## 2 Vibration Signal Processing Method Based on VMD
### IPSO-VMD Algorithm Design
\tag{3}\]  
where,  
* \(\omega_{\min}\) and \(\omega_{\max}\) are the preset minimum and maximum values of inertial weight \(\omega\), taking \(\omega_{\min}=0.4\), \(\omega_{\max}=0.9\);
* \(f_{\text{average}}^{k}=\sum\limits_{i=1}^{n}f\Big{(}x_{i}^{k}\Big{)}/n\) is the average value calculated by the fitness of the whole population at k-th iteration;
* \(f_{\min}^{k}=\min\Big{\{}f\Big{(}x_{1}^{k}\Big{)},f\Big{(}x_{2}^{k}\Big{)}, \cdots,f\Big{(}x_{n}^{k}\Big{)}\Big{\}}\) is the minimum fitness of the whole population at iteration. 2. Compressibility factor method  
Different from inertia weight \(\omega\), the individual learning factor \(c_{1}\) and social learning factor \(c_{2}\) represent the influence of the historical information of particles themselves and other particles on the later behavior of particles, while the learning factor describes the information interaction between populations. Therefore, individual learning factor \(c_{1}\) and social learning factor \(c_{2}\) have great influence on the PSO method. In order to balance the global search ability and local search ability of the algorithm, the PSO method is optimized with compression factor, which can ensure that the PSO algorithm has strong convergence in solving, and can reduce its speed limit [31]. If \(c_{1}=c_{2}=2.05\), then \(C=c_{1}+c_{2}=4.1\), compression factor \(\Psi=2/\Big{|}\Big{(}2-C-\sqrt{C^{2}-4C}\Big{)}\Big{|}\). For the speed in Equation (3), it can be updated as follows:  
\[v_{id}^{k+1}=\Psi\Big{[}\omega v_{id}^{k}+c_{1}r_{1}\Big{(}p_{id}-x_{id}^{k} \Big{)}+c_{2}r_{2}\Big{(}p_{gd}-x_{id}^{k}\Big{)}\Big{]} \tag{4}\]
3. Elite Learning Strategies  
Elite learnt strategy is an improved algorithm that enhances search efficiency and accuracy by introducing an elite guidance mechanism. In each iteration, the best-performingparticles or a small number of good particles are selected as 'elites', and the other particles in the population are allowed to take into account the position information of these elites in addition to the \(p_{Best}\) and \(g_{Best}\) when updating themselves. Non-elite particles can update their velocity and position by introducing the influence of the positions of the elite particles:  
\[v_{i}(t+1)=\omega\cdot v_{i}(t)+c_{1}\cdot r_{1}\cdot\left(p_{Best_{i}}-x_{i}(t )\right)+c_{2}\cdot r_{2}\cdot\left(g_{Best}-x_{i}(t)\right)+c_{3}\cdot r_{3} \cdot\left(Elite_{i}-x_{i}(t)\right) \tag{5}\]  
where \(Elite_{i}\) is the position of the \(i\)-th elite particle, \(c_{3}\) is the elite learning factor, and \(r_{3}\) is a random number between [0,1]. 4. Fitness Distance Ratio Optimization Strategy  
The Fitness Distance Ratio (FDR) is a measure of the relative strengths and weaknesses of particles in relation to other particles. It combines the fitness value of a particle with information about the distance between particles and is used to guide the search behaviors of the particles. The fitness distance ratio is defined as follows:  
\[FDR_{i}=\frac{f(x_{i})}{\left\|x_{i}-P_{Best_{i}}\right\|} \tag{6}\]  
where \(f\left(P_{Best_{i}}\right)\) is the fitness value of particle \(i\) and \(\left\|x_{i}-P_{Best_{i}}\right\|\) is the Euclidean distance between the current position of the particle and its individual optimal position. \(FDR_{i}\) is a factor that adjusts the speed of the particle as it moves towards b to enhance the algorithm to pay more attention to particles with high adaptation whose distance is far away, and the updated formula for the speed is as follows:  
\[v_{i}(t+1)=\omega\cdot v_{i}(t)+c_{1}\cdot r_{1}\cdot\left(p_{Best_{i}}-x_{i}( t)\right)+c_{2}\cdot r_{2}\cdot\left(g_{Best}-x_{i}(t)\right)\cdot FDR_{i} \tag{7}\]  
This approach improves the ability of the particle swarm to concentrate towards more optimal solution regions, as the elite particles represent the currently searched optimal solution regions, and guiding the other particles towards these regions helps to avoid premature convergence and accelerates the finding of the global optimal solution.