# Effect of Coriolis Force on Vibration of Annulus Pipe
## 3 Methodology
### Analytical Approach  
From the studies reported in [29; 30; 31], the analytical expression for the motion equation can be written. Let us start by assuming that the curvature of a pipe in which a fluid flows with a velocity \(v\) depends on the transversal displacement \(w\) caused by its weight,the external forces and the internal forces. Let us assume that \(m\) is the total mass of fluid and pipe and \(\rho\) is the mass of fluid per length span. If the pipe is immersed in the fluid, it is possible to sum the mass of the external fluid to the mass of the pipe, and it can be calculated by multiplying the density of the fluid itself times the area of the considered section. The forces acting on the element \(dx\) of the pipe are:  
* \(m\tilde{w}=\) inertia force due to the vertical acceleration of the pipe;
* \(E\)\(fw^{\prime\prime\prime}=\) elastic forces due to [PERSON]'s theory. \(E\) is the Young module of the material, \(J\) is the inertia moment of the geometry, and the derivatives over time and space of the displacement \(w\) are defined as follows:  
\[\dot{w}=\frac{dw}{dt}\quad,\quad w^{\prime}=\frac{dw}{dx}\quad,\quad w^{ \prime\prime}=\frac{d^{2}w}{dx^{2}}\quad,\quad\omega=\dot{w}^{\prime}=\frac{ d^{2}w}{dtdx}\]  
When a fluid is flowing in a pipe, and it is subjected to the Coriolis acceleration [32] through the mechanical introduction of the apparent rotation into the pipe, the amount of deflecting force generated by the Coriolis inertial effect will be a function of the mass flow rate of the fluid. If a pipe is rotated around a point while a liquid is flowing through it, that fluid will generate an inertial force acting on the pipe that will be at right angles to the direction of the flow. Figure 1 shows the entire methodology, and Figure 2 shows a particle \(dm\) traveling at a velocity \(v\) inside a tube. The tube is rotating around a fixed point \(P\), and the particle is at a distance of one radius \(r\) from the fixed point. The particle moves with an angular velocity \(\omega\) under two components of acceleration, a centripetal acceleration directed toward \(P\), and a Coriolis acceleration acting at right angles  
* \(a_{r}\) (centripetal) = \(\omega^{2}r\)
* \(a_{t}\) (Coriolis) = 2\(\omega v\)  
Moreover, when the pipe deflects, a centripetal force acts on an element \(dx\) of mass \(dm\) that tends to move the pipe back to the equilibrium position. If \(R\) is the curvature radius, the result is:  
\[dF=dm\frac{v^{2}}{R}\approx dmv^{2}w^{\prime\prime}dx\]  
Substituting \(dm=\rho dx\)  
\[dF=-\rho v^{2}w^{\prime\prime}dx\]  
The sign '-' appears because the force \(dF\) with positive curvature points to the opposite direction with respect to the displacement \(w\). Because of that, the forces imposed by the fluid to the pipe per unit length neglecting \(a_{r}\) are:  
\[F_{t}=2\rho v\dot{w}^{\prime}\]  
\[F=\rho v^{2}w^{\prime\prime}\]  
and the equation of motion becomes:  
\[E\frac{\partial^{4}w}{\partial x^{4}}+m\frac{\partial^{2}w}{\partial t^{2}}+ \rho v^{2}\frac{\partial^{2}w}{\partial x^{2}}+2\rho v\frac{\partial^{2}w}{ \partial t\partial x}=0 \tag{1}\]  
Before reducing Equation (1) to its dimensional form, it is possible to write down the integration form that represents the potential energy equation without considering Neumann boundary conditions  
\[\Pi=\frac{1}{2}\int_{0}^{L}\biggl{(}E\biggl{(}\frac{\partial^{2}w}{\partial x ^{2}}\biggr{)}^{2}+\frac{\rho v^{2}}{2}\biggl{(}\frac{\partial w}{\partial x} \biggr{)}^{2}+2\rho v\frac{\partial^{2}w}{\partial t\partial x}+m\frac{ \partial^{2}w}{\partial t^{2}}w\biggr{)}dx+W \tag{2}\]where \(W\) is the work of external loads. Interpolating the displacement field using the Hermitian shape function, separating the variables  
\[ \begin{cases}w(x,t)=\Delta^{T}(t)N(x)\\ \Delta(t)=(w_{1},\phi_{1},w_{2},\phi_{2})\end{cases}  \tag{3}\]  
where \(\Delta(t)\) is the vector of displacements and rotation referred to each node of the finite element.