Resonant beam vibration: A wave evolution analysis

In an attempt to verify the concept of wave-mode duality and to understand the wave-train closure principle in resonant beam vibration, an elastically supported beam subjected to a swept sinusoidal force is investigated through a simulation of wave evolution from initial to final steady-state response. Wave evolution is simulated in the superposition and the degenerate states according to the nature of wave reflection at the boundaries of the beam. From the results, resonance condition is established, and the natural frequencies and mode shapes are calculated. Using a simply supported beam and a cantilever beam as examples, the concept of wave-mode duality at resonance is verified. An examination of the wave-train closure principle shows that, although the frequency equations obtained by using the principle are identical to those derived from the wave evolution results, the principle does not describe the realistic process of the formulation of vibration modes in general. The waves described in the wave-train closure principle are not physical but “virtual” waves that represent final steady-state waves, which can only be asymptotically approached in reality. The principle can be used as a convenient technique to obtain vibration modes from the point of view of wave propagation.