Paliy O. Stability of nonlinear vibrations of thin shells under periodic loads

The thesis is devoted to investigation of the stability of nonlinear oscillations of thin elastic axisymmetric shells subjected to periodic loads and to the estimation of the influence of geometrical parameters of shells on critical dynamic values of loads and forms of stability loss. Applying the geometrically nonlinear relations of the moment theory of thin elastic shells, formulated in tensor form and based on the Kirchhoff-Love hypotheses, the system of calculated equations of steady-state forced nonlinear oscillations is formed. A mathematical model of the dynamic stability of steady-state forced nonlinear oscillations of thin elastic shells is constructed using the projection method. The discretization of differential calculated relations of the theory of thin shells in the problems of steady-state forced nonlinear oscillations and their stability is done on the basis of the modified method of finite differences - the method of curvilinear grids. The solutions to new applied problems of stability of nonlinear oscillations of thin axisymmetric elastic steel shells subjected to periodic force or kinematic excitations are solved using combination of method of continuation the solution by a parameter, Newton-Kantorovich method and the theorems of stability in the sense of Lyapunov. The influence of geometrical parameters of thin shells on frequencies and forms of natural oscillations without and taking into account loading, amplitudes of steady-state forced nonlinear oscillations, critical values of dynamic loads and corresponding forms of loss of stability is estimated. The numerical approach to the study of the stability of nonlinear oscillations of thin shells is implemented in the form of software that provides the development of the computational complex of the curvilinear grids method.