Various standing waves in a Timoshenko beam

Abstract

Vibration modes of a finite-length Timoshenko beam are studied as standing waves using the wave-train closure principle, in order to obtain a complete picture of various vibration modes in a beam and to understand the mechanism of their formulations. In particular, the existence of degenerate modes in a beam is investigated. Firstly, it is shown that the two degenerate flexural waves accommodated by an infinite Timoshenko beam are derived from the in-phase and the out-of-phase relations between transverse vibrations due to bending and shear deformations, respectively. A wave representation of beam vibration is thus developed. Secondly, wave reflection behavior at an elastically supported boundary is analyzed. It is shown that while these two waves are degenerate in an infinite beam, they have to be superposed at the boundary in general, but remain degenerate for certain special boundary conditions. Based on these results, the expression of wave-train closure principle for a finite-length Timoshenko beam is derived, and used to study different standing waves in a beam. It is shown that three types of standing waves (vibration modes) exist in Timoshenko beams, namely, superposed, degenerate, and single. A condition of space synchronization must be satisfied for superposed standing waves. For the other two types of standing waves, this condition is satisfied naturally. While the superposed standing wave is the most general form of vibration mode, vibration modes of elastically supported beams at specific frequencies or beams with sliding and/or simply supported boundary conditions are single standing waves. When additional conditions are satisfied, two single standing waves could exist at the same natural frequency to formulate degenerate standing waves.