# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 5 Flutter Analyses
### Simplified Formulations and Comparisons  
Given the absence of reference data for a direct validation, different simplified formulas available in the literature for predicting the flutter wind velocity were employed for a comparison. The results are collected in Table 2. The simplified formulas adopted are in Equations (5a)-(5f). The first one is based on the reduction of the divergence wind velocity (\(U_{d}\)) and is the only one involving the aerodynamics of the deck by means of the derivative of the moment coefficient (\(C_{m}^{\prime}\)). Equations (5b)-(5d) are empirical formulas fitted for thin airfoils, while Equations (5e) and (5f) contain a coefficient fitted for suspension bridge decks.  
Figure 15: Variation of complex eigenvalues vs. wind velocity for the double-deck configuration: second method (labels within parentheses refer to mode shapes under wind action).  
\[\begin{split}\text{Frandsen}\end{split} \tag{5a}\] \[\begin{split} U_{f}=2.623f_{t}B\sqrt{\left(1-\frac{1}{\gamma_{ \omega}^{2}}\right)r_{\alpha}\mu_{1}}.\end{split}\] (5b) \[\begin{split} U_{f}=6.282f_{t}B\sqrt{\left(1-\frac{1}{\gamma_{ \omega}^{2}}\right)\frac{r_{\alpha}^{2}\mu_{1}}{1+8r_{\alpha}^{2}}}.\end{split}\] (5c) [PERSON] [43]: \[\begin{split} U_{f}=3.71 Bf_{t}\sqrt{\frac{\sqrt{Im}}{\rho B^{3}} \left[1-\left(\frac{f_{0}}{f_{t}}\right)^{2}\right]}.\end{split}\] (5d) [PERSON] [44] \[\begin{split} U_{f}=\eta\left[1+\left(\frac{\omega_{t}}{\omega_{v}}-0.5\right)\sqrt{\frac{2r}{B}0.72\mu_{2}}\;\right]\frac{B}{2}\omega_{v}.\end{split}\] (5e) [PERSON] and [PERSON] [45] : \[\begin{split} U_{f}=\eta\left(2.5\sqrt{\frac{2r}{B}\mu_{2}}\right) Bf_{t}.\end{split} \tag{5f}\]  
where \(\gamma_{\omega}=\frac{\omega_{t}}{\omega_{v}}\), \(r_{\alpha}=\sqrt{\frac{I}{mB^{2}}}\), \(\mu_{1}=\frac{2m}{\rho B^{2}}\), \(\mu_{2}=\frac{4m}{\pi\rho B^{2}}\), \(r=\sqrt{\frac{I}{m}}\). \(I\) is the polar mass moment of inertia per unit span length, \(m\) is the mass per unit span length, \(\omega_{v}\), \(\omega_{t}\) are the angular frequencies of the first vertical mode and the first torsional mode, \(f_{v}\), \(f_{t}\) are the respective frequencies and \(\eta\) is an empirical coefficient representing the difference between the flat plate and a generic profile (equal to 0.7 for suspension bridges [44]). Other simplified formulations based on flutter derivatives can be found in [46].  
Results provided by the different formulations for the single-deck configuration are similar to the ones obtained in the present work within the range \([-10\%;15\%]\). Larger variations are found for the double-deck configuration. In particular, formulas fitted for the thin airfoil provide a considerable overestimation of the critical speed with respect to the formulas fitted for the suspension bridges. The flutter limit provided by the first method is close to the prediction obtained by Equations (5b)-(5d), whereas the flutter velocity provided by the second method is closer to the predictions of Equations (5e) and (5f), and is the most conservative prediction.  
Hence, although for a single-deck configuration having the aerodynamic coefficients similar to those of the airfoil it is possible to obtain reasonable results using simplified formulations, for a double-deck configuration it is necessary to perform more detailed analyses.