ДИНАМИЧЕСКАЯ УСТОЙЧИВОСТЬ ПОЛОГИХ ОБОЛОЧЕК НА ПРЯМОУГОЛЬНОМ ПЛАНЕ С УЧЕТОМ ГЕОМЕТРИЧЕСКОЙ И ФИЗИЧЕСКОЙ НЕЛИНЕЙНОСТИ

Abstract

Proposed and developed a dynamic criterion of stability loss of rectangular spherical shells under the action of time-varying lateral loads taking into account two types of nonlinearities - geometric and physical. A mathematical model for thin shallow shells, based on the kinematic hypotheses of the Kirchhoff - Love. Geometric nonlinearity is accounted on the basis of the ratios of T. Karman. Physical nonlinearity is described by the deformation theory of plasticity by A.A. Ilyushin. The developed approach to the calculation of the dynamics of shells under time-varying lateral load, which is based on the method of finite differences of second order accuracy, the method of Bubnov - Galerkin in higher approximations and Runge - Kutta of fourth order accuracy. The results obtained according to the methods of finite differences and the method of Bubnov-Galerkin fit together well. For a number of geometric parameters of the built area of the character of the oscillation depending on the amplitude of the driving load and frequency. Analysed scenarios of transition of harmonic oscillations to chaotic, obtained for various geometrical and physical parameters. It is noted that a single script is not, and it depends on the control parameters. Perhaps the existence of three classic scenarios (Feigenbaum, Ruelle - Takkens of Newhouse and the Aid of Mandevilla) with minor modifications. Keywords: chaos, harmonic oscillations, geometrical nonlinearity, physical nonlinearity, shallow shells, finite difference method, method of Bubnov − Galerkin.