## 4 Validation and Results
The equation mean squared error (MSE) = \(\frac{1}{d}\sum_{k=1}^{d}x\left(k\right)^{2}\) is implemented to calculate the average vibration minimization results and is displayed in Table 2. In the equation, \(x\left(k\right)\) stands for chatter and \(d\) illustrates the total amount of data. In the Table 2, T1 is type-1 and T2 is type-2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Units** & **Parameter** & **Value** & **Units** \\ \hline \(m_{x}\) & 20 & kg & \(m_{y}\) & 20 & kg \\ \(c_{x}\) & 1200 & Ns/m & \(c_{y}\) & 4300 & Ns/m \\ \(k_{x}\) & \(7.2\times 10^{6}\) & N/m & \(k_{y}\) & \(6.48\times 10^{7}\) & N/m \\ \(\xi_{1}\) & \(6700\times 10^{9}\) & N/m\({}^{3}\)Figure 4: Tool vibration along \(x\)-direction using DSMC. Figure 5: Tool vibration along \(y\)-direction using DSMC. Figure 6: Tool vibration along \(x\)-direction using D-PID. Figure 7: Tool vibration along \(y\)-direction using D-PID. Figure 8: Tool vibration along \(x\)-direction using DSMC-T1 fuzzy. The results of percentage vibration minimization of all the controllers are calculated utilizing the MSE indicator. The percentage vibration mitigation using DSMC is 40.88% along the \(x\) component and 36.46% along the \(y\) component, whereas D-PID reduces the vibration by 51.58% along the \(x\) component and 52.89% along the \(y\) component. The percentage vibration mitigation using DSMC-T1 fuzzy is 70.20% along the \(x\) component and 70.63% along the \(y\) component. Finally, the percentage vibration mitigation using DSMC-T2 fuzzy is 79.02% along the \(x\) component and 78.2% along the \(y\) component. So it validated that the implementation of the type-2 fuzzy system in DSMC made it perform better than the DSMC-T1 fuzzy controller. The outcome of percentage vibration suppression depicts that the type-2 fuzzy system performed better than the type-1 fuzzy system. In general, the DSMC-T2 fuzzy controller performed better than all the controllers used in this research. A type-2 fuzzy PID controller was used to mitigate the chatter vibration in the milling process [56]. The plots depicting the vibration attenuation using the type-2 fuzzy PID controller are illustrated in Figures 12 and 13. The MSE results reveal that the average vibration attenuation with the type-2 fuzzy PID controller along the \(x\)-direction is 80% in comparison to 79.02% with DSMC-T2 fuzzy, whereas \begin{table}
\begin{tabular}{l c c c c c} \hline
**Direction** & **No ctrl.** & **DSMC** & **D-PID** & **DSMC-T1 Fuzzy** & **DSMC-T2 Fuzzy** \\ \hline \(x\)-direction & 0.3823 & 0.2260 & 0.1851 & 0.1172 & 0.0802 \\ \(y\)-direction & 0.3702 & 0.2352 & 0.1744 & 0.1087 & 0.0807 \\ \hline \end{tabular}
\end{table} 
Table 2: Average vibration attenuation, mean squared error (MSE) indicator. Figure 11: Tool vibration along \(y\)-direction using DSMC-T2 fuzzy. Figure 10: Tool vibration along \(x\)-direction using DSMC-T2 fuzzy. Figure 9: Tool vibration along \(y\)-direction using DSMC-T1 fuzzy. along the \(y\)-direction with the type-2 fuzzy PID controller is 76.3% in comparison to 78.2% with the DSMC-T2 fuzzy. So it is clear that both the controllers performed well and their performances are almost equal. The advantages of using DSMC-T2 fuzzy over type-2 fuzzy PID controller are: (a) It is effective in terms of robustness against the changes in the parameters with external disturbances and can be implemented without the knowledge of system parameters; (b) the computational cost of the type-2 fuzzy PID controller is bigger than the DSMC-T2 fuzzy. For superior vibration mitigation, it is very important to place the damper in a proper position. This work demands that the vibration attenuation should be accomplished along the \(x\) and \(y\) components and hence suitably two AVDs need to be installed along the \(x\) and \(y\) axes, separately. However, in this research, the setup is made cost effective by installing a single AVD in an inclined position to control the forces along the \(x\) and \(y\) axes.