Symplectic Eigensolutions Research Articles

Symplectic analysis for periodical electro-magnetic waveguides

Symplectic analysis is introduced into electro-magnetic waveguide theory, by using Hamiltonian system theory in which the transverse electric and magnetic field vectors are the dual vectors. The method can accommodate arbitrary anisotropic material and includes the interface conditions between adjacent segments of the waveguide. An electro-magnetic stiffness matrix is introduced which relates to the two ends of each segment of the waveguide. Both the pass- and stop-band stiffness matrices for plane waveguides with constant cross-section are given analytically and also a transformation matrix is given to permit abrupt changes of cross-section to occur. The variational principle is applied to obtain the segment combination algorithm needed to generate the electro-magnetic stiffness matrix related to the two ends of the fundamental periodical segment. Then the Wittrick–Williams algorithm is used to extract the eigenvalues. Thereafter, an energy band analysis is performed for a periodical waveguide, e.g., a grating, by using the symplectic eigensolutions.

Localization investigation of stationary random wave transmission along damped ordered substructural chains

Based on the symplectic eigensolution and the pseudo-excitation method, the localization of structural chains subjected to stationary random point excitation is investigated. The stiffness, mass and the hysteretic damping distributions are all continuously variable. The cause of the displacement and internal force localization is explained in terms of the symplectic eigenvalues and eigenvectors of a typical substructure of the structural chain. When duplicate symplectic eigenvalues occur, the degeneration of the stopband as well as the corresponding degeneration of the mode localization phenomenon are discussed.

Computation of gyroscopic systems and symplectic eigensolutions 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.