Combination resonances in the response of the Duffing oscillator to a three-frequency excitation

The response of a weakly nonlinear, single-degree-of-freedom system with cubic nonlinearities to multifrequency excitations is studied analytically and numerically. The method of multiple scales is used to obtain uniformly valid, approximate solutions of the governing equation for various combination resonances. The analytical and numerical solutions are in virtually perfect agreement for all cases considered, but differ markedly from the exact solution of the linearized equation of motion. The peak amplitudes in the solutions of the nonlinear equation can be several times those in the solutions of the linearized equation, and they can occur rather more often; moreover, the addition of a static load can affect the natural frequency and, hence, either tune or detune a resonance, producing profound changes in the response. The present results demonstrate that the actual response of a structure can lead to a fatigue life that is much shorter than what is predicted by linear analysis. Hence, the conventional structural engineering practice of considering the structure to be safe from resonant responses when none of the frequencies of the excitation matches the natural frequency is shown to be fraught with danger; a practicing engineer, therefore, cannot afford to be ignorant of nonlinear phenomena.