Free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines

Free vibrations of layered conical shell frusta of differently varying thickness are studied using the spline function approximation technique. The equations of motion for layered conical shells, in the longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. Assuming the displacement components in a separable form, a system of coupled equations on three displacement functions are obtained. Since no closed form solutions are generally possible, a numerical solution procedure is adopted in which the displacement functions are approximated by cubic and quintic splines. A generalized eigenvalue problem is obtained which is solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two-layered conical shells, made up of several types of layer materials and supported differently at the ends are considered. Linear, sinusoidal and exponential variations in thickness of layers are assumed. Parametric studies are made on the variation of frequency parameter with respect to the relative layer thickness, cone angle, length ratio, type of thickness variation and thickness variation parameter. The effect of neglecting the coupling between bending and stretching is also analysed.