Hopf分岔的代数判据及其在车辆动力学中的应用  

AN ALGORITHM CRTERION FOR HOPF BIFURCATION AND ITS APPLICATIONS IN VEHICLE DYNAMICS

作　　者：张继业[1] 杨翊仁[2] 曾京[1] 

机构地区：[1]西南交通大学牵引动力国家重点实验室,成都610031 [2]西南交通大学应用力学与工程系,成都610031

出　　处：《力学学报》2000年第5期596-605,共10页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金!(19672052);西南交通大学牵引动力国家重点实验室开放基金&&

摘　　要：利用Ｈｕｒｗｉｔｚ行列式，给出平衡点失稳而发生Ｈｏｐｆ分岔的代数判定准则和计算方法，这一方法将Ｈｏｐｆ分岔点的求解转化为一个非线性方程的求解问题，从而克服了以前方法在计算Ｈｏｐｆ分岔点时，对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量．应用这一方法，我们进行了非线性车辆系统蛇行运动稳定性的研究，得到了轮对系统发生蛇行运动的临界速度的解析表达式．The stability of the railway vehicle system is an important dynamic problem for determining the maximum operating speed of vehicle on tracks. Instability should be avoided for any vehicle in its normal operating speed range otherwise severe hunting oscillations may occur and thus worsen the dynamic performance of the vehicle or even cause derailment. Therefore, stability problems especially for high-speed passenger cars have been paid much attention to by researchers. When the vehicle speed increases beyond a certain limit called critical speed, the steady motion loses its stability and hunting starts. The hunting is a periodic motion that bifurcates from the steady motion and can be described by Hopf bifurcation theory. The Critical speed happens to correspond to Hopf bifurcation point. The traditional Hopf bifurcation criterion is stated in terms of eigenvalues of Jacobian matrix. That is to say, finding the Hopf bifurcation point is by solving the characteristic equation and judging when a pair of complex conjugate pass through the imaginary axis while all other eigenvalues have negative real parts. The method turns out difficult and tedious. Even numerical computation of eigenvalues is feasible, it is more ideal to have a criterion stated in terms of the coefficients of the characteristic equations. The main objective of this paper is to establish a method for searching the Hopf bifurcation point and hunting motion. Using the Hurwitz determinants, an algebraic criterion and corresponding computational method for determining the Hopf bifurcation point are proposed. The searching of Hopf bifurcation point is now equivalent to solving a nonlinear algebraic equation for bifurcation parameter. If the bifurcation value is exactly obtained this a pair of the purely imaginary eigenvalues and the Hopf bifurcation are determined by this approach. This method does not need to calculate all the eigenvalues of Jacobian matrix of the system for any parameter and saves computer time demand. Using the method, the critical sp

关 键 词：车辆系统 HOPF分岔 蛇行运动 机车 铁路 车辆动力学 代数判据 

分 类 号：U260.11[机械工程—车辆工程]