In order to further observe the magnetization characteristics of the surface, simulation models of the inner and outer bearing rings are established by COMSOL Multiphysics 5.4 finite element simulation software. As shown in Figure 2, the yoke is set to 60 mm × 100 mm × 100 mm, and the distance from the bottom of the yoke to the bearing ring is 5 mm. The surface diameter of the outer bearing ring is 200 mm, and the axial width is 65 mm. The wall thickness of the small end is 5 mm, and the cone angle of the conical surface is 25. The diameter of the inner surface of the bearing inner ring is 50 mm, ignoring the shoulders of the fixed rollers. The thickness of the small end is 5 mm, and the conical angle is 25. In order to ensure a distance of 5 mm between the bottom of the yoke and the outer surface of the bearing ring, the bottom of the yoke is set as a conical surface. The coil is set built-in material copper, magnetic yoke and bearing ring material is set soft iron (without losses), the rest is set air. When the coil current is set 10 A and the number of coil turns is 500 turns, the inner and outer rings of the bearing can be saturated and magnetized. For convenient observation, the conical surfaces of the outer and inner rings are expanded into a plane, and the cloud diagram of magnetic field distribution is shown in Figure 3.

It can be seen that the magnetization distribution of the inner and outer bearing rings is non-uniform. The magnetic flux density in the middle part is greater than that at the upper and lower ends, and the magnetic flux density along the centerline on the left side is smaller than that on the right side.

In order to make the study more rigorous and representative, it is necessary to investigate the tapered roller bearing rings with different sizes, establish the outer ring simulation models of four types in Table 1 and the inner ring simulation models of four types in Table 2, keeping the yoke, conical angle and magnetizing current unchanged.

In order to further observe the magnetization characteristics of the near-surface of the bearing ring, the near-surface magnetic induction at the centerline is extracted along scanning path one in Figure 3. The internal magnetic flux density tangent component Bx is obtained from .1 mm to the surface, as shown in Figure 4.

It can be seen that the inner and outer rings have the same magnetization law for the four model sizes. The surface curvature radius along scanning path one continuously increases, and the corresponding magnetic flux density decreases. Therefore, the circumferential magnetization of the tapered roller bearing ring has significant non-uniform magnetization at different curvature radius. A smaller curvature radius indicates a greater magnetic induction intensity.

In order to further explore the influence of the surface curvature radius on the axial crack signal, the subsequent analysis is conducted on the inner and outer rings of tapered roller bearings with an axial width of 65 mm. Based on the simulation model in Figure 2, axial cracks with a width and a depth of .2 mm are set on the centerline of the inner and outer bearing rings, respectively. The MFL signal of the outer bearing ring is extracted along scanning path two at Locations 1, 2, and 3. Moreover, the tangent component Bx of MFL signal of the inner bearing ring is extracted along scanning path two at Locations 4, 5, and 6. The lift-off is .1 mm, and the results are shown in Figure 5.

It can be seen that for cracks of the same size, there are significant differences in the peak and baseline MFL signals at different positions. The outer bearing ring has the smallest radius of curvature at position 1, and the peak of the MFL signal of the crack is the largest. The inner bearing ring has the smallest radius of curvature at position 5, and the peak value of the MFL signal of the crack is the largest.

Afterward, the crack width is kept at 2 mm, and axial cracks with a crack depth of 3 mm, 4 mm, and 5 mm are added. The peak-valley values of MFL signals at three positions of four sizes of cracks are extracted, as shown in Figure 6.

It can be seen that the MFL signals are inconsistent for the four depths of cracks at different positions. A smaller radius of curvature indicates greater peak and valley values in MFL signals of cracks obtained by scanning.

Based on the above research, the crack MFL signals obtained by the array sensor scan from different curvature radius are different in the axial crack detection of bearing rings. A smaller curvature radius indicates a larger crack MFL signal. The actual contour size of the defect cannot be accurately inverted through the detection signal, affecting the accuracy of crack detection and subsequent quantitative research.

In this study, the relative change of signal amplitude is used as the index to evaluate the signal consistency: ∆ B = B max − B min B max × 100 % (1) where B _max is the maximum value of signal amplitude; B _min is the minimum value of signal amplitude.

A larger amplitude value suggests a worse signal consistency. Based on Eq. 1, the relative change of the signal amplitude of the outer bearing ring with the crack MFL signal of .2 mm is ∆B = 30.43%, and the relative change of signal amplitude of the inner ring is ∆B = 35.03%.

According to the above analysis, the key to ensuring the consistency of the detection signal is to establish a uniform magnetic field near the surface of the bearing ring. In Figure 2, the magnetization part of the circumferential DC coil of the bearing ring consists of a DC excitation coil and yoke. Considering the symmetry of the magnetic circuit and structure, half of the magnetizer structure is taken for analysis. Its corresponding equivalent magnetic circuit model can be obtained, as shown in Figure 7.

Taking a closed magnetic line in the magnetic circuit composed of the yoke and bearing ring, the following equation can be obtained: ∮ H → ∙ d l → = ∑ H i ∙ L i = N I = ε M (2) where N is coil turns; I is coil current; ε _M is magnet-motive force; R _y , R _g , R _b represents the magnetic resistance of the yoke, air, and bearing ring, respectively. Afterward, the following equation is obtained: ε M = N I = Φ R y + R g + R b (3)

During the magnetization of the bearing ring, the magnetic line of force follows the direction of the annular tube wall. In order to analyze the distribution characteristics of the magnetic field inside the bearing ring, the cross-section of the outer bearing ring is subdivided into n equal parts with an area of ∆S, and the corresponding magnetic line loop radius is r, as shown in Figure 8. Therefore, the total magnetoresistance of the bearing ring can be divided into n parallel magnetoresistance.

The corresponding magnetoresistance of each branch can be calculated as: R i = l i μ ∆ S = π r i μ ∆ S (4) where l is the equivalent length of the magnetic circuit; μ is the relative permeability of bearing ring; r is the radius of curvature per unit area.

It can be seen from Eq. 4 that the branch magnetoresistance R is proportional to the corresponding radius of curvature r. A larger radius indicates a smaller magnetoresistance. Taking the unit area S _i and S _j at different positions on the inner surface of the outer ring of the bearing, the corresponding radius of curvature is r _i and r _j , then: r i > r j (5)

According to Eq. 4, the following relation can be obtained: R i > R j (6)

Since the total magnetic flux through the bearing ring is constant, it can be obtained by combining the parallel Ohm’s law of the magnetic circuit: ϕ i < ϕ j (7)

It can be further concluded from B = ϕ/S: B i < B j (8)

There are differences in the size of the magnetic field on the inner surface of the outer bearing ring. A larger radius of curvature suggests a smaller magnetic induction intensity, consistent with the simulation results in the previous section.

Based on the same analysis method, there are also curvature differences on the conical surface of the outer wall of the bearing inner ring, and its radius r _i and r _j also meet Eq. 5. Therefore, there is a difference in the size of the magnetic field on the inner surface of the bearing ring. A larger radius of curvature indicates a smaller magnetic induction intensity, consistent with the simulation results in the previous section.

Based on the above analysis, it can be found that the uneven magnetization of the conical surface of the bearing ring is caused by the different radius of curvature. Therefore, a compensation method based on the normalization of surface magnetization is proposed to achieve a consistent MFL signal for the same crack.

Because the smaller curvature radius of the bearing ring indicates the greater magnetic induction intensity, the minimum curvature radius of the conical surface of the inner and outer rings of the bearing is selected as the reference point for signal compensation. In this way, a greater magnetic induction intensity can be obtained.

As shown in Figure 9, the difference in curvature radius from each position of the conical surface of the inner and outer bearing rings to the reference point can be expressed as: R d i f = x ∙ tan ⁡ θ (9) where x is the distance between the compensation position and reference point; r is the radius of curvature of the datum end face; θ is the conical angle.

The compensation coefficient based on the radius of curvature is set as K. Since the magnetization at the crack is linearly corresponding to the amplitude of the MFL signal, the consistency of the magnetization at all parts of the bearing ring through the compensation coefficient is equivalent to the consistency of the MFL signal of the crack, so there are: K B i = B (10) where B _i is compensated magnetic flux density; B is magnetic flux density at the reference.

By performing bisection on the cross-section in front, the following equation can be obtained: K = R i R (11)

By applying Eq. 4 and 9 to the above equation, the following equation can be obtained: K = 1 + x ⁡ tan ⁡ θ r (12)

Since the difference in curvature radius can be expressed by Eq 9 and 12 can be used as the compensation coefficient of the inner and outer bearing rings. The cone angle θ and the radius of curvature R of the reference end face are determined by the model of the bearing ring. Therefore, for the single bearing ring, only the distance x from the sensor to the reference end face needs to be identified for calculating the compensation signal.

In order to verify the effectiveness of the compensation method, the signal amplitude after compensation is calculated by using the amplitude of the crack MFL signal of the outer bearing ring. Furthermore, the compensation coefficients at position 1, position 2, and position three are calculated as 1.146, 1.235, and 1.326, respectively. The compensated crack MFL signal is plotted by the .2 mm crack MFL signal, as shown in Figure 10A. The peak and valley values of the four crack MFL signals after compensation are shown in Figure 10B.

It can be seen that the MFL signals of the same crack at different positions are basically the same after signal compensation, and the baseline drift is also improved. According to Figure 6A and Figure 10B, the MFL signals of defects in different sizes at different positions are separated after signal compensation, and the contour size of defects can be inverted through the MFL signals.

Taking the .2 mm crack as an example, the relative change of the signal amplitude of the inner wall crack of the outer ring of the bearing is ∆B = .65%. As a result, the consistency of the MFL signal of the bearing ring crack is improved, and the validity of the signal compensation method based on the curvature radius is validated.