Jump Phenomena, Bifurcations, and Chaos in a Pressure Loaded Spherical Cap Under Harmonic Excitation

This paper investigates the dynamic non-linear behaviour of a pre-loaded shallow spherical shell under a harmonic excitation. For this, the Marguerre partial differential equations of motion for an imperfect, pre-loaded cap is reduced to a finite degree of freedom system using the Galerkin method. The displacements and stress functions are described by a linear combination of Bessel functions and modified Bessel functions that satisfy all the relevant boundary and continuity conditions. The resulting differential equations of motion are solved by the Galerkin-Urabe procedure, or, alternatively, by numerical integration. To study the response of the shallow cap under harmonic excitation, phase plane portraits, Poincare´ maps, resonance curves, and bifurcation diagrams are plotted for a number of loading conditions. Results indicate that, for static load levels between the upper and lower limit point loads, the shell may display jumps due to the presence of competing potential wells and the presence of non-linear resonance curves within each well. Additionally, different physical situations are identified in which period-doubling phenomena and chaos can be observed.