有界噪声激励下MATHIEU-DUFFING方程的稳定性

Approximate Solution of Mathieu-Duffing Equation with Bounded Noise Excitation

研究了M ATH IEU-DU FFING方程在随机激励下的主共振响应和系统的稳定性问题,用多尺度法分离了系统的快变项,讨论了非线性项、随机项对系统的影响。求出了随机M ATH IEU-DU FFING系统的不变测度和最大LYAPUNOV指数,由最大LYAPUNOV指数得到系统的零解几乎必然稳定的充要条件。利用摄动法研究了系统的非零稳态响应,并进行了数值模拟。

Mathieu-Duffing equation has been and is still used for parametric nonlinear vibrations in engineering. To our best knowledge, we have not found in the open literature any paper concerning the solution of Mathieu-Duffing equation with bounded noise excitation. We aim to obtain such solution. As is customary in vibration engineering, we use eos（Ωt＋γW（t）＋δ） as the bounded noise excitation term in Mathieu-Duffing equation, where W（t） represents the Wiener process, γ represents the intensity of noise, and δ represents uniformly distributed stochastic variable in the interval （0,2π）. In the full paper, we explain our solution in much detail; here we give only a briefing. We obtain an approximate theoretical solution with the method of multiple scales; this approximate theoretical solution is quite good as the nondimensional coefficients in the Mathieu-Duffing equation are very small quantities. We also obtain solution with perturbation method, in which reasonable numerical values of the coefficients in Mathieu-Duffing equation need to be and can be easily assumed. The numerical solution agrees quite well with the approximate theoretical solution. Just how good is the agreement can be seen in Fig. 2 of the full paper for the following values of the quantities in Mathieu-Duffing equation： h = 0. 1, ω0= 1.0, γ= 0. 5, β=0. 2, α = 0. 1 （ε〈〈1, so coefficients εh,εβ, and εα are very small; ω0^2 is the natural frequency, and γ is the intensity of noise）. This agreement shows preliminarily that our approximate theoretical solution obtained with the method of multiple scales is reliable.