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.
This paper presents a numerical model to simulate the initial stress stiffening effect, induced by radial pressure and/or axial load on the dynamic behaviour of axisymmetric shells. This effect is particularly important for thin shells since their bending stiffness is very small compared to membrane stiffness. The theoretical formulation is based on a combination of the finite element method and classical shell theory. For a perfect geometrical consistency, two semi-analytical finite elements, conical and cylindrical, are used to model axisymmetric shells. The displacement functions are derived from exact solutions of Sanders' shell equilibrium equations. The results obtained using this approach are remarkably accurate. The potential energy is calculated to estimate the initial stiffening effect using direct membrane forces per unit width and rotations about the orthogonal axes. The final stiffness matrix of each finite element is composed of the regular stiffness matrix and the added stiffness matrix generated by membrane loads. The frequencies of vibration are compared with those obtained in other experimental and theoretical research works and very good agreement is observed.
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