# Fault Feature Extraction of Parallel-Axis Gearbox Based on IDBO-VMD and t-SNE
## 1 Introduction  
Parallel-axis gearboxes are one of the most common components in mechanical transmissions and are widely utilized in various rotating mechanical equipment, serving as crucial transmission structures in several fields such as aerospace, metallurgy, chemical engineering, shipping, and the automotive industry [1]. However, due to high loads and severe operating conditions, gearboxes are prone to various issues including pitting, wear, cracking, and even tooth breakage. Improper selection of surface coatings and lubricants can also accelerate gear wear, potentially leading to accidents [2]. To prevent accidents, it is crucial to conduct research on the durability, noise, and vibration of gears. Durability-related studies primarily focus on dynamic tooth forces and dynamic stress coefficients, while noise studies concentrate on dynamic transmission errors and gearbox vibration [3]. Vibration signals are easier to collect and utilize for fault diagnosis, providing an effective means to prevent accidents. Due to the structural characteristics and complex operating conditions of parallel-axis gearboxes, gear meshing can result in impact or collision when gears experience faults [4]. This non-steady force or torque input leads to the nonstationary nature of vibration signals, which exhibit nonlinear and nonstationary properties [5]. Therefore, preprocessing of raw signals is necessary before feature extraction. In 1998, [PERSON] et al. [6]introduced the Hilbert-Huang transform (HHT) algorithm, incorporating the empirical mode decomposition (EMD) algorithm, and successfully applied it to the decomposition of mechanical vibration signals with certain effectiveness. However, EMD has drawbacks such as mode mixing and endpoint effects [7; 8]. Subsequently, [PERSON] proposed variational mode decomposition (VMD) [9], an adaptive signal processing method. This method's decomposition approach differs from EMD, using a non-recursive variational mode that effectively addresses endpoint effects and mode mixing issues. Due to its ability to determine the number of decomposed modes and having mathematical theoretical support, VMD is widely applied in the field of fault diagnosis [10]. However, a drawback of this method is that the decomposition quantity \(K\) and penalty factor \(\alpha\) significantly impact the decomposition results, necessitating further improvements to the VMD algorithm. [PERSON] et al. [11] explained VMD principles and optimized it using the particle swarm optimization (PSO) algorithm, applying it to fault diagnosis. Test results indicate that this method can identify faults more rapidly and accurately. However, the PSO algorithm is sensitive to population initialization, and significant differences in the initial population distribution may affect the quality of feature information extraction methods. The effectiveness of feature extraction methods determines the success of fault diagnosis, especially in the process of extracting features from nonlinear and non-stationary signals. Entropy, as a sensitive feature, is widely applied in the field of fault diagnosis [12]. Analyzing signals using entropy helps measure signal complexity [13]. [PERSON] et al. [14] extracted permutation entropy for four types of ship signals as a fusion feature vector, inputting it into a support vector machine (SVM) model for classification and recognition. Experimental results show that this method has a higher recognition rate compared to existing methods. However, a single entropy characteristic may not fully reflect the feature information of the signal. Therefore, it is necessary to extract different entropy features and time-frequency domain features. However, when the extracted feature dimensions are high, they may contain redundant information. Therefore, dimensionality reduction methods are needed for secondary feature extraction to eliminate redundant features.