# Inspection of Cracks in the Piston Rod of a Hydraulic Cylinder Using Injected Alternating Current-Field Measurement

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## Abstract

**:**

## 1. Introduction

## 2. Theories of IAC-FM

#### 2.1. Theoretical Model

**J**has only z component J

_{z}. According to Maxwell’s equations and Ohm’s law, the magnetic vector potential

**A**in the conductor conforms to the following equation [17]

**A**and current density

**J**satisfy:

**J**= σ

**E**= −jωσ

**A**, where

**E**is the electric field. Thus,

**A**only has only z component Az. Then, Equation (2) can be simplified as:

_{z}can be expressed as

_{0}(.) is the zero order Bessel function of the first kind and Y

_{0}(.) is the zero order Bessel function of the second kind. According to the property of Bessel functions, Y

_{0}(x) goes to infinity when x approaches 0. To guarantee the convergence of A

_{z}at r = 0, C

_{2}must be equal to 0. Then, Equation (4) can be written as:

_{R}, then the boundary condition is ${J}_{z}{\left.\right|}_{r=R}={J}_{R}$, and J

_{z}can be expressed as

_{R}can be obtained by integrating the Equation (6):

**B**satisfies $B=\nabla \times A$, thus

**B**has only φ component B

_{φ}, which can be expressed as

#### 2.2. Distribution of Eddy Current and Magnetic Field

^{7}S/m. The frequency of alternating current was set to 10 kHz, 50 kHz and 100 kHz, and the total current was 1 A. The variations of J

_{z}and B

_{φ}with the radius are shown in Figure 2. It can be concluded from Figure 1 and Figure 2 that the magnetic field and current are both concentrated on the surface of the rod, which is beneficial for the inspection of surface cracks. In addition, the magnetic field and current are in orthogonal directions, making the detection of both longitudinal and transverse cracks possible.

## 3. Finite Element Simulation

_{0}) into the steel bar, an alternating magnetic field B

_{0}will be produced around the workpiece in the air. In the area without cracks, the current density is uniform. When the current flows around the vicinity of the transverse crack, it will not only be disturbed but also change the exterior magnetic field, with Δj and ΔB produced, which can be obtained by coils scanning axially over the crack.

_{t}) and normal magnetic field (B

_{n}) in three paths is shown in Figure 8 and Figure 9. In the region away from the crack, the current distributed uniformly, so the extracted magnetic field was a constant value. In the vicinity of the crack, the currents passed around the crack from its ends and bottom and caused perturbation to the magnetic field above the crack. In the center of the crack, the currents mainly bypassed the crack from the bottom, thus were still in the axial direction. Accordingly, there was no magnetic field perturbation for Path No. 2.

## 4. Experiments and Discussion

_{n}and B

_{t}in the experiment, two coils were used. One coil had its axis along the normal direction to obtain B

_{n}and the other had its axis along the tangential direction to obtain Bt. Similar to the simulation condition, three scan paths were set as comparisons. Path B was above the center of the crack. Path A and Path C were, on the other hand, both 2.5 mm away from Path B. The lift-off values of all paths were the same as 0.4 mm. The results are shown in Figure 13 and Figure 14. From the experiment results and simulation results, the signal features agree with each other. Through these features of the magnetic field, cracks can be quantitatively evaluated.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Distribution of eddy current and magnetic flux density along radius in the analytical model: (

**a**) variation of current density with radius; (

**b**) variation of magnetic flux density with radius.

**Figure 3.**Distribution of eddy current and magnetic flux density along radius in the finite element model: (

**a**) variation of current density with radius; (

**b**) variation of magnetic flux density with radius.

**Figure 8.**Simulation signal of B

_{n}in three paths: (

**a**) path No. 1; (

**b**) path No. 2; (

**c**) path No. 3.

**Figure 9.**Simulation signal of B

_{t}in three paths: (

**a**) path No. 1; (

**b**) path No. 2; (

**c**) path No. 3.

**Figure 11.**Testing signals of longitudinal and transverse cracks: (

**a**) longitudinal crack; (

**b**) transverse crack.

**Figure 12.**Change of testing signal amplitude with frequencies: (

**a**) longitudinal crack; (

**b**) transverse crack.

**Figure 13.**Experimental testing signal of B

_{n}for transverse cracks at different paths: (

**a**) path A; (

**b**) path B; (

**c**) path C.

**Figure 14.**Experimental testing signal of B

_{t}for transverse cracks at different paths: (

**a**) path A; (

**b**) path B; (

**c**) path C.

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**MDPI and ACS Style**

Zhang, J.; Huang, Y.; Tang, J.; Zhou, F.; Kang, Y.; Feng, B. Inspection of Cracks in the Piston Rod of a Hydraulic Cylinder Using Injected Alternating Current-Field Measurement. *Water* **2022**, *14*, 2736.
https://doi.org/10.3390/w14172736

**AMA Style**

Zhang J, Huang Y, Tang J, Zhou F, Kang Y, Feng B. Inspection of Cracks in the Piston Rod of a Hydraulic Cylinder Using Injected Alternating Current-Field Measurement. *Water*. 2022; 14(17):2736.
https://doi.org/10.3390/w14172736

**Chicago/Turabian Style**

Zhang, Jikai, Yuewen Huang, Jian Tang, Fangfang Zhou, Yihua Kang, and Bo Feng. 2022. "Inspection of Cracks in the Piston Rod of a Hydraulic Cylinder Using Injected Alternating Current-Field Measurement" *Water* 14, no. 17: 2736.
https://doi.org/10.3390/w14172736