## 2 Vibration Signal Processing Method Based on VMD  
### IPSO-VMD Algorithm Design  
VMD algorithm analyzes the original signal according to the number of modal decompositions, obtains \(K\) modal functions \(u_{k}(t)\) with center frequency \(\omega(t)\), and performs Hilbert transform on \(u_{k}(t)\) to obtain the bandwidth and spectrum information of each IMF component. By adding an exponential term to \(u_{k}(t)\), and determining the bandwidth of the IMF component according to the \(L^{2}\) norm, the solution formula is as follows [22,23]:  
\[\left\{ \begin{array}{l}\underset{\{u_{k}\},\{\omega_{k}\}}{\min}\left\{ \underset{k}{\sum}\bigg{\|}\partial_{t}\bigg{[}\bigg{(}\delta(t)+\frac{j}{\pi t }\bigg{)}*u_{k}(t)\bigg{]}e^{-j\omega_{k}t}\bigg{\|}_{2}^{2}\right\}\\ s.t.\underset{k=1}{\sum}K\,u_{k}=f(t),u_{k}=u_{1},u_{2},\cdots,u_{K}\end{array} \right. \tag{1}\]  
where, \(u_{k}\) is the IMF component obtained after VMD decomposition, \(\{\omega_{k}\}=\{\omega_{1},\omega_{2},\cdots,\omega_{K}\}\) is the center frequency of each IMF component, and \(\delta(t)\) is the pulse function. PSO [24] is widely used in parameter selection as a global search algorithm. Suppose that the domain of definition to be searched is a D dimensional space, where there is a population of m particles, and each particle itself contains information such as position and velocity. The position of the \(i\)-th particle in this space is represented by vector \(X_{i}=\left(x_{i1},x_{i2},\cdots,x_{iD}\right)^{T}\), and its velocity is represented by vector \(V_{i}=\left(v_{i1},v_{i2},\cdots,v_{iD}\right)^{T}\). The best position of particle \(i\) is \(P_{best}=\left(p_{i1},p_{i2},\cdots,p_{iD}\right)^{T}\), and the best position of all particles in the whole population can be expressed as \(Pgbest=\left(p_{g1},p_{g2},\cdots,p_{gD}\right)^{T}\). In the PSO algorithm, each particle can update its speed and position information iteratively according to the information such as the best position of the individual and the best position of the population:  
\[\left\{ \begin{array}{l}v_{id}^{k+1}=\omega v_{id}^{k}+c_{1}r_{1}\Big{(}p_{id}- x_{id}^{k}\Big{)}+c_{2}r_{2}\Big{(}p_{gd}-x_{id}^{k}\Big{)}\\ x_{id}^{k+1}=x_{id}^{k}+v_{id}^{k+1}t=x_{id}^{k}+v_{id}^{k+1}\end{array} \right. \tag{2}\]  
where, \(i=1,2,\cdots,m\); \(d=1,2,\cdots,D;k\) is the current number of evolutions; \(c_{1}\) and \(c_{2}\) are learning factors; \(\omega\) is the inertia weight; \(r_{1}\) and \(r_{2}\) are random numbers in \([0,1]\), \(t\) is the time of each step of movement, and \(t=1\) is taken here. In the particle swarm optimization algorithm, the parameters to be designed mainly include the following: population size \(m\), learning factors \(c_{1}\) and \(c_{2}\), maximum speed \(V_{max}\), inertia weight \(\omega\)[25; 26; 27; 28], among which the learning factor and inertia weight are the key objects of optimization. 1. Adaptive inertia weight strategy  
First, the most important inertia weight \(\omega\) in the PSO algorithm is optimized by using the adaptive inertia weight strategy, which can realize the automatic optimization of the inertia weight, and then make the algorithm converge to the optimal solution quickly. In the adaptive inertia weight strategy, if the minimum value of the objective function is solved, the more is the fitness and the closer is the distance to the optimal solution. At this time, a smaller weight is needed to facilitate local search. On the contrary, a larger weight should be adopted to improve the global search ability of particles [29; 30]. The adjustment method is as follows:  
\[\omega_{i}^{k}=\left\{ \begin{array}{l}\omega_{\min}+(\omega_{\max}-\omega_{ \min})\frac{f(x_{i}^{k})-f_{\min}^{k}}{f_{\text{average}}^{k}-f_{\min}^{k}}&, f\Big{(}x_{i}^{k}\Big{)}\leq f_{\text{average}}^{k}\\ \omega_{\max}&,f\Big{(}x_{i}^{k}\Big{)}>f_{\text{average}}^{k}\end{array} \right.