Tkachenko V. Nonlinear vibration and stability of laminated plates of complex shape

The focus of the research is on multilayered plates with the symmetric and antisymmetric structure of the layers. The mathematical formulation is carried out in the framework of two theories: classical theory and refined theory of the first order by S. P. Timoshenko. The proposed method consists of several stages. Firstly, it is necessary to determine the heterogeneous subcritical state of the plate. To do this, a variational statement is offered for symmetric and antisymmetric structure of the layers. It is the first time that the corresponding functionals for plates of an antisymmetric structure in the framework of the classical and refined theory of the first order have been obtained. The second stage of the algorithm is the solution of a linear vibration problem. This problem is solved by using RFM and the Ritz methods. The third stage of the algorithm is the determination of the critical load value. For this purpose, two methods are applied: energy and dynamical. The dynamical method is a more general one. The critical load corresponds to the smallest nonnegative value of eigenfrequency. At the fourth stage, reduction of the initial nonlinear system of PDEs to ODEs of Mathieu type is fulfilled. It is necessary to solve some additional boundary problems in order to realize this stage. Examining dynamical stability and finding the main instability zones are carried out applying the Bolotin approach.