free vibration of layered annular circular plate of...

Indian Journal of Engineering & Materials Sciences

Vol. 16, December 2009, pp. 433-448

Free vibration of layered annular circular plate of variable thickness

using spline function approximation

K K Viswanathan

a,b,* & Dongwoo Sheen

a

aDepartment of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea bDepartment of Naval Architecture and Ocean Engineering, Division of Mechanical Engineering,

Inha University, Incheon 402- 751, South Korea

Received 28 July 2008; accepted 28 October 2009

Free vibration of layered annular circular plate of variable thickness, made up of isotropic or specially orthotropic

materials, is studied using spline function approximation by applying the point collocation method. Three different thickness

variations are considered, namely, linear, exponential and sinusoidal, along the radial directions. The equations of motion

are derived by extending Love’s first approximation theory. A system of coupled differential equations, in terms of

longitudinal, circumferential and transverse displacement functions is obtained by assuming the solution in a separable form.

These functions are approximated by using Bickley-type splines of suitable orders. A generalized eigenvalue problem is

obtained by applying a point collocation technique with suitable boundary conditions, from which the values of a frequency

parameter and the corresponding mode shapes of vibration, for specific values of various parameters, are obtained.

Keywords: Free vibration, Annular plate, Variable thickness, Spline, Eigenvalue

Extensive literature is available on the vibration of

isotropic, orthotropic and laminated annular plates of

uniform thickness. Timoshenko and Woinowsky-

Krieger1 frequently quote an exact solution for the

static analysis of radially tapered disc springs

originally developed by Conway2. Conway et al.

3

extended the analysis to the study of free vibration of

tapered circular plates. These exact analyses,

however, are of limited use, being applicable to

materials with Poisson ratio of 1/3. Vogel and

Skinner4, have presented natural frequencies for

annular plates of constant thickness for various

combinations of clamped, simply supported and free

boundary conditions. A large number of studies have

also been made on homogeneous annular plates of

variable thickness. Leissa’s5 monograph on vibration

of plates and surveys by Bert6,7

and Reddy8 contain

discussion of these plates. Raju et al.9 studied

axisymmetric vibrations of linearly tapered annular

plates.

Layered plates of constant thickness have been

analyzed in a few studies. The simplest case of a plate

of homogeneous and isotropic layers was analyzed by

Pister10

. Yu’s11

theoretical works on sandwich plates,

incorporating shear deformation and rotary inertia are

particularly significant. In the work of Vodika12

,

radial non-homogeneity in circular plates is

accommodated by treating the plate as a composite of

homogeneous, isotropic annuli and enforcing

continuity conditions across the internal junctions.

Srinivasan et al.13

used an integral equation technique

to analyze the vibration of composite annular plates.

Baharlou and Leissa14

extended the Ritz method,

utilizing a strain energy functional containing both

bending and stretching effects, and developed a

method for the analysis and buckling of generally

laminated composite plates. Displacement functions

were taken in the form of polynomials and the plates

considered were rectangular.

Sankaranarayanan et al.15

analyzed free vibrations

of laminated annular plates, whose thickness varied

directly as the radius. Isotropic and specially

orthotropic layer materials were considered. An

energy method based on the Rayleigh-Ritz technique

was used for the analysis. Polynomial displacement

functions satisfying only the geometric boundary

conditions were assumed. This work, nevertheless,

has restricted application because of the particular

kind of variation in thickness considered, viz.,

thickness being proportional to distance from the

center. Singh and Saxena16

and Lal and Sharma17

studied the vibrations of annular plates of ________________

*For correspondence (E-mail: [email protected])

INDIAN J. ENG. MATER. SCI., DECEMBER 2009

434

exponentially varying thickness. Lal et al.18

also

studied transverse vibration of orthotropic rectangular

plates of linearly varying thickness, without

lamination, using the spline method. Wu and Liu19

have used a differential quadrature technique for the

analysis of circular plate with variable thickness. An

exact element method has been used by Eisenberger

and Jabareen20

to study the axisymmetric vibration of

circular and annular plates with variable thickness.

Zhou et al.21

applied the Chebyshev-Ritz method for

analyzing the vibration of circular and annular paltes.

Selmane and Lakis22

found the frequencies of

transverse vibrations of non-uniform circular and

annular plates using the finite element method. Lal

and Sharma17

, in their study of the free axisymmetric

vibrations of non-homogeneous polar orthotropic

annular plates of exponentially varying thickness on

the basis of classical theory of plates, used the

Chebyshev collocation technique to solve the

differential equation governing the transverse motion

of such plates. The non-homogeneity of the plate

material was assumed to arise due to the variation of

Young’s moduli and density which were assumed to

vary exponentially with the radius. Free vibration

analysis of thin annular plate with thickness varying

monotonically in arbitrary power form is presented by

Duan et al.23

. After a transformation of variable,

analytical solutions in terms of generalized

hypergeometric function, taking either logarithmic or

non-logarithmic forms, are proposed and illustrated.

In the present study, the general linear, exponential

and sinusoidal variation in thickness of laminae along

the radial direction of the plate are considered and the

versatile spline function technique is used. Here, a

chain of lower-order approximation is used which can

yield greater accuracy than a global higher order

approximation. This conjecture was made and tested

by Bickley24

over a two point boundary value problem

with a cubic spline. Many researchers25-30

have also

demonstrated this technique, along with its attractive

features of elegance in handling and convergence.

Both the axisymmetric and asymmetric vibrations

are considered. Extensive parametric studies are made

to provide insight into the individual and interactive

influence of various geometric and material

parameters. The effects on the frequency parameter,

with respect to the relative layer thickness, the radii

ratio, and the other parameters characterizing the

nature of variation of thickness, the number of

circumferential modes and boundary conditions are

analyzed. The effect of neglecting the coupling

between bending and stretching is also investigated.

Significant mode shapes are presented in each case.

Extensive convergence tests and comparative studies

with the available literature are made.

Formulation of the Problem and the Method of

solution The geometry parameters of layered annular

circular plate of exponentially varying thickness are

shown in Fig. 1. Each individual layer is considered to

behave macroscopically as a homogeneous

orthotropic and linearly elastic material. The layers

are assumed to be perfectly bonded. Rotatory inertia

and transverse shear deformation are neglected. The

plate is assumed to be thin, so that the angular shifting

of the axes of material symmetry of each layer due to

the variation in thickness is considered to be

negligible. The line RO is the section of the

reference surface. The thickness of the kth layer of the

plate is taken in the form

Fig. 1–Geometry of layered annular circular plate of exponentially

varying thickness

VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE

435

0( ) ( )k kh r h g r= … (1)

where 0kh is a constant. The thickness becomes

uniform when ( ) 1g r = . The elastic coefficients

corresponding to layers of uniform thickness, with

superscript c, can be defined as

( ) , ( ), ( )c c c

ij ij ij ij ij ijA A g r B B g r D D g r= = = … (2)

where cijA ,

cijB , and

cijD are extensional, extensional-

bending, and bending elastic coefficients,

respectively.

In this study, the thickness variation of each layer

is assumed the form

0( ) ( )h r h g r= … (3)

where

( ) 1 expa ae

r r r rg r C C

− − = + +

( )

sin as

r rC

π − +

… (4)

Here b a= − is the width of the plate and ar is

the radial distance from the origin to when ar = .

The stress resultants and moment resultants are

expressed in terms of the radial, circumferential and

transverse displacements vu , and w of the reference

surface. The displacements are assumed in the

separable form given by

( , , ) ( )cos i tu r t U r n e

ωθ θ=

( , , ) ( )sin i tv r t V r n e

ωθ θ= … (5)

( , , ) ( )cos i tw r t W r n e

ωθ θ=

where r and θ are the polar coordinates to describe

the radial and rotational directions, t is the time, ω

is the angular frequency of vibration and n is the

circumferential mode number. When 0,n = the

vibration becomes axisymmetric. Using Eq. (5) in the

constitutive equations and the resulting expressions

for the stress resultants and the moment resultants in

the equilibrium equations, (given in Appendix A), the

governing differential equations of motion are

obtained in the form

11 12 13

21 22 23

31 32 33

0

L L L U

L L L V

WL L L

=

… (6)

The operators ( , 1, 2, 3)ijL i j = are defined in

Appendix B.

The differential equations on the displacement

functions in Eq. (6) contain the derivative of third

order in ,U second order in V and fourth order in

.W The order of U is not suitable for the existing

derivative orders in Eq. (6). Therefore, the equations

are combined within themselves and a modified

system of equations on the displacement functions, of

order 2 in ,U order 2 in V and order 4 in W are

obtained, as given by

11 12 13

21 22 23

* * *

31 32 33

0

L L L U

L L L V

WL L L

=

… (7)

where the operators 3,2,1,3* =jL j , are given in

Appendix C.

The length parameters, frequency parameters, radii

of the plate and thickness parameters are non-

dimensionalized and are given in Appendix D. Spline

collocation procedure is adopted to solve this problem

assuming in the same way as in Viswanathan and

Navaneetakrishnan29

.

2 13

0 0

*( ) ( ) ( )N

i

i j j j

i j

U X a X b X X H X X−

= =

= + − −∑ ∑

2 13

0 0

*( ) ( ) ( )N

i

i j j j

i j

V X c X d X X H X X−

= =

= + − −∑ ∑

4 15

0 0

*( ) ( ) ( )N

i

i j j j

i j

W X e X f X X H X X−

= =

= + − −∑ ∑

… (8)

The boundary conditions are used as follows:

(i) both the edges clamped (C-C), (ii) both the edges

hinged (H-H) and (iii) the inner edge clamped and the

outer edge free (C-F). The resulting field and

boundary conditions give rise to the generalized

eigenvalue problem of the form

[ ] [ ] 2M q P qλ= … (9)

where [ ]M and [ ]P are matrices of order

(3 7) (3 7)N N+ × + , q is a matrix of order

(3 7) 1N + × , and 1N + is the number of knots of the

splines on radial direction. The parameter λ is the


436

eigenparameter and q the eigenvector whose

elements are the spline coefficients. Only two-layer

plates are considered with =δ ratio of thickness of

the first mentioned layer, to the total thickness, at the

inner circular edge.

Results and Discussion

Certain analyses of the convergence studies for

C-C boundary conditions are carried out and the

results are presented in Table 1. In a similar way, the

convergence results are studied for other two types of

boundary conditions H-H and C-F, whose results are

not presented here due to space constraints. Based on

these findings it is decided to set ,N number of

subintervals of the width of the annular plate, equal to

16. Table 2 compares the fundamental frequency

parameter values obtained for homogeneous plates of

linearly varying thickness (treated as special cases of

St-Al (steel-aluminium) layers with δ = 1) with those

of Raju et al.9. Five taper ratios ( )τ and two radii

ratios ( )β are considered. The agreement is quite

good. An FEM result of Pardeon31

and Leissa5 on an

isotropic plate of different radii ratio β and the

corresponding result obtained in the present study are

shown in Table 3. Table 4 gives the comparison of

fundamental frequency for axisymmetric vibration of

Table 1–Convergence study of annular circular plate of linear variation of thickness for axisymmetric case using

HSG-SGE material combination under C-C boundary conditions

β = 0.5, γ = 0.05, δ = 0.4, η = 0.75, Ce = Cs= 0

N λ1 % change λ2 % change λ3 % change

4 0.360826 1.072209 2.220066

6 0.363484 0.734 1.040225 -2.983 2.153058 -3.018

8 0.364137 0.180 1.025160 -1.448 2.076124 -3.573

10 0.364407 0.074 1.017884 -0.709 2.036935 -1.888

12 0.364545 0.037 1.013880 -0.393 2.015331 -1.061

14 0.364626 0.022 1.011457 -0.239 2.002343 -0.644

16 0.364678 0.143 1.009883 -0.156 1.993888 -0.422

Table 2 – Comparative study with Raju et al.9

Boundary conditions: C-C ; ν =0.3

β Ω τ

0.1 0.3 0.5 0.7 0.9

0.3 Ωp 14.54 18.74 21.61 24.08 26.33

ΩR 13.66 17.26 20.48 23.41 26.15

0.5 Ωp 27.01 33.50 40.71 46.56 51.67

ΩR 26.09 33.46 39.98 45.89 51.40

Ω : (2ωb2/ h b) √ ρ/E, fundamental frequency parameter

Ωp : Present analysis

ΩR : From Raju et al.9

Table 3– Comparative study of axisymmetric vibration of homogeneous isotropic plates of constant thickness

2 32

(1 )1 / 3 , 0 .0 5 ,

λβ β γ

ν γ ζ−

= = =

β λ ζ 2

Present Pardoen % Change Leissa % Change

0.3 0.13735 45.3137 45.3707 0.1256 45.36 0.1021

0.5 0.32254 89.3849 89.2993 -0.0958 89.49 0.1174

0.7 0.75445 248.9039 248 -0.3645

0.9 2.91280 2242.2744 2237 -0.2358


437

annular plate of linear variation in thickness with

different boundary conditions. The differences are not

significant. All these confirm the validity of the

analysis and the results.

The influence of the relative layer thickness of two

layered plates is first studied. Figures 2-4 pertain to

such studies on HSG-SGE (High strength graphite – S

glass epoxy) plates of the three types of variation in

thickness. All three types of boundary conditions are

considered for each type of thickness variation. The

condition C-F, for example, implies that the inner

circular boundary of the annular plate is clamped and

the outer boundary is free. The plates are of medium

annular width for β = 0.5. The thickness-to-inner

radius ratio γ has the value 0.05. When δ = 0 or 1,

the plate becomes homogeneous, made up of the SGE

or HSG, accordingly. In Fig. 2, two types of linear

variation in thickness are considered, with the taper

ratio =η 0.75 (outer thickness larger) and =η 1.50

(outer thickness smaller). It is clearly seen that the

frequency parameter values are the same for =δ 0

and =δ 1 (homogeneous) and vary for 0 < δ < 1.

The range of variation of mλ is the least for m =1

and increases with the increasing value of m . These

ranges are 0.038, 0.108 and 0.215 for m = 1, 2 and 3,

respectively, for the case of Fig. 2a. It is possible to

attain frequencies higher and lower than those of

homogeneous plates made up of either of the two

materials. The boundary conditions do affect the

frequencies. The values of λ are highest for C-C

conditions, lower for C-F conditions under the same

other conditions and λ is lower for higher value of η

which is expected.

The continuous and dotted lines correspond

respectively, to the inclusion and omission of the

coupling effect between the radial and transverse

displacements. The effect of neglect of this coupling

is seen to raise frequencies for all modes. The effect is

more significant for higher modes. The percent

change is significant for higher modes. The percent

change is however so small (11.04%, 11.12% and

Fig. 2– Effect of relative layer thickness, coupling and boundary conditions on the frequency parameter of annular circular plates of

linearly varying thickness

Table 4 – Comparative study of axisymmetric vibration of annular

plate of linear variation in thickness

0 .2 5 , 1 , 0 .2 5a b β= = =

Outer Boundary Inner Boundary Ω

Ωp ΩE

Clamped Clamped 5.647141 5.740721

Simply Supported Simply Supported 3.918508 4.012092

Free Clamped 2.328780 2.422364

Ω : Normalized fundamental frequency

Ωp : Present value

ΩE : Eisenberger and Jabareen20


438

Fig. 3– Effect of relative layer thickness and boundary conditions on the frequency parameter of annular circular plates of exponential

variation in thickness

Fig. 4 – Effect of relative layer thickness and boundary conditions on the frequency parameter of annular circular plate of sinusoidal

variation in thickness


439

11.21% for m =1, 2, 3 for η = 0.75, for example) that

ijB can be set equal to zero without introducing

appreciable error, but resulting in some computational

advantage. This is in agreement with Ashton’s32

findings that the reduced stiffness matrix

approximation, obtained by neglecting coupling,

yields reasonably accurate results. However, the

coupling effect is not excluded throughout this study.

Layering with different materials of layers affect the

frequency parameter differently. The two types of

exponential variation in thickness of layers considered

in Fig. 3, correspond to Ce = + 0.2 and Ce = − 0.2. The

thickness increases and decreases, respectively, as the

radius increases, in these two cases. This explains

why the values of mλ for the same m are lower for

Ce = + 0.2 than those for Ce = − 0.2. The charac-

teristics of λ ∼ δ relation are similar, in general, to

the case of linear variation in thickness. The ranges of

variation of mλ for m =1, 2, 3, for the case of Fig. 3a

are 0.064, 0.177 and 0.351, respectively. Figure 4

pertains to sinusoidal variation in thickness of layers

corresponding to Cs = 0.25 and Cs = − 0.25. The

thickness of the plate is same at ar = and br = ; the

surface of the plate is convex or concave for a < r <

b , for Ce<

> 0. This is in keeping with the observation

that the values of mλ , for all the boundary conditions

considered, are lower for Cs = − 0.25 than for Cs =

0.25. The influence of δ and the boundary

conditions on λ for the sinusoidal variation in

thickness are same qualitatively, as in the other two

cases considered. The ranges of variation of mλ for

m =1, 2, 3, for the cases of Fig. 4a are 0.050, 0.145

and 0.294 respectively. The layer materials also affect

the λ ∼ δ relation. The maximum percent-changes

in 3λ with respect to the corresponding minimum

3λ in the cases of the material combinations St-SGE

(Steel – S glass epoxy), HSG-SGE and HSG-PRD

(high strength graphite-PRD), for the case of linear

variation in thickness with η = β = 0.5 under C-C

conditions are found to be 33.19%, 17.82% and

8.83%. Thus the maximum effect of layering is felt

for St-SGE and the minimum for HSG-PRD.

The effect of the annular width of the plate on the

circular frequency of axisymmetric vibration is

depicted in Figs 5 and 6. The figures bring out also

the effects of the boundary conditions and the

thickness variation of layers. The reason for

considering ω , instead of λ , is that it may not be

meaningful to study the variation of λ with β, since

λ is the function of the length . The plates are of

HSG-SGE layers, with γ = 0.05 and δ = 0.4. All the

three types of boundary conditions are considered.

Figure 5 corresponds to linear change in thickness; for

Fig. 5 (a)-(c), η = 0.75 and for Fig.5 (d)-(f) η = 1.5.

The thickness of the plate at the inner boundary is 1.

The circular frequencies mω are very low for small

values of β (i.e., for large width of the plate) and

increase rapidly with β , assuming very large values

for very short widths of the plate. The change of rate

of increase occurs somewhere in 0.4 < β < 0.6

considering the first three modes of vibration. The

steep increase starts earlier for higher modes. With

increasing η , the curves tend to come closer to β =

0.9 ordinate, agreeing with the expected behaviour.

This happens for all the boundary conditions

considered. In the lower range of β, the boundary

conditions do not appreciably affect the frequencies,

whereas in the higher range the effects are

considerable. This is in complete agreement with the

observations of Sankaranarayanan et al.15

, which was

a study of the special case corresponding to β = η .

Figure 6 (a)-(c) correspond to the exponential

variation in the layer thickness, corresponding to Ce =

0.2, and Fig. 6 (d)-(f) to the sinusoidal variation given

by Cs = 0.2. The other parameters and boundary

conditions are explained in the figures. The

characteristic patterns of the curves are, of course,

similar. However, from the same scale of diagrams, it

is easy to see how the different boundary conditions

affect the patterns. It is seen, for example, that for the

same plate width (i.e., for a particular value of β ),

mω values are the highest for C-C conditions, the

lowest for C-F conditions and between the two for H-

H conditions. This is true for both the types of

thickness variations shown. Figures 7-9 present the

important study of influence of the coefficients of the

three types of variation of thickness of the layers of

the plate on the frequency parameter for a typical

plate. The plate is of HSG-PRD lamination. For each

type of variation in thickness, all the three boundary

conditions are considered with β = 0.5, γ = 0.05.


440

Fig. 5–- Variation of frequencies with radius ratio of annular circular plates of linear variation in thickness and effect of boundary

conditions

Fig. 6 – Variation of frequencies with radius ratio of annular circular plates of exponential and sinusoidal variation in thickness with radii

ratio and effect of boundary conditions


441

Figure 7 corresponds to linear variation in thickness

with the taper ratio ranging from 0.5 to 2.1. For η < 1

the outer thickness is larger than the inner thickness.

For η > 1, it is the other way, and for η = 1, the

thickness is constant. For small values of η

considered, mλ decreases rapidly with increase of η

and tends to assume constant values as η assume

large values. Though this property is seen commonly

for all the three boundary conditions, there are

differences in the range of variation of mλ , for any

particular m , under the different boundary

conditions. The percent changes for the first three

modes are 110.19%, 106.51%, 104.93%; 95.62%,

104.53%, 104.02% and 17.76%, 55.51%, 79.11% for

the C-C, H-H and C-F boundary conditions,

respectively. Thus, the order of these conditions,

according as their decreasing influence on the

frequency changes with respect to taper, is as C-C, H-

H and C-F.

In Figs 8 and 9, the influence of the coefficient of

exponential variation of thickness Ce and the

coefficient of sinusoidal variation Cs on mλ are

depicted, along with the combined effect of the three

types of boundary conditions. As Ce or Cs increases,

the stiffness increases and, hence, the frequency

parameter values mλ increase; the increase is seen to

be almost steady for all the values of m and for all the

three boundary conditions. (The cases η = 1, Ce = 0

and Cs = 0 correspond to constant thickness and,

hence, corresponding mλ for any m is therefore seen

to be the same). The percent changes in 321 ,, λλλ

due to variation of Ce over the range [-0.2,0.2] are

119.47%, 113.75%, 111.29%; 101.03%, 107.56%,

107.92%, and 28.59%, 78.40%, 98.49%; for C-C, H-

H and C-F boundary conditions respectively. The

corresponding percent changes in the case of

sinusoidal variation in thickness are 32.04%, 62.85%,

77.38%; 152.34%, 108.01%, 98.62%, and 73.89%,

106.65%, 98.05%. As for the case of linear variation

in thickness, in these two cases also , the C-C

boundary conditions contribute the maximum to the

influence of the coefficients of thickness variation on

frequencies, the H-H conditions contributing less, and

the C-F conditions contributing the least.

Certain mode shapes of axisymmetric vibrations of

a St-HSG plate are presented in Fig. 10. All the three

types of boundary conditions and all the three types of

variation in thickness of layers are considered as

indicated. The transverse displacements predominate

in all of them. Both the transverse and radial

displacements are normalized with respect to the

maximum transverse displacements. Up to the third

modes of vibration are presented. The displacement

curves of any particular mode for all the three types of

thickness variation are seen close to each other. The

radial displacement has an additional node compared

to the transverse displacements for the C-C

conditions. The plates considered are of moderate

Fig. 7– Variation of frequency parameter of plates of linear variation in thickness with taper ratio and the effect of boundary conditions


442

Fig. 8 – Variation of frequency parameter of plates of exponential variation in thickness and the effect of boundary conditions

Fig. 9 –Variation of frequency parameter of plates of sinusoidal variation in thickness with sC and the effect of boundary conditions


443

width (β= 0.5). In the case of symmetric boundary

conditions for vary small values of β, it is seen that U-

and W-displacements are not appreciable for low

values of r; for large values of β, these displacements

are almost symmetric or skew symmetric. It is also

seen that the maximum values of U increase with β

for symmetric boundary conditions.

The frequencies of asymmetric vibrations of plates

are affected by the circumferential mode number n.

Figure 11 describes the influence of n on mλ ( m =1,

2, 3) of HSG-SGE plates whose layers are linearly,

exponentially and sinusoidally varying in thickness

and which are supported at both the edges either

clamped or hinged. The other parameters are as

described in the figure. Three typical values for the

coefficient of variation are considered. It is seen that

as n increases mλ increases for all values of m

considered, in all the cases. Thus the frequencies for

axisymmetric vibrations ( n =0) are the least.

The influence of the width of the annular plate on

the frequencies of vibration for a typical asymmetric

(n = 8) vibration is depicted in Fig. 12. The layers are

Al-SGE (aluminium-S glass epoxy), γ = 0.05,

δ = 0.4. The boundary conditions considered are C-C

and H-H. Three typical values for the coefficients of

variation of the three types of variation in thickness

are considered. The patterns of influence are similar

to those in the axisymmetric case. However, the

frequencies for this asymmetric case are higher than

those of the corresponding axisymmetric case of

vibration, as expected. Though the case of C-F

boundary conditions is not presented, the behavioural

pattern is similar. The frequencies at C-C conditions

are higher than those at H-H conditions, which again

are higher than those at C-F conditions, under

identical other conditions.

Mode shapes of asymmetric vibrations for n = 8 of

St-SGE plates under C-C and H-H conditions are

presented in Fig. 13. Normalization is done as usual.

Transverse displacements are seen to predominate in

all the cases considered. The extensional and torsional

displacements follow in that order. The neglect of

coupling reduces the extensional and torsional

Fig. 10 – Mode shapes of axisymmetric vibrations of plates of linear, exponential and sinusoidal variations in thickness under different

boundary conditions


444

Fig. 11 –Variation of frequency parameter of plates of linear, exponential and sinusoidal variation in thickness with circumferential mode

number under different boundary conditions

Fig. 12 –Variation of frequencies of asymmetric vibration of plates of linear, exponential and sinusoidal variations in thickness with

length ratio under different boundary conditions


445

Fig. 13 – Mode shapes of asymmetric vibration of plates of linear,

exponential and sinusoidal variations in thickness under different

boundary conditions

displacements to zero. The nature of variations in the

thickness of layers introduces differences in mode

shapes. However, the general patterns of mode shapes

are similar for all types of thickness variations under

the same boundary conditions for any particular

circumferential mode number n. It is observed that

with increasing n the maxima of U and V get

reduced.

Conclusions Layering and variation in thickness are two factors

which independently influence the vibrational

behaviour of a thin plate. When the plate is both

layered and of variable thickness, its behaviour is

naturally more complex due to the interactive

influence of these aspects. The natural frequencies of

vibration of such a plate are influenced by the

materials of the constituent layers, their relative

thickness, the nature of variation of thickness, the

value of the corresponding thickness parameter,

radius ratio, circumferential mode number and the

boundary conditions.

The frequencies, in general, vary with the relative

layer thickness for the same layer materials and the

same type of variation in thickness under the some

other geometric and boundary conditions. This

provides scope for one to choose suitable thickness

proportions among the given materials to achieve the

desired vibrational behaviour. The effect of neglecting

coupling between the extensional and flexural

displacements is to raise the frequencies in general for

all circumferential mode numbers. This effect is least

for axisymmetric vibrations and can be ignored with a

view to simplify the analysis.

In the case of asymmetric vibrations, the natural

frequencies of vibration generally increase with the

increase of circumferential mode number. The neglect

of coupling suppresses torsional and longitudinal

displacements. When the thickness of layers vary

linearly, the effect of taper on frequencies is

considerable for both axisymmetric and asymmetric

vibrations for very small and very high values of the

taper ratio (since the stiffness is then higher) for larger

value of the radius ratio (smaller width of plate), for

higher modes of vibration and for clamped-clamped

boundary conditions (in comparison with the other

boundary condition considered). The thickness

variation coefficients in the cases of exponential and

sinusoidal variations affect the frequency values

almost linearly, but differently for different material

combinations and boundary conditions.

Acknowledgement

The authors thankfully acknowledge the financial

support from Brain Korea 21 Project , NRF-2008-

C00043 and NRF-2009-0080533, South Korea.

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Appendix A

The equilibrium equations for annular plate of variable

thickness.

( ) 2

0

1( ) 0r

r

d N NrN n g r R U

r dr r r

θ θ ω+ − + = … (A.1)

( ) 2

0

1( ) 0r

r

N d Nn rN g r R V

r r dr r

θ θθ ω− + + + = … (A.2)

( )2

2

2 2

1 2 1( ) ( )r r

d M n d drM n M M

r dr r r dr r dr

θθ θ− + −

2

02

2( ) 0r

nM g r R W

rθ ω+ + = … (A.3)

Where

( , , ) ( , , )r x r r

z

N N N d zθ θ θ θσ σ σ= ∫ … (A.4)

( , , ) ( , , )r r r r

z

M M M z d zθ θ θ θσ σ σ= ∫ … (A.5)

and ( )

0 1( ) ( ) ( )k

k k k k

k k

R z a z a h aρ ρ−= − = ∑ ∑ is the inertial

coefficient. … (A.6)

Appendix B

The differential operators ijL appearing in Eq. (1) are

22 2

11 2 3 102 2

' 1 ' 1( ) '

d g d gL s s s n

dr g r dr rg rλ

= + + + − + +

… (B.1)

10 312 2 10 2 2 2

'( )

n d g s sL s s n s

r dr rg r r

= + + − −

… (B.2)

3 2

13 4 43 22 '

d g dL s g s g

dr r dr

= − − +

2

6 5 11 52 2

'( 2 ) 2

g g g ds s s n s

r r r dr

+ + + −

2 2 2

5 6 112 3 3 3

2 '2

g g g gs n s n s n

r r r r

+ − − −

… (B.3)

2 10 3 1021 102 2

's s d s s gL n n s

r r dr r r r g

=− + − + + … (B.4)

2 22

22 10 10 3 102 2

1 ' ' 1'

d g d n gL s s s s

dr r g dr r r g rλ

= + + − − + +

… (B.5)

2

623 5 11 112

2 4 'g d n s g d

L n s s s gr dr r r dr

= + + +

2

11 62 3

'4

g gn s s n

r r

− + … (B.6)

( )3 2 2

4 431 4 3 2

2 '4 ' 2 4 ' 2 "

d s d s rgL s g rg g g rg

dr r dr r g

= + + + + +

2

2 55 6 5 112 2

' 2 '2 ( 2 2 "

g g g d s gs s n s s g

r r r dr r g

+ − − + + +

2

66 112 2

2 ' 4 's g n g

g s s gr r r r

− − − +

… (B.7)

2

32 5 11 12 5 112 2

22 4 4 '

g d n gL n s s s s g s

r dr r r

= + + + +

2

3

11 11 6 6 52 3

2 2 '2 ' 2 ''

g g d g n gs g s s s n s g

r r r dr r r g

+ − − − + +

11 6 2

' 2 '4

g g gs s

r r r

− + −

… (B.8)

4 3 22 27 8 9

33 7 4 3 2

3 '(6 ' 2 )

d s d s gg s gL s g rgg g

dr r dr r r

= − − + + +

2

2 278 2

6 '(6 ' 3 '' 6 ' )

s g gggg rgg rg s n

r r r

− + + + −


447

2 2 2

2

8 12 92 2 2 3

3 '( 4 )

g d gg gs s n s

r dr r r

+ + + −

2 2

2 28

2

2 '3 ' 6 ' 6

s n g gggg g n

r r r

− + + −

2

212

2

4 6 's g gg dn

r r r dr

− −

2 2 2

2 2

12 94 3 4 3 4

4 6 ' 2 3 '2

g gg g gg gs n s n n

r r r r r

+ − + − −

2 2 2

2 2 2812 4 2 2

2 6 '4 3 '' 6 ' '

g s n g ggs n gg g

r r r rλ

− + + + − +

… (B.9)

in which

12 22 11 12 222 3 4 5 6

11 11 11 11 11

, , , ,c c c c c

c c c c c

A A B B Bs s s s s

A A A A A= = = = =

11 12 22 66 667 8 9 10 11

11 11 11 11 11

, , , ,c c c c c

c c c c c

D D D A Bs s s s s

A A A A A= = = = =

6612

11

c

c

Ds

A= … (B.10)

22 0

11

'c

R

A

ωλ = is a frequency parameter … (B.11)

Appendix C

The operators *3 jL ( j =1,2,3 ) defined in Eq.(7) are given by

2 '* ' 2

31 4 4 2 4 3 4 102 2 23

g d g g gL s g s s s s s s n

r dr r r r

= − + + − −

' 2

5 6 11 52 22 (2 )

g g n gs s s s

r r r− + + −

' '

'' 2

4 42

34

g g g ds g s g

g r r drλ − + + + +

2 ' ' 2 ' 2

''52 4 3 42 3

' 22

g g g g s gs s s s g

r r rg r r gr

+ − − + − +

' 2

2

4 10 6 6 113 2 3 2

22 4 '

g g g n gs s n s s s g

r r r r r

+ + − + + … (C.1)

2*

32 4 2 10 5 11 12 2( ) 2 4

ng dL s s s s s s

r dr= + − − −

6 11 4 2 10 5 11( ) 4 ' 2n g g g

s s s s s s g sr r r r

+ − + − −

2

10 34 2 4 2

' '' ''

s g s d n g gs s g s s g

r r dr r r g

+ − − + − −

( )''

224 5

10 3 63

2 '2 2 2

s g s gs s n s g

r r g

+ + + − +

11 62 2 3

' 2 '4

g g gs s

r r r

+ + −

… (C.2)

4 2* 2 2 2

33 4 7 44( ) 3 '

d gL g s s s gg

dr r

= − − − +

3

2 2743 2

'(6 ' 2 ) 2 ''

s d g grgg g s g g

r dr r r

− + + − −

( )2 2

2 2

4 6 5 11 4 5 82 2

' 3 '( 2 ) 2

g gg g ggs s n s s s s s n

r r r r

+ + + − − −

2 2 2

2 279 8 122 2 2

(6 ' 3 '' 6 ' ) ( 4 )g s g d

s gg rgg rg s s nr r r dr

− + + + − +

2 2

2

4 6 5 112 3 2 3

' 2 ' 2( 2 )

gg g gg gs s s s n

r r r r

+ − + + −

2 2

2 2

5 5 6 112 2 3 3

' '' 2 '2 ( 2 )

gg g g gg g gs s n n s s

r r r r r

+ − + − − +

2 2

2 289 2 3 2

3 ' 6 ' 23 '' 6 '

gg g s gg gs gg g n

r r r r r

− − + + − −

2 2 2

2

12 4 52 2 3 4

4 2 '' 5 ' 312 '

n g d gg g g gs gg s n s

r r dr r r r

+ − + − +

2 2 2

2 126 4 113 4 4 3 3

' 3 6 2 ' 2 4g g g gg s n gs s s n

r r r r r r

− − + − −

2 2 2

2 2 2

9 122 4 2 4 4

6 ' 2 3 '4

gg g gg g gs n n s n

r r r r r

− − − − +

2 2

2 28

2 2

2 6 '3 '' 6 ' '

s n g gggg g

r r rλ

− + + − −

… (C.3)

Appendix D The parameters are non dimensionalized by writing

λ = ',λ a frequency parameter

,a

bβ = radii ratio

0 ,a

h

rγ = ratios of thickness to radius of inner circle

,kk

h

hδ = a relative thickness ratio

,r a

X a x b−

= ≤ ≤

, the meridional coordinate of reference

… (D.1)


448

where kh is the thickness of the k -th layer , h is the total

thickness of the shell, 0h is the constant thickness and

ar is the

radius of inner circular plate. When there are only two layers, we

define 1δ δ= , and hence

2 1δ δ= − and X ∈ [0,1].

For a two-layered annular circular plate, the independent

geometric material parameters are , ,β γ δ , C, Ce and Cs.

The thickness hk (X) of the k -th layer at the point distant X

from the smaller radius of the annular circular plate, already

explained, can be expressed as

0( ) ( )k kh X h g X= … (D.2)

Where

( )g X = 1 + C

X + Ceexp ( X ) + Cs sin π X . … (D.3)

Case (i) : If Ce = Cs = 0, the thickness variation becomes linear. In

this case it can be easily shown that

11C

η= −

… (D.4)

where η is the taper ratio (0) / (1)k kh h .

Case (ii) :If C

= Cs = 0, the excess thickness over uniform

thickness varies exponentially.

Case (iii) : If C

= Ce = 0, the excess thickness varies

sinusoidally.

It may be noted that the thickness of any layer at the

end X = 0 is 0kh for the cases (i) and (iii), but is 0kh

(1+Ce) for the case (ii).

free vibration of layered annular circular plate of...

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