Extended D'Alembert solution of finite length second order flexible structures with damped boundaries

The paper considers the problem of deriving the exact response to initial conditions of flexible structures governed by the wave equation with boundary conditions of pure damping. The aim is to develop a wave oriented solution for this non-conservative case that is hardly considered in classical vibration theory. The celebrated D'Alembert approach, which applies to systems of infinite length, is extended to systems with finite medium and non-conservative boundary conditions. Displacement and velocity responses of the structure are developed in terms of propagating waves with decreasing amplitude. It is shown that additional waves exist as a result of non-zero initial displacement at the ends. An equivalent infinite structure and its corresponding initial conditions are then defined so that the solution is given in a D'Alembert like fashion, using single progressive and regressive waves.