# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 5 Conclusions  
This article has established the initial system for the time-frequency analysis on the bridge resonant response in a train-track-bridge interactive system based on continuous wavelet transform and its computational properties. The direct integration through an intersystem iteration solution is adopted on the translation factors and realizes the conversion from excitation to response in the wavelet domain, which is proved feasible and precise in time-frequency evolutionary characteristic analysis. Relevant conclusions can be drawn as follows:  
1. The bridge resonant response is stationary while the train is crossing the bridge. The resonance-triggering band is included within each order of loading frequency, while the triggered resonance band dominates the bridge response only when the signal energy of the corresponding triggering band is significant. Typically, the resonance triggering band, which is located in the first-order loading frequency and equals the natural frequency of bridge, will lead to the most intense bridge resonant response.
2. The bridge's resonantly transitional response is stationary while the train is crossing the bridge. The quasi-resonance-triggering band is included within each order of loading frequency and will dominate the bridge response. Typically, the closer the quasi-resonance-triggering band is to the natural frequency of bridge, the more the bridge response will be intensified.
3. The nonresonant response of the bridge is mainly composed of the steady state response contributed by the loading frequency component and the transient response contributed by the bridge natural vibration frequency. The response is steady state when the train moves on the bridge forming periodic loading, while transient at the initial stage and later stage of the train entering and leaving the bridge. Staggering domination between the steady state and the transient response is the main phenomenon for the nonresonant bridge.
4. Within the low-frequency band, the resonant bridge will have more significant impact on the track subsystem, especially for the substructure closer to the bridge deck, than the nonresonant one, while the impacts from the bridges almost neglect the car body even under the resonant bridge condition.  
[PERSON] came up with the concept, realized the simulation, analyzed the data and edited the draft of manuscript. [PERSON] conducted the literature review, checked the computations, wrote the draft of the manuscript, replied to reviewers' comments and revised the final version. [PERSON] verified the simulated data and polished the article. [PERSON] corrected the model of simulation and debugged the programming. All authors have read and agreed to the published version of the manuscript.  
National Natural Science Foundation of China, Grant number: 52178101.  
Not applicable.  
The data presented in this article are available on request from the corresponding author.  
Thanks for the support for this work from the National Natural Science Foundation of China.  
The authors declare no conflict of interest.