## 5 Conclusions  
In this paper, the mathematical model of CTEHS is established considering the gravity effect of the magnet, and the influences of the end magnet mass and the initial vibration point on the system's static bifurcation and output characteristics are investigated by the method of numeric calculation, and experiments are carried out to validate the theoretical analysis. The major findings of the research are described below:  
(1) The gravity of the end magnet has a considerable influence on the system's static bifurcation behavior. With the increase of the end magnet mass \(M_{\rm e}\), the downward shift of the stable middle position of the system becomes more and more obvious, and the upper and lower stable positions also show significant asymmetry. In addition, the saddle and bifurcation points in the bifurcation diagram move under the gravity of the end magnets, which makes the range of different systems' stable states change, and the system's response characteristics change accordingly.  
(2) When the gravity of the end magnet is considered, the system generates additional gravitational potential energy, which causes the system's potential energy graph to show asymmetry. When the mass \(M_{\rm e}\) of the end magnet is increased, the potential well on one side of the system gradually moves down and the depth increases, and the potential well on the other side gradually moves up and decreases in depth, and the asymmetric characteristics of the potential energy curve are more obvious.  
(3) The end magnet mass \(M_{\rm e}\) has a remarkable impact on the system's response characteristics. Considering the gravity effect of the magnets can significantly reduce the excitation acceleration required for the system to realize higher output capability. As the end magnet mass \(M_{\rm e}\) increases, the acceleration threshold of the system gradually decreases, and the response displacement and output voltage gradually increase. The gravity effect of the magnet can make it easier for the system to perform large inter-well oscillations at low excitation intensities, thereby increasing the system's energy capture efficiency.  
(4) When the initial vibration point \(u_{0}\) of the system is different, the system's output characteristics are also different. When the system starts to vibrate from a stable equilibrium  
Figure 15: Phase graph, time course graph of the experiment under different \(u_{0}\): (**a**) \(u_{0}=\)\(-\)14 mm; (**b**) \(u_{0}=\) 12 mm.  
point with a deeper potential well, the acceleration threshold for the system to achieve a large response is high. When the system starts from a stable equilibrium point with a shallower potential well, the acceleration threshold for a large response is relatively low. The suitable initial vibration point is beneficial to reduce the system's requirement for external excitation intensity, thus greatly enhancing the system's performance in harvesting energy.  
Conceptualization, X.Z. and H.X.; methodology, H.X.; software, J.P. and X.C.; validation, F.Z. and Y.G.; writing--original draft preparation, H.X.; writing--review and editing, H.T.; visualization, Y.C.; supervision, H.X.; project administration, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No.51974228), Shaanxi Coal Joint Fund Project (Grant No.2021 JLM-03), Shaanxi Innovative Talent Plan Project (Grant No.2018 TD-032), and Key R&D project in Shaanxi (Grant No.2018 ZDCXL-GY-06-04).  
Not applicable.  
Authors acknowledge the support from the students [PERSON], [PERSON], and [PERSON] for their assistance in the modeling and experiments.  
The authors declare no conflict of interest.