Braking force characteristics and multi-objective optimization of eddy current brake under intensive impact load

Abstract

As a new type of permanent magnet arrangement in the eddy current brake (ECB), the Halbach array is gradually gaining attention thanks to its excellent magnetic field utilization. However, the absence of an accurate theoretical model causes great distress in the design phase of the ECB. To tackle this, this article establishes a subdomain model that can intuitively observe the influence of structural and material parameters on the magnetic field and eddy current braking force. The accuracy of the analytical model is verified by the finite element method (FEM). The calculation results show that the eddy current braking force is the largest when the relative velocity of the primary and secondary reaches the critical velocity value. The recoil equation of motion is introduced to analyze the change law of the braking force under intensive impact load. A small prototype impact test platform was set up to analyze the change of braking characteristics of the ECB under the impact load. To reduce the weight of the ECB, the multi-objective optimization of the ECB parameters is carried out. The vital structural mass of the ECB is reduced by 19.44% while meeting the recoil requirements.

Keywords

Eddy current brake subdomain model braking force intensive impact load multi-objective optimization

1. Introduction

Eddy current brake (ECB) is an emergency braking system with the advantages of good controllability, no pollution and no contact between the primary motivation and secondary motivation. For decades, eddy current brakes have been used in many applications, such as vibration suppression [1,2], high-speed train [3,4], aerospace applications [5,6] and bridge construction [7,8]. ECB is based on the interaction between a conductor and a magnetic field in relative motion. According to Lenz’s law, the induced magnetic field generated by eddy currents is opposite to the direction of the source magnetic field and always has a tendency to impede relative motion. The braking characteristics of ECB are affected by many factors such as structural and material parameters. In some special occasions such as aerospace and weaponry, it is necessary to reduce the overall volume and vital structural mass of the ECB while satisfying the braking force. Therefore, it is significant to analyze the braking characteristics and perform multi-objective optimization.

The magnetic field and braking force characteristics of ECB can be researched by analytical method, finite element method (FEM) and experimental method. The analytical methods are less time-consuming and dependent on design parameters, so it can provide theoretical support in the initial design stage of the ECB. The commonly used analytical methods are subdomain model method, magnetic equivalent circuit (MEC) method and equivalent current method. Jin et al. [9] established a subdomain analytical model of permanent magnet linear eddy current brake considering static and dynamic transverse edge effects. The accuracy of analytical model was verified by the FEM, and it was also found that the 2D and 3D finite element results were similar. Mohammadi et al. [10] described the magnetic circuit characteristics of the permanent-magnetic couplers considering the magnetic saturation condition by the MEC. Bae et al. [11] used the equivalent current method to analyze the movement of the permanent magnet in the conductor tube. This method could observe the relationship between electromagnetic parameters and braking characteristics, but could not perform analytical integration. Li et al. [12] established the subdomain model of ECB, and concluded that the Halbach array has better braking characteristics under intensive impact load. The subdomain model method was based on a simplified layering of the structure, and the permanent magnet and the conductor tube were assumed to be infinite long in the direction of motion. This assumption made the calculated results deviate from the actual results, so the analytical method inevitably has limitations. Safaeian et al. [13] analyzed the influence of the conductivity and permeability of the back iron on the stiffness and damping of the configuration. Because ECB has inherently nonlinear characteristics, the subdomain model that doesn’t take into account the saturation has a limited accuracy. Djelloul-Khedda et al. [14] and Kou et al. [15] proposed an analytical model based on the magnetic saturation technique to overcome this limitation. But it would increase the computation time to calculate the nonlinear constitutive model by iteration. The nonlinear constitutive model of the ECB calculated by FEM could intuitively observe the magnetic field and braking characteristics, and could also observe the change of the magnetic field under different motion conditions [16,17]. In this paper, each component of ECB is divided into independent layers based on its unique cylindrical structure using the subdomain model method. The analytical expression of magnetic field in each region is solved using Maxwell’s theory and the boundary conditions between each layer. The subdomain model can directly observe the effects of key parameters on the braking force characteristics of ECB. The analytical model can reduce the workload in the pre-design of ECB and provide a theoretical basis for the subsequent study of the braking force characteristics of ECB under intensive impact load.

Most of the current literature researched on eddy current brakes were based on the braking characteristics at low speeds, such as the calculation of braking coefficients in suppressing vibrations of sensors [18] and low speed vibrations in heavy vehicles [19]. Applications of eddy current brakes for intensive impact loads [20–22] have been widely investigated for several years. Intensive impact load is a high-acceleration and time-varying motion process with large impact force and short duration. Li et al. [23] investigated the effect of irreversible demagnetization under high temperature and high velocity during the electromagnetic buffering. Ge et al. [24] proposed a double-layer permanent magnet combined with a spring buffer. Unfortunately, it could only be applied to low-acceleration movements due to structural limitations. Under the action of intensive impact load, the ECB moves at high speed. Regarding the relationship between velocity and eddy current braking force, Thompson et al. [25] found that there was a speed-related peak point in the braking force curve. The speed corresponding to the peak eddy current braking force was called the critical speed. When the speed exceeds the critical speed, the braking force would no longer increase after the speed exceeded the critical speed. When designing the ECB, ensure that the maximum speed would not exceed the critical speed under intensive impact load. Under the action of intensive impact load, ECB inevitably suffer from magnetic saturation and eddy current demagnetization due to the nonlinear constitutive relations of the material when generating braking force. The influences of magnetic saturation and eddy current demagnetization on braking force characteristics generated by ECB in the recoil process are analyzed. To reduce the overall mass of ECB and improve the utilization of permanent magnets, a multi-objective optimization of ECB is performed.

Although the FEM and analytical method can observe the change of the eddy current braking force of ECB, it is time-consuming to directly optimize the ECB. A surrogate model could map a complex problem to an approximate model that can be solve quickly [26]. A surrogate model is essentially a simple mathematical model. Its main feature is to construct a computational result that can replace the original model by computing a small number of model samples. By combining the surrogate model with the optimization algorithm, the optimal solution of the ECB could be quickly solved. Xu et al. [27] proposed a multi-objective optimization design method for electromagnetic buffer based on Nash game theory in order to stabilize the buffering process. Asl et al. [5] applied a multi-objective genetic algorithm optimization to find the ECB structural parameters and geometry to achieve the desired torque-speed characteristic while minimizing the overall weight of the ECB. Mohammadi et al. [28] applied a genetic algorithm to maximize the torque produced by an eddy current coupler while trying to reduce the permanent magnet volume and the moment of inertia of the conductive sheet. The braking characteristics of ECB are influenced by several key structural parameters. To obtain the optimal braking characteristics of ECB within a certain rang in a short time, a surrogate model is built to replace the FEM computational process for multi-objective optimization. Therefore, the multi-objective optimization of ECB can obtain better braking characteristics, and the surrogate model can improve the optimization efficiency.

This article mainly analyzes the eddy current braking characteristics of ECB and performs multi-objective optimization. The remainder of this article is divided into the following sections. In Section 2, the subdomain model and finite element model of ECB are established to analyze its magnetic field and braking characteristics. The change of the magnetic field during the recoil process of ECB is introduced, which provide a theoretical basis for its application under intensive impact load. In Section 3, the magnetic field and eddy current braking force of ECB under intensive impact load are studied. The B-H curve is introduced to analyze the influence of the nonlinear constitutive model on the eddy current braking force. A small prototype is set up to observe the changing law of the eddy current braking force under intensive impact load. In Section 4, a multi-objective optimization algorithm is employed to minimize vital structural mass of the ECB while an attempt is made to reduce the braking force and recoil displacement. Finally, Section 5. will summarize the full text. The structural of the paper is shown in Fig. 1.

Fig. 1.

The structure of the paper.

2. Eddy current brake model

2.1. Structure and principle

The structure of the tubular linear permanent magnet of eddy current brake with Halbach array is shown in Fig. 2. ECB can be divided into primary part and secondary part. The primary part is composed of Halbach permanent magnet array, thermal insulation cylinder, inner cylinder and some fixed parts (such as nuts). The secondary part is composed of conductor cylinder and back iron. The conductor cylinder is combined with the back iron to form a composite cylinder. The material of the conductor cylinder is aluminum which has high conductivity. The material of the back iron is magnetically conductive steel which is used to provide the external loop of the magnetic field, improving the utilization rate of the source magnetic field. Halbach permanent magnet array can increase the flux density on the working side and decrease the flux density on the other side compared with the ordinary radial permanent magnet array. At the same time, it can avoid the influence of tape leakage in the non-working area. The permanent magnets are mounted on the inner cylinder.

As the composite cylinder moves, eddy currents are created in the conductor cylinder, which will generate a strong magnetic field. This strong magnetic field will interact with the magnetic field generated by the permanent magnets. According to Lenz’s law, this magnetic field will hinder the movements of the primary part. To analyze the magnetic field of ECB, subdomain analytical model and finite element model are established.

Fig. 2.

Structure of the ECB with Halbach array.

2.2. Subdomain model

The simplified model of eddy current brake is shown in Fig. 3. When applying the subdomain model method to solve the magnetic field performance, the analytical model is divided into six solving regions.

Fig. 3.

2D simplified eddy current brake model.

In establishing analytical solutions for the magnetic field distribution, the following assumptions are made:

The ECB in the Fig. 3 is a cylinder with z-axis as rotation axis. Region III is the Halbach permanent array, and the direction is the magnetization direction of the permanent magnet.

The model is infinite long in the z direction. The magnetic field distribution is periodically varying in the z-direction.

The magnetic induction intensity of air gap is constant in the direction of 𝜙, and the end effects is not considered.

The relative permeability of the conductor cylinder and the PM is supposed unity, μ_r = 1.

Consequently, the analytical model can be confined to six regions along the r-direction, namely, the inner cylinder layer (region I) and the thermal insulation cylinder layer (region II) where the permeability is μ_1,2, the back iron layer (region VI) where the permeability is μ₆. Assuming that eddy currents are generated in the relevant region, so the magnetic field variation is described by introducing vector magnetic potential A (using a Coulomb gauge 𝛁 ⋅ A = 0), namely, B = 𝛁 × A , where B is the magnetic intensity. The governing equations are obtained by Maxwell’s equations: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times \boldsymbol{H}=\boldsymbol{J}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{B}={\mu}_{0}{\mu}_{r}\boldsymbol{H}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{J}={\sigma}(\boldsymbol{E}+v\times \boldsymbol{B})\end{eqnarray}$ (4) where H , J , B and E denote the magnetic field intensity, electric current density, magnetic flux density and electric field intensity, μ₀, μ_r, v and σ are the air permeability, relative permeability, velocity (Assume that the conductor cylinder and back iron are moving at a relative velocity v) and conductivity.

In addition, the constitutive relation of PM can be described by residual magnetism. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{B}={\mu}_{0}\boldsymbol{H}+\boldsymbol{B}_{\mathit{res}}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{B}_{\mathit{res}}=\boldsymbol{B}_{\mathit{resr}}+\boldsymbol{B}_{\mathit{resz}}\end{eqnarray}$ (6) where B _resr and B _resz denote the remanence of B in the radial and axial directions, respectively. Figure 4 shows the relationship of B _r and B _z, which can be expanded by Fourier series of the forms: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \boldsymbol{B}_{\mathit{resr}}(z)=\mathop{\sum }_{n=1}^{\infty }\boldsymbol{B}_{\mathit{resr}}^{n}\cdot e^{jmz}\quad \\[6.0pt] \displaystyle \boldsymbol{B}_{\mathit{resz}}(z)=\mathop{\sum }_{n=1}^{\infty }\boldsymbol{B}_{\mathit{resz}}^{n}\cdot e^{jmz}\quad \end{array}\right.\end{eqnarray}$ (7) where m = nπ∕τ, τ is the pole pitch of the ECB.

Fig. 4.

Distribution of B _r and B _z.

Therefore, the governing equation of ECB’s magnetic field distribution can be described as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}^{2}\boldsymbol{A}^{\mathbf{I},\mathbf{II},\mathbf{IV },\mathbf{V I}}=0\quad \text{Region I,II,IV,VI},\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}^{2}\boldsymbol{A}^{\mathbf{III}}=-({\nabla}\times \boldsymbol{B}_{\mathit{res}})\quad \text{Region III},\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}^{2}\boldsymbol{A}^{\text{V}}={\mu}_{0}{\sigma}v\frac{\partial \boldsymbol{A}}{\partial z}\quad \text{Region V}.\end{eqnarray}$ (10)

The model of ECB is axisymmetric and the vector magnetic potential ( A can be expressed by Fourier series $A(r,z)=\sum _{n=1}^{\infty }A_{n}(r)\cdot e^{jmz}$ ) is only function of r and z, so Eqs (8)–(10) can be written in cylindrical coordinate system as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{d^{2}A_{n}^{\text{I},\text{II},\text{IV},\text{VI}}(r)}{dr^{2}}+\frac{dA_{n}^{\text{I},\text{II},\text{IV},\text{VI}}(r)}{rdr}-m^{2}A_{n}^{\text{I},\text{II},\text{IV},\text{VI}}(r)-\frac{A_{n}^{\text{I},\text{II},\text{IV},\text{VI}}(r)}{r^{2}}=0\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{d^{2}A_{n}^{\mathbf{III}}(r)}{dr^{2}}+\frac{dA_{n}^{\mathbf{III}}(r)}{rdr}-m^{2}A_{n}^{\mathbf{III}}(r)-\frac{A_{n}^{\mathbf{III}}(r)}{r^{2}}=-jmB_{\mathit{resr}}^{n}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{d^{2}A_{n}^{\mathbf{V}}(r)}{dr^{2}}+\frac{dA_{n}^{\mathbf{V}}(r)}{rdr}-m^{2}A_{n}^{\mathbf{V }}(r)-\frac{A_{n}^{\mathbf{V}}(r)}{r^{2}}=jm{\mu}_{0}{\sigma}vA_{n}^{\mathbf{V }}(r).\end{eqnarray}$ (13)

The analytical expression of the Laplace operator in cylindrical coordinates can solve the Eqs (11)–(13): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{I},\mathbf{II},\mathbf{IV },\mathbf{V I}}(r,z)=(a_{n}^{\mathbf{I},\mathbf{II},\mathbf{IV },\mathbf{V I}}I_{B}(r^{\prime })+b_{n}^{\mathbf{I},\mathbf{II},\mathbf{IV },\mathbf{V I}}K_{B}(r^{\prime }))e^{jmz}\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{III}}(r,z)=\left(a_{n}^{\mathbf{III}}I_{B}(r^{\prime })+b_{n}^{\mathbf{III}}K_{B}(r^{\prime })-\frac{jB_{\mathit{resr}}^{n}{\pi}S(r^{\prime })}{2m}\right)e^{jmz}\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{V }}(r,z)=(a_{n}^{\mathbf{V }}I_{B}(wr^{\prime })+b_{n}^{\mathbf{V }}K_{B}(-jwr^{\prime }))e^{jmz}\end{eqnarray}$ (16) where r ^′ = mr, $w=\sqrt{(ju_{0}v{\sigma}+m)/m}$ . $a_{n}^{i}$ and $b_{n}^{i}∼(i=\text{I},\text{II},\text{III},\text{IV},\text{V},\text{VI})$ are constants determined by boundary conditions, I _B and K _B are different classes of Bessel functions, and S (r ^′) is the solution of nonhomogeneous Bessel equation. At the boundary of adjacent regions, the normal component of flux density B _r is continuous, and the tangential component of magnetic field strength H _z is continuous. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\text{I}}(r_{0})=0\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{q}(r)=A_{n}^{q+\text{I}}(r)|r=r_{p}\quad (q=\text{I}\cdots \text{V},p=1\cdots 5)\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\text{VI}}(r)=0|r=+\infty\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{A_{n}^{\mathbf{I}}(r)+r{\displaystyle \frac{\partial A_{n}^{\mathbf{I}}(r)}{\partial r}}}{{\mu}_{1}}=\frac{A_{n}^{\mathbf{II}}(r)+r{\displaystyle \frac{\partial A_{n}^{\mathbf{II}}(r)}{\partial r}}}{{\mu}_{2}}|r=r_{1}\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{A_{n}^{\mathbf{II}}(r)+r{\displaystyle \frac{\partial A_{n}^{\mathbf{II}}(r)}{\partial r}}}{{\mu}_{2}}=A_{n}^{\mathbf{III}}(r)+r\frac{\partial A_{n}^{\mathbf{III}}(r)}{\partial r}-rB_{\mathit{resz}}|r=r_{2}\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{III}}(r)+r\frac{\partial A_{n}^{\mathbf{III}}(r)}{\partial r}-rB_{\mathit{resz}}=A_{n}^{\mathbf{III}}(r)+r\frac{\partial A_{n}^{\mathbf{III}}(r)}{\partial r}|r=r_{3}\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{IV }}(r)+r\frac{\partial A_{n}^{\mathbf{IV}}(r)}{\partial r}=A_{n}^{\mathbf{V }}(r)+r\frac{\partial A_{n}^{\mathbf{V}}(r)}{\partial r}|r=r_{4}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n}^{\mathbf{V }}(r)+r\frac{\partial A_{n}^{\mathbf{V}}(r)}{\partial r}=\frac{A_{n}^{\mathbf{VI}}(r)+r{\displaystyle \frac{\partial A_{n}^{\mathbf{VI}}(r)}{\partial r}}}{{\mu}_{6}}|r=r_{5}.\end{eqnarray}$ (24)

The magnetic flux density, braking force, and eddy current density in the conductor cylinder can be obtained by solving the magnetic vector potential. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle J_{n}^{\mathbf{V }}(r,z)=jm{\sigma}v[a_{n}^{\mathbf{V }}I_{B}(wr^{\prime })+b_{n}^{\mathbf{V }}K_{B}(-jwr^{\prime })]e^{jmz}\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{rn}^{\mathbf{V }}(r,z)=-jm[a_{n}^{\mathbf{V }}I_{B}(wr^{\prime })+b_{n}^{\mathbf{V }}K_{B}(-jwr^{\prime })]e^{jmz}\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F^{\mathbf{V }}=2{\pi}\int _{-{\tau}}^{{\tau}}\int _{r_{4}}^{r_{5}}B_{rn}^{\mathbf{V }}\times J_{n}^{\mathbf{V }}rdrdz.\end{eqnarray}$ (27)

The subdomain analytical model reveals the various key structural and material parameters that affect the magnetic field and braking characteristics of ECB. To further verify the accuracy of the analytical model, a finite element model is built to analyze the magnetic field.

2.3. Numerical simulation

The finite element method (FEM) is a numerical simulation method based on the differential principle, which divides the whole model solution region into many sub-regions. Then, according to the solution equation and boundary conditions, each sub-region is solved to obtain the solution of the whole solution region.

Fig. 5.

Simplified 2-D numerical analysis model of ECB.

The FEM analysis is carried out by COMSOL Multiphysics software. Compared with 3-D simulation, 2-D simulation consumes less computer resources and requires shorter computing time. Since ECB is composed of multiple rotators, there is no significant difference between the 3-D and 2-D simulation results. Figure 5 shows the two-dimensional axisymmetric mesh model of ECB. Set the boundary condition of electromagnetic field and give the conductor cylinder and back iron a small acceleration to find its magnetic field variation in steady state. Ensure that material parameters and constitutive model settings are consistent with the analytical model. The structural parameters of FEM and analytical models are shown in Table 1.

Table 1

Parameters of ECB

Symbol	Description	Value
c	Axial length of radially magnetized PMs	20.0 mm
d	Axial length of axially magnetized PMs	15.0 mm
r ₀	Inner radius of inner cylinder	74.5 mm
r ₁	Inner radius of thermal insulation cylinder	133.5 mm
r ₂	Inner radius of PM cylinder	137.5 mm
r ₃	Outer radius of PM cylinder	167.0 mm
r ₄	Inner radius of conductor cylinder	169.0 mm
r ₅	Inner radius of back iron	170.5 mm
τ	Pole pitch	35.0 mm
B _res	Residual flux density of PMs	1.45 T
σ₁	Conductivity of conductor cylinder	38.74 MS/s
σ₂	Conductivity of back iron	4.40 MS/s
μ_1,2,6	Relative permeability	600

Fig. 6.

Magnetic flux density and Magnetic induction line distribution at static state.

Figure 6 shows the distribution of magnetic flux density and magnetic induction line of ECB at static state. ECB with Halbach array has a low flux density on the non-working side and a strong flux density on the working side, which can avoid serious magnetic leakage on the non-working side. The strong flux density is mainly concentrated in the radially magnetized PMs and its radial corresponding position. The magnetic induction line passes through the air gap, the conductor cylinder and the back iron to form a closed loop.

2.4. Model validation

It can be observed form Fig. 7 that the solution results of FEM and Analytical method at r = (r ₅ − r ₄)∕2 coincide well. The FEM can accurately describe the motion characteristics of ECB at static and moving conditions. Since the magnetic flux density at the conductor cylinder does not change much and the magnetic induction line is perpendicular to the conductor cylinder at static state, the magnetic flux density will present a platform effect in the radial direction. When the ECB is moving at a relative speed of 10 m/s, the static platform effect becomes an extreme point. An eddy current magnetic field will be generated when the eddy current is induced from the conductor cylinder. The eddy current magnetic field will react with the source magnetic field and synthesize a new magnetic field, which will change the flux density at the conductor cylinder.

Fig. 7.

Comparison of radial magnetic flux density B _r as a function of z, at r = (r ₅ − r ₄)∕2.

Fig. 8.

Comparison of braking force as function of v.

The computational results of the analytical method and the FEM of braking force changing with velocity are shown in Fig. 8. The braking force increases first and then decreases with the increase of recoil velocity, which can be divided into three regions from the whole curve. In the linear growth region, when the recoil velocity of ECB increases at a low speed, the braking force increases greatly. As the velocity increases, the eddy current in the conductor cylinder increases, so the braking force increases linearly. In the slow growth region, with the gradual increase of recoil speed, the increase speed of braking force becomes slow and finally reaches the peak value. When braking force reaches the peak value, the corresponding value of the recoil speed is called the critical speed. In the high-speed reduction area, when the recoil velocity exceeds the critical velocity, the eddy current density in the conductor cylinder increases continuously. The increase of eddy current makes the current demagnetization enhanced, which reduces the magnetic field intensity at the air gap and leads to the decrease of braking force. The edge effect has little influence on the braking force at low speed. Because the braking force generated by the weak magnetic flux density can meet the recoil requirements. But at high speed, the braking force needs stronger magnetic flux density to produce greater braking force, so the side effect should be considered at high speed.

Braking force and different recoil velocity corresponding to every harmonic values as shown in Fig. 9. When the harmonic number is higher, the braking force is smaller. The braking force generated by high harmonics is negligible compared with that generated by fundamental magnetic field, so only fundamental wave action is considered in the calculation of braking force.

Fig. 9.

Comparison of braking force as function of v and harmonic order.

This section analyzes the magnetic field and braking force characteristics of ECB to provide a theoretical guidance for the pre-design of ECB. To meet the braking requirements of ECB under intensive impact load, the recoil process of ECB needs to be studied to ensure that the peak force does not exceed the designed peak force and the maximum velocity does not exceed the critical velocity.

3. ECB under intensive impact load

The force analysis of ECB under intensive impact load is shown in Fig. 10. The impact load F _p is mainly derived from the breech combined force in the artillery weapon firing system. F _f is mainly derived from the recuperator in the artillery weapon firing system. The braking force F _e generated by ECB during recoil is analyzed by the recoil equation of motion. The function of the recuperator is to restore ECB to its original position. When subjected to intensive impact load F _p, the ECB generates eddy current braking force F _e and is subjected to recoil force F _f at the same time. The curve of force is shown in Fig. 11, and the equation of recoil process can be written as: $\begin{eqnarray}m\frac{dv}{dt}=F_{p}(t)-F_{e}(t)-F_{f}(t)\end{eqnarray}$ (28)

3.1. Braking characteristic analysis

According to the above analytical method and FEM results, the accuracy of the model has been verified. But the analytical method mainly adopts the linear constitutive model, which has high accuracy in steady state and low speed. To further analyze the influence of magnetic saturation on the braking force, the B-H curve of nonlinear constitutive relation is introduced, which reflects the change rule of magnetization degree of material with magnetic field. The eddy current generated in the back iron is not evenly distributed, but occurs on the surface close to the conductor cylinder due to the skin effect. Figure 12(a) shows that when the velocity reaches 10 m/s, the magnetic field lines are distorted due to the interaction between the magnetic field generated by the eddy current and the original magnetic field, and the direction of the distortion is opposite to the direction of the velocity. Figure 12(b) shows the change of eddy currents, which are mainly generated at the conductor cylinder. At the same time, the eddy currents are generated at the back iron due to the magnetic saturation of the back iron. The eddy current area in the magnetic saturation region varies with the relative velocity, as shown in Fig. 13. Therefore, the FEM is used to study the variation of eddy current barking force under intensive impact load.

Fig. 10.

Force analysis of ECB device.

Fig. 11.

Curves of Impact load and Recuperator load.

Fig. 12.

Magnetic flux density and Current density considering magnetic saturation.

Fig. 13.

Magnetic saturation under intensive impact loads.

Under the action of intensive impact load, the barking force generated by the ECB is shown in Fig. 14, and the peak value of the analytical braking force is about 530 kN. The maximum error of the results generated by FEM method and analytical method is 2.59%, which is mainly caused by edge-side effect and demagnetization effect. The subdomain analytical model does not consider edge-side effect, but assumes that the model has infinite length in z-direction. The edge-side effect causes the magnetic field at the end of ECB permanent magnet array to change, thus affect the performance of the braking. The demagnetization effect means that eddy currents generate a magnetic field, which couples with the original magnetic field. The new magnetic field reduces the magnetic flux density through the conductor, which in turn reduces the braking force. However, after introducing nonlinear constitutive model, the peak value of braking force is about 480 kN, which is caused by the increase of eddy current area, leading to the enhancement of demagnetization effect. During the recoil process, the maximum recoil displacement of the nonlinear constitutive model is about 744 mm, while the linear constitutive model is about 738 mm, and the maximum recoil speed is basically the same. The maximum recoil velocity generated by ECB is 12.4 m/s, which doesn’t exceed the critical velocity. It can be observed that the magnetic saturation has a great influence on the eddy current braking force, so the magnetic saturation should be considered under intensive impact load.

Fig. 14.

The braking force varying with the time.

3.2. Experiment

To verify the accuracy of the established finite element model and analytical model under intensive impact load, a small ECB prototype with Halbach array was made. The small ECB prototype was mainly composed of permanent magnet, stainless steel guide rod and aluminum alloy outer cylinder. The impact load simulation experiment platform is mainly composed of three parts: impact load generator, data processing unit and the perform unit, as shown in Fig. 15.

Fig. 15.

Impact load simulation experiment.

The working principle of impact experiment is that the air compressor pushes the air hammer to produce impact load to impact the mass block, and then the mask block moves along the rail. At the same time, the small ECB prototype is connected to the mass block to provide braking force. A force sensor is installed at the back of the mass block to detect the force received by the mass block and the ECB prototype during the impact load. The displacement sensor is used to detect the change of the displacement with time in the recoil process. The finite element model of small ECB prototype is established, and the modeling method is consistent with that of ECB. The model parameters of the small ECB prototype are shown in Table 2.

Table 2

Parameters of small ECB prototype

Description	Value	Description	Value
Conductor cylinder thickness	2 mm	Conductivity of conductor cylinder	28 MS/m
Air gap	2 mm	Residual flux density of PMs	0.9 T
Inner radius of PMs	12 mm	Back iron thickness	8 mm
Outer radius of PMs	39 mm	Radius of moving rod	12 mm
Magnet thickness	13.5 mm	Number of PMs	9

Fig. 16.

Analytically, numerically and experimentally calculated force-time relationship.

Subdomain model, FEM and experimental results of ECB under impact load are shown in Fig. 16 and Fig. 17. Since the force sensor is directly affected by the force when it receives the impact, the experimental value is not zero initially, and the data received by the force sensor is processed when ECB work. When ECB is subjected to the impact load, the braking force reaches the peak value in a short time, and the variation rule of the experimental values are consistent with the FEM and the analytical values. Through the above experiments, the FEM results and analytical results are consistent with the experimental results, and it can be considered that the established model can accurately describe ECB braking characteristics under intensive impact loads.

This section analyzes the braking force characteristics of ECB under intensive impact load. The accuracy of the analytical and FEM models are verified using experiments. A nonlinear constitutive model is introduced to study the variation of braking characteristics in the case of magnetic saturation. It can be seen that ECB do not reach the peak corresponding to the critical velocity under the intensive impact load. Therefore, to improve the utilization of permanent magnets and further reduce the overall mass of ECB, the multi-objective optimization of ECB.

Fig. 17.

Analytically, numerically and experimentally calculated force-velocity relationship.

4. Optimization of ECB

In this study, the optimization model is established by back-propagation (BP) neural network surrogate model. Because it requires a lot of simulation calculation to calculate the optimal braking force, recoil displacement and vital structural mass of ECB by the FEM. The surrogate model is a simple mathematical model, whose main characteristic is to construct a calculation result which can replace the original model through a small number of calculation model samples. The computation amount of this surrogate model is much less than that of the FEM, but the calculation accuracy can keep a high level. The optimization flowchart of the ECB surrogate model is shown in Fig. 18.

Fig. 18.

Flowchart of the ECB surrogate model.

The establishment of the surrogate model requires a large number of data training samples. To make the accuracy of the established surrogate model, the selection of samples should cover all the optimization scope. The selection of training samples should be purposeful and can represent sample information. At present, the commonly used sample selection methods mainly include DOE (Design of Experiment). The variation range of samples of ECB selected with consideration of machining difficulty and design rationality is shown in Table 3. The optimized variables are key structural parameters of ECB.

Table 3

Design variables

Symbol	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
Range/mm	25.5-33.5	15-25	0.5-5	0.5–5	12-18	12–18

Where x ₁ is the thickness of PMs; x ₂ is the axial length of radially magnetized PMs; x ₃ is the thickness of conductor cylinder; x ₄ is the air gap length; x ₅ is the thickness of back iron; x ₆ is the axial length of axially magnetized PMs;

Table 4

Correlation coefficient of the ECB surrogate model

Correlation coefficient	F _max	S _max	m _max
R ²	0.999	0.973	0.982

60 sets of samples are selected using the optimal Latin hypercube experiment design method, and are analyzed by FEM. Set up the structure and parameters of BP neural network. The training algorithm selected in this paper is Levenberg-Marquardt, which is characterized by large memory consumption but short training calculation time. Correlation coefficient R ² is commonly used in engineering to verify the accuracy of the surrogate model, and its calculation formula is (29). The closer R ² is to 1, the higher the accuracy of the surrogate model is. Table 4 shows the verification results of BP neural network surrogate model. $\begin{eqnarray}R^{2}=1-\frac{\displaystyle \mathop{\sum }_{i=1}^{n}(y^{i}-{\hat{y}}^{i})^{2}}{\displaystyle \mathop{\sum }_{i=1}^{n}(y^{i}-\overline{y}^{i})^{2}}\end{eqnarray}$ (29) where n is the number of test samples; y ⁱ is the result of FEM; $\bar{y}^{i}$ is the average of y ⁱ; ${\hat{y}}^{i}$ is the predicted values for the surrogate model.

Multi-objective Ant Lion Optimizer(MOALO) is a population intelligent optimization algorithm studied and proposed by Mirjalili [29]. The algorithm has the advantages of less adjustment parameters and easy implementation. MOALO searches the planned space by simulating the behavior of antlion capturing ants in nature. Antlions use their jaws to dig a conical trap for random ants roaming the nature world. After the ants enter the trap and are captured successfully, the optimal position of the trap is adjusted according to the number of captured ants. When the number of ants captured by one traps is more than that of the other optimal trap, a better trap location is considered, and the optimal trap location is finally obtained by constantly changing the trap location.

Table 5

Input value optimization results

Parameter	Initial	Optimization	Parameter	Initial	Optimization
x ₁ (mm)	29.50	27.50	x ₂ (mm)	20.00	17.9
x ₃ (mm)	1.50	1.58	x ₄ (mm)	2.00	1.12
x ₅ (mm)	15.00	12.19	x ₆ (mm)	15.00	12.96

The optimization objective of this paper is to reduce the vital structural mass of ECB to meet the requirements of lightweight, and at the same time meet the recoil requirements while reducing the braking force peak to reduce the reduce braking force curve fluctuation and reduce recoil displacement. Taking the six key parameters in the Table 3 as design optimization variables, the multi-objective optimization model is established as follows: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \min [f_{\mathbf{F}}(X),f_{\mathbf{S}}(X),f_{\mathbf{m}}(X)]\quad \\[3.0pt] \displaystyle s.t.x_{\text{min}}^{i}\leq x_{i}\leq x_{\text{max}}^{i}\quad \\[1.5pt] \displaystyle X=(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})\quad \end{array}\right.\end{eqnarray}$ (30)

Pareto optimal solution can be obtained after the above optimization. According to the preferences of the actual work needs and on the premise of meeting the recoil requirements, the main optimization objective is to take the lightweight of ECB as the main optimization objective. Table 5 and Table 6 show the optimization results obtained by ECB through MOALO algorithm. Table 5 shows the comparison between the initial input values and the optimized values of the ECB model. The results show that except for the increase in the thickness of the conductor cylinder, other parameters decrease to varying degrees, resulting in a decrease in the overall volume of ECB. Table 6 shows the result analysis of the initial value, FEM optimization value and BP neural network predicted value of the optimization target. The vital structural mass decreased by 19.44% and the overall volume is reduced by 22.45% compared with that before optimization. However, the braking force characteristics of ECB remain basically unchanged, with the braking force and recoil displacement decreasing by 1.74% and 0.33% compared with those before optimization. Figure 19 shows the comparison of braking force before and after optimization. It can be seen that after optimization, the peak value of braking force and the fluctuation decreases, resulting in reduced vibration of ECB during recoil.

Table 6

Output value optimization results

Parameter	Initial	FEM optimization	BP optimization	Error	Diff
m _max (kg)	473.2	381.2	394.6	3.40%	19.44%
F _max (kN)	482.6	474.2	477.9	0.77%	1.74%
S _max (mm)	744.8	742.3	764.3	2.88%	0.33%

Fig. 19.

Braking force before and after optimization.

5. Conclusion

To provide a theoretical basis for the design of the ECB in the early stage, the magnetic field and the braking characteristics of ECB under intensive impact load have been studied by the combination of experiments, theories and numerical simulations. The specific conclusions acquired in this article are as follows:

The magnetic field and braking characteristics of the eddy current are obtained by using the subdomain model method. The subdomain model method can intuitively observe the influence of the structure and material parameters of ECB on magnetic field and the eddy current braking force. By analyzing the subdomain model method, it can be concluded that a higher eddy current braking force can be obtained by keeping the recoil speed lower than the critical speed during the recoil process.

The B-H curve is introduced to consider the influence of the magnetic field and mechanics. The research shows that the magnetic induction line is distorted due to the effect of the eddy current demagnetization in the magnetic field at high speed.

Multi-objective optimization is carried out on the ECB. The vital structural mass of the ECB is taken as the main objective of optimization, while reducing the peak value of the braking force and the recoil displacement. The optimization results show that the vital structural mass of the ECB is reduced by 19.44% after optimization, and the designed recoil requirements are met at the same time.

Footnotes

Acknowledgements

The work was primarily supported by the National Natural Science Foundation of China (grant number 52105106).

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