Asymptotic differential quadrature solutions for the free vibration of laminated conical shells

Three-dimensional (3-D) elasticity solutions for the free vibration analysis of laminated circular conical shells are presented by means of an asymptotic approach. The formulation begins with the 3-D equations of motion in circular conical coordinates. After proper non-dimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple time scales is used to eliminate the secular terms and make the asymptotic expansion feasible. The method of differential quadrature (DQ) is adopted for solving the problems of various orders. The present asymptotic formulation is applicable to the analysis of laminated cylindrical shells by vanishing the semivertex angle (α). The natural frequencies, modal stresses of cross-ply cylindrical and conical shells with simply supported - simply supported (S-S) boundary conditions are studied to demonstrate the performance of the present asymptotic theory. It is shown that the asymptotic DQ solutions of the present study converge rapidly. The present convergent results are in good agreement with the accurate solutions obtained from the approximate 2-D shell theories in the cases of thin shells. Furthermore, these present results may serve as the benchmark solutions for assessment of various 2-D shell theories in the cases of moderatively thick shells.