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Michigan State
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This is to certify that the
dissertation entitled

THE FUNDAMENTAL FREQUENCIES OF PLATES WITH A
CORE

presented by
HUSEYIN YUCE

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in _ MATHEMATICS

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The Fundamental Frequencies of Plates With a Core
By

Hiiseyin Yiice

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

2005

ABSTRACT

The Fundamental Frequencies of Plates With a Core
By

Hiiseyin Yiice

A boundary perturbation method is developed to extract the fundamental eigen-
value of the biharmonic boundary value problem. The method then applied to wavy
boundary, elliptical, and polygonal plates with clamped and simply supported bound-
ary conditions. Approximate analytical formulations of the fundamental frequency
for such plates with and without clamped core are obtained. A numerical pb-2 Ritz

method is used for comparison.

ACKNOWLEDGMENTS

I am very grateful to have known Prof. C.Y. Wang both as a person and as a
mathematician. It has been a privilege to work with him, and this thesis would have
not been possible without his guidance. I would like to thank my co-advisor Prof. C.
Wei for his help and support in the numerical part of this thesis. I would also like to

thank Prof. M. Miklavéic’: for his support and helpful discussions.

iii

In memory of my father, Halil

iv

TABLE OF CONTENTS

LIST OF TABLES .............. q ................... vii
LIST OF FIGURES .................... ‘ .' .......... ix
1 Introduction .......... ' ........................ 1
2 Fundamental. hequency of a Circular Plate with a Core 6
2.1 Formulation ........... , ....................... 6
2.2 Free - Clamped Boundary Condition ..................... 9
2.3 Clamped - Clamped Boundary Condition .‘ ................. 15
2.4 Simply Supported - Clamped Boundary Condition ............. 19
3 Fundamental Frequencies of Plates .................... 23
3.1 Perturbed Boundary: General Case ..................... 23
3.1.1 Clamped Boundary Condition ....................... 24
3.1.2 Simply Supported Boundary Condition .................. 26
3.2 Wavy Boundary Plates ............................ 27
3.2.1 Clamped Boundary Condition ....................... 28
_.3 2. 2 Simply Supported Boundary Condition .................. 32
3. 3 Elliptic Plates ............. , ................. '. . . 36
3.3.1 Clamped Boundary Condition ....................... 39
3.3.2 Simply Supported Boundary Condition . . .1 ..... ” .......... 43
3.4 Polygonal Plates ............................... 47
3.4.1 Clamped Boundary Condition ....................... 48 '
3.4.2 Simply Supported Boundary Condition .................. 53
4 Fundamental Frequencies of Plates With a Core ------------ 58
4.1 Wavy Boundary Plates With a Core .......... ' .......... 58
4.1.1 Clamped Boundary Condition ....................... 59
4.1.2 Simply Supported Boundary Condition .................. 65
4.2 Elliptic Plates With a Core ......................... 70
4.2.1 Clamped Boundary Condition ....................... 71
4.2.2 Simply Supported Boundary Condition .................. 78
4.3 Polygonal Plates With a Core ........................ 83
4.3.1 Clamped Boundary Condition ....................... 84
4.3.2 Simply Supported Boundary Condition .................. 89

V

5 The Rayleigh.Rjtz Method ......................... 94

5.1 Circular Plates With a Core ......................... 95
5.1.1 Clamped Boundary Condition ....................... 96
5.1.2 Simply Supported Boundary Condition .................. 97
5.2 Polygonal Plates With a Core ........................ 99
5.2.1' Clamped Boundary Condition ....................... 101
5.2.2 Simply Supported Boundary Condition ................. . . 103
6 Conclusions and Discussions ........................ 105
A Perturbed Boundary Condition Coefficients ........... l. . . 109
B Plate Coemcients . . . . ............................ 111
B] Coefficients of the Clamped Plate With a Core ............... 111
B.2 Coeficients of the Simply Supported Plate With a Core ......... 113
BIBLIOGRAPHY 117

vi

LIST OF TABLES

2.1 The fundamental frequencies of free circular plate with a core ......
2.2 Fundamental frequency of free edge plate with a core ...........
2.3 Fundamental frequency of clamped plate with a core ...........
2.4 Fundamental frequency of simply-supported circular plate with a core . .

3.1 Fundamental frequency of clamped wavy plate ...............
3.2 Fundamental frequency of simply-supported wavy plate ..........
3.3 Fundamental frequency of clamped elliptic plate ..............
3.4 Fundamental frequency of simply supported elliptic plate (12 = 0.3) . . .
3.5 Fundamental frequency of regular clamped hexagon ............
3.6 Fundamental frequency of regular clamped hexagon ............
3.7 Fundamental frequency of regular simply-supported hexagon .......
3.8 Fundamental frequency of regular simply-supported hexagon .......

4.1 Fundamental Frequency of clamped wavy plate with a core ........
4.2 Fundamental frequency of simply supported wavy plate with a core . . .
4.3 Fundamental frequency of clamped elliptic plate with a core .......
4.4 Fundamental frequency of clamped elliptic plate with a small core . . . .
4.5 Fundamental frequency of clamped elliptic plate with a core (having the

same area as the annular plate) .....................

4.6 Fundamental frequency of simply supported elliptic plate with a core . .

4.7 Fundamental frequency of simply-supported elliptic plate with a small core

4.8 Phndamental frequency of clamped hexagonal plate with a core .....
4.9 Fundamental frequency of clamped hexagonal plate with a small core . .
4.10 Fundamental frequency of clamped hexagonal plate with a core .....
4.11 Fundamental frequency of simply supported hexagonal plate with a core

4.12 Fundamental frequency of simply supported hexagonal plate with a core
(with the same area as the annular plate)

5.1 Natural frequencies of clamped circular plate ................
5.2 Fundamental frequencies of clamped circular plate with a core ......
5.3 Natural frequencies of simply supported circular plate ...........

5.4 Fundamental frequencies of simply supported circular plate with a core .
5.5 Natural frequencies of clamped hexagonal plate ..............

5.6 Fundamental frequencies of clamped hexagonal plate with a core .....

vii

15
18
22

32
36
43
47
52
52
57
57

65
70
75
76

77
82
82
87
88
88
92

93
97

97
98
99
102
103

5.7 Natural frequencies of simply supported hexagonal plate ......... 103
5.8 Fundamental frequencies of simply supported hexagonal plate with a core 104

viii

2.1
2.2
2.3
2.4

LIST OF FIGURES

Free edge circular plate with a core .....................
Free circular plate with a core: Asymptotic vs. Numerical Frequency
Clamped circular plate with a core: Asymptotic vs. Numerical Frequency

Simply-supported circular plate with a core: Asymptotic vs. Numerical
Frequency .................................

ix

14
19

22

CHAPTER 1

Introduction

Transverse vibration of plates has extensive applications in civil, mechanical,
aerospace, and material engineering as well as vibration of piezoelectric and acoustic
devices. The governing equation of a vibrating plate in Cartesian coordinate system

is first given by Sophie Germain [12]
mm + pa. = 0 (M)

where w = w(:c, y, t) is the vertical displacement and

Eh3

D = 12(1 - V2)

D: flexural rigidity, E: Young modulus, u: Poisson ratio, h: half-thickness, p:
density.

The biharmonic equation (1.1) reduces to
V4W — k4W = 0, (1.2)

where k4 = pa)2 / D, W is the transverse displacement, and w is the natural frequency,
assuming that the modes are simple harmonic (i.e., the solution is assumed to be
periodic), w(:r, y, t) = W(a:, y)e“‘". The fundamental frequency, which is the smallest

eigenvalue of the equation (1.2), plays a crucial role when designing plates in applied

sciences, since there is no displacement before the fundamental frequency, in other
words, vibration starts at the fundamental frequency.
The biharmonic eigenvalue problem has a general analytical solution in a circular

domain [8]

W(7‘, 6)

M8

[Aan(kr) + B..Y.,(kr) + 0.141”) + D..K,.(kr)] cos n6

3
II
G

M8

[A;J..(kr) + B;Y,.(kr) + C;In(kr) + D;K..(kr)] sin n0

3
II
H

(1.3)

where the coefficients A", B", C", D", A; B;, 0;, D;, which determine the mode
shapes, are solved using the given boundary conditions, and Jn, Ya, In, K7, are the
Bessel functions. However, the difficulty in finding solutions arises when the domain
is no longer circular. For rectangular domains Navier’s double series solution and
Levy’s single series solution for certain boundary conditions are known [8].

The lack of analytical solutions for several other domains with various boundary
conditions led many researchers to use numerical methods. Galerkin’s method [7],
Rayleigh-Ritz method [9], [21], [10], [17], the finite element method [2], and the
collocation method (a special case of Galerkin’s method) are among such numerical
methods. However, in some cases, the numerical methods often encounter the problem
of singularity, scaling, and sensitivity to the boundary conditions. This leads us to a
special formulation of perturbation theory to improve accuracy and reliability of the
fundamental frequency.

The purpose of the present work is to provide approximate analytical formulation of
the fundamental frequency for clamped and simply supported plates of various shapes.
We adopt a boundary perturbation method to extract the fundamental eigenvalue of

the problem.

In Chapter 2, we consider the circular domain with a concentric circular core.

Free, clamped, and simply supported outer boundary conditions are considered. Since
the general analytic solution to the problem is available (1.3), we have adopted an
asymptotic expansion [26] of the solution for “small” argument to obtain analytical
approximate solution to the fundamental frequency. A similar argument has been
done for polygonal membranes [26] and membranes with arbitrary shapes [27]. In the
free boundary case, sensitivity of the frequency to the radius of the core is presented
and it is shown that the frequency rises very rapidly as the radius of the clamping
circle increases from zero. We improve the result of Southwell [22]. The frequency
values are simulated and tabulated for both asymptotic and numerical solutions.

In Chapter 3, we develop a boundary perturbation method in its general case
to extract the fundamental eigenvalue of the problem for wavy boundary, elliptical,
and polygonal plates with clamped and simply-supported boundary conditions. The
method is the generalization of work by C.Y. Wang [25] for the Helmholtz equation on
circular and polygonal waveguides. Most of the papers on the vibration of elliptical
and polygonal plates present numerical solutions and many authors use the Rayleigh-
Ritz method [9], [21], [10], [17], [13], [18]. In the case of clamped elliptical plates,
Parnes [15] used a perturbation scheme based on perturbing the boundary of a circle
circumscribing the plate. For a better estimate, we apply the boundary perturbation
method to the “averaging circle” for both clamped and simply-supported boundary
conditions. An analytical method for free vibration of fully clamped polygonal plate
was developed by Iris et a1. [3]. In the case of simply-supported boundary condition,
the dynamical analogy between plates and membranes has been used by Iris et a1. [5]

to simplify the problem to
V2W + k2W = 0 (1.4)

where k4 = ,0ch2 / D, since the equation V2W — k2W = 0 produces imaginary frequen-

cies. However, this analogy and all the numerical results use the simply-supported

boundary conditions to be
W = VZW = 0, (1.5)

taking the radius of curvature R —-) oo in the moment M (W)

2
a W + 16W + V__a?w = o. (1.6)

MW): (1.2 ”Ea—n 3.2

 

Here we shall note that the tangential derivatives 8—5;”; in the moment (1.6) vanish,
since there is no tangential deflection on the boundary of the plate. By taking (1.6), we
also consider the contribution from the Poisson ratio V and we adopted the boundary

perturbation method for the “averaging circle”. The boundary of the plates are

perturbed by
r=1+ef(6l), (1'7)

where f < 1 and c are the separation variable of orders. The averaging radius of the
circle is taken to be 1. The plates are perturbed about the circular plate of radius
1 to obtain wavy boundary, elliptical, and polygonal plates. Corresponding radii of
inscribing circles of polygonal plates, perturbed about the circular plate of radius 1,
are tabulated in [24]. Since the frequencies of elliptical and polygonal plates with
clamped and simply-supported boundary conditions are available in the literature,
we have tabulated and compared the results in Chapter 3. For the wavy boundary
plates, there are no published results for comparison.

In Chapter 4, we have adopted the boundary perturbation method of Chapter 3 to
extract the fundamental eigenvalue of the problem for wavy boundary, elliptical, and
polygonal plates with a concentric circular core in the case of clamped and simply-
supported outer boundary conditions. There are no published results for this problem
except in the case of circular plate (discussed above). The fundamental frequencies

of wavy boundary, elliptical, and polygonal plates with a concentric circular core in

the case of clamped and simply-supported outer boundary conditions are tabulated
for various radii of the core.

In Chapter 5, we use the pb-2 Ritz method to calculate the fundamental frequencies
of circular, elliptical, and polygonal plates with and without a core. The frequency

values have been tabulated for each case.

CHAPTER 2

Fundamental Frequency of a

Circular Plate with a Core

The biharmonic eigenvalue problem has a general analytical solution in a circular
and annular domain [8]. The free circular plate with circular clamped core problem
has been investigated, however as the core shrinks down to 0, the singularity causes
problems due to rapid increase in SIOpe of the fundamental frequency and asymptotic
approximations to this problem have not been investigated. This leads us to a special
formulation of perturbation theory to improve accuracy and reliability of the funda-
mental frequency. Besides free outer boundary condition, we also consider clamped
and simply supported outer boundary conditions. We start with the formulation of

the problem.

2.1 Formulation

Consider a circular plate with a concentric circular core. The PDE which governs the

undamped free transverse vibration of a circular plate in polar coordinates is given

by

Dv4w + pwtt __._ 0 (2.1)

where w = w(r, 6,11) is the vertical displacement and

Eh3

D : 12(1— V2)

D: flexural rigidity, E:Young modulus, u: Poisson ratio, )1: half-thickness, p: den-
sity.

The motion in a particular mode is simple harmonic, so with w = W(r,0)e“"‘,
where W(r, 6) is the shape function describing modes of the vibration and w is the

natural circular frequency of the plate, (2.1) becomes
V4W — k4W = 0 (2.2)

where k4 = pwz/D.

The differential equation (2.2) can be written in symbolic form as
(V2 — k2)(V2 + k2)W = 0 (2.3)
We assume solution to equation (2.2) in the form of the general Fourier series

W(r, 6) = Z Wn(r) cos n6 + Z W; (T) sin n0.

n=0 n=1

Then each term of the series must satisfy (2.3). Thus the general solution is given by

the Bessel functions Jn(kr), Yn(kr) and the modified Bessel functions 1,,(kr), and
Kn(kr)

oo

W(r,0) = Z [A J n(kr) + 3.1/"(1a) + CnIn(kr) + D.K,.(kr)] cos n9
n=0
+ i [A‘Jn) (kr) + B;Y ”(kr) + 0*] n(kr) + D;K n(kr‘)] sin 716

(2.4)

where the coefficients An, B", C", D", A;, B;, 0;, D;, which determine the mode

shapes, are solved using the boundary conditions.

Since the boundary conditions are symmetrically applied about a diameter of the
plate, the second expression in (2.4) containing sin n0 is not needed to represent the

solution. Then, the n-th term of the equation (2.4) becomes
W..(r, 6) = [Aan(kr) + 3.1/"(m + 0.1.,(kr) + D..K.,(kr)] cos n6 (2.5)

for n=0,1,~-.
A circular plate with a fixed circular core is considered with three sets of boundary

conditions:

(a) Free-clamped

02W 1 0W 1 WW
l 81'2 ”(F 81‘ + r3 662 )] r=l - 0 (2.6)
l6r(v W) + r2 562( 6r 7)] r=1 - 0 (2'7)
6W
W(b,6) — 79—1”— r=b — 0 (2.8)

 

which represent no—moment and no—shear force respectively.

(b) Clamped-clamped

 

aw , _
W(1,6) _ 7,9;- r=1 _ 0 (2.9)
W(b,0) = 5%“?! r=b = o (2.10)

 

which represent no—displacement and no-slope respectively.

(0) Simply supported-clamped

 

 

W(1,6) = 0 (2.11)
82W 16W 1 02W
[7.2 V m? :5 392 )l - 0 (2.12)
6W
W(b,0) _ 797 M _ o (2.13)

 

which represent no—displacement and no moment respectively. Here in the sim-
ply supported case the 9-derivative Q33”; in the moment (2.12) vanish, since

there is no tangential deflection on the boundary of the plate.

2.2 Free - Clamped Boundary Condition

Throughout this chapter concentric circular core has clamped boundary condition.

Let inner radius be b and outer radius be 1, as indicated in Figure 2.21.

 

Figure 2.1: Free edge circular plate with a core

Noting the following recurrence formulae for Bessel functions

O

42(2) = 3 M — J...<z>

1.202) = {32(2) + I,.,(.)

(formulae for Y; and K,’, are the same as J}, above) (see McLachlan 1955)[14], we

impose the first set of boundary conditions (2.6) and (2.7):

Atr=1

[Jn(k) — n{aJn(k)+%Jn+1(k)}]
_. Bn [Yn(k) —n{aYn(k) +1-1Yn+l(k)}]
[1,,(k) +n{aIn(k) - E n+l(k)}]

+ D.[K.(k) + 77{01Kn(k)+% ...(k)}] = o
and

- An [an(k) — kJn+1(k) + i6{\7n}]

_ Bn [nYn(k) — kYn+l(k) + fl{yn}]

+ Cn[n1n(k) + kIn+l(k) - IB{In}]

+ D. [nK,,(k) — H.110.) — [3{IC.,}] = 0
where a = n(n —1)/k2, 77 =1— V, [3 = nnz/kz, and
jn = (Tl _1)Jn(k) _ kJn+l(k) yr; = (71. _1)Yn(k) — kYn+1(k)
1,. = (n —1)I,.(k) + kIn+1(k) IC. = (n —1)Kn(k) — kKn+1(k).

At r = b
AanUcb) + BnYnUcb) + CnInUcb) + DnKn(kb) = O

and

A.[£5Jnk(_bJ)-J).+1(kbi+B[-%Yn(kb)-Y.+1(kb)i

+ C.%1(kb) +1....Wb]+0.[1{.(kb)— Kn+1(kb)]=0.

The antisymmetric case (for n = 0) corresponds to modes of vibration in which

nodal diameters do not occur.

10

For n = 0, the determinant equation becomes
A B ' C” D
J1(k) Y1(k) 11(k) -K1(k)
Jo(kb) Y0(kb) 10W?) K0095)
-J1(kb) —Y1(kb) 11095) "K1(kb)

= 0 (2.14)

 

 

where

11>:
||

—kJo(k) + MW 3 = -kYo(k) + nYlUc)

Q:
ll

ho(k)- 77110C ) D = kKo(k) +nK1(k)

For n = 1 and b —-> O, rigid body rotation about the diameter occurs due to
zero frequency which can be verified by setting k = O in the equation and the free
boundary conditions.

For n = 1, the determinant equation, with the aid of recursion formulae for Bessel

functions, becomes

A B C" D
-J2(k) -Y2(k) 12(k) -K2(k)
—J1(kb) -Y1(kb) 11065) K1(kb)

—Jo(kb) -Yo(kb) 10W?) -Ko(kb)

= 0 (2.15)

 

 

where

in
II

kJ1(k)— nJ2(k ) B = kY1(k) — mac)

C):
II

k11(k) — 77120:) D = mac) + was)

Table 2.1 gives the fundamental frequencies for modes n = 0,1,2, which improves
the frequency values of Southwell [22]. Here Poisson’s ratio u is taken to be 0.3.
For b > 0.3 the smallest frequency of the plate is in the case of n = 0 where there

is no nodal diameter; however, for b S 0.3 the gravest frequency of the plate occurs

11

 

n=0 n=1 n=2

b k b k b k
0.1 2.05849 0.1 1.86498 0.1 2.37122
0.2 2.27620 0.2 2.19386 0.2 2.53911
03 2.58077 0.3 2.55975 0.3 2.82072
0.4 3.00343 0.4 3.01919 0.4 3.23489
0.5 3.60891 0.5 3.64552 0.5 3.83456
0.6 4.53009 0.6 4.57558 0.6 4.74356
0.7 6.07887 0.7 6.12359 0.7 6.26713
0.8 9.19238 0.8 9.22832 0.8 9.33772

 

 

 

 

 

 

 

 

 

 

Table 2.1: The fundamental frequencies of free circular plate with a core

in the case of n = 1 where there is one nodal diameter and as b ——> 0 there is no
vibration k = 0 which corresponds to a rigid body rotation about the diameter.

Starting with the following asymptotic expansions for Bessel functions for small

arguments [14]

3’00“”) ~ [7 + In (533)] Jo(kb) + 0(kb)?

@_§+mw)
o+§+mw)

7 + 1n (%)] 10(kb) + O(kb)2,

J10“) N

tolzrtola‘aIw

1109) N

K0095) N

f—_l

where 7 = 0.57721 is Euler’s constant, and imposing them to the determinant equa-

tion (2.15), we obtain

an 012 013 014

021 022 023 024

0 2/7r 0 1

 

 

—1-21nb/7r 1 lnb

12

where

k k3 k2
a“ ’ k(2-16) s
2 k 4 1 3k2
- k(';;‘§;)+"(;15+;+1‘a)
k k3 k2
- “rial-"g
1 k 2 1 3k2
- k(rz)+"(z§‘§+3§)
k2 k4
0.21 = ——8—+§6
_ .3. 1+3’i
G22 _ 7rk2 7r 167r
— k2+k4
“23 _ 8 96
2 1 3k2
“2“ = ’F '2‘— 32’
whichgives
01(16 + k41n b) + 021:4 + 031:8 lnb = 0, (2.16)

where 01 = 17 — 4, 02 = (1377 — 44)/12, and 03 = —(n — 6)/144. Since we do not

know how I: approaches zero with b, set

where 61(b), 62(b) are asymptotic sequences (i.e. 6,(b) —> O as b —> 0 for 2' = 1,2)
and k1, k2 are constants.

Balancing the leading terms of (2.16) yields
16 + k‘l‘é‘,‘ ln b = o (2.17)

which gives

1
61 = m , k1 = 2. (2.18)

The next order gives
021.3: + C3kf6‘f ln b + 4Clki‘5fk262 1n b = 0 (2.19)

13

 

 

1.9....1114V16

1.8 - 1_45.. , C

1.7 .. 1,4 .. .1.

1.6 - .35.
k 1

1.5...) 1.3.

1.4 . 1.25... .i.

1.3 ~ :- 1,2.

   

 

 

 

 

1.2. B»: - ? « 1.15-.--—Z-1~-.

 

 

.1 i 1 1 40 1 g 1 1 1 1.1 1 1 1 1 1 I 1 L ‘
0 0.01 0-02 0-03 0.04 0-05b0-06 0-07 0-08 0.09 0.1 0 0.001 0.002 0003 0.004 00050006000700060009001
b

Figure 2.2: Free circular plate with a core: Asymptotic vs. Numerical Frequency

 

 

or
_ Cg e 1603
16252 — 2C1(—lnb)5/4°
Then
1 431/ + 113
= k = —- 2.2
62 (_ 1n b)5/4 1 2 72(1) + 3) ( 0)
Thus the correction to the fundamental frequency is
2 43V + 113 1
k = --- 2.21
(- 11113)“4 + (72(u + 3) ) (-ln b)5/4 + ( )

and hence, as b —1 0, k -—> 0; however, the slope becomes

fill:—>oo as b—>O.

db
Figure 2.2 shows the graphical comparison between the asymptotic frequency and the
numerical frequency solution.
Table 2.2 shows the comparison between numerical and asymptotic values of fun-
damental frequencies, 16.
Thus, for small values of b, the asymptotic result for the fundamental frequency is

accurate and avoids the singularity and costly numerical calculations. In other words,

14

 

 

 

 

 

 

b k (numerical) k (Rayleigh-Ritz Method) 16 (asymptotic)

0.1 1.86498 1.8677 1.8104
0.01 1.45243 1.6474 1.4438
0.001 1.28356 1.6420 1.2810
0.0001 1.18204 1.6426 1.1811
0.00001 - 1.6462 1.1108
1 x 10‘8 - 1.6413 0.9793
1 x 10-12 - 1.6426 0.8807
1 x 10‘24 - 1.6459 0.7371
1 x 10"50 - NaN 0.6120
1 x 10-100 - NaN 0.5140
1 X 10'200 - NaN 0.4325
0 - NaN 0

 

 

Table 2.2: Fundamental frequency of free edge plate with a core

to do the calculation numerically, one needs to apply the boundary conditions to the
solution, then find the eigenvalues of the resulting matrix whose entries are Bessel
functions. This would require programming. Table 2.2 shows the comparison of fun-
damental frequencies with numerical methods. While the first numerical method (di-
rect computation by programming) fails to give any result, the Rayleigh-Ritz method
fails to converge. In fact the method even fails to give the right convergence pattern
i.e., after b = 0.001 the frequency becomes inconsistent which contradicts the plate

having the zero frequency as b —+ 0.

2.3 Clamped - Clamped Boundary Condition

Applying the clamped boundary conditions (2.9) and (2.10) to the solution (2.5), on

the outer boundary 7' = 1 we have

AJ.(k) + BY..(k) + CI..(k) + DKn(k) = 0

15

and
/1%[J-.(k)J.+1(k)]+B[%Yn(k)—Y.+1(k)]
+ c[~l;1.k( )+I.+1(k)] +DE K—.(k) 70.10)] =0.

On the inner boundary r = b we have
AJ.(kb) + BY.(kb) + CI.(kb) + DK.(kb) = 0

and

Ag.[ J .(kb)-— 1.106)] + B [7.7” (kb) — Y.+1(kb)]
+ 615,; —1 .06) + 1.100] + 0%1060— K=o...(kb)]

For axisymmetric case n = 0, the determinant equation becomes
Jo(k) Yo(k) 1006) K606)
-J1(k) -Y1(k) 111(k) -K1(k)

10099) 3’00“”) 1009(3) K009”)
—J1(kb) -Yl(kb) 11(kb) —K1(kb)

= 0. (2.22)

 

 

When b = 0, determinant equation (2.22) shows that the fundamental frequency is

the first root of

(J0(k) - 10(k))l2K1(k) + 7TY1(k)l
-(11(k) + 11(k))[2Ko(k) + 7rYo(k)] = 0 (223)
which is k = [to = 4.76831. Note that, a clamped edge plate without a core constraint
(characteristic equation 11(k)Jo(k) + Io(k) J1(k) = 0) has a fundamental frequency of

k0 = 3.19622 in the axisymmetric case n = 0.

Now, we perturb the fundamental frequency 16 about [to = 4.76831 so that
k = k0 + 6(b)k1+ o(6(b)),

16

where 6(b) —1 0 as b —> 0.

To this end, we start with following asymptotic expansions:

70(1) = 700:. + 61:1 + . . .) = .7001.) — 67.1.7100) + . ~ (2.24)
and
51131
J10?) = J1(ko) + 7010060) — J2(k0)) + ° " - (2-25)

1606), 10(16), and K006) are expanded similarly. Then the determinant equation

(2.22) with this expansion becomes

Jo“) 141(k) 1000 KOUC)
-J1(k) —Y1(k) 1101) —K1(k)

1 Zlnb/fl' 1 -lnb
—bko/2 2/7rbko 0160/2 —1/bk0

= 0, (2.26)

 

 

which gives
Co + C1b2 + 0202111 b + C'3b21n2 b
+6k1(C4 + 05b2 + Cab2 1n b + C'7b2 1n2 b) = 0 (2.27)
where
Co = "(110901 + J1(k0))(2K0(k0) + 7rY0(’¢O))

+(Jo(ko) — 10(k0))(2K1(k0) + 70106)) = 0
C3 = 430100600) 4000600))
c. = k30[(11(ko)+J.(ko))(2Ko(ko) +0.06)
+ (1006) —- Jo(ko))(2K1(ko) + mum]
- 8J0(ko)Ko(k0) - 47r10(ko)Yo(k0)-

Balancing the leading orders in (2.27),

(5k1 = “-9302 11120 (2.28)
C4

17

which is

£2

4

6(b) = 6211126, k1 = = 14.0279. (2.29)
This gives the correction to the fundamental frequency

k = 4.76831 + 140279621112 b + . . .. (2.30)

Table 2.3 shows the comparison between the numerical frequency values and the fre-

quency values of asymptotic expansion. Figure 2.36 shows the graphical comparison

 

 

b k (numerical) k (asymptotic)
0.1 5.22308 5.5120
0.01 4.78791 4.7980
0.001 4.76881 ~ 4.7690
0.0001 4.76832 4.7683
0 4.7683 4.7683

 

 

 

 

 

Table 2.3: Fundamental frequency of clamped plate with a core

between the asymptotic frequency and the numerical frequency solution.

Thus, for small values of b, the asymptotic result for the fundamental frequency
is accurate and avoids costly numerical calculations. In other words, to do the cal-
culation numerically, one needs to apply the boundary conditions to the solution,
then find the eigenvalues of the resulting matrix whose entries are Bessel functions.
This would require programming. Table 2.3 shows the comparison of fundamental

frequencies with a numerical method (direct computation by programming).

18

 

 

 

 

 

 

 

 

 

 

5 4.8
4.95- 4.795 _.
4.9» ....
4.79
.>,~ , ; -. g
5 4,8- ...... . . ., , ,. Asymptotic . 0 4-785
3 71/57. 8-
&4,75_ ..... Hmnm g 4.78
x 47 .2 '
' 7 4.775 ..
4.65-~--~ :
: 4 4.77 ' "
4.6......_..:.. .. ...... : ...... .
4,55L......,,. . ...... ....... . 4W765" ‘ _
4.5 A # 1 4 4.76 - i i i
0 0.002 0.004 0.006 0.008 0.01 o 0.002 0.004 0.006 0.008 0.01

0 (core radius) b (core radius)

Figure 2.3: Clamped circular plate with a core: Asymptotic vs. Numerical Frequency

2.4 Simply Supported - Clamped Boundary Con-
dition
Applying the simply supported boundary conditions (2.11) and (2.12) to the solution
(2.5), on the outer boundary 7' = 1 we have
AJ.(k) + BY.(k) + 01.06) + DK.(k) = O
and
A -— 7.27.0.) + (u — 1)(n(1 — n)J.(k) - kJn+1(k))]

B l — k?1/.(k)+ (u — 1)(n(1 — Tin/.0.) — kY.+1(k))]

c121...) + (u _ 1)(n(1 — n)I.(k) + k1..1(k))]

 

+ D 7.210(1) + (u -— 1)(n(1 — n)K.(k) — kK.+1(k))] = 0.
On the inner boundary 7‘ = b we have
AJ.(kb) + BY.(kb) + CI.(kb) + DK.(kb) = O

19

and

455.14%) — Janka] + B [31’ (kb) — Yuma]

kb "
+ C [%1n(kb) + In+l(kb)] + D [fianm — Kn+1(kb)] = o.

For axisymmetric case n = 0, the determinant equation becomes

~

 

 

A B (3* D
Jove) Yeas) love) KoUc) : O (2.31)
Jo(kb) Yo(kb) 10(kb) Ko(kb) _
7J1(kb) "Y1(kb) 11(kb) ‘K1(kb)
where T]=1—V and
A = —kJo(k) + MW 3 = —kYo(k) + nY1(k)

6' = “00“) - 011(k) D = “(009) + 77K1(k)

Determinant equation (2.31) shows that the fundamental frequency is the first root

of

2k(2Jo(k)Ko(k) + 7rIo(k)Yo(k)) — n(J1(k) + 1109)) (2Ko(k) + 7rYo(k))

+n(Jo(k) — 10(k))(2K1(k) + 7FY1(k)) = 0

which is k = k0 = 3.84899 as b —+ 0. Note that, a clamped edge plate without a core
constraint (characteristic equation 2ho(k)Jo(k)—(1—1/)(11(k)Jo(k)+Io(k)J1(k)) = O)
has a fundamental frequency of k = 2.22152 in the axisymmetric case n == 0.

We perturb the fundamental frequency k about k0 = 3.84899 so that
k = k0 + 6(b)k1+ o(6(b)),
where 6(b) -> O as b —-> 0. Using the asymptotic expansions (2.24) and (2.25) (Y0(k),

10(k), and K006) are expanded similarly). The determinant equation (2.31) with this

20

 

expansion becomes

 

kJoUC) " 715(k) kYo(k) '— 77Y1(k) (910(k) - 011(k) “(009) + UKIUC)
—J k —Y I»: I k K k
0( ) o( ) o( ) o( ) = 0, (2.32)
—1 —2ln b/vr 1 — lnb
O -2/7r O —1
which gives
Co +C1b2 + C2b21n b + C3b2 1112 b +
5a (0., + 05b2 + (:61)2 In b + C7b21n2 b) = o (2.33)

where

00 = 21:3 (2Jo(lc0)K0(k0) + 410(k0)Y0(k0))
— 17k0(J1(ko) + 1104.)) (2Ko(ko) + 7rY0(k0))
+ nko(Jo(k0) — Io(ko)) (2K1(k0) + 7rY1(ko))
03 = —4k3Jo(ko)Io(ko) + 2nngo(k0)11(ko) + 2nk§J1(ko)Io(ko)
0.. = $5227 - 4kg)Jl(ko)Ko(k0) — kioon + 4k§)Jo(ko)K1(ko)
+ 24(1 - n>Io(ko>Yo<ko> + {3014+ 77)II(’€0)Yo(ko) + 4Jo<ko)Ko(ko)
— {301:3 — 77)Io(ko)Y1(ko) + %(J1(ko)yo(ko) — 480mm»

klo(11(ko)Ko(ko) + Io(ko)K1(ko)>-

+
'Balancing the leading orders in (2.33) by

5k1 = «931221112 b (2.34)
04

we obtain

6(b) = 32 in2 b, k1 = —%‘- = 9.75534 (2.35)

4

which gives the correction to the fundamental frequency

1: = 3.84898 + 9.755341)2 1n2 b + . . - . (2.36)

21

 

 

 

k (trequency)
5"
(D
m

 

3.87 a .
3365- . ..gAsymptotic.
Asymptotlc g 3.86-... ..- ...... .1
:’ Numerical‘ g . :
i i "‘ 3.855 .. Numerical.
3.85 ....... 4
3.845

 

 

Figure 2.4:

Fiequency

0.002 0.004 3.606 0.608 0.01

 

    
  

 

 

 

0 0.002 0.004 8.606 0.008 0.01

Simply-supported circular plate with a core: Asymptotic vs. Numerical

Table 2.4 shows the comparison of k between the numerical values and the asymp-

totic expansion. Figure 2.4 shows the graphical comparison between the asymptotic

 

b

k (numerical)

k (asymptotic)

 

0.1
0.01
0.001
0.0001
0

 

 

4.21774
3.86398
3.84936
3.84899
3.84898

 

4.36621
3.86968
3.84945
3.84899
3.84898

 

 

Table 2.4: Fundamental frequency of simply-supported circular plate with a core

frequency and the numerical frequency solution. Thus, for small values of b, asymp-

totic result for the fundamental frequency gives an accurate result avoiding costly

numerical calculations.

22

CHAPTER 3

Fundamental Frequencies of Plates

This chapter presents a perturbation method to study free vibration of wavy bound-
ary plates, elliptic plates, and polygonal plates, with clamped and simply-supported
boundary conditions. The natural frequencies of these plates are determined by cal-
culating the eigenvalues of the governing equations using perturbation methods. The
perturbation solutions yield analytic approximate solutions of the fundamental fre—

quencies for these plates.

3.1 Perturbed Boundary: General Case

The governing Biharmonic equation of a plate in polar coordinates (r, 0) is given by
(2.2). The fundamental frequency A2 = 1:273 = wrzm is normalized with the
radius r of a circular plate of which we perturb about. Normalization of r as follows:

For a given any arbitrary wavy boundary plate, let F09) = a + b f (6) Set the

normalized mean radius to be 1
1 211' ~
—— d6 = .
ZWCA a+bf(0) 1 (31)
where C is the normalization constant. C may be written as
C = a + b f..(9) (3.2)

23

where

1 21r~

173(9) — f (9) d9- (33)

= 27l- 0
Set the normalized r to be 1 + e f (0) ,

_ a+bf(6) =

_ 0+ bva(9) 1+ 4(6). (3.4)

nor

Let 6 = 1, then the normalization of f (0) becomes

b(f(6) - we».

“6) = a + bfavw)

 

(3.5)
Then without loss of generality we can take a plate with perturbed boundary as
r = 1 + ef(0), (3.6)

where f < 1 and e is separation variable of orders (amplitude). We consider two
types of boundary conditions: Clamped and simply supported. In this section, general
case of these two boundary conditions analyzed and then it is applied to wavy, elliptic,

and polygonal plates.

3.1.1 Clamped Boundary Condition

We perturb the solution W(r, 9) and the fundamental frequency I: about the circular

state as follows. Let

W(r, 0) = W0(r) + 6W1(T', 6) + €2W2(r, 6’) + - .. (3.7)
k4=k3(1+e2b+---) (3.8)
where b is the correction to the fundamental frequency. The second term in (3.8)

is of order 0(62) due to invariant frequency for any change in 0, so the perturbed

frequency is even.

24

At 1' = 1 + e f (0), clamped boundary conditions are that

W(1 + €f(6), 9) = o (3.9)

6W
—— = 0 3.10
6n r=1+¢f(9) ( )

where n is the unit normal vector. The first boundary condition (3.9) can be expanded

to in Taylor series as

aw
W(1+€f(9)i9) = W(li9)+€f(9)-67(1i9)
02W
5T2
Applying (3.7) into (3.11) and collecting all terms of the same order in 6, yields

 

+ $62749) (1,e)+0(e3) =0. (3.11)

W(1+ ef(6),0)

W0(1,6) + 6[W1(1,6) + f(6)Wo,(1,6)]
+ €2[W2(1,0) + f(6)W1,(1,0)
+ §f2(6)wo..(1.9)] + . . - = o. (3.12)
For simplicity, let
W(1+ef(9),6) = W0(1,6) +6[W1(1,6) -- <I>§(1,6)]
+ o2 [W2(1,9) — <I>§(1,6)] + . -- = 0. (3.13)
Where ‘I’i(1i9)= -f(9)Wor(1i9) and 93(1i9) = -f(9)W1.(1i9) - f2(9)Wo..(1i9)/2~
For the second boundary condition (3.10), let
F(r, a) = r — 1 — em) = 0. (3.14)
The normal derivative being zero gives
VF - vw = aw. + $17M),
-_— w0,(1,9) + 6[W1,(1,6) - f’(9)Wo,(1,9)
+ f(6)wo..(1, 6)] + «22 [Wine + 2f<6)f'(6)wo.<1, 0)
- f’(0)W10(170)+ f(9)W1.-.(1i9)
- f(6)f'(9)wo..(1. 9) + §f2(6)wo...(1,9)] + . ~ = o. (3.15)

25

For simplicity, let

0W

— = W 6 _ c
an r=1+£f(0) °'(1’ )+€[W1,(1,6) ‘I’1(1i9)]

+ e2 [WNW — 93(1, 9)] + - -. = 0 (3.16)
where
@1016) : f’(6)W09(1) 9) — f(6)W0rr(1’ 9)
‘1’§(1i9) = -2f(9)f'(9)Woo(1i9) + f’(6)W19(1)6) - f(9)W1..(1i9)

+ f(9)f'<6)wo..(1,9) — §f2<6)wo...<1.9).

The equations (3.13) and (3.16) give the perturbed clamped boundary conditions for

an arbitrary function f (19).

3.1.2 Simply Supported Boundary Condition

We perturb the solution W(r, 0) and the fundamental frequency 1: about the circular
state as in equations (3.7) and (3.8).

At 1' = 1 + e f (6) , simply supported boundary conditions are that

W(1 + 4(9), 9) = o (3.17)
82W u 8W
( 8n? + B67?) r=1+cf(6) — 0 (3.18)

 

where n is the unit normal vector, V is the Poisson ratio, and R is the normalized
radius of curvature. Since there is no tangential deflection on the boundary of the
plate, the tangential derivatives in the moment 923%- + V%% + ”<29;ng = O vanish in
the boundary condition (3.18). The first boundary condition (3.17) has the same

expansion as in equation (3.13). Then the second boundary condition (3.18) becomes

VF VF u VF
-——- —- W — —- W = .1
|VF| V(|VF| V )+R(|VF| V ) 0 (3 9)
1 1 V
W (EC. + 518909) + EG _ 0 (3.20)

26

where F(r,9) = r — 1 — ef(0) = 0 and G = (FTW, + F9Wg/T2)/IVF| . Using the
radius of curvature in polar coordinates

[(9')2 + r2] 3”

 

13(0) = T, + 2(7)), _ Tr”. (3.21)
we obtain the asymptotic expansion of the second boundary condition as
[W0,,(1,9) +uW0,(1,9)] +o[W1,,(1,9) +uW1,(1, 9) —\115( 1 9]
+52 [W2,,(1, 9) + uWn.(1, 9)— 93(1, 9)]+ =0 (3.22)

where \Il‘l9 and q»; are given respectively, by equations (A7), (A8) in the Appendix.
Thus the equations (3.13) and (3.22) give the perturbed simply supported bound-

ary conditions for an arbitrary function f (0)

3.2 Wavy Boundary Plates

Consider a wavy boundary plate. The governing biharmonic equation in polar coordi-
nates (r, 6) is given by (2.2) and corresponding general solution, in polar coordinates,

is given by

W(r, t9) = i Wn(r, 0) cos 710 (3.23)

n=0

where
Wn(7‘, 9) = [Cn1Jn(kT) + anynUCT) + Cn31n(k7‘) + Cn4Kn (kT)] COS n6 (3.24)

where Cal, Caz, 0,,3, CM are arbitrary constants as before (2.5). In order to have
a finite solution, asymptotic behavior of Yn(kr) and Kn(kr) as r —-> 0 leads us to
Caz = CM = 0 for all n = 0, 1,2, - - - . Then the n-th term of the solution can be

written as

Wn(r,9) = [CnlJn(k'r) + Cn31n(kr)] cos (720). (3.25)

27

A nodal line is one which has zero deflection (Wu = 0). For a circular plates nodal
lines are either concentric circles or diameters. The nodal diameters are determined
by n6 = 7r/2, 37r/ 2, - -- . For each value of n, we have infinitely many frequency.
The fundamental frequency is the smallest frequency of 811. Therefore we consider
the first frequency of the equation (3.25) for n = O in the case of clamped and simply
supported boundary conditions. However, the fundamental frequency for the plate

with free boundary condition occur for n = 2.

3.2.1 Clamped Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency k about the circular
state as follows. We let
W(r, 6) = W0(7‘) + 6W1(7‘, 6) + €2W2(r, 6) + 0(63) (3.26)
k4 = kg(1+ 62b + - - -) (3.27)
where b is the correction to the fundamental frequency as before.

Consider wavy circular boundary by taking f (6) = cos (M6) in the previous sec-

tion, then
r = 1 + 6cos (M6) (3.28)

where 6 << 1 is a small amplitude and M 2 2 is the number of circumferencial waves.
The clamped boundary conditions are given by
W(1 + ecos (M6), 6) = O (3.29)
6W
.517.- r=l+ccos(M6) _ 0 (330)

where n is the unit normal vector. Using the boundary expansions (3.13), (3.16)
with f (6) = cos (M6) and substituting (3.26) and (3.27) into (2.2), and collecting

the similar order terms we obtain the zeroth order equation as
V4W0 - kgwo = 0 (3.31)

28

with the boundary conditions
W0(1,6) = O Wo,(1,6) = O (3.32)

which corresponds to the circular plate (for n = O). The boundary conditions (3.32)
are the zeroth order terms of the expansions (3.13), (3.16). The solution of the

equation (3.31) with the boundary conditions (3.32) is given by
Wo(7‘, 6) = 001 J0(k07‘) + Cozlo(k07‘) (3.33)

where the constants C are rearranged. Imposing the clamped edge boundary con-
ditions and using the recursion formulas for the Bessel functions, we obtain the fre-

quency equation
Jo(>\)11(/\) + J1(/\)Io(/\) = 0 (334)

where /\ = k0 since 7‘ = 1. The solution of (3.34) gives the fundamental frequency
k0 = 3.1937 of the clamped circular plate. Then the solution (3.33) may be written

as
W0(r, 6) = 001(Jo(kor) + aoIo(k0r)) (3.35)
where do = J1(ko) / 11(ko). Let Cm = 1 without loss of generality. Thus,
Wo(r, 6) = Jo(kor) + aolo(kor). (3.36)
The first order 0(6) equation is
V4W1 — k3W1 = 0 (3.37)

subject to the following boundary conditions that are the first order terms of the

expansions (3.13), (3.16)
W10. 9) = 0 (3.38)
W1,(1,6) = — cos(M6)Wo,.,(1,6), (3.39)

29

since Wo,(1,6) = O and W0 = Wo(r) in the boundary condition of order 0(1).

Solution of the equation (3.37) with above boundary conditions is

 

W1(r, 6) = [CuJM(koT) + 0121M(k0r)] cos M6 (3.40)

where
Cu = WWOJU (3.41)
012 = 2J¢k°)Wo..(1) (3.42)

where H = kol‘JM(k0)(IM—l(k0) + IM+1(k0)) + IMUCOXJM—rUCo) — JM+1(k0))l-

The second order 0(62) equation is
V4W2 — k3W2 = bkgwo (3.43)

subject to following boundary conditions

1
W2(1,6) = iwo"(1)+ ZW°"(1) cos (2M6) (3.44)
M2 A 1A 1
W2.(1,9) = —2—W1(1)— §Wln(1)— 3W0...(1)
+ (- “—2—W1(1) — §W1"(1)- 1W0rrr(1)) COS (2M6) (3.45)

where W1(r) = CnJM(kor) + 0121M(k07‘) and W1(r, 6) = 1(7‘) cos (M6).
The second order boundary conditions suggest that we have a solution of type

W2(r, 6) = U (r) +V(r) cos (2M 6) for (3.43). Upon substitution into (3.43) we obtain
v4v — kgv = 0 (3.46)
the solution of which is given by
V0") = A1J2Mfko7") + A212M(ko7‘) (3.47)

where A1 and A2 are arbitrary constants. Since the solution of (3.46) has no effect

on the correction b to the fundamental frequency, instead we solve
V4U — ng = bkéWo. (3.48)

30

The solution of the homogenous part of (3.48) is
Uh(7‘) = BlJo(ko7") + BQIoUCo’I‘) (3.49)

where 81 and B2 are arbitrary constants. The particular solution of (3.48) is

bkoT‘

(1,3) = ——4—— (J1(kor) — 0.011(3)»). (3,50)

Thus the general solution of (3.48) is

U (7‘) Uh(7‘) + U100”)

= BlJ0(koT') + 82100607") — @1: (J1(k07‘) — 01011(k07‘)) . (3.51)

The boundary conditions for (3.48) are

Um) = iwoflu)
v.0) = Mfmn—ngn—fiwow) (3.52)

Imposing the boundary conditions (3.52) and (3.52) into (3.51),

1
B1J0(ko) + BzIoUCo) = 2W0"(1)
BlJ1(kO) "' 3211090) = UK + 5 (3-53)

A

where R = -k() [(JoUCo) — J2(ko)) — 0(Io(ko) + 1206(0)] /8 and 5 '3 —M2W1(1)/2ko +
mvzko + We...(1)/4ko.

In order for the system (3.53) to have a nontrivial solution, we need

Rank «10090) 10090) = Rank Jo(ko) 10(ko) Warr(1)/4 . (3.54)
J1(ko) —11(k0) J1(k:0) —11(k0) (JR + 8
It follows that
10(ko) W0rr(1)/4 : 0 (3.55)
—11(k0) bR + s

31

 

e 0.1 0.08 0.06 0.04 0.02 0.01 0
k 3.35659 3.30152 3.25671 3.22352 3.20311 3.19795 3.19622

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.1: Fundamental frequency of clamped wavy plate

which gives us a unique solution for b. Hence

411(ko)5 + 11(ko)W0"(1)
- 410(kolR

which is the first correction to the fundamental frequency of a clamped plate with

b:

 

(3.56)

wavy boundary. Table 3.1 below lists the values of the fundamental frequencies kg
with various 6 for M = 6. The frequency itself depends on the amplitude e and is

given by equation (3.27).

3.2.2 Simply Supported Boundary Condition

We perturb the solution W(1“, 6) and the fundamental frequency 1: about the circular

state as in (3.26) and (3.27) respectively. Consider wavy circular boundary,
7' = I + 6 cos (M6) (3.57)

where 6 << 1 is a small amplitude and M Z 2 is the number of circumferencial waves.

Then, as before, the simply supported boundary conditions are that

W(1+ ecos (M6), 6) = 0 (3.58)
62W V (9W
( 672.2 + FEE) r=1+ecos(M6) — 0 (359)

 

The first boundary condition (3.58) can be expanded into an asymptotic series (3.13),
as before, for f (6) = cos (M 6).
For the second boundary condition (3.59), we let f (6) = cos (M 6) in the equation

(3.22). Note that here 6-derivatives of W00", 6) are zero.

32

Then collecting the same order terms we obtain the zeroth order equation as
V4W0 — kgwo = 0 (3.60)
with the boundary conditions
Wo(1, 6) = 0, W0"(1, 6) + uWo,(1, 6) = 0 (3.61)
which corresponds to simply supported circular plate (for n = 0). The solution of
the equation (3.60) with the boundary conditions (3.61) is given by

W0(r, 6) = C01Jo(kor) + CogIo(k0T) (3.62)

where C01 and 002 are constants. Imposing the simply supported edge boundary
conditions and using the recursion formulas for the Bessel functions, we obtain the

frequency equation
(u — 1>[J6(A)I.(A> + J1<A>Ioo>1 + 2koJoo>Io<A> = o (3.63)

where A = 160 since 1‘ = 1. The solution of (3.63) gives the fundamental frequency
k0 = 2.22152 of the simply supported circular plate. Then the solution (3.62) may

be written as
Wo(r, 6) = C01[Jo(kor) + (1110(k0r)] (3.64)
where (11 = —Jo(k0)/Io(ko). Let us take Cm = 1 without loss of generality. Thus,
Wo(r, 6) = Jo(k0r) + 011100607"). (3.65)
The first order 0(6) equation is given by
V4W1 — [€ng = 0 (3.66)
subject to following boundary conditions upon substituting f (6) = cos(M 6) into
<I>f(1,6) and \Ills(1,6)
W1(1,6) = —W6.(1,6) cos (M0) (3.67)
w...<1.0)+ vii/1.0.6) = [141+ M2)Wo.(1,0> — uwo..<1.6>
— W0,,,(1,0)] cos (M6). (3.68)

33

The solution of the equation (3.66) is
W1(T', 6) = [CnJM(k0T) + 0121M(k07")] COS M9. (3.69)

Imposing the boundary conditions (3.67) and (3.68) and denoting the moment oper-
ator by M(JM(kor)) = Jg’d(kor) + 11])",(1601') we obtain

IM(k0)[W0r1-r(1) + VWo...(1)] — K1Wo.(1)

JM(ko)M(IM(koT)) - [M(ko)M(JM(koT))

—JM(kO) [Wo...(1) + VWO..(1)] + 1C2Wo.(1)
JM(ko)M(IM(koT)) “ [M(k0)M(JM(koT))

where K1 = M(IM(]€0))+V(1+M2)IM(IC0) and [C2 = M(JM(ko))+l/(1+M2)JM(ko).

Now, let W1(r, 6) = Vim cos (M 6) where W1(r) = C'nJM(kor)+C12]M(kor) in order

 

Cu

 

012 =

to make further calculations simpler.

The second order 0(62) equation is
V4W2 — kSWQ = bkgwo (3.70)
subject to following boundary conditions

W2(1,6) = —R—’Rcos(2M6)
W2"(1,6) + VW2,(1,6) = -:-(8 + C) + %( — 8 + C) cos (2M6) (3.71)

where

1A 1
R = 2W1'(1)+ZW0"(1)

5 = M2(—2v’171(1)+2vT/Z(1)+Wo..(1)+z/Wl(1))

c = —u(1 — 2M2)Wo,(1) + u(1 — M2)W1\,(1)+ u(1— M2)W0,,(1)
A V A 1
- VW1"(1) — 5W0...(1) ‘ W1...(1) - 5W0....(1)-

Assuming a solution of type W2(r, 6) = U (7‘)+V(r) cos (2M 6) for (3.70), we obtain

V4V - kgv = 0. (3.72)

34

The solution of equation (3.72) is given by
V(r) = A1J2M(kor) + A212M(kor) (3.73)

where A1 and A2 are arbitrary constants. Since the solution of (3.72) has no effect

on correction b to the fundamental frequency, we solve

V4U — ng = bkgwo. (3.74)
The solution of the homogenous part of (3.74) is

Uh(r) = B1J0(kor) + 32100607) (3.75)

where BI and B2 are arbitrary constants. The particular solution of (3.74) is

01607”

Up(r) = ——4—(J1(k0r) — 0111(k0r)). (3.76)

Thus the general solution of (3.74) is

U”) = Uh“) + Up”)
= B1Jo(k07‘) + BzIoUCoT‘) — b—IZ):(J1(IC0T) — 0111(k07")). (3.77)

The boundary conditions for (3.74) are

0(1) = —(§WZ(I>+§WO..(1)) (3.78)
U,.(1)+VU,.(1) = $(8+C) (3.79)

Imposing the boundary conditions (3.78) and (3.79) into (3.77), We obtain

BlJo(ko)+B210(ko) = b.7-‘(k0) -72

B.M(Jo(ko)> + 8.1700(6)) = bM(F(1)) + as + C) (3.80)

where f(r) = kor(J1(kor) —a111(k0r)) / 4 and M () is the moment operator as before.

35

 

 

0.1

0.08

0.06

0.04

0.02

0.01

0

 

 

 

2.57748

 

2.46656

 

2.36862

 

2.29041

 

2.23934

2.22602

2.22151

 

 

 

 

Table 3.2: Fundamental frequency of simply-supported wavy plate

In order for the system (3.80) to have a nontrivial solution, the rank of the coeffi-
cient matrix has to be equal to the rank of the augmented matrix. Therefore we have

the following equation

Io(ko) bf(1) - 7?.
= 0 (3.81)
M(Io(ko)) bM(f(1)) + (S + C)/2
which gives us a unique solution for b. Hence
10066) (S + c) + 2M(Io(ko))7z (3.32)

 

= 2[M(Io(k6))f-(ko) - 16(ko)M(f(1))]
which is the first correction to the fundamental frequency of a simply supported plate
with wavy boundary. Table 3.2 below lists the values of the fundamental frequencies
kg with various 6 for M = 6. The frequency itself depends on the amplitude «sand is

given by equation (3.27).

3.3 Elliptic Plates

We consider an elliptic plate in the my plane with semi-major and semi-minor radii

a and b respectively

2 2

a2 ()2 a>b.

(3.33)

This problem has no direct analytic and tractable solution yielding the required eigen-
values. Therefore, we proceed with a perturbation scheme adopting the boundary

perturbation method presented in section (3.1).

36

The solutions of the eigenvalue problem (2.2) with boundary conditions (2.6)-
(2.12) are obtained by considering ellipse as the perturbed boundary with respect to
the circle with an averaging radius 1'. Generic points on the ellipse may be defined

by the coordinates
:1: = acos 6, y = bsin6 0 _<_ 6 g 27r. (3.84)

Then the variable radial distance from the origin to a generic point on the ellipse is

 

r = \/0.2 cos2 6 + b2 sin2 6. (3.85)

In order to calculate the averaging radius, we perturb the semi—major and semi-

minor radii as follows

a = 1 + cal + 6202 + 0(63) (3.86)

b = 1 — ebl — 6202 + 0(3). (3.87)
Then the asymptotic series expansion of r in 6 becomes

1
T(9) = 1+ 5 [(0.1 — 01) + ((11 + 01) COS 26]€
1
+ E [(a1 - b1)2 + 8(a2 — b2) + 8(a2 + b2) cos 26

— (a1 + b1)2 cos 46]e2 + 0(63). (3.88)

We determine a1 , b1, a2, and b2 such that the mean radius of the perturbed circle is

1. To this end, set

211'
1 = 51; 0 7(6) d6 (3.89)
which gives
1 1 2 2
5(01 — (906 + 1—6 (0.1 + 0.2) '1' 8(02 — 02)]6 =3 O (3.90)

for the first two orders.

37

On the other hand, for moderately elliptic plates define the eccentricity parameter

by 82 = 1 — bz/az. Then the asymptotic expansion of it inc becomes
e2 = 2(a1 + a2)e + (—3a'f — b? + 2a2 + 2b; — 4a1b1)e2 + 0(63). (3.91)

Since we consider moderately elliptic plates, eccentricity e is small. Therefore we
take eccentricity to be e2 / 4 = e, which gives first order term to be 2 and the higher
order terms to be 0 in (3.91). Collecting the same order terms in (3.90) and (3.91)
results a1 = bl = 1, which give the following perturbed averaging radius in orders of

eccentricity e
62
r(6) = 1 + 4 cos 46 + 0(e4). (3.92)

The boundary is then expanded in Fourier series:

7' = 1 + f (6)
f(0) = Z c. cos (2nM6) (3.93)
n=1
where c1 = (32/4, -- - , and M = 2 is the number of circumferencial waves. Thus the

boundary conditions in section (3.1) apply to r = 1 + f (6).
The ellipticity 5 = a/ b — 1 of the plate, however can be also used to obtain f (6),

in which case we have the asymptotic expansion of the ellipticity
a = (a1 + b1)e + (a2 + b2 + albl + b¥)os2 + 0(63). (3.94)

In moderately elliptic plates, we take ellipticity e to be small; therefore, let 5 = e,
which gives first order term to be 1 and the higher order terms to be 0 in (3.94).
Collecting the same order terms in (3.90) and (3.94) results a1 = bl = 1/2, a2 =

—5/ 16, b2 = —3 / 16 which give the following perturbed averaging radius
2 2

1(6) = 1+ (g — 53-) cos 26 — 52—005 46 + 0(64). (3.95)

38

Then similarly, the boundary is expanded in Fourier series:

7‘ = 1 + f (6)
f(6) = Z 6,. cos (266) (3.96)
n=1
where c1 = 6/ 2 - 62/4, c2 = —€2 / 2, . Thus the boundary conditions in section

(3.1) again apply to r = 1 + f(6).

3.3.1 Clamped Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency k about the circular

state as follows. We let

W(r, 6) = W0(r) + 6W1(T, 6) + €2W2(r, 6) + 0(63) (3.97)

k4 = k3(1+ £26 + - - -) (3.98)

where c is the correction to the fundamental frequency. Notation for the correction
is taken to be c in order not to confuse with the minor-axis b of the ellipse.
We only consider the first two orders of f (6) in (3.96). The clamped boundary

conditions are that

W(1 + f(0), 6) = 0 (3.99)
W
6n" r=1+m> = (3.100)

where n is the unit normal vector. Then the equations (3.13) and (3.16) give the
boundary conditions of the perturbed boundary value problem up to the second order.

Then the zeroth order equation is given by
V4Wo — kgwo = 0 (3.101)
with the boundary conditions

won, 6) = 0 mu, 6) = 0 (3.102)

39

which corresponds to the circular plate (for n = 0). The solution of the equation
(3.101) with the boundary conditions (3.102) is given before by (3.35)

The first order 0(6) equation is given by
V4W1 — k3W1 = 0 (3.103)
subject to following boundary conditions

W1(1, 6) = 0 (3.104)

W1.(1,9) = _f(6)W0rr(1)1 (3-105)

since Wo,(1,6) = 0 and W0 = Wo(r) in the boundary condition of order 0(1).

Solution of the equation (3.103) is

 

 

W1('r, 6) = 5°: [0‘1"1 J2nM(koT) + C?212nM(kor)] cos 271M 6 (3.106)

where ":1
Oil = _CnI2nA;-E:(()])C:I)IOTT(1) (3.107)
{.2 = CnJ2nA;-l(f?]::;/0rr(l) (3.108)

where Hn(ko) = Ian(ko)JénM(ko) — IénM(ko)J2nM(k0) and M = 2. due to two
circumferencial waves in elliptical plates.

The second order 0(62) equation is
V4W2 — 16ng = ckgwo (3.109)
subject to following boundary conditions

W2(1,6)

<I>g(1,6) (3.110)

W2.(1,6) = \I'§(1,6) (3.111)

Where 950.9) = éf2(9)Wo..(1) and \I’§(1.9) = f’(9)W1.(1,9) - f(9)W1..(1,9) -

% f2(6)Wom(1) are simplified version of the equations (A2) and (A.4) respectively in

the appendix.

40

Assuming a solution of type W2(r, 6) = U (r) + 2:11 Vn(r) cos (271M 6) for (3.109),

we obtain following homogenous and non-homogenous equations

V4V — kSV = 0 (3.112)
and

WW — ng -_— ckgwo (3.113)
respectively. The solution of the homogenous equation (3.112) is given by

V(r) = Z A'fJgnM(kor) + 4312.M(kor) (3.114)

n=1

where A1 and A2 are arbitrary constants. Since the solution of (3.112) has no effect

on correction c to the fundamental frequency, we solve

V4U - ng = ckéWo. (3.115)
The solution of the homogenous part is

Uh(r) = BlJo(ko'r) + B210(kor) (3.116)

where BI and Bg are arbitrary constants. The particular solution is

Up(r) = —£]:—0:(J1(kor) — 0011(kor)) (3.117)

where (10 as in (3.160) and Cm = 1. Thus the general solution of (3.115) is

ckor

U(r) = BlJo(kor) + 3210066) — —4— (J1(kor) — 0011(koT‘)). (3.118)

where BI and 32 are arbitrary constants. The boundary conditions (3.110) and

(3.111) can be written as

<I>§(r, 6) = 903(1) + Z cpf,(r) cos 471.6 (3.119)
n=l

‘II§(7', 6) = {8(1") + Z {:(1‘) cos 4116 (3.120)
n=1

41

respectively. Then the boundary conditions for (3.115) become
U0) = 798(1) (3121)
U.(1) = €80) (3-122)

Where cp3(1) and {5(1) are the constant terms of the second order perturbed clamped

boundary conditions <I>§(1,6) and \IJ§(1,6) respectively. These constant terms are

that
C 1 w
i60(1) = ZWoAl) C3:
n=1
C 1 w m n
6.0) = -ZW0...(1)ZC:+4ZCnn2Wf (1)
n=1 n=1
1 °° n
— ézc‘nwfrr(1)'
n=1

Where W1(r, 6) = 2:11 Wf"(r) cos (4716) with
wfn(r) = [03.14.0163 + Cf214n(rko)]. (3.123)

Imposing the boundary conditions (3.121) and (3.122), into (3.118) we obtain,

BlJ0(k0) + 3210090) = CFO) + 903(1) (3-124)
—Bl.]1(ko) + 3211060) = ng'LI) + gig—1) (3.125)
0 0

where F (r) = kor[J1(kor) — a011(kor)] /4. Then the following determinant equation

gives us a unique solution of c:

10660) 017(1) + $80)

 

= 0 (3.126)
11666) cF’(1)/ko + 58(1l/ko
and hence
__ k011(k0)<P3(1) - Io(ko)€8(1)
C "’ 16(ko)F'(1) — 3611(6))4‘0) (3‘12?)

which is the first correction to the fundamental frequency of a clamped elliptic plate.

All the published values of the fundamental frequencies are normalized with respect

42

 

 

e k Prasad et 01. Jones Leissa
0.8 3.44216 3.66424 3.66424 3.65145
0.6 3.28013 3.28478 3.28557 3.28363
0.5 3.23750 - - -

0.4 3.21332 - - -
0.3 3.20166 - - -
02 3.19730 - - -
0.1 3.19629 - - -
0 3.19622 3.19624 3.19609 3.19624

 

 

 

 

 

 

 

 

Table 3.3: Fundamental frequency of clamped elliptic plate

to either major or minor axes of an elliptical plate. The present result however is
normalized w.r.t. the averaging circle. In order to obtain the frequencies w.r.t. the
normalized radius, we use the following relationship between the averaging radius and

the axas (e.g., for b)

 

b 21f \/ a 2 2 . 2
r — E 0 (3) cos 6 + sm 6 d6. (3.128)
Table 3.3 below lists the values of the fundamental frequencies 16 with various eccen-

tricity e and compares the frequencies with published results. The last row of Table

3.3 gives the fundamental frequency k of a clamped circular plate.

3.3.2 Simply Supported Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency 11: about the circu—
lar state as in (3.97) and (3.98) respectively. Then the simply supported boundary

conditions are that

 

W(1+ f(6),6) = 0 (3.129)
62W u 6W
( 6712 + BB—n) r=1+f(6) _ O (3.130)

 

where n is the unit normal vector, u is Poisson ratio, and R is the normalized

radius of curvature. We form the boundary conditions from the perturbed boundary

43

condition equations (3.13) and (3.22) for each order. Similarly collecting the same

order terms we obtain the zeroth order equation as

V4Wo — kgwo = 0 (3.131)
with the boundary conditions

Wo(1,6) = 0, Wo,,(1,6) + VW0,(1,6) = 0 (3.132)

which corresponds to simply supported circular plate (for n = 0). The solution of

the equation (3.131) with the boundary conditions (3.132) is given by
W0(T, 0) = 001[J0(k07') + (1110(k07“)] (3.133)

where al = —Jo(k0)/Io(ko) and k0 = 2.22152 is the fundamental frequency of the
simply supported circular plate (solution of (3.63)). Taking Cm = 1 for simplicity,

we have

Wo(r, 6) = Jo(kor) + a110(k0r). (3.134)
Then the first order 0(6) equation is given by

V4W1 —- kSWI = 0 (3.135)
subject to following boundary conditions

W1(1, 6) = <1>f(1, 6), W1,,(1, 6) + uWI.(1, 6) = 015(1, 6). (3.136)

where the first order 0(6) perturbed boundary conditions (I);9 (1,6) and q}? (1,6) are
given by the equations'(A.5) and (A.7) respectively. The solution of the equation

(3.135) with two circumferencial waves M = 2 is

W1(7‘, 6) = Z [031.]4n(k07") + 3214n(k07‘)] COS 477.6. (3.137)
n=1

44

Imposing the boundary conditions (3.136), we obtain
n _ N3W0,(1) + Vf(9)14n(ko)Wo..(1) + f(9)14n(ko)Wo...(1)

 

 

05.2 = N3W6.(1> — Vf(9)J4n(ko)V§1:(1)- f(9)J4n(ko)Wo...(1)
where n
N: = 461131.06) — vf(6)14..(k6) — 10(6) + f"<6))1..(ko),
M: = f(9)J4’..(ko) + uf(6)J;.(ko> + 10(6) + f"(6))J..(ko>,
and

D. = I4h(k0)J4n(k0) — 146(ko)J£'n(ko) + V(1£n(ko)J4n(ko) - 146(ko)J£..(ko))-
The second order 0(62) equation is
V4W2 — k3W2 = ckSWo (3.138)
subject to following boundary conditions
W2(1,6) = <I>§(1,6), W2,,(1,6) + uW2,(1,6) = 913(1, 6). (3.139)

where the second order 0(62) perturbed boundary conditions <I>§ (1, 6) and 1113 (1,6)
are given by the equations (A.6) and (A8) respectively. We assume a solution of
type W2(r, 6) = U (1')+Z:°___1 Va (7) cos (4n6) for (3.137), then we obtain the equations
(3.112) and (3.115) as before. Since the solution of (3.112) has no effect on correction
c to the fundamental frequency, we solve (3.115). The general solution of (3.115) is
given by (3.118).

Let W1(r, 6) = 2:1 W14"(r) cos (4716) where

14/1470) = [03,J4..(rko) + 03214,.(rko)]. (3.140)
The boundary conditions for U (r) can be written as
@303 6) 2 993(7) + 2 6:50) cos 4716 (3.141)
n=1
\Il§(r, 6) = {5(1") + 2630") cos 4716. (3.142)
71:1

45

Since we consider the cos (4n6)-free part, the boundary conditions of of U (1‘), which

are the constant terms of (3.139) become

U0) = 630) (3.143)

0.0) + uU.0) = 660). (3.144)

The constant terms in (3.143) and (3.144) are

«63(1) = -iWo..(1):Ci-%§:anf?(l)
6.50) = we..(1):jlci(8n2—u$‘33§:i)—-’iw.,,,(1):;cz,
— i 0.... :31 +204 (8")nu—(4n )")W‘”‘()
+ i900)— “ill—3 )0W14z‘) 324w
=1
- g 00 anfMl)

=1

Imposing the boundary conditions (3.143) and (3.144) into (3.118), We obtain

BiJo(k0)+leo(ko) = 0F(1)+<P§(1) (3.145)

3.4/10606)) + 82440606» cM(F0)) + 450) (3.146)

where F(r) = kor(J1(k0r) - a111(kor))/4, M(Jo(ko)) = J6’(ko) + VJ6(k0), and
M(Io(ko)) = 16’(ko) + V16(k0). Here M is the moment operator.

Then the following determinant equation gives us a unique solution of c:

I0090) 017(1) + 906(1) = 0 (3.147)
M(10(k0)) CM(F(1)) + 56(1)

and hence

M(Io(k0))906(1)- 10(kolf050)

= M(F0))1.06)— 00440.06) (3148)

.46

 

 

 

a / b 6 k Prasad et 611. Jones
1.5 0.74535 2.34804 2.40981 2.44784
1.4 0.6998 2.32159 2.34563 2.38660
1.3 0.639 2.29246 2.28993 2.33250
1.2 0.5528 2.26207 2.24519 2.28744
1.1 0.4166 2.23484 2.21479 2.26098
1.05 0.3 2.22513 - -
1.02 0.2 2.22223 - -
1.005 0.1 2.22156 - -
1.0 0 2.22152 2.2033 2.24572

 

 

 

 

 

 

 

 

Table 3.4: Fundamental frequency of simply supported elliptic plate (12 = 0.3)

which is the first correction to the fundamental frequency of a simply supported
elliptic plate. Table 3.4 below lists the values of the fundamental frequencies k with
various eccentricity 6 and compares the frequencies with published results.

In order to obtain the frequencies w.r.t. the normalized averaging radius, we use
the relationship (3.128) between the averaging radius r and the minor-axes b. The
last row of Table 3.4 gives the fundamental frequency of simply supported elliptical

plate.

3.4 Polygonal Plates

Let a be the radius of the inscribing circle of a regular polygon with M sides. Then

in polar coordinates (r, 6) one side is given by

a

 

1- = -—6 s 6 s 6 (3.149)

0036’

where 6 is the half central angle vr/M . We determine a such that the mean radius

of the polygon is 1 by
1 fl

a=———-= .
1f03c0_:9d9 111[tan(§+ §)]

 

(3.150)

47

We then expand the boundary in Fourier series:

7. =1+ f(g) (3.151)
f(6) = chcos (nM6) (3.152)

where

__ 2a 5 cos (nM6)

Cn-F 0 cos6

66. (3.153)

For tabulated values of an ’s (see Wang 1998)[24]. Thus the boundary perturbation in
section (3.1) applies to r 2 1+ f (6) We consider polygonal plates with two different
boundary conditions: clamped and simply supported. These boundary conditions
are given in equations (2.9)-(2.12). The more terms in the Fourier series (3.152) will
result a better approximated polygon. Since the corners of the polygon are rounded
by this process, the radius of curvature R in the boundary conditions contributes to

the solution.

3.4.1 Clamped Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency 1: about the circular

state as follows

W(1", 6) = Wo(r) + €W1(T, 6) + 62W2(7‘, 6) + 0(63) (3.154)

k4 = k3(1+ 620+ - - -) (3.155)

where b is the correction to the fundamental frequency. The clamped boundary

conditions are that

W(1+ 1(6), 6) = 0 (3.156)
3W
6: r=1+f(6) — 0 (3.157)

where n is the unit normal vector.

48

The equations (3.13) and (3.16) give the boundary conditions of the perturbed
boundary value problem up to the second order. Then the zeroth order equation is

given by
' V4W0 - kgwo = 0 (3.158)
with the boundary conditions
Wo(1,6) = 0 W0r(1,6) = 0 (3.159)

which corresponds to the circular plate (for n = 0). The solution of the equation
(3.158) with the boundary conditions (3.159) is given before by (3.35).

The first order 0(6) equation is given by
V4W1 — k3W1 = 0 (3.160)
subject to following boundary conditions

W1.(119) = _f(0)W0rr(]-)1 (3'162)

since Wo,(1,6) = 0 and W0 = W0(r) in the boundary condition of order 0(1).

Solution of the equation (3.160) is

co

W1(r, 9) = Z: [CI'IJnM(koT') + szlnM(kor)] cos nM6 (3.163)

n=1

where M is the number of circumferencial waves in a polygon. Imposing the boundary

conditions (3.161) and (3.162), we obtain

 

 

_anInM(k0)W0 (1)
n = " 3.164
11 JnM(k0)Ir’1M(k0) - 1’1M(k0)InM(k0) ( )
"aanM(k0)W0 (1)
C"1 = " 3.165
‘2 J.M(ko)1:.1.(ko) — 11.061.11.00) ( )
The next order 0(6") of the boundary value problem is
V4W2 — 11ng =.- 663% (3.166)

49

subject to following boundary conditions

W209) §12(6)w1..0) (3.167)

W2.(1.0) = f'<6)w1.0,6)—f<6)wl..0.6)—$120)Wo...0) (3168)

obtained from 4>§(1,6) and \I'§(1,6) in the appendix.
Assuming a solution of type W2(r, 6) = U (7‘) + 2111610") cos(nM6) for the

equation (3.166), we obtain
V‘V — kgv = 0 (3.169)
which has a solution of

V(’") = ZAniJnM(k07“) + AnzlnMUCOT), (3.170)

n=1

where Am and Ang are arbitrary constants, which has no effect on correction b to

the fundamental frequency. So we solve

V4U — ng = bkgwo. (3.171)
The solution of the homogenous part is

Uh(r) = BlJo(kor) + 82100607“) . (3.172)

where BI and 82 are arbitrary constants. The particular solution is

Up(r) = —b—l:9r-(J1(kor) — aoll(kor)) (3.173)

where do as in (3.160) and Cm = 1. Thus the general solution of (3.171) is

U(T') = B1J0(ko7‘) + BgIoUCoT’) — 21%(J1(k07‘) — CloIIUCoT». (3.174)

The boundary conditions for (3.174) are the cos (nM6)-free part of (3.167) and
(3.168). To this end, let
W1(r, 6) = Z Wf‘M(r) cos nM6 (3.175)
n=1

50

where

Wf‘M(r) = C?1JnM(kor) + 01121,...(1101) (3.176)
and let
<I>§(r, 6) = 908(7) + :0: (02(1) cos nM6 (3.177)
n=1
@130”, 6) = 68(7‘) + :0: £30") cos nM6. (3.178)
n=1

Then the boundary conditions for (3.174), which are the constant terms of (3.167)
and (3.168), become

11(1) = 808(1) (3-179)
(11(1) = 68(1) (3.180)
where
1113(1) = iWo"(r):ci (3.181)
680‘) = éMzgn291WI‘M (T)-%:anfily(7‘)
— iWo...(T)§:;C§1 (3.182)

Thus, the frequency correction b can be determined by imposing the boundary

conditions (3.179) and (3.180) into (3.174). Then we have the following equation 1

100170) 908(1)+ bF(1) = 0 (3.183)
11016) 68(1) + bF’(1))
where F (r) = 1cJifr-(JlUcor) -— a011(kor)), and hence

b = 110909030) - 10050580)
10(kolF'(1) ‘ 11030170),

which is the first correction to the fundamental frequency of a clamped polygonal

(3.184)

plate. The fundamental frequency 16 for M = 6 (hexagon) with n = 12 is found to

51

 

N 3 6 12 24
16 3.26493 3.26715 3.26784 3.26803

 

 

 

 

 

 

 

 

Table 3.5: Fundamental frequency of regular clamped hexagon

be 3.26784. In practice the corners may not be mathematically sharp and a finite N
would be desirable. For determination of N, see Wang 1998 [24]. Boundary function
f (6) gives an acceptable approximation to an hexagon for N = 12. Table 3.5 shows
the frequency approximation to an hexagon inscribed by a circle of radius a as N
increases.

All the published values of the fundamental frequencies are normalized with respect
to either inscribing or circumscribing radius of a circle for a polygonal plate. The
present result however is normalized w.r.t. the averaging circle. In order to obtain ‘
the frequencies w.r.t. the averaging radius, we use the following relationship between

the averaging radius and the inscribing radius a for a polygonal plate

 

 

6
r = % 0 301:5 d6 where 6 = 7r/M. (3.185)
Authors Fundamental Frequency
Present 3.26803
Irie(2) et al. 3.24905
Irie(l) et al. 3.21863
Shahady et al. 3.25639
Walkinshaw—Kennedy 3.25010
Liew—Lam 3.44329
Yu 3.30990

 

 

 

 

Table 3.6: Fundamental frequency of regular clamped hexagon

The result obtained for fully clamped hexagonal plate is given in Table 3.6 together

52

with the published values.

3.4.2 Simply Supported Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency 16 about the cir-
cular state as in (3.154) and (3.155) respectively. The simply supported boundary

conditions (vanishing displacement and moment) are that

 

02W V 6W
( 8n" + RE) r=1+f(0) _ 0 (3.187)

 

where n is the unit normal vector, V is Poisson ratio, and R is the normalized
radius of curvature. We form the boundary conditions from the perturbed boundary
condition equations (3.13) and (3.22) for each order. Collecting the same order terms

we obtain the zeroth order equation as

V4Wo — kgwo = 0 (3.188)
with the boundary conditions

Wo(1,6) = 0, Wo,,(1, 6) + VW0,(1,6) = 0 (3.189)

which corresponds to simply supported circular plate (for n = 0). The solution of

the equation (3.188) with the boundary conditions (3.189) is given by
Wo(7‘, 6) = J0(koT) + 0110(k07‘) (3.190)

where 611 = —Jo(ko)/Io(ko) and k0 = 2.22152 is the fundamental frequency of the
simply supported circular plate (solution of (3.63)).

Then the first order 0(6) equation is given by

V4W1 — k3W1 = 0 (3.191)

53

subject to following boundary conditions

W1(1,6) .—_ <I>f(l,6) (3.192)

W1"(1, 0) + VW1r(1, 0)

111? (1, 6) (3.193)

where (Pf (1, 6) and 1119(1, 6) are given by the equations (A5) and (A.7) respectively.

Then the solution of the equation (3.191) with M circumferencial waves is

co

W1(r, 6) = Z [CflJnM(k0r) + C?2InM(kor)] cos nM6. (3.194)

n=1

We write the boundary conditions (3.192) and (3.193) as

 

(P190; 6) = Z (0,5;(7‘) cos nM6 (3.195)
n=1
1115(1‘, 6) = 2103(7) cos nM6 (3.196)
n=1
where
03(7) = k0(.]1 (1907”) — (1111(k07‘)) Z 6,; (3.197)
n=1
1613(7) = [— ékgalVUOUCOT) + 12(k07‘)) (3.198)
— [6001(3160 — V + M2V)Il(k0r) - 211-1631111461) (3.199)
+ ék3V(Jo(kor) — J2(kor)) + 211-330(67) (3.200)
_ (3163 + 1601/ — k0 M21.) 3001)] Z c. (3.201)
n=1
Imposing the boundary conditions (3.192) and (3.193), we obtain
0n = InM(k0)z/}S(1) _ ¢S(1)M(InM(k0))
11 JnM(ko)M(-’nM(ko)) - InM(ko)M(JnM(ko))
n _ 111103011150) - ¢.S.(1)M(JnM(ko))

 

12 _ JnM(ko)M(InM(ko)) - 1nM(ko)M(JnM(ko))
where M is the moment operator, e.g. M(JnM(kor)) = ,’,’M(kor) + VJ,’,M(kor).

The second order 0(6") equation is
V4W2 — k3W2 = bkgwo (3.202)

54

subject to following boundary conditions

W2(1,6) = <03" (1, 0) (3.203)

W2,,(1,6)+VW2,(1,6) = \II§(1,0) (3.204)

where (P59 (1,6) and \115'; (1,6) are given by the equations (A.6) and (A.8) respectively
in the appendix.

We assume a solution of type W2(r, 6) = U (r) + 22:, Vn(r) cos (nM6), then
(3.202) results homogenous biharmonic equation in Vn(r) which has no contribution
to the correction b of the fundamental frequency, therefore we solve non-homogenous
biharmonic equation (3.171). The general solution of (3.171) is given earlier by
(3.174). The boundary conditions for U (r) are the cos (nM6)-free part of (3.203)
and (3.204). So, we let

W1(r, 6):: Wf‘M(r )cos nM 6 (3.205)

where Wf‘M(r) = CH J:M(kor) + CIZInM(kor).
We write the boundary conditions (3.203) and (3.204) as

<I>§(r, 6) = (pS()r )+Zcp,3, (r) cosnM6 (3.206)
\II§(1',6) = £5(r)+2€f(r)cosnM6 (3.207)
n=1
where
103(1) = —§Wo..(1)2ci—§Zc.wl':“<1)
n=1
1505(1) = :Wo..(1)zci[( nM) <+1—(nM))] ——wo...(1)Zc3.
n=1 n=1
— :- w.....(1)2c3.—-Zc.<nM) (2—u)W"M(1)
n=1
+ "2:;Cn[(nM)2+)§(1—(nM))] W12“ 1)— 2::an 1"..(
1 oo
“ 5n an 1n...(

=1

55

 

N 3 6 12 24
k 2.39704 2.45201 2.50631 2.54853

 

 

 

 

 

 

 

 

Table 3.7: thdamental frequency of regular simply-supported hexagon

Thus we have the following boundary conditions for U (r)

11(1)

1.03 (1) (3.208)

U..(1) + uU.(1) = {5(1) (3.209)
Imposing the boundary conditions (3.208) and (3.209) into (3.174), We obtain

B1Jo(ko)+3210(k0) = bF(1)+so€(1) (3.210)

BrM(J0(k0))+BzM(Io(ko)) = bM(F(1))+5€(1) (3.211)

where F(r) = kor(.]1(kor) -— 0111(k01‘))/4 and M is a moment operator acting as
M(Jo(ko)) = J6’(ko) + VJ6(ko). Then the following determinant equation gives us a

unique solution of b:

s
10(k0) 1917(1) + $00 (1) ___ 0 (3.212)
M(Io(ko)) bM(F(1)) + 56(1)

and hence

b = M(Io(k0))90§(1)- 10(k0)§09(1)
M(F(1))10(k0) - F(ko)M(Io(ko))

(3.213)
which is the first correction to the fundamental frequency of a simply supported
polygonal plate. The frequency 10 for M = 6 (hexagon) with n = 12 is found to
be 2.50631. Table 3.7 shows the frequency approximation to a simply supported

hexagonal plate with various N.

Table 3.8 below compares the values of the fundamental frequencies for a simply

supported hexagon with published values.

56

 

 

 

 

Authors . Fundamental Frequency
Present 2.50631
Irie(2) et al. ' 2.43705
Irie(l) et al. 2.43495
Shahady et al. 2.43495
Walkinshaw-Kennedy 2.44439
Liew-Lam 2.56179
Yu 2.46223
Conway 2.42341

 

 

 

Table 3.8: Fundamental frequency of regular simply-supported hexagon

The same factor (3.185) for converting the frequencies with respect to the averaging

radius has been used.

57

CHAPTER 4

Fundamental Frequencies of Plates

With a Core

This chapter presents a perturbation method to study free vibration of wavy bound-
ary plates, elliptic plates, and polygonal plates, all with concentric circular core. The
core has the clamped boundary condition throughout this chapter. Two different
outer boundary conditions are considered: clamped and simply supported. The nat-
ural frequencies of these plates are determined by calculating the eigenvalues of the
governing equations using perturbation methods. The perturbation solutions yield

analytic approximate solutions of the fundamental frequencies for these plates.

4.1 Wavy Boundary Plates With a Core

Consider a wavy boundary plate with a concentric circular core of radius c. We
perturbed the outer boundary of an annular plate to obtain wavy boundary plate
with a concentric circular core. The governing Biharmonic equation of a circular

plate with a concentric circular core in polar coordinates (r, 6) is given by

v4W(r, 0) — k4W(r, 0) = 0 (4.1)

58

and corresponding general solution, in polar coordinates, is given by

W(r, 6) = i[Cn1Jn(k'r) + Cn2Yn(kT) + Cn31n(kr) + Cn4Kn(k'r)] cos 716 (4.2)

n=0
where Cal, Caz, 07.3, CM are arbitrary constants as discussed in chapter 2. The
solution (4.2) is finite, since 1' can not be zero due to the concentric core.

We proceed with a perturbation scheme adopting the boundary perturbation
method prasented in section (3.1). Then the solution of the eigenvalue problem with
clamped and simply supported boundary conditions will be obtained by considering
wavy boundary plate as the perturbed boundary with respect to the circle with an

averaging radius 1' given in (3.96).

4.1.1 Clamped Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency k about the circular

state as follows. We let

W(r, 6) = Wo(r) + 6W1(T, 6) + €2W2(T, 6) + 0(63) (4.3)

k4 = k3(1+ 80 + - - -) (4.4)

where b is the correction to the fundamental frequency as before.
Consider wavy circular boundary by taking f (6) = cos (M 6) as in the section 3.1,

and let
T = 1+ ecos(M6) (4.5)

where 6 << 1 is a small amplitude and M 2 2 is the number of circumferencial waves.

59

Then the clamped boundary conditions (both inner and outer boundary) are that

 

W(1 + ecos (M6), 6) = 0 (4.6)
BW
37:- r=1+ccos (M6) — 0 (47)
W(c, 6) = 0 (4.8)
6W
6:; - 0 (4-9)

 

where n is the unit normal vector. The first two boundary conditions (4.6) and
(4.7) can be expanded into asymptotic series (3.13) and (3.16) as before. Using these

asymptotic boundary conditions and substituting (4.3) and (4.4) into
V4W(r, 6) — k4W(r, 6) = 0 (4.10)
and collecting the similar order terms we obtain the zeroth order equation as
V4Wo(r, 6) — k3Wo(r, 6) = 0 (4.11)

with the boundary conditions

W1,6 =0 ,Wc,6 =0
0( l o( ) (4.12)
Wo,(1,0) = 0 , Wor(C,6) = 0

which corresponds to the circular plate (for n = 0) with a concentric circular core of
radius c. The solution of the equation (4.11) with the boundary conditions (4.12) is

given by
W0(7‘,9) = C'01Jo(k07") + 002360907") + C0310(ko7‘) + C04Ko(ko7‘)- (4-13)

Imposing the clamped edge boundary conditions and using the recursion formulas for

60

the Bessel functions, we obtain the characteristic equation

 

—C:kg + 11(ck0)[1$(cko)(J1(ko)Ko(ko)— J0( ((ko)K1 (1%))
+Jo(Ck0)(K1(ko)Y0(ko)“K0(ko)Y1(ko))]
+11(k0) [Ko(cko)(J1(cko)Yo(ko) J0 (ko)Y1(cko))

+K1(cko)(Jo(ko)Yo(cko)— J0 (cko)Yo(ko))] (4.14)
+Io(k0)[K1(Cko)(J1(k0)Yc)(Cko)— Jo(Cko)Y1(ko))
+Ko(cko)(J1(cko)Y1(ko)— J1(ko)Y1(cko) )]
+Io(Cko)[J1(Ck0)(K1(k0)Yo(ko) — Ko(k0)Y1(k0))
+Y1(cko)(J1(ko)Ko(ko) — Jo(ko)Kr(ko))] = 0

By taking limit of the characteristic equation as c _. 0, we obtain the frequency

equation
ko(2J1(ko)Ko(ko) — 2Jo(ko)K1(k0) + «11(k0)Y0(k0) + r10(1r0)1/1(1c0)) = —4 (4.15)

the first zero of which gives the fundamental frequency k0 = 4.76831 of the clamped
circular plate with a concentric circular core as c -—> 0. Then the solution (4.13) may

be written as
W0(T, 0) = 001[J0(7‘k0) + Gall/()(Tko) '1' 0821007790) + (183K0(Tk0)] (4.16)

where 0:31, 0:32, and £133 are given in the appendix by (B.1), (B2), and (B3) respec-
tively; and without loss of generality we can take Cm = 1.

Then the first order 0(5) equation is
V4W1(T, 6) — k3W1(T,6) = 0 (4.17)

subject to following boundary conditions

W1(1,6) = 0 , Wo(c, 6) = 0
Wi.(1i9) = -COS(M9)W0rr(1.9) , W0.(C.9) =0

(4.13)

61

since Wo,(1, 6) = 0 and W0 2 Wo(r) in the outer boundary condition of order 0(1).

Solution of the equation (4.17) with boundary conditions (4.18) is
W1(r, 6) = [aflJM(rko) + aizYMUko) + a§3IM(rko) + 054KM(rk0)] cos (M6) (4.19)

where C151 , 0152, 0:53, 0254 are given in the appendix by the equation (B.7).

Let W10, 6) = 1371(7) cos (M6) where
v’v‘1(r) = [name/co) + o52vM(rko) + o§31M(rko) + name/cw]. (4.20)

Then the second order 0(62) equation is
V4Wr)(r, 6) — k3W2(r, 6) = bk3W0(r) (4.21)

subject to following boundary conditions

1 1
W2(1r9) = 2W0..(1)+ZW0..(1)005(2M9)

wane) = $1710) — $1731) - imam)

+ (—M;@(1)-%W:(1)—iWorrr(1))COS(2M9) (4.22)
W2(c,6) = 0
W2,(c,6) = 0

Assuming a solution of type W2('r, 6) = U (r)+V(r) cos (2M 6) for (4.21), we obtain
V4V(r, 6) — k3V(r, 6) = 0 (4.23)

the solution of which has no effect on correction b to the fundamental frequency.

Therefore, we solve _
V4U(r) — k3U(r) = bkgwo(r). (4.24)
The solution of the homogenous part of (4.24) is

Uh(7‘) = BlJ0(koT) + Bgl/oUiloT) + B310(k07‘) + B4Ko(k07‘) (4.25)

62

where B1, B2, B3, and B4 are arbitrary constants. The particular solution of (4.24)
is

bk 1‘
0,0) = "40— (J1(k0r) + a81Y1(kor) — a8211(k0r) + 053K1(k0r)). (4.20)

Thus the general solution of (4.24) is

U(r) = Ua(r)+Ur(r)

= BlJo(k07') + lefoUmT) + B310(k07‘) + B4Ko(ko'l‘)

b r
—]:O— (J1(ko7‘) + QSIKUCOT) — 08211(k07‘) + 033K1(k0r)) . (4.27)
The boundary conditions for (4.24) are the cos (2M 6)-free part of (4.22)

11(1) = lWino)

4
0.0) = §fi(1)—§W;(1)-§wa-<1) (4.28)
U(c) = 0
U,(c) = 0.

Imposing the boundary conditions (4.28) into (4.27),

1
81.10060) ‘1' BgYoUCo) + B3]o(ko) + B4Ko(ko) == bF(1) + "Wo"(1)

4
b ,
B1J1(ko) + BzYlUco) - 3311(k0)+ 34K1(k0) = _EF (1) + G (4-29)
Bljo(Cko) + 32%(Cko) + B3Io(Cko) + B4Ko(Cko) = bF(C)
b ,
31J1(Cko) + 822/1 (Cko) — 8311(Ck0) + B4K1(Cko) = —;;F (C)
where
1:
F11) = +1? (Jitter) + airYrtkor) — aialrtkrr) + 083K1(kor))
M2 A l A l
G — -2k—0W1(1)+ 2—k0W1"(1) + Ewan“)-

63

In order for the system (4.29) to have a nontrivial solution, we need

( «10090) 3,0090) 10090) KOUCO) \
Rank J1(k0) 3,1090) —11(k0) K1090) ___
Jo(Cko) %(Cko) 10(Cko) Ko(Cko)
\J1(ck0) 13(ck0) —11(ck0) K1(ck0) )

( Jo(ko) Yo(ko) Io(ko) Kouco) bF(1)+wo,,(1)/4)
J1(ko) Y1(ko) —Il(ko) K1000) —bF’(1)/ko+G
Jo(cko) Yo(cko) 10(ck0) K0(cko) bF(c)
(Metro) Y1(ck0) —Il(ck0) K1(cko) —bF’(c)/ko )

 

 

 

 

It follows that

YOUCO) I0(k0) K0090) bF(1) + Wo..(1)/4
“((C0) ‘11(k0) K1090) —bF'(1)/k0 + G
Yo(Cko) 10(Cko) K0(Cko) bF(C)
Y1(cko) —Il(cko) K1(cko) —bF’(c)/ko

= 0 (4.30)

 

 

which gives us a unique solution for b. Therefore we obtain the first correction to the
fundamental frequency of a clamped wavy boundary plate with a concentric circular
core. Table 4.1 lists the values of the fundamental frequencies k with various 6 and c
for M = 6. The frequency itself depends on the amplitude 15 and is given by equation
(3.27).

The last row of Table 4.1 gives the the fundamental frequency of the circular plate
(when 6 = 0) with a concentric circular core of radius c and the valuas agree with
those of Table 2.3 in chapter 2. There is no known work has been done in wavy
boundary plates in the literature to compare with. It is possible only in the pertur-
bation method to take c —+ 0, because numerical calculations can be problematic for
a small c. Note also that unlike membranes the fundamental frequencies of clamped

wavy plate without a core, given in Table 3.1, are not the same as those that with a

64

 

 

c = 0.2 c = 0.1 c = 0.01 c = 0.001 c = 0.0001 c --> 0

e k k k k k k
0.1 6.14575 5.45469 4.99320 4.97268 4.97214 4.97213
0.09 6.09839 5.41293 4.95613 4.93585 4.93532 4.93531
0.07 6.01603 5.34034 4.89173 4.87191 4.87139 4.87138
0.05 5.95197 5.28388 4.84171 4.82224 4.82173 4.82172
0.03 5.90809 5.24521 4.80749 4.78825 4.78775 4.78774
0.01 5.88577 5.22555 4.79009 4.77099 4.77049 4.77048

0 5.88296 5.22308 4.78791 4.76881 4.76832 4.76831

 

 

 

 

 

 

 

 

 

Table 4.1: Fundamental Frequency of clamped wavy plate with a core

core as c —> 0. In the limiting case, when the clamping circle is indefinitely reduced,
the frequency does not reduce to the fundamental frequency of the plate without a

core.

4.1.2 Simply Supported Boundary Condition

We perturb the solution W(r, 6) and the fundamental frequency 1: about the circular

state as follows. We let

W(r, 6) = Wo(r) + 6W1(T', 6) + €2W2(T, 6) + 0(63) (4.31)

k4 = k3(1+ c212 + - . -) (4.32)

where b is the correction to the fundamental frequency as before.
Consider wavy circular boundary by taking f (6) = cos (M 6) in the section 3.1,

and let
7' = 1 + ecos (M6) (4.33)

where 6 << 1 is a small amplitude and M 2 2 is the number of circumferencial waves.

65

The simply supported boundary conditions are that

 

W(1+ 6 cos (M6), 6) = 0 (4.34)
82W V 6W
( (3n? + T2677) r=1+ccoe(M6) — 0 (4°35)
W(c, 6) = 0 (4.36)
6W
:9;- r=c _ 0 (4.37)

 

where n is the unit normal vector and R is the normalized radius of curvature.
Asymptotic expansions of the first two boundary conditions (4.34) and (4.35) are
given earlier by (3.12) and (3.22) respectively.

Now, using the perturbation (4.31) and (4.32) and collecting the similar order

terms lead us to the following zeroth order boundary value problem:
V“W0 - k3Wo = 0 (4.38)

with the boundary conditions

W0(1,6) = 0 , Wo(C,6) = 0
Wo"(1,6) + VWor(1,6) = 0 , Wor(C,0) = O

(4.39)

which corresponds to simply supported circular plate with a concentric clamped cir-
cular core. The solution of the equation (4.38) with the boundary conditions (4.39)

is given by
W0(Ti9) = C'nJono’") + C12Y0(ko7') + C43100607) + C14Ko(ko")- (4-40)

where C11, 012, 013 and CM are constants to be determined. Imposing the boundary

conditions (4.34)-(4.37) give us the characteristic equation

JOUCO) Y0(ko) 10(ko) K0090)
M(Jo(k0)) M(YoUcoll M(IoUColl M(Ko(ko))
J0(Ck0) Yo(cko) 10(Ck‘o) Ko(cko)
—J1(Cko) —Yo(cko) 10(cko) —Ko(cko)

= 0 (4.41)

 

 

66

where M( )— — -—-—- is the moment operator. Taking the limit as c —-> 0 gives

us the frequency equation

31/ + 2k3(12(k0)Ko(ko)10(ko)K2(ko) + 4Jo (ko)Ko(ko)
-4ko(1 — u)(J1(ko)Ko(ko)—J0(ko)K1(k0
—27rko((1 — V)Io(ko)Y1(ko) — uIr (ko)Yo(ko

+k§7r(12(ko)Yo(ko) + Jo(ko)Y2(ko)— J2(ko)Yo(ko) + 310(ko)Yo(ko))= 0

)
))
)) (4.42)

the first zeros of which gives the fundamental frequency 100 = 3.84899 of the simply
supported circular plate with a concentric circular core as c —> 0. Then the solution

(4.40) may be written as
Wo(r, 6) = 011 [Jo(rko) + egg/00kg) + agzlofiko) + ag3Ko(rko)] (4.43)

where (1591, (132, and 01% are given in the appendix by (8.13), (8.14), and (B.15)
respectively; and without loss of generality we can take Cu = 1.

Balancing the first order 0(6) terms, we have
V4W1(r, 6) — k3W1(r, 6) = 0 (4.44)
subject to following boundary conditions

W1(1, 6) = —Wo,(1, 6) cos (M6)

W1"(1,6) + VW1r(1,6) = [V(1+ M2)Wor(1, 6) - VWO"(1, 6)

— W0...(1i 6)] cos (M6) (4.45)
W1 (C, 6) = 0
W1, (C, 9) = 0

Solution of the equation (4.44) with boundary conditions (4.45) is

W1(r, 6) = 1471(1) cos (M 6). (4.46)

67

where 1471(1) = aflJMUko)+af2YM(rko)+af31M(rko)+als4KM(rko) where (115,, 04192,
0153, 0?; are given in the appendix by (B.18).

The second order 0(62) equation is
V4W2(r, 6) — k3W2(7', 6) = bk3W0(r) (4.47)
subject to following boundary conditions

W2(1,6)

{QT/1(1) + imam)

— ($117741) + 4W°"(1)) cos (2M6)
W2,,(1,6)+VW2,(1,6) = -:-(S+C)+%(—S+C)cos(2M6) (4.43)
W2(c,6) = 0
W2,(c,6) = 0

where

s = )W(—2W1)+2vTIT.(1)+Wa.(1)+z/Wi(1)).
c = —u(1-— 2M2)W0..(1) + u(1- M2)W1\,(1) + l/(l — M2)Wo,,(1)

A V A 1
- VWi..(1) " -2-Wo...(1) — W1...(1) — 5W0..."(1)-

Assuming a solution of type W2(r, 6) = U (r) + V(r) cos (2M 6) for (3.70) suggested

by the second order boundary conditions (4.48), we obtain

V4V(r, 6) — k3V('r, 6) = 0 (4.49)
and

V4U(r) — k3U(r) = bk8W0(r). (4.50)

The solution of (4.49) has no effect on correction b to the fundamental frequency.

Therefore, we solve (4.50) and the general solution of which is given by (4.27). The

68

simply supported outer boundary, clamped inner boundary conditions for (4.50) are

0'0) = —(-;-vT/I;(1)+§,-W;(1))
U...(1)+z/Ur(1) = —;—(s+c)

U(c) = 0

U.(c) = 0

Imposing the boundary conditions (4.51) into (4.27), We obtain

BlJoUco) + Bel/()(ko) + B3Io(ko) + BaKoUco)

BIM(J0(kO)) + B2M(Y0(k0)) + BsM(10(k0)) + B4M(Ko(ko))

BIJ0(Cko) + le/()(Ck0) + B310(Cko) + B4K0(Ck0)

BlJ1(Cko) + BzY1(Cko) — B311(Ck0) + B4K1(Cko)

(4.51)

= bF(1)—T
= bM(FUD
+ $(8+C)
= bF(c)

b r
= —-k—O'F (C)

where F(r) = kor(J1(rko) + aleIO'ko) — 0.321101%) + ag3K1(rko))/4 and ’1’ =

(WT,(1)/2 + Wo"(1)/4).

In order for the above system to have a nontrivial solution, the rank of the coeffi-

cient matrix has to be equal to the rank of the augmented matrix. Therefore we have

the following equation

Y0(ko) 10030) K0090) bF(1) — T
M(Y0(k0)) 6400090)) M(Ko(ko)) bM(F(1)) + (8 +C)/2

Y0(Cko) 10(Ck0) Ko(Cko) MIC)

Y1(Ck0) ‘1 1(Ck0) K1(Ck0) *bF'(C)/k0

 

= 0 (4.52)

 

which gives us a unique solution for b. Therefore we obtain the first correction to the

fundamental frequency of a simply supported wavy boundary plate with a concentric

circular core. Table 4.2 lists the values of the fundamental frequencies k with various

6 and c for M = 6. The frequency itself depends on the amplitude 15 and is given by

equation (3.27).

69

 

 

 

 

 

 

 

 

 

 

c = 0.2 c = 0.1 c = 0.01 c = 0.001 c = 0.0001 c -—1 0

e k k k k k k
0.1 5.02189 4.47292 4.11459 4.09955 4.09917 4.09916
0.09 4.97624 4.42775 4.07045 4.05548 4.05510 4.05510
0.07 4.89639 4.34840 3.99268 3.97785 3.97748 3.97747
0.05 4.83383 4.28590 3.93122 3.91650 3.91613 3.91612
0.03 4.79073 4.24266 3.88859 3.87393 3.87357 3.87356
0.01 4.76874 4.22053 3.86674 3.85212 3.85175 3.85174

0 4.76597 4.21774 3.86398 3.84936 3.84899 3.848988

 

 

Table 4.2: Fundamental frequency of simply supported wavy plate with a core

The last row of Table 4.2 gives the the fundamental frequency of the circular plate
(when 6 = 0) with a concentric circular core of radius c and the values agree with
those of Table 2.4 in chapter 2. There is no known work has been done in wavy bound-
ary plates in the literature to compare with. It is possible only in the perturbation
method to take c —> 0, because numerical calculations can be problematic for a small
' c. Note also that unlike membranes the fundamental frequencies of simply-supported
wavy plate without a core, given in Table 3.2, are not the same as those that with a
core as c —> 0. In the limiting case, when the clamping circle is indefinitely reduced,
the frequency does not reduce to the fundamental frequency of the plate without a

core.

4.2 Elliptic Plates With a Core

Consider an elliptic plate, having semi-major and semi-minor radii a and b respec-
tively (3.83), with a concentric circular core of radius c. We perturbed the outer
boundary of an annular plate to obtain moderately elliptic plate with a concentric

circular core. The governing Biharmonic equation of a circular plate with a concentric

70

circular core in polar coordinates (r, 6) is given by
V4W(r, 6) - k4W(r, 6) = 0 (4.53)
and corresponding general solution of which is given by

W(r, 6) = i[Cn1Jn(kr) + 0.01/"()W) + Cn3In(kr) + Cn4Kn(kr)] cos 71.6 (4.54)

n=0
where CM, Cng, 0713, CM are arbitrary constants as discussed in chapter 2. The
solution (4.54) is finite, since r can not be zero due to the concentric core.

We proceed with a perturbation scheme adopting the boundary perturbation
method presented in section (3.1). Then the solution of the eigenvalue problem with
clamped and simply supported boundary conditions will be obtained by considering
ellipse as the perturbed boundary with respect to the circle with an averaging radius

1' given in (3.96).

4.2.1 Clamped Boundary Condition

Since concentric circular core considered as a clamped boundary, we have a doubly
connected region with clamped inner and outer boundary. We perturb the solution

W(r, 6) and the fundamental frequency 1: about the circular state as follows. We let

W(r, 6) = Wo(r) + 5W1(r, 6) + €2W2(r, 6) + 0(3) (4.55)

k4 = k3(1+c2d+-~) (4.56)

where d is the correction to the fundamental frequency as before. Notation for the
correction is taken to be (1 in order not to confuse with the minor-axis b of the ellipse.

We consider the boundary function f (6) given by (3.96) in chapter 3. Then the

71

clamped boundary conditions (both inner and outer boundary) are that

 

W(1 + gm, 6) = 0 (4.57)
BW
51? r=1+ef(6) _ O (4.58)
W(c, 6) = 0 (4.59)
6W
‘5; — 0 (45°)

 

where n is the unit normal vector and f (6) = 22:, cncos (211M 6) as in (3.96). The
first two boundary conditions (4.57) and (4.58) can be expanded into asymptotic
series (3.13) and (3.16) as before.

Using these asymptotic boundary conditions and substituting (4.55) and (4.56)
into V4W(r, 6) - k4W(r, 6) = 0 and collecting the similar order terms we obtain the

zeroth order equation as
V4Wo(7‘,6) — k3W0(T, 6) = 0 (4.61)

with the boundary conditions

W1,6 =0 ,Wc,6 =0

o( ) o( ) (4.62)
Wor(1,9) = 0 , Wor(C,0) = 0

which corresponds to the circular plate (for n = 0) with a concentric circular core of

radius c. The solution of the equation (4.61) with the boundary conditions (4.62) is

given by
Wo(7‘, 6) = J0(Tko) + 081Y0(Tk0) + (18210(7‘ko) + 083Ko(7‘k0). (4.63)

Imposing the clamped edge boundary conditions and using the recursion formulas
for the Bessel functions, we obtain the characteristic equation (4.14). The frequency
equation is given by (4.15) as c —> 0. The first zero of frequency equation gives the ‘
fundamental frequency 100 = 4.76831 of the clamped circular plate with a concentric

circular core as c —+ 0.

72

Now, consider the first order 0(6) equation
V4W1(r, 6) — k3W1(r, 6) = 0 (4.64)

subject to following boundary conditions

W1(1,6) = 0 , W1(C, 6) =
W1.(1r9) = _f(6)W0rr(110) 1 W1r(610) = O

(4.65)

since Wo,(1, 6) = 0 and W0 = Wo(r) in the outer boundary condition of order 0(1).

Solution of the equation (4.64) with boundary conditions (4.65) is

z: [7'11 J4"(rk0) )+ 7n2Y4n(Tk0) ‘1' 77131411 (Tko)

n=1

+7§4K4n(rko)] cos (4716) (4.66)

where 75,1 , 732, 7:3, 7:4 for n = 1, 2, - - - are given in the appendix by the equation
(8.9) and even orders of (4.66) are due to two circumferencial waves for an ellipse.
Let W1(r, 6) = 22:, Wf"(r) cos (4n6) where
W60) -—- [rer..(rko) + 7514.040) + 65.3mm.) + 7:64.046] (46?)
The second order 0(62) equation is
V4W2(T, 6) — k3W2(T, 6) = dk3W0(T‘) (4.68)
subject to following boundary conditions

W2(116) = (1)3010) 1 W2(C10) : 0 (4.69)
W2,(1,6) = ‘I’§(1r9) 1 W2.(Cr9) = 0-

Assuming a solution of type W2(r, 6) = U (r) + 2:, Vn(r) cos (4716) for (4.68), we

obtain

V4V(r, 6) — k3V(r, 6) = 0 (4.70)

73

the solution of which has no effect on correction d to the fundamental frequency.

Therefore, we solve

V4U(r) -— k3U('r) = bk3W0(r). (4.71)
The solution of the homogenous part of (4.71) is

Uh(r) = BlJ0(kor) + BzYoUCoT) + B310(k0r) + B4Ko(kor) (4.72)

where BI, 82, B3, and B4 are arbitrary constants. The particular solution of (4.71)
is

__ dkoT‘

U,(r) = 4 (J1(k07‘)+081Y1(k07‘)—(18211(k07‘)+083K1(k07‘)). (4.73)

Thus the general solution of (4.71) is
U(r) = Uh('r) + Up(r). (4.74)

The boundary conditions for (4.71) are the cos (4n6)-free part of (4.69)

U(1)=<P8(1) .U(c)=o (4.75)
U.(1)=€8(1) .U.<c)=o

where

156.6) = tos<r)+ztoi(r)cos4n6

n=1

{6(7) + 2: £30") cos 4n6.
n=1

\II§(1‘, 6)

The constant terms cp8(1) and 55(1) are that

1 °° 1 °°
(P30) = —ZW0..(1)ZCE. — '2' ZCanTU)
n=1 n=1
C 1 m m n
50(1) = —ZW0...(1)nZ=:ICEr+4n=ICnn2W14 (1)
1 0° 4n
_ 5:6"W1"(1)°
n=1

74

 

 

 

 

 

 

 

 

c = 0.2 c = 0.1 c = 0.05 c = 0.01 c = 0.001
e k k k k k
0.5 5.91873 5.26684 4.99845 4.83129 4.81206
0.4 5.89769 5.24114 4.97261 4.80582 4.78667
0.3 5.88764 5.22881 4.96021 4.79360 4.77449
0.2 5.88389 5.22421 4.95558 4.78903 4.76994
0.1 5.88302 5.22315 4.95451 4.78798 4.76888
0.05 5.88297 5.22308 4.95445 4.78791 4.76882
0 5.88296 5.22307 4.95444 4.78790 4.76881

 

 

Table 4.3: Fundamental frequency of clamped elliptic plate with a core

Imposing the boundary conditions (4.75) into (4.74),

BlJo(ko) + Bel/0(6)) + 8310(k0) + BaKoUco) = dF(1) + 63(1)

31.71040) + 32min.) — 331104,) + B,K1(Ic0) = "ISBN” + 53(1)
BiJ0(cko) + Bomcko) + Ban(ck0) + BrKo(cko) = dF(c) (4.76)
B,J.(c1c0) + Bng(cko) — B3Il(cko) + B4K1(cko) —_- —%F'(c)

where F(r) = kor(J1(kor) + alel(kor) — 05211(k0r) + a33K1(kor))/4.
In order for the system (4.76) to have a nontrivial solution, the rank of the coeffi-
cient matrix has to be equal to the rank of the augmented matrix. Therefore we have

the following equation, which gives us a unique solution for d

 

Y0(k0) 10030) K0060) (W(1) + $30)

Y1(ko) —11(k0) K1(ko) "dF’(1)/k0 + {8(1) = 0. (4.77)
Y0(Cko) 10(Ck0) K0(Cko) (117(6)

Y1(Cko) -11(Cko) K1(Cko) 'dF’(C)/k0

 

Therefore we obtain the first correction to the fundamental frequency of a clamped
elliptic plate with a concentric circular core. Table 4.3 lists the values of the funda-

mental frequencies k with various 6 and c.

75

 

The last row of Table 4.3 gives the the fundamental frequency of the circular plate
with a concentric circular core of radius 6 and the values agree with those of Table 2.3
in chapter 2. There is no known work has been done in elliptical plates with a core in
the literature to compare with. It is possible only in the perturbation method to take
c —> 0 (Table 4.4), because numerical calculations can be problematic for a small c.
Note also that unlike membranes the fundamental frequencies of clamped elliptical
plates without a core, given in Table 3.3, are not the same as those that with a core
as c —+ 0. In the limiting case, when the clamping circle is indefinitely reduced, the

frequency does not reduce to the fundamental frequency of the plate without a core.

 

 

 

 

 

 

c = 0.0001 c = 0.00001 c = 1 x 10-“1

e k k k
0.5 4.81156 4.81155 4.81155
0.4 4.78617 4.78616 4.78616
0.3 4.77399 4.77398 4.77398
0.2 4.76944 4.76943 4.76943
0.1 4.76839 4.76838 4.76838
0.05 4.76832 4.76831 4.76831

0 4.76831 4.76830 4.76830

 

 

Table 4.4: Fundamental frequency of clamped elliptic plate with a small core

The calculation of the fundamental frequency, keeping the area fixed as follows:
Given the eccentricity 6, major and minor axis of the plate are a = 1 + 62/4 and
b = 1 - £2 / 4 respectively. The new averaging radius R can be defined with respect

to the new major—axis a’ and the minor axis b’ as

 

1 21f
=_ r2 2 2 - 2 .
R 277 0 fl) cos 6+(b’) Sln 6d6

76

 

Note also that a / b = a’ / b’ , then

 

_b’l 2

R——-— Jazcosz6+b2sin26d6
b2” 0

which implies b’ = bR by (3.89). Then the area of the circular plate is 7r and it is

equal to the area 776’ b’ of the ellipse with the axes a’ and b’. Thus, we have
77 = 7ra’b’ = 770sz

and hence R2 = 1 / ab is the multiplying factor for the frequency. The fundamental

 

 

c = 0.2 c = 0.1 c = 0.05 c = 0.01 c = 0.001
e k k k k k
0.5 5.93033 5.27716 5.00824 4.84075 4.82149
0.4 5.90242 5.24533 4.97659 4.80967 4.79051
03 5.88913 5.23014 4.96147 4.79481 4.77570
0.2 5.88418 5.22447 4.95583 4.78927 4.77018
0.1 5.88304 5.22317 4.95453 4.78799 4.76890
0.05 5.88297 5.22308 4.95445 4.78791 4.76882
0 5.88296 5.22307 4.95444 4.78790 4.76881

 

 

 

 

 

 

 

 

Table 4.5: Fundamental frequency of clamped elliptic plate with a core (having the
same area as the annular plate)

frequencies given in Table 4.5 are normalized by the averaging radius 7

Then 122/?”2 = 1 / ab gives a factor for the fundamental frequency of the plate. Table
4.5 lists the fundamental frequencies of the clamped elliptical plate with a core which

has the same area as the annular plate.

77

...-n .— ~‘—. mm'fi'ku v. _- -. n

4.2.2 Simply Supported Boundary Condition

On the doubly connected region with clamped inner and simply supported outer
boundary, we perturb the solution W(r, 6) and 10 about the circular state as follows.

We let
W(r, 6) = Wo(r) + 6W1(7‘, 6) + €2W2(r, 6) + 0(63) (4.78)
k4 = k3(1+ e2d+ - - .) (4.79)
where d is the correction to the fundamental frequency as before.

We consider the boundary function f (6) given by (3.96) in chapter 3. Then the

boundary conditions are that

 

W(1 + ef(6), 6) = 0 (4.80)
02W V 6W
( 6112 + REL.) r=1+cf(6) - O (4.81)
W(c, 6) = 0 (4.82)
0W
a r=c — 0 (4.83)

 

where n is the unit normal vector and f (6) = Efficacos (271M 6) for M = 2 as
in (3.96). The first two boundary conditions (4.80) and (4.81) can be expanded into
asymptotic series (3.13) and (3.22) as before.

Using these asymptotic boundary conditions and substituting (4.78) and (4.79)
into V4W(r, 6) - k4W(r, 6) = 0 and collecting the similar order terms we obtain the

zeroth order equation as
V4Wo(r, 6) — k3Wo(r, 6) = 0 (4.84)

with the boundary conditions

W0(1) = 0 i W0(C) = 0 (4.85)
W0..(1) + 1/W0.(1) = 0 1 W0.(C) = 0

78

 

which corresponds to the circular plate (for n = 0) with a concentric circular core of
radius 0. The solution of the equation (4.84) with the boundary conditions (4.85) is

given by
Wo(7', 6) = J0(rk0) + 615911130160) + 632106140) + 0083Ko(7'k0). (4.86)

Imposing the boundary conditions (4.85) and using the recursion formulas for the
Bessel functions, we obtain the characteristic equation (4.14). The frequency equation
is given by (4.42) as c -—1 0. The first zero of frequency equation gives the fundamental
frequency [to = 3.84899 of the clamped circular plate with a concentric circular core
as c -—1 0.

Now, consider the first order 0(6) equation
V4W1(r, 6) — k3W1(7‘, 6) = 0 (4.87)

subject to following boundary conditions

W1(1,6) = <I>§(1,6) , W1(c,6) = 0

(4.88)
W1,(1,6) = \IIf(l,6) , W1,(c,6) = 0
Solution of the equation (4.87) with boundary conditions (4.88) is
W10, 6) = 2 [7516.019 + 752Y4n(rko) + 73314.04)
n=1
+754Kr..(r1co)] cos (4n6) (4.89)

where 751, 752, 753, 754 for n = 1,2, - -- are given by the equation (B21) in the
appendix and even orders of (4.89) are due to two circumferencial waves for an ellipse.

Let W1(7', 6) = 22:, W14”(7‘) cos (4n6) where
Witt“) = [751J4n(rk0) + 752K1n(rk0) + 75314110190) + 754K4n(rk0)] -(4-90)
The second order 0(62) equation is

V4W2(r, 6) — k3W2(r, 6) = dk3wo(r) (4.91)

79

 

subject to following boundary conditions

W2(1,6) =¢g(1,6) , W2(C, 6) = 0

(4.92)
W2r(1’6) = \I’§(119) a W2r(C16) = 0

Assuming a solution of type W2(r, 6) = U (r) + 22:, 14,0”) cos (4716) for (4.91), we

obtain
V4V(r, 6) — k3V(r, 6) = 0 (4.93)

the solution of which has no effect on correction d to the fundamental frequency.

Therefore, we solve

V4U(r) — k3U(r) = bk3W0(r). (4.94)
The solution of the homogenous part of (4.94) is

Uh(7‘) = BlJo(kor) + BgY0(k0r) + B310(kor) + B4K0(kor) (4.95)

where Bl, Bg, B3, B4 are arbitrary constants. The particular solution of (4.94) is

die 7
Up(r) = ”743- (J1(kor) + cam/tor) — 63211(1c0r) + 553K1(1c0r)). (4.96)

Thus the general solution of (4.94) is
U(r) = Uh(r) + Up(r). (4.97)
The boundary conditions for (4.94) are the cos (4n6)-free part of (4.92)

U(1)=<P§(1) iU(C)=0

(4.98)
U.r(1) + VUr(1) = {5(1) . Ur(c) = 0

where

«166,0 = eitr)+za.i(r)cos4ne
n=1

{5(7) + 2: {3(7) cos 472.6.

n=1

‘1’50‘. 9)

80

 

The constant terms 960(1) and {0 (1) are that

48(1) = —1wr..(1)\;c3. ~1anWfiT‘0)
560) = we..(1)Zc3.(8n2—u(4———- ME) )——wo...(1) Z.

- Ewaanzciistsniu—66017171)

+ "Zen“ 4102 -V(——-—l4n):) )(UWff‘ )--::an1..-(

— 126%:11)

Imposing the boundary conditions (4.98) into (4.97),

3110090) + 321/0090) + 3310060) + B4Ko(k0)

M(Jo(k0)) + B2M(Y0(k0)) + 33M(10(k0)) + B4M(Ko(ko))

Bljo(Cko) + BzYo(Cko) + B3Io(Cko) + B4K0(Ck0)

—B1J1(Cko) — BQK(C,€0) + B311(Cko) — B4K1(Cko)

where

— dF(1) + i66(1)

dM(F(1))
63(1) (499)

F(r) = a} (J1(kor) + ale1(kor) — agzIfiko'r) + agaKl(kor)).

Since the determinant equation on the left hand side of (4.99) is zero, we have the

following equation, which leads to a unique solution for d

Yo(cko) Io(cko) Ko(cko) dF(c)

 

1,0090) 10090) K0090) dF(1)+ 906(1)
M(Y0(k0)) M(Io(ko)) M(Ko(k0)) dM(F(1))+€0g(1)

-Y1(Cko) 11(Cko) -K1(Cko) (”WW/($0

= 0. (4.100)

 

Therefore we obtain the first correction to the fundamental frequency of a simply

supported elliptic plate with a concentric circular core. Table 4.6 below lists the

values of the fundamental frequencies 1: with various 6 and c.

81

 

 

 

 

 

 

 

 

 

c = 0.2 c = 0.1 c = 0.05‘ c = 0.01 c = 0.001
e k k k: k k
0.5 4.78703 4.24795 4.02931 3.89676 3.88217
0.4 4.77463 4.23019 4.01045 3.87751 3.86290
0.3 4.76871 4.22169 4.00141 3.86828 3.85366
0.2 4.76651 4.21852 3.99804 3.86483 3.85021
0.1 4.76600 4.21779 3.99726 3.86403 3.84942
0.05 4.76597 4.21775 3.99721 3.86498 3.84937
0 4.76596 4.21774 3.99720 3.86398 3.84936

 

 

Table 4.6: Fundamental frequency of simply supported elliptic plate with a core

 

 

 

c = 0.0001 c = 0.00001 c = 1 x 10‘14

e k k k
0.5 3.88181 3.88180 3.88180
0.4 3.86254 3.86253 3.86253
0.3 3.85330 3.85329 3.85329
0.2 3.84985 3.84984 3.84984
0.1 3.84905 3.84904 3.84904
0.05 3.84900 3.84899 3.84899

0 3.84899 3.84898 3.84898

 

 

 

 

 

 

Table 4.7: Fundamental frequency of simply-supported elliptic plate with a small core

The last row of the the Table 4.6 gives the the fundamental frequency of the simply
supported circular plate with a concentric circular core of radius 0 and the values
agree with those of Table 2.4 in chapter 2. There is no known work has been done in
elliptical plates with a core in the literature to compare with. It is possible only in
the perturbation method to take c —+ 0 (Table 4.7), because numerical calculations
can be problematic for a small c. Note also that unlike membranes the fundamental
frequenciae of simply-supported elliptical plates without a core, given in Table 3.4,
are not the same as those that with a core as c —> 0. In the limiting case, when

the clamping circle is indefinitely reduced, the frequency does not reduce to the

82

fundamental frequency of the plate without a core.

An increase in eccentricity causes a change in the area. In order to find the

frequency of the plate with same area as the annular plate, we multiply the frequencies

in table 4.6 with a factor 1/ab where a = 1 + 62/4 and b = 1 — 62/4 for a given

eccentricity.

4.3 Polygonal Plates With a Core

Let a be the radius of the inscribing circle of an M sided regular polygon with a
concentric circular core of radius c. The radius a has been determined such that the
mean radius of the polygon r is 1 by (3.150). We perturbed the outer boundary of an
annular plate to obtain an M sided regular polygonal plate with a concentric circular
core. The governing Biharmonic equation of a circular plate with a concentric circular
core in polar coordinates (7', 6) is given by

V4W(r, 6) — k4W(r, 6) = 0 (4.101)

and corresponding general solution of which is given by

W(r, 6) = anlwr) + 0,214.06) + Cn3In(kr) + C...,K,,(kr)] cos 716 (4.102)
n=0

where Cal, Cng, Cn3, CM are arbitrary constants as discussed in chapter 2. The
solution (4.102) is finite, since 7' can not be zero due to the concentric core.
We proceed with a perturbation scheme adopting the boundary perturbation
method presented in section (3.1). Then the solution of the eigenvalue problem with
clamped and simply supported boundary conditions will be obtained by considering

the polygon as the perturbed boundary with respect to the circle with an averaging

radius 1' given in (3.152).

83

 

4.3.1 Clamped Boundary Condition

Let b be the correction to the fundamental frequency. We perturb the solution W(r, 6)
and the fundamental frequency k about the circular state as before (3.154), (3.155).
We take the boundary function f (6) as given in chapter 3 by (3.152). Then the

clamped boundary conditions (both inner and outer boundary) are that

 

W(1 + cf(6), 6) =-. 0 (4.103)
6W
E7.— r=1+ef(6) — 0 (4.104)
W(c, 6) = 0 (4.105)
6W
7,5; 7:6 _. 0 (4.106)

 

where n is the unit normal vector and f (6) = 23;, cncos (nM 6) as in (3.152). The
first two boundary conditions (4.103) and (4.104) can be expanded into asymptotic
series (3.13) and (3.16) as before.

Using these asymptotic boundary conditions and substituting (3.154) and (3.155)
into V4W(r, 6) — k4W(r, 6) = 0 and collecting the similar order terms we obtain the

zeroth order equation as
V4W0(T, 6) — k3W0(T, 6) = O (4.107)

with the boundary conditions

W1,6 =0 ,Wc,6 =0

o( ) o( ) (4.108)
Wo,(1,6) = 0 , Wo,(c, 6) = 0

which corresponds to the circular plate (for n = 0) with a concentric circular core of

radius c. The solution of the equation (4.107) with the boundary conditions (4.108)

is given by

W00", 6) = Jo(7‘k0) + Gall/()(Tko) + 048210(7‘k0) + 083K0(Tk0). (4.109)

84

 

Imposing the clamped edge boundary conditions and using the recursion formulas
for the Bessel functions, we obtain the characteristic equation (4.14). The frequency
equation is given by (4.15) as c —1 0. The first zero of frequency equation gives the
fundamental frequency k0 = 4.76831 of the clamped circular plate with a concentric
circular core as c —-1 0.

Now, consider the first order 0(6) equation
V4W1(r, 6) — k3W1(r, 6) = 0 (4.110)
subject to following boundary conditions
W1(1,6) = 0 , W1(c,6) = 0

Who-10) = "f(0)WOrr(]-10) 1 er(c16) = 0

since Wo,(1, 6) = 0 and W0 = Wo(r) in the outer boundary condition of order 0(1).

(4.111)

Solution of the equation (4.110) with boundary conditions (4.111) is

W10», 6) = Z [same/co) + 6§2YnM(rko) + 6:31.,M(r1co)
n=1
+6;,K..M(rko)] cos (nM6) (4.112)
where 631, 65,2, 6,0,3, 65,4 for n = 1, 2, - . ~ are given in the appendix by the equation

(8.12) and M orders of (4.112) are due to M circumferencial waves for an M sided
regular polygon. Let W1(r, 6) = 22:, W?“ (7‘) cos (71M 6) where

Wf‘M(r) = [cifilJnM(rko) + 6§2YnM(rko) + 6:31,M(r1co) + 6§4KnM(rko)]. (4.113)
The second order 0(62) equation is
V4W2(r, 6) — k3W2(r, 6) = bk3W0(r) (4.114)
subject to following boundary conditions

W2(1,6) = <I>§(1,6) , W2(c, 6) = 0
W2r(179) = ‘1’2(116) 1 W2r(C1 0) = 0'

(4.115)

85

A

 

film-P'w.

Assuming a solution of type W2(1', 6) = U (r) + 2:, V, (1*) cos (71M 6) for (4.114),
we obtain
V4V(r, 6) — k3V(r, 6) = 0 (4.116)

the solution of which has no effect on correction b to the fundamental frequency.

Therefore, we solve

V4U(r) —- k3U(r) = bk3Wo(r). (4.117)
The solution of the homogenous part of (4.117) is

Uh(r) = BlJo(kor) + BgYOUcor) + B3Io(kor) + BaKo(kor) (4.118)

where B1, 82, B3, B4 are arbitrary constants. The particular solution of (4.117) is

“if
4
Thus the general solution of (4.117 ) is

Up(r) = — (J1(kor) + a311’1(kor) — 032I1(k0'r) + 633144106). (4.119)
U(7‘) = Uh(r) + Up(r). (4.120)
The boundary conditions for (4.117 ) are the cos (nM6)-free part of (4.115)
0'0) =<PS(1) , U(C) =0

(4.121)
Ur“) = 65(1) 1 Ur(c) = 0
where
930.6) = 4301+ 2145.0) cos we
n=1
\II§(7‘, 6) = {3(7‘) + Zéf,(r)cosnM6.
n=1
The constant terms 908(1) and {3(1) are that
c 1 0° 2 1 00 nM
900(1) = ’ZW0"(1);=:1% — 52011144. (1)
1 °° 1 °° ,,
58(1) = —ZW0".(1)ZC3. + '2" 2%(RM)2W1M(1)
n=1 n=1
1 °° ,,
" 52911145340)-
n=1

86

 

Imposing the boundary conditions (4.121) into (4.120),

BlJo(k0) + BzYoUCo) + B3Io(ko) + B4Ko(k0) = bF(1) + <p3(1)

BlJ1(k0) + 821/1040) — BsI1(k0) + Bamako) = —%F’(1) + {3(1)
Bl J0(Cko) + BzYo(C/C0) + B3Io(Cko) + B4K0(Cko) = bF(C) (4.122)
81.]1 (Cko) + BzY1(Ck0) — B311(Ck0) + B4K1(Ck0) = —%F’(C)

where F(7‘) = k0T(J1(koT) “’1‘ (1611/1 (1007‘) — 082110607) ‘1“ (183K1(k07‘))/4.
Since the determinant equation on the left hand side of (4.122) is zero, we have

the following equation, which leads to a unique solution for b

141090) 10090) K0000) bF(1)+ 903(1)
Y1(ko) “11060) K1060) —bF’(1)/ko + 58(1)
Y0(Cko) 10(Cko) Ko(Cko) 517(0)
Y1(cko) —I1(Cko) K1(cko) -F’(C)/ko

= 0. (4.123)

 

 

Thus we obtain the first correction to the fundamental frequency of a clamped polygo-

nal plate with a concentric circular core. Table 4.8 lists the values of the fundamental

frequencies 11: with N = 3,6,12 and various c.

 

. 4‘ AJ' Invi—

 

c=0.2

c=0.1

c = 0.05

 

 

S©WZ

 

k
6.00014
6.00502
6.00657

 

k
5.32464
5.32854
5.32978

 

k
5.04868
5.05219
5.05330

 

 

Table 4.8: Fundamental frequency of clamped hexagonal plate with a core

In practice the corners may not be mathematically sharp and a finite N would be

desirable. For determination of N, given corner curvature, see Wang 1998 [24]. For

87

 

 

c c=0.01
Lie 4.88138

6 = 0.001
4.86159

0 = 0.0001
4.86107

c = 0.00001
4.86106

c = 1 x 10'10
4.86106

 

 

 

 

 

 

 

 

 

Table 4.9: Fundamental frequency of clamped hexagonal plate with a small core

N = 12 boundary function f (6) gives good approximation to an hexagon. Calculat-
ing the fundamental frequency, keeping the area fixed, gives us the frequency factor

of R2/r2 = 11/362. With this factor, the actual frequencies are listed in Table 4.10

 

 

 

c=0.2 c=0.1 c=0.05
N k k k
3 6.44157 5.71638 5.42010
6 6.44681 5.72056 5.42388
12 6.44847 5.72189 5.42507

 

 

 

 

 

Table 4.10: Phndamental frequency of clamped hexagonal plate with a core

Liew and Lam [10] calculated the fundamental frequency of an hexagonal plate
without a core using Rayleigh-Ritz method to be k = 3.44329. There is no known
work has been done in polygonal plates with a core in the literature to compare
with. It is possible only in the perturbation method to take c -—1 0 (Table 4.9),
because numerical calculations can be problematic for a small 6. Note also that
unlike membranes the fundamental frequencies of clamped polygonal plates without
a core, given in Table 3.6, are not the same as those that with a core as c -> 0.. In
the limiting case, when the clamping circle is indefinitely reduced, the frequency does

not reduce to the fundamental frequency of the plate without a core.

88

4.3.2 Simply Supported Boundary Condition

Let b be the correction to the fundamental frequency to the simply supported polyg-
onal plate with a core. On the doubly connected region with clamped inner and
simply supported outer boundary, we perturb the solution W(r, 6) and k about the
circular state as before (4.78), (4.79). We take the boundary function f (6) as given

in chapter 3 by (3.152). Then the boundary conditions are that

 

W(1+ cf(6), 6) = 0 (4.124)
62W V 6W
( 6712 + R7371-) r=1+ef(6) _ O (4.125)
W(c, 6) = 0 (4.126)
3W
E m _ 0 (4.127)

 

where n is the unit normal vector and f (6) = 22:, on cos (71M 6) as in (3.152). The
first two boundary conditions (4.124) and (4.125) can be expanded into asymptotic
series (3.13) and (3.22) as before.

Using these asymptotic boundary conditions and substituting (4.78) and (4.79)
into V4W(r, 6) — k4W(r, 6) = 0 and collecting the similar order terms we obtain the

zeroth order equation as
V4W0(T, 6) — k3Wo(7‘, 6) = 0 (4.128)
with the boundary conditions

W00") = O 1 W0(C) = 0
W0"(1)+ VWO,(1) = 0 , W0,(C) = 0

(4.129)

which corresponds to the circular plate (for n = 0) with a concentric circular core of
radius c. The solution of the equation (4.128) with the boundary conditions (4.129)
is given by

W06, 6) = Jo(rko) + egg/(,(rko) + 632106140) + 633K0(rk0). (4.130)

89

 

Imposing the boundary conditions (4.129) and using the recursion formulas. for the
Bessel functions, we obtain the characteristic equation (4.14). The frequency equation
is given by (4.42) as c —1 0. The first zero of frequency equation gives the fundamental
frequency [to = 3.84899 of the simply-supported circular plate with a concentric
circular core as c —-> 0.

Now, consider the first order 0(6) equation
V4W1(7‘, 6) — [Cal/V10", 6) = O (4.131)

subject to following boundary conditions

W1(1,6)=<I>f(1,6) , W1(c,6)=0

(4.132)
W1,(1,6) = \Ilf(1,6) , W1,(c, 6) = 0
Solution of the equation (4.131) with boundary conditions (4.132) is
W1(r, 6) = Z [éflJnM(rko) + asp/"Mum + 65,1,M(rko)
n=1
+654K,M(rlco)] cos (nM6) (4.133)

where 531, 652, 6:3, 654 for n = 1,2, - -- are given by the equation (B.24) in the
appendix and M orders of (4.133) are due to M circumferencial waves for an M

sided regular polygon. Let W1(r, 6) = 22:, W?” (1') cos (71M 6) where
pr(r) —_- [55,JnM(rko) + 552Y,,M(rk0)
+ 6§3InM(rko) + 654KnM(Tko)]. (4.134)
The second order 0(62) equation is
V4W2(r, 6) — k3W2(r, 6) = dk3W0(r) (4.135)

subject to following boundary conditions

W2(1,6) = <I>§(1,6) , W2(c,6) = 0
Wr.(1,6)=v§(1.6) , Wa(c,6)=0.

(4.136)

90

Assuming a solution of type W2(r, 6): U (r) + 23:, V" (7‘) cos (11M 6) for (4.135),
we obtain V4V(r, 6) — k3V(r, 6) = 0 the solution of which has no effect on correction

b to the fundamental frequency. Therefore, we solve
4U(r~) — IcgU(r) = bk3W0(r). (4.137)
The solution of the homogenous part of (4.137) is
Uh(r) = BlJo(kor) + BgYo(kor) + B310(k07') + B4Ko(kor) (4.138)
where Bl, Bz, B3, B4 are arbitrary constants. The particular solution of (4.137) is
bkor

Up(7') = —T (J1(k07') + ale1(kor)— 05211(k07‘) + (153K1(k07‘)). (4.139)

Thus the general solution of (4. 137) 1s U (7‘) Uh(r)+Up (r ) The boundary conditions
for (??) are the cos (nM6)-free part of (4.136)

U(1) =‘Pos(1) 111(6) =0

(4.140)
Ur(1)=£€(1) 1 Ur(C)=0
where
<I>§(r,6) = (p0( (r)+Z<p§() c)osnM6
n=1
111590, 6) = gg(r)+Zg§(r)cosnM6.
n=1
The constant terms (03(1) and {5(1) are that
1 °° 1 °° n
«80) = 716.1026. - §Zawr,M<1)
n=1 n=1
1 oo 00
16(1) = 5W0..(1)):c3.[(nM)2 +111 — (nM))] — —wo...(1)Z..
n- n=1
1 0° 1 0° 2 nM
— 1W0....(1)n=1c3.—§;ca(nM) (2—v)Wr (1)

1. ;a[<nM)2+ )2(§<1-<nM))]WcM1)——29W
1 00
" 5:91:14] 1:110

n=1

91

Imposing the boundary conditions (4.140) into U
BlJO(k0) + 321/0016) + 3310630) + B4K0(ko) = bF(1) '1' 903(1)
B1M(Jo(ko))+ BzM(Y0(ko)) + BsM(Io(ko)) + B4M(K0(k0)) = b4407(1))
+ 43(1) (4141)

B1J0(Cko) + BzYo(Cko) + 8310(Ck0) + B4K0(Cko) = bF(C)

b ,
—B1J1(Ck0) — 821/1(Ck0) “I" B3I1(Ck0) — B4K1(Ck0) = EF (C)
where
_ (907‘ s s s
F(7‘) — T (J1 (1607') + 001Y1(k07') — 002[1(k07‘) + 003K1(k07')) .

Since the determinant equation on the left hand side of (4.141) is zero, we have

the following equation, which leads to a unique solution for b

YOUCO) 10000) Kofko) bF(1) '1’ (P30)
M(Y0(ko)) M(IO(k0)) M(Ko(k0)) bM(F(1))+§6(1)
Y0(Ck0) 10(Ck0) Ko(Cko) bF(C)
—Y1(cko) I1(cko) ——K1(cko) . bF’(c)/ko

= 0. (4.142)

 

 

Therefore we obtain the first correction to the fundamental frequency of a simply
supported M sided polygonal plate with a concentric circular core. Table 4.11 below

lists the values of the fundamental frequencies 11: with various c for N = 12.

 

c 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 1x10-74
k 4.99993 4.49103 4.27008 4.13574 4.12084 4.12046 4.12045 4.12045

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.11: Fundamental frequency of simply supported hexagonal plate with a core

Calculating the fundamental frequency, keeping the area fixed, gives us the fre-

quency'factor of R2/7'2 = 7r / 3112. With this factor, the actual frequencies become

92

 

c 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 1x10-14
It 5.36770 4.82143 4.58423 4.44001 4.42401 4.42360 4.42359 4.42359

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.12: Fundamental frequency of simply supported hexagonal plate with a core
(with the same area as the annular plate)

Liew and Lam [10] calculated the fundamental frequency of an hexagonal plate
without a core using Rayleigh-Ritz method to be k = 2.56179. There is no known
work has been done in polygonal plates with a core in the literature to compare
with. It is possible only in the perturbation method to take c —+ 0 (Table 4.11),
because numerical calculations can be problematic for a small c. Note also that
unlike membranes the fundamental frequencies of simply-supported polygonal plates
without a core, given in Table 3.8, are not the same as those that with a core as
c -1 0. In the limiting case, when the clamping circle is indefinitely reduced, the

frequency does not reduce to the fundamental frequency of the plate without a core.

93

CHAPTER 5

The Rayleigh-Ritz Method

There are many numerical methods for solving eigenvalue problems. The variational
methods of approximation presented here are limited to the Rayleigh-Ritz method.
Because the method is one of the most direct methods that bypasses the derivation
of the Euler-Lagrange equations and go as directly from the variational statement
of the problem to the solution of the Euler-lagrange equations, unlike Galerkin’s
method, the plate (trial or approximation) functions in the Rayleigh-Ritz method do
not have to satisfy all the boundary conditions (only geometric boundary conditions
need to be satisfied), the method does not require the same differentiability as in the
differential equation (unlike Galerkin’s method) and it is used extensively by many
authors. The method was proposed first by Lord Rayleigh [12] then generalized
by Ritz [20], [19], [23]. Recent publications on the subject are improvements of the
Rayleigh-Ritz method. One such method is pb2-Ritz method discussed here [10]. The
pb2-Ritz method is a special case of the Rayleigh-Ritz method, which uses the pb-2
Ritz functions. The pb-2 Ritz functions consist of the product of a two-dimensional
polynomial function (p-2) and a basic function (b). The basic function is defined
by the product of equations specifying the boundary conditions. It is well-known
that variational approach gives an upper bound solution to the exact value [20].

Therefore frequency values obtained here are slightly bigger than the actual values.

94

Convergence can be made better with more iterations. The derivation of potential
and kinetic energies, required by the variational form, of plates are carried out by

[19], [23]. We start with the variational form of the vibrating plate.

5.1 Circular Plates With a Core

The variational (or weak) from of the vibrating circular plate (2.2) is given by [10]

[{D 32W525W 1(QVKQ’AVZAMQ’1K
61“2 a 2 6T 6T2 87' 6T2
19212281129121)

7' 862 07'2 1' 862 672

D (16W 1 WW) (106W 1626W)

 

7.97 75—662 ; or “La—642
162W 161W 162W 1 now
+ 2(1’”)D(76r66 ”13%) (33 6r66 _ F5 66 )

— )2 (pW 6W) }, (5.1)

where Q is domain of the plate, with the assumption of no dissipation (i.e., kinetic

energy = potential energy). Consider N -parameter Rayleigh-Ritz approximation
W(r,) 6)~ WN(r,) 6): Zc,¢,( 7‘, 6) (5.2)
Substituting (5.2) into (5.1), we obtain
N
2(Kij —)\2M¢'J’) Cj =0, 2: 1,2,°'° ,N. (5.3)

i=1

In matrix notation equation (5.3) has the form of an eigenvalue problem
(1K1 — 120141) {c} = 0. (5.4)
For a non-trivial solution, the coefficient matrix in equation (5.4) should be singular

|[K] — X"[M]| = 0. (5.5)

95

 

In equation (5.3) the coefficient matrices are given by
_ 82¢,- 62¢j 1 6d), 1 62¢,- 1 64),- 1 L265,-
Kw — /D{D 6:2 6:2 +1) (ii—£97 + :36‘) (:3: + :2 >672
6‘15: 62¢j 6453' 6245:" 1 821:; a_2_¢j 1 62¢j 62%
T 0(6? 61'2 + 6r 8T2 + r 662 6:2 1"; 662 6r2 (5'6)

1 62¢, 1 29¢,- 1 6%,- 1 aqs,
+ 20— ”)D (?W ‘ F266) (:6166 ‘ :26?) }" ”d",

Mij = /p¢,¢jrdrd9. (5-7)
0

 

 

 

 

 

Above formulation is valid also for annular plate. Now, we only need to specify the

assumed modes 45,- ’s satisfying the geometric boundary conditions.

5.1.1 Clamped Boundary Condition

In the case of clamped circular plate, the geometric boundary conditions are

BWN

WNU‘) 6) = 5T

 

= O at r = a (5.8)

where a is the radius of the plate. For N -parameter Rayleigh-Ritz solution we can
assume ¢,-(r, 6): f,(r) cos(n6), where n is the number of nodal diameters. Since
we are considering the fundamental frequency which occurs in the case of no-nodal
diameter (i.e., n = O), we only need to specify f,-(r). In order to avoid logarithmic
singularity in the equation (5.6), we take enough differentiability of f satisfying the
geometric boundary conditions f’ (O) f’ (a) f-a( ) = 0. The generating functions

f,- ’s can be chosen as

f,(r) = (1— (2)2)”l i=1,2,.~ ,N (5.9)

to guarantee the integrability and the completeness in the process of the Rayleigh-
Ritz method. The less power in (5.9) will result non-integrability, more power makes
the sequence incomplete which leads to errors. Stable convergence of the solution is
reached for N = 12 iterations. Table 5.1 gives the values of the natural frequencies

of the clamped circular plate.

96

 

n 0 1 2 3 4
A 3.1962 4.6109 5.9057 7.1435 8.3466

 

 

 

 

 

 

 

 

 

Table 5.1: Natural frequencies of clamped circular plate

In Table 5.1 n is the number of nodal diameters and A2 = wazm the nor-
malized frequency. The frequency values agree with the published results in the
monograph of Leissa [8].

Consider a clamped circular plate with a core of radius b. Then the generating

functions f,- ’s can be chosen as

f,(:) = (1— (2)2)2 (1 — (92)) (2)71. i=1,2,--- ,N. (5.10)

Table 5.2 gives the fundamental frequency ( for n = 0, no-nodal diameters) of clamped

circular plate with a core of various radii using N = 12 iterations

 

b 0.2 0.1 0.05 0.01 —> 0
A 5.8830 5.2231 4.9689 4.8556 4.7688

 

 

 

 

 

 

 

 

 

Table 5.2: Fundamental frequencies of clamped circular plate with a core

The values of the fundamental frequencies given in Table 5.2 verifies those that

are given by the asymptotic approximation in Table 2.3 in Chapter 2.

5.1.2 Simply Supported Boundary Condition

In the case of simply supported circular plate, the boundary conditions are given by

WN(r,6) = (%+;Qg‘£) =0 at r=a (5.11)

97

For N -parameter Rayleigh-Ritz solution we can assume $1030) = f,(r)cos(n6),

where n is the number of nodal diameters and

fi(7') = 1 " (:)i+1- _ (5'12)

(1

Here f,('r)’s satisfy the geometric boundary conditions, which are f,-’(0) = f,(a) =

0. Since the moment (3%} + $939) is the natural (not essential (i.e., geometric))

boundary condition, fi(r) ’5 don’t have to satisfy the moment equation (5.11).
Stable convergence of the solution is reached for N = 12 iterations. Table 5.3

givas the values of the natural frequencies of the clamped circular plate.

 

n 0 1 2
A 2.2215 3.7280 5.0610

 

 

 

 

 

 

 

Table 5.3: Natural frequencies of simply supported circular plate

In Table 5.3 n is the number of nodal diameters and A2 = wazm the nor-
malized frequency. The frequency values agree with the published results in the
monograph of Leissa [8].

Consider a simply supported circular plate with a core of radius b. Then the

generating functions f,’s can be chosen as

f.(r) = (1 — g) (1— (92)! (2)“. 1:12,... ,N. (5.13)

Table 5.4 gives the fundamental frequency ( for n = O , no—nodal diameters) of clamped
circular plate with a core of various radii using N = 12 iterations
The values of the fundamental frequencies given in Table 5.4 verifies those that

are given by the asymptotic approximation in Table 2.4 in Chapter 2.

98

 

 

b 0.2 0.1 0.05 0.01 —> 0
LA 4.7660 4.2187 4.0113 3.9265 3.8489

 

 

 

 

 

 

 

 

 

Table 5.4: Fundamental frequencies of simply supported circular plate with a core

5.2 Polygonal Plates With a Core

Consider a regular hexagonal plate situated in the cartesian coordinate system with
an inscribing circle of radius a = 0.9532 at which we have averaging circle of radius 1.
A set of two dimensional orthogonal plate functions is used to describe the deflected
shape of the plate and the Rayleigh-Ritz method (pb2-Ritz) is used to obtain the
frequencias [10].

Since there is no dissipation, the maximum potential equals the maximum kinetic

62W 62W 62W62W 62W 2
—/{<.—2 —2-> [—<>]}

—-2-pc62/ W2(x,y) dxdy = 0. (5.14)
0

energy:

 

 

The displacement W(1, y) can be exprassed as

W(z.y)~ W~( (a: y) =Zq¢.(x y) (5215)

where c,- ’s are the coefficients to be minimized by the Rayleigh-Ritz method and 43,- ’s
are the orthogonal assumed mode shapes (plate functions). The assumed modes <15,-

are generated by the Gram-Schmidt recurrence procedure:

¢n($ay)= fn($7 y )(¢1 (I? 1y) _Z¢n,i¢i( 23 y) (5.16)

for n=2,3,--- ,N,where

f0 fn($2 y)¢1(x2 y)¢2(:v2 y) dxdy
f9 ¢?($2 y) dwdy

 

wnfl = (5.17)

99

 

in which fn(:r,y)’s are generating functions (i.e., approximating basis:
1, :r, y, my, $2, 312, fly, ---) chosen to make sure that the higher order
orthogonal plate functions (25,- satisfy the geometric boundary conditions. The

generating functions are produced in following manner:

Let
r = [m]
t = (n — 1) - r2. (5.18)

where f 1 denotes the greatest integer function and n is the total number of terms
(iterations) used in the deflection series.

If t is even then
p=U% OSpSr
fn($2y) = wry”. (5.19)
If t is odd then
p=0-¢V% OSpSr-l
fn(-’L‘2 y) = xpy’- (5-20)

For a polygon with m sides, the starting plate function, satisfying the geometrical

boundary conditions, 451(x, y) is given by

451(17231) = II «62(252 y) (521)
p=l
where (6, ’s are the edge functions of the polygon.
Substituting equation (5.15) into (5.14) and minimizing the Rayleigh quotient with
respect to the undetermined coefficients c, , we obtain

6 ” 6%.- 2 N 6%.- N 6%.-
0 = 5C:{D/n{ gay] +211 [Zq?§q yz (5.22)

._ . 0
N ”l l” ”l l” l2

+ - ' +2(1—) - ‘ — 2 <6? dd
[gc'ayz V 2:616:60}; pw 26' } :1: y}

100

 

 

 

 

which gives us the eigenvalue problem

 

 

 

 

 

2 (K2,- — AzMij) c, = 0 (5.23)
where
K.,- = R,- + Q2,- + 12(R,,- + 521') + 2(1- V)Ti‘ (524)
M2. = [.2 2.12, y)¢j($2y) dxdy (5.25)
P.) = n 626656226) 32055:? y) dxdy (5.26)
Q2,- — n 62%;? 3”) 62%;? 3’) dandy (5.27)
12., = a 32%; 3”) 62%;? y) dxdy (5.28)
5.,- = a 62%;? y) 62%;? 3’) dzdy (5.29)
T5 = (9262-0236) Wat-($211) duly (5.30)

n BxBy away
The eigenvalues of equation (5.23) gives the natural frequencies of the plate. We
will only consider a regular hexagonal plate in this section to compare our frequency
values with those that are obtained by the boundary perturbation method in Chapter
3 and Chapter 4.

5.2.1 Clamped Boundary Condition

The starting plate function for a clamped hexagonal plate can be specified by the
terms <p,,(a:, y) in equation (5.21. The edge functions cpp(x, y) for an hexagonal plate
are obtained according to the individual edge conditions. To ensure that at least the
geometric boundary conditions are satisfied, we take
451013211) = (II: - 0/2)2($ + a/2)2(y - mm: - b/2)2(y - mzx - b/2)2
(y — mlzz: + b/2)2(y — mga: + b/2)2 (5.31)
where m1 = —b/2a and m2 = b/2a.

101

Numerical results are obtained by using the pb2-Ritz method is tabulated in Table
5.5 with N = 5, N = 10, and N = 15 iterations. The more the iteration is the better

the approximations become.

 

 

 

 

 

 

 

 

 

 

N = 5 N = 10 N = 15
n A n A n A
0 3.35885 0 3.27845 0 3.27845
1 4.82053 1 4.71163 1 4.69954
2 4.82054 2 4.78899 2 4.70972
3 6.07767 3 6.07490 3 6.00887
4 6.26642 4 6.07768 4 6.07490

 

 

Table 5.5: Natural frequencies of clamped hexagonal plate

Here n is the number of nodal diameters. The values of the natural frequencies given

in Table 5.5 verifies those that are given by the asymptotic approximation in Table

3.5 in Chapter 3 and published values.

Consider a fully clamped hexagonal plate with a core of radius c. Then the starting

plate function can be chosen as

45105.?!) = (a? — 0/2)2($ + a/2)2(y - mix - b/2)2(y - mzx - b/2)2

(y — mm: + b/2)2(y — 777.21? + b/2)2(:1:2 + y2 — C?)2 (5.32)

Table 5.6 gives the fundamental frequency of clamped circular plate with a core of
various radii using N = 10 iterations

The values of the fundamental frequencies given in Table 5.6 are higher than those
that are given by the asymptotic approximation in Table 4.8 in Chapter 4. Because
the pb2—Ritz method gives the upper bound for the frequency and it converges to the

true frequency as N gets arbitrarily large.

102

 

 

 

c 0.2 0.1 0.05 0.01 0.001
A 6.48446 6.10927 6.02703 6.00135 6.00029

 

 

 

 

 

 

 

 

 

Table 5.6: Fundamental frequencies of clamped hexagonal plate with a core

5.2.2 Simply Supported Boundary Condition

The starting plate function, satisfying at least the geometric boundary conditions, for

a simply supported hexagonal plate can be specified by
¢1(I2 y) = (:c - 0/2)($ + a/2)(y - mil: - b/2)(y - mzx - b/2)
(y — mg: + b/2)(y - "12$ + b/2) (5.33)

Numerical results are obtained by using the pb2-Ritz method is tabulated in Table
5.7 with N = 5, N = 10, and N = 15 iterations.

 

 

 

N = 5 N = 10 N = 15
n A n A n A
0 2.53362 0 2.50228 0 2.50228
1 3.91994 1 3.91793 1 3.90364
2 3.91994 2 3.91926 2 3.91784
3 5.31728 3 5.29025 3 5.21592
4 5.56188 4 5.31728 4 5.29026

 

 

 

 

 

 

 

 

Table 5.7: Natural frequencies of simply supported hexagonal plate

Here n is the number of nodal diameters. The values of the natural frequencies given
in Table 5.7 verifies those that are given by the asymptotic approximation in Table

3.7 in Chapter 3 and published values.

Consider a simply supported hexagonal plate with a core of radius c. Then the

103

 

c 0.2 0.1 0.05 0.01 0.001
A 5.88091 5.63531 5.57653 5.55800 5.55724

 

 

 

 

 

 

 

 

 

Table 5.8: Fundamental frequencies of simply supported hexagonal plate with a core

starting plate function is given by

$10526) = (in - a/2)(x + a/2)(y - mix - b/2)(y - mza: - b/2)

(y — mlx + b/2)(y — mgr + b/2)(:I:2 + y2 - 62)2 (5.34)

Table 5.8 gives the fundamental frequency ( for n =0, no—nodal diameters) of simply
supported hexagonal plate with a core of various radii using N = 10 iterations. The
values of the fundamental frequencies given in Table 5.8 are higher than those that
are given by the asymptotic approximation in Table 4.11 in Chapter 4. Because the
pb2-Ritz method gives the upper bound for the frequency and it converges to the

true frequency as N gets arbitrarily large.

104

CHAPTER 6

Conclusions and Discussions

An accurate perturbation method to find the fundamental frequency of various plates
with clamped and simply-supported boundary conditions is presented and compared
with numerical methods (mainly the Rayleigh-Ritz method). The fundamental fre-
quencies of plates are formulated and tabulated using perturbation methods. The lack
of analytical solutions for several domains (other than the circular domain), and the
problem of singularity, scaling, and sensitivity to the boundary conditions in the nu-
merical solutions lead us to a special formulation of a perturbation theory to improve
accuracy and reliability of the fundamental frequency in vibrating plates. Therefore,
an approximate analytical formulation of the fundamental frequency for clamped and
simply supported plates is presented by adopting a boundary perturbation method
[25] to extract the fundamental eigenvalue of the problem.

In Chapter 2, we consider the circular domain with a concentric circular core.
The frequency values are simulated and tabulated for both asymptotic and numerical
solutions; and the comparisons showed that both methods are in good agreement in
clamped and simply-supported cases. However, in the case of free boundary condition,
due to singularity at 0, the numerical method fails to give any frequency result as
the radius of the core I) —> 0. The comparison of fundamental frequencies between

numerical methods and perturbation method has been given in Table 2.2. While the

105

first numerical method (direct computation by programming) fails to give any result,
the Rayleigh-Ritz method fails to converge. In fact the method even fails to give
right convergence pattern i.e., after b = 0.001 the frequency becomes inconsistent
which contradicts the zero frequency of the plate as b —> 0. Therefore we have
adopted an asymptotic expansion [26] of the solution for “small” argument to obtain
analytical approximate solutions to the fundamental frequency. A formula for the
fundamental frequency has been obtained for free, clamped, and simply-supported
boundary conditions in (2.21), (2.30), and (2.36) respectively. The formula (2.21) is
an improvement of Southwell’s result [22].

In Chapter 3, a boundary perturbation method [25] has been developed in its gen-
eral case to extract the fundamental eigenvalue of the problem for wavy boundary,
elliptical, and polygonal plates with clamped and simply-supported boundary condi-
tions. There have been several papers on elliptical and polygonal plates [18] [17] [15]
[13] [11] [10] [9] [5] [4] [1], however no work has been done on wavy boundary plates.
Most of the papers on the vibration of elliptical and polygonal plates present numeri-
cal solutions. In the case of clamped elliptical plates, Parnes [15] used a perturbation
scheme based on perturbing the boundary of a circle circumscribing the plate. In this
work a better estimate has been developed by applying the boundary perturbation
method to the “averaging circle” for both clamped and simply-supported boundary
conditions. For wavy boundary plates, there are no published results for comparison.
For elliptical and polygonal plates, the results of our method agreed with those by
other authors. The relative errors between the proposed asymptotic formulae in this
work and the existing raeults are found to be within 10%. The numerical calculation
in Chapter 5 using pb—2 Ritz method also agreed with the perturbation method and
the relative error between the two methods is found to be less than 2%. The funda—

mental frequency formulae suggested for clamped and simply-supported plates (wavy

106

boundary, elliptical, polygonal) are reliable and easy to use in design.

In Chapter 4, we have adopted the boundary perturbation method of Chapter 3 to
extract the fundamental eigenvalue of the problem for wavy boundary, elliptical, and
polygonal plates with a concentric circular core in the case of clamped and simply-
supported outer boundary conditions. There are no published results for this problem
except in the case of circular plate (novel asymptotic formulae given in Chapter 2).
The fundamental frequencies of the above plates are tabulated for various radii of the
core. The numerical calculation in Chapter 5 using pb—2 Ritz method gave bigger
frequency values than the perturbation method. This is consistent, because the pb—2
Ritz method gives the upper bound for the frequency and it converges to the true
frequency as the number of iterations N gets arbitrarily large. Although N = 10
iterations were enough to achieve the desired convergence in the case of plates with
no core, more iterations are needed for the plates with a core.

The boundary perturbation method is applied to the biharmonic eigenvalue prob-
lem up to the second degree of approximation. This is a new approach to the problem,
and our results are consistent with the existing literature (on elliptical and polygonal
plates).

The wavy boundary plates and plates with a concentric circular core are considered
for the first time in this work, therefore there is no existing literature for comparison
of results. An attempt for a numerical calculation has been made. The program used
in this work uses symbolic calculation in both Mathematica and MatLab, and could
only run up to N = 10 iterations. We observed that as the radius of the clamping
core gets smaller, the frequency values did not converge to the perturbed frequency
values. Note that the error in the perturbation method is of order 0(6‘). So for small
6 the error is negligible. Therefore our perturbed frequency gives a better estimate

for the frequency values.

107

For future work, free, discrete, and mixed boundary conditions can be considered
using the same perturbation method. Furthermore, it would be interesting to study
plates with more cores, and the optimal locations of such cores in order to achieve

the maximum possible fundamental frequencies.

108

 

Appendix A

Perturbed Boundary Condition

Coefficients

Perturbed clamped boundary conditions appearing in equations (3.13) and (3.16) are

determined by the following equtions:

2:026) = -f(9)Wo.(129) (A.1)
250.0) = —f(6)W2.(126)—§12(6)Wo..(126) (A2)
@1029) = f'(9)Woo(129)-f(9)Wor.(129) (A3)
250, 0) = -2f(9)f’(9)Wo.(120) + f’(9)W1.(120) — f(9)W1..(12 6)

+ f<0)f'(0>wo..(1,e)-$12<0>Wo...<1,6>. (A.4)

109

 

Perturbed simply supported boundary conditions appearing in equations (3.13) and

(3.22) is determined by the following equations:

<I>f(1,6)
630.6)

251,6)

530,9)

-f(9)Wo.(129) (62.5)
—f(9)W1.(12 6) — $12(6)Wo..(1,6) (A.6)
—2f'(6)wo.(1, 6) + 60(9) + f"(9))Wo.(12 6)

2f'<6)Wo..(1. 6) + Vf’(0)Wo.(12 6) — Vf(9)Wo.2.(12 6)
f(9)Wo...(129) (A.7)
6f(9)f’(9)Wo.(12 6) — f’(9)f”(9)Wo.(12 6)

2f’(0)W1.(12 6) —- f’(9)2Wo..(12 6)

6(7f(6)2 - 3f(6) (2f(9) + f"(6)) + f(0)f”(9))Wo.(12 6)
6f(9)f’(9)Wo..(12 6) + 21'(6)W2..<1. 6) + f’(9)2Wo.-.(12 6)
me) + f”(0))( — f’(9)Wo.(12 6) — W211, 6) + f(9)Wo..(12 6))
2f(6)f'<6)wo...(1. 6) + Vf’(9)W1.(12 6) - 26f(9)f’(9)Wo.(12 6)
Vf(9)f’(0)Wo..(12 6) — Vf(9)W1..(129) — §VI<6)2wo...<1, 6)

f(9)W1...(12 6) - §f(6)2wo....(1,6). (A.8)

110

 

Appendix B

Plate Coefficients

B.1 Coefficients of the Clamped Plate With a

Core

The coefficients of the zeroth order solution (4.13) are that

C
001

C
0‘02

C

0‘03

where

601
fioz
.303
604
605
floe

fimKoUCo) - 302Ko(Cko) - floaKloco)

 

3043/0090) " 3053603190) - fiOGYIUVO) (Bi)
3:; [J1(ko)Ko(ko) — Jo(ko)K1(ko)

as.(K2(ko)Yo(ko) “Ko(ko)Y1(ko))] (B6)
1 C C

-W [10090) + 0013/0090) + “0210(k0)[° (B3)

11(ko)Jo(CkZo) + Io(Cko)J1(ko)

'11(ko)~]o(ko) + Io(’€o)JI(ko)

Io(cko)Jo(ko) — 10(k0)J0(cko)
11(k0)Ko(cko) + I0(cko)K1(ko)
11(ko)Ko(ko) + Io(ko)K1(ko)
Io(cko)Ko(ko) — Io(ko)Ko(cko)

111

 

The coefficients of the first order solution (4.19) can be determined by Cramer’s

rule. Let

'JMUco) YMUCO) 12.020) K2205)?
A. = 61.622) 361.060) 11.622) K5622) (3.4)
JM(Cko) YM(Ck0) [M(Cko) KM(Ck0)
[2714660) make) 114(660) Kt<cko)_

BC : [0 —Wo,,(1)o 0]T (3.5)

 

 

and let Af be the matrix obtained by replacing the i-th column of A6 by BC. Then

the coefficients are given by

c _ deth)
011,- -— det(AC) (B.6)

 

for i: 1,2,3,4.
The coefficients of the first order solution (4.66) for an elliptic plate with a core

can be determined by Cramer’s rule. Let

"12.65) 162.060) 122(6)) 15.620)
«7522090) Y2'n(ko) [infkol K5460)
J2n(Ck0) Y2n(Cko) IM(Ck0) KM(Ck0)
_ 211(Ck0) 3340160) 1522(0160) K2n(Ck0)_
[o -c..wo,,(1) o o]T (B.8)

 

 

"A
II

for n = 1, 2, - - - and let E3,- be the matrix obtained by replacing the i-th column of

E; by F: . Then the coefficients are given by

c _ det(E1cu')
7m — det(E,°,) (13.9)

for i=1,2,3,4 and n=1,2,---.

112

 

The coefficients of the first order solution (4.110) for an M sided regular polygonal

plate with a core can be determined by Cramer’s rule. Let

p

J.M(ko) 1522(6)) 1.2.2.020) KnMUCo) l
G: = 122(6)) £12.60) 12.620) K1126.) (5.10)
JnM(Ck0) YnM(Ck0) InM(CkO) KnM(CkO)
[J;M(Ck0) YdM(Ck0) I;M(Ck0) LMW‘O) .

Hf, : [0 -c,,Wo,,(1) o 0]T (B.11)

 

 

for n = 1, 2, - - - and let G5,,- be the matrix obtained by replacing the i-th column of

Of, by Hg. Then the coefficients are given by

c _ det(GEzi)
"1 _ det(G$,)

 

(13.12)

for i=1,2,3,4 and n=1,2,---.

B.2 Coefficients of the Simply Supported Plate

With a Core

The coefficients of the zeroth order solution (4.43) are that

s 311M(Io(k0)) + 312M(Jo(ko)) — 303M(Ko(ko))

 

 

(2,, = 622M(Io(ko)) + 6..M(Ko(ko)) - 602M056.» (3'13)

as = 315M(Ko(ko)) -' Ko(ko)M(515) (B 14)
02 K0(ko)M(Io(ko)) " 10(k0)M(K0(k0)) .

0163 = —K—o(ko_) [Jo(ko) + agiyoUCo) + agzloUcol] (B.15)

113

where

(in : Jo(ko)Ko(ck0)—Jo(cko)Ko(ko)
612 : Io(ck0)Ko(ko)—Io(ko)Ko(cko)
613 = Ko(ko)Yo(cko)-Ko(cko)Yo(ko)
[314 = Io(cko)Yo(ko) -Io(ko)Yo(cko)

515 = Jo(ko)+aglyo(ko)

The coefficients of the first order solution (4.46) can be determined by Cramer’s

rule. Let

F -

JMUCO) YMUCO) [M(kO) KMfko)
AS ___ M(JM(ko)) M(YM(ko)) M(IM(kO)) M(KM(kO)) (B.16)
JM(ck0) YM (cko) I M(cko) K M(cko)

1 JMCko) Yuck.) 114(660) KMCko)
" 'l

 

—Wo.(1)
BS = ”(1 + M2)Wor(1) " VWo.:(1) "‘ Worn-(1) (13.17)
0

0

 

 

b -

and let A? be the matrix obtained by replacing the i-th column of A5 by BS . Then

the coefficients are given by

det(AS)
S _ z
011' _ det(AS) (B.18)

for 2' = 1,2,3,4.

The coefficients of the first order solution (4.89) for an elliptic plate with a core

114

can be determined by Cramer’s rule. Let

' J2.(ko) 152(60) 12.160) K2n(ko) l
E: = M(J2n(k0)) MO/ZnUCO» M(I2n(k0)) M(K2n(k0)) (B19)
J2n(Ck0) an (Cko) 12n(Cko) K2n(Ck0)

[_ J21: (CkO) 1[2’1'1(Ck:0) 1211(Ck0) K2n(Ck0)

 

' —c.Wo.(1) 1

F: = —c.(wo...(1)+ on...(1)+ 6666 — 1)wo.(1)) (13.20)
0

 

 

L 0

.l

for n = 1, 2, - - - and let E5,- be the matrix obtained by replacing the i-th column of

Eff by F: . Then the coefficients are given by

det(ES-)
S _ m

for i: 1,2,3,4 and n =1,2,---.
The coefficients of the first order solution (4.133) for an an M sided regular polyg-

onal plate with a core can be determined by Cramer’s rule. Let

 

' 1.222(6)) 16.225.) 1.22020) 16.2260) '
Cf, = M(JnM(k0)) M(YnM(k0)) M(InM(k0)) M(KnM(k0)) (B.22)
JnM(Ck0) YnM(Ck0) InM (CkO) KnM (CkO)
«7.114(6’50) YJMC’CO) LM(Cko) LMCko)
- —CnWo.(1) q
H3 = -cn(Wo...(1) + VWor.(1) + V0114")2 — 1)Wo.(1)) (B23)
2. 0
1 6 .

 

 

for n = 1, 2, - -- and let Ci,- be the matrix obtained by replacing the i-th column of

0;? by HE . Then the coefficients are given by

det(Gs-) .
S. : _EL 2 = e e e . 8.24
m det(G§,) for z 1, 2, 3, 4 and n 1, 2, ( )

115

 

BIBLIOGRAPHY

116

 

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