Transcendental Mode Shape Development for Vibration Analysis of a Pinned–Pinned Double-Overhung Beam with End Lumped Masses

Abstract

This study presents a transcendental mode shape formulation for the vibration analysis of a pinned–pinned double-overhung beam carrying lumped masses at both free overhung ends. The beam is modeled as a slender Euler–Bernoulli beam, with the effects of shear deformation and rotary inertia neglected. The governing differential equation for free transverse vibration is solved analytically by enforcing boundary conditions at the pinned supports, along with continuity and equilibrium conditions at the locations of the lumped masses. This procedure leads to the derivation of transcendental frequency equations and the corresponding mode shape functions. The first three bending natural frequencies obtained from the analytical solution are 4.9396 Hz, 7.016 Hz, and 71.9266 Hz, respectively. To validate the analytical formulation, numerical simulations are carried out using a finite element–based model. The corresponding natural frequencies from the numerical analysis are 7.4583 Hz, 7.0189 Hz, and 81.537 Hz. A close agreement between the analytical and numerical results is observed, with minor deviations attributed to modeling assumptions and numerical discretization effects. The proposed analytical formulation provides an effective framework for the vibration analysis of double-overhung beams with lumped masses and can serve as a reliable reference for further analytical and numerical investigations in beam dynamics.