# Influence of Dry Friction on the Dynamics of Cantilevered Pipes Conveying Fluid
## 5 Results
### Effect of Friction Force \(f_{n}\)  
For certain friction surfaces, the values of the friction force mainly depend on the normal pressure acting on the friction surfaces. In this subsection, we will explore the effect of different friction forces on the stability of the pipe system. Thus, the friction force \(f_{n}\) is respectively taken as 0.1, 0.3, and 0.5 for analysis, with \(\xi_{d}=0.5\). The other two key parameters of the pipe system are \(\alpha=0.001\) and \(\beta=0.213\). The results of the bifurcation diagrams are shown in Figure 4. The blue and red dots represent the bifurcation diagrams without dry friction and with dry friction, respectively.  
It is seen from Figure 4 that the dimensionless critical flow velocity of the pipe for flutter instability is about 5.78 without dry friction. When the applied dry friction force is 0.1, as shown in Figure 4a, the critical flow velocity of the pipe system will increase to 6.29. Moreover, the dry friction has almost no effect on the vibration amplitude of the pipe when the pipe system becomes unstable. It is also seen that, when the flow velocity is slightly  
Figure 4: Bifurcation diagrams for the tip\(-\)end responses of the pipe with different dry friction forces when \(\alpha=0.001\), \(\beta=0.213\), and \(\xi_{d}=0.5\): (**a**) \(f_{n}=0.1\); (**b**) \(f_{n}=0.3\); (**c**) \(f_{n}=0.5\).  
Figure 3: Convergence analysis of the [PERSON]’s truncation approximation.  
higher than the critical value, the displacement amplitude at the end of the pipe has a jump with a discontinuous change, which may be related to the applied dry friction force because the expression of the dry friction force is a discontinuous function in mathematical form. When the direction of the lateral velocity at \(\xi_{d}\) changes sign, the sign of the friction force will become opposite, and thus, a jump change in the bifurcation diagram occurs.  
It can be seen from Figure 4b that when the friction force increases to 0.3, the critical flow velocity of the pipe system will increase to 7.28. The results given in Figure 4c are for \(f_{n}=0.5\), showing that the critical flow velocity increases to 8.11. Therefore, as the friction force acting on the pipe increases, the critical flow velocity of the pipe system will also increase, which is beneficial to improving the stability of the system. It can also be understood that applying a friction force to the cantilevered pipe is somehow equivalent to adding a loose constraint or a form of support to the pipe. The larger the friction force, the greater the energy consumed by the system can be achieved. Hence, the stability of the pipe with dry friction would be enhanced.  
When the flow velocity in the pipe exceeds the critical value, the pipe would produce relatively large-amplitude vibration. Figure 5 shows the time history curve, oscillation shapes, and phase portrait of the response of the pipe when the flow velocity is 7.5 under the condition defined in Figure 4b. It can be seen from Figure 5a that the vibration of the pipe is steady in this case, and the amplitude is about 0.5. Figure 5b shows the oscillation shapes of the cantilevered pipe at this flow velocity, indicating that the response of the pipe mainly contains the first- and second-mode components of a cantilevered beam. The phase diagram given in Figure 5c shows that the pipe vibrates in the form of a limit cycle motion.