## 4 The Vehicle-to-Bridge Mass Ratio on Frequency Deviation Rate
### Probability of Accurately Identifying Bridge Frequencies  
Here, we define that when \(|\Delta_{b}|<0.05\), the bridge fundamental frequency is considered to be accurately identified. Since vehicle speed and mass mentioned in the above analysis process are accurately known, they are considered as controllable parameters. Within the specified range of parameters in the previous section, \(\alpha_{mv}\) and \(\alpha_{v}\) are selected as variables. The probability of accurately identifying the bridge fundamental frequency was obtained, as shown in Figure 7. In Figure 7, when \(\alpha_{mv}<0.05\) and \(\alpha_{v}<0.3\), the probability of accurately identifying the bridge fundamental frequency is greater than 80%. However, the probability of the bridge fundamental frequency being accurately identified is less than 50% when \(\alpha_{mv}>0.25\). If the standard for accurately identifying fundamental bridge frequencies is \(|\Delta_{b}|<0.1\) and the results are shown in Figure 7b. It can be seen from Figure 7b that the resulting probability has increased significantly. The probability of the bridge fundamental frequency being accurately identified is greater than 80% when \(\alpha_{mv}<0.1\) and \(\alpha_{v}<0.6\). From the conclusion drawn from Figure 7, when \(0.3<\alpha_{v}<0.6\) and \(\alpha_{mv}<0.1\), selecting the value of \(\alpha_{mv}\) carefully can improve the probability of accurately identifying the bridge fundamental frequency to a certain extent. To ensure the accuracy of identifying the bridge fundamental frequency, the vehicle-to-bridge mass ratio should not be selected as too high and the vehicle speed should not be too fast. If either of them is too large, it will cause a large deviation in the identified frequency.  
In the parameter analysis, in order to avoid resonance between the vehicle and bridge, \(\mu_{1}\) is not selected as a value between 0.9 and 1.2. However, for short and rigid bridges, when vehicles move at a relatively low speed on the bridge, vehicle frequencies may overlap  
Figure 7: Probability of identifying bridge fundamental frequencies. (**a**) \(|\Delta_{b}|<0.05\), (**b**) \(|\Delta_{b}|<0.1\).  
with the bridge frequencies. Faced with this problem, the most direct way is to change the frequency of vehicles or bridges so that the two are no longer close. Obviously, for bridges that must be used continuously, as only the frequency of the vehicles can be adjusted. Adjusting the frequency of the vehicle can be accomplished by increasing or decreasing the vehicle's mass, or by changing the vehicle's suspension stiffness. In comparison, the method of changing vehicle mass is relatively simple. In order to make the frequency change obvious, it is recommended that the adjusted vehicle mass should not be less than 2 times the original mass or more than half of the original mass. After adjusting the vehicle frequency, try to identify the bridge frequency again.  
It should be noted that due to the limitation of the simplified VBI model, the results of the numerical simulation in this paper should not be directly applied to the actual bridge. In addition, for the actual bridge, the value of \(\alpha_{v}\) may not reach the upper limit of the parameter analysis. Different bridges have different \(\alpha_{v}\) value ranges. Therefore, for vehicle speed, we can only give advice not to be too fast. For large-span bridges, due to the large self-weight, the problem of the obvious shift in identified frequency is generally not considered. Instead, the focus should be on how to obtain a stronger bridge response. For small-span bridges or bridges with small self-weight, attention should be paid to the frequency deviation when the vehicle mass is large or there are heavy vehicles, such as buses or trucks, passing through the bridge. Therefore, it is recommended that the vehicle-to-bridge mass ratio \(\alpha_{mv}\) be less than 0.1, or even 0.05, to ensure the success of accurately identifying bridge frequencies.