## 3 Calibration of ATMD with Viscous Damping-Stiffness Control Force
### Equations of Motion and Dynamic Amplification  
The equations of motion of the two masses in Figure 1a are:  
\[ \begin{split} m_{1}\ddot{x}_{1}+k_{1}x_{1}+k_{2 dh}(x_{1}-x_{2})& =f_{exc}-f_{a},\\ m_{2}\ddot{x}_{2}-k_{2 dh}(x_{1}-x_{2})&=f_{a}.\end{split}  \tag{14}\]  
where \(\ddot{x}_{1}\), \(\dot{x}_{1}\), and \(x_{1}\) are the absolute acceleration, velocity, and displacement of the primary structure with its mass \(m_{1}\) and stiffness \(k_{1}\), while \(\ddot{x}_{2}\), \(\dot{x}_{2}\), and \(x_{2}\) are the absolute acceleration, velocity, and displacement of the ATMD with its mass \(m_{2}\) and passive spring stiffness \(k_{2 dh}\) (1), \(f_{exc}\) is the harmonic force excitation of amplitude \(F_{exc}\) and varying angular frequency \(\omega\), and \(f_{a}\) is the control force (12). After accounting for the control force (12), we can express these equations as:  
\[ \begin{split} m_{1}\ddot{x}_{1}+k_{1}x_{1}+c_{a}\dot{x}_{d}+(k_{ a}+k_{2 dh})x_{d}&=f_{exc},\\ m_{2}\ddot{x}_{2}-c_{a}\dot{x}_{d}-(k_{a}+k_{2 dh})x_{d}& =0.\end{split}  \tag{15}\]  
Based on this system of equations, the steady-state displacement amplitude of primary structure mass and the relative displacement amplitude of the ATMD can be expressed in a similar way as previously for the case of the TMD, see (8) and (9), but after replacing \(k_{2 dh}\) with \(k_{a}+k_{2 dh}\), and \(c_{2 dh}\) with \(c_{a}\):  
\[\frac{X_{1}(\omega)}{F_{exc}}=\Bigg{|}\frac{k_{a}+k_{2 dh}-\omega^{2}m_{2}+i \omega c_{a}}{\left(k_{1}-\omega^{2}(m_{1}+m_{2})\right)\left(k_{a}+k_{2 dh}- \omega^{2}m_{2}+i\omega c_{a}\right)-\omega^{4}m_{2}^{2}}\Bigg{|}, \tag{16}\]  
\[\frac{X_{d}(\omega)}{F_{exc}}=\Bigg{|}\frac{\omega^{2}m_{2}}{\left(k_{1}- \omega^{2}(m_{1}+m_{2})\right)\left(k_{a}+k_{2 dh}-\omega^{2}m_{2}+i\omega c_{ a}\right)-\omega^{4}m_{2}^{2}}\Bigg{|}. \tag{17}\]  
Later in this paper, (16) and (17) will be used to determine the frequency responses for the primary structure with the ATMD. Calibrating Viscous Damping-Stiffness Control Force to Mimic the Optimal Acceleration Feedback Control by [PERSON] et al. To determine the optimal parameters for the control force (12) that fulfill the stated control objective, we can use the well-established theory of the so-called equal peak design of TMDs [5; 18; 19; 20] that has also been adopted to ATMDs [17; 21; 22; 23]. In the work by [PERSON] et al. [17], this approach was applied to derive the optimal acceleration feedback control law, with a variant that also includes relative velocity feedback. The approach presented here aims to calibrate the parameters of the control force (12) so that it mimics the optimal control force outlined by [PERSON] et al. Let us consider the control force \(f_{Nis}\) discussed in [17], which takes the form of a sum of inertial and viscous damping forces:  
\[f_{Nis}=-g_{Nis}m_{1}\ddot{x}_{1}+c_{Nis}\dot{x}_{d}, \tag{18}\]  
where \(g_{Nis}m_{1}\) and \(c_{Nis}\) are the negative acceleration and relative velocity feedback gains. Our goal is to find \(c_{a}\) and \(k_{a}\) such that the control force (12) is equal to the force (18). Since we assume that both forces should be equal, we can substitute (18) into the equation of motion for the ATMD mass in (14):  
\[m_{2}\ddot{x}_{2}-k_{2 dh}x_{d}=-g_{Nis}m_{1}\ddot{x}_{1}+c_{Nis}\dot{x}_{d}. \tag{19}\]  
Considering that the acceleration of the ATMD mass can be expressed as \(\ddot{x}_{2}=\ddot{x}_{1}-\ddot{x}_{d}\), we arrive at the following:  
\[\ddot{x}_{1}(m_{2}+g_{Nis}m_{1})=m_{2}\ddot{x}_{d}+c_{Nis}\dot{x}_{d}+k_{2 dh}x_ {d}. \tag{20}\]  
In harmonic motion, the acceleration is proportional to the displacement, as expressed by \(\ddot{x}_{d}=-\omega^{2}x_{d}\).