## 3 Overview on Antiresonance Assignment through the Active or Semi-Active Tuned Mass Dampers
### The Delayed-Resonator  
Another interesting application of active TMD is the delayed resonator (DR), also called the time-delayed vibration absorber, that has been introduced in Reference [28]. This device, sketched in Figure 4 and attached to a primary system, is an ATMD with a displacement feedback control force defined through a control gain \(k_{c}\) and a tunable time delay \(\tau_{c}\), such that its equation of motion is:  
\[\left[\begin{array}{cc}m&0\\ 0&m_{a}\end{array}\right]\left\{\begin{array}{c}\ddot{q}(t)\\ \ddot{q}_{a}(t)\end{array}\right\}+\left[\begin{array}{cc}c+c_{a}&-c_{a}\\ -c_{a}&c_{a}\end{array}\right]\left\{\begin{array}{c}\dot{q}(t)\\ \dot{q}_{a}(t)\end{array}\right\}+\left[\begin{array}{cc}k+k_{a}&-k_{a}\\ -k_{a}&k_{a}\end{array}\right]\left\{\begin{array}{c}q(t)\\ q_{a}(t)\end{array}\right\}=\left\{\begin{array}{c}F_{0}\sin(\omega_{f}t)\\ 0\end{array}\right\}+\left\{\begin{array}{c}-k_{c}q_{a}(t-\tau_{c})\\ +k_{c}q_{a}(t-\tau_{c})\end{array}\right\} \tag{10}\]  
The DR must be designed such that a proper tuning of the time-delay enables the damped absorber composed by \(m_{a}\), \(c_{a}\) and \(k_{a}\) to operate as an undamped absorber tuned at the prescribed antiresonance frequency, i.e., the DR exhibits a non-dissipative (conservative) behavior. In this scenario, complete vibration absorption is achieved.  
The DR is tuned in two stages: first, \(m_{a}\), \(c_{a}\) and \(k_{a}\) are computed as for passive TMDs to assign a prescribed antiresonance \(\omega_{f}\) (see, e.g., the tuning rules proposed in Table 1). Secondly, the control gain and the time-delay are computed as follows:  
\[\begin{array}{c}\tau_{c}=\frac{1}{\omega_{f}}\tan^{-1}\left(\frac{c_{a} \omega_{f}}{m_{a}\omega_{f}^{2}-k_{a}}\right)\\ k_{c}=\pm\sqrt{c_{a}^{2}\omega_{f}^{2}+\left(k_{a}-m_{a}\omega_{f}^{2} \right)^{2}}\end{array} \tag{11}\]  
The characteristic equation of the main linear system coupled with a delayed resonator is transcendental. It follows that infinite poles are obtained through this controller. Further, the complex conjugated imaginary pole pair introduced by the absorber is marginally stable to ensure complete vibration absorption. Hence, the authors in [28] deeply discussed the stability and the robustness issues of the system controlled through the DR. Two main stability conditions are obtained with respect to the additive perturbations \(\Delta k_{c}\) and \(\Delta\tau_{c}\) (of the control gain and of the time delay, respectively). The closed-loop system is stable if:  
\[\left\{\begin{array}{c}\Delta k_{c}<0\\ \Delta\tau_{c}>0\end{array}\right.\quad\text{ if }\quad\omega_{f}<\omega_{0} \sqrt{1-2\xi^{2}}\text{ and }\left\{\begin{array}{c}\Delta k_{c}<0\\ \Delta\tau_{c}<0\end{array}\right.\quad\text{ if }\quad\omega_{f}>\omega_{0} \sqrt{1-2\xi^{2}} \tag{12}\]  
where \(\omega_{0}\) and \(\xi\) are the natural frequency and the damping ratio of the primary system.  
The delayed resonator has been adapted to assign two [29] or more antiresonance frequencies [30]. Analytical and experimental results in the design and stability analysis of time-delayed vibration absorbers have been proposed recently in [31,32].  
Figure 4: Sketch of the delayed resonator.