Dynamic effective mass of granular media and the attenuation of structure-borne sound

Abstract

We report a theoretical and experimental investigation into the fundamental physics of why loose granular media are effective deadeners of structure-borne sound. Here we demonstrate that a measurement of the effective mass,M (ω), of the granular medium is a sensitive and direct way to answer the question: What is the specific mechanism whereby acoustic energy is transformed into heat? Specifically, we apply this understanding to the case of the flexural resonances of a rectangular bar with a grain-filled cavity within it. The pore space in the granular medium is air of varying humidity. The dominant features ofM (ω) are a sharp resonance and a broad background, which we analyze within the context of simple models. We find that: a) On a fundamental level, dampening of acoustic modes is dominated by adsorbed films of water at grain-grain contacts, not by global viscous dampening or by attenuation within the grains. b) These systems may be understood, qualitatively, in terms of a height-dependent and diameter-dependent effective sound speed [∼ 100 − 300 (m · s −1 )] and an effective viscosity [∼ 5 × 10 4 Poise]. c) There is an acoustic Janssen effect in the sense that, at any frequency, and depending on the method of sample preparation, approximately one-half of the effective mass is borne by the side walls of the cavity and one-half by the bottom. d) There is a monotonically increasing effect of humidity on the dampening of the fundamental resonance within the granular medium which translates to a non-monotonic, but predictable, variation of dampening within the grain-loaded bar.

Acoustical properties of double porosity granular materials

“…50 A similar shift has been documented in loudspeakers 51 partially filled and resonators 52 fully filled with activated carbon. This effect has been however attributed to sorption processes that occur in the small pores within the grains.…”

Effect of granular media on the vibrational response of a resonant structure: Theory and experiment

“…5 Here, we briefly describe the technique we use to measure M ͑͒. Then, we outline the concomitant flex-bar measurements.…”

“…Our approach is to take M c to be a constant, equal to its average value. 5 This value is slightly larger than the inertial mass of the cup reflecting the fact that some fraction of the force gauge is moving with the cup. Because the motion of the grains inside the cavity is partly in phase and partly out of phase with the drive, M ͑͒ is complex valued and frequency dependent.…”

“…Previously we have established that the measured effective mass of the empty cup is nearly constant as the frequency is varied. 5 Although one would expect M c to be a frequency independent constant equal to the inertial mass of the cup, the connectors between the cup and the force gauge and between the force gauge and the shaker are not literally rigid. Consequently, we see some small structure in the measured frequency dependence of the empty cup.…”

“…In a previous publication 5 we demonstrated that our approach is capable of computing, accurately, the complexvalued resonance frequencies of a flex bar having a cylindrical cavity filled with either a simple liquid or a granular medium, as the case may be. While accurate, this approach does not account for the fact that some of those resonances may be too small in amplitude to be easily measurable in a resonance-type experiment.…”