A Comprehensive Analytical Dynamic Model of a T-Beam

Abstract

This paper derives a comprehensive analytical dynamic model of a T-shaped beam that includes in-plane and outof-plane vibrations for mid-frequency range analysis, defined here as approximately 1 kHz to 10 kHz. The web, right part of the flange, and left part of the flange of the T-beam are modelled independently with two-dimensional elasticity equations for the in-plane motion and the classical flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions, which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to solutions from Bickford beam theory and finite element analysis. Higher order branch waves are discussed and a simplified symmetric model is presented.