## 2 Modeling of the Milling Process with Active Control
#### Active Vibration Damper (AVD) for Active Control Mechanism  
As seen from Figure 2, AVD is placed on the top of the spindle to minimize the tool chattering generated by the external force. The main mechanism behind AVD is that it works as a linear servo actuator. A linear actuator converts the rotary motion of the motor to the required linear motion. The position of the AVD is at the center of mass (CM) and it makes an inclination \(\varphi\) with CM. It is considered to be an effective placement due to the mitigation of spacing problem of the AVD. The efficient placement of damper is cost effective, thus mitigating the requirement of two dampers. The combination of the modeling equation and control force \(\mathbf{u_{c}}\) yields:  
\[M_{m}\ddot{\mathbf{x}_{c}}(t)+C_{m}\dot{\mathbf{x}_{c}}(t)+K_{m}\mathbf{x}_{c }(t)=\mathbf{F}_{m}(\mathbf{t})+\mathbf{u_{c}}-\mathbf{d_{c}} \tag{7}\]  
the control signals impinged on the damper along both axes and is illustrated by \(\mathbf{u_{c}}=\left[u_{cx},u_{cy}\right]^{T}\in\Re^{2\times 1}\); \(\mathbf{d_{c}}=\left[d_{cx},d_{cy}\right]\in\Re^{2\times 1}\) is the combined damping-fricton effect resolved along two axes. Now with the control signals the closed loop Equation (1) can be illustrated as:  
\[\begin{array}{l}m_{mx}\ddot{x}+c_{mx}\dot{x}+k_{mx}x=F_{fx}+u_{cx}-d_{cx}\\ m_{my}\ddot{y}+c_{my}\dot{y}+k_{my}y=F_{fg}+u_{cy}-d_{cy}\end{array} \tag{8}\]  
The damper force \(f_{d}\) is:  
\[f_{d}=m_{d}(\ddot{g}_{i}+\dot{\gamma}_{i}) \tag{9}\]  
where \(m_{d}=\) mass of the damper, \(\ddot{g}_{i}=\) acceleration of the damper, \(\dot{\gamma}_{i}=\) tool acceleration. Moreover, \(\dot{\gamma}_{i}^{2}=a_{i,x}+a_{i,y}\) implies the relative accelerations of the tool associated with both the axes. Now, \(f_{d}\) can be resolved along the \(x\) and \(y\) components as follows:  
\[\begin{array}{l}f_{d}=\frac{m_{d}(\ddot{g}_{i}\cos\varphi+a_{i,x})}{\cos \varphi}\text{ or }f_{d}=\frac{m_{d}(\ddot{g}_{i}\sin\varphi+a_{i,y})}{\sin\varphi}\\ \dot{\gamma}_{i}=\frac{a_{i,x}}{\cos\varphi}=\frac{a_{i,y}}{\sin\varphi}\\ \ddot{x}_{i,x}-a_{i,x}=\ddot{g}_{i}\cos\varphi\text{, }\dot{x}_{i,y}-a_{i,y}= \ddot{g}_{i}\sin\varphi\end{array} \tag{10}\]  
where \(\varphi,\dot{x}_{i,x}\) and \(\ddot{x}_{i,y}\) represent the angle of the damper, relative acceleration of the damper along the \(x\) component and the relative acceleration of the damper along \(y\) component, respectively. Now, \(f_{d}\) is represented mathematically:  
\[f_{d}=\frac{m_{d}}{\cos\varphi}\left(\cos\varphi\ddot{g}_{i}+a_{i,x}\right)= \frac{m_{d}}{\sin\varphi}\left(\sin\varphi\ddot{g}_{i}+a_{i,y}\right) \tag{11}\]  
The control action along the \(x\) and \(y\) directions represented by \(\mathbf{u_{c}}=\left[u_{cx}\ u_{cy}\right]^{T}\) is given by:  
\[\mathbf{u_{c}}=\left[m_{di}\ddot{x}_{i,x}\ m_{di}\ddot{x}_{i,y}\right]^{T} \tag{12}\]  
It is very important to consider the damper friction, which can be resolved as:  
\[\begin{array}{l}d_{cx}=\Lambda\dot{x}_{i,x}+\Gamma m_{d}g\tanh\left[\dot{Y} \dot{x}_{i,x}\right]\\ d_{cy}=\Lambda\dot{x}_{i,y}+\Gamma m_{d}g\tanh\left[\dot{Y}\dot{x}_{i,y}\right] \end{array} \tag{13}\]  
where \(\Lambda\), \(Y\) and \(\Gamma\) are termed as damping coefficients associated with the Coulomb friction [44]. Moreover, \(\mathbf{d_{c}}=[d_{cx}\ d_{cy}]^{T}\). Taking into consideration Equations (8) and (13), the combination of the control methodology with the closed loop system is given by:  
\[\begin{array}{l}m_{mx}\ddot{x}+c_{mx}\dot{x}+k_{mx}x=F_{fx}+u_{cx}-\Lambda \dot{x}_{i,x}-\Gamma m_{d}g\tanh\left[\dot{Y}\dot{x}_{i,x}\right]\\ m_{my}\ddot{y}+c_{my}\dot{y}+k_{my}y=F_{fg}+u_{cy}-\Lambda\dot{x}_{i,y}-\Gamma m _{d}g\tanh\left[\dot{Y}\dot{x}_{i,y}\right]\end{array} \tag{14}\]In Equation (14), the nonlinear terms are \(\Lambda\dot{x}_{i,x}+\Gamma m_{d\mathcal{G}}\tanh\left[\Upsilon\dot{x}_{i,x}\right], \Lambda\dot{x}_{i,y}+\Gamma m_{d\mathcal{G}}\tanh\left[\Upsilon\dot{x}_{i,y} \right],F_{fx}\) and \(F_{fy}\). The intelligent technique is incorporated to deal with the involved nonlinearities.