## 4 Hand-Transmitted Vibration Prediction Model  
### Prediction Model Establishment  
The vibration exposure time of the subjects operating the PPT tool increased with the increase in the number of the non-detaching fastening device. Taking the 300 mm operating rod as an example, after drawing the scatter diagram of the test data, it is found that the vibration exposure increases monotonously with the number of the non-detaching fastening device, and the vibration exposure of the palm and opisthenar tends to increase gradually, as shown in Figure 4.  
To predict the vibration exposure produced by different types of operating rods, the length of the operating rod and the number of the non-detaching fastening device are considered independent variables to characterize the vibration exposure. The test data are processed by nonlinear and polynomial fitting. In the polynomial model, it is assumed that the influence of the length of the operating rod \(L\) and the number of the non-detaching fastening device \(n\) on the vibration exposure is nonlinear. The mathematical model is composed of three parts, the influence of constant term, the influence of the change of each variable, and the influence of the combination of each variable. Therefore, the polynomial full model can be expressed as Equation (4):  
\[\widehat{A_{k}}=a_{0}+\sum_{i=1}^{N+M}a_{i}n_{k}^{i}+\sum_{j=1}^{N+M}b_{j}L_{k }^{j}+\sum_{i=1}^{N}\sum_{j=1}^{M}c_{ij}n_{k}^{i}L_{k}^{j} \tag{4}\]  
where \(\widehat{A_{k}}\) is the predicted vibration exposure after the \(k\)-th operation (m/s\({}^{2}\)); and \(a_{0}\), \(a_{i}\), \(b_{j}\), \(c_{ij}\) are constant terms to be solved; and \(N\), \(M\) are positive integers, representing the highest order of \(n\) and \(L\) in the combined variable \(n_{k}^{i}L_{k}^{j}\), respectively; and \(n_{k}\) is the cumulative number of the non-detaching fastening device after the \(k\)-th operation; \(L_{k}\) is the length of the operating rod used in the \(k\)-th operation. According to the maximum likelihood principle, data fitting is a process of minimizing the sum of squares of residuals. To obtain the optimal fitting condition, the objective function \(\chi^{2}\) should be minimized. Its calculation is shown in Equation (5):  
\[\chi^{2}=\sum_{k=1}^{K}(A_{k}-\widehat{A_{k}})^{2} \tag{5}\]  
where \(K\) is the number of tests, and \(A_{k}\) is the actual vibration exposure of the hand after the \(k\)-th operation. Figure 4: Scatter plot of test data for the variation of the vibration exposure with the number of the non-detaching fastening device. Make the partial derivative of the objective function equal to zero, as shown in Equation (6):  
\[\frac{\partial\chi^{2}}{\partial a_{i}}=\frac{\partial\chi^{2}}{ \partial b_{i}}=\frac{\partial\chi^{2}}{\partial c_{ij}}=0 \tag{6}\]  
The parameter matrix is expressed as Equation (7):  
\[\mathbf{Y}=[K,\sum_{k=1}^{K}n_{k},\ldots,\sum_{k=1}^{K}n_{k}^{N+M}, \sum_{k=1}^{K}L_{k},\ldots,\sum_{k=1}^{K}L_{k}^{N+M},\ldots,\sum_{k=1}^{K}n_{k }^{N}L_{k},\ldots,\sum_{k=1}^{K}n_{k}^{N}L_{k}^{M}] \tag{7}\]  
The constant coefficient matrix is expressed as Equation (8):  
\[\mathbf{X}=[a_{0},a_{1},\ldots,a_{N+M},b_{1},\ldots,b_{N+M},c_{11}, \ldots,c_{1m},c_{21},\ldots,c_{2M},c_{N1},\ldots,c_{NM}]^{\mathsf{T}} \tag{8}\]  
The relationship between the parameter matrix, the constant coefficient matrix, and the vibration exposure is shown in Equation (9):  
\[\sum_{k=1}^{K}A_{k}=\mathbf{Y}\mathbf{X} \tag{9}\]  
The calculation of the constant coefficient matrix is shown in Equation (10):  
\[\mathbf{X}=\sum_{k=1}^{K}A_{k}\mathbf{Y}^{-1} \tag{10}\]  
When the highest order of the independent variable is greater than 5, the Runge phenomenon of the polynomial difference will occur when the polynomial is used for surface fitting, i.e., the error at the edge of the interpolation interval is large, so the ideal fitting effect cannot be achieved. Therefore, only \(2\leq N+M\leq 5\) is considered [20].