谐和与随机噪声作用下二阶参激系统的Lyapunov指数

Discovering Instability Conditions for Second-Order Linear System under Combined Harmonic and Random Parametric Excitations

研究了二阶线性系统在谐和与随机噪声联合作用下的主共振响应和稳定性问题。用多尺度法分离了系统的快变项,讨论了系统的阻尼项、随机项等对系统响应的影响。用路径积分法求出了系统的稳态概率密度,从而求出了系统的最大Lyapunov指数,由最大Lyapunov指数可得系统几乎必然稳定的充分必要条件。数值模拟表明文中提出的方法是有效的。

Our method is different from that used by Namachchivaya in Ref. 5, even though the subject studied is the same. More importantly, we believe that our method can discover certain instability conditions that cannot be discovered by Namachchivaya's method. The principal resonance of a second-order stochastic oscillator under combined harmonic and random parametric excitations is investigated. We use the method of multiple scales to derive the equations of modulation of amplitude and phase. Then we analyze the effects of damping, detuning, bandwidth, and magnitudes of random excitation. We use the method of path integration to obtain the steady state probability density function of the system. We also obtain the explicit formulas for the maximum Lyapunov exponent λ. The numerical results show that the system will become unstable under one of the following three conditions: (1) when A is greater than zero due to increase of magnitudes of harmonic and 1 or random excitations; (2) when the center frequency of harmonic and random excitation is close to twice the natural frequency of the system; (3) when the bandwidth of random excitation is sufficiently narrow (what is sufficiently narrow is quantitatively determined by using the equations presented). The theoretical analyses are preliminarily confirmed by numerical results.