Numerical analysis of thin cylindrical shell vibrations with a weak nonlinearity

Abstract

In this paper, nonlinear vibrations of an infinite thin cylindrical shell as a limiting case of a nanotube are studied. The main relations of Sanders-Koiter's nonlinear shell theory and the Hamilton variation principle are applied to obtain a nonlinear mathematical model of the shell vibrations and allow fully accounting for the influence of nonlinear effects. Using the method of multiple scales with specification of fast and slow times, high-order asymptotic relations taking into account quadratic and cubic nonlinearities are found. Based on the solution of the asymptotic scheme at fourth order and application of the inextensibility condition for the semi-membrane shell theory, the numerical analysis of tangential and radial displacements of the cylindrical shell at leading order relative to the Fourier coefficients is conducted. The impact of the wave number, polar angle, radius, and wall thickness of the shell on the amplitude and period of the arising vibrations is investigated. Numerical illustrations of the obtained solutions are presented for several cases.