A note on analytical solutions for vibrations of beams with an attached large mass

In this work, we investigate analytical solutions for beams with sizeable mass attachments under externally induced base motions. Specifically, a point arises because the boundary conditions involving the presence of a large mass attached to the beam complicates the underlying eigenvalue problem and a straightforward separation of variables in the governing differential equation of motion becomes problematic. In order to address this point, as well as accuracy issues, three analytical methods of solution are presented, where the selection of eigenfunctions for the dependent variable representation is discussed, as well as the appropriate normality rules for the ensuing modal analysis. Next, a numerical example is presented for a cantilevered pylon undergoing axial vibrations due to sinusoidal base motions, where the accuracy and convergence of the aforementioned methods of solution when the magnitude of the attached mass comparable to that of the supporting pylon is investigated. In closing, we note that these issues would be absent if the supported mass was small and could have been considered as a secondary system.