## Appendix B Granular Vibration Transition Curves and Vibration State  
Based on the traditional interaction between particles, the vibration forms of granular flows can be classified into three types: (1) quasistatic granular flow, which exhibits solid-like behavior; (2) collisional or rapid granular flow, resembling liquid behavior; and (3) dense flow, showing gas-like behavior [31; 32; 33; 34; 35]. Under different excitation conditions, granular materials exhibit non-equilibrium energy dissipation phenomena and display rich and complex dynamic behaviors. While there have been numerous studies on the vibrational forms of granular materials, there is currently no widely accepted classification system. However, there is still much to be understood regarding the vibration patterns and behavior of granular assemblies. Based on extensive experimental research, scholars believe that granular materials exhibit diverse vibrations within different particle containers, including solid-like states [36; 37], fluidization [38; 39; 40], bouncing bed behavior [41], convection [42; 43], the Leidenfrost effect [44; 45; 46; 47; 48], and buoyancy-driven convection [49; 50]. * [PERSON] and [PERSON] (2022) [PERSON]; [PERSON] Energy dissipation in horizontally driven particle dampers of low acceleration intensities. _Nonlinear Dyn._**2022**, _108_, 3009-3024. [CrossRef]
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