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.

翻译：在强非线性乃至非光滑振动系统中计算精确的周期响应，始终是非线性动力学领域的一项基础性挑战。现有的数值方法，如谐波平衡法（HBM）和打靶法（SM），虽已取得显著成功，但在应用于复杂、高维或非光滑系统时仍面临固有局限。一个关键瓶颈在于相关代数方程的雅可比矩阵构造；尽管数值近似可避免显式解析推导，但对于大规模或非光滑问题，这些近似方法变得不可靠且计算成本高昂。为克服这些挑战，本研究提出摄动函数迭代法（PFIM），这是一种基于摄动理论构建的新颖框架。PFIM将非线性方程转化为时变线性系统，并通过分段常数近似方案求解其周期响应。与HBM不同，PFIM采用无基底的迭代形式，避免了非光滑问题中傅里叶截断误差与吉布斯现象之间的权衡，同时显著简化了雅可比矩阵的计算。广泛的数值研究，包括自激系统、参数延拓、光滑度变化的系统以及高维有限元模型，表明PFIM在光滑系统中实现了二次收敛，并在高度非光滑情况下保持了稳健的线性收敛。此外，对比分析显示，对于高维非光滑系统，PFIM能以比HBM低达两个数量级的计算成本，获得精度相当的解。这些结果表明，PFIM为复杂非线性动力系统的周期响应分析提供了一种稳健且高效的替代方案，在工程实际应用中具有巨大潜力。