Analysis method of permanent magnet eddy current brake under impact load

Abstract

Permanent magnet eddy current brake (PMECB) with high damping performance is widely used in engineering vibration suppression and braking. In this study, based on the braking dynamics of PMECB under impact load, the analysis method related to the damping characteristics are established, including a static magnetic model with flux leakage, a uniform damping force model with demagnetization effect and skin effect, and an acceleration damping force model with magnetic field distortion. The comparison of the analysis method, numerical simulation and experimental results verifies that the analysis method can reproduce the damping law under impact load. The results show that the maximum displacements of the analysis method and numerical simulation deviate from the experimental results within 3%. The analysis method can complete the calculation of the damping characteristics within a few seconds. The variations of the thickness, conductivity of the conductive cylinder, and air gap thickness have significant effects on the nonlinear and critical characteristics of the velocity-damping force curve, which can be corrected by changing the coefficients in the analysis method. In summary, the proposed analysis method can provide insights for rapid engineering design and optimization calculation of the PMECB by its completeness, accuracy, adaptability and rapidity.

Keywords

Analysis method eddy current brake permanent magnets impact load numerical simulation

1. Introduction

In the past decades, the eddy current brakes have made considerable progress in the application of high-speed train braking, aerospace equipment, civil engineering, weapons launch, and other fields with the advantages of non-contact, no fluid leakage, no energy loss, and extended system life [1–4].

This paper focuses on the braking performance of the permanent magnet eddy current brake (PMECB) under the impact load of weapon launch. Compared to other PMECB application studies, the most significant problems when applied in weapon systems are high loads and substantial impacts. In addition to the conventional skin effect, the demagnetization effect has a significant influence on the damping characteristics of the PMECB. Scholars have used numerical simulation and experimental investigations to verify the feasibility of the PMECB. Ge et al. [5] proposed the design of double-layer PMECB. They verified that its braking efficiency is higher than single-layer PMECB and hydro-pneumatic buffer through numerical simulation and buffering experiments. Li et al. [6] gave test results of a PMECB equipped with artillery and analyzed the law of demagnetization effect. Wang et al. [7] used a combination of experiments and simulations to build an experimental system of vortex buffers for aerodynamic impact. They demonstrated the possibility of PMECB application in weapon systems through a reliable damping braking effect. The above studies show that the PMECB is a promising alternative for the modern artillery recoil device, with the ability to provide reliable braking for high-acceleration, large-inertia mechanical equipment. However, the current related research mainly focuses on long-term numerical simulation and high-cost experiments, leading to the limitation of rapid engineering applications. Therefore, it is urgent to establish a complete set of analysis method to calculate damping characteristics.

In engineering fields, researchers have established calculation formulas such as magnetic flux density and damping force under different configurations of eddy current dampers and theoretical analysis of mechanisms such as skin effect and demagnetization effect. All these provide a reference for establishing the theoretical model of the PMECB. Sodano et al. [8,9] proposed the developing, modelling, and testing an eddy current damper device for suppressing flexible beam vibrations. More accurate eddy current density calculations can be derived from this model. Ebrahimi et al. [10,11] developed a new eddy current damper for the automotive suspension system and gave the calculation formulas of magnetic flux density and eddy current damping force under the influence of skin effect. The accuracy of the braking force expression of the permanent magnet cylindrical linear eddy current damper was verified by theoretical analysis and test experiments. Li et al. [12] established the quasi-static analytical model of the PMECB and the analytical expression of the magnetic vector equation and compared the braking force characteristics under quasi-static and high acceleration. Cheng et al. [13] studied the nonlinear theory of eddy current force and velocity in high-frequency vibration damping in civil engineering, and presented that the eddy current force under time-varying motion can be approximated as the sum of a dissipative force, a pseudo-inertial force, and a pseudo-dissipative force, which is proportional to the relative velocity, acceleration, and jerk between the magnet and the conductor respectively. The Magnetic Equivalent Circuit (MEC) is an effective method for magnetic field analysis that provides a good balance between accuracy and computational cost. Zhao et al. [14] utilized the MEC method and eddy current loss principle to derive the braking torque of the permanent magnet eddy current retarder for automotive braking and verified the model accuracy by experiment. Kou et al. [15] developed an analytical model of a hybrid excitation linear ECB considering core saturation using the MEC method and the layer theory approach. Fan et al. [16] established an analytical model based on the MEC method, introduced magnetic saturation and magnetic momentum, and thus proposed a method to calculate the braking force by approximating the electric field cross-section with the mean value method. Krämer et al. [17] used the MEC method in their study of modeling permanent magnet linear synchronous motors, allowing the model to respond to local magnetic saturation and nonlinear characteristics of the motor quantities, as well as the variation of the air gap distance. In the above studies, the braking performance-related calculation equations were given more by discussing the microscopic properties of the magnetic field. At the same time, a complete set of analysis method were seldom established.

This paper discusses the magnetic circuit laws and damping characteristics under static, uniform velocity and high acceleration conditions are discussed through dynamic analysis of PMECB applied to weapon systems, respectively. As a result, the static magnetic force model with flux leakage is developed by the MEC method, and the uniform damping force model with demagnetization effect and skin effect, and the acceleration damping force model with magnetic field distortion are derived analytically. By comparing the results of the analysis method, numerical simulation, and validation experiments, it is verified that the analysis method can reproduce the damping law under high impact load. Meanwhile, the calculation accuracy and calculation time of the analysis method and simulation models are compared and discussed. Finally, the effects of some structural parameters on the velocity-damping force curves are analyzed.

2. Analysis method of the PMECB

The PMECB mainly consists of the primary mover and the secondary stator, as shown in Fig. 1. The primary mover includes a moving bar, permanent magnets, yokes, and non-magnetic connection parts. The secondary stator is a damping cylinder composed of a conductive inner cylinder and a magnetically conductive outer cylinder. The permanent magnets and the yokes are staggered to form a magnetic group. The adjacent permanent magnets are arranged opposite each other with the same pole, making more magnetic lines of force pass through the primary and secondary to form a closed circuit. When working, the mover is rapidly accelerated by the impact load, cutting magnetic lines between the mover and the stator. The damping force comes from the induced eddy currents in the damping cylinder. The impact braking ends when the displacement of the mover reaches its maximum.

Fig. 1.

Schematic diagram of the PMECB.

The electromagnetic field distribution generated during the impact braking process is complex. Herein, the magnetic circuit problem is simplified to the magnetic field problem within a specific path, which can reduce the analysis difficulty and ensure the solution’s accuracy. The effect of heat generated by the eddy current loss in the inner cylinder on the permanent magnet performance, as well as the elastic deformation generated by the primary and secondary during the motion, is neglected in the analysis. According to the movement law of PMECB under impact load, three models of static magnetic model, uniform damping force model, and acceleration damping force model should be established in the calculation of damping force.

2.1. Static magnetic model with flux leakage

Among the methods to simulate the real effects of magnetic fields, building static magnetic models can help researchers quickly determine the basic properties of electromagnetic structures. The total magnetic flux provided by the permanent magnets to the external magnetic circuit in the PMECB includes the leakage flux that cannot be effectively utilized at positions such as the interpoles, air gaps and ends. Therefore, it is necessary to consider the flux leakage in establishing the static magnetic model.

The magnetic field of the PMECB is simplified as a model, which employs permanent magnets as the magnetic source. The equivalent magnetic circuit is illustrated in Fig. 2. The total reluctance of the magnetic field lines generated by a permanent magnet can be described as: $\begin{eqnarray}R_{m}=R_{0}+R_{1}+\frac{(2R_{2}+2R_{3}+R_{4})\cdot (2R_{5}+R_{6})}{2R_{2}+2R_{3}+R_{4}+2R_{5}+R_{6}}.\end{eqnarray}$ (1) Each reluctance is defined as follows: R ₀ is the axial reluctance of the permanent magnet, R ₁ is the axial reluctance of the yoke, R ₂ is the outward radial reluctance of the yoke, R ₃ is the radial reluctance of the air gap and conductor cylinder, R ₄ is the axial reluctance of the magnetically conductive cylinder, R ₅ is the inward radial reluctance of the yoke, and R ₆ is the axial reluctance of the moving bar.

The general formula for calculating the reluctance in a magnetic circuit is $\begin{eqnarray}R_{k}=\frac{l_{k}}{{\mu}_{k}S_{k}}.\end{eqnarray}$ (2) Where, l _k is the magnetic circuit length, 𝜇_k is the magnetic permeability, and S _k is the magnetic circuit area. The simplified formulas for calculating the reluctance under the equivalent magnetic circuit are $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}R_{0}={\displaystyle \frac{{\tau}_{m}}{{\pi}{\mu}_{m}(r_{0}^{2}-r_{z}^{2})}}\quad \\[12.0pt] R_{1}={\displaystyle \frac{{\tau}-{\tau}_{m}}{{\pi}{\mu}_{i}(r_{1}^{2}-r_{z}^{2})}}\quad \\[12.0pt] R_{2}={\displaystyle \frac{r_{1}-l}{2{\pi}{\mu}_{i}(r_{1}+l)({\tau}-{\tau}_{m})/4}}\quad \\[12.0pt] R_{3}={\displaystyle \frac{r_{{\delta}}-r_{1}}{2{\pi}{\mu}_{0}(r_{{\delta}}+r_{1})({\tau}-{\tau}_{m})/4}}+{\displaystyle \frac{r_{c}-r_{{\delta}}}{2{\pi}{\mu}_{c}(r_{c}+r_{{\delta}})({\tau}-{\tau}_{m})/4}}\quad \\[12.0pt] R_{4}={\displaystyle \frac{{\tau}}{{\pi}{\mu}_{s}(r_{j}^{2}-r_{c}^{2})}}\quad \\[12.0pt] R_{5}={\displaystyle \frac{l-r_{x}}{2{\pi}{\mu}_{i}(l+r_{x})({\tau}-{\tau}_{m})/4}}\quad \\[12.0pt] R_{6}={\displaystyle \frac{{\tau}}{{\pi}{\mu}_{z}r_{z}^{2}}}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (3) Where 𝜇₀, 𝜇_z, 𝜇_i, 𝜇_m, 𝜇_c, 𝜇_s are the permeability of the air, the moving bar, the yoke, the permanent magnet, the conductive cylinder, and the magnetically conductive cylinder. The length parameters are defined as follows: r _z, r ₀ are the inner and outer radii of the permanent magnet, r _𝛿, r _c are the inner and outer radii of the conductor cylinder, l is the radius of the confluence of the two magnetic circuits, r ₁ is the outer radius of the yoke, r _j is the outer radius of the magnetically conductive cylinder, 𝜏 is the pole pitch, and 𝜏_m is the thickness of the permanent magnet.

Fig. 2.

(a) Magnetic circuit model (b) Equivalent magnetic circuit.

The magnetic flux through the permanent magnet is expressed as: $\begin{eqnarray}{\Phi}_{1}={\displaystyle \frac{F_{pm}}{R_{m}}}.\end{eqnarray}$ (4) Where, F _pm is the equivalent magnetic potential of the permanent magnet potential source F _pm = H _c𝜏_m, H _c is the coercivity of the permanent magnet.

According to Kirchhoff’s current laws (KCL) and Kirchhoff’s voltage laws (KVL), the magnetic circuit model can be expressed as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\Phi}_{2}(2R_{2}+2R_{3}+R_{4})={\Phi}_{3}(2R_{5}+R_{6})\quad \\ {\Phi}_{1}={\Phi}_{2}+{\Phi}_{3}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (5)

The ratio of the total magnetic flux provided by the permanent magnets to the effective main magnetic flux of the external magnetic circuit is defined as the flux leakage coefficient 𝜎0. To the best of our knowledge, there is no unified method to establish its analytical mathematical model. Therefore, the flux leakage coefficient 𝜎0 through inverse calculation is estimated after determining the geometric parameters of the PMECB. Therefore, the external magnetic circuit flux is $\begin{eqnarray}{\Phi}_{2}={\displaystyle \frac{2R_{5}+R_{6}}{2R_{2}+2R_{3}+R_{4}+2R_{5}+R_{6}}}K{\Phi}_{1}.\end{eqnarray}$ (6) Where, K is the leakage correction factor, and it can be defined as: K = 1∕𝜎₀. It is deduced that the air gap flux density of the PMECB at static state is $\begin{eqnarray}B_{0}={\displaystyle \frac{{\Phi}_{2}}{S_{0}}}={\displaystyle \frac{2{\Phi}_{2}}{{\pi}(r_{1}+r_{{\delta}})({\tau}-{\tau}_{m})}}.\end{eqnarray}$ (7)

2.2. Uniform damping force model with demagnetization effect and skin effect

The schematic diagram of the eddy current magnetic field is depicted in Fig. 3. A vortex magnetic field is associated with the magnetic yoke between like-poled permanent magnets. At the leading edge of the motion, the yoke exerts a demagnetizing effect on the main magnetic field, while at the trailing edge, it has a magnetizing effect, resulting in the deformation of the air gap magnetic field. When the magnetic circuit is designed to be in a state of magnetic saturation, the air gap magnetic field at the magnetizing edge weakens, whereas at the demagnetizing edge, it remains nearly unchanged. Consequently, the PMECB exhibits a demagnetizing effect during uniform motion. Therefore, conditions such as high speed, high temperature, and material magnetic properties cause demagnetization effects in the PMECB. The radial distribution of the current is close to the surface layer inside the conductive cylinder due to the skin effect induced in the conductive cylinder. Therefore, it is necessary to establish a more accurate uniform damping force formula based on considering these two effects.

Fig. 3.

Eddy current schematic.

The influence of the eddy current demagnetization magnetic field under dynamic operation is summarized as the combined action of the eddy current conversion coefficient K _d and the effective value of the eddy current I _d. The eddy current conversion coefficient K _d is a function of the velocity, and its function expression can be defined as $\begin{eqnarray}K_{d}=N_{1}v^{q}.\end{eqnarray}$ (8) Where N ₁ and q are constant values in the same PMECB, which affect the slope and critical value in the relationship between uniform damping force and velocity, respectively.

For the conductive cylinder, the micro-element generates an electromotive force: $\begin{eqnarray}|{\varepsilon}|={\displaystyle \frac{{\Delta}{\Phi}}{{\Delta}t}}=Blv=2{\pi}Brv.\end{eqnarray}$ (9) Due to the skin effect, the eddy current density decreases exponentially from the inner surface to the outer surface of the conductive cylinder, introducing an equivalent penetration depth as $\begin{eqnarray}{\delta}_{p}=\sqrt{{\displaystyle \frac{1}{{\sigma}{\pi}{\mu}_{c}f}}}=\sqrt{{\displaystyle \frac{2r_{1}}{{\sigma}{\mu}_{c}v}}}.\end{eqnarray}$ (10) The effective instantaneous induced electromotive force is $\begin{eqnarray}E=\int _{0}^{{\tau}-{\tau}_{m}}\int _{r_{{\delta}}}^{r_{{\delta}}+{\delta}_{p}}2{\pi}Bv\cdot rdrdz.\end{eqnarray}$ (11) Where, 𝜎 is the conductivity of the conductive cylinder, f is the angular frequency of the magnetic field change. The effective resistance is $\begin{eqnarray}R={\displaystyle \frac{L}{{\sigma}S}}={\displaystyle \frac{2{\pi}(r_{{\delta}}+{\delta}_{p}+r_{{\delta}})/2}{{\sigma}{\delta}_{p}({\tau}-{\tau}_{m})}}={\displaystyle \frac{{\pi}(2r_{{\delta}}+{\delta}_{p})}{{\sigma}{\delta}_{p}({\tau}-{\tau}_{m})}}.\end{eqnarray}$ (12) The effective value of the eddy current is $\begin{eqnarray}I_{d}={\displaystyle \frac{E}{R}}={\displaystyle \frac{({\tau}-{\tau}_{m})^{2}{\sigma}{\delta}_{p}[(r_{{\delta}}+{\delta}_{p})^{2}-r_{{\delta}}^{2}]Bv}{2r_{{\delta}}+{\delta}_{p}}}.\end{eqnarray}$ (13) Then the synthetic air gap flux density after considering the demagnetization effect can be calculated as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}F_{pm}=H_{c}{\tau}_{m}-K_{d}I_{d}\quad \\[12.0pt] {\Phi}_{1}=K{\displaystyle \frac{H_{c}{\tau}_{m}-K_{d}I_{d}}{R_{m}}}\quad \\[12.0pt] {\Phi}_{2}={\displaystyle \frac{2R_{5}+R_{6}}{2R_{2}+2R_{3}+R_{4}+2R_{5}+R_{6}}}{\Phi}_{1}\quad \\[12.0pt] B_{r}={\displaystyle \frac{{\Phi}_{2}}{S_{0}}}={\displaystyle \frac{2{\Phi}_{2}}{{\pi}(r_{1}+r_{{\delta}})({\tau}-{\tau}_{m})}}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (14) The eddy current density is $\begin{eqnarray} J=J_{s}e^{\frac{-ds}{{\delta}_{p}}}. \end{eqnarray}$ (15) Where, J _s is the eddy current density on the surface of the conductive cylinder, ds is the depth from the surface.

Since only the magnetic field cut vertically by the conductive cylinder contributes to the damping force, the uniform damping force formula with demagnetization effect and skin effect can be given as $\begin{eqnarray}F_{N}=\int _{V}(J\times B)\text{d}V=-N{\sigma}v\int _{0}^{2{\pi}}\int _{0}^{{\tau}-{\tau}_{m}}\int _{r_{{\delta}}}^{r_{c}}rB_{r}^{2}e^{\frac{-(r-r_{{\delta}})}{{\delta}_{p}}}\text{d}r\text{d}z\text{d}{\theta}.\end{eqnarray}$ (16) The negative sign indicates that the direction of damping force is opposite to the velocity direction, and N indicates the number of permanent magnets. Since the equivalent magnetic circuit is simplified as only the magnetic flux density exists at the position corresponding to the thickness of the yoke, and for the two edge yokes, magnetic flux density exists only at half of the thickness, the N yoke thicknesses are integrated along the axial direction in the uniform damping force equation. Under the dynamic operation, the damping force of the PMECB has a constitutive relationship with the speed of motion.

2.3. Acceleration damping force model with magnetic field distortion

The PMECB operates under impact load, and the speed rises rapidly to a great value in a short time. During this starting acceleration section, the enormous acceleration makes the eddy current distortion complicated, as shown in Fig. 4. Hence, it is necessary to establish a separate calculation model for the acceleration damping force.

Fig. 4.

Vortex schematic (a) Under 4 m/s during acceleration (b) Under uniform speed 4 m/s.

The damping force correction coefficient for the starting acceleration section is defined as K _m, and the value is related to the velocity and acceleration. In studies where it is difficult to directly determine the analytical relationship between the variables and the results, researchers usually obtain the corresponding relationship by inverse calculation through experiments or simulations. Currently, no consistent research exists on the specific causes of the acceleration damping force. Therefore, it is feasible to establish the acceleration damping force by collecting simulation samples for inverse calculation and fitting. The method is as follows: firstly, take several groups of impact loading forces with different values, and obtain dynamic data samples through numerical simulation. Secondly, take the acceleration and the theoretical correction coefficient value at different velocity points in each simulation result (the theoretical correction coefficient value is the numerical simulation damping force divided by the analysis method damping force). Next, take the velocity value as the grouping standard, and fit all samples’ theoretical correction coefficient values into a linear function related to the acceleration a. From this, the resulting correction value functions at different speeds are obtained. The slope k and the intercept b of the correction value function are fitted to a three-term function related to the velocity. The damping force correction coefficient is obtained as $\begin{eqnarray}K_{m}=ka+b.\end{eqnarray}$ (17) When the size and the structure of the PMECB is determined, the function coefficients of the slope k and the intercept b of this formula are also determined values.

The acceleration damping force is $\begin{eqnarray}F_{A}=K_{m}\cdot F_{N}.\end{eqnarray}$ (18)

3. Numerical simulation and theoretical calculation

3.1. Numerical simulation

As shown in Fig. 5, the PMECB electromagnetic finite element dynamics model is established using a model coordinate system with the center of rotational symmetry about the Z-axis. In addition to the solid geometry model, solution domains such as Band and Outer domain need to be established. The main dimensions and material parameters are shown in Table 1.

Fig. 5.

Numerical simulation model of the PMECB.

Table 1

Model parameters of the PMECB

Item/Symbol	Value/Unit	Item/Symbol	Value/Unit
Number of poles N	8	Conductor cylinder’s inner radii diameter r _𝛿	20 mm
Pole pitch 𝜏	15 mm	Conductor cylinder’s outer radii diameter r _c	22 mm
Magnets’ thickness 𝜏_m	10 mm	Magnetically conductor cylinder’s outer radii diameter r _j	29 mm
Magnets’ inner radii diameter r _x	6 mm	Magnets’ material	NdFeB N35
			H _c = 890000A/m
Magnets’ outer radii diameter r ₀	17.5 mm	Conductor cylinder’s material	①aluminum
			②Copper
Yokes’ outer radii diameter r ₁	19.5 mm	Conductor cylinder’s conductivity	①36 × 10⁶ S/m
			②55 × 10⁶ S/m

After the material setting, the infinity boundary condition Balloon Boundary is defined, which can make the plotted solution domain range not too large, reduce the computation time, and also examine the model leakage magnetic properties. In the motion options settings, the values measured by the pressure transducer in Section 4 are used as the loading force in the model. Next, the dynamics model is pre-processed with mesh partitioning and setup, and then the simulation can be started to calculate the damping force, velocity, displacement and other dynamics parameters.

3.2. Theoretical calculation

Based on the analysis method of the PMECB established above, the parameters of the example model are assigned, and the nonlinear permeability of the ferromagnetic materials is defined. At this point, the missing parameter values in the analysis method are the flux leakage coefficient 𝜎₀ in the static magnetic model, the eddy current conversion coefficient K _d in the uniform damping force model, and the damping force correction coefficient K _m in the acceleration damping force model. The values of these coefficients in different specific structures of the PMECB are not fixed. Furthermore, a suitable analysis method for these coefficients is also lacking. Herein, a simple uniform point inverse calculation of PMECB using an aluminum conductive cylinder is performed using numerical simulation.

In Fig. 6, when no eddy current conversion coefficient K _d is applied, the damping force increases continuously with the increase of velocity, which is inconsistent with the actual situation. By estimating the flux leakage coefficient 𝜎₀ of 1.02 in the static magnetic model and applying the eddy current conversion coefficient K _d = 32000 v ^2∕3, the maximum deviation of the velocity-damping force curve between the analysis method and the numerical simulation is obtained as 4.79%. It indicates that the uniform damping force model with demagnetization effect and skin effect can reproduce the velocity-damping force curve of PMECB under uniform motion, which has implications for the design of other PMECB at uniform speeds.

Fig. 6.

Comparison diagram of uniform damping force.

By collecting numerical simulation samples and performing inversion calculations, the damping force correction factor K _m is calculated as follows $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}k=-(2\times 10^{-8})v^{3}+(1\times 10^{-6})v^{2}-(1\times 10^{-5})v+(8\times 10^{-6})\quad \\[3.60004pt] b=-0.0004v^{3}+0.0012v^{2}+0.1532v\quad \\ K_{m}=ka+b.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (19)

As shown in Fig. 7, the recoil motion state of the PMECB is: the impact load of the breech forces on PMECB results in high acceleration in the starting acceleration section, which starts to decelerate when the speed reaches its maximum value until the speed returns to zero. Theoretically, in the initial acceleration state, the eddy current loop at the damping cylinder exhibits noticeable eddy current distortion after moving lag. The permanent magnet has a demagnetization effect, reducing the effective radial air gap flux density. In this way, the presence of acceleration inhibits the growth of the damping force, so that the damping force generated instantaneously cannot reach the damping force at that speed in the uniform state. In the deceleration state, the demagnetization effect of the permanent magnet is weakened and the magnetic circuit shape under the transient is close to that of the speed in the uniform state, so the damping force at deceleration is nearly equal to the uniform damping force. Consequently, the value of the damping force correction coefficient K _m related to the velocity and acceleration is reasonable, making the acceleration damping force model with magnetic field distortion predict the damping force of high acceleration with higher precision.

Fig. 7.

Comparison diagram of damping force under working condition.

4. Experimental verification

Experiments for validating the accuracy of the proposed approximate analysis method are shown in this section. Specifically, these experiments are designed to verify:

The accuracy of the analysis method in predicting the eddy current damping force, velocity and displacement of the PMECB under impact load.

The accuracy of the analysis method in predicting differentf impact loads and different conductive cylinder materials, i.e., the analysis method adaptability.

The damping characteristics of the PMECB can be investigated by analytical, finite element and experimental methods [18]. In the previous chapters, when analysis method is established by analytical means and numerical simulation is performed by the finite element method, the models are inevitably idealized and simplified, ignoring the influence of secondary factors, and thus the uncontrollable factors that exist in reality cannot be completely avoided. The experimental method is the ultimate means to verify the accuracy of the analytical and finite element methods. Therefore, to verify the correctness of the analysis method and the numerical simulation to calculate the damping characteristics of the PMECB under impact load, it is necessary to compare the results of the analysis method and numerical simulation with the experimental results.

4.1. Impact load experiment

The stator and mover of the PMECB are shown in Fig. 8. As depicted in Fig. 9, an impact load experimental platform is employed in the impact load evaluation. Specifically, the stator is fixed on the base plate, and the mover is screwed with the pressure sensor and the mass slider. The air compressor is released after a certain pressure is applied, so that the hammer protrudes from the impact components to impact the mass slider, which drives the PMECB mover to move. Under the action of the damping force, the mover decelerates gradually until it stops. The dynamic response performance of PMECB can be analyzed by collecting the impact loading force and mover displacement signals through the data acquisition system.

Fig. 8.

(a) PMECB’s stator (b) PMECB’s mover.

Fig. 9.

Impact load experimental platform.

To investigate the damping performance of the PMECB under different impact loads, univariate comparison groups with different loading energies regarding the same conditions were constructed in the experimental study. In addition, impact braking experiments with aluminum and copper conductive cylinder materials were conducted to investigate the degree of influence of the conductive cylinder material on the damping characteristics.

Fig. 10.

Impact loading force curves at different aeration pressures and conductive cylinder materials.

The first peak value of the impact loading force generated by the inflatable hammer released after pressurization is less than 50 kN, the first peak pulse width is less than 2 ms, and the pulse width of the impact segment is less than 4 ms, which meets the requirements of transient impact load. Due to the different conductive cylinder materials and impact energy, as well as the existence of interference noise in the data dynamic acquisition, the four groups of loading force data have slight difference. It can be clearly seen from Fig. 10 that the first peak value of the loading force under 0.64 MPa aeration pressure all exceeds 40 kN, while the first peak value of loading force under 0.50 MPa aeration pressure is lower than 35 kN. After a large oscillation of less than 4 ms, the loading force gradually returns to zero. Different aeration pressures construct different loading energies, but the impact pulse width is almost the same.

Fig. 11.

Experimental displacement results diagram.

From Fig. 11, it can be seen that the PMECB mover was effectively braked under all experimental conditions. The displacement of the copper conductive cylinder is smaller than that of the aluminum conductive cylinder, and the displacement under 0.5 MPa aeration pressure is smaller than that of 0.64 MPa, which reflects that the conductive cylinder material and the loading energy have a considerable effect on the braking effect. Since the correspondence between the damping force and velocity of PMECB is not entirely linear, the higher the velocity, the higher the degree of nonlinearity. Therefore, the subsequent discussion focuses on the PMECB damping at a higher speed of 0.64 MPa aeration pressure.

4.2. Results comparison

The loading force of 0.64 MPa/aluminum shown in Fig. 10 is loaded into the analysis method and numerical simulation respectively, and the damping characteristics results in Fig. 12 can be obtained.

Fig. 12.

Comparison diagram of the damping characteristics under 0.64 MPa/aluminum.

Fig. 13.

Comparison diagram of the damping characteristics under 0.64 MPa/copper.

Table 2

Results deviation

		Maximum damping force	Maximum displacement
0.64 MPa/aluminum	Numerical simulation	1.48462 kN	95.44867 mm
	Analysis method	1.63165 kN	96.59065 mm
	Experiment	/	98.14662 mm
	Deviation of numerical simulation and experiment	/	2.75%
	Deviation of analysis method and experiment	/	1.59%
0.64 MPa/copper	Numerical simulation	1.84207 kN	65.63177 mm
	Analysis method	2.20976 kN	64.40426 mm
	Experiment	/	68.19872 mm
	Deviation of numerical simulation and experiment	/	3.76%
	Deviation of analysis method and experiment	/	5.56%

Combined with Fig. 12 and Table 2, the agreement between the analysis method and the numerical simulation is exceptionally high, while the maximum displacements of both methods deviate from the experimental results within 3%, which indicates the reliability of both methods. Since the conductivity of the conductive cylinder has minimal effect on the magnetic circuit and the relative permeability difference between copper and aluminum is very small, only the conductive cylinder conductivity in the analysis method is modified and used as the analysis method of the copper conductive cylinder to develop the calculation, and the results are shown in Fig. 13.

As seen from Fig. 13, the modified analysis method can reproduce the damping condition of PMECB with copper conductive cylinder, which indicates that the magnetic circuit calculation developed in this paper is accurate and the established analysis method has high adaptability. If only the conductive cylinder material is changed, a certain accuracy can be achieved by changing only the conductivity of the analysis method.

In the numerical simulation using the finite element method, improving the solution accuracy and reducing the computation time are two critical objectives, which are frequently contradictory to each other. In other words, the reduction of computation time is often at the expense of computation accuracy. In this paper, the number of meshes is reduced to about 15000 based on ensuring the accuracy of numerical simulation. The computational step size is set to 0.00001 s for both numerical simulation and analysis method. It can be seen from Table 3 that, disregarding the pre-processing time, the analysis method can complete the calculation of the damping characteristics in the braking process within a few seconds, while the numerical simulation requires more than 20 hours under the same calculation step. This analysis method can be applied to the trial calculation stage of the PMECB engineering design to realize the estimation of the structure size scheme and loading excitation. Compared with numerical simulation, the analysis method has outstanding advantages in the rapid engineering design and optimization calculation as it saves much time for the subsequent possible work such as parameter matching and optimization.

Table 3

Calculation time comparison

		Calculation time
0.64 MPa/aluminum	Numerical simulation	25 h 58 min 22 s
	Analysis method	6.46s
	Theoretical/Numerical	0.0069%
0.64 MPa/copper	Numerical simulation	25 h 20 min 52 s
	Analysis method	6.67 s
	Theoretical/Numerical	0.0073%

5. Parametric analysis

The loading energy is limited to a specific range because of the existence of limit values for the experimental aeration pressure. In other studies of the PMECB [3,12], we can easily find that a critical velocity occurs when the loading energy is immense. At this speed, the damping force reaches a maximum. In ((16)), if dF _N∕dv = 0, the critical speed v _c can be derived, and the maximum damping force F _Nmax at the critical speed can be obtained by substituting v _c. The determination of the critical velocity v _c can provide an indicator for the design of the PMECB to maximize the utilization of magnetic field energy. Since the eddy current conversion factor K _d = N ₁ v ^2∕3 under the aluminum cylinder structure has been back-calculated, the critical velocity can be derived from ((16)): $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}v_{c}=\left({\displaystyle \frac{6R_{m}}{7s}}\right)^{\frac{3}{5}}\quad \\[14.39996pt] s=kq\quad \\ k={\displaystyle \frac{2}{{\pi}{\sigma}_{0}(r_{1}+r_{d})({\tau}-{\tau}_{m})}}{\displaystyle \frac{(2R_{5}+R_{6})}{(2R_{2}+2R_{3}+R_{4}+2R_{5}+R_{6})}}\quad \\[18.0pt] q={\displaystyle \frac{N_{1}{\sigma}{\delta}_{p}({\tau}-{\tau}_{m})^{2}((r_{d}+{\delta}_{p})^{2}-r_{d}^{2})}{2r_{d}+{\delta}_{p}}}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (20)

In this section, several design parameters are studied to understand how they affect the velocity-damping force curve and investigate the analysis method’s coefficient variation law under different structural dimensions.

5.1. Conductive cylinder thickness

As shown in Fig. 14 (related parameters are summarized in Table 4), the velocity-damping force relationship is almost the same for different conducive cylinder thicknesses in the low-speed section. As the conductive cylinder thickness increases, the critical speed decreases, and the maximum damping force also decreases. At low speed, the larger the conductive cylinder thickness, the stronger the total magnetic field is suppressed by the induced magnetic field, and also more eddy currents are distributed on the inner side of the conductive cylinder. Both factors maintain the low sensitivity of the conductive cylinder thickness to the damping coefficient at low speed. At high speed, the induced magnetic field strongly suppresses the total magnetic field, making the thicker the conductive cylinder, the lower the damping force. The increase of the thickness of the conductive cylinder does not affect the flux leakage coefficient 𝜎₀ of the analysis method, but causes a decrease in the N ₁ associated with eddy currents.

Fig. 14.

Velocity-damping force curves for different conductive cylinder thickness 𝛿.

Table 4

Critical case of motion and analysis method coefficients for different conductive cylinder thickness 𝛿

Conductive cylinder thickness 𝛿	Critical speed v _c	Maximum damping force F _Nmax	Flux leakage coefficient 𝜎₀	N ₁ in eddy current conversion coefficient K _d
2.0 mm	10.65 m/s	2.05 kN	1.02	32000
2.5 mm	9.41 m/s	1.85 kN	1.02	28000
3.0 mm	8.56 m/s	1.69 kN	1.02	25000

5.2. Conductive cylinder conductivity

The effect of conductive cylinder conductivity is also evaluated, as the related parameters are summarized in Table 5. It can be seen from Fig. 15 that as the conductive cylinder conductivity increases, the critical speed decreases, and the maximum damping force remains almost constant. The damping force decreases at low speed and increases at high speed. The ratio of the damping coefficient at low speed is approximated to the ratio of conductive cylinder conductivity. The increase of conductive cylinder conductivity not only has a linear effect on the value of damping force, but also weakens the main magnetic field strength. Under the combined action of the two, the maximum damping force is almost stable. The conductive cylinder conductivity increases, and the flux leakage coefficient 𝜎₀ of the analysis method remains the same, but the N ₁ increases due to the increase in eddy current density.

Fig. 15.

Velocity-damping force curves for different conductive cylinder conductivity 𝜎.

Table 5

Critical case of motion and analysis method coefficients for different conductive cylinder conductivity 𝜎.

Conductive cylinder conductivity 𝜎	Critical speed v _c	Maximum damping force F _Nmax	Flux leakage coefficient 𝜎₀	N ₁ in eddy current conversion coefficient K _d
24 × 10⁶ S/m (6061-T6)	>15 m/s	/	1.02	25000
36 × 10⁶ S/m (1060)	10.65 m/s	2.05 kN	1.02	32000
55 × 10⁶ S/m (copper)	7.13 m/s	2.09 kN	1.02	41000

5.3. Air gap thickness

The influence of air gap thickness is also conducted, and the related parameters are provided in Table 6. From Fig. 16, it can be seen that the increase in air gap thickness causes an overall decrease in damping force while the critical speed increases slightly. The increase of air gap thickness reduces the magnetic flux density, so the damping coefficient will decrease. Since the conductive cylinder size and conductivity are not changed, N ₁ in the analysis method remains the same. However, the increase in air gap thickness makes the leakage increase, so the flux leakage coefficient 𝜎₀ increases.

Fig. 16.

Velocity-damping force curves for different air gap thickness x _ag.

Table 6

Critical case of motion and analysis method coefficients for different air gap thickness x _ag

Air gap thickness x _ag	Critical speed v _c	Maximum damping force F _Nmax	flux leakage coefficient 𝜎₀	N ₁ in eddy current conversion coefficient K _d
0.5 mm	10.65 m/s	2.05 kN	1.02	32000
1.0 mm	11.71 m/s	1.68 kN	1.06	32000
1.5 mm	12.74 m/s	1.38 kN	1.22	32000

6. Conclusion

In this study, design method of PMECB under impact load is developed by comprehensively analyzing the effects brought by flux leakage, demagnetization effect, skin effect and acceleration on the magnetic field distribution and eddy current damping force. They contain a static magnetic model with flux leakage, a uniform damping force model with demagnetization effect and skin effect, and an acceleration damping force model with magnetic field distortion. The analysis method is complete and adaptable, which allows the damping law of the PMECB during braking to be reproduced. In addition, an experimental platform for impact load was established, and laboratory experiments were performed to verify the velocity, displacement, and damping force. The agreement between the analysis method and the numerical simulation is very high, while the maximum displacements of both methods deviate from the experimental results within 3%, which indicates the reliability of both methods. The analysis method can complete the calculation of the damping characteristics in the braking process within a few seconds, while the numerical simulation requires more than 20 hours under the same calculation step. Parametric studies show that the thickness, conductivity of the conductive cylinder, and air gap thickness have significant effects on the nonlinear and critical characteristics of the velocity-damping force curve: the thicker the conductive cylinder, the stronger the suppression of the total magnetic field at high speed and the lower the damping force; the conductive cylinder conductivity increases, the critical speed decreases, while the maximum damping force remains almost constant; the air gap thickness increases, the magnetic flux density decreases, leading to an overall decrease in the damping force. The variation of the above structural parameters in the analysis method is also manifested in the variation of two parameters, the flux leakage coefficient 𝜎₀ and the N ₁ in eddy current conversion coefficient K _d. The analysis method proposed in this paper can achieve sufficient computational accuracy with reference to only a small number of numerical simulation inversions, and is characterized by completeness, adaptability and rapidity, which has outstanding advantages in the rapid engineering design and optimization calculation of such PMECB.

Footnotes

Acknowledgements

This work is supported by the Chinese Defense Advance Research Program of Science and Technology under Grants 301071904.

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