Effect of plate’s parameters on vibration of isotropic tapered rectangular plate with different boundary conditions

Abstract

A mathematical model is presented to analyse the vibration of a tapered isotropic rectangular plate under thermal condition. Tapering in the thickness of rectangular plate is considered bi-parabolic. Here, temperature variation is assumed bi-linearly, i.e. temperature varies linearly in the x-direction and linearly in the y-direction. First two modes of frequency of rectangular plate are calculated for five boundary conditions such as C-C-C-C, SS-SS-SS-SS, C-SS-SS-SS, C-SS-C-SS and C-C-C-SS, where C and SS stand for clamped boundary and simply supported boundary, respectively. The plate is considered homogeneous and made up of a visco-elastic isotropic material. Rayleigh Ritz technique is applied to get the first two modes of frequencies at different values of plate’s parameters. Numeric results are presented in tabulated and graphical forms.

Introduction

In recent years, tapered plates are being increasingly used in modern engineering structures or space vehicles due to their wide technical importance. Consideration of tapered plates not only helps to reduce the weight of structural elements but also improves the utilisation of the material. Almost every structure or vehicle works under the influence of elevated temperature field due to which a heat flux is created and it directly affects the efficiency of a vehicle or structure.

In the available literature, many authors have discussed the vibration of plates having one or two directional tapering with one-dimensional thermal effect but little work has been done in the field of two-dimensional thermal effects. Therefore, the authors of this paper discuss vibration problem of visco-elastic rectangular plate with bi-parabolic thickness variation along with bi-linear thermal condition.

A survey of research papers, monographs and books published in the last six decades is given as follows.

A monograph on vibration of plates with different shapes and boundary conditions is given by Leissa. ¹ Leissa ² studied the effect of non-homogeneity on the free vibrations of rectangular plate of various combinations of clamped, simply supported, and free boundary conditions. Jain and Soni ³ discussed the free vibrations of rectangular plate with parabolic varying thickness using classic theory of plate. In this paper, two parallel edges of the plate are considered as simply supported and different boundary conditions are considered for the remaining two edges. Tomar and Gupta ⁴ studied the effect of thermal gradient on the free vibrations of an orthotropic rectangular plate with bi-linearly thickness variation for various boundary conditions. Leissa ⁵ analysed the effect of thermal gradient on the vibration of parallelogram plate with bi-directional thickness variation in both directions for various combinations of boundary conditions. Qiu et al. ⁶ presented some experiments on the active control of circular disk vibration. Lal ⁷ studied transverse vibrations of an orthotropic non-uniform plate with continuously varying density. Two parallel edges are considered simply supported and different combinations of boundary conditions are considered for the other two parallel edges. Li ⁸ has given a vibrational analysis of rectangular plate with general elastic boundary support. Leissa ⁹ discussed the historical bases of Rayleigh and Ritz’s methods to minimise the frequency of the vibrations of different structures of beams, bars, plates, etc. Gupta and Kumar ¹⁰ studied the thermal effect on vibrations of non-homogeneous rectangular plate with bi-linearly varying thickness variations. Lal and Dhanpati ¹¹ analysed the free transverse vibrations of non-homogeneous plate of varying thickness. Hota et al. ¹² analysed the free vibration of perforated plates. Gupta and Kumar ¹³ analysed the effect of thermally induced vibration of orthotropic trapezoidal plate with linearly varying thickness. Shooshtari and Razavi ¹⁴ worked on non-linear vibration of laminated fiber-reinforced rectangular plates. Abu Bakar et al. ¹⁵ analysed axisymmetric vibration of circular plate with attached annular piezoceramic plate. Khanna and Arora ¹⁶ analysed the effect of sinusoidal varying thickness on the vibrations of non-homogeneous parallelogram plate with bi-linearly temperature variation. They also discussed the effect of non-homogeneity of the material. Khanna and Kaur ¹⁷ studied the effect of exponentially varying thickness, temperature, and Poisson ratio on the free vibration of visco-elastic rectangular plate.

The main endeavour of the present investigation is to study the effects of varying plate’s parameters, i.e. thermal gradient (due to temperature variation), taper constants (due to thickness variation) and aspect ratio (ratio of length and breadth of the plate) on the vibration of visco-elastic isotropic rectangular plate. Here, the first two modes of frequency of the vibration of isotropic rectangular plate are calculated with the help of Rayleigh Ritz technique for five different boundary conditions. Results are given in the form of tables and graphs.

Analysis of equation of motion

The differential equation describing the motion of a visco-elastic isotropic rectangular plate may be written as ¹⁸

$\frac{\partial 2M_x}{\partial x2}+2\frac{\partial 2M_{\mathrm{xy}}}{\partial x\partial y}+\frac{\partial 2M_y}{\partial y2}=\rho h\frac{\partial 2w}{\partial t2}$

(1)

where x and y are the coordinates of the plate geometry, M_x and M_y are the bending moments, M_xy is the twisting moment per unit length of plate, ρ is the mass per unit volume, h is the thickness of plate and w is the displacement at time t.

The expressions for M_x, M_y and M_xy are given by ¹⁹

$M_x=-\tilde{D}{D}_1[\frac{\partial 2w}{\partial x2}+\upsilon \frac{\partial 2w}{\partial y2}],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{M}_y=-\tilde{D}{D}_1[\frac{\partial 2w}{\partial x2}+\upsilon \frac{\partial 2w}{\partial y2}]\text{and}{M}_{\mathrm{xy}}=-\tilde{D}{D}_1(1-\upsilon )\frac{\partial 2w}{\partial x\partial y}$

where $\tilde{D}$ is visco-elastic operator.

Here, D₁ is the flexural rigidity of the plate’s material and it is expressed as ²⁰

$D_1=\frac{\mathrm{Eh}3}{12(1-\upsilon 2)}$

(2)

Substituting the values of M_x, M_y and M_xy in equation (1), one gets

$\tilde{D}[D_1(\frac{\partial 4w}{\partial x4}+2\frac{\partial 4w}{\partial x2\partial y2}+\frac{\partial 4w}{\partial y4})+2\frac{\partial D_1}{\partial x}(\frac{\partial 3w}{\partial x3}+2\frac{\partial 3w}{\partial x\partial y2})+2\frac{\partial D_1}{\partial y}(\frac{\partial 3w}{\partial y3}+2\frac{\partial 3w}{\partial y\partial x2})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\frac{\partial 2D_1}{\partial x2}(\frac{\partial 2w}{\partial x2}+\upsilon \frac{\partial 2w}{\partial y2})+\frac{\partial 2D_1}{\partial y2}(\frac{\partial 2w}{\partial y2}+\upsilon \frac{\partial 2w}{\partial x2})+2(1-\upsilon )\frac{\partial 2D_1}{\partial x\partial y}\frac{\partial 2w}{\partial x\partial y}]+\rho h\frac{\partial 2w}{\partial t2}=0$

(3)

Using variable separation method, deflection w may be considered as the product of two functions as ²¹

$w(x,y,t)=W(x,y).T(t)$

(4)

where W(x, y) is the deflection function in x and y and T(t) is a time function for the vibration of rectangular plate.

Substituting equation (4) into equation (3), one obtains

$[D_1(\frac{\partial 4W}{\partial x4}+2\frac{\partial 4W}{\partial x2\partial y2}+\frac{\partial 4W}{\partial y4})+2\frac{\partial D_1}{\partial x}(\frac{\partial 3W}{\partial x3}+2\frac{\partial 3W}{\partial x\partial y2})+2\frac{\partial D_1}{\partial y}(\frac{\partial 3W}{\partial y3}+2\frac{\partial 3W}{\partial y\partial x2})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\frac{\partial 2D_1}{\partial x2}(\frac{\partial 2W}{\partial x2}+\upsilon \frac{\partial 2W}{\partial y2})+\frac{\partial 2D_1}{\partial y2}(\frac{\partial 2W}{\partial y2}+\upsilon \frac{\partial 2W}{\partial x2})+2(1-\upsilon )\frac{\partial 2D_1}{\partial x\partial y}\frac{\partial 2W}{\partial x\partial y}]/\rho \mathrm{hW}=-(\frac{\partial 2T/\partial t2}{\tilde{D}T})$

(5)

Equating both sides of equation (5) to a constant p², one obtains

$[D_1(\frac{\partial 4W}{\partial x4}+2\frac{\partial 4W}{\partial x2\partial y2}+\frac{\partial 4W}{\partial y4})+2\frac{\partial D_1}{\partial x}(\frac{\partial 3W}{\partial x3}+2\frac{\partial 3W}{\partial x\partial y2})+2\frac{\partial D_1}{\partial y}(\frac{\partial 3W}{\partial y3}+2\frac{\partial 3W}{\partial y\partial x2})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\frac{\partial 2D_1}{\partial x2}(\frac{\partial 2W}{\partial x2}+\upsilon \frac{\partial 2W}{\partial y2})+\frac{\partial 2D_1}{\partial y2}(\frac{\partial 2W}{\partial y2}+\upsilon \frac{\partial 2W}{\partial x2})+2(1-\upsilon )\frac{\partial 2D_1}{\partial x\partial y}\frac{\partial 2W}{\partial x\partial y}]-\rho p2\mathrm{hW}=0$

(6)

and

$\frac{\partial 2T}{\partial t2}+p2\tilde{D}T=0$

(7)

Equations (6) and (7) are differential equations of motion and time function for visco-elastic rectangular plate.

Since the research area of vibration of plates is too wide to discuss at once, the authors proceed with a few limitations.

Limitation 1

Thickness of the rectangular plate is considered non-uniform, i.e. thickness of the plate varies bi-parabolic ¹⁸ as (shown in Figure 1)

$h=h_0(1+{\beta }_1\frac{x2}{a2})(1+{\beta }_2\frac{y2}{b2})$

(8)

where a and b are the length and breadth of rectangular plate, respectively, β₁ and β₂ are the taper parameters in the x-direction and y-direction, respectively, and h₀ is the thickness of the plate at x = y = 0.

Figure 1.

Rectangular Plate with 2-directional thickness variation.

Limitations 2

The structures of high-speed space vehicles, i.e. supersonic flights, missiles, etc. are subjected to high surface temperature and large thermal gradient. These conditions can seriously affect or damage the structure of these vehicles. Variations in the structure’s characteristics as a result of thermal effect can be observed by the changes in frequency of vibration. Therefore, the authors assumed bi-linear temperature variations as ²²

$\tau ={\tau }_0(1-\frac{x}{a})(1-\frac{y}{b})$

(9)

where τ denotes the temperature excess above the reference temperature at any point on the plate and τ₀ denotes the temperature excess above the reference temperature at x = y = 0.

The temperature dependence of the modulus of elasticity for most of the engineering materials can be expressed as follows ²⁰

$E=E_0(1-\gamma \tau )$

(10)

where E₀ is value of the Young’s modulus at reference temperature, i.e. τ = 0 and γ is slope of variation of E and γ. Using equation (9) in equation (10), one obtains

$E=E_0[1-\alpha (1-\frac{x}{a})(1-\frac{y}{b})]$

(11)

where α = γ τ₀ (0 ≤ α < 1) is the thermal gradient.

Using equations (8) and (11) in equation (2), one gets

$D_1=\frac{E_0[1-\alpha (1-\frac{x}{a})(1-\frac{y}{b})]h_0^3(1+{\beta }_1\frac{x2}{a2})3(1+{\beta }_2\frac{y2}{b2})3}{12(1-\upsilon 2)}$

(12)

Limitation 3

In order to fulfil the practical aspects for using in structures or vehicles, the authors discuss five different boundary conditions for the rectangular plate, i.e. C-C-C-C, SS-SS-SS-SS, C-SS-SS-SS, C-SS-C-SS and C-C-C-SS^23,24as follows:

When the boundary of plate is C-C-C-C, the boundary conditions are

$W=\frac{\partial W}{\partial x}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=0,a\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}W=\frac{\partial W}{\partial y}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=0,b$

To satisfy these boundary conditions, the two-term deflection function is defined as

$W=[(\frac{x}{a})2(\frac{y}{b})2(1-\frac{x}{a})2(1-\frac{y}{b})2]\times [A_1+A_2(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]$

• When the boundary of the plate is SS-SS-SS-SS, the boundary conditions are

  $W=\frac{\partial 2W}{\partial x2}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=0,a\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}W=\frac{\partial 2W}{\partial y2}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=0,b$
  To satisfy these boundary conditions, the two-term deflection function is defined as
  $W=[(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]\times [A_1+A_2(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]$

• When the boundary of the plate is C-SS-SS-SS, the boundary conditions are

  $W=\frac{\partial W}{\partial x}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=0,W=\frac{\partial 2W}{\partial x2}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=a\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}W=\frac{\partial 2W}{\partial y2}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=0,b$
  To satisfy these boundary conditions, the two-term deflection function is defined as
  $W=[(\frac{x}{a})2(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]\times [A_1+A_2(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]$

• When the boundary of the plate is C-SS-C-SS, the boundary conditions are

  $W=\frac{\partial W}{\partial x}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=0,a\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}W=\frac{\partial 2W}{\partial y2}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{at}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=0,b$
  To satisfy these boundary conditions, the two-term deflection function is defined as
  $W=[(\frac{x}{a})2(\frac{y}{b})(1-\frac{x}{a})2(1-\frac{y}{b})]\times [A_1+A_2(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]$

• When the boundary of plate is C-C-C-SS, the boundary conditions are

  $W=\frac{\partial W}{\partial x}=0\text{at}x=0,a;W=\frac{\partial W}{\partial y}=0\text{at}y=0\text{and}W=\frac{\partial 2W}{\partial y2}=0\text{at}y=b$
  To satisfy these boundary conditions, the two-term deflection function is defined as
  $W=[(\frac{x}{a})2(\frac{y}{b})2(1-\frac{x}{a})2(1-\frac{y}{b})]\times [A_1+A_2(\frac{x}{a})(\frac{y}{b})(1-\frac{x}{a})(1-\frac{y}{b})]$

Here, A₁ and A₂ are the arbitrary constants occurred due to the first two modes of vibration.

Solution of frequency equation

Rayleigh Ritz technique is applied to solve the frequency equation. In this method, one requires that the maximum strain energy (S _E ) must be equal to the maximum kinetic energy (K_E). So it is necessary for the problem under consideration that ²⁵

$\delta (S_E-K_E)=0$

(13)

Here

$K_E=\frac{1}{2}\rho p2{\int }_0^a{\int }_0^b\mathrm{hW}2\text{d}y\text{d}x$

and

$S_E=\frac{1}{2}{\int }_0^a{\int }_0^b{D}_1\times [(\frac{\partial 2W}{\partial x2})2+(\frac{\partial 2W}{\partial y2})2+2\upsilon \frac{\partial 2W}{\partial x2}\frac{\partial 2W}{\partial y2}+2(1-\upsilon )(\frac{\partial 2W}{\partial x\partial y})2]\text{d}y\text{d}x$

Assuming the non-dimensional variables as

$X=\frac{x}{a},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}Y=\frac{y}{a},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\overline{W}=\frac{W}{a},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\overline{h}=\frac{h}{a}$

(14)

Using equation (14), the kinetic energy (K_E) and strain energy (S_E) become

$K_E^{*}=\frac{1}{2}\rho p2\overline{h_0}a5{\int }_0^1{\int }_0^{b/a}[(1+{\beta }_{1}X2)(1+{\beta }_2\frac{a2}{b2}Y2)\overline{W}2]\text{d}Y\text{d}X$

(15)

and

$S_E^{*}=\frac{E_0{\overline{h}}_0^{3}a3}{24(1-\upsilon 2)}{\int }_0^1{\int }_0^{b/a}\{1-\alpha (1-X)(1-\frac{a}{b}Y)\}(1+{\beta }_{1}X2)3(1+{\beta }_2\frac{a2}{b2}Y2)3\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\times [(\frac{\partial 2\overline{W}}{\partial X2})2+(\frac{\partial 2\overline{W}}{\partial Y2})2+2\upsilon \frac{\partial 2\overline{W}}{\partial X2}\frac{\partial 2\overline{W}}{\partial Y2}+2(1-\upsilon )(\frac{\partial 2\overline{W}}{\partial X\partial Y})2]\text{d}Y\text{d}X$

(16)

Using the modified values of kinetic and strain energy in equation (13), one gets

$\delta (S_E^{**}-\lambda 2K_E^{**})=0$

(17)

where

$K_E^{**}={\int }_0^1{\int }_0^{b/a}[(1+{\beta }_{1}X2)(1+{\beta }_2\frac{a2}{b2}Y2)\overline{W}2]\text{d}Y\text{d}X$

and

$S_E^{**}={\int }_0^1{\int }_0^{b/a}\{1-\alpha (1-X)(1-\frac{a}{b}Y)\}(1+{\beta }_{1}X2)3(1+{\beta }_2\frac{a2}{b2}Y2)3\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\times [(\frac{\partial 2\overline{W}}{\partial X2})2+(\frac{\partial 2\overline{W}}{\partial Y2})2+2\upsilon \frac{\partial 2\overline{W}}{\partial X2}\frac{\partial 2\overline{W}}{\partial Y2}+2(1-\upsilon )(\frac{\partial 2\overline{W}}{\partial X\partial Y})2]\text{d}Y\text{d}X$

Here, $\lambda 2=\frac{12\rho p2a2(1-\upsilon 2)}{E_0{\overline{h}}_0^2}$ is the frequency parameter.

Equation (17) consists of two unknown constants, i.e. A₁ and A₂ arising due to the substitution of W corresponding to the different boundary conditions. These two constants are to be determined as follows

$\frac{\partial }{\partial A_n}(S_E^{**}-\lambda 2K_E^{**})=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}n=1,2$

(18)

Simplifying equation (18), one gets

${\mathrm{bn}}_1{A}_1+{\mathrm{bn}}_2{A}_2=0,n=1,2$

(19)

where b_n1 and b_n2 include parametric constants, i.e. taper constants, thermal gradient, aspect ratio and frequency parameter. For a non-trivial solution, the determinant of the coefficient must be zero. So, one gets the frequency equation as

$|b_{11}{b}_{12}{b}_{21}{b}_{22}|=0$

(20)

With the help of equation (22), one can obtain a quadratic equation in λ² from which the two values of frequency parameter for both the modes of vibration can be evaluated easily.

Results and discussion

Computations were made for the first two modes of frequency at different values of thermal gradient (α), taper constants (β₁ and β₂) and aspect ratio (a/b) for the five different boundary conditions. In calculations, Poisson ratio is considered as constant, i.e. 0.345, and thickness of the plate h₀ at X = Y = 0 is taken as 0.01 m.

Frequency for the first two modes of vibration were calculated for increasing values of thermal gradient at various combinations of taper parameters and fixed value of aspect ratio, and the results are shown in Table 1. It is evident that the frequency for both the modes of vibration of plate with different boundary conditions decrease continuously as thermal gradient (α) increases corresponding to each paired value of β₁ and β₂. It is interesting to note that the first mode of frequency is maximum for C-C-C-SS plate but minimum for C-C-C-C plate. For second mode, frequency is maximum for C-C-C-C plate, while minimum for SS-SS-SS-SS.

Table 1.

Frequency versus thermal gradient α at fixed a/b = 1.5.

		β1 = β2 = 0.0		β1 = β2 = 0.4		β1 = β2 = 0.8
α	B.C's	Mode I	Mode II	Mode I	Mode II	Mode I	Mode II
0	C-C-C-C	256.20	64.83	339.26	86.10	450.47	113.58
	SS-SS-SS-SS	256.92	34.25	351.22	44.99	480.69	59.51
	C-SS-SS-SS	260.97	39.75	378.40	55.56	535.30	76.38
	C-SS-C-SS	288.81	41.94	392.21	54.96	532.36	72.17
	C-C-C-SS	357.65	55.24	554.18	74.97	794.72	100.11
0.4	C-C-C-C	243.05	61.50	327.12	82.90	438.91	110.24
	SS-SS-SS-SS	243.73	32.50	340.30	43.27	471.12	57.76
	C-SS-SS-SS	252.60	38.37	371.31	54.09	528.79	74.79
	C-SS-C-SS	273.99	39.79	379.64	52.84	521.15	70.00
	C-C-C-SS	353.70	53.12	549.95	72.71	789.85	97.65
0.8	C-C-C-C	229.15	57.98	314.53	79.54	427.05	106.76
	SS-SS-SS-SS	229.79	30.64	329.01	41.47	461.36	55.95
	C-SS-SS-SS	243.95	36.92	364.09	52.57	522.20	73.16
	C-SS-C-SS	258.32	37.51	366.64	50.63	509.69	67.76
	C-C-C-SS	349.71	50.91	545.69	70.37	784.95	95.13

Table 2 shows the frequency for the first two modes of vibration for various values of taper parameters at a fixed aspect ratio and thermal gradient. The authors conclude that the frequency continuously increases as one of the taper parameter increases for a fixed value of another. It is clear from Table 2 that the frequency increases faster when β₂ increases from 0.0 to 0.8 as compared to increasing β₁.

Table 2.

Frequency versus taper constants at fixed a/b = 1.5, α = 0.2.

β 1		0		0.4		0.8
β 2	B.C's	Mode I	Mode II	Mode I	Mode II	Mode I	Mode II
0	C-C-C-C	249.71	63.18	281.18	71.64	316.63	81.12
	SS-SS-SS-SS	250.41	33.39	284.62	38.36	324.28	44.10
	C-SS-SS-SS	256.82	39.07	314.26	47.77	377.57	57.05
	C-SS-C-SS	281.50	40.88	314.83	47.10	351.36	54.33
	C-C-C-SS	355.68	54.19	396.42	61.50	440.64	69.75
0.4	C-C-C-C	296.47	74.78	333.24	84.52	374.53	95.37
	SS-SS-SS-SS	305.39	38.37	345.80	44.14	392.38	50.83
	C-SS-SS-SS	309.34	44.76	374.87	54.83	447.23	65.61
	C-SS-C-SS	345.64	46.82	385.98	53.91	430.03	62.14
	C-C-C-SS	495.89	65.10	552.07	73.85	612.77	83.72
0.8	C-C-C-C	353.25	88.23	396.43	99.47	444.73	111.92
	SS-SS-SS-SS	372.85	44.09	420.89	50.82	475.93	58.64
	C-SS-SS-SS	373.74	51.34	448.93	63.04	532.06	75.59
	C-SS-C-SS	424.47	53.60	473.46	61.69	526.78	71.10
	C-C-C-SS	641.91	76.87	714.27	87.21	792.29	98.89

Frequency for the first two modes of vibration at different values of aspect ratio for the fixed values of plate’s parameters are calculated and presented in Table 3. It is noted that the frequencies for both the modes of vibration increase rapidly as the aspect ratio (a/b) increases from 0.5 to 1.5 for the five different boundary conditions at each paired value of α, β₁ and β₂.

Table 3.

Frequency versus aspect ratio.

		α = β1 = β2 = 0.0		α = β1 = β2 = 0.4		α = β1 = β2 = 0.8
a/b	B.C's	Mode I	Mode II	Mode I	Mode II	Mode I	Mode II
0.5	C-C-C-C	104.94	26.14	134.25	33.54	175.58	43.17
	SS-SS-SS-SS	105.91	13.17	140.28	16.45	190.08	20.96
	C-SS-SS-SS	135.30	20.00	210.61	26.14	303.96	34.11
	C-SS-C-SS	86.90	25.38	112.18	32.69	148.62	42.51
	C-C-C-SS	85.71	26.19	118.62	35.48	163.45	47.51
1	C-C-C-C	150.10	38.35	191.34	49.06	219.30	63.21
	SS-SS-SS-SS	149.67	21.08	198.25	26.84	230.95	35.03
	C-SS-SS-SS	169.42	27.19	253.40	36.70	304.74	49.35
	C-SS-C-SS	151.36	30.95	197.75	39.60	228.63	51.50
	C-C-C-SS	176.67	35.78	266.02	47.93	318.97	63.76
1.5	C-C-C-C	256.20	64.82	327.12	82.90	383.36	106.76
	SS-SS-SS-SS	256.92	34.25	340.30	43.27	396.35	55.95
	C-SS-SS-SS	260.97	39.75	371.31	54.09	453.81	73.16
	C-SS-C-SS	288.81	41.94	379.64	52.84	452.66	67.76
	C-C-C-SS	357.65	55.24	549.95	72.71	691.74	95.13

For better understanding of the results in terms of boundary conditions, the results in Tables 1 to 3 are presented in a graphical form. It is clear from Figures 2 and 3 that the frequency for both the modes of vibration decreases continuously at fixed values of taper constant and the aspect ratio for different boundary conditions. One can clearly observe from Figures 4 to 7 that the frequency increases continuously for the first two modes of vibrations as the taper constants (β₁ and β₂) increase for the fixed values of the thermal gradient and aspect ratio. The authors conclude from the graphs in Figures 8 and 9 that the frequency for both the modes of vibration increases very fast as the aspect ratio (a/b) increases from 0.5 to 1.5 for all the five combinations of boundary conditions.

Figure 2.

Frequency (Mode 1) versus thermal gradient.

Figure 3.

Frequency (Mode 2) versus thermal gradient.

Figure 4.

Frequency (Mode 1) versus taper constant (β₁).

Figure 5.

Frequency (Mode 2) versus taper constant (β₁).

Figure 6.

Frequency (Mode 1) versus taper constant (β₂).

Figure 7.

Frequency (Mode 2) versus taper constant (β₂).

Figure 8.

Frequency (Mode 1) versus aspect ratio (a/b).

Figure 9.

Frequency (Mode 2) versus aspect ratio (a/b).

Comparison and conclusions

Table 4 shows a comparison of the results of frequency for both the modes of vibration for C-C-C-C boundary condition only with Khanna and Singhal ²² at the corresponding values of the plate’s parameters. Here, the authors found that the frequency for the both modes of vibration is less than that in the present paper (shown in italic) than ²² (shown in bold) at the corresponding values of the plate’s parameters. It shows that the frequency of the rectangular plate changes with the change in tapering of the plate.

Table 4.

Frequency of present paper versus ¹⁵ at a/b = 1.5.

α	β1 = β2 = 0.4		β1 = β2 = 0.8
	Mode 1	Mode 2	Mode 1	Mode 2
0.2	333.24 370.94	84.52 93.77	444.73 529.98	111.92 133.59

The results, in light of the comparison, can thus be summarised in the following points:

Frequency can be controlled by appropriate tapering, i.e. by assuming suitable values of taper parameters. Desired values of frequency can be obtained by choosing appropriate taper parameters.

Boundary conditions directly affect the vibrational characteristics of the rectangular plate.

First mode of frequency is maximum at C-C-C-SS boundary condition but minimum at C-C-C-C boundary condition. But second mode is maximum at C-C-C-C boundary condition and minimum at SS-SS-SS-SS boundary condition.

With increasing values of thermal gradient, frequency decreases for both the modes of vibration of plate. It implies that temperature variations also directly affect the vibration of plate.

Vibration can be controlled effectively by bi-parabolic thickness variation (present paper) as compared to the bi-linear thickness variation. ²²

Therefore, scientists and design engineers who are working in designing space vehicles or structures are advised to analyse the findings of the present study in a practical manner to make more efficient and authentic mechanical structures and safe designs of vehicles.