## 6 Conclusions  
In this paper, the one degree-of-freedom oscillator and a simply-supported Euler-Bernoulli beam model were used to establish a vehicle-bridge interaction system. With a large vehicle-bridge mass ratio, dimensionless parameters, mainly from the three aspects, vehicle speed, vehicle-bridge frequency ratio, and mass ratio, were used to discuss the patterns of an extracted bridge fundamental frequency from the acceleration signal of a  
Figure 15: Vertical acceleration response and spectrum of vehicle: (**a,b**) light vehicle passing through single-piece steel beam with \(v=0.027\); (**c,d**) heavy vehicle passing through single-piece steel beam with \(v=0.031\); (**e,f**) light vehicle passing through double-piece steel beam with \(v=0.065\); and (**g,h**) heavy vehicle passing through single-piece steel beam with \(v=0.066\).  
passing vehicle. In general, under the influence of vehicle mass, the bridge fundamental frequency identified from the vehicle acceleration signal shows a different pattern than that of when the vehicle mass is ignored. An intuitive display of the accuracy of the extracted bridge fundamental frequency is given. Through the model test in the laboratory, the theoretically calculated patterns are verified. In addition, the corresponding screening principles and methods are proposed on how to select the appropriate peak frequency from the spectrogram to achieve successful bridge frequency identification. The main conclusions are summarized as follows:  
1. The right-biased frequency does not always occupy the main advantage in the identified bridge fundamental frequencies, especially when \(\mu_{1}>1\), the identified bridge fundamental frequency is dominated by the left-biased frequency.
2. When \(\mu_{1}>1\), \(\Delta_{b}\) is greatly affected by \(\alpha_{mv}\) and \(\alpha_{v}\), and the interaction between \(\mu_{1}\) and the other two parameters is small; when \(\mu_{1}<1\), \(\mu_{1}\), \(\alpha_{mv}\), and \(\alpha_{v}\) have a greater impact on \(\Delta_{b}\), and there is a similar degree of interaction between them.
3. To ensure the accuracy of identifying the bridge fundamental frequency, the vehicle-bridge mass ratio and vehicle speed should not be too large. It is recommended that the vehicle-bridge mass ratio should not exceed 10%, or even 5%.
4. This paper proposed a few basic principles and reference methods for effective bridge frequency identification using spectrograms of vehicle acceleration, which provide greater accuracy and rigorous for bridge frequency extraction.  
The conclusions of this paper can provide useful suggestions for the application of bridge indirect monitoring techniques. Due to the limitations of the simplified model and the scale of the experimental model, further investigations are required in order to understand the pattern of bridge frequency deviation. First, the models of vehicles and bridges should be refined. In future parametric analyses, the effects of vehicle damping and bridge damping on identifying bridge frequencies from a passing vehicle's acceleration will be considered. Second, field tests will be carried out. For a real bridge, the conditions under which the frequency deviation phenomenon occurs should be studied.  
Conceptualization, B.H.; methodology, Q.W. and H.X.; software, Q.W. and W.Y.; validation, Q.W., B.H. and J.Y.; formal analysis, Q.W.; investigation, Q.W. and W.Y.; writing--original draft preparation, Q.W.; writing--review and editing, B.H., J.Z. and H.X.; visualization, Q.W.; supervision, B.H. and J.Z.; project administration, B.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities under grant no. 2018 YSI02; the Science and Technology Project of Beijing Transportation Commission under grant no. KJ-2017-5; and the Fundamental Research Funds for the Central Universities under grant no. 2019 RC012.  
Not applicable.  
The authors declare no conflict of interest.