Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

The aim of this work is to study the free dynamic response of thin plates characterized by geometrical nonlinearities. To achieve this task, the equation of motion of the plate is first carried out through modeling by hierarchical finite element method whose interpolating shape functions are sinusoidal. Then, the study of the nonlinear vibrations was carried out by the development of asymptotic linearization and equivalent linearization methods in modal space. The nonlinear angular frequencies are successively deduced by exciting the corresponding vibrating mode of the structure. The confrontation of these results to those obtained by the iterative method in the physical space and to those found in the literature, showed a very good agreement between the various methods. From the elementary nonlinear frequencies we showed that there exists an equivalent linear dynamical system characterized by only one equivalent linear stiffness matrix. Numerical experiments were carried out on beams and thin plates of various dimensions ratios and boundary conditions. These numerical test simulations, whether in time space or frequency space, have showed that the nonlinear elastic energy is restored by the equivalent linear dynamical system. Nevertheless, we have to say that the dynamic effects of modes above the excited one are neglected.