Sampling statistics for vibrating rectangular plates

Abstract

A thin, uniform, rectangular, elastic plate is freely supported in air, and driven so that it vibrates flexurally at a resonance frequency. The modal function, which describes the vibrational displacement over the surface of the plate, is approximated as the product of two sine functions. From this modal function the probability density function (pdf), P(s), for random sampling over the plate surface of the rms acceleration, is calculated to be 4π−2 K(1−s2), where K is a complete elliptic integral. Using this expression and experimental values sampled randomly over the plate surface, estimates of known precision can be made of the mean vibrational level of the plate. Experimental results are given for a steel plate measuring 4 ft×3 ft×1/4 in., resonating at a single frequency; they agree quite well with the theoretical cumulative function derived from the pdf given above. Also considered are cases where two overlapping modes are excited at a given frequency. The corresponding pdfs are found, by Monte Carlo calculations, for five different values of phase difference between two modes excited with equal amplitudes. As the pdfs are known for the case where a pure tone excites many overlapping modes (exponential for the mean-square acceleration, Rayleigh for the rms value), an outline of the sampling statistics is now available for all modal densities.