Traveling and standing waves in structures with damped boundaries

The paper considers the free response of second order flexible structures with viscous damping boundary conditions, from wave motion point of view. The problem is solved using infinite dimension Laplace transfer functions. Our previous research viewed the solution as traveling waves that are reflected from the boundary. This led also for a generalization of the D'Alembert principle to finite length non-conservative systems. In this paper we focus on obtaining an explicit infinite series solution based on separation of variables. In conservative systems the approach is well established with closed form modal solutions. On the other hand, state of the art results for the damped case still fall short of providing a complete solution for the series, in particular its coefficients. The paper presents the full explicit modal solution of a free response of damped structures, which consists of a sum of uniformly decaying standing waves. It is also shown how the traveling waves can be identified from the series.