Chaotic vibrations of spherical and conical axially symmetric shells

摘要

Chaotic vibrations of deterministic, geometrically nonlinear, elastic, spherical and conical axially symmetric shells, subject to sign-changing transversal load using the variational principle, are analysed. The paper is motivated by an observation that variational equations of the hybrid type are suitable to solve many dynamical problems of the shells theory. It is assumed that the shell material is isotropic, and the Hooks principle holds. Inertial forces in directions tangent to mean shell surface and rotation inertia of a normal shell cross section are neglected. A transition from PDEs to ODEs (the Cauchy problem) is realized through the Ritz procedure. Next, the Cauchy problem is solved using the fourth-order Runge-Kutta method. Qualitative and quantitative analysis is carried out in the frame of both nonlinear dynamics and quantitative theory of differential equations. New scenarios from harmonic to chaotic dynamics are detected. Various vibration forms development versus control parameters (rise of arc; amplitude and frequency of the exciting force and number of vibrational modes accounted) are illustrated and discussed.