Approximate fully analytical Fourier series solution to the forced and damped Helmholtz–Duffing oscillator

Abstract

• A new method to obtain approximate fully analytical steady state solutions of NLODEs is presented. • The method is applied to the forced and damped Helmholtz–Duffing oscillator. • We obtain improved approximate fully analytical Fourier series solutions valid for strong nonlinearities . • The solutions describe very well the principal and superharmonic resonances. • The method allows obtaining explicit expressions for the coefficients of the Fourier series solution. We obtain improved approximate fully analytical steady state solutions for the ordinary differential equation of the forced and damped anharmonic oscillator with quadratic and cubic nonlinearities. The solutions, are obtained in the form of a truncated Fourier series using the harmonic balance method (HBM). In order to obtain explicit analytical expressions for the Fourier coefficients we introduce the hypothesis of weak interdependence between the equations in the system of nonlinear algebraic equations produced by the HBM. Comparison with the numerical solution shows that the analytical results for the first harmonics which we investigated are valid in comparatively strong nonlinear regimes, where previously obtained expressions are completely inaccurate. Our expressions are correct over a wide range of the physical parameters, such as excitation frequency , nonlinearity and forcing. The proposed method of solution is expected to be suitable for other nonlinear ordinary differential equations.