## 3 Numerical Analysis
### Multi-Span Bridges  
As the next step, the numerical study was repeated for the two- and three-span bridges. Figure 11 presents the variation of the natural vibration frequencies of the two-span bridge for the first vertical modes. As expected, the two-span bridge has higher frequencies for each mode compared to its single-span counterpart (Figure 7). Similar to the single-span bridge, vibration frequencies of the first two modes of the two-span bridge remain relatively identical for \(k_{v}\geqslant 1\times 10^{8}\) kN/m while a significant drop can be observed in the third and higher mode frequencies when the bearing stiffness is \(k_{v}=1\times 10^{8}\) kN/m and continue to drop as the bearing stiffness continues to reduce. Figure 11: Variation of the eigenvalues of the two span bridge with bearing stiffness. Figure 10: Mode shapes of the bridge with different bearing stiffness values. Figure 12 presents the maximum accelerations at different locations of a two-span bridge for different bearing stiffness values and train speeds of 50, 80, and 130 km/h, respectively. The results for the two-span bridge are similar to those obtained for a single-span bridge that are presented in Figure 8 in many aspects. For example, the location of the absolute maximum acceleration along the length of the bridge shifts from the middle of the span to the ends of the bridge. However, the accelerations at the middle support remain the lowest along the length of the bridge for all the bearing stiffness values investigated. For \(k_{v}\geqslant 1\times 10^{9}\) kN/m, the bearings remain virtually rigid, and the vibrations are minimal although the shift in the location of maximum acceleration location can already be observed for \(k_{v}=1\times 10^{9}\) kN/m. As the bearing stiffness decreases, the accelerations at the middle support increase but remain less than those at the other locations of the bridge. This is unlike the supports at either end of the bridge, which become the point of highest acceleration for lower bearing stiffness values. This difference can be explained by the discrepancy in the boundary conditions for these locations. The train traverses the relatively soft bridge before arriving at the middle support and after leaving it. As such, the vibrations at the middle support are dominated by the bridge stiffness and frequency. On the contrary, the train travels over an infinitely rigid surface before crossing the bearing, which is stiffer compared to the bridge itself even for the lower bearing stiffness values. Only after traveling some meters, the stiffness and the frequency of the bridge start to dominate the behavior. Therefore, the vibrations close to the end bearings are dominated by high frequency motion that leads to higher acceleration amplitudes compared to the bearing at the middle. To further study the impact of the bearing stiffness on the frequency content of the dynamic response, Fourier Amplitude Spectrum of the acceleration response at the middle of the first span was created and presented in Figure 13 for speeds of 50 km/h and 80 km/h. For both speeds, the sensitivity of the frequency content of the vibrations to the bearing stiffness is evident. Particularly for the speed of 50 km/h, the vibrations are dominated by the first mode for \(k_{v}=1\times 10^{4}\) kN/m and \(k_{v}=1\times 10^{5}\) kN/m; see Figure 13a. For the higher bearing stiffness values, \(k_{v}\geqslant 1\times 10^{8}\) kN/m, the effect of the second mode at approximately 20 Hz is evident from Figure 13a. The most complicated behavior as far as frequency content is concerned can be observed for \(k_{v}=1\times 10^{6}\) kN/m and \(k_{v}=1\times 10^{7}\) kN/m. For the former, the behavior is dominated by the first mode, but the contributions of the third and fourth modes at approximately 25 Hz and 40 Hz are not negligible. For the speed of 80 \(km/h\) (Figure 13b), for higher bearing stiffness values of \(k_{v}\geqslant 1\times 10^{8}\) kN/m, the contribution of the second mode becomes more prominent.