# Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction
## 3 Results and Discussion
### The Effect of Forcing Frequency \(\omega\) on Dynamic Characteristics  
The forcing frequency \(\omega\) has a significant effect on the dynamic response of the drill string system. In order to investigate the forcing frequency of the drill string system in detail, all parameters except the forcing frequency are taken as constants. The corresponding bifurcation diagram is shown in Figure 3, with \(\omega\) as the control parameter in the interval [0, 120] at \(k_{\text{a}}=0\) and \(k_{\text{b}}=0\). Figure 4 shows the bifurcation diagram of forced frequency \(\omega\) when \(k_{\text{a}}=k_{\text{b}}=0\), \(k_{\text{a}}=2.8\times 10^{5}\) and \(k_{\text{b}}=2\times 10^{3}\) in the range of [40, 60] respectively. In the bifurcation diagram, the abscissa is the forcing frequency, and the ordinate is the displacement amplitude of the vibration at the center of the drill string. As shown in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Notation** & **Value** \\ \hline Density of cutting fluid & \(\rho_{\text{f}}\) & \(8.65\times 10^{2}\) kg/m\({}^{3}\) \\ Density of drilling shaft & \(\rho_{x}\) & \(7.8\times 10^{3}\) kg/m\({}^{3}\) \\ Young’s modulus & \(E\) & \(2.14\times 10^{11}\) Pa \\ Internal diameter & \(d_{1}\) & \(7.68\times 10^{-2}\) m \\ External diameter & \(d_{2}\) & \(1.27\times 10^{-2}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the drill string system. Figure.3: Bifurcation diagram \(\omega\) = 8 \(\left(0,\ 120\right)\), \(k_{\text{a}}=k_{\text{b}}=0\) Figure.4a: Bifurcation diagram \(\omega\) = 8 \(\left(40,\ 60\right)\), \(k_{\text{a}}=k_{\text{b}}=0\) Figure.5a: Chaotic motion \(\omega\) = 89.5a: Chaotic motion \(\omega\) = 89.5b: Chaotic motion \(\omega\) = 89. Figure 3, quasi-periodic and chaotic motions can be observed in the range of [0, 10.55], when \(\omega=8\), the corresponding dynamic response of the system is shown in Figure 5(a\({}_{1}\)-a\({}_{3}\)). It can be observed that the system shows chaotic characteristics, the Poincare map appears scattered points. As the forcing frequency is increased, namely 10.55 \(<\omega<\) 33.1, a series of inverse period-doubling bifurcations appears in the system, where the system exists periodic-6, periodic-2 motion and apparent jump discontinuity. The related periodic-2 motion at \(\omega=21\) is shown in Figure 5(b\({}_{1}\)-b\({}_{3}\)) with waveform, phase diagram and Poincare map, which indicates that the system experiences periodic-2 motion and only two points are shown in the Poincare map. Figure 5(c\({}_{1}\)-c\({}_{3}\)) show the dynamic characteristics of the system at \(\omega=29.9\), The Poincare map is a closed circle, and the system features quasi-periodic motion. To further analyze the dynamics of the drill string system, the local bifurcation diagram of the forcing frequency \(\omega\) in the range of [40, 60] is given in Figure 4a. As the forcing frequency is increased, the chaotic responses are observed in the interval [33.1, 43.5]. The related chaotic motion at \(\omega=35\) is shown in  
Figure 4: Bifurcation diagram of system response with \(\omega\), (a) \(k_{\rm a}=k_{\rm b}=0\); (b) \(k_{\rm a}=2.8\times 10^{5}\), \(k_{\rm b}=2\times 10^{3}\) (A: periodic motion; B: quasi-periodic motion; C: chaotic motion). Figure 3: Bifurcation diagram of system response with \(\omega\), \(k_{\rm a}=k_{\rm b}=0\) (A: periodic motion; B: quasi-periodic motion; C: chaotic motion). Figure 5(d\({}_{1}\)-d\({}_{3}\)) by a waveform, phase diagram and Poincare map. By increasing \(\omega\) from 43.5 to 57.75, a series of inverse period-doubling bifurcations appears in the system, and the system response presents the phenomenon of period-doubling reverse bifurcation, the period number of vibration changes exponentially, and the chaotic motion of the system is transformed into periodic motion in the form of period-doubling reverse bifurcation. Figure 5: \(Cont.\)  
Figure 5: \(Cont\).