延时反馈振荡器在弱正弦外激下的强制振动

Forced Vibration of a Self-Excited System with Time Delayunder Weak Harmonic Excitation

讨论了一类延时反馈振荡器对弱正弦外激的动态响应，用KBM渐近法与Poincare－Dulac定性分析方法研究了系统中可能存在的主共捧（频率俘获现象）与组合共根（拍现象）及其稳定性。清晰地描述了在小解谐情形下系统中发生强制振动的各种情景。主共振曲线图显示出：在一定条件下系统中存在双稳态主共根，在某些情形下还明显伴随有回滞振动现象（跳跃现象）。根据奇点特性，对双稳态情形下各个主共振的发生概率进行了定性讨论。另一方面，通过van der Pol平面上的极限坏分析．定性地证明了系统在频率俘虏区外附近存在唯一稳定的组合共振。

The resonant responses of a self-excited system with time delay under weak harmonic excitations were explored. Existence of principal resonance, or frequency entrainment, or phase locking phenomenon, was examined by the well-known KBM asymptotic method. Principal resonance curves on amplitude-detuning parametric plane are obtained, which reveal that bistable principal resonance, together with associated vibration hysteresis, or jump phenomenon, may exist under certain conditions. This kind of vibration hysteresis resulting from saddle-node bifurcation often occurs in driven Duffing oscillator, but seldom occurs in driven van der Pol oscillator. The probabilities for occurrence of either one of the bistable principal resonances were discussed qualitatively. On the other hand, through qualitative analysis of limit cycles on the van der Pol plane by the Poincare´-Dulac theorems, the existence of a unique stable combined resonance, or a beat phenomenon, in the neighborhood outside the frequency entrainment band is confirmed. The above results clearly describe the various situations of forced vibrations in the system considered and deepen one's understanding of nonlinear phenomena.