v13

VARIATIONAL-PROBLEM AND GREEN’S. *

FUNCTIONS FOR THE IMPROVED THEORY V_

OF ELASTIC PLATES

Thesis for the Degree of P11 .. D.
MICHIGAN STATE UNIVERSITY
PETER E. OORNWELL
1 9 7 2

     
    

  

LIB RA R Y
MiChigan Sta te
University

This is to certify that the

thesis entitled

VARIATIONAL PROBLEM AND GREEN '8
FUNCTIONS FOR THE IMPROVED THEORY
OF ELASTIC PLATES

I

presented by

Peter E . Cornwell

has been accepted towards fulfillment
of the requirements for

Ph . D. degree in Mathematics

 

 

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Major professes

 

Date 11-1-72

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ABSTRACT

VARIATIONAL PROBLEM AND GREEN'S FUNCTIONS
FOR THE IMPROVED THEORY
OF ELASTIC PLATES

BY

Peter E. Cornwell

This thesis is divided into three chapters. The first chapter
is a historical review of works that are concerned with the dynamic
response of thin elastic plates within the improved Timoshenko plate
theory and related theories.

Within the classical plate theory, the existence of a discrete
spectrum of natural frequencies is well known for plates with clamped,
simply supported, or free edges. Chapter two of this thesis investigates
the existence of a discrete spectrum within the improved Timoshenko
theory for arbitrarily shaped plates with any of the above mentioned
edge conditions. The investigation is done through the formulation of
a variational problem using the elliptic system of equations that is
characteristic of the improved theory. The discrete Spectrum for
plates with torsional springs attached is established, and the dif-
ficulties encountered upon omitting the torsional springs are mentioned,

indicating a possible difficiency of the improved theory.

Peter E. Cornwell

Little work has been done with forced vibrations within the
improved theory. In Chapter three the forced vibration of a circular
plate on a Winkler-type foundation is considered. The impressed force,
applied normal to the surface of the plate, is assumed to be continu-
ously distributed over a portion of the plate and each of the above
mentioned edge conditions is considered. The problem is solved by
constructing Green's functions using the well known fundamental sin—
gularity of the improved theory. Certain curves in the frequency-
stiffness plane are considered separately, and corresponding Green's
functions and frequency equations are established. Finally, the
influence of torsional springs on the construction of the Green's

functions is considered throughout the frequency—stiffness plane.

VARIATIONAL PROBLEM AND GREEN'S FUNCTIONS
FOR THE IMPROVED THEORY
OF ELASTIC PLATES

BY

.\ 1"

Peter EJ"Cornwell

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1972

61- 7:; 3x :3

ACKNOWLEDGEMENT

The author wishes to thank Dr. David Yen of the Department of
Mathematics and the Department of Metallurgy, Mechanics and Materials
Science at Michigan State University for suggesting the tapic of this
thesis and for his guidance and assistance in all phases of this
research.

Sincere appreciation is also extended to Mrs. Frances Abel of
the Mathematics and Engineering Mechanics Department at General Motors

Institute for her many hours of typing in the preparation of this final

thesis.

LIST

LISI

II.

II

([1

ACKNOWLEDGEMENT .

LIST OF TABLES

LIST OF FIGURES

I. HISTORICAL REVIEW

TABLE

II. THE VARIATIONAL PROBLEM

2.1.
2.2.
2.3
2.4.
2.5.
2.6.
III.
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
BIBLIOGRAPHY

Introduction .
The Variational Problem .

Existence of G

Complete Continuity . .

Existence of G
Remarks and Conclusions .

Introduction .

Formulation

OF CONTENTS

FOR THIN ELASTIC PLATES

(1)

(2) (35

and C

Construction of U . . . .

Construction of V

(1) .

Boundary Conditions . . .
Remarks and Conclusions. .

iii

GREEN'S FUNCTIONS FOR CIRCULAR PLATES

Page
ii

iv

12
12
19

22
24

30
32

37

37
37
4O

44
55
84

9O

LIST OF TABLES

Table Page
1 Summary of of and p: . . . . . . . . . . . . . . . . . . . ll
2 Resonant Radii at T for the Plate with Free Edges . . . . 70

iv

LIST OF FIGURES

Figure

Page

1 Dependence of of and p2 on k and w2 .

. . . . . . . . 10

2 Position of First Zero of Bil) to the Left of the Point T.. 75

 

’b
In+1(p r) 3’2 r
3 Relationship Between-——————§—— and (k—lZ-v ) . . 76
In(ozr)

I. HISTORICAL REVIEW

The classical two-dimensional theory of flexural motions of

elastic plates leading to Lagrange's equation

D V”w + p1h§3!-- q(x,y,t)

31:2

is well known and presented in most elementary vibration texts [22],
[38], [42]. This classical theory neglects shear deformation. From
1944 through 1947, Eric Reissner presented a series of papers intro—
ducing the effect of transverse shear deformation on the bending of
elastic plates [33], [34], [35]. His theory leads to a system of
equations that requires the specification of three boundary conditions
along the edge of the plate, in contrast to the two edge conditions of
Kirchoff in the classical theory.

R. D. Mindlin presented a paper in 1951 developing a two-
dimensional theory deduced from the three-dimensional equations of
elasticity [25]. This theory includes the effects of rotatory inertia
and shear in the same manner as Timoshenko's one-dimensional theory of
bars [41], [42]. At various stages of the development of his theory,
Mindlin directs attention to the very close similarities between his
theory and E. Reissner's theory mentioned above. Of special interest
is the conclusion that, as in Reissner's theory, three boundary conditions
are to be satisfied rather than the two of classical plate theory. We

refer to Mindlin's theory as the improved Timoshenko plate theory.

In his above mentioned paper, Mindlin tests his theory for the
case of straight-crested flexural waves by using the exact solution of
the three-dimensional equations of elasticity. It can be seen from the
three-dimensional theory that as the wave length diminishes, the wave
velocity has as its upper limit the velocity of Rayleigh surface waves.
The classical theory is not in agreement with this result. In fact,
Mindlin points out that when the wave length becomes less than five or
ten times the plate thickness the classical plate theory departs
markedly from the three-dimensional theory. By introducing rotatory
inertia, Mindlin shows that the limiting wave velocity is brought into
closer agreement with the results of the three-dimensional theory;
however, through the further introduction of shear deformation, he is
able to modify the shear modulus, C, so that the limiting wave velocity
for very short waves is made identical with the velocity of Rayleigh
surface waves. His modification of C also guarantees that the velocities
intermediate between very long and very short waves are in close agree-
ment with the three-dimensional theory. Of special interest is his
observation that it is the transverse shear deformation that accounts
almost entirely for the discrepancy between the classical plate theory
and the three-dimensional theory.

In 1954 and 1955, Mindlin and H. Deresiewicz published a
collection of papers investigating the frequency equations within the
improved theory for rectangular and circular plates with various edge
conditions prescribed [6], [26], [27]. Some of the results of these
papers will be used in chapter three of this thesis.

T. C. Huang published a paper in 1961 applying the variational

methods of Ritz and Galerkin to the vibration of plates including

3

rotatory inertia and shear deformation [17]. It should be noted,
however, that he departed from Mindlin's theory by assuming horizontal
plate rotations to be zero. The variational problem in chapter two of
this thesis makes no such assumption.

There have been several studies on Green's functions in the
dynamic theory of elastic plates [20], [45]. Kalnins approached the
problem within the improved theory by first deriving the exact form of
the fundamental singularity of the Green's function, making use of the
reciprocity theorem in the theory of elastic plates, and then con-
structing the Green's function by separation of variables and super-
positions.

The work of D. Yen, [45], is regarded as a generalization of
the work of Kalnins. He obtains the Green's function for plates of
infinite extent with the inclusion of a Winkler-type foundation,
proceeding in a manner more direct than that of Kalnins by introducing
the Fourier transform. The Green's function so constructed reveals the
nature of its fundamental singularity. D. Yen also determines certain
curves in the frequency-stiffness plane along which large amplitudes
of the plate vibration occur. Since the results of D. Yen are pertinent
to the discussion of chapter three of this thesis, we present his
results within the next few pages.

Consider a thin elastic plate, infinite in extent, and supported
by a Winkler-type elastic foundation. Assume a time harmonic point load
at the origin.

According to the improved plate theory [25], the transverse
deflection of the plate, w - w(x,y,t), satisfies the linear fourth-order

partial differential equation

2 91 2 olh3 o h
[V - E'— att] (v - 12D act w + D art '7
h2
1 DV2 pl
3 B'(1 ETE'+ 126' art) f (1'1)

where V2 E a + 3 , p is the density of the plate, h and D are,

xx yy 1
respectively, the plate thickness and plate modulus, f = f(x,y,t) is
the transverse force per unit area on the plate, and G' is the modified

shear modulus discussed by Mindlin [25]. Let

w

f(x,y,t) = 6(X.y) e.1 t - k W(X.y.t) (1.2)

where 6(x,y) is the Dirac delta function. The 6(x,y) e-iwt term corres-

ponds to a concentrated force at the origin that is simusoidal in time;

w being the frequency of the impressed force. The second term on

the right-hand side of equation (1.2) represents the pressure due to

the elastic foundation, with k corresponding to the foundation stiffness.
Seeking steady state response, the displacement is assumed of

the form

wt

w(x,y,t) = g(x,y) e“1 (1.3)

After substituting (1.2) and (1.3) into (1.1) and introducing

the dimensionless quantities

 

— x — —

x=g.y=§.g=§

- 6 - kh - t

5 = 67', k 3 67', t ='E; (1.4)
12D G' h

VI = 3 ’ v: - E_" to =‘3_
0h 1 2

the equation
(V2 + BZ)(V2 + 3&2); — 12332§ + k[123 — (v? + s5?)] g
= [123 — (v2 + 352)] §(§,§) (1.5)

is obtained. In order to simplify notation we drop all overbars in
(1.5) and realize that all results are dimensionless and must be sub-
jected to the transformations (1.4) for their final form.

A solution of (1.5) is called the Green's function for the
infinite plate provided it satisfies the radiation condition for out-
going waves. Due to the Dirac delta function in (1.5), this Green's
function must be regarded as a distribution. D. Yen established the
Green's function by the method of Fourier transforms. In applying this
technique, he noticed that various regions in the (k,w2) plane must be
considered and that the expression for the Green's function depends on
these regions. Figure 1 is a reproduction of the various regions con-

sidered in the (k,w2) plane. In this figure,

p(k,w2) (1 + s) m2 - k (1.6)

q(k,w2) 3(12 - m2)(k - wz) (1.7)

Case I:

_ ___!L__.
- (1 + s)

at the origin and at the point T, the Green's function was determined as

a function of r a /x2 + y2 to be

In regions I through VII and on the line wz , except

8(r) = 81(r) + g2(r) (1.8)

 

where
81(r) = %'--l--'[offigl) (plr) - pgflgl) (02r)] (1.9)
of - pg
_ 2
82(r) = %-S:12 2w ) [H§1) (olr) - H51) (02r)] (1.10)
01 - 02

/p2 - 4q) (1.11)

l l
of = 5-(p + (P2 - 4Q). a: = 3-(9

02 + 02 = p . ofpg = q (1.12)
(1)
In (1.9) and (1.10) above, the H0 (pir), i = l, 2, are Hankel functions
of order zero, and are of the first kind.
. 232.
Case II. p1 92 0

At the origin and at the point T it can be seen from (1.6), (1.7),

(1.11), and (1.12) that p? 3 0% = 0. At these two points

_ .1.
g1(r) - - 2" log r (1-13>
o if (k.w2) = T
g (r) = (1.14)
2 -%;-s r2 log r if (kswz) = (0,0)

Case III: 02 8 0, p1 # 0

It can be seen that 02 = 0 and p1 ¥ 0 along the lines w2 = k

and w2 = 12. In this case

' H(1)

81(r) = fi- 0 (olr) (1.15)

_ 2
521; 8(12p2 (DJ [Hgl) (plr) + 32:1? 108 1'] if (”2 = k
32(r) ' 1 (1.16)

o if w2 = 12
Case IV: of = p: i 0

Finally, we note that of = p: # 0 along the parabola p2 = 4q.

In this final case

 

D I'
310:) = % tag“ (91:) - + Hf“ (.3er (1.17)
-i_ 3(12 - wz) (1)
g2(r) a 8[ pl r H1 (plr)] (1.18)

where Hfl) (plr) is the Hankel function of order one and is of the first
kind.

Table l and Figure 1 show the dependence of pi and p3 on w and k
and are instrumental in obtaining the results of the above four cases.

They will be referred to at various points of the paper that follows.

Remarks Concerning the Above Results

Besides establishing the Green's function for the various regions

mentioned, D. Yen made the following observations:

1. Noting that Hél) (pit) a - fii-log (pir) for small r, it is
clear, after examining g(r) in each of the above four cases, that
g(r) = C1 log r, where C1 is a non—zero constant. This logarithmic

singularity comes from the g1(r) term in each case.

2. Except where g2(r) = 0, an investigation of the Hankel
functions leads to a singularity in g2(r) equal to that of the classical

plate theory; i.e., g2(r) = Czr2 log r, where C2 is a non-zero constant.

3. As seen from (1.1) and the analysis leading to (1.5), the
V26(x,y) term in (1.5) is present because of the effect of shear defor—
mation. It is also seen that this term is responsible for gl(r); hence,
it is the V26(x,y) term, or equivalently the effect of shear deformation,
that is responsible for the infinite deflection under the point load

in the improved plate theory.

4. Although the point load singularity of the improved plate
theory leads to infinite deflection under the load, and that of the
classical plate theory does not, the results above are, nevertheless,
correct. In a physical problem the Dirac-delta function in (1.2) is
replaced by a function, say ¢, with compact support in a region around
the load. This function represents a continuously distributed load
rather than a point load. To obtain the solution to such a problem
one simply integrates ¢ over its support with the Green's function

serving as the kernel of integration.

5. For the infinite plate, we say that resonance occurs when
g(r) tends to infinity as r tends to infinity. Investigation of Cases I
and IV shows that resonance occurs along the parabola p2 - 4q, and along

the line w2 a k.

6. The resonance conditions for the infinite plate are related
to those of a finite plate in the following way. Consider a large, but

finite plate, supported along its edge. The total deflection consists

of two parts. The first is a singular part given by the Green's

function for the infinite plate, and the second is a regular part
satisfying (1.5) with the right hand side set equal to zero. The

regular part is equal to the Green's function in magnitude along the

edge of the plate, but is of opposite sign, so that the total deflection
along the edge of the plate is zero. At resonance the regular part should
transfer the large deflections from the edge of the plate to under the

load.

7. It is possible to obtain the regular part of the Green's
function mentioned in remark (6) by separation of variables provided
the shape of the plate is geometrically simple. For more arbitrarily
shaped plates the problem reduces to solving integral equations with
the Green's function for the infinite plate serving as a principal

fundamental solution.

8. Finally, we mention that the Winkler-type foundation does
not influence the singularity of the Green's function. The foundation
does, however, contribute to complicated plate resonances. To be
specific, when the foundation stiffness and frequency of oscillation
are such that (k,w2) lies on one of the resonance curves, p2 = 4q or

2

w - k, the Green's function for the infinite plate becomes unbounded

away from the point singularity.

10

N3 mam x no ma new we no mocmvamaom "H ouswwm

3 CM ON OH 0
do v Na

UQHN.

HH>

OH
NH I N3 >H

 

 

HHH

03

HH

Aw + Hv\x u N3
ON

 

11

Table 1

Summary of pi and p3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 2
Region pl p2
I Real, Positive Real, Positive
not Equal to 0%
11 Real, Positive Real, Negative
III Real, Positive Real, Negative
IV Real, Positive Real, Positive
not Equal to pg
V Complex, Complex Conjugate of p?
Positive Real Part
VI Real, Negative, Real, Negative
Greater than 9%
VII Complex, Complex Conjugate of p?
Negative Real Part
(0,0) and T Zero Zero
w2 = k Real, Positive Zero
w2 = 12, Left of T Real, Positive Zero
w2 = 12, Right of T Zero Real, Negative
p2 - 4q Real, Equal to g- Real, Equal to.§
w2 = k/(l + s), Imaginary Complex Conjugate
w2 < 12
w2 - k/(l + 5), Real, Positive Real, Negative

2>12

 

 

 

 

II. THE VARIATIONAL PROBLEM FOR THIN ELASTIC PLATES

 

2.1 Introduction

Consider the free vibrations of a thin elastic plate on a
Winkler-type elastic foundation. Assume that torsional springs are
attached to the surface of the plate. Equivalently, we could assume
that the torsional springs are built into the foundation in some way
so that potential energy due to rotation is present in addition to the
potential energy normally contributed by a Winkler-type elastic
foundation. When referred to an x, y, 2 system of rectangular coor-
dinates, the faces of the plate are assumed to lie in the planes
2 8 igu Letting ux, uy, and uz represent plate displacements in the
x, y, and 2 directions respectively, it is assumed that ux and uy are

proportional to 2, while uz is independent of 2. Based on this

assumption we write

ux = z¢x(x,y,t), uy = z¢y(x,y,t), u2 = w(x,y,t) (2.141)

According to the improved (Timoshenko) plate theory [25], the motion of
the plate is governed by the three plate-displacement equations of

motion:

12

13

 

 

3 2
P- (1‘ )Vzd) +(l+p)-a$ - ' (¢ +fl)+Cfl-plh 3¢X
2 u x ax G h x ax ay 12 étz
o h3 32¢
D 2 32 3W aw 1
_ - + + —- "' ' +_ - ——--
2 [(1 u)V ¢y (1 u) By] G h(¢y 3y) Cax 12 3:22
2
G'h(v2w+¢)-kw=plh3—K
8t2
(2.1—2)

where V2 5 a + 8 , p1 is the density of the plate, D is the plate

xx yy
modulus, G' is the modified shear modulus, u is Poisson's ratio,
3¢x Bo a¢x 8¢
¢ - ———-i--Jl , W = ——- - __Z_’ k is the Winkler-type foundation con-
Bx 3y 3y 3x

2
stant, and C is-%-times the torsional spring constant. Seeking steady

state response of the plate, we assume w(x,y,t), ¢x(x,y,t), and

¢y(x,y,t) of the form

W(X.y.t) = w(x,y) e"iwt
¢x(x.y.t) = wx(x.y) e'i‘”t . (2.1-3)
¢y(x,y.t) = wy(X.y) e-iwt

where m is the frequency of vibration. Substituting (2.1-3) into

(2.1—2); letting C = g-: introducing the dimensionless quantities (1.4);

and dropping the overbars for brevity; we obtain the time-reduced

system:

ae'
[(1 - u)V2wx + (l + u) —;'

8w 3? _
a 0

+ _
3X Cay

kflh‘

+ 8(w2 - 12).)x - 123

 

hflhd

 

(1-u)V2W +(1+u)‘.3__.+ 2- - EH- .322.
y 3y 8(w 12)wy 128 3} Cax 0

ll
0

VZW + ((1)2 - k)w + d)

(2.1-4)

l4

awx aw aux aw
where<l>-——+—Xandw-_-_l
3x 8y 3y 3x

Write (2.1-4) in the form

3
f z (D) u = o 1 = 1,2,3 (2.1-5)

where D E (g- -i—

X 9 3y), u]. = WK, UZ = lily, and U3 3 W. Den0te by

£11(D) the sum of the terms in £13 of order exactly two. It is seen that
the system (2.1—4) is strongly elliptic in the sense that for a real
vector E = (51,52) and a complex vector n = (nl,n2,n3),
3 ' _
X 1: (SM n 7‘ 0 when E 1‘ 0. na‘ 0 (2.1-6)
1.1=1 13 i j

The above definition of strong ellipticity agrees with that given by L.
Nirenberg in his discussion of the differentiability of weak solutions

of elliptic systems [31].

 

 

 

 

Let
r
2 | 2 l
-l (1-u)v2+(1+u)§-—--1zs (1 + u) a | -1Zs§—- 1
2 3 2 I 2 axay 3x
-_._.___...x....._n--..._...-__.__-._- -......
2 ' '
1(1) = (1 + u) a | l-(1-u)v2+(1+u)§—---1zs' -1293—
2 3y3x I 2 3y? ' 3y
—————————— rh———au-—‘---r—---
%;’ I_%_ ] v2 - k
it. I y I J
s 0 0

14

aw aw aw aw
where<l>=—)£-l-—Xand‘l‘”—33-—z
Bx 3y 3y 8x

Write (2.1-4) in the form

3
K D U = O i = 1,2,3 2.1-5
3:1 13 < ) j ( >
h D = (§—- 3—9 u — u — d u = D c b
w ere - 3x , 8y , l - wx’ 2 — Dy, an 3 w. eno e y

(D) the sum of the terms in £1 of order exactly two. It is seen that

£1
13 J
the system (2.1-4) is strongly elliptic in the sense that for a real

vector 6 = (51,52) and a complex vector n = (n1,n2,n3),

X z' (a)nifij # 0 when a # 0.11% o (2.1-6)

The above definition of strong ellipticity agrees with that given by L.
Nirenberg in his discussion of the differentiability of weak solutions

of elliptic systems [31].

 

 

 

 

Let
F I
1_ _ 2 32 _ I (1 +;p) 32 a I
2 (l u)V +(l+u)—-—- 123. 2 axay ] 1283;
3?.. _._l- -..._...-._._...--._.. _..._.-
2 ' '
L(l) = (1 + “) a |-— (1-u)v2+(1+u)§——--1zs' -1299—
2 8y3x ' ayz ' 8y
—————————— r—b———-——-—-r——--
in. I ' ll
5 0 0

 

 

 

 

 

 

"’ 32 32 '1
' a 3x 0
3Y2 y
2 2
Lm " " 3:3 3— 0
Y 3x2
0 0 0
h- "" Pd) ‘
x
and let GT denote the transpose of G = W
y
kw -
The system (2.1—4) now takes the form
L(l) G + w2L(2) G + CL(3) G = 0 (2.1-7)

from which we obtain

I I GT L(l) G dA + C I I GT L(3) G dA
A A

2 = - (2.1-8)
[JCT L(2) G dA
A

 

(A)

where A denotes the area occupied by the surface of the plate. The
numerator of (2.l~8) represents the maximum potential energy of the
system while the denominator represents the maximum kinetic energy.

They are given by

-[ GT L(l) c dA - c I J GT L(3) c dA = P + p + P + P
A A s B F T

and (2.1-9)

T (2)
G L c dA = + K
[A] KR T

respectively, where

2
.ldA (2.1-10)

 

 

16

is the maximum potential energy due to shear;

aw 3D 2 aw aw 2
P =§H[<1+u)(—x+—Y—) +(1-u)(~—x-—X]
A

B ax ay ax 8y
3% 311"-
+ (1 - u) (23—y— + 8x) dA (2.1-11)

is the maximum potential energy due to bending;

PF = 123k I

I w2 dA (2.1—12)
A

is the maximum potential energy contributed by the Winkler-type

foundation;
aux 3D 2
Pr = c [A] (5;- - ‘13:. ) dA (2.1-13)

is the maximum potential energy introduced by the torsional springs;

KR __. s [I [1:2 1 $21“ (2.1-14)
x Y
A
is the maximum kinetic energy due to rotation; and

KT = 123 I J w2 dA (2.1-15)

A
is the maximum kinetic energy due to translation.

We are interested in the system (2.1-4) for plates with clamped,
simply supported, or free edges. The boundary conditions for each of

these three edge conditions are

X Y
8;: 22 = 21 _ 3_X = =
(2) Mx 8n + (Myx + C?) an My 3n + (M.yx Cw) an w 0
3X 21 21 3x 3x 2):
_ + a -— _ = —— =
(3) Mx an (Myx + CW) an My 3n + (Myx CW) an Qxan + ann 0

17

respectively, where

awx aw
Mx = $— + 1] 3y (2.1-16)
and
8W 3W '
a ._XL __5. _
My 3y + u 8x (2.1 17)

represent the bending moments,

3W 3W
l .-
2 U) [Dy—X 4" 23—222. ) (2.1-l8)

 

Myx = (

represents a twisting moment, and the quantities

_ 2!, -
Qx _ 3x + DX (2.1 19)
and
Q = 33-+ D (2 1-20)
V 8y y '

represent transverse shearing forces.

In the above mentioned vibrating system, subject to any one of
the boundary conditions (1), (2), or (3), the plate is vibrating under
the action of forces inherent in the system, and in the absence of
external impressed forces. Such vibrations are referred to as free or
natural vibrations. A freely vibrating plate will vibrate at one of
its natural frequencies which is a property of the dynamical system and .
is given by (2.1-8), where G represents a nontrivial solution of (2.1—7)
satisfying one of the three sets of natural boundary conditions given
above. If the plate is subjected to a time harmonic force, it will

vibrate, in its steady state, at a frequency equal to the frequency of

18

the impressed force. This frequency is completely independent of the
natural frequencies of the plate. As the frequency of the impressed
force approaches a natural frequency, the amplitudes of the vibrations
become dangerously large and a condition of resonance or unstable
vibrations will occur. It is therefore necessary to stay away from
natural frequencies when solving forced vibration problems. Hence, a
discussion of natural frequencies is in order. To be more specific, it
is the purpose of this chapter to verify that under relatively mild
conditions placed upon the region occupied by the surface of the plate,
the natural frequencies for the clamped, simply supported, and free
plates form a discrete Spectrum in the sense that they form an infinite
collection with no finite accumulation point. This will be done for
plates with torsional springs attached. The difficulties encountered
with the absence of torsional springs will be considered under remarks.
It is noted that both Mikhlin [23] and Weinberger [44] approach
the problem within the realm of the classical plate theory for the
clamped and simply supported plates. They use variational procedures
which are identical except in verifying the existence of minimizing
functions. Mikhlin uses integral representations with non-essential
singularities to represent the functions in the minimizing sequence.
Through well known results concerning compactness of such sequences, the
minimizing function with the desired properties is guaranteed.
Weinberger establishes the existence of a minimizing function by com-
pleting the space in which the functions of the minimizing sequence
belong. The desired properties of the limiting function are then

considered.

19

The question of a discrete spectrum will be discussed here
within the realm of the improved plate theory and will be answered
through the formulation of a variational problem, with the existence of
a minimizing function being established by a procedure that is
essentially a combination of the procedures used by Mikhlin and

Weinberger.

2.2 The Variational Problem

 

In the remainder of this chapter we let A denote the region
occupied by the surface of the plate, and assume this region can be
decomposed into a set of subregions, each of which can be mapped into
a square by a one-to-one transformation with bounded derivatives and a

bounded, non-vanishing Jacobian.

Definition: Let FT 8 (Yx’Yy’Y) and HT 8 (nx’ny’n) be continuous
vector functions over the region A. We define the two symmetric.

positive definite bilinear forms

_ 3Y 8n 8 .gfl
P(F’H) - 128 [A] [5;I+ Yx][3;-+ nx] +'[3%.+ YY][3Y + ”Y + kyn dA

By By an an
(1 + )
+ JA'[ 11 (Bxx + y) (axx + J)

 

 

 

 

2 3y By
By 8y an an
0-1:) X_ y X_ Y
+ 2 (8x 3y )(ax 8y )
By an 3n
(1 - u) Yx y x y
+ 2 (8y + 3x )(By 3x )

 

By By an 3n
+ c (. " - 3,311.." - 4)] «m <2.2-1>

20

and
K(F H) = s I I [Y n + y n + lZyn] dA (2.2-2)
’ A x x y y

Assuming any one of the three sets of natural boundary conditions (1),

(1) (2)

(2), or (3), the operators L and L are self—adjoint implying that

the eigen functions of (2.1-7) are real-valued.

Definition: A nonzero, real-valued vector function satisfying one
of the three sets of natural boundary conditions specified in section 2.1
is called admissible provided its components belong to C2 in A and C1

in A, the closure of A.

Letting Q be the set of admissible functions satisfying the ith set of

1

natural boundary conditions listed in section 2.1, where i = 1, 2, or 3,

we obtain

Theorem 1: Let mg i 8 32b RE: 2) . Suppose there exists an
’ G60 ’

i

(1)6(1)
admissible function, C(i), in Q , with w2 t P<G ) ; then
i 0,1K(G<1>:(1))

 

mo 1 is the lowest eigen (natural) frequency associated with the eigen
9

value problem posed by equation (2.1-7), together with the ith set of

C(1) is an associated eigen

natural boundary conditions. In addition,
solution.

The proof of the above theorem is established by considering the

function

P(G(i) + tH(1), C(1) + tH(i))
K(G(i) (i ) G(1 ) + tH(1))

 

f(t) = (2.2-3)

+ tH

21

where H<1is an arbitrary element of Q and t is a real variable. Clearly,

i
f(t) is a continuously differentiable function of t having a relative

minimum at t - 0, f'(0) - 0. Rewriting f(t) as

P(G(i),c(i)) + 2t P(G(i), 3(1)) + t2 P(H(i), H(1))

 

f(t) = . . (2.2—4)
K(G(i), 6(1)) + 2t K(G(i), H(i)) + t2 K(H(1), H(i))
differentiating, and setting f'(0) = 0, we obtain the equation

P(G(i), H(i)) - w: 1 K(G(i), H(i)) = o (2.2-5)

Applying (2.2-l) and (2.2-2), equation (2.2-5) is seen to be

equivalent to the system (2.1-7), implying that w
(1)

0,1 is an eigen

frequency and G is an associated eigen solution corresponding to the
ith set of natural boundary conditions. The above theorem constitutes
the variational problem for the clamped, simply supported, and free plates,
and will be complete provided we can verify the existence of
c‘i), 1 = 1, 2, 3.

Note that “1 forms a normed linear vector space under either of
the two norms (G,G) or MRZG:G) . For simplicity in what follows we

set
Humps/Wan
and
IIGIIKEW
and call II [IP and [I IIK the potential and kinetic energy norms

respectively, for the space of admissible functions, 91.

22

2.3 Existence of 0(1)

Definition: Let P and Q be points in A and let r be the distance

between these points. Let 0 < o < 2 and assume B(P,Q) is a bounded

B P
function. The operator, Kv - I I -£-:92'V(Q) dA , is called an integral
A r
operator with a non-essential singularity, and its kerne1,B(: ) , a
r

kernel with a non-essential singularity. Note that dA means

Q

integration over A with respect to the variable Q.

Theorem 2: An integral operator with a non—essential singularity
transforms any set of functions with bounded norms into a set of

functions which is compact in the sense of convergence in the mean.

Since the above theorem is a well known theorem presented by Mikhlin

[23]. [24], the proof will be omitted. Using Theorem 2 we are able

to verify the existence of 6(1) as follows:
Let wz - glb P G G) and let {G } be a minimizing sequence
0,1 G89 K(G,G) n
l
. 2 .1.
in $21 with K(Gn, on) 1 and Men, on) _<_ “0,1 + n . Letting P(x.y) and

Q(€,n) be any two points in A, by virtue of the fact that

 

 

 

 

G: - (win), mg“), w(n)) - 0 on BA, we are able to write
DD“) 3 1n —1- 311:“) a 1n l
w(n) (x ) _ l;_ x r‘+ x r dA
x ’3’ 21: A 85 BE: an an Q
(n) 1 3);“) 3 ln-% awgn) 3 ln-%
WY (Kay 8 .2—17 [A 38; 8E + an an dAQ (2.3-1)

 

 

1 1
(n) 3 1n - (n) a 1n -
w(n) (KW . 1 I][3w r+ 8w r] dAQ
A

23
1

fi 8-—
where r = le - £)2 + (y - n)2 . Noting that 3E£-= (£;%_§ .1; and
r

taking B(P,Q) = fi—Z-é , it is clear that the C have integral represent-
r n

 

ations with non-essential singularities. Investigating (2.2—1) and

noting that 0 §_P(Gn, Gn) :_w2 + l, we guarantee the existence of a
9

constant Cl with the property that

 

 

and ' (2.3-2)
3¢§n) 2 3w(n) 2 3w(n) 2 8w(n) 2

LJ [Ia—I +(-.-,='-—J +1.; ) MEL) [11.1

 

Now apply Theorem 2 to (2.3-1) and obtain a subsequence, {Gn }, of {Cu}
1

that converges with respect to the Kinetic energy norm to an admissible
function, G; that is, IIGn - GIIK '+ 0 as n +*w. It is clear that if H
1
is any element of 01 and t is any real number,
2
IIGn1+ tHIIP

u)2 i (2.3-3)
0’1 IIG + tall2
:11 K

which in turn implies that

 

 

P c H - 2 K c n
(n s ) 0301 (n1, ) 0’1

2
:IWHPUWme-fi (L%0
1 9

Assuming ||H||P :_C for some constant C, (2.3-4) implies that

P(c;n , H) - wg’l K(cn , H) -+ o (2.3-5)
1 l n+oo

For H = Gn — Gm we can replace C by 2(wg 1 + 1) and obtain
1 l ’

24

_ _2
P(G ,G Gm) w01K(Gn,Gn

.6 )—->0 (2.3-6)
n1 n1 1 ’ 1 1 m

l n+0!)

Interchanging n and m in (2.3—6) and adding, we get

2
Han -GmI|2-w2 ||c;n -c ||K —»o (2.3—7)

1 1 n,m + w

Hence, the sequence,{Gn },is also Cauchy with respect to the potential
1

energy norm II "P' Completing the space 91 with respect to

II 11,,K = II II, + II IIK

1
we obtain an element G( ) in the completion with

(2.3-8)
llGnl - C(l) ||P+K

l 1
Since (2.3—8) implies that IIGn1 — G( )IIP + O and IIGn - G( )"K + 0

as n + m, we obtain, after using the triangle inequality and the Schwarz

inequality,
2
l
llcnlllp Ilc‘ ’II;
w31= lim -———————2— = “'73—; (2.3-9)
’ n + m C
II n1H1. IIG IIK
1 l
with HG( )1 [K = 1. Clearly, Haul — c‘ m1. :3 .. and “an cllK—r: 3 ..
(1)

together imply that G = G and is therefore admissible.

2.4 Complete Continuity

(2) (3)

In preparation for verifying the existence of’G and G , we

introduce the idea of complete continuity of quadratic functionals.

25

Definition: Let B(F,F) and C(F,F) be two quadratic functionals.
C(F,F) is called completely continuous with respect to B(F,F) if from
every sequence, {In}, that is uniformly bounded with respect to B, a
subsequence, {F5 },can be extracted that is Cauchy with respect to

1

C(F,F); that 18 , C(Fn1 - Pml, Pnl — le) + 0 as m, n + m.

We refertfiuareader to Weinberger [44] for a more complete discussion
of the following lemmas, which will prove crucial in establishing

(3)

the existence of 6(2) and G .

Lemma 1: Let U = {(x,y): 0 :_x :_l and 0 :_y §_l}. Let

V = {v(x,y): v(x,y) is a twice differentiable real-valued function

in 0}.
Defining

B(v,v) = '04 ( [grad V]2 + v2) dx dy
and

C(v,v) = [0' v2 dx dy,

C(v,v) is completely continuous with respect to B(v,v).

The proof of the above lemma begins by dividing the region 9 into K2
squares of side %:= S. Denote the subregions by 01' i = l, 2, ... K2.
It can be shown that

I I v2(X.y) dx dy :L (J I V(X.y) dx dy)2
D 2 D

1 S 1

+ 32 I! Igrad v|2 dx dy (2.4-1)
01

26

for each subregion Di’ 1 = l, 2, ... K2, and each element, v(x,y), in
V. Consider the sequence {vn} in V, defined on 9. Assume {vn} is

uniformly bounded with respect to B, that is B(vn, vn)§_d for some
positive constant d. Applying (2.4-l) and summing, we obtain the

inequality
1 K2
- _ ._ _ 2
C(vn vm, vn vm) :' 2 E (I I (vn vm) dx dy)
S i—l U
i
2 _ _ -
+ S B(vn vm, vn vm) (2.4 2)
l 2 l
Letting p be any integer, choose K such that -;-= S (853" then the
K
last term of (2.4-2) is less that l—- since B(v — v , v - v ) < 4d.
2p n m n m -

Applying Schwarz's inequality to [I I vn dx dy]2, it is seen that the
sequences of real numbers, I I vn dx dy, are uniformly bounded for each
0

i = l, 2, ... K2. A subsequence, {vé}, therefore exists with the

property that I I v; dx dy + a1, for some a1, 1 = l, ... K2. Hence,
0

n

there exists an Np such that

 

2 .1
| I I (v1 - vl ) dx dy §_ ( S )2 for n, m :_N
Di pn pm 2pK2 p
Summing over i, we obtain
1 K2 1
—_' I [I J (V1 - v1 ) dx dy]2 < —- for n, m :.N (2.4-3)
32 i=1 0 Pa Pm — 2" 1’

Using (2.4-3) and the fact that SZB(vn - vm, v (2.4-2)

reduces to

27

1
C v1 — v1 v1 - <-”
( p p . p )__

for m, n > N (2.4-4)
n m n p

Vl
pm

'0

Now construct the collection of sequences {vi } , i = 1,2,3, ...,

p+i>n
where {v%p+i)n} is a subsequence of {V[p+i-l)n}’ and all have the
property that
C(vl +' - VI +') ’ v1 +2) _ VI +') ) 5- 1' for m’" > N +'
(p 1)n p 1 m (p n (p 1 m p 1 p 1
(2.4—5)

This is possible since (2.4-4) was verified for an arbitrary positive

integer, p. Now letting p = l, we construct the new subsequence,

1 = 1
{v(p+1)N }, i 0, l, 2, ..., where v(P+i)N is the N(p+i) th
(p+i) (p+i)

element of the subsequence {v1 } . The lemma is proved since

(p+i)n
\VIP+1)N } is Cauchy with respect to C.

(p+i)
Lemma 2: Let 0' be a region that can be decomposed into a set of
regions each of which can be mapped into a square by a one—to-one
transformation with bounded derivatives and a bounded, non-zero
Jacobian. Defining V, B, and C as in Lemma 1 with 0 replaced by D',

C(v,v) is completely continuous with respect to B(v,v).

In order to verify Lemma 2, we suppose Dl.can be subdivided into the K

subregions 01, i = l, 2, ... K. Introduce the transformations

Xe) .. x11) (x (i) ___ Xe) (x

2 2 ,Y). 1 = 1,2, ... K, in such a way that

1) (i) (i)
1 X2

.y). x

D; is the square E1: 0 :_x:(i):_l, 0 §_x§ :_l in the x , plane.

1

Let Bi(v,v) and Ci(v,v) denote the quadratic functionals B(v,v) and

28
C(v,v), respectively, restricted to the subregion 0%. Upon transform-

ation, Bi(v,v) and C1(v,v) become

(2.4—6)
2 a
31(st) = I Z ajl avi -!-+ p(§i),x (1)) v2) dxfi), dxél)
E1 392:1 axj l
and
Ci(v,v) = I I q(xfi), X§1)) dxfi) dxgi) (2.4_7)
Ei
where

 

 

 

aj£(Xfi). Xéi)) = aQj (Xii), x(i)). 3.1 = 1. 2
f ail 32;. 3V’ 3 2 + 3 2
_ 1) (1) :,d ( (lies) ( V . )
3,2—1 aij ax2 3x(1) 3X(1)
2
for some d > 0, and
(1) (1) = (1) <1) 3 a(x,y)
p(x1 ’ X2 ) q(xl ’ x2 ) 3(x(i), x(1))
l 2
Assuming the derivatives of if) and xii), i = l, 2, ... K, are bounded and
the Jacobians-§£§%§l-—- , i = 1, 2, ... K, have a positive lower bound,
3(x ), x(1))

it can be shown using Lemma 1 that C1(v,v) is completely continuous with

respect to Bi(v,v), i = l, 2, ... K. Now over the original region 01,

K K
B(v,v) lffi Bi (v, v) and C(v, v) < 2 Ci (v ,v), from which it can be
i=1 _'i= 1

concluded that C(v,v) is completely continuous with respect to B(v,v).

Lemma 3: Let D1 be the region of Lemma 2 and let v(x,y) be any
twice-differentiable real-valued function in 0‘. Given 5 > 0, there

exists a constant, K(e,Dl), such that

I I v: dx dy :_e I [lgrad vxl2 dx dy + K(g,vl) [ J v7 dx dy
01 01 D'
(2.4-8)

29

The above Lemma is a direct consequence of the inequality (2.4-1). The

reader is referred to Weinberger for the details of the proof.

Lemma 4: Let 0' be the region of Lemma 2 and assume the quadratic

functionals B(v,v) and C(v,v) are given as

= 2 2 2 2

B(v,v) [ J [vxx + vXy + vyy + v ] dx dy
01

and

C(v,v) = J I v2 dx dy
91

where veC2 in 01, then C(v,v) is completely continuous with respect

to B(v,v).
Applying (2.4-8) and its analogue for the y-derivative it can be shown

that J I [Igrad VIZ + v2] dx dy is bounded by a multiple of B(v,v).
D1

Lemma 4 now follows upon applying Lemma 2.

Lemma 5: Let 01 be the region of Lemma 2. Assume veCm and let
B(v,v) be a quadratic functional that is bounded below by the
integral of the sum of the squares of all derivatives of v of order
m. Let C(v,v) be a quadratic functional containing only derivatives
of order less than m, then C(v,v) is completely continuous with

respect to B(v,v).

The above lemma is a consequence of Lemmas 1—4, and is established by

applying (2.4-8) repeatedly to various derivatives of v.

30

2.5 Existence of 62) and C(3)

 

Let w2 = gib i = 2, 3, and let {Gél)} be the

0,1

corresponding minimizing sequence in Q

. (i) _ ,
1 w1thHGn I'K - 1 and

P(G§1), Géi)) §_w2 + 1_ Now if the kinetic energy norm were completely
0,1 n'
continuous with respect to the potential energy norm the existence of

C(2) and 6(3) would be well known [36], [44]. Investigating

I|é§l|§=P +p +1» +12 withP p pr...“

3 B F T S’ B’ defined by (2.1—10)

T
through (2.1-l3) respectively, we see that this complete continuity is

too much to ask for; however, we see that the potential energy involves

quadratic functionals of the type
a 2 2
B(v, v) = (l) + (3‘1) dA (2.5-1)
A ax By
and the kinetic energy involves quadratic functionals of the type

C(v,v) = I I v2 dA (2.5-2)

A

From Lemma 5 of section (2.4) we see that C(v,v) is completely continuous

with respect to B(v,v). We claim that this information will be enough to
(3)

verify the existence of C(2)and G . We proceed as follows.

T
(i) _ n n n _
Let G n (wx,i’ Dy,i, wi), i l, 2. As was done for the

clamped plate, investigate (2.1-10) through (2.1-15) to obtain a constant

C. with
1

31

n n n n n

aw 2 aw 2 31 2 94» 2 aw 2

i 1 i i 1 1

I I [ (_) + .3;— + (3%2— + (3—yx.2__) + (3)—(L) + (__lL.) dA < C

(2.5-3)

n

Replacing v in (2.5-1) and (2.5-2) by w1

and using the complete
continuity of C with respect to B, we guarantee the existence of a
(i) (1) n1 2-
subsequence, {Gn1 }, of {Gn } with {w1 }Cauchy with respect to the L norm.
Using the complete continuity of C with respect to B twice more, first
with v replaced by w: i’ then with v replaced by W; i’ we obtain a
S ’

subsequence, {G§:)}, of {Gii)} that is Cauchy with respect to II 'lK'

Applying (2.3-4) to the sequence {Géi)} we see that it is also Cauchy with
3

respect to II IIP. Letting 51 be the completion of 91 with respect to
(i) - 2 ‘ (i) 2 (i) 2 =
II IIP + II IIK we obtain a G e 91 with wOai IIG HP and IIG IIK 1.
It is clear that the 6(1), 1 a 2, 3, are at least weak solutions

of (2.1-7) in the sense that equation (2.2-5) is satisfied for all B(i)

in 5 For strongly elliptic systems of second order, the smoothness of

1.
solutions up to the boundary has already been demonstrated [28]. Hence,
the 6(1), i = 2, 3, are in fact solutions to (2.1-7) in the classical
sense.

(1), i = 2, 3, satisfy the boundary

It remains only to show that G
conditions (2) and (3), respectively. In the case of the clamped plate
the boundary conditions were stable in the sense that they were pre-
served in the process of completing Q1. The boundary conditions
corresponding to 92 and 93, however, are unstable in the sense that
they may not be satisfied by every element in 52 and 53, respectively.

(1)

To guarantee that the G , i = 2, 3, are elements of 92 and 93,

respectively, we need only remark that the boundary conditions (2) and

32

(3) turn up in equation (2.2-5) and therefore must be satisfied by the

(3)

solution vectors 6(2) and G , respectively [44].

2.6 Remarks and Conclusions

 

The results of sections 2.3 and 2.5 clearly indicate that mo 1
9

is the lowest natural frequency for the system (2.1-7) corresponding

to an eigen solution satisfying the ith set of natural boundary con-

2
IIGII
ditions. Let w: i be the greatest lower bound of —————%- for all G
’ IIGIIK
in 91 with the supplementary condition that K(G, 6(1)) = 0, Clearly,
By repeating the previous arguments we find that w is

(31,1 1 000,1-
the 2nd eigen value for the system (2.1-7) and that this value

1’1

corresponds to a normalized eigen function, 9:1), orthogonal to 9(1). Con-
tinuing this process we are able to construct a sequence of eigen

values w :_w < ... < w , and an associated set of

0,1 1,1 5-”2,1 - —- n,i "'

orthonormal eigen vectors. To establish a discrete spectrum we need
only verify that each eigen value is of finite multiplicity. This

results from observing that w + w as n + m. Indeed, if ”n 1 < C
, ._

n,i i

for some constant Ci’ letting 6:) be the orthonormal eigen solution

(1)
corresponding to ”n,i’ we have HGn IIP a mi, §_C2, n = 0,1,2, ... .

i
This, however, implies the existence of a subsequence, {6(1)}, that is
Cauchy with respect to the K-norm. Thus,

lim ||c(1)- cm||K = o

k,2, + co nk n9,

This is a clear contradiction since the orthogonality and normality of

the eigen solutions implies that ||G§:) - 6:1)IIK = 2 for every K and 2.
l

33

It is interesting to note the difficulties encountered when
we remove the effect of the torsional springs from the vibrating system.
This is done by setting the torsional spring constant, C, equal to zero
in the expressions for the natural boundary conditions and in equations
(2.1-4). By letting C - 0, we are admitting the possibility of
rotation but are neglecting any existing potential energy due to

(3)

2 .
rotation. Problems arise in verifying the existence of G( ) and G which

indicate a possible difficiency in the improved Timoshenko plate theory.

Let

|lc||§ -PS + PB + PF (2.6-1)

where P P , and PF are given by (2.1-10), (2.1-11), and (2.1-12)

S’ B
respectively. Constructing the minimizing sequence {G

l (i)
wj 1 ___IIGn ||2< wj i +-and IIGn 'IK - l, a constant Cj(wj 1),
independent of n, can be found so that
31231“, 2
_LI". ._ _
JAJ (Byx __ZLBX dA : C101):l i) (2.6 2)

except when lim I [(M;X)Z dA = 0 under every possible rotation of
A

n + w

coordinates. The inequality (2.6-2) gives an upper bound on the integral
of the square of the plate rotation, and is needed to guarantee the

total boundedness of

awn 2 awn 2
(__Xsl) + (_Li) M (2.6-3)
A By ax
which in turn allows us to extract the subsequence {0(1)}, of section 2.5-

n3

34

It can be shown that lim I I (ng)2 dA = 0 under every rotation
A

n+ao

of coordinates implies that I

J (MD - Mn)2 dA tends to zero as n + m,
A y x

2
which further implies that lim ||c(i)|| = m2 g
n+m n P j,i

system (2.1-4), it can be shown that all eigen solutions for the simply

12. Investigating the

supported plate that have Myx - My - Mx - 0 and that correspond to

2

j 1 - 12 are of the form

0)

wx = ay + b L (2.6—4)

 

which amounts to a rigid body rotation in every cross section of the
plate parallel to the midsurface. Hence, since the improved theory
does not restrict the magnitude of the rotation, either interior to the
plate or along the boundary, we are unable to show the convergence of

the minimizing sequence to an admissible function when w 12.

2 a
1,1
Although this problem exists for the simply supported plate and for the
plate with free edges, it does not appear in the discussion of the
clamped plate where the rotation is automatically bounded due to the
boundary conditions wx - wy . 0.

Clearly, in discussing the variational problem for the plate
with the torsional springs attached, if we could prove that the eigen
frequencies were analytic in C near C = O, we could obtain the discrete
spectrum for the plates without the torsional springs by taking a

limit. In the case of the circular plate, the results of Chapter III

do yield this analyticity; however, in the case of arbitrarily shaped

35

plates, we must rely on results already developed within the theory of
analytic perturbations of symmetric operators [36]. All results

(3)

developed require the operator L defined in section 2.1 to be

bounded in the sense that

HL‘” c‘i’ll :M(IIG(“II + In”) a“) + we”) c‘i’m

. i .
fOI 811 G( ) in 91 and for some constant M.: Here, II II refers to the

LZ-norm. There seems to be no way of imposing this restriction on

3
L( ) applied to the class, 91, of admissible functions.

If one is willing to depart from the improved theory slightly
and assume that the plate rotation is zero as was done by T. C. Huang
[17], the variational problem can be solved. In this theory it is

assumed that the vertical deflection can be written as

w = wb + wS (2.6-5)
with
3w .
...;h _
wx ax (2.6 6)
and
3w
b
a ___. (2.6—7)
WY 3y

In (2.6—5), (2.6—6), and (2.6-7), wb represents the vertical deflection
due to bending and wS represents the vertical deflection due to shear.

In this case, the system (2.1-4) can be replaced by the system

V2(V2wb + S(w2 — 12) wb - 123 w) = 0
(2.6—8)

II
C

vzwb + v2 w + (m2 - k) w

36

with the natural boundary conditions of section 2.1 left unchanged.

In conclusion, the variational problem for plates with clamped,
simply supported, or free edges has been solved within the realm of the
improved plate theory provided torsional springs are attached to the
surface of the plate. In general, the existence of a minimizing
function for plates with simply supported or free edges is not
guaranteed within the realm of the improved plate theory, without the
inclusion of torsional springs, suggesting a possible defect in the
theory. Something should be said concerning the rotation present in
the vibrating system. Assuming the rotation is zero seems to be too
strong since the variational problem can be solved provided we
require the rotation to be zero only along the boundary. Research along

these lines is currently under way.

III. GREEN'S FUNCTIONS FOR CIRCULAR PLATES

3.1 Introduction

 

Consider the forced vibration of a thin, elastic, circular
plate supported by a linear, Winkler-type elastic foundation. The plate
is subjected to a point load whose intensity varies sinusoidally in
time, and the edges of the plate are clamped, simply—supported, or
free. For each of these three edge conditions, a fundamental solution,
called the Green's function, will be obtained for the elliptic system of
partial differential equations that is characteristic of the improved
Timoshenko plate theory.

As mentioned in Chapter I,D. Yen constructed the Green's
function for the infinite plate with the aid of the Fourier transform.
The Green's function was analyzed along certain curves in the frequency-
stiffness plane. Using his results, a similar analysis will be carried
out for the circular plate under each of the three edge conditions
mentioned above. Suitable frequency equations will result from the

analysis.

3.2 Formulation

 

Assume the surface of the plate occupies a circular region
A: x2 = y2 :_R. The plate is subjected to a time harmonic point load

concentrated at an arbitrary point, P0 (€,n), interior to the plate.

37

38

There is no loss in generality to assume n = 0 so that the load is
concentrated along the x-axis. A rotation may be performed, if necessary,
to assure this condition.

Letting ux, uy, and uz represent plate displacements in the
x, y, and 2 directions, respectively, we use the relations (2.1—l) of
Chapter II in order to obtain the three plate-displacement equations

of motion:

 

 

 

o h3 82¢
D 2 3¢ 3w 1 x
__ _ v ___! __=
2 [(1 u) ¢x + (1 + u) By] G h(¢x + ax) 12 at?
D h3 32¢
D 2 Bo 3w 1
—- 1 — n V ¢ + 1 + u -' ‘ G'h + —- = 3.2—1
2 [( ) y ( ) 3y] (¢y ay) 12 3:2 ( )
32
G'h(V2w + ¢) + f = olh-—Ji
3:2

where f = f(x,y,t) is the normal pressure on the plate. In the present

study

f(x,y,t) = 6(x - E. y)e_iwt - k w(x,y,t)

where 5(x - E, y) is the Dirac—delta function corresponding to a con-
centrated force at the point (5,0), with frequency w. Again, k
corresponds to the foundation stiffness. Note that the system

(3.2-l) is the same as the system (2.1-2) of Chapter II, except for the
introduction of the Dirac-delta function, and the omission of torsional
springs. Again, seeking steady state response, we assume w(x,y,t),
¢x(x,y,t), and ¢y(x,y,t) in the form (2.1-3). Proceeding as in Chapter II

with the introduction of the dimensionless quantities (1.4), we obtain

39

the time-reduced, dimensionless system:

 

1
-§ [(1 — p)v2¢x + (l + u) gfil + 5(0)2 - 12MX - 12$ %¥-= 0
1 23¢. 3w
—- 1 - 2 + 1 + ——- + 2 — 12 - 12 ——-= 0 3.2-2
2 [( u)v my ( u) ayj 8(w )wy 3 3y ( )
VZW + (w2 - k)W + ¢ = -6(X ' £9 Y)
where
3 3
3x 8y
In matrix form, the system (3.2—2) has the representation
L(1) G + w2L(2)G = A (3.2—3)
(1) (2)

where L , L , and G are as defined in section 2.1 of Chapter II,
and AT = (0,0,6(x - E, y)). We remark that a solution, C, of (3.2—3)
should be regarded as an element of C2 in A and C1 in A, except at

(5,0) where an integrable singularity exists.

Definition: Let 6(1), 1 = l, 2, 3, be the solutions of (3.2-3)
satisfying the edge conditions (1), (2), and (3), respectively, of

section 2.1, Chapter II. The C(i)

, i = 1, 2, 3, are called the
Green's functions for the clamped, simply supported, and free plates,

respectively.

40

The problem is to determine 6(1), 1 = l, 2, 3, for the
circular region A and the regions discussed by D. Yen in the frequency-

stiffness plane. We let

(i) 1 = 1, 2, 3 (3.2-4)

(1)

where the transposes of U and V are given by

UT = (ax, my, u) (3.2—5)

and

T

(1) _ <1) (1) (1) _
V - (BX , By 9 V ) (302 6)

The vector functions U and V(1) represent a singular solution of (3.2-3)
and a regular solution of (2.1-7), respectively, with their sum satis-
fying the ith set of natural boundary conditions. Of course it is

assumed that C is set equal to zero in (2.1-7).

3.3 Construction of U

 

Consider the system

m

.1. _ 2 ~93: 2 _ _ 1L“:
2 [(1 U)V 01x + (1 + U) 3x]+ 5(w 12)ax 128 EX 0
1 33 au
_ _ 2 _ 2 _ _ ___=.
2 [(1 U)V 0y + (1 + u) ay.]+-s(w 12)ozy 128 8y 0

m (3.3—1)
V2u + (w2 - k) u + d> -—- -6.(x,y)
m Ba Ba
(p = —-—x_+__.l

41

We are momentarily assuming that the point load is at the origin. Upon
solving this system we simply replace x by x - E to obtain the desired
singular solution of (3.2-3).

To simplify the equations of the above system, we attempt to

find a solution with the property

3a 60
X = 2 (3.3-2)
8y 3x

The system reduces to

Vza + s(w2 - 12)a - 123 -—-= O
x x 3x

v2a + s(w2 - 12)a - 123 333 = o (3.3-3)
y y 3y

Bax 3a 2 2

§;—-+45§1-+ V u + (w - k)u = -6(x,y)

It is clear that a solution of (3.3-3) with the property (3.3-2)
is also a solution of (3.2-3) with the point load at the origin,
provided its partials up to order two have integrable singularities,

in two dimensions, at the origin.

Letting

aX(X.y) = %;- J_m [_m a€(€.n) ei(x€ + y”) d: dn (3.3-4)
_.l_ m m -i(x§ + yn) _

ag(£,n) - 2n J_m [_max(x,y) e dx dy (3.3 5)

and using similar expressions for my and u, we determine the Fourier

transform of the system (3.3.3), Let 92 = g2 + n2 and obtain

42

 

 

 

-p2ag + 8(w2 - 12)a€ - 12sg1 E = o
-pzan + so»2 - 12)an - 12sni G = o (3.3-6)
_ 2’ 2 _ ' = _.l_
igag + inan p u + (w k) u 2“
Solving the system (3.3-6) for ca, an, and G, we obtain
631g 1
= 3.3—7
“5 n (Q(o) ) ( ’
_ 6sin l _
an n (Q(p)) (3.3 8)
G = §1(o) + é2(p) (3 3—9)
where
_ 2 \
81(0) = ZWQ(o)
- _ 8(12 - wz) _
0(0) = [02 - ofJIDZ - 0%] A

 

We remark that of and 0% are given by (1.11) of Chapter I. The
Fourier inverses of (3.3-7), (3.3-8), and (3.3—9) determine the
components ax, my, and u, respectively, of U. We consider these

inverses in each of the four cases discussed by D. Yen [45].

Case I: of ¥ 9% # 0

 

In regions I through VII and along the line w? =

43

except at the origin and at the point T, the Fourier inverses of

(3.3-7), (3.3-8), and (3.3-9) are

12 8

ax = m 5; 820:) (33-11)
_ 12 g__ .

0y - i§_:—EY 3y g2(r) (3.3-12)

u = gl(r) + g2(r) = 8 (3.3-13)

respectively, where g1(r) and g2(r) are given by (1.9) and (1.10).

0

Ca e II: 2 = 2

S 01 02
At (k,w2) = (0,0) and at the point T, Q(p) = p“. The Fourier

inverse of ifi-is well known in the study of the classical plate theory.

The components, ax and ay, become

3 3
ax = Eg-s 5;-r2 log r (3.3-14)
_ L 3. 2
(1y - 21! S 3y r log r (3.3-15)

We note that gl(r) and g2(r) are given by (1.13) and (1.14), respectively.

Case III: p2 = 0, p1 # 0

Along the lines, w2 = k and w2 = 12, ax and my are given by

 

_ 351 3 (l) 2 .

ax —';¥—-§; [H0 (plr) +11; log r (3.3-16)
_ 331 a (1) 2 '

ay — B¥—'5;-[HO (plr) +~IF log r (3.3-l7)

while u is defined by (3.3-l3) with gl(r) and g?(r) given by (1.15) and

(1.16), respectively.

44

Case IV: pf = pg # 0

Finally, we note that along the parabolap2 = 4q, ax, my, and u
are given by (3.3-11), (3.3—12) and (3.3—13), respectively, with g1(r)
and g2(r) defined by (1.17) and (1.18).

Replacing r = /x2 + y2 in each of the above cases by

 

E = /Qx — g)2 + y2 , we obtain
U = U(x.y;&) (3.3-18)

It is clear from analyzing each component that U and its derivatives
up to order two have integrable singularities at (g, 0) in each of the
four cases mentioned above. It is also clear that “X and ay have the

property (3.3-2).

3.4 Construction of v(i)

 

Case I: p2 # p2 # 0
1 2
Referring to Mindlin's paper [25], it is seen that in the

absence of surface loading the plate displacements can be expressed in

(1) v(i) (1) 1(1) (1)
’ 2

terms of three functions, vl , and H , where v and v2 are

components of the displacement perpendicular to the middle surface of the

(1)

plate, and H is the potential function giving rise to twist about
the normal to the plane of the plate. Working with the dimensionless

system (2.1—4), where C is set equal to zero and the foundation is

included, the plate displacement equations may be replaced by

(v2 + pf) vfi) = 0 (3.4—1)
(v2 + 03) véi) = o (3.4-2)

(3.4~3)

I
O

(V2 + 0%) B(i) —

45

with of and 0% given by (1.11), and

 

 

 

 

2 = 2 - ._
Q3 1 _ u s(w 12) (3.4 4)
The plate displacements, v(i), 8:1) , and 851) take the form
v(i) = vfi) + vgi) (3.4—5)
. 8v(i) 3v(i) . (i)
em = (o - 1) 1 + (o — 1) —-—2-— + 211—. (3.4-b)
x 1 3x 2 3x 8y
9(1) = ( - 1) 3Vfi) + ( — 1) avéi) —-1fl:ii (3 4-7)
’y 01 8y 02 8y ax ’
where
= 2 2 2 _ ‘1 _
(01. 02) (02, pl)[8(w 12)] (3.4 8)

(i)

Noting that the plate displacement, v , is symmetric with respect to the

x—axis, we assume solutions of (3.4—1), (3.4—2), and (3.4-3) of the

form

vfi) = X Aéi) Jn(plr) cos n6 (3.4-9)
n=0

véi) = 2 8:1) Jn(02r) cos n6 (3.4-10)
n—O

B(i) = nil Céi) Jn(p3r) sin n6 (3.4-11)

respectively.

46

Case II: of = p: = 0

At the point (k,w2) = (0,0),the equations for the determination of

v(i) reduce to
. . m l
_l _ 2 (1) 8¢* _ (1) _ av _
2[(1 u) V Bx + (1 + 11) _3x ] lZsBx 123 ————3x - 0
1-(1 - ) v73(i)+(1 + )-93: - 129;(i) - 125 ”v(j) = o
2 H Y H 8y ‘Iy 8y

v2v(i) + ¢* = 0

 

38:1) 8851)
¢* = 8x + By

 

Combining the equations in the system (3.4-12), it is easily seen

(i)

that v

V”v(i) = 0

must satisfy the biharmonic equation

Due to the point load along the x-axis, we assume a Fourier cosine

(i). (1)

expansion of v Requiring V

vs) = E [Am rn

+ B(i)
n=0 n n

n+2?‘
r cos n0

f(3.4-12)

 

to be bounded at the origin, we obtain

(3.4-13)

We now consider (3.4—12) as a system of three equations in the two unknowns,

8:1) and 8:1), and determine these components in terms of v(l). The
most general expressions for 8:1) and 851) will be found by first f

”(1)

a particular solution, (Bx

v(i) given by (3.4—13); then 8:1)

and B

“(1) .
, By ), of the system (3.4—12) With

(1)

y will be given by

inding

47

 

. . (i)
,(1) _ “(1) 8H
bx - “X + By (3.4-14)
. (i)
(1) _ %(i) _ 8H _
By - By 3x (3.4 15)
. (i) . . . . q .
respectively, where H satisfies equation (3.4-3), with p} equal to
-24s
2 = ______ -
03 1 _ u (3.4 16)
Letting
g(i) = 88(1) W
x 3x
and ) (3.4-17)
gm = as“)
y 3y /

 

and replacing 8:1) and Bil) in (3.4-12) by 3:1) and fiél) , respectively,

we obtain the system

§—-[v28(i) — 1253(1) - 12sv(i)] = 0

3x
§;-[v28(i) - 1258(1) - 12sv(i)]= 0 (3.4-18)
V28(i) + V2v(i) = o

(i) (i)

for the determination of 8 . We finally let 8 satisfy the system

v23(i) - 1258(1) - 123v(i)

ll
0

(3.4—19)
V?B(i) + v2v(i)

ll
0

48

from which 8(1) is determined as
B(i) = _ [v(i) + 1%: V2v(i)] (3.4-20)

As in Case I, H(1) is given by (3.4-11).

At the point T, we are led to consider the system

 

l
O

. (i)
%[(1-11)V28)((1)+(l+p)-aib:]-IZSg: '

 

 

 

(1)

l (i) 3¢* 0v _
-2- [<1 — 11)V728y + (1 + u) 37-] - 123 '3‘), ‘ 0 7(3'4‘21)
V2 (1) lst(i) + ¢* = 0

38:1) 38(1) J
M -

8x 3y

(i) (i)

It is again seen that v is biharmonic; hence, assuming v is given
by (3.4-l3), we attempt to find a particular solution, (kii), §;1)),

0f the system (3-4-21). considered as a system of three equations in

the two unknowns, 6:1) and Eél) . The determination of 3:1) and $51)
(1)

is now a bit more involved. Indeed, if (3.4-l7) were assumed, v
would either have to be harmonic or restricted to satisfy an equation

of the form

V2v(i) = C
where C is an arbitrary constant. Such restrictions on v(l) are not
desired, hence we approach the problem of determining Eil) and fifii) in

a more direct manner.

49

Solving the third equation of (3.4-21) for ¢* and substiLuting the
result into the first two equations of (3.4-21), we obtain the following

. ”(1) ”(i)
three equations for the determination of 8x and By

V7?;(i) = 31(i)(x,y) (3.4-22)
X X

v2§<i) 82(i)(x,y>

 

y = 3y (3.4—23)
£(i)(x,y) = lst(i) + (%—§L§) V2v(i) (3.4-24)
Let
Kii) = X f(i)(r) cos n0 (3.4-25)
n=0 n
and
“(1) X h(i) (r) sin nO (3.4-26)
y n=l n

Substituting (3.4-25) and (3.4-26) into (3.4—22) and (3.4-23),
respectively, with v(i) given by (3.4—13), we determine Eii) and Eél)
as particular solutions of (3.4—22) and (3.4-23), respectively. The

functions féi) (r) and héi)(r) are then given by

 

 

 

 

' ' l + i 3 ° \
£31) = [3SA1(1) + 2 [—-—-l __ :) Bf )]r2 + ~2- s 31(1) r”
(1) <1) <1) n+4 P (3'4‘27’
(i) (An + Yn ) +2 C r
f = rn -+-Jl—————*- n > 1
n 4(n + l) 8(n + 2) —- I
and
(i) (i) (i) n+4
h(i) = (An - Yn ) n+2 _ gn r (3 4—28)
n 4(n + 1) r 8(n + 2) ' L

50
where

Afi) = 243 8(1)

 

O
. (3.4-29)
K(l) = 123 8(1) n> 2
n n-l -—
yéi) = 12s(n + 1) Agii
+ 4 (i 1‘ {jun + 1)(n + 2) BS1: n 1 1 (3.4-30)
C511) = 12s(n + 2) BS3 n _>_ 1 (3.4-31)

In order to assure that (Ail), E;I)) satisfies the third equation of

(3.4-21), we modify (3.4-25) and (3.4-26) to read

(i) n

g(l) = Z f(1)(r) cos n8 + 2 D r cos ne (3.4-32)
x n n
n=0 n=l
g(i) = Z h(1)(r) sin n0 + X D(i)rn sin n0 (3.4-33)
Y n=l “ n=l “
%(i) %(i) . .
Indeed, Bx and By remain particular solutions of (3.4-22) and

(3.4-23), respectively, since the modifications only involve the
addition of harmonic functions. The constants Dn’ n=1, 2, 3, ... are now
determined so that the third equation of (3.2-2), with the right hand

side set equal to zero, is satisfied. They take the following form:

(1)

(i) _ 213(1)
1 o

D = 63A
0

(3.4-34)

(i) _ is; (i) _ 1 + U (1)
Dn ’ n An-1 (1 - u + 2) Bn-l “ 3-2

We remark that (3.4—32) and (3.4-33) now represent the particular

51

solution we are looking for. Taking into account the symmetry of the

vibrations and requiring the solution to be bounded at the origin, the

components of the most general solution, v(i), of (3.1-4) are v(i),
8:1), and 8:1), where v(i) is given by (3.4—13) and 8:1) and fiéi)

take the form

 

(i)
(i) _ w(i) 8H
Bx - Bx +'ng*-— (3.4—35)
(1)
{5(1) = "1(1) - 9—H ' -'
y (’y 3x (3.4 36)
. (i) . .
with H being a solution of
v23(i) = 0 (3.4-37)
of the form
H(i) = 2 C(1) rn sin n0 (3.4-38)

Case III: p2 = 0, pl # O

 

Along the line w2 = 12 the plate-displacement equations become
(1)

l 2 (i) 3¢* 3V

... _ + + —— .. =

2 [(1 U) V Bx (l u) 3x ] 128 8x 0 \

 

(1)
§ [(1 - u) V2851) + (1 + u) 393-] - 12s 3: 0 > <3-4-39>

 

V2v(i) + (12 - k)v(i) + ¢*

ll
0
\

Combining the equations (3.4-39), we see that v(l) satisfies the equation

v2(v2 + of) v(i) = 0 (3.4-40)

52

with

of = 12(s + 1) - k (3.4-41)

(1)

We therefore assume v of the form

v(i) = Z [A(i) rn + B(i) J (p r)] cos n0 (3.4—42)
n n n 1
n=0
Proceeding as we did at the point T with (3.4-42) replacing (3.4-13),

we obtain a particular solution, (3:1), §;1)),of the system (3.4-39),

 

 

 

considered as a system of three equations in the two unknowns, 3(1)
and $51). The equations (3. 4- 32) and (3. 4- -33) with f(i)(r), h(i)(r),
and Déi) given by
1
lAfi) r2 3:1)
f0 = 4 - 201 (128) Jo(p1r)
A(1) (1)
f - jl—Z—————-- l—-(EE——-- 3(1)) (12 ) J ( ) (3 4—43)
1 - 4 pl 2 o S 1 plr > o
A(i)
V1 An+1 n+2
f = ——————- r
n 4
_ 1 B(i)_ B(i)
2-0—1 (B 11- BM) <12s) Jn<plr) n12 J
_, Au) B<1)
h = ——l————-r3 + l—-(—3——-+ 3(1)) (123) J (p r)
l 4 oz 2 o 1 l
(i)
-v A
_ 1 n+1 n+2 _
“n - ‘—‘7r-—‘ r (3.4 44)
(i) (i)
+— 1(Bn+1 + Bn _1 (123) Jn(plr) n 3.2
and
(i) ,
A
(i) a _ 0
D1 (k 12) 2
(3.4-45)

 

<1).___1_ _31 (1)
Dn n (k - 12 2 ) An-l

53
(i

x
(3.4-43) through (3.4-45),

m .
respectively, determine 8 ) and 3:1) correctly. Note that in

 

.4-46
243 - (1 + u) (k - 12) (3 )
V1: 1‘11
Using (3.4-38) for B(i) , B (1) and 8(1) can now be expressed through
3 y

the use of (3.4-35) and (3.4-36).

Along the line wz - k, except when k = 12 and k - O, we assume
3:1) and 351) have the form (3.4-17). The plate-displacement equations

become

g—x- 0728(1) + s(k - 12m“) - 12s ‘7‘”) = o
%;-(V28(i) + s(k - 12)B(i) - 123 v(i)) = 0 (3-4-47)
V2v(i) + v26”) = o

In (3.4-47), it is assumed that v(i) is given by (3-4-42). We deter-

mine B(i) so that

V28(i) + s(k - 12)B(i) - lZs v(i) = 0 \
(3.4-48)
V2v(i) + V28(1) = 0
We write
v(i) = vfi) + vgi)
where
vfi) = E Aéi) rn cos n9 (3-4-49)

n=0

54

and

v(i) = 2 3(1)
2 _ n
n-O

Jn(plr) cos n0 (3.4-50)
A solution of (3.4-48) is now given by

8(1) = -v:i) + (o - 1) vfi) (3.4—51)
where

(3.4-52)

Along the line w2 = k, with k = 12 and k = 0 being exceptions, the com-
ponents, v(i), 8:1), and B;i), of the solution of (2.1-4), with C set

equal to zero, are given by (3.4—42),

 

 

(i) (i) .
8(1) = _ avz + ( _ 1) avl 3H(1) (3 4-53)
x 3x 0 3x 8y °
and
, (i) (i)
B(i) = :33?— + (O _ 1) 311—. _ iii) (3 4_54)
Y 3y 8y 3x '

respectively. In (3.4—53) and (3.4-54), the H(1) are again given by

(3.4—38).

Case IV: of = p2 # 0

2
Finally we observe that along the parabola p2 = 4q, v(l) is
required to satisfy
(V2 + of)2 v(i) = 0 (3.4-55)

Noting that the two bounded solutions of (3.4-55) take the form

.9) =

and

v(i) =

2

we take as the

components v

v(i)

B(i) g

X

B(1) =

Y

where

(i)

0% =

Again, it is

3 . 5 Boundarj

(i) _

 

55

Z A(1) J (p r) cos n0
=0 n n 1

E B(l) r a—J (p r) cos n6
n 8r n 1
n=0

regular solution of (2.1-7),
(1) (1)

 

 

 

 

 

 

1
the vector, V

 

, Bx , and By , given by
- vii) + véi) ‘
a8(1) + 3H(i)
3x 3y ?
38(i) _ 8H(1)
3y 3x 1
1 [(123 + w2 - k) v(i) + V7v(i)] \
s(w2 - 12)
Z Cii) Jn(p3r) sin n9
n=1
2 2 -
l _ U s(w 12) 1

assumed that C = O in (2.1-7).

Conditions

 

In each of the four

express

G<1) .

will)
,(i) 1 1

Y
w<1)

(3.4-56)

(3.4—57)

with

(3.4-58)

(3.4-59)

cases discussed above, we are now able to

(3.5-1)

56

in the form (3.2-4) with

wii) = ax<r.e;e) + 8:1) (r,0) (3.5—2)
(1) = . (i) _
W), ay(raeag) + By (r90) (3.5 3)
w(i) = u(r,6;§) + v(i) (r,6) (3-5-4)

6(1), 1 = l, 2, 3, to represent the Green's functions for

In order for
the clamped, simply supported, and free circular plates, we must impose
the boundary conditions (1), (2), and (3), respectively. Equivalently,
for each i, i = 1,2,3, we must determine the three sets of constants,

1(1), 3(1), and C(1) (1)
n n n

, which force G to satisfy the ith boundary con—

dition. The first step is to determine the frequency equations yielding

the natural frequencies of vibration, then, assuming w is not a

natural frequency, we determine the final expressions for 6(1),

1 = 1,2,3. Each case, I through IV, will be discussed separately.
Introducing a polar coordinate system more suitable for

applying the boundary conditions to the circular plate, we let

8:1) (r,9) = 8:1) (r.6) cos 0 + Béi) (r,0) sin 0 (3.5—5)

and

8&1) (r,6) Béi) (r,6) cos 6 - 8:1) (r,0) sin 0 (3-5-6)

These transformations are simple rotations determining plate displace—

ments measured in the r and 6 directions, respectively. Similar

. (i) (i)
expreSSions are obtained for or, a wr , and $0 .

a,

57
The boundary conditions (1), (2), and (3), become
19:1) (Rae;€) = Wé1)(Rae;€) = w(1)(R:O;€) = 0 (Clamped)

Mr(w(2)) = Mre(w(2)) = w(2)(R,6;g) = 0 (Simply supported)

 

 

 

 

 

(3.5-7)
and
(3) - (3) _ (3) _
Mr(w ) - r6<¢ ) - Qr(w ) - 0 (free)
respectively, where
(1) _ 3 <1) (1) a__ (1) ,
Mrw ) {37111, (R e a) + Rh), (R e a) + 30 we (R.0.a))]
Mre(¢(i)) = [fiigow ¢:1)(R, 0 ,€)- w(i)(R, 0 ,g))+ 3r ¢51)(R,o;g)]
(i) _ (i) . L (i) .
Qr(W ) - Wr (R,0,€) + 3r W (R9695)
(3.5-8)
Case I: of # p: # 0
In regions I through VII and along the line w2 = 1~§fE-,
except at the origin and at T, 8:1) and B(i 1) are given by
(i) (i)
a“) = (o — 1) avl + (o - 1) 3V? + -1— ”(1) (3 5-9)
r 1 8r 2 3r r 80 °
(1) avfi) avfi) au(i)
86 = (01 _ l) r80 + (02 ‘ l) ESE—"" ar (3'5'10)

where o and 02 are given by (3.4-8); vfl) and vgl) are given by

1

(3,4,9) , and (3.4-10), respectively; and H(i) is given by (3.4—11),

Tine frequency equations for the clamped, simply supported, and free

58

plates, respectively, are given below [17].
A(1)JO (0 1R) + 3(1)J0(02R) =
(1) _ . (1) _ . _
A0 (01 l)J0(le) + B0 (02 1)J0(92R) -

(l)

(1)
An Jn(le) + Bn

Jn(02R)

A§1)(ol - 1) an(le) + 3:1)(02 — l)%Jn(02 R) + C(1)J n(93R) =

Aé1)(ol - 1)J$(le) + 3:1)(02 ~ 1)J;(02R) + cél) EJD(03R)

J (01R) + BS2)

(2)
A0 0

J0(02R) =

A(2)(o - l)[d3(le) + fia5(le)]

+ 1382M)2 - 1)[J3(p2R) + E-J6(02R)] =
(2) (2) _
An Jn(le) '1' En Jn(02R) -

A512) (0 — 1)[{{- Jn (o R>'P‘;Jn(01R>]

 

R
+ 8(2) (02 —l) [—-Jn (p R) --—— J n(p R)]
n R?
(2) .
C 1n)
+ n J" (0 3R)— — J' n(o 3R) +_ J (0 R) =
2 R2
Aéz) (01 -1)[J"(p R) +— R J' (o R) --:;-n J n(o R)]

(2) _ n g . _ un?
+ Bn (o2 1)[Jn(sz) + RJn(p?R) R? Jn(07R)

(2) 2. . .2_ _
+ Cn (1 - u)[ R Jn(p3R) - R2 Jn(o3R)] -

1

(3.5—11)

01

0 ) (3.5-12)

 

) (3.5-13)

 

0‘) (3.5-14)

O

 

J

59

 

 

(3) (3) . _
A0 0 1J' o(p lR) + BO 0 2J0(p2R) - OW
A(3)(o - 1) [J"(p R) +— RJ 0(p R)] ) (3.5—15)
+ 3(3)(02 - 1) [$0pr R) +— R'J 0(p 2m] = 0)
A§3)01Jé(le) + 8:3)02J;(p2R) = O1
(3) n . n
An (01 - 1) [RiJn(le) -.RE Jn(olR)]
(3)
+ Bn (02 -l)[%.1n(sz) - iE-Jn (9 2R)]
C(3)
+ 3 [J;(p3R)- l-J n(p3R) +— R2 J “(93R)] = 0 }(3.5—16)
Rf’wl - 1) [J"(o1R)§ +— J (p R) — P— J (p m]
R2
Ema: - 1) [mp R) +— J (o R) — —- J (p R)
n 2 R R2
+C§3)(1-u)[-Jn(o3R)-—Jn(p3nZR)] =0
R2
J

 

We remark that as a result of introducing the dimensionless quantities,
(1.4), R now represents the radius of the plate divided by the plate
thickness. The prime (') denotes differentiation with respect to r.
The systems were obtained by applying the boundary conditions, (3.5-7),
‘with Wii) , wéi), and w(i) replaced by 8:1), 851), and v(i),
respectively. The equations, (3.4-5), (3.4—9), (3.4-10), (3.5-9), and
(i) (i)

6

(3.5-10),define 8:1), 8 , and v .

In matrix form, (3.5-11), (3.5—13), and (3.5—15) become
E51) ng) = o, 1 - 1,2,3 (3.5-17)

RJhere B(l), Eéz), and E53) are the coefficient matrices of (3.5-11),

60

(3.5-13), and (3.5—15), respectively, and

Au)
F(i) . 0 1 = 1 2 3 (3 5-18)
0 (i) 9 3 0
B0

The systems, (3.5-12), (3.5-14), and (3.5-16), written in matrix form,
are given by

Eii) Féi) = o 1 = 1, 2, 3, n 3_1 (3.5-19)

(1)

n ,

where E Biz), and EiB) are the coefficient matrices of (3.5-12),

(3.5-14), and (3.5-16), respectively, and

 

Au)
n
F(1) = 3(1) 1 = 1, 2, 3, n :_1 (3.5—20)
n n
C(i)
1‘1
Letting
E = /Q; - §)2 + y2 (3.5-21)

replace r, we use (3.3—11) and (3.3-12) to define ax and ay. The

corresponding expressions, in polar coordinates, for or and 06 become

Q
II

a
r 3r (3.5-22)

8
a9 "' F56- 82(raeag)

where g2 is defined by (1.10). The function, u, is defined by (3.3-13)

with r replaced by 2. Let us now define

61

TI
3(1) = - l-J u(R,6;§) cos n6 d6
n n _"
b(1) = +-1 W a (R 6'5) sin n6 d6
n n _fl 6 ’ ’
<1)- .1." .
cn - - H I-" ar(R,6,£) cos n6 d6
a<2) = 3(1)
n n

(2) 1__ " .
bn + 2n I-" Mre(a) Sin n6 d6

 

n
C(Z) = — l-[ M (a) cos n6 d6
n n _fl r
a(3) = - l-J" Q (a) cos n6 d6
n n _n r
b<3) g b(2)
n n
C<3) = c<2)
n n
Let
()
:03.
(j) 2
IO -(C(j)) J 1, 2, 3
0
2
I(J) _ a(j) = 1,2,3; n

“V
H

) (3.5-23)

 

? (3.5-24)

 

62

Assuming Eéi), n = 0,1,2, ... , has an inverse, that is to say, w is

not an eigenfrequency corresponding to the ith boundary condition, we

are a

The Green's functions, G

ble to write

-1
F(1) z (1) I<1)
n n

E
n

(1)

through VII and the line m2 =

n:

1 + s

T, are now completely determined.

C(i) = U(r.9;€) + V<i)(r.6;€)

where

U

V

8

U

8

B

B

T
= (ax. my. U)

. T . .
(1) 2 (8(1), 8(1), v(i))
X y

a (x y' ) = 12
x x ’ ’5 12 - wz

 

= ( ~a) = ———33L1r—
y my x,y, 12 - w
, _ 1_s(12 - m2) (1
2(X9Y9g) ‘ 4 [*1

2 _ 2
pl 02

0,1,2,...-

To summarize,

1 = 1,2,3

82(x,y;€)

2.. .
3y 82(X,Y,€)

0

’ (01*) - nél) (92%)]

= u(x.y;€) = gl(x,y;€) + 82(x.y;€)

1(x,Y;€) = %‘;?f%jgg [:pfnél) (pl?) - pingl) (02%)]
(i) (1) ,

(i) = ( _ 1) 3:i___.+ ( _ 1) 3:g___.+.§fl::l_

x 01 3x 02 3x 3y
(i) (i) .

(i) = (O _ 1) 3v1 + ( _ 1).3:3___ _ 3H(1)

y 1 ay- 02 3y ax

 

(3.5-25)

, i = 1,2,3, corresponding to the regions I

, except at the origin and the point

 

r (3.5-26)

 

63
and

vfi) = nZo Aéi) Jn(plr) cos n0

véi) = ; 3:1) Jn(pzr) cos n9
n-O
B(i) = Z Céi) Jn(03r) sin n6

n=1

(1) _ V1(1) + v<1)
2

(1)
.. F0
(1)
B0

A<1)
n

 

B<1) = F(1)
n

C(1)}

 

1 (p + 1’132 - 4Q )
02 = é-(p - “P2 - 4Q )

U
5—:

II

N”

p = (l + S)!»2 - k

8(12 - w2)(k - w2)

.0
ll

 

- 2 _
oz 1 _ u s(w 12)

(01, 02) = (03.0%)[8(w2 - 12)]—

l

? (3.5—27)

 

3 ’

‘ 'l‘ .o'
a"

 

 

if": M

64

We remark that for fixed k, R, and s, the natural frequencies
for the clamped, simply supported, and free plates form a discrete

spectrum in the sense that

O < w :_w

1 :.w < ... < w :_ ... ; lim w = +w;

2 3 —- —- n

n + m

This is seen through an investigation of the Bil), i = 1,2,3, defined by

(3.5-11) through (3-5-16). and is a consequence of the fact that the
(l) B(2) (3)
n

, , tend to infinity as n tends toward
n n

first zeros of E , and E

infinity. Further discussion of the natural frequencies corresponding

to Case I appears in Chapter II.

Case II: of = 0% = 0

At the origin, (k,w2) = (0,0), Bii) and 8(1) are defined by

y
(3.4—14) and (3.4-15). In polar coordinates we obtain

 

 

r

(i) (i) (i)
(i) _ _ 3V 1 2 3V 1 3H
8 - [3r + 125 V 3r ] + r 36

(3.5—28)

 

 

r36 lZsr 80

8(1) = _ [3V(i) + 1 V2 3V(i) ] - 3H(i)
8 3r

where v(i) (i)

is given by (3.4-14) and H is given by (3.4-11) with

(3.4-16) defining 0%. Using (3.3—14) and (3.3-15) to define ax and my,

with r replaced by 2, we obtain the expressions

. 3S 3 “2 “ 1
ar(r,6,g) - 2n 3r [r log r]

1 (3.5—29)

 

._3S_3__“2 "
ae(r,e,€) an 36 [r log r]

 

a»?

‘tftmui—gn Ari-5mm ..fi

)4

a. ,y

65

The Green's functions, C(i), i = 1,2, are now uniquely determined

through the use of (3.5-25), with the Eéi), i = 1,2, defined by:

 

 

 

 

 

 

 

2 W
(1) [1 R ]
Eo -
0 -2R
F n n+2 j
R R . 0 (3. 5-30)
(1) g n-l n+1 n n-l '
En nR [nR + 33 (n + l) R ] Jn (03R) 1
I
n-l n+1 n n—l n 5
nR -[(n + 2)R + 33 (n + l) R ] ‘R Jn(p3R) E"
h .4
I
E(2) _ l R) W
a 0 -2(1 + u)
r 1 . 1
R“ : a"+2 ' u
n 5
5:2) - -n(n - 1)R"'2 :-[n(n 4» UR" + "—(lLfiU—Lll an'z] “[1" + 'L ,1 - ii]
' '- n R“ n
-(1 - u)n(n - 1)R"‘2 kl}. + ”(2(1) + 1) + n(l - y))Rn + (.1...:.);>.r1(.rj.:-.I_,>-(.n.:.,l). R"‘3:II (1 - HE J' - E‘- J]
I 55 ' R n R" n J
_ I .

By expanding det (Eéi)), i = 1,2, and using the properties of Bessel
functions, it is indeed seen that the above matrices are nonsingular;
hence, the static problem for the clamped and simply supported plates

in the absence of a foundation has a unique solution. The static problem
for the circular plate with free edges is not well posed in the absence

of a foundation. This is recognized mathematically by the vanishing of

det (Eg3)).

66

Summarizing, we obtain the following representations for the

G<1)

Green's functions, , i = 1,2, corresponding to the static problem

in the absence of a foundation:

 

 

 

 

 

0(1) = U(r,6;§) + v(i) (r,6;€) 1 = 1,2
where 'h]
T W 1‘; m
U = (ax,ay,u)
(1)T <1) (1) <1) 1.
= l)
v (BX 9 By 9 V ) E j]
3 a a .
ox = ax(x,y;€) =‘2; 3'5; [r2 log r]
= . = 3__ .3. “2 “
ay ay(X.y,€) 2“ 8 3y [r 108 r]
u = u(r 6°E) = - l—-lo E +-;— s 22 10 E
. . 2" g 2" g } (3.5-32)
v(i) = z [A(i) rn + B(i) rn+2] cos n6
n=0 n n
8(1) = 3H“) _ av(1) _ __1_ v2 3v(1)
X By 3x 123 EX
B(i) _ 3H(1) _ 8v(i) _ l V2 8v(i)
y 3x By 125 By
(1) °° (1)
H “:1 CD Jn(p3r) sin n8
24$
2 = _, __
O3 1 - u

 

67

and Aéi), Bfii) and Céi) are defined through the use of (3.5-25) with
Eéi), i = 1,2, given by (3.5-30) and (3.5-31), respectively.

Turning now to the point T, we again let or and a6 be defined
by (3.5-29). Using equations (3.4-31) through (3.4-37), we determine

the following expressions for 8:1) and Bgi):

8:1) = Z dii) (r) cos n6 ,
n=0

and ) (3.5—33)

 

(i) _
Be -

n

1 eéi) (r) sin n6 J

"MB

where

(i) _ n + 2 n+1 (1) 2s_ n + A n+3
dn _ 38(n + 1 r A + 2 (n + 2) r

 

+ (1 f u n - 2) rn+1']B§i) + nrn-l Céi)

 

l u
and (3.5-34)
(i) = _ 3sn n+1 (i) §§_ n n+3
en {n+1 r An + 2 (n+2) r

 

+ (i J: 3 (n + 2) + 2) rnfl] 3:1) + nrn-l Grin}

Applying the boundary conditions for the clamped plate, we determine the

(1)

Green's function, G . The vectors Fn’ n = 0, 1, 2, ..., are given by

(3.5'24) with the Bil), n = 0,1,2, ... , given by

‘1: 'r-fi

 

IFIf'fl’l __-

68

n+2

2

(3.5—35)

E(l) . lsn ,n4l

'3
i:
I
I
I
z:
s
+
w
+
a
A
=
+
N
V
”T..
l;+
‘m
w
4.
N
L._._J
x
a
+
3
x
:J
l
:

 

 

 

3
3
+
H
N
A
3 ,
+
N
V

3.8!"..15 “*3 l__+___y_ n4»!
2 ‘n i R + [nlI _ ) ~ 2] R

Realizing that the Bil), n = 0,1,2, ... , are nonsingular for all real R,

we conclude that when k = 12(1 + s), w = /12' is not an eigen frequency
for the clamped circular plate of arbitrary radius.

Upon determining Biz), n = 0,1,2, .. , it is noticed that Efz)
is singular for all real R, indicating that when k = 12(1 + s),
w = /12' is an eigen frequency for the simply supported plate, inde-
pendent of the radius of the plate. It is interesting to note that it
is the matrix Efz) that contributes mode shapes corresponding to pure

shear in the x-direction.

The matrices, Eé3), n = 0,1,2, ... , are given as

 

 

 

. ‘ ‘3 '1
5(3) . 68R 35R
° 63(u + 1) MR1 + (u + 1)(35R* - 2)
r. n + 2 n+1 n l : 39 n + a n+3 2n n+1 ' l —1
‘ ~ u-
38fn + l) R + nR ' 2" (8:3)R + ‘1‘“:j R : HR
1
I
(3) n . 3- + '3 +2 1 + I -.
En ' lsnR : 313“}: 2)R" + u(n + ”(I _ 31R”. u(n ~ HRH-h u I (3.5-36)
I I
l -’
[3sn(1 — u) + 65(1 + u)]Rn . ' u(n - 1)() u)R" ‘
1.. 9 I

where

r ' (gfix } [(u * l)n + (u + 3)] ‘ 3” [(l ‘ “)H * (V * 1"}""*2

0(1). ”(II .’)(x. O l)|\’n

".

 

o‘o‘fiA ...-n

"7’ In

69

The resonant radii can easily be determined from (3.5—36).

Let us assume 8 = .5 and u = .312. Table 11 gives the resonant

(3)

0 through E(3). The

radii for the point T that are contributed by E 20

data from the table suggests that the resonant radii contributed by

Eé3) tend to infinity as n tends toward infinity. The data also suggests
a certain interlacing of resonant radii. Letting Réi) denote the ith F
zero of det (Eé3)), we obtain the inequalities
(l) (2) (2)
Rn+1 < Rn < Rn+1 n :_1 (3.5-37)

 

[writ—_- 1

(3)

The resonant radii contributed by En

, n :_l, are the positive zeros

of a fourth degree polynomial, as can be seen by direct evaluation of
det (Eé3)). Since the behavior of this polynomial is rather insensitive
to small changes in s and u as n gets large, the above observations

hold for the values of s and u that are usually encountered in vibration
problems.

We again remind the reader that the resonant radii spoken of
above are dimensionless quantities equal to the radii of the plate
divided by the plate thickness. The reader should keep in mind, in
evaluating the data of Table II, that our plate theory is inaccurate
for radii less than five or ten times the thickness of the plate.

Summarizing, if we assume that R is not a resonant radius for

(l)

the clamped plate or the plate with free edges, the Green's functions G

(3)

and G , corresponding to the point T, have the following representa-

tions:

1) 1,3

G( = U(r,6;€) + v(i) (r,6;£) i

70

TABLE 2
Resonant Radii at T for the Plate with Free Edges

s = .5, p = .312

Resonant Radii

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E 0.93524
Ef37 0.00000 2.76186
3:3) 0.64106 5.05379
E§3) 1.07652 7.77046
R53) 1.49216 10.85583
Eg3) 1.89938 14.27075
3:3) 2.30207 17.98521
a§3) 2.70202 21.97771
Ré3) 3.10014 26.22768
£33) 3.49702 30.72440
Efg) 3.89306 35.44832
Bf?) 4.28837 40.39534
Rf?) 4.68321 45.55545
Rf?) 5.07766 50.88148
8:3) 5.47174 56.41859
Rf?) 5.86529 62.38499
Efg) 6.25917 68.02516
Bf?) 6.65253 74.13402
Efg) 7.04565 80.66260
Efg) 7.43899 86.87961
Egg) 7.83197 93.55298

 

 

‘lmfl.fl mam-u—

where

and

71

UT = (0x,0y.U)
(1)T (1) <1) (1)
V =(B ,8 ,v)
x y
3 A .
ax = ax(x,y;£) = %;-s-§; [r2 log r]
g . =_3_§_“2 “
my ay(x,y,€) 2" 8 3y [r log r]

u = u(r,6;£) = - %;~log E + %% E2 log r
v(i) = X [A§i) rn + Béi) rn+2] cos n0
n=0
. (i)
(i) _ (1) 3H
Bx - gx + By
B<1) g g(i) _ au(i’
y y 3x
gii) = Z f(i) (r) cos n0 + z D(i) rn cos n0
_ n _ n
n—0 n—1

g(i) = Z h(i) (r) sin n0 + Z D(i) rn sin n0
y n=1 “ n=1 “

1

 

 

(3.5—38)

 

1 [ft W'm-tmu -

(1)
D1
D01)

(1)
f0
f(1)

n
hm

1'1

1‘“

A<1)
n

Y(i)

C(i)

Ha)

Fu)

(1)

n

= 6sA(i) - 2B(i)
0 0

1 +
p____E

31 <1)-
A 1 _ u

n n-l

72

+ 2) 3(1) ,
n-

1 n 3_2

3 [38Af1) + 2 G%T§";) Bfi)] r21+-% s Bfi) r1+

(i) (1)
(1n + Yn )

 

 

(1)
Cu

= rn+2 + —~———*~—-rn+4 n >
4(n + l) 8(n + 2) ’ -
(i) _ (i) (i)
= (An Yn ) rn+2 _ Ln rn+4 n >
4(n + 1) 8(n + 2) ’ -
= 243B(i)
0
= 123 B(i) , n > 2
n-l -

- (i)
— 123(n + 1) An+1 +

= 123(n + 2) Béii

(1)
= i) = E
.0 [o J
Au)

C<1)
n

4 f%€;—E)(n + l)(n + 2) Béii

, n :_l

I(1)
0

-1
_ n a (i) (i) >
_ ( 3(1)) [En J In n __1
n

n)

l

 

(3.5-39)

Ii TILE}. rd|Aftr m d““.‘ wu- 1.1..1

Case III: p2

Consider first the line w2 = 12.

73

= 0, p1# 0

Along this line

 

 

 

 

331 3 (l)
= -— + '—
ar 2 3r [H0 (plr) in log r 1
‘31
si 8 (l) ‘
a =-——~-—* H + ——-
6 2 86 [ 0 (plr) in log r] {-
r01 .
. . a
'3
and '
9(1) _ ; 1:13.”--.(k _ 12 + n )A<1) _ 126 B(1) . n-l .(l)
l’ ”-4) 2(n + I) 3 V] n UT n Jn(“lr) + I" Ln ('03 n"
(3.5-41)
"- n+1
Pf” - n2] F2717??? “ - ,2 -9122) 0') AS” * {13:3 ”1:”J11("Ir) ' "tn—l (19)]7‘" "H
I

J

We analyze each of the three sets of boundary conditions separately.

Clamped Plate

Using

(3.4-43) and (3.5-41) and applying the boundary

conditions, we obtain

(1)
so -

E(1) ;
n

 

 

 

2 0 1
pl
n I n '-
R : Jn(le) : 0
Rn+1 (n + 2) : lan | n 1
- - , -
2(n + 1) (k ' 12 ' 2 1) ' 2 Jn( 1R) ank
I le ,
n+1 ' a
R _ g a -123 . n-l
2(n + 1) (k 12 + 2 “1) . 2 Jn (91“) :“R
01 4
(3.5-42)

74

Expanding det (E51)) and restricting k so that O < k < 12(s + 1) it is

(l)

0 contributes an infinite number of resonant radii

easily seen that E

with no finite accumulation point. For the same values of k, the

(1)

resonant radii contributed by En

, n :_l, are the roots of the

transcendental equations

 

J (D R) v
n+1 1 1 R
Jn(le) - pl(k - 12 - 2 ) n + l - O n :_1 (3.5 43)
Jn+1 (01R) 91 v1
Noting that the slope of Jn(le) is larger than ail—I (k — 12 - §—)
when R = 0, we are able to determine the behavior of the resonant radii

contributed by each Bil)

(1)

n

through an investigation of Figure II. It
is seen that each E , n :_l contributes an infinite number of
resonant radii, and since the first zero of Jn+l(le) tends to infinity
as n tends to infinity, the total collection of resonant radii for the

clamped plate, corresponding to the segment of the line w2 = 12 that

lies to the left of T, has no finite accumulation point.

For the segment of the line, wz = 12, lying to the right of T,
it is understood that the 01 in (3.5-42) must be replaced by p2, since
pl = 0 and p2 # 0 along this segment. The equation (3.5-43) is now

replaced by

 

'L 'b
In+1 (92R) 02 91 R 44
10,11) "126(k‘12”2—)m=° “:1 (3'5"
n 2
where p2 = 132. It is seen through the use of Figure III that along

the above mentioned segment there are no resonant radii for the clamped

plate.

ERNIE-rm“;

75

14:4?

IL

 

"H uafiom
mnu mo umoq mnu ou AHWM mo oumN umuwm mo aowufimom "N muswfim
. cm mo oumn uma msu ma .
AHV
am new AquVcw mo oumu Low mcu mH ch
AHV AHV

c- H+c :

3”. 3“ 3h

    

. Ao.ov

 

 

 

76

+cH
WI;

II
II
n

w
5-2-éiw

H

80

was

t.

N
c as

G
mvH

a
H+ H

cmmsumm magmaowumaom

"m muawfim

 

 

 

 

 

 

Ao.ov

77

Simply Supported Plate
Applying the boundary conditions for the simply supported plate

we obtain

 

n I
I R . J(HR) I 0
I " I
I I
.n) n n 'lhn 1 1 ' 7
L - J R "°-~v" *" - -— ' I - "__‘h
n 4 I : “1 R" Jn(DlR) R .In(I IR) :n(n I) R
l I
1 "V1 n ' 12' 7 -
-( +101-12Hn-1. ., I--.” “1‘. . .2 ~. - ' - "‘3
2[U ) ) 2 JR .“7 R! JnII 1“) R v’nII IR) Jn(I lR) ' (I Iu)n(n - ”R
I I

 

 

_ ‘ J j
(3.5-45)

(2) is

Analyzing (3.5-45) it is immediately seen that El singular

indicating that w = #12 is an eigen frequency for the simply supported

(2)

1 that contributes mode shapes

plate. As with the point T, it is E
corresponding to pure shear in the x direction.

Free Plate

 

Applying the boundary conditions for the free plate along the

line m2 = 12, we obtain

 

 

 

 

 

(k - 12) 123 _
— R (1 - J,' R
(3) 2 “—7“p1 ) «'(“I )
EU I
1L2). _ '123 u u .
2 (k 12) 71‘ (J0({.1R)+ Il- J0(r)lR))
P ' ‘
n+1 n n l I 12.1 I
-.__._. _ ,_ - - _ ‘_ o H"
2’“ ‘_ 1) (k 12 + 2 3’) + “R I (l {.T) JOOIXK) “R
I ( R)
.(3) n n 12 " VI J. _.,
I.‘n - 1. '1R I ’g’nl‘Pi-b‘” ‘ _n_ (I‘ R) n(n - |)Rn ‘
5' - R I
I I R
1 ( + l)(k - 12) + (I ~ ) “I! R" I ‘2“ En? “ . u "-1
2 U U 2 | ET “I JHOIIR) ‘ l‘l .I"(IIIR) — .In(I IR) "(n - |)(| - )3)“

(3.5-46)

 

”IF"

78
Analyzing (3.5-46), it can be seen that a discrete spectrum of resonant
radii exists for each R > 0 along the line w2 8 12.

Considering the line w2 = k, we obtain

 

 

 

 

 

 

 

— 1
, R
5(1) . 1 .100l )
o O -J'(o R)
o I
L.
I I 1
F R“ I J (o R) | o
I n l I
(I) 1 ' ' 1
. - n" "E I ““
En (o 1) all 1 R Jn(olR) ' n R
-1 ' I
(o - 1)n a“ I ~J' (o a) I n n““
I n l '
b ' A
5(2) - I J0(olR)
0 n E I
0 -[Jo("IR) + R Jo(ulfl)]
F “ : I O T
R ' Jn(DlR) |
' l
- -2
5:2) . u(n - l)(o - 1) R“ 2 : g[% Jn(olR) - J;(9'R)] : n(n - l)Rn
n-2 ' I n-2
n(n - l)(o - l)(l - u) R I uof Jn(an) - (1 - u)J;(ulR) . u(n - I)(I - u)R
L I I _
0
(3)
80 I 0 JH( R) + E J'( R
0 0‘ R 0 ol )
I- -1 l I -1 q
noRn I 0 : nu“
I l
(3) _ - “‘2 ' 2 l - . l -. “-2
En - n(n l)(0 l) R ' R [It Jaw!!!) Jn(olR)] : Mn 1) R
l
n-2 I 2 .. ' n-2
n(n - l)(o - l)(l - u) R ’ uol Jn(olR) - (1 - u)Jn(oxR) : u(n -1)(1- U) R
. .1

(3.5-47)

. .1 '(v 1.5,
."
.

 

79

where 0 is given by (3-4'53) and pi = swz = sk. Investigation of Bil)
and Biz), n :_0 yields the existence of a discrete spectrum of natural

frequencies along w2 = k for the clamped and simply supported plates.

(3)

0 that w2 . k is a natural frequency for

It is clear by the form of E
the free plate for all k > 0, independent of the radius of the plate.
(3)

It is E0 that contributes fundamental mode shapes corresponding to

rigid body motion.

’ "Z-A I- 62.; (fl

Summarizing, if we assume R is not a resonant radius along

w2 = 12, excluding T, the Green's functions become

 

 

 

 

 

6(1): u(r,e;g> + v”) (r.e;€) i = 1. 2. 3
where 1
UT —
‘ (axaays u)
T
(i) _ (i) (i) (i)
V ‘ (BX 9 By , V )
_ 331 a ' (1) a 2 C
ox — E?—-5;- H0 (plr) + I;-log r
_ 351 A. ' (1) a _2__ a”
ay — 3¥— 3y H0 (plr) + in log r
u = %-Hé1)(plf)
v(i) = n20 [A§i) rn + 3:1) Jn(plr)] cos n6 }(3.5-48)
(i)
(i) _ “(1) 33
8x - 8x + 3y
(i)
(i) _ ”(1) _ 33
8y - By 3x
g(i) = X f(i) (r) cos n6 + z D(i) rn cos n6
x n _ n
n=0 n-l
g(i) = Z [h(i) (r) + D(i) rn] sin n6
Y _ n n
n-l J

 

80

and

B(i) = 2 C(1) rn sin n6

n=l n
(i) _ (1) '1 (1)

F0 — [E0 ] Io (3.5-49)

F(1) = [B(i)]'1 I(i) n > 1

n n n -
and the féi), hi1), Dii), and the Eéi) are given by (3.4—44), (3.4—45),

(3.4-46), (3.5—42), (3.5-45), and (3.5-46).

”(1) ”(i) (i)
Bx , By

n

Replacing the expressions for , and the E in

(3.5-48).by (3.4-54), (3.4-55), and (3.5-47), respectively, we obtain

the Green's functions for the clamped and simply supported plates along

the line w2 = R.

Case IV: p2 = p2 # 0.

Finally we consider the parabola 02 = 4q. Along this parabola

 

 

Bi“ = 50.2 .1. 12) .2. “A? you» + mum + Maw» ‘
- 20f J;(01r)] Béi)} cos n6 + nil g-céi) Jn(o3r) cos n8
and I(3.s—50)
Béi) = _n:l{ rs(w: _ 12) [AA§1) Jn(°1r) + (Ar J$(91r)
-2pf Jn(plr))Béi)] + Céi) J; (93r)} sin n6 J

 

tJhere A = 02- s(w2 - 12), p2 = [(1 + S)w2 - k]/2, and
1 1

 

:3; = 1 E u [80»2 - 12)]. Using {3.5—50), the Bil), i = 1,2,3, are

determined as

 

81

 

 

 

I
JO(D|R) : R J;(le)
AJf(le) I ’ 20f Jg(ulR)
..xr.-___.._ I [J;(le) + a J"(le)l - ,
s(w’ - 12) I s(w’ -12) ” s(s’ - 12)
l
I
JOIIIR) : R J3(o,R)
Eu)_ I
f) ' .‘,'
--~--i-~—- -(--“ ‘ ” J;I. lIII - ..‘fJUII.|I<I . —-——---- (2.9; - m, + I).-1'I.I”(IIIIII - (\y‘l‘fl - 2II - I.) §‘--I.I;\(.~,RI
ah’-1D ‘ Ism’-In
I
v I
of I
-—~— J' R - »2 - .9 ' .
30.7 -12) 0(91 ) : (A + l)(clR Jo(le) + 2I IJ00 IR))
(3) I
E0 - I v?
A (p -1) J'(o R) - on (o R) '-—l—" (20“ - HI: + l)I=") J (v R) ' (INK - ?(| - I») *1) ."(I‘ R)
s(w’ - 12) R 0 ‘ ’ ° ' : s(m? - 12) ' ' " l ‘ R “ 1
(1) (1) (1)
all all dII
8:1) - 3;?) ail) 35:) l - 1.2.3
(I) (I) (I)
a}! "II “I!
(3.5-51)
where
(i)
a — J R i = l 2
11 u(pl ) ’ 9
2i
p
(3) 1
a1 1 = Jr'1(olR)
s(w2 - 12)
(i)
- Fl ' I! ' ==
(3) n2 2 2
a A + l —-— R J R - 2 J' R
12 ( )(R 01) n(01) 0111(01 )
(1)
a3]. = 0 ’ i = 192
(3) n
= —-J R
a31 R n(D3 )

 

‘

 

82

nAJ (p R)
as? n I

 

 

 

 

 

 

 

Rs(w2 - 12)
J'(D R) J (p R)
3:1) = nA n R1 _ n Rl ’ i = 2,3
s(w2 - 12)
202

(1) 1 n
a = [XJ'(o R) - ———J (p R)]

22 n 1 R n 1 8(w2 _ 12)

J (o R) J'(p R)

85:) = 20 n[-B-;;L-* - iii-L] + An J;(01R) , i = 2,3

,(1) = I
323 Jn (p3R)

2 I

(i) _ n2 03 JnG33R) _
823 - (‘R—z- 2—)J (0310- E- , 1 - 2,3

(l) A I
a = J (o R)

13 s(wz - 12) n 1

(I)= 1 A _ _rfi_2 _<_1.:_IJ_2.. 1=2
a13 s(wz _ 12) [( <1 10R2 o1> JnolR) A R Jump]. ,3

(1) = 1 I II _ 2 I
832 s(wz _ 12) [K(Jn(olR) + RJn(olR)) ZolJn(olR)]
3(1) = [20'+ - (l - I1) £13 - (Mu + 1) +233le J (o R)

32 1 R2 R2 l n 1

n2 2o?
+ [K(f-ofR)+ (l-u)-R—] Jr'l (01R) . i= 2.3

egg) = E Jn(p3R)

agé) = (1 ‘ U) % (Jn(03R) - R Jn(p3R)) ’ 1

2,3

 

I
I
y
‘l;‘? Ales .

83

(1)

Due to the complicated components of En

, i = 1,2,3, an investigation
of the resonant radii along the parabola is best done with the aid of
a digital computer for a specific vibration problem, the results of
course depending on the values of k, w?, and s.

In summary, for fixed k, wz, s, and R, not equal to a resonant

radius, the Green's functions for the clamped, simply supported, and

free plates corresponding to the parabola, p2 - 4q, become

C(i) = U(r,6;€) + V(i) (r,9;€) i = 1,2,3
where T
T
U = (ax.ay.U)
(UT (1) <1) (I)
V = (B I B I V )
X y
a=——“12 L80A”) ‘552
X 12 _ (1)2 3X 2 f‘ (3. - )
_. 12 _a_ A
“y _ 12 - wz 3y 82(r)

01r (1)

gl<fr> =-}; mg“ (01;) - —2— H1 (91%)]

 

A —i s(12 — wz) . (l) A
82(r)- §[ 91 rH1 (olr)]

 

 

84

and where

 

 

 

 

T
i m i i 2
v( ) = nEO [As ) Jn(olr) + Bi ) r-g; Jn(plr)] cos n0
8(i) _ 38(1) an'i)
x - 3x 3y
(i) (i)
(i) _ 88 an
By ' 3y ' ax ? (3.5-53)
3
8(1) = l [(128 + w2 - k) v(i) + V2v(i)] E
s(w2 - 12) E
J _I
I:

The Aéi) and Béi) are determined from equations (3.5-17) through

(3.5-20) with the Eéi) defined by (3.5—51) .
3.6 Remarks and Conclusions

In a physical problem, the point load of sections 3.2 and 3.3
is replaced by a load, f(x,y), that is continuously distributed over
some region A1 :_A, the region occupied by the plate. Consider the
vibration problems posed in section 3.2 for the clamped, simply
supported, and free circular plates, where now the Dirac—delta function
is replaced by the continuously distributed load, f(x,y). For the
time being assume f(x,y) has continuous partials up to order four in A,
and assume f(x,y) vanishes outside A1 < A; i.e., assume
f(x,y) e C3(A1). The Green's functions G(i), i = 1,2,3, established
in section 3.5 are used in the following way to solve the above

mentioned vibration problems.

85
(i)

We first move the singularity contributed by G to an

arbitrary point (€,n) interior to the region A. It is remembered that

(i)

, as constructed in section 3.3, is along the

0(1)

the singularity of G
x-axis. With its singularity at (&,n). dvlorminvs doIlvvlluns
due to a point load at (£,n). Assume (§,n) has the polar representation
(p,¢), then these deflections contributed by C(i) are symmetric with f‘
respect to the ray 6 = ¢. This symmetry indicates that the components

ox(r), ay(r), and u(r) of the particular solution, U, determined in

 

section 3.3, should be altered only by replacing E = ’Qx - C)2 + y2

 

 

by /Qx - g)2 + (y - n)2 , thereby causing ax = nx(x,y;g,n),

{5}.”

my = oy(x,y;€,n) and u = u(x,y;£,n). Equivalently, we obtain the
expressions
= -( ...
ax ax(r.o.’ ¢)
ay = ay(r.o;6 - ¢)
U = u(r,p;6 - ¢>

in polar coordinates. The components, 8:1) and le), of v(') are now

considered even and odd functions of 6 - ¢, respectively, while the

v(l), of v(l) is considered to be even in 6 — d. As such,

(1) B(1) (1)
x ’ y

third component,

we assume the Fourier series expansions of B and v in 8 - ¢.

This amounts to replacing cos n6 and sin n6 in the expressions established

in section 3.4 for 3(1), 8(1) (1)

x by cos [n(0 - ¢)] and

and v
sin [n(6 - ¢)], respectively. It is clear that the frequency equations

established in section 3.5 are not and should not be effected by such
(1)

a replacement. With G , i = 1,2,3, altered in the above way,

86

w(i) (XIYIt) = e—iwt I 1' E(€,n)G(i)(XIY;€m) dg dn 1 = 1:293
A

represents the solutions to our vibration problems for the clamped,
simply supported, and free plates, respectively. It is remarked that
the above solutions are in dimensionless form and are subject to the
changes implied by (1.4).

We remark that it is not necessary for f(x,y) e C3(A1). Indeed,
if we require f(x,y) to be continuous in A1 with continuous partial
derivatives up to order four in Al, the above results still follow
provided we do not require the solution to satisfy the system, (3.2-l),
along the boundary of A'. It is clear that the solution is valid in A1
and in A - A1. By the continuity of the solution across the boundary of
A1, it must be a valid solution to the physical problem for all (x,y)
in A.

The Green's functions along the various curves in the frequency-
stiffness plane could have been obtained by taking limits of the Green's
functions established for regions I through VII of Figure 1. It was
easier, however, to consider each curve separately, establishing the
plate displacement equations along these curves, and solving the resulting
equations.

As remarked earlier, the procedures of this chapter can be
extended to include five additional natural boundary conditions dis—
cussed by W. R. Callahan. Our procedures discussed in sections 3.3
through 3.5 also allow us to solve the corresponding dynamic and

static problems for the annular plate with or without a foundation.

 

87
It is interesting to observe how the results of this Chapter
are modified with the introduction of the torsional springs of

Chapter II. The system (3.3-l) becomes

 

m m
l. 2 .QQ. 2 _ _ £22 EEK =
2 El. u)V ax + (l + u) 3x]'+ s(w 12)0Ix 123 8x + C By 0
1 33 au BI Ii
—- - 2 +- .+ -—- +- 2 - - -—- - -—— = .
2 El u)V my (1 u) 3y s(w l2)oy 128 By C 3X 0 g
E
2 2 m I
v u + (w - k) u + ¢ = -6(x.y) i
(3.6—1) 3
m Sax 3a m Bax 8a 5
where ¢ = ———-+-——1-and W = -——-- __Z. . Since the function .1
3x 3y 3y 3x
a
X ’\.I
U = ay of section 3.3 was constructed so that W = 0, we see that it
u

is a particular solution of (3.6-1) as well as (3.3-l). Investigating

the general solution of (3.6-1) with the right hand side set equal to
(1)

takes on the same form as it did in

(i)

X

zero, it is seen that V

section 3.4. As in section 3.4, the plate displacements B and

851) involve a potential function B(i) giving rise to rotation or

twist about the normal to the plane of the plate. It is in the

(i) (i)

that Bx and 8(1) for (3.6-1) differ from the
(1)

Y
still remains a solution of (3.4-3)

determination of H

(i) (1)
Bx and By for (3.3-l). H

provided we replace the pi defined by (3.4-4) by
2 25(w2 - 12)

p3 = 1 - u + 2c (3°6'2)

 

Hence, the introduction of torsional springs does not complicate the

(1)

problem of determining the Green's functions G , i = 1, 2, 3, for the

vibrating plate. In fact, the frequency equations developed in this

 

 

88

Chapter remain valid provided we define Q; by (3.6-2). Upon
investigation of the results of this Chapter, it is easily seen that
the eigen functions and eigen values for circular plates with torsional
springs attached are analytic in the torsional spring constant C.

In conclusion, considering a finite plate, we are forced to
consider the system (3.2-l) determined by Mindlin [25]. In this Chapter
the Green's function for this system corresponding to a circular region
A, with three types of boundary conditions imposed, was determined.
Particular attention was paid to certain curves in the frequency-

stiffness plane that were discussed by D. Yen [45]. Following the method

It. '. AIL“: m. ninja—“q—P
I

of D. Yen, namely, Fourier transforms, we were able to obtain a
particular solution, U, of the time reduced system (3.2-2) with the
result that the first component of U, representing transverse deflection,
contains the correct logarithmic singularity of the improved plate
theory. The singularities of the ox and ay components of U are deter-
mined for the first time. It is seen that the singularities of ax and

ay under the point load at (g,n) are of the form

 

c1<x — a) log /Q¥ - g)2 + (y - n)2

and

 

C2(y - n) 10g /Qx - €)2 + (y - n)2

respectively, where C1 and C2 are constants. These singularities are
the same as those appearing in the ax and ay components of the classical
plate theory. The regular part of the solution to the system, (3.2-2),
was obtained using Fourier Series techniques, and an investigation of

the frequency equations along various curves in the frequency-stiffness

89

plane was possible. The static problem for the clamped and simply
supported plates was solved within the realm of the improved plate
theory as a limiting case of the dynamic problem. Although this static
problem was solved in the absence of a foundation, the effect of a
foundation may be included and the solution obtained by letting w tend
to zero in the solution (3.5-26) of section 3.5. The methods of
sections 3.3 through 3.5 apply to annular plates with suitable edge

conditions on each boundary.

 

. III}

h I.
FHuCuWZIflI L511}. $1.- .. ~

BIBLIOGRAPHY

10.

ll.

BIBLIOGRAPHY

Ake Pleijel, "0n the Eigenvalues and Eigenfunctions of Elastic
Plates," Comm. Pure, Appl. Math., Vol. 3, No. 1, (1950),
pp. 1-10.

Browder, F. E., "Dirichlet Problem for Linear Elliptic Equations
of Arbitrary Even Order with Variable Coefficients," Proc.
Nat. Acad. Sci. U.S.A., Vol. 38, (1952), pp. 230-235.

Callahan, W. R., "On the Flexural Vibrations of Circular and
Elliptical Plates," Q. Appl. Math., Vol. 13, No. 4,
pp. 371-380, (1956).

Courant and Hilbert, "Methods of Mathematical Physics,"
Interscience, Vol. 1 and 2.

Courant, R., "fiber die Schwingungen Einges Pannter Flatten,"
Mathematische Zeitschrift, Vol. 15, (1922), pp. 195-200.

Deresiewicz, H. and Mindlin, R. D., "Axially Symmetric Flexural
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