转子支承系统频率方程伴随矩阵的平衡变换

Balancing Transformation on the Companion Matrix of RotorBearings System's Frequencies Equation

传递矩阵－多项式方法在转子支承系统动力特性分析中得到了成功的应用，但对于超柔性超高速转子的数值计算困难依然存在。为获得转子支承系统的特征频率多项式，需引入比例换算、高次项缩减的计算技巧；求解频率方程时采用求特征多项式伴随矩阵全部特征值的方法。应用矩阵相似变换理论，对矩阵进行了一系列相似变换的平衡过程，大大减轻了伴随矩阵Ｅｕｃｌｉｄ范数大、各元素量级相差巨大的数值病态特征。一个典型的数值算例表明了平衡变换的重要性。

The transfer matrixpolynomial method is successfully used in finding the stability threshold and damped critical speeds of rotorbearing systems, but the numerical difficulties still exist in computation for ultraflexible and ultrahighspeed rotors. Its necessary to take the scaling multiple process and the polynomial order reduction technique for getting the characteristic polynomial of rotorbearings systems. In the programming for finding the eigenvalues of the companion matrix of the characteristic polynomial equation, a new balancing transformation technique is developed. Balancing transformation on the companion matrix can greatly reduce its numerical illconditioned characteristics including huge Euclid norm and great differences among matrix elements. A numerical example validates the importance of balancing transformation technique. Two other examples are presented to confirm the analysis and programs in the case of actual rotorbearing systems.