Wave Problems for Repetitive Structures and Symplectic Mathematics

Abstract

Based on the analogy between structural mechanics and optimal control theory, the eigensolutions of a symplectic matrix, the adjoint symplectic ortho-normalization relation and the eigenvector expansion method are introduced into the wave propagation theory for sub-structural chain-type structures, such as space structures, composite material and turbine blades. The positive and reverse algebraic Riccati equations are derived, for which the solution matrices are closely related to the power flow along the sub-structural chain. The power flow orthogonality relation for various eigenvectors is proved, and the energy conservation result is also proved for wave scattering problems.