## 4 Method Validation
### HLSSA-VMD Algorithm Validation  
Variational mode decomposition (VMD) is one of the most commonly used methods in the field of signal decomposition, and its decomposition performance is often heavily influenced by the number of modes (\(K\)) and the quadratic penalty term (\(\alpha\)). Improper parameter settings can lead to information loss and frequency aliasing. Researchers typically employ optimization algorithms for parameter tuning, yet different optimization algorithms yield varying degrees of effectiveness in optimization. This section presents two sets of simulated signals for validating the effectiveness of the HLSSA-VMD algorithm. A comparative analysis was conducted with the PSO-VMD, WOA-VMD, MVO-VMD, and SSA-VMD to further highlight the superior performance of the HLSSA-VMD algorithm. Figure 13: (**a**) Three-dimensional plot of \(F_{12}\); (**b**) convergence curve. The first set of simulated signals we constructed is the simulated signal representing an outer race fault in a bearing, denoted as \(x(t)\). The construction formula is as follows [33]:  
\[\left\{ \begin{array}{l}x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)+n(t)\\ x_{1}(t)=Ae^{-\xi 2\pi f_{n}t}\sin(2\pi f_{n}\sqrt{1-\xi^{2}t})\\ x_{2}(t)=\sum\limits_{i=1}^{M}l_{i}\sin(2\pi f_{i}^{1}t+\theta_{i})\\ x_{3}(t)=\sum\limits_{i=1}^{N}k_{i}\sin(2\pi f_{i}^{2}t+\varphi_{i})\end{array} \right. \tag{12}\]  
where \(f_{n}\) is the natural frequency; \(\xi\) is the damping coefficient; \(A\) is the amplitude of the pulse fault; and \(n(t)\) is Gaussian white noise with an SNR of \(-10\) dB. The detailed information of the simulated signal is shown in Table 6, with a sampling frequency of 20 KHz and a number of 4096 data samples. Figure 14a,b display the initial state and the state after adding Gaussian white noise to the simulated bearing fault signal, respectively. We employed envelope entropy minimization as the fitness function for optimizing VMD parameters through the HLSSA. Envelope entropy reflects the sparsity traits of the original signal. When the obtained intrinsic mode functions (IMFs) after decomposition have more noise and less feature information, the envelope entropy value is larger. Conversely, when there is less noise and more feature information in the decomposed IMFs, the envelope entropy value is smaller. Currently, many studies use envelope entropy as the objective function for optimizing VMD parameters. The search range for \(K\) was [1, 10], and the search range for \(\alpha\) was [100, 3000]. Figure 15 illustrates the iterative optimization process for the five algorithms, and Table 7 presents the optimal solutions obtained with these algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Signal** & \multicolumn{4}{c}{**Parameters**} \\ \hline \(x_{1}(t)\) & \(A\) & \(\xi\) & \(f_{n}\) (Hz) & \\  & 3 & 0.02 & 2600 & \\ \hline \(x_{2}(t)\) & \(M\) & \(l\) & \(f_{1}^{1}\) (Hz) & \(\theta_{1}\) \\  & 1 & 0.025 & 70 & 0 \\ \hline \(x_{3}(t)\) & \(N\) & \(k\) & \(f_{1}^{2}\) (Hz) & \(\varphi_{1}\) \\  & 1 & 0.035 & 100 & 0 \\ \hline \end{tabular}
\end{table} 
Table 6: The detailed parameters of the simulated signal. Figure 14: Two states of the bearing outer ring fault signal: (**a**) original state; (**b**) state after adding noise. From Table 7, it is evident that both the SSA and HLSSA demonstrated superior performance in terms of convergence accuracy. However, as shown in Figure 15, the HLSSA outperformed the SSA regarding initialization and convergence speed. This indicates that the proposed improvement to the SSA is effective, and the HLSSA demonstrates higher-quality initialization, faster convergence speed, and superior convergence accuracy. The second set of simulated signals consisted of three signals with different frequencies that were superimposed at different time intervals. To simulate real-world signals more accurately, Gaussian white noise was added to the signal \(s\). The construction process of signal s is as follows:  
\[