Natural Frequencies and Mode Shapes of Timoshenko Beams with Attachments

The Laplace transform is used to obtain a solution for a Timoshenko beam on an elastic foundation with several combinations of discrete in-span attachments and with several combinations of attachments at the boundaries. These attachments include translation and torsion springs, masses, and undamped single degree-of-freedom systems. The Laplace transform technique, which apparently has not been used previously to solve for the eigenvalues of coupled systems of equations, produces a solution in which the boundary conditions can be considered independently of the number and type of in-span attachments. Taking advantage of this independence, many specific combinations of in-span attachments and boundary attachments are examined. It is shown that the Laplace transform method removes some of the drawbacks of the most frequently used methods used to solve Timoshenko beams with in-span attachments: beam partitioning, Green's functions, and Lagrange multipliers. Excellent agreement is found upon comparing previously reported results for a wide range of boundary conditions and attachments. In addition, several sets of new results are given.