Lansing.

II.. 1983.

96

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

97

Dix. 3.0.. "A Users lanual for IEDUSA (IEchanism-Dynamics
-Universal-System-Analyzer)", lechanics. Iechanical and Aerospace
Engineering Department. Illinois Institute of Techn010gy. Chicago.
111.. Jan.. 1975.

Goudreau. G.L.. ”Evaluation of Discrete Methods for the Linear
Dynamic Response of Elastic and Viscoelastic Solids". SESI Ept
69-15. University of California at Berkeley (1970).

General References

Rayleigh, J.I.S., The Theory of Sound, V01. I, Dover
Publications. Inc.. 1945.

Hildebrand, F.B., Finite Difference Equations and Simulations.
Prentice-Hall. Inc.. 1968.

Parlett. B.N.. The Symmetric Eigenvalue Problem. Prentice-Hall,
Inc.. 1980.

Wilkinson, J.E.. The Algebraic Eigenvalue Problem, Oxford
University Press. 1965.

Sandell. N.E., Varaiya, P., Athens, M., and Safonov. I.G.. "Survey
of Decentralized Control lethods for Large Scale Systems". IEEE
Trans-AC, 23(1978). pp108-128.

Kwatny. 3.6.. and Mablckos. V.E.. "The Modelling of Dynamical
Processes". IEEE Decision and Control Conference. Houston. 1975.

lilne. R.D.. "The Analysis of Weakly Coupled Dynamical Systems".
Int. J. Control, 2(1965). pp171-199.

Dovell. E.B., "On Some General Preperties of Combined Dynamical
Systems", Trans ASHE, J. Appl. lech.. 40(1970). pp206-209:

Belytschko, T;. and lindle, I.L., "Flerural lave Prepagation

Behavior of Lumped lass Approximations”. Camp. and Struc..
12(1980). pp805-812.

Crede. C.E., and Harris. C.M.. Shock and Vibration Handbook.
Vol. I. IcGrav-Hill, 1961.