## 3 Multi-Scale Coupled Vibration Analysis of the TTB System
### Field Test Validation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Track Class**} & \multicolumn{3}{c}{**Parameter**} & \multicolumn{3}{c}{**Allowed Speed (km/h)**} \\  & \(\mathbf{A_{\mathrm{a}}}\) & \(\mathbf{A_{\mathrm{v}}}\) & \(\Omega_{\mathrm{c}}\) & \(\Omega_{\mathrm{b}}\) & \\  & **(cm\({}^{2}\) rad/m)** & **(cm\({}^{2}\) rad/m)** & **(rad/m)** & **(rad/m)** & **Passenger Train** & **Freight Train** \\ \hline
1 & 3.3634 & 1.2107 & 0.8245 & 0.6046 & 24 & 16 \\
2 & 1.2107 & 1.0181 & 0.8245 & 0.9308 & 48 & 40 \\
3 & 0.4128 & 0.6816 & 0.8245 & 0.8520 & 96 & 64 \\
4 & 0.3027 & 0.5376 & 0.8245 & 1.1312 & 128 & 96 \\
5 & 0.0762 & 0.2095 & 0.8245 & 0.8209 & 144 & 128 \\
6 & 0.0339 & 0.0339 & 0.8245 & 0.4380 & 176 & 176 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Power spectral density parameters and allowed train speed proposed by the U.S. Federal Railroad Administration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & **Locomotive** & **Wagon** \\ \hline Mass of the car-body & kg & 9.05e4 & 1.56e4 \\ Moment of inertia of car-body around longitudinal axis & kg m\({}^{2}\) & 8.88e4 & 2.66e4 \\ Moment of inertia of car-body around lateral axis & kg m\({}^{2}\) & 2.88e6 & 2.66e5 \\ Moment of inertia of car-body around vertical axis & kg m\({}^{2}\) & 2.88e6 & 2.84e5 \\ Mass of the bogie & kg m\({}^{2}\) & 1.02e4 & 1.13e3 \\ Moment of inertia of bogie around longitudinal axis & kg m\({}^{2}\) & 5e3 & 1.92e2 \\ Moment of inertia of bogie around lateral axis & kg m\({}^{2}\) & 2.26e4 & 4.19e2 \\ Moment of inertia of bogie around vertical axis & kg m\({}^{2}\) & 2.26e4 & 4.19e2 \\ Mass of the wheelest & kg & 4.52e3 & 1.2e3 \\ Moment of inertia of wheelest around longitudinal axis & kg m\({}^{2}\) & 3.26e4 & 7.4e2 \\ Moment of inertia of wheelest around vertical axis & kg m\({}^{2}\) & 3.26e4 & 7.4e2 \\ Primary vertical spring stiffness & N/m & 1.47e6 & 1.7e7 \\ Primary lateral spring stiffness & N/m & 2.65e6 & 3.6e6 \\ Primary longitudinal spring stiffness & N/m & 0 & 3.6e6 \\ Secondary vertical spring stiffness & N/m & 6.62e6 & 5.32e6 \\ Secondary lateral spring stiffness & N/m & 5.5e7 & 4.14e6 \\ Secondary longitudinal spring stiffness & N/m & 0 & 6.5e6 \\ Primary vertical damping & N s/m & 1.2e5 & 3e3 \\ Primary lateral damping & N s/m & 0 & 0 \\ Primary longitudinal damping & N s/m & 0 & 0 \\ Secondary vertical damping & N s/m & 7.2e4 & 3e3 \\ Secondary lateral damping & N s/m & 1e5 & 0 \\ Secondary longitudinal damping & N s/m & 0 & 0 \\ Half span of the primary suspension system & m & 1.025 & 1 \\ Half distance between two wheelsets & m & 1.8 & 0.875 \\ Half distance between two bogies & m & 6.0 & 4.35 \\ Half distance between two car-body couplers & m & 21.1 & 13.438 \\ Wheel radius & m & 0.525 & 0.42 \\ Distance between the car-body and the secondary suspension system & m & 1.744 & 0.74 \\ Distance between the secondary suspension system and bogie & m & 0.010 & 0.59 \\ Distance between the bogie and wheelsets & m & 0.55 & 0.015 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main parameters of the test train. Figure 6 shows the computed and measured stress time-histories associated with the instrumented cross-sections of the lower chord, post, stringer, and floor beam (Figure 2). The two sets of stress time-histories are processed with a low-pass filter of 30 Hz, considering modal analysis results where the natural frequencies of the first 100 bridge modes of vibration are below about 9.06 Hz. Moreover, the sampling frequency adopted in the field test is 128 Hz for stress data collection. It can be seen from the figure that good agreement is achieved between the two sets of stress data. In addition, stress ranges and their cycle counts generated from stress time-histories of the stringer (Figure 6c) by using the rain-flow counting method