首系数奇异的二阶系统解耦

针对首系数奇异的二阶系统的解耦问题,提出了一种奇异二阶系统的同谱构造解耦方法。引入了对逆二阶系统,将难以描述的二阶系统无穷大特征值转化为其对逆二阶系统的零特征值来加以描述,分析了奇异二阶系统可对角化应该满足的条件及其相应的证明,并且给出了一种具体的奇异二阶系统的同谱构造解耦方法,从而实现了奇异二阶系统解耦,即将奇异二阶系统的3个参数矩阵同时对角化。数值试验中将一个三自由度的奇异质量弹簧二阶系统解耦成3个无关的单自由度二阶子系统,而且解耦前后系统不仅保持了相同的有限特征值及其重复度结构,还保持了相同的无限特征值及其重复度结构。工程应用前景十分广泛。

Research of Quadratic System Decoupling with Singular Leading Coefficient

In order to solve the quadratic system decoupling problem with singular leading coefficient,a constructive method for decoupling the singular quadratic system is proposed.A reverse quadratic pencil is introduced to describe eigenvalue at infinity of quadratic pencil which is transformed into zero eigenvalue of the reverse quadratic pencil.Some necessary and sufficient conditions on decouplization are established.An isospectral constructive method for decoupling the singular quadratic system is also proposed to implement the proposed theoretical results.Finally,some numerical experiments on reducing a three-degree-of-freedom singular mass-spring quadratic system to a three totally independent single-degree-of-freedom second-order subsystems are reported.It has been observed that the decoupled system sharing the same finite eigenvalue structure and the finite eigenvalue structure with original system.