Chaotic dynamics size-dependent flexible rectangular flat shells

Abstract

In this paper was constructed mathematical model nonlinear vibrations flexible size-dependent spherical rectangular shell. Sheath viewed as a continuum Cosserat with constrained rotation of the particles (pseudo-continuum). The equations of motion of the shell element and the boundary conditions are obtained from the Ostrogradski-Hamilton energy principle on the basis of Kirchhoff-Love kinematic hypotheses. The geometric nonlinearity is taken into account by the model of T. von Karman. The equations of motion of the shell element in the work are written in a mixed form. A system of nonlinear partial differential equations reduces to an ODE system by the Bubnov-Galerkin method in higher approximations. The system is regarded as a system with an infinite number of degrees of freedom. The Cauchy problems are solved by methods of Runge-Kutta type from the second to the eighth order of accuracy. The convergence of each method depending on the time step and the number of terms in the expansion series of functions in the Bubnov-Galerkin method. The influence of the size-dependent parameter to the nonlinear dynamics of the rectangular plane into a spherical shell. The largest Lyapunov exponent is determined using three methods: Wolf, Kantz, and Rosenstein to prove the truth of chaos.