## 4 Active Antiresonance Assignment with No Additional Masses
### The Fully-Active, Virtual Active Tuned Mass Damper
#### 4.5.1 Numerical Example: Design of a VATMD and Effect of the Actuator Dynamic  
In this numerical test case, a simple oscillator with one DOF is considered. The primary system is assumed undamped and composed by a mass (\(m=1\) kg) and a grounding spring (\(k=1\) Nm\({}^{-1}\)). The goal is to design a VATMD and then to consider the effect of the dynamic of the actuator that exerts the control force, which is often neglected for simplicity (see, e.g., [19]). Let us assume that an antiresonance \(\omega_{f}=\sqrt{\frac{k_{c}}{m_{c}}}\) should be assigned, for example, at 0.950 rads\({}^{-1}\) (which corresponds to 0.95 \(\omega_{0}\) with \(\omega_{0}=\sqrt{\frac{k}{m}}\)). The passive TMD design formulas proposed by [PERSON] can be applied [10; 80] to design the second order compensator, rather than using the undamped TMD as in Equation (36). In practice, Equations (35) and (36) are replaced by the following ones:  
\[u(t)=-k_{c}(q(t)-q_{a}(t))-c_{c}\big{(}\dot{q}(t)-\dot{q}_{a}(t)\big{)} \tag{37}\]  
\[m_{c}\ddot{q}_{a}(t)=k_{c}(q(t)-q_{a}(t))+c_{c}\big{(}\dot{q}(t)-\dot{q}_{a}(t) \big{)} \tag{38}\]  
A definition of the control forces has been proposed in [79] for the AMTMD. By considering the scenario of a disturbance force applied to the mass of the primary system, the control gains are obtained through the following relations commonly used for passive TMDs:  
\[\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\ \ \ \xi_{d}=\sqrt{\frac{3\mu}{8(1+\mu)}} \tag{39}\]  
where \(\mu=\frac{m_{c}}{m}\) and \(\xi_{d}\) is the damping ratio of the TMD alone, assumed as grounded.  
The control loop to implement the control forces in Equations (37) and (38) is sketched in Figure 14. The receptance of the open-loop primary system in the Laplace \(s\)-domain of the main system is denoted G(s), while C(s) denotes the transfer function of the controller in Equations (37) and (38) that allows us to compute the force to be exerted by the actuator:  
\[\begin{array}{c}\text{G}(s)=\frac{1}{s^{2}m+\bar{k}}\\ \text{C}(s)=\frac{s^{2}m_{c}(s_{c}+k_{c})}{s^{2}m_{c}+s_{c}+k_{c}}\end{array} \tag{40}\]  
Figure 14: A sketch of the closed-loop system with VATMD.  
To propose a more comprehensive analysis of the actual effectiveness of VATMDs, which is usually neglected in most of the literature, the effect of the actuator bandwidth \(\omega_{A}\) is investigated in this paper by representing it through a first order low-pass filter A(s):  
\[\text{A}(s)=\frac{\omega_{A}}{s+\omega_{A}} \tag{41}\]  
The transfer function between the disturbance force\(f_{d}\) and the displacement of the main system \(q\) is therefore:  
\[\frac{q(s)}{f_{d}(s)}=\frac{\text{G}(s)}{1+\text{C}(s)\text{A}(s)\text{G}(s)} \tag{42}\]  
The amplitude of the response of the primary system, for the same tuning of the VATMD made through Equation (39), is compared in Figure 15 for different bandwidths of the actuator. While the VATMD features the same amplitude reduction of the passive TMDs if \(\omega_{A}>>\omega_{f}\), the effectiveness of the active controller vanishes whenever the actuator dynamic is comparable with \(\omega_{f}\). This result has been here shown in the case of VATMD; however, the same concept can be applied to all the approaches discussed and tested in this paper, also including the case of multi-DOF primary systems.  
The model adopted in Figure 14 allows us to explain the effect of the sensor bandwidth by considering C(s) as the transfer function of the sensor and by leading again to the same closed-loop system of Equation (42).  
Therefore, as a simple rule-of-thumb, the bandwidth of the actuator and of the sensors should be approximately 10 times higher than the desired antiresonance frequency, to make their effect almost negligible compared to the \"ideal\" ATMD and TMD.