## 3 Design Concept and Characteristics
### Simulation of the System
#### 4.2.2 Time History Analysis  
Through the time domain analysis, the displacement of the structure excited by resonance frequency (10 (rad/s)) is shown in Figure 10a. The ATMID and PTMD devices can substantially suppress the vibration of the primary structure in comparison with the uncontrolled resonated structure. The optimal inertance generated by the ATMID is \(b\) = 32 (kg) and results in reducing the displacement of the main structure almost twice as much as the PTMD. In a frequency of \(\omega_{\rm e}\) = 7.13 (rad/s), the PTMD behaves intolerably compared with  
Figure 9: Damping properties of the ATMID based on: (**a**) inherent structural damping coefficient, (**b**) ATMID’s damping coefficient, (**c**) ATMID’s mass ratio. the uncontrolled case. However, the considered ATMID performs effectively by generating an inertance of \(b=84\) (kg), as is evident in Figure 10b. The relationship between the maximum displacement and the maximum acceleration of the main structure for the PTMD, ATMID, and uncontrolled cases in both frequencies is depicted in Figure 11. According to the results, an excellent structural response reduction of the ATMID in contrast with the PTMD is observed. In addition, Table 4 shows the structural displacement denoted by D and its acceleration (denoted by A) for the uncontrolled (U), passively controlled (P), and adaptively controlled (A) structures. In this table, a vibration reduction effect (VRE) is attained for the passive and adaptive control methods compared to the uncontrolled case. For instance, at the frequency of 7.31 (rad/s), the passive TMD  
Figure 10: Structural displacement time-history at: (**a**) resonance frequency of 10 (rad/s), (**b**) frequency of 7.13 (rad/s). is destructive for the structure as it increases the structural displacement by 76% relative to the uncontrolled structure. In contrast, the performance of the ATMID in the reduction of the displacement is superior where it reduces the displacement by 23% relative to the uncontrolled case. The efficiency of both devices is comparable in resonance frequency (10 rad/s). Figure 12 shows the correlation between the inertial force and the displacement produced by the variable inerter of the ATMID. The generated inertance, that is, the gradient of the line in the resonance frequency is less than that of the frequency of 7.13 (rad/s) while its deformation is more than the deformation at the frequency of 7.13 (rad/s). The structural deformation for both cases is represented by dashed lines and dotted symbols for frequencies of 10 and 7.13 (rad/s), respectively, in Figure 12. It is worth noting that the disparity between the two inertances shows the ability of the ATMID to generate different frequencies in various excitation frequencies. Moreover, the displacement-force curves resulting from the PTMD and ATMID are contrasted in Figure 13 where the structural displacement boundaries are displayed by dashed lines for the ATMID and dotted lines for the PTMD. It is apparent from Figure 13 that substantial reduction in structural displacement is obtained by using the ATMID at the frequency of 7.13 (rad/s), while the displacement of the structure by using the PTMD is more than that of the uncontrolled case as shown by dashed-dotted lines.