陀螺系统与反对称矩阵辛本征解的计算

Computation of Gyroscopic System and the Symplectic Eigensolution of Anti-Symmetric Matrix

用状态向量法,引出陀螺线性系统的广义本征问题,证明了本征向量之间的加权共轭辛正交关系,以及用本征向量对任意状态向量的展开定理。运用反对称矩阵胞块组成的LDL^T分解,将本征方程导向辛本征问题的标准型。这套方法适用于陀螺系统K阵不正定的情形。对于辛本征问题用SH变换将矩阵化为半边三对角线胞块阵或三对角线胞块阵,然后再求解其全部本征解。为陀螺系统的模态分析打下了基础。

Based on the state vector method, the generalized eigenproblem of linear gyroscopic system, which can be the case of non-positive definite stiffness matrix K,is derived. The weighted adjoint symplectic orthogonality between the eigenvectors and the expansion theorem for an arbitrary state vector are proved. By applying the cell LDLT decomposition for an anti-symmetric matrix, the generalized eigenproblem is reduced to the standard form of a symplectic eigenproblem of an antisymmetric matrix.The symplectic eigenproblem of an anti-symmetric matrix is tranformed to the form of cell semi-tridiagonal matrix by means of the orthogonal SH transformation, and then all its eigensolutions are solved, which gives the foundation of modal analysis for the gyroscopic system.