1 Introduction
Rotating machinery is an integral part of modern industry, which has many applications in gas turbines, aero-engine, wind turbines, and other critical machinery equipment. Because the working environment of rotating machinery is harsh, it is easy to generate mechanical failure. The crack fault is one of the most common failure forms in the rotor system [
1], which can seriously threaten the reliability of rotating machinery operating. Therefore, timely and accurate fault diagnosis of the crack fault is of great significance in improving the operation reliability of rotating machinery. Since the 1970s, many researchers have studied the crack fault diagnosis of the rotor system in many aspects. Mayes et al. [
2] first used a more accurate cosine breathing function to describe the breath behavior of the crack and studied the dynamic response of the rotor system containing a breathing crack. Then Al-Shudeifat et al. [
3] proposed a new type of breathing function using the Fourier series and compared it with Mayes’ results. The comparative results show that this respiration function could more accurately represent the breathing process of crack. Besides, it is found that the super-harmonic resonance can be regarded as the vibration characteristics caused by the crack, and the phenomenon of the natural frequency change with the increase of the crack depth is reported. Darpe et al. [
4] analyzed the influence of the interaction of the two cracks on the breathing behavior and the dynamic response of the Jeffcott rotor based on the concepts of fracture mechanics. Significant transformations in the rotor’s dynamic response are observed when the angle between the two cracks’ directions changes. Xiang et al. [
5] considered an asymmetric rotor-bearing system with crack and rub-impact coupling faults under oil-film forces. The effects of crack depth on the onset of instability and nonlinear responses of the rotor-bearing system are studied. Hou et al. [
6] found the super-harmonic resonance phenomenon of the cracked rotor due to maneuver load. Lu et al. [
7] studied the dynamic response of a hollow shaft dual-rotor system with breathing crack and unbalanced excitation. In the spectrum, it is found that the peak value of the super-harmonic resonance is related to the dimensionless depth of the crack. Fu et al. [
8] reported similar results in the dynamic response of a cracked rotor system with uncertain crack parameters. Wang et al. [
9] considered the anisotropy in a cracked rotor system and analyzed its parameter instability phenomena. Unlike traditional dynamic analysis methods, Liu et al. [
10] developed a novel Nonlinear Output Frequency Response Functions (NOFRFs) based criterion and discussed its application to the cracked rotor system. Through simulations and experiments, they found that some specific index is sensitive to the degree of crack propagation. Most of these studies focus on the dynamic response characteristics of the cracked rotor system or only qualitative analysis of the influence of the crack parameters (such as the depth and location of the crack) [
11‐
13]. However, quantitatively identifying the fault properties based on the dynamic response is necessary for many practical situations. This type of problem is known as fault diagnosis [
14].
In recent years, various machine learning methods, such as the artificial neural network (ANN) [
15], support vector machine (SVM) [
16], and decision-making tree (DT), have been widely applied in various fields. Many researchers also adopt these methods to study the fault diagnosis of the rotor system. Munoz-Abella et al. [
17] used the ANN and a large number of simulation data to make the crack fault diagnosis for a simple Jeffcott rotor and achieved good results. Guo et al. [
18] proposed a fault diagnosis method for a Jeffcott rotor with a breathing crack at the early stage of crack propagation based on the empirical mode decomposition (EMD) technology combined with ANN and conducted experimental verification. Vashisht et al. [
19] investigated the effect of cracks on a complex rotor system with ball bearing and advanced a crack detection method using the switching control strategy and Short Time Fourier Transform. Yan et al. [
20] extracted multi-domain features from the vibration signals by combining multiple signal processing technologies (such as statistical analysis, Variational Mode Decomposition, and Fast Fourier Transform). Then, a novel optimized SVM is adopted to study the fault diagnosis problem of the rotor system. Fault types include a crack in the outer race, an inner race with the spall, and pitting in balls. Comparative test results show that the proposed method is better than the traditional SVM. Bin et al. [
21] proposed a new approach for rotating machinery fault diagnosis based on wavelet packet decomposition (WPD)-EMD fault feature extraction and the neural network. Ma et al. [
22] presented a diagnosis method for rotor and bearing faults of rotating machinery based on ensemble learning. In this study, the method of weighting and integrating the Convolution Residual Network (CRN), Deep Belief Network (DBN), and Deep AutoEncoder (DAE) obtains a significant effect on the problem of multi-fault classification. Wang et al. [
23] determined the crack parameters, including crack location, depth, and angle for a solid shaft by using Kriging Surrogate Model and improved Nondominated Sorting Genetic Algorithm-III (NSGA-III), which has high parameter identification accuracy. Wang et al. [
24] used the K-means clustering method to classify crack faults for a planetary gearbox. Li et al. [
25] studied multiple crack identification based on the three steps meshing, and experimental verification was also carried out. Most researchers’ objects are simple Jeffcott rotors [
26‐
28] or solid shaft rotors [
29‐
31]. Nevertheless, to improve the rotor operating efficiency in practical problems, most large complex rotor systems adopt the hollow shaft. There is relatively little research on crack fault diagnosis of hollow shaft rotor systems.
The crack model of the hollow shaft is more complex than the solid shaft, which leads to a more complex dynamic response, making the crack fault diagnosis more difficult. Besides, many researchers focus on the pattern recognition of the crack fault in the various faults of the rotor system or calculating the crack depth with the fixed crack position. However, in practice, the crack failure may occur at any shaft position. The depth and position of the crack can have a coupling effect on the system’s dynamic response, which may confuse the crack diagnosis results. Therefore, identifying the hollow shaft crack’s parameters based on the system’s dynamic response when the crack depth and position are both unknown remains challenging.
The motivation of this study is to develop a novel crack fault diagnosis method for a two-disk hollow shaft rotor system. In which both the crack depth and location are uncharted. Considering the crack’s periodic opening and closing pattern and different degrees of crack depth, we establish the hollow shaft crack model with the breathing function. The dynamic response of the cracked rotor system is obtained by adopting the Harmonic Balance Method, and some dynamic characteristics related to the crack properties are summarized. Based on this, the Radial basis function (RBF) neural network and pattern recognition network are utilized to solve the crack fault diagnosis problem when the crack’s depth and location are both unknown. The effectiveness of the proposed method is verified by simulation.
The paper is organized as follows, in Section
2, the motion equations of the dual-disk hollow shaft rotor system with a breathing crack are constructed by the finite element method. Secondly, in Section
3, the harmonic balance method (HBM) is used to solve the dynamic response of the rotor system, and the effect of crack depth and position is analyzed, respectively. The Runge-Kutta method is used to verify the results of HBM. The crack depth estimation problem is discussed based on the RBF neural network, and the problem of crack location is solved using the pattern recognition network in Section
4. Finally, Section
5 summarizes the primary results of this paper.
5 Conclusions
In this paper, a dual-disk hollow shaft rotor system model with a breathing crack is established, where two different crack forms (non-penetrating and passthrough crack) are considered. Then, the dynamic response of the cracked rotor system is obtained through the HBM, and the dynamic characteristics related to the crack parameters are summarized by analyzing the amplitude-frequency curve and waterfall plot. Based on this, a novel crack fault diagnosis and location method based on the RBF network and PRNN is proposed. The main conclusions are as follows.
(1)
Due to introducing the time-varying stiffness, the crack can cause the super-harmonic resonance phenomenon in the rotor system near 1/n (n = 2,3,4) first critical speed. Besides, the crack can reduce the stiffness of the system, resulting in a decrease in the system’s critical speed.
(2)
The analysis results of the cracked rotor system’s amplitude-frequency curves and waterfall plots with different crack parameters reveal the dynamic characteristics related to the crack depth and position. The first critical speed, first subcritical speed, first critical speed amplitude, and first subcritical speed amplitude can be utilized to detect the crack.
(3)
Based on the RBF network and PRNN, the quantitative crack fault diagnosis method is proposed. In the case where both the crack depth and position are uncertain, Adopting the analyzed dynamic characteristics as input, the maximum percentage error between the trained RBF network’s output results and the ground truth is 7.56%. Besides, the approximate recognition accuracy of the crack position obtained by the PRNN can reach 98.2%. The requirements of crack fault diagnosis are satisfied preliminarily.
(4)
Several alternative machine learning-based crack fault diagnosis methods are considered in the comparison experiment. The results show that the approach developed in this paper achieves the optimal fault diagnosis performance, further demonstrating its effectiveness.
In future work, further research should focus on the following aspects. Firstly, the proposed method relies on the analyzed dynamic characteristics, such as the super-harmonic resonance peak, which maintain its theoretical basis and interpretability. However, at the same time, noise and nonlinearities can introduce disturbances to these dynamic characteristics and thus confuse the diagnosis results. The robustness of the proposed method regarding interference needs to be further improved. Combining the signal processing methods with the proposed approach may be a good solution. Secondly, limited by the finite element model, the proposed method can only give the approximate crack position. Modeling technology that can accurately describe the crack’s location should be further studied. Finally, experimental verification is necessary. In this paper, the validation of the proposed method is based on simulation data. In the future, we will establish the cracked rotor system experiment bench and validate our proposed method with experimental data.