# Effect of Coriolis Force on Vibration of Annulus Pipe
## 3 Methodology
### FEM Approach
#### 3.2.1 Fixed Pipe  
Let us now consider a pipe fixed to its end nodes and defined by two finite elements. These two elements have the same length \(L\) and the total amount of degrees of freedom is two, as shown in Figure 3.  
Figure 1: Methodology flow chart.  
Figure 2: Fluid moving through a pipe.  
The motion equation becomes  
\[m\left(\begin{matrix}\frac{26L}{35}&0\\ 0&\frac{2L^{3}}{105}\end{matrix}\right)\left(\begin{matrix}\tilde{w}_{2}\\ \dot{\phi}_{2}\end{matrix}\right)+2\rho v\left(\begin{matrix}0&-\frac{L}{5}\\ \dot{\phi}_{2}\end{matrix}\right)\left(\begin{matrix}\tilde{w}_{2}\\ \dot{\phi}_{2}\end{matrix}\right)+\left(EJ\left(\begin{matrix}\frac{24}{L}&0\\ 0&\frac{8}{L}\end{matrix}\right)-\rho v^{2}\left(\begin{matrix}\frac{12}{5L}&0 \\ 0&\frac{4L}{15}\end{matrix}\right)\right)\left(\begin{matrix}w_{2}\\ \dot{\phi}_{2}\end{matrix}\right)=\left(\begin{matrix}F_{2}\\ M_{2}\end{matrix}\right) \tag{8}\]  
The system is now ready to be solved by determining the values of natural frequencies while the velocity changes. In particular, the load vector is set equal to zero while the displacement \(w\) and the rotation \(\phi\) are defined as an exponential function in the complex domain (\(\mathbb{C}\)) multiplied to a unknown constant (\(W\) for displacement and \(\Phi\) for rotation):  
\[\left\{\begin{matrix}w_{2}=We^{i\omega t}\\ \dot{w}_{2}=\partial_{t}w_{2}\\ \dot{w}_{2}=\partial_{t,t}w_{2}\end{matrix}\right.\left\{\begin{matrix}\phi_{ 2}=\Phi e^{i\omega t}\\ \dot{\phi}_{2}=\partial_{t,t}\phi_{2}\\ \dot{\phi}_{2}=\partial_{t,t}\phi_{2}\end{matrix}\right. \tag{9}\]  
Substituting Equation (9) in Equation (8) and dividing everything by \(e^{i\omega t}\) results in:  
\[\left(\begin{matrix}-\frac{2(-420 EJW+42L^{2}v^{2}W\rho+L^{4}\omega(7i\rho\Phi+ 13 mW\omega))}{3L^{3}}\\ \frac{8 EJ\Phi}{L}-\frac{2}{105}L(14v^{2}\rho\Phi-21 ivW\rho\omega+L^{2}m\Phi \omega^{2})\end{matrix}\right)=0\]  
that can be transformed in the following homogeneous system collecting the coefficients of \(w\) and \(\phi\)  
\[\left(\begin{matrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{matrix}\right)\left(\begin{matrix}w\\ \phi\end{matrix}\right)=0 \tag{10}\]  
excluding the trivial solution of the Equation (10), it is possible to find the eigenvalues substituting \(\omega^{2}\) with \(\lambda\) and solving  
\[Det[\mathbf{A}^{fix}]=0 \tag{11}\]  
where  
\[\mathbf{A}^{fix}=\left(\begin{matrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{matrix}\right)\]  
leading to  
\[\left\{\begin{matrix}\lambda_{1}=-\frac{1}{26L^{8}m^{2}}7(-840 EJL^{4}m+32L^{6} mv^{2}\rho-21L^{6}v^{2}\rho^{2}+(L^{8}(518400 EJ)^{2}m^{2}\\ -720 EJL^{2}mv^{2}(40m-49\rho)\rho+L^{4}v^{4}\rho^{4}(400m^{2}-1344m\rho+441\rho ^{2})))^{1/2})\\ \lambda_{2}=\frac{1}{26L^{8}m^{2}}7(840 EJL^{4}m-32L^{6}mv^{2}\rho+21L^{6}v^{2} \rho^{2}+(L^{8}(518400 EJ)^{2}m^{2}\\ -720 EJL^{2}mv^{2}(40m-49\rho)\rho+L^{4}v^{4}\rho^{4}(400m^{2}-1344m\rho+441\rho ^{2})))^{1/2})\end{matrix}\right. \tag{12}\]  
The corresponding eigenvector associated to these eigenvalues describes the vibration modes of the structure. For a better representation, it is worth to transform the characteristic polynomial in its dimensionless form, introducing  
\[\theta=\frac{L^{2}v^{2}\rho}{EJ}\]  
Figure 3: Element representation of the pipe fixed to its boundary nodes.  
so the power of the frequency is  
\[\omega^{2}=\frac{EJ}{L^{4}m}(226.2+\theta(-8.165+5.654\eta)\pm 0.0385(2.5 4e^{7}+\] \[10^{6}\theta(-1.410+1.729\eta)+\theta^{2}(19600-65856\eta+21609 \eta^{2})^{1/2}))\]  
and finally:  
\[\Psi=\sqrt{\frac{L^{4}m}{EJ}}\omega=\sqrt{*} \tag{13}\]