谐和与随机噪声联合参数激励下Mathieu系统的矩稳定性

Moment Stability of Damped Mathieu Oscillator to Combined Harmonic and Random Noise Parametric Excitation

研究了确定性谐和与随机噪声联合参数激励下Mathieu系统的矩稳定性问题。通过适当的坐标变换和随机平均法,将系统转化为一阶线性伊藤随机微分方程组。利用伊藤法则给出了系统一、二阶矩满足的常微分方程,根据微分方程的稳定性理论得到了系统一阶矩稳定充分必要条件的解析表达式和二阶矩稳定充分必要条件的数值算法,并对理论结果用数值方法进行了仿真计算。理论和数值结果表明,无论是相对于一阶矩还是二阶矩的稳定性,随着随机噪声强度变大、确定性谐和激励振幅变大,系统的稳定性区域变小从而变得不稳定。而当调谐参数趋于零系统达到参数主共振情形时,系统的稳定性区域变得最小。当随机噪声强度逐渐变小趋于零时,由二种矩稳定性给出的稳定性区域变得一致。

The moment stability of a damped Mathieu oscillator to combined deterministic harmonic and random noise of the form of a stationary Gaussian process parametric excitation is investigated. The analysis is based on a suitable coordinate transformation and stochastic averaging method, which reduces the system to two linear Ito＇s stochastic differential equations. By using the Ito＇s differential rule, differential equations ruling the time evolution of the first and second order response moments are obtained. The necessary and sufficient conditions of stability for the first and second order moments are that the matrix of the coefficients of the differential equations ruling the moments have complex eigenvalues with negative real parts. The analytical expression of the stability condition of the first order moment is obtained, while resuits of the second order moment stability are given numerically. Some numerical simulations and graphs are presented for representative cases. It is founded that, when the intensity of the random noise and the amplitude of the deterministic harmonic excitation increase, the stability regions will reduce whether for the first order moment or the second order moment stability. The stability regions will reduce to the minimum value if the detuning parameter tend to zero. The stability regions based on different order moments will become identical when the intensity of the random noise increases to zero.