## 3 Formulation of a d-MTFDI-Controlled System  
For WIV of a high-rise building, the basic modal response is dominative [12]. Furthermore, for an inerter-enhanced damper-controlled system, the error in vibration control performance based on a generalized single-degree-of-freedom (sDOF) model can be considered negligible [31]. Therefore, the \(F\)-floor high-rise building will be modeled as a generalized sDOF model to streamline the analysis process. For a generalized sDOF system, the equation of motion (EOM) subjected to wind excitation \(P(t)\) without control is as follows:  
\[\ddot{u}+2\zeta_{s}\omega_{s}\dot{u}+\omega_{s}^{2}u=p(t) \tag{1}\]  
where \(\omega_{s}=\sqrt{K/M}\) and \(\zeta_{s}=C/(2\sqrt{KM})\) are the basic circular natural frequency and damping ratio of the primary structure, respectively; \(M\), \(C\) and \(K\) are the generalized mass, damping, and stiffness of the generalized primary structure, respectively; \(u\) is the generalized displacement of the primary structure; and \(p(t)=P(t)/M\) is the generalized wind excitation normalized by mass. For a d-MTFDI-controlled system, the EOM excited by \(p(t)\) can be derived from the d'Alembert's principle and expressed as follows:  
\[\left\{ \begin{array}{l}M\ddot{u}+C\dot{u}+Ku=p(t)-\sum\limits_{i=1}^{R}[ \varphi_{ai}F_{di}(t)+\varphi_{bi}F_{bi}(t)]\\ m_{di}\ddot{u}_{di}(t)-F_{bi}(t)+F_{di}(t)=0\quad i=1,2,\ldots,R\end{array} \right. \tag{2}\]  
where \(\varphi_{ai}\) and \(\varphi_{bi}\) are the normalized basic modal value of the \(a_{i}\)th floor and \(b_{i}\)th floor, respectively; \(u_{di}(t)\) is the relative motion of the \(i\)th TFDI; and \(F_{di}(t)\) and \(F_{bi}(t)\) are the damping force and inertance force produced by the \(i\)th TFDI; and, as follows:  
\[\left\{ \begin{array}{l}F_{di}(t)=k_{di}u_{di}(t)+c_{di}\dot{u}_{di}(t)\\ F_{bi}(t)=-b_{di}\{\varphi_{ai}\ddot{u}(t)-[\varphi_{bi}\ddot{u}(t)+\ddot{u}_{ di}(t)]\}\end{array} \right. \tag{3}\]  
For the \(i\)th TFDI, the circular frequency is \(\omega_{di}=\sqrt{k_{di}/(m_{di}+b_{di})}\), the circular frequency ratio is \(\upsilon_{di}=\omega_{di}/\omega_{s}\), the damping ratio is \(\xi_{di}=c_{di}/2\omega_{di}(m_{di}+b_{di})\), the mass ratio is \(\mu_{i}=m_{di}/M\) and the inertance ratio is \(\beta_{i}=b_{di}/M\). By substituting the above-defined  
Figure 4: Elevation view of a super high-rise building with a d-MTFDI system. parameters and Equation (3) into Equation (2) and applying the Fourier transform, the following is obtained:  
\[ \begin{array}{l}-\omega^{2}u(\omega)+2i\omega\varsigma_{s}\omega_{s}u(\omega)+ \omega_{s}^{2}u(\omega)=p(\omega)\\ -\sum\limits_{i=1}^{R}\{[\varphi_{ai}(\mu_{i}+\beta_{i})(v_{Ai}^{2}\omega_{s}^{ 2}+2i\omega\varsigma_{di}v_{d}\omega_{s})+\varphi_{bi}\beta_{i}\omega^{2}]u_{di }(\omega)+\varphi_{bi}(\varphi_{ai}-\varphi_{bi})\beta_{i}\omega^{2}u(\omega)\} \end{array}  \tag{4}\]  
where  
\[u_{di}(\omega)=\frac{[\mu_{i}\varphi_{ai}+\beta_{i}(\varphi_{ai}-\varphi_{bi})] \omega^{2}}{(\mu_{i}+\beta_{i})(-\omega^{2}+v_{di}^{2}\omega_{s}^{2}+i\omega^{ 2}\xi_{di}v_{di}\omega_{s})}u(\omega) \tag{5}\]  
Based on Equation (4), the normalized transfer function (NTF) of generalized acceleration with d-MTFDI can be derived by substituting Equation (5) into Equation (4) and can be expressed as follows:  
\[H_{a}^{\rm d-MTFDI}(\omega)=\omega^{2}H^{\rm d-MTFDI}(\omega)=\omega^{2}\frac{ u(\omega)}{p(\omega)} \tag{6}\]  
For the uncontrolled structure, the corresponding NTF of generalized acceleration can be derived from Equation (1):  
\[H_{a}^{\rm uncontrolled}(\omega)=\frac{\omega^{2}}{\omega_{s}^{2}-\omega^{2}+2i \varsigma_{s}\omega_{s}\omega} \tag{7}\]  
Thus, the response of the standard deviation (std) of generalized acceleration can be obtained as follows:  
\[\sigma_{a}^{\rm w}=\int_{0}^{\omega_{c}}|H_{a}^{\rm w}(\omega)|^{2}S_{p}( \omega)d\omega\ \ \ \ \ {\rm w}={\rm d}-{\rm MTFDI}\ {\rm or}\ {\rm uncontrolled} \tag{8}\]  
where \(S_{p}(\omega)\) is the power spectra density (PSD) of \(p(\omega)\) and \(\omega_{c}\) is the cut-off circular frequency.