Non-linear axisymmetric response of closed spherical shells to a radial harmonic excitation

Abstract

A numerical-perturbation approach is used to study the axisymmetric dynamic response of closed spherical shells to an external harmonic excitation having a frequency near one of the natural frequencies of a flexural mode (i.e. primary resonance of a flexural mode) in the presence of a two-to-one internal (autoparametric) resonance between the excited mode and a lower flexural mode. The method of multiple-time scales is used to derive four first-order non-linear equations describing the evolution (modulation) of the amplitudes and phases of the interacting modes. A numerical scheme that combines a shooting technique and a Newton-Raphson procedure is used to calculate limit cycles of the evolution equations. The stability of these limit cycles is determined by using a Floquet analysis. As the excitation amplitude varies, the fixed-point solutions of the evolution equations exhibit the jump and saturation phenomena. They also undergo supercritical and subcritical Hopf bifurcations as the frequency of excitation varies; the supercritical Hopf-bifurcation frequency is lower than the subcritical one. As the excitation frequency increases above the supercritical Hopf-bifurcation value, the fixed-point solutions lose their stability and limit cycles are born. These limit cycles experience a symmetry-breaking (pitchfork) bifurcation followed by a cascade- of period-doubling bifurcations culminating in chaos. On the other hand, as the excitation frequency decreases below the subcritical Hopf-bifurcation value, the resulting limit cycle deforms and eventually loses its stability through a cyclic-fold bifurcation causing a transition to chaos. In the frequency range in between the above two chaotic regions the evolution equations possess limit-cycle solutions.