# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 3 Time-Frequency Interaction Analysis on the Train-Track-Bridge System
### Time-Frequency Analysis from Excitation to Bridge Response
#### 3.3.3 On the Nonresonant Bridge  
In order to study the time-frequency evolutionary characteristics on the nonresonant bridge from excitation to response, the speed conditions of 320, 350 and 400 km/h were selected, and the corresponding first-order loading frequencies are 3.56, 3.89 and 4.44 Hz, respectively. The CWT scalograms of mid-span bridge excitation and vertical acceleration response are shown in Figure 9 and it can be seen that:Figure 7: Scalogram from excitation to vertical vibratory resonant acceleration response at mid–span bridge: (**a**) Excitation scalogram at 195 km/h; (**b**) Bridge response scalogram at 195 km/h; (**c**) Excitation scalogram at 292.5 km/h; (**d**) Bridge response scalogram at 292.5 km/h; (**e**) Excitation scalogram at 585 km/h; (**f**) Bridge response scalogram at 585 km/h.  
Figure 8: _Cont._  
1. For the excitation scalogram in Figure 9a,c,e, the time-frequency band distributive regulations are almost identical to Sections 3.3.1 and 3.3.2. Owing to the train speed which is situated in the nonresonant bridge condition according to Figure 6 in Section 3.2, the quasi-resonance and resonance-triggered band disappears from the excitation scalogram.
2. For the response scalogram in Figure 9b,d,f, there is a \"peak staggering\" phenomenon between the transient response band and the steady state response band. During this period, the first wheel set of the vehicle acts on the bridge which has no time to produce a steady state response, so free vibration whose frequency equals the bridge natural frequency will occur within the transient response band. When the moving load becomes stably periodic, the bridge presents a steady state response, and the scalogram value will increase to the first-order response frequency contributed by the first-order loading frequency, while the scalogram value is attenuated in the transient response band. At the moment when the last wheel set leaves the bridge, with the end of wheel set periodic loading, the steady state response weakens and is gradually replaced by the transient response. Therefore, a higher scalogram value is reintensified in the transient response band after the train leaves the bridge. With the increase in train running speed, the scalogram values of steady state response and transient response are slightly higher than those under low-speed conditions.
3. In general, the nonresonant response of the bridge is mainly composed of the steady-state response contributed by the loading frequency component, while the transient response is contributed by the bridge natural vibration frequency. The response is steady state when the train moves on the bridge and transient at the initial stage and later stage of the train entering and leaving the bridge. In addition, the intensity of nonresonant response of the bridge is related to the train speed.  
Figure 8: Scalogram from excitation to vertical vibratory resonantly transitional acceleration response at mid–span bridge: (**a**) Excitation scalogram at train speed of 175 km/h; (**b**) Bridge response scalogram at train speed of 175 km/h; (**c**) Excitation scalogram at train speed of 210 km/h; (**d**) Bridge response scalogram at train speed of 210 km/h; (**e**) Excitation scalogram at train speed of 550 km/h; (**f**) Bridge response scalogram at train speed of 550 km/h; (**g**) Excitation scalogram at train speed of 680 km/h; (**h**) Bridge response scalogram at train speed of 680 km/h.  
Figure 9: Scalogram from excitation to vertical vibratory resonant acceleration response at mid–span bridge: (**a**) Excitation scalogram at train speed of 320 km/h; (**b**) Bridge response scalogram at train speed of 320 km/h; (**c**) Excitation scalogram at train speed of 350 km/h; (**d**) Bridge response scalogram at train speed of 350 km/h; (**e**) Excitation scalogram at train speed of 400 km/h; (**f**) Bridge response scalogram at train speed of 400 km/h.