*4.3. Particle Damper (PD)*

The conventional computing model of the PD uses the discrete element method (DEM), which is very time consuming and complicated. Based on correlational studies by Papalou and Masri [35], Lu et al. [3] proposed a simplified analytical method for the PD. The simplified model has been verified to have computational efficiency and a satisfactory degree of accuracy in practical applications. Therefore, Lu et al.'s model is adopted in this study.

The essence of the model is to transform multiple particles into an equivalent single particle, as shown in Figure 7. The mass of the single particle equals the total mass of the multiple particles, assuming that the collisions between the particles can be neglected; then, the damping forces of the PD mainly originate from the collisions between the particles and the container wall.

**Figure 7.** Equivalent single particle of multiple particles (adapted from [3]).

In Figure 7, the clearance *d* of the single particle is a critical parameter governing the damping force and can be determined by the following equation:

$$\left(\frac{1}{\rho\_{PD}} - 1\right) \frac{M\_{PD}}{\rho} = \frac{M\_{PD}}{2\rho} + \frac{\pi}{4} \left(\frac{6M\_{PD}}{\pi\rho}\right)^{\frac{3}{9}} d\tag{27}$$

where *MPD* is the mass of the PD; *ρ* is the density of the particle material; *ρPD* is the packing density of the multiple particles. Hales et al. [36] suggested that *ρPD* should not exceed 0.74; a value of 0.6 is assigned to *ρPD*.

The mass *MPD* of PD is the product of the mass ratio *γM* and the mass *Ms* of the primary structure. Four mass ratios ranging from 1% to 4% are selected. The stiffness *KPD* of the equivalent single PD is determined as *KPD* = *MPD*(<sup>2</sup>*π fPD*)2. Masri and Ibrahim [37] recommended that *fPD* ≥ 20 *f*1, so the frequency of PD is set as 20 *f*1 ≈ 20 Hz. The damping coefficient *CPD* of the equivalent single PD can be written as *CPD* = <sup>2</sup>*MPDξPD*(<sup>2</sup>*<sup>π</sup> fPD*), where the damping ratio *ξPD* is related to the coefficient of restitution *e* [38]. In this study, a steel particle with *e* = 0.5 is adopted; thus, *ξPD* is determined to be 0.2. The damping force of the single PD can be expressed as:

$$f\_{PD} = \mathbb{C}\_{PD} H\left(\mathbf{x}\_{PD\prime}^{r} \dot{\mathbf{x}}\_{PD}^{r}\right) + K\_{PD} G\left(\mathbf{x}\_{PD}^{r}\right) \tag{28}$$

where *xrPD* and .*xrPD* are the relative displacement and velocity of the PD with respect to the primary structure. *<sup>H</sup><sup>x</sup>rPD*, .*xrPD* and *<sup>G</sup>*#*xrPD*\$ are two nonlinear functions with the expressions of:

$$H\left(\mathbf{x}\_{\rm PD}^{r}, \dot{\mathbf{x}}\_{\rm PD}^{r}\right) = \begin{cases} \dot{\mathbf{x}}\_{\rm PD}^{r} \, for \, \mathbf{x}\_{\rm PD}^{r} \le -d/2 \, and \, \mathbf{x}\_{\rm PD}^{r} \ge -d/2\\ 0, for -d/2 < \mathbf{x}\_{\rm PD}^{r} < d/2 \end{cases} \tag{29}$$

$$G(\mathbf{x}\_{PD}^{r}) = \begin{cases} \mathbf{x}\_{PD}^{r} + d/2, \text{for } \mathbf{x}\_{PD}^{r} \le -d/2\\ 0, \text{for } -d/2 < \mathbf{x}\_{PD}^{r} < d/2\\ \mathbf{x}\_{PD}^{r} - d/2, \text{for } \mathbf{x}\_{PD}^{r} \ge d/2 \end{cases} \tag{30}$$
