Vibration of rectangular and circular plates: The effect of a concentrated mass.

Abstract

The problem of vibration of plates loaded with a concentrated mass has not been extensively studied. In this study therefore, the free vibration and harmonically forced vibration of rectangular and circular thin plates under various support systems carrying a single concentrated mass at various positions was studied. This study was done analytically by use of the finite element method utilizing the discrete Kirchoff triangle element. The mass was treated as lumped for both the plate elements and the concentrated mass. The convergence test was used to find the best possible discritization of the surfaces. For some few cases, the comparison of the values obtained herein with those available in literature showed this analysis to be accurate. It was found that in general the presence of the concentrated mass lowers the natural frequency of vibration. The reduction is more pronounced at high frequencies and this variation stabilizes asymptotically at high magnitude of the concentrated mass. This variation however depends on the magnitude and placement position on the plate of the concentrated mass at any particular frequency of vibration. By studying the modes of vibration of the plate at various frequencies, it was found that the concentrated mass had the effect of lowering the frequency if and only if there exists an amplitude in the modal shape at the position of the mass application. In case of forced excitation. it was found that resonance occurs if the natural frequency of the plate-mass system is lowered by the presence of the mass to a value equal to the excitation force frequency. It was also Ioud that for non-resonance excitation the effect of the concentrated mass on the amplitude of vibration of any point on the plate was minimal at low frequencies.