A general approach to formulating the frequency equation for a beam carrying miscellaneous attachments

Beams carrying various lumped attachments have been studied extensively over the years. In this paper, a simple approach is proposed that can be used to readily determine the eigenvalues of an arbitrarily supported single-span or multi-span beam carrying any combination of lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, an undamped or damped oscillator with or without a rigid body degree of freedom. Rather than solving a generalized eigenvalue problem to obtain the eigenvalues of the system, a frequency equation is formulated instead whose solution can be easily solved either numerically or graphically. The proposed scheme is easy to code, and can be easily modified to accommodate beams with arbitrary supports and carrying any number of miscellaneous lumped attachments.