## 3 Design Concept and Characteristics  
Through the accurate adjustment of the CVT ratio (\(\phi\)) to a proper value along its ratio band, the required output angular acceleration and its associated inertance can be acquired. The variability in the flywheel's angular acceleration and inertance affects the frequency of the ATMID and its inertial force. Thus, the natural frequency of the ATMID can be seamlessly changed and controlled by using the CVT core. The frequency of the ATMID can be formulated in a simple way in which there ought to be few negligible parameters:  
\[\omega_{\text{d}}=\sqrt{\frac{\text{k}_{\text{d}}}{b+\text{m}}} \tag{2}\]  
It is well known that the design concept of passive devices comprising an attached mass or an inerter relies on the tuning of their components so as to adjust their frequencies close to that of the structure through the fixed-point theory [47]. Nevertheless, the CVT allows the ATMID calibrates its parameters, resulting in the frequency of the ATMID (\(\omega_{\text{d}}\)) to equate to that of the current excitation (\(w_{\text{e}}\)). Substantially, the ATMID is capable of adapting its mechanical characteristics such as inertance and frequency based on the detected current force's frequency. Furthermore, the retuning of the device can be easily implemented using CVT during excitation in addition to the elimination of the detuning effect. In order to demonstrate the alterability in generated inertance by AID, the device is excited by the harmonic displacement of \(\text{X}(\text{t})=X_{0}\text{e}^{\text{i}\omega_{\text{e}}t}\), where the amplitude and frequency of excitation are denoted by \(X_{0}\) and \(\omega_{\text{e}}\). The proposed AID under harmonic excitation is able to produce the inertial force of \(\text{F}_{\text{I}}(\text{t})=b\dot{\text{X}}(\text{t})\). By substituting the acceleration in the equation, the relationship between the displacement and the variable inertial force is given as:  
\[\text{F}_{\text{I}}(\text{t})=-b\omega_{\text{e}}{}^{2}\text{X}(\text{t}), \tag{3}\]  
The correlation between the inertial force of the AID and its displacement is shown in Figure 4 in which the gradient of the curve can be altered as the CVT ratio is changed. The slope of the line is the value of the inertance. Through the CVT ratio alteration, a group of the curves with various gradients is provided, showing the variability in inertance (\(b_{\text{h}}\)) and inertia force generated by the ATMID. In fact, alterations in CVT ratio (\(\phi\)) are directly proportional to the generated inertance and its associated resistance force. However, there is an inverse relationship between the CVT ratio and the device frequency. Damping Effect of the ATMID for Harmonic Excitation with Various Frequencies 4.1. Model of the ATMID-Controlled SDOF Structure  
The adaptive TMD-controlled structure was compared with a passive TMD-controlled structure so as to evaluate the damping property of the considered ATMID. The adaptive  
Figure 4: Force-displacement correlation curve of the proposed AID. and passive devices were employed in a single degree of freedom (SDOF) structure induced by harmonic ground excitation as shown in Figure 5. The frequency of the excitation was varied through this research scenario. The harmonic ground motion of \(\mathrm{x_{g}(t)}=\mathrm{x_{g_{0}}e^{i\omega_{\mathrm{e}}t}}\) induces the SDOF structures equipped with passive TMD (PTMD) and adaptive (ATMID) devices. Moreover, the amplitude and frequency of the ground excitation are denoted by \(\mathrm{x_{g_{0}}}\) (m) and \(\omega_{\mathrm{e}}\) (rad/s), respectively.