Van der Pol-Duffing单边约束系统的随机响应

 研究了Van der Pol-Duffing单边约束系统在谐和与随机噪声联合激励下的响应问题。用多尺度法分离了系统的快变项，讨论了系统的阻尼项、非线性项和随机项等参数对系统响应的影响。在一定条件下，当约束距离较大时对应于不同的初始条件，系统具有两个非碰撞的稳态响应；而当约束距离不大时，对应于不同的初始条件，系统也可以有两个不同的稳态响应，其中一个是发生碰撞的响应，而另外一个则不发生碰撞。随机扰动可以使得系统的响应从一个极限环变为一扩散的极限环。数值模拟表明本文提出的方法是有效的。

Response of Van Der Pol-Duffing One-sided Constraint Oscillator to Combined Deterministic Harmonic and Random Excitation

The resonance of single degree of freedom Van der Pol-Duffing oscillator impact oscillator to combined deterministic and random excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the intensity of the random excitation increase, the nontrivial steady state solution may change form a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state responses, one is a non-impact response, and another is an impact one.