# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 2 Dynamic System Analysis
### The Vibrations' Composition
If we denote \(\omega_{m}=\frac{\omega+\omega_{x}}{2}\) and \(\omega_{2}(t)=\omega_{m}t+\theta\), then differential Equation (24) can be written as follows:  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]2  
For the right section of Equation (24), a particular solution of the equation when using composed forces can take the following form:  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]3  
When the successive derivations are introducing into differential Equation (24), provided that it is identical, we obtain  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]4  
Similarly, we can write  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]5  
where  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]6  
In conclusion, taking into account non-homogeneous Equation (2) and forming the general solution to Equation (3) we obtain the following:  
\[x(t)= \begin{cases}e^{-nt}X_{1}\cdot\sin\Bigl{(}t\sqrt{p^{2}-n^{2}}+\theta_{1} \Bigr{)}+X_{2}\sin[\omega_{2}(t)+\theta_{2}],&n^{2}-p^{2}<0\\ e^{-nt}\biggl{(}x_{1,0}\cosh\sqrt{n^{2}-p^{2}}t+\frac{V_{1,0}+nX_{1,0}}{\sqrt {n^{2}-p^{2}}}\sinh\sqrt{n^{2}-p^{2}}t\biggr{)}+X_{2}\sin[\omega_{2}(t)+\theta _{2}],&n^{2}-p^{2}>0\\ e^{-nt}[(V_{1,0}+nx_{1,0})t+x_{1,0}]+X_{2}\sin[\omega_{2}(t)+\theta_{2}],&n^{2} -p^{2}=0\end{cases}  \tag{30}\]  
For case 1, the following numerical values were used: \(F_{0}=500\) N, \(P_{r}=300\) N, \(\omega=50\) s\({}^{-1}\), \(\omega_{r}=60\) s\({}^{-1}\), \(\varphi=\varphi_{r}=0\), \(m=200\) Kg, \(k=9\cdot 10^{4}\) N/m, \(c=4\cdot 10^{2}\) Ns/m, with initial conditions of \(x_{1,0}=0.002\) m and \(V_{1,0}=0.001\) m/s. The graphic representations of displacement versus time variations, shown in Figure 3, are for case 1, when \(n^{2}-p^{2}<0\). The displacement graphs show the amplitude displacement in meters and time in seconds. For Case 2, only the damping was changed, \(c=2\cdot 10^{4}\) Ns/m, with the same initial conditions. The graphical representations of the variations in the displacement function of time, shown in Figure 4, are for Case 2, where \(n^{2}-p^{2}>0\). For Case 3, only the damping was changed, \(c=2\sqrt{\text{km}}\) Ns/m, with \(\xi=1\). The graphic representations of the variations in displacement, depending on the time, are shown in Figure 5 for Case 3 when \(n^{2}-p^{2}=0\). Some special equipment, such as electron microscopes and laser wave equipment, include operating limitations in their technical operating specifications, which are specific to the small displacements in the structural elements on which they are mounted. On the same structural elements/floors, there may be other equipment that, due to vibrations, can lead to significant displacements of the building structures during operation; this can intervene with vibration control solutions. Figure 4: Displacement versus time representation for Case 2: (**a**) solution of the free response equation without excitation; (**b**) solution of the particular equation with composed force; (**c**) general solution of displacement. Figure 3: Displacement versus time representation for Case 1: (**a**) solution of the free response equation without excitation; (**b**) solution of the particular equation with composed force; (**c**) general solution of displacement. Based on the results of the time variations in the displacements, obtained in each case shown in Figures 3-5, and their successive derivatives, the time variations in the velocity or accelerations can be easily obtained.