An Analytical Approach for the Nonlinear Free Vibration Analysis of Thin-Walled Circular Cylindrical Shells

Abstract

This paper presents a new formulation combining the nonlinear theory of Novozhilov with the classical finite element method for the purpose of evaluating the vibratory characteristics of thin, closed and isotropic cylindrical shells. The theory developed in this paper is able to include the shell curvature effect in the circumferential direction of the orthogonal displacements and considers the impact of initial geometric imperfections on the dynamic response of the system. The formulation first takes a general form by expressing the shell displacements as an alliance between the generalized coordinates and spatial functions. Nonlinear kinematic relationships are inferred from Novozhilov’s theory. The equations of motion as well as the expressions of the mass, linear and nonlinear stiffness matrices are derived through the Lagrange method by considering the coupling between the different modes. An application of this model is illustrated in a further step, by adopting the displacement functions derived from exact solutions of linear Sanders’ theory equilibrium equations for thin cylindrical shells. The governing equations of motion are solved with the help of a direct iterative method. Linear and nonlinear frequencies are validated by comparison with the results in the literature. The relative nonlinear frequencies are determined as a function of vibration amplitudes and then compared with published results for several cases of shells. Excellent agreement is observed between the results derived from this theory and those found in the literature. The effect of different parameters including axial and circumferential wave number, length-to-radius ratio, thickness-to-radius ratio and various boundary conditions, on the nonlinear frequencies of cylindrical shells is investigated.