# Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments
## 3 SBOA-Optimized FMD  
The SBOA uses the envelope entropy value of the reconstructed signal after FMD as the fitness function to optimize the parameters of FMD. At the same time, this paper proposes the Integrated Signal Evaluation Index (ISEI) as the selection criterion for the best mode. The detailed steps are as follows:  
1. Initialize the population: The maximum number of SBOA iterations \(P\) is set to 70 and the population size \(Q\) is set to 6. \(P\) candidate solutions are randomly generated, and each candidate solution represents an eigenmode selection scheme. Combined with the influence of FMD parameters on the noise reduction effect in the literature, this paper sets the range of the mode number \(n\) as 3 to 20, the filter length \(L\) as 10 to 300, the frequency band cutting number \(K\) as 5 to 40, and the cycle period \(m\) as 5 to 30, where the frequency band cutting number \(K\geq\) the mode number \(n\).
2. Fitness calculation: The signal is decomposed according to the FMD parameters corresponding to the candidate solution, the envelope entropy value of the reconstructed signal is calculated, and \(F(x)\) is constructed as the fitness function as the criterion for stopping the SBOA.
3. Update location: According to the parameter search and position+n update process outlined in the literature [26], the exploration phase employs a differential evolution strategy to introduce population diversity. This phase combines Brownian motion and Levy flight strategies [26] to expand the search range, assisting the algorithm in overcoming local optimality. In the utilization phase, the SBOA mimics the secretary bird's behavior of evading natural predators through two strategies: camouflage and rapid movement. The camouflage strategy is implemented using a dynamic perturbation factor \(\left(1-T_{t}\right)^{2}\), which enables the search agents to disperse more effectively within the solution space, helping to avoid local optima. The rapid movement strategy, on the other hand, adjusts the position and velocity of the search agents to simulate the secretary bird's swift movement, thus improving the efficiency of the search and accelerating convergence. Specifically, the rapid movement strategy incorporates a dynamic perturbation factor \(\left(1-T_{t}\right)^{2}\), which allows the search agents to move quickly across the solution space, facilitating a faster approach to the global optimal solution. Additionally, the search direction and step size are dynamically adjusted to guide the algorithm towards efficiently approaching the global optimal solution, thereby ensuring accurate determination of the FMD parameters during the optimization process.
4. Individuals with better fitness values (FMD combined solutions) are selected according to the criterion of the minimum envelope entropy value, individuals with poor fitness are replaced, and the optimal solution is retained to ensure that the algorithm converges to the global optimum.
5. Iteration: The fitness calculation, position update, local search and selection, and replacement steps are repeated until the termination condition is met (the maximum number of iterations \(P\) is reached).
6. Output the optimal solution: When the algorithm terminates, the optimal eigenmode combination solution and FMD parameters are output.