## 2 Dynamic Model of Drilling System
### Taylor Series Expansion of the Nonlinear Cutting Force  
Stability analysis of system with state-dependent delay is more complicated than the system with constant time delay, because the solution of the system is non-differentiable. At the same time, the traditional numerical method cannot be used directly due to the implicit form of the state-dependent delay. The SDD model cannot be completely linearized. So a linear differential equation needs to be sought that has a common dynamic characteristic with SDD. [PERSON] and [PERSON] (2007) linearized the regular SD-DDEs, [PERSON] (2007) linearized the SD-DDEs of the time period. In this paper, the Taylor series was used to linearize the implicit SDD.  
To analyze the Hopf bifurcation characters in the SDD and CD model of the drilling system, the implicit SDD was successfully converted into an explicit form through a series of expansion procedures. Assuming that small displacement disturbances \(\ddot{\zeta}\) and \(\eta\) occur in the \(z\) and \(\phi\) directions of the drilling system, respectively, and the displacements in the dynamic equations can be expressed as:  
\[z=\varepsilon\ddot{\zeta},\phi=\varepsilon\eta. \tag{25}\]  
The series expansion for the SDD at the equilibrium point (\(\varepsilon=0\)) can be expressed as ([PERSON] and [PERSON], 2007):  
\[\begin{array}{ll}\tau_{n}(\tau)&=\tau_{n0}[1-\alpha(\phi(\tau)-\phi_{\tau}) ]=\tau_{n0}-\epsilon(\tau_{n0}\alpha\eta_{t})+\varepsilon^{2}\big{(}\tau_{n 0}^{2}\alpha^{2}\eta_{t}\dot{\eta}_{\tau}\big{)}\\ &-\varepsilon^{3}\Big{(}\tau_{n0}^{3}\alpha^{3}\eta_{t}\dot{\eta}_{\tau}^{2}- \frac{1}{2}\tau_{n0}^{3}\alpha^{3}\eta_{t}^{2}\ddot{\eta}_{\tau}\Big{)}\end{array}, \tag{26}\]  
where \(\varepsilon\) is a small, positive parameter, \(\alpha=\frac{N}{2\pi}\), \(\eta_{\tau}=\eta(\tau-\tau_{n0})\), \(\eta_{t}=\eta(\tau)-\eta(\tau-\tau_{n0})\).  
Substitution of Equations (25) and (26) into Equations (20) and (21) yields:  
\[\ddot{\xi}+Q_{A}^{-1}v^{-1}\dot{\xi}+(1+\psi k_{r})v^{-2}\xi-\rho \psi k_{r}v^{-2}\eta-\psi k_{r}v^{-2}\xi_{\tau}+\psi k_{r}\rho v^{-2}\eta_{\tau} =-\psi k_{r}v^{-2}f \tag{27}\] \[\ddot{\eta}+Q_{T}^{-1}\ddot{\eta}+(1-\rho\psi)\eta+\psi\ddot{\xi}- \psi\xi_{\tau}+\psi\rho\eta_{\tau}=-\psi f \tag{28}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Quantities** & **Variable** & **Value** & **Unit** \\ \hline Feed velocity & \(V_{0}\) & (2–5) \(\times 10^{2}\) & mm/s \\ Rotational velocity & \(\Omega_{0}\) & 15–50 & rad/s \\ control parameter & \(\rho\) & 5–35 & mm/rad \\ \hline \end{tabular}
\end{table}
Table 2: Control parameters of the deep hole drilling.  
\[\begin{array}{ll}f&=\varepsilon\Big{[}\rho\tau_{n0}\alpha\eta_{t}\dot{\eta}_{\tau}- \tau_{n0}\alpha\eta_{t}\dot{\xi}_{\tau}\Big{]}+\varepsilon^{2}\Big{[}\rho\Big{(} \frac{1}{2}\tau_{n0}^{2}\alpha^{2}\eta_{t}^{2}\ddot{\eta}_{\tau}-\tau_{n0}^{2} \alpha^{2}\eta_{t}\dot{\eta}_{\tau}\Big{)}\Big{]}\\ &-\varepsilon^{2}\Big{[}\frac{1}{2}\tau_{n0}^{2}\alpha^{2}\eta_{t}^{2}\ddot{ \xi}_{\tau}-\tau_{n0}^{2}\alpha^{2}\eta_{t}\dot{\xi}_{\tau}\dot{\eta}_{\tau} \Big{]}\end{array}. \tag{29}\]  
Defining a state vector \(\mathbf{x}=\big{[}x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\big{]}^{\mathrm{T}}=\Big{[}\dot{\xi}\), \(\ddot{\xi}\), \(\dot{\eta}\), \(\eta\Big{]}^{\mathrm{T}}\) and rewriting the equations in vector form:  
\[\dot{\mathbf{x}}(\tau)-\mathbf{A}_{\mathbf{SDD}}\mathbf{x}(\tau)-\mathbf{B}_{\mathbf{SDD}}\mathbf{x}(\tau- \tau_{n0})=\mathbf{F}_{\mathbf{NS}}. \tag{30}\]  
The state vector equation of the CD model is:  
\[\dot{\mathbf{x}}(\tau)-\mathbf{A}_{\mathbf{CD}}\mathbf{x}(\tau)-\mathbf{B}_{\mathbf{CD}}\mathbf{x}(\tau- \tau_{n0})=\mathbf{F}_{\mathbf{NC}}, \tag{31}\]  
where the coefficient matrices \(\mathbf{A}_{\mathbf{SDD}},\ B_{\mathbf{SDD}},\ A_{\mathbf{CD}},\ B_{\mathbf{CD}}\) and the non-linear forcing vector are given in Appendix A.