一类半整数自由度级的非线性共振及其在电力系统中的应用

本部分讨论下列两自由度实系统:(?) (1)其中“·”代表 d/(dt),λ_1>0,λ_2>0,ε为正小参数,k_1(t) 和 k_2(t) 是周期为(2x)/v 的周期函数.同时假设,f_1(x),f_2(x),k_1(t),k_2(t)都是足够光滑的.系统(1)包括双机电力系统为其特例.在§4中将叙述本文的结果在双机电力系统上的应用.本文关心的问题是在状态 x_1和 x_2之间,是否会发生周期与 k_1(t),k_2(t) 的周期相同的振动现象,以及具体的计算方法.为此,需要有 x 的方程.在 λ_1=λ_2时问题特别简单,

THE NONLINEAR RESONANCE OF A CLASS OF HALFINTEGER DEGREE OF FREEDOM AND ITS APPLICATION IN POWER SYSTEMS

This part considers the nonlinear resonance problem for a class of two degrees of free-dom systems,including the two-machine power system.The main feature of this class of sy-stems is that the state variables appearing in the equations are only in the form of relative dif-ferences and their derivatives.By means of nonlinear perturbation methods,especially Friedrich'smethod,it is discovered that there are one and a half degree of freedom resonant phenomenahidden in these systems.The main conditions which cause the resonance are obtained.A directcomputation method is presented for the backbone curve and maximum resonance amplitude andis performed for near equilibrium situation.The applications of the above results to the two-machine power system are discussed and compared with previous results on this problem.