# Influence of Dry Friction on the Dynamics of Cantilevered Pipes Conveying Fluid
## 5 Results
### Effect of Installation Position \(\xi_{d}\)  
Next, we will discuss the effect of the installation position of the dry friction restraint on the stability of the pipe system. The location parameter \(\xi_{d}\) is set to be 0.3, 0.5, 0.7, and 0.9. The other key parameters are chosen to be: \(\alpha=0.001,\ \beta=0.213,\) and \(f_{n}=0.5\). The obtained bifurcation diagrams are shown in Figure 6. When the installation position of the dry friction restraint is \(\xi_{d}=0.3\), it can be seen that the system is subjected to flutter instability at \(u=7.72\). As the flow velocity is in the range of [7.73,7.87], compared with the pipe without dry friction, the displacement amplitude at the free end of the pipe is significantly reduced. When the flow velocity continues to increase, however, the effect of dry friction on the vibration amplitude of the pipe is not pronounced. Figure 5: Dynamic response of the pipe with dry friction when \(u=7.5\): (**a**) Time history curve of the tip\(-\)end of the pipe; (**b**) oscillation shapes of the pipe; (**c**) phase portrait of the tip\(-\)end of the pipe. When the installation position is \(\xi_{d}=0.5\), the critical flow velocity increases to 8.11, as shown in Figure 6b. However, when the installation position is \(\xi_{d}=0.7\), Figure 6c shows that the critical flow velocity of the pipe system is reduced to 6.36. As the installation position \(\xi_{d}\) continues to increase to 0.9, it is found that the critical flow velocity of the pipe system decreases obviously if compared with the pipe without the dry friction, indicating that the stability of the pipe is weakened in this case. For \(\xi_{d}=0.9\), the pipe begins to lose stability at the flow velocity \(u=4.89\), which is much lower than the critical flow velocity \(u_{\sigma}=5.78\) for a pipe without dry friction. As the flow velocity continues to increase, the displacement amplitude at the tip-end of the pipe would gradually increase. Therefore, the result for \(\xi_{d}=0.9\) shows that the application of the dry friction might be detrimental to the stability of the pipe system in some special cases. In order to better observe the dynamical behavior of the pipe after flutter instability, we further calculated the time history curve of the pipe with \(\xi_{d}=0.3\) and \(u=7.8\), as shown in Figure 7a. It is noted that the pipe vibrates periodically, but the vibration amplitude of the pipe with the dry friction constraint is much smaller than that of the pipe without dry friction, again, showing that the application of dry friction restraint can suppress the vibration of the pipe. The phase portrait and power spectral density (PSD) curve of the pipe with dry friction at this flow velocity are given in Figures 7b and 7c, respectively. The phase portrait is shown as an ellipse-like shape. It is noted that the phase portrait of Figure 7b is not a strictly periodic motion. This is because that the maximum displacements of the pipe in this case might fluctuate slightly, resulting in a relatively disordered PSD curve. However, since the fluctuation of the displacement of the pipe is relatively small, the pipe may be still considered to be in a periodic-like motion. Figure 6: Bifurcation diagrams for the tip–end responses of the pipe for different installation positions, with \(\alpha=0.001\), \(\beta=0.213\), and \(f_{n}=0.5\): (**a**) \(\xi_{d}=0.3\); (**b**) \(\xi_{d}=0.5\); (**c**) \(\xi_{d}=0.7\); (**d**) \(\xi_{d}=0.9\). Thus, through numerical calculations, we have analyzed the influence of dry friction on the stability and vibration response of the pipe. It can be seen that the critical flow velocity of the system can be greatly increased by installing a suitable dry friction support, which provides a new idea for the passive control method in the dynamical system of pipes conveying fluid.