## 3 Methodology
### Segmentation of Peak Region of Acceleration Spectrum  
Adaptive recognition and segmentation of peak regions is the basis of the algorithm applied to real-time measurement systems. Considering the noise characteristics of environmental micro-vibration signals and the real-time requirements of the measurement system, an efficient and automatic signal contraction peak detection algorithm is adopted [24]. The  
Figure 4: IMF Component of acceleration signal based on VMD algorithm.  
Figure 5: The spectrum of IMF components based on VMD algorithm.  
signal contraction peak detection algorithm can effectively suppress the fluctuation noise interference in low-amplitude signals. The algorithm can avoid the influence of \"burr\" and \"false peak\" phenomena in the observed acceleration spectrum curve on the segmentation of the peak region [25].  
Assuming the given acceleration signal time series is \(S(n)=\{S_{1},S_{2},S_{3},\ldots,S_{N}\}\), \(n=1,2,3,\ldots,N\). Firstly, we need to find the local minimum and local maximum values of the acceleration signal \(S(n)\) and their corresponding positions. The condition for wave peak points is as follows:  
\[P(n)=S(n)\quad if\ S(n-1)<S(n)>S(n+1) \tag{1}\]  
\[L_{p}(i)=n\quad if\ S(n-1)<S(n)>S(n+1) \tag{2}\]  
where \(P(n)\) is the wave peak at the time series \(S(n)\) of the \(n\)th acceleration signal, \(L_{p}(i)\) is the position corresponding to the local wave peak \(P(n)\), \(i=1,2,3,\ldots,m\), and \(m\) is the number of local wave peaks (valleys).  
The condition for wave valley points is as follows:  
\[V(n)=S(n)\quad if\ S(n-1)>S(n)<S(n+1) \tag{3}\]  
\[L_{v}(j)=n\quad if\ S(n-1)<S(n)>S(n+1) \tag{4}\]  
where \(V(n)\) is the wave valley at the time series \(S(n)\) of the \(n\)th acceleration signal, \(L_{v}(j)\) is the position corresponding to the local wave valley \(V(n)\), and \(j=1,2,3,\ldots,m\);  
Secondly, the value of valley peak difference (\(VPD\)) was calculated. The false peak points were screened out by judging the value of \(VPD\). When calculating the value of \(VPD\), make sure that the processing starts from the wave valley. The position of the first wave peak and the first wave valley are compared. If the wave peak appears first, the data are discarded to ensure that the vibration signal is obtained from the wave valley first. This means that the discarded wave peaks have no corresponding wave valleys to ensure that the numbers of wave peaks and wave valleys are the same. The expression of \(VPD\) are as follows:  
\[VPD(k)=P(k)-V(k) \tag{5}\]  
where \(P(k)=\{P_{1},P_{2},P_{3},\ldots,P_{m}\}\) is the set of all local wave peaks, \(V(k)=\{V_{1},V_{2},V_{3},\ldots,V_{m}\}\) is the set of all local wave valleys, \(VPD(k)\) is the difference between the value of the \(k\)th wave peak and the value of the \(k\)th wave valley, and \(k=1,2,3,\ldots,m\).  
After the calculation of \(VPD\)s, the algorithm searches the value of \(VPD\)s. If the value of \(VPD\)s satisfies the relationship shown in Equation (6), this wave peak is removed. Then, the corresponding local wave peak \(P(n)\) and the position information corresponding to the wave peak \(L_{p}(i)\) is removed. Repeating this \(VPD\) processing until the number of wave peaks in two consecutive iterations remains the same.  
\[VPD(k)<\theta\{VPD(k-1)+VPD(k)+VPD(k+1)\}/3 \tag{6}\]  
where \(\theta\) is the weight parameter, the value of \(\theta\) is confirmed by reference or experimental data.  
In order to facilitate the subsequent analysis and reduce the influence of noise fluctuations on the fitting results, the amplitude and frequency of the micro-vibration in the N-segment peak region were normalized, respectively, to obtain the normalized acceleration spectrum \(\overline{A}_{i}(\overline{f})\), \(\overline{A}_{i}=\frac{A_{i}}{\sum_{i=1}^{m}A_{i}\in(0,1)}\), \(i=1,2,3,\ldots,n\), and \(\overline{f}\) is the normalized frequency.