# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 2 Dynamic System Analysis
### The Vibrations' Composition  
Usually, in practice, when a sinusoidal disturbing force acts on a system with viscoelastic support, the force response measured on the system is modified by the system; the response can be modeled as a frequency-modulated force or as a combination of two vibrations. For the right section of Equation (2), in order to find a particular solution \(x_{2}\), it is necessary to obtain the composition of two vibrations: \(F_{m}\sin(\omega t+\varphi)+P_{m}\sin(\omega_{r}t+\varphi_{r})=F_{T}(t)\). Assuming that the angular frequency values of \(\omega\) and \(\omega_{r}\) are different, to simplify the calculation, a complex formulation is considered:  
\[\widetilde{x}_{2}=F_{m}e^{i(\omega t+\varphi)}+P_{m}e^{i(\omega_{r}t+\varphi_ {r})} \tag{14}\]  
where we considered  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]  
Inserting the notations  
\[\Omega_{1}=\omega+\omega_{r}\,,\ \Omega_{2}=\omega-\omega_{r}\,\text{thus}\ \omega=(\Omega_{1}+\Omega_{2})\,/2\,,\ \ \omega_{r}=(\Omega_{1}-\Omega_{2})\,/2 \tag{16}\]  
will result in the general form  
\[\widetilde{F}_{T}=\widetilde{F}_{m}+\widetilde{P}_{m}=F_{m}e^{i(\frac{\Omega_ {1}+\Omega_{2}}{2}t+\varphi)}+P_{m}e^{i(\frac{\Omega_{1}-\Omega_{2}}{2}t+ \varphi_{r})} \tag{17}\]  
or  
\[\widetilde{F}_{T}=F_{m}e^{i\frac{\Omega_{1}}{2}t}e^{i\frac{\Omega_{2}}{2}t}e^ {i\varphi}+P_{m}e^{i\frac{\Omega_{1}}{2}t}e^{i-\frac{\Omega_{2}}{2}t}e^{i \varphi_{r}}=\left[F_{m}e^{i\frac{\varphi+\Omega_{2}}{2}t}+P_{m}e^{i\frac{ \varphi-\Omega_{2}}{2}t}\right]e^{i\frac{\Omega_{1}}{2}t}=X_{2}e^{i(\frac{ \Omega_{1}}{2}t+\theta_{2})} \tag{18}\]where, in the mathematical operation, unknowns \(X_{2}\) and \(\theta_{2}\) are finally obtained in the following forms:  
\[A=\sqrt{F_{m}^{2}+P_{m}^{2}+2F_{m}P_{m}\cos[(\omega-\omega_{r})t+(\varphi-\varphi _{r})]} \tag{19}\]  
and  
\[\tan\theta=\frac{F_{m}\sin\bigl{(}\varphi+\frac{\omega-\omega_{r}}{2}t\bigr{)}+ P_{m}\sin\bigl{(}\varphi_{r}+\frac{\omega-\omega_{r}}{2}t\bigr{)}}{F_{m}\cos \bigl{(}\varphi+\frac{\omega-\omega_{r}}{2}t\bigr{)}+P_{m}\cos\bigl{(}\varphi_ {r}+\frac{\omega-\omega_{r}}{2}t\bigr{)}} \tag{20}\]  
In this case, the particular solution in a complex formulation for the right member of Equation (2) is:  
\[\widetilde{F}_{T}=Ae^{i(\frac{\Omega_{1}}{2}t+\theta)} \tag{21}\]  
Because the \(F_{m}\) and \(P_{m}\) parameters have imaginary axis directions, for the resulting force, we obtain  
\[F_{T}=\text{Im}\widetilde{F}_{T}=A\sin\biggl{(}\frac{\Omega_{1}}{2}t+\theta \biggr{)} \tag{22}\]  
For the right section of Equation (2) the total force produced by the two forces can be in the following form:  
\[F_{T}=A\sin\biggl{(}\frac{\omega+\omega_{r}}{2}t+\theta\biggr{)} \tag{23}\]  
For calculation, the following 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. For the composed force, the variations in time \(F_{T}(t)\) and frequency (single-sided amplitude spectrum) \(F_{T}(f)\) presented in Figure 2 were taken into account, as in Equation (23) and the notations \(F_{m}=F_{0}/\,m\) and \(P_{m}=P_{r}/\,m\). The results obtained using Equation (23) are very close to the direct numerical calculation of the amount \(F_{m}\sin(\omega t+\varphi)+P_{m}\sin(\omega_{r}t+\varphi_{r})\). Using the right section of the amount obtained in (23), differential Equation (2) can be written as follows:  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]0  
\[\widetilde{F}_{m}=F_{m}e^{i(\omega t+\varphi)};\quad\widetilde{P}_{m}=P_{m}e^ {i(\omega_{r}t+\varphi_{r})} \tag{15}\]1 Chart of the excitation forces and the results of their \(F_{T}\) composition: (**a**) time representation; (**b**) frequency representation.