# Influence of Dry Friction on the Dynamics of Cantilevered Pipes Conveying Fluid
## 2 Theoretical Model  
In this paper, the dynamical system under consideration is shown in Figure 1. The cantilevered pipe is placed vertically and connected to the ground by a dry friction restraint at \(s=s_{d}\). This dry friction restraint would generate transversal friction force at the connection when the pipe is deformed at \(s_{d}\). Suppose that the length of the pipe is \(L\); the mass per unit length of the pipe is \(m^{\prime}\) the bending stiffness is \(EI\), with \(E\) being the Young's modulus and \(I\) the moment of inertia; the Kelvin-Voigt damping coefficient of the pipe is \(E^{*}\), and the acceleration of gravity is \(g\). There is a fluid passing through the slender pipe. The mass per unit length of the fluid is \(M\), and the average flow velocity is \(U\). For simplicity, it is assumed that the pipe moves in a plane. The lateral displacement of the cantilevered pipe along the \(y\) axis is represented by \(y(s,t)\), which is a function of the curvilinear coordinate s and time \(t\)The following assumptions are introduced: (1) the pipe centerline is inextensible; (2) the fluid is incompressible; (3) the moment of inertia and shear deformation of the pipe are ignored; (4) the fluid velocity is uniform and constant; (5) the pipe strain is small, although its lateral displacement may be large. Now introduce the generalized [PERSON]'s principle [9]:  
\[\delta\int_{t_{1}}^{t_{2}}{(T-V)dt}+\int_{t_{1}}^{t_{2}}{\delta W_{f}dt}=\int_{t _{1}}^{t_{2}}\{MU[(\partial t_{L}/\partial t)+U\tau_{L}]\cdot\delta r_{L}\}dt \tag{1}\]  
where the Lagrangian of the system includes the kinetic energy \(T\) and potential energy \(V\) of the cantilevered pipe system, and \(W_{f}\) represents virtual work done by the dry friction. The term on the right side of the Equation (1) represents the influence on the system when the fluid flows into and out of the pipe. The position vector and tangential direction of the pipe are denoted as \(r\) and \(\tau\), respectively, and they are given by  
\[r=xi+yj\,\tau=\frac{\partial x}{\partial s}i+\frac{\partial y}{\partial s}j \tag{2}\]  
where \(i\) and \(j\) are the unit vectors in the \(x\)- and \(y\)-direction, respectively. In addition, the velocities of the pipe and the fluid are as  
\[v_{p}=\frac{\partial r}{\partial t}=\dot{x}i+\dot{y}j\,v_{f}=\big{(}\dot{x}+Ux ^{\prime}\big{)}i+\big{(}\dot{y}+Uy^{\prime}\big{)}j \tag{3}\]  
The total kinetic energy of the pipe and the fluid is expressed as  
\[T=T_{p}+T_{f}=\frac{1}{2}m\int_{0}^{L}{v_{p}^{2}ds}+\frac{1}{2}M\int_{0}^{L}{ v_{f}^{2}ds} \tag{4}\]  
The potential energy due to the deformation of the pipe is  
\[V=\frac{1}{2}EI\int_{0}^{L}{\kappa^{2}\text{d}s}=\frac{1}{2}EI\int_{0}^{L}{ \left(\frac{\partial^{2}y/\partial s^{2}}{\sqrt{1-\left(\partial y/\partial s \right)^{2}}}\right)^{2}\text{d}s} \tag{5}\]  
The dry friction force exists as a non-conservative force of the system, and its work is given by  
\[W=-\int_{0}^{y(s_{d},t)}{F(\dot{y})\text{d}y(s_{d},t)} \tag{6}\]  
Figure 1: Schematic diagram of the cantilevered pipe with a dry friction restraint. Substituting Equations (4)-(6) into Equation (1), one can obtain the governing equation for the cantilevered pipe with dry friction:  
\[