Control of Resonance by Chaos-Elucidation of Fundamental Mechanism

为了研究混沌振子是否能抑制特定频率的周期性外力作用引起的共振，基于简单的非线性振子模型，研究了混沌振子抑制共振的基本机理：在通常的动力吸振器中，通过在主振子上附加一个子振子来移动共振频率，从而避免共振。相反，插入的振荡器模型，即，在主系统和周期外力之间插入一个次非线性振子，从理论上分析了当周期外力变为宽带谱的混沌振动时，系统在某一特定频率处的共振振幅可能会降低。数值计算表明，由于混沌引起的能量展宽实际上可以抑制共振。两种插入的非线性振子模型（即，非耦合和耦合的情况下）在各种条件下的数值求解和谐振振幅以及振荡的频谱进行了详细检查。主要研究结果如下：1.非线性振荡器的插入导致的非线性饱和显著降低了主振子的谐振幅度，无论是非耦合还是耦合情况.当非线性插入振子将外力改变为谐波、次谐波或宽谱混沌振动时，共振振幅进一步减小.动力吸振器与插入非线性振子模型的不同之处在于，插入非线性振子模型在弱非线性情况下使共振频率发生大的偏移，而在强非线性情况下，非线性饱和能有效地抑制共振，从而至少从理论上证明了插入非线性振子可以作为共振的有效衰减器。

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.