# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 2 Methods
### Dynamics of Rocking Frames  
For this study, the rigid rocking frame modeling assumption is adopted [10]. The examined typical rocking viaduct is depicted in Figure 1. Since rocking initiation depends on ground acceleration, the examined system is assumed to remain inactive until the uplift criterion is fulfilled. The uplift condition derived from the static equilibrium depends on the slenderness \(\alpha\) of the rocking pier and ground acceleration \(\alpha_{\text{g}}\). The following equation gives the minimum acceleration value required to uplift the examined system:  
\[\frac{\left|\alpha_{\text{g}}\right|}{\text{g}}\geq\tan\alpha \tag{19}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**Seismic**} & \multirow{2}{*}{**Description**} \\  & **Parameter** & & \\ \hline Amplitude-Based & PGA & Peak Ground Acceleration & [57] \\  & PGV & Peak Ground Velocity & [57] \\  & PGD & Peak Ground Displacement & [57] \\  & EPA & Effective Peak Acceleration & [58] \\  & EPA\({}_{\text{max}}\) & maximum EPA & [58] \\ \hline Duration-Based & t\({}_{\text{sig}}\) & Strong motion duration & [59] \\ \hline Frequency-Based & PGV/PGA & Period parameter & [56] \\  & PGD/PGV & Period parameter & [39] \\  & T\({}_{\text{m}}\) & Mean Period & [60] \\  & CP & Central Period & [61] \\ \hline Energy-Based & a\({}_{\text{rms}}\) & Acceleration’s Root Mean Square & [62] \\  & v\({}_{\text{rms}}\) & Velocity’s Root Mean Square & [62] \\  & d\({}_{\text{rms}}\) & Displacement’s Root Mean Square & [62] \\  & CAV & Cumulative Absolute Velocity & [63] \\  & CAD & Cumulative Absolute Displacement & [63] \\  & SED & Specific Energy Density & [57] \\  & I\({}_{\text{A}}\) & Arias Intensity & [64] \\  & Ic\({}_{\text{C}}\) & Characteristic Intensity & [65] \\  & I\({}_{\text{F}}\) & Fajifar–Vidic–Fischinger Intensity & [66] \\  & I\({}_{\text{H}}\) & Housner Intensity & [67] \\  & ASI & Acceleration Spectrum Intensity & [68] \\  & DPA\({}_{\text{s}}\) & Araya–Saragoni Destructiveness Potential & [69] \\  & P\({}_{\text{90}}\) & Seismic Power of Jennings & [70] \\  & Lm\({}_{\text{a}}\) & Characteristic length scale of acceleration & [71] \\  & Lm\({}_{\text{v}}\) & Characteristic length scale of velocity & [31] \\  & EFP & Energy–Frequency Parameter & [54] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conventional IMs. After the uplift of rocking piers, the equation of motion is expressed as follows [10]:  
\[\ddot{\theta}=-\hat{\rho}^{2}\left[\sin(\text{sgn}(\theta)\alpha-\theta)+\frac{ \alpha_{\text{g}}}{\text{g}}\cos(\text{sgn}(\theta)\alpha-\theta)\right] \tag{20}\]  
where \(\text{sgn}()\) is the signum function of the ground acceleration and \(\hat{\rho}\) the frequency parameter, defined as follows:  
\[\hat{\rho}=\sqrt{\frac{\text{g}}{\text{I}_{\text{o}}\text{R}}}\sqrt{\frac{1+2 \gamma}{1+4\gamma/\text{I}_{\text{o}}}} \tag{21}\]  
where R is the semi-diagonal length of the piers, \(\gamma=\text{m}_{\text{b}}/\text{N}\text{m}_{\text{c}}\) is the relative mass of the deck, \(\text{m}_{\text{b}}\) is the mass of the deck, N is the number of rocking piers, \(\text{m}_{\text{c}}\) is the mass of the rocking pier, \(\text{I}_{\text{o}}=\text{I}_{\text{o}}/(\text{m}_{\text{c}}\text{R}^{2})\), and \(\text{I}_{\text{o}}\) is the rotational inertia of the pier about the pivot point. For rectangular rocking piers, \(\text{I}_{\text{o}}=4/3\). During the rocking response, energy dissipation is instantaneous and occurs only at impact instants when the rocking rotation of the piers reverses the sign. In the case of rocking frames, the impulse-momentum theorem yields the coefficient of restitution (COR) as follows [10]:  
\[\text{COR}=\frac{\dot{\theta}^{+}}{\dot{\theta}^{-}}=1-2\sin^{2}\alpha\frac{1 +4\gamma}{\text{I}_{\text{o}}+4\gamma} \tag{22}\]  
where \(\dot{\theta}^{-}\) and \(\dot{\theta}^{+}\) are the rotational velocities before (\(-\)) and after (\(+\)) impact, respectively.