Development of transcendental mode shape functions of a pinned–pinned overhung beam with a lumped mass at the free overhung and an intermediate lumped mass

Abstract

This study presents the development of transcendental mode shape functions for a pinned–pinned overhung beam carrying a lumped mass at the free overhung end and an additional lumped mass located between the pinned supports. The beam is modeled as a slender Euler–Bernoulli beam, neglecting the effects of shear deformation and rotary inertia. The governing differential equation for free transverse vibration is solved analytically by imposing boundary conditions at the pinned supports and continuity and equilibrium conditions at the locations of the lumped masses, leading to the formulation of transcendental frequency equations and corresponding mode shapes. The first three bending vibration natural frequencies obtained from the analytical solution are 3.253 Hz, 13.769 Hz, and 103.329 Hz, respectively. To validate the analytical formulation, numerical simulations are performed using a finite element–based model. The corresponding simulated natural frequencies are found to be 4.167 Hz, 16.941 Hz, and 103.250 Hz.