# Dynamic Analysis of Bolted Cantilever Beam with Finite Element Method Considering Thread Relation
## 3 Results and Discussion  
Physical parameters of bolted cantilever beam are listed in Table 1, and the numerical results are presented in Figures 6-11. It can be seen from Figure 6(a) that in the first order generalized coordinate system, with the dimensionless frequency \(\Omega\) increasing from 1 to 2, the generalized transverse coordinate value \(q\) slowly increases from \(8.2\times 10^{-7}\), and the maximum value is \(1.402\times 10^{-5}\), and the corresponding dimensionless resonance frequency is 1.642. Then, the value of the generalized horizontal coordinate decreased rapidly to \(4.827\times 10^{-7}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Symbol** & **Value** & **Unit** \\ \hline Elastic modulus & \(E\) & 210 & GPa \\ Total length of bolted cantilever & \(L\) & 434 & mm \\ Length of single beam & \(l\) & 234 & mm \\ Distance from bolt axis to wall & \(s\) & 217 & mm \\ Thickness of beam & \(w\) & 54 & mm \\ Thickness of beam & \(h\) & 8 & mm \\ Coefficient of linear translational viscous damping & \(c_{l}\) & 0.27 & N-s/m \\ Coefficient of linear torsional viscous damping & \(c_{0}\) & 0.18 & N-s/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of bolted cantilever beam. Figure 6: 1 st order generalized coordinate of the frequency response curve (**a**) in transvese direction (**b**) in rotation direction. Figure 7: Influence of \(\beta\) in 1 st order generalized coordinate of the frequency response curve (**a**) in transvese direction (**b**) in rotation direction. Figure 6b displays that in the first order generalized coordinate system, with the dimensionless frequency \(\Omega\) increasing from 1 to 2, the generalized bending coordinate value \(p\) slowly enlarges from \(2.216\times 10^{-6}\) to a maximum value of \(2.853\times 10^{-5}\), and the corresponding dimensionless resonance frequency is \(1.660\). Then, the value of the generalized horizontal coordinate decreased rapidly to \(6.846\times 10^{-7}\). Figure 8: Influence of \(\eta\) in 1 st order generalized coordinate of the frequency response curve (**a**) in transvese direction (**b**) in rotation direction. Figure 10: Influence of \(k_{ln}\) in 1 st order generalized coordinate of the frequency response curve (**a**) in transvese direction (**b**) in rotation direction. Figure 9: Influence of f\({}_{0}\) in 1 st order generalized coordinate of the frequency response curve (**a**) in transvese direction (**b**) in rotation direction. Figure 6a,b describe that the peaks of the frequency response curve are inclined to the right. This asymmetry reflects the existence of a certain degree of nonlinearity in transverse and bending vibration. Specifically, near the right side of the resonance frequency, there is a frequency corresponding to two amplitude of vibration, which is to say, there is a certain degree of uncertainty. Furthermore, no matter the transverse vibration or the bending vibration, there will be typical bifurcation phenomenon. Of course, if the bifurcation phenomenon is further developed, it may cause strong uncertainty chaos. Figure 7a,b show that in the first-order generalized coordinate system, as the value of dimensionless frequency \(\Omega\) increases from 1 to 2, the generalized transverse coordinate values \(q\) and the generalized rotation coordinate values \(p\) initially slowly rise to their respective resonance peaks, and then rapidly drop to around 0. All of them will display a certain scale of bifurcation. Moreover, with the increase of \(\beta\), the resonance frequencies in both transverse direction and rotation direction increase in turn, and the formant decrease in turn. It may be that the natural frequency will increase as soon as it is too large. Under the action of small external force, there is not enough time for the bolt connected cantilever system to make response, so as to increase the amplitude.