Internal magnetic field tests and magnetic field coupling model of a three-coil magnetorheological damper

Abstract

Magnetorheological damper is a typical semi-active control device. Its output damping force varies with the internal magnetic field, which is a key factor affecting the dynamic performance of the magnetorheological dampers. Existing studies about the magnetic field of magnetorheological dampers are limited to theoretical analysis; thus, this study aims to experimentally explore the complicated magnetic field distribution inside the magnetorheological dampers with multiple coils. First, the magnetic circuit of a three-coil magnetorheological damper was theoretically analyzed and designed, and the finite element model of the three-coil magnetorheological damper was set up to calculate the magnetic induction intensities of the damping gaps in different currents and numbers of coil turns. A three-coil magnetorheological damper embedded with a Hall sensor was then manufactured based on the theoretical and finite element analysis, and internal magnetic field tests under different conditions were carried out to obtain the actual magnetic induction intensities. At last, the magnetic field coupling model of the three-coil magnetorheological damper was proposed by introducing a coupling coefficient to describe the complex magnetic field distribution due to the strong coupling effect of the three coils, and the results calculated by the proposed model agreed well with the finite element analysis and magnetic field test data. The proposed model lays a foundation for the optimal design of the magnetic circuit and the mathematical model of multi-coil magnetorheological dampers.

Keywords

Three-coil magnetorheological damper magnetic induction intensity internal magnetic field test magnetic field coupling model

1. Introduction

Magnetorheological (MR) fluid is one of the most promising smart materials. In the absence of the magnetic field, its rheological properties are similar to Newtonian flows, but with the magnetic field, its rheological properties can turn into semi-solid or even solid instantaneously (in millisecond) (Ashour et al., 1996; Goncalves et al., 2006). MR dampers can be designed to control the seismic vibration of civil engineering structures utilizing the reversible solid–liquid conversion property of MR fluid (Spencer et al., 1996; Yoshida et al., 2003). The seismic vibration of the structure can be dissipated by the relative movement of the piston in the MR damper. For the advantages of simple structure, fast response speed, low energy consumption, and controllability compared with viscoelastic dampers (Xu et al., 2020), MR dampers have been widely applied in many fields, such as civil engineering structures (Xu et al., 2013; Yang et al., 2002), the automotive manufacture (Sun et al., 2015; Yu et al., 2012), machinery (Fang et al., 2009), the medical industry (Vicente et al., 2011), and so on.

Since its appearance, MR dampers have attracted many scholars to carry out in-depth research and achieved significant results. Chen et al. (2010) proposed an external coil MR damper. The excitation coil of this damper is designed outside the cylinder. Due to the reduction in piston volume, the working stroke of the damper can be effectively increased. Xu et al. (2013) designed and manufactured a five-stage coil shear valve MR damper with a maximum stroke of ±50 mm, an effective piston length of 250 mm, and a maximum output of 260 kN. Bai et al. (2015) designed a built-in bypass MR damper using concentric tubes, and it was found that this structure can greatly improve the adjustable coefficient of the damper by experiment. Zhang and Xu (2012) connected the MR damper with a lead squeeze damper to form a lead-MR composite damper, which could still provide sufficient damping with the failure the MR damper, and the maximum output of the damper reached 472 kN. A bunch of mechanical models were proposed to describe the dynamic properties of MR dampers. Spencer et al. (1997) used the Bouc–Wen hysteresis operator to describe the force–velocity hysteresis of the MR damper. This model is called phenomenological model. Xu et al. (2005) proposed a temperature phenomenological model with mass elements based on the Bouc–Wen model, considering the effect of temperature rise effect on the working performance of the damper and the inertial effect of the MR fluid. Zhou et al. (2006) proposed a simple and effective improved model. He replaced the Bouc-Wen operator with a Dahl hysteresis operator to simulate the Coulomb force of the MR damper, thereby reducing the number of parameters to be identified in the model and avoiding solving differential equations, which greatly reduces the difficulty of model parameter identification. One of the key issues of semi-active control is to choose an appropriate control algorithm to effectively control the amount of current input to the MR damper. Yoshida and Dyke (2005) proposed a clipped-optimal control algorithm based on state space LQG control theory using acceleration feedback to control the MR damper to generate a semi-active control force that is close to the optimal control force. It can basically reach the level of active control. Xu and Shen (2003) used neural networks to predict the vibration state of the structure at the next moment and proposed intelligent dual-state control, which can effectively reduce the impact of time-delay systems on vibration control. Guo et al. (2019) developed the particle swarm optimization (PSO) algorithm to control the displacement and acceleration responses simultaneously, and analysis results demonstrated that the PSO algorithm can reduce the displacement responses and the acceleration responses of the structure obviously.

As mentioned above, for MR dampers, there are many in-depth studies on the design, experiments, mechanical models, and control algorithms. However, the internal magnetic field of MR dampers is also a key parameter in the dynamic performance of MR dampers, especially for multi-coil MR dampers with a more complex magnetic field distribution. The magnetic induction intensities inside the MR damper determines the shear yield stress of the MR fluid, which in turn affects the damping force provided by the MR damper. Parlak et al. (2012) optimized the magnetic field strength and maximum output at the damping gap of the MR damper using the magnetic field analysis module and the fluid analysis mode in the general finite element software ANSYS. Yazid et al. (2014) performed the experimental test under quasi-static loading to observe the performances of MR damper, and results showed that the increment of forces was obtained with the increased current due to higher magnetic flux density generated by electromagnetic coils. Krishna et al. (2017) analyzed optimization of electromagnetic circuit of an MR damper in order to maximize the magnetic ﬂux density. The optimization procedure was proposed by genetic algorithm and design of experiments techniques. Elsaady et al. (2020) combined the finite element analysis of the MR damper magnetic circuit with the computational fluid dynamics analysis of the fluid flow field to analyze the non-linear flow characteristics. Paul et al. (2014) analyzed the effect of magnetic field on damping ability of MR damper and cutting performances, and it was observed that direction of magnetic field parameter on MR damper reduces tool vibration effectively and brought forth better cutting performance. Zheng et al. (2014) designed a novel MR damper with a multistage piston and independent input currents, and the distribution of magnetic field through the MR ﬂuid region was analyzed by finite element method (FEM). Li et al. (2017) designed a new structure of rotary MR damper and ANSYS was used to analyze the magnetic field of the MR damper. The relevant physical quantity of the damper in the magnetic field was obtained. Cheng et al. (2018) proposed a novel MR damper with a meandering magnetic circuit. A theoretical model of the MR damper was set up, and a finite element analysis was performed to verify the principle of the damper. Li et al. (2019) designed and a single-coil MR damper embedded with a Hall sensor to obtain the actual magnetic field inside and to verify the proposed proportional–integral–derivative (PID) control of hysteresis compensation method.

Based on the above literature review, the existing studies about the internal magnetic field of MR dampers are limited to theoretical and finite analysis. What’s more, studies on the magnetic field models to describe the internal magnetic field distribution are rare. There is a lack of experimental tests to obtain the actual internal magnetic field of MR damper to verify the correctness of the theoretical and finite element analysis. Therefore, experimental study about the internal magnetic field of MR dampers with multiple coils are necessary and vital for the investigation of the magnetic field distribution of multi-coil MR dampers. In addition, a magnetic field model is important for describing the magnetic field distribution of MR dampers, especially for multi-coil MR dampers.

This article aims to research the actual magnetic field distribution of multi-coil MR dampers experimentally. First, the magnetic circuit of a three-coil MR damper was theoretically analyzed and designed, and the finite element model of the three-coil MR damper was set up to calculate the magnetic induction intensities of the damping gaps in different currents and numbers of coil turns. A three-coil MR damper embedded with a Hall sensor was then designed and manufactured to explore the actual magnetic field distribution inside the MR damper. Internal magnetic field tests were conducted in different excitation currents and coil combinations. A magnetic field distribution model for this three-coil MR damper was proposed by introducing a coupling coefficient to describe the complex magnetic field distribution due to the strong coupling effect of the three coils, and the magnetic induction intensities calculated by proposed model were compared with the finite element model set up in ANSYS.

2. Design of the magnetic circuit

2.1. Structural design of the MR damper

The magnetic circuit of the MR damper depends on the structural parameters of the damper. Thus, the structural design of the MR damper should be analyzed first. The whole magnetic circuit of a single coil in an MR damper consists of the magnetic core, the magnetic yoke, the damping gap, and the outer cylinder, as shown in Figure 1. The magnetic induction lines start from the magnetic core, then the magnetic yoke A, going through the damping gap to the outer cylinder, and then through the damping gap to the magnetic yoke B, and finally turns back to the magnetic core. The two principles in the designing of the magnetic circuit to make the full use of the MR fluid in the damping gap are as follows: (1) the magnetic induction intensities in the damping gap should be larger than the saturation magnetic induction intensity of the MR fluid and (2) the other components of the magnetic circuit cannot achieve the magnetic saturation earlier than the damping gap. As can be seen in Figure 1, the structural dimensions of the MR damper related to the magnetic circuit are the thickness of the outer cylinder t, the damping gap h, the diameter of the piston D, the diameter of the magnetic core d, and the effective working length L (L = 2*L₂).

The DT4-E magnetic pure iron is chosen as the material for the piston and the piston rod and 45# steel is chosen for the outer cylinder, for their high permeability and low coercivity. MR fluid is the important working medium in the magnetic circuit, and MR fluid with higher yield shear stress and lower sedimentation rate can contribute more to the performance of the MR damper. The MR fluid used in this study was prepared by our group and the yield shear stress remains constant after the magnetic induction intensity increases to 330 mT; thus, the saturation magnetic induction intensity of the MR fluid can be determined to 330 mT (Sun et al., 2019). The volume fraction is the rate of the volume of ferromagnetic particle to that of the carrier liquid, and the MR fluid used in this study has the volume fraction of 35%.

2.2. Theoretical analysis of the magnetic circuit

The magnetic circuit in Figure 1 can be simplified as the magnetic circuit diagram in Figure 2. In Figure 2, R₁ is the reluctance of the magnetic core, R₂ and R₆ are the reluctance of the two magnetic yokes, R₃ and R₅ are the reluctance of the damping gaps, and R₄ is the reluctance of the outer cylinder. According to the Ampere circuit rule (Hu et al., 2014; Liao et al., 2012), the closed magnetic circuit can be expressed in equation (1)

NI = \oint \underset{H}{\to} \cdot d \underset{r}{\to} = \sum_{i = 1}^{6} H_{i} l_{i} = \sum_{i = 1}^{6} \frac{B_{i}}{μ_{ri} μ_{0}} l_{i} = \sum_{i = 1}^{6} \frac{φ_{Bi}}{μ_{ri} μ_{0} S_{i}} l_{i}

(1)

where N is the number of the coil turns, I is the current value of the coils, $μ_{0}$ is the permeability of vacuum, $μ_{0} = 4 π \times 10^{- 7} N / A^{2}$ . $H_{i}$ is the magnetic field intensity of the ith magnetic circuit, $B_{i}$ is the magnetic induction intensity of the ith magnetic circuit, $μ_{ri}$ is the differential permeability of the ith magnetic circuit, $l_{i}$ is the length of the ith magnetic circuit, $S_{i}$ is the section area of the ith magnetic circuit, $φ_{Bi}$ is the magnetic flux of the ith magnetic circuit, and $φ_{Bi} = B_{i} S_{i}$ .

The magnetic flux $φ_{Bi}$ around the magnetic circuit is equal; thus, equation (1) can be expressed as in equation (2)

NI = φ_{B} \sum_{i = 1}^{6} \frac{l_{i}}{μ_{ri} μ_{0} S_{i}} = φ_{B} \sum_{i = 6}^{6} R_{i}

(2)

where $R_{i}$ is the reluctance of the ith magnetic circuit.

According to the basic theories of magnetic theories, the reluctance of the annular radial element and the annular axial element can be calculated by equations (3) and (4), respectively

R_{r} = \frac{1}{2 π h μ_{0} μ} \ln (\frac{r_{2}}{r_{1}})

(3)

R_{a} = \frac{l}{π μ_{0} μ (r_{2}^{2} - r_{1}^{2})}

(4)

where $μ$ is the differential permeability of the element; $r_{2}$ and $r_{1}$ are the outer diameter and the inner diameter of the annular element, respectively; l is the axial length of the axial element; and h is the width of the radial element.

The reluctance of the damping gap is much larger than the other parts; thus, the total reluctance of the magnetic circuit can be approximately regarded as the reluctance of the damping gap, and equation can be written as in equation (5)

NI = φ_{B} (R_{3} + R_{5}) = 2 φ_{B} R_{3}

(5)

The damping gap is the annular radial element, the reluctance of the damping gap can be calculated by equation (3)

R_{3} = \frac{1}{2 π μ_{0} μ L_{2}} \ln [\frac{(D + 2 h)}{D}]

(6)

Therefore, the magnetic circuit in Figure 2 can be expressed as

NI = 2 φ_{B} \cdot \frac{1}{2 π L_{2} μ_{0} μ} \ln [\frac{(D + 2 h)}{D}] = \frac{B \cdot π (D + h) L_{2}}{π L_{2} μ_{0} μ} \ln [\frac{(D + 2 h)}{D}] = \frac{B \cdot (D + h)}{μ_{0} μ} \ln [\frac{(D + 2 h)}{D}]

(7)

Based on the above analysis and the structural dimensions of the damper, the reluctance and the magnetic flux area of each part in the magnetic circuit can be calculated. At initial design, the structural dimensions of the MR damper related to the magnetic circuit are listed in Table 1.

Table 1.

The structural dimensions of the MR damper related to the magnetic circuit.

Thickness of the outer cylinder, t (mm)	Diameter of the piston, D (mm)	Diameter of the magnetic core, d (mm)	Effective working length, L = 2L₂ (mm)	Damping gap, h (mm)
15	176	106	60	2

MR: magnetorheological.

The order of the magnetic saturation needs to be calculated to ensure the damping gap is the first to achieve magnetic saturation. The calculation processes are listed in Table 2. The saturation magnetic induction intensities of the DT4-E and 45# steel are set as 2.5 T, and the saturation magnetic induction intensity of the MR fluid is set as 0.33 T. The magnetic flux area of the magnetic core, the magnetic yoke, the damping gap, and the outer cylinder can be calculated by the following equations respectively

S_{1} = \frac{π}{4} d^{2}, S_{2} = \frac{π}{2} (D + d) L_{2}, S_{3} = π (D + h) L_{2}, S_{4} = \frac{π}{4} [{(D + 2 h + 2 t)}^{2} - {(D + 2 h)}^{2}]

(8)

Table 2.

The order of the magnetic saturation for the elements in the magnetic circuit.

Element	R1	$R 2$	R3	R4
Saturation magnetic induction intensity (T)	2.50	2.50	0.33	2.50
Magnetic flux area ×10⁻⁴ (m²)	88.24	132.89	167.76	91.89
Saturation magnetic flux 10⁻⁴/W_b	220.60	332.23	55.36	229.73
Order of saturation	2	4	1	3

As can be seen in Table 2, the saturation magnetic flux of the damping gap is smaller than the other three elements, that is the damping gap is the first element to achieve the magnetic saturation in the magnetic circuit, which satisfies the second principle in the designing of the magnetic circuit to make the most of the MR fluid. As to the first principle, the magnetic induction intensity of the damping gap can be calculated in the finite element analysis.

According to equation (7), the magnetic induction of the damping gap can be expressed as

B = \frac{NI μ_{0} μ}{(D + h) \cdot \ln [\frac{(D + 2 h)}{D}]}

(9)

Equation (9) is the theoretical magnetic induction intensity of the damping gap for a single coil. For the magnetic field distribution of a multi-coil MR damper, the magnetic field generated by each coil has a strong coupling at the damping gap. The simple superposition principle is no longer applicable. Thus, finite element analysis and the internal magnetic field tests are necessary for the research of the actual internal magnetic field.

The magnetic circuit of MR damper is designed and analyzed, which provides basic data for the finite element analysis of the magnetic field of the damper and the manufacture of the multi-coil MR damper.

3. Finite element analysis of the magnetic circuit

3.1. Finite element simulation of the magnetic circuit model

Due to the non-linear characteristics of the magnetic circuit material and the coupling effect of different coils, it is difficult to propose an accurate model to describe the magnetic field distribution of the multi-coil MR damper in different currents and coil combinations. Therefore, in this section, the FEMs were applied to simulate the magnetic field distribution of a three-coil MR damper. The magnetic field distribution of a three-coil MR damper can reflect the basic characteristics of the coupling of multiple coils of the MR damper.

Before the finite element analysis, the magnetic properties of the materials used in the magnetic field analysis should be obtained. Figure 3 is the magnetization curves of the DT4-E, 45#steel, and MR fluid. It can be seen from Figure 3 that the magnetization process of the three materials has non-linear characteristics.

Since the damper model is an axisymmetric model, it can be simplified as a two-dimensional (2D) model with finite element analysis, with the axial direction of the MR damper as the symmetric axis. Figure 4 is the schematic diagram and the finite element model of the full-component MR damper. The eight-node element PLANE53 was used to build the MR damper model in the preprocessor module of ANSYS. According to the symmetry of the model and the direction of the magnetic line, the boundary condition of parallel magnetic flux was applied to the model outer nodes. The “BFE” commands were used to apply a current density load to excitation coil elements, and the current density is the area of the coil divided by the total current. The current density of the middle coil is opposite to the current density of the coils on both sides, which represents the current direction of the three coils. The differential permeabilities of the MR fluid, the outer cylinder, and the propulsion shaft are defined by the magnetization curves in Figure 3, and the cover plates and the coils are treated as the air, with the differential permeabilities of 1.

The magnetic induction lines and the magnetic induction intensity distributing were obtained in the ANSYS/EMAG postprocessor, as shown in Figure 5. As can be seen from Figure 5(a), the magnetic induction lines all go perpendicularly through the damping gap, except the two lines on two ends, which demonstrates that the magnetic field excited in the magnetic core is made full use of, without magnetic flux leakage. Figure 5(b) is the magnetic induction distribution with the excitation current of 2 A and the coils of 1500 turns, and it can be seen from that the magnetic induction intensity of the damping gaps is approximately uniform distribution. The maximum magnetic induction intensities appear in the magnetic core and the outer cylinder. The magnetic induction intensities in the four effective damping gaps are uniformly distributed, with nearly 0.8 T, which is much larger than the saturation magnetic induction intensity of the MR fluid.

3.2. The analysis of the magnetic induction intensities in the damping gaps

For the finite element model of the three-coil MR damper, the parameters affecting the magnetic field distribution are the currents and the coil turns. In order to calculate the magnetic induction intensities in the damping gap, the “PATH” command was applied to defined a path in the ANSYS/EMAG postprocessor, and the “PDEF” command was applied to map the magnetic induction intensities into this path. The defined path was set in the middle of the damping gap, with the length of 360 mm, and the 73 measuring points are uniformly distributed with the interval of 5 mm.

Figure 6 shows the distribution of magnetic induction intensities of the damping gap in different numbers of coil turns with the excitation current of 2 A. As can be seen in Figure 6, the magnetic induction intensity of the damping gap increases with the numbers of the coil turns in the 2 A excitation current, which agreed well with equation (9). The magnetic induction intensities are approximately the same in each damping gap, and the magnetic induction intensities of the middle two damping gaps are much larger than the two damping gaps of the ends, which reflects a strong coupling effect of the magnetic field excited by the three coils. The coupling effect of the three coils results in that the magnetic induction intensities of the middle damping gaps are reinforced, and the magnetic induction intensities of damping gaps of the ends are weakened. Because the three coils of the MR damper are evenly arranged, the magnetic induction intensities are symmetrically distributed in the vertical direction.

Two specific points were selected to show the relationship between the magnetic induction intensities and the number of coil turns. Point A is in the middle of the exterior damping gap, and point B is in the middle of the interior damping gap. The exact positions of points A and B can be seen in Figure 13, where point A is point 2, and point B is point 5. Figure 7 shows the magnetic induction intensities of points A and B in different numbers of coil turns with the excitation current of 2 A. The magnetic induction intensities have an approximate linear relationship with the number of the coil turns. In order to increase the magnetic induction intensities in the damping gap within the determined structural dimensions of the MR damper, the number of coil turns was set as 1500. What’s more, as can be seen from Figure 7, with the increase in the coil turns, the gap between the magnetic induction intensities of the exterior damping gap and the interior damping gap gradually widens. This means that the more turns of the coil, the stronger the coupling between the magnetic fields generated by the three coils.

Figure 8 is the magnetic induction intensities of the damping gap in different excitation currents with 1500 coil turns. The magnetic induction intensity reaches nearly 0.9 T when the applied current is 2.0 A, which satisfies the first principle in the designing of the magnetic circuit that the magnetic induction intensities in the damping gap should be larger than the saturation magnetic induction intensity of the MR fluid. As can be seen in Figure 8, the magnetic induction intensities of the four effective damping gaps increases with the excitation currents. The magnetic induction intensities of the interior damping gap were “V-shaped,” showing a “high on both sides, low in middle” phenomenon in the 0.2 A current, and this phenomenon gradually weakens with the increase in the current.

Figure 9 shows the magnetic induction intensities of points A and B in different currents. The magnetic induction intensities of the four damping gaps increase with the excitation current, but the growth trend gradually slows down. In order to make the most of the magnetic field excited by the current, the range of the applied current was set as 0–2 A. In addition, as can be seen from Figure 9, with the increase in the excitation current, the gap between the magnetic induction intensities of the exterior damping gap and the interior damping gap gradually widens. This means that the larger the excitation current, the stronger the coupling between the magnetic fields generated by the three coils.

Through the finite element analysis of the magnetic field distribution of the three-coil MR damper, a more accurate and intuitive understanding of the magnetic induction intensities in the damping gaps was obtained. What’s more, the finite element analysis provides a good foundation for the manufacture of the three-coil MR damper.

4. Internal magnetic field tests

4.1. Experimental setup

In order to obtain the actual internal magnetic field distribution of the three-coil MR damper and to verify the accuracy of the finite element model, the MR damper embedded in a Hall sensor was manufactured especially for the research of the internal magnetic field, as shown in Figure 10. According to the finite element analysis, the number of turns per coil was chosen as 1500. The upper and lower cover plates were made of stainless steel, which is non-magnetic material. The connecting lines of the Hall sensor and the three coils were combined together and were led out of the damper from the top. As can see from Figure 11, a groove was dug in the surface of the piston on the edge of the second stage, and the Hall sensor was placed in the groove and pasted firmly in the cylinder surface of the piston. The Hall sensor was small in size with the three dimensions of 0.7 mm × 1.5 mm × 6 mm and the groove was the same shape of the Hall sensor with the depth of 1 mm. Machining grooves in the surface of the piston will inevitably affect the distribution of the actual magnetic field; thus, in order to evaluate the influence of the groove on the original magnetic field distribution, the magnetic induction intensities obtained by the embedded Hall sensor and the Hall probe were compared, as can be seen in Table 3. The magnetic field test object was the MR damper without the outer cylinder, and the electrify condition was the coil 2 energized. The Hall probe was placed at the same circumferential site as the embedded Hall sensor. Table 3 shows that the errors between the magnetic induction intensities obtained by the embedded Hall sensor and the Hall probe were limited to 10%. Therefore, the influence of the groove on the magnetic field distribution can be ignored. The reason for placing the Hall sensor in this position was that the magnetic induction intensities were relatively higher and evenly distributed in this area, according to the finite element analysis. The Hall sensor and the three coils of the MR damper were sealed by epoxy resin. Figure 12 is the devices needed for the magnetic tests. The currents are input to the three coils by the DC power, and the magnetic induction intensities of each measuring point were obtained by putting the Hall probe onto the surface of the measuring point. The Hall probe is connected to the Tesla meter for the value display of the magnetic induction intensities.

Table 3.

The comparison of magnetic induction intensities obtained by the embedded Hall sensor and the Hall probe.

Excitation current, I (A)	Tested magnetic induction intensities (mT)
Excitation current, I (A)	Hall sensor	Hall probe	Error (%)
0.4	24.93	22.91	4.75
0.8	46.90	43.38	8.11
1.2	65.21	60.11	8.48
1.6	85.37	84.8	0.67
2.0	106.43	102.81	3.52

4.2. Magnetic field tests of the propulsion shaft of the MR damper

After the experiment was set up, the magnetic field tests of the MR damper without the outer cylinder were conducted for a better understanding of the magnetic field distribution of the four damping gaps. Figure 13 shows the selected measuring points. The propulsion shaft of the MR damper has four stages, corresponding to the four effective damping gaps. According to the finite element analysis, the magnetic induction intensities of the effective damping gaps are much larger than the other areas, thus 12 measuring points were selected on the surface of the four stages, with three points evenly distributed on each stage. Points 1–3 present the magnetic field distribution of damping gap 1, points 4–6 present the magnetic field distribution of damping gap 2, points 7–9 present the magnetic field distribution of damping gap 3, and points 10–12 present the magnetic field distribution of damping gap 4. In order to reduce the experimental error, each measurement point is tested in the circular direction for four sets of data, and then the average value is taken. Figure 14 shows the photo taken on site when the magnetic field tests of the MR damper without the outer cylinder were conducted. As can be seen in this photo, the propulsion shaft of the MR damper after the three coils were winded was placed horizontally on the workbench, and the connecting wires of the three coils were connected to the DC power. With different excitation currents, the Hall probe was placed on the surface of each measuring point, and the values of the magnetic induction intensities were obtained in the screen of the Tesla meter.

First, the magnetic induction intensities of the MR damper without the outer cylinder with single coil energized were analyzed. Figure 15 shows the magnetic induction intensities of the 12 measuring points with coil 1, coil 2, and coil 3 energized, separately. It can be seen from these three pictures that the magnetic induction intensities of all the damping gaps were “V-shaped” and are much higher for the two side points than for the middle point. The reason for this phenomenon was due to the point discharge of the two sharp edges of each stage of the piston. Point discharge is a discharge phenomenon that occurs in the sharp part of an object under the action of a strong field. Under the action of a strong field, where the surface curvature of the object is large (such as the top of sharp, small objects), the field strength increases sharply, causing the air near it to be ionized to produce a gas discharge. Because when the magnetic field tests of the propulsion shaft of the MR damper was conducted exposed to the air, the point discharge phenomenon was obvious. While the MR damper was assembled and filled with MR fluid, the measuring points were no longer exposed to the air, and the “V-shaped” magnetic field distribution can be greatly attenuated, as can be seen in Figures 6 and 8.

What’s more, the magnetic induction intensities of all the 12 measuring points had a linear relationship with the excitation currents. Compared with the finite element results in Figure 6 and Figure 8, the “V-shaped” magnetic field distribution was much larger. This is due to the incomplete magnetic circuit of the MR damper without the outer cylinder.

As can be seen in Figure 15(a), when only coil 1 was energized, the magnetic induction intensities of damping gap 1 and 2 were higher than the other two damping gaps, especially points 3 and 4 with over 100 mT. Figure 15(b) was the magnetic induction intensities with only coil 2 energized. Because coil 2 is in the middle of the MR damper, the distribution of the magnetic induction intensities was symmetrical. When only coil 3 was energized, the maximum magnetic induction intensities appeared in points 9 and 10. It can be found that points near the energized coil had higher magnetic induction intensities, and the magnetic induction intensities of the points far from the energized coil were relatively lower, about 40 mT.

In order to analyze the magnetic induction intensities of the four damping gaps, the average values of the three measuring points of each damping gap were obtained, and the results are shown in Figure 16. For a specific damping gap, the average magnetic induction intensities increased linearly with the excitation current. The damping gaps near the energized coil had higher average magnetic induction intensities, and the magnetic induction intensities dropped approximately linearly with the increase in distance between the energized coil and the damping gap, as shown in Figure 16(a) and (c). In addition, the single energized coil could only affect the magnetic induction intensities of the adjacent two damping gaps, while the magnetic induction intensities of the further damping gap had no change. Figure 16(b) is the distribution of the magnetic induction intensities with coil 2 energized, and the distribution is symmetrical in the axial direction of the damper.

Then, the magnetic field tests of the propulsion shaft with three coils energized were conducted. The electrify conditions of three coils energized could be divided into “+–+” and “+++.”“+–+” means that the current direction of coil 2 is opposite to that of coils 1 and 3, while “+++” means that the current direction of coil 2 is the same as that of coils 1 and 3. Figure 17 is the magnetic induction intensities of the 12 measuring points with three coils energized. Compared with the distribution of the magnetic induction intensities with single coil energized, the magnetic induction intensities with three coils energized were not the simple combination of those with single one energized. The complex coupling effect of the three coils resulted in the increase and decrease of the magnetic induction intensities of the four damping gaps. Figure 17(a) shows the distribution of the magnetic induction intensities with “+–+” electrify condition, and it can be seen from this figure that the magnetic induction intensities of the middle damping gaps increased and those of the damping gaps on both ends decreased due to the coupling effect with “+–+” electrify condition. On the other hand, with “+++” electrify condition, the magnetic induction intensities of the middle damping gaps decreased and those of the damping gaps on both ends increased, as shown in Figure 17(b).

Table 4 lists the averages and the variances of the magnetic induction intensities in different currents with “+–+” and “+++” electrify conditions. It can be clearly seen from Table 4 that the average magnetic induction intensities of the 12 measuring points with “+++” electrify condition were slightly higher than those with “+–+” electrify condition. While the variances of magnetic induction intensities with “+++” electrify condition were much larger than those with “+–+” electrify condition. This means that the magnetic induction intensities with “+–+” electrify condition were considerably high and much more evenly distributed. Therefore, when three coils of the MR damper are energized, the “+–+” electrify condition is preferred in practical.

Table 4.

The comparison of magnetic induction intensities with the “+++” and “+–+” three coils energized.

Electrify conditions	0.4 A		0.8 A		1.2 A		1.6 A		2.0 A
Electrify conditions	Average $\bar{B} (mT)$	Variance $(mT)^{2}$	Average $\bar{B} (mT)$	Variance $(mT)^{2}$	Average $\bar{B} (mT)$	Variance $(mT)^{2}$	Average $\bar{B} (mT)$	Variance $(mT)^{2}$	Average $\bar{B} (mT)$	Variance $(mT)^{2}$
+++	22.3	232.8	38.7	604.0	57.5	1389.7	72.3	2128.9	81.6	2397.0
+–+	16.6	38.9	31.1	183.1	48.5	518.0	60.7	681.8	70.8	805.1

After the magnetic field tests of the single propulsion shaft were finished, the propulsion shaft of the damper was placed into the outer cylinder, and the magnetic field tests of the propulsion shaft with the outer cylinder were performed. As can be seen in Figure 18, the Hall sensor was pasted on the edge of the second stage of the piston, that is the measuring point 7. Due to the limited length of the Hall probe, the magnetic field tests of all the 12 measuring points could not be conducted. Only the magnetic induction intensities of points 7 and 12 could be obtained. The magnetic induction intensities of point 7 were obtained by the embedded Hall sensor, and the magnetic induction intensities of point 12 could be obtained by putting the Hall probe on the surface.

Figure 19 is the magnetic induction intensities of points 7 and 12 of the propulsion shaft with the outer cylinder. As can be seen from Figure 19, with the installation of the outer cylinder, the magnetic induction intensities increased sharply, from about 100 mT to nearly 1000 mT. This was resulted from a more complete magnetic circuit, with the outer cylinder. Figure 19(a) is the comparison of magnetic induction intensities of point 7 with “+–+” and “+++” electrify conditions. It can be seen from this figure that the magnetic induction intensities of point 7 when the electrify condition was “+–+” had an approximately linear relationship with the excitation current, but the magnetic induction intensities of point 7 when the electrify condition was “+++” were slightly decreased with the increase in the excitation current. This is due to the coupling effect of the three coils. Figure 19(b) is the comparison of magnetic induction intensities of point 12 with “+–+” and “+++” electrify conditions. The magnetic induction intensities of point 12 with both electrify conditions increased with the excitation current, and the growth rate gradually slowed, which resulted from the saturation of the magnetic circuit. What’s more, the magnetic induction intensities of point 12 with “+–+” electrify condition were lower than those with “+++” electrify condition. This is the coupling effect of the three coils that the magnetic induction intensities of the middle damping gaps were enhanced, and those of the damping gaps on both sides were weakened.

4.3. Internal magnetic field tests with the assembled MR damper

After the magnetic field tests of the propulsion shaft, the gaps of the MR damper were filled with MR fluid, and the MR damper was assembled finally. When the upper cover plate was installed, the Hall probe could not be placed on the surface of the measuring point, and the internal magnetic field tests could only be conducted by the embedded Hall sensor. Thus, only the magnetic induction intensities of point 7 were obtained in different currents and coil combinations, as can be seen in Figure 20. The connecting wires of the three coils were connected to the DC power, and the embedded Hall sensor was connected to the Tesla meter. With different excitation currents and coil combinations, the magnetic induction intensities of point 7 were acquired by the Hall sensor and can be read on the screen of the Tesla meter. To minimize test errors, each data were measured four times and averaged.

Figure 21 shows the magnetic induction intensities of point 7 with different currents and coil combinations. First, compared with Figure 19(a), the magnetic induction intensities of point 7 with the electrify condition of both “+–+” and “+++” of the assembled MR damper were slightly higher than those of the propulsion shaft with outer cylinder. This means that the assembled MR damper formed a complete and closed magnetic circuit, and the magnetic leakage was further reduced. Moreover, with the electrify condition of “+–+,” the magnetic induction intensities of point 7 increased linearly in the current range 0–1.6 A, and then gradually stabilized. In the 2 A current, the magnetic induction intensities of point 7 reached over 900 mT, and the saturation current of the three-coil MR damper could be treated as 2 A. Most importantly, when comparing the different coil combinations, the magnetic induction intensities with the “+–+” electrify condition were distinctly higher than other electrify conditions, about twice as the coil 2 energized and coils 1 and 3 energized. The magnetic induction intensities of point 7 with only coil 2 energized and coils 1 and 3 energized electrify conditions were almost the same, which means that the coupling effect of the two coils on both sides was similar to the effect of the middle coil. The “+++” electrify condition had the lowest magnetic induction intensities among the four electrify conditions, and the magnetic induction intensities decreased slightly with the increase in current. This reflects that when the current direction of the three coils was the same, the magnetic induction intensities in the middle were greatly attenuated, while when current direction of the middle coil was different from the coils on two ends, the magnetic induction intensities in the middle were highly improved. This is the basic principle of the coupling effect of the magnetic field generated by the three coils.

Since the “+–+” electrify condition had the highest magnetic induction intensities, this electrify condition was chosen as the working mode for this three-coil MR damper. In order to verify the effectiveness of the finite element analysis of the MR damper, the results of the finite element analysis were compared with the internal magnetic field tests on the same working condition. The number of coil turns was 1500, the current range is 0–2 A, and the electrify condition is “+–+ .”Figure 22 shows the comparison of the magnetic induction intensities obtained by these two methods. As can be seen from this figure, the magnetic induction intensities calculated by the finite element analysis were close to the results obtained by the internal magnetic field tests in every excitation current. This comparison shows that the FEM can basically describe the distribution of the magnetic induction intensities for this three-coil MR damper, and the saturation property of the magnetic circuit.

5. Magnetic field coupling model

Based on the comparison between the internal magnetic field tests and the finite element analysis, the finite element model can basically describe the distribution of the magnetic field of the MR damper. Therefore, the analysis and proposing of the magnetic field coupling model can be based on the finite element results.

The theoretical analysis of the magnetic circuit above gives the magnetic induction intensity of the damping gap for the magnetic circuit with a single coil, as seen in equation (9). For a three-coil MR damper, this equation cannot be applied. When multiple coils are energized, the average magnetic induction intensity generated at all effective damping gaps does not increase linearly with the number of coils. As can be seen in the above finite element analysis and internal magnetic field tests, the magnetic field generated by the three coils will cause a strong coupling effect in the four effective damping gaps. For example, with the “+–+” electrify condition, the magnetic induction intensities of the internal damping gaps will be strengthened, and the magnetic induction intensities of the external damping gaps will be weakened. Therefore, a coupling coefficient $α$ was introduced in equation (9) to represent the coupling effect of the three coils, as shown in equation (10). $α$ is related to the excitation current and the position of the damping gap

B = \frac{α NI μ_{0} μ}{(D + h) \cdot \ln [\frac{(D + 2 h)}{D}]}

(10)

where $μ$ is the differential permeability of the MR fluid, which is related to the excitation current I. $μ$ is not a fixed value, and it changes with the excitation current. It belongs to an inherent characteristic of the MR fluid itself. The relationship between $μ$ and the excitation current I can be expressed as (Xu, 2017)

μ = 1.933 I^{- 1.567} + 1.944

(11)

Taking the structural parameters of the MR damper, the number of the coil turns, and equation (11) into equation (10), this equation can be turned into

B = α (0.29 I^{- 1.567} + 0.2916)

(12)

The distribution of magnetic induction intensities of the defined path can be obtained by finite element analysis. As can be seen in Figure 7 and 8, the magnetic induction intensities of the four damping gaps were symmetrical, that is the two internal damping gaps had the same magnetic induction intensities, so did the two external damping gaps. Thus, by calculating the magnetic induction intensities of the internal and external damping gap, the whole distribution of the magnetic induction intensities can be obtained.

As can be seen in equation (12), the relationship between the coupling coefficient $α$ and the current I can be calculated by the magnetic induction intensities B divided by $(0.29 I^{- 1.567} + 0.2916)$ . Thus, the relationship between the coupling coefficient $α$ and the current I can be plotted in Figure 23. As can be seen in Figure 23, the relationship between the coupling coefficient $α$ and the current I was approximately linear; thus, the relationship can be fitted by the polynomial of first degree

α_{E} = 0.916 I - 0.048, α_{I} = 1.264 I - 0.177

(13)

In equation (13), $α_{I}$ is the coupling coefficient for the internal damping gap, and $α_{E}$ is the coupling coefficient for the external damping gap. Taking the coupling coefficients into equation (10), the magnetic induction intensities for the internal damping gap and external damping gap can be expressed as

B_{E} = (0.916 I - 0.048) \frac{NI μ_{0} μ}{(D + h) \cdot \ln [\frac{(D + 2 h)}{D}]}

(14)

B_{I} = (1.264 I - 0.177) \frac{NI μ_{0} μ}{(D + h) \cdot \ln [\frac{(D + 2 h)}{D}]}

(15)

where $B_{E}$ is the magnetic induction intensities for the external damping gap, and $B_{I}$ is the magnetic induction intensities for the internal damping gap.

Therefore, the magnetic field coupling model for the three-coil MR damper was proposed in equations (14) and (15). This model can calculate the average magnetic induction intensities of the four damping gaps in this three-coil MR damper. In order to verify the effectiveness of this model, the errors between the proposed model and the finite element analysis are listed in Table 5. As can be seen in Table 5, the average magnetic induction intensities obtained by the proposed magnetic field distribution model were in good agreement with the finite element simulation results. The proposed magnetic field model with a coupling coefficient can finely describe the magnetic field distribution and the coupling effect of the three-coil MR damper.

Table 5.

The comparison of magnetic induction intensities calculated by the proposed model and the FEM.

Excitation current, I(A)	Magnetic induction intensities of internal damping gap (T)			Magnetic induction intensities of external damping gap (T)
Excitation current, I(A)	Model	FEM	Error (%)	Model	FEM	Error (%)
0.4	0.4971	0.4467	11.29	0.4814	0.4134	16.45
0.8	0.5868	0.5793	1.29	0.4818	0.5087	5.29
1.2	0.6829	0.7020	2.70	0.5359	0.5668	5.43
1.6	0.7946	0.8099	1.89	0.6106	0.6168	1.01
2.0	0.9159	0.8906	2.84	0.6952	0.6625	4.95

FEM: finite element method.

6. Conclusion

In this article, a three-coil MR damper embedded with a Hall sensor was designed and manufactured specially to experimentally obtain the actual internal magnetic induction intensities, based on the theoretical and finite element analysis of the magnetic circuit. Then, internal magnetic field tests of the single propulsion shaft, the propulsion shaft with the outer cylinder, and the assembly MR damper were conducted in turn. Finally, the magnetic field coupling model for the three-coil MR damper was proposed by introducing a coupling coefficient. Some conclusions can be drawn as follows:

Based on the theoretical analysis and optimization of the magnetic circuit, the magnetic field of the three-coil MR damper and the properties of the MR fluid are made full use of. The magnetic induction intensities in the damping gap were larger than the saturation magnetic induction intensity of the MR fluid, and the other components of the magnetic circuit achieved the magnetic saturation later than the damping gap, according to the order of the magnetic saturation.

The finite element model of the MR damper can finely reflect the magnet field distribution of the three-coil MR damper. According to the finite element analysis, the magnetic field has a strong coupling effect due to the three energized coils. The magnetic induction intensities increase with the excitation currents and the number of the coil turns. The larger the excitation current and number of coil turns, the stronger the coupling effect between the magnetic fields generated by the three coils.

According to the magnetic field tests, the “+–+” electrify condition can generate the highest magnetic induction intensities, and the saturation current for the MR damper is about 2 A. The basic principle of the coupling effect for the three energized coils is that when the current directions of the three coils are the same (+++), the magnetic induction intensities in the middle are greatly attenuated and the magnetic induction intensities at the ends are increased, while when current direction of the middle coil is different from the coils on two ends (+–+), the magnetic induction intensities in the middle were highly improved and the magnetic induction intensities at the ends are decreased. What’s more, the comparison between the finite element analysis and the internal magnetic field tests showed that the FEM can basically describe the distribution of the magnetic induction intensities for the three-coil MR damper.

A magnetic field coupling model was proposed by introducing a coupling coefficient to describe the coupling effect of the magnetic field generated by the three energized coils. The average magnetic induction intensities obtained by the proposed magnetic field distribution model were in good agreement with the finite element simulation results and magnetic field tests data. The proposed magnetic field coupling model with a coupling coefficient can finely describe the complex magnetic field distribution and the coupling effect of the three-coil MR damper. This model can be applied to describe the magnetic field distribution of other multi-coil MR dampers and lays a foundation for the optimal design of the magnetic circuit and the mathematical model of multi-coil MR dampers.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by National Science Fund for Distinguished Young Scholars with Grant No. 51625803, the Program of Chang Jiang Scholars of Ministry of Education, the Tencent Foundation through the XPLORER PRIZE, Ten Thousand Talent Program (Innovation Leading Talents), National Natural Science Foundation of China with Grant No. 51878355.

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