Analysis of a cubic-quintic nonlinear oscillator using modified harmonic balance method

This study develops an enhanced version of the Modified Harmonic Balance Method (MHBM) for accurately analyzing strongly nonlinear, externally forced, and heavily damped oscillatory systems with cubic–quintic nonlinearities. The proposed formulation addresses the reduced accuracy and convergence limitations of conventional harmonic balance techniques under strong nonlinear conditions. The enhanced MHBM incorporates a systematic power-series expansion of the Fourier coefficients within the harmonic balance framework, improving convergence while maintaining computational efficiency. A representative cubic–quintic nonlinear oscillator is analyzed under different damping ratios, nonlinearity strengths, forcing amplitudes, and initial conditions. The analytical predictions are validated through high-resolution numerical simulations. Results: The proposed method exhibits excellent agreement with numerical solutions over a broad range of system parameters. Compared with the classical MHBM, the enhanced formulation provides improved analytical accuracy, faster convergence, and a more faithful representation of nonlinear dynamic behaviors, including amplitude and phase modulation, particularly in strongly nonlinear regimes. The enhanced MHBM significantly extends the applicability of the classical harmonic balance framework without increasing computational complexity. It provides a reliable, accurate, and computationally efficient semi-analytical approach for investigating strongly nonlinear dynamical systems encountered in engineering and applied physics.