## 3 Overview on Antiresonance Assignment through the Active or Semi-Active Tuned Mass Dampers
### The Active Tuned Mass Damper
For example, in [17], a control algorithm based on the classical optimal feedback control theory has been proposed to reduce both the energy consumption of the actuator and the vibrations of the main system. Eigenvalue assignment exploiting pole placement techniques for the control of the steady-state vibration of tall buildings by means of an ATMD has been proposed in [18]. Fuzzy control logics are adopted too, such as in [19]. An iterative adjoint-based optimization method has been exploited in [20] to design the control law for a vibration absorber. The effectiveness of the techniques is assessed for the control of a two-store frame subject to harmonic base excitation. Most of the approaches have been developed for systems with symmetric matrices. The optimum parameters of ATMD for an asymmetric system have also been discussed in [21]. Numerical Examples of ATMD with Different Tuning Methods  
Let us consider, with no lack of generality, a sample spring-mass-damper system with \(m=1\) kg, \(c=10^{-3}\) Nsm\({}^{-1}\) and \(k=1\) Nm\({}^{-1}\). The goal of this section is to assign an antiresonance \(\omega_{f}=\sqrt{\frac{k_{c}}{m_{a}}}\) at \(\omega_{f}=0.950\) rads\({}^{-1}\), which corresponds to \(\omega_{f}=0.950\omega_{0}\) being \(\omega_{0}=\sqrt{\frac{k}{m}}=1\) rads\({}^{-1}\), by means of an ATMD which employs just a physical mass, while the stiffness and the damper of the TMD are fully active, i.e., are exerted by the actuator control force \(u\)(t):  
\[u(t)=-k_{c}(q(t)-q_{a}(t))-c_{c}\big{(}\dot{q}(t)-\dot{q}_{a}(t)\big{)}. \tag{8}\]  
The system is sketched in Figure 2 and its equation of motion are:  
Figure 1: A sketch of an ATMD. \[\left[ \begin{array}{cc}m&0\\ 0&m_{a}\end{array} \right]\left\{ \begin{array}{c}\ddot{q}(t)\\ \dot{q}_{a}(t)\end{array} \right\}+\left[ \begin{array}{cc}c&0\\ 0&0\end{array} \right]\left\{ \begin{array}{c}\dot{q}(t)\\ \dot{q}_{a}(t)\end{array} \right\}+\left[ \begin{array}{cc}k&0\\ 0&0\end{array} \right]\left\{ \begin{array}{c}q(t)\\ q_{a}(t)\end{array} \right\}=\left\{ \begin{array}{c}F_{0}\sin(\omega_{f}t)\\ 0\end{array} \right\}+\left\{ \begin{array}{c}u(t)\\ -u(t)\end{array} \right\}. \tag{9}\]  
The control gains in Equation (8) are computed in this example exploiting different approaches, proposed for passive TMDs, to fulfill different control specifications. The selected tuning methods are briefly reported here in Table 1. Each method has been denoted by a number that spans from 1 to 5. The optimization goal fulfilled through each tuning method is here reported (for a more deep analysis see [16,22]):  
Tuning 1. Minimize the maximum displacement of the primary mass. Tuning 2. Minimize the total displacement of the primary mass over all frequencies. Tuning 3. Minimize the duration of the system transient vibration (the settling time), i.e., maximize the absolute value of the pole real part. Tuning 4. Minimize the primary mass displacement and the relative displacement. Tuning 5. Minimize the total kinetic energy of the primary mass over all frequencies. The mass ratio and the damping ratio are denoted by \(\mu=\frac{m_{a}}{m}\) and \(\xi_{d}\) respectively. The response of the primary mass without the absorber and with the ATMDs tuned by different techniques is shown in Figure 3, where the receptance amplitudes are normalized through \(k\), i.e., \(|kx/F_{0}|\) is shown, so that the proposed result does not depend on \(m\) and \(k\) of the primary system.