An asymmetrical bias magnetic field force and stiffness characteristic equivalent calculation method considering magnetic flux leakage

Abstract

This invention belongs to the field of hybrid magnetic bearing technology and discloses an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux. This invention reduces the nonlinearity of force and stiffness when the rotor is offset in a static asymmetric bias magnetic field by enlarging the radial air gap domain, effectively expanding the linear operating range of the hybrid magnetic bearing while improving the rotor's dynamic response speed. Furthermore, due to the increased radial air gap, the leakage flux problem cannot be ignored when analyzing the rotor's stress state. Therefore, an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux is proposed. This method has advantages such as low computational load and convenient operation in practical engineering applications, and can provide technical support for predicting the static bearing characteristics of asymmetric bias magnetic fields applicable to hybrid magnetic bearings.

Description

Technical Field

[0001] This invention belongs to the field of hybrid magnetic bearing technology and relates to an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field that takes into account leakage magnetic flux. Background Technology

[0002] Hybrid magnetic bearing technology is a novel bearing technology that utilizes the magnetic field generated by permanent magnets to replace the static bias magnetic field generated by electromagnetic windings, thereby reducing the heating and power consumption of the electromagnetic windings. The presence of the static bias magnetic field provides a linearized operating point for the system, allowing mature linear control theories (such as PID control) to be directly applied to electromagnetic bearing systems, simplifying controller design. Furthermore, the bias magnetic field generated by the permanent magnets can significantly improve the force-current gain of the system, enabling the control system to achieve faster force adjustment with smaller current changes and improving the system's dynamic response speed. Therefore, the rational and effective design of the bias magnetic field in hybrid magnetic bearings plays a crucial role in improving the overall performance of the system.

[0003] Existing technologies for improving the load-bearing capacity of hybrid magnetic bearings mostly rely on increasing the strength of the bias magnetic field generated by the permanent magnet in the air gap. In their paper "Application Design and Characteristic Analysis of Halbach Array in Hybrid Radial Magnetic Suspension Bearings," Liu Jun, Lin Xiaojun, and Dong Dengwen proposed a 2-pole, 3-segment Halbach magnetic ring array bias magnetic field structure to improve the single-sided magnetization effect and the utilization rate of the permanent magnet. However, since the bias magnetic field generated by the permanent magnet is uniformly distributed along the rotor circumference, the magnetic attraction forces on the rotor cancel each other out, and the strong bias magnetic field is prone to causing magnetic yoke saturation, thus failing to achieve the desired bearing performance. Therefore, proposing an asymmetric static bias magnetic field is expected to be an effective way to solve the limitation of improving the load-bearing capacity of hybrid magnetic bearings.

[0004] Numerous patents have addressed the issue of generating asymmetric bias magnetic fields. In their patent "A Marine Permanent Magnet Suspension Support Water-Lubricated Bearing" (CN201922020392.9), Ma Zhongwei, Chen Demin, and Ma Xiao proposed a single-sided arrangement of permanent magnets that generates an asymmetric bias magnetic field, thus producing an upward magnetic attraction force on the shaft. This reduces the pressure of the shaft on the water-lubricated bearing body and improves the bearing's service life. This patent demonstrates the feasibility of using asymmetric bias magnetic fields to improve bearing load performance. However, the proposed bearing structure still relies on contact friction between the composite bushing and the shaft for load bearing. The stiffness of the bias magnetic attraction force is highly nonlinear, making it impossible to achieve full shaft suspension and directly applicable to hybrid magnetic bearing systems. Furthermore, the patent does not provide a method for analyzing the force state of the rotor moving in an asymmetric bias magnetic field.

[0005] The magnitude and linearity of the forces and stiffness experienced by the rotor when moving in the static bias magnetic field generated by the permanent magnet directly affect the control difficulty and dynamic response speed of the hybrid magnetic bearing system. Therefore, accurate analysis of the force state of the rotor in the bias magnetic field is of great significance for guiding the initial design of hybrid magnetic bearings and predicting their dynamic control performance. Currently, most analyses of the force state of the rotor in the bias magnetic field use the finite element method. However, due to limitations in mesh generation, boundary condition settings, and other factors, the solution time and accuracy vary considerably. Furthermore, existing analytical methods mostly ignore the influence of factors such as leakage flux. Therefore, proposing a fast analytical calculation method that considers the influence of leakage flux is of great significance.

[0006] In summary, an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering the influence of leakage flux is proposed. This method is of great significance for solving the problems of limited load capacity, difficulty in linearization control, and poor dynamic response speed of the bias magnetic field in existing hybrid magnetic bearings, as well as for providing initial guidance for rapid analytical calculation of the force and stiffness characteristics of the asymmetric bias magnetic field and for accurate prediction of bearing dynamic performance in the later stage. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, this invention provides an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field that considers leakage flux. The aim is to improve bearing capacity while reducing system control complexity by proposing a low-negative-stiffness, linearized design method for an asymmetric bias magnetic field suitable for hybrid magnetic bearing systems. Furthermore, it combines the equivalent magnetic circuit method to analytically calculate the force and stiffness characteristics of the rotor moving in the static asymmetric bias magnetic field generated by the permanent magnet. This method does not rely on finite element simulation and offers good speed, practicality, and convenience in practical engineering applications.

[0008] The technical solution of the present invention: An equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux is proposed, with the following steps: The first step is to construct an asymmetric bias magnetic field structure suitable for hybrid magnetic bearings; To address the limitations of existing hybrid magnetic bearings in terms of limited load capacity, difficulty in linearization control, and poor dynamic response speed due to their static bias magnetic fields, an asymmetric bias magnetic field structure suitable for hybrid magnetic bearings is proposed. This asymmetric bias magnetic field structure is symmetrical along the axial direction and includes a bearing housing 1, bias magnets 2, a plastic support 3, and a bushing 4. The bias magnets 2, arranged axially adjacent to each other with opposite magnetization directions and a tile-shaped outline, are embedded in the upper side of the bearing housing 1 and fixed by the plastic support 3. The asymmetric bias magnetic field generated by the bias magnets 2 within the bushing 4 produces a bias magnetic attraction force on the bushing 4, counteracting the gravity of the bushing 4. The bias magnets 2, with the same magnetic flux, are symmetrical about the axial direction. The axes are divided into three groups of two. The two bias magnets 2 located at the two ends of the axes are divided into group one, the two bias magnets 2 adjacent to them and close to the axis of symmetry are divided into group two, and the two bias magnets 2 close to the axis of symmetry are divided into group three. The radial air gap domain of this asymmetric bias magnetic field structure includes the radial air gap domain between the plastic support 3 and the bushing 4, the radial air gap domain between the bearing housing 1 and the bushing 4 for non-contact levitation of the bushing 4, and the radial air gap domain between the bias magnet 2 and the bushing 4, which is equivalent to the plastic support 3 and ignores the influence of the plastic support 3 on the magnetic field distribution. The radial air gap domain equivalent to the plastic support 3 is used to weaken the nonlinearity of the force and stiffness generated by the bias magnet 2 on the bushing 4. The second step is to discretize the air gap domain based on the magnetic field distribution. Based on the characteristics of the magnetic field distribution, the radial air gap domain of the asymmetric bias magnetic field structure is discretized into air gap solution domains according to different types.① ~ ⑤, where the air gap solution domain ① is the solution domain of the radial magnetic field generated between the bearing housing 1 and the bushing 4 at the radial air gap region on both sides of the shaft end, and the magnetic reluctance of the air gap solution domain ① is R g1 The air gap solution domain ② is the solution domain for shaft end leakage magnetic flux generated in the radial air gap region between the bias magnet 2 located at both shaft ends and the bearing housing 1, and the magnetic reluctance of the air gap solution domain ② is... R glf1 The solution domain for the radial magnetic field generated between any bias magnet 2 and the bushing 4 in the radial air gap region is uniformly divided axially into four parts. Air gap solution domain ③ is located on both sides of the radial magnetic field solution domain and occupies one-quarter of the width. Air gap solution domain ④ is located in the middle of the radial magnetic field solution domain and occupies half the width. The magnetic reluctance of air gap solution domain ③ is... R g2 The magnetic reluctance of the air gap solution domain ④ is R g3 The air gap solution domain ⑤ is the solution domain for the inter-pole leakage magnetic flux generated by any two adjacent bias magnets 2 in the radial air gap region, and the magnetic reluctance of the air gap solution domain ⑤ is... R glf2Since the leakage flux of the air gap solution domain ② and air gap solution domain ⑤ does not exert a force on the bushing 4, the influence of the air gap solution domain ② and air gap solution domain ⑤ is ignored when solving for the force and stiffness of the bias magnet 2 on the bushing 4. The third step is to establish an equivalent magnetic circuit model of the asymmetric bias magnetic field that takes into account the effects of shaft end leakage flux and inter-pole leakage flux. Any two adjacent bias magnets 2 arranged close together along the axial direction have opposite magnetization directions. Therefore, any two adjacent bias magnets 2 can form a closed magnetic circuit, and the total magnetic flux of the two bias magnets 2 in group one is... The total magnetic flux of the two bias magnets 2 in group 2 is The total magnetic flux of the two bias magnets 2 in group 3 is Due to the presence of leakage flux at the shaft end, the total magnetic flux... Less than the total magnetic flux and total magnetic flux And the total magnetic flux when the leakage flux at the shaft end is stable Since the plastic support 3 is equivalent to a radial air gap region, the total magnetic flux With total magnetic flux and total magnetic flux There is inter-electrode leakage flux; An equivalent magnetic circuit model of the asymmetric bias magnetic field, considering the effects of shaft-end leakage flux and inter-pole leakage flux, is established using the equivalent magnetic circuit method: First, the bias magnet 2 is equivalent to a voltage source with internal resistance, and the magnetomotive force of the bias magnet 2 is... F pm The magnetic reluctance of bias magnet 2 is R pm Secondly, the total magnetic flux of bias magnet 2 in group two... Starting from the bias magnet 2 in group 1, establish the total magnetic flux. The total magnetic flux of bias magnet 2 in group 3 The corresponding main magnetic circuit, total magnetic flux First, it passes through the reluctance formed by two air gap solution domains ③ and one air gap solution domain ④ connected in parallel. R / / 2 It then enters the bushing 4 and splits into two paths, one of which passes through the magnetic resistance of the bushing 4. R as Entering the reluctance formed by the parallel connection of an air gap solution domain ③ and an air gap solution domain ④ R / / 1 Then the total magnetic flux is fed in. And through the magnetic resistance of bearing housing 1 R bh Finally, it flows into the starting point's total magnetic flux. And the magnetic resistance of the bearing housing 1 R bh Then it is re-infused into the starting point's total magnetic flux. The magnetic flux becomes The other path passes through the magnetic reluctance of bushing 4. R as Entering the reluctance formed by the parallel connection of two air gap solution domains ③ and one air gap solution domain ④ R / / 2 Then the total magnetic flux is fed in. And through the magnetic resistance of bearing housing 1 R bh Finally, it flows into the starting point's total magnetic flux. And the magnetic resistance of the bearing housing 1 R bh Then it is re-infused into the starting point's total magnetic flux. The magnetic flux becomes the total magnetic flux. Based on the conservation of magnetic flux, the magnetic flux relationship is as follows: Then, the total magnetic flux of bias magnet 2 in group one is used. Establish a leakage magnetic circuit at the shaft end as the starting point, with total magnetic flux Partial magnetic flux First, the leakage magnetic reluctance at the end of the bearing housing 1 R e4 and end leakage magnetic reluctance R e3 It then splits into two paths, one of which passes through the end leakage magnetic reluctance of bearing housing 1. R e2 Then, the magnetic reluctance is calculated in the air gap solution domain ②. R glf1 Ultimately, it flows into the total magnetic flux at the starting point. The other path passes through the air gap solution domain ① and measures the magnetic reluctance. R g1 The leakage magnetic reluctance at the end of the bushing 4 then enters R e1 And through the air gap solution domain ③, the magnetic reluctance R g2 Finally, it flows into the total magnetic flux at the starting point. According to the law of conservation of magnetic flux, the magnetic flux relationship is as follows: Finally, establish the total magnetic flux. With total magnetic flux Total magnetic flux The corresponding inter-pole leakage magnetic circuit is obtained by solving the magnetic reluctance of domain ⑤ using the air gap. R glf2 It is obtained by directly connecting it in parallel with the main magnetic circuit in the radial air gap domain; thus, the equivalent magnetic circuit model of the asymmetric bias magnetic field considering the influence of shaft end leakage magnetic field and inter-pole leakage magnetic field is established; Step 4: Calculate each magnetic reluctance and solve the magnetic circuit based on Kirchhoff's laws for magnetic circuits; The reluctances of each element in the equivalent magnetic circuit model of the asymmetric bias magnetic field established in the third step are calculated. First, the reluctance of the main magnetic circuit is calculated, including the reluctance of the bias magnet 2. Rpm Expressed as follows: In the formula, μ 0 is the permeability of free space, and μ 0 = 4π × 10 -7 H / m; μ pm The relative permeability of bias magnet 2; α The sector angle of bias magnet 2; l m The axial width of the bias magnet 2; r pmo The outer radius of the bias magnet 2; r pmi Let be the inner radius of the bias magnet 2; Magnetic resistance of bearing housing 1 R bh Expressed as follows: In the formula, μ i The relative permeability of the bearing housing 1 and the bushing 4 is given. r bho The outer radius of bearing housing 1; Magnetic resistance of bushing 4 R as Expressed as follows: In the formula, r aso The outer radius of bushing 4, r asi The inner radius of bushing 4; Secondly, the magnetic reluctance of the air gap solution domains ① to ⑤ is calculated and expressed by the following formula: In the formula, μ a The relative permeability of air; l n The thickness of the shaft end of bearing housing 1; r bhi Let be the inner radius of bearing housing 1; δ y This is the offset when bushing 4 moves in the vertical direction; g yThe thickness of the radial air gap region along the vertical direction is obtained from the following formula: R / / 1 and R / / 2 The parallel reluctance in the solution domain of the air gap is obtained from the following equation: In the formula, / / indicates the parallel relationship of magnetic reluctance; Then, the leakage magnetic reluctance at the end of the bushing 4. R e1 and the end leakage magnetic reluctance of bearing housing 1 R e2 , R e3 and R e4 The calculation is performed using the following formula: Total magnetic reluctance of end leakage R elf Expressed as follows: Finally, establish magnetic flux. , , , , and The relationship equations between the two sides are used to solve the equivalent magnetic circuit model of the asymmetric bias magnetic field established in the third step, based on Kirchhoff's law for magnetic circuits, to obtain the magnetic flux. , and The analytical expression, where the magnetomotive force of bias magnet 2 is... F pm Expressed as follows: In the formula, H c The coercivity of the bias magnet 2; The bias magnet 2 is equivalent to a voltage source with internal resistance. The magnetic flux in the equivalent magnetic circuit model of the asymmetric bias magnetic field is obtained according to Kirchhoff's laws for magnetic circuits. , , , , and The system of relational equations is expressed by the following formula: The magnetic flux is obtained by solving the above system of equations. , and The analytical expression is represented by the following formula: In the formula, m and n To simplify the system of coefficient equations, it is expressed as follows: Step 5: Integrate the magnetic attraction force generated in each air gap solution domain to obtain the expression for the bias magnetic attraction force; Ignoring the magnetic attraction force generated by the leakage flux on the bushing 4 at the air gap solution domains ② and ⑤, according to Maxwell's stress tensor method, the bias magnetic attraction force can be obtained by integrating the magnetic attraction force generated in each air gap solution domain. F m ( δ y The expression for ) is as follows: In the formula, B gk For the first k The air gap magnetic flux density is calculated in the solution domain of the air gap. k Take 1, 3, and 4, and obtain the following formula: A gk For the first k The cross-sectional area of ​​the air gap is calculated using the following formula: Step 6: Obtain the stiffness and linear control equation of the bushing when it is offset by fitting. By fitting the bias magnetic attraction force F m ( δ y The expression is simplified linearly to obtain the linear governing equation of the bias magnetic attraction force on the bushing 4 when it is offset vertically in the asymmetric bias magnetic field generated by the bias magnet 2: In the formula, K m For linear fitting values ​​of displacement stiffness, F b0The value is the linear fitting value of the static bias magnetic attraction force at the center position where the bushing 4 and the plastic support 3 are coaxial; thus, the equivalent calculation of the asymmetric bias magnetic field force and stiffness characteristics considering leakage magnetic field is completed.

[0009] The beneficial effects of this invention are that it proposes an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux. The asymmetric bias magnetic field structure employs a radially equivalent large air gap design, which weakens the nonlinearity of the force and stiffness generated by the permanent magnet, widens the linear operating range of the hybrid magnetic bearing, helps improve the dynamic response speed of the rotor, and reduces the design difficulty of the controller. Furthermore, addressing the problem that the increased radial air gap leads to leakage flux affecting the solution accuracy, an equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux is proposed. Through analytical analysis, the linear control equations for the asymmetric bias magnetic field suitable for hybrid magnetic bearings are obtained, providing technical support for the initial design of the bias magnetic field for hybrid magnetic bearings and the subsequent prediction of their dynamic performance. In practical engineering applications, it has advantages such as low computational load and convenient operation. Attached Figure Description

[0010] Figure 1 It is a simplified three-dimensional model of an asymmetric bias magnetic field structure suitable for hybrid magnetic bearings; Figure 2 yes Figure 1 3D simplified model along XY A sectional view of a plane; Figure 3 yes Figure 1 3D simplified model along YZ A sectional view of a plane; Figure 4 It is an equivalent magnetic circuit analysis model for an asymmetric bias magnetic field structure that takes into account the influence of leakage magnetic field. Figure 5 This is a schematic diagram of the discretization of the air gap domain at an asymmetric bias magnetic field structure. Figure 6 This is an equivalent calculation flowchart for the force and stiffness characteristics of an asymmetric bias magnetic field that considers leakage flux; In the diagram: 1-bearing housing, 2-bias magnet, 3-plastic support, 4-shaft sleeve. Detailed Implementation

[0011] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0012] Example An asymmetric bias structure with a static bias magnetic attraction force design value of 1000N was selected for equivalent calculation of force and stiffness characteristics. Its key parameters are as follows: Figure 2 and Figure 3 The equivalent calculation process is as shown below. Figure 6As shown, the specific parameter values ​​are set as follows: r bho =0.075m r bhi =0.0505m r pmo =0.062m r pmi =0.054m r aso =0.05m r asi =0.03m l m =0.015m l n =0.01m、 α =2.094 rad; the material properties for each computational domain are set as follows: μ a =1、 μ i =4000、 μ m =1.05、 H c =890kA / m; the specific solution process is as follows: The first step is to construct an asymmetric bias magnetic field structure suitable for hybrid magnetic bearings; A simplified 3D model of the asymmetric bias magnetic field structure for hybrid magnetic bearings is shown below. Figure 1 The bearing housing includes a bearing housing 1, a bias magnet 2, a plastic support 3, and a bushing 4. The bias magnet 2 is embedded in the upper side of the bearing housing 1 and fixed by the plastic support 3. The asymmetric bias magnetic field generated by the bias magnet 2 in the bushing 4 produces a bias magnetic attraction force on the bushing 4 to counteract the gravity of the bushing 4. The influence of the plastic support 3 on the magnetic field distribution is negligible. The radial air gap domain equivalent to the plastic support 3 is used to weaken the nonlinearity of the force and stiffness generated by the bias magnet 2 on the bushing 4. The second step is to discretize the air gap domain based on the magnetic field distribution. The results of the radial air gap domain discretization at the asymmetric bias magnetic field structure are shown below. Figure 5 Among them, air gap solution domain ① is the solution domain of the radial magnetic field generated between the bearing housing 1 and the bushing 4 located at both shaft ends in the radial air gap region; air gap solution domain ② is the solution domain of shaft end leakage magnetic field generated between the bias magnet 2 and the bearing housing 1 located at both shaft ends in the radial air gap region; air gap solution domain ③ is the solution domain of the radial magnetic field occupying one-quarter of the width between the bias magnet 2 and the bushing 4; air gap solution domain ④ is the solution domain of the radial magnetic field occupying half the width between the bias magnet 2 and the bushing 4; air gap solution domain ⑤ is the solution domain of the inter-pole leakage magnetic field generated between any two adjacent bias magnets 2 in the radial air gap region; The third step is to establish an equivalent magnetic circuit model of the asymmetric bias magnetic field that takes into account the effects of shaft end leakage flux and inter-pole leakage flux. The equivalent magnetic circuit model of the asymmetric bias magnetic field considering the effects of shaft end leakage flux and inter-pole leakage flux is as follows: Figure 4 As shown, where, F pm For the magnetomotive force of bias magnet 2, R pm For the magnetic reluctance of bias magnet 2, R bh For the magnetic resistance of bearing housing 1, R as For the magnetic resistance of bushing 4, R e1 ~ R e4 For end leakage magnetic reluctance, R / / 1 and R / / 2 Parallel reluctance in discrete air gap domain, and For the magnetic flux of the magnetic circuit, R g1 For the magnetic reluctance of the discrete air gap domain ①, R glf1 For the magnetic reluctance of the discrete air gap domain ②, R g2 For the magnetic reluctance of the discrete air gap domain ③, R g3 For the magnetic reluctance of the discrete air gap domain ④, R glf2 The magnetoresistance of the discrete air gap domain ⑤; Step 4: Calculate each magnetic reluctance and solve the magnetic circuit based on Kirchhoff's laws for magnetic circuits; The forces acting on bushing 4 and plastic support 3 when they are coaxial are calculated. δ y Substituting =0m into the calculation yields the result. R pm =3.3328×10 6 H -1 , R bh =1.6×10 3 H -1 , R as =1.781×10 3 H -1 , R g1 =3.7807×10 5 H -1 , R g2=1.339×10 7 H -1 , R g3 =4.1774×10 6 H -1 , R glf1 =5.6172×10 6 H -1 , R glf2 =1.1052×10 7 H -1 , g y =0.004m R / / 1 =3.1841×10 6 H -1 , R / / 2 =2.5724×10 6 H -1 , R e1 =1.4842×10 3 H -1 , R e2 =2.6281×10 3 H -1 , R e3 =1.3123×10 3 H -1 , R e4 =2.201×10 3 H -1 , R elf =3.9945×10 6 H -1 , F pm =7120At、 , , , m =3.7874×10 6 H -1 , n =3.513×10 6 H -1 ; Step 5: Integrate the magnetic attraction force generated in each air gap solution domain to obtain the expression for the bias magnetic attraction force; Substituting the result from step four into the calculation yields the result. B g1=0.1508T B g2 =0.3503T B g3 =0.6416、 A g1 =0.001m 2 , A g2 =4.1888×10 -4 m 2 , A g3 =7.3304×10 -4 m 2 Bias magnetic attraction F m (0) = 984.6 N; Step 6: Obtain the stiffness and linear governing equations of the bushing when it is offset by fitting. Take respectively δ y =±0.0004m, ±0.0003m, ±0.0002m, ±0.0001m, repeat steps four and five, and the calculated bias magnetic force on bushing 4 under different offsets is: F m (-0.0004) = 847.8 N F m (-0.0003) = 880.5 N F m (-0.0002) = 914.2 N F m (-0.0001) = 948.9 N F m (0.0001) = 1021.5 N F m (0.0002) = 1059.7 N F m (0.0003) = 1099.1 N F m (0.0004) = 1139.9 N. The stiffness linear fitting value can be obtained through linear fitting. K m =3.65×10 5 N / m, the fitted value of the static bias magnetic attraction force at the center position of the coaxial position of the bushing 4 and the plastic support 3. F b0 =988.47N, so the linear governing equation for the bias magnetic attraction force is: F m =3.65×10 5 δy +988.47, the calculation is now complete.

[0013] This calculation method takes into account the influence of leakage magnetic field on the solution accuracy and proposes a discrete equivalent calculation method for the air gap domain to solve the force and stiffness characteristics of asymmetric bias magnetic field. It can provide technical support for the prediction of static bearing characteristics of asymmetric bias magnetic field applicable to hybrid magnetic bearings. It has the advantages of small calculation amount and convenient operation in practical engineering applications.

Claims

1. A method for equivalent calculation of the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux, characterized in that, The steps are as follows: The first step is to construct an asymmetric bias magnetic field structure suitable for hybrid magnetic bearings; The asymmetric bias magnetic field structure is symmetrical along the axial direction and includes a bearing housing (1), bias magnets (2), plastic supports (3), and bushings (4). The bias magnets (2), which are arranged close together along the axial direction, have opposite magnetization directions, and have a tile-shaped outline, are embedded in the upper side of the bearing housing (1) and fixed by the plastic supports (3). The asymmetric bias magnetic field generated by the bias magnets (2) in the bushings (4) produces a bias magnetic attraction force on the bushings (4) to counteract the gravity of the bushings (4). The bias magnets (2) with the same magnetic flux are divided into three groups of two about the axis of symmetry along the axial direction. The two bias magnets (2) located at the two ends of the shaft are divided into two groups. First, the two bias magnets (2) adjacent to and close to the axis of symmetry are divided into group two, and the two bias magnets (2) close to the axis of symmetry are divided into group three; the radial air gap domain of the asymmetric bias magnetic field structure includes the radial air gap domain between the plastic support (3) and the bushing (4), between the bearing housing (1) and the bushing (4) for non-contact suspension of the bushing (4), and the radial air gap domain between the bias magnet (2) and the bushing (4) which is equivalent to the plastic support (3) ignoring the influence of the plastic support (3) on the magnetic field distribution, wherein the radial air gap domain equivalent to the plastic support (3) is used to weaken the nonlinearity of the force and stiffness generated by the bias magnet (2) on the bushing (4); The second step is to discretize the air gap domain based on the magnetic field distribution. The third step is to establish an equivalent magnetic circuit model of the asymmetric bias magnetic field that takes into account the effects of shaft end leakage flux and inter-pole leakage flux. Step 4: Calculate each magnetic reluctance and solve the magnetic circuit based on Kirchhoff's laws for magnetic circuits; Step 5: Integrate the magnetic attraction force generated in each air gap solution domain to obtain the expression for the bias magnetic attraction force; Step 6: Obtain the stiffness and linear control equation of the bushing when it is offset by fitting.

2. The equivalent calculation method for the force and stiffness characteristics of the asymmetric bias magnetic field considering leakage flux as described in claim 1, characterized in that, The specific implementation process of the second step is as follows: Based on the characteristics of the magnetic field distribution, the radial air gap domain of the asymmetric bias magnetic field structure is discretized into air gap solution domains according to different types.① ~ ⑤, where the air gap solution domain ① is the solution domain of the radial magnetic field generated between the bearing housing (1) and the bushing (4) at the radial air gap region on both sides of the shaft end, and the magnetic reluctance of the air gap solution domain ① is R g1 The air gap solution domain ② is the solution domain for shaft end leakage magnetic flux generated in the radial air gap region between the bias magnet (2) located at both shaft ends and the bearing housing (1), and the magnetic reluctance of the air gap solution domain ② is... R glf1 The radial magnetic field solution domain generated between any bias magnet (2) and the bushing (4) in the radial air gap region is uniformly divided into four parts along the axial direction. Among them, the air gap solution domain ③ is located on both sides of the radial magnetic field solution domain and occupies one-quarter of the width, and the air gap solution domain ④ is located in the middle of the radial magnetic field solution domain and occupies half of the width. The magnetic reluctance of the air gap solution domain ③ is R g2 The magnetic reluctance of the air gap solution domain ④ is R g3 The air gap solution domain ⑤ is the solution domain for the inter-pole leakage magnetic flux generated by any two adjacent bias magnets (2) in the radial air gap region, and the magnetic reluctance of the air gap solution domain ⑤ is... R glf2 Since the leakage flux of the air gap solution domain ② and the air gap solution domain ⑤ does not exert a force on the bushing (4), the influence of the air gap solution domain ② and the air gap solution domain ⑤ is ignored when solving the force and stiffness of the bias magnet (2) on the bushing (4).

3. The equivalent calculation method for the force and stiffness characteristics of the asymmetric bias magnetic field considering leakage flux as described in claim 2, characterized in that, The specific implementation process of the third step is as follows: Any two adjacent bias magnets (2) arranged close together along the axial direction have opposite magnetization directions. Therefore, any two adjacent bias magnets (2) can form a closed magnetic circuit. The total magnetic flux of the two bias magnets (2) in Group 1 is 1. The total magnetic flux of the two bias magnets (2) in group two is The total magnetic flux of the two bias magnets (2) in group three is Due to the presence of leakage flux at the shaft end, the total magnetic flux... Less than the total magnetic flux and total magnetic flux And the total magnetic flux when the leakage flux at the shaft end is stable Since the plastic support (3) is equivalent to a radial air gap domain, the total magnetic flux With total magnetic flux and total magnetic flux There is inter-electrode leakage flux; An equivalent magnetic circuit model of the asymmetric bias magnetic field considering the effects of shaft-end leakage flux and inter-pole leakage flux is established using the equivalent magnetic circuit method: First, the bias magnet (2) is equivalent to a voltage source with internal resistance, and the magnetomotive force of the bias magnet (2) is... F pm The magnetic reluctance of the bias magnet (2) is R pm Secondly, the total magnetic flux of the bias magnet (2) in group two. Starting from the bias magnet (2) in group 1, establish the total magnetic flux. The total magnetic flux of the bias magnet (2) in group 3 The corresponding main magnetic circuit, total magnetic flux First, it passes through the reluctance formed by two air gap solution domains ③ and one air gap solution domain ④ connected in parallel. R / / 2 It then enters the bushing (4) and splits into two paths, one of which passes through the magnetic resistance of the bushing (4). R as Entering the reluctance formed by the parallel connection of an air gap solution domain ③ and an air gap solution domain ④ R / / 1 Then the total magnetic flux is fed in. And through the magnetic resistance of the bearing housing (1) R bh Finally, it flows into the starting point's total magnetic flux. And the magnetic resistance of the bearing housing (1) R bh Then it is re-infused into the starting point's total magnetic flux. The magnetic flux becomes The other path passes through the magnetic resistance of the bushing (4). R as Entering the reluctance formed by the parallel connection of two air gap solution domains ③ and one air gap solution domain ④ R / / 2 Then the total magnetic flux is fed in. And through the magnetic resistance of the bearing housing (1) R bh Finally, it flows into the starting point's total magnetic flux. And the magnetic resistance of the bearing housing (1) R bh Then it is re-infused into the starting point's total magnetic flux. The magnetic flux becomes the total magnetic flux. Based on the conservation of magnetic flux, the magnetic flux relationship is as follows: Then, the total magnetic flux of the bias magnet (2) in group one is used. Establish a leakage magnetic circuit at the shaft end as the starting point, with total magnetic flux Partial magnetic flux First, the leakage magnetic reluctance at the end of the bearing housing (1) R e4 and end leakage magnetic reluctance R e3 It then splits into two paths, one of which passes through the end leakage magnetic reluctance of the bearing housing (1). R e2 Then, the magnetic reluctance is calculated in the air gap solution domain ②. R glf1 Ultimately, it flows into the total magnetic flux at the starting point. The other path passes through the air gap solution domain ① and measures the magnetic reluctance. R g1 The end leakage magnetic reluctance of the bushing (4) then enters R e1 And through the air gap solution domain ③, the magnetic reluctance R g2 Finally, it flows into the total magnetic flux at the starting point. According to the law of conservation of magnetic flux, the magnetic flux relationship is as follows: Finally, establish the total magnetic flux. With total magnetic flux Total magnetic flux The corresponding inter-pole leakage magnetic circuit is obtained by solving the magnetic reluctance of domain ⑤ using the air gap. R glf2 It is obtained by directly connecting it in parallel with the main magnetic circuit in the radial air gap domain; This completes the establishment of the equivalent magnetic circuit model of the asymmetric bias magnetic field that takes into account the effects of shaft end leakage flux and inter-pole leakage flux.

4. The equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux as described in claim 3, characterized in that, The specific implementation process of the fourth step is as follows: The magnetic reluctances of each component in the equivalent magnetic circuit model of the asymmetric bias magnetic field established in the third step are calculated. First, the magnetic reluctance of the main magnetic circuit is calculated, including the magnetic reluctance of the bias magnet (2). R pm Expressed as follows: In the formula, μ 0 is the permeability of free space, and μ 0 = 4π × 10 -7 H / m; μ pm The relative permeability of the bias magnet (2); α The sector angle of the bias magnet (2); l m The axial width of the bias magnet (2); r pmo The outer radius of the bias magnet (2); r pmi Let be the inner radius of the bias magnet (2); Magnetic resistance of bearing housing (1) R bh Expressed as follows: In the formula, μ i The relative permeability of the bearing housing (1) and the bushing (4) is given by... r bho Let be the outer radius of the bearing housing (1); Magnetic resistance of bushing (4) R as Expressed as follows: In the formula, r aso The outer radius of the bushing (4) r asi Let be the inner radius of the bushing (4); Secondly, the magnetic reluctance of the air gap solution domains ① to ⑤ is calculated and expressed by the following formula: In the formula, μ a The relative permeability of air; l n The thickness of the shaft end of the bearing housing (1); r bhi Let be the inner radius of the bearing housing (1); δ y The offset of the bushing (4) when it moves in the vertical direction; g y The thickness of the radial air gap region along the vertical direction is obtained from the following formula: R / / 1 and R / / 2 The parallel reluctance in the solution domain of the air gap is obtained from the following equation: In the formula, / / indicates the parallel relationship of magnetic reluctance; Then, the end leakage magnetic reluctance of the bushing (4) is... R e1 The end leakage magnetic resistance of the bearing housing (1) R e2 , R e3 and R e4 The calculation is performed using the following formula: Total magnetic reluctance of end leakage R elf Expressed as follows: Finally, establish magnetic flux. , , , , and The relationship equations between the two sides are used to solve the equivalent magnetic circuit model of the asymmetric bias magnetic field established in the third step, based on Kirchhoff's law for magnetic circuits, to obtain the magnetic flux. , and The analytical expression, in which the magnetomotive force of the bias magnet (2) is... F pm Expressed as follows: In the formula, H c The coercivity of the bias magnet (2); The bias magnet (2) is equivalent to a voltage source with internal resistance. The magnetic flux in the equivalent magnetic circuit model of the asymmetric bias magnetic field is obtained according to Kirchhoff's law of magnetic circuits. , , , , and The system of relational equations is expressed by the following formula: The magnetic flux is obtained by solving the above system of equations. , and The analytical expression is represented by the following formula: In the formula, m and n To simplify the system of coefficient equations, it is expressed as follows: 。 5. The equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux as described in claim 4, characterized in that, The specific implementation process of step five is as follows: Ignoring the magnetic attraction force generated by the leakage flux at the air gap solution domains ② and ⑤ on the bushing (4), according to Maxwell's stress tensor method, the bias magnetic attraction force can be obtained by integrating the magnetic attraction force generated in each air gap solution domain. F m ( δ y The expression for ) is as follows: In the formula, B gk For the first k The air gap magnetic flux density is calculated in the solution domain of the air gap. k Take 1, 3, and 4, and obtain the following formula: A gk For the first k The cross-sectional area of ​​the air gap is calculated using the following formula: 。 6. The equivalent calculation method for the force and stiffness characteristics of an asymmetric bias magnetic field considering leakage flux as described in claim 5, characterized in that, The specific implementation process of step six is ​​as follows: By fitting the bias magnetic attraction force F m ( δ y The expression of ) is simplified linearly to obtain the linear control equation of the bias magnetic attraction force on the bushing (4) when it is offset vertically in the asymmetric bias magnetic field generated by the bias magnet (2): In the formula, K m For linear fitting values ​​of displacement stiffness, F b0 The static bias magnetic attraction force at the center position of the bushing (4) and the plastic support (3) is the linear fitting value; thus, the equivalent calculation of the asymmetric bias magnetic field force and stiffness characteristics considering leakage magnetic field is completed.

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