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 the mesh and the boundary and initial conditions are obtained from the Hamilton–Ostrogradsky energy principle taking into account the Kirchhoff–Love kinematic hypotheses and the von Karman theory. 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 ribs of the same material, which makes it possible to average the ribs on the panel surface using the Pshenichnov theory. To reduce the partial derivative problem to a system of ordinary differential equations in spatial coordinates, two fundamentally different methods: the finite difference method with the second-order approximation and the Bubnov–Galerkin method with higher approximation are used. The obtained Cauchy problem has been solved by the Runge–Kutta-type methods with different orders of accuracy. The results obtained by different numerical methods are compared. The nonlinear dynamics of the examined systems is investigated depending on the mesh geometry. The necessity of studying the propagation of longitudinal waves has been justified.