# Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction
## 2 The Dynamic Model and Numerical Methodology
### Discretization of the System Model  
In this study, the dynamic model is discretized by [PERSON]'s method, and the vibration displacement \(\eta(\xi,\tau)\) of the drill string can be expressed as:  
\[\eta(\xi,\tau)=\sum_{r=1}^{N}\phi_{r}(\xi)q_{r}(\tau) \tag{7}\]  
where \(\phi_{r}(\xi)\) is the appropriate set of base functions, \(q_{r}(\tau)\) represents the corresponding generalized coordinates, and \(N\) is the total number of models used for the beam consideration. Substituting Equation (7) into Equation (5), multiplying by \(\phi_{l}(\xi)\) and integrating throughout [0, 1], the dimensionless non-linear vibration equation of the drill string system is obtained:  
\[\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}+\mathbf{f}(\mathbf{q})+\mathbf{h}\big{(}\mathbf{q },\dot{\mathbf{q}}\big{)}=0 \tag{8}\]  
where \(\mathbf{C}\) represents stationary damping and \(\mathbf{K}\) represents stiffness matrices. It can be seen from reference [39] that \(N=2\) can obtain a higher calculation accuracy. The elements of \(\mathbf{C}\), \(\mathbf{K},\mathbf{f}\) and \(\mathbf{h}\) are composed of:  
\[\begin{array}{c}C_{ij}={C_{ij}}^{\prime}+\big{(}2\sqrt{\beta}\mu u_{0}\sin \omega\tau\big{)}b_{ij},\\ K_{ij}={K_{ij}}^{\prime}+u_{0}^{2}\big{(}2\mu\sin\omega\tau+\mu^{2}\sin^{2} \omega\tau\big{)}c_{ij}+\big{(}\sqrt{\beta}\mu u_{0}\omega\cos\omega\tau\big{)} \big{(}c_{ij}-d_{ij}\big{)},\\ f_{i}=k_{\text{a}}\bigg{[}\sum\limits_{r=1}^{N}\phi_{r}(\xi_{\text{a}})q_{r} \bigg{]}^{3}\phi_{i}(\xi_{\text{a}})+k_{\text{b}}\bigg{[}\sum\limits_{r=1}^{N }\phi_{r}(\xi_{\text{b}})q_{r}\bigg{]}^{3}\phi_{i}(\xi_{\text{b}}),\\ \bigg{(}\begin{array}{c}h_{1}\\ h_{2}\end{array}\bigg{)}=2\alpha\kappa(q_{1}\ q_{2})\bigg{(}\begin{array}{c}a _{11}\\ a_{22}\end{array}\bigg{)}\bigg{(}\begin{array}{c}\dot{q}_{1}\\ \dot{q}_{2}\end{array}\bigg{)}\bigg{(}\begin{array}{c}a_{11}\\ a_{22}\end{array}\bigg{)}\bigg{(}\begin{array}{c}q_{1}\\ q_{2}\end{array}\bigg{)}\\ +\kappa(q_{1}\ q_{2})\bigg{(}\begin{array}{c}a_{11}\\ a_{22}\end{array}\bigg{)}\bigg{(}\begin{array}{c}q_{1}\\ q_{2}\end{array}\bigg{)}\end{array}\end{array} \tag{9}\]  
where  
\[{C_{ij}}^{\prime}=\alpha\lambda_{j}^{4}\delta_{ij}+2\sqrt{\beta}\mu_{0}b_{ij},{K_{ij}}^{\prime}=\lambda_{j}^{4}\delta_{ij}+u_{0}^{2}c_{ij} \tag{10}\]  
Furthermore, some constants are defined as follows:  
\[\begin{array}{c}a_{11}=-\pi^{2},\ \ \ a_{22}=-4\pi^{2},b_{ij}=\left\{ \begin{array}{cc}\frac{2\lambda_{j}\lambda_{j}}{\left(\lambda_{j}^{2}- \lambda_{j}^{2}\right)}\Big{[}(-1)^{i+j}-1\Big{]},&i\
eq j\\ 0,&i=j\\ \frac{4\lambda_{i}\lambda_{j}^{3}}{\left(\lambda_{j}^{2}-\lambda_{j}^{2} \right)}\Big{[}1-(-1)^{i+j}\Big{]},&i\
eq j\\ \frac{\lambda_{j}}{2},&i=j\end{array}\right\}\end{array} \tag{11}\]  
For the purpose of numerical simulations, the state vector is introduced as follows:  
\[\begin{array}{c}\mathbf{z}=\big{\{}q_{1},q_{2},\dot{q}_{1},\dot{q}_{2}\big{\}}\\ \dot{\mathbf{z}}=\mathbf{A}\mathbf{z}+\mu(\omega\cos\omega\tau\mathbf{B}_{1}+\sin\omega\tau\bm {B}_{2})\mathbf{z}+\mu^{2}\sin^{2}\omega\tau\big{[}\mathbf{B}_{3}]\mathbf{z}+\mathbf{G}(\mathbf{z}) +\mathbf{Q}(\mathbf{z})\end{array} \tag{12}\]  
where  
\[\begin{array}{c}\mathbf{A}=\left[\begin{array}{cc}0&I\\ -{K^{\prime}}_{ij}-{C^{\prime}}_{ij}\end{array}\right]\ \ \,\ \ \ \mathbf{B}_{1}=\left[\begin{array}{cc}0&0\\ -u_{0}\sqrt{\beta}\big{(}c_{ij}-d_{ij}\big{)}&0\end{array}\right]\ \ \,\ \ \ \mathbf{B}_{2}=\left[\begin{array}{cc}0&0\\ -2u_{0}^{2}c_{ij}&-2\sqrt{\beta}\mu_{0}b_{ij}\end{array}\right]\\ \mathbf{B}_{3}=\left[\begin{array}{cc}0&0\\ -u_{0}^{2}c_{ij}&0\end{array}\right],\mathbf{G}=\{0\ \ \ 0\ \ -f_{1}\ \ \ -f_{2}\}^{\text{T}},Q=\{0\ \ \ 0\ \ -h_{1}\ \ -h_{2}\}^{\text{T}}\end{array} \tag{13}\]