# Identification of Mode Shapes Based on Ambient Signals and the IA-VMD Method
## 3 The Iterative Adaptation VMD Method
### The Estimation of Mode Number \(K\)  
Compared to the EMD algorithm, the VMD algorithm not only retains the advantages of the EMD algorithm but also offers better fitting accuracy, improved decomposition results, and a faster running speed. However, the traditional VMD method has its own disadvantages. On the one hand, the preset value of \(K\) is mainly set according to operational experience. This parameter affects the accuracy of the oscillation decomposition of the VMD algorithm. However, modal confusion may occur in the case of high-noise signals. The traditional VMD method can only distinguish the ambient signal from the environmental noise and cannot extract the IMF directly. To make the VMD algorithm better adapted to ambient signals, this paper improves the VMD algorithm in two aspects. On the one hand, to increase the accuracy of the decomposition results, the self-adaptability of the algorithm is improved to choose a suitable value of \(K\); on the other hand, the signal energy method is introduced to screen the intrinsic modal components obtained from the first decomposition so as to further reduce the influence of high-noise signals. This paper combines the VMD algorithm with frequency domain analysis by introducing a frequency domain analysis method. The steps for estimating the number of modal components \(K\) are as follows:  
Step 1: Obtain the frequency diagram of the measured signal by fast Fourier transform (FFT). Step 2: Take the data of 0.2 Hz-2 Hz in the spectrum and calculate the average magnitude of the signal in this frequency band. Step 3: Extract data segments where the magnitude is higher than 3 times the average magnitude. When there is no oscillation, the signal will only contain direct current components and noise components. So, there will be many low amplitude peaks in the spectrum that do not correspond to oscillations. By this step, the influence of noise and interference in the nonoscillation period can be effectively removed. Step 4: Count the peak number of each extracted data segment by evaluating whether the data are monotonically increasing. The sum of the peak numbers of all the extracted data segments is the mode number \(K\). Step 5: If the LFO characteristic of the modes is not obvious, decompose the mode by the VMD method again; then, extract the dominant mode. Aiming at the iterative improvement of the traditional VMD method, this paper introduces the signal energy method to screen the oscillation modes obtained by the first mode decomposition and then improves the VMD algorithm's decomposition process as follows:  
(1) The signal energy \(E_{inf}\left(i\right)\) of each modal component is calculated according to [PERSON]'s theorem. The formulas are as follows:  
\[E_{inf}(i)=\int_{0}^{T}\bigl{|}f_{i,j}\bigl{(}t_{j}\bigr{)}\bigr{|}^{2}dt=\sum _{j=1}^{m}\bigl{|}x_{ij}\bigr{|}^{2} \tag{10}\]  
In the formula, \(i\) represents the number of intrinsic modal components, \(j\) represents the discrete sampling points of the signal, \(f_{i,j}(t_{j})\) represents the sampling signal, \(x_{ij}\) represents the discrete sampling points of the signal, and \(E_{inf}\left(i\right)\) is the signal energy of the \(i\)th intrinsic modal component. IA-VMD of the sampled signal can obtain \(i\) inherent modal components, in which the modal component with higher signal energy is called the dominant modal component. This component can reflect the operation characteristics of the system. The specific steps of extracting dominant modal components are as follows:  
(2) The signal energy value \(E_{inf}\left(i\right)\) of each intrinsic modal component can be calculated by Formula (10). The sum of the signal energy values of each modal component is the total energy value \(E_{IMF}\) of the sampled signal oscillation signal.