## 4 The Vehicle-to-Bridge Mass Ratio on Frequency Deviation Rate
### Parameters Sensitivity Analysis  
The Morris method is used to qualitatively analyze the effect of the vehicle-bridge mass ratio \(\alpha_{mv}\), vehicle-bridge frequency ratio \(\mu_{1}\), and vehicle speed \(\alpha_{v}\) on the bridge frequency deviation rate \(\Delta_{b}\). The [PERSON] method was first proposed by [PERSON] in 1991 [50]. It is a discrete search method based on parameter space, which studies model parameters on a global scale. Its main idea is to assume elementary effects (\(EE\)) that measure the sensitivity of the parameter \(x_{i}\) obeys probability distribution \(f_{i}\), and the global sensitivity of the parameter \(x_{i}\) can be determined by measuring the mean and standard deviation of the distribution. If the mean value of the parameter \(x_{i}\) is larger, it has a greater impact on the model output. The standard deviation \(\sigma\) is used to measure the interaction between parameters. The larger the value, the greater the interaction and vice versa.  
For a model with \(k\) parameters, assume that the range of parameters is [0, 1]. For a given parameter vector:  
\[X=(x_{1},x_{2},\cdots,x_{k}). \tag{21}\]  
where each \(x_{i}\) may take on value \(\left\{0,\frac{1}{p-1},\frac{2}{p-1},\cdots,1\right\}\) and \(p\) is the number of levels. The elementary effect of the \(i\)-th input is calculated by the formula below:  
\[EE_{ij}=\frac{Y\big{(}x_{1},\cdots,x_{j}+\Delta_{j},\cdots,x_{k}\big{)}-Y(X)} {\Delta_{j}} \tag{22}\]  
where \(Y(X)\) is the model output corresponding to the initial parameter vector and  
\(Y\big{(}x_{1},\cdots,x_{j}+\Delta_{j},\cdots,x_{k}\big{)}\) is the model output after the \(j\)-th parameter of the initial parameter vector is changed; \(\Delta\) is the magnitude of the step between 0 and \(1-1/(p-1)\), which is a multiple of \(1-1/(p-1)\). To calculate \(EE\) of the remaining \(k\)-1 parameters, the above procedure is repeated. Then, randomly generate \(n\) initial vectors and repeat the above process to calculate the \(n\) elementary effects of the \(k\) parameters, which is \(EE_{ij}\big{(}\ j=1,2,\cdots k\ \ \ i=1,2,\cdots n\ \ \big{)}\). The mean and standard deviation of \(EE_{ij}\) can be expressed as:  
\[\mu_{i}=\sum_{j=1}^{n}EE_{ij}/n \tag{23}\]\[\sigma_{i}=\sqrt{\sum_{j=1}^{n}\left(EE_{ij}-\mu_{i}\right)^{2}/n}. \tag{24}\]  
Combined with the actual situation, the frequency of the vehicle is generally between 0-15 Hz [2,49]. Therefore, during parameter analysis, the value range of each dimensionless parameter are \(\mu_{1}=0.2{\sim}\)\(0.9,1.2\sim 15\), \(\alpha_{mv}{=}\)\(0.01{\sim}\)\(0.5\), and \(\alpha_{v}=0.03{\sim}\)\(0.75\). Take the identified bridge fundamental frequency increment \(\Delta_{b}\) as the model output. There are three variables, and the parameter space is divided into five equal parts. The sampling calculation results are shown in Table 2 to discuss the global sensitivity of each parameter by using the Morris method.  
It can be seen from Table 2 that when \(\mu_{1}>1\), \(\alpha_{mv}\) and \(\alpha_{v}\) have great impact to \(\Delta_{b}\). Moreover, the interaction between \(\mu_{1}\) and the other two parameters is not obvious. When \(\mu_{1}<1\), \(\mu_{1}\), \(\alpha_{mv}\), and \(\alpha_{v}\) all have a great impact on \(\Delta_{b}\). There is also a similar degree of interaction between them. The above conclusion is mutually verified with the pattern described in the previous section. For instance, when \(\mu_{1}>1\), the value of \(\Delta_{b}\) tends to be stable as \(\mu_{1}\) increases; when \(\mu_{1}<1\), the rule of change of \(\Delta_{b}\) caused by changing the values of \(\mu_{1}\) and \(\alpha_{mv}\) will also change accordingly as \(\alpha_{v}\) takes on a different value.