## 3 Overview on Antiresonance Assignment through the Active or Semi-Active Tuned Mass Dampers
### The Active Tuned Mass Damper  
The basic idea of an ATMD (Figure 1) is to place an actuator between the primary system to control and the auxiliary mass of the TMD and to make it exert a control force \(u\) according to predefined control logic. It is, however, common to place the actuator in parallel with the spring (whose stiffness is \(k_{a}\)) and the damper (\(c_{a}\)) connecting the primary system to the mass of the TMD [13; 14]. In the simple case of a single-DOF ATMD attached to a damped single-DOF primary system, the equation of motion can be easily inferred from the formulation of the passive TMD for a damped system:  
\[\left[ \begin{array}{cc}m&0\\ 0&m_{a}\end{array} \right]\left\{ \begin{array}{cc}\tilde{q}(t)\\ \tilde{q}_{a}(t)\end{array} \right\}+\left[ \begin{array}{cc}c+c_{a}&-c_{a}\\ -c_{a}&c_{a}\end{array} \right]\left\{ \begin{array}{cc}\dot{q}(t)\\ \dot{q}_{a}(t)\end{array} \right\}+\left[ \begin{array}{cc}k+k_{a}&-k_{a}\\ -k_{a}&k_{a}\end{array} \right]\left\{ \begin{array}{cc}q(t)\\ \dot{q}_{a}(t)\end{array} \right\}+\left\{ \begin{array}{cc}u(t)\\ -u(t)\end{array} \right\}, \tag{7}\]  
where it is assumed that an external harmonic force with amplitude \(F_{0}\) and frequency \(\omega_{f}\) excites the primary system. The force exerted by the actuator tries to replicate the effect of the mass, the stiffness, and the damping of the classical passive TMD on the basis of the sensed displacement (and in case velocity too) of the masses of the primary system and of the absorber by means of a control law \(u(t)=u\big{(}q(t),q_{a}(t),\dot{q}(t),\dot{q}_{a}(t),\ddot{q}_{a}(t)\big{)}\). In particular, position feedback emulates the passive spring of the TMD, speed feedback emulates the passive linear damper, and acceleration feedback emulates the TMD mass. The crux is the definition of the suitable control law and the tuning of the parameters, to absorb vibration while accomplishing other tasks such as increasing robustness or decreasing the oscillation of the absorber mass. Due to the similarity between passive and active spring, dampers and masses, the criteria developed to tune passive TMDs can be adopted for ATMD, too, and easily tuned through gain selection. This versatility offered via a wise design of the control law is an advantage of ATMDs over classical passive TMDs. Additionally, since the gains of the control law can be varied during operation, ATMDs can be adapted to variable excitation frequencies. On the other hand, a relevant drawback of ATMDs compared to passive TMDs is that their effectiveness decreases when \"high-frequency\" antiresonances are to be created in the case of low actuator bandwidths [15] due to the limitations of the controlled actuator itself (related to electrical and mechanical issues). Such an issue that is usually neglected in the literature is analyzed in Section 4.5.1. A similar issue is the use of sensors with reduced bandwidths, or with time delay in the measurements, as well as the presence of noise (that downgrades bandwidths due to the need of filtering). Another obvious limitation is that the actuator should be properly sized to exert the desired control forces and ensure adequate displacements for effectively damping the unwanted vibrations of the primary system. It should be pointed out that the sizing the system to properly account for the feasible displacements of the TMD mass is an issue that affects passive TMDs, too. As already mentioned, an effective idea to tune an ATMD is through the various approaches proposed for passive TMDs, such as the one proposed by [PERSON] [10] or other ones optimizing different goals ([16]), by treating the actuator as a spring plus a damper. The degrees of freedom in the choice of the variables to be sensed and in the gain tuning, however, permit further enhancement of the performances and have led to various approaches specifically targeted to ATMDs and SATMDs.