Localization of a Crack in Moving Cylindrical Ferromagnetic Rods by Measuring the Fourier Coefficients of the Leakage Magnetic Flux

This article presents a novel leakage magnetic flux (LMF) method to estimate crack positions in moving cylindrical ferromagnetic rods. The crack positions are estimated by the circumferential first-order Fourier coefficients of the LMF by utilizing the dc excitation and modeling the moving crack as a magnetic dipole. The Fourier coefficients are directly measured by using only two coils. In addition, numerical experiments were conducted to evaluate the effect of motion-induced eddy current (MIEC). The demonstration of locating hole-shaped cracks in moving S45C cylindrical rods was performed by using the proposed method.


I. INTRODUCTION
F ERROMAGNETIC rods used in various mechanical parts, such as coil springs, are often processed by induction hardening to increase their hardness and friction resistance.Before the hardening, nondestructive testing of the rods is necessary, because cracks in the rods, if they existed, would extend in the process.For manufacturing efficiency, localizing cracks in moving rods is crucial.For nondestructive testings, magnetic flux leakage testing (MFLT) [1], eddy current testing [2], and ultrasonic testing [3] have been used.MFLT is a noncontact method that detects cracks through measurement of the leakage magnetic flux (LMF) caused by cracks.
Conventionally, an array of magnetic sensors is used to measure the LMF to estimate crack positions [1].However, such arrays have resolution limitations due to the number of sensor elements and difficulty calibrating many sensor elements.On the other hand, a method using only two coils ("axial Fourier coils") was proposed in [4].This method enables easy calibration and provides high-resolution estimation of crack axial and circumferential positions.However, the coil liftoff was significant due to the measurement of the Fourier coefficients of the axial LMF, resulting in reduced sensitivity.In contrast, we proposed using the "radial Fourier coils" and the ac excitation [5].This method can estimate high-resolution crack positions using only two coils with higher sensitivity than the previous coils in [4] by measuring the Fourier coefficients of the radial LMF with a smaller liftoff.However, ac excitation was utilized in both methods [4], [5]; the methods are unsuitable for inspecting moving specimens.Various nondestructive testing methods to inspect moving ferromagnetic materials have been developed so far.Lei et al. [6] proposed a probe consisting of permanent magnets and Hall sensors for a moving plate.They used a radial basis function neural network to reduce the effect of motion-induced eddy current (MIEC) and obtain the static LMF signals.Lu et al. [7] proposed a method that utilizes permanent magnets with an array of Hall sensors and eddy current sensors to inspect a moving pipe.Tohara and Gotoh [8] proposed the thinning detection method for a moving pipe employing a sensor consisting of dc excitation and detection coils.
However, a method for estimating the crack position, especially the crack circumferential position in moving rods, has not been proposed.For inspection efficiency and manufacturing purposes, locations of cracks in the rod circumferential positions provide helpful information in addition to their axial positions.This study aims to estimate the crack positions in moving cylindrical ferromagnetic rods with only two coils.Using the dc excitation, the crack positions are estimated based on the circumferential first-order Fourier coefficients of the LMF by modeling the moving crack as a magnetic dipole.We developed a sensor unit consisting of a permanent magnet and the radial Fourier coils with which the circumferential position of a crack in a moving ferromagnetic rod is identified.
The rest of this article is organized as follows.Section II describes the method to locate a crack in moving ferromagnetic rods using the Fourier coefficients of the radial LMF.In Section III, the distributions of MIEC and magnetic flux density and the effect of MIEC on crack signals are numerically evaluated.In Section IV, the estimations of the locations of hole-shaped cracks using the proposed method are demonstrated for S45C cylindrical rods.Section V concludes this article.

II. METHOD A. Setting
As shown in Fig. 1, for a rod with a radius of ρ 0 , we set the z-axis along the axis of the rod.Let v = ve z be the velocity of the rod and r 0 (t) = ρ 0 cos ϕ 0 e x + ρ 0 sin ϕ 0 e y + (vt + z 0 )e z be the center of a crack, where z 0 is the initial z position of the crack.We assumed that the LMF due to the crack is considered to be generated by a magnetic dipole p = p z e z at the crack center position on the rod surface, as in [4].Subsequently, the radial component of the leakage field on a circle with a radius of ρ s (> ρ 0 ) at z = 0 can be modeled as follows: where r s = ρ s cos ϕe x + ρ s sin ϕe y ∈ and µ 0 is the permeability of vacuum.In this article, we consider the inverse problem of estimating the circumferential position of the crack ϕ 0 , by measuring the radial LMF B r (r s , t, ϕ 0 ) on .

B. Estimation of ϕ 0
Under this setting, we showed that the crack circumferential position ϕ 0 can be estimated by using the circumferential first-order Fourier coefficients of B r (r s , t, ϕ 0 ) as follows: where This formula is derived by applying the lemma determining ϕ 0 by the argument of the first-order, complex Fourier coefficient of f (cos(ϕ − ϕ 0 )) in [5] to B r (r s , t, ϕ 0 ).Our previous study [5] showed a similar result for a fixed specimen to which an ac magnetic field was applied.The main theoretical result of this article is that we showed that (2) also holds even for a crack on moving rods.It is remarkable under the moving condition that, although B r depends on time at the right side of (2), ϕ 0 can be estimated by calculating the right-hand side for arbitrary time t.In this article, we estimate ϕ 0 at the time when the sum of the squares of the Fourier cosine and sine coefficients takes a peak value, which will be shown in Section IV-B.
From (2), it was shown that ϕ 0 can be estimated using only the circumferential first-order Fourier cosine and sine coefficients of the radial LMF.Thus, in Sections III and IV, we used only two coils to measure the radial Fourier coefficients.The outputs of those two coils, which we call the Fourier cosine and sine coils, are given by where N and 2w are the number of turns and the maximum width of the Fourier coils.Supposing that 2π 0 B r (r s , 0, ϕ 0 ) cos ϕdϕ = 2π 0 B r (r s , 0, ϕ 0 ) sin ϕdϕ = 0, the flux penetrating the coils are given by its time integral which are the circumferential first-order Fourier coefficients of the radial magnetic flux density.Then, the crack circumferential position can be estimated by ψ c and ψ s as

III. NUMERICAL EVALUATION OF THE EFFECT OF MIEC
A. Setting MIEC is generated in the specimen when it moves relative to an excitation device and affects crack signals.Mandayam et al. [9] found that crack signals are distorted by MIEC when the specimen's relative velocity exceeds 0.9 m/s.Lei et al. [6] and Lu et al. [7] found that MIEC affects crack signals when the relative velocity exceeds 1.0 m/s.To evaluate the effect of the MIEC in our setup, in this section, we performed simulations using a finite-element method software, COMSOL Multiphysics. 1 A cylindrical rod with a diameter of 9.0 mm containing a hole-shaped crack with a diameter of 2.0 mm and a depth of d c = 2.0 mm, respectively, as shown in Fig. 2(a), was used as a specimen.It was placed such that the crack center was at ϕ 0 = 0 • and moved to the axial direction with a velocity of 0.6 m/s, which is a typical speed of practical operation.Assuming that the specimen was made of S45C, the conductivity was 4.95 × 10 6 S/m.The magnetic property was set, as shown in Fig. 2(b).A ring-shaped permanent magnet (length: 38.1 mm, outer diameter: 38.0 mm, inner diameter: 17.0 mm, and residual magnetic flux density: 1.28 T) was used as an excitation device.The radial Fourier coils for which the number of turns is 20, 2w = 6.0 mm, and ρ s = 5.5 mm were placed at the magnet's center.Fig. 3(a) shows half of the domain of the analysis model geometry.In this setup, simulations considering and ignoring MIEC were conducted.

B. Results
Fig. 3(b) shows the distribution of the norm of MIEC in the rod.We observe that MIEC is significant near the magnetic poles, while it is relatively small at the magnet's center.This MIEC distribution can be generated, because the spatial gradient of the magnetic flux density is larger around the magnetic poles than that at the magnet's center.Fig. 4 shows the distribution of the magnetic flux density norm with and without MIEC.The magnetic flux density around the poles spreads in the direction of motion when considering MIEC, whereas that with MIEC and without MIEC around the magnet's center, far from the poles, looks similar.Fig. 5 compares the magnetic flux penetrating the Fourier cosine coil placed at the magnet's center with and without MIEC.Although we observe that the amplitude with MIEC is slightly larger than that without MIEC, the effect of MIEC is not significant under the slow-moving speed in our setting.From these results, we set the Fourier coils at the magnet's center to detect the LMF without not being much affected by MIEC.

A. Setting
Fig. 6 shows the illustrations of the radial Fourier coils.We developed the radial Fourier coils for which the number of turns is 20, 2w = 6.0 mm, and ρ s = 5.5 mm, as shown in   Fig. 7(a).The sine coil was wound on the cosine coil.Note that they were two disconnected coils.The jig for winding the Fourier coils was developed using an FFF 3-D printer (FLASHFORGE, Creator Pro2).A ring-shaped permanent magnet (length: 38.1 mm, outer diameter: 38.0 mm, inner diameter: 17.0 mm, and residual magnetic flux density: 1.28 T) was used as an excitation device.In the measurement, the Fourier coils were inserted in the direction indicated by the green arrow in Fig. 7(a) and placed at the magnet's center.Fig. 7(b) shows the measurement setup: a specimen was penetrated through the coils and moved in the axial direction with v = 0.6 m/s by a driving stage.As specimens, on the surface of S45C rods with a diameter of 9.0 mm, hole-shaped cracks with a diameter of 2.0 mm and a depth of d c = 2.0 or 0.5 mm, respectively, were created, as shown in Fig. 2(a).The cracks were created every 30 • in the circumferential and 40 mm in the axial directions; namely, 12 cracks were created at (ϕ 0 , z 0 ) = (30 • n, 40 n mm), (n = 0, 1, . . ., 11), in the rods.An axial scan can inspect all cracks.We measured the outputs nine times for each crack position.Low-pass and high-pass filters with 120-and 30-Hz cutoff frequencies, respectively, were used to reduce high-frequency noise and offset signals.

B. Results
Figs. 8 and 9 show the time dependence of ψ c and ψ s when d c = 2.0 and 0.5 mm, respectively.The obtained signals are explained as follows: in Fig. 8, when the crack at ϕ 0 passes by the coil, ψ c is larger than ψ s , because the width of the Fourier cosine coil at ϕ = ϕ 0 is maximum, whereas that of the Fourier sine coil is zero.When ϕ 0 varies from 0 • to 90 • , Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and ψ s (T ) exhibit approximately sinusoidal dependence on ϕ 0 .ϕ 0 was estimated as ϕ 0 = arg(ψ c (T ) + iψ s (T )) from ψ c (T ) and ψ s (T ).Fig. 12 shows the mean and standard deviation of the estimation errors.For d c = 2.0 mm, the maximum mean error was 14.8 • when the true value was 240 • , and the maximum standard deviation was 0.6 • when the true value was 90 • .For d c = 0.5 mm, the maximum mean error was 14.1 • when the true value was 240 • , and the maximum standard deviation was 2.5 • when the true value was 300 • .The results show that the circumferential position of the cracks can be estimated stably even when the rods are in motion.When d c = 0.5 mm, although the outliers were observed at 0 • and 180 • , the estimation error was small, because ψ s (T ) is approximately zero at these angles.

C. Discussion
The mean estimation error of 14.1 • for the crack with a depth of 0.5 mm obtained from this experiment provides sufficient information to expedite visual inspections performed by stopping the motion of the rods.The causes of the errors are discussed below to improve the precision.
As shown in Fig. 11, regarding ψ c (T ), we observed that the following hold: 1) a negative bias is included and 2) the amplitude of ψ c (T ) is larger than that of ψ s (T ).To examine the effects of the errors of ψ c (T ) in 1) and 2) on the errors of the estimation of ϕ 0 using (2), we conducted the following.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The estimation error is expressed as the sum of the errors by 1) and 2), which well explains the estimation errors in Fig. 12(a) and (b).Thus, it is said that the bias of a coil and the amplitude difference of the two coils are dominant causes of the estimation errors.Shifting the center of the rod, Fourier coils, and permanent magnet can cause a negative bias.The larger amplitude of ψ c (T ) is caused by the smaller liftoff of the cosine coil than that of the sine coil.
Improving the precision of centering the rod, Fourier coils, and permanent magnet and reducing the difference in the liftoffs of the cosine and sine coils are important aspects for better accuracy of the estimations.A supporting mechanism for the rod is needed to suppress the rod's bending for centering precision.Winding the coils at different positions along the axis can reduce the difference in the liftoffs.
The other significant aspects of further studies include detection of shallower cracks.For that purpose, increasing the number of turns for enhancement of sensitivity is necessary.Taking the differential of the outputs of the two same-type Fourier coils (cosine or sine) by setting them axially is the other option to suppress the high-frequency noise and obtain high SN ratio.

V. CONCLUSION
In this study, we demonstrated that the circumferential positions of cracks can be estimated by the circumferential first-order Fourier coefficients of the LMF moving with the cracks when applying the dc excitation.The numerical simulations showed that the MIEC distributes dominantly near the magnetic poles, far from the magnet's center; thus, the MIEC does not much influence the crack signals measured at the magnet's center.We developed a sensor unit consisting of a ring-shaped permanent magnet and the radial Fourier coils.Through experimental verification for S45C cylindrical rods with a velocity of 0.6 m/s and a diameter of 9.0 mm, we found that the positions of hole-shaped cracks with the depths of 2.0 and 0.5 mm can be estimated.The localization formula is derived by modeling the LMF due to a crack generated by a magnetic dipole at the crack center position.Thus, theoretically, the crack position is estimated at the crack center position regardless of the crack shape.Also, the crack position can be estimated continuously because of the continuous localization formula and measurement.However, experimental verifications of these aspects are future work.Detection of smaller cracks and verification of detection accuracy for various velocities are other areas of further studies.
Fig. 10.Dependence of ψ c , ψ s , and ψ on time when ϕ 0 = 0 • and d c = 2.0 mm, and definition of T .

Fig. 13 .
Fig. 13.Simulated evaluation of influences of the bias and amplitude of ψ c (T ) on the estimation errors.(a) Influence of the negative bias.(b) Influence of the larger amplitude of ψ c (T ) than that of ψ s (T ).

1 )
We simulated estimation errors when ψ c includes a negative bias by arg((a cos ϕ 0 − b) + ia sin ϕ 0 ) − ϕ 0 where a = 7.9 and b = 1.2, which are determined from Fig.11(a).As shown in Fig.13(a), the error is positive if ϕ 0 is at the first or second quadrants and negative at the third or fourth quadrants.2)We simulated estimation errors when the amplitude of ψ c is larger than that of ψ s by arg(a ′ cos ϕ 0 +ia sin ϕ 0 )− ϕ 0 where a ′ = 9.1 and a = 7.9, which are determined from Fig.11(a).As shown in Fig.13(b), the error is negative if ϕ 0 is at the first or third quadrants and positive at the second or fourth quadrants.