## 4 Results and Analysis
### Response Characteristics and Parameter Influence Analysis
#### 4.2.1 Influence Analysis of NES Nonlinear Stiffness  
By keeping the mass and damping of an NES constant and changing the nonlinear stiffness coefficient \(\kappa\), the amplitude-frequency response curve of the system was obtained (according to Equation (20)), and the stability of the trivial solution was analyzed (by the method described in Section 3.2. The results are shown in Figures 5-7. To facilitate the analysis, the pipe motion amplitude and the pipe-NES relative motion amplitude were compared. In Figures 5-7, the solid line indicates that the system motion is stable, while the imaginary line indicates that the system motion is unstable, and saddle-node bifurcation or Hopf bifurcation occurs. The black cube represents the Hopf bifurcation point. At this point, the response shifts from periodic to quasi-periodic. The linear definition in the subsequent amplitude-frequency response curve is the same. Figure 5 shows that when \(\kappa\) = 97.97, the response of the pipe exhibits a stable solution and unstable solution. Hopf bifurcation occurs in the region of \(\sigma\) = [0, 0.024], and the response shifts from a periodic solution to a quasi-periodic solution, as shown in Figure 5a. In this frequency range, the relative motion of the pipe-NES increases significantly, as shown in Figure 5b. Most of the energy of the pipe is transferred to an NES, and the vibration of an NES is suppressed. Figure 4: Response displacement spectrum. (**a**) Pipe displacement and (**b**) relative displacement of pipe-NES. Figure 5: Amplitude frequency response curve when \(\kappa\) = 97.97, \(\xi_{2}\) = 0.0085, \(\rho_{2}\) = 0.02, (**a**) pipe displacement amplitude and (**b**) relative displacement amplitude. When \(\kappa\) decreases to 24.425, the pipe and NES are both periodic steady-state solutions without nonlinearity, as shown in Figure 6a. Compared with the results in Figure 5, the extreme value of the amplitude of the pipe in Figure 6 is approximately 1.7 times that in Figure 5. When \(\sigma=0\), the maximum amplitude of the pipe is 0.0557, and the maximum amplitude of the relative displacement of the pipe-NES is 0.0493. Obviously, the vibration suppression of the NES is unsatisfactory at this time. When \(\kappa\) increases to 367.38, the amplitude-frequency response curve exhibits multiple response branches, as shown in Figure 7. In the range of \(\sigma=[-0.164,\,-0.1]\), there is a saddle-node bifurcation, which indicates that the displacement of the system has multiple solutions. Under some initial conditions, the vibration of the pipe will suddenly increase. The system response in the region of \(\sigma=[-0.012,\,0.13]\) is quasi-periodic motion, and the saddle-node bifurcation occurs from \(\sigma=0.068\), as shown in Figure 7a. Under different initial conditions, the system response jumps between different equilibrium points of the period and quasi-period. Compared with the results in Figure 5, the unstable region of the system at \(\kappa=367.38\) is larger. The time-domain curves of the system response under different initial conditions are given in Figure 8. The parameters selected in the calculation process are the same as those in Figure 7, and the frequency is \(\sigma=-0.144\). Figure 6: Amplitude frequency response curve when \(\kappa=24.425\), \(\xi_{2}=0.0085\), \(\rho_{2}=0.02\), (**a**) pipe displacement amplitude and (**b**) relative displacement amplitude. Figure 7: Amplitude frequency response curve when \(\kappa=367.38\), \(\xi_{2}=0.0085\), \(\rho_{2}=0.02\), (**a**) pipe displacement amplitude and (**b**) relative displacement amplitude.