# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 4 Result and Conclusions
To see the exact effect of applying the minimum function method, the amplitude variations obtained after applying the vibration control--with the color orange denoting \"with control\"--are added to the same graph. The limitation of this method is that too many broader areas of the duration of coefficients \(P_{r},\omega_{r},\varphi_{r},t_{r}\) response can be delayed, requiring a large number of computations with matrices of large dimensions. In Figure 10, we can see that this search method obtains effective adjustment coefficients, especially for \(\zeta\leq 1\); that is, (a) and (c). As can be seen in Figure 10b, for \(\zeta>1\), the active control has a low efficiency and should be avoided. As previously demonstrated, function \(g_{i,\min}(P_{r},\omega_{r},\varphi_{r},t_{r})\) is time-dependent, which means that if time has a minimum value, \(t_{r}\to 0\) is viewed as the read range of displacement (range of read/sampling rate) of \(x(t)\) from the closed loop in Figure 6, the control adjustment coefficients are in real-time, and active control is achieved. A wide range of experimental tests were carried out within the Department of Mechanics at the National University of Science and Technology Politehnica Bucharest based on this control method, through which this control method was validated, but the experimental results will be presented in a future study. In the present study, a detailed analysis was conducted of the magnitudes of the displacements obtained in a dynamic system using the Voight-Kelvin model. The minimum function method presented for vibration control can be applied in the same way to identify the adjustment parameters necessary to obtain certain minimum speeds or minimum accelerations, depending on the requirements. This numerical method used in this paper can be successfully applied to other dynamic systems with different rheological models [24], such as the Maxwell model, Hooke model, Newton model, and Zener model. This study demonstrates that an electrodynamic shaker (ED) can be used in vibration control. Of course, depending on the mass of the dynamically excited system and the magnitude of the obtained force \(F(t)\), as shown in Figure 1, capable electrodynamic actuators that can produce the control force \(F_{c}(t)\) are required. Such electrodynamic shakers are available on the market, with excitation force values ranging from a few newtons to several tons of force. Conceptualization, O.V. ; methodology, O.V. ; software, O.V. and M.B. ; validation, O.V. ; formal analysis, O.V. ; investigation, O.V. ; resources, O.V. and M.B. ; data curation, O.V. and M.B. ; writing--original draft preparation, O.V. ; writing--review and editing, O.V. ; visualization, O.V. ; supervision, O.V. ; project administration, O.V. and M.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author. The authors of this article are thankful to the National University of Science and Technology Politehnica Bucharest for providing a serene environment and the facilities for carrying out this research. The authors declare no conflicts of interest.