Applied Mathematics and Mechanics >
Periodic response and stability analysis of vibro-impact systems by an enriched harmonic balance method
Received date: 2024-11-19
Revised date: 2025-03-21
Online published: 2025-05-07
Supported by
Project supported by the National Natural Science Foundation of China (No. 12372028) and the Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011809)
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A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering. In general, the alternating frequency-time harmonic balance (AFT-HB) method can be used to solve elastic collision. However, since the system is non-smooth, the required Fourier/harmonic truncation order is high in order to achieve the theoretical convergence rate, resulting in expensive computational cost. Furthermore, for rigid body collision, the periodic response of the system cannot be solved with the AFT-HB method due to the discontinuous velocity of the system. In order to accelerate the convergence and solve highly non-smooth systems, an enriched harmonic balance (HB) method is proposed, which is derived from the AFT-HB method in the framework of event-driven Gauss quadrature. The basic idea is to augment the Fourier bases by introducing a non-smooth Bernoulli base such that the non-smooth Bernoulli base compensates for the non-smooth part of the solution and the smooth part of the solution is approximated by the Fourier bases, thus achieving accelerated convergence. Based on the enriched HB method, gear pair systems with gear backlash and oscillator systems with rigid impact are solved, and the dynamic response characteristics are analyzed in this work. Then, based on the Floquet theory, the event-driven monodromy matrix method for non-smooth systems is used to analyze the stability and bifurcation of the periodic solutions. The numerical example shows that the results obtained from the enriched HB method are consistent with those from the Runge-Kutta method, which proves that the presented method is an effective method for analyzing the dynamic response characteristic of the vibro-impact system.
Yu ZHOU, Li WANG, Jianliang HUANG . Periodic response and stability analysis of vibro-impact systems by an enriched harmonic balance method[J]. Applied Mathematics and Mechanics, 2025 , 46(5) : 907 -926 . DOI: 10.1007/s10483-025-3253-8
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