A study on vibration of tapered rectangular plate under non-uniform temperature field

In modern technology, sufficient temperature is produced in most of engineering structures i.e. rockets, submarines etc. Also, it is obvious that plates undergo some vibration due to non-uniform temperature field. Therefore, scientists and engineers are keen interested to know that how non-uniform temperature field affects the vibrational characteristics of non-homogeneous plates of variable thickness due to their utility in constructions of bridges, buildings, wings, tails & fins of rockets & missiles etc. Non-homogeneous visco-elastic tapered plates are mainly used for two-fold requirements of safety and economy due to their high strength, high temperature resistance characteristics, low cost and high durability. Due to this, vibration of plates had become one of the most interesting research area in last few decades. Khanna & Kaur [1] worked on thermally induced vibrations of non-homogeneous tapered rectangular pate. Gupta and Khanna [2] studied the effect of linear thickness variations in both directions on vibration of visco-elastic rectangular plate having clamped boundary conditions on all the four edges. Khanna & Sharma [3] calculated frequencies for first two modes of vibration with parabolic thickness variation and bi-parabolic temperature variation. Khanna & Kaur [4] obtained first two modes of frequencies with exponential thickness and temperature variation. Khanna and Sharma [5] investigated free vibrations of non-homogenous square plate with exponential thickness variation. The authors had considered bi-parabolic temperature variations along with linear density variation. Chakraverty [6] introduced new concepts of boundary characteristic orthogonal polynomials on vibration of plates along with a discussion of various plate geometries and boundary conditions. Leissa [7] provided excellent data for vibration of plates of different shapes with different boundary conditions in his monograph. Avalos and Laura [8] discussed transverse vibrations of a simply supported plate of generalized anisotropy with an oblique cutouts. Bambill et. al. [9] carried out an experiment on transverse vibrations of an orthotropic rectangular plate of linearly varying thickness with free edges. Chyanbin et. al. [10] gave results on vibration suppression of composite sandwich beams. Gutirrez et. al. [11] investigated vibrations of rectangular plates of bi-linearly varying thickness with general boundary conditions. R. Lal et. al. [12, 13] evaluated transverse vibrations of non-homogeneous rectangular plates with thickness variation. Liessa [14] discussed vibrations of rectangular plate with general elastic boundary supports. Before finalizing any mechanical design or structure, researchers and engineers are always in searching of first few modes of vibration so that they would provide more authentic and reliable structures. In order to accomplish this purpose, authors provide a collection of numeric data for first two modes of frequency and deflection at various values of plate parameters i.e. taper constant, thermal gradient, aspect ratio and non-homogeneity constant. Kelvin type model is taken for consideration and plate is assumed clamped on the boundary.


Introduction
In modern technology, sufficient temperature is produced in most of engineering structures i.e. rockets, submarines etc.Also, it is obvious that plates undergo some vibration due to non-uniform temperature field.Therefore, scientists and engineers are keen interested to know that how non-uniform temperature field affects the vibrational characteristics of non-homogeneous plates of variable thickness due to their utility in constructions of bridges, buildings, wings, tails & fins of rockets & missiles etc.
Non-homogeneous visco-elastic tapered plates are mainly used for two-fold requirements of safety and economy due to their high strength, high temperature resistance characteristics, low cost and high durability.Due to this, vibration of plates had become one of the most interesting research area in last few decades.
Khanna & Kaur [1] worked on thermally induced vibrations of non-homogeneous tapered rectangular pate.Gupta and Khanna [2] studied the effect of linear thickness variations in both directions on vibration of visco-elastic rectangular plate having clamped boundary conditions on all the four edges.Khanna & Sharma [3] calculated frequencies for first two modes of vibration with parabolic thickness variation and bi-parabolic temperature variation.Khanna & Kaur [4] obtained first two modes of frequencies with exponential thickness and temperature variation.Khanna and Sharma [5] investigated free vibrations of non-homogenous square plate with exponential thickness variation.The authors had considered bi-parabolic temperature variations along with linear density variation.Chakraverty [6] introduced new concepts of boundary characteristic orthogonal polynomials on vibration of plates along with a discussion of various plate geometries and boundary conditions.Leissa [7] provided excellent data for vibration of plates of different shapes with different boundary conditions in his monograph.Avalos and Laura [8] discussed transverse vibrations of a simply supported plate of generalized anisotropy with an oblique cutouts.Bambill et.al. [9] carried out an experiment on transverse vibrations of an orthotropic rectangular plate of linearly varying thickness with free edges.Chyanbin et.al. [10] gave results on vibration suppression of composite sandwich beams.Gutirrez et.al. [11] investigated vibrations of rectangular plates of bi-linearly varying thickness with general boundary conditions.R. Lal et.al. [12,13] evaluated transverse vibrations of non-homogeneous rectangular plates with thickness variation.Liessa [14] discussed vibrations of rectangular plate with general elastic boundary supports.
Before finalizing any mechanical design or structure, researchers and engineers are always in searching of first few modes of vibration so that they would provide more authentic and reliable structures.In order to accomplish this purpose, authors provide a collection of numeric data for first two modes of frequency and deflection at various values of plate parameters i.e. taper constant, thermal gradient, aspect ratio and non-homogeneity constant.Kelvin type model is taken for consideration and plate is assumed clamped on the boundary.

Differential equation of motion
Differential equations of motion and time function for visco-elastic rectangular plate are [1]: and here D 1 is flexural rigidity of rectangular plate i.e.
( ) Deflection w(x, y, t) is expressed as the product of deflection function W(x, y) and time function T(t) [2]: , , , w x y t W x y T t = . (4)

Assumptions
Few assumptions are taken by authors to justify practical applications of the present study and to make calculations easy and convenient as well.
Assumption 1: It is assumed that variation in temperature field is bi-linear i.e.: ( )( ) where τ denotes the temperature excess above the reference temperature at any point on the plate and τ 0 denotes the temperature excess above the reference temperature at The temperature dependence of the modulus of elasticity for most of engineering materials can be expressed as [3]: ( ) where E 0 is the value of the Young's modulus at reference temperature and γ is the slope of the variation of E with τ .
After substituting the value of τ in Eq. ( 5), it becomes: where α = γτ 0 (0 ≤ α < 1), is thermal gradient.Assumption 2: It is assumed that thickness varies linearly in x − direction i.e.: where β is taper constant in x − direction and g = g 0 at x = 0. Assumption 3: Also, It is assumed that poisson ratio of plate's material varies exponentially in x − direction as [4]: where, v 0 denotes poisson ratio at reference temperature and α 1 is non-homogeneity constant.Since maximum value of poisson ratio is less than equal to 1/2, numeric value of α 1 (as it varies exponentially in this paper) can not be greater than 0.16 (approximately).Hence, variation in poisson ratio is taken from 0.0 to 0.15 (at most) in calculation.

Methodology
To obtain frequency equation for vibration of rectangular plate, authors used Rayleigh Ritz method.This method is based on principle of conservation of energy i.e. maximum strain energy (E P ) must be equal to the maximum kinetic energy (E K ).So it is necessary for the problem under consideration that [6]: where and ( ) Here, authors introduced two non-dimensional variables X and Y i.e.: After using Eq.(15) in Eq. ( 13) and Eq. ( 14), one gets: ( ) ) ( ) ( ) ( ) where, E from Eq. ( 16) and Eq. ( 17) in Eq. ( 12), one obtains: ( ) where is frequency parameter.
Eq. (18) consists two unknown constants i.e.A 1 and A 2 arising due to the substitution of W. These two constants are to be determined as follows: On simplifying Eq.( 19), one gets: , involve plate parameters and frequency parameter.
Eq. ( 20) is a set of two simultaneous homogeneous equations of variables A 1 and A 2 which has infinite numbers of solutions.Choosing 1 1 A = , one can easily evaluate 2 A i.e.
Equation ( 21) is a quadratic equation in λ 2 from which two values of λ 2 can be found.
Time period of the vibration of visco-elastic plate is given by:

Formulation of deflection
With the help of the values of A 1 and A 2 , one can obtain deflection function W as: Time function for non-homogeneous rectangular plate can be obtained by solving Eq. ( 2) as [3]: where .

Results and discussion
In calculations, following parameters are used: In Table 1, frequency for first two modes of vibration is reported at fixed aspect ratio It is obvious to note that frequency increases for both the modes of vibration with increasing α 1 as well as increasing values of α and β (from 0.0 to 0.6).

Table 1
Frequency Vs non-homogeneity constant at fixed aspect ratio Again first two modes of frequency at different values of aspect ratio are shown in Table 2 for the following combinations of thermal gradient α, taper constant β and non-homogeneity constant α 1 : α = β = α 1 and α = β = 0.2, α 1 = 0.1.
Authors noticed that both the modes of frequency increase with increasing value of aspect ratio for both combinations of α, β and α 1 .Also, a small increment is found in both the modes of frequency when combined values of α, β and α 1 increases.

Table 2
Frequency Vs Aspect Ratio Since deflection function assumed in Eq. ( 11) is symmetrical for X and Y, it shows same values for X = 0.2 and X = 0.8 as well as X = 0.4 and X = 0.6.Same case is valid for Y. Also for X = 0.0 and X = 1.0 orY = 0.0 and Y = 1.0, deflection becomes zero.Therefore, in calculation of deflection, authors reported the values of deflection only at X = 0.2, 0.4 and Y = 0.2, 0.4.
At different values of time function i.e.T = 0K and T = 5K, deflection for both modes of vibration with different α 1 i.e. α 1 = 0.0, 0.1 are calculated for different values of X and Y for the following cases: In Tables 3-5, an acute increment is noticed in both modes of deflection at each paired value of X and Y along with different values of α 1 i.e. 0.0 and 0.1 for T = 0K.In Table 3, for T = 5K, an acute decrement is noticed in both modes of deflection at each paired value of X and Y with α 1 = 0.0 and 0.1.At T = 5K, variation in both modes of deflection is different in Tables 4 and 5 as compared to Table 3.Here, very small decrement (but not negligible) is found in first mode of deflection and again very sensitive increment is found in mode of deflection at each paired value of X and Y with α 1 = 0.0 and 0.1.
For T = 0K and T = 5K, first two modes of deflection to increasing aspect ratio are tabulated in Table 6 at fixed α = β = 0.2, α 1 = 0.1 at two paired values of X and Y.
At X = Y = 0.2 & T = 0K, deflection increases for both the modes of vibration as aspect ratio increases from 0.5 to 1.5.
First mode of deflection increases continuously with increasing aspect ratio but second mode first increases and then decreases with increasing aspect ratio for the following cases: i) X = Y = 0.2 and T = 5K; ii) X = 0.4, Y = 0.2 and T = 0K; iii) X = 0.4, Y = 0.2 and T = 5K.The present analysis is about to study the vibration of tapered rectangular plate made up of visco-elastic non homogeneous material.Tapering in the plate is considered in one direction.Exponential variation in poisson ratio is assumed for non homogeneous nature of plate's material.Non uniformity in temperature field is considered bi-linear i.e. linear in x-direction and linear in y-direction.Frequency and deflection for first two modes of vibration are calculated and tabulated for various values of plate parameters.A comparison of the results of present study with those available in literature is given.

Table 3
*Values written in bold and {} brackets show deflection for both the modes of vibrations for T = 5K