Closed-Form Transient Response of One-Dimensional Continua

Exact and closed-form transient response of general one-dimensional distributed dynamic systems subject to arbitrary external, initial and boundary disturbances is determined. Non-self-adjoint operators characterizing damping, gyroscopic and circulatory effects, and eigenvalue-dependent boundary conditions are considered. Through introduction of augmented operators, a closed-form modal expansion of the displacement and internal forces of the distributed system is derived. The eigenfunction expansion is realized in a spatial state-space formulation, which systematically yields exact eigensolutions, eigenfunction normalization coefficients, and modal coordinates. The proposed method is illustrated on a cantilever beam with end mass, damper and spring.