Free vibration of a uniform beam with multiple elastically mounted two-degree-of-freedom systems

The free vibration of a beam with one or more elastically mounted two-degree-of-freedom systems that translate and rotate is considered in this note. The assumed-modes method is applied to formulate the equations of motion, and the natural frequencies of the system are found by solving for the roots of a given characteristic determinant. If the number of attached spring–mass systems is small, one can exploit the Sherman–Morrison–Woodbury determinant formula and reduce the characteristic determinant to one of smaller size, which will be easier to code and more computationally efficient to solve.