BQ-05 Localization of multiple cracks in a cylindrical ferromagnetic rod based on the Fourier coefficients of the radial leakage magnetic flux
Creators: Kanta Shiku
and Takaaki Nara
Publisher: name: IEEE
Year: 2025

Magnetic flux leakage testing (MFLT) is necessary before induction hardening of cylindrical ferromagnetic rods because cracks would extend in the process. Although conventional MFLT uses an array of magnetic sensors to measure the leakage magnetic flux (LMF) [1], such arrays have difficulty in calibration. In contrast, we proposed a crack localization method using only two coils (“Fourier coils”) [2], which measure the first-order Fourier coefficients. However, estimating multiple crack circumferential positions was impossible if they were at the same axial positions. This study proposes a method to estimate multiple crack circumferential positions in a rod based on measuring the circumferential multiple-order Fourier coefficients of the radial LMF. As shown in Fig. 1, for the case when two cracks are located at the circumferential positions φ1 and φ2 with the same z-axis coordinates, we derived algebraic equations cm = ∑2k=1pke-imφk, m = 0,1,2 where cm and pk are, respectively, the m th-order Fourier coefficient and a magnetic dipole moment relevant with the crack size and shape. This form of equations are called a moment problem [3], and φk with pk are algebraically obtained from measurements of cm. Theoretically, this formulation can easily be extended to any number of cracks. Since c1 can be measured using the coils in [2], we developed sensors that directly measure the zeroth (c0) and the second order (c2) Fourier coefficients of the radial LMF in this research. The magnified photo in Fig. 1 shows the proposed sensor that measures Re c2, the second-order Fourier cosine coefficient, by winding the coil in the repeated figure-of-eight form, whose width is 2w|cos 2φ| at circumferential position φ, where 2w is the maximum width. Im c2 can be measured using the same coil rotated with 90 degrees. c0 is measured by the coil whose width is 2w at any φ. Fig. 2 shows the estimation results of the two crack locations using the proposed sensors. The maximum error was 22.5ο, showing that the proposed method can estimate two crack positions.References: [1] Z. Zhou and Z. Liu, IEEE Transactions on Industrial Electronics., vol. 68, pp. 2543–2553 (2021) [2] K. Shiku, T. Nara, and Y. Gotoh, IEEE Transactions on Magnetics., vol. 59, pp. 1–5 (2023). [3] T. Nara and S. Ando, Journal of Inverse and Ill-posed Problems., vol. 15, pp. 403–418 (2007)
Magnetic flux leakage testing (MFLT) is necessary before induction hardening of cylindrical ferromagnetic rods because cracks would extend in the process. Although conventional MFLT uses an array of magnetic sensors to measure the leakage magnetic flux (LMF) [1], such arrays have difficulty in calibration. In contrast, we proposed a crack localization method using only two coils (“Fourier coils”) [2], which measure the first-order Fourier coefficients. However, estimating multiple crack circumferential positions was impossible if they were at the same axial positions. This study proposes a method to estimate multiple crack circumferential positions in a rod based on measuring the circumferential multiple-order Fourier coefficients of the radial LMF. As shown in Fig. 1, for the case when two cracks are located at the circumferential positions φ1 and φ2 with the same z-axis coordinates, we derived algebraic equations cm = ∑2k=1pke-imφk, m = 0,1,2 where cm and pk are, respectively, the m th-order Fourier coefficient and a magnetic dipole moment relevant with the crack size and shape. This form of equations are called a moment problem [3], and φk with pk are algebraically obtained from measurements of cm. Theoretically, this formulation can easily be extended to any number of cracks. Since c1 can be measured using the coils in [2], we developed sensors that directly measure the zeroth (c0) and the second order (c2) Fourier coefficients of the radial LMF in this research. The magnified photo in Fig. 1 shows the proposed sensor that measures Re c2, the second-order Fourier cosine coefficient, by winding the coil in the repeated figure-of-eight form, whose width is 2w|cos 2φ| at circumferential position φ, where 2w is the maximum width. Im c2 can be measured using the same coil rotated with 90 degrees. c0 is measured by the coil whose width is 2w at any φ. Fig. 2 shows the estimation results of the two crack locations using the proposed sensors. The maximum error was 22.5ο, showing that the proposed method can estimate two crack positions.References: [1] Z. Zhou and Z. Liu, IEEE Transactions on Industrial Electronics., vol. 68, pp. 2543–2553 (2021) [2] K. Shiku, T. Nara, and Y. Gotoh, IEEE Transactions on Magnetics., vol. 59, pp. 1–5 (2023). [3] T. Nara and S. Ando, Journal of Inverse and Ill-posed Problems., vol. 15, pp. 403–418 (2007)