Abstract
Fabry-Perot resonator (FPR) is an important concept for the study of optical and microwave systems. But it is a relatively alien concept for mechanical engineers. This paper reveals that a lossless bar (i.e. a bar whose Young’s modulus is a real number) undergoing longitudinal vibrations is essentially a Fabry-Perot resonator. First, the harmonic response of a longitudinal bar with generic mass-damper-spring (MDS) boundaries is derived to show that it is identical to the Airy function of an optical FPR. The derivation confirms that the resonance of a longitudinal bar is due to the superposition (i.e. interference) of the longitudinal waves travelling back and forth between the two boundaries. Next, the transmission and phase shifts at the MDS boundaries are derived. Based on which, the resonant frequencies of the longitudinal bar with different boundary conditions are determined directly from the phase matching condition. A perfect agreement with those obtained from conventional approaches was achieved. Finally, the concepts of whitelight interference (WLI) and fringe analysis are applied to determine the bar length from the calculated frequency response function (FRF). The results presented demonstrate that the data processing algorithms developed for optical and microwave FPRs are directly transferable to the study of longitudinal bars. This work lays the theoretical foundation for future developments of mechanical FPR sensors.
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Published Version
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