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 \(\eta \propto \varGamma ^{-1}\).

我们通过离散元方法模拟研究了附着在其上的颗粒阻尼器的表观质量（m，驱动力与产生的加速度之比）和损耗因子（\（\ eta \），耗能与储能之比）。垂直驱动的单自由度机械系统。颗粒阻尼器（或颗粒阻尼器）包含在容纳宏观粒子的容器中，这些粒子由于受到非弹性碰撞和摩擦而在受到振动时会耗散能量。尽管许多研究都集中在\（\ eta \）上，但用于m的工作却很少。。颗粒阻尼器的表观质量是一个重要的特征，因为在其容器内自由移动或碰撞的谷物起着非恒定且随时间变化的作用，从而在不平凡的情况下改变了主振动系统的质量。办法。特别地，近来已经证明（马斯穆迪等人在Granul物18:71，2016），该米的非线性地依赖于驱动加速度\（\ varGamma \）根据幂定律，\（M \ propto \ varGamma ^ k \）。使用粒子的三维（3D）堆积的实验表明\（k = -2 \）。但是，在振动板上用一维（1D）粒子列进行的模拟以及基于非弹性弹跳球模型（IBBM）的理论预测表明，\（k = -1 \）。这些发现提出了一个问题，即视在质量是否取决于线性动量如何从阻尼器传递到初级系统，是取决于填料的尺寸还是取决于壁与颗粒之间的横向相互作用。有趣的是，\（\ eta \）被证明遵循通用曲线\（\ eta \ propto \ varGamma ^ {-1} \），无论晶粒运动的尺寸和约束如何。在这项工作中，我们考虑了在颗粒运动（1D，准1D，准2D和全3D）的不同限制条件下不带盖的粒状阻尼器。我们发现粒状阻尼器（m和\（\ eta \））的机械响应对横向约束或尺寸不敏感。但是，根据驱动频率是高于还是低于初级系统的谐振频率，我们已经观察到两种不同的状态。在惯性状态下，\（\ eta \）根据所有尺寸的IBBM衰减，\（\ eta \ propto \ varGamma ^ {-1} \），而m下降为\（\ varGamma ^ {-2} \ ）在各个方面都与Masmoudi的实验一致。但是，m的幂定律仅对中等加速度有效，然后在高加速度下变为负。在准静态状态下，m和\（\ eta \）都显示出与激励幅度有关的复杂行为，但平均趋势与IBBM预测一致，即\（m \ propto \ varGamma ^ {- 1} \）和\（\ eta \ propto \ varGamma ^ {-1} \）。