Asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness with same boundary conditions

In the present paper, asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness resting on Winkler elastic foundation is studied by using boundary characteristic orthonormal polynomials in Rayleigh-Ritz method. Convergence of the results is tested and comparison is made with results already available in the existing literature. Numerical results for the first ten frequencies for various values of parameters describing width of annular plate, thickness profile, material orthotropy and foundation constant for all three possible combinations of clamped, simply supported and free edge conditions are shown and discussed. It is found that (a) higher elastic property in circumferential direction leads to higher stiffness against lateral vibration; (b) Lateral vibration characteristics of F -F plates is more sensitive towards parametric changes in material orthotropy and foundation stiffness than C-C andS-S plates; (c) Effect of quadratical thickness variation on fundamental frequency is more significant in cases of C-C andS-S plates than that ofF -F plates. Thickness profile which is convex relative to plate center-line tends to result in higher stiffness of annular plates against lateral vibration than the one which is concave and (d) Fundamental mode of vibration of C-C andS-S plates is axisymmetrical while that of F -F plates is asymmetrical.


Introduction
Annular circular plate is the simplest and widely used structural element in various engineering fields.The vibration of such plates has been the subject of various studies.Leissa [1][2][3][4][5][6][7][8] summarized the information in his well-known monograph and six comprehensive review articles.For the orthotropic plates, except for a few cases, no closed form solution exits.Researchers have used different approximation methods.Among them Vijaya Kumar and Ramaiah [9], Narita [10,11] and Gutierrez et al. [12] used Rayleigh-Ritz method, Greenberg and Stavsky [13] used finite difference method and Ginesu et al. [14] and Gorman [16] used finite element method.
A lot of information on annular circular plates having varying thickness is also available in the existing literature.The Chebyshev collocation method was used by Soni and Amba-Rao [17] to study the axisymmetric vibration of an annular plate with linearly varying thickness.Gupta and Lal [18] have extended the above paper to include the effect of in-plane forces.Lal and Gupta [19] solved the same problem for polar orthotropy.The axisymmetric vibration of such plates was further considered by Gupta et al. [20] using spline technique.Gorman [16] employed finite element method to compute natural frequencies of axisymmetric and asymmetric modes of polar orthotropic annular plates of linearly varying thickness, Raju et al. [21] used the same technique to analyze axisymmetric vibration of linearly tapered isotropic annular plates.Exact closed form solutions have been presented by Conway et al. [22] for linearly tapered isotropic annular plates and Lenox and Conway [23] for polar orthotropic plates of parabolic thickness variation.Kim and Dickinson [24] have analyzed composite circular plates as a particular case of annular plates by taking inner radius very small but only few results are given on circular plates and that too are for uniform thickness only.Wang et al. [25] studied free vibration analysis of annular plates by differential quadrature method.Laura et al. [26] have analysed annular circular plates having cylindrical anisotropy and non-uniform thickness using polynomial coordinates functions.Chen and Ren [27] studied lateral vibration of isotropic and orthotropic thin annular and circular plates of arbitrarily varying thickness along radius using finite element method and obtained natural frequencies and mode shapes of the axisymmetric and asymmetric modes.Gutierrez et al. [12] have analyzed annular plates of polar orthotropy using Rayleigh -Ritz method.Recently, Neeraj et al. [32] have studied effect of elastic foundation on the vibration of orthotropic elliptic plates with varying thickness.In all of the above papers, variation of thickness depends only on one taper parameter and there is no mention of nodal lines and mode shapes.
In the present paper, asymmetric vibration of annular plates of polar orthotropic material having quadratically varying thickness along radial direction and resting on Winkler elastic foundation is analyzed by using boundary characteristic orthonormal polynomials in Rayleigh-Ritz method.Two taper parameters are used for the quadratic thickness variation, which give more flexibility to study thickness variation.Many thickness variations can be approximated by it by suitably choosing the values of taper constants.Frequencies for the first ten normal modes of vibrations are computed for various values of inner radius, taper, orthotropy and foundation parameters for all three possible combinations of clamped-clamped, simply supported-simply supported and free-free conditions for inner and outer edges respectively.Convergence of frequencies at least upto five significant figures is observed.Comparison of frequencies in particular cases are made with the results already available in the literature.Apart from close agreement, it is also found that the present results are better in almost all the cases.Figures are shown for nodal lines and their corresponding three dimensional mode shapes.

Equation of motion
A thin annular plate of outer radius a, inner radius b, variable thickness h(r), is made up of orthotropic material and resting on Winkler elastic foundation is considered.Figure 1 shows a sketch of the plate problem treated in this paper.The plate is referred to cylindrical coordinate by taking the axis of the plate along the z-axis and the middle plane of the plate in the r − θ plane.The displacement components u, v and w in the directions of r, θ and z axes respectively, are taken as and w(r, θ, z, t) = w(r, θ, t).

Energy considerations
The strain energy due bending of the plate is given by The kinetic energy of the plate is given by where ρ is the density of the plate.The potential energy due to Winkler elastic foundation [31] can be taken as where k f is vertical stiffness of the foundation per unit area.
Introducing the following non-dimensional variables: a , and T = (t/a) (E r /ρ) and for harmonic vibration taking w(r, z, t) = a W (R, θ) cos ωT (5) and where m is the number of nodal diameters, Eqs (2), ( 3) and (4) could be reduced to In the above expressions, E r , E θ and ν r , ν θ are the Young's modulii and Poission's ratio's in r and θ directions respectively, G is the shear modulus, K f is the foundation constant and ω is the natural frequency of harmonic vibration.
The above formulation is well known and can be found for instance in [32].

Thickness variation
The non-dimensional thickness of the plate is taken as where H a = h a /a, h a is the thickness of the plate at the outer periphery and where α and β are the taper constants.Figure 2 shows a sketch of the plate variation considered in this paper.The quadratically varying thickness has two taper parameters, which give more flexibility in thickness variation.By suitably adjusting the taper parameters, many thickness variations can be approximated.As can be seen from Fig. 2, value of β will produce thickness variation which is convex (β < 0) and concave (β > 0) with respect to plate centerline.The special case of linear thickness variation when β = 0 is also shown.
The functional J (W )obtained by subtracting the maximum kinetic energy from the sum of the maximum strain energy and the maximum potential energy due to foundation is

R dθ dR,
where For applying the Rayleigh-Ritz method the functional J (W ) is to be minimized.

Generation of boundary characteristic orthonormal polynomials
The N -term approximation of the deflection function is taken as where Φ j are the boundary characteristic orthonormal polynomials satisfying at least the geometric edge conditions of the plate.Using three terms recurrence relation given by Chihara [30], Φ j are generated as The value of p is equal to 0,1 or 2 for free (F ), simply supported (S) or clamped (C), condition of the inner and outer edges of the plate, respectively.Substitution of W m (R) from Eq. ( 12) into energy Eq. ( 11) and then minimization of J (W )as a function of the coefficients c m j leads to the following standard eigenvalue problem: where and The integrals involved in Eqs ( 15) and ( 17) are evaluated by using the formula The eigenvalues (Ω) and the eigenvectors (c m j ) are computed using Jacobi method.The mode shapes are computed from Eq. ( 12) and the nodal lines are computed from the same equation by putting W m (R) = 0 .
As can be seen from the description above, Rayleigh-Ritz method with orthonormally generated boundary characteristic polynomials requires a set of deflection shapes that satisfy at least the geometrical boundary conditions of the vibrating structures.It reduces the problem into eigenvalue problem.Also, the method in combination with the expression for thickness variation used, can be applied to practically any thickness variation provided the integrals can be evaluated accurately.For a polynomial variation, it is possible to evaluate them in closed form, so there is no loss of accuracy on that account but for other type of thickness variation some numerical methods have to be used.

Results and discussion
The following sets of computations have been carried out: i) Convergent characteristics of the formulation used in this current study ii) Comparison with available results by other researchers Table 3 Comparison of Ωm,n for isotropic annular plates of linearly varying thickness when α = 1.0, β = 0.0 , er = 1.0, gr = 0.384,  iii) Vibration characteristics of annular circular plate resting on elastic foundation considering different combinations of the following parameters: material polar orthotropy e r , foundation stiffness K f , geometry of annular circular plate R 0 and taper parameters for thickness variation (α, β).
Table 1 shows the convergence of the first ten frequencies Ω m,n of at least upto five significant figures for all possible three combinations C-C, S-S and F -F of edge conditions at inner and outer edges when α = β =0.4,e r = g r = 5.0, R 0 = 0.5 and K f = 500.The suffixes m and n in Ω m,n denote number of nodal diameters and nodal circles, respectively.m = 0 corresponds to axisymmetric modes whereas m = 1, 2, 3, . . .corresponds to asymmetric modes of vibration.It can be seen that 7 terms are required to get the accuracy of upto five significant figures in all the cases.
The values of ν r and g r are taken as 0.3 and 5.0, respectively, for all computations except for the results presented in Tables 2, 3, 4 and 5, where other values are used for the purpose of comparison to known results.The variations in parameters for all possible three combinations of edge conditions for inner and outer edges are taken as follows: 1. e r from 0.25 to 8.0 by doubling the value at each steps; 2. K f from 0 to 1000 in steps of 200; 3. R 0 from 0.1 to 0.6 in steps of 0.1; 4. α and β from −0.4 to 0.4 in steps of 0.1.
Figure 3 shows the physical meaning of variations in R 0 and e r .Comparison of Ω m,n for C-C edge condition for isotropic annular plates of uniform thickness with Chen and Ren [27], Vogels [28] and Kim and Dickinson [24] is given in Table 2 when ν r = 0.3 and for isotropic annular plates of linearly varying thickness with Chen and Ren [27], Kim and Dickinson [24], Conway's exact solution [22] and Sankarnarayanan et al. [29] for clamped periphery is given in Table 3 when α = 1.0, β = 0.0 , e r = 1.0, g r = 0.384 , K f = 0and ν r = 1/ 3 and 0.3.
Table 4 shows the comparison of Ω m,n with results of Gorman [15], Narita [10] and Kim and Dickinson [24] for polar orthotropic annular plates of uniform thickness with simply-supported peripheries when e r = 5.0 , g r = 0.356 , ν r = 0.06 , R 0 = 0.5, K f = 0. Comparison of Ω m,n with Kim and Dickinson [24] and Chen and Ren [27] for polar orthotropic annular plates of linearly varying thickness with clamped peripheries is given in Table 5 when β = 0.0, e r = 50, g r = 0.6652, v r = 0.0052 and K f = 0.
From the computational results of convergence checking (Table 1), the main advantage of Rayleigh-Ritz method in monitoring rate of convergence through comparison of consecutive approximations is clearly seen.Increasing the order of approximation can increase accuracy of the result and the process can be terminated when the required number of frequencies has converged to the desired accuracy.Comparisons with available results (Tables 2 to 5) confirm that Rayleigh-Ritz method possesses faster rate of convergence than other methods such as Frobenious method, Chebyshev Collocation method, Spline method, differential quadrature method etc.Besides close agreement with the results compared, results obtained using the Rayleigh-Ritz formulation with the use of boundary characteristic orthonormal polynomials are found to be better even with lesser number of terms in almost all the cases.From the analysis, it is found that in the generation of boundary characteristic orthonormal polynomials, there is a loss of accuracy unless the precision is increased when proceeding to higher order polynomials.If precision is not increased, the results tend to show convergence up to a certain degree and subsequent divergence thereafter due to accumulation of rounding off errors.Variation in Ω * m,n with increasing e r for C-C, S-S and F -F plates on elastic foundation is given in Table 6 when α = β = 0.4 , g r = 5.0, R 0 = 0.5 and K f = 5 00.To show the full effect of e r on frequencies, variation of Ω * m,n instead of Ω m,n are taken in Table 6 because the parameter Ω m,n contains e r whereas Ω * m,n is free from e r as can be seen from Eq. (3).It can be seen in Table 6 that all frequencies increase monotonically with increasing e r .This is due to the fact that, stiffness of the plate increases with the increase in e r .The rate of increase is however higher when e r > 1.Since e r = E θ /E r , the results show that for the particular combination of parameters used, higher elastic property in circumferential direction tends to produce higher stiffness with respect to lateral vibration Table 8 Variation in Ωm,n with R 0 for C-C, S-S and F -F plates when α = β = 0.5, er = gr = 5.0 and K f = 500 of annular plate on elastic foundation.Variation in Ω m,n with K f for C-C, S-S and F -F plates is given in Table 7 when α = β = 0.4 , e r = g r = 5.0 and R 0 = 0.5.It is observed that all frequencies increase monotonically with increasing K f and the rate of increase falls in higher modes.This is due to the fact that, lateral stiffness of the annular plate-foundation combined increases with the increase in K f .Comparison in terms of mode shapes shown in Tables 6 and 7 show that, there is no change in the first 10 mode shapes for cases of C-C and S-S plates.However for the case of F -F plates, other than the fundamental mode shape, all higher mode shapes are not coincident.This indicates that lateral vibration characteristics of F -F plates is more sensitive to parametric changes in material orthotropy and foundation stiffness.
Variation in frequencies with inner radius R 0 for C-C, S-S and F -F plates is given in Table 8 when α = β = 0.5 , e r = g r = 5.0 and K f = 500.It is observed that as R 0 increases, all frequencies increase for C-C and S-S plates.For F -F plate, frequencies Ω 1,1 , Ω 0,2 , Ω 2,1 , Ω 0,1 and Ω 3,1 first decrease and then increase; Ω 1,2 increases and Ω 2,0 , Ω 3,0 , Ω 4,0 and Ω 5,0 decrease.Closer examination of the results shown in Table 8 shows that change in width of annular plates has no significant influence on the fundamental frequencies of F -F plates.The effect of increasing R 0 on fundamental frequencies of C-C and S-S plates is more pronounce.This later fact is especially true for R 0 greater than 0.3.As R 0 increases, width of annular plate becomes narrower physically (refer Fig. 3(a)).Under such circumstance, boundary condition of plates tends to exert higher influence on lateral stiffness resulting in stiffer response to lateral vibration.
Variation in Ω m,n with taper parameters α and β for C-C, S-S and F -F plates are given in Tables 9 to 11 when e r = g r = 5.0 , R 0 = 0.5 and K f = 500.It is observed that all Ω m,n decrease continuously for C-C, S-S and F -F plates.However, close examination reveals that effect of thickness variation on fundamental frequencies of F -F plates is practically zero.Although the effect seems to be more pronounce for higher modes, nevertheless it is deemed to be not significant.Comparing to the case of F -F plates, effect of thickness variation on fundamental frequencies is more significant in the case of C-C plates than S-S plates.For both cases of C-C and S-S plates, thickness profile which is convex relative to plate center-line (both α < 0 and β < 0, see Fig. 2(b)) tends to result in higher stiffness of annular plates against lateral vibration than the profile which is concave (both α > 0 and β > 0, see Fig. 2(a)).The same observation applies to the case of F -F plates although the effect is very small relative to C-C and S-S plates.
Nodal lines and their corresponding three dimensional mode shapes for the first ten normal modes of vibration for all three combinations are shown in Fig. 4 to 9 when α = β = 0.4, e r = g r = 5.0, R 0 = 0.5 and K f = 500.It can be seen from Figs 4, 5, 6 and 7 that the fundamental mode of vibration for both C-C and S-S plates are axisymmetrical mode, Ω 0,0 .Vibration modes for both cases are similar up until the 5 th mode.The case of F -F plates however exhibits fundamental mode of vibration which is asymmetry, Ω 2,0 .

Conclusion
Rayleigh-Ritz method with orthonormally generated boundary characteristic polynomials has been used to determine natural frequencies and mode shapes of annular circular plates resting on elastic foundation with C-C, S-S and F -F edge conditions.Comparisons with available results have shown that the above formulation possesses faster rate of convergence.With regards vibration characteristics, it is found that:

Table 1
Convergence of Ωm,n when

Table 2
Comparison of Ωm,n for polar isotropic annular plates of uniform thickness when er = 1.0 , gr = 0.384 , νr

Table 7
Variation in Ωm,n with K

Table 9
Variation in Ωm,n with α and β for C-C plate when er = gr = 5.0 , R 0 = 0.5 and K f = 500