Analytical approximations to resonance response of harmonically forced strongly odd nonlinear oscillators

Abstract

New and expressive analytical approximate solutions to resonance response of harmonically forced strongly odd nonlinear oscillators are proposed. This method combines Newton’s iteration with the harmonic balance method. Unlike the classical harmonic balance method, accurate and explicit analytical approximate solutions are established because linearization of the governing nonlinear differential equation is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations which have no analytical solution. With carefully constructed corrective measures, only one single iteration is required to obtain very accurate analytical approximate solutions to resonance response. It is found that since determination of stability of the initial approximate solution that resulted from the single-term harmonic balance can lead to erroneous conclusions, correction to the solution is necessary. Three examples are presented to illustrate the applicability and effectiveness of the proposed technique. Specially, for oscillations in high-energy orbits of the bistable Duffing oscillator, the proposed method can also give excellent analytical approximate solutions.