# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 4 Validation and Results  
The drilling process parameters illustrated in [48] were implemented for drilling process simulation to verify the efficacy of the created theory and to confirm the notion of the new fault detection technique. The parameters are shown in Table 1: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline \(m_{xx}\) & 27 & kg \\ \hline \(m_{zz}\) & 172 & kg \\ \hline \(m_{\theta\theta}\) & 0.13 & kgm\({}^{2}\) \\ \hline \(k_{yy}\) & \(1.3\times 10^{6}\) & N/m \\ \hline \(k_{z\theta}\) & \(1.2\times 10^{12}\) & N/rad \\ \hline \(c_{xx}\) & 2500 & Ns/m \\ \hline \(c_{zz}\) & 5000 & Ns/m \\ \hline \(c_{\theta\theta}\) & 5000 & Nms/rad \\ \hline \(m_{yy}\) & 28.7 & kg \\ \hline \(m_{z\theta}\) & 2.6 & kgm \\ \hline \(k_{xx}\) & \(1.4\times 10^{6}\) & N/m \\ \hline \(k_{zz}\) & \(7.69\times 10^{8}\) & N/m \\ \hline \(k_{\theta\theta}\) & \(5.79\times 10^{9}\) & Nm/rad \\ \hline \(c_{yy}\) & 2500 & Ns/m \\ \hline \(c_{z\theta}\) & 5000 & Ns/rad \\ \hline \(\psi\) & 200 & rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters. _Simulation of the Drilling Process for Verification_  
Equation (7) can be represented along \(x,y,z\) and \(\theta\) components as follows:  
\[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\] \[-\frac{k_{xx}-m_{xx}p^{2}}{m_{xx}}x+2\dot{y}\dot{y}+\dot{y}\frac{c_{ xx}}{m_{xx}}y\] \[\ddot{y}=\frac{F_{y}}{m_{yy}}+\frac{u_{y}}{m_{yy}}-\frac{c_{yy}}{m _{yy}}\dot{y}\] \[-\frac{k_{yy}-m_{yy}p^{2}}{m_{yy}}x-2\dot{y}\dot{x}-\dot{y}\frac{c_ {yy}}{m_{yy}}x\] \[\frac{y}{z}=\frac{F_{z}}{m_{zz}}+\frac{u_{z}}{m_{zz}}-\frac{c_{ zz}}{m_{zz}}\dot{z}\] \[-\frac{k_{zz}-m_{zz}\dot{y}^{2}}{m_{zz}}z-\frac{m_{\theta}}{m_{zz} }\dot{\theta}-\frac{c_{\theta}}{m_{zz}}\dot{\theta}-\frac{k_{\theta}-m_{ \theta}\dot{y}^{2}}{m_{zz}}\theta\] \[\ddot{\theta}=\frac{F_{\theta}}{m_{\theta\theta}}+\frac{u_{ \theta}}{m_{\theta\theta}}\frac{c_{\theta\theta}}{m_{\theta\theta}}\dot{\theta}\] \[-\frac{k_{\theta}-m_{\theta\theta}\dot{y}^{2}}{m_{\theta\theta}} \theta-\frac{m_{\theta z}}{m_{\theta\theta}}\dot{z}-\frac{c_{\theta z}}{m_{ \theta\theta}}\dot{z}-\frac{k_{\theta z}-m_{\theta z}\dot{y}^{2}}{m_{\theta \theta}}z\]  
where \(\dot{x},\dot{y},\dot{z}\) and \(\ddot{\theta}\) are the accelerations along \(x,y,z\) and the \(\theta\) component. In Figures 2 and 3, the block diagrams depict the simulation of the drilling process without and with faults, respectively. By utilizing Equation (78), the acceleration signals along \(x,y,z\), and \(\theta\) directions are obtained. These accelerations are then fed to the numerical integrator for the generation of velocity and position signals. These signals are used for the vibration analysis of the drill bit. The vibration signals along \(x\), \(y\), \(z\), and \(\theta\) directions will vary in the events of the fault in the drill tool. For simplicity, the analysis results along \(x\) and \(y\) directions are demonstrated. The nonlinear cutting forces implemented for the simulation process along \(x\) and \(y\) directions are given by [56]:  
\[F_{x} = +\frac{N}{2\pi}\{\zeta_{1}\Delta x^{3}+\eta_{1}\Delta y^{3}+\zeta _{2}\Delta x^{2} \tag{79}\] \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]0 \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]1 \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]2 \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]3 \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]4 \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]5  
Equation (79) is utilized to generate the cutting forces for the drilling process. The cutting forces for a period of 2s along the components x and y and are represented using Figures 4 and 5, respectively.