Control of Resonance by Chaos-Elucidation of Fundamental Mechanism

基于简单的非线性振子模型，研究了能否用混沌振子抑制某一特定频率处的外加周期激振引起的共振。在通常的动力吸振器中，通过在主振子上附加一个下沉振荡器来避免共振。相反，插入振子模型，即在主系统和外周期力之间插入一个下沉的非线性振子，理论上期望如果外周期力变为具有宽带谱的混沌振荡，则在特定频率处的共振幅度可能发生减小。数值结果表明，混沌导致的能量展宽实际上可以抑制共振。对两种插入的非线性振子模型(即非耦合和耦合情况)在不同条件下进行了数值求解，并详细分析了振荡的共振幅度和频谱。研究结果总结如下：1.在非耦合和耦合情况下，由于插入的非线性振荡器引起的非线性饱和都显著降低了主振荡器的共振幅度。当非线性插入振子将外周期力改变为以谐波、亚谐或宽谱为混沌的振荡时，共振幅度进一步减小。动力吸振器与插入式非线性振子模型的不同之处在于，插入式非线性振子模型在弱非线性情况下产生较大的共振频率漂移，而在强非线性情况下可有效抑制共振，从而至少在理论上证明了插入式非线性振子可以作为一种有效的共振衰减器。

To investigate whether the resonance due to external periodic forcing at a particular frequency can be suppressed or not by using chaotic oscillator is the purpose of this research and fundamental mechanisms are examined based on simple nonlinear oscillator models.In the usual dynamical vibration absorber, resonance is avoided by a shift of resonance frequency due to a subsidary oscillator attached to the main oscillator. On the contrary, the inserted oscillator model, i.e., a subsidary nonlinear oscillator is inserted in between the main system and the external periodic force, is theoretically examined with an expectation that if the external periodic force is changed to chaotic oscillations with broad band spectra, reductions of the resonant amplitude at a particular frequency might take place. It was observed numerically that broadening of energy due to chaos can actually suppress the resonance.Two types of inserted nonlinear oscillator models(i.e., uncoupled and coupled cases)are solved numerically under various conditions and resonant amplitudes as well as spectra of oscillations are examined in details. Results are summarized as follows :1. Nonlinear saturations due to the inserted nonlinear oscillator reduce significantly the resonant amplitudes of the main oscillator for both uncoupled and coupled cases.2. Further reductions of the resonant amplitudes are observed when the nonlinar inserted oscillator changes the external periodic force to oscillations with harmonics, sub-harmonics, or broad spectra as chaos.3. The differences between the dynamical vibration absorber and the inserted nonlinear oscillator model exhibit that the latter gives large shift of resonance frequency for weakly nonlinear case and nonlinear saturation effectively suppress resonance for strongly nonlinear case.Thus it is shown at least theoretically that the inserted nonlinear oscillator can work as an effective attenuator of resonance.