Magnetic Force Equations Based on Computer Simulation and the Effect of Load Line

C. Chen, H. Meng, M. Fan
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Abstract

The Maxwell magnetic force equation $\mathrm {F}= \mathrm {B}^{2}\mathrm {A}/ ( 2 \mu _{0})$[1–6] can be used for determining the magnetic force of magnetic components, where F is the force in newton (N), B is the flux density in tesla (T), A is the area of cross-section in square meter (m2), and $\mu _{0}$ is the permeability of the vacuum $( 4 \pi \times 10 ^{-7}\mathrm {H} /\mathrm {m})$. The formula can be converted to an easy to remember expression of F = 40B2A, in which the unit of A is cm2. This equation says that if the field is 1T, and the area is 1cm2, then the magnetic force is 40N or 4kgf. However, it is somehow difficult to determine the B value in many practical cases, and the accuracy is usually not satisfactory. Computer simulation using finite element method can determine the magnetic forces with various boundary conditions, but usually it is not convenient for industrial users. In this paper, we report several simple equations, which are established based on the large database generated by using 3D computer simulation. The users can use the equations to obtain the force by simply inputting the magnet’s$\mathrm{B}_{\mathrm{r}}$, area and thickness. The effect of load line is also analyzed in this paper. Infolytica’s MagNet software was chosen for the simulation. Parameterization function with newton tolerance 0.1% was used to systematically solve the problems for NdFeB cylinders, rings, and rectangular blocks interacting with CR1010 steel. The steel plates are both thicker and larger than the magnets. The maximum sizes for the magnets are shown in Table 1. The result database for each gap in a single boundary condition includes 62500 data points for rectangular blocks, 30625 data points for rings, and 1250 data points for cylinders. The gaps between the magnets and steel plates are in the range of 0.01 – 15mm with 23 unequal intervals. The itemized data were then plotted and analyzed to establish the force equations for the magnets with relative high load lines. Figure 1 shows the magnetic force vs the area of N52 magnet rings with gap = 0.01 mm to steel plates. Fig. 1a and 1b have different boundary conditions: 1a has CR1010 steel on both ends of the magnets, and 1b has the steel only on one end. The load line of a standalone magnet can be estimated by using the equations described in Parker’s book[7], but the magnets in this project have much higher load lines compared to the standalone magnets since steel plates are associated with these magnets. Boundary condition 1a obviously gives much higher load line compared to boundary condition 1b. For these ring magnets with higher load line in condition 1a, the force value vs area for each thickness can generate 2nd degree polynomial formulas, which has R-squared R2>0.9997 as shown in Figure 1. (R2 of 1.0000 was obtained for all the thicknesses of rectangular blocks). These formulas were then analyzed to establish a general equation $F = B_{r}^{2} {(aA}^{2}+ {bA)}$. Using the equation, the magnetic force for any $B_{r}$ value can be determined by inputting magnet’s $B_{r}$, area, and thickness. As shown in Table 1, the factor a is a function of thickness in 2nd degree polynomial, and the factor b is also a function of thickness but in power form. The effect of boundary condition is tremendous. Condition1b has much lower load line compared to condition 1a, hence the magnetic force values vs the area cannot generate satisfactory equations. As shown in the Fig. 1b, for the same magnet area, the magnetic force values are in a range with various values due to different load lines. For example, the ring magnets with exact the same thickness 0.1cm and area 2.8cm2, the force values range from 18.4N to 74N for ID/OD values from 0.1/1.9cm to 4.3/4.7cm. Details for all the magnet shapes with two boundary conditions will be reported in this paper, and the effect of load line will be analyzed.
基于计算机仿真的磁力方程及载重线的影响
麦克斯韦磁力方程$\mathrm {F}= \mathrm {B}^{2}\mathrm {A}/ ( 2 \mu _{0})$[1-6]可用于确定磁性元件的磁力,其中F为力,单位为牛顿(N), B为磁通密度,单位为特斯拉(T), A为横截面面积,单位为平方米(m2), $\mu _{0}$为真空的磁导率$( 4 \pi \times 10 ^{-7}\mathrm {H} /\mathrm {m})$。这个公式可以转换成一个容易记住的表达式F = 40B2A,其中A的单位是cm2。这个方程说,如果磁场是1T,面积是1m2,那么磁力是40N或4kgf。然而,在许多实际情况下,确定B值有些困难,而且精度通常不令人满意。利用有限元方法进行计算机模拟可以确定各种边界条件下的磁力,但通常不方便工业用户使用。在本文中,我们报告了几个简单的方程,这些方程是基于三维计算机模拟生成的大型数据库建立的。用户可以通过简单地输入磁铁的$\mathrm{B}_{\mathrm{r}}$、面积和厚度来使用公式来获得力。本文还分析了载重线的作用。我们选择Infolytica的MagNet软件进行模拟。参数化功能,牛顿公差0.1% was used to systematically solve the problems for NdFeB cylinders, rings, and rectangular blocks interacting with CR1010 steel. The steel plates are both thicker and larger than the magnets. The maximum sizes for the magnets are shown in Table 1. The result database for each gap in a single boundary condition includes 62500 data points for rectangular blocks, 30625 data points for rings, and 1250 data points for cylinders. The gaps between the magnets and steel plates are in the range of 0.01 – 15mm with 23 unequal intervals. The itemized data were then plotted and analyzed to establish the force equations for the magnets with relative high load lines. Figure 1 shows the magnetic force vs the area of N52 magnet rings with gap = 0.01 mm to steel plates. Fig. 1a and 1b have different boundary conditions: 1a has CR1010 steel on both ends of the magnets, and 1b has the steel only on one end. The load line of a standalone magnet can be estimated by using the equations described in Parker’s book[7], but the magnets in this project have much higher load lines compared to the standalone magnets since steel plates are associated with these magnets. Boundary condition 1a obviously gives much higher load line compared to boundary condition 1b. For these ring magnets with higher load line in condition 1a, the force value vs area for each thickness can generate 2nd degree polynomial formulas, which has R-squared R2>0.9997 as shown in Figure 1. (R2 of 1.0000 was obtained for all the thicknesses of rectangular blocks). These formulas were then analyzed to establish a general equation $F = B_{r}^{2} {(aA}^{2}+ {bA)}$. Using the equation, the magnetic force for any $B_{r}$ value can be determined by inputting magnet’s $B_{r}$, area, and thickness. As shown in Table 1, the factor a is a function of thickness in 2nd degree polynomial, and the factor b is also a function of thickness but in power form. The effect of boundary condition is tremendous. Condition1b has much lower load line compared to condition 1a, hence the magnetic force values vs the area cannot generate satisfactory equations. As shown in the Fig. 1b, for the same magnet area, the magnetic force values are in a range with various values due to different load lines. For example, the ring magnets with exact the same thickness 0.1cm and area 2.8cm2, the force values range from 18.4N to 74N for ID/OD values from 0.1/1.9cm to 4.3/4.7cm. Details for all the magnet shapes with two boundary conditions will be reported in this paper, and the effect of load line will be analyzed.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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