The modes of non-honogeneous damped beams

This short note is concerned with computing the eigenvalues and eigenfunctions of a continuous beam model with damping, using the separation of variables approach. The beam considered has different stiffness, damping and mass properties in each of two parts. Although applications are not considered in detail, one possible example is a thin beam partly submerged in a fluid. The fluid would add considerable damping and mass to the beam structure, and possibly some stiffness. Both the overdamped and the underdamped eigenvalues and associated eigenfunctions have been computed for two different sets of parameters. For high damping the lower underdamped modes seem to be local to the undamped part of the beam. The procedure given assumed the boundary conditions to be pinned. In the general case the spatial solution would require four unknown parameters for each beam section, and the boundary conditions equivalent to pinned-pinned could not be incorporated explicitly. The result would a search for a zero determinant of an 8x8 matrix rather than a 4x4.