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

Three-dimensional (3-D) elasticity solutions for the free vibration analysi s of laminated circular conical shells are presented by means of an asympto tic approach. The formulation begins with the 3-D equations of motion in ci rcular conical coordinates. After proper non-dimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differen tial equations at various levels. The method of multiple time scales is use d to eliminate the secular terms and make the asymptotic expansion feasible . The method of differential quadrature (DQ) is adopted for solving the pro blems of various orders. The present asymptotic formulation is applicable t o the analysis of laminated cylindrical shells by vanishing the semivertex angle (alpha). The natural frequencies, modal stresses of cross-ply cylindr ical and conical shells with simply supported - simply supported (S-S) boun dary conditions are studied to demonstrate the performance of the present a symptotic theory. It is shown that the asymptotic DQ solutions of the prese nt study converge rapidly. The present convergent results are in good agree ment with the accurate solutions obtained from the approximate 2-D shell th eories in the cases of thin shells. Furthermore, these present results may serve as the benchmark solutions for assessment of various 2-D shell theori es in the cases of moderatively thick shells.