Modal representation of second order flexible structures with damped boundaries

The problem of obtaining a modal (i.e., infinite series) solution of second order flexible structures with viscous damping boundary conditions is considered. In conservative boundary systems, separation of variables is well established and there exist closed form modal solutions. However, no counterpart results exist for the damped boundary case and previous publications fall short of providing a complete solution for the series, in particular, its coefficients. The paper presents the free response of damped boundary structures to general initial conditions in the form of an infinite sum of products of spatial and time functions. The problem is attended via Laplace domain approach, and explicit expressions for the series components and coefficients are derived. The modal approach is useful in finite dimension modeling, since it provides a convenient framework for truncation. It is shown via examples that often few modes suffice for approximation with good accuracy.