Effect of lateral confinement on the apparent mass of particle dampers

We study, via DEM simulations, the apparent mass $m$ and loss factor $\eta$ of particle dampers (PD) attached to a vertically driven, single degree of freedom mechanical system. Although many studies focus on $\eta$, less work has been devoted to $m$. It has been recently demonstrated [M. Masmoudi \textit{et al}. Granular Matter 18 (2016) 71.] that $m$ non-linearly depends on the driving acceleration $\gamma$ according to a power law, $m\propto\gamma^k$. Experiments using 3D packings of particles suggest $k=-2$. However, simulations with 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 left open questions whether m may depend on the dimensionality of the packing or on lateral interactions between walls and grains. In turn, $\eta$ was shown to follow a universal curve, $\eta\propto\gamma^{-1}$, whatever the dimensionality and the constraints in the motion of the grains. In this work, we consider PD under different confinement conditions in the motion of the particles (1D, quasi-1D, quasi-2D and 3D). We find that the dynamical response of the PD ($m$ and $\eta$) is not sensitive to the lateral confinement or dimensionality. However, we have observed two distinct regimes: (i) In the inertial regime, $\eta$ decays according to the IBBM for all dimensions, $\eta\propto\gamma^{-1}$, while $m$ falls with an apparent power law behaviour that matches Masmoudi's experiments, $m\propto\gamma^{-2}$, for all dimensions but only in the range of moderate acceleration, before becoming negative for very high accelerations. (ii) In the quasi-static regime, both $m$ and $\eta$ display a complex behavior as functions of the excitation amplitude, but tend to the IBBM prediction, $m\propto\gamma^{-1}$ and $\eta\propto\gamma^{-1}$.