On the formation of bubbles in the frequency response curves of nonlinear oscillators

Abstract eng:
When a multi degree-of-freedom nonlinear oscillator is subject to harmonic excitation, complex dynamics may occur, leading to a non-harmonic, quasi-periodic or even a chaotic response. For light nonlinearities, the system response may be still dominated by the harmonic solution at the excitation frequency, while the other components at other different frequencies may be negligible. In this case, an approximate closed-form solution for the amplitude-frequency equation can be derived and the system response may be conveniently described in terms of the corresponding frequency response curve. For particular values of the system parameters, these curves are characterized by multi-valuedness, which means that at a single excitation frequency the system may respond at different amplitudes, depending on the initial conditions or on some external perturbation. Multi-valuedness can also lead to closed detached resonance curves (bubbles), which appear inside the main continuous frequency response curve. For such a reason, their detection may be hidden by numerical continuation methods or experimental analysis. In this paper, a combination of an analytical and numerical approach is adopted to predict their appearance for the specific case of a light damped two degreeof-freedom system with cubic stiffness nonlinearity. The relation between the frequency response curves and the bifurcation curves is illustrated, and the effect of the system parameters is investigated. The analytical findings are validated by direct numerical integration of the equations of motion of the coupled oscillator.

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