Integral transform approach for solving dynamic thermal vibrations in the elliptical disk

Application of Fractional Order Theory of Thermoelasticity in an Elliptical Disk and Associated Thermal Stresses

“…The temperature of the disk ( , , , ) z t    satisfies the time fractional order differential equation (Following [8] and [21]…”

“…The expressions of thermally induced resultant force and resultant moment are given as [21] ( , , ) ( , , , ) , ( , , ) ( , , , )…”

“…To obtain the expression for the temperature function ( , , , ) z t    , we use the finite Fourier cosine transformation with respect to the variable z and then introduce the finite integral transform ( [22] and [21]) involving Mathieu functions as , ,…”

“…McLachlan [19,20] solved the heat conduction problem for an elliptical cylinder in the form of an infinite Mathieu function series. Dhakate [21] determined thermally induced transverse vibration of a uniform thin elliptical disk with elastic supports at both radial boundaries by using the integral transform technique. Gupta [22] determined the boundary value problem for elliptic cylinders by introducing a finite transform involving Mathieu functions.…”

Temperature and stress distribution in a non-simple elastic elliptical plate due to point impulsive heat sources

“…Roy Choudhary [3], Khobragade and Deshmukh [4], Varghese and Khalsa [5] and many others. Similarly, the present author [6][7][8][9][10][11][12][13] have investigated various thermoelastic problems in elliptical objects of simple materials due to interior heat generation or sectional heat supply with different solid objects. Even one-dimensional thermoelastic problems made up of non-simple elastic material due to heat sources has been studied by various authors, viz.…”