## 4 Hand-Transmitted Vibration Prediction Model
### Prediction Model Establishment
The coefficient of determination (R-square) and root mean square error (RMSE) are used as indicators to evaluate the goodness-of-fit, and the fitting results of the polynomial model of the opisthenar are shown in Table 2.  
The eigenvalue determination method is used to test the multicollinearity among the independent variables of the model. It is found that the maximum condition number is much higher than 100 when \(N+M>2\), and there is serious multicollinearity among the independent variables. The results show that the fitting accuracy of the complex model is very high, but the generalization performance becomes worse and the prediction effect decreases due to the over-fitting phenomenon of higher order. It is found that the model deviates from the real situation by drawing the images of each order with \(N+M>2\). Therefore, it is considered to establish a second-order polynomial model for \(N=M=1\). In the nonlinear 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 \begin{table}
\begin{tabular}{c c c c c} \hline
**Evaluation Index** & \(N=1,M=1\) & \(N=1,M=2\) & \(N=1,M=3\) & \(N=1,M=4\) \\
**R-square** & 0.9940 & 0.9956 & 0.9938 & 0.9950 \\
**RMSE** & 0.001034 & 0.009200 & 0.001154 & 0.001086 \\ \hline
**Evaluation Index** & \(N=2,M=1\) & \(N=2,M=2\) & \(N=2,M=3\) & / \\
**R-square** & 0.9911 & 0.9961 & \(-\)34,780,000 & / \\
**RMSE** & 0.001317 & 0.0009260 & 94.41 & / \\ \hline
**Evaluation Index** & \(N=3,M=1\) & \(N=3,M=2\) & / & / \\
**R-square** & 0.9933 & \(-\)2895,00 & / & / \\
**RMSE** & 0.001193 & 8.614 & / & / \\ \hline
**Evaluation Index** & \(N=4,M=1\) & / & / & / \\
**R-square** & \(-\)25,920 & / & / & / \\
**RMSE** & 2.480 & / & / & / \\ \hline \end{tabular}
\end{table} 
Table 2: Goodness-of-fit for polynomial models of different orders. equal and the combined influence of the independent variables is in a power form. The mathematical model consists 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 nonlinear full model can be expressed as Equation (11):  
\[\widehat{A_{k}}=d_{0}+\sum_{i=1}^{Q}e_{i}n_{k}^{f_{i}}+\sum_{i=1}^{Q}g_{i}L_{k} ^{h_{i}}+\sum_{i=1}^{Q}p_{i}n_{k}^{q_{i}}L_{k}^{r_{i}} \tag{11}\]  
where \(d_{0}\), \(e_{i}\), \(f_{i}\), \(g_{i}\), \(h_{i}\), \(p_{i}\), \(q_{i}\) and \(r_{i}\) are the constant terms to be solved, and the cumulative term \(Q=1,2,3,4,5\), where the maximum number of cumulative terms 5 indicates that the overdetermined problem condition that the number of parameters to be solved is less than the number of tests is satisfied. Similarly, the constant coefficient matrix is obtained by minimizing \(\chi^{2}\), and the fitting results of the nonlinear model on the opisthenar are shown in Table 3.  
According to the result of goodness-of-fit and the image of each cumulative term model, the quadratic cumulative nonlinear model is established by using \(Q=2\). The nonlinear and polynomial vibration exposure parametric equations can be expressed as Equations (12) and (13):  
\[A_{N}=d_{0}+e_{1}n^{f_{1}}+e_{2}n^{f_{2}}+g_{1}L^{h_{1}}+g_{2}L^{h_{2}}+p_{1} n^{q_{1}}L^{r_{1}}+p_{2}n^{q_{2}}L^{r_{2}} \tag{12}\]  
\[A_{P}=a_{0}+a_{1}n+a_{2}n^{2}+b_{1}L+b_{2}L^{2}+c_{11}nL \tag{13}\]  
where \(A_{N}\) and \(A_{P}\) are the vibration exposure of the nonlinear model and the vibration exposure of the polynomial model, respectively. The constant coefficients are shown in Table 4.  
The coefficient of determination and root mean square error of the two models are shown in Table 5.