## 2 Mathematical Modeling of the Drill String System
### Energy Principle and Expression  
The drilling system in this study considers only the directions of axial strain, bending, and torsion in the generalized shaft coordinates (\(X\), \(Y\), \(Z\), \(\theta\), and \(\psi\)). These include two horizontal displacements of the drill at the disk location (\(X\) and \(Y\)), one axial displacement (\(Z\)), one rigid-body rotation (\(\theta\)), and two torsional deflection angles (\(\psi\)). The kinetic energy of the drill string disc is expressed as the sum of the translational and rotational mass imbalance and the kinetic energy of the motor system. The full expression for the derivation step appearing in this article is extensively developed in [14] and is therefore omitted for brevity and can be written as follows:  
\[\begin{array}{l}T=\frac{1}{2}(m_{u}+M)\bigg{(}\dot{X}^{2}+\dot{Y}^{2}+\dot{Z} ^{2}\bigg{)}+\frac{1}{2}I_{D}\big{(}\dot{\theta}+\dot{\psi}\bigg{)}^{2}+\frac{ 1}{2}J_{M}\dot{\theta}^{2}+\frac{1}{2}m_{u}e^{2}\dot{\psi}^{2}+m_{u}e^{2}\dot{ \theta}\dot{\psi}\\ +\frac{1}{2}m_{u}e^{2}\dot{\theta}^{2}\big{(}1+\psi^{2}\big{)}-m_{u}\dot{X} \dot{\theta}\big{[}(e_{x}-\psi e_{y})\sin\theta+\big{(}e_{x}\psi+e_{y}\big{)} \cos\theta\big{]}+m_{u}\dot{Y}\dot{\theta}\big{[}(e_{x}-\psi e_{y})\cos\theta \\ -\big{(}e_{x}\psi+e_{y}\big{)}\sin\theta\big{]}-m_{u}\dot{X}\dot{\psi}\big{(}e_ {x}\sin\theta+e_{y}\cos\theta\big{)}+m_{u}\dot{Y}\dot{\psi}\big{(}e_{x}\cos \theta-e_{y}\sin\theta\big{)}\end{array} \tag{1}\]  
Figure 1: Typical rotary drilling rig. (**a**) Drill string model. (**b**) Simplified schematic model of the drill string.  
where \(J_{M}\) is the inertia of the top rotary system, \(M\) is the drill string mass, \(J_{D}\) is the shaft-disc mass moment of inertia, \(m_{u}\) is the imbalance mass, and the pairs, \(e_{x}\) and \(e_{y}\), are the components of \(e\) in \(x\) and \(y\) coordinates. The inertias are interconnected by linear springs (\(K_{XX}\), \(K_{YY,}\) and \(K_{ZZ}\)) which consist of three translational deformations (\(X\), \(Y\), and \(Z\)). Referring to a large bending deflection, axial movement, and torsional deformation \(\psi\), the potential energy is expressed as:  
\[V=\frac{1}{2}K_{XX}X^{2}+\frac{1}{2}K_{YY}Y^{2}+\frac{1}{2}K_{ZZ}Z^{2}+\frac{1 }{2}K_{\psi\psi}\psi^{2} \tag{2}\]  
where \(K_{\psi\psi}\) is the system torsional stiffness, and \(K_{XX}\), \(K_{YY}\), and \(K_{ZZ}\) are the stiffness coefficients associated with the system degrees of freedom given by:  
\[K_{XX}=K_{YY}=K_{ZZ}=K_{0}-\frac{P(t)\pi^{2}}{2L} \tag{3}\]  
where \(K_{0}\) is the modal drill string stiffness value, and \(P(t)\) is an axially compressive force which is assumed to fluctuate harmonically around a constant mean value of the weight of the drill string \(P_{O}\) and is written as [5]:  
\[-P(t)=P_{o}+P_{f}\sin\Omega_{f}t \tag{4}\]  
where \(P_{f}\) and \(\Omega_{f}\) are, respectively, the amplitude and the frequency of the axial force. The frequency of this periodic component is related to the frequency of the drilling system by \(\Omega_{f}=n\Omega\), where \(n\) is an integer indicating the bit factor, and \(\Omega\) represents the rotary speed of the drill string. The Rayleigh dissipation function defined in the case of a viscous damper and the torsional damping attached to the system can be expressed as:  
\[R=\frac{1}{2}C_{T}\dot{\psi}^{2}+\frac{1}{2}C_{XX}\dot{X}^{2}+\frac{1}{2}C_{ YY}\dot{Y}^{2}+\frac{1}{2}C_{ZZ}\dot{Z}^{2} \tag{5}\]  
where \(C_{T}\) is the system torsional damping, and \(C_{XX}\), \(C_{YY}\), and \(C_{ZZ}\) are the respective degree of freedom damping coefficients. Furthermore, viscous damping torque is taken into account at the upper drive system (\(T_{on}\)) and the bit (\(T_{bt}\)). Dry friction torque (\(T_{Rb}\)) is considered at the drill bit. The extended Lagrange formulation can be therefore used to derive the equation of motion of the damped drill string suspension system.