Effect of lateral confinement on the apparent mass of granular dampers

Abstract

We study, via discrete element method simulations, the apparent mass (m, the ratio of a driving force to a resulting acceleration) and loss factor ($\eta$, the ratio of dissipated to stored energy) of granular dampers attached to a vertically driven, single degree of freedom mechanical system. Granular dampers (or particle dampers) consist in receptacles that contain macroscopic particles which dissipate energy, when they are subjected to vibration, thanks to the inelastic collisions and friction between them. Although many studies focus on $\eta$, less work has been devoted to m. The apparent mass of granular dampers is an important characteristic since the grains, which are free to move or collide inside their receptacle, act as a non-constant and time-dependent mass which alters the mass of the main vibrating system in a non-trivial way. In particular, it has been recently demonstrated (Masmoudi et al. in Granul Matter 18:71, 2016) that m non-linearly depends on the driving acceleration $\varGamma$ according to a power law, $m\propto \varGamma ^k$. Experiments using three-dimensional (3D) packings of particles suggest $k=-2$. However, simulations with one-dimensional (1D) columns of particles on a vibrating plate and theoretical predictions based on the inelastic bouncing ball model (IBBM) suggest that $k=-1$. These findings opened questions as to whether the apparent mass, which relies on how linear momentum is transferred from the damper to the primary system, depends on the dimensionality of the packing or on lateral interactions between walls and grains. Interestingly, $\eta$ was shown to follow a universal curve, $\eta \propto \varGamma ^{-1}$, whatever the dimensionality and the constraints in the motion of the grains. In this work, we consider granular dampers without a lid under different confinement conditions in the motion of the particles (1D, quasi-1D, quasi-2D and full 3D). We find that the mechanical response of the granular damper (m and $\eta$) is not sensitive to the lateral confinement or dimensionality. However, we have observed two distinct regimes, depending on whether the driving frequency is above or below the resonant frequency of the primary system. In the inertial regime, $\eta$ decays according to the IBBM for all dimensions, $\eta \propto \varGamma ^{-1}$, while m falls as $\varGamma ^{-2}$ for all dimensions, in agreement with Masmoudi’s experiments. However, the power law for m is valid only for moderate acceleration, before becoming negative at high accelerations. In the quasi-static regime, both m and $\eta$ display a complex behavior as functions of the excitation amplitude, but the mean trend is consistent with the IBBM predictions, i.e., $m\propto \varGamma ^{-1}$ and $\eta \propto \varGamma ^{-1}$.