# Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction
## 2 The Dynamic Model and Numerical Methodology  
### The Equation of Motion  
A dynamic model for a drill string system, conveying pulsating fluid on a simply supported pipe, is developed in this section. The equations of motion, without taking into account the effect of motion constraints, are provided in [34; 35; 36]. As shown in Figure 1, based on the motion equation, considering the effect of support constraints [37], the cubic nonlinear spring forces \(F_{\mathrm{a}}(y)\) and \(F_{\mathrm{b}}(y)\) are used to simulate the reaction force [38], and the fluid-structure coupling motion differential equation of the drill string system is obtained, as follows:  
\[\begin{array}{c}(m_{\mathrm{f}}+m_{\mathrm{d}})\frac{\partial^{2}\gamma}{ \partial\xi^{2}}+2m_{\mathrm{f}}U\frac{\partial^{2}\gamma}{\partial\alpha \partial t}+\left(1+a\frac{\partial}{\partial t}\right)EI\frac{\partial^{4}y }{\partial x^{4}}+\\ \left[m_{\mathrm{f}}U^{2}+m_{\mathrm{f}}\frac{\partial U}{\partial t}(l-x)- \left(1+a\frac{\partial}{\partial t}\right)\frac{EA_{\mathrm{b}}}{2T}\int_{0} ^{1}\left(\frac{\partial y}{\partial x}\right)^{2}dx\right]\frac{\partial^{2 }y}{\partial x^{2}}\\ +F_{\mathrm{a}}(y)\delta(x-x_{\mathrm{a}})+F_{\mathrm{b}}(y)\delta(x-x_{ \mathrm{b}})=0\end{array} \tag{1}\]  
where \(m_{\mathrm{f}}\) represents the fluid mass per unit length, \(m_{\mathrm{d}}\) is the mass of the drill string per unit length, \(U\) is the velocity of the fluid in the drill string, \(E\) is Young's modulus, \(A\) is the cross-sectional area of the drill string and \(\delta(x-x_{0})\) is the Dirac delta function. Furthermore, \(l\) is the length of the drill string and \(l\) denotes the movement in the cross-sectional area. \(a\) is the viscoelastic damping coefficient, \(x_{\mathrm{a}}\) and \(x_{\mathrm{b}}\) represent the coordinates along the centerline of the drill string, \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) represent the support stiffness and \(y(x,t)\) is the lateral vibration displacement function of position \(x\) and time \(t\). \(F_{\mathrm{a}}(y)\) and \(F_{\mathrm{b}}(y)\) represent the effects of the motion constraints on the drill string.  
\[F_{\mathrm{a}}(y) =K_{\mathrm{a}}y^{3} \tag{2}\] \[F_{\mathrm{b}}(y) =K_{\mathrm{b}}y^{3} \tag{3}\]  
Furthermore, several dimensionless parameters can be defined as follows:  
\[\begin{array}{c}k_{\mathrm{a}}=\frac{K_{\mathrm{a}}l^{5}}{EI},\tau=\left( \frac{EI}{m_{\mathrm{f}}+m_{\mathrm{d}}}\right)^{\frac{1}{2}}\frac{l}{l^{2}},u=\left(\frac{m_{\mathrm{f}}}{EI}\right)^{\frac{1}{2}}Ul,\beta=\frac{m_{ \mathrm{f}}}{m_{\mathrm{f}}+m_{\mathrm{d}}}\kappa=\frac{AI^{2}}{2l}k_{\mathrm{ b}}=\frac{K_{\mathrm{b}}l^{5}}{EI},\\ \xi=\frac{\chi}{l},\eta=\frac{\psi}{l}\alpha=\left(\frac{EI}{m_{\mathrm{f}}+m_ {\mathrm{d}}}\right)^{\frac{1}{2}}\frac{a}{l^{2}},\xi_{\mathrm{a}}=\frac{\chi _{\mathrm{a}}}{l},\xi_{\mathrm{b}}=\frac{\chi_{\mathrm{b}}}{l}\end{array} \tag{4}\]  
Substituting Equation (4) into Equation (1), the dimensionless equation form of nonlinear dynamics can be rewritten as:  
\[\begin{array}{c}\alpha\frac{\partial^{5}\eta}{\partial x^{4}\partial\tau}+ \frac{\partial^{4}\eta}{\partial x^{4}}+\left[u^{2}+\beta^{\frac{1}{2}}\frac{ \partial u}{\partial t}(1-\xi)\right]\frac{\partial^{2}\eta}{\partial x^{2}}+ 2\beta^{\frac{1}{2}}u\frac{\partial^{2}\eta}{\partial x^{2}\tau}\\ +\frac{\partial^{2}\eta}{\partial x^{2}}-\frac{\partial^{2}\eta}{\partial x^{2 }}f_{0}^{2}\left(\frac{\partial\eta}{\partial x^{2}}\right)^{2}d\xi-2\alpha \frac{\partial^{2}\eta}{\partial x^{2}}f_{0}^{1}\frac{\partial\eta}{ \partial x^{2}}\frac{\partial^{2}\eta}{\partial x^{2}}d\xi\\ +k_{\mathrm{a}}\eta^{3}\delta(\xi-\xi_{\mathrm{a}})+k_{\mathrm{b}}\eta^{3} \delta(\xi-\xi_{\mathrm{b}})=0\end{array} \tag{5}\]  
Moreover, assuming that the flow of fluid through the drill string is sinusoidal,  
\[u=u_{0}(1+\mu\sin\omega\tau) \tag{6}\]  
where \(\omega\) is the forcing frequency, \(u_{0}\) is the flow velocity, and \(\mu\) is the perturbation amplitude.  
Figure 1: Schematic diagram of a simply supported drill string with constraints.