# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 2 Type-2 Fuzzy Logic Modeling Technique  
The vibration in the drilling tool system can be resolved along three components (4 DOF), which is shown in Figure 1.  
The lateral vibrations along two components (\(x,y\)), axial vibration (\(z\)) as well as torsional vibration (\(\theta\)) associated with the drilling tool are modeled as [48; 49]:  
\[M_{dr}\ddot{X}+C_{dr}\dot{X}+K_{dr}X=F_{ct} \tag{1}\]  
here, the vibration is represented by \(X\), and the cutting force is illustrated by \(F_{ct}\), stated as:  
\[X=\left\{x\ y\ z\theta\right\}^{T} \tag{2}\]  
Figure 1: Three modes of vibration (4 DOF) in drilling tools. In addition,  
\[M_{dr}=\left[ \begin{array}{cccc}m_{xx}&0&0&0\\ 0&m_{yy}&0&0\\ 0&0&m_{zz}&m_{\varnothing}\\ 0&0&m_{\varnothing}&m_{\varnothing\theta}\\ \end{array} \right]\]  
\[C_{dr}=\left[ \begin{array}{cccc}c_{xx}&0&0&0\\ 0&c_{yy}&0&0\\ 0&0&c_{zz}&c_{\varnothing}\\ 0&0&c_{\varnothing z}&c_{\theta\theta}\\ \end{array} \right]\]  
\[+2\psi\left[ \begin{array}{cccc}0&-m_{xx}&0&0\\ m_{yy}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array} \right] \tag{3}\]  
\[K_{dr}=\left[ \begin{array}{cccc}\bar{K}_{xx}&0&0&0\\ 0&\bar{K}_{yy}&0&0\\ 0&0&\bar{K}_{zz}&\bar{K}_{z\theta}\\ 0&0&\bar{K}_{\theta z}&\bar{K}_{\theta\theta}\\ \end{array} \right]\]  
\[+\psi\left[ \begin{array}{cccc}0&-c_{xx}&0&0\\ c_{yy}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array} \right]\]  
In addition, \(\bar{K}_{xx}=k_{xx}-m_{xx}^{2}\psi^{2}\bar{K}_{yy}=k_{yy}-m_{yy}^{2}\psi^{2}\), \(\bar{K}_{zz}=k_{zz}-m_{zz}^{2}\psi^{2}\), \(\bar{K}_{z\theta}=k_{z\theta}-m_{z\theta}^{2}\psi^{2}\), \(\bar{K}_{\theta z}=k_{\theta z}-m_{\theta z}^{2}\psi^{2}h\) and \(\bar{K}_{\theta\theta}=k_{\theta\theta}-m_{\theta\theta}^{2}\psi^{2}\). Again, mass \(M_{dr}\), damping matrix = C\({}_{dr}\), stiffness matrix \(K_{dr}\), and spindle speed dependent = \(\psi\). Again, (\(m_{xx}\), \(m_{yy}\) ), (\(k_{xx}\), \(k_{yy}\) ) and (\(c_{xx}\), \(c_{yy}\)) are the modal masses, stiffness and damping constants of the most flexible mode along two main components (\(x\),\(y\)) of the drill bit, respectively. The vector associated with the cutting force can be represented as follows:  
\[F_{ct}=\left\{ \begin{array}{c}F_{xd}\\ F_{yd}\\ F_{zd}\\ F_{\theta d}\\ \end{array} \right\}=\bar{G}X(t)+\bar{G}_{T}X(t-T)+\bar{G}_{P}\dot{X}(t) \tag{4}\]  
In addition,  
\[\left\{ \begin{array}{c}\bar{G}=\sum\limits_{i=1}^{2}\sum\limits_{j=1}^{N_{c }}T_{\varphi}G^{e}\\ \bar{G}_{T}=\sum\limits_{k=1}^{2}\sum\limits_{j=1}^{N_{c}}T_{\varphi}G_{T}^{e }\\ \bar{G}_{P}=\sum\limits_{k=1}^{2}\sum\limits_{j=1}^{N_{c}}T_{\varphi}G_{P}^{e }\\ \end{array} \right\} \tag{5}\]  
The diagonal coefficient matrices are \(\bar{G},\bar{G}_{T}\), and \(\bar{G}_{P}\)[50]. The nonlinear nature of the cutting forces is an important factor and has to be handled in an appropriate manner. The governing equation associated with the drilling process can be achieved as follows using Equations (1) and (4):  
\[ \begin{array}{c}M_{dr}\ddot{X}(t)+C_{dr}\dot{X}(t)+K_{dr}X(t)=\bar{G}X(t)\\ +\bar{G}_{T}X(t-T)+\bar{G}_{P}\dot{X}(t)\end{array}  \tag{6}\]Here, time delay is signified by \(T\). A controller is used to generate the control forces required to control the drilling operation. Discretizing Equation (6) with a control force is:  
\[ \begin{array}{c}M_{dr}\ddot{X}(t)+C_{dr}\dot{X}(t)+K_{dr}X(t)=\bar{G}X(t)\\ +\bar{G}_{T}X(t-T)+\bar{G}_{P}\dot{X}(t)+U_{ctrl}(t)\end{array}  \tag{7}\]  
and the control is illustrated as \(U_{ctrl}=\left\{ \begin{array}{c}u_{xd}\\ u_{yd}\\ u_{zd}\\ u_{\theta d}\end{array} \right\}\). The state space form of Equation (7) is illustrated as:  
\[\dot{z}(t)=A_{i}z(t)+B_{i}lL_{ctrl}(t)+D_{i}v(t) \tag{8}\]  
where  
\[ \begin{array}{c}z=\left[X^{T}\ \dot{X}^{T}\right]^{T},v(t)\in z(t-T)\\ A_{i}=\left[\begin{array}{cc}0&I\\ M_{dr}^{-1}\bar{G}-M_{dr}^{-1}K_{dr}&M_{dr}^{-1}\bar{G}_{P}-M_{dr}^{-1}C_{dr} \end{array}\right]\\ D_{i}=\left[\begin{array}{cc}0&0\\ M_{dr}^{-1}\bar{G}_{T}&0\end{array}\right]\end{array}