# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 3 Result Analysis and Discussion
### Response Under Different Contact Pressure Conditions  
In the connection force expression, the contact pressure \(F_{N}\) is an important variable whose physical meaning is the preload force in bolt connections and the normal force in other connection situations, which significantly affects the tightness of the connection. In this section, the structural frequency response under different levels of contact pressure will be discussed to analyze the influence of contact pressure on the structural dynamic characteristics. As shown in Figure 3, the frequency response curves of the connection beam structure on both sides of the connection point under different preload conditions are presented, where the connection stiffnesses are \(k_{u}=10^{7}\) N/m, \(k_{b}=10^{7}\) N\(\cdot\)m/rad; the friction coefficient is \(\mu=0.3\); the excitation amplitude \(F_{0}\) of 50 N is applied at the midpoint of beam \(L_{1}\); and the boundary conditions are fixed support or simply supported. In this section and Sections 3.3-3.5, the structural dimensions are \(L_{1}=0.5\) m, \(L_{2}=0.8\) m; height is \(H=20\) mm; and width is \(B=20\) mm. The excitation frequency range is [\(\omega_{1}-5\), \(\omega_{1}+5\)], where \(\omega_{1}\) represents the first natural frequency under the framework of linear theory. It is evident that with an increase in the preload, the system response curve tends towards the linear frequency response curve, indicating that the structure can be regarded as a linear system under sufficient preload force. As the preload force decreases, the frequency response curve gradually flattens, eliminating the resonance peak and resulting in a range of resonance frequencies. In the low-preload state where the peak is eliminated, the maximum response frequency always remains lower than the linear natural frequency, which is situated at the left end of the resonance interval. \begin{table}
\begin{tabular}{l l l l l l l l} \hline  & & \multicolumn{2}{l}{**Case I**} & \multicolumn{4}{l}{**Case II**} \\ \hline  & & **Experiment** & **Present** & **Error** & **Experiment** & **Present** & **Error** \\ \hline \multirow{3}{*}{Clamped} & 1 st & 81.055 & 83.322 & 2.8\% & 83.008 & 85.279 & 2.7\% \\  & 2 nd & 245.117 & 241.488 & \(-\)1.5\% & 231.445 & 231.747 & 0.1\% \\  & 3 rd & 439.453 & 443.556 & 0.9\% & 460.938 & 458.237 & \(-\)0.6\% \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 3: Frequency response curves under different preload conditions. (**a**) Fixed support boundary, \(L_{1}\) connection point; (**b**) fixed support boundary, \(L_{2}\) connection point; (**c**) simply supported boundary, \(L_{1}\) connection point; (**d**) simply supported boundary, \(L_{2}\) connection point. In response to the 'clipping' phenomenon of the response peak in the resonance range, Figures 4 and 5 illustrate the variation of equivalent stiffness and damping with excitation frequency under different preload conditions. It is evident that the maximum equivalent stiffness equals the connecting transitional stiffness \(k_{u}\), which experiences a sharp decrease in the resonance region, where the magnitude of stiffness reduction and the frequency range affected are negatively correlated with the contact pressure. In other words, lower contact pressure leads to a greater reduction in equivalent stiffness and a wider affected frequency range, which clearly corresponds with the frequency response curve. Within a certain frequency range, due to the reduction in equivalent stiffness, the resonance frequency will shift to the left, resulting in changes to the peak frequency in the frequency response curve corresponding to different pressures. The more the stiffness is reduced, the more the resonance peak frequency will shift to the left and decrease.