# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 4 Conclusions  
Aiming to resolve the stick-slip problem that may occur in the connection joints of multi-beam structures, this study employed the Coulomb friction model to develop a vibration analysis framework incorporating contact friction. This framework systematically examines how connection characteristic parameters influence the vibrational behavior of multi-beam structures. Systematic dynamic response analysis yielded the following primary conclusions:  
1. The alterations in the state of structural connections primarily manifest as a decrease in connection stiffness near the resonance frequency, with the reduction being more pronounced in the simply supported condition than the fixed condition. The \"tracking\" phenomenon caused by the abrupt change of damping and the variation in resonance frequency results in a relatively stable vibration response amplitude over a larger frequency range.
2. As the contact pressure and friction diminish, the frequency response curve progressively flattens, indicating a resonance frequency range. The peak response frequency is observed at the lower end of this resonance range, which is situated below the linear natural frequency.
3. When the connection stiffness is reduced, the resonance peaks broaden and diverge, exhibiting the characteristics of internal resonance in nonlinear systems. This feature becomes increasingly pronounced as the stiffness decreases.
4. The magnitude of external excitation serves as a direct factor in inducing nonlinear softening of the connection. When the excitation amplitude is minimal, its impact on the connection state remains limited. As the external excitation escalates, the structure's frequency response curve exhibits a resonance region.
5. Various sub-beam combinations display distinct nonlinear characteristics when used in multi-beam structures. In asymmetric configurations, alterations in connection states are primarily induced by excitation and predominantly manifest at the junctions of shorter beams. As the excitation point approaches the structural center, it is more likely to cause significant changes in connection states, thereby broadening the range of resonant intervals.  
Conceptualization, P.L.; Methodology, Y.X. and P.L.; Validation, X.Z., D.N. and Z.L.; Writing--original draft, X.Z.; Writing--review & editing, Y.X., P.L., D.N. and Z.L. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the Fundamental Research funds for the Central Universities (Grants no. 202213033), National Natural Science Foundation of China (Grants no. 52471352). And the APC was funded by National Natural Science Foundation of China (Grants no. 52471352).  
Figure 19: The changes in the equivalent stiffness of the connection at different excitation positions, for which the coordinates were (1) \(x_{1}=0.2\) m, (2) \(x_{1}=0.4\) m, (3) \(x_{2}=0.2\) m, (4) \(x_{3}=0.2\) m, (5) \(x_{3}=0.4\) m. (a) Joint 1; (b) joint 2.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.