转子进动定理的扩充和完善

Improvement of Theorems on Precessional Motion of Rotors

为更深入地揭示转子进动的轨迹特征,提出了关于转子进动轨迹的2个定理。定理1表明,转子进动的任一阶轨迹分量所围面积可直接由同阶频率成分构成的轨迹方程系数矩阵行列式求得。定理2证明,转子进动轨迹矢径在单位时间内所扫过的面积是恒定的,而与起始点无关。同时,提出了欧拉复向量的内积法则,即2个欧拉复向量的内积等于2个复向量的幅值与欧拉角之差的余弦三者之积,且符合交换律。利用2个定理和内积法则对转子进动理论进行了补充和完善。结果表明,在转子的运动中,作用力所做的功关于转子的进动是正交的。所得结论对于揭示转子的进动规律和机理、诊断转子故障具有参考价值。

In order to reveal the features of precessional motion of rotors,two theorems on the precessional orbits of rotors are developed.Theorem 1 shows that the area surrounded by an elliptical orbit of rotor precession can directly be obtained in terms of the determinant of matrix constituted by components of rotor precession at the same frequency.Theorem 2 stats that within a same time the precessional vector of a rotor running at constant speed sweeps always a same area,regardless of the origin of the time,that means,the time rate of change of area swept by precessional vector is constant.Furthermore a new operation of inner product of complex vectors is suggested,that is,the inner product of two complex vectors in Euler-form is equal to the product of two magnitudes with cosine of their phase angles.In terms of both theorems and the new operation,the theory of precessional motion of rotors is improved and extended.The result of this paper is an interesting improvement for rotor dynamics.