奇异摄动方法在输电线非线性振动问题中的应用

Singular Perturbation Method and Its Application in Nonlinear Systems with Fast and Slow Variables Coupling

同一系统内部快变量和慢变量的同时存在往往引发相异于一般系统的特殊效应,比如输电线的松弛振荡。本文推导了架空输电线具有初始垂度的非线性动力学模型,发现该模型是具有快慢变量耦合的数学模型,应用求解周期运动的奇异摄动方法,得到系统的近似解析解,考察了快慢变量对系统周期运动的影响规律。结果表明解析解较数值解略微偏小,但仍有很好的吻合度,说明本文结果的有效性和正确性。进一步计算表明,随着摄动方法应用过程中近似次数的增加,两解逐次接近。

The co-existence of different scale variables in one system may lead to special effects different from the general control systems, such as the relaxation oscillation. A nonlinear dynamical equation was deduced to model the nonlinear vibration of a suspended transmission line with initial degree relaxation. It is found that there exist some fast and slow variables coupling in the obtained model. Thus, the singular perturbation method was employed. An approximate periodic solution was obtained analytically to observe the effect of the fast and slow variables on the periodic motion of the system under consideration. The results show the analytical soultion is in a good agreement with that derived from the numerical simulation, though comparing to be a little smaller. It aleo presents that the provided result is valid and correct. Further computation shows that with the increase in the order of approximation, those two solutions approach gradually.