高维非线性转子系统周期分岔解的数值计算

利用Poincare映射原理,提出了求高维非线性系统周期解及其分岔的方法．将从初值至稳态解的整个积分长度分成若干积分子段,设定每个积分子段中的最大循环数,并使周期数按一定规律增加．在每个子段中应用直接积分法求解,根据Poincare截面上映射点的距离判断周期解的收敛精度．由于每个积分子段中的周期数是递增的, 故求周期解所用的总积分长度趋于最小,从而耗时较少．同时,通过对Poincare映射数据矩阵中的元素排序、差分和筛选,可以计算出周期分岔解的周期数以及周期解的分岔点．应用该方法计算了2个非线性转子模型的周期分岔解:一个是考虑非线性油膜力和非线性内阻力作用的4DOF单跨转子,发现由于油膜失稳可导致内阻失稳;另一个是考虑非线性油膜力作用的16DOF双跨转子,发现了双跨转子系统失稳后的双低频现象．

Calculations of the Periodic Solution and Its Bifurcations of High-Dimensional Nonlinear Rotor Systems

A method determining the periodic solution and the corresponding period number of high-dimensional nonlinear systems is proposed, based on investigation of the numerical solution of the periodic responses and their bifurcations by using the Poincar6 Map technique. In this method, the whole integrated time length, from the beginning to the end when the steady state is obtained, is divided into many segments. The maximum number of integral cycle for every segment is set in advance. The enacted period number of every segment will increase in a certain rule. The ODEs are then numerically simulated within every integral segment. At the same time, the Poincare Map is adopted to acquire a series of mapping points. The distance between any two sequential points is used to judge the convergence of the periodic solution. Calculation will be carried out in the following integral segment until the converging precision is satisfied. Because the period number is increased along with the integral segments, the total integral length of time is effectively minimized. So the presented numerical method is less time-consuming. Simultaneously, the period number of the solution can be determined through the procedures of sorting, difference and filtration to the elements of mapped matrix of solution. The bifurcation points can also be calculated accurately according to the period numbers. This method is applied to two nonlinear rotor systems to calculate the periodic bifurcation solutions. The results show that for the 4DOF Jeffcott rotor with nonlinear oil film forces and nonlinear internal damping, the internal damping instability can be induced by the oil instability. For the 16DOF one that consists of two shafts and with nonlinear oil film forces only, there exists two whirl/whip frequencies in spectra of response simultaneously after oil instability.