Distribution of the Vibration Amplitude and of the Energy Density over Complex Vibratory Systems

Abstract

The geometric mean between the maxima and the minima of the frequency curve of the velocity amplitude of a vibrator is independent of the damping, and equal to the amplitude of the wave that propagates from the driver to the boundaries. It is described by the characteristic admittance and is a simple function of the mass of the system and the narrow-band average of the frequency difference between successive resonances. Discontinuities in the density function of the resonances generate an imaginary part that exceeds the real part only if the forced frequency is close to the frequency of the discontinuity. The characteristic admittance is independent of the energy dissipation; in contrast, the energy density is a function of the energy dissipation. The energy density is composed of that of the outgoing wave and that of the reverberant vibration field; it can be computed by methods standard in reverberation theory. Measurements with noise bands always lead to a determination of the energy density, because of the incoherence of the phases of the noise components. The theoretical predictions are compared with experimental results for plates and cylindrical shells.