## 3 Design Concept and Characteristics
### Simulation of the System
#### 4.2.1 Results in Domain of the Frequency  
The frequency-domain analysis of the structure was conducted to appraise the damping properties of the ATMID as well as its effectiveness on the dynamics of the system. In this manner, the frequency response curve (FRCs) of the PTMD-controlled, ATMID-controlled, and uncontrolled structures were contrasted to specify their dynamic amplification factors (DAFs). We assumed the ratio of the structural deformation (\(\chi_{0}\)) to the acceleration of the ground (\(\alpha_{\rm g_{0}}\)) to be the dynamic amplification factor (DAF). Through the diverse CVT ratio, various sets of inertance (\(b_{\rm i}\)) and frequency (\(\omega_{\rm d}\)) can be provided by the ATMID, therefore, a set of diverse curves in the frequency domain \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mass** & **Stiffness** & **Damping** & **Damping** & **Damping Ratio** & **Frequency** \\
**M (kg)** & **K (N/m)** & **C (N-s/m)** & **C (N-s/m)** & \(\xi=\frac{C}{2\sqrt{\text{KM}}}\) & **\(\omega_{\rm s}\) (rad/s)** \\ \hline
200 & 20,000 & 6 & 0.15\% & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics for the primary structure. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Device** & **Auxiliary Mass** & **Mass Ratio** & **Damping** & **Stiffness** & **Flywheel Mass** & **Variable Inertance** & **Frequency** \\  & **m (kg)** & **\(\mu=\frac{m}{M}\)** & **cd (N-s/m)** & **kd (N/m)** & **mf (kg)** & **\(b\) (kg)** & **\(\omega_{\rm d}\) (rad/s)** \\ \hline PTMD & 40 & 0.2 & 166 & 2500 & - & - & 7.9 \\ ATMID & 20 & 0.1 & 32 & 5000 & 0.66 & 8–128 & 5.8–13.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Device parameters for TMD and proposed ATMID. is provided as depicted in Figure 6. Each curve possesses its own specifications (\(b_{\text{i}}\),\(\omega_{\text{d}_{\text{i}}}\)) which were achieved by tilting a set of the planets at different angles. In each frequency of the excitation, the minimum DAF is achieved as such that the minimal ordinate is selected between all DAFs curves. Each of the opted minimal DAF is an expression of an ATMID with a particular frequency (\(\omega_{\text{d}_{\text{i}}}\)) at which the topmost DAF reduction occurs in the corresponding force frequency (\(\omega_{\text{e}_{\text{i}}}\)). The optimal operation of the ATMID was acquired by connecting the entire obtained minimum DAF points in each frequency as referred to in Figure 6. The remarkable damping performance of the ATMID was identified by comparing its optimal curve (red line) with the uncontrolled case (black dashed line). As shown in Figure 7 and summarized in Table 3, the ATMID behaves very well in displacement-DAF suppression in comparison with the PTMD. A significant displacement-DAF reduction was achieved using the devices at the resonance frequency of 10 (rad/s). This reduction in the ATMID with a mass ratio of 0.1 is more than in the PTMD with a mass ratio of 0.2. At the frequency of 9.22 (rad/s) with maximal DAF of the ATMID, the DAF in the ATMID is much less than the PTMD while their generated inertance is 42 (kg). Furthermore, the widest DAF-variation between the PTMD and uncontrolled cases happens at 7.13 (rad/s) where the ordinate of the PTMD is roughly increased by 46% compared to uncontrolled cases. However, in this frequency (7.13), the ATMID's ordinate is 56% less than that of the PTMD. Furthermore, it is worth noting that the slight disparity between the excitation frequency and the calculated natural frequency of the ATMID (Table 3) is due to the simplification in the formulation of the device frequency, as presented in the former section. The optimal inertance and its associated device frequency was attained such that the device frequency was adjusted to that of the excitation (\(\omega_{\text{e}}=\omega_{\text{d}}\)).