Frequency analysis of strongly nonlinear generalized Duffing oscillators using He’s frequency–amplitude formulation and He’s energy balance method

In this paper, He’s Frequency–Amplitude Formulation (HFAF) and He’s Energy Balance Method (HEBM) are employed to solve the generalized Duffing equation in the form of ẍ+x+α3x3+α5x5+α7x7+⋯+αnxn=0. For any arbitrary power of nn, a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HFAF, HEBM and exact integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results.