# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
[10] used the wavelet transform as a time-frequency representation for system identification purposes and confirmed the accuracy of this method by applying it to a numerical example and the acceleration responses from a real bridge under ambient excitation (the Queensborough Bridge in Vancouver, Canada). [PERSON] et al. [11] described the dynamic tests performed on a simply supported bridge in Northern Italy under traffic excitation and prove that the ability of wavelet transforms to detect abrupt changes, gradual change beginnings and ends of events make them well suited for the analysis of bridge health monitoring data. [PERSON] et al. [12] present the application of wavelet analysis to identify events and changes in structural state in a bridge during and after its construction. [PERSON] et al. [13] showed the capabilities of the real-time kinematic (RTK) global positioning network system (GPS) to measure the low-frequency vibration of a medium-span suspension bridge. The traditional method of attaching strain transducers to the softt of the bridge and placing axle detectors on the road surface has been replaced here by using additional transducers underneath the bridge for axle detection and nothing-on-the-road (NOR). [PERSON] et al. [14] presented a wavelet-based analysis of strain signals and showed the efficacy of using wavelets in pattern recognition of these signals. An algorithm based on a plot of wavelet coefficients versus time to detect damage appears to be very sensitive to noise. [PERSON] et al. [15] addressed these questions by: (a) using the acceleration signal, instead of the deflection signal, (b) employing a vehicle-bridge finite element interaction model, and (c) developing a novel wavelet-based approach using wavelet energy content at each bridge section, which proves to be more sensitive to damage than a wavelet coefficient line plot at a given scale as employed by others. [PERSON] et al. [16] described a completely new method based on modal analysis to locate and also measure the length and orientation of crack-type damage features in thin-walled composite beams (TWCB). It was found that the method is useful for the measurement of damage features in a variety of thin-walled composite beams such as aircraft wings and wind turbine blades, among others. In spite of plenty of research mentioned above, there is still a lack of comprehensive time-frequency characteristics analysis on bridge resonance for how the frequency component for the input excitation evolves as time goes by, how the bridge response transits from the nonresonant to resonant condition in the time-frequency domain and how the resonant bridge reacts on the track and the train subsystems. In this case, the article will establish a 2D train-track-bridge interactive model based on the wavelet computational properties, clarify the time-frequency characteristics from the input excitation to the output response through the comparison among a nonresonant, resonantly transitional, and resonant bridge, and analyze the resonant reaction from the bridge on the track and the train subsystem. Establishment and Solution of Motion Equation for Train-Track-Bridge Interactive System Based on Wavelet Computational Properties