## 3 Multi-Scale Coupled Vibration Analysis of the TTB System
### Field Test Validation  
A dynamic load field test was performed on the Baihe Bridge in 2011 [19] (Figure 2). Detailed information about the field measuring campaign including the test train and measuring point layout can be found in [PERSON] et al. [17]. Stress time-histories of the several truss members (i.e., lower chord, post, stringer, and floor beam) marked in Figure 2 located in the center of the middle span obtained from the field test are compared with the results produced by multi-scale coupled vibration analysis of the TTB system. Track irregularities adopted in the analysis are generated using the power spectral density functions \(H(\,)\) fitted by the U.S. Federal Railroad Administration as:  
\[\left\{ \begin{array}{l}H_{\mathrm{a}}(\Omega)=\frac{0.25A_{\mathrm{a}}\Omega ^{2}}{\Omega^{2}\cdot\left(\Omega^{2}+\Omega_{\mathrm{c}}^{2}\right)}\\ H_{\mathrm{v}}(\Omega)=\frac{0.25A_{\mathrm{v}}\Omega^{2}_{\mathrm{c}}}{\Omega ^{2}\cdot\left(\Omega^{2}+\Omega_{\mathrm{c}}^{2}\right)}\\ H_{\mathrm{c}}(\Omega)=\frac{A_{\mathrm{v}}\Omega^{2}_{\mathrm{c}}}{\left( \Omega^{2}+\Omega_{\mathrm{c}}^{2}\right)\cdot\left(\Omega^{2}+\Omega_{ \mathrm{c}}^{2}\right)}\end{array} \right. \tag{4}\]  
where \(\Omega\) (rad/m) is the spatial angular frequency; \(H_{\mathrm{a}}\), \(H_{\mathrm{v}}\), and \(H_{\mathrm{c}}\) are, respectively, the lateral, vertical, and torsional irregularities in cm\({}^{2}\)/(rad/m); \(\Omega_{\mathrm{c}}\) (rad/m) and \(\Omega_{\mathrm{s}}\) (rad/m) are the cut-off frequencies; and \(A_{\mathrm{a}}\) (cm\({}^{2}\) rad/m) and \(A_{\mathrm{v}}\) (cm\({}^{2}\) rad/m) are the surface roughness constants. The Federal Railroad Administration proposed that track irregularities can be categorized into six classes (i.e., class 1 to class 6) based on the measured data from various lines [31]. The four parameters \(A_{\mathrm{a}}\), \(A_{\mathrm{v}}\), \(\Omega_{\mathrm{c}}\), and \(\Omega_{\mathrm{s}}\) in Equation (4) for calculating the power spectral density and the allowed travel speed are reported in Table 2. It can be seen that tracks of class 6 have the highest allowed speed, and accordingly, the best track smoothness. Samples of the track irregularities are subsequently numerically produced through trigonometric series summation. In this study, power spectral densities associated with class 5 are adopted to simulate track irregularities on the Baihe Bridge since class 5 can characterize well the level of irregularity related to the regular main railway lines in China, e.g., the Beijing-Tongliao line [32]. Herein, the test train, composed of two locomotives, six loaded wagons, and 14 empty wagons with the static axle weights = 23 t, 20.6 t, and 5.6 t, respectively, travelled over the Baihe Bridge at a designated speed of 80 km/h. The main parameters of both the locomotive and wagon, such as geometric dimension, mass, moment of inertia, spring stiffness and damping, are reported in Table 3. In order to calculate the two parameters in the Rayleigh damping formulation, an identical modal damping ratio is considered as 1% for the fundamental modes of the Baihe Bridge involved in the calculation [6; 33]. The time step selected to solve the equation of motion of the TTB system (i.e., Equation (1)) is 0.005 s. A time step of 0.01 s has been used to perform train-bridge dynamic analysis focusing on acceleration and displacement response of the bridge structure [29]. Since the high-frequency contents of the stress data are usually fewer than that of the accelerations, 0.005 s is considered sufficient to capture the high-frequency stresses considering the loading frequency (e.g., train speed), low natural frequencies of the structure, and frequency characteristics of the stress response.