Raw vibration signals are susceptible to environmental noise, electromagnetic interference, and baseline drift during acquisition. Direct feature extraction will degrade diagnostic accuracy. Therefore, standardized preprocessing is performed on the signals prior to multi-domain feature extraction, with the procedure as follows:

(1) Wavelet threshold denoising

The db4 wavelet is used for 3-level decomposition, and soft threshold filtering is adopted to remove high-frequency random noise while retaining fault impact features.

(2) Zero-mean normalization

Zero-mean processing is applied to the denoised signal to eliminate the effects of sensor zero offset and amplitude dimensions, with the formula as follows:(1)xi′=xi−μwhere μ denotes the mean value of the signal.

(3) Fixed-length segmentation and resampling

Each 6-s signal is segmented into a fixed length of 2048 points to ensure consistent input feature dimensions and avoid errors caused by inconsistent sample lengths.

After the above steps, feature extraction in the time domain, frequency domain, and envelope domain is performed.

Time-domain features directly reflect the statistical characteristics of vibration signals. By extracting the statistical and morphological features of signals, they reflect the amplitude distribution, fluctuation law and mutation characteristics of signals, and are sensitive to the transient impact of early faults in gearboxes. The time-domain feature vector is defined to include the following indicators:

(1) Root Mean Square (RMS) (2)RMS=1N∑i=1Nxi2where N is the signal length and x_i is the i-th sampling point. The RMS is used to measure the overall energy level of the signal. Gearbox faults will lead to an increase in vibration energy, and the RMS value will rise accordingly.

(2) Kurtosis (K) (3)K=1N∑i=1N(xi−μ)4σ4where μ is the mean value of the signal and σ is the standard deviation. Kurtosis reflects the sharpness of the signal distribution. When the impact caused by faults thickens the tail of the distribution, the kurtosis value increases.

Frequency-domain features reveal the periodic components of signals through Fourier transform. As a core mathematical tool in the field of signal processing, Fourier transform can convert signals in the time domain to the frequency domain, revealing the composition of signal frequency components, which breaks the limitation that time-domain analysis is difficult to intuitively present the frequency characteristics of signals. (4)F(ω)=∫−∞+∞f(t)e−jωtdtwhere ω is the angular frequency and j is the imaginary unit. Through integral operation, the time-domain signal is decomposed into a superposition form of sine/cosine components with different frequencies.

The spectral centroid is an indicator reflecting the concentration trend of signal energy distribution in the frequency domain. The more the energy shifts to high frequencies, the larger the value of the spectral centroid. (5)C=∑n=0N−1f(n)⋅|X(n)|2∑n=0N−1|X(n)|2where C is the spectral centroid, f(n) is the frequency at the n-th point, |X(n)|2 is the power spectrum corresponding to the frequency, and the denominator is the total power.

Envelope analysis is an important method for extracting amplitude variation features in the field of signal processing [24]. Its core lies in separating the slowly varying amplitude envelope that reflects key information from complex signals modulated by high-frequency carriers. This method can effectively eliminate redundant high-frequency components [25] and highlight the core variation trend of signals. (6)A(t)=|z(t)|=x2(t)+x^2(t)where the original signal is denoted as x(t), the signal obtained via Hilbert transform is denoted as x^(t), and the analytic signal is denoted as z(t)=x(t)+jx^(t).

The envelope root mean square is an indicator reflecting the concentration degree of signal envelope energy, which can effectively highlight the amplitude fluctuations caused by faults and is often used in equipment fault diagnosis. (7)RMSenv=1N∑i=1Nei2

RMSenv is the envelope root mean square value, N is the number of sampling points of the envelope signal, and B is the sampling value of the i-th envelope signal.

In this paper, 12 effective features are extracted from three dimensions: time domain, frequency domain, and envelope domain, and a comprehensive feature matrix is constructed for model training and testing. The dimension of the feature matrix finally input into the random forest model is 32 × 12, where 32 is the total number of samples and 12 is the dimension of the feature vector. The detailed composition is shown in Table 1.

In this paper, multi-dimensional features are constructed from the time domain, frequency domain, and envelope domain, with weak correlation among these features. Random Forest (RF) is insensitive to high-dimensional features and does not require complex feature selection, enabling direct fusion of multi-domain information. Meanwhile, given the limited sample size in the gearbox experiment, RF reduces the risk of overfitting through ensemble and randomization strategies, and exhibits strong robustness to environmental noise in vibration signals, which conforms to actual industrial field conditions. In contrast, SVM relies on appropriate kernel function selection, and MLP requires a large amount of data and careful parameter tuning. RF imposes no strict assumptions on data distribution, has fewer parameters and is easier to optimize, making it more suitable for engineering applications.

Random Forest (RF) is a powerful ensemble learning algorithm composed of M decision trees [26]. Each decision tree is constructed based on a different training subset, which is obtained from the original training set through random sampling with replacement [27]. This sampling method enables each decision tree to be trained on relatively independent and differentiated data, thereby increasing the diversity of the model. The workflow diagram of the RF model is shown in Fig. 1.

For classification problems, each decision tree T_m independently performs classification prediction based on the input gearbox feature vector x [28]. The final prediction result is determined by a voting mechanism, and the specific formula is as follows:(8)y^=arg⁡maxc∑m=1MI(Tm(x)=c)where y^ is the final predicted class of the RF model; c denotes the fault class; I(⋅) is the indicator function.

The selection of parameters has a crucial impact on the performance of the RF model. To improve the accuracy and stability of the model, this paper adopts the method of grid search combined with cross-validation to optimize its key parameters.

Number of decision trees: The number of decision trees determines the scale of the ensemble model. A larger number of decision trees can enhance the stability and generalization ability of the model, but it will also increase the computational cost [29]. The value range considered in this paper is {50, 100, 200, 300}.

Maximum depth: The maximum depth limits the growth of decision trees. A smaller maximum depth can prevent overfitting, but may lead to underfitting; a larger maximum depth can enable the model to better fit the training data, but may increase the risk of overfitting. The value range is set as {5, 10, 15, 20, None}, where None means no depth limit, the decision tree will grow until all leaf nodes are pure or other stopping conditions are met.

Number of split features: When splitting at each node, a certain number of features are randomly selected for evaluation. This increases the diversity among decision trees and helps improve the overall performance of the model. The value range is {n,log2⁡(n),n}, where n is the total number of features.

Minimum number of samples at leaf nodes: This parameter limits the minimum number of samples in leaf nodes and prevents decision trees from being overly complex. The value range is {1, 2, 4}.

The objective function adopts the average accuracy of 5-fold cross-validation, and the calculation formula is as follows:(9)Score=15∑i=15Acci

By traversing all possible hyperparameter combinations and calculating the objective function value corresponding to each combination, the optimal hyperparameter combination of the random forest model is finally determined in this paper as follows:(10){n_estimators=100max_depth=10max_features=nmin_samples_leaf=1}

This set of parameters ensures the classification accuracy of the model while effectively avoiding overfitting, enabling the model to achieve better stability and generalization ability on small-sample datasets.

To comprehensively evaluate the performance of the model on the four-classification problem, the following evaluation metrics are adopted in this paper:

Accuracy: As the most commonly used evaluation metric, accuracy refers to the proportion of correctly predicted samples to the total number of samples. The calculation formula is as follows:(11)Acc=TP+TNTP+TN+FP+FN

Macro-Precision: Macro-average precision takes into account the precision of each category and avoids the impact of class imbalance on the evaluation results. The calculation formula is as follows:(12)Pmacro=1k∑i=1kPi

Macro-Recall: Macro-average recall measures the model’s ability to identify faults of each category. The calculation formula is as follows:(13)Rmacro=1k∑i=1kRi

F1-Score: F1-score is the harmonic mean of precision and recall, which comprehensively considers the precision and recall of the model. The calculation formula of macro-average F1-score is as follows:(14)F1=2×Pmacro×RmacroPmacro+Rmacrowhere TP, TN, FP, and FN denote true positive, true negative, false positive, and false negative, respectively; k is the number of categories; P_i and R_i represent the precision and recall of the i-th fault category, respectively.

The experimental platform consists of two main parts: a gear reducer and a signal acquisition system, whose structure is shown in Fig. 2. The core components include a base, a gear reducer, a three-phase asynchronous motor, an electromagnetic speed-regulating motor, a magnetic powder brake, a speed regulator, and a signal acquisition device. In this experiment, the JZQ250 type two-stage parallel shaft gearbox is taken as the research object. This gear reducer is widely used in various mechanical equipment and has important research significance.

For the vibration signal acquisition process, the Donghua DH5981 dynamic signal acquisition instrument was selected, paired with the IEPE low-impedance voltage-output general-purpose vibration sensor (Model 14100), with the sampling frequency set to 20 kHz.

An industrial-standard vibration sensor arrangement is adopted. The sensors are installed vertically on the bearing housing of the gearbox, close to the shortest transmission path of vibration energy, which can effectively capture gear meshing vibration and fault impact signals, ensuring that the acquired vibration data have high signal-to-noise ratio and high representativeness. This installation position follows the general specification layout for rotating machinery fault diagnosis, which can stably obtain fault-sensitive features and meet the requirements of fault feature extraction and diagnostic identification.

The specific connection method is as follows: after installing the vibration sensor at the designated position of the reducer, it is connected to the Donghua DH5981 dynamic signal acquisition instrument via a dedicated connecting cable. The acquisition instrument then transmits the vibration signals to the supporting signal acquisition software of Donghua. This software can implement core functions such as signal storage, format conversion and preliminary analysis, providing fundamental support for subsequent data processing.

Common fault modes of gearboxes include gear wear, gear cracks, gearbox tooth breakage, and tooth surface scuffing. This experiment mainly takes gear cracks as an example for research and analysis. First, to explore how to identify gear crack faults of varying degrees, four working conditions were set for the intermediate gear under the same operating conditions, namely normal state, 2 mm crack, 5 mm crack, and 8 mm crack.

Vibration signal acquisition should be initiated after the experimental platform operates stably, so as to avoid the interference of factors such as insufficient equipment preheating and voltage fluctuations in the initial stage of startup on signal quality. The experimental parameters are set as follows: the motor input speed is 1200 r/min, the sampling frequency of the vibration sensor is 20 kHz, the data acquisition duration of each group is 6 s, and the next group of acquisition is carried out after an interval of 5 s between groups, with a total of 8 groups of data obtained. The specific signal acquisition scheme is shown in Table 2.

To fully verify the diagnostic performance and stability of the proposed method, experiments are carried out in this paper under the premise of consistent controlled variables and strictly identical working conditions. Eight groups of high-quality, high-signal-to-noise ratio valid samples are collected under each working condition to ensure good representativeness and consistency of the samples. A stratified 5-fold cross-validation strategy is adopted in the experiment to comprehensively evaluate the model. Through multiple random divisions of the training set and test set, the value of data is fully exploited, and the reliability and statistical significance of experimental results are effectively improved.

For the collected dataset samples, the stratified 5-fold cross-validation strategy is adopted for data partitioning. According to four working condition categories, all samples are randomly divided into five mutually exclusive subsets via proportional stratified sampling. In each iteration, four subsets are used as the training set and one subset as the test set. The process is repeated five times and the average results are obtained.

Afterwards, comparative analysis of signal waveforms is conducted, followed by multi-dimensional feature extraction using the method proposed above. Finally, distribution boxplots are applied, and dimensionality reduction and visualization are performed on the multi-dimensional feature space based on PCA and t-SNE.

As shown in Fig. 3: The vibration amplitude of the healthy state S1 remains stable within ±0.2 g, with a steady signal and no obvious impacts; the amplitude of the 2 mm crack state S2 expands to ±0.3 g, and periodic impacts begin to appear; the amplitude of the 5 mm crack state S3 further increases to ±0.4 g, with shortened impact intervals and enhanced energy; the maximum amplitude of the 8 mm crack state S4 exceeds ±0.5 g, with both impact intensity and frequency reaching the highest levels. As the crack size increases, the vibration amplitude, impact characteristics, and energy level show a stepwise increasing trend, which is consistent with the gear crack propagation mechanism.

As illustrated in Fig. 4, the energy distribution of the normal state S1 is uniform across the full frequency band without distinct peaks. Obvious prominent peaks emerge at the gear meshing characteristic frequencies for S2, S3 and S4, and the peak amplitudes increase by 2 to 5 times with the expansion of cracks. Meanwhile, the vibration energy gradually concentrates in the high-frequency band, which is positively correlated with the fault severity.

It can be observed from Fig. 5 that envelope analysis can effectively separate high-frequency carriers and highlight the slowly varying characteristics of fault impacts. The envelope curve of S1 is stable with slight amplitude fluctuations. For S2–S4, the envelope amplitude gradually increases and the fluctuations become more severe as the crack propagates, and the root mean square value of the envelope rises by more than three times, which clearly reflects the evolution law of fault severity.

Fig. 6 presents the box plot of feature distribution, a statistical chart for displaying data distribution that enables rapid identification of data features and anomalies. Significant differences exist in the distribution of various statistical features of vibration signals, such as root mean square (RMS), kurtosis, spectral centroid, and envelope root mean square, when the gearbox operates normally and under different fault conditions. Specifically, in the normal state, the RMS values show concentrated distribution with relatively low magnitudes; the kurtosis values are low and centralized; the spectral centroid values are high and stable; and the envelope RMS values are low and concentrated. In contrast, under fault conditions, especially with an 8 mm gear crack, the RMS values exhibit more dispersed distribution with higher magnitudes; the kurtosis values increase sharply with scattered distribution; the spectral centroid values are low with fluctuations; and the envelope RMS values are high and dispersed. These differences indicate that various statistical features can serve as effective indicators for distinguishing between the normal and fault states of the gearbox as well as for identifying different fault types.

As can be seen from the multi-dimensional feature space analysis in Fig. 7, in the feature space distribution of Principal Component Analysis (PCA), the sample points of S1 state (green dots), S2 state (blue dots), S3 state (red dots) and S4 state (orange dots) have a certain degree of separability. The normal sample points are relatively concentrated in a specific area, while the fault sample points are distributed in different ranges, indicating that PCA can partition the feature space of gearbox samples under different states to a certain extent. The principal component contribution chart shows that the explained variance ratio of PC1 reaches 0.400, making it the most dominant principal component. With the increase in the number of principal components, the cumulative contribution degree rises gradually, which indicates that the first several principal components can explain most of the data variance. In the t-SNE feature space distribution, the clustering effect of sample points under different states is more obvious. The normal sample points and various types of fault sample points form relatively independent clusters, respectively, suggesting that t-SNE has a stronger ability to distinguish gearbox samples under different states after dimensionality reduction, which is more conducive to subsequent fault identification and classification.

Finally, the multi-dimensional features including time-domain, frequency-domain and envelope analysis are integrated into a comprehensive feature matrix, forming a complete feature set that can reflect the multi-domain characteristics of gearbox faults.

For the extracted fault feature set, the previously constructed fault diagnosis model based on Random Forest (RF) was trained, and a comprehensive comparative analysis was conducted. The models involved in the comparison included Random Forest (RF), Gradient Boosting (GB), Extremely Randomized Trees (ET), Support Vector Machine (SVM), Multi-Layer Perceptron (MLP), Logistic Regression (LR), K-Nearest Neighbors (KNN), and Gaussian Naive Bayes (GNB). Through the training and evaluation of these models on the same dataset, quantitative analysis was carried out from multi-dimensional metrics such as accuracy, precision, recall, F1-score, and cross-validation stability. Ultimately, a comprehensive performance comparison result was formed, which demonstrated the effectiveness and superiority of the selected fault diagnosis model.

From the perspective of model characteristics, methods such as SVM and MLP usually have stronger nonlinear fitting capabilities in large-scale data scenarios, but their performance is sensitive to sample size. The multi-dimensional feature and random forest framework constructed in this paper is originally designed to adapt to industrial fault diagnosis scenarios with small samples and high-dimensional features. Through ensemble learning and random attribute selection mechanisms, random forest can maintain strong feature learning and classification ability under limited sample conditions. All comparative models in this paper adopt a unified grid search strategy to complete parameter optimization, ensuring consistent comparison conditions. The experimental results can effectively reflect the applicability and performance differences of different methods in gearbox fault diagnosis tasks with small samples.

Based on the comprehensive evaluation results in Table 3, the Random Forest (RF) algorithm exhibited the optimal performance in the fault diagnosis task, achieving an accuracy and F1-score of 95.83% each. It outperformed other algorithms significantly in both the recognition accuracy of various fault categories and overall stability, demonstrating excellent adaptability to the data characteristics of this task. The performance of Gradient Boosting (GB) was extremely close to that of RF, with its core metrics being basically on par. However, due to its serial iterative training mechanism, GB might be slightly less efficient when processing large-scale datasets, resulting in relatively poor adaptability to real-time diagnosis scenarios. Both Extremely Randomized Trees (ET) and Support Vector Machine (SVM) delivered good overall performance, with an accuracy of 91.67% for each. Among them, ET was slightly more sensitive to noise than RF because of its more extreme random feature splitting strategy. Although SVM performed stably on medium-dimensional features, the parameter tuning difficulty and computational complexity of its kernel function would increase significantly when dealing with high-dimensional complex nonlinear relationships. Constrained by the sample size and feature dimensions, the Multi-Layer Perceptron (MLP) failed to give full play to the complex pattern fitting capability of deep learning models, achieving an accuracy of 87.50%, and its generalization performance in small-sample scenarios needs to be improved. Although the K-Nearest Neighbors (KNN) algorithm achieved an accuracy of 91.67%, its prediction speed would decrease significantly with the increase of data volume due to its mechanism relying on inter-sample distance calculation, making it difficult to meet the real-time requirements of large-scale diagnosis tasks. In contrast, Gaussian Naive Bayes (GNB) achieved an accuracy of only 66.67%, performing relatively poorly in this task, because its strict assumption of feature independence was inconsistent with the strong correlation among features in the actual fault data.

Based on the multi-metric performance comparison chart of models in Fig. 8, the Random Forest (RF) and Gradient Boosting (GB) models exhibited excellent performance across accuracy, precision, recall and F1-score, with their metric values remaining close and at high levels, which indicates that these two models have outstanding comprehensive performance in classification tasks. The Extremely Randomized Trees (ET) and Support Vector Machine (SVM) models also achieved relatively high values in all metrics, showing good performance. The Multi-Layer Perceptron (MLP) model performed moderately, while the Logistic Regression (LR) and Naive Bayes (NB) models yielded relatively low metric values. In particular, the NB model showed significantly lower accuracy, precision, recall and F1-score than other models, reflecting poor classification performance. In terms of the comparison of model performance radar charts, the RF and GB models stood out in all metrics, with their radar charts covering a wide area, which demonstrates their strong performance in multiple aspects. The ET model also reached a favorable level with balanced performance across all metrics. The radar charts of models such as SVM and K-Nearest Neighbors (KNN) covered a relatively smaller area, indicating slightly inferior performance. The NB model had the smallest coverage area in its radar chart, which further reflects its weak performance in these evaluation metrics. Overall, the above results are consistent with those in Table 2. Ensemble learning models including RF, GB and ET achieved superior performance in this gearbox fault classification task, whereas simple models such as NB showed relatively insufficient performance.

As can be seen from Fig. 9, the Random Forest (RF) model demonstrated remarkable advantages. In terms of performance, its F1-score reached a relatively high level, close to 0.95, indicating that the RF model could accurately identify various types of faults and achieve excellent classification results in the gearbox fault classification task. Meanwhile, in terms of training time, the RF model required a relatively short training duration, which was much lower than that of the Gradient Boosting (GB) model. This characteristic of ensuring high performance while maintaining high training efficiency enables the RF model to not only complete the model training process rapidly in practical applications, but also provide accurate results for gearbox fault diagnosis, thus achieving a favorable balance between training efficiency and classification performance.

As can be clearly observed from Fig. 10, the Random Forest (RF) model exhibited exceptionally outstanding performance. The narrow box of its box plot indicates that the RF model had minimal performance fluctuations during the cross-validation process, demonstrating excellent stability. This means that the RF model could maintain stable and favorable classification results across different data subsets, with strong generalization ability. Meanwhile, the mean F1-score of the RF model stayed at a high level, which shows that in the gearbox fault classification task, the model could not only perform stably but also ensure high classification accuracy, enabling precise identification of various gearbox faults. Therefore, it serves as an excellent model choice that balances stability and accuracy for gearbox fault diagnosis tasks.

According to the analysis results of the confusion matrix of the RF model in Fig. 11, the overall performance of RF is excellent, with an F1-score of 0.958, indicating good comprehensive classification performance of the model. In the raw count matrix, all 8 samples of state S1 are correctly predicted; all 8 samples of state S2 are accurately classified; and the 8 samples of state S3 are also correctly identified. However, one sample of state S4 is misclassified as S1, with the remaining seven samples correctly predicted. From the row-normalized proportion matrix, the prediction accuracies of classes S1, S2, and S3 are all 1.00, meaning all samples of these three classes are correctly classified. The prediction accuracy of class S4 is 0.88, indicating a certain degree of misclassification for this class, with 12% of S4 samples incorrectly predicted as other categories.

Based on the comprehensive evaluation results of all categories, the RF model demonstrated significant superiority in the source domain fault diagnosis task. In terms of performance, its accuracy and F1-score both reached 95.83%. In the multi-metric performance comparison, all core indicators maintained high and balanced levels, leading to remarkable classification results for gearbox faults and high overall classification accuracy. In terms of stability, the RF model showed minimal performance fluctuations in cross-validation, with a narrow box in the box plot, which indicates that it could maintain stable and favorable classification results across different data subsets and possessed strong generalization ability. In terms of efficiency, the RF model required a short training time, which was much lower than that of the GB model. It could complete training rapidly and be put into practical applications, achieving a sound balance between training efficiency and classification performance. Therefore, it is an ideal model choice for gearbox fault diagnosis tasks.