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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
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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.
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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
− + + + −
VISWANATHAN & SHEEN: FREE VIBRATION OF LAYERED ANNULAR CIRCULAR PLATE
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)
INDIAN J. ENG. MATER. SCI., DECEMBER 2009
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).