Analytical study and modeling of the effect of high acceleration on the electromagnetic resistance of eddy current brake

Abstract

With the development of application and research of eddy current brakes (ECB) in various fields, the application of ECB in large braking machines targeting strong impact load is beginning to be explored. When braking against strong impact load, a high acceleration phase will exist, and sometimes under this special condition, a high peak of electromagnetic resistance will appear, which can disturb the braking effect of the ECB. The effect of high acceleration on the electromagnetic resistance characteristics of ECB under such special application conditions is of concern. In this paper, the change of electromagnetic field under high acceleration is analyzed. Combining the working principle of eddy current brake with the data obtained from FEM model, the principle equation and explanation analysis are given. By comparing the electric field distribution with and without high acceleration, effects of high acceleration at different velocity stages on the eddy current electric field are discussed. In order to be able to better predict the resistance of the ECB under strong impact loads, an electric field strength incremental considering high acceleration was introduced according to a simplified analytical model combining the MEC model with the equivalent electric field approach. Numerical modeling was performed based on the data obtained from FEM models, and the final model obtained is verified by comparison with experimental results. In this paper, combining analytical modeling, finite element modeling and numerical modeling, an analytical model of electromagnetic resistance and braking process considering high acceleration is proposed, and the braking process can be simulated quite accurately. Future research can use this model to optimize the design of the ECB mechanism to obtain a more efficient braking force and a smoother braking process.

Keywords

analytical modeling eddy current brake electromagnetic resistance characteristic high acceleration strong impact load

Introduction

Eddy Current Brake (ECB) is a braking device that derives its resistance from eddy currents generated in a conductor according to the Lenz's law. This new braking method has been applied and studied in many different areas, in addition to the relatively mature applications in automotive and maglev train braking, there are some recent studies in tuned mass damper, aerospace mechanisms and neurosurgical instruments.^1–5

There has been many research on the structure and properties of ECB, and the commonly used research methods are finite element method (FEM) and analytical modeling method. Although FEM^6–8 can help researchers to understand and analyze the electromagnetic field distribution inside the ECB, it is still time-consuming if one wants to model and calculate multiple models with different parameters.

The magnetic equivalent circuit (MEC) model⁹ is a commonly used approach in analytical modeling of magnetic field. Recently, Amrhein et al. further investigated 3D MEC modeling to seek a wider application of this method.¹⁰ However, the MEC model is in general difficult to consider the hindrance generated by eddy currents in the analytical model. Until in recent years, some studies have introduced the eddy currents as a magnetomotive force (MMF) into the MEC model and obtained the analytical model that can consider the effect of eddy currents.¹¹

In order to predict the electromagnetic braking force, most researchers choose to use Maxwell's equations and hierarchical method for analytical modeling of ECB.^12–14 To study ECB structure with slotted conductors, Kou et al. proposed an analytical model that considers slotted conductor disk, magnetic saturation of primary core and actual path of eddy currents in the conductor disk.¹⁵ However, such models usually have complex formulas and are still time-consuming to compute.

Most of the current research focused on resistance-velocity characteristics of ECB under quasi-static conditions.¹⁶ There are some studies for ECB under high-speed condition but there are rare studies for ECB under high acceleration condition. For an ECB under strong impact load, the influence of high acceleration on braking force performance has been analyzed by Li Jiahao et al. There are some basic principle formulas been presented in the discussion, but the effect of high acceleration on ECB under different parameters are what was mainly discussed. Moreover, it is difficult to incorporate the effect of high acceleration in the analytical model because the subdomain model and Maxwell's equations was applied in the ECB model.^17,18

In this paper, a simplified numerical model combining the MEC model with an equivalent electric field strength method has been used to predict the eddy currents as well as the braking force of ECB in quasi-static condition. This model contains both formulas that represent clear physical meanings and direct performance of the electromagnetic field characteristics, which allows for a simple yet clear indication of the relationship between the electromagnetic parameters and the braking characteristics. Although slightly less accurate, it is computationally fast, can be used in the preliminary design of electromagnetic buffers,¹⁹ and can easily incorporate other influences into the model.

In order to study a ECB under strong impact load, where a high acceleration phase will exist during the braking, this paper investigates how the high acceleration will affect the original electromagnetic field and resistance, with equations and illustrations. Firstly, the FEM model of the ECB model under strong impact load as well as under high acceleration conditions are established. Then, based on the FEM model, which has the advantage of results visualization, how the high acceleration will affect the eddy current electric field at different velocities is comparatively analyzed, and it is discussed and verified corresponds to the principle equations. According to the simplified analytical model mentioned previously and the numerously results obtained from FEM models, an analytical model considering high acceleration is proposed. By incorporating an electric field strength increment, the model can simulate the affection of high acceleration on braking force prediction. Lastly, the proposed method is verified through experiments. Results shows that the analytical model is able to predict the braking process with the better expression of the high acceleration phase.

ECB model

Structure and kinematic model of the ECB

Figure 1 shows the ECB model discussed in this paper, where multiple sets of permanent magnet (PM) rings with opposite magnetization directions are set on the shaft, and with iron poles separating the PMs while aids in the formation of magnetic circuit. The conductor cylinder and iron back cylinder have no contact with PMs and iron poles, and form a magnetic flux path with them across the air gap. The shaft is attached to the structure that receives strong impact loads, and all moving parts are collectively referred to as the primary. The iron back is attached to the base to keep it stationary, and the conductor cylinder and iron back are referred to as secondary. During braking, the primary and secondary will move relative to each other, and changes of the magnetic field in the secondary will produce eddy currents and electromagnetic resistance. Electromagnetic resistance is generated by the phenomenon of electromagnetic induction as well as eddy current losses.

Figure 1.

Schematic structure of ECB model.

According to the principle of electromagnetic induction, the changing magnetic field produces eddy currents. And the eddy currents in the secondary of ECB arise from the induced electromotive force (e.m.f.) generated by the relative motion of the primary and the magnetic field in the secondary as well as the change of the magnetic field.

\begin{aligned} ζ = ζ_{m} + ζ_{i} & = \int (v \times B) \cdot d l + \oint E_{i} \cdot d l \\ = \int (v \times B) \cdot d l - \int \int \frac{\partial B}{\partial t} \cdot d S \end{aligned}

(1)

where ε is e.m.f., and B is magnetic flux density.

When the relative motion is uniform, the eddy current in the secondary consists only of the kinetic electric potential, and the following equation for eddy current density and eddy current loss can be obtained.

J = σ E = σ (v \times B)

(2)

\begin{aligned} F = 2 π \int \int (B \times J) r d z d r = 2 π \int \int (B \times σ (v \times B)) r d z d r \\ = 2 π v σ_{c c} \int \int_{c c} B_{r}^{2} (r, z) r d z d r + 2 π v σ_{i b} \int \int_{i b} B_{r}^{2} (r, z) r d z d r \end{aligned}

(3)

where $σ_{c c}$ $σ_{i b}$ refers to the conductivity of conductor cylinder and iron back cylinder.

The ECB model will be applied to a heavy braking device, which will be subjected to a strong impact force F_load as shown in Figure 2 and will reach an acceleration of 1000–2000 m/s² in a short time. At the same time, there is a spring-loaded reentrant mechanism that creates a resistance F_f during braking, the resistance is related to the braking distance, as shown in Figure 3. Assuming that there is no friction, and that the braking motion is completely parallel to the ground, i.e., there is no influence of gravity, the motion model of the braking process of this machine can be obtained as

m \frac{d^{2} x}{d t^{2}} = F_{l o a d} - F_{f} - F

(4)

Figure 2.

Two sets of loading force.

Figure 3.

Reset mechanism force with displacement.

FEM modeling of the ECB

With the advantage of results visualization, FEM model can significantly assist researchers to understand and analyze the electromagnetic field distribution inside the ECB. FEM model of the discussed ECB was established in a 2D Cartesian coordinate system. Since there is relative motion between the primary and secondary, the meshes cannot be continuous. In the middle of the air gap, an EMF-consistent boundary pair was set up so that the meshes on both sides of the boundary can be separated. In order to ensure the calculation speed and accuracy of the FEM model, the minimum value of the mesh was set to 1/4 of the air gap in the mesh size setting. The PM number of this ECB model was 36, and the one end of ECB mesh model was presented in Figure 4 for the full model being relatively too long to display.

Figure 4.

Partial view of the finite element mesh model.

In order to investigate the effect of high acceleration on the drag characteristics of the ECB using the finite element model, in addition to the motion model described above, motion models were set up that allows the ECB to accelerate at a specified acceleration starting from 0 m/s.

Effect of acceleration on ECB resistance characteristics

When the ECB is subjected to a strong impact force, its acceleration will reach 1000 m/s²–2000 m/s² in a short period of time. And the electromagnetic resistance will increase speedily and then decrease dramatically after it reaches the maximum value. When the acceleration phase is completely stopped, it will return to a uniform resistance curve, as is shown in the resistance during actual braking curve in Figure 5. This electromagnetic resistance fluctuation was noticed in our experiments and can be simulated with relative consistency in the finite element model. The curves here are only meant to demonstrate such phenomena, and the data are the results of a FEM model.

Figure 5.

Electromagnetic resistance curve during braking.

The importance of analytical models in performing ECB studies is unquestionable, and therefore, in order to obtain an accurate prediction of the resistance, several analytical model modeling methods for ECB are presented in the introduction. However, in the existing studies, there is no analytical model that can take into account the change in resistance due to acceleration.

In Figure 5, the electromagnetic resistance results during actual braking process were compared with the results calculated with the analytical model which did not take acceleration into account, and it can be noticed that there was a significant peak difference. It is mainly due to the effect of high acceleration on the eddy current electric field and electromagnetic resistance (Figure 6). If the high acceleration conditions in practice were not reasonably matched to the design of the ECB, this resistance peak could be very high and affect the stability and reliability of the braking process.

Figure 6.

Acceleration during braking.

Set the horizontal coordinate as velocity, the two curves in Figure 5 were transformed into Figure 7, it can be seen that the electromagnetic resistance during acceleration is first smaller and then larger than that at quasi-static conditions, while it is essentially the same during deceleration. The absolute value of deceleration during braking is an order of magnitude smaller than the absolute value of acceleration, and it is assumed preliminarily that high acceleration has a more pronounced effect on the EMF results.

Figure 7.

Resistance versus velocity curves.

The electromagnetic resistance versus velocity curves of an ECB model at different accelerations were given in Figure 8, where the data without acceleration were the FEM results for given velocities under quasi-static conditions. And the data with acceleration were obtained from the FEM model with its motion models were set up that makes the ECB to accelerate at specified accelerations starting from 0 m/s.

Figure 8.

Electromagnetic resistance versus velocity curves at different accelerations.

At the same time, set the initial speed as 25 m/s, the acceleration as −500 m/s², and the electromagnetic resistance versus velocity curve during deceleration was calculated. It can be seen that when the acceleration is −500 m/s² and 500 m/s², the difference between the electromagnetic resistance curve and the resistance curve without acceleration is insignificant and almost symmetrical. However, such deceleration can hardly be reached in the actual braking process, so the effect of deceleration on electromagnetic resistance can be considered negligible. Moreover, as the acceleration becomes larger, the effect of acceleration on electromagnetic resistance becomes more and more significant, and it can be assumed that high acceleration (a ≥ 1000 m/s²) will have a significant effect on electromagnetic resistance.

Correlation study of the effect of high acceleration on electromagnetic resistance and ECB design parameters

In order to analyze the effect of high acceleration on electromagnetic resistance, it should first be clear that the original magnetic field generated by the PMs will not be affected much by high acceleration, and the difference is more manifested in the eddy current electric field in the secondary. Therefore, the first thing analyzed in this paper is how the conductivity and thickness of the cylinders of the secondary correlate with the effect of high acceleration on electromagnetic resistance.

According to the FEM model of the ECB braking process established earlier, the electromagnetic resistance versus velocity curves (F-v curve) of the braking process are given for different values of the four parameters while keeping the other parameters constant. The four parameters are $σ_{c c}$ , $σ_{i b}$ , $t h_{c c}$ and $t h_{i b}$ (the thickness of conductor cylinder and iron back). The peak acceleration of this model is approximately 1200 m/s². Comparing the computational results of this model, it is possible to observe not only the pattern of the effect of high acceleration on the F-v curve under different parameters, but also the pattern of the effect of high acceleration on the braking process under different parameters.

From Figures 9 and 10, it can be seen that with the change of the conductivity of conductor cylinder, the variation of the F-v curve during the braking process is relatively linear, which is because the conductivity of conductor cylinder is one of the main factors in the calculation of the eddy current of the ECB. The electromagnetic resistance with high acceleration obviously increases with the increase of the conductivity of conductor cylinder, while increases less with the increase of the thickness of conductor cylinder. The conductor cylinder thickness mainly affects the performance of the F-v curve at uniform speed. The conductor cylinder thickness mainly affects the performance of the F-v curve at uniform speed. It has little effect on the resistance when there is high acceleration. The analysis found that this is caused by the skin effect of the eddy current. When the thickness of the conductor cylinder is small, the distribution of the eddy current field can still be more concentrated and uniform. However, as the thickness increases, the skin effect becomes more and more obvious, and the eddy current will be concentrated on the inner side, so even if the cross-sectional area increases, the overall eddy current will not increase. The same applies to the change in eddy current field due to high acceleration.

Figure 9.

F-v curves of braking process at different $σ_{c c}$ .

Figure 10.

F-v curves of braking process at different $t h_{c c}$ .

Meanwhile, as can be seen from Figures 11 and 12, the effect of the conductivity and thickness of iron back on the F-v curve of the braking process is not homogeneous. First, the thickness of iron back obviously has an optimal size, above or below which cuts down the electromagnetic force, and the change of electromagnetic resistance for high acceleration also has little correlation with the thickness of the outer barrel. Secondly, according to the F-v curves of different iron back conductivity, it can be found that conductivity smaller than a certain value drastically reduce the effect of high acceleration on the F-v curve.

Figure 11.

F-v curves of braking process at different $σ_{i b}$ .

Figure 12.

F-v curves of braking process at different $t h_{i b}$ .

According to the above discussion, it can be considered that there is a large correlation between the conductor cylinder and iron back conductivity and the effect of high acceleration on the F-v curve.

The ECB FEM model with constant acceleration is used to calculate the F-v curves for different conductor cylinder and iron back conductivity with different accelerations. The F-v curves for different conductor cylinder and iron back conductivity with acceleration of 2000 m/s² and 0 m/s² are given in Figures 13,14.

Figure 13.

F-v curves with high acceleration at different $σ_{c c}$ .

Figure 14.

F-v curves with high acceleration at different $σ_{i b}$ .

It can be seen that the high acceleration amplifies the effect of the inner cylinder conductivity on the F-v curve, and the F-v curves at high acceleration all have certain hysteresis compared to the curves with no acceleration. At the same time, the effect of high acceleration on the F-v curve is gradually lagged to the higher speed region as the iron back conductivity decreases, and the value does not change much. This explains the Figure 11, in which the effect of high acceleration on the electromagnetic resistance would be insignificant if the high speed and high acceleration motion state is not reached when the iron back conductivity is small.

Electromotive force induced by motion and magnetic field variation

It is generally accepted that the eddy currents in the secondary of ECB arise from the induced electromotive force (e.m.f.) generated by the relative motion of the primary and secondary, i.e., the induced e.m.f. generated by the motion of a conductor in a uniform magnetic field. It is considered that the eddy currents in the secondary under uniform speed condition are generated only by the e.m.f. induced by relative motion. And when there is acceleration during the relative motion, the magnetic field will change with the variation of speed, and there will be eddy currents generated by the e.m.f. induced by the variation of magnetic flux density in the secondary at the same time, as shown in the following equation. However, it is only when the acceleration is very large that this kind of e.m.f. has a large effect on the eddy currents, causing significant discrepancies in the calculation of the electromagnetic resistance considering only the first kind of induced e.m.f.

ε = \int (v \times B) d l - \int \int \frac{\partial B}{\partial t} d S

(5)

The calculation formula of e.m.f. induced by the variation of magnetic flux density was given by Eq. (6), i.e., the induced e.m.f. can be obtained by calculating the partial derivative of the magnetic flux or magnetic vector potential with respect to time.

ε = \oint E_{i} d l = - \int \int \frac{\partial B}{\partial t} d S - \frac{\partial}{\partial t} \int \int (\nabla \times A) d S = - \oint \frac{\partial A}{\partial t} d l

(6)

However, the distinction and combined calculation of the two kinds of induced e.m.f. in ECB is not so simple and clear. It can be said that the eddy currents in the ECB all originate from the e.m.f. induced by the variation of magnetic flux density. In fact, when we look at an electromagnetic induction problem, the first kind of induced e.m.f. can become the second kind, or vice versa, as the reference system is transformed.

In this study, it depends on whether the observer's perspective is stationary relative to the primary or the secondary. For the sake of study and calculation, the perspective in this paper remains relatively stationary with respect to the primary.

Figure 15 shows the distribution of the radial and axial components of the magnetic flux density (B_r/B_z) at the middle position of the conductor cylinder of an ECB FEM model in the axial direction when the relative motion velocity is constant. The z-coordinate in the graph can be considered as time in order to show the change of the magnetic flux density at a certain position, and the induced e.m.f. can be calculated this way. However, to use this method one needs to calculate the magnetic flux or magnetic vector potential and then solve for the derivative, which is more complicated to calculate, especially when compared to calculating the product of magnetic flux density and velocity.

E = v \times B

(7)

where E is electric field intensity.

Figure 15.

Axial distribution of B_r and B_z.

Considering that the magnetic flux density does not change with time, by multiplying the magnetic flux density vector with the relative motion velocity vector, the electric field intensity of the induced e.m.f. was obtained. The magnetic flux density has only radial and axial components, and the axial direction is parallel to the direction of the velocity, so only the product of B_r and v was calculated. In Figure 16, E stands for the actual electric field strength, and E_m stands for the electric field strength calculated using only Eq. (7). The black part is under uniform speed, and the gray part is with acceleration, and it can be seen that the two electric field strength are almost identical under uniform speed, and have some small differences when there is acceleration.

Figure 16.

Axial distribution of B_r E_m and E(black: a = 0 m/s²; gray: a = 1000 m/s²).

During the actual braking of an ECB, there is no doubt that acceleration exists, i.e., B_r and B_z will change with respect to z, as shown in Figures 17,18. However, when the acceleration is small, the change in B before and after d t is almost negligible until the acceleration becomes gradually larger and the difference will slowly become nonnegligible.

Figure 17.

B_r under different velocity.

Figure 18.

B_z under different velocity.

By extracting and calculating the electromagnetic field data in the ECB FEM model, the physics principle and formula derivation discussed previously can be verified. Equations (5)-(6) calculates the induced e.m.f. and is derived by considering the primary as stationary. However, the physical quantity that is more convenient for the calculate and analyze is the induced electric field strength, moreover, the FEM motion model was set up to be consistent with that of the actual braking machinery, with the primary in motion and the secondary at rest. Therefore, the induced electric field strength E_i induced from the variation of the magnetic field when the primary is considered to be at rest was needed, i.e., the variated magnetic field should be located at the corresponding positions before and after dt.

E_{i} = - \frac{\partial A}{\partial t} = \frac{A_{x - d x} - A_{x}}{d t}

(8)

E = v \times B + E_{i} = - v B_{r} + E_{i} = E_{m} + E_{i}

(9)

Based on one ECB FEM model, the magnetic flux density, magnetic potential vector and electric field strength data of the middle layer of the conductor cylinder at a certain moment (v = 8 m/s, a = 2000 m/s²) were extracted and calculated to obtain the results as shown in Figure 19, which can verify the above discussion.

Figure 19.

Axial distribution of E_m, E_i, E_m+ E_i and E_FEM (v = 8 m/s, a = 2000 m/s²).

However, it can be seen in the figure that there is a clear positional and numerical difference between the electric field strength under acceleration and the electric field strength calculated by considering only the uniform velocity (E_m), but the difference is not large. In fact, compared with the obvious numerical difference, high acceleration actually has an overall offset effect on the eddy current electric field, which is difficult to be illustrated completely and obviously by the expressions. Therefore, the analysis will continue with the electromagnetic field results obtained from FEM models.

Comparative analysis of eddy current electric field strength distribution using FEM models calculation results

In order to have a clearer view at the effect of high acceleration on the overall eddy current more clearly, contour plots of the electric field strength module (normE) and current density module (normJ) in the secondary of an ECB model at a velocity of 9 m/s for accelerations of 2000 m/s² and 0 m/s² were given. According to the FEM results, the maximum value of the normE did not change with the high acceleration, which were both 18.2 V/m, but it can be clearly seen that both the electric field and the current are somewhat different in terms of their sparsity and overall shape. This can also correspond to the analysis at Figure 19.

Comparing the two contour plots in Figure 20, it can be seen that the electric field strength in the lower right region of the iron back increases significantly when there is high acceleration, while the region of the stronger electric field in the iron back also becomes larger. In addition, in the region where the electric field strength is concentrated higher in the conductor cylinder, the distribution of contours at the acceleration of 2000 m/s² is slightly denser compared to that at a uniform speed. Although it does not appear to be obvious, this small change in the electric field strength in the conductor cylinder can be hugely amplified in the electric current distribution, which is because there is an item of σ·E in the calculation of it.

Figure 20.

Contour plots of normE with high acceleration.

Comparing the current density contour plots, it can be seen that the variation in shape is similar to that of the electric field strength (Figure 21). It can also be seen that since the conductivity of the iron back is less than that of the conductor cylinder, the current density in iron back is relatively small compared to conductor cylinder. Therefore, in the discussion of analytical modeling, other region in iron back besides the approximate triangular region with the stronger electric field strength can be neglected.

Figure 21.

Contour plots of normJ with high acceleration.

In addition, in order to analyze the effect of high acceleration on the eddy currents at different velocities, comparison of the contours of normE at velocities of 3 m/s and 15 m/s were also given in Figures 22 and 23. For the velocity of 15 m/s, the highest value of the normE in conductor cylinder is 24.8 V/m with high acceleration and 23.5 V/m at a uniform velocity. The variation is similar to that at the velocity of 9 m/s, but the effect caused by the acceleration is significantly reduced. This result can also be corresponded to Figure 8, where the electromagnetic resistance at higher velocities with high accelerations is only slightly larger than that at uniform velocities.

Figure 22.

Contour plots of normE (v = 3 m/s).

Figure 23.

Contour plots of normE (v = 15 m/s).

For the velocity of 3 m/s, it can be seen that the electric field at the acceleration of 2000 m/s² is significantly reduced compared to that at uniform velocity, with a maximum value of 7.11 A/m at uniform velocity and 4.5 A/m at the acceleration of 2000 m/s². This result corresponds to the low velocity period in Figure 8.

This is because to calculate the e.m.f. induced by the variation of magnetic field at a given position, it is necessary to take the radius from that position to the corresponding axis, rotate it to obtain a circular cross-section, and calculate the change in magnetic flux (axial) passing through that circular cross-section. And the distribution of the magnetic field in the ECB is gradually tilted to the direction opposite to the direction of motion as the velocity becomes larger from the velocity of zero until the strength and tilted shape of the magnetic field are gradually fixed (as in Figure 25) after the velocity reaches about 9 m/s. That is, when the velocity is small, the high acceleration leads to a greater change in the magnetic field, and the resulting this kind of induced e.m.f. has a relatively greater effect on the reduction of the overall eddy current electric field. After velocities approximately greater than 9 m/s, the effect of this kind of induced e.m.f. on the primary induced electric field becomes more minor and presents and more of the deformation of the overall electric field and then some increase in the local electric field strength.

Analytical and numerical modeling of ECB considering high acceleration

Analytical model of ECB without considering acceleration

A common approach to analyzing this type of electromagnetic problem is the MEC model, in which the segments of the magnetic circuit are combined and simplified into uniform flux tubes, and then the electromagnetic field is transformed into a model similar to the current diagram, as is shown in Figure 24.

Figure 24.

Schematic diagram (a) and the MEC model and (b) of ECB.

Each R_x element refers to the reluctance of the corresponding part x of magnetic circuit. R_a refers to the reluctance of air gap. Ф_m, Ф_δ, and Ф_r are the magnetic flux pass through PM, air gap and the shaft. F_c is the magnetomotive force generated by PM. F_I is the magnetomotive force (MMF) generated by eddy current. When the relative velocity of the primary and secondary is zero, F_I =0, and this state is called no-load condition.

R = l/(sμ₀μ_r) is used to calculate the reluctance in the MEC, l is the average path, and s can be calculated by dividing the volume of the flux path by the length of the average path. According to the magnetic equivalent circuit and the Kirchhoff magnetic potential difference law, equations can be get:

{\begin{aligned} F_{A B} = F_{c} - R_{m} Φ_{m} = (4 R_{1} + R_{5}) Φ_{r} = R_{m m} Φ_{r} \\ F_{A B} = (4 R_{2} + 2 R_{n} + 4 R_{3} + R_{4}) Φ_{δ} + F_{I} = R_{r} Φ_{δ} \\ Φ_{m} = Φ_{r} + Φ_{δ} \\ F_{c} = H_{c} d_{m} \end{aligned}

(10)

where H_c is coercive force of permanent magnet. The air gap magnetic flux density under no-load condition can be derived from (10):

B_{a g} = \frac{Φ_{δ}}{π d_{s} (r_{i p} + r_{i})} = \frac{(R_{r} + R_{m m}) F_{c}}{π d_{s} (r_{i p} + r_{i}) (R_{r} R_{m m} + R_{r} R_{m} + R_{m} R_{m m})}

(11)

The magnetic flux density at other position in conductor cylinder or the back iron can be calculated similarly.

Calculation of eddy currents and electromagnetic resistance in the secondary needs to obtain the electric field strength and current density in the conductor cylinder and back iron corresponding to each iron pole. An equivalent electric field method is used to simplify the calculation of the electric field and current in the secondary, i.e., the eddy current electric field is equated to three regions of uniform field strength.

As shown in Figure 25, the parts of the conductor cylinder and the back iron in which the electric field strength is stronger and more uniform are designated as part1 and part2 (orange region), and their mean values of electric field strength are obtained by the B_r value in the center of the region and the calculation of Eq. (7). The upper and lower parts of the conductive cylinder are combined and viewed as part3 (yellow region), in which the electric field strength is uniformly attenuated to zero, and its average value of electric field strength is obtained through multiplying the average value of electric field strength in region 1 by k_E. With (12)–(14), eddy current can be obtained.

E_{2} = k_{E} E_{1}

(12)

J_{1} = σ_{i} E_{1} J_{2} = σ_{b} E_{2} J_{3} = σ_{i} E_{3}

(13)

I = J_{1} S_{1} + J_{2} S_{2} + J_{3} S_{3}

(14)

Figure 25.

Schematic diagram of the equivalent electric field method.

Figure 26.

Electromagnetic resistance versus velocity obtained by FEM and proposed model.

where, σ_i, σ_b are conductivity of conductor cylinder and back iron, J₁, J₂, J₃ are average current density of three regions. The velocity-dependent parameters k_E and k_I were obtained from reference.¹⁹ The electromagnetic resistance of ECB under no-acceleration condition can be calculated through (15)–(17) (Figure 26).

F_{I} = k_{I} I

(15)

P_{1} = \frac{J_{1}^{2}}{σ_{i}} V_{1} P_{2} = \frac{J_{2}^{2}}{σ_{b}} V_{2} P_{3} = \frac{J_{3}^{2}}{σ_{i}} V_{3}

(16)

F = \frac{n_{p m} (P_{1} + P_{2} + P_{3})}{v}

(17)

where, n_pm is the number of PM.

Numerical modeling of the effect of acceleration on eddy current electric fields

After studying the principle and performance of high acceleration on eddy current electric field and electromagnetic resistance, an acceleration-related variable was proposed to be added into the simplified analytical model of ECB to represent the effect of high acceleration in the high impulse braking process of ECB. It was expected to be able to predict the peak perturbation of electromagnetic resistance due to high acceleration at the beginning of the impact braking process. If an effective numerical model can be obtained, the ECB can be optimized to achieve a better braking effect.

According to Figure 27, it can be seen that in the small speed, the value of electromagnetic resistance with high acceleration is smaller than the value of uniform speed; in the approximate range of speed 5 m/s–10 m/s, the value of electromagnetic resistance with high acceleration will be larger than the value of uniform speed; and in the continuous increase of speed, the value of electromagnetic resistance with high acceleration gradually tends to be slightly higher than the value of electromagnetic resistance with uniform speed. At the same time, a trend can be seen that the junction point between F-v with high acceleration and F-v with uniform velocity increases with increasing acceleration.

Figure 27.

F-v curves with different high accelerations.

Subtracting the F-v curve with acceleration from the F-v curve with uniform velocity gave the difference curve as in Figure 28. A variable ΔE considering high acceleration was proposed in this paper, which is added to (10), as shown in (15). This increment can represent the change in electric field in the secondary analyzed in section Ⅲ.C, and can also indicate the effect of the conductivity of the conductor cylinder and iron back in conjunction with the high acceleration on the electromagnetic resistance. The ΔE was assumed to be fitted to the sinusoidal curve in Figure 28, as shown in (16). The portion of the sinusoidal curve for velocities greater than one cycle of the sinusoidal curve is assumed to be ignored because it is difficult to achieve such a high acceleration and high velocity braking scenario in practice.

Figure 28.

Difference curve and schematic of the fitted sinusoidal curve for the increments.

With other parameters kept constant, FEM models were established for different high accelerations with different $σ_{c c}$ and $σ_{i b}$ , as shown in Table 1. The data of amplitude A_a and phase B_a of ΔE were obtained. Correlation analysis showed that these two parameters are not only strongly correlated with the critical velocity and high acceleration, but also have a high correlation with the conductivity of iron back. The equation for their fitting will be given in the appendix.

J_{1} = σ_{i} E_{1} (1 + Δ E), J_{2} = σ_{b} E_{2} (1 + Δ E), J_{3} = σ_{i} E_{3} (1 + Δ E)

(18)

Δ E = A_{a} \sin (\frac{2 π v}{3 B_{a}} - \frac{2 π}{3})

(19)

A_{a} = f (a, v_{c}, σ_{i p})

(20)

B_{a} = g (a, v_{c}, σ_{i p})

(21)

where v_c refers to the critical velocity the speed at which the electromagnetic resistance is maximal.

Table 1.

Parameters for FEM models.

Parameter	value	Parameter	value	Parameter	value
Thickness of PM	20 mm	Thickness of iron pole	8.5 mm	$t h_{c c}$	1.5 mm
Radius of the shaft	32 mm	Outer radius of PM	86.5 mm	$t h_{i b}$	8.5 mm
Outer radius of iron pole	88 mm	Air Gap Width	0.5 mm	Pole number	18
Remanence of PM	1.45 T

Experiments and validation of the proposed model

A prototype of ECB brake machine was constructed for experimental validation with the material and structural parameters shown in Table 2. The force sensor was installed at the end of the moving rod in the direction of motion, and the eddy current resistance of the ECB was obtained by subtracting the inertia force and friction force from the data measured by the force sensor. The displacement, velocity and acceleration during braking were also obtained using high-speed photography. The resistance of reentry mechanism and impact loads of the experiments are consistent with those given in section Ⅱ.A. A FEM model was established using the same material and structural parameters. The computational results of the experiments, FEM model, and proposed model are obtained for comparison and analysis, and the data comparisons of electromagnetic resistance, acceleration, velocity, and displacement are given as shown in the following figures.

Table 2.

Parameters for the prototype.

Parameter	value	Parameter	value	Parameter	value
Thickness of PM	20 mm	Thickness of iron pole	8.5 mm	$σ_{c c}$	27 MS/m
Radius of the shaft	23 mm	Outer radius of PM	87 mm	$σ_{i b}$	8 MS/m
Outer radius of iron pole	89.5 mm	Air Gap Width	0.5 mm	Remanence of PM	1.45 T
$t h_{c c}$	1.5 mm	$t h_{i b}$	8 mm	Pole number	37

Comparing the experimental, FEM and proposed model results of the braking process for two sets of load force, it can be seen that the results are in good agreement. Comparing the speed and displacement results, as in Figures 29–32, the data obtained from the experiments have numerical fluctuations, and the experimental displacement is slightly smaller than that of the other two models results at load 2. But in general, the FEM model and proposed model can simulate a more realistic and reasonable motion process.

Figure 29.

Electromagnetic resistance and acceleration results for load 1.

Figure 30.

Electromagnetic resistance and acceleration results for load 2.

Figure 31.

Velocity and displacement results for load 1.

Figure 32.

Velocity and displacement results for load 2.

The data of acceleration over time were not measured in the experiment, but the results of the proposed model can be verified with the FEM model results. In addition, the velocities in the experimental data were obtained by deriving the data acquisition results for the displacements. However, the experiment was subjected to a strong shock load, and both the shock of the actual machinery and the shock of the sensor results are possible, and neither the finite element model nor the theoretical model can describe this situation at present.

The electromagnetic resistance under two loads shows that the experimentally obtained force is still slightly larger than that of the FEM and the proposed model in the early stage, but the overall value and the tendency are basically the same. At the same time, the computational results of both models also show the resistance peak due to high acceleration in the early stage of braking, but still do not fully show the same resistance peak as the experimental results. To analyze this peak gap, it may be due to the overall large size of the ECB, which generates a large additional friction force when subjected to strong impact loads and cannot be predicted using a simple friction force formula. Secondly, the force sensor output results may fluctuate due to the short time but large impulse during the experimental measurements, especially during the high impact phase. Nevertheless, this proposed analytical and numerical model as well as the FEM model are still considered to be able to simulate the braking process of the ECB when subjected to strong impact loads in a good degree.

Conclusion

In this paper, the change of electromagnetic field under high acceleration was analyzed. Under the operating condition of high acceleration, the eddy current in the ECB not only came from the relative motion between the primary and the secondary, but also affected by the electromotive force induced by the change of magnetic flux due to the change of speed.

FEM models were established to compare the electric field strength distribution with and without acceleration. The comparison shows that when the electromagnetic resistance nearly grows linearly with velocity, the effect of high acceleration on eddy currents is more in the form of weakening of the original eddy current by the second type of induced e.m.f. When the velocity is around the critical velocity, the effect of high acceleration on the eddy currents is more manifested in the compression and deformation of the eddy currents in the secondary. And at higher speeds, the effect of high acceleration on the eddy currents is weaker and only slightly increases the electromagnetic resistance.

In order to be able to better predict the braking force of the ECB under strong impact loads, an incremental ΔE considering acceleration was introduced according to a simplified analytical model combining the MEC model with the equivalent electric field approach. The increment can represent both the influence of high acceleration and the influence of the conductivity of the conductor cylinder and iron back on the electromagnetic resistance. Numerical modeling was performed based on the data obtained from FEM models, and the final model obtained was verified by comparison with experiments results.

Although the model does not use the Maxwell's equations for analytical modeling, it can present the effects of each design parameter on the braking force and the results of the braking process, and the model is fast to compute. Future research can use this model to optimize the design of the ECB mechanism to obtain a more efficient braking force and a smoother braking process. In addition, the model can also be analyzed and modeled in a more refined way to obtain more accurate prediction results.

Footnotes

Ackhowledgement

This research was financially supported by the “China National Postdoctoral Program for Innovative Talents” [Grant No. BX20230493], the “National Natural Science Foundation of China” [Grant No. 52305155], the “Jiangsu Province Natural Science Foundation” [Grant No. BK20230904], the “State Key Laboratory Open Fund” [Grant No. 2024-JSS-GF-095-02], the “Postdoctoral Fellowship Program of CPS” [Project no. GZB20240979], the “Jiangsu Province Excellent Postdoctoral Program” [Project no. 2024ZB064]. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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.

Appendix

A_{a} = C 20 a^{2} + C 10 a + C 00 + C 02^{'} v_{c}^{2} + C 11^{'} a v_{c} + C 01^{'} v_{c} + C 02^{″} σ_{i p}^{2} + C 11^{″} a σ_{i p} + C 01^{″} σ_{i p}

B_{a} = D 20 a^{2} + D 10 a + D 00 + D 11^{'} a v_{c} + D 01^{'} v_{c} + (σ_{i p} - D D) (D 11^{″} a + D 02^{″} σ_{i p} + D 01^{″})

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