Green’s function approach to frequency analysis of thin circular plates

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BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL SCIENCES, Vol. 64, No. 1, 2016
_DOI:_ _10.1515/bpasts-2016-0020_

# Green’s function approach to frequency analysis of thin circular plates

K.K. ŻUR ∗

Faculty of Management, Bialystok University of Technology, 2 Ojca Stefana Tarasiuka St., 16-001 Kleosin, Poland

Abstract. The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green’s functions is considered. The
formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions
of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach
allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal
diameters. The first ten dimensionless frequencies for eight different natural modes of circular plates are calculated. A part of obtained
results have not been presented yet in open literature for thin circular plates. The results of investigation are in good agreement with selected
results obtained by other methods presented in literature.

Key words: free vibration, circular plates, Green’s function.

1. Introduction

In recent years, lightweight plate structures have been used in
many engineering applications. Components of circular plates
are commonly used in aerospace industries and aviation as
well as in marine and civil engineering applications. Circular
plates are the most critical structural elements in high speed
rotating engineering systems such as circular saws, rotors, turbine flywheels, etc. The natural frequencies of the plates have
been studied extensively for more than a century, if only because the frequency of an external load matches the natural
frequency of the plate, destruction may occur.

Researchers have used various methods of analysis of dynamic behavior of plates with different boundary conditions.
The work of Leissa [1] is an excellent source of information
about methods used for free vibration analysis of plates. The
free vibration analysis has been carried out by using a variety of weighting function methods [1] such as the Ritz
method, the Galerkin method or the finite element method.
In many works [1, 2] natural frequencies of circular plates
are expressed in terms of the Bessel functions. Chakraverty et al. [3, 4] have studied the free vibration analysis of
plates of various geometries by using two-dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz
method. Wu and Liu [5, 6] proposed the generalized differential quadrature rule (GDQR) to a free vibration analysis of
circular thin plates. Jaroszewicz and Zoryj [7] have studied
free vibration of circular thin plates of constant and linearly
variable thickness using the method of partial discretization
(MPD). Ebrahimi and Rastgo [8] investigated the natural vibration behavior of circular functionally graded plates with
clamped edges based on classical plate theory. Yalcin et al. [9]
have studied free vibration of circular plates by using differential transformation method (DTM). Zhou et al. [10] applied

∗e-mail: k.zur@pb.edu.pl

the Hamiltonian approach to a solution of the free vibration
problem of circular and annular thin plates. Kukla and Szewczyk [11] applied the Green’s functions (influence function)
and Bessel functions to the solution eigenvalue problem of
annular thin plates with discrete elements such as oscillators.
Similarly, Sorokin and Peake [12] have studied the free vibration analysis of sandwich plates with concentrated springs
and mass by using Green’s functions. An area of application
of Green’s functions in a free vibration analysis of isotropic
beams and plates with constant and variable distribution of
parameters is presented in the monograph of Kukla [13]. The
application of Green’s functions in free axisymmetric vibration of circular thin plates with clamped edges is presented in
the book of Jaroszewicz and Zoryj [14].

In the present study, Green’s functions are used to obtain ten lower natural frequencies for eight different natural
modes of circular plates with different boundary conditions.
A formula of construction of influence functions for different
modes of uniform circular thin plates is obtained in closed
form. The characteristic equations for different boundary conditions and different number of nodal lines of thin circular
plates are defined. The numerical results of investigation are
compared with results presented in literature.

2. Statement of the problem

Consider an isotropic, homogeneous circular thin plate of constant thickness h in cylindrical coordinate (r, θ, z) with the
z-axis along the longitudinal direction. Geometry and coordinate system of considering plate as shown in Fig. 1. The
partial differential equation for free vibration of thin plates
has following form

∇ 4 W + ρh

∂ 2 W

= 0, (1)
∂t 2

181

K.K. Żur

where ρ is the mass density, D = Eh 3 /12(1 - ν 2 ) is the flexural rigidity, E is Young’s modulus, ν is the Poisson ratio,

M (w)|ξ=1 = 0, (8a)

ddξ 3 w 3 + 1 ξ

∇ 2 = ∂r ∂2

∂r ∂22 + 1 r

∂ 2

∇ 2 = ∂r ∂22 + 1 r ∂r + r 12 ∂θ 2 is the Laplacian and W (r, θ, t) is

the small deflection compared with the thickness h of plate.

Free:

V (w)|ξ=1 =

ddξ 3 w 3

∂r∂ + r 12

r 2

dξ

(8b)

1 + 2n 2 - νn2

ξ 2

d 2 w

dξ 2 −

ξ 3

- 3n2 − νn2

= 0.

ξ=1

Sliding supports:

Elastic supports:

ξ=1

)����|ξ=1

dξ

= 0, (9a)

V (w)|ξ=1 = 0. (9b)

+φ dw

dξ

ξ=1

ν dw dξ − νn ξ 22

νn ξ 22 w

=0, (10a)

Fig. 1. Geometry and coordinate system of the circular plate

The deflection of a circular plate may be expressed as
follows

W (r, θ, t) = w(r) cos(nθ)e iωt, (2)

where n is the integer number of diagonal nodal lines, w(r) is
the radial mode function, ω is natural frequency, and i 2 = −1.
Substituting Eq. (2) into Eq. (1) using the dimensionless coordinate ξ = r/R the governing differential equation of the
circular plate becomes:

L(w) − λ 2 w = 0, (3)

where

Φ(w)|ξ=1 =

�� ddξ2w 2 +ν dw dξ

(10b)

Ψ(w)|ξ=1 =

dξ 3 + d dξ 2w2 + (νn2 − 2n2 − 1)
�� d3w

- ψw

- dw dξ + (3n2 − νn2)w

ξ=1

= 0.

L(w) ≡ d dξ 4w4

d 2 w

dξ 2

(4)

d dξ 4w4 + 2 ξ

d 3 w

ddξ 3 w 3 − (1 + 2 ξ 2 n2)

ξ 2

M (w) and V (w) are the normalized radial bending moment and the normalized effective shear force, respectively.
φ = KφR/D and ψ = KψR 3 /D are normalized parameters
of elastic supports. Kφ and Kψ are rotational and translational
spring constants (Fig. 2), respectively.

Fig. 2. The cross-section of uniform circular plate with elastic sup
ports

3. Mathematical background

The formula of construction of the Green’s function (general
solution) for homogeneous differential equations

+ (1 + 2n2)

ξ 3

dξ + (n4 − ξ 44n2)

w
ξ 44n2)

is differential operator and

frequencyλ = ωRof 2�

ρh/D (5)

is dimensionless frequency of vibration.

The boundary conditions at the outer edge (ξ = 1) of the
circular plate may be one of the following: clamped, simply
supported, free, sliding supports and elastic supports. These
conditions may be written in terms of the radial mode function
w(ξ) in the following form:
Clamped:

w(ξ)|ξ=1 = 0, (6a)

L(y) ≡

k=0

pk(x) dky a < x < b (11)

dx k = 0,

k=1

Ck ′ yk(x), (12)

Ck ′

dξ

= 0. (6b)

has following form

K(x, α) =

Simple supports:

M (w)|ξ=1 =

dξ 2 + ν ξ

- d2w

ξ=1
����

w(ξ)|ξ=1 = 0, (7a)

dwdξ − νn ξ 22

ξ 22 w

where coefficients pk(x) are continuous functions, pn(x)̸ = 0
for x ∈ [ab] and α is a position where all discrete elements
could be mounted [11, 14].

ξ=1

= 0. (7b)

182 Bull. Pol. Ac.: Tech. 64(1) 2016

_Green’s_ _function_ _approach_ _to_ _frequency_ _analysis_ _of_ _thin_ _circular_ _plates_

Constants Ck ′ have form [13]

(−1) n+k
k ′ =

Ck ′

pn (α) Wn(α)

The roots of Eq. (20) have form

s1 = n, s2 = −n, s3 = 2 − n, s4 = 2 + n. (21)

The linear independent solutions of Eq. (15) are

The Green’s function (solution of homogeneous Euler equation L(K(ξ, α)) = 0) for different nodal diameters may be
received from Eq. (15) presented in the following form

|An|
Kn(ξ, α) = W (α)npn(α) , (23)

where pn(α) = 1 is coefficient placed on before highest order
of derivative of Euler differential Eq. (19) and

yk+1(α)
yk ′ +1 (α)

k ′ +1 (α)

n ′ (α)

...
yn (n−2)

n (n−2) (α)

yn(α)
yn ′ (α)

(22)

����������

y1 ′ (α) - · · yk ′ −1 (α)

... - · · ...
y1 (n−2) (α) - · · yk (n - − 1 2)(α

y1(α) - · · yk−1(α)
y1 ′ (α) - · · yk ′ −1 (α)

1 ′ (α) - · · yk ′

w1(ξ) = ξ n, w2(ξ) = ξ −n,

w3(ξ) = ξ 2−n, w4(ξ) = ξ 2+n .

...
yk (n - − 1 2)

k (n - − 1 2)(α)

1 (n−2) (α)

- · ·

...
yk (n +1 −2)

k (n +1 −2)(α)

- · ·

- · ·
...
. . .

����������

k = 1, 2, . . ., n,

(13)
where Wn is Wronskian

Wn(x) =

k (x)

independent

���y(i−1)

(14)
1≤i, k≤n

solutions of Eq. (1), then

���

and yk(x) are linear independent solutions of Eq. (1), then
Wronskian satisfies condition Wn(α)̸ = 0.

The other formula of construction of the Green’s function
for homogeneous differential equations has form [14]

|A|
K (x, α) = W (α)pn (α) , (15)

where

α n α −n α 2−n α 2+n

d(α 2+n )

dα

d 2 (α 2+n )

dα 2

d(α −n )

dα

d 2 (α −n )

dα 2

d(α 2−n )

dα

d 2 (α 2−n )

dα 2

|An| =

����������������

d(α n )

dα

d 2 (α n )

dα 2

, (24)

ξ n ξ −n ξ 2−n ξ 2+n

y1(α) y2(α) . . .
y1 ′ (α) y2 ′ (α) - · ·

yn(α)
yn ′ (α)

n ′ (α)

α n α −n α 2−n α 2+n

����������������

y1 ′ (α) y2 ′

d(α 2+n )

dα

d 2 (α 2+n )

dα 2

d 3 (α 2+n )

dα 3

�����������������

d(α 2−n )

dα

d 2 (α 2−n )

dα 2

d 3 (α 2−n )

dα 3

d(α −n )

dα

d 2 (α −n )

dα 2

d 3 (α −n )

dα 3

|A| =

�������������

y1 ′ (α) y2 ′ (α) - · ·

... ... . . .
y1 (n−2) (α) y2 (n−2) (α) . . .

...
yn (n−2)

n (n−2) (α)
yn(x)

, (16)

�����������������

d(α n )

dα

d 2 (α n )

dα 2

d 3 (α n )

dα 3

, (25)

�������������

W (α)n =

. . .
. . .
. . .

1 (n−2) (α)
y1(x)

...
y2 (n−2)

2 (n−2) (α)
y2(x)

yn(α)
yn ′ (α)

. (17)

After calculations, the Green’s function (GF) for different
number nodal diameters n has following form

W (α)=

����������

y1 ′ (α) y2 ′ (α) - · ·

... ... ...
y1 (n−1) (α) y2 (n−1) (α) . . .

y1(α) y2(α) . . .
y1 ′ (α) y2 ′ (α) - · ·

1 ′ (α) y2 ′

...
yn (n−1)

n (n−1) (α)

n ′ (α)

����������

1 (n−1) (α)

...
y2 (n−1)

2 (n−1) (α)

...
. . .

Additionally, the functions K(x, α) must satisfy conditions
presented in the following form [13]

Kn(ξ, α) = αn+1ξ−n+2 − ξnα−n+3

8n 2 - 8n

(26)

(27a)

∂ i K(x, α)

∂x i

x=α

n−1

����

= 0, i = 0, 1, . . ., n − 2, (18a)

+ α−n+1ξn+2 − ξ−nαn+3, n ≥ 2

8n 2 + 8n

∂ n−1 K(x, α)

1
= pn(α) . (18b)

and satisfies conditions from Eq. (18)

Kn(α, α) = ∂Kn(ξ, α) |ξ=α

∂ξ

= ∂2Kn(ξ, α) |ξ=α = 0,

∂ξ 2

∂x n−1

x=α
����

4. Green’s function

The characteristic equation of homogeneous differential Euler
equation

L(w) ≡ d dξ 4w4

d 3 w

(19)

d dξ 4w4 + 2 ξ

ξ 2

d 2 w

dξ 2

ddξ 3 w 3 − (1 + 2 ξ 2 n2)

∂ 3 Kn(ξ, α)

|ξ=α = 1. (27b)
∂ξ 3

The function Kn(ξ, α) is indeterminate for n = 0 and n = 1,
then the Green’s functions for this values have following form

+ (1 + 2n2)

ξ 3

dξ + (n4 − ξ 44n2)

w = 0

− ξ 44n2)

, (28)

α 2 - ξ 2 + (ξ 2 + α 2 )ln ξ

lim
n→0 Kn(ξ, α) = α 4

has following form

s 4 - 4s 3 + (4 − 2n 2 )s 2 + 4n 2 s − n 2 (4 − n 2 ) = 0. (20)

nlim→1 Kn(ξ, α) =

ξ 4 - α 4 + 4ξ 2 α 2 ln α ξ

ξ . (29)
16ξ

Bull. Pol. Ac.: Tech. 64(1) 2016 183

K.K. Żur

In Table 1 the examples of formulas of Green’s function for
seven nodal diameters are presented.

K0(α)u and K0(α)v are the limited independent solutions depending on number of nodal lines presented in the
following form

K0(α)u = α n, (32a)

K0(α)v = α n+2 . (32b)

The characteristic equations ∆n = 0 for different boundary
conditions and different values of parameter n are obtained
from well known the characteristic determinants given by:
Clamped:

Table 1
The formulas of Green’s functions for the different number of nodal

diameters

n Kn(ξ, α)

0 K0(ξ, α) = α

formulas of Green’s
presented.

Table 1
functions for the different number

diameters

Kn(ξ, α)

α 2 - ξ 2 + (ξ 2 + α 2 )ln ξ

1
K1(ξ, α) =

ξ 4 - α 4 + 4ξ 2 α 2 ln α

ξ
16ξ

α 3 − α ξ 47

α 4 − α ξ 58

α 5 − α ξ 69

ξ9

α 6 − α ξ 107

∆n(λ) ≡

Free:

∆n(λ) ≡

2 K2(ξ, α) = (ξ2 − α2)3

∂ξ

Kn(ξ, λ)u Kn(ξ, λ)v
∂Kn(ξ,λ)u ∂Kn(ξ,λ)v

∂ξ ∂ξ

�����

ξ=1
�����

; (33)

)

2 + (ξ 2 +

4 + 4ξ 2 α

16ξ

2 − α2)3

48ξ 2 α

α2)3(ξ2 +

96ξ 3 α 2

1
+
160

4 + 4ξ 2 α

16ξ

2 − α2)3

48ξ 2 α

α2)3(ξ2 +

96ξ 3 α 2

1
+
160

1
+
240

16ξ

2 − α2)3

48ξ 2 α

α2)3(ξ2 +

96ξ 3 α 2

1
+
160

1
+
240

1
+
336

48ξ 2 α

α2)3(ξ2 +

96ξ 3 α 2

1
+
160

1
+
240

1
+
336

1
+
448

∆n(λ) ≡

Simply supported:

∂Kn(ξ,λ)v

48ξ 2 α

α 2 )

2 ln α

α2)

- ξ6

- ξ7

- ξ8

2 ln α

α2)

- ξ6

α 3

- ξ7

- ξ8

- ξ9

α2)

- ξ6

α 3

- ξ7

α 4

- ξ8

- ξ9

α2)

α 3

- ξ7

α 4

- ξ8

α 5

- ξ9

∂ξ

3 K3(ξ, α) = (ξ2 − α2)3(ξ2 + α2)

96ξ 3 α 2

ξ=1
�����

4 K4(ξ, α) = 1

α α

) =

2(ξ, α

) =

- α5

- α6

- α7

) =

2(ξ, α

) =

- α5

ξ 2

- α6

- α7

- α8

) =

2(ξ, α

) =

- α5

ξ 2

- α6

ξ 3

- α7

- α8

) =

ξ 2

- α6

ξ 3

- α7

ξ 4

- α8

Kn(ξ, λ)u Kn(ξ, λ)v
M [Kn(ξ, λ)u] M [Kn(ξ, λ)v]
�����

M [Kn(ξ, λ)u] M [Kn(ξ, λ)v]
V [Kn(ξ, λ)u] V [Kn(ξ, λ)v]
�����

; (34)

; (35)

ξ 4

1
5 K5(ξ, α) =
160

ξ 2 − ξ α 4

ξ 3 − α ξ52

ξ 4 − α ξ63

α8

ξ 5 − α ξ74

α 2

ξ 5

6 K6 (ξ, α) = 2401

7 K7(ξ, α) = 3361

5. Solution

∆n(λ) ≡

α 3

ξ 6

α 4

ξ 7

Sliding supports:

∂Kn(ξ,λ)u

∂Kn(ξ,λ)v

ξ=1
�����

∂ξ ∂ξ

V [Kn(ξ, λ)u] V [Kn(ξ, λ)v]

The ordinary differential equations with constant or variable
coefficients can be transformed to the Fredholm or the Volterra integral equations using e.g. Fubini’s method [17]. The
solutions of this equations are solutions of the transformed
ordinary differential equation. If the Green’s function (kernel
of integral equation) is well known, the linear independent solutions can be expanded in the Neumann power series rapidly
convergent to eigenvalues (spectrum of integral kernel) based
on the method of successive approximations [18].

The limited (for ξ = 0) independent solutions of Eq. (19)
are w1(ξ) = ξ n and w2(ξ) = ξ n+2 . This solutions were expanded in the Neumann power series _in_ _the_ _following_ _form_

�����

Elastic supports:

ξ=1
�����

; (36)

. (37)

∆n(λ) ≡

Φ[Kn(ξ, λ)u] Φ[Kn(ξ, λ)v]
Ψ[Kn(ξ, λ)u] Ψ[Kn(ξ, λ)v]

For all boundary conditions formula of ∆n

�����

For all boundary conditions formula of ∆n has following form

∆n = a0 +

i=1

(−1) i aiλ 2i, (38)

i=1

Kn(ξ, λ)u = ξ n +

Ki(ξ)u · λ 2i, λ ∈ R + (30a)

where a0, a1, . . ., aη are coefficients of characteristic equations depending on boundary conditions and number of nodal
lines n.

Kn(ξ, λ)v = ξ n+2 +

i=1

Ki(ξ)v · λ 2i, (30b)

6. Numerical results

The numerical results for the first ten dimensionless frequencies for different boundary conditions are presented in Tables 2–8. The nodal diameters are considered for the values
between 0 and 7. The Neumann power series Eq. (30) were expanded only for η = 25. Poisson ratio is taken as ν = 0.3 for
all considered cases. The numerical results for free, clamped
and simply supported circular plates are presented in Tables
2–4 with comparison to Refs. [1, 6, 9, 10, 16]. The numerical
results for plates with sliding supports are shown in Table 5
with comparison to Refs. [6, 15]. The eigenvalues for plates
with elastic supports and different values of elastic parameters
on the edges are presented in Tables 6–8 with comparison to
Refs. [6, 15].

i=1

where Ki(ξ)u and Ki(ξ)v are iterated kernels [18] presented
in the following form

Ki(ξ)u =

Ki(ξ)v =

Kn(ξ, α)Ki−1(α)udα, (31a)

�0

Kn(ξ, α)Ki−1(α)vdα (31b)

�0

and η is the degree of approximations.

184 Bull. Pol. Ac.: Tech. 64(1) 2016

_Green’s_ _function_ _approach_ _to_ _frequency_ _analysis_ _of_ _thin_ _circular_ _plates_

Table 2
The first ten lower dimensionless frequencies λ of free vibration of clamped circular plates

Table 3
The first ten lower dimensionless frequencies λ of free vibration of free circular plates

Bull. Pol. Ac.: Tech. 64(1) 2016 185

K.K. Żur

Table 4
The first ten lower dimensionless frequencies λ of free vibration of simply supported circular plates

Table 5
The first ten lower dimensionless frequencies λ of free vibration of circular plates with sliding edge

Table 6
The first ten lower dimensionless frequencies λ of free vibration of circular plates with elastic supports (ϕ = 0.1; ψ = 100)

186 Bull. Pol. Ac.: Tech. 64(1) 2016

_Green’s_ _function_ _approach_ _to_ _frequency_ _analysis_ _of_ _thin_ _circular_ _plates_

Table 7
The first ten lower dimensionless frequencies λ of free vibration of circular plates with elastic supports (ϕ = 10; ψ = 100)

Table 8
The first ten lower dimensionless frequencies λ of free vibration of circular plates with elastic supports (ϕ = 1000; ψ = 100)

7. Conclusions

In this paper, the Green’s functions have been employed to
solve natural vibration of circular thin plates with different
boundary conditions. The universal Green’s function for different nodal diameters is defined in a closed form. The ten
lower natural frequencies are calculated for eight different natural modes. The limited independent solutions of differential
Euler equations were expanded into the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as
power series rapidly convergent to exact eigenvalues for a different number of nodal diameters. The obtained results are
in good agreement with results obtained by other methods
presented in literature. The obtained numerical results can be
used to validate the accuracy of other numerical methods as
benchmark values. The calculations are evaluated with the
help of Mathematica v10, which is a symbolic calculation
software.

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188 Bull. Pol. Ac.: Tech. 64(1) 2016

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