## 3 Application Example
### Axis Motion Trajectory Feature Extraction  
The original signals are segmented into samples, with each sample containing 3000 data points. The axis trajectory signals are then purified and denoised. Using the maximum dimension method, a Hankel matrix is constructed and SVD decomposition is performed on the sample signals.  
To achieve effective denoising, the effective singular value sequence is selected based on the magnitude relationship of the singular values of the original signal. The results of denoising and purification are presented in Figure 11.  
The SVD-processed axis trajectory sample signals yield clear and smooth curves, retaining the main geometric features while eliminating ineffective components. Subsequently, the axis trajectory signals are divided into regions categorized as general curvature regions and abrupt curvature change regions. This division informs the selection of feature points, completing the representation of the axis trajectory pattern using a distance sequence. The distance sequence representation results for each experimental condition are shown in Figure 12.  
As depicted in Figure 12, with changes in speed and weight, the rotor system's imbalance worsens and becomes more complex. These changes are captured by the distance sequences and reflected in the final feature signals. It is evident that for different experimental conditions, the distance sequences effectively represent the axis trajectory patterns, reflecting the shape characteristics of the axis trajectory while significantly reducing the data volume.  
Figure 11: Axis trajectory purification and denoising results. (**a**) Experimental condition 1; (**b**) experimental condition 2; (**c**) experimental condition 3; (**d**) experimental condition 4; (**e**) experimental condition 5; (**f**) experimental condition 6; (**g**) experimental condition 7; (**h**) experimental condition 8; (**i**) experimental condition 9.  
Figure 13a illustrates the peak-to-peak values of distance sequences under various experimental conditions. It can be observed that, regardless of changes in speed or weight, rotor arrangement type 1 consistently exhibits the highest peak-to-peak values. This indicates that the unbalanced state of rotor arrangement type 1 is more severe and highly sensitive to changes in speed. Similarly, rotor arrangement type 3, which is also in an extreme arrangement position, shows a similar trend to type 1, with changes in speed and weight further exacerbating its unbalanced state. In contrast, rotor arrangement type 2, positioned in the middle, presents an opposite scenario. As seen in Figure 10, the axis trajectory patterns of type 2 tend to be elliptical, suggesting a more stable operating state.  
Figure 12: Results of axis trajectory feature extraction. (**a**) Experimental condition 1; (**b**) experimental condition 2; (**c**) experimental condition 3; (**d**) experimental condition 4; (**e**) experimental condition 5; (**f**) experimental condition 6; (**g**) experimental condition 7; (**h**) experimental condition 8; (**i**) experimental condition 9.  
Figure 13: Analysis of distance sequence representation results. (**a**) Peak-to-peak values of various experimental conditions; (**b**) percentage of peak-to-peak values for each experimental condition.  
By setting the peak-to-peak value of 2.0 as 100% and using this as a baseline to calculate the percentage of each peak-to-peak value, as shown in Figure 13b, an optimal series of matches can be obtained. Considering the different impacts of speed and weight on the peak-to-peak values, the weight proportions for speeds of 800 RPM, 1200 RPM, and added weight are set at 40%, 40%, and 20%, respectively. The total percentages for each rotor arrangement are calculated as 96.74%, 86.1%, and 67.1%. It is evident that rotor arrangement type 3 has the best general applicability.