Scanning Induction Thermography for Bearing Ring Under AC-DC Composite Magnetization

Induction thermography is a promising nondestructive testing technique with the advantages of high efficiency, lift-off tolerance, and defect imaging. To fulfill scanning detection and quantify the defect in bearing rings, this paper proposes scanning eddy current thermography under AC-DC composite magnetization. With DC magnetization, the defect will cause permeability disturbance in the bearing ring surface. Under high-frequency AC magnetization, disturbed permeability will generate a non-uniform thermal response captured by the infrared camera. To investigate the detection principle, a multi-physics coupling simulation model for permeability and thermal distribution in the scanning process is built, involving DC-AC composite magnetization, nonlinear ferromagnetic material, heat generation, heat conduction, and scanning movement. Then, a thermal data post-processing method for scanning imaging and crack quantification is proposed. Finally, the scanning induction thermography system for bearing rings is designed and developed. The experimental results show that the proposed scanning method can quantify bearing ring cracks with depths up to 2.0 mm.


Introduction
Bearings, as commonly used machine elements, utilize the rolling action to permit minimum friction and constrain the motion of one body relative to another [1]. As an essential bearing component, the bearing ring is often subjected to heavy impact and alternating stress in motion [2]. Any defects of bearing rings in production and operation will influence the performance of machines and even lead to a B Jianbo Wu wujianbo@scu.edu.cn 1 breakdown [3]. To ensure the quality of bearing rings, nondestructive testing (NDT) techniques are usually applied to inspect the defects. So far, there are several NDT methods used for bearing rings, including vibration testing (VT), ultrasonic testing (UT), computer vision (CV), magnetic particle testing (MPT), magnetic flux leakage (MFL) testing, and eddy current testing (ECT). VT diagnoses and locates bearing failures by monitoring and analyzing vibration signals of bearings during rotating [4,5], which is a real-time detection method. Still, this method cannot accurately quantify the size of defects. UT provides a scanning method to quantify inner defects [6], but it is not sensitive to surface defects, and liquid couplants are required. CV takes a camera instead of human eyes to visualize cracks with high detection accuracy and efficiency [7,8]. However, it can only identify surface defects, and it is hard to quantify defect depth. MPT exerts magnetic particles forming apparent magnetic marks that indicate defections and then complete auto-detection based on CV [9][10][11]. It shows high sensitivity in detecting defects, while the evaluation result is easily influenced by many factors. In addition, it cannot quantify defect depth. MFL testing can pick up the weak leaked magnetic field generated by tiny cracks [12,13], in which the sensors need to contact the ring surface to avoid the lift-off effect. ECT is also a contact testing method, and it can be easily affected by non-uniform permeability [14,15]. To sum up, these testing methods still have some shortcomings for bearing rings. Hence, a novel method for efficient and quantitative detection of bearing ring defects is in great need.
Induction thermography (IT) is a promising nondestructive testing technique with the advantages of high efficiency, lift-off tolerance, and defect imaging [16][17][18][19]. It is based on the interaction between the eddy current and the defect. When the eddy current flows perpendicular to the defect, its distribution will be significantly changed by the defect, leading to an apparent non-uniform thermal distribution. However, with their included angle decreasing from 90°to 0°, the interaction gradually weakens. Especially when the eddy current flows parallel to the defect, the defect has little effect on eddy current distribution [20]. In this situation, the detection capability is much diminished. The defect of the bearing ring has a random orientation, and parallel defect detection is a great challenge. Besides, defect depth quantification is another challenge. The induced eddy current can flow below the defect, making deep defects detectable. However, when the defect depth increases to a great value, the thermal response difference among deep defects becomes small, which is not helpful for depth quantification. To improve the detection capability, scanning induction thermography (SIT) under AC-DC composite magnetization is proposed [21][22][23]. These studies show that the detection performance of SIT is improved under DC magnetization, better to find and quantify deeper defects, and suitable for various defect orientations from 0°to 90°. However, these studies did not reveal the multi-physics coupling mechanism. Besides, the studied specimens are steel plates, which are simple and regular. In this paper, a new AC-DC composite magnetization and thermal data post-processing method is proposed and designed to scan and quantify defects for bearing rings. Specifically, a multi-physics coupling simulation model is built to investigate permeability and thermal distributions in the scanning process for detection principle revealing. In the proposed method, the bearing ring is placed in a yoke for DC magnetization, which changes the magnet field distribution and the permeability around the defect. And then, the induction coil is used to detect the permeability in the skin layer. Then, the proposed method can overcome the challenge of depth quantification.
The rest of this paper is structured as follows: Sect. 2 introduces the theoretical fundamentals of the proposed method for bearing rings. Section 3 investigates the permeability and thermal distribution caused by bearing ring defects by performing a multi-physics coupling simulation, and a scanning thermal data post-processing method is proposed to quantify

Theoretical Fundamentals
The schematic diagram of SIT for bearing rings under AC-DC composite magnetization is illustrated in Fig. 1. The DC magnetization coil generates a uniform magnetic field to magnetize the bearing ring. To enhance the magnetization intensity, a C-type yoke fitting the bearing ring shape is designed to form a magnetization loop. At the same time, the high-frequency alternating current passing the AC magnetization coil induces eddy currents and generates heat in the bearing ring. The bearing ring rotates at a constant speed, and the infrared camera captures the thermal response for defect detection and quantification. The placement of the AC coil in this way can form uniform heating in the axial direction and realize the scanning inspection with full coverage for the bearing ring.
According to the electromagnetic induction principle and Joule's law, the AC magnetization coil generates eddy current and heat in the bearing ring. The generation and diffusion of heat in the bearing ring can be described as Eqs. (1) and (2) [24]: where Q is the Joule heat. I coil and f denote the current and frequency in the AC magnetization coil. μ and σ are the permeability and electrical conductivity. ρ, C p and k represent the density, specific heat capacity, and thermal conductivity. From Eqs. (1) and (2), any defect, changing the μ and σ , will generate a different thermal response. This is the detection principle of induction thermography. However, the  detection capability is much diminished when the eddy current flows parallel to the crack, and it is challenging to quantify defect depth when the defect depth increases to a great value. In order to quantify the defect in the bearing ring, an AC-DC composite magnetization is proposed. Under DC magnetization, the defect will distort magnetic flux distribution, as shown in Fig. 2. Referring to [25], B-H and μ r -H curves for the bearing ring material GCr15 are plotted in Fig. 3. The magnetic flux in the defect region (R defect ) will leak into the air, resulting in a smaller magnetic flux density and greater magnetic permeability than that in the defect-free region (R defect-free ), i.e., H defect < H defect-free and μ defect > μ defect-free . According to Eqs. (1) and (2), the thermal response in the defect region is greater than that in the defect-free region, i.e., Q defect > Q defect-free . Thus, bearing ring defects could be quantified under AC-DC composite magnetization.

Simulation Studies
To investigate the detection principle of the proposed method, a multi-physics coupling simulation model for permeability and thermal distribution in the scanning process is built, involving DC-AC composite magnetization, nonlinear ferromagnetic material, heat generation, heat conduction, and scanning movement. A 3D finite element SIT model under AD-DC composite magnetization for bearing rings is built in COMSOL Multiphysics 5.5. The flow chart of the simulation is illustrated in Fig. 4. First, the simulation model is initialized, including defining simulation parameters, importing the geometry model, specifying material properties, and meshing. Next, permeability distribution is studied by solving DC magnetization stationary field, and then thermal distribution is studied by solving the induction heating transient field. The motion of the bearing ring is simulated by coordinate rotation transformation. Finally, the thermal data post-processing method is used to quantify the defects.

Simulation Configuration
The main simulation parameters are listed in Table 1.
The simulation geometry model is built and imported as Fig. 5. The bearing ring specimen to be studied is shown in Fig. 6. It is a part of the large-scale bearing suitable for heavy-duty transmission. It has four notches labeled N 1,1 , N 1,2 , N 1,3 , N 1,4 , with the depth varying from 0.5 to 2.0 mm with a 0.5 mm interval. These defects' length is 10.0 mm, and the width is 1.0 mm.
The materials of components are specified in Table 2.
Physics-controlled mesh automatically generates a mesh for the simulation. The maximum and minimum mesh sizes of the specimen in the ROI are set at 1.0 and 0.1 mm. In order to resolve the Joule heating, for the bearing ring under the AC coil, the boundary layer grid is divided. The number of boundary layers is 8, the stretching factor of the boundary is 1.2, and the thickness of the first layer is 0.01 mm. After meshing, this model consists of about 1.2 million elements, as shown in Fig. 7.

Permeability Distribution Study
Permeability distribution is studied using the 3D Magnetic Fields interface by solving DC magnetization stationary field. DC magnetization coil parameters are defined in Table 1. The magnetization model of the specimen is based on the B-H curve plotted in Fig. 3. The magnetic field distribution of the specimen at each moment is solved in the stationary study, and the permeability distribution is calculated by Eq.
(3) [26]. where B and H donate the magnetic flux density and the magnetic strength, and the relationship between permeability μ and relative permeability μ r is described as follows: where μ 0 is the vacuum permeability. Taking the bearing ring defect N 1,4 in the ROI as an example, the relative permeability distribution is illustrated in Fig. 8. For traditional SIT, i.e., I DC 0A, the relative permeability μ r of the bearing ring is constant. Under AC-DC   composite magnetization, increasing I DC to 5A, the μ r distribution of the defect region changed significantly. The μ r distributions caused by the defects with different depths are displayed in Fig. 9. Along the mid-section line, μ r is picked up and plotted, as shown in Fig. 10a. In this study, defect depth is quantified by peak value and valley value, and the linear fitting results are shown in Fig. 10b. The fitting performance evaluation is listed in Table 3. It is better to quantify the defect depth by peak value, with R 2 up to 98.6% and RMSE to 0.1778. There is a significant linear relationship between the μ r peak value and defect depths. The simulation results show that the defect causes permeability disturbance in the bearing ring surface with DC magnetization, and more importantly, the defect with increasing depth causes a greater μ r . Based on Eq. (1), the greater μ r will generate a higher thermal response, and this effect can be used for defect depth quantification.

Thermal Distribution Study in the Scanning Motion
In this part, thermal distribution is studied using the induction heating multiphysics interface by solving the induction heating transient field. AC magnetization coil parameters are defined in Table 1. The magnetization model of the specimen is based on the relative permeability imported from the permeability distribution study. The thermal field distribution of the specimen at each moment is solved in the frequency-transient field. The motion of the bearing ring specimen around the y-axis is simulated differentially. The sample interval Δt and the rotation speed are both listed in Table 1. The initial temperature is 20°C. The solved thermal distribution is transformed by coordinate rotation as the initial value of temperature at the next moment, which is formulated as Eqs. (5) and (6) [27].
where θ denotes the differential rotation angle, and T (x, y, z) is coordinate rotation transformed to T (x', y', z') as the initial temperature value of the next moment. Taking the bearing ring defect N 1,4 in the ROI as an example, the thermal distribution is illustrated in Fig. 11. It can be seen that the defect causes a non-uniform thermal response in the bearing ring surface. In addition, the thermal distribution is also influenced by bearing shape and rotation.

Scanning Thermal Data Post-Processing
The raw thermal data is set as T orig (x, y, t), where 0 ≤ x ≤ x max , 0 ≤ y ≤ y max , and 0 ≤ t ≤ t max . T orig can be considered as a three-dimensional (3D) matrix, where x and y represent horizontal and vertical pixels, and t denotes time. The axis of rotation is the y-axis. In order to fulfill scanning induction thermography for bearing rings, it is necessary to build an efficient scanning thermal data post-processing method. Common methods include background subtraction [21], data reconstruction [22], Fourier transforms [28] and Discrete Wavelet Transform (DWT) [29]. However, they are more suitable for static detection, performing poorly in scanning detection. To overcome the heating and conduction interference caused by bearing shape and rotation, this work Fig. 11 The thermal distribution of the bearing ring specimen  . 12 The schematic diagram of the thermal data post-processing process proposes a three-step thermal data post-processing method, as illustrated in Fig. 12. y a ) and (x b , y b ) are the two diagonal points that specify the bounding box of the ROI. In this paper, the ROI is selected at the region of the bearing ring under the AC magnetization coil. Let v denote the linear velocity of the bearing ring, and then v x ≈v 2πrn, where r denotes the radius of the bearing ring, and n represents the rotating speed. The bearing ring rotates from x x a to x x b through time (x b -x a )/v x . In order to evaluate the bearing ring temperature change in the ROI, T orig (x, y, t) is reconstructed into a new matrix T reco (x,y,t). Data reconstruction expression is described as Eq. (7) Fig. 13 The bearing ring thermal data when I DC 0A: a the raw thermal distribution, b the processed thermal distribution, c the raw thermal distribution along the mid-section line, and d the processed thermal distribution along the mid-section line

Data Normalization
Due to the edge effect, there is a difference in the temperature of the edge and the middle area after heating for bearing rings. It is necessary to normalize the thermal data to reduce the interference caused by background temperature. The reference background temperature is denoted as T bg T reco (x,y,t bg ), assuming no defect in the ROI at t t bg . Data normalization expression can be described as follows: where k t is a time-varying coefficient around 1, fine-tuning the fluctuation of the background temperature at different Fig. 18 The raw thermal data when I DC 0A: a the overall thermal distribution of S 1 , b the overall thermal distribution of S 2 , c mid-section line-scan plot in S 1 , and d mid-section line-scan plot in S 2 bearing angles. This work uses the interior point method [30] to calculate k t . The normalized data T norm has the same variable scope as T reco .

Data Dimension Reduction
To better visualize and quantify defects of the bearing ring, the 3D matrix T norm is reduced to a new 2D matrix T redu (x,y) by taking the average. Data dimension reduction expression can be described as Eq. (10) /v x and 0 ≤ y ≤ y b -y a . Further, time t can be transformed into horizontal distance x and bearing rotation angle θ as follows: )/r and 0 ≤ y ≤ y b -y a . The processed thermal data T is better for quantifying defects of the bearing ring. In the following parts, the raw data after dimensionality reduction in step 3 is compared with the processed data to show improvements. For traditional SIT, i.e., I DC 0A, the raw thermal distribution of the bearing ring defect region is shown in Fig. 13a. The mid-section line is taken to explore the relationship between temperature and defect depth, as shown in Fig. 13c. It can be seen that the thermal change caused by the defect is only reflected at the defect bottom, and the temperature decreases with increasing depth. After thermal data postprocessing, the processed thermal distribution is shown in Fig. 13b, and along the mid-section line, the thermal distribution is presented in Fig. 13d. Under AC-DC composite magnetization, increasing I DC to 5A, the raw thermal distribution of the bearing ring defect region is shown in Fig. 14a  Fig. 19 The processed thermal data when I DC 0A: a the overall thermal distribution of S 1 , b the overall thermal distribution of S 2 , c mid-section line-scan plot in S 1 , and d mid-section line-scan plot in S 2 and c. After thermal data post-processing, the processed thermal distribution is shown in Fig. 14b and d. It can be seen that the thermal change caused by defects is consistent with the permeability distribution, as shown in Fig. 9. It demonstrates that the permeability distortion caused by defects will lead to the thermal contrast in the defect region. The proposed data post-processing method can amplify the difference in thermal effect, which benefits defect depth quantification. Additionally, due to the asymmetry of heat conduction for two sides of the defect in the scanning process, the curves are asymmetric in Fig. 14c and d. From Figs. 13d and 14d, when I DC 0A defect depth could be quantified by the feature of valley value, and when I DC 5A the optimal feature is peak value, as presented in Fig. 15. The linear fitting performance evaluation is listed in Table 4, showing that the evaluation performance is better under DC magnetization, with R 2 up to 98.5% and RMSE to 0.0010. Thus, the proposed method is more effective for defect quantification than traditional SIT, especially with the application of the proposed thermal data post-processing method.

Experimental Validations
In this section, experimental studies are conducted to validate the feasibility of the proposed method, using different DC magnetization currents and different-depth defects.

Specimens
Two bearing ring specimens with artificial defects are tested to validate the proposed method, as pictured in Fig. 16. Several notches at the surface were manufactured by electrical discharge machining (EDM). Specimen details are listed in Table 5. The defect dimensions of S 2 share the same length and depth as S 1 , but the width is reduced from 1.0 to 0.5 mm to evaluate the performance on narrower defects. Figure 17 pictures the SIT system configuration for bearing rings. The fixed DC magnetization coil generates the static magnetic field, and then the magnetic field transmitted by the C-type yoke magnetizes the bearing ring specimen to a Fig. 20 The raw thermal data when I DC 5A: a the overall thermal distribution of S 1 , b the overall thermal distribution of S 2 , c mid-section line-scan plot in S 1 , and d mid-section line-scan plot in S 2 near saturation state. The bearing ring is heated by eddy currents induced by the AC magnetization coil. The diameter of the AC magnetization coil is 6 mm, and the lift-off distance from the bearing ring is 4 mm. The heating module is Ambrell EASYHEAT 0112, which keeps the switch-on state in testing. The motor drives the bearing ring rotating at the speed of 6 r/min. The infrared camera FLIR A655sc captures thermal data of the bearing ring in the scanning. The spatial resolution of the infrared camera is 640 × 120 pixels. To improve the emissivity of the bearing ring, a black PVC tape was pasted on the surface of the bearing ring before testing [31]. The main experiment parameters are listed in Table 6. Experiment parameters are mainly the same as the simulation listed in Table 1.

SIT Results without DC Magnetization
For I DC 0A (traditional SIT), the raw and processed thermal data for S 1 and S 2 are shown in Figs. 18 and 19, respectively. The x-axis coordinate corresponds to the bearing rotation angle θ . It can be seen that the defect is indicated by the low temperature at the defect bottom, and the bottom temperature decreases with the increase of defect width. It also can be demonstrated that the proposed thermal data post-processing method can effectively reduce the effect of inhomogeneous heating. However, the relationship between temperature valley value and defect depth is not apparent.

SIT Results Under AC-DC Composite Magnetization
For I DC 5A, the raw and processed thermal data are shown in Figs. 20 and 21, respectively. It can be seen that an apparent high-temperature zone appeared in the defect region under DC magnetization. According to the simulation results, it corresponds to the high magnetic permeability zone caused by defects. The disturbance due to uneven heating under AC-DC magnetization is more obvious in the experiment, but the data post-processing method can eliminate it. The peak value temperature at the defect region is positively correlated with the defect depth after magnetization. In addition, the thermal Fig. 21 The processed thermal data when I DC 5A: a the overall thermal distribution of S 1 , b the overall thermal distribution of S 2 , c mid-section line-scan plot in S 1 , and d mid-section line-scan plot in S 2

Fig. 22
The processed thermal data linear fitting curve with error bar: a S 1 and b S 2 contrast of S 1 is greater than that of S 2 due to the greater permeability disturbance effect by the wider defect.

Defect Depth Quantification
Defect length and width can be directly visualized in the thermal distribution, and defect depth can be quantified by linear fitting. Similarly to Fig. 15, for I DC 0A defect depth is quantified by valley value, and with DC magnetization the depth is quantified by peak value. The fitting performance evaluation for S 1 and S 2 are displayed in Fig. 22 and Table 7. It can be seen that SIT under AC-DC composite magnetization is more effective than traditional SIT for defect quantification, especially with the proposed thermal data post-processing method. The valley temperature of defects can be easily disturbed in the scanning experiment, and the peak value is a better feature to quantify defect depth. When I DC 5A, linear regression R 2 and RMSE for S 1 and S 2 have the best performance. The reason is that when the I DC increases to 5A, the specimen is approaching the saturation situation. After that, the DC magnetization current increase has little contribution to the permeability distribution but causes an unstable movement. By comparing Fig. 22a and b, it can be seen that the narrower defect is more difficult to quantify. Hence, the width should be taken into consideration in the depth quantification.

Conclusions
IN this study, a new SIT under AC-DC composite magnetization method is proposed to scan and quantify defects for bearing rings. A multi-physics coupling simulation model for permeability and thermal distribution in the scanning process is built to reveal the multi-physics coupling mechanism. A thermal data post-processing method is proposed to overcome the disturbance of in-homogeneous heating caused by bearing ring shape and motion. The SIT system for bearing rings is designed and developed. The experiment shows that the proposed method can quantify bearing ring cracks with depths up to 2.0 mm. The depths of defects are effectively quantified by linear regression with R 2 up to 96.9% and RMSE to 0.0017. The future work will focus on natural defects detection and evaluation.
Author Contributions ZX performed the main study and wrote the main manuscript text. QJ helped conduct the experiment and simulation, programming for thermal data post-processing analysis. YZ developed the detection system. JW provided the method concept, acquired funding, and supervised the research. LL, FQ and ZW provided research resources, validated the method, and gave constructive advice. All authors reviewed the manuscript.
Funding This work is supported by the National Natural Science Foundation of China under Grant 92060114 and the Sichuan Science and Technology Program (No. 2021YFG0039, 2022YFS0524, and 2022YFG0044).

Data Availability
The data that support the findings of this study are available from the corresponding author, Jianbo Wu, upon reasonable request.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.