# Active Vibration Control Study of Harmonic Excitation for Voigt-Kelvin System
## 2 Dynamic System Analysis  
The Kelvin-Voigt system presented in Figure 1 shows the known parameters of the system \(k\), \(m\), \(c\) under the dynamic action of known disruptive forces \(F(t)=F_{0}\sin(\omega t+\varphi)\), at which point the force amplitude \(F_{0}\) and harmonic parameters of angular frequency \(\omega\) and phase shift angle \(\varphi\) are known. The following question arises: if a disruptive action is taken, what control force would need to intervene to reduce the amplitude of the vibration system? To control the system, an additional control force is used: \(F_{c}(t)=P_{r}\sin(\omega_{r}t+\varphi_{r})\). This will be considered when forces \(F(t)\) and \(F_{c}(t)\) have the same line of action at baseline and the same direction. The essential problem is identifying the control parameters \(P_{r}\), \(\omega_{r}\), and \(\varphi_{r}\), which define the function of force control.  
Force control is produced via an electrodynamic shaker (ED) that can produce sinusoidal harmonic variations, where the size of the force amplitude control \(P_{r}\) is directly proportional to the supply shaker's voltage (in volts). Each parameter control has a limited range of adjustment according to the type and capacity of the electrodynamic shaker. A control loop can be achieved with a non-contact transducer, which measures the total displacement of a moving mass \(m\) from the fixed base (ground). The control unit takes over the signal recorded based on the size of the displacement \(x(t)\), corrects the sinusoidal voltage waveform depending on the parameters \(\omega_{r}\), \(\varphi_{r}\), and controls the amplitude of the required force \(P_{r}\). As it is possible to set the three parameters using separate repetitive loops, the question arises as to whether correlation coefficients can be controlled using the disturbing force, as well as whether the relationships connecting the loops' control reports can be identified using the parameters of disruptive force and force control to obtain an optimal result in a short time.  
To determine this intrinsic connection between the system's response to a disruptive force and a force control system for linear viscoelastic damping, forced vibration is considered an analytical equation with the following form [5; 9; 10; 26; 27]:  
\[m\ddot{x}+c\dot{x}+kx=F_{0}\sin(\omega t+\varphi)+P_{r}\sin(\omega_{r}t+\varphi _{r}) \tag{1}\]  
where the index \"\(r\)\" denotes the control parameters of the electrodynamic shaker; \(F_{0}\) and \(P_{r}\) represent the force amplitudes (\(N\)); \(\omega\), and \(\omega_{r}\) represent the angular frequency (s\({}^{-1}\)); and \(\varphi\), and \(\varphi_{r}\) represent the phase angle (_radian_).  
Figure 1: Voigt–Kelvin system with active control.  
In the previous relation, by dividing the mass \(m\) and noting \(F_{m}=F_{0}/m\) and \(P_{m}=P_{r}/m\), we obtain  
\[\ddot{x}+\frac{c}{m}\dot{x}+\frac{k}{m}x=F_{m}\sin(\omega t+\varphi)+P_{m}\sin( \omega_{r}t+\varphi_{r}) \tag{2}\]  
For this differential equation, the general form of the considered solution is as follows:  
\[x=x_{1}+x_{2} \tag{3}\]  
where \(x_{1}\) is the free response; \(x_{2}\) is the particular solution of the equation for a forced response in steady state condition.  
The free response equation has the following form:  
\[\ddot{x}+\frac{c}{m}\dot{x}+\frac{k}{m}x=0 \tag{4}\]  
If we note \(\frac{c}{m}=2n\), \(p^{2}=\frac{k}{m}\), Equation (4) becomes  
\[\ddot{x}+2n\dot{x}+p^{2}x=0 \tag{5}\]  
which has the characteristic equation of the form  
\[r^{2}+2 nr+p^{2}=0 \tag{6}\]  
with the roots \(r_{1,2}=-n\pm\sqrt{n^{2}-p^{2}}\).  
The general solution of a free response equation is obtained, in the following form:  
\[x_{1}=A_{1}e^{(-n-\sqrt{n^{2}-p^{2}})t}+B_{1}e^{(-n+\sqrt{n^{2}-p^{2}})t} \tag{7}\]  
For parameters \(k\), \(m\), and \(c\) of the system, a further amortization regime can be enacted: \(\left(n^{2}-p^{2}\right)\).