## 3 Design Concept and Characteristics
The governing equation of the motion of the SDOF structure incorporating with the ATMID induced by periodic ground motion is written as follows:  
\[\mathrm{M\ddot{x}}+\mathrm{C\dot{x}}+\mathrm{Kx}-b\left(\mathrm{\ddot{x}_{T}}- \mathrm{\ddot{x}}\right)-\mathrm{c_{d}}\left(\mathrm{\dot{x}_{T}}-\mathrm{ \dot{x}}\right)-\mathrm{k_{d}}\left(\mathrm{x_{T}}-\mathrm{x}\right)=-\mathrm{ M\ddot{x}_{g}}, \tag{4}\]  
\[\mathrm{m\ddot{x}_{T}}+b\left(\mathrm{\ddot{x}_{T}}-\mathrm{\ddot{x}}\right)+ \mathrm{c_{d}}\left(\mathrm{\dot{x}_{T}}-\mathrm{\dot{x}}\right)+\mathrm{k_{d }}\left(\mathrm{x_{T}}-\mathrm{x}\right)=-\mathrm{m\ddot{x}_{g}}, \tag{5}\]  
where the notation of the structure's parameters such as mass, stiffness, and damping coefficient is written as M (kg), K (N/m), and C (N\(\cdot\)s/m), respectively. The natural frequency of the structure is given as \(\omega_{\mathrm{s}}=\sqrt{\frac{\mathrm{K}}{\mathrm{M}}}\). The deformation of the auxiliary mass and device components are denoted by \(\mathrm{x_{T}(t)}\) and \(\mathrm{x_{T}(t)}-\mathrm{x(t)}\), respectively; where \(\mathrm{x(t)}\) is the deformation of the structure. The governing equations of the system can be transformed into the matrix equation as follows:  
\[\left[ \begin{array}{cc}\mathrm{M}+b&-b\\ -b&\mathrm{m}+b\end{array} \right]\left[ \begin{array}{c}\mathrm{\ddot{x}(t)}\\ \mathrm{\ddot{x}_{T}(t)}\end{array} \right]+\left[ \begin{array}{cc}\mathrm{C }+\mathrm{c_{d}}&-\mathrm{c_{d}}\\ -\mathrm{c_{d}}&\mathrm{c_{d}}\end{array} \right]\left[ \begin{array}{c} \mathrm{\dot{x}(t)}\\ \mathrm{\dot{x}_{T}(t)}\end{array} \right]+\left[ \begin{array}{cc}\mathrm{K} +\mathrm{k_{d}}&-\mathrm{k_{d}}\\ -\mathrm{k_{d}}&\mathrm{k_{d}}\end{array} \right]\left[ \begin{array}{c} \mathrm{x(t)}\\ \mathrm{x_{T}(t)}\end{array} \right]=-\left[ \begin{array}{c}\mathrm{M}\\ \mathrm{m}\end{array} \right]\mathrm{\ddot{x}_{g}(t)} \tag{6}\]  
The displacement, velocity, and acceleration of the structure induced by the ground acceleration of \(\mathrm{\ddot{x}_{g}(t)}=\mathrm{a_{g_{0}}e^{i\omega_{\mathrm{e}}t}}\) with the amplitude of \(\mathrm{a_{g_{0}}}=-\omega_{\mathrm{e}}{}^{2}\mathrm{x_{g_{0}}}\) are achieved as follows:  
\[\mathrm{x(t)}=\mathrm{x_{0}e^{i\omega_{\mathrm{e}}t}}, \tag{7}\]  
\[\mathrm{\dot{x}(t)}=\mathrm{i}\omega_{\mathrm{e}}\mathrm{x_{0}e^{i\omega_{ \mathrm{e}}t}}, \tag{8}\]  
\[\mathrm{\ddot{x}(t)}=-\omega_{\mathrm{e}}{}^{2}\mathrm{x_{0}e^{i\omega_{ \mathrm{e}}t}}, \tag{9}\]  
The displacement, velocity, and acceleration of the attached mass that was periodically excited are respectively attained as follows:  
\[\mathrm{x_{T}(t)}=\mathrm{x_{T_{0}}e^{i\omega_{\mathrm{e}}t}}, \tag{10}\]  
\[\mathrm{\dot{x}_{T}(t)}=\mathrm{i}\omega_{\mathrm{e}}\mathrm{x_{T_{0}}e^{i \omega_{\mathrm{e}}t}}, \tag{11}\]  
Figure 5: Model of the SDOF structure incorporated with: (**a**) passive TMD (**b**) considered adaptive TMID. \[\ddot{x}_{T}(t)=-{\omega_{\rm e}}^{2}x_{T_{0}}e^{i\omega_{\rm e}t}, \tag{12}\]  
Also, using little mathematical manipulation in the equations of the system, the output to input ratio (\(\frac{\lambda_{0}}{\alpha_{\rm g_{0}}}\)), that is, deformation response of the structure to ground acceleration is written as follows:  
\[\frac{\lambda_{0}}{\alpha_{\rm g_{0}}}=\frac{\omega_{\rm e}e^{2}(b(M+m)+Mm)-i \omega_{\rm e}c_{\rm d}(M+m)-k_{\rm d}(M+m)}{\omega_{\rm e}e^{4}(b(M+m)+Mm)-i \omega_{\rm e}e^{3}(c_{\rm d}(M+m)+C(m+b))-\omega_{\rm e}e^{2}(k_{\rm d}(M+m)+K( m+b)+C_{\rm d})+i\omega_{\rm e}(K_{\rm d}+C_{\rm Kd})+K_{\rm Kd}} \tag{13}\]  
According to the current excitation frequency (\(\omega_{\rm e}\)) exerted on the structure, inertance would be altered through the proper calibration of CVT ratio by implementing the ideal tilting in axels of planets. Afterward, the device equates its frequency (\(\omega_{\rm d}\)) to that of the excitation (\(\omega_{\rm e}\)).