A novel variational numerical method for analyzing the free vibration of composite conical shells

This paper proposes an efficient numerical method in the context of variational formulation and on the basis of Rayleigh-Ritz technique to address the free vibration problem of laminated composite conical shells. To this end, the energy functional of Hamilton's principle is written in a quadratic form using matrix relations first. Displacements are then approximated via a linear combination of base functions, by which the number of final unknowns reduces. After that, the strain tensor is discretized by means of matrix differential quadrature (DQ) operators. In the next step, using Taylor series and DQ rules, a matrix integral operator is constructed which is embedded into the stiffness matrix so as to discretize the quadratic representation of energy functional. Finally, the reduced form of mass and stiffness matrices are readily obtained from the aforementioned discretized functional. To obtain the natural frequencies of conical shell, hybrid harmonic-beam base functions are employed as modal displacement functions. The accuracy of the present numerical method is examined by comparing its results with those from the published literature. It is revealed that the method is capable of accurately solving the problem with a little computational effort and ease of implementation.