# Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction
## 3 Results and Discussion
### The Effect of Perturbation Amplitude \(\mu\) on Dynamic Characteristics  
Under different support stiffness conditions, the influence of the perturbation amplitude \(\mu\) on the dynamics characteristics of the system is analyzed. The corresponding bifurcation diagrams, with \(\mu\) as the control parameter at [0.2, 0.6], are shown in Figure 6. The response of the drill string system at \(k_{\rm a}\) = 3.1 \(\times\) 10\({}^{6}\) and \(k_{\rm b}\) = 0 is demonstrated in detail in Figure 6a. The system shows chaotic characteristics at [0.2, 0.5225], and the intensity of chaotic motion is increased. Figure 7(a\({}_{1}\)-a\({}_{3}\)) shows the waveform, phase diagram and Poincare map of the system at \(\mu\) = 0.426, at which point the system has chaotic motion. With the increasing perturbation amplitude \(\mu\), the system shows periodic motion and quasi-periodic motion. The jump discontinuous non-linear phenomenon appears at (0.5225, 0.5745), and the system dynamic response is shown in Figure 7(b\({}_{1}\)-b\({}_{3}\)) at \(\mu\) = 0.571. As can be seen in this figure, the system exhibits quasi-periodic motion. With the further increase  
Figure 5: Waveform (1), phase diagram (2) and Poincaré map (3) with \(\omega\) = 8 chaotic motion (**a**), \(\omega\) = 21 Periodic-2 motion (**b**), \(\omega\) = 29.9 quasi-periodic motion (**c**), \(\omega\) = 35 chaotic motion (**d**), \(\omega\) = 46.35 Periodic-8 motion (**e**), \(\omega\) = 47.5 Periodic-4 motion (**f**), \(\omega\) = 59.5 chaotic motion (**g**), \(\omega\) = 89.5 quasi-periodic motion (**h**), \(\omega\) = 110 chaotic motion (**i**) and \(\omega\) = 119 Periodic-2 motion (**j**).  
in the perturbation amplitude, the motion of the system is a narrow period-2 at [0.5475, 0.5805]. As shown in Figure 7(c\({}_{1}\)-c\({}_{3}\)), period-2 motion is observed at \(\mu\) = 0.576.  
Figure 6: Bifurcation diagram of system response with \(\mu\): (**a**) \(k_{\rm a}\) = 3.1 \(\times\) 10\({}^{6}\), \(k_{\rm b}\) = 0; (**b**) \(k_{\rm a}\) = 2.6 \(\times\) 10\({}^{6}\), \(k_{\rm b}\) = 8.6 \(\times\) 10\({}^{6}\) (A: periodic motion; B: quasi-periodic motion; C: chaotic motion).  
Figure 7: Waveform (1), phase diagram (2) and Poincaré map (3) with \(\mu\) = 0.426 chaotic motion (**a**), \(\mu\) = 0.571 quasi-periodic motion (**b**) and \(\mu\) = 0.576 periodic motion (**c**).  
Increasing the perturbation amplitude from 0.5805 to 0.6, the system returns to chaotic motion.  
Figure 6b presents the system bifurcation diagram with \(\omega\) in the range of [0.2, 0.6] at \(k_{\text{a}}\) = 2.6 \(\times\) 10\({}^{6}\), \(k_{\text{b}}\) = 8.6 \(\times\) 10\({}^{6}\). In the interval [0.2, 0.258], the system shows the non-linear characteristics of periodic, quasi-periodic and chaotic motion. As \(\mu\) is further increased, it can be observed that the system exhibits periodic-4 motions, periodic-2 motions, periodic-10 motions, jump discontinuity phenomenon and period-doubling bifurcations phenomenon in the interval (0.258, 0.519). With the further increase in the perturbation amplitude, the system markedly exhibits chaotic motion at [0.519, 0.6]. Compared with Figure 6a, it can be seen that the chaotic motion region of the system can be reduced, to a certain extent, by adjusting the combination of support stiffness coefficients. This is mainly because the change in support stiffness causes the change in the inherent characteristics of the system. It can be seen that the support stiffness has a complex impact on the non-linear dynamic behavior of the system.