# Fault Feature Extraction of Parallel-Axis Gearbox Based on IDBO-VMD and t-SNE
## 6 Experimental Verification
### The Extraction of Fault Signal Features  
Using envelope entropy as the fitness value, we compared the fitness curves of the IDBO, DBO, SSA, and GWO algorithms in optimizing VMD. To ensure fair testing, each algorithm was configured with 30 individuals, a maximum iteration count of 50, and a range of values for K set to [3; 10], all of which are integers. The penalty factor \(\alpha\) had a value range of [200, 5000]. In Figure 5, each curve represents the optimization iteration process for VMD using the four algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **IDBO** & **DBO** & **GWO** & **SSA** \\ \hline \(F_{1}\) & & & & \\ Optimal value & 0 & \(2.4081\times 10^{-176}\) & \(5.5438\times 10^{-29}\) & \(7.1886\times 10^{-100}\) \\ Mean value & 0 & \(6.0809\times 10^{-107}\) & \(6.7022\times 10^{-28}\) & \(3.7487\times 10^{-60}\) \\ Standard deviation & 0 & \(1.9229\times 10^{-106}\) & \(1.3445\times 10^{-27}\) & \(1.1854\times 10^{-59}\) \\ \hline \(F_{2}\) & & & & \\ Optimal value & 0 & \(1.4802\times 10^{-76}\) & \(4.9103\times 10^{-17}\) & \(5.0108\times 10^{-81}\) \\ Mean value & 0 & \(1.0308\times 10^{-58}\) & \(8.2908\times 10^{-17}\) & \(5.4129\times 10^{-32}\) \\ Standard deviation & 0 & \(3.2597\times 10^{-58}\) & \(2.9568\times 10^{-17}\) & \(1.6937\times 10^{-31}\) \\ \hline \(F_{3}\) & & & & \\ Optimal value & 0 & \(3.2393\times 10^{-76}\) & \(2.2534\times 10^{-7}\) & \(2.8709\times 10^{-93}\) \\ Mean value & \(3.6372\times 10^{-90}\) & \(1.8364\times 10^{-54}\) & \(7.4764\times 10^{-7}\) & \(7.0487\times 10^{-31}\) \\ Standard deviation & \(1.1299\times 10^{-89}\) & \(5.8014\times 10^{-54}\) & \(5.1431\times 10^{-7}\) & \(2.2202\times 10^{-30}\) \\ \hline \(F_{4}\) & & & & \\ Optimal Value & 0 & \(4.3236\times 10^{-304}\) & \(3.0912\times 10^{-53}\) & \(5.4264\times 10^{-243}\) \\ Mean Value & 0 & \(4.1335\times 10^{-210}\) & \(1.7039\times 10^{-50}\) & \(3.8374\times 10^{-124}\) \\ Standard Deviation & 0 & 0 & \(3.7227\times 10^{-50}\) & \(1.2135\times 10^{-123}\) \\ \hline \(F_{5}\) & & & & \\ Optimal value & 0 & \(1.2767\times 10^{-162}\) & \(7.9735\times 10^{-105}\) & \(6.3863\times 10^{-119}\) \\ Mean value & 0 & \(6.2664\times 10^{-114}\) & \(3.5699\times 10^{-100}\) & \(4.1615\times 10^{-44}\) \\ Standard deviation & 0 & \(1.9816\times 10^{-113}\) & \(9.4946\times 10^{-100}\) & \(1.316\times 10^{-43}\) \\ \hline \(F_{6}\) & & & & \\ Optimal value & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & \(7.5051\times 10^{-14}\) & \(4.4409\times 10^{-16}\) \\ Mean value & \(4.4409\times 10^{-16}\) & \(4.4409\times 10^{-16}\) & \(1.0134\times 10^{-13}\) & \(4.4409\times 10^{-16}\) \\ Standard deviation & 0 & 0 & \(3.8454\times 10^{-14}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test results comparison. From Figure 5, it can be observed that GWO has the highest fitness values in the signal decomposition of the four types of fault signals. This indicates that GWO has poor  
Figure 4: Experimental equipment: (**a**) gearbox; (**b**) surface pitting; (**c**) surface cracking; (**d**) surface wear; (**e**) tooth root breakage. Figure 5: Fitness curves for optimizing VMD using different algorithms: (**a**) normal condition; (**b**) tooth root breakage; (**c**) surface cracking; (**d**) surface pitting. global search capability and is prone to becoming stuck in local optima. DBO, the SSA, and IDBO have lower fitness values than GWO, indicating stronger global search capabilities. However, with the increase in iteration count, GWO exhibits strong convergence ability. Through comparison, it is evident that IDBO has lower fitness values, attributed to the cubic chaos mapping's ability to quickly locate global optimal solutions. With iterations, IDBO can rapidly find the lowest fitness value. This is due to the inclusion of the Levy flight strategy and adaptive weighting, which enhance global optimization capability and convergence performance.