## 2 Discrete Element Modeling and Energy Consumption Calculation
### Energy Consumption Calculation  
Energy consumption values are aggregated based on PFC principles. The reliability of these values is confirmed through experiments conducted on vibrating powder beds, with simulations aligning with the experimental results [23,24]. The formulations used are as follows:  
(1) Sliding strain energy \(E_{\text{st}}\)  
Sliding strain energy (\(E_{\text{st}}\)) is the energy resulting from the elastic forces between contact points and stored within the system as potential energy. The formula for computing the sliding strain energy (\(E_{\text{sst}}\)) is:  
\[E_{\text{sst}}=\sum_{i=1}^{k}\frac{1}{2}\Bigg{(}\frac{f_{\text{n,i}}^{2}}{k_{ \text{n}}}+\frac{\left\|f_{\text{s,i}}\right\|^{2}}{k_{\text{s}}}\Bigg{)}|_{t} \tag{7}\]  
where \(f_{\text{n,i}}\) and \(f_{\text{s,i}}\) are the normal and tangential contact forces of the \(i\)-th contact; and \(t\) is the real time. (2) Slip energy \(E_{\mu}\):  
\[E_{\mu}=\sum_{i=1}^{k}\frac{1}{2}\Big{[}\left(f_{\text{s,i}}\right)_{0}+f_{ \text{s,i}}\Big{]}\Delta\delta_{\text{s,i}}^{\mu}|t \tag{8}\]  
where \(f_{\text{s,i}}\) is the tangential force of the \(i\)-th contact at the time \(t\); and \(\Delta\delta_{\text{s}}\)u is the relative tangential displacement increment \(\Delta\delta_{\text{s}}\) in the sliding direction. (3) Dashpot energy \(E_{\beta}\):  
Dashpot energy (\(E_{\beta}\)) denotes the overall energy dissipated by the damper, and its formula for calculation is as follows:  
\[E_{\beta}=F^{\text{i}}\cdot\left(\dot{\delta}\Delta t\right) \tag{9}\]  
\[\dot{\delta}=\dot{x}_{\text{c}}^{\text{i}}-\dot{x}_{\text{c}}^{\text{j}} \tag{10}\]  
\[\dot{x}_{\text{c}}^{\text{i}}=\dot{x}^{\text{i}}+\omega^{\text{j}}\times(x_{ \text{c}}-\dot{x}^{\text{j}}) \tag{11}\]  
where \(\dot{\delta}\) is the relative velocity of particles; \(\dot{x}_{\text{c}}^{\text{i}}\) and \(\dot{x}_{\text{c}}^{\text{j}}\) are the velocities of the two particles at the contact point \(c\), respectively; \(\dot{x}^{\text{i}}\) and \(\omega^{\text{j}}\) are the velocity and angular velocity of the \(i\)-th particle, respectively; \(x_{\text{c}}\) and \(\dot{x}^{\text{i}}\) are the positions of contact point \(c\) and the center of gravity of the particle \(i\), respectively. (4) Rolling strain energy \(E_{\text{sst}}\) \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Particle density (kg/m\({}^{3}\)) & 2650 \\ Local dashpot \(D_{\text{p}}\) & 0.01 \\ Normal viscous dashpot \(\beta_{\text{n}}\) & 0.70 \\ Tangential viscous dashpot \(\beta_{\text{s}}\) & 0.70 \\ Sliding friction \(\mu_{\text{b}}\) & 0.60 \\ Rolling friction \(\mu_{\text{r}}\) & 0.25 \\ Effective contact modulus \(E_{\text{cb}}\) (Pa) & \(1.24\times 10^{8}\) \\ Effective contact modulus of loaded plate \(E_{\text{cl}}\) (Pa) & \(3.31\times 10^{9}\) \\ Normal-to-tangential stiffness ratio \(k^{\star}\) & 4.0/3.0 \\ Coarse particle diameter & 40–60 \\ Fine particle diameter & 5–15 \\ \hline \end{tabular}
\end{table} 
Table 1: Mesoscopic parameters used in numerical simulations. Rolling strain energy (\(E_{\rm rst}\)) signifies the energy generated from particle contact and accumulated within the rolling spring. The formula for calculating this energy is as follows:  
\[E_{\rm rst}=\frac{1}{2}\frac{\left\|M_{\rm r}\right\|^{2}}{k_{\rm r}} \tag{12}\]  
(5) Rolling slip energy \(E_{\rm\mu r}\):  
Rolling slip energy (\(E_{\rm\mu r}\)) pertains to the energy dissipation resulting from the occurrence of rolling slip between particles. The formula for calculating this energy is as follows:  
\[E_{\rm\mu r}=\frac{1}{2}[(\mathbf{M}_{\rm r})_{0}+\mathbf{M}_{\rm r}]\cdot\Delta\theta ^{\rm\mu r} \tag{13}\]  
where \(\Delta\theta^{\rm\mu r}\) is the increment component bending \(\Delta\theta\) in the sliding direction.