Closed-Form Solution for the Eigenmodes of Euler-Bernoulli Beams in Pinned-Pinned-Free Configuration

Abstract

Euler-Bernoulli beams are commonly used elements in modeling the dynamics of mechanical structures. For static boundary conditions, the beam dynamics are well understood and analytical expressions for the normalized eigenmodes and characteristic equations are available. Until now, cantilever beams are modeled as a series connection of two general beams and solved numerically. This paper provides an analytical solution for the normalized eigenmodes of cantilever beams with inner pinned support at an arbitrary position. The eigenfrequency can be computed by finding the roots of an analytically given characteristic equation. Using a provided solution interval for each root, it is possible to compute each eigenfrequency within a fixed number of iterations for any given precision in a deterministic way.