## 2 Aerodynamic Admittance Identification Methodology  
Although buffeting forces are essentially unsteady, they are conventionally modelled based on quasi-steady theory and then modified by AAFs to consider unsteady behaviours [21; 22; 23]. Much research has shown that the average buffeting force remains constant, but the fluctuating aerodynamic force caused by turbulent wind is dependent on the dimensionless reduced frequency. Because the normal wind case is generally considered for most periods of bridge service, the AAFs related to the drag force (\(D_{b}\)), lift force (\(L_{b}\)) and pitch moment (\(M_{b}\)) are focused on and investigated most often, which was also done in this paper. According to [PERSON] [24], the buffeting force is often expressed by six AAFs:  
\[D_{b}(x,t)=\frac{1}{2}\rho U^{2}B\bigg{[}2C_{D}\frac{u(x,t)}{U}\chi_{Du}(t)+C_ {D}^{\prime}\frac{w(x,t)}{U}\chi_{Dw}(t)\bigg{]} \tag{1}\]  
\[L_{b}(x,t)=\frac{1}{2}\rho U^{2}B\bigg{[}2C_{L}\frac{u(x,t)}{U}\chi_{Lu}(t)+(C _{L}^{\prime}+C_{D})\frac{w(x,t)}{U}\chi_{Lw}(t)\bigg{]} \tag{2}\]  
\[M_{b}(x,t)=\frac{1}{2}\rho U^{2}B^{2}\bigg{[}2C_{M}\frac{u(x,t)}{U}\chi_{Mu}( t)+C_{M}^{\prime}\frac{w(x,t)}{U}\chi_{Mu}(t)\bigg{]} \tag{3}\]  
where \(\chi_{ij}(t)\) (\(i=D\), \(L\), or \(M\), corresponding to the lateral, lift and pitch moment loads, respectively, and \(j\) represents \(u\) or \(w\), the wind direction components) is the component of the aerodynamic admittance of the buffeting loads in the respective wind direction. Based on the frequency dependence of \(\chi_{ij}(t)\), Equations (1)-(3) are transformed by the Fourier transform while neglecting the effect of cross spectra, and the sectional buffeting force spectra are expressed as  
\[S_{D}(\omega)=\bigg{(}\frac{\rho LIB}{2}\bigg{)}^{2}\bigg{[}4C_{D}^{2}S_{u}( \omega)\big{|}\chi_{Du}(\omega)\big{|}^{2}+C_{D}^{\prime}{}^{2}S_{w}(\omega) \big{|}\chi_{Dw}(\omega)\big{|}^{2}\bigg{]} \tag{4}\]  
\[S_{L}(\omega)=\bigg{(}\frac{\rho LIB}{2}\bigg{)}^{2}\bigg{[}4C_{L}^{2}S_{u}( \omega)\big{|}\chi_{Lu}(\omega)\big{|}^{2}+(C_{L}^{\prime}+C_{D})^{2}S_{w}( \omega)\big{|}\chi_{Lw}(\omega)\big{|}^{2}\bigg{]} \tag{5}\]  
\[S_{M}(\omega)=\bigg{(}\frac{\rho LIB^{2}}{2}\bigg{)}^{2}\bigg{[}4C_{M}^{2}S_{u} (\omega)\big{|}\chi_{Mu}(\omega)\big{|}^{2}+C_{M}^{\prime}{}^{2}S_{w}(\omega) \big{|}\chi_{Mw}(\omega)\big{|}^{2}\bigg{]} \tag{6}\]  
Usually, the equivalent AAF method, or the autospectral method, is applied to identify the AAF for a bridge segment model based on fluctuating aerodynamic force and fluctuating wind velocity [16; 19; 25]. This method assumes that two AAFs (\(\chi_{iu},\chi_{inv},i=D,L,M\)) in each \(S_{i}\) related to the same force component and wind velocities along \(u\) and \(w\) are equal to each other, i.e.,\(\chi_{iu}=\chi_{inv}=\chi_{i}\). In the frequency domain, this method can obtain the modulus squared value of equivalent AAFs for each force, i.e., \(|\chi_{i}|^{2}\), which can ensure that the reproduced force auto spectrum is equivalent to the real or tested force autospectrum. \(|\chi_{i}|^{2}\) is a weighted average of \(|\chi_{iu}|^{2}\) and \(|\chi_{iw}|^{2}\), and is often close to \(|\chi_{Dw}|^{2}\), \(|\chi_{Lw}|^{2}\) or \(|\chi_{Lw}|^{2}\) for common bridges because the weight factors of these three AAFs in the spectral \begin{table}
\begin{tabular}{c c} \hline \(U\) & Mean velocity of turbulence \\ B, H, L & Width, height and length of segment model \\ z & Relative elevation of DDTG \\ \(u,v,w\) & Velocity components of turbulence \\ \(\alpha\) & Wind attack angle \\ \(F_{D},F_{L},F_{M}\) & Measured drag force, lift force and pitch moment \\ \(C_{D},C_{L},C_{M}\) & Drag, lift and pitch moment coefficients \\ \(C_{D}^{\prime}C_{L}^{\prime},C_{M}^{\prime}\) & Derivatives of drag, lift and pitch moment coefficients \\ \(S_{u},S_{w}\) & Power spectra of velocity components in the \(u\) and \(w\) directions \\ \(S_{D},S_{L},S_{M}\) & Power spectra of drag, lift and pitch moment \\ \(Ixx,Iyy,Izz\) & Inertia moments of an equivalent girder \\ \hline \end{tabular}
\end{table}