## Appendix A Discrete Element Vibration Energy Characteristics Model  
When the adhesion of the granular surface is not considered, the normal force between the granules is generally used. Normal contact force \(F_{c}\) is the soft ball contact model, which is a soft ball dry contact model based on the improved discrete element method [28; 29]. Deformation will occur, and the normal contact force will be obtained \(F_{n}\):  
\[F_{n}=\frac{4}{3}E^{*}\sqrt{R^{*}}\delta_{n}^{\frac{3}{2}} \tag{1}\]  
where \(R^{*}\) is the equivalent radius; \(E^{*}\) is the equivalent elastic modulus, given by the following formula; and \(\delta_{n}\) is the normal overlap.  
\[\frac{1}{R^{*}}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \tag{2}\]  
\[\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}} \tag{3}\]  
The energy loss in this process is mainly based on the research of [PERSON] et al. [30]. The normal damping force \(F_{n}^{d}\) is represented in the following equation:  
\[F_{n}^{d}=2\sqrt{\frac{5}{6}}\zeta\sqrt{k_{n}m^{*}}v_{n}^{\overset{\rightarrow} {\rightarrow}} \tag{4}\]  
where \(\zeta\) is the damping ratio, \(\overset{\rightarrow}{v_{n}^{\overset{\rightarrow}{\rightarrow}}}\) is the normal component of the relative velocity of the two particles, \(m^{*}\) is the equivalent mass, and \(k_{n}\) is the normal stiffness; its expression is as follows:  
\[\frac{1}{m^{*}}=\left(\frac{1}{m_{j}}+\frac{1}{m_{i}}\right)^{-1} \tag{5}\]  
\[\beta=\frac{-\ln e}{\sqrt{\ln^{2}e+\pi^{2}}} \tag{6}\]  
\[k_{n}=2E^{*}\sqrt{R^{*}\delta_{n}} \tag{7}\]  
where \(m_{i},m_{j}\) is the mass of the two collision particles. The particles selected in this experiment are particles of the same material and volume. Therefore, the mass of the particles that collide is the same.  
The tangential contact force model of the particle is based on the Mindlin-Deresiewicz contact theory. The calculation of the tangential contact force is as follows:  
\[F_{l}=-k_{l}\delta_{l} \tag{8}\]  
where \(\delta_{l}\) is the tangential overlap and \(k_{l}\) is the tangential stiffness, derived from Equation (9):  
\[k_{l}=8G^{*}\sqrt{R^{*}\delta_{n}} \tag{9}\]  
In Equation (9), \(G^{*}\) is the equivalent shear modulus is as follows:  
\[G^{*}=\frac{2-v_{1}}{G_{1}}-\frac{2-v_{2}}{G_{2}} \tag{10}\]where \(G_{1}\) and \(G_{2}\) are the shear modulus of the two particles, which is determined by the elastic modulus and Poisson's ratio of the material.  
\[G_{1}=\frac{E_{1}}{2(1+v_{1})},G_{2}=\frac{E_{2}}{2(1+v_{2})} \tag{11}\]  
Based on the study of [PERSON] et al. [30], the tangential damping force can be defined as follows:  
\[F_{t}^{d}=2\sqrt{\frac{5}{6}}\zeta\sqrt{k_{t}m^{*}}v_{t}^{rel} \tag{12}\]  
where \(v_{t}^{rel}\) is the tangential component of the relative velocity of the two particles.  
In the forced vibration friction model, the particle is in contact with other particles or container walls and is subjected to vertical positive pressure \(F_{n}\) and horizontal thrust \(F_{l}\). In the discrete element method, the shareholder friction of the particle body is considered to apply a moment to the surface of the particle as follows:  
\[\tau_{i}=-\mu_{r}F_{n}R_{i}\omega_{i} \tag{13}\]  
where \(\mu_{r}\) is the rolling friction coefficient, \(R_{i}\) is the distance from the particle contact point to the center of mass, and \(\omega_{i}\) is the unit angular velocity vector of the object at the contact point.  
The forced vibration model of the granular assemblies adopts the Hertz contact theory model, and the normal contact force adopts the soft ball contact model as the soft ball dry contact model of the improved discrete element method.