## 4 The Vehicle-to-Bridge Mass Ratio on Frequency Deviation Rate
### Results Description  
Without considering the mass of the vehicle, the bridge fundamental frequency deviation rate \(|\Delta_{b}|\) is 0.72%, 3.6%, 7.2%, 10.8%, 14.4%, and 18.0%, their corresponding value of \(\alpha_{v}\) is 0.03, 0.15, 0.30, 0.45, 0.60, and 0.75, respectively. However, as shown in Figures 5 and 6 even when \(\alpha_{mv}\) is 0.01, the \(\Omega_{bex}\) will change with the change of \(\mu_{1}\), and both the left-biased and right-biased frequencies will appear. Therefore, it should be noted that introducing vehicle mass also brings the influence of the vehicle-to-bridge frequency ratio.  
As shown in Figures 5 and 6, at different speeds (take \(\alpha_{v}\) as 0.03, 0.15, 0.30, 0.45, 0.60, and 0.75, respectively) \(\Delta_{b}\) varies with \(\mu_{1}\) and \(\alpha_{mv}\) in inconsistent patterns.  
In the case of \(\mu_{1}>1\), the recognition accuracy becomes increasingly worse with the increase of \(\alpha_{v}\). When \(\alpha_{v}\) is small \(\Delta_{b}<0\), and \(\Delta_{b}\) increases with the increase of \(\mu_{1}\) and gradually stabilizes. As \(\alpha_{v}\) increases, \(\Delta_{b}\) will become greater than zero when \(\alpha_{mv}\) and \(\mu_{1}\) are small, and the range of this situation becomes larger. \(\Delta_{b}\) gradually decreases as \(\mu_{1}\) increasesunder the condition of \(\Delta_{b}<0\). Generally speaking, when \(\mu_{1}>1\), \(\Omega_{bx}\) is mainly left-biased frequency and \(|\Delta_{b}|\) increases as \(\alpha_{v}\) and \(\mu_{1}\) decrease. Specifically, \(\Delta_{b}\) decreases as \(\alpha_{v}\) and \(\alpha_{mv}\) increase when \(\mu_{1}\) is constant, and \(\Delta_{b}\) decreases as \(\alpha_{mv}\) increase when \(\alpha_{v}\) is constant.  
Compared to the case where \(\mu_{1}\) is greater than 1, the changing pattern of \(\Delta_{b}\) is relatively complicated when \(\mu_{1}\) is less than 1. Similar to the previous case, as \(\alpha_{v}\) increases, the recognition accuracy becomes increasingly worse. As shown in Figure 4, the shaded part indicates that the fundamental frequency of the bridge cannot be identified. When \(\alpha_{v}\) is small, \(\Delta_{b}>0\). There will be a situation where \(\Delta_{b}\) is less than zero when \(\alpha_{mv}\) or \(\mu_{1}\) is small as \(\alpha_{v}\) increases. Especially, within specific range of \(\mu_{1}\), the area where \(\Delta_{b}<0\) will extend to where the value of \(\alpha_{mv}\) is comparatively large (as shown in the red frame of Figure 5c,d). Generally speaking, \(\Omega_{bx}\) is mainly a right-biased frequency when \(\mu_{1}<1\). Specifically, as \(\alpha_{mv}\) and \(\alpha_{v}\) increases, \(\Delta_{b}\) increases and decreases, respectively when \(\mu_{1}\) is constant, however when \(\alpha_{v}\) continues to increase, the above pattern shows the opposite trend. In most cases, \(\Delta_{b}\) increases as \(\mu_{1}\) increases while \(\alpha_{mv}\) remains constant, however the pattern is different in the area where \(\Delta_{b}<0\); \(\Delta_{b}\) increases as \(\mu_{1}\) and \(\alpha_{mv}\) increases while \(\alpha_{v}\) remains constant, however decreases when \(\alpha_{v}\) becomes larger.