Chaotic Vibrations of Conical and Spherical Shells and Their Control

This chapter is aimed on investigation of non-linear dynamics of conical and spherical shells. The variational equations are derived, and then the problem is reduced to a set of non-linear ordinary differential and algebraic equations. Since further axially symmetric deformation of closed shallow rotational shells and circled plates subjected to uniformly distributed periodic load being normal to the middle plate/shell surface are studied, the polar co-ordinates are used and four types of boundary conditions are investigated. The obtained equations are solved numerically, and the results reliability and validity are discussed in either regular, bifurcation or chaotic regimes including constant and non-constant thickness of the mentioned structural members, taking into account an initial imperfection/deflection. The classical approaches (time histories and frequency power spectra) are used to monitor different transitions from periodic to chaotic vibrations. Novel non-linear dynamical phenomena exhibited by the studied plates/shells are detected and discussed versus the chosen control parameters. In particular, the so called vibration type charts (amplitude-frequency of excitation planes) versus the different shell slopes are reported, which are of a particular importance for direct engineering applications. Finally, it is demonstrated how one may control non-linear dynamics of the studied continuous systems by using their thickness, in order to avoid buckling and stability loss phenomena.