# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 3 Analysis of Control Parameters  
The control loop and response were considered, adjusting the control force \(F_{c}(t)=P_{r}\sin(\omega_{r}t-\varphi_{r})\) by parameters \(P_{r}\), \(\omega_{r}\), and \(\varphi_{r}\) using an electrodynamic shaker. Typically, depending on the size and working capacity allowed by the shaker, the control force has a limitation, \(P_{r,\max}\). According to the technical specifications set by the manufacturers, when the coil moves in the magnetic field, a voltage is generated in proportion to the velocity of motion. The \(E(x)\) voltage across the coil may be written in terms of the dynamic shaker's current and velocity parameters. The coil produces an axial force, acting on the shaker's mechanical elements in proportion to the applied current. The control loop can be achieved with the acquisition/control of software programs such as LabView or other specialized programs. Another way is to include a hardware unit for the \"controller\"; this has some advantages but may be limited in terms of the number of loops (coefficients control). Therefore, it is possible to define a sinusoidal voltage signal in the form \(P_{r}\sin(\omega_{r}t-\varphi_{r})\), which can be transmitted to the shaker through the appropriate adjustment of voltage and current. The movement of \(m\) mass can be measured using a displacement inductive or non  
Figure 5: Displacement versus time representation for Case 3: (**a**) solution of the free response equation without excitation; (**b**) solution of the particular equation with composed force; (**c**) general solution of displacement. contact transducer depending on the frequency of interest. The proposed control scheme is shown in Figure 6.  
Depending on the recorded signal \(x(t)\), the desired corrections can be made using displacement, velocity, or acceleration via successive derivations in time \(dx/dt\) or \(d^{2}x/dt^{2}\). In the case of the Voigt-Kelvin-type system, displacement amplitude control can be obtained by modifying the force's parameters \(F_{c}(t)=P_{r}\sin(\omega_{r}t-\varphi_{r})\). The conditioning required for the motion control system may be applied to the overall solutions obtained from the relationship (30). Depending on the known parameters of the system \(k\), \(m\), and \(c\), obtained using known force excitation \(F(t)=F_{0}\sin\omega t\), and the imposed initial conditions, \(x_{1,0}\) and \(V_{1,0}\), if we know the obtained general solutions, then we can define a function of the type  
\[g(P_{r},\omega_{r},\varphi_{r},t_{r})=|x_{1}+x_{2}| \tag{31}\]  
where \(x_{1}\) and \(x_{2}\) are the solutions, taking into account Equations (2) and (30). To achieve effective control and the maximum reduction in the amplitude of the mass \(m\), theoretically, we can put forward the following condition:  
\[g(P_{r},\omega_{r},\varphi_{r},t_{r})=0 \tag{32}\]  
This condition can be implemented by three control loops that can be operated simultaneously or independently, depending on the type of damping system \(\left(n^{2}-p^{2}\right)\): subcritical, critical, or supercritical damping. According to Equations (30) and (32), the control conditions may be expressed as follows:  
\[ \begin{cases}g_{1}=\left|e^{-nt}X_{1}\cdot\sin\left(t\sqrt{p^{2}-n^{2}}+\theta _{1}\right)+X_{2}\sin\left[\omega_{2}(t)+\theta_{2}\right]\right|=0,&n^{2}-p^ {2}<0\\ g_{2}=\left|e^{-nt}\left(x_{1,0}\cosh\sqrt{n^{2}-p^{2}}t+\frac{V_{1,0}+nx_{1,0 }}{\sqrt{n^{2}-p^{2}}}\sinh\sqrt{n^{2}-p^{2}}t\right)+X_{2}\sin\left[\omega_{2 }(t)+\theta_{2}\right]\right|=0,&n^{2}-p^{2}>0\\ g_{3}=\left|e^{-nt}[(V_{1,0}+nx_{1,0})t+x_{1,0}]+X_{2}\sin\left[\omega_{2}(t)+ \theta_{2}\right]\right|=0,&n^{2}-p^{2}=0\end{cases}  \tag{33}\]  
where functions \(g_{1}\), \(g_{2}\), and \(g_{3}\) are dependent on the control parameters/adjustable parameters \(P_{r},\omega_{r}\), \(\varphi_{r}\) and \(t\), such that \(\left[g_{i}(P_{r},\omega_{r},\varphi_{r},t)\right]_{i=\overline{13}}\to 0\).