Superposed standing waves in a Timoshenko beam

Abstract

A wave mechanics approach is used to solve the Timoshenko beam equation, revealing that two waves exist. One is called the s a –wave and the other the s b –wave. These two waves are found to be the basic constituent components of the mode shapes of the beam. An experiment was carried out and the measured mode shapes of a free–free beam are shown to consist of one s a –wave and one s b –wave in superposition for each of the modes. The measured s a –wave and s b –wave exhibit the Rayleigh–Lamb first (with anomalous dispersion) and Rayleigh–Lamb second (with normal dispersion) flexural modes, respectively. The issue of the second spectrum is addressed and it is shown that, within the measurable frequency range, Rayleigh–Lamb second flexural modes are present in the free–free beam. The s b –wave is identified as the second–spectrum mode. The role of shear deformation is also investigated in explaining the basic difference in the behaviour of the s a – and s b –waves. This paper also contributes to a physical interpretation of the hyperbolic functions in the classical solution of beam vibration problems.