## Model
### Equilibrium Equation  
To derive the dynamic equation for the vibration process of a drill-string, a small drill-string element \(ds\) is selected and analyzed. The element is considered in natural curvilinear coordinates (\(\stackrel{{\rightarrow}}{{e}}_{t}\), \(\stackrel{{\rightarrow}}{{e}}_{n}\), \(\stackrel{{\rightarrow}}{{e}}_{b}\)), and is subjected to various forces, such as internal tension force \(\stackrel{{\rightarrow}}{{T}}\), normal contact force \(\stackrel{{\rightarrow}}{{F}}\), friction force \(F_{f}\), damping force, and buoyant weight of drilling (as displayed in Figure 1). By balancing these forces, we can arrive at an equation that describes the dynamic equilibrium condition [29]:  
\[ \begin{split}\stackrel{{\rightarrow}}{{T}}(s+ds,t )-\stackrel{{\rightarrow}}{{T}}(s,t)+\stackrel{{ \rightarrow}}{{F}}\Bigl{(}s+\frac{ds}{2},t\Bigr{)}ds-[F_{f}(s,t)+c(s+ \frac{ds}{2},t)\frac{\partial n(s+\frac{ds}{2},t)}{\partial t}]ds\stackrel{{ \rightarrow}}{{e}}_{t}\\ +\delta s\Bigl{(}s+\frac{ds}{2}\Bigr{)}ds\stackrel{{ \rightarrow}}{{e}}_{g}=\rho(s+\frac{ds}{2})A(s+\frac{ds}{2})ds\frac{ \partial^{2}u(s+\frac{ds}{2},t)}{\partial t^{2}}\stackrel{{ \rightarrow}}{{e}}_{t}\end{split}  \tag{1}\]where \(T_{t}(s,t)\), \(T_{n}(s,t)\), and \(T_{b}(s,t)\) are internal forces acting on cross-section in \(\overset{\rightarrow}{e}_{t}\), \(\overset{\rightarrow}{e}_{n}\), and \(\overset{\rightarrow}{e}_{b}\) direction, respectively; \(F_{n}\), \(F_{b}\) are the lateral contact force acting on drill-string in principal normal \(\overset{\rightarrow}{e}_{n}\) and binormal direction \(\overset{\rightarrow}{e}_{b}\), respectively. The following simpler equilibrium equation can be derived from the Taylor expansion in (\(s\), \(t\)) of variables by omitting higher order terms:  
\[\frac{\partial\overset{\rightarrow}{T}(s,t)}{\partial s}+\overset{ \rightarrow}{F}(s,t)-[F_{f}(s,t)+c(s,t)\frac{\partial u}{\partial t}]\overset{ \rightarrow}{e}_{t}+g_{s}(s)\overset{\rightarrow}{e}_{g}=\rho(s)A(s)\frac{ \partial^{2}u(s,t)}{\partial t^{2}}\overset{\rightarrow}{e}_{t} \tag{2}\]  
Separating the internal force \(\overset{\rightarrow}{T}\) and distributed lateral contact force \(\overset{\rightarrow}{F}\) into components in natural curvilinear coordinates (\(\overset{\rightarrow}{e}_{t}\), \(\overset{\rightarrow}{e}_{n}\), \(\overset{\rightarrow}{e}_{b}\)):  
\[\overset{\rightarrow}{T}(s,t)=T_{t}(s,t)\overset{\rightarrow}{e}_{t}+T_{n}( s,t)\overset{\rightarrow}{e}_{n}+T_{b}(s,t)\overset{\rightarrow}{e}_{b} \tag{3}\]  
\[\overset{\rightarrow}{F}(s,t)=F_{n}(s,t)\overset{\rightarrow}{e}_{n}+F_{b}( s,t)\overset{\rightarrow}{e}_{b} \tag{4}\]  
where \(T_{t}(s,t)\),\(T_{n}(s,t)\), and \(T_{b}(s,t)\) are internal forces acting on cross-section in \(\overset{\rightarrow}{e}_{t}\), \(\overset{\rightarrow}{e}_{n}\), and \(\overset{\rightarrow}{e}_{b}\) direction, respectively; \(F_{n}\), \(F_{b}\) are the lateral contact forces acting on drill-string in principal normal \(\overset{\rightarrow}{e}_{n}\) and binormal direction \(\overset{\rightarrow}{e}_{b}\), respectively. The Frenet-Serret formulas can be used to describe the centerline of the borehole [15]:  
\[ \begin{array}{c}\overset{\rightarrow}{e}_{g}\cdot\overset{\rightarrow}{e }_{t}=\cos\overset{\rightarrow}{\alpha}\\ \overset{\rightarrow}{e}_{g}\cdot\overset{\rightarrow}{e}_{n}=\frac{k_{b}} {k_{b}}\sin\overset{\rightarrow}{\alpha}\\ \overset{\rightarrow}{e}_{g}\cdot\overset{\rightarrow}{e}_{b}=-\frac{k_{b} }{k_{b}}(\sin\overset{\rightarrow}{\alpha})^{2}\\ \overset{\rightarrow}{e}_{g}=-\cos\overset{\rightarrow}{\alpha}\overset{ \rightarrow}{e}_{t}+\frac{k_{b}}{k_{b}}\sin\overset{\rightarrow}{\alpha} \overset{\rightarrow}{e}_{n}+\frac{k_{b}}{k_{b}}\sin^{2}\overset{\rightarrow} {\alpha}\overset{\rightarrow}{e}_{b}\end{array}