Residue harmonic balance analysis for the damped Duffing resonator driven by a van der Pol oscillator

The residue harmonic balance is developed for coupled systems exemplified by the damped Duffing resonator driven by a van der Pol oscillator. This technique combines the features of harmonic balance and parameter bookkeeping to obtain approximate solutions to any desired accuracy. For the two degrees of freedom system, the zeroth-order approximation uses just one Fourier term resulting in a set of two nonlinear algebraic equations that can be solved analytically. The unbalanced residue due to Fourier truncation is improved by solving linear equations that automatically increase the number of terms to balance the previous truncation completely. Highly accurate bifurcation frequencies for various parameters are obtained and are verified by numerical integration results. The effects of the coupling stiffness on the angular frequency and amplitude of steady state response are investigated. The effects of the nonlinear damping are also studied.