Vibration of ViscoElastic Parallelogram Plate with Parabolic Thickness Variation

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies parabolically. It is assumed that the plate is clamped on all the four edges and that the thickness varies parabolically in one direction i.e. along length of the plate. Rayleigh-Ritz technique has been used to determine the frequency equation. A two terms deflection function has been used as a solution. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection and linear visco-elastic properties of “Kelvin” type are taken. We have calculated time period and deflection at various points for different values of skew angles, aspect ratio and taper constant, for the first two modes of vibration. Results are supported by tables. Alloy “Duralumin” is considered for all the material constants used in numerical calculations.


Introduction
The materials are being developed, depending upon the requirement and durability, so that these can be used to give better strength, flexibility, weight effectiveness and efficiency.So some new materials and alloys are utilized in making structural parts of equipment used in modern technological industries like space craft, jet engine, earth quake resistance structures, telephone industry etc. Applications of such materials are due to reduction of weight and size, low expenses and enhancement in effectiveness and strength.It is well known that first few frequencies of structure should be known before finalizing the design of a structure.The study of vibration of skew plate structures is important in a wide variety of applications in engineering design.Parallelogram elastic plates are widely employed nowadays in civil, aeronautical and marine structures designs.Complex shapes with variety of thickness variation are sometimes incorporated to reduce costly material, lighten the loads, and provide ventilation and to alter the resonant frequencies of the structures.Dynamic behavior of these structures is strongly dependent on boundary conditions, geometric shapes, material properties etc. Dhotarad and Ganesan [1] have considered vibration analysis of a rectangular plate subjected to a thermal gradient.Amabili and Garziera [2] have studied transverse vibrations of circular, annular plates with several combinations of boundary conditions.Ceribasi and Altay [3] introduced the free vibration analysis of super elliptical plates with constant and variable thickness by Ritz method.Gupta, Ansari and Sharma [4] have analyzed vibration analysis of non-homogenous circular plate of non linear thickness variation by differential quadrature method.Jain and Soni [5] discussed free vibrations of rectangular plates of parabolically varying thickness.Singh and Saxena [6] discussed the transverse vibration of rectangular plate with bi-directional thickness.Free vibrations of non-homogeneous circular plate of variable thickness resting on elastic foundation are discussed by Tomar, Gupta and Kumar [7].Yang [8] has considered the vibration of a circular plate with varying thickness.Gupta, Ansari and Sharma [9] discussed the vibration of non-homogeneous circular Mindlin plates with variable thickness.Bambill, Rossit, Laura and Rossi [10] have analyzed transverse vibration of an orthotropic rectangular plate of linearly varying thickness with a free edge.
Sufficient work [11,12] is available on the vibration of a rectangular plate of variable thickness in one direction, but none of them done on parallelogram plate.Recently Gupta, Kumar and Gupta [13] studied the vibration of visco-elastic parallelogram plate of linearly varying thickness.A simple model presented here is to study the effect of parabolic thickness variation on vibration of viscoelastic parallelogram plate having clamped boundary conditions on all the four edges.The hypothesis of small deflection and linear visco-elastic properties are made.Using the separation of variables method, the governing differential equation has been solved for vibration of visco-elastic parallelogram plate.An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function.It is assumed that the visco-elastic properties of the plate are of the "Kelvin Type".Time period and deflection function at different point for the first two modes of vibration are calculated for various values of taper constant, aspect ratio and skew angle and results are presented in tabular form.

Equation of Transverse Motion
The parallelogram (skew) plate is assumed to be nonuniform, thin and isotropic and the plate R be defined by the three number a, b and θ as shown in Figure 1.
The skew coordinates are related to rectangular coordinates are The boundaries of the plate in skew coordinates are The governing differential equation of transverse motion of visco-elastic parallelogram plate of variable thickness, ξ-and η-coordinates is given by [13]:  1 cos ( ) 2(1 ) 2 sec ( , , , ) and 2 Ď 0 A comma followed by a suffix denotes partial differential with respect to that variable.Here p 2 is a constant.
Here solution w(ξ, η, t) can be taken in the form of products of two functions as for free transverse vibration of the parallelogram plate so that where T(t) is the time function and W is the maximum displacement with respect to time t.Assuming thickness variation of parallelogram plate parabolically in ξ-direction as where β is the taper constant in ξ-direction and The flexural rigidity D of the plate can now be written as

Solution and Frequency Equation
In using the Rayleigh-Ritz technique, one requires maximum strain energy be equal to the maximum kinetic energy.So it is necessary for the problem consider here that for arbitrary variations of W satisfying relevant geometrical boundary conditions.
Using Equations ( 5) and ( 6) in Equations ( 1) and ( 2), one obtains and ) 24( 1) Using Equations ( 10) and (11) in Equation ( 7), one obtains where and is a constant.But Equation ( 12) involves the unknown A 1 and A 2 arising due to the substitution of W(ξ, η) from Equation (9).These two constants are to be determined from Equation (12), as follows: where b n1 , b n2 (n =1,2) involve parametric constants and the frequency parameter.For a non-trivial solution, the determinant of the coefficient of Equation ( 17) must be zero.So one gets the frequency equation as (18) Here, where F 1 , F 2 , F 3 , B 1 , B 2 , B 3 involves parametric constants, skew angle and aspect ratio and given as .

Differential Equation of Time Function and its Solution
Time functions of free vibrations of viscoelastic plates are defined by the general ordinary differential Equation (3).Their form depends on the viscoelastic operator Ď.For Kelvin's model, one has where, ň is viscoelastic constant and G is shear modulus.The governing differential equation of time function of a parallelogram plate of variable thickness, by using Equation (20) in Equation (3), one obtains as Equation ( 21) is a differential equation of order two for time function T. Solution of Equation (21) comes out as where, and Let us take initial conditions as Using initial conditions from Equation (25) in Equation (22), one obtains Thus, deflection w may be expressed, by using Equations ( 26) and (19) in Equation ( 4), to give Time period of vibration of the plate is given by where p is frequency given by Equation (18).
All the results are presented in the Tables 1-11.
The value of time period (K) for β = 0.6, θ = 45˚ have been found to decrease 35.89% for first mode and 34.74% for second mode in comparison to rectangular plate at fixed aspect ratio (a/b = 1.5).
The value of time period (K) for β = 0.6, θ = 45˚ have been found to decrease 20.54% for first mode and 21.23% for second mode in comparison to parallelogram plate of uniform thickness at fixed aspect ratio (a/b = 1.5) .
Table 1 shows the results of time period (K) for different values of taper constant (β) and fixed aspect ratio (a/b = 1.5) for two values of skew angle (θ) i.e. θ = 0˚ and θ = 45˚ for first two mode of vibration.It can be seen that the time period (K) decrease when taper constant (β) increase for first two mode of vibration.
Table 2 shows the results of time period (K) for dif-ferent values of skew angle (θ) and fixed aspect ratio (a/b = 1.5) for two values of taper constant (β) i.e. β = 0.0, β = 0.2 for first two mode of vibration.It can be seen that the time period (K) decrease when skew angle (θ) increase for first two mode of vibration. .Table 3 shows the results of time period (K) for different values of aspect ratio (a/b) and fixed taper constant (β = 0.0 and β = 0.6) for two values of skew angle (θ) i.e. θ = 0˚ and θ = 45˚ for first two mode of vibration.It can be seen that the time period (K) decrease when aspect ratio (a/b) increase for first two mode of vibration.
The value of deflection (w) for β = 0.6 and θ = 45˚ have been found to increase 14.19% for first mode and 1.07% for second mode in comparison to parallelogram plate of uniform thickness for initial time 0.K at X = 0.2, Y = 0.4 and a/b = 1.5.
The value of deflection (w) for β = 0.6 and θ = 45˚ have been found to decrease 4.76% for first mode and 0.53% for second mode in comparison to rectangular plate for initial time 0.K at X = 0.2, Y = 0.4 and a/b = 1.5.
The value of deflection (w) for β = 0.6 and θ = 45˚ have been found to increase 11.91% for first mode and decrease 6.03% for second mode in comparison to parallelogram plate of uniform thickness for time 5.K at X = 0.2, Y = 0.4 and a/b = 1.5.
The value of deflection (w) for β = 0.6 and θ = 45˚ have been found to decrease 7.91% for first mode and 11.96% for second mode in comparison to rectangular plate for time 5.K at X = 0.2, Y = 0.4 and a/b = 1.5.
Tables 4-11 show the results of deflection (w) for different values of X, Y and fixed taper constant (β = 0.0 and β = 0.6), and aspect ratio (a/b = 1.5) for two values of skew angle (θ) i.e. θ = 0˚ and θ = 45˚ for first two mode of vibration with time 0.K and 5.K.It can be seen that deflection (w) start from zero to increase then decrease to zero for first two mode of vibration (except second mode at Y = 0.2 and 0.4) and second mode of vibration deflection (w) at (Y = 0.2 and Y = 0.4) start zero to increase, decrease, increase, decrease and finally become to zero for different value of X.

Conclusions
The Rayleigh-Ritz technique has been applied to study the effect of the taper constants on the vibration of clamped visco-elastic isotropic parallelogram plate with parabolically varying thickness on the basis of classical plate theory.On comparison with [13], it is concluded that: Time period K is more for non-uniform thickness in case of parabolic variation as comparison to linear variation.
Deflection w is less for non-uniform thickness in case of parabolic variation as comparison to linear variation.
In this way, authors concluded that parabolic variation is more useful than linear variation.
For a parallelogram plate, clamped (c) along all the four edges, the boundary conditions are the corresponding two-term deflection function is taken as 18), one can obtains a quadratic equation in p 2 from which the two values of p 2 can found.After determining A 1 & A 2 from Equation (17), one can obtain deflection function W. Choosing A 1 = 1, one obtains A 2 = (-b 11 /b 12 ) and then W comes out as

Table 2 . Time period K (in second) for different skew angle
(θ) and a constant aspect ratio (a/b = 1.5).

Table 3 . Time period K (in second) for different aspect ratio
(a/b).