Perturbation Function Iteration Method: A New Framework for Solving Periodic Solutions of Non-linear and Non-smooth Systems

Abstract

Computing accurate periodic responses in strongly nonlinear or even non-smooth vibration systems remains a fundamental challenge in nonlinear dynamics. Existing numerical methods, such as the Harmonic Balance Method (HBM) and the Shooting Method (SM), have achieved notable success but face intrinsic limitations when applied to complex, high-dimensional, or non-smooth systems. A key bottleneck is the construction of Jacobian matrices for the associated algebraic equations; although numerical approximations can avoid explicit analytical derivation, they become unreliable and computationally expensive for large-scale or non-smooth problems. To overcome these challenges, this study proposes the Perturbation Function Iteration Method (PFIM), a novel framework built upon perturbation theory. PFIM transforms nonlinear equations into time-varying linear systems and solves their periodic responses via a piecewise constant approximation scheme. Unlike HBM, PFIM avoids the trade-off between Fourier truncation errors and the Gibbs phenomenon in non-smooth problems by employing a basis-free iterative formulation, while significantly simplifying the Jacobian computation. Extensive numerical studies, including self-excited systems, parameter continuation, systems with varying smoothness, and high-dimensional finite element models, demonstrate that PFIM achieves quadratic convergence in smooth systems and maintains robust linear convergence in highly non-smooth cases. Moreover, comparative analyses show that, for high-dimensional non-smooth systems, PFIM attains solutions of comparable accuracy with computational costs up to two orders of magnitude lower than HBM. These results indicate that PFIM provides a robust and efficient alternative for periodic response analysis in complex nonlinear dynamical systems, with strong potential for practical engineering applications.