## 4 Results Analysis  
This section aims to illustrate the effect of fastener rubber pad's temperature dependent Young's modulus to the vibration responses of the vehicle-rail-viaduct system. Via the dynamic flexibility method and the constructed vehicle-rail-viaduct coupling model for vertical vibration analysis, the track irregularity spectrum of the rail from ISO 3095:2005 was considered as excitation, and according to the distinct fastener rubber pad's stiffness under different temperatures, five scenarios were studied for investigating the performances of the vehicle-rail-viaduct coupling vibration, in order to draw out the effect of the fastener rubber pad's temperature-dependent Young's modulus. Figure 7 shows the dynamic flexibility amplitude of the rail-viaduct for all five scenarios. From Figure 7, the dynamic flexibilities of the rail-viaduct for all five scenarios showed peaks around 5  Hz, and the dynamic flexibilities increased as then temperature increased, while the peak frequency shifted leftward; in addition, the peak frequency was the first order resonant frequency of the rail-viaduct vertical coupling system, which implies that the temperature dependent performance of the fastener rubber pad had little effect to such first order resonant frequency; in frequency range [0, 107] Hz, the dynamic flexibilities of the rail-viaduct increased as the temperature increased; in frequency range [107, 144] Hz, the ratios of the peaks of the rail-viaduct's dynamic flexibilities to the peak under first scenario were 100%, 389.3%, 506.6%, 586.1%, 624.6%. As the temperature increased for the fastener rubber pad, the peaks of the dynamic flexibilities of the rail-viaduct increased, and the resonant frequency peaks shifted leftward. Table 4 gives the exact first order resonant and dominant frequencies for five scenarios. The dominant frequencies change more than the first order resonant frequencies for the same scenario in comparison to the Case #1. Figure 7: Dynamic flexibility amplitude of the rail under \(-40\), \(-20\), \(0\), \(20\), and \(40\)\({}^{\circ}\)C.  
Figure 8 shows the wheel-rail contact forces under five scenarios. From Figure 8, it can be found that: (1) below 20 Hz, the temperature dependent Young's modulus of the fastener rubber pad of the fastener had little effect to the wheel-rail force; (2) in frequency range [20; 42] Hz, the amplitude of the wheel-rail force increased sharply as the temperature increased; (3) as the temperature increased, the peaks of the wheel-rail forces for all five scenarios decreased, and so did the dominant frequencies; the dominant frequencies for five scenarios were 74, 51, 47, 43 and 42 Hz, respectively; (4) In frequency range [56; 100] Hz, as the temperature decreases, the wheel-rail forces increased. Figures 9 and 10 illustrate the accelerations of the vehicle body and the wheelset, respectively. Because the responses of the vehicle mainly occupy the frequency range below 20 Hz, this study takes the frequency range [0; 20] Hz for further investigation. From Figure 9, for all five scenarios, the vertical accelerations of the vehicle had little difference, which suggests that the temperature dependent Young's modulus of the fastener rubber pad had little influence to the vertical responses of the vehicle on such occasion. In Figure 10, for all five scenarios, regarding to the frequency range below 20 Hz, the temperature dependent Young's modulus of the fastener rubber pad had little influence to the vertical responses of the wheelset; while as temperature increased, the peaks of the vertical accelerations decreased gradually, and the dominant frequencies also decreased.