Abstract-This paper is concerned with elastic vibration in an elliptical plate to determine the vibration of plate at for time (t > 0) with the help of finite mathieu transform technique.
Keywords – Elliptical plate, Mathieu transforms, Marchi-Fasulo transform, Laplace transform, Elastic vibration.
In this paper, an attempt has been made to determine the elastic vibration in a elliptical plate with known radiation type boundary conditions, using Mathieu transform, Laplace transform and Marchi-Fasulo transform technique.
STATEMENT OF THE PROBLEM
Heat conduction equation in elliptical co-ordinates (x,h) for elliptical cylinder define as
Subject to the interior condition
The boundary conditions are
where k1 and k2 are radiation constants on the plane surfaces of the cylinder. Equations (1) to (5) constitute the mathematical formulation of the problem under consideration.
SOLUTION OF THE PROBLEM
We define the transform with respect to t as
Its inverse transform is
Applying the transform defined in (6) to the equations (1),(2), (5) and using (3),(4) one obtains
Where the eigen values are the solutions of the equation
Where denotes the Marchi-Fasulo transform of and denotes the Marchi-Fasulo transform parameter.
From equation (8), we obtain
If temperature is symmetric about both axes of the ellipse, the appropriate solution of equation (10) is
Where and are defined as modified and ordinary Mathieu function of
Using equations (9) and (12), we get
In order to get the value of constant, multiply (13) by, integrate with respect to from 0 to and making use of the following result:
Substituting equation (15) in equation (12), one obtains,
Applying inverse transform defined in (7) to the equation (16) and using condition (5), one obtain the temperature distribution and unknown temperature gradient as,
The Mathieu transformation of
is the root of the equation
Where k = q and Y is Bessel function
Thus integral transform possess the property
Inversion of Mathieu transform is
Now by applying the finite Mathieu transform (19) and using the equation (17) ,(21),(22) & (23) we get,
Now this equation is solved by Laplace transform and the following result is obtained
Using inversions of Mathieu Transform we get
Hence from equation (27) and (28) we get,
Equation (29) is the desired solutions of the problem
In this paper, we have investigated the elastic vibration of elliptical plate on the outer curved surface with the help of the finite Mathieu transform techniques. The results are obtained in the form of infinite series. The expressions that are obtained can be applied to the design of useful structures or machines in engineering application.