Applied Mathematics and Mechanics >
Dynamic analysis of asymmetric piecewise linear systems
Received date: 2024-11-05
Revised date: 2025-01-25
Online published: 2025-04-07
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12272242 and U1934201)
Copyright
Piecewise linear systems are prevalent in engineering practice, and can be categorized into symmetric and asymmetric piecewise linear systems. Considering that symmetry is a special case of asymmetry, it is essential to investigate the broader model, namely the asymmetric piecewise linear system. The traditional averaging method is frequently used for studying nonlinear systems, particularly symmetric piecewise linear systems, with the harmonic response of the oscillator serving as a key prerequisite for calculating steady-state solutions. However, asymmetric systems inherently exhibit non-harmonic, asymmetric responses, rendering the traditional averaging method inapplicable. To overcome this limitation, this paper introduces an improved averaging method tailored for an oscillator characterized by asymmetric gaps and springs. Unlike the traditional method, which assumes a purely harmonic response, the improved averaging method redefines the system response as a superposition of a direct current (DC) term and a first harmonic component. Herein, the DC term can be regarded as the offset induced by model asymmetry. Furthermore, the DC term is treated as a slow variable function of time, with its time derivative assumed to be zero when calculating the steady-state solution, akin to the amplitude and phase in the traditional averaging method. Numerical validation demonstrates that the responses computed in both time and frequency domains with the improved averaging method exhibit greater accuracy compared with those derived from the traditional method. Leveraging these improved results, the study also examines the parameter effect, stability, and bifurcation phenomena within the amplitude-frequency responses.
Key words： nonlinear vibration; piecewise linear system; asymmetry; averaging method; offset
Ruiliang ZHANG, Yongjun SHEN, Xiaotong YANG . Dynamic analysis of asymmetric piecewise linear systems[J]. Applied Mathematics and Mechanics, 2025 , 46(4) : 633 -646 . DOI: 10.1007/s10483-025-3234-9
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