Transitions from regular to chaotic vibrations of spherical and conical axially-symmetric shells

By the variational principle, the chaotic vibrations of deterministic geometrically nonlinear elastic spherical and conical axially symmetric shells with non-homogeneous thickness subjected to a transversal harmonic load are analyzed. The material of the shells is assumed to be isotropic and of the Hookean type. Inertial forces tangent to the averaged surface and inertia of rotation of the cross-section are neglected. By the Ritz procedure, the original PDEs are transferred to the ODEs (Cauchy problem), which are then solved by the fourth-order Runge-Kutta method. In the numerical studies, scenarios of transitions from harmonic to chaotic states for vibrations of flexible spherical and conical shells are detected. Various vibrational states for different combinations of the following control parameters: shell's deflection arrow, the amplitude and frequency of the exciting force, number of modes considered, boundary conditions, and the thickness and shape of the shell cross-section are studied. By adjusting the above parameters, we can detect the transition of a continuous system to the lumped one, and the transition from the harmonic to chaotic vibrations.