Computation of Gyroscopic Systems and Symplectic Eigenproblems of Skew-symmetric Matrices

The generalized eigenproblem of a linear gyroscopic system with a non-positive definitive stiffness matrix K is derived based on the state vector method. The expansion theorem for an arbitrary state vector and the weighted adjoint symplectic orthogonality between eigenvectors are also proved in the paper. By using the symplectic LDL^T factorization, the generalized eigenproblem is reduced to the standard form of symplectic eigenproblem of skew-symmetric matrices. The symplectic orthogonal transformations are then used to tri-diagonalize the coefficient matrix for the solutions of eigenvalues and eigenvectors which is the central part of modal analysis for gyroscopic systems.