一類非自治滯后 - 自激系統的主共振與鎖模現象

Abstract

研究一類受外激勵作用的滯後 - 自激系統 (van der Pol系統) 的振動。對於主共振情況，用平均法求出了穩態響應方程，揭示了響應與系統參數的關係。結果表明，外激勵的影響占主導地位，滯後因素對非線性共振頻率有影響，而自激因素使得周期解在偏離非線性共振頻率後發生Hopf分岔，轉變為概周期運動。通過用環映射法求得旋轉數，揭示了主共振前、後的概周期運動中存在的鎖模現象，其階數分布符合Farey樹規律。研究表明，鎖模振動因滯後非線性因素而產生。隨著滯後參數值 (代表滯後程度) 增大，各階共振特別是亞諧振動的存在能力增強，對系統結構會帶來較大影響。

The nonlinear dynamic behaviours of a van der Pol system with hysteretic nonlinearity and acted by external force were studied in this paper. The results show that the first approximate analysis using the averaging method is quite adequate for determining the steady state responses of the system in primary resonant case. The external force is found playing the leading role in primary resonant responses, while the hysteretic nonlinearity plays its effect on the nonlinear resonant frequency. Under the effect of the van der Pol damping, the steady-state periodic response loses its stability through Hopf bifurcation and then the system undergoes quasi-periodic motions. The circle maps was used to get the winding numbers to reveal the super- and subharmomic resonances in various orders known as mode-locking, which take place according to the Farey number tree as having been revealed in many other systems. The study shows that as the increase of the hysteretic parameters' values, which represents increase of the level of such a nonlinearity, it's easier for occurrence of the subharmonic resonances of order 2 as well as 3, known as strong resonances and can often damage the mechanical structures.