二类复特征值表达式的适用性研究

RESEARCHES ON TWO EXPRESSIONS ABOUT COMPLEX EIGENVALUE

对于线性系统，根据其质量、刚度或阻尼等矩阵的性质了解特征值，是十分有意义的问题．对于有阻尼陀螺系统，本文首先通过模态正交性推导了有关矩阵的若干关系式．然后分析了复特征值的两类表达式的适用性，着重讨论了通过一元二次方程研究特征值性质的现行处理方法存在的困难．提出一种确定特征值的补充方法，指出可能存在虚数特征值的系统的性质．

It is very interesting to study eigenvalues of a linear system by analyzing the properties of its mass matrix, stiffness matrix, damping matrix and so on. After a brief review on the complex mode theory, this paper developed a new set equations about corresponding matrices by recalling the orthogonality between the complex modes of a damped gyroscopic linear system. Based upon this, the applicability of two popular expressions about complex eigenvalues was investigated. The emphasis was given to the questionable aspects of one expression. The expression is a one variable two order algebraic equation and had been used to reflect the complex eigenvalues of a linear system qualitatively. A complementary approach to determine the realistic eigenvalues was then put forward. The systems were pointed out, which possibly possess imaginary eigenvalues. Examples show that the conclusions of present paper are all right.