LINEAR AND AOTOPARAMETRIC ^ ODAL ANALYSIS

OF AEROELASTIC STRUCTURAL SYSTEMS

by

TOMffY DALE VTOOOALL, B . S . i n M.E.

A THESIS

IN

MECHANICAL ^GINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

MECHANICAL ^GINEERING

Approved

Accepted

May, 1984

he-"7— '

tJa.Jt^

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to Dr.

R.A. Ibrahim for his patient guidance of this thesis and to

the other members of my committee. Dr. H.J. Carper and Dr.

J.W. Oler, for their helpful criticism. I am also grateful

for the assistantship provided by Dr. J.H. Lawrence through

the Department of Mechanical Engineering and for the support

provided by the ASME Auxilary. I would also like to thank

Mr. Hun Heo for his technical contributions to this thesis.

My thanks is also given to Mrs. Rebecca Yochcun for her help

in preparing this manuscript.

11

CONTENTS

ACKNOWLEDGMENTS ii

ABSTRACT iv

LIST OF FIGURES vi

I. BACKGROUND AND SURVEY OF RECENT LITERATURE 1

Parametric Vibration 1

Nonlinear Coupling (Autoparametric Resonance) 9

Scope of Present Research 19

II. FORMULATION OF THE PROBLEM 23

III. LINEAR MODAL ANALYSIS 29

Eigenvalues 29

Mode Shapes 36

IV. AUTOPARAMETRIC MODAL INTERACTION 41

Asymptotic Approximation Solution 41

Resonance Conditions 44

Analytical Solutions 46

Numerical Solutions 53

V. CONCLUSIONS AND RECOMMENDATIONS 63

LIST OF REFERENCES 65

APPENDICES 69

A. KINETIC ENERGY DERIVATION 69

B. COEFFICIENTS 7 2

C. CSMP PROGRAM LISTINGS 76

111

ABSTRACT

This investigation deals with the linear modal emalysis and auto-

parcunetric interaction of aeroelastic systems such as an airplane

fuselage emd wing with fuel storage. The mathematical modeling is

derived by applying Lagrange's equations taking into consideration the

Christoffel symbol of the first kind to account for the nonlinear

coupling of the system coordinates, velocities, euid accelerations.

The linear modal amalysis will be obtained by considering the

linear, conservative portion of the equations of motion. The normal

mode frequencies eind the associated mode shapes are obtained in terms

of the system parameters. The main objective of the linear analysis

is to explore the critical regions of autoparametric (or internal)

resonance conditions, 2ku)j| = 0 (where k^ are integers and o) are the

normal mode frequencies). The results show that for certain system

parameters the condition of internal resonance is satisfied.

The dynamic behavior of the structure in the neighborhood of

internal resonance conditions is obtained by considering the nonlinear

coupling of the normal modes. The asymptotic approximation technique

due to Struble is employed. Three groups of internal and normal reso­

nance conditions are obtained from the secular terms of the first-

order perturbational equations.. The transient cuid steady-state

responses cure obtained numerically by using the IBM Continuous System

Modeling Program (CSMP) with double precision Milne integration. The

transient response shows a build up in the interacted modes to a level

IV

which exceeds the steady-state response. In addition, the excited

mode is suppressed by virtue of the nonlinear feedback of other modes.

Under certain conditions, the steady-state response is derived cuialy-

tically.

It is concluded that the nonlinear modal analysis reveals certain

types of response characteristics which cannot be interpreted within

the framework of the linear theory of small oscillations.

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1,5

Figure 1,6

Figure 1.7

Figure 1.8

Figure 1 .9

Figure 1.10

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Simple pendulum subject to parametric excitation,

Stability diagram for a system with parametric excitation.

Mode shape of a cantilever under combination resonance S = o) + au.

Stability boundaries of a structure under com­bination resonance.

An elastic pendulum as an example of an auto­parametric system.

Two degree-of-freedom autoparametric system.

Chaotic response of a system with periodic excitation.

Fluid filled structure on an elastic support.

The autoparametric vibration absorber.

Schematic diagram of aeroelastic system with its coordinates.

Dependence of ov, on ^']-\/'^2 ^ ° ^ various values of a)22/< 33-

Dependence of 0)3 on ^-^^/^^ ^ ° ^ various values

of W22/'**33-

Dependence of 0)3 on w^^/uij^ for various values

of t»)22/' 33-

Variation of natural frequency combinations with (0. ./o)-- for S2/'**33 = ^'^ ^^ compared to m^.

Variation of natural frequency combinations with

'**11/ 33 ^ ° ^ ' 22/'* 33 ">-O as compared to in^*

Linear mode shape for system excited at ^ = cu ; ^ = 2 5 , * = 5 , 0 = .402, a = .112.

2

4

11

13

15

18

20

21

31

32

33

34

35

38

VI

Figure 3.7

Figure 3.8

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Linear mode shape for system excited at il = 0^2' 5 = 25, * = 5, e = .402, a = .112.

Linear mode shape for system excited at J = 0)3; C = 25, $ = 5, 3 = .402, a = .112.

Steady-state response for r 3 + r23 = r^^ = nv.

Steady-state response for r23 ~ ^13 ~ ^ 33 ~ ^^'

Steady-state response for 2r-|3 = r33 = nv.

CSMP time history for fi = r33 = Ji 3 + ^23 (ni/n2 * r^/r2).

CSMP simulation of quasi-steady-state response for r 3 + r23 = r33 = fl (n^/n2 + r^/r2).

CSMP time history for i2 = r33 = 123 " ^13 (m/n2 + r^3/r23).

CSMP simulation of two mode interaction, r33 = 2r^3 = nv.

Response of normal coordinate equations motion as determined by CSMP with ^ = r33 = Ji 3 + 23' q = 2 = •0'"' = 'O ' = •°'' •

Response of general coordinate equations motion as determined by CSMP with Q = r33 = r 3 + r23, 5 = i;2 = .01, C3 = .05, e = .01 .

39

40

49

52

54

55

57

58

59

61

62

VI1

CHAPTER I

BACKGROUND AND SURVEY OF RECENT LITERATURE

Parametric Vibration

In the field of structural dynamics, many problems exist in which

an oscillating system possesses time-varying parameters. A simple pen­

dulum subject to vertical, harmonic motion of its support is an

example of such a system (see Figure 1.1). The governing equation of

motion for this system is (sin0 = 0):

0 + [0) 2 _ (yQj ,2/L) cosoJt] 0 = 0 (1.1)

where dots indicate differentiation with respect to time, L is the

length of the pendulum, 0 is the angular displacement of the pendulum

from the vertical, U) = (g/L)V2 is the natural frequency of the pen­

dulum, Yn ^^ ^® amplitude of the support motion, and o) is the fre­

quency at which the support oscillates.

For the system described by equation (1.1), the time-variation of

the stiffness term is an explicit function of time, and the system is

said to be subject to "parametric excitation" because of the explicit

time dependence of the system parameters. Unlike forced excitation,

where the excitation appears on its own on the right-hand side of the

system equation of motion, the excitation in system (1.1) appears as a

coefficient of the system coordinate in the equation of motion. It

should be noted that the explicit function of time need not be har­

monic for parametric excitation to occur. The instabilities asso­

ciated with the parametrically excited, simple pendulum and other

Figure l.i. simple pendulum subject to parametri c excitation.

dulum and other systems under parametric excitation are well docu­

mented [1-5]. The main feature of the dynamic behavior of such sytems

is that their equilibrium position becomes unstable if the excitation

frequency, u), is twice the natural frequency of the system, oi . The

conditon O) = 200^ is referred to as the principal parametric resonance,

which is different from normal resonance, u) = ui^, in forced vibra­

tions. For parameteric instabilities to occur, a small initial con­

dition must be given to the system. Figure 1.2 shows the stability-

instability boundaries for harmonic, parametric resonances.

For multi degree-of-freedom systems, like the cantilever beam

shown in Figure 1.3, combination parametric resonance can occur for

the conditions o) = l | uj jf u). I (i + j), where ui^ and u). are the n

natural frequencies of the ith and jth modes, respectively, and n is

an integer. Figure 1.4 gives the stability boundaries of a structure

under combination parametric resonance.

A linear analysis defines the regions of instability at which the

amplitudes of parametric systems increase exponentially. Such an

analysis is not adequate in determining the steady-state amplitude-

frequency response (if it exists) for parametric systems. The

amplitude-frequency response can only be determined by including the

inherent nonlinearities of the system.

An extensive survey on parametric and autoparametric vibrations

is presented by Ibrahim and Barr [4,6] . This survey includes the

mechanics of linear problems [4] as well as nonlinear problems [6]. A

continuation of this work by Ibrahim [7,8] contains reviews of the

CN

Figure 1.3. Mode shape of a cantilever under combination resonance 2 « ut " ^ 2*

Unstable Reaions

rr / r

l \

1/3 1/2 •£L CJi + CJ2

Figure 1.4. St2ibility boxindaries of a structure under com­bination resonsmce.

current problems of free surface of liquids in closed containers,

rods, beams, pipes, plates, shells, pendulum systems, shafts, mecha­

nisms, machine components, missiles, satellites, and hydroelastic and

aeroelastic systems. In the fifth part of the review, Ibrahim and

Roberts [9] considered stochastic problems in parametric vibrations.

The trend of recent investigations is to develop more refined analyti­

cal models along with numerical solutions. A summary of recent work

is presented below.

The response of a two degree-of-freedom system with multifrequency,

parametric excitations was considered by Nayfeh [10]. The method of

multiple scales was utilized to investigate the stability regions of

the system under combination summed and principal resonances, com­

bination difference and principal resonances, and combination summed

and difference parametric resonance conditions. For the cases in

which only one of the parametric resonance conditions was fulfilled,

the predicted results were confirmed by a previous work by Nayfeh and

Mook [11],

Yamamoto, et al [12] investigated the cases of summed-and-

differential type harmonic oscillation in a simply supported beam.

Theoretical results indicated that only the summed-type oscillation

should occur, but the experimental work showed that both summed-and-

differential harmonic oscillations occurred.

Subharmonic oscillations in a beam supported at both ends was

investigated theoretically and experimentally by Yamamoto, et al [13],

Subharmonic oscillations of orders 1/2 and 1/3 were predicted ana-

lytically. The existence of these oscillations was confirmed experi­

mentally,

A study of unsymmetrical shafts was presented by Yamamoto, et al

[14] . They concluded the following: the characteristics of summed-

and-differential harmonic oscillation were similar to those of subhar­

monic oscillation; oscillation phenomena which differ from those of a

round shaft occurred; and varying system parameters gave both hard and

soft spring-type solutions. The experimental results were qualitati­

vely consistent with the theoretical results,

Yamamoto, et al [15] investigated subharmonic and summed-and-

differential harmonic oscillations in an unsymmetrical rotor. The

presence of nonlinearities in the unsymmetrical case caused the

results to be much different than those found in the symmetrical case.

The qualitative characteristics of the oscillations of the unsym­

metrical rotor were both theoretically and experimentally the same as

the previously mentioned unsymmetrical shaft system.

Zajaczkowski [16] used the Galerkin method to obtain the solution

to the general Mathieu equation. The analytical solutions were con­

firmed by numerical integration. The results were in agreement with a

previous work by Zajaczkowski and Lipinski [17],

Takahashi [18] investigated the instability of multi degree-of-

freedom, dynamic systems under parametric excitation. A vector solu­

tion was expanded in a Fourier series, and the harmonic balance

technique was used to develop the characteristic equations. Numerical

results were presented for the damped and undamped Mathieu equation.

and comparison was made to results obtained by Szemplinska-Stupnicka

[19], Hsu [20], and Bolotin [5]. The results were in agreement with

those obtained from Floquet Theory, which involves the transformation

into a set of linearly independent solutions of a function which can

be represented in the form of a converging Fourier trigometric series

[5],

Struble's method was used by Stanisic [21] to study the stability

of the generalized Hill's equation with three independent parameters.

The solution, which is valid for a system equation with periodic coef­

ficients, gave an explicit form of the stability regions to any order

of approximation. The first- and second-order approximations were

reasonably comparable,

Sato, et al [22] investigated the parametric response of a hori­

zontal beam carrying a concentrated mass. An approximate solution was

obtained by using the Galerkin method and the harmonic balance tech­

nique. The results showed that the parametric resonance occurred

along with the forced resonance because of the initial static deflec­

tion of the beam,

Szemplinska-Stupnicka [19] generalized the harmonic balance

method to multi degree-of-freedom, parametric, dynamic systems. The

method was applied to a two degree-of-freedom system. The bound«ries

of the principal and combination parametric resonances were calculated

and verified by analog computer simulation.

The method of Bogoliubov-Mitropolski was used by Hsu [23] to eva­

luate the response amplitude of parametric systems. Hsu considered a

system which involved time-varying damping and stiffness terms and was

subject to forced, periodic excitation. The first- and second-order

approximation of the stability regions were in agreement with the

results obtained by numerical integration.

Nonlinear Coupling (Autoparametric Resonance)

For the systems considered thus far, the parametric instabilities

have been associated with equations whose variable coefficients are

explicit functions of time, but in many instances the time variation

of system parameters is induced through implicit time dependencies,

i,e,, the variation depends on the motion of another normal mode. For

a system with two degrees-of-freedom, the equations of motion can take

the form

X + f^^ ^X-\ = ^ 1 1 ' 1 ' 1 ' 2' ^2' ^2^ (. . . . . . . , \ \ »Z)

^2 •*• ' 2 2 ^^2^^1' ^1' 1' ^2' ^2' ^2^

where, again, dots indicate differentiation with respect to time, X-j

and Xo are two normal coordinates of the system, u) and oi^ are the

corresponding linear natural frequencies, F and F2 are nonlinear

functions of the normalized coordinates, and e is a small parameter of

the system. The function, F , may contain terms of the form X.,X2/

etc, and Fo ^^V involve terms like X2X^ , etc. Both X., and X2 are

implicit functions of time and act as parametric excitations to the

first and second modes, respectively, A system whose motion can be

described by equations like equations (1 .2) is said to possess auto­

parametric coupling [24], since the time variation of the system para-

10

meters is induced by the nonlinear terms involving the system coor­

dinates and their derivatives with respect to time, Nonlinearities

can enter into a system model through inertial terms and elastic

restoring forces, Inertial nonlinearities can be introduced into a

system through the kinetic energy in the Lagrangian formulation of the

equations of motion. Elastic nonlinearities originate in nonlinear

strain-displacement relations.

The presence of the nonlinear terms can lead to a certain type of

instability which is referred to as internal, or autoparametric, reso­

nance of the form:

n 1

kj w = 0 (1,3)

i = 1

where the k- are integers and the OJ. are the normal mode natural fre­

quencies of a system of n degrees-of-freedom. These instabilities may

result in various forms of dynamic phenomena, such as amplitude jumps

and energy exchange between modes, which cannot be predicted by the

classical theory of small linear oscillations. In nonautonomous

systems, autoparametric resonance occurs when the conditions of exter­

nal and internal resonance occur simultaneously. The elastic pendulum

shown in Figure 1.5 is a classic example involving autoparametric

coupling. Other structures exhibiting phenomena associated with auto­

parametric resonance include a fluid filled tank on an elastic sup­

port, the autoparametric vibration absorber, and an aircraft wing in

flutter [6,25,26] .

11

Figure 1.5, An elastic pendulum as an example of an auto­parametric system.

12

As previously mentioned, the elastic pendulum is an excellent

example of a system which can exhibit the phenomena associated with

nonlinear (thus autoparametric) coupling, Breitenberger and Mueller

[27] used the slow fluctuation technique and obtained conditions for

the energy exchange between the r and 0 modes, which is dependent on

the system parameters. The suspension mode (r) behaved as a driven

harmonic oscillator, and the pendulum mode (0) exhibited charac­

teristics similiar to the simple pendulum described by equation (1,1),

Breitenberger and Mueller noted the resemblance between the elastic

pendulum and a ship in heave, pitch, and roll. The dominant feature

of the results was the autoparametric energy transfer,

Hatwal, et al [28,29] investigated the two degree-of-freedom

system with autoparametric coupling shown in Figure 1.6. For the

forced system with the harmonic excitation, the harmonic balance

method correctly predicted the existence or absence of stable harmonic

and near harmonic solutions, as verified by numerical integration.

For certain frequency ranges, the steady-state response ceased to be

harmonic. For decreasing excitation frequency, the transition from

regions of stable, harmonic solutions to regions of unstable solutions

occurred without any amplitude jumps. In these regions more than one

amplitude-modulated, steady-state solution was possible, depending on

the initial conditions given to the system. For increasing frequency,

this transition was associated with sharp amplitude increases, and

no periodic solutions existed [28] . For the gravity controlled pen­

dulum, when the excitation parameter was increased beyond a critical

13

/ // / ^ y / / / /

h P« cos

r

CJt l"^l 1

Figure 1 .6 . Two degree-of-freedom autoparsuaetric system.

14

value, the responses were found to be nonperiodic. To first-order

approximations, the solutions were found to be inadequate to describe

the stablity of the system, it was noted that the pendulum could be

used to act as a vibration absorber for the primary system when the

system was excited near the resonance condition of oc = 2a) where, o

was the natural frequency of the main mass system and OL was the

natural frequency of the pendulum system [29].

As a continuation of the analyses of the system shown in Figure

1 .6, Hatwal, et al [30] investigated the chaotic responses of the

autoparametric system. Chaotic motion is a form of motion (system

response) which is nonperiodic in nature as shown in Figure 1,7,

They showed that for certain combinations of forcing amplitude and

frequency the responses became random. The existence of this chaotic

motion was verified experimentally. The results of the numerical

integrations were used to obtain mean square values and frequency con­

tent of the response. The statistical parameters were shown to be

independent of the initial conditions and the numerical integration

step size, but dependent on the values of the system parameters. The

response of the primary mass showed a strong periodic component at the

forcing frequency, and the pendulum response had a wider spectrum.

The theory of chaotic responses of deterministic, nonlinear

systems is currently being developed. The classical techniques, such

as averaging, fail to detect these motions [31], Various qualitative

analyses have shown the existence and characteristics of chaotic

motions in deterministic, nonlinear systems, but at the present, there

15

u •s u I 4J

e

09

CO

O

00

c a (0 • 4) C u 0

•H o .u •H (« •U 4J O -H IQ U u <0

0)

9

•H

16

is no theory to predict the range of system parameters for which these

motions can occur. The goal of current investigations is to develop a

chaotic motion parameter (analogous to the Reynolds number in fluid

flow) below which periodic motions would occur and above which

chaotic, nonperiodic motions would be insured. It is thought that

this "chaotic Reynolds number" is a function of driving frequency,

forcing amplitude, and damping for most mechanical systems [32],

The nonlinear response of bowed structures to combination reso­

nances was investigated by Nayfeh [33], The results showed that the

difference-type combination resonances could never be excited for the

system considered. The physical conditions for the occurrence of the

summed-type internal resonance were given. The quenching of com­

bination resonances was verified by numerical integration for cases

including and excluding internal resonance,

Singh [34] used a two-scale perturbation analysis to investigate

quenching in a system of van der Pol oscillators with nonlinear

coupling. The suppression of the excited mode predicted by the analy­

tical technique was confirmed by digital computer simulation,

Yamamoto, et al [35-37] investigated the internal resonance in a

two degree-of-freedom, nonlinear system, Nonlinearities up to fourth

order were considered. When the natural frequencies were in the ratio

1:2, they concluded that the first and second harmonics were strong in

the vicinity of the first internal resonance condition; the response

curves, which consisted of three branches, \</ere slowly varying

periodic functions which were greatly influenced by damping. The

17

coefficients of the third-order terms were found to have a significant

effect on the form of the response curves. In the vicinity of the

higher resonance condition, when the natural frequencies were in the

ratio 1:2, they found that the type of response the system follows was

determined by the initial conditions. The amplitude response for a

single mode was similiar to that of ordinary forced oscillation; and

the response curves were, again, greatly influenced by damping. These

theoretical results were verified by analog computer simulation

[36,37], When the natural frequencies were in the ratio 2:3, they

showed that the vibratory state taken on by the system was dependent

upon the initial conditions. The response curves,in this case, con­

sisted of two branches and were greatly influenced by damping. The

fourth-order term had a significant effect on the character of the

response. These results were verified by analog computer analyses, as

in the previous cases [35],

The effects of autoparametric resonance in a structure containing

a liquid (see Figure 1,8) has been investigated theoretically and

experimentally by Ibrahim and Barr [25,38]. For two-mode interaction

only, the coupling between liquid sloshing and vertical vibration was

shown to be weak. The first-order perturbation solution was not ade­

quate to predict the response of the system. The second-order solu­

tion gave the response of the main mass as that of a single degree-of-

freedom system. The experimental results were in agreement with the

second-order perturbation solution [38] . The experimental results for

three-mode interaction confirmed the internal resonance phenomena

18

K^/4

• 0 COCCUt

Figure 1.8. Fluid filled structure on an elastic support.

19

(amplitude jumps and modal energy exchange) predicted analytically by

Struble's asymptotic approach [2]; however, the experimentally deter­

mined amplitudes were not as great as those determined analytically

[25] .

Haxton and Barr [26] analyzed the resonant response of an auto­

parametric vibration absorber which is shown in Figure 1.9. Their

results indicated that when the main mass was externally excited at a

frequency close to its natural frequency, the coupled beam suppressed

the main mass motion if ta = 2u), where, o), is the natural frequency of

the main mass sytem and (x is the natural frequency of the cantilever

portion of the system. The asymptotic approximation method of Struble

[2] was used to develop the theoretical response of the system, and a

reasonable comparison was observed between theoretical predictions and

experimental results. The main discrepancy was in that the experimen­

tal amplitudes were, in general, greater than those predicted analyti­

cally.

Scope of Present Research

The present investigation deals with linear and nonlinear modal

analyses of aeroelastic systems such as an airplane wing with fuel

storage. Figure 1.10 shows a typical model under vertical, harmonic

excitation. The mathematical modeling of the system is derived by

applying Larange's equations. The linear modal analysis will he

obtained by considering the linear, conservative portion of the

equations of motion. The normal mode frequencies and the associated

20

F(t)

.J

TTnTTTTTTTT

Pigur. 1.9. Th« autoparamatric vibration absorb«r.

21

es a*

(S 6

\ r

CVJ es

cvj

M O u

u CM

s

o

s

CN

cs ;^

^

i^

u

I' CO

a*

;r Csl

s

CO

-*VW— —f

L J OJ CO

u.

/ /

/ e /

s 0) 4-) <n >^ m

o •H 4-1 CO (0

i H V

o 0) ns

<4.| O

E (0

(0 •r-( • "O 03

•H (0 •U C (0 - H S T3 <U

0

o CO o

u 3 0^

22

mode shapes will be analyzed. The main objective of the linear modal

analysis is to explore the critical regions of internal resonance for

which relation (1,3) is satisfied. Should the system properties

fulfill relation (1,3), the dynamicist must conduct a nonlinear ana-

lyis to examine the associated dynamic behavior which may influence the

safe operation of the system.

The nonlinear coupling of the normal coordinates will be exa­

mined by considering quadratic nonlinearities. The asymptotic

approximation technique due to Struble [2] will be employed. Three

groups of internal and external resonance conditions will be obtained

from the secular terms of the first-order perturbational equations for

typical autoparametric resonance conditions. The response of the

system will be given in both the frequency and time domains with ana­

lytical and numerical results presented for both two- and three-mode

interactions.

CHAPTER II

FORMULATION OF THE PROBLEM

The response of the structure in Figure 1,10 can be determined by

considering only one-half of the model with the generalized coor­

dinates q^, q2, and q as shown. The system consists of the main mass

m3, linear spring K^, and dashpot C^. The main mass carries two

coupled beams with stiffnesses K-i and K2 (lengths 1 and I2) and tip

masses m-] and m2. The kinetic energy of the structure can be derived

assuming a static deflection curve for the beams where the axial

displacement at the end of a beam is given by 3qj /51i. The kinetic

energy for the system is given by (details of the derivation are shown

in APPENDIX A):

T = 1/ [mi + m2(1 + (312/21^)2)] q 2 _^_ ^/2m2^2^

+ 1/2[m + m2 + m3] q^'^ + 3/2 m2 ^2 ^^^2

I1

+ [m-, + 1113] c[ q3 + 9m2 I2 f<?i 2^ + 5qiqiq3]

20 1 2

+ 3m2 t q i C [ t q 2 / ^ ••• ^1°^2^3 ••" ^1*^2^3 " ^1^q2^ ( 2 . 1 )

21-

+ 6m2 fq2^2^3 " '^1^2^2^ " • " —

5 1 ' 2 1 , 2

4 (m.| + m2) + 9m2l22

25 16 l i ^ .

'^^W'

+ 9m2 tq i ^2° f2^ " q i * '2^q2^

511^2

23

24

where, terms up to fourth-order are r e ta ined . Neglecting g r a v i t a ­

t i o n a l e f f e c t s , the p o t e n t i a l energy i s given by

V = V2[KiqT2 + K2q2^ + K3^3^1 (2.2)

Applying Lagrange's equations in terms of the generalized

coordina tes

3 3 3 8V

[jk,i]q^qj^ + 8q^ = Q.

j = 1 j=1 k=1

(2 .3)

where, [ j k , i ] i s the Chr i s to f fe l symbol of the f i r s t kind and i s given

by the expression

[ j k , i ] = 1 K j " ^ i k • ^ j k

2 L3qk ^^j ^ i -I

( 2 . 4 )

The metric tensor m^^ and the Christoffel symbol are generally func­

tions of the q^r and for motion about the equilibrium configuration

they can be expanded in a Taylor series about that state. Thus from

inertial sources, quadratic, cubic, and higher-order nonlinearities

can arise. The resulting equations of motion are

"•ll " 12 " 13

n>12 " 22 0

m-| 3 0 m 33.

K 0 0

0 K2 0

0 O K 3J (2,5)

4*.

™2^ ''2

"i-

25

where,

™11

™22

™33

™12

= m 1 + m2 [1 + 2 , 2 5 ( 1 2 / 1 ^ ) 2 ]

= m.

m 13

m + m2 + m^

= 1 , 5 ( l 2 / l ^ ) m 2

™1 + ™2

(2 ,6)

and

^

(2 ,7)

^

4' = . 4 5 ( l 2 ) ( q ^ 2 ^ 2 q i q T ) / l i 2 ^ 1 ,2(q2q2 + q 2 2 ) / l 2

+ 3 (q iq2 + q2qi + q i q 2 / i o ) A i

+ 2,25(12) q iq3 /1^2 ^ 1 .5q2q3/ l i

= , 3q^q^ / l . , + 1 .2 (q2q^ / l2 + q2q3A2 " " 1 ^ / ^ l ^

+ 1 ,5q^q3/ l^

= 2,25(12) (4^2 + q^q^) /1^2 + i .2(q22 + q2q2 )A2

+ 1,5(q-,q2 + 2q., q2 + q 2 q i ) / l i

For t he l i n e a r modal a n a l y s i s , the homogeneous cases of equa t ions

(2 ,5 ) a r e c o n s i d e r e d . In order to e l i m i n a t e the e f f e c t s of l i n e a r

c o u p l i n g , equa t i ons (2 ,5) a re t ransformed i n t o normal c o o r d i n a t e s ,

{p}T _ {p - , P^, P^} where T denotes t r a n s p o s e , such t h a t

{q} = [R] {P} (2 ,8)

where [R] i s the modal mat r ix which c o n s i s t s of the normalized e i g e n ­

v e c t o r s

[R]

1 1 1

" l " 2 " 3

. P i P2 P3-

(2 ,9)

26

This transformation results in the equations of motion in terms of the

normalized coordinates

Xo' 3

M 11 0

0 M22 0

0 0 M 33j

^P. K ^ 0 0

0 K22 0

0 K 33J

F(t)

m-

(eo«Q(U32)

(2,10)

where.

F ( t )

P i

T

XQ

e

s

=

=

=

=

FQ COS n t

P i / ^ 0

(i^t

F0/K3

X Q A I

and

M 11

"11

a

b

a

(2,11)

1 + a(a + 2n^b + n^2) + 2(1 -f- a)pi + (1 + a

1 + n^2 0 (n/e)3 + p^2^

1 .+ (1 .5 l2Ai)^

1.5 I2/I1

m2/n*i

m3/in-j

$)p^2

(2,12)

27

and

3

0

n

i

r

Ki(i=

and

* i

=

=

=

s

=

»1 ,2)

=

I 2 / I 1

Wj/W^

t 2 / t i

K3A1

[ 4 m ^ ( l ^ / t ^ ) 3 ] / ( j ^ ^ )

= E Wi t i3 /4 i^3

^ l ^ ^ i l l P l " 1^121^2

(2,13)

P2f^112^1 - 122^2 - 132^3^-^

P3f^i13Pl + ^123^2 - 33^3^ " ^2,14)

^ill^l^ + Mi22P2^ ^ ^133^3^ +

Mil 2 1 2 •*• ^113^^3 " ^123^2^3

The L^j]^ and M .j terms of equation (2,14) are given in APPENDIX B, In

equations (2,13), E is Young's modulus and w^ and t^ are the widths and

thicknesses of the two beams, respectively. Equations (2,10) may now

be written as

P + i 13 Pl = 1 cos NT - eaV^/M^^ - 2e^^P^

P2 "*" 23^P2 ' ^2 ^°^ "" ~ °''*'2/M22 " 2^^2^2 (2,15)

P3 + r332p3 = f3 cos NT - ea'i'3/M33 - 2eC3P3

where,

(2,16)

•^13

u^i

N

f i

I S

=

s

=

0)./a)3

K i i / ^ i i

^2/0)3

CPi / (Mi^ 0132 e)

28

and damping has been inserted into normal mode equations of motion to

account for energy dissipation. It should be noted that the nonli­

nearities of equations (2,15), i,e,, the H'i's of equations (2.14), are

physically attributed to the non-vanishing axial displacement of the

tip masses.

CHAPTER III

LINEAR MODAL ANALYSIS

Eigenvalues

For the linear modal analysis, equations (2.5) are written in the

form:

[m]{q} + [K]{q} = {f} - W (3.1)

In order to determine the eigenvalues and eigenvectors for the

system, the right-hand side of equation (3.1) is set to zero. This

gives

[m]{q} + [K]{q} = 0 (3.2)

Assuming a solution of the form q = Q sin u>t gives

UK] - X[m] {Q} = 0 (3.3)

where, X^ = 0) 2 .g j g squares of the eigenvalues of the system (i =

1,2,3), which cause the matrix which is multiplied by {Q} to be singu­

lar. The eigenvalues are determined by the solution of the charac­

teristic equation, which is given by

det[D(X)] = 0 (3.4)

where [D(X)] = [K] - X[m] . For the system under consideration, the

characteristic equation is

("»13 m22 - ™12^™33 " " 11"*22'"33 ^ +

tK-,in22™33 " 2 '"l1™33 " ""13 ) + K3 "*1l"'22 " "»12 >1 ^ - ^3,5)

^ 1 2"'33 " 1 3"*22 " 2 3"'l1 ^ •*" ^1^2^3 = °

where, the mij's and K^'s are given by equations (2,6) and (2.13),

respectively.

29

Equations (3.5) may be written in the form

30

y( 10/(1)33) 6 •»• 0)

L<^33J "^2

T 2

+ 1 - mi22

"'11™22

0)

L'^3j

•err - mi3'

'"ll'"33

"^3

(3.6)

"^22"

."^33. • _'^33.

2 " ^ 2 '

_ ' ^ 3 _

= 0

where.

0). '11 = Ki/miT

2 _ "^2 = ^2/ m 22

0)332 = K3/ m 33 (3.7)

P = -1 + m 12 + m^32

"»11 ""22 ™11 ""33

The frequency parameters, (n^^, are the natural frequencies if each

component of the system acts as a single degree-of-freedom. Figures

3,1 - 3,3 show the variation of the natural frequencies of the system

as functions of ^'\']/^23 *"^ ^^22/^3,

Figures 3,4 and 3,5 show the variation of several combinations of

the natural frequencies with ^•\/^2 ^"^ ***22/' 33' "^^^ critical points

are located where *-he curves (co, + a)2)/ui33, 20) /0)33, and 20)2/0)33

intersect the curve for (^y^^^S ^^ Figure 3,4, These intersections

indicate the possibility of combination resonances in the structure

when the forcing frequency is near 0)3, In Figure 3,5, the critical

points are located at the intersections of 2aV|/o»33 and (0)3 - ui^)/(ji^^

31

m

CN

CD

S

09 3 O

•H

u o

cn

c 0

O CO C 3 <U \

•O CM «= -T* <0 3

a 0) (w

o o

en <0 U 3

CO

o vO

• o m m

^ *

o

CN

.

o

32

3

n

- CN

a 9

OS 9 0

•H U

o

m en

c 0

i %4 0

0) 0 c 4)

T3 c 0) a <u Q

. pn

- P 3 \

OJ

r %i 0

(N

3

•H

33

m

3

- m

9) 9

CO 3 0

•H

ki

o

m

a c 0 m

3 lU 0

0) o e 0) T3 e 4) a 0)

a

• pn m

3 '«v

CN CN

3

<u 0

. m

V U 3

34

CO C T3 0 9>

40 « C

jQ

U

a 6 0 o

O CO O (0

o • C r-Oi 3 II

cr 4) pn

ft CN

2/ 3 ••0 U (« 0 C <4-l

%4 pn O pn

C \

o .-40 3 •

•H JS U 40 (0 -H > IS

pn

3

•H

pr 3

35

pn n

r-l (H

3

ID a

rH

o a

rH

in a

o

CD C TJ 0 «

-H V4 ^ (Q (T3 a c s

•-1 0 £ 0 e 5 en o nj >-o o • c .-0) 3 II cr 0) pn

i:i ^v r-< CM

Si^ ( J u <T3 0 C (4-1

u-i pn 0 ^

c ^. 0 f-

•J i-(C •-1 X V4 .»J (0 -H > 5

• I T

• PO

<r V4

£"

fT'

36

with 0)2/0)33. This shows the possibility of combination resonance when

the forcing frequency is near 0)2. When combination resonances like

those shown in Figures 3.4 and 3.5 exist in a system, a nonlinear ana­

lysis is necessary to determine the response characterisitics of the

system.

Mode Shapes

A normalized eigenvector gives the shape of the motion of the

system at one of the eigenvalues after dividing each amplitude by the

first element. The matrix, [D(X)], of equation (3,4) may be written

ro^a^^) °ab^\)

[D(X)] = (3,8)

where, 0,^ is a 1 x 1 matrix, Dw is a b x 1 vector, D j is a 1 x b

vector, DKK is a b x b matrix, and $. is a vector containing the b

remaining elements of the normalized eigenvector. For distinct eigen­

values, Dwvj is nonsingular, therefore

This gives

(*b>i = - f 5 b b < \ " - ' t £ b a < \ "

The normalized eigenvector for the i t h mode i s tiien given by

$. =

(3,9)

(3,10)

(3,11)

For this analysis, the International Math and Science Library

37

(IMSL) subroutine EIGZS was used to determine the eigenvectors as

defined by equations (3,10), For the condition where ^ = 25, $ = 5, 3

= .402, and a = ,112, the normalized eigenvectors are

1 ,ooo'

{$ } = ^-0,599

0,000

1 ,000^

{*2> = <-3,903 (3,12)

0,046

1 ,000

{$3} = < 3.117

0.444

These mode shapes are shown in Figures 3,6 - 3,8

38

Tj

<l 40 —1

0 X

v E

a; 40 CO >> M

U 0

(t-i

V CL 03

x: CO y *

0) T 0 s u fC 0)

c •r-(

^

• «£

• pn

0)

u

« CM T -

^ ,

II

a ^

CN

o ^ ,

II

•a %

in

II

•€•

% in CN

II

UJ»

39

rvj 3

II

->. "" «>

• • ^

T T

i W ~ l 0 X

a; £ <i; ^ CO >, OT

U 0

14

0) a TJ

£ 7)

u 0 £

u T3 3) C

rsj

•

II

3

« CM O ^

.

II

IX

^ in

II

^

^ in PM

!l

J '^

u

40

^^^

4J 10

T3 07 40 •H 0 X 0)

e (0 40 CO > CO

u 0 (l-l

<u a (T3

JC CO

0) TJ 0 E

U (B 0) C

•H K1

i

CN T -^ •

• II

3

^ (N

o rr .

II

T l

^ in

II

B>

% in CN

II

UJ«

00 .

pn

u 3

CHAPTER IV

AUTOPARAMETRIC MODAL INTERACTION

Asymptotic Approximation Solution

The nonlinear dynamic response of the system shown in Figure 1.10

can be determined by employing the asymptotic method of Struble [2].

This method has distinct advantages in that "it exhibits not only

resonant and non-resonant responses but also other characteristics of

a nonlinear system, including the interplay of the system parameters

and the entrainment of harmonic, subharmonic, and other responses.

In addition, since the method of Struble incorporates two classical

techniques, i,e,, the variation of parameter method and a perturbation

method, it has the advantages of both methods while avoiding the main

difficulties of each" [21], The method is outlined below and then

applied to equations (2,15),

Assume solutions of the normal modes expressed in the form

P = A(T) cos[r^3T + *^(T)] + ea + e2a2 + ...

P2 = B(T) cos[r23T + $2 * ^ + 6b + e2b2 + ... (4,1)

P3 = D(T) cos[r33T + ^^{T)] + 6d + £2^2 + ...

where, the amplitudes A, B, D, and phases *v *2' ^^ *3 ^^® slowly

varying functions of the dimensionless time parameter, T. Performing

the time derivatives of the first of equations (4.1) as indicated by

equations (2.15) gives

41

42

P-, = -A(r^3 "*• *i)sin(r^3T + $ ) + Acos(r^3T + <I» ) + ea

P = [A - A(r^3 + *T)2]cos(ri3T + ^^) + ea - (4.2)

[A*-, + 2A(r^3 + *^)]sin(r^3T + * )

Similarly

P2 = -B(r23 + *2^si'^^^23^ "*" *2^ "*" Bcos(r23T + $2^ " 1

P2 = [B - B(r23 + *2^^1cos(r23'^ + *2^ " 1 "

[B*2 + 2B(r23 + *2)]sin(r23T + $2^ ^^'^^

P3 = -D(r33 + $3)sin(r33T + $.) + Dcos(r33T + $3) + ed

P3 = [D - D(r33 + *3)2]cos(r33T^ +^3) + d' "

[D*3 + 2D(r33 + $3)]sin(r33T + $3)

where, only the first-order perturbations are retained, and r33 is

equal to unity by equation (2.16).

Equations (4.1-4.3) are substituted into equations (2.15), and

like harmonics are collected based on the order of e. The equations of

order e^ are referred to as the "variational equations". Neglecting

the slowly varying terms (A, B, b, ^^, eA, £2^, A$, etc), the

variational equations are

-2ArT3< i = A(S^2^ _ ^132)

-2Ar^3 = 0

-2Br23*2 = B(S22v2 - r232) (4,4)

-2Br23 = 0

-2Dr33*3 = D(S32v2 - r332)

-2Dr33 = 0

where, the forcing frequency, ^, has been replaced by nv and v is

43

near one of the normal mode frequencies. The S^'s are positive real

numbers such that] S^^v^ - ri32 | < £.

The terms of order £ are called the "first-order perturbational

equations," Neglecting the slowly varying terms, as before, gives

• • ^ 1 3 ^

0

0

0 0

^23^ 0

0 ^^33^.

f2 \ cos(nVT) +

^1 ^2 ^3

^1 ^2 M3

i2>

B

3 N2 N3J ( D2

'L4 L5 Lg

M4 M5 Mg

_N4 N5 Ng^

L7

A2 C O S (2ri3T + 2$^)

B 2 C O S (2r23'^ + 2*2)

D 2 C O S (2r33T + 2*3)

'2^1^13 0 °

0 2^2^23 °

0 0 2^31^33

Mo Mc M7 Mg

N7 ^8 ^<

L1O ^11 ^12'

M10 ^11 ^12

L^IO ^11 ^12.

A sin (r-,3T + $ )

B sin (^23^ + ^2^

D sin (r33T + $3)

AB cos [(r^3 - r23)T + $ - $2^

AD cos [(r^3 - r33) T + $ - $3]

ED cos [(r23 - ^33) T + *2 " *3^

'AB cos [(r^3 + r23) T + ^ + ^2^

AD cos [(r^3 + r33) T + $ + $3]

BD cos [(r23 + ^33^"^ + *2 "*" " 3

(4,5)

44

with the Li's, M^'s, and N^'s given in Appendix 2,

Resonance Conditions

The first-order perturbational equations contain secular terms.

Secular terms are those terms which give rise to a resonance condition

such that the denominator of these terms approaches zero as the reso­

nance condition is satisfied. These terms result in two types of

resonance conditions: external and internal. Inspection of equation

(4,5) shows that three groups of resonance conditions are possible.

Group I

When the first mode is externally excited at a frequency close to

its natural frequency (i.e,, nv = r-13), the following internal reso­

nance conditions are possible

^13 " I 23 i ^33 I

^ 13 = ''/2 r23

^13 = ''/2 33 (4,6)

^13 " ^ 23

^ ^ 1 3 ^ ^ ^33

Group II

For external excitation of the second mode such that nv = r23r

the following internal resonance conditions are found to exist

45

^23 = I 13 ± r33

^23 = V2 ri3

^23 - V2 r33

^23 = 2 r^3

r^rj = 2 r 23 33

Group III

If the third mode is externally excited at nv = r33, the internal

resonance conditions are

^33 = I 13 ± ^23!

^33 = V2 rT3

^33 = V2 r23 (4,8)

^33 = 2 r^3

^33 = ^ 23

The above internal resonance conditions are classified in two

major groups: combination internal resonance, r = | r + r^ | , and

principal internal resonance, r = nr^ (where n = 2 or 0,5), it has

been shown that the resonances of each group possess their own speci­

fic properties [39]; therefore, the presentation of analysis will be

restricted to the numerical and analytical solutions for the summed

and difference cases of combination resonance and one case of prin­

cipal resonance where the external forcing frequency is in the neigh­

borhood of r33 (i,e,, nV = r33 = 1 ) .

46

Analytical Solutions

Case 1: Summed Internal Resonance r33 = ^13 + ^23 ^^^ External

Resonance nv = r33

The secular terms of equations (4,5) corresponding to the summed

internal resonance condition are transferred to the variational

equations (equations (4,4)) which become

-2Ar^3*^ = A(S^2^ _ ri32) + £BDL9 cos 0-,

-2Ari3 = £{2CTr^3A + BDLg sin 0 }

-2Br23*2 " B(S22v2 - r23^) + EADMQ COS 0 (4,9)

-2Br23 = ^124^^23^ " ^8 ^^" ®1 1

-2Dr33*3 = D(S32v2 - r332) + £{ABNIO ^ ° ^ ® 1 " ^3 ^ ° ^ "^3!

-2Dr33 = ^(2^3r33D - ABN Q sin0^ + f3 s i n $3}

where, ©i = *i + *2 " * 3 ' I n t roduc ing the fol lowing t r ans fo rma t ions

•

0 )

T

Y

^ i

S i

f 3

33 1 ^10 1

2T

^J^3 1 ^10 1

^ 3 3 ^ - v2

^33SA3 1 10 1

2 q ri3

^ ^ 3 ! '^lol

^13 / ^33

y2

^3

( 4 , 1 0 )

47

g i v e s e q u a t i o n s ( 4 , 9 ) a s

-*^ = -S^Y + y2y3 Lg c o s 0^

y i ^ i | N ^ O |

-Yl = Tl^yi/S^ + y2y3 Lg s i n 0^

Si N^o

_ $ 2 = -S2Y + y i y 3 MQ COS0^

y2S2 I N10

- y 2 = ^2^2 /52 + y i y 3 ^ sin0^ (4,11)

S2 I ^10

-$3 = -Y + cos $3 + y^y2 N Q COS0^

yi y3 I N10

-y3 = n3y3 + sin$3 - y^y2 N^Q sin0^

N10I

The steady-state solution of equations (4,11) can be obtained by

setting the left-hand sides to zero. This results in six nonlinear

algebraic equations which are inconsistent since the actual number of

unknowns is five (y- » y2f y3 f ©1 and $3); however, these equations can

be consistent for two conditions

n = n2 = 0

or (4,12)

ni/n2 = S^/S2

The second condition of equation (4.12) is more physically realizable

than the first one. For this case, the amplitudes are given as

48

yi = ^2 / S 2 11.9

Si I Mg

2 = [y^J^^'^' ( 4 , 1 3 )

^ 3 ^ = S2N10 2 (Y2g^4 ^ ^^2)

Sl^s

Another solution is found for yi = yo = 0

y3 ^ 1

rsP^+n^^ (4,14)

which is the normal, single degree-of-freedom response.

The amplitude response, as predicted by equations (4,13 - 4,14),

is shown in Figure 4,1, There are two vertical tangencies on either

side of n = 1, One occurs (frequency increasing) at the intersection

of the two solutions for y3 (n = 0,985), At this point, the values of

y- and y2 jump from zero to the values defined by the first two of

equations (4,13), and y3 follows the curve defined by the last of

equations (4,13) which shows the typical absorbing effect of systems

with autoparametric coupling. When the forcing frequency is increased

to the point where n = 1,195, the values of Y-\ and y2 collapse to

zero, and y3 collapses to the single degree-of-freedom response

defined by equation (4,14),

For decreasing frequencies, the amplitude jumps again occur at the

intersection of the two solutions for y3 (n = 1,104), and the values of

y. , Y2' ^^^ y3 follow the response curves defined by equations (4,13),

49

0.98 1,02

Figure 4 . 1 . S t e a d y - s t a t e response for r^^ '^ ^23 ' ^33 t, = 19 .8 , « = 8, P = .549, a = .629, Q^ =

.008, (,2 = -Q^' ^3 = • ° ^ ' ^ = • ° ' *

= nv.

50

At the point where n = 0.979, the values of Y^ and y2 collapse to zero,

and y3 follows the single degree-of-freedom responses curve. The

values of n where the amplitudes jump or collapse depend primarily on

the system parameters such as damping and internal detuning.

For the case where n = ri2 = 0, the response curves are similar to

those of Figure 4,1, The only major difference is that the value of y3

is zero at n = 1, which implies that damping has a destabilizing effect,

Case 2: Difference-Type Internal Resonance ^33 ~ ^23 ~ ^13 ^^^ External

Resonance nv = r33

Following the same procedure as before, the secular terms of

equations (4,5) corresponding to the difference internal resonance con­

dition are transferred to the variational equations. The resulting

equations are

-^1 = -s^Y + y2y3 L^2 ^°^ ®2

y i S i I N7

- y i = ^ y i / S i + y2y3 ^12 ^ ^ " ®2

Si I N7

$2 = - S 2 ^ + y i y 3 "11 ^ ° ^ ®2

Y2'=2 I ^7 I ( 4 , 1 5 )

- y 2 = ^2^2 /82 - y i y 3 "11 s ^ " ®2

• ^ 3

S2 I N7

- Y + cos $3 + y i y 2 N7 cos 0.

y3 ^3 I ^7

- y 3 = n3y3 + s i n $3 + yi y2 N7 s i n 0,

N7

51

where, ©2 = *i ••" *3 ~ *2* " ^ solution for y-j = y2 = 0 is the same as

that given by equation (4,14),

Equations (4,15) are similar to equations (4,13), The steady-

state response curves for the difference case with ^1/^2 = Si/S2 are

shown in Figure 4,2, The jump and collapse phenomena, seen in Figure

4,1 for the summed case, are also exhibited in the response for the

present case. For Hi = ^2 = 0, the value of y3 is zero at n = 1 ,

which is the same as for the summed condition.

Case 3: Principal Internal Resonance r33 = 2r-3 and External

Resonance nv = r33

For this case of internal resonance, the interaction takes place

between the first and third modes. Transferring the secular terms from

equation (4,5) to equations (4,4) gives

-$^ = -S^ Y + y3 Ls cos 03

Si I N 4 I

- y i = Ti^yi/Si + YyY2 L3 s i n Q^

Si I N4 I ( 4 , 1 6 )

_ $ - = -Y + c o s $ 3 + y i 2 N4 c o s 0 3

y3 ^3 I ^4

- y 3 = n3y3 + s i n * 3 - yi 2 N4 s i n 0 3

^4

where, 0 = 2 $ - *-a. I^ ^® left-hand side of equations (4.16) are

set to zero, the resulting algebraic equations are found to be com­

patible, and the steady-state solutions are

52

0 , 9 6 0 . 9 8 1.0 n

1.02 1,04

F i g u r e 4 . 2 . S t e a d y - s t a t e r e s p o n s e for r23 " ^i 3 = 1 33 = - ^ . 4 = 25 , <P = 5 , ^ = . 402 , a = . 1 1 2 , Ci = . 0 0 4 , ^2 = •Ol t ^2 = •05 , £ = .01 .

53

yi 2 _ N.

8 J 2x /^2 Y^/16 + m^) {y^ + Ti3 ) i 1

(4.17)

^3 N.

L8 ^ Y 2 / 1 6 + m^

For yi = 0, the solution for y3 is the same as given by equation

(4.14).

The theoretical, steady-state response for y and y- is shown in

Figure 4.3. The jump and collapse phenomena seen in the summed and

difference internal resonance cases are also exhibited by the response

for principal internal resonance. The system response is similar to

that of the autoparametric vibration absorber [26].

Numerical Solutions

Case 1: Summed Internal Resonance Too = r-3 + r23 and External

Resonance nV = r33

When one of the conditions of equations (4.12) is not satisfied,

a numerical integration of equations (4.11), including the left-hand

terms, is required to determine whether the system achieves a steady-

state response in the time domain. The six equations can be simulated

utilizing the IBM Continuous System Modeling Program (CSMP) [40] (to

avoid division by zero, small non-zero initial conditions are assumed).

Sample programs are listed in APPENDIX C. The time history response

reveals that the system is quasi-steady-state as shown in Figure 4.4.

The modal interaction is seen in the form of energy exchanges between

54

4 _

2 -

0 . 9 6

F i g u r e 4 .3

0 . 9 3

y3( y i = 0 )

1.0 n

1,02 1,04

S t e a d y - s t a t e r e s p o n s e for 2r i 3 = ^33 = nv , % -• 1 5 , * = 5 , ti = .601 , a = . 574 , ; i = .01 , C,^ = . 0 5 , £ = . 0 1 .

55

pn CM

u

+ m

vT n pn pn

u

u 0

%t

> 1

u 0 40 CD

•H X

«

. ^-*

CM U

^«. t —

u s •«• n cu 2 CO u

CM c V , ^ c ^-'

0) V4 3

• H Gu

56

the first two modes and the third mode. Another important feature is

that the peak cimplitudes during the transient period are greater than

twice the quasi-state amplitudes. This is critical in the analysis of

maximum stresses.

The response of the system as a function of internal detuning, n, is

shown in Figure 4.5. The peak amplitudes for yi and y2 are not found

in this case since there are so near the regions of parametric instabi­

lity. The jump and collpase phenomena both occur at the points where

the behavior of y3 changes to the autoparametric response from the nor­

mal linear resonance response and vice versa.

Case 2: Difference Type Internal Resonance ^33 ~ ^ 23 ~ ^13 ^^^ External

Resonance nv = r33

Equations (4.15), for this case of internal resonances, are simu­

lated using CSMP. Figure 4.6 shows the time history for the response

of the three amplitudes yi, ^2' and y3. The response characteristics

display the same features of the previously examined summed internal

resonance condition.

Case 3: Principal Internal Resonance r^^ = 2r-3 and External

Resonance nv = r33

For this case of two mode interaction, the response is also deter-

ined by using CSMP. Figure 4.7 gives the frequency response of the

system under principal internal resonance. The jump and collapse phe­

nomena occur at the transition point for y3 as in the summed case.

m

57

2.0

1.0

0.98 0.99 1.0

n

1.01 1.02

F i g u r e 4 . 5 . CSMP s i m u l a t i o n of T u a s i - s t e a d y - s t a t e r e s p o n s e f o r r i 3 + r23 = r33 = d (n^ /n2 + r ^ / r 2 ) . ^ = 1 9 . 8 , «i? = 8, c5 = . 5 4 9 , a = . 112 , (,^ = ^2 = • 0 ' ' ' C3 = . 0 5 , £ = .01 ,

58

pn

J" I

en CN

U

II

m f) u II

u 0

U -n C CN

cn "*< —I m j = »-

u « E -H-

•H •^ CN

C 04 \ r -CO -

V u 3

•H

Ln . r-i

o •

rH

LO

• o

CN

O O •

o

"g* •

o M

o m

59

4.0 -

Figure 4.7 CSMP Simulation of two mode interaction, r33 = 2r = nv. C = 15, ® = 5, 0 = .601, a = .574, ;

= !oi, 3 = -O^. = -01•

60

Case 4: Numerical Integration of Equations of Motion

Another means of evaluating the response of the system is by

direct integration of the equations of motion. Upon inspection of the

nonlinear terms (the H' 's given by equations (2,14)) of equations

(2,15), it is seen that they involve acceleration terms, P^, which can

be replaced approximately by - ^ 32 p^, if these approximate P.'s are

substituted into the nonlinearly coupled terms of equations (2,15), the

equations of motion are easily integrated with the aid of CSMP,

Figure 4.8 shows a time history of the normal mode coordinates

for the summed internal resonance condition r33 = r 3 + r23 = ^

(program listing given in APPENDIX C), Since the forcing term appears

in all three equations of motion, the normal modes oscillate at the

forcing frequency, r33. The first two modes are 180® out of phase and

appear to be exchanging energy with the third mode during the first 60

seconds of motion. Damping has little effect on the maximum amplitudes

as long as 3 is greater than Ci and i;^' ^^ ^® forcing terms for the

first two modes. Pi and P2, are set to zero, the system takes on the

normal single degree-of-freedom response, as shown by the dashed lines

in Figure 4.8.

The generalized coordinate response can be determined from the

transformations of equations (2.9) and (2.11), Figure 4,9 gives the

amplitude-time response of the general coordinates for the summed

internal resonance condition r33 = ^^3 + r23 = «. The effects of the

nonlinear coupling can be seen in the significant amplitudes of qi and

for external forcing in the q3 coordinate direction, "2

61

o c 0

•H •U 0 £ CO c 0

II

(C -n 3 m . c r »M . -0) o

I! • (V

4J -y: II

c ^ u • o - o

p-t

o ID

•H ^ u ^ in 0 o C o . . o r CD It

-H O (TJ pn e N o» VJ £ 0 c -r ^

a; o «*J c . 0 .^

E II V U CO 0! CN C -u o* 0 (D a T II CO a; to , -a: rr ^

. cr • '*

V u D

04 CN on 04

62

o in r-l

O o t H

o in

c 0

•H

mot

CO

c 0

•H ;J (C5 1"^

$

V V ITJ

c •H TJ u 0 0 o

rH fl u <0 c 4) tr

UJ 0

(U CO c 0 c:* CO (U a:

-. m CN

+ PO T —

'u

M

m rr .

O 11 •

c: II

— (*» _) —4 k 3 IP

O ^ 4 .

s en II c

pn > . vj»

-» «

c . Till

U V CN

J vJ> ••1

r II X » -- u»

(T

V U 3

-H cr

b

pn

CN

cr

m

m cr

Pu

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

The dynamic behavior of aeroelastic structural systems with three

degrees-of-freedom of the type shown in Figure 1,10 is determined.

The investigation includes linear and nonlinear modal interaction. In

connection with the normal mode analysis, the normal mode frequencies

and mode shapes are obtained for various system parameters such as

masses and stiffnesses. The results reveal certain critical regions

for which the normal mode frequencies satisfy the internal resonance

conditions Skj w. = 0 , Under the autoparametric conditions (when the

external normal resonance and the internal resonance are satisfied),

the system dynamic response is obtained by including the nonlineari­

ties inherent to the system and by employing the asymptotic approxima­

tion technique due to Struble, The analysis results in three groups

of internal resonance conditions, each corresponding to normal reso­

nance for one of the three normal modes.

In the present study, the steady-state and quasi-steady-state

responses are obtained in the frequency and time domains. The steady-

state response reveals amplitude jump and collapse phenomena as well

as suppression of the motion of the externally excited mode. These

results also give the range of excitation frequency for which auto­

parametric instabilities can occur. In the time domain, numerical

integration shows an energy exchange between the externally excited

mode and the other modes with suppression of the externally excited

63

64

mode. The results of these numerical integrations also give the peak

amplitudes during the transient period. These peak amplitudes are

greater than twice the values of the quasi-steady-state amplitudes,

A linear modal analysis is usually adequate to predict the

response of dynamic systems. However, this study shows that if the

linear eigenvalues of the system satisfy one or more of the internal

resonance conditions, a nonlinear analysis must be conducted to exa­

mine the dynamic behavior of the system.

In order to obtain more refined results, a continuation of this

investigation should include cubic and higher-order nonlinearities in

the equations of motion for which further resonance conditions may be

derived. Futhermore, there could be a possibility of multiple inter­

nal resonance conditions, A more realistic analysis is to replace the

harmonic forcing function, Fgcosi t, with a broad band, random forcing

function. Finally, experimental models should be constructed for the

verification and improvement of the mathematical modeling and to

investigate the possibilities of chaotic responses of the system.

LIST OF REFERENCES

1 Struble, R.A,, "Oscillations of a pendulum under parametric excitation," Quarterly of Applied Math 21, 121-131, 1963.

2 Struble, R.A,, Nonlinear Differential Equations, New York, McGraw-Hill, 1962,

3 Skalak, R., Yarymovych, M.I., "Subharmonic oscillations of a pendulum," J. Applied Mechanics 27, 159-164, 1960.

4 Ibrahim, R.A., Barr, A.D.S,, "Parametric vibration. Part I: Mechanics of linear problems," The Shock and Vibration Digest 2£ , 1, 15-29, 1978,

5 Bolotin, V,V,, The Dynamic Stability of Elastic Systems, San Francisco, CA, Holden-Day, Inc,,1964,

6 Ibrahim, R,A,, Barr, A.D,S., "Parametric vibration. Part II: Mechanics of nonlinear problems," The Shock and Vibration Digest 10, 2, 9-24, 1978,

7 Ibrahim, R,A,, "Parametric vibration. Part III: Current problems (1)," The Shock and Vibration Digest 10, 3, 41-57, 1978,

8 Ibrahim, R,A,, "Parametric vibration. Part IV: Current problems (2)," The Shock and Vibration Digest 10, 4, 19-47, 1978.

9 Ibrahim, R.A., Roberts, J.W,, "Parametric vibration. Part V: Stochastic problems," The Shock and Vibration Digest 10, 5, 17-38, 1978,

10 Nayfeh, A,H,, "Response of two-degree-of-freedom systems to multifrequency parametric excitations," J. of Sound and Vibration 88, 1, 1-10, 1983,

11 Nayfeh, A.H,, Mook, D,T,, "Parametric excitation of linear systems having many degrees of freedom," J, of the Acoustical Society of America 62, 375-381, 1977,

12 Yamamoto, T,, Yasuda, K., Tei, N,, "Super summed and differen­tial harmonic oscillations in a slender beam," Bulletin of the JSME 25, 204, 959-968, 1982,

65

66

13 Yamamoto, T., Yasuda, K., Aoki, K., "Subharmonic oscillations of a slender beam," Bulletin of the JSME 24, 192, 1011-1020, 1981.

14 Yamamoto, T., Ishida, Y., Ikeda, T., "Summed-and-differential harmonic oscillations of an unsymmetrical shaft," Bulletin of the JSME 24, 187, 183-191, 1981.

15 Yamamoto, T., Y., Ikeda, T., Yamada, M., "Subharmonic and Summed-and-Differential Harmonic Oscillations of an Unsymmetrical Rotor," Bulletin of the JSME 24, 187, 192-199, 1981,

16 Zajaczkowski, J., "An approximate method of analysis of para­metric vibration," J. of Sound and Vibration 79, 4, 581-588, 1981 .

17 Zajaczkowski, J., Lipinski, J,, "Instability of the motion of a beam of periodically varying length," J. of Sound and Vibration 63, 1, 9-18, 1979,

18 Takahashi, K,, "An approach to investigate the instability of the raultiple-degree-of freedom parametric dynamic systems," J. of Sound and Vibration 78, 4, 519-529, 1981,

19 Szemplinska-Stupnicka, W,, "The generalized harmonic balanace method for determining the combination resonance in the para­metric dynamic systems," J. of Sound and Vibration 58, 3, 347-361, 1978,

20 Hsu, C , "On the parametric excitation of dynamic systems having multiple degrees of freedom," J, of Applied Mechanics 30, 363-372, 1963,

21 Stanisic, M.M., "On the stability of generalized Hill's equation with three independent parameters," International J, of Non-linear Mechanics 15, 6, 485-496, 1980.

22 Sato, K,, Siato, H,, et al, "The parametric response of a hori­zontal beam carrying a concentrated mass under gravity," J, of Applied Mechanics, Transactions of the ASME 45, 3, 643-648, 1978.

23 Hsu, C,S,, "On nonlinear parametric excitation problems," Advances in Applied Mechanics 17, 245-301, 1977,

24 Minrosky, N,, Nonlinear Oscillations, Van Nostrand, 1962,

67

25

26

27

Ibrahim, R.A., Barr, A.D.S,, "Autoparametric resonance in a structure containing a liquid. Part ll: Three mode inter-action," J, of Sound and Vibration 42, 2, 181-200, 1975,

Haxton, R.S,, Barr, A,D,S., "The autoparametric vibration absorber," J, of Engineering for Industry, Transactions of the ASME 94, 1 , 119-125, 1972, ~ ~

Breitenberger, E,, Mueller, R,D,, "The elastic pendulum: A nonlinear paradigm," J, of Mathematical Physics 22, 6, 1196-1210, 1981, "~"

28 Hatwal, H., Mallik, A.K., Ghosh, A., "Forced nonlinear oscilla­tions of an autoparametric system - Part 1: Chaotic Responses," J. of Applied Mechanics 50, 3, 657-662, 1983,

29 Hatwal, H,, Mallik, A.K,, Ghosh, A,, "Nonlinear vibrations of a harmonically excited autoparametric system,' J, of Sound and Vibration 81, 2, 153-164, 1982,

30 Hatwal, H., Mallik, A.K., Ghosh, A., "Forced nonlinear oscilla­tions of an autoparametric system - Part 2: Chaotic Responses," J. of Applied Mechanics 50, 3, 663-668, 1983.

31 Moon, F.C., Holmes, P.J., "A magnetoelastic strange attractor," J. of Sound and Vibration 65, 2, 275-296, 1979.

32 Moon, F.C., "Experiments on chaotic motions of a forced nonlinear oscillator: Strange attractors," ASME J. of Applied Mechanics 47, 638-644, 1980,

3 3 Nayfeh, A,H,, "Combination resonances in the nonlinear response of bowed structures to a harmonic excitation," J, of Sound and Vibration 9£, 4, 457-470, 1983,

34 Singh, Y,P,, "Quenching in a system of van der Pol oscillators with non-linear coupling," J, of Sound and Vibration 77, 4, 445-453, 1981 ,

35 Yamamoto, T,, Yasuda, K,, Nagasaka, I,, "On the internal resonance in a two-degree-of-freedom system (when the natural frequencies are in the ratio 2:3)," Bulletin of the JSME 22, 171, 1274-1283, 1979,

36 Yamamoto, T,, Yasuda, K,, Nagasaka, I., "On the internal resonance in a nonlinear two-degree-of-freedom system (forced vibrations near the higher point when the natural frequencies are in the ratio 1:2)," Bulletin of the JSME 20, 147, 1093-1100, 1977.

68

37 Yamamoto, T,, Yasuda, K,, Nagasaka, I,, "On the internal resonance in a nonlinear two-degree-of-freedom system (forced vibrations near the lower resonance point when the natural frequencies are in the ratio 1:2)," Bulletin of the JSME 20, 140, 168-175, 1977,

38 Ibrahim, R.A,, Barr, A.D.S,, "Autoparametric resonance in a structure containing a liquid. Part I: Two mode interaction," J, of Sound and Vibration 42, 2, 159-179, 1975,

39 Kovalchuk, P,S,, "Nonlinear resonances in mechanical self-oscillatory systems," Soviet Applied Mechanics 12, 10, 1040-1046, 1977,

40 Speckha r t , F ,H , , A Guide to Using CSMP - The Continuous Modeling Program, P r e n t i c e - H a l l , Englewood C l i f f s , N , J . , 1976,

APPENDIX A

KINETIC ENERGY DERIVATION

1) Position vectors of mi and m2 (See Figure 1,10 for reference)

Pi = ^ 1 + ^3) i + h^' i.

P2 = (q3 + qi + AV + q2 sin 0 + h2' cos 0) j_

- (Ah + q2 cos 0 - h2' sin 0) £

where

hi ' = 22i!.

51l

h2 ' = 3q22

5I2

Av = I2 (1 - cos 0)

Ah = I2 s i n 0 - 3q^2

51l

s i n 0 i 0 = 3qi

2I1

cos 0 = 1 - ^ + . . . = 1 - 9qi 2

2 81i2

2) V e l o c i t y Vectors

Vi = Pi = ^^1 + ^3) 1 + 6q^qi i

r ^2 = ?2 = ± - J 1 3 + ^1 + 3q^q2 + 9l2qi2 ±_J U2 - 1 - 3q^q2 +

d t L 2I1 SI , 2

69

70

^ q i ^ 2 + 2q iq2q2) > i

I O I 1 I 2

^3 - 4 l + _ ! . ( q i q 2 + q 2 q i ) + 9 I2 q^c^^ + 6q2q 2

2 I1 41^2 5^^

" 27 ( q i q i q 2 ^ + q i ^ q 2 ^ 2 ^ ( i

201i2i2 J

3) K i n e t i c Energy

T = 1; mi [V^2^^^^2 j^ j ^ 1 ^ m 2 [ V 2 2 j . + ^^^jj + ^/2m2^2^

S u b s t i t u t i n g the exp re s s ions for Vi and V2 i n t o T and r e t a i n i n g

up t o f o u r t h o rde r terms g i v e s :

T = 1/2[mi + m2(1 + 2 ,25 l22 / i ^ 2 j j ^ ^ 2 ^ ^/2m2Cf2^

+ 1/2(mi + ni2 + m3)q32 + 2 "*2 "'•2 * 1 ^2 •*" ^"'l "*• " '3^ '^ l^3

2 l l

+ ^ m2 ^2 ^ ^ 1 ^ 2 ^ + 5 q i q i q 3 )

20 l i 2

+ 3m2 ( 1 / 5 q i q i q 2 + ^ i ^ 2 ^ 3 " ^1^2^3 " ^1^q2^

21 1

71

+ 12 m2 ^<l2'^2^3 " ^1^2^2^

10 l 2

+ 9 4 (m^ + itij) + 9 ^2 I2 U 2

lll^ L25

^^'i^' 16 li^J

+ 9 nij (q iq2^2^ + qi^2^q2^

5 1 , 1 1- 2

APPENDIX B

COEFFICIENTS

L. ., and M. ., for equa t ions 2,14 ^ i j k i:]k ^

' i j k = I - 9 ^ + 1 1 1 n^n^^ + 3n. + ,3n^

+ 2 .25p^3 + LSn^p^ j

+ n i [ . 3 + I : : ! n . (1 + p^) + 1.5Pkj 3

+ p ( 2 . 2 5 3 + 1 . 5 ( n j + n ^ ) + l i i n ^ n ^ ^ j

M i l l = {'^^^ + l i 2 n3 2 + 3nA - 1 .2ni

+ P i f 2 . 2 5 3 + 3n^ + J j d ^ l ^ j

"^ijk = (-^^ "" 1 : 1 ^j"k - 3(n. + nj^)j

- 2 . 4 n i - P i ( 4 . 5 3 . 3(n . . n , ) . 1 ^ n -n^)

coefficients for Equations (4,5)

L l = cxr^32 [L^ i i - M^ill

2Mii

•23" '"^122 ^ = <xro,2 [ L , „ - M^22^

2Mii

L3 = Q^ SS" '^133

2Mii

= a r , , 2 [ L , . ^ - M^33^

72

73

2M11

L5 « 0^23^ ^^122 + ^ 2 2 ^

2Ml1

2M 11

r^-^rool L7 « g t ^ 1 2 l ' ' l 3 + Ll 12^23 - Ml 12^13^23

2Mii

L8 2 - Mi i - j r - i - j r , , ]

Lg -

g t L ^ B i ^ n ^ + Ll 13^33^ - Ml 13^131:33

2Mii

g [Li32^23^ + Li23^33^ " ^i 23^23^33^

2Mii

rool L i e = « t L l j i ' n ^ + Ll 12^23^ + Ml 12^13^23

2Mi i

L11 = a t L i 3 i r ^ 3 ^ + Li13^33 r-,^2 + Ml 13^13^33]

2Mii

L12 a tLi32r232 + Li 23^33^ + Mi 23^^23^33^

Mi

M'

M3

2Mii

^ 2 3 ^ ^^211 - ^211

2M22

gr 2 [L.,,0 - M222J 23 ^""222

2M22

= (xr 23^ ^^233 ^23 3^

2M22

74

M4

Ms

M,

M-

°^23^ ^^211 + ^^^^

2M 22

M, 8

°°^23^ fL222 + ^222]

2M22

^ 2 3 ^ [L233 + M233]

2M22

- 2 _ ^ ^ 2 1 ^ 1 3 ^ + L2i2 '^23^ - M2i2J^i3^23^

2M22

- 2 L - t L 2 3 i r ^ 3 2 + L 2 i 3 ^ 3 3 ^ " M 2 i 3 r i 3 r 3 3 ]

2M22

Mg = _ a _ [ L . 3 , r „ 2 ^ 232 ' '23 + L223^33 - M223^23'^33^

2M 22

M 10

M 11

M 12

Ni

-^L-_ fL22 i r^32 + L212^23^ + M212^13^23^

2M22

-5L— fL23 i r^32 + L 2 i 3 r 3 3 2 + M2i3 r i3 r23 ]

2M22

- ^ i — ^^232^23^ + L223^33^ + M223^23^33^

2M22

^ 3 3 ^ f ^ l l - M311]

N-

N3

2M33

°^33^ ^ ^ 2 2

2M 33

°^33^ f^333

^322^

^333^

2M33

75

N4 =

N5

N,

°^^33^ f ^ 3 l 1 + ^ 3 ^ ^ ]

2M33

^ ^ 3 3 ' tL322 H- M322]

2M33

°^33^ f^333 + M333]

2M

N 7 =

N8 =

33

_ o c _ [ L 3 2 i r i 3 2 + L 3 i 2 r 2 3 2 - M 3 i 2 r i 3 r 2 3 ]

2M33

_ 2 L - fL33 i r^32 + L 3 i 3 r 3 3 2 . M3i3r^3r33]

2M33

Ng = _ a _ _ [ L . . , r , ^ 2 ^ 332^23 + L323r33 " M323^23^33^

2M 33

N 10

N 11

N 12

- 2 — fL32 i r 32 + L312^232 + M3i2 r i3 r23 ]

2M33

_ a _ _ [ L 3 3 ^ r , 3 2 + L 3 i 3 r 3 3 2 + M3i3r^3r33]

2M33

- 2 — ^^332 ' '23^ + L323^33^ + M323^23^33^

2M33

APPENDIX C

CSMP PROGRAM LISTINGS

CSMP Program L i s t i n g f o r s i m u l a t i o n of e q u a t i o n s ( 4 . 1 1 )

/ / JOB CWP$TTT,3259tt30, l5) ,»W000ALL« //CSMP EXEC CSMP PARAMETER Rl« .4^96553 ,R2« .5503A488 PARAMETER GAM « ( - 3 f - 2 . 8 , - 2 . 7 , - 2 . 6 , - 2 , - l , 0 , 1 , 2 , 2 . 6 , 2 . 7, 2 . 8 . 3) PARAMETER L 5 l « - . 0 1 6 2 7 6 1 8 1 2 , L 5 2 « - . 0 0 7 4 9 8 3 3 1 1 , L 5 3 = - , 0 0 6 0 0 2 7 2 7 1 PARAMETER NNl = . 1 0 0 0 0 0 , NN2 « . 1 0 0 0 0 0 , NN3 =*200000 INCON AlO « O.OIOO, A20 » O.OIOO, A30 » O.OIOO INCON PIO - 0 . 0 0 0 0 , P20 > 0 . 0 0 0 0 , P30 » 0.0000 THTA s PI • P2 - P3 A l 3 INTGRL(A10,A10) A2 " INrGRLCA20,A20) A3 = INTGRL(A30,A3DJ P I « INTGRL(P10,P1D) ?2 * INTGRL(P20,P2D) P3 » INTGRL«P30,P30J

AID = - f N N l * A l / R l • A2*A3«L51*SIN(THTA) / (ABS(L53) *RU} A20 = - (NN2*A2 /R2 • A1*A3*L52*SINiTHTA) / (ABStL53) • R 2 n A3D = - (NN3*A3 • S IN(P3) - A l *A2* l .53*SIN( THTA) / I ABS (L 53) IJ PIO = - ( -R1*GAM • A2*A3*L51*COS(THIA) / (A l *R l *ABS(L53 ) ) ) P20 = - l -R2»GAM • A1*A3*L52*C0S(THTA)/(A2*R2»ABS(L53))J P3D = - ( -GAM *• C0S(P3) /A3 • Al*A2*L53*CUS(THrA ) / ( A3*ABS ( L53) ) ) *

TIMER OELT "0.0010, FINTIN « 50.00, OUTOEL ».2000 PRTPLOT AllAl) PRTPLOT A2(A2) PRTPLOT A3(A3) ENU STOP ENOJOB //

76

CSMP Program L i s t i n g for s imulat ion of equations ( 2 . 1 5 ) .

77

JOB CWPSTTT,3259,,30,15J,«WCQ0ALL» EXEC CSMP

. 4 4 9 6 5 5 3 , R2 » .5503448f 1 9 . 7 9 9 9 9 2 9 , EP « . 0 1 1 . 9 3 1 2 0 5 9 , M2 » 4 . 1 9 1 9 2 0 6 , • 6 2 8 8 3 1 6 , B » .5489877 - . 0 2 7 6 1 8 5 , N2 « - . 0 8 6 3 5 6 2 , N3 - . 0 3 2 5 8 4 7 , 0 2 =» . 3 4 0 0 7 2 7 , 03 *

I! II PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER INCON PIO INCON PDIO

Rl « EX = Ml « AL » Nl « QI « ETl L I U « M i l l = L121 = M121 = L131 = M131 = L211 = M211 = L221 = M221 = L231 = M231 = L311 = M311 = L321 = M32i = L331 = M331 = = . 0 1 0 ,

= 0 . 0 0 ,

0 . 0 5 , ET2 . 3 2 2 7 8 9 , • 1 6 1 3 9 5 , . 1 5 9 1 7 9 , . 0 7 9 5 8 9 , 4 . 4 1 9 9 4 9 , 2 .209975 . 7 4 1 5 2 9 , . 6 6 1 9 3 9 , . 5 5 3 7 0 2 , . 5 4 8 6 2 2 , 5 .445121 3 .499651 . 7 4 8 9 3 6 ,

= 0.05, L112 =« M112 = L122 ' M122 = L132 «

, M132 = L212 a M212 » L222 » M222 « , L232 = 9 M232 « L312 =

0 .50 L113 M113 L123 M123

ET3 « ( . 7 4 1 5 2 9 , . 0 7 9 5 8 9 , . 5 5 3 7 0 2 , • 0 0 5 0 3 0 ,

5 . 4 4 5 1 2 1 , Ll 1 .945470 , Ml

1 .097244 , L21 . 5 4 8 6 2 2 , H213 . 8 9 0 8 2 1 , L223 . 4 4 5 4 1 1 , M223

6 . 2 6 6 5 1 4 , L2 3 . 1 3 3 2 5 7 , M2

1 .946354 , L31

GAMl = - ( P1*R1**2 ) • ( - ( P2*R2**2 ) • ( - ( P3 ) • (

M l l l * P D l * * 2 + ¥

- 1 . 4 6 1 0 3 8 , M312 =* - 1 . 5 5 3 2 9 7 , • 392172 , L322 = U 4 9 5 9 2 8 , L32 - 1 ^ 5 5 3 2 9 7 , M322 « - 1 ^ 6 3 7 3 2 8 , 9 . 6 8 3 0 8 1 , L332 « 1 3 . 2 2 5 9 9 1 , L . 8 4 9 3 3 4 , M332 = . 5 5 1 0 2 7 , M333

P20 a . 0 1 0 , P30 * . 0 1 0 P020 ' 0 . 0 0 , PD30 = 0 .00

L111*P1 + L121«P2 *• LI L112*P1 • L122*P2 • L l L113*P1 • L123»P2 + Ll

M122*PD2**2 > M133*P03*

W3 » 1.6207

M3 s 5.2363615

1.4433017 .0910431 , . 2 5 , . 7 5 ) =» .748936 = 2.2099 75 « .392172 = 1.9454 70

33 = 9.683081 33 ' 8.833748 3 ' 1.946354

= 3.4996 51 = 1.495928 ^ 3.133257

33 = 13.225991 33 = 12.674964 3 = 1.698666 M313 « .849334 3 ^ 1.102054 M323 =« .551027 333 ' 16.639141

= 8.319571

( (

CAM2 = - ( - ( - ( •f • f

•••

M112 *• M121 )*P01*PD2 M123 > M132 ) *P02*P03 P1*R1**2 ) • ( L211*P1 • L221*P2 P2*R2**2 ) • ( L212*P1 • L222*P2 P3 ) • ( L213*P1 • L223*P2

M 2 1 l * P 0 1 * * 2 • M222«PD2**2 • ( M212 * M221 ) *PDl*PD2 • ( ( M223 • M232 J*PD1*P03

3 i *P3 » . . . 32*P3 ) . . . 33*P3 ) . . . • 2 . . .

• ( M113 *• M131 )*P01*P03

> L2 ^ L2 • L2

M233*PD3* M213 * M2

3 l * P 3 ) . . . 32*P3 ) . . . 33«P3 } . . . •2 . . . 31 J*PD1*PD3

78

GAM3 « - ( P1*R1*«2 ) • ( L 3 l l * P l f L321*P2 • L331«P3 I . . . - ( P2«R2^*2 ) • ( L312*P1 • L322*P2 • L332*P3 ) . . . - ( P3 ) • ( L313*P1 • L323*P2 *• L333*P3 ) . . . • M311*PDl**2 • M322*P02**2 • M333*PD3**2 . . . • ( M312 «- M321 )*PDl*PD2 • C M313 • M331 )*PD1*P03 . *• [ M323 «• M332 )*P02*PD3

P I « INTGRL(P10,PD1) P2 « INTGRL(P20,P02) P3 = INTGRHP30,P03) PDl = INTGRL(PD10,PDD1) P02 = INTGRL(P020,P002) P03 * INTGRL(P030,P0D3) F l a EX*QI / (M1*W3**2 ) F2 « EX*Q2/(M2*W3**2) F3 « EX*Q3/(M3*W3**2) PODl = - P 1 * R 1 * * 2 • F1*C0S(TIME) - EP*AL*CAM1/M1 - 2«ET1*P01 PDD2 ' - P 2 * R 2 * * 2 f F2*C0S(TIME) - EP*AL*GAM2/M2 - 2*ET2*P07 PD03 = - P 3 • F3«C0S(TIME) - EP*AL*GAM3/M3 - 2*ET3*P03 TIMER OELT = . 0 0 0 1 , FINTIM = 5 0 . 0 , QUTDEL = .250 PRTPLOT P l ( P l ) PRTPLOT P2(P2) PRTPLOT P3(P3) END STOP ENDJOB