An asymptotic approach for large amplitude motions of generic nonlinear systems

Starting from the observation that classical asymptotic methods fail to correctly describe the resonance peak of the frequency response curve of a nonlinear oscillator under moderate and large excitation amplitudes, an alternative approach is proposed to overcome this problem. The differences between the multiple time scale method (one of the most performant classical methods) and numerical simulations are initially shown with reference to on the paradigmatic Duffing equation. They are also shown some characteristics of the near peak behavior. Then, the proposed asymptotic approach is illustrated. The basic idea is that of having the zero-order problem nonlinear, while in classical methods it is linear. Thanks to the energy conservation, the zero-order problem is solved exactly. Also, the exact solution of the higher-order problems is obtained in closed-form, thus providing a fully analytical approach. Although the proposed method is valid for any kind of motion, special attention is dedicated to periodic nonlinear oscillations, because of their interest in practical applications. A simple formula for determining the exact intersection of the frequency response and backbone curves is obtained, and it is shown that it can be computed without the need of solving explicitly not even the zero-order problem. Some illustrative examples are finally reported.