Nonlinear oscillation of circular cylindrical shells (Q1094920)

The method of multiple scales is used to analyze the nonlinear forced response of circular cylindrical shells in the presence of a two-to-one internal (autoparametric) resonance to a harmonic excitation having the frequency \(\Omega\). If \(\omega_ f\) and \(a_ f\) denote the frequency and amplitude of a flexural mode and \(\omega_ b\) and \(a_ b\) denote the frequency and amplitude of the breathing mode, the steady-state response exhibits a saturation phenomenon when \(\omega_ b\approx 2\omega_ f\) if the excitation frequency \(\Omega\) is near \(\omega_ b\). As the amplitude f of the excitation increases from zero, \(a_ b\) increases linearly whereas \(a_ f\) remains zero until a threshold is reached. This threshold is a function of the damping coefficients and \(\omega_ b-2\omega_ f\). Beyond this threshold \(a_ b\) remains constant (i.e. the breathing mode saturates) and the extra energy spills over into the flexural mode. In other words, although the breathing mode is directly excited by the load, it absorbs a small amount of the input energy (responds with a small amplitude) and passes the rest of the input energy into the flexural mode (response with a large amplitude). For small damping coefficients and depending on the detunings of the internal resonance and the excitation, the response exhibits a Hopf bifurcation and consequently there are a no steady-state periodic responses. Instead, the responses are amplitude- and phase-modulated motions. When \(\Omega \approx \omega_ f\), there is no saturation phenomenon and at close to perfect resonance, the response exhibits a Hopf bifurcation, leading again to amplitude- and phase-modulated or chaotic motions.