Axisymmetric free vibration of layered conical shells using chebyshev polynomial with collocation method

Abstract

Free axisymmetric vibrational characteristics of layered truncated conical shells are studied. The method of collocation with Chebyshev polynomial approximation is used for solving the problem. The formulation of the problem is based on an extension of Love’s first approximation theory. The governing partial differential equations of motion are obtained in terms of the reference surface displacements. The equations are coupled in the longitudinal and transverse displacements. Assumption of the solution in separable form leads to a pair of ordinary differential equations in the assumed pair of displacement functions, which are functions of only a meridional coordinate. These equations are still coupled and have to be solved only numerically, in their general form. The displacements are assumed in series of Chebyshev polynomials. Collocation leads to a set of homogeneous equations in the unknown coefficients in the two series assumed and become as a generalized eigenvalue problem solving which the frequency parameter values and the corresponding mode shapes of vibration are obtained. Parametric studies are made to find the influence of the many geometric and material parameters available on the frequencies. The effect of layering and that of neglecting the coupling between extensional and flexural displacements are studied, in particular. The results are presented in terms of graphs and are discussed.