## 6 Conclusions and Prospects  
Based on the investigation of the thermal loss percentage and mechanism of forced-vibration particles with a focus on 2 mm particles, the following conclusions can be drawn:  
(1) Differences in temperature changes exist for different vibration states of granular assemblies. The temperature-increasing effect of particles is significant in strong and gas-like vibration states. There is a certain temperature increase effect over a long time scale when particles are in micro-vibrational and intermediate vibrational states. In contrast, when particles are in solid-like states, the temperature increase effect is not obvious even under prolonged excitation. The temperature increase characteristics of particles are related to their vibration states, which reflect the strength of collisions and frictional interactions between particles.  
(2) The forced-vibration heating characteristics of particles are related to frequency and amplitude. As the frequency increases, the heating effect of particles decreases, while as the amplitude increases, the heating effect of particles strengthens.  
(3) For the same reduced acceleration, as frequency increases, the thermal loss of particles increases. At the same frequency, as the reduced acceleration increases, the thermal loss of particles decreases. Particles in strong and gas-like vibration states vibrate intensely and exhibit significant heating effects. However, because of the high input energy, the thermal loss percentage is relatively small. Particles in micro-vibrational and intermediate vibrational states have weaker vibrations and lower input energy, which results in higher thermal loss percentages due to intense particle collisions and frictional interactions.  
In summary, granular assemblies mainly dissipate input energy through collision and friction during forced vibration, which is reflected in an increase in the temperature of particles. Through experimentation, it was found that particles do experience an increase in temperature under forced vibration conditions, but the corresponding thermal loss percentage does not significantly increase. This suggests that there may be more effective ways for particles to dissipate energy during energy consumption, such as sound loss caused by collisions or other forms of energy loss.  
[PERSON], [PERSON], [PERSON] and [PERSON] proposed the idea. [PERSON], [PERSON] and [PERSON] completed the experiments. [PERSON], [PERSON], [PERSON] and [PERSON] wrote the paper. All of the authors analyzed the data, discussed the results, and reviewed the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Major Special Basic Research grant number J2019-II-0013-0033, the Basic Strengthening Plan Key Basic Research grant number 2020-JCJQ-ZD-204, and the Discrete Mass Damping Nonlinear Energy Dissipation Mechanism and Application Research grant number No. 1227020353.  
Not applicable.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.