## 4 SVM Analysis
### Variable Amplitude Continuous Impulse Load  
The previous section predicted the acceleration responses obtained by continuous impulses of equal amplitude using machine learning and SVM, and good prediction results were obtained. However, the load often has a variable amplitude in practical engineering applications. Therefore, a certain number of sample points were obtained based on the finite element model of variable amplitude load calculation and our designed method was used to verify the accuracy of the prediction. As shown in Table 3, the prediction accuracy of the SVM, based on the acceleration sample data of the random variable amplitude (10% random amplitude fluctuation, as in Equation (12)), for the bolt preload state is 88.9%, which is lower than the prediction accuracy of the model with equal amplitude shown above, but is still high. This proves the method's effectiveness, especially for the stochastic load variable amplitude model, which still has a prediction accuracy of nearly 90% under the small-sample learning amount. Figure 18: Multi-condition prediction accuracy comparison chart (continuous impulse load of equal amplitude); (**a**) 70% samples learning; (**b**) 60% samples learning. As shown in Table 3, when the CEEMD decomposition algorithm is used to construct the feature vectors, the prediction accuracy of the bolt preload state under variable amplitude load drops to 77.78%. In comparison, the method using the VMD algorithm in this paper can still maintain close to 90% accuracy. We continued to add Gaussian white noise with different signal-to-noise ratios to the acceleration response signal generated by 10% random amplitude fluctuation excitations (Equation (12)), which is consistent with the above signal-to-noise ratio, to simulate the signal fluctuations in actual engineering and increase the interference of the signal. It is evident from Figure 19 that this method can maintain a high prediction accuracy under the conditions of 10% excitation random amplitude fluctuation. Even under the interference of white Gaussian noise with a signal-to-noise ratio of 20 dB, the accuracy rate remains above 70%. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Accuracy** & \
\begin{tabular}{c} **Feature Vector by VMD and** \\ **PE** \\ \end{tabular} &
\begin{tabular}{c} **Feature Vector by CEEMD** \\ **and PE** \\ \end{tabular} \\ \hline No loosening (5000 N) & 83.33\% & 83.33\% \\ Slight loosening (3000 N) & 83.33\% & 50\% \\ Serious loosening (300 N) & 100\% & 100\% \\ Total & 88.99\% & 77.78\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Prediction accuracy under the conditions of variable-amplitude pulse load. Figure 19: Multi-condition prediction accuracy comparison chart (10% variable amplitude continuous impulse load): (**a**) 70% samples learning; (**b**) 60% samples learning. To further test the performance of this method, we continued to increase the fluctuation range of the excitation amplitude. The range of the random variation in the amplitude was expanded to 30%. This was divided into three levels for comparison: a 10% random fluctuation in amplitude, a 20% random fluctuation, and a 30% random fluctuation. This expands (0,0.1) in Equation (12) to (0,0.2) and (0,0.3). The labels for these three were rand1, rand2, and rand3. It is evident from Figure 20 that the increase in the random amplitude fluctuation range does not affect the effectiveness of the method proposed in this paper. The change in the accuracy rate caused by the number of samples is that the base of the small sample is small, and the slight modifications to the numerator caused the numerical fluctuation. This section comprehensively analyzes the effectiveness and universality of the method proposed in this paper through various working conditions, various equipment forms, different random amplitude changes, and artificially added Gaussian white noise, with varying signal-to-noise ratios, and proves the method.