Mathematical modeling of complex oscillations of flexible micropolar mesh cylindrical panels

A new mathematical model of oscillations of mesh micropolar geometrically nonlinear cylindrical panels under the action of a normal alternating distributed load has been constructed. The equations of motion for an element of a smooth panel equivalent to a mesh, the boundary and initial conditions are obtained from the Hamilton-Ostrogradsky energy principle, taking into account the Kirchhoff-Love kinematic hypotheses and Theodore von Karman’s theory. In order to take into account the size-dependent behavior, a non-classical continual model based on a Cosserat medium is used in the work, where, along with the usual stress field, momentary stresses are also taken into account. The panel consists of n sets of densely arranged edges of the same material, which makes it possible to average the edges on the panel surface using the Pshenichnov G.I. theory. To reduce the partial differential problem to the system of ordinary differential equations using spatial coordinates, we use two fundamentally different methods: the finite difference method with the second order accuracy approximation and the Bubnov - Galerkin method in higher approximations. The obtained Cauchy problem is solved by Runge-Kutta-type methods of different order of accuracy. Comparison of the results obtained by various numerical methods. A study of the nonlinear dynamics of the systems under consideration, depending on the geometry of the grid. Justified the need to study the propagation of longitudinal waves.

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