FREE PERIODIC VIBRATIONS 0F
CONTINUOUS SYSTEMS GOVERNED BY
CONPLED NONLINEAR PARTIAL
DIFFERENTIAL EQUATIONS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
TCHUOCV WEI LE
1969'

 

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ABSTRACT

FREE PERIODIC VIBRATIONS OF CONTINUOUS SYSTEMS GOVERNED
BY COUPLED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

BY

Tchuoc Wei Lee

A perturbation method for obtaining approximate solu—
tions of coupled nonlinear partial differential equations
is developed. The nonlinear partial differential equations
are first converted into a sequence of linear partial dif-
ferential equations, in which the zeroth order equation
corresponds to a homogeneous linear problem and can be
solved by the method of separation of variables. The
higher order equations correspond to inhomogeneous linear
problems and are solved by suitable eigenfunction expansions.

The method is applied to study the free periodic
vibrations of continuous systems such as beams, circular
membranes and circular plates with immovable boundary sup—
ports. One essential feature of all these governing equa-
tions of motion is that they incorporate effects of the
so-called second invariant of the middle surface strains
as well as that of the in-plane inertia. These effects are
usually neglected in more elementary nonlinear theories

(such as under the Berger's hypothesis) so that uncoupled

 

Tchuoc Wei Lee
equations of motion will result. More accurate explicit
solutions for the frequency—amplitude relations, the in-
plane as well as the out—of-plane displacements are ob—
tained.

Numerical results are obtained using a CDC 6500
digital computer. Comparisons and discussions of these
results with those previously obtained using more elementary

nonlinear continuum theories are presented.

FREE PE

BY CC

FREE PERIODIC VIBRATIONS OF CONTINUOUS SYSTEMS GOVERNED

BY COUPLED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

BY

Tchuoc Wei Lee

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and
Materials Science

1969

The ;
Yen who a.
helpful 8‘
Than
Drs.
591"?ng 0
Dr.
Metallurg
Encourage
The
State Uni
the Ccurs

Mrs,

6 0/74,
3 ~/ 25"; 7o

ACKNOWLEDGMENTS

The author sincerely wishes to thank Dr. David H. Y.
Yen who acted as his research advisor and offered many
helpful suggestions during the course of this investigation.

Thanks are also due to:

Drs. R. W. Little, R. K. Wen, and P. K. Wong for
serving on his guidance committee,

Dr. Donald J. Montgomery, Chairman of Department of
Metallurgy, Mechanics and Materials Science, for his
encouragement and awarding a graduate assistantship,

The personnel of the Computer Laboratory of Michigan
State University for their advice and c00peration during
the course of this study,

Mrs. R. M. Wheaton for typing this thesis.

ii

ACKJCWLEI

LIST OF

LIST OF

C~APTER
I. I:
II . M;
III E
IV_ N
V’- s

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . .

LIST OF TABLES

O O O O O O O O O O O O O O 0 0 9

LIST OF FIGURES . . . . . . . . . . . . . . . . .

CHAPTER

I.

II.

III.

IV.

INTRODUCTION . . . . . . . . . . . . . .

1.1.
1.2.
1.3.

Historical Background . . . . . . .
Purpose of Investigation . . . . . .
Organization of Report . . . . . .

METHOD OF ANALYSIS . . . . . . . . . . . .

2.1.
2.2.

Equations of Motion . . . . . . .

Method of Solution . . . . . . . . .

EXAMPLES . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . .
Elastic Beams with Immovable Supports
Elastic Circular Membrane with Initial
Tension and Immovable Edge . . . . .
Elastic Circular Plate with Immovable
Edge . . . . . . . . . . . . . . . .

NUMERICAL RESULTS AND COMPARISON STUDIES .

4.1.
4.2.
4.3.

4.4.

Introduction . . . . . . . . . . . .
Beams with Various Boundary Conditions

~Clamped Circular Membrane with

Initial Tension . . . . . . . . . .
Circular Plate with Clamped Edge .

SUMMARY AND CONCLUSIONS . . . . . . . . . .

LIST OF REFERENCES . . . . . . . . . .

iii

Page
ii
iv

Vi

H

(Owl-l

21

21
25

31
36
42

42
43

58
68

79

83

TABLE

4.2.1.

LIST OF TABLES

TABLE Page

4.2.1° Numerical values of the expansion coef—
ficients Cég) and Céé) of hinged—
hinged beams . . . . . . . . . . . . . . . 45

4.2.2. Numerical values of F(K) of hinged—
hinged beams . . . . . . . . . . . . . . . 46

4.2.3. Numerical values of the expansion coef-
1

ficients Con of clamped—clamped beams

withany%............... 48
4.2.4. Numerical values of the expansion coef-

ficients Céé) of clamped—clamped beams . 48
4.2.5. Numerical values of the expansion coef—

ficients Céé) of clamped—hinged beams

with any %- . . . . . . . . . . . . . . . 49
4.2.6. Numerical values of the expansion coef—

ficients Cég) of clamped-hinged beams . 49

4.2.7. Convergence of F(%) of clamped—clamped

beams . . . . . . . . . . . . . . . . . . 51

4.2.8. Convergence of F(x) of clamped-hinged

beams . . . . . . . . . . . . . . . . . . 51

4.3.1. Numerical values of the expansion coef-
ficients Cég) of clamped circular membranes
with any initial strain . . . . . . . . . 60

4.3.2. Numerical values of the expansion coef—

ficients Céé) of clamped circular membranes 60

iv

4.4.1.

4.4.2.

4.4 3.

LIST OF TABLES (continued)

TABLE

4.3.3.

Page
Convergence of F(eo) of clamped circular
membranes . . . L . . . . . . . . . . . . 61
Numerical values of the expansion coef-
ficients C§;) of clamped circular plates
with any %- . . . . . . . . . . . . . . . 71
Numberical values of the expansion coef-
ficients Céfi) of clamped circular plates 72
Convergence of F(%nu) of clamped circular
plates . . . . . . . . . . . . . . . . . . 73

FIGURE

4.2.1.

4.2.2,

4.2.3.

4.3.1.

4.4.1,

4.4.2.

4'4.3.

4.494

FIGURE

4.2.1.

4

.4

.4.

LIST OF FIGURES

Frequency—amplitude curves for hinged—

hinged beams . . . .

Frequency-amplitude curves

clamped beams . . .

Frequency—amplitude curves

hinged beams . . . .

for clamped-

for clamped-

Ratio of nonlinear to linear period vs.

nondimensional amplitude for clamped

circular membranes with initial strain

Ratio of
ratio of

circular

Ratio of
ratio of

circular

Ratio of
ratio of

circular

Ratio of
ratio of

circular

nonlinear to
amplitude to
plates with

nonlinear to
amplitude to
plates with

nonlinear to
amplitude to
plates with

nonlinear to
amplitude to
plates with

linear period XE:
thickness for clamped

u = 0.1 . . . . .

linear period 2&-
thickness for clamped

u = 0.2 . . . . .

linear period Kg.
thickness for clamped

u = 0.3 . . . . .

linear period gs,
thickness for clamped

u = 0.4 . . . . .

vi

Page

54

55

56

65

74

75

76

77

 

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The inte.
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I. INTRODUCTION

1.1. Historical Background

Problems of finite deflection of continuous systems
lead to nonlinear partial differential equations. For ex-
ample, the governing equations of motion of an axisymmetric
circular plate executing large amplitude vibrations are a
pair of coupled nonlinear partial differential equations.
The interaction between the middle surface forces in the
plane of the plate and the out-of-plane deflection is a
main source of the nonlinearity in the system.

This type of nonlinear problem seems to be first
recognized in 1910. Von Karman [1] extended the small de—
flection plate equation, introduced by Lagrange in 1810, to
include the straining of the middle surface of the plate.
He obtained the well-known pair of nonlinear partial dif—
ferential equations now bearing his name for plates under
static loadings. The difficulties presented in obtaining
solutions to this pair of coupled nonlinear equations have
led to approximations proposed by many researchers.

In 1955, Berger [2] suggested the neglect of the strain
energy due to the second invariant of the middle surface

strains. He solved the static uncoupled nonlinear

different
under var

sinc

linear st
tended tl
plates.

fa

.rme I:

solved t}

2
differential equations for the problem of circular plates
under various boundary conditions.

Since then, the Berger's hypothesis has been used by
many researchers to solve nonlinear static problems as well
as dynamic problems. Nash and Modeer [3] studied the non-
linear static problem of shallow shells. They also ex—
tended the Berger equations to non-linear vibrations of
plates. Sinha [4] investigated the static problem of uni-
formly loaded plates rested on elastic foundation. Wah [5]
solved the pair of uncoupled nonlinear equations of motion
for plates by a modified Galerkin method using a one term
approximation. Gajendar [6] followed the same method of
Wah and solved the problem of large vibrations of plates on
elastic foundations. Recently, Yen and Blotter [7] studied
free periodic vibrations of continuous systems governed by
nonlinear partial differential equations. Both the first
and second order approximations to the solutions of the
uncoupled nonlinear equations were obtained by a perturba-
tion method.*

The Berger's hypothesis in the dynamic case consists in
neglecting the strain energy due to the second invariant of
the middle surface strains as well as the in—plane inertia
of the system. Then the pair of nonlinear partial differ-
ential equations is uncoupled. Although it is fairly easy
to solve a single equation by the perturbation method or by
the Galerkin method, it should be pointed out that a satis-
factory justification of the Berger's hypothesis has not yet
been available in the literature.

*Many additional references on related works on beams, mem-
branes and plates may be found at the end of [7].

 

3

1.2. Purpose of Investigation

To include the strain energy of the second strain in-
variant as well as the in—plane inertia effects in the
formulation of dynamic problems, it is necessary to solve
a pair of coupled nonlinear partial differential equations.
The purpose of this thesis is to develop a perturbation
method for solving this pair of coupled nonlinear partial
differential equations and to apply it to the study of free,
periodic vibrations of continuous systems, such as beam,
circular membrane, and circular plate, which are governed

by equations of the type mentioned above.
1.3. Organization of Report

In Chapter II, a method of solution of a pair of
coupled nonlinear partial differential equations is devel-
Oped in general terms. This may be regarded as an extension
of the work carried out recently by Yen and Blotter [7]
from single nonlinear partial differential equations to
coupled equations. A perturbation expansion is used to con-
vert the nonlinear partial differential equations into a
sequence of linear partial differential equations. The
zeroth order solution is that of the corresponding linear
problem and is well-known, or, say, can be obtained by the
method of separation of variables. The higher order re—
sults are solutions of the inhomogeneous linear problems

which are obtained by suitable eigenfunction expansions.

for
cula

usir

(7

Z)‘

In
'0

4

The governing partial differential equations of motion
for the continuous systems studied here such as beam, cir-
cular membrane and circular plate are derived in Chapter III
using energy approach without the Berger's assumption. Both
first and second order approximations to frequency-amplitude
relations, longitudinal and transverse displacement are
then found by using the general expressions developed in
Chapter II.

Numerical results are obtained using a CDC 6500 digital
computer. It is found that usually the first few terms in
the eigenfunction expansions are sufficient to give satis—
factory results. These results are compared with those ob-
tained using Berger's approximation and are presented in
graphical form. Assessments and discussions of the results
are given in Chapter IV.

Chapter V contains the conclusions.

2.1. =

“m

(ii) a

EEC-tion

II. METHOD OF ANALYSIS

2.1. Equations of Motion

 

Consider the free, undamped, large amplitude, periodic
motion of certain continuous systems governed by the follow—
ing pair of nondimensional coupled equations:

(i) an equation governing the out—of-plane or transverse

motion of the system

I?"

L17] +1? 2 + 5 c1 M(n,y,w2,g) = 0 (2.1.1a)
T

0/

(ii) an equation governing the in—plane or longitudinal

motion of the system

52y
T2 + {3. fl N(T],y,(b2,s:) = O (2.1.1b)

 

LzY + “3

0/

Where n = n(C,r) and y = y(C,T) are the two dependent
functions of the independent variables C and T. Q is
the spatial variable which is assumed to be defined over
the domain from 0 to 1 and T is the time variable. w2

is a frequency parameter and e is a small parameter which
is introduced into the problem either naturally or arti-
ficially. The coefficients a and B are dependent upon
C. L1 and L2 are two linear differential operators with

respect to C of order 2m and Zn respectively. M and

MIII.

I. .
‘\I’I

.+
F1
0
'1
U)

A—l

6
N are two nonlinear differential operators of order not
exceeding 2m and Zn reSpectively, and defined as
2m 2n
2 u llll°°° I u (Tu/h
MINIYMD .3) 3 f(T]IT]'IT] I°°"IT] IYIY :V :°°°°:Y' I
wfi.fi.?.e) (2.1.2)

Lm 3L
g(n’n‘n"’ oooo'n""00°ly’y“' 0000'y5"I.°°‘

N(n.y.w2.c)

(0231,13) (2.1.3)

where f and g are polynomials of finite degree in
q,n',n",....,y,y',y",....,w¢,fi,§,c. The primes here denote
partial differentiation with reSpect to C, and the dots
stand for partial derivatives with respect to T.

It will be assumed that the linear operators L1 and
L2 are self—adjoint for every T in the space of func—

tions defined by the homogeneous boundary conditions:

Diq(0,r) - 0 1 = 1,2 ...... k.

(2.1.4a)
Djn(1 T) = 0 j = k+1, ..... 2m
Six/(0,1) = o 1 = 1,2,.. ,k'
L (2.1.413)
Djy(1,’r) - 0 j = k'+1,. . , 2n .

where the D‘s and 5's are linear differential operators,
of order less than 2m and Zn respectively, with respect
to the spatial variable C.

The periodicity and initial conditions are chosen as

WIC T) = n(C.T+2W) (2.1.5a)
Y(C 1) - Y(C.T+27) (2.1.5b)
fiIC 0) = 0 (2.1.6a)
'(I: 0)= 0 (2.1.6b)

This means that the system has zero initial velocities and
periodic motions are initiated by releasing the system from
rest in an as yet unspecified initial configuration.

Setting 8 = 0 in (2.1.1), the pair of coupled, non-
linear equations become linearized as

2

O/

 

W

LIT] "I‘ (Dz 5T2 = 0 (2.1.73)
52y

Lzy + 002 5—1"; = 0 (2.1.7b)

The corresponding linear solutions are easily found.

A method for solving the pair of coupled, nonlinear
equations (2.1.1) in the vicinity of a set of linear solu-
tions subject to the boundary conditions (2.1.4), periodic—
ity conditions (2.1.5), and the initial conditions (2.1.6)

will be presented in the next section.
2.2. Method of Solution

To solve the pair of coupled nonlinear partial differ—
ential equations (2.1.1) subject to the given boundary,
periodicity and initial conditions, the functions n and
y as well as w2 are expanded into power series in e

as

I ‘ “((4
collec'

results

('I

r;
t—c

KN'

“3 i
n(C.T) = 2 e n (2.2.1)
-_ 1
1-0 .
(c ) 02° '
V IT — H C y
i=0 1 (2.2.2)
00 '
w2 = z slufi (2.2.3)
- 1
1:0

These eXpansions are substituted into the pair of equations
(2.1.1a) and (2.1.1b). Upon expanding f and 9 about
n = no, y = yo, w2 = mg and 5 = 0 as Taylor's series and

collecting like powers of e, the following set of equations

results;
to: Llno + (r3310 = 0 (2.2.4a)
Lzyo + (”3&0 = 0 (2.2.4b)
:1. qul + agfil : -a€fio - a? (2.2.4c)
LzYi + “SN: ‘ ‘QENO ‘ 52 (2°2-4d)
..2 . __ 2.. 2 -
v . Liflz + @022 " ‘Ubno ‘ @221 ‘ aIUifn
+ If ,+ I’ u + ...+ f + 'f ,
WI W W: W Y1 Y 71 Y
M" 2- .. - .. -
+ f " +....+<n f + f“ + f.
Vi y 1 ma U1 W 71 V
+ E ] (2.2.4e)

9

L272 + w5V2 = "$270 ‘ ¢g71 - SIn1§n
+ '— , + "f " +....+ ' + " ,
”lgn U1 W 7’19Y Y19y
+ n— .... 2- .. — .. -
V19y" + + wlng + Ulgfi + Ilgv
+ <38] (2.2.4f)
83: —————————————————————————————————————————
where the notations f, f , etc. mean that f, fn, etc.
are evaluated at (n0, n5, n3,...., yo, y5, y3,...., mg,
Clo: 70: 0)°
The set of equations can be solved recursively for the
unknowns ni, Yi’ and mi, i = 0,1,2,...., assuming that

”i and Vi individually satisfy the boundary conditions as
given in equations (2.1.4), the periodicity condition (2.1.5)
and the initial condition (2.1.6). As usual, for i = 0,

the first two equations of the set are homogeneous and are
just the equations of motion in the linear theory as given
by (2.1.7). For i 2.1, equations (2.2.4c), (2.2.4d),....
have same homogeneous parts as the first two equations but
with inhomogeneous terms consisting of the lower order co-
efficients in the expansions (2.2.1) through (2.2.3). An
essential feature of the perturbation method to be developed

here consists in an eigenfunction expansion procedure for

2

solving these inhomogeneous equations for ni, yi, and wi'

i.: 1.

10
Let us first consider the homogeneous equations
(2.2.4a) and (2.2.4b). The equation for No can be
solved easily by the method of separation of variables.

It has periodic solutions of the form

20(C,T) = Alka(C)cos T (2.2.5)
(1% = a: (2.2.6)
k = 1,2,3, .........

where the constant A1 is a nondimensional amplitude

k
parameter. vk(C) and Qi are the k—th eigenfunction and

eigenvalue of the following problem:

2 _
lek " @ka - 0 (2 .2.7)
Din(0) - 0, 1 = 1,2, .,h (2.2.8)
Dij(1) = 0, j = h+1,....,2m (2.2.9)

The orthogonality condition for the set of eigenfunctions

Vk(C) is

r(c)vk(c)vp(c)dc = ékp (2 2 10)

cg“:

where r(C) is some weighting function and okp the
Kronecker delta.
The equation for yo may have periodic solutions of

the form

VOICIT) = F£(C)cos T (2.2.11)
as: =A2£ . (2.2.12)

11

where P2 is the E—th eigenfunction of the following
problem:
LZFE - AZFE = 0 (2.2.13)
Birg(o) = o, i = 1,2,....,h' (2.2.14)
fijrg(1) = 0, j = h'+1,....,2n (2.2.15)
and [\Z is the corresponding eigenvalue. The orthogonality

condition for the set of eigenfunctions F£(C) is

1

F F = 0 . .
g s(c) )(c) q(:)dc ,q (2 2 16)
where s(§) is some weighting function and 5 the

IQ

Kronecker delta.

Now (2.2.6) determines the values of w2 With one of

0.
these values for a%, the equation (2.2.4b) for yo will
in general have no nontrivial solution satisfying its
periodicity and boundary conditions unless [\Z = 9:. It is

assumed here that

/\2 ¢ 9: (2.2.17)

Therefore with the periodic solution (2.2.5) for no, one

has

VOICIT) s 0 (2.2.18)

This implies that the oscillations take place primarily in
the transverse direction of the system.
The inhomogeneous equations (2.2.4c), (2.2.4d),....,

yield the higher order terms that are demanded by nonlinear

interac

( 1
F‘
C)
Im’
U]

where t

()1
(D
I 1
(D
I"
:3
L4

12
interactions with the zeroth order solution. For the equa—

tions for ”i (i 3.1), solutions are sought in the form

.- I (i) A
qi(C,.) 2 2 Amn Vn(C)cos mr (2.2.19,
m-o n-1 .
1 = 1,2,3,

where the coefficients of expansion Aéi) are as yet to be

i.e. there is no Vk(C)cos r in all ni for 1.: 1.
Similarly, for equations for Vi (i.: 1), solutions
are sought in the form

(C ) = C? 03 C(i)P (C)cos m (2 2 21)
vi ’T- m:o n:1 mn n T ° °

1 = 1,2,3,....

. . a o i u o
where the coeff1c1ents of expan51on Cén) are likeW1se yet

to be determined.

Now, to obtain the i-th order solution for mi and

mi, the procedure consists of substiuting the ni series

given in (2.2.19) into the 81 equation given in (2.2.4).

At this stage expressions for nj in terms of A£g), y.

3
in terms of ng) and w; are already known for j __i—1.

The orthogonality of both the spatial eigenfunctions and

Au)

trigonometric time functions is used to determine mn

and mi. The same procedure is then applied to determine

7. in terms of C(l).
1 mn

 

v-

 

13

In what follows, let us illustrate how to determine
the solution for ni, Vi and m: up to the second order
by equations (2.2.4). The process of solving subsequent
higher order equations can be carried as far as is desired,
but the results will not be presented.

Having determined mg, mg, and yo as given in equa—
tions (2.2 5), (2.2.6) and (2.2.18), one can calculate f
and g in equation (2.2.4c) and (2.2.4d). Substituting

q.(t,i) as given by (2.2.19) into (2.2.4c) yields

8
8

,(Q: - mZQ:)Aé;)Vn(C)Cos m1 =

‘7.
rub

II M
(IN

C
:3

wiAlka(C)cos T - of (2.2.22)

1
To find w2 and A( I
1 mn

, multiply each side of the above
equation by r(C)Vq(C)cos pr and integrate between limits
0 to 1 with respect to C and between limits 0 and 2?
with reSpect to 1. Using the orthogonality conditions one

obtains the following equation:

2
of r(C)Vq(§)cos pr drdfi

cufim

2 . 1 . 1 1
(Qq p S7kIqu ULfiAlkékqélp I I

(2.2.23)
Setting p = 1, q = k, one determines the first order

frequency—amplitude relation as follows:

2 1 (2? _
ml - WA f df r(§)Vk(Q)cos 1 d1 dc (2.2.24)

1k 00 .

 

For p = m (m being some fixed integer other than 1), and

q = n (n being some fixed integer other than k), equation

14

(2.2.23) yields:

A(1I = 1 1 2W 3 ( ) ( I d d 2 2 2
m“ WIQE- mzflfi) I I a r C Vn 4 cos mT T C I ° . 5)

 

Thus the amplitude parameters Aég) are determined provided
._2_22
that an m 9k # 0.

Let us now turn to equation (2.2.4d). Upon substituting

V1(C T) as given by(2.2.21) into equation (2.2.4d), and

recalling that yo has only trivial solution, it follows

that
CD CD
a 2 2 1 _ _
mic r§.(fi‘n - mznk)c( )rn(c)cos mT - -8g (2.2.26)

Multiply both sides of the above equation by s(C)Pq(C)cos p1
and integrate between limits 0 and 1 for C and between
limits 0 and 2v for r. Making use of the orthogonality

conditions, one determines

c‘l) = - 1 I 7” as s(c)r (c)cos m. did:
m ”(Ag—magi) 0 o n (2.2.27)

 

>1 , . .
In the above expression, Cén) lS determined prOV1ded that
fit: — mZQ; # 0. Thus, the first order nonlinear correction

for the k-th linear mode is completely determined as

- C; 0; A(1)V (§)cos m (2 2 28)
W ‘ r . .
1 m=o n-1 mn n

m # 1, n # k

V1

"818
ut48
0A

rn(c)cos mT (2.2 29)

15
The next or second order nonlinear correction for the
k-th linear mode shape and the corresponding frequency are
determined in a similar manner. The inhomogeneous terms on
the right hand side of equations (2.2.4e) and(2.2.4f) are

v, 2 2
found W1th the known no, we, yo, n1, ml,

and Y1. Sub-
stituting n2(:,r) as given by (2.2.19) for i = 2 into

(2.2.4e) one obtains

00 a) 2 \2) 2
Y s _ 2 2 = .
f : (9n m 9kmmn Vn(C)cos mt QgAlka(C)COS T
m—o n-l
oo 03 oo on
+ w: 2 Z p2A(1)V (C)cos pr — u[f Z Z A(1)V (C)cos p1
p=1 q=1 Pq q p=o q=1 Pq q
_ co co dv (c)
+ f , Z Z A(l)——%E—— cos pT
W p=o q=1 Pq
o) oo d2V (Q)
~ f H 2 2 .A(1) ___Q§__ cos pr + ..
.11 pzo qzl pq d:
00 o:
+ f 2 E C(1)P (C)cos pr
7 p30 q=_ Pq q
_ CD 00 (1) df‘ (C)
+ f , 2 Z Cpq —-%Z—— cos p7 +
y p.10") qzi.
2% E a; 0?. 2A(1') (2;)
.4. (D —- " V COS T
1 0J2 T] p21 q:1 p Pq q p
_ (I) (I) (1) _
- f- 2 2 pZC r (C)cos pT + f 1 (2.2.30)
y p=1 q;1 pq q 8

Multiplying (2.2.30) by r(C)V£(C)cos jr, integrating with
respect to C from 0 to 1 and T from 0 to 27, and

using the orthogonality properties one obtains

 

/ 2 _ 2 2 (2) = (1)
‘92 j Qijz sziké kflé + U.) j jg
IIM' 00 00 I) I)
- —- f [f E Z A V C cos pr
W 0 o W P=0 q=1 pq q

CD 00 A(1).__J1£EZ cos p1 +

W p: —o q=1qu dCZ

[88

02411.)

+ f
y p=0 q=l pq Fq(§)cos pr
_ a) a) (1) d? (C)
+ f i E 2 C —_%C— cos p1 + .....
Y p=0 q— 1 Pq
(D 00
+ wzf 2 - f 2 Z p2A(1)V (C)cos pr
1 (D p=1 q- pq q

7 p“. q 1p pq q
+ ff] a r(§)VZ(C)cos jT d1 d: (2.2.31)
Setting j = 1, Z = k, and using the conditions of A(k) = 0
for i = 1, the second order frequency-amplitude relation is

f f [f 2 z A
0 O T] p20 q:1 pq vq

 

VIC)COS p?

00 oo d2V (C)
_ 1
+f,,z 2A()—-—‘L——cospT+ .........
p=o q-1 pq dC2

CDCI)

V 2 (1)
1 q:1p Cpq Fq(C)cos pr

+ fc]a r(C)Vk(C)cos T dr dc (2.2.32)

For j = m, and 2 = n, equation (2.2.31) yields

 

 

A(2) _ 2 12 2 [wim2A(1)]
mn (an-m 9k) mm
1 III" 00 m (1) ()
[f 2 Z A V C cos pr
W(92~m20§) o o n p=o q=1 pq q
00 oo dV (C)
+ f , Z Z A(1) -—£%f—— cos pr
TI p=0 q=1 pg-
00 oo d2V (C)
+ f n Z Z A(1)———9;——-cos p1 + .........
CD 00
+ E. 2 2 C(1)P (§)cos pr
y p_o q_1 pq q

00 oo
2- 1‘
+ m f f 2 V 2 ( )
1 2 u p A V (C)cos p1

18

- f CE 03 p2C(1)F (Q)cos p1
Y p=1 q=1 Pq q
_ m # 1
+ f‘]a r(§)v (§)cos mT deC, (2.2.33)
t q n # k

Similarly, substituting Y2(C,T) as given by (2.2.21)
into equation (2.2.4f) and using the results of (2.2.18),

it follows that

a) 00 2 2 2 (2)
2 Z (ftn - m 9kmmn P (C)cos mT
m=0 n=1
“0 00 (1)
= mi 2 Z pzc P (C)cos pT
p:1 q-zl pq q
- 0° 0° (1)
- B[g Z Z qu Vq(C)cos p1

+ g H 2 Z A

000000000

(I) 00 (1)

+ Q 2 E C P (C)cos pT

_ CD 00 (1) d? (C)
+ 9 . Z 2 C cos p1 + ........
p=o q=0 pq dc
+ wzéw? - g" é; 0; p2A(1)V (C)cos pT
’ 9y 2 Z pchq P (C)cos pT + gs] (2.2.34)
p=1 q=1 .

Multiplying the above equation by s(C)P£(§)cos jr, inte-

grating with respect to Q and T over the interval (0,1)

19

and (0,2w), and using the orthogonality properties one

 

determines
2.
cl.) = naively]
('An"m 9k)

1 1 271' _ (I) (I) (1)
- Z Z A V
7T(.I\:'m2§2]2<) if of [gr] p=0 q=1 pq q(C)COS pT

 

_ 03 oo dv (C)
n pq dC
p=o q=1

0:) <1) d2V (C)
+ 5 M Z Z A(1)-—-Jl—-— cos p1 + .......

p=o q=1 pq dCZ

CD

a) \
- 1;
+ g 2 2 C( 'F (§)cos pT
V p=o q=1 qp q

_ C0 00 df‘ (C)
+ g 2 2‘, (3(1) J—d COS p'r + .....
p=0 q=1 C
CD CD

2- - (1)
+ w, . - N Z Z 2A V s
19 2 9n p=1 q=1P pq q(C)Co PT

g pZCé;)Fq(§)cos pT

+ §fi15 s(c)rn(g)cos m1 deg (2.2.35)

Thus the second order nonlinear correction for the k—th

linear mode is given by

0° 00 (2)
n2 = Z Z: Amn Vn(C)cos mT, m # 1, n # k (2.2.36)
m=o n=1
(I) (I) (2)
72 - Z Z: Cmn Pn(C)cos m1 (2-2-37)
m=o n=1

The complete solutions to the problem up to terms 52

can be written by adding the zeroth, first and second

20
approximate solutions. The final results are explicit once
a knowledge of the spatial eigenfunctions of the associated
linear problem, along with the linear frequencies is avail-
able. Applications of the general results will be made to
problems involving structural elements such as beams,1myfiranes,

and plates in the next chapter.

 

.4.:
.lr‘u‘vv“

III. EXAMPLES

3.1. Introduction

 

The energy method will be used here to derive the
equations governing the nonlinear behavior of continuous
systems such as beams, circular membranes and circular
plates without the Berger's hypothesis.

A continuous system possesses both the strain energy
and the kinetic energy for deflections of the same order
of magnitude as the thickness of a prismatic beam or a
circular plate. The strain energy is composed of that of

bending and that of stretching, namely,

2 2 \
[(3)721 +§le21dx (3.1.1)

for a beam and

2W a
3 f f {[(vzw2 +3522]
..0 O 1']

2(1)(—- + “ML—w) cum (3 1 2)
_ "LL e2 —§—a-—r'— 2 jI' I“ ' o .

for a plate in axisymmetric motions. The strain energy
of a circular membrane in axisymmetric motions is due

solely to the stretching of its middle surface

21

22

aVa
V = f f {Noeo + Noe + 2 g?“ [e2-2(1—u)e2]}r drde (3.1.3)
o o ’

In the above expressions, E is the elastic modulus of
the material, h the thickness, u the Poisson's ratio.
The origin of the x, z coordinate is located at the

left hand end of the beam. The x—axis coincides with the
median line and L is the undeformed length of the beam.
The z-axis is normal to the median line. S is the cross—
sectional area and I the second moment of area of the
beam. For the axisymmetric circular membrane and plate,
the origin of the r, z coordinate is at the center. The
r—axis coincides with the middle surface and a is the
radius. The z—axis is normal to the middle surface. N0

and e0 are, respectively, the initial stress and strain

Eh3
12(1-u2)
rigidity of the plate. (‘72)2 is the biharmonic operator

of the membrance. D = denotes the flexual

defined as follows

 

 

\2 _ 52 1 5 52 1 5
(V2) _ (51:2 +ES-f)(ar2 +175?) (3'1°4)

The first strain invariant expressed in x, z and r,

coordinate system is, respectively,

2
e: CX_%E+_21.(§¥) (3.1.5)
and
e = 5r + 59 (3.1.6)

where the strains in the radial and tangential directions

23

are taken to be

Bu 1 5w 2
€r=§;+-2-( r) (3.1.7)
_ u

Here u and w denote the components of displacement of

a point in the middle surface.

The second strain invariant for membrane and plate

with circular symmetry in r, z coordinate system is

e2 = arse : [37E +-;-(%¥) ]9r- (3.1.9)

It should be noted that there is no second strain invariant
in the case of beam because only the bending and stretching

plane are considered in the expression (3.1.1).

in the x,z

The expression for the kinetic energy is

L
S Bu 2 aw 2
T =-%7 g [( t) + (5?) ]dx (3.1.10)
for a beam, and
ZW’ a 2 2
T = 9;) jug—LE1) + (3%) ]r drde (3.1.11)
0 0 ~
for a membrane or a plate. In the above expressions, p

denotes the mass density per unit volume and t the time.

It is now possible to form the Hamilton's integral, f\,

for beam, membrane and plate, namely,

t2

L
/\= f g F(ut, w

dxdt (3.1.12)

11 u W W W
I I XI I X, XX)

+ t‘.

.24
.1.

and

24

[0

f\=

t’ wt, u, ur, w, wr, wrr)dr d9 dt (3.1.13)

fikar+
Ogle

F(u
1 .

Subscripts in the above expressions denote partial deriva-

tives and the integrand, F, is defined as

F(ut, wt. u, u . w, w , wxx)
= 11 S(u2 + wz) — EI(w2 + §-e2) (3 1 14)
2 P t t xx I . ° '

for a beam,

Fallt, wt: up url wl wr! wrr)
9h 2 2
= r{2 (ut + wt) — [Noeo + Noe
. Eh 2
2(1_u2) ( ( u) 2)]} < )

for a membrane, and

F(ut, wt, u, ur, w, w , w )

r rr

_ r22. 2 2

— rLz (ut + wt)

D 2 12 12 1

- §[(‘72w) + EE-e2-2(1-u)(33e2 +varwrr)]] (3.1.16)

for a plate.
According to Hamilton's principle
5A=o (3.1.17)

The corresponding Euler equations expressed respectively

in x, z and r, z coordinate system are then

25

 

gg-§?§%t-%£ggx+§gg%;=o (3.1.18)
33'-%;%E‘-§‘;55 :0 (3.1.19)
and
3%— 3E3%Z— 3;3%;+3%§%rr=0 (3.1.20)
§%_F.§F_t_ 375: :0 (3.1.21)
r

By carrying out the differentiations, equation (3.1.18)
and (3.1.19) yield a pair of partial differential equations
of motion for beams vibrating at large amplitudes. Simi-
larly, equations (3.1.20) and (3.1.21) yield two pairs of
partial differential equations of axisymmetric motion for
circular membranes and plates. It should be pointed out
that all three pairs of partial differential equations in-
clude the longitudinal inertia effects and the last two
pairs include also the so-called second invariant of the
middle surface strains. One characteristic feature of all
these sets of equations is that they are coupled and non—
linear. The method developed in Chapter II is then used
to obtain approximate solutions of these equations. Prob-
lems of beams, membranes and plates will be considered in

the next three sections.
3.2. Elastic Beams with Immovable Supports

Let us first consider the large amplitude, free, un-

damped, periodic vibrations of an elastic beam in the x, z-

26
plane with an extensible median line. By (3.1.14) ,(3.1.18)

and (3.1.19) one has

éiy._ S 52w §_5w Bzu 5w 52w wBu 1 6w _
' 3.. 31 W + 13%? x37) 37%.. 5(3):) 11- 0
(3.2.1a)
S azu S azu S 5w 52w _
IW-fiiw+35§m-° (”'11”)

To nondimensionalize the equations, let us introduce the

following quantities:

' = E. - = E. C =.§
T) L V L’ L
(3.2.2)
2
- QKLZ —2 L
1.:th (‘02: E ’ ?\=(-IT1

is the radius of gyration of the cross

section and %- is the slenderness ratio of the beam.
Equations (3.2.1a) and (3.2.1b) then take the following
form

—IV . a; . " ’a; ... -31 ll 1-

W - wzn + bin (V' 1 n'n ) + n (v + 53'2)] 1 0 (3 2 3a)

1?? - a?) + ;,'fi“ : o (3.2.3b)

In order to obtain a perturbation solution of the above
pair of equations, the small parameter s is now introduced
into the formulation of the problem through the following

change of variables:

a = an . 3 = EV (3°2°4)

The nondimensional equations of motion then become

IV
U

+ 0.211 — 81(1):)" + q")- +3—511'2nu) = o (3.2.5a)

_)\y" + 0323’, _ Ekq'n" : O (3.2.5b)

A comparison of the above equations with (2.1.1a) and

(2.1.1b) shows that they are identical provided that

The

The

and

Let

4
L1 = 3%? , a = -% (3.2.6)
(n'IWHIV‘I XII, E.) _ nlyll + quay: +gen12nn (3.2.7)
52
L2 = -113E; E>= -1 (3.2.8)
9(1'. n") = n'n" (3.2.9)

corresponding linear equations are found by setting

0 in (3.2.5a) and (3.2.5b), namely,
qfiv + 3330 = o (3.2.10a)

-1y3 + w§§o = o (3.2.10b)

solution to equation (3.2.10a) is

no = A1k Vk(C)cos T

w2 = 9: (3.2.11)
the equation (3.2.10b) has only the trivial solution,

Y0 = 0 (3.2.12)

us consider perturbation in the neighborhood of the

first linear mode. By taking no = A11V1(C)cos T,

91, yo = 0 and a = 0, one can calculate f and §

28

as follows:

E

f(T]'I Tl"! Ya: Y": E ) I u I H = 0 (302013)
flatflotYoflolo -
9(n', n") ' = Ailviv'l'cos2 T (3.2.14)

30,36

LQ
II

In view of equation (3.2.13) the first order frequency—
amplitude relation as given by equation (2.2.24) in

Chapter II follows as
w = 0 (3.2.15)

One also finds that all the amplitude parameters Aéi)
as given by equation (2.2.25) vanish identically. Con-
sequently,
W1(C: T) = 0 (3.2.16)
Upon substitution of 5 as given by equation (3.2.14)
into equation (2.2.27) and integrating with respect to 7,

the expansion coefficients Céi) are found to be

 

 

2
(1) All 3 2 17
Con = Z/\g I ( . . )
2
C(1) All I (3 2 18)
2n ’2<A"£—4ai> .'°
where
1
I = f xvi v; s rn dc (3.2.19)
0 .

and n is an interger equal to or greater than unity.
1
All other Cén) are equal to zero.
The above results immediately lead to the first order

solution for the in-plane displacement

29

31(c. r) = :1 (egg) + egg) cos 2.) Pn(§) (3.2.20)

To determine the second order approximation, deriva—
tives of the nonlinear functions f and g are needed.

These are:

 

En. = y3 = O \

fn“ = y; = O

fv' — n; > (3.2.21)
Ev" = n6

E. = 3‘ 35233 ,

93‘ = ”3 } (3.2.22)
6”” = 3;

Upon substitution of equation (3.2.21) into equation
(2.2.32) and integrating with respect to T, the second

order frequency-amplitude relation is

1 a)
.2 = _ - (1) 1 (1) u I 1 II
“E x) Lqil <C0q +'§C2q )(Vqu + Vqu)
+ 9A2 ‘2 ” d 3 2 23
- §'11V1 V1]r V1 C ( ' ° )

The amplitude parameters follow from equation (2.2.33)

1 (D
{2) A11 g‘ (1) 1 (1) I I II
A\ = fM~C +-C v" +VI‘
w W. .=‘ «q 2 23 1‘ ..>
+- g— Aiiv‘12v'1']r vn dC. n _>_ 1 (33-24)

 

2
(2) All If (I) 1 (1) I n
- M 2 - c (v r + v I‘ )
3n 2 2 2 1 1
(an—991) o q=1 q q
+gAilvizv'l'1r Vn dc, n : 1 (3.2.25)
with all other Aéi) = 0. Thus the second order correction

for the transverse displacement is of the form of equation
(2.2.36) with the coefficients determined above.

Substituting (3.2.16) and (3.2.22) into (2.2.35), one
finds that the expansion coefficients Céi) vanish

identically. Consequently,

72(Cn) = 0 (3.2.26)

Thus the expansion coefficients for the first order
longitudinal displacement, the second order frequency—
amplitude relation, and the amplitude parameters for the
transverse displacement are readily obtained by evaluating
the integrals (3.2.19), (3.2.23), (3.2.24), and (3.2.25).
The integrands of all these integrals involve the products
of three or four spatial eigenfunctions. ~It is, of course,
the boundary conditions such as hinged-hinged, clamped-
clamped or clamped—hinged that determine the set of allowed
eigenvalues and the corresponding eigenfunctions. A CDC
6500 digital computer is then used to perform the integra—
tions numerically. Several particular examples involving
immovable boundary conditions will be studied and the

numerical results will be discussed in the next chapter.

31

3.3. Elastic Circular Membrane with Initial Tension and

Immovable Edge

 

Let us next consider the free, undamped, periodic,
axisymmetric vibrations of an elastic circular membrane
with initial tension and immovable edge. By equations
(3.1.15), (3.1.20) and (3.1.21) one obtains the following
pair of coupled nonlinear partial differential equations

of motion

 

 

 

 

 

l-uz 83w 1.5w gig; 5w gig 1+H Bu
Eh [N0(5r2 + r '51?) - phatz] 537% 2 + r '5?
5w 52w 1. 8w 52w Bu 1 5w 2 u _
”($5.2 2r(§;)1+3r2[r+2(r>+?u1-°
(3.3.1a)
azu l Bu _ E _ 9(1-(12) 5% 5w 52w L11. 5w 2
3.2+]: r .2 E 5.2”???“ 2J5) '0
(3.3.1b)

The above equations may be nondimensionalized so that the
membrane is of unit radius and the period of vibration is

fixed at 2v by introducing

- ._ W - _ u __ r
*1 * 2: , V - a ' C - a
(3.3.2)
_ 932 _ NO

Equations (3.3.1a) and (3.3.1b) then take the form
. 2_ 2 (U2: -' _" 41+ -' -l-li
(1+u)eo V'n - (1-u ) n + n (v + C v + n n + fiz'n

_ 1 _
+71 / +‘%Y+'§fl

32

2 — \ 2 -'-’N - -'
( -%2>v- (1—u2)wzy+nn +:_21_C_£n2 =0 (3-3o3b)
where
62 1

Performing the following change of variables

5:8” , §=ev (3.3.5)

one finally obtains the two coupled, nonlinear, nondimen-

sional equations of motion

2 5 1+.
_ __._... .4. .2 _ __5. I II __E I II I E
Luvnvw 1_‘2'H[T](Y +1; v)+n(v +CY)

. 2 1 .
+ 3(3ql T)" + C 1.13)] = 0 (3.3.63)
1 2 1 v. a I .' 1— l
"fiflv -Ez)v+wzv-f;2'(nn +—§%n'2)=0

(3.3.6b)
which are similar to the two equations of motion described

in Chapter II. The analogous quantities are

e0 2 1
L1 = -i_—uv , a: —1_’L_L-2- (3.3.7)
' H i 'I I M 1+ l‘ I. I
f(Tl'IT1:YuV.-Y:€)=U(Y+_‘CHY)+W(Y+%Y)
£1 2 H 1 '3
+ é(3q' T] + E T] ) (3°3-8)
and
1 2 1 1
L2=-j:p'(V -E2)» B=-fi2 (3.3.9)
II .. I II 1— -
9(n'ln ) - n n + U" n'2 (3.3.10)

33
The corresponding linear equations are found by setting

8 = o in (3.3.6a) and (3.3.6b), namely.

 

e0 2 ..
1 2 1 u
- 1-(1:2 (V _ E2))0 + 0533/0 = 0 (3°3°11b)

Their solutions are then taken to be

We = A11 V1 (C)COS T

(3.3.12)
2 _ 2
mo 01
It then follows from (3.3.8) and (3.3.10) that
f = f(n'. n“. y. 7'. v". a) ' H a u = 0(3.3.14)
WOIUOIY0:Y0:YOIO
i = gin”. n") ‘ n = A§.(vivf + l§%v;2)coszT (3.3.15)
30:30

By (3.3.14), the first order frequency—amplitude relation

as given by equation (2.2.24) is
2
w = 0 (3.3.16)

and the amplitude parameters as given by equation (2.2.25)

are
A(1) - o (3 3 17)
mn ‘ ° '
I: then follows that
31(C, T) = 0 (3.3.18)

Upon substitution of (3.3.15) into (2.2.27) and

integrating with respect to r lead to z

%|§3[ai ‘44: I ”(an

34

 

 

(1) A11 > 1
C0n EKQ'I, n...1 (3.3. 9)
n .
2
(1) _ A11 >
- I n _ 1 (3.3.20)
2n 2mg; 49")
1
where
1
1 I I II 1_- g
I = f l-uz (V1V1 + 7% V12)s Pn dc, n : 1 (3.3.21)
0 .

All other C£i> are equal to zero. The first order radial
displacement is then
(C T) = 0; (C(l) +C(1) cos 21') I" (C) (3 3 22)
Y1 ’ ‘ on an n ° °
n=1 . .
The second order result may be determined in a similar

manner. The following derivatives of f and g are first

evaluated.

 

E , -~ E ,, == 0 )
n W
E) :Ali 1% V: COS T
.. ' +v II
fy, = A11(123=v£ + V1)cos 1 )(3.3.23)
E)” = All Vi COS T
— A3! a I
f = —§;-’3V12V1 + %-V1 )cos3 T j
and
6“. = A11(V; + lEE‘V1)COS T (3 3 24)
g . = A1] V1 cos T ‘

35

By the first order amplitude parameter results (3.3.17),
substituting (3.3.23) into the general expression given by
equation (2.2.32) and integrating with respect to time,

one obtains the second order frequency-amplitude relation as

 

1 (I)
2 _ _ 1 (1) .1 (1) .2 '
wz — 1_u2 g {qil (coq + 2 c2q )[C vqu
, (1+)! VI + II . + VI ..
+ g-A§1(3vi2vf +~% v13)}r v1 dc (3.3.25)

The second order amplitude parameters follow from the

general expression given by equation (2.2.33)

 

 

(1) _ A11 1 (1) 1 (1) u u
— 2 + -C -—
Aln (Q: - a?) g 1:37{q=1 (Coq 2 2q )[C Vqu
1+ . I II I I II
+ (—E-Li-v1 + v1)Fq + Vqu]
3 2 :2 u 1 .3
+ g A} 1(3V1 V1 + 2" V1 )}I' Vn dc: n > 1 (3 °3 ‘26)
(2) A11 } 1 OD 1- (1) E; '
, = 4 —:—§-{ 2 -c [ v1?
3n (9: _ 90%) 0 1 u q=1 2 2g C q
1+ . , " I I u
a" (‘TLL' V1 1" V1)Fq + Vqu]
_- 1 2 I II 1 t3\ >
...9. E. A11(3V1V1 + E V1 /} r Vn dC’ n _ 1 (3 03 027)

2
All other Aén) are equal to zero.
The second order transverse displacement is then
00

- _ v (2) ) £2 (2) ) 3 3 3 28)
32(CvT) - n22 A1n Vn(C COST'+ n=1 A3n Vn(C cos T ( . .

36
In view of (3.3.16), (3.3.17), and (3.3.24), one con-

cludes that equation (2.2.35) gives
C(2) = 0 (3.3.29)
mn V

Consequently,

720;, I) = 0 (3.3.30)

A representative example of a circular membrane with
a clamped immovable edge will be studied in the next

chapter.
3.4. Elastic Circular Plate with Immovable Edge

Let us finally consider the free, undamped, periodic,
axisymmetric vibrations of an elastic circular plate of
radius a with immovable supports at the outer rim. By

equations (3.1.15), (3.1.20) and (3.1.21)

1 Bu ow 52w 1 5w 2
'WW) D Traffic?! r E+$§F+§E(5?H

62 a 1 a 2
+ vii-5% +1 + Eva—11> n = 0 (3.4.13
2 l. _ 1- 2 azu §y_52w 1’“ §w_2 _
(V —r2)u E W+ar5757+ 2r(or) _0 (3'4°1b)

Nondimensionalizing this pair of coupled nonlinear partial
differential equations of motion in the same manner as was

done previously, namely,

’ —.E " = E. g _.£
T] — a ' y a ’ D _ a
(3.4.2)
_, 4... 2 _
1mm, wzzagiwz’ (212%)

37

equations (3.4.1a) and (3.4.1b) become

2- 2 - -' -u 1+ -I -I-H 1 -I2
-(Vn) -wzn+%[1(v +7Ev+nn +5771)
—II —I _ 1 _‘2
+ .1 h .3, +51. )1 (3.4.3.)
2 1 ’ 2 “I-" 1- -I
MV -Ez)y-wzy+Mnn +*§%nz)=0 (3.4.313)

Let us now introduce the perturbation parameter 8

through the following change of variables

5- cn: §= a) (3.4.4)
The pair of nondimensional, governing equations of motion

in a final form used to obtain solutions is

4 1+ .
V n + 0023 - eMn'h" + 7& 7') + n"(v' + if 7)

J

1 3
+‘% (BHIT‘u +1.1]. )] = 0 (3.4.53)

The identification of equations (3.4.5) with (2.1.1)

becomes evident if one lets

N

2 (E) a )2 , CL —)\ (3.4.6)

1
FEE-+333
f(.q:’ T1", y, ya! y", E) : Tll(yl0 +%yl) + nu(yl +‘LEL'Y)

+ 3 (33‘3" + 'jén'a) (3'4”)

and

L2 = -%(V .. 31:), B = _ x (3.4.8)

38

I II - I II 1- I
9(3 . n ) - n n +'-§% 1 2 (3-4-9)

Setting 5 = o in (3.4.5a) and (3.4.5b) yields the

following linear equations of motion
4 2..

2 1 "
"MV ‘ :2) Yo + 0333’0 = 0 (3-4-10b)

The solution of which is taken as

30 = A11V1(C)COS T '\
r (3 .4.11)
$2 = 02 “I
o 1
YO : 0 (3.4.12)

i.e. perturbations in the vicinity of the first linear

mode are considered. Thus f and 6 can be evaluated as

follows:
E = f(n'. n".- 7. y'. y", 3) . .. ' .. = 0 (3.4.13)
WOIWolYoIYo.0 .
- = ' ”l - u = 2 v'v" +-12E v'2 s2 7 3.4.14
g 9“] I 7] TIOITIO A1111 1 2C 1 )CO ( )

 

In view of (3.4.13), equations (2.2.24) and (2.2.25)

then yield reSpectively
mi = 0 (3.4.15)
1)
A = 0 (3.4.16)

It follows that

31(c.1) = 0 (3.4.17)

39
Upon substituting (3.4.14) into (2.2.27) and inte-

grating with respect to T, one obtains

 

(1) - A31 > 1 3 4 18)
Con - Z—I-C: I I n _ ( . .
2
(1) All >
c 2 .47- 1 , n._ 1 (3.4.19)
21‘- 2(An - 491) .
where
1
_ “ I gs 1— I2
I - f )(viv1 + —§%év1 ) s rn dc, n 2.1 (3.4.20)
0 .

All other Céi) are equal to zero.
The first order result for the radial displacement

then follows as

00 (1)
“(CH0 = Z (c “
n=1

+ C(1)

on 2n cos 21) Fn(C) (3.4.21)

To determine the second order solution, the following

derivatives of f and g are evaluated.

 

f . == E .. = 0 \
W W
— = .. E II
fy A11 C V1 cos T
1 E

f). = A11(—+E‘E' V; + V£)COS T >(3 .4.22)
fy" = Allvi cos T

A2 1
E8 =-j%l (3Vizvi + Z'V13)COS3 T J

éna' A11(V; + liE'Vikos T
(3 .4 .23)

 

Q”. = Allv; cos T

40
Inserting equation (3.4.22) into (2.2.32) and upon

integrating with respect to T, the second order frequency-

amplitude relation follows as

)

1
_ _- (1
A f 2 {(CO Oq

o q=1
'1"? 1 II I ..
+x—gi v. + v1) Pg; + V1111]

+%A11(3v12 v1 +-%-Vi3)}r vldg (3.4.24)

Similarly, the amplitude parameters for the second

order correction are obtained from equation (2.2.33)

 

 

(2) _ A11 1 CD (1) 1 (1) .2 -
A _ f A{ Z (c + —C )[ V1?
in (9% _ 92) o q=1 oq 2 2q C q
+ (lifi v1 + v1)ré + virg]
2 1 II I
+‘g'A11(3V12V1 + 2 V13)}r VndC: n > 1 (3°4°25)
.(2) A11 1 00 1 (1).3 '
A = fx{ 2 —-c [ v1?
a; 2 2 2

1 + I H | I II
+ (—EE V1 + v1)rq + vqu]

+ é-Ai1(3V1V: + %-V13)}r VndC. n-1 1 (3-4o25)

1311 other Aéi) are equal to zero.

The second order result for the transverse displacement

tflen follows as

CD

Th2 (C T) = mi Aii) Vn(C)cos T + Z HAg:

(c)cos 31 (3.4.27)
2 n==1vn

 

go, illltlgz 1 .. ‘ ..n '
(r ,

41
In view of (3.4.15), (3.4.16), and (3.4.23), equation

(2.2.35) yields

C(z) = 0 (3.4.28)
mn .

It follows that
72(CIT) = 0 (3.4.29)

An illustrative example for a circular plate with
specific immovable boundary conditions will also be pre—

sented in the next chapter.

IV. NUMERICAL RESULTS AND COMPARISON STUDIES

4.1. Introduction

In the preceding chapters, general expressions for
frequency-amplitude relations as well as for the mode
shapes of vibrations for a wide class of nonlinear vibra—
tion problems have been presented. These expressions apply
to beams, circular membranes and circular plates subject
to a variety of boundary conditions. In this chapter,
problems with specific boundary conditions are solved.

When the boundary conditions are given, the spatial
eigenfunctions Vk, Pn and the corresponding eigenvalues
9:. f\: appearing in the general expressions are there—
fore known. As is noted previously, the determination of
the frequency—amplitude relations and of the coefficients
of the series expansions for the mode shapes of vibrations
is then reduced to the evaluation of integrals involving
such known eigenfunctions. The integrals are evaluated
numberically on a CDC 6500 digital computer. The computer
program proceeds as follows: ‘

The interval of integration (0,1) is divided initially
into NP = 10 equal intervals and a six~point Newton-Cotes
formula is applied to each of the NP subdivisions. A

42

43

sum over NP represents an approximation to the desired
integral over (0,1). NP is then doubled and the process
repeated until the relative error between two approximations
is less than or equal to 10-6.

An additional subroutine is required for calculating
Bessel functions which arise as the spatial eigenfunctions
of the circular membrane and the circular plate problems.
This subroutine was obtained from the Program.Library of
Michigan State University Computer Laboratory.

Numerical results are presented in the next three
sections. These results are then discussed and compared

with those previously obtained by solving single nonlinear

partial differential equations.
44?. Beams with Various Boundary Conditions

Prismatic beams with immovable supports as described
in Section 3.2 are considered here for hinged—hinged,
clamped-clamped, and clamped-hinged boundary conditions.

Let us first consider a beam having both ends hinged.
The transverse displacement, the bending moment, and the
longitudinal displacement are zero at either end. The

boundary conditions then follow as

Tl(O,T) :- T]"(0,T) = 0
., (4.2.1)
T] (111') = T)" (1.1:) = 0 .

7(0.T) 7(1,T) = 0 (4.2.2)

44
The differential equation (2.2.7) together with boundary
conditions (4.2.1) are identical to that of the Euler-
Bernoulli beam. By the orthogonality condition (2.2.10)
with r(§) = 1, the normalized transverse spatial eigen-

functions and its corresponding eigenvalues are

Vk(C) = J2 sin kw: ‘
n: = (5k)4 = (kv)4 )(4.2.3)
k = 1,2,3, ...... J

 

Also, the differential equation (2.2.13) together with
boundary conditions (4.2.2) are identical to that of a beam
executing longitudinal vibrations. By the orthogonality
condition (2.2.16) with s(c) = 1, the normalized longitudinal

spatial eigenfunctions and their corresponding eigenvalues

are

 

rn(g) =nf2 sin nWC \
f\: = 1(nw)2 >(4.2.4)
n = 1,2,3, ..... a

. . 2 . .
Substituting of V1, 9:, Pn and j\n into equations

(3.2.17), (3.2.18) and integrating with respect to C give

 

respectively
1 2
C32) : -1—7é-‘J—2 A11 (4.2.5)
_ F 2
egg) - - W2 J2 A11 (4.2.6)
16(1— §—-)

45

with all other Céi) being equal to zero. It is seen

that Cég) depends on the parameter I which is defined
2
as (E) .

r Numerical results of these two coefficients are

given in Table 4.2.1 as a function of %~.
Table 4.2.1. Numerical values of the expansion coeffici-

1
ents C52) and Céé) of hinged-hinged beamsf

 

 

 

{:- 43 4?
40 -0.27768 -O.2794O
80 -O.27768 -O.27811
120 -0.27768 —O.27787
160 -O.27768 —O.27779
200 -O.27768 -0.27775

 

*-
All values multiplied by Ail.

The second order frequency—amplitude relation is found

from equation (3.2.23) to be

§=§oimwéAnf wan)
where

F”) = 2(7T:_>\)(?\ - '3' ”2) +2 (4'23)
and

61 = W.

Numverical results of F(x) over a range of %- are shown

in Table 4.2.2.

46

Table 4.2.2. Numerical values of F(k) of hinged—hinged
beams.

 

 

%- 4o 80 120 160 200

 

F(%) 0.99897 0.99974 0.99989 0.99994 0.99996

 

Two other beam examples of practical interest are
a beam having both ends clamped and a beam clamped at one
end (Q = O) and hinged at the other end (Q = 1). The
transcendental equation for the eigenvalues of the clamped—

clamped beam is

cosh 6 cos B = 1 (4.2.9)
and the transcendental equation for the eigenvalues of the
clamped—hinged beam is

tanh B = tan B (4.2.10)

The normalized transverse spatial eigenfunctions and their

corresponding eigenvalues for.both cases follow in the form

 

Vk(C) = cosh Bk: - cos BkC - sinh Bk _ sin Bk(sinh Bk:

9: = (5k) (4.2.11)
k

Numerical values of Bk for both end conditions are tabu—

lated in [3].

47

As the boundary conditions for the longitudinal dis—
placement here are the same as in the hinged-hinged beam
case. The normalized longitudinal spatial eigenfunctions
Fn and their corresponding eigenvalues f\: remain the
same as those given in equation (4.2.4).

After substituting V1, 0:, Fn and ft: into (3.2.17),
(3.2.18) and performing the integrations on the computer
the results of the first nine expansion coefficients for
both cases are given in Tables 4.2.3 through 4.2.6.

One may note that in the hinged-hinged and the clamped-
clamped cases, the product of the derivatives, vivf, in
equation (3.2.19) is anti—symmetrical about the mid-section
of the beam and that the shapes sin ZFC, sin 4WC, etc. are
also anti-symmetrical about this mid-section. By constrast

the shapes sin WC, sin 3WC, etc. are symmetrical about the

mid-section. It is then obvious that coefficients like

1 .
C31), C03), ...., céi), Cég), ...., etc. must vanish for
these two cases. For the case of clamped—hinged beam, of

course, these symmetry considerations are no longer appli-
cable.
Using equation (3.2.23) one obtains the second order

frequency-amplitude relation as

2
.2 = 3 s1 Fmé A1.) (4.2.12)

where

48

.Hmd 00 UmamfiuHSE mmsHm> Ha<*

 

 

50000.0 0H000.0 00000.0 00000.0 00H00.0 00000.0 005H0.0 0000fi.0 50H00.0I 000

50000.0 0H000.0 v0000.0 00000.0 00H00.0 00000.0 005H0.0 0000H.0 00H00.0I 00H

50000.0 0fi000.0 V0000.0 00000.0 00H00.0 00000.0 005H0.0 #500H.0 HvH00.0I 00H
50000.0 0H000.0 00000.0 00000.0 00H00.0 00000.0 505H0.0 0000H.0 00H00.0I 00
50000.0 0H000.0 #0000.0 00000.0 00H00.0 00000.0 H05H0.0 0000H.0 00000.0I 0v

H

m

0H 0H VH 0H 0H 0 0 v 0 Gnu
0

 

 

 

*.mEmmn GmQEmHU
IUmmEmao mo Aflwo mucmfloflmwwoo coamcmmxm map 00 mmdam> HmUflHmEsz .v.0.v magma

.me an Umflamfluane mmsam> HH¢
.x.

 

C0
50000.0 0H000.0 00000.0 00000.0 00H00.0 00000.0 005fi0.0 0000H.0 00H00.0I \HVU
/

 

0H 0H «a 0H 0H 0 0 v 0 c

 

 

H .
so *.m mam sues mfimwn UmmEmHo

ImeEmHo mo AHVU mucmfloflmmmoo soamcmmxm mpu mo mmSHm> Havaumfisz .0.0.v magma

49

.Hwé an 0mHmHuHSE mmsHm> 0H0
*

 

 

 

 

 

00000.0 05000.0 00000.0 00000.0 00000.0 00000.0 0000H.0 0H500.0I 50000.0 000
00000.0 05000.0 00000.0 00000.0 00000.0 00000.0 0000H.0 00500.0: 05000.0 00H
00000.0 05000.0 00H00.0 00000.0 00000.0 00000.0 0000H.0 .00000.0I 05000.0 00H
00000.0 05000.0 00H00.0 00000.0 00000.0 00000.0 0000H.0 00500.0: 05000.0 00
00000.0 05000.0 00H00.0 00000.0 00000.0 00000.0 0005H.0 HH0H0.01 05050.0 00
m.
A
0 0 5 0 0 0 0 0 H Gnu
3
c
an *.mEme 0m0GH£
IwmmEmHo mo AHVU mucmfloflmmmoo coamcmmxm map 00 mmsHm> HMUHHmEdz .0.0.0 magma

.fimm >9 Umaamfluasfi madam> Had
*

 

0
00000.0 05000.0 00000.0 00000.0 H0000.0 00000.0 H000H.0 00500.0I 00000.0 Afivu

 

0 0 5 0 0 0 0 0 H c

 

 

IH

* mam Sufi? mammn Ummcfln
mucmfloflmmmoo coflmcmmxw msu mo mmsHm> Hmoflumfisz

..'.|

C0
-ommsoflo 00 “H00 .o.m.v manna

 

 

1 OO
.. 8 (1) (1)
F7\ -- E c +—c VF
( ) i [geiAii q=1 ( °q 2 2g )( 1
, I" 3 12"
+ V1Fq)+ 4 V1 V1:IV1 dC (4'2'13)
1
and
51 = 4.7300408 (4.2.14)

for clampednclamped,

B1 = 3.9266023 (4.2.15)
for clamped—hinged end conditions. It is to be noted
that the integration of (4.2.13) involves a series. A
sufficient number of longitudinal spatial eigenfunctions
must be taken to insure convergence. Tables 4.2.7 and
4.2.8 show how closely the truncated series represents the
function F(%) for different values of N where N repre-
sents the number of terms used in the truncation.

With the information of F(x) from Table 4.2.2 for
hinged-hinged, 4.2.7 for clamped-clamped, and 4.2.8 for
clamped-hinged beams, nonlinear frequency-amplitude rela-
tions including the second order corrections are now in

the form

2
.32 = w: [1 + g Fm @5211.) 1 (43-16)

Blotter [7] and Evensen {9] have studied nonlinear
vibrations of beams with various end conditions using a
single nonlinear partial differential equation. Before
one compares (4.2.16) with the corresponding results of

Blotter and Evensen, it will be of interest to rederive

51

Table 4.2.7. Convergence of F(k) of clamped—clamped beams.

 

 

 

 

 

 

 

( >N
F A
E. 3 6 9
r
40 0.30235 0.30196 0.30196
80 0.30266 0.30227 0.30227
120 0.30271 0.30233 0.30232
160 0.30273 0.30235 0.30234
200 0.30274 0.30236 0.30235
Table 4.2.8. Convergence of F(x) of clamped-hinged beams.
( )N
F K
E. 3 6 9
r
40 0.56483 0.55496 0.55485
80 0.56687 0.55701 0.55689
120 0.56723 0.55737 0.55726
160 0.56735 0.55750 0.55738
200 0.56741 0.55756 0.55744

 

 

52
their equation of motion using the energy approach, so as

to see what simplifications have been made. Application

of equations (3.1.16) and (3.1.17) to (3.1.12) yields

82w 52w 8w 5e 84w _
_pS SEE+ES(6W+ X X.) - EI SF— 0 (4.2.17)
2
-pS S:%-+~ES 33-: 0 (4.2.18)

When the longitudinal inertia term is neglected, equation
(4.2.18) becomes

%§= 0 (4.2.19)

Thus, e is independent of x. It is recalled that the

first strain invariant, e, is defined as

e = 5X = 3% + $43—32 (3.1.5)

In view of the assumed constancy of e with respect to x,
it is possible to multiply by dx and integrate over the

length of the beam to find

L L 5 L L
u 1 8w 2 1 5w 2 4.2.20)
f GdX = f [5; + ..2—(675) ]dX = u + '2-f(~5;) dX (
0 0 0 o
The vanishing of u at the boundary leads to
L
1 5w 2
e = 12"]: f (5;) dX (4.2.21)
0 .
Equation (4.2.17) now becomes
84w 02w 02w
EI -— ‘- T -——— + S = 0 14.2.22
3x4 5x2 9 5t§ ( )

where the total axial tension in the beam is defined as

53

L
T(t) = g—ff f ($92.51.. (4.2.23)
0 .

Equation (4.2.22) is exactly the same equation of motion
used by Blotter and Evensen. It was Woinowsky—Krieger [10]
who first established this equation using the balance of
forces approach and studied the effect of axial force on
the vibration of hinged bars.

It must be pointed out that the single equation of
motion (4.2.22) results as the consequence of neglecting
both the effects of longitudinal inertia and that of the
first spatial derivative of the first strain invariant.
This theory yields only the total axial force of the beam
by equation (4.2.23). To determine the distribution of the
axial force in the longitudinal direction, the displacement
u cannot be ignored. It is necessary to return to the full
pair of coupled nonlinear partial differential equations
of motion as given by (3.2.1a) and (3.2.1b).

The frequency-amplitude relationships as given by
(4.2.16), along with those of Blotter and Evensen, are
shown in Figures 4.2.1 through 4.2.3. In these figures A

is related to A11 by

A = A11 | v1 (4.2.24)

lmax

All these curves show the same feature that the nonlinear
frequencies increase with increasing amplitudes. It is to
be stressed that the results of Blotter and Evensen do not

depend on the slenderness ratio %- while the results of

.9

Nonlinear—Linear Frequency Ratio

54

 

 

1.6-
Blotter and Evensen's
uncoupled equation result
1.4-4
.5
1.2)-
I1:200
/, r
1.0
£=40
<? r
l l l l l
0 0.4 0.8 1.2 1.6 2.0 2.

 

Nondimensional Amplitude LEA

Figure 4.2.1. Frequency—amplitude curves for hinged-hinged
beams.

1.15_

Nonlinear—Linear Frequency Ratio

 

55

Blotter and Evensen's
uncoupled equation result

 

Figure 4.2.2.

 

1.05-
L-=200
r
1.00
31:40
r
T 1 lo I I l I
0 0.4 0.8 1.2 1.6 2.0 2.4
Nondimensional Amplitude LEA

Frequency-amplitude curves for clamped—
clamped beams.

56

1.3-

Blotter and Evensen's
uncoupled equation result

313°

1.1“

Nonlinear-Linear Frequency Ratio

 

 

 

£=200
r
1 0 2.: 40
r
if 1 1 1 I L I
0 0.4 0.8 1.2 1.6 2.0 EFFI—
Nondimensional Amplitude LEA

Figure 4.2.3. Frequency-amplitude curves for clamped-
hinged beams.

57

(4.2.16) do. In all three cases, results based on (4.2.16)
are shown for %-= 200, representing a thin beam and for
%-= 40, representing neither a thin nor a thick beam. It

is to be noted that for small %- values, curves based on

the more accurate analysis lie below those of Blotter and
Evensen. It is interesting to note also that Blotter and
Evensen's results coincide with those of the present analy—
sis only for beams with large slenderness ratio (%-2:200).
This fact can also be established analytically as follows.
For large I, the expansion coefficients for the longitudinal

displacement of a hinged—hinged beam as given in equations

(4.2.5) and (4.2.6) can be approximated by

2
03;) = cg) = - {gfz A11, (4.2.25)

Also, F(x) as given in equation (4.2.8) approaches unity.
The second order frequency-amplitude relation of a hinged-

hinged beam as given in equation (4.2.7) becomes

0: =§6§<If A...)2 (42-26)
which is identical with Evensen‘s expression.
On the basis of the above discussion, the following
conclusion can therefore be drawn;
It is reasonable to neglect the longitudinal inertia
effect and to assume the first strain invariant independent
of the spatial variable of the beam only for slender beams

vibrating transversely principally.

58

4.3. Clamped Circular Membrane with Initial Tension

 

As an example for circular membranes with initial ten—
sion and immovable edge, let us consider a circular membrane
clamped at the boundary Q = 1. This implies that both the
transverse and the radial displacements at the rim are zero.

The boundary conditions are therefore

0 is bounded as Q-—> 0+

(4.3.1)
"1(10) = 0

y is bounded as C —+ 0+
(4.3.2)

7(1)?) = 0
By the orthogonality condition (2.2.10) with the
weighting function r(§) equal to C. the normalized
transverse spatial eigenfunctions and their corresponding
eigenvalues satisfying the differential equation (2.2.7)

and the boundary conditions (4.3.1) follow as

vk(c) =3-{3—7 J0(j C) .
.1 30k 0k

e0
9: = 1 _ L1j:k )(4.3.3)

 

 

1,2,3, .....

k

A
where J0(jokC) and J1(j0k) are the Bessel functions of
the first kind, of order zero and order one respectively.

j0k is the k—th positive zero of J°(j0k) = 00

59
Similarly by the orthogonality condition (2.2.16) with
the weighting function s(C) equal to C, the normalized
radial spatial eigenfunctions and their corresponding
eigenvalues satisfying the differential equation (2.2.13)

and the boundary conditions (4.3.2) are

_ J2 .
1"n(C)-m Jib1n C)
/\2 = -l—7-j2 (4 3 4)
n 1—0‘ in ' ’
n = 1,2,3 ......
where j1n is the n—th positive zero of J1(j1n) = 0.
Numerical values of j0k and j1n have been calcu—

lated and tabulated in [11].

Upon substituting equations (4.3.3) and (4.3.4) into
(3.3.19) and (3.3.20), and performing the integrations on
the computer, one obtains the first order expansion coef-
ficients for the radial displacement. The results of the
first nine coefficients are tabulated as a function of ea
in Tables 4.3.1 and 4.3.2.

From equation (3.3.25), the second order frequency—
amplitude relation follows as

2 — 2 All 2
db - db F(e0)(JZ%) (4.3.5)

where

60

.HH
0¢

an ooflamfluase nmoam> HH¢
.x.

 

 

moooo.o ofiooo.o oHooo.o HHooo.o m~ooo.o mfiooo.o mmooo.o oeooo.o venoo.o1 .ooH
nmooo.o1 noooo.o1 mfifioo.o1 vaoo.o1 ooHHo.o womoo.o 0mmoo.o mmwoo.o oomeo.o1 oH
mmooo.o1 mmooo.o1 omooo.o1 amooo.o1 nvfioo.o1 momoo.on wmmoo.o1 mnono.o vomnm.o1 H
mmooo.o1 mmooo.o1 mwooo.o1 wuooo.o1 nmfioo.o1 mbmoo.o1 oeuoo.o1 Hommo.o mooofi.mfiu n.o
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o m u o n o m m H AMWU

 

 

 

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Ca *.mQCMHQEmE
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N

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.x.

 

00000.01

00000.01

50000.01

05000.01

00H00.01

00000.01

50000.01

H0500.01

00000.0

CO
3.

 

0

0

0

 

0

0

0

0

H

C

 

 

HMHSUHHU cmmfimao mo

1

C0

F

*.cflmuum Hmfluflufl mam £UH3 mmcmHQEmE

V0 mwcmfloflmmmoo coflmcmmxm 000 00 mmsam> HMUHHOEDZ .H.0.0 mHQmB

1 1 1 00 (1.) 1
We ) - “—22— {-—- >3 (c e-
\ 0 (1‘+'Ll)301 3 A31} q=1 oq 2
. (13.1 . 1.3.).q
+ 3(3V5.2vu + % V13)]C V) dc

(4.3.6)

Table 4.3.3 shows how closely the truncated series repre-

 

 

 

 

 

The ratio

as

a? e 02

« (All 2
)Jéo)]

sents the function F(e0) with the Poisson“s ratio 0
equal to 0-3.
Table 4.3.3. Convergence of F(eo) of clamped circular
membranes.

7 N

F(eo) 3 6 9

e0

0.01 0.35245 0.35225 0.35223

0.1 0.34164 0.34144 0.34141

0.4 0.14744 0.14724 0.14722

0.5 2.27625 2.27605 2.27602

1 0.43980 0.43960 0.43957

10 0.40288 0.40292 0.40289

100 0.40046 0.40034 0.40032

From equation (2.2.3). the frequency—amplitude relations

including terms up to 52 are given by

(4.3.7)

of the nonlinear to the linear period then follows

62

 

max

*-
1% = 3 1 1 (4.3.8)
3 ...0) 2 £2
1 + 2 ...f‘A.._
L 1V1 Imax( “(90)
where
A = All IVll (4.3.9)

Equation (4.3.8) again shows that tne nonlinear frequencies
increase (nonlinear periods decrease) with increasing
amplitudes.

Chobotov and Binder [13} have obtained the pair of
coupled nonlinear equations of motion for an axisymmetric
clamped membrane with initial tension by summing the changes
in the membrane forces due to radial and transverse dis-
placements in the respective directions. Without consider—
ing the effects due to the radial inertia term, a perturba—
tion procedure and the Ritz—Galerkin technique was employed
to solve approximately the pair of coupled nonlinear par-
tial differential equations. Their results can be used for
the purpose of comparison with the present analysis. Also,
it is interesting to investigate how the Berger's hypothesis
will effect these results. Upon neglecting terms containing
the second strain invariant e2, equation (3.1.15) is

modified as follows

“[Non + Nge + {Ell—w 82]} (403.10)

63

Application of equations (3.1.20) and (3.1.21) to (4.3.10)

 

 

yields
2 __ E}; 52W Eh 1' 2 @317 Be 3 _
V w N. 5:7 + (1-2).. .. V .. .. 1:1- 0 (11-3-11)
-,2 22 ‘
_ 911E ) :11 g5; : 0 (403.12)

When the radial inertia term is negleCted. equation

(4.3.12) becomes

g; = 0 (4.3.13)
Thus, the first strain invariant. e. is independent of r.

It is recalled that e for an axisymmetric membrane is

defined as

.2 -
3.1.1. + 1(5‘111) . :1

e = a): (3.1.6)

Multiplying both sides of equation (3.1.6) by rdrde and

integrating over the area. one obtains

a
a \ .
2 1 11w 3
e = —2 ru + —2 f 1&1, dr (4.3.14)
3 o a CI"

The vanishing of u at the boundary leads to

1 a 3:2
e = a? g r(§¥) dr (4.3.15)

In View of equations (4.3.13) and (4.3.15), equation

(4.3.11) now becomes uncoupled as follows

:¥-+ Eh v2 w f r<5w)2 dr = 0 (4.3.16)

(l-LLZ )N032

 

Z/lO/

vw-%

Although (4.3.16) is much easier to solve than the fully

64
coupled pair of equations (3.3.1), the hypothesis that the
term containing e2 in equation (3.1.15) may actually be
neglected lacks a physical justification. It is expected
that, for a more accurate calculation of the membrane
stresses from the displacements. the pair of coupled non—
linear partial differential equations as given by equations
(3.3.1a), (3.3.1b) instead of (4.3.11) and (4.3.12) must
be employed.

Numerical results for the ratio of the nonlinear to
the linear period are plotted in Figure 4.3.1 versus the
dimensionaless amplitude (fi-). With initial strain eo
less than 0.487, the curves lie above those of Chobotov and
Binder. It is seen that, as eo becomes small, they ap-
proach the latter. Physically, this is fairly easy to
justify. The results of Chobotov and Binder were obtained
on neglecting the radial inertia term. Since e0 = 0.1,
say. corresponds to an axisymmetric membrane with a small
initial strain. The contribution of the radial inertia term
in equation (3.3.6b) is thus negligible. This fact can be
established as follows. For small initial strain. the ex—
pansion coefficients C(1) for the radial displacement as

2n

given in equation (3.3.20) can be approximated by

2
(1.) = All .. __ l1)
c2n 2 if: 1 — can (4.3.17)

Thus the results for the ratio of the nonlinear to the

linear period obtained by (4.3.8) are found to be reasonably

”4...:

— ‘w-

 

Figure 4.3.1.

65

Ratio of nonlinear to linear period vs.
nondimensional amplitude for clamped
circular membranes with initial strain.

Bua

Nonlinear—Linear Period Ratio

0.4

0

0.2

 

.05
0

66

Chobotov and Binder's
coupled equations result

e0 = 100

 

Uncoupled equation's
result

Figure 4.3.1-

Nondimensional Amplitude eA/Jeo
l L l

L.

 

2 4 6 8 10

67
close to those of Chobotov and Binder. On the other hand,
the curves for eO greater than 0.478 lie below those of
Chobotov and Binder. This is because, as ea becomes large,
say, e0 = 100, the expansion coefficients Cég) for the
radial displacement change sign (see Table 4.3.2) and can

be approximated by

C(l) = -§-—:2-1 (4.3.18)

The function F(e0) as given by (4.3.6) does not seem to
be sensitive to the variations of the large initial strain
(see Table 4.3.3). Thus the differences between the results
obtained by (4.3.8) and those of Chobotov and Binder are

about 2.5 per cent.

2
1.

When e0 = 0.487, it can be shown that J\: = 49
The expansion coefficient Céi> for the radial displace—
ment becomes unbounded because the provision 11: - mzfli # 0
fails. Physically, this means that the frequency of the
radial oscillation induced by transverse motions approaches
to that of the purely radial oscillation of the membrane.
The coefficients of the radial displacement become very
large. This case must be treated by a modified perturba—
tion method and will not be considered here.

The results obtained by neglecting the effects of the
second strain invariant and the radial inertia are found

to be reasonably close to that from the present analysis

only when e0 is near unity.

68

4.4. Circular Plate with Clamped Edge

 

In this section the problem of a circular plate with
clamped edge is considered as an illustrative example of
Section 3.4. The pair of coupled nonlinear partial dif—
ferential equations of motion including the effects of the
so—called second strain invariant as well as the radial
inertia term is given by equations (3.4.5a) and (3.4.5b).

The boundary conditions are

 

n is bounded as C —+ 0+ ~
q(1 T) = 0 )(4.4.1)
q‘(1,T) = O J

y is bounded as Q —+ 0

+ 1

y(1,T) = O J.

The differential equation (2.2.7) together with the boundary
conditions (4.4.1) yield the follOWing transcendental

equation for the eigenvalues:

. J ( .
31(5) + 0:3) 11(6) = 0 (4.4.3)
onfi

where Jn and In are the Besel and the modified Bessel

functions of the first kind. The subscript n refers to

the order of these functions. By the orthogonality condi-
tion (2.2.10) with the weighting function r(§) equal to

C, the normalized transverse Spatial eigenfunctions and

their corresponding eigenvalues are

 

. J (B .3
vkm = c [30(BkC) - 3:751]? I.(ekc)1 x
9i = Bi >(4.4.4)
k = 1,2 3 ..... J

where the normalized constant C is defined as

C =. J? (4.4.5)

2 2 J2(B B)k I2 yz
[31(Bk) + ZJQ-(Bk) — 10(Bk)1(fik)]

 

Numerical values of Bk have been calculated and tabulated
in [14].

The differential equation (2.2.13) together with the
boundary conditions (4.4.2) yield the following tran-

afimdental equation for the eigenvalues:
J1(j) = 0 (4.4.6)
By the orthogonality condition (2.2.16) with the
weighting function s(§) equal to Q, the normalized

radial Spatial eigenfunctions and their corresponding

eigenvalues are

 

;. _ J2 . . .
Pnké) -m J1.(31n C.) x
/\: = ijin )(4.4.7)
n = 1,2,3,.... J

Upon introducing equations (4.4.4) and 4.4.7) into
(3.4.18) and 3.4.19), and performing the integrations on

the computer, one obtains the first order expansion

70
coefficients for the radial displacement. The results of
the first nine coefficients are tabulated as a function of
g and u, in Tables 4.4.1 and 4.4.2.

From equation (3.4.24), the second order frequency—

amplitude relations follow as

2 _ 2 a . 2
(D2 — (1)0 F(}—]' 5 LLJMII (4 04 08)
where
a 1 1 1 0:) ' (1) 1 (1) LL 1
F _i = _ 1 ! 2 1C . + _ .—
(-—1+}i vi + v: )1“ + V{F"]
C ' q q
1‘ "7' :3
+ %(3V12V1 + % V]; )} CV1 d2; (4'4'9)

Table 4.3.3 shows how closely the truncated series repre-

sents the function F(%nu).
Using equation (2.2.3) and recalling that A is

.2
equal to 12(%) , the frequency—amplitude relations up to

terms including 52 are given by

(1.2 = (1): {1+ 123(§.u)(% £111.13) 1 (4.4.10)

~

The ratio of the nonlinear to the linear period then follows

 

as
1
T*
E = ‘ y" (4.4.11)
a 2 .
1 12“?” ’a A)2
+ ‘ —-s
2 \
lvllmax .h
where

(4 .4 .12)

71

 

 

 

 

 

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AHVU mucmflofimmwoo

72

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*

 

 

 

 

 

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Table 4.4.3. Convergence of F(% u) of clamped circular

73

 

 

 

 

plates. .
N
a .

NF”) 3 6 9
.3
h
H20.1

10 0.02908 0.02905 0.02905
100 0.02913 0.02910 0.02910
u;0.2

10 0.03031 0.03028 0.03028
100 0.03035 0.03032 0.03032
u:0.3

10 0.03129 0.03127 0.03127
100 0.03133 0.03130 0.03130
urO.4

10 0.03204 0.03202 0.03202
100 0.03207 0.03205 0.03205

 

Wah [53 and Blotter [7] have both studied nonlinear
vibrations of circular plates using the Berger's hypothesis.
Their results along with those calculated from equation
(4.4.11) are plotted in Figures 4.4.1 through 4.4.4. The
curves again reveal the general feature that the nonlinear
periods decrease with increasing amplitudes. It should be
noted that the results of the present analysis are displayed
in terms of the Poisson's ratio u. whereas those of Wah
and Blotter are independent of this ratio. This is because
the last term (l—u)Dw w in equation (3.1.16) has no con-

rrr

tribution to the equation governing the transverse motion of

t.

 

 

 

 

 

 

 

 

 

74

:10

100

/
:rIm

Wah's uncoupled
equation result for \
all values of n

Blotter‘s uncoupled _J////
equation result for

all values of u

l l l l I

 

1.000

,.. 0.950-
Elli-4
O
'7'!
4.»
(U
m
'0
O
H
H
(D
D4
H

8 0.900—
C
H
'7‘
H
(U
(D
C‘.
H
H
(:3
0
Z

0.850—

0.825

0

Figure 4.4.1.

0.2 0.4 0.6 0.8 '1.0
Amplitude-Thickness Ratio -;%5

Ratio of nonlinear to linear period gs,
ratio of amplitude to thickness for clamped
circular plates with u = 0.1.

1.000
0.950
E45
0
1*!
"6
m
"G
O
-H
H
m
Ch
H 0.900
m
m
c
-H
g
I
p
(U
m
C.
-a
a
c
0
Z
0.850
0.825

Figure 4.4.2.

I

 

75

cm:

= 100

Wah's uncoupled
equation result for
all values of u

Blotter°s uncoupled
equation result for
all values of u

l l l l l
0.2 0.4 0.6 0.8 1.0
Amplitude-Thickness Ratio Efié
Ratio of nonlinear to linear period vs,
ratio of amplitude to thickness for clamped
circular plates with u = 0.2.

76

 

 

 

1.000
a —
3" 10
a .—
h'— 100
0.950-—
*
E48
0
H
4..)
m
0:.
c
o
«a
s
(D
m
:3 0.900 r \
04‘.
c
~r-l
*3
S Wah‘s uncoupled
g equation result for
3 all values of u \
c
0
Z
Blotter's uncoupled
0°850F' equation result for
all values of u
0 .825 l 1 l J l
0 0.2 0.4 0.6 0.8 1.0‘
Amplitude-Thickness Ratio 2&5-

Figure 4.4.3. Ratio of nonlinear to linear period XE:
ratio of amplitude to thickness for clamped
circular plates with u = 0.3

 

77

 

Figure 4.4.4.

1.000*
a ‘=
F 10
\
319
I 0.950 t \
E g = 100
m
96
U
o
.2 \
H
w
m
1.4
m
m
c
H
.3 0.900 r \
14
m
w
c
H
E Wah's uncoupled
o equation result for
z all values of n \
Blotter's uncoupled
_ equation result for
0°850 all values of u
00825 I l l l l
0 0.2 0.4 0.6 0.8 1.0
Amplitude—Thickness Ratio -%%&

Ratio of nonlinear to linear period gs,
ratio of amplitude to thickness for clamped
circular plates with u 2 0.4

78

the clamped circular plate. The neglecting of the second
strain invariant of the middle surface leads their ratios
of the nonlinear to the linear period to be independent of
u- In addition, the term corresponding to the radial
inertia is ignored, their first strain invariant is con-
stant throughout the plate. This yields only the approxi-
mate sum of the membrane stresses. To determine the varia-
tions of the membrane stress in the radial direction, the
pair of coupled nonlinear partial differential equations
as given by (3.4.1a) and (3.4.1b) must then be used.

The ratio 2-= 100 represents a very thin circular

h

plate while %-= 10 corresponds to a plate that is neither
too thin nor too thick. The frequency-amplitude relations
do not seem to be sensitive to the variations of this ratio.
This fact is seen in Table 4.4.3. Also, the results of

Wah and Blotter do not depend on this ratio. 'The curves

for clamped plates with %-= 10 and %-= 100 are undistin-
guishable and are plotted in Figures 4.4.1 through 4.4.4.
For u equal to 0.1, 0.2, or 0.3, they lie above those of
Wah and Blotter. For n equal to 0.4, they lie above

those of Wah but below those of Blotter. The periods ob—

tained by Blotter are reasonably close to the present ones

only when the Poisson's ratio u is near 0.3 to 0.4.

V. SUMMARY AND CONCLUSIONS

In this thesis a method of solution of two coupled
nonlinear partial differential equations is presented. It
is one form of the perturbation technique. The pair of
coupled nonlinear partial differential equations are first
converted into a system of linear partial differential
equations. The linear equations are then solved recur-
sively using the method of eigenfunction expansions. The
method is an extension of the one used in [7] for single
partial differential equations and is motivated by the fact
that many vibration problems for continuous structures are
actually governed by coupled nonlinear partial differential
equations, unless simplifying assumptions are made. -Equa-
tions governing free vibrations of nonlinear continuous
systems such as beams, circular membranes and circular
plates are derived by means of Hamilton's principle. This
results in equations that are both coupled and nonlinear
and are of the type considered here. Using the method
developed, approximate solutions for frequency-amplitude
relations, out-of-plane as well as in-plane displacements
up to second order are then obtained.

It is shown that, in general, uncoupling of these
equations of motion is achieved by neglecting the effects

79

80
of both the second invariant of the middle surface strains
and the in—plane inertia. With the availability of the
present results obtained from the solutions of the coupled
equations, it is possible to assess the accuracy and valid—
ity of the various previously published results, especially
those contained in [7], on beams, circular membranes and
circular plates obtained from solving simplified, uncoupled
equations of motion.

In the case of elastic beams with large slenderness
ratio. the frequency-amplitude relations predicted by [7]
are identical to those of the present analysis, while small
deviations exist in the case of moderately thick beams
(%-= 40). Thus there is a maximum error of only 0.027 per
cent for hinged-hinged beam (see Figure 4.2.1), a maximum
error of only 0.014 per cent for clamped-clamped beam (see
Figure 4.2.2), and a maximum error of only 0.081 per cent
for clamped—hinged beam (see Figure 4.2.3). These observa-
tions tend to confirm the correctness of the assumptions of
[7].

In the case of the elastic clamped membrane with small
initial strain, say e0 = 0.1 or smaller, the results agree
with those of Chobotov and Binder. For large initial strain,
say e0 = 100 or larger, there is a difference of 2.5 per
cent. For e0 between 0.4 and 0.5, the results deviate
substantially from those of Chobotov and Binder. The present

perturbation method does not apply. The results obtained

81
by using Berger's assumption are found to be reasonably close
to that of the present analysis only when e0 is near
unity.

In the case of the thin elastic clamped plate, the
results of [7] do not depend on the Poisson's ratio nor the
ratio of fin while the results of the present more accurate
analysis do. The periods predicted by [7] are reasonably
close to the present ones when the Poisson‘s ratio u is
near 0.3 to 0.4. For n equal to 0.1, there is a maximum
error of 1.07 per cent.

The present work represents an extension of [7], in
which the same perturbation method was applied only to a
single nonlinear partial differential equation. The method
is extended here to solve pairs of coupled nonlinear partial
differential equations. It is rather remarkable that such
an extension is accomplished without requiring too greater
an effort than that required for solving single nonlinear
partial differential equations. The pair of coupled non-
linear partial differential equations enable us to take
into account the second invariant of the middle surface
strains as well as the effects of the in—plane inertia of
the continuous systems and thus lead to more accurate results.

Whereas the nonlinear frequency—amplitude relationships
obtained here do not seem to deviate substantially from
those previously obtained using simplified theories in gen—
eral, it is clear, however, that solutions of the coupled

equations do reveal more information on the motions of the

82
continuous systems, such as that on the radial displace—
ments and the distributions of the in-plane stresses in

the radial direction.

REFERENCES

LIST OF REFERENCES

Von Karman, T., "Festigkeitsprobleme im Maschinenbau,"
Encyklopaedie der Mathematischen Wissenschaften.
Teuber-Verlag, Leipzig, Germany, Vol. 4, 1910,
pp. 348—352.

Berger, H. M., "A New Approach to the Analysis of Large
Deflections of Plates," J; Appl. Mech., Vol. 22,
1955, pp. 465-472.

Nash, W. A. and Modeer, J. R., "Certain Approximate
Analyses of the Nonlinear Behavior of Plates and
Shallow Shells," Proc. of the Symposium on the
Theory of Thin Elastic Shells,Delft, The Netherlands,
August, 1959, Interscience Publishers, Inc., New
York. N.Y., 1960, pp. 331-354.

Sinha, S. N., "Large Deflections of Plates on Elastic

Foundations," grog. ASQE g. Engng, Megh, Dig. EM1,
February 1963, pp. 1-24

Wah, T., "Vibration of Circular Plates at Large
Aplitudes," Proc. ASCE J. Enqng. Mech. Div. EMS,

 

October 1963, pp. 1-15.

Gajendar, N., "Large Amplitude Vibrations of Plates

on Elastic Foundations," Int. J. Nonlinear Meghanigg,
Vol. 2, 1967, pp. 163-172.

Blotter, P. T., "Free Periodic Vibrations of Continuous
Systems Governing by Nonlinear Partial Differential

Equations," Ph.D. Dissertation, Michigan State
University, 1968.

Young, D. and Felgar, R. P., "Tables of Characteristic
Functions Representing Normal Modes of Vibration of

a Beam," Publication 4913, July 1949, Univ. of
Texas, Austin, Texas.

Evensen, D. A., "Nonlinear Vibrations of Beams with
Various Boundary Conditions," AIAA Journal, Vol.

6, No. 2, February 1968, pp. 370—372.

83

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

84

Woinowsky-Krieger, S., "The Effect of an Axial Force
on the Vibration of Hinged Bars," J. Appl. Mech.,

 

Vol. 17, No. 1, March 1950, pp. 35—36.

Abramowitz, M. and Stegun, I. A., Hanghggk_gj_M§;h§:
matigal Eungtigns, Dover Publications, Inc., New
York, 1965.

Wylie, C. R., Advanced Engineering Mathematics, 3rd
Edition, McGraw—Hill, New York, 1966.

Chobotov, V. A. and Binder, R. c., "Nonlinear Response
of a Circular Membrane to Sinusoidal Acoustic
Excitation," Wm... Vol. 36, No. 1..

January 1964, pp. 59-73.

Volterra, E. and Zachmanoglou, E. C., Dynamics 9f
Vibrations. Merrill Books Inc., Columbus, Ohio,
1965.

Meirovitch, L., Analytical Methods in yibrations,
Macmillan, New York,1967.

Timoshenko. S. and Woinowsky-Krieger, S., Th§91y_gf
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