Abstract: In this paper, we will compute the transfer matrices to find the eigenfrequenciesfor the vibrations of the general non-collinear Euler-Bernoulli or Timoshenko beamstructure with dissipative joints. We will allow the structure to be three dimensional,and thus we must consider all types of vibrations simulaneously, including longitudinaland torsional vibrations. The general structure considered will consist of any number ofbeams joined end to end to form a chain. Many, different kinds of dampers areallowed, even within the same structure. We also will allow different materials withinthe structure as well as different beam widths. We then will show. that asymptotic estimates can be used to find the eigenfrequencies approximately.