## 1 Introduction  
Fiber Bragg gratings (FBGs) have attracted more and more attention due to their small size, high resolution, multiplexing capability, immunity to electromagnetic fields, and other interesting features. Recently, FBG sensors have been considered as promising in structural health monitoring (SHM) [1]. When the central wavelength of the reflected light shifts with the introduced structure strain/stress, the FBG sensor performs as an optical strain gage for strain/stress measurements [2]. Furthermore, FBG sensors have shown great potential for monitoring applications in aluminum fatigue crack by analyzing the deformation spectrum signal with crack propagation [3]. However, the truthful signal data contains various noises caused by the environment, personal operation, and other reasons. In this paper, to identify and abstract valuable information from the initial signal, combined with background noise, especially under a high-noise condition, we propose wavelet thresholding in a variational mode decomposition (VMD) domain signal denoising method, called the VMD-DWT denoising algorithm. The VMD was first proposed by [PERSON] and [PERSON] in Reference [4]. It is an ensemble, non-recursive variation decomposition mode, the basic mode being named band-limited intrinsic mode functions (BLIMFs) \(u_{k}\), where the modes are extracted concurrently and their central frequencies \(w_{k}\) are estimated online using the alternate direction method. The research shows that the frequency of BLIMFs is more compact than that of intrinsic mode functions (IMFs), and the VMD outperforms the empirical mode decomposition (EMD) in signal-noise separation and robustness based on some experimental data. Thus, the VMD can be used to remove the noise from non-linear and non-stationary signal. In Reference [5], the VMD was used to denoise the vibration signals caused by rotor-to-stator rubbing, and the analysis results show its superiority over empirical wavelet transform (EWT) [6], ensemble empirical mode decomposition method (EEMD) [7], and EMD [8]. In Reference [9], VMD was successfully used to denoise a biomedical image, and in Reference [10], a typical EGG signal was denoised using only VMD. Moreover, [PERSON] et al. [11] studied chatter detection in a milling process based on the entropy of the basic modes decomposed using VMD and wavelet packet decomposition (WPD). [PERSON] et al. [12] demonstrated a new coarse-to-fine decomposing strategy of the VMD that can detect the weak repetitive transients of heavy noise signals at a high quality. Recently, researchers have verified the basic decomposition modes of VMD related to spectrum energy. The decomposition level \(K\) decides the energy distribution in each basic mode [13]. However, in the literature, there are few studies on the definition method of decomposition level number \(K\). In Reference [6], the decomposition level of VMD coincides with that of the EMD decomposition number. Moreover, in Reference [13], VMD combined with detrended fluctuation analysis (DFA) is proposed to denoise noisy signals corrupted by white Gaussian noise (WGN), and the criterion based on DFA is designed to select the number \(K\). [PERSON] et al. [14] applied simulated Acoustic emission (AE)signals to its wave packages adaptively, and the number of wave packages is seen as the value \(K\). However, the reasonability of the operability of the decomposing strategy of VMD for the experimental FBG signals are seldom discussed because it is difficult to establish the basic components of the signals. Obviously, these predefined methods are disproportionate, and the selection method of the best decomposition number \(K\) is worth investigating. FBG signals detected under fatigue loading are a non-stationary time series. The degree of noise interference in damage detection processing is changed with the diverse external load environment.