# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 3 Result Analysis and Discussion
### Multi-Beam Situation
#### 3.6.1 Beams of Different Lengths  
Firstly, this summary presents various combinations of beam structures with different lengths and analyzes the frequency response curves of each beam segment, as well as the changing connection state at each connection joint. The total length of the structure is maintained at 1.6 m, including two connection joints and three beams. Considering the comprehensive structural symmetry, the lengths of the beams in the five combinations are as follows: (1) \(L_{1}\) = 0.6 m, \(L_{2}\) = 0.4 m, \(L_{3}\) = 0.6 m; (2) \(L_{1}\) = 0.65 m, \(L_{2}\) = 0.4 m, \(L_{3}\) = 0.55 m; (3) \(L_{1}\) = 0.7 m, \(L_{2}\) = 0.4 m, \(L_{3}\) = 0.5 m; (4) \(L_{1}\) = 0.75 m, \(L_{2}\) = 0.4 m, \(L_{3}\) = 0.45 m; (5) \(L_{1}\) = 0.8 m, \(L_{2}\) = 0.4 m, \(L_{3}\) = 0.4 m. An external excitation force \(F_{0}\) = 50 N is applied to the midpoint  
Figure 14: The curves of equivalent stiffness and excitation frequency under different excitation amplitudes. (**a**) Fixed support boundary; (**b**) simply supported boundary.  
Figure 15: The curves of equivalent damping and excitation frequency under different excitation amplitudes. (**a**) Fixed support boundary; (**b**) simply supported boundary.  
of beam \(L_{2}\). The preloads \(F_{N1}\) and \(F_{N2}\) on the two connection joints are 500 N, the friction coefficients \(\mu_{1}=\mu_{2}=0.3\), the connecting transitional stiffness \(k_{u1}=k_{u2}=1\times 10^{7}\) N/m, and the connecting rotational stiffness \(k_{\theta 1}=k_{\theta 2}=1\times 10^{7}\) N\(\cdot\)m/rad. Three frequency response observation points were selected to be located on their respective segment beams, and the coordinates were \(P_{1}\): \(x_{1}=0.4L_{1}\); \(P_{2}\): \(x_{2}=0.3L_{2}\); \(P_{3}\): \(x_{3}=0.6L_{3}\).  
The frequency response curves at each observation point are depicted in Figure 16. It is evident from the figures that variations in the beam length (reflecting the position of the connection joint in the overall structure) significantly impact the structural frequency response curve under external excitation. The response at location \(P_{3}\) is more sensitive to changes in beam length, indicating a higher sensitivity to the symmetry of the structure, while the response at location \(P_{3}\) is less affected by variations in the position of the connection joint. When \(L_{1}=L_{3}\), the structure is symmetrical, and it can be observed that the vibration response at each observation point is the'sharp', close to the ideal linear situation. As the length of beam \(L_{1}\) increases, the overall structure becomes asymmetric, and the vibration response of each observation point is in a straight line within the resonance frequency range.  
When the structure is symmetrical (Combination 1), the equivalent stiffness of both joints (Figure 17) decreases by the same amount as the resonance frequency. In contrast, when the structure is asymmetric, the equivalent connection stiffness at the joint between the longer beam \(L_{1}\) and the intermediate beam \(L_{2}\) becomes independent of frequency; meanwhile, at the joint on the left side of the shorter beam (L3), there is still a decrease in equivalent stiffness depending on the resonance frequency, with a greater reduction compared to symmetric cases, and the frequency range affected increases with the degree of asymmetry. This suggests that in asymmetrical configurations, the overall structural stiffness diminishes and becomes concentrated towards the joint of the short beam, which plays an important role in predicting changes in structural strength.