Asymptotic Transfer Function Analysis of Conical Shells

An asymptotic transfer function method is presented for modeling and analysis of conical shells. The displace- ment functions are first expanded in Fourier series in the circumferential direction, and the motion equations are decoupled into a group of partial differential equations with one space variable and one time variable. Introducing a small perturbation parameter and using the Laplace transformation and perturbation technique, the partial differential equations with variable coefficients are reduced to ordinary differential equations with constant co- efficients, which are solved by the transfer function method. The method is used to perform analysis of stepped conical shells with different conical angle or thickness and subjected to various initial and boundary conditions. Numerical methods are presented and compared with the finite element method. ONICAL shells have wide applications in aeronautic, astro- nautic, civil, and chemical engineering. The research on their mechanical behavior under various external excitations and bound- ary restrictions has great importance in engineering practice. As one type of revolutionary thin shells, conical shells have been studied by many researchers, and a lot of modeling and analysis methods have been developed. Chang1 gave a literature review of the vibration of conical shells. Liew2 reviewed recent developments in the free vibration analysis of thin, moderately thick shallow shells. Com- pared with cylindrical shells, conical shells are difficult to analyze in exact and closed form because of the mathematical complex- ity in geometry and variable surface curvature.2 Wan3-4 obtained a closed-form solution of the variable coefficient differential equa- tions of conical shells in terms of generalized hypergeometric func- tions. Tong5 obtained the solution of laminated conical shells in the form of power series. However, their solutions are very com- plicated and are difficult to use for complex loads, boundary con- ditions, and geometric configurations. Therefore, approximate or numerical methods, such as Raleigh-Ritz, Galerkin, finite differ- ence, and finite element methods, have been widely used in the analyses. Teichmann6 presented an approximate solution of funda- mental frequencies and buckling loads of cylindrical and conical shell panels. Srinivasan and Krishnan7 provided the free vibration frequencies of fully clamped open conical shells by using an inte- gral equation approach. Cheung et al.8 employed a spline finite strip method to investigate the natural frequencies of fully clamped singly curved shells, and design charts for specific fully supported conical shell configurations were presented. Xi et al.9 studied free vibra- tion of composite shells of revolution by the finite element method. Sivadas and Ganesan10 conducted vibration analysis of laminated conical shells with variable thickness. These methods provide ef- fective ways for engineering analysis in most cases. However, their defaults are obvious in some specific situations, such as analysis concerned with stress concentration, high-frequency response, etc. Based on the method proposed in Refs. 11 and 12, an asymp- totic distributed transfer function method for the analysis of conical shells is presented in this paper. First, the displacements, external excitations, and boundary conditions are expanded in Fourier se- ries in a circumferential direction. Because of the orthogonality of trigonometric functions, the governing equations for different wave numbers are decoupled and Laplace transformation is used to transform the time t to obtain ordinary differential equations with complex parameter s. Second, introducing the perturbation pa- rameter £ = Lsma/r(), those ordinary differential equations with variable coefficients are reduced to a group of ordinary differential