# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 4 Result and Conclusions  
The minimum value of the function in Equation (35) can easily be identified with normal MATLAB (R2022a) operations to find the minimum of a 4D matrix: \(\min[g_{i,\min}]_{4D}\). Once the minimum value \(g_{i,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})\), which corresponds to total displacement defined by Equation (31) has been found, the coefficients that define the minimum value of the amplitude of displacement can easily be identified an indexed, and the combination of values of control parameters containing this number can be noted as \(P_{\min}\), \(\omega_{\min}\), \(\varphi_{\min}\), and \(t_{\min}\). The chosen points within the defined ranges of the 4D matrix are limited only by the amount of computer memory used in the analysis. For the analysis and to search for the necessary control values, the following ranges of reference were considered: \(P_{r}=[0...1000]\ N\), \(\omega_{r}=[-300...300]\ s^{-1}\), \(\varphi_{r}=[-27...27]\ radian\), \(t_{r}=[0...5]\ s\). The minimum value found in this situation in **Case 1** (with the numerical values considered above), was \(g_{1,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})=3.26\cdot 10^{-12}m\). The corresponding control values obtained in this case were \(P_{\min}=212.12\ N\ \omega_{\min}=293.94\ s^{-1}\), \(\varphi_{\min}=4.25\ radi\), and \(t_{\min}=2.27\ s\). A representation of the \(g_{1,\min}\) function, according to the control parameters \(P_{r},\omega_{r},\varphi_{r},t_{r}\), is shown in Figure 7 for four combinations of variables. The minimum value found in this situation in **Case 2** (with the same numerical values as previously considered) was \(g_{2,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})=1.99\cdot 10^{-13}m\). The corresponding control values obtained in this case were \(P_{\min}=20.20\,N\), \(\omega_{\min}=124.24\)\(s^{-1}\), \(\varphi_{\min}=2.73\,rad\), and \(t_{\min}=4.55\) s. A representation of the \(g_{2,\min}\) function according to the control parameters \(P_{r},\omega_{r},\varphi_{r},t_{r}\) is shown in Figure 8 for two combinations of variables. Figure 8: Variations in control parameters in Case 2, for two different situations and total displacement \(g_{2,\min}\) depending of the variation of: (**a**) parameters \(\varphi_{r}\) and \(P_{r}\); (**b**) parameters \(\varphi_{r}\) and \(\omega_{r}\). Figure 7: Variations in control parameters in Case 1, in four different situations and total displacement \(g_{1,\min}\) depending on the variation of: (**a**) parameters \(\omega_{r}\) and \(P_{r}\); (**b**) parameters \(\varphi_{r}\) and \(P_{r}\); (**c**) parameters \(\varphi_{r}\) and \(\omega_{r}\); (**d**) parameters \(t_{r}\) and \(P_{r}\). The minimum value found in this situation in **Case 3** (with the previously considered numerical values) was \(g_{3,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})=8.87\cdot 10^{-12}\)m. The corresponding control values obtained were \(P_{\min}=20.20\) N, \(\omega_{\min}=293.94\) s\({}^{-1}\), \(\varphi_{\min}=-0.063\) rad, and \(t_{\min}=0.91\) s. A representation of the \(g_{3,\min}\) function according to the control parameters \(P_{r},\omega_{r},\varphi_{r},t_{r}\) is shown in Figure 9 for two combinations of variables. The control coefficients found in each case were included in the analyses presented above in Figures 3c-5c. Figure 10 shows the effect of the weighting control by highlighting curves with and without the control. Figure 10: The effect of the weighting control on the amplitude of displacement variation, showing results with and without the control for (**a**) Case 1, (**b**) Case 2, and (**c**) Case 3. Figure 9: Variations in control parameters in Case 3 for two situations and total displacement \(g_{3,\min}\) depending of the variation of: (**a**) parameters \(\varphi_{r}\) and \(P_{r}\); (**b**) parameters \(\varphi_{r}\) and \(\omega_{r}\). The blue curves, \"without control\", are the ones obtained previously (see Section 2).