An enhanced incremental harmonic balance method to improve the computational efficiency and convergence for systems with non-polynomial nonlinearities

The incremental harmonic balance (IHB) method is a semi-analytical and semi-numerical method widely applied to solve periodic responses of strongly nonlinear systems. However, it suffers from deficiencies in convergence and computational efficiency, particularly for systems with non-polynomial nonlinearities where time-consuming numerical integration is needed. The solution process of the IHB method is transformed into a nonlinear least squares optimization problem to elucidate the fundamental reasons for its convergence deficiencies in this work. An enhanced incremental harmonic balance (EIHB) method that incorporates the fast Fourier transform (FFT) to obtain the residuals of nonlinear algebraic equations, Broyden’s method to approximate the Jacobian matrix of these equations, and an improved Levenberg–Marquardt (L–M) method with adaptive search direction adjustment and restart steps to improve convergence is then proposed. The introduction of the FFT and Broyden’s method significantly reduces computation time and greatly simplifies the process of formula derivation and programming. The proposed improved L–M method enhances convergence by introducing a new negative gradient search direction beyond the Gauss-Newton search direction of the traditional IHB method and adjusting the step sizes adaptively. It can also utilize restart steps to reinitiate iterations and escape from local minima. The proposed EIHB method, which boosts both computational efficiency and convergence, is particularly suited for systems with non-polynomial nonlinearities, where the traditional IHB method is time-consuming and often fails to ensure convergence. The effectiveness of the EIHB method is demonstrated through three examples: a dielectric elastomer balloon with negative exponent nonlinearity, an inverted rod pendulum with trigonometric nonlinearity, and a non-smooth gear transmission system with piecewise linear functions. Comparative results with those from the fourth-order Runge–Kutta method, the traditional IHB method, and a modified IHB method combining the FFT and Broyden’s method indicate that the EIHB method significantly improves convergence and computational efficiency while maintaining the same accuracy as the traditional IHB method.