CN117477889A - Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method - Google Patents

Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method Download PDF

Info

Publication number
CN117477889A
CN117477889A CN202311338692.6A CN202311338692A CN117477889A CN 117477889 A CN117477889 A CN 117477889A CN 202311338692 A CN202311338692 A CN 202311338692A CN 117477889 A CN117477889 A CN 117477889A
Authority
CN
China
Prior art keywords
permanent magnet
permanent magnets
axis
magnetized
permanent
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202311338692.6A
Other languages
Chinese (zh)
Inventor
曾理湛
刘凯
赵烁
陈福祥
胡傲奇
张昊
陈学东
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Huazhong University of Science and Technology
Original Assignee
Huazhong University of Science and Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Huazhong University of Science and Technology filed Critical Huazhong University of Science and Technology
Priority to CN202311338692.6A priority Critical patent/CN117477889A/en
Publication of CN117477889A publication Critical patent/CN117477889A/en
Pending legal-status Critical Current

Links

Classifications

    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K41/00—Propulsion systems in which a rigid body is moved along a path due to dynamo-electric interaction between the body and a magnetic field travelling along the path
    • H02K41/02—Linear motors; Sectional motors
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K2213/00—Specific aspects, not otherwise provided for and not covered by codes H02K2201/00 - H02K2211/00
    • H02K2213/03—Machines characterised by numerical values, ranges, mathematical expressions or similar information

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Computer Hardware Design (AREA)
  • Electromagnetism (AREA)
  • Power Engineering (AREA)
  • Combustion & Propulsion (AREA)
  • Evolutionary Computation (AREA)
  • Geometry (AREA)
  • General Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Chemical & Material Sciences (AREA)
  • Particle Accelerators (AREA)

Abstract

The invention discloses a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor and a modeling method, belonging to the field of permanent magnet arrays and electromagnetic modeling, wherein the permanent magnet arrays are p-row q-column arrays formed by alternately and periodically arranging first permanent magnet units and second permanent magnet units; the first permanent magnet unit comprises vertical magnetized permanent magnets arranged according to a Halbach arrangement mode and horizontal magnetized permanent magnets distributed on the upper part, the lower part, the left part and the right part of the vertical magnetized permanent magnets; the vertical magnetized permanent magnet is a polygonal prism with 4 alpha sides, which is formed by chamfering four right-angle parts of a square column body; the horizontal magnetization permanent magnet is a rectangular cylinder or a 4 alpha-side polygonal prism formed by chamfering four right-angle parts of the rectangular cylinder; the second permanent magnet unit has the same structure as the first permanent magnet unit and has opposite polarization directions. And a modeling method of the permanent magnet array is provided. The invention solves the problems of larger electromagnetic force fluctuation and time consumption in calculation in the running process of the traditional long-stroke magnetic levitation planar motor.

Description

Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method
Technical Field
The invention belongs to the field of permanent magnet arrays and electromagnetic modeling, and particularly relates to a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor and a modeling method.
Background
The long-stroke magnetic levitation planar workbench is a core component for realizing large-stroke six-degree-of-freedom precise movement, becomes an urgent need for development of semiconductor manufacturing equipment and ultra-precise intelligent manufacturing equipment represented by an IC photoetching machine, and mainly comprises a permanent magnet array and a coil array as key components, and utilizes Lorentz force generated by an electrified conductor in an air gap magnetic field to drive a moving component. Compared with the traditional contact type bearings such as an air bearing, a ball screw bearing, a flexible bearing and the like, the long-stroke magnetic levitation planar motor has the characteristics of simple structure, vacuum compatibility and large stroke. In the electromagnetic force control process of the long-stroke magnetic levitation planar motor, in order to improve the control frequency, a fourier series method is generally adopted, only the fundamental component of the air gap magnetic field is taken, and the influence of higher harmonics is ignored, so that the following two problems are mainly caused:
(1) In order to reduce higher order harmonic waves, the topological structure of the traditional permanent magnet array generally adopts oblique magnetized permanent magnets, and dimensional parameters such as the height, the mounting gap, the width and the like of the permanent magnets are modified, so that the difficulty of a processing technology is improved.
(2) In the permanent magnet array, when the hypotenuse of the changeable cylinder does not pass through the origin of coordinates, the equation for solving the bias derivative of the boundary magnetic field is too complex, and the process of calculating the electromagnetic force by adopting the triple volume fraction is time-consuming.
The first problem described above causes an increase in the manufacturing cost of the permanent magnet, and in the installation process of the permanent magnet array, an increase in the accumulated error of the size chain, while increasing the difficulty of the processing process. The second problem causes complex solving process of the air-gap field of the permanent magnet array, complex expression of components of each direction of the magnetic field, and can not clearly and rapidly show the distribution rule of the air-gap field in the design process of the permanent magnet array, so that the calculation speed can be improved only by adopting a mode of reducing the control frequency or only calculating the Lorentz force fundamental wave in the real-time control process, errors exist between the actual Lorentz force and the expected Lorentz force, and the driving precision is reduced.
Therefore, the problems of larger electromagnetic force fluctuation and time consumption in calculation in the operation process of the conventional long-stroke magnetic levitation planar motor become the technical problem in the field.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor and a modeling method, and aims to provide a topological structure of the low-harmonic permanent magnet array and design a rapid modeling method of magnetic induction intensity in an air gap magnetic field, thereby solving the problems of larger electromagnetic force fluctuation and time consumption in calculation in the operation process of the conventional long-stroke magnetic levitation planar motor.
In order to achieve the above object, according to one aspect of the present invention, there is provided the following technical solution:
a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor is a p-row q-column array formed by alternately and periodically arranging a first permanent magnet unit and a second permanent magnet unit, wherein p is more than or equal to 2, and q is more than or equal to 2;
the first permanent magnet unit comprises vertical magnetized permanent magnets arranged according to a Halbach arrangement mode and horizontal magnetized permanent magnets distributed on the upper, lower, left and right of the vertical magnetized permanent magnets; the vertical magnetization permanent magnet is a polygonal prism with 4 alpha sides, which is formed by chamfering four right-angle parts of a square column body, wherein alpha is more than or equal to 2; the horizontal magnetization permanent magnet is a rectangular cylinder or a 4 alpha-side polygonal prism formed by chamfering four right-angle parts of the rectangular cylinder;
the second permanent magnet unit and the first permanent magnet unit have the same structure and opposite polarization directions.
Preferably, the first permanent magnet unit comprises first negative vertical magnetized permanent magnets arranged according to a Halbach arrangement mode, and first positive upper magnetized permanent magnets, first negative lower magnetized permanent magnets, first negative left magnetized permanent magnets and first positive right magnetized permanent magnets which are distributed on the upper, lower, left and right horizontal magnetized of the first negative vertical magnetized permanent magnets.
Preferably, the second permanent magnet unit comprises second positive vertical magnetized permanent magnets arranged according to a Halbach arrangement mode, and second negative lower magnetized permanent magnets, second positive upper magnetized permanent magnets, second positive right magnetized permanent magnets and second negative left magnetized permanent magnets which are horizontally magnetized and distributed on the upper, lower, left and right of the second positive vertical magnetized permanent magnets.
According to another aspect of the present invention, the following technical solution is also provided:
the modeling method of the low-harmonic permanent magnet array of the long-stroke magnetic levitation planar motor comprises the following steps of:
(S1) discretizing the permanent magnet subjected to chamfering into a superposition of a rectangular cylinder without chamfering and a triangular permanent magnet in a chamfering area, discretizing the triangular permanent magnet into enough right-angle trapezoidal permanent magnets, and equivalent the right-angle trapezoidal permanent magnets into rectangular permanent magnets;
(S2) superposing magnetic field component square wave functions projected by all equivalent rectangular permanent magnets in the permanent magnet array along the x axis to obtain a progression expression A of the magnetic field distribution model on the x axis, superposing magnetic field component square wave functions projected by the permanent magnets along the y axis to obtain a progression expression B of the magnetic field distribution model on the y axis, and adding the A and the B to obtain the progression expression of the magnetic field distribution model
(S3) obtaining the series expression of the vertical z-direction magnetization square column magnetic field distribution model of all the chamfering areas in the permanent magnet array by adopting a Fourier series method m M zs And the series expression of the rectangular column permanent magnet magnetic field distribution model of all horizontal x-direction magnetization m M x Series expression of rectangular cylinder permanent magnet magnetic field distribution model with all horizontal y-magnetization m M y ；
(S4) will m M zs 、 m M x 、 m M y And performing deflection guide on the air domain above the permanent magnet and the magnetic field boundary of the permanent magnet domain to obtain an expression of the magnetic induction intensity in the air gap magnetic field.
Preferably, the center of a gap position of the upper surface of the permanent magnet array without the permanent magnets is taken as an origin, the right direction in the paper surface is taken as the x-axis forward direction, the vertical x-axis direction in the paper surface is taken as the y-axis forward direction, the vertical paper surface is taken as the z-axis forward direction outwards, and the axes of the permanent magnets are parallel to the z-axis, so that a magnet coordinate system is established.
Preferably, in the step (S1), the triangular permanent magnets are equally spaced apart along the x-axis to form enough rectangular trapezoidal permanent magnets, and when the number of the rectangular trapezoidal permanent magnets is more than one hundred, the rectangular trapezoidal permanent magnets are equivalent to rectangular permanent magnets.
Preferably, in step (S1), the rectangular column includes square column permanent magnets magnetized vertically and rectangular column permanent magnets magnetized horizontally.
Preferably, when the permanent magnet magnetized vertically adopts an octagonal prism obtained by chamfering four right-angle portions of a square column, and the permanent magnet magnetized horizontally adopts a rectangular column, the expression of the magnetic induction intensity in the air-gap field obtained in the step (S4) is as follows:
wherein the intermediate variable K 1 P and Q are as follows,
in the method, in the process of the invention,for the component of the magnetic induction in the air-gap field in the x-axis,/->For the component of the magnetic induction in the air-gap field on the y-axis,/->Mu, the component of the magnetic induction intensity in the air gap field in the z-axis 0 Is vacuum permeability, mu r K is the harmonic order of the positive direction of the x axis, l is the harmonic order of the positive direction of the y axis and B is the relative magnetic permeability h To magnetize the permanent magnet horizontally, B v For the magnetization intensity of the vertical magnetization permanent magnet, e is the installation gap between the horizontal magnetization permanent magnet and the vertical magnetization permanent magnet, C is 1/2 of the width of the vertical magnetization permanent magnet, f is 1/2 of the length of the horizontal magnetization permanent magnet, A (l) is calculated by replacing k in A (k) with l, B (l) is calculated by replacing k in B (k) with l, and C (l) is calculated by replacing k in C (k) with l; τ is the magnetic pitch of the permanent magnet array, g is the rectangular side length of the chamfer part, N is the number of N right-angle trapezoidal permanent magnets which are obtained by equally dividing the triangular permanent magnets into N right-angle trapezoidal permanent magnets along the x axis, namely the number of the right-angle trapezoidal permanent magnets which are scattered, and N is the nth right-angle trapezoidal permanent magnet in the N right-angle trapezoidal permanent magnets; z t And z b Coordinates of projections of the upper surface and the lower surface of the permanent magnet array on the z axis; m x、 m and y and z are the coordinates of projection points of any point above the permanent magnet array on the x axis, the y axis and the z axis respectively.
In general, the above technical solutions conceived by the present invention, compared with the prior art, enable the following beneficial effects to be obtained:
1. the invention provides a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor, wherein a first permanent magnet unit and a second permanent magnet unit comprise vertical magnetized permanent magnets and horizontal magnetized permanent magnets which are arranged according to a Halbach arrangement mode, the vertical magnetized permanent magnets adopt 4 alpha-side polygonal prisms formed by chamfering four right-angle parts of square columns, and the horizontal magnetized permanent magnets adopt rectangular columns or 4 alpha-side polygonal prisms formed by chamfering four right-angle parts of the rectangular columns, namely eight prisms, twelve prisms and the like. Compared with square prism and rectangular prism permanent magnets, the polygonal prism permanent magnet can effectively reduce the higher harmonic duty ratio of the air gap field, especially the higher harmonic of the air gap field in a chamfer area under the condition of ensuring that the fundamental wave amplitude of the air gap field is almost unchanged, so that the duty ratio of the higher harmonic of electromagnetic force can be obviously reduced in the real-time control process, electromagnetic force fluctuation of a long-stroke magnetic levitation planar motor in the operation process is further reduced, and compared with an oblique magnetized permanent magnet and a special-shaped permanent magnet, the difficulty of a processing technology can be obviously reduced, and the manufacturing cost is reduced.
2. The low-harmonic permanent magnet array of the long-stroke magnetic levitation planar motor provided by the invention comprises polygonal cylindrical prismatic permanent magnets, wherein the heights and the magnetization intensities of the vertical magnetized permanent magnets and the horizontal magnetized permanent magnets can be different, the gap and the installation fall can be adjusted according to the requirement in the installation process, the applicability is strong, and the manufacturing cost is reduced.
3. According to the modeling method of the low-harmonic permanent magnet array of the long-stroke magnetic levitation planar motor, provided by the invention, the chamfer part of the polygonal prism column-shaped permanent magnet array is processed based on a Fourier series method and a discrete method, so that the chamfer part is equivalent to a rectangular permanent magnet, and unified calculation and superposition processing are performed, so that the calculation complexity can be reduced, the calculation speed is improved, the calculation precision is ensured, and the modeling method is beneficial to being applied in a real-time control process.
Drawings
FIG. 1 is a flow chart of a modeling method according to a first embodiment of the present invention;
FIG. 2 is a top view of a permanent magnet array of a polygonal prism according to a first embodiment of the present invention;
fig. 3 is a side view of a vertical permanent magnet of a polygonal prism permanent magnet array according to a first embodiment of the present invention;
FIG. 4 is a side view of a permanent magnet array of a polygonal prism in a horizontal direction according to a first embodiment of the present invention;
fig. 5 is a schematic diagram of discretization of a triangular permanent magnet according to a second embodiment of the present invention;
FIG. 6 is a graph showing the air-gap field contrast calculated by the discrete method and the finite element method according to the second embodiment of the present invention;
FIG. 7 is a side view of a permanent magnet array with a horizontally magnetized permanent magnet down in accordance with the present invention;
FIG. 8 is a side view of a permanent magnet array with a horizontally magnetized permanent magnet up according to the present invention;
FIG. 9 is a top view of a permanent magnet array of a polygonal prism according to a third embodiment of the present invention;
fig. 10 is a top view of a polygonal permanent magnet array according to a fourth embodiment of the present invention;
FIG. 11 is a top view of a permanent magnet array of a polygonal prism according to a fifth embodiment of the present invention;
FIG. 12 is a square wave projection of an x-direction magnetized permanent magnet on an x-axis in a polygonal prismatic permanent magnet array according to a second embodiment of the present invention;
FIG. 13 is a square wave projection of a y-direction magnetized permanent magnet on an x-axis in a polygonal prismatic permanent magnet array according to a second embodiment of the present invention;
fig. 14 is a square wave projection of a z-magnetized square permanent magnet on an x-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention;
fig. 15 is a square wave projection of a rectangular permanent magnet obtained after z-direction magnetization triangle permanent magnet in a polygonal prism permanent magnet array is discretized on an x-axis according to a second embodiment of the present invention;
FIG. 16 is a square wave projection of an x-direction magnetized permanent magnet on a y-axis in a polygonal prismatic permanent magnet array according to a second embodiment of the present invention;
FIG. 17 is a square wave projection of a y-direction magnetized permanent magnet on a y-axis in a polygonal prismatic permanent magnet array according to a second embodiment of the present invention;
FIG. 18 is a square wave projection of a z-magnetized square permanent magnet on the y-axis in a polygonal prismatic permanent magnet array according to a second embodiment of the present invention;
fig. 19 is a square wave projection of a rectangular permanent magnet on a y axis, which is obtained after z-direction magnetization triangle permanent magnets in a polygonal prism permanent magnet array are scattered, according to a second embodiment of the present invention.
The same reference numbers are used throughout the drawings to reference like elements or structures, wherein:
1-a first permanent magnet unit, 11-a first negative vertical magnetization permanent magnet, 12-a first negative lower magnetization permanent magnet, 13-a first positive right magnetization permanent magnet, 14-a first positive upper magnetization permanent magnet, and 15-a first negative left magnetization permanent magnet;
the permanent magnet unit comprises a first permanent magnet unit, a first positive vertical magnetization permanent magnet, a first negative lower magnetization permanent magnet, a first positive right magnetization permanent magnet, a first positive upper magnetization permanent magnet and a first negative left magnetization permanent magnet, wherein the first permanent magnet unit comprises a first permanent magnet unit, a first positive vertical magnetization permanent magnet, a first negative vertical magnetization permanent magnet, a first positive right magnetization permanent magnet, a first positive upper magnetization permanent magnet and a first negative left magnetization permanent magnet;
3-x-axis projection points, 31-first x-axis projection points, 32-second x-axis projection points, 33-third x-axis projection points, 34-fourth x-axis projection points, 35-x-axis magnetic pitch punctuation;
4-y-axis projection points, 41-first y-axis projection points, 42-second y-axis projection points, 43-third y-axis projection points, 44-fourth y-axis projection points, 45-y-axis magnetic pitch punctuation;
5-right trapezoid, 51-first right trapezoid, 52-second right trapezoid, 53-third right trapezoid and 54-fourth right trapezoid;
61-vertical magnetization permanent magnet non-chamfer side length, 62-vertical magnetization permanent magnet width, 63-permanent magnet array magnetic pitch, 64-permanent magnet height, 65-horizontal magnetization permanent magnet width, 66-horizontal magnetization permanent magnet length, 67-permanent magnet installation gap, 68-right trapezoid height.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
The invention provides a low-harmonic permanent magnet array topological structure and designs a rapid modeling method of an air gap field, and the high-order harmonic of the air gap field of a variable cylinder permanent magnet array is relatively low, so that the electromagnetic force harmonic generated by a coil array after being electrified is relatively low, the electromagnetic force linearity can be improved, and the electromagnetic force fluctuation can be reduced. Meanwhile, the method is based on a Fourier series method and a discrete method, the air gap magnetic field generated by the variable cylinder permanent magnet array is accurately modeled, the calculation speed is improved, the control frequency in the real-time control process is further improved, and the motion control precision is improved.
The embodiment of the invention provides a low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor, which is a p-row q-column array formed by alternately and periodically arranging a first permanent magnet unit and a second permanent magnet unit, wherein p is more than or equal to 2, and q is more than or equal to 2.
The first permanent magnet unit comprises vertical magnetized permanent magnets and horizontal magnetized permanent magnets distributed on the upper, lower, left and right of the vertical magnetized permanent magnets, wherein the vertical magnetized permanent magnets are arranged according to a Halbach arrangement mode; the vertical magnetized permanent magnet is a 4 alpha-side polygonal prism formed by chamfering four right-angle parts of a square column body, and alpha is more than or equal to 2; the horizontal magnetized permanent magnet is a rectangular cylinder or a 4 alpha-side polygonal prism formed by chamfering four right-angle parts of the rectangular cylinder, such as an eight-square prism, a twelve-square prism and the like.
The second permanent magnet unit and the first permanent magnet unit have the same structure and opposite polarization directions.
The two magnetized permanent magnets are arranged in the base groove in a Halbach arrangement mode, so that an air gap field at one side is enhanced, electromagnetic force driving constants are improved, and system power consumption is reduced.
Compared with square prism and rectangular prism-shaped permanent magnets, the polygonal prism permanent magnet array structure provided by the invention can obviously reduce the high-order harmonic duty ratio of the air gap field, especially the high-order harmonic duty ratio of the magnetic induction intensity in the air gap field in the chamfer area under the condition of ensuring that the amplitude of the air gap field fundamental wave is almost unchanged. Therefore, in the real-time control process, the duty ratio of electromagnetic force higher order harmonic waves can be obviously reduced, and further, the position fluctuation in the control process is reduced.
The embodiment of the invention also provides a modeling method of the air-gap field, which mainly adopts a Fourier series method and a discrete method to accurately and rapidly model the air-gap field of the permanent magnet array.
As shown in fig. 1, the modeling method of the air-gap magnetic field includes the following steps:
(1) All vertically magnetized permanent magnets in the permanent magnet array are decomposed into square permanent magnets and triangular permanent magnets cut off by chamfering in a superposition mode, each triangular permanent magnet is equally spaced along the x-axis and divided into a plurality of right-angle trapezoidal permanent magnets, and when the number of the divided triangular permanent magnets is enough (more than one hundred), the right-angle trapezoidal permanent magnets can be equivalent to rectangular permanent magnets. After the polygonal prism permanent magnets are segmented, the polygonal prism permanent magnets can be regarded as a mode of superposing a plurality of rectangular permanent magnets, and then the included angle between the hypotenuse of the triangular permanent magnets and the coordinate axis is converted into the angle between the rectangular permanent magnet sides and the coordinate axis in parallel or perpendicular. If the horizontally magnetized permanent magnets are also subjected to chamfering, transformation is performed in the same manner.
(2) Superposing magnetic field component square wave functions of rectangular permanent magnets corresponding to triangular permanent magnets in all chamfer areas and projected along the x axis to obtain a progression expression A of a magnetic field distribution model in the x axis, superposing magnetic field component square wave functions of rectangular permanent magnets corresponding to triangular permanent magnets in the chamfer areas and projected along the y axis to obtain a progression expression B of the magnetic field distribution model in the y axis, and adding A and B to obtain the progression expression of the magnetic field distribution model
(3) Obtaining a series expression of magnetic field components projected on a z-axis by all square permanent magnets with chamfer areas in a permanent magnet array by adopting a Fourier series method m M zs And the series expression of the magnetic field component projected on the z axis by all the rectangular cylinder permanent magnets magnetized in the horizontal direction and the x direction m M x Series expression of magnetic field component projected on z axis by all rectangular cylinder permanent magnets magnetized in y direction horizontally m M y 。
(4) Will be m M zs 、 m M x 、 m M y After the deviation is calculated on the boundary between the air field above the permanent magnet and the magnetic field of the permanent magnet field, the expression of the air gap magnetic field can be obtained rapidly and accurately. The saidThe triangular permanent magnets in the chamfer areas comprise triangular permanent magnets in all chamfer areas of the permanent magnets which are magnetized vertically.
The low-harmonic permanent magnet array of the long-stroke magnetic levitation planar motor and the modeling method thereof provided by the invention are further described in detail below with reference to the accompanying drawings and the embodiment.
Example 1
As shown in fig. 2, 3 and 4, the first embodiment provides a polygonal prism permanent magnet array of a long-stroke magnetic levitation planar motor, which includes a first permanent magnet unit 1 and a second permanent magnet unit 2. The center of a gap position of the upper surface of the permanent magnet array without a permanent magnet is taken as an origin, the right side in the paper surface is taken as an x-axis forward direction, the vertical x-axis direction in the paper surface is taken as a y-axis forward direction, the vertical paper surface is outward taken as a z-axis forward direction, and the axes of the permanent magnets are parallel to the z-axis, so that a magnet coordinate system is established.
The first permanent magnet unit 1 comprises a vertically magnetized permanent magnet, namely a first negative vertical (i.e., -z) magnetized permanent magnet 11, and a horizontally magnetized permanent magnet, comprising a first negative downward (i.e., -y) magnetized permanent magnet 12, a first positive right (i.e., + x) magnetized permanent magnet 13, a first positive upward (i.e., + y) magnetized permanent magnet 14, and a first negative left (i.e., + x) magnetized permanent magnet 15. The second permanent magnet unit 2 comprises a vertical magnetized permanent magnet, namely a second positive vertical (i.e., + z-direction) magnetized permanent magnet 21, and a horizontal magnetized permanent magnet, comprising a second negative downward (i.e., -y-direction) magnetized permanent magnet 22, a second positive right (i.e., + x-direction) magnetized permanent magnet 23, a second positive upward (i.e., + y-direction) magnetized permanent magnet 24, and a second negative left (i.e., -x-direction) magnetized permanent magnet 25.
The permanent magnets in the first permanent magnet unit 1 and the second permanent magnet unit 2 are arranged according to a Halbach arrangement mode, and it is known that the air-gap field is reinforced at one side in the +z direction, namely, above the permanent magnet array.
According to the working stroke of the magnetic levitation planar workbench, the first permanent magnet unit 1 and the second permanent magnet unit 2 can be expanded in the x direction or the y direction in the horizontal plane. The first permanent magnet units 1 and the second permanent magnet units 2 are alternately and periodically arranged in 2 rows and 2 columns, and the adjacent permanent magnet units in each column and each row are the first permanent magnet units 1 and the second permanent magnet units 2.
The first and second positive perpendicular magnetized permanent magnets 11 and 21 are octagonal, and are obtained by chamfering square permanent magnets, wherein the chamfering angle is 45 degrees. The permanent magnet array magnetic pitch 63 is τ. The straight side length 61 of the parallel side of the vertical magnetization permanent magnet and the x axis is 2 (c-g), g is the right-angle side length of the chamfer part, and the width 62 of the vertical magnetization permanent magnet is 2c. The horizontal magnetized permanent magnets are rectangular columns, the width 65 of the horizontal magnetized permanent magnet is 2d, the length 66 of the horizontal magnetized permanent magnet is 2f, and the size of an installation gap 67 between the horizontal magnetized permanent magnet and the vertical magnetized permanent magnet is e.
The magnetization intensity of the vertical magnetization permanent magnet is B v And the magnetization intensity of the horizontal magnetization permanent magnet is B h 。
The permanent magnet height 64 in the two magnetization directions is h, and the upper surfaces are flush.
The permanent magnet array with the chamfer provided by the embodiment of the invention can reduce the duty ratio of the high-order harmonic wave of the air-gap magnetic field, especially the high-order harmonic wave of the chamfer area, and simultaneously keep the amplitude of the fundamental wave component of the air-gap magnetic field almost unchanged. When the long-stroke magnetic levitation planar motor adopts the topological structure, the fluctuation of electromagnetic force generated by the electrified coil in the breath magnetic field can be effectively reduced, and meanwhile, the amplitude of the electromagnetic force is ensured, so that the system power consumption is reduced.
Example two
According to the magnetization direction of the permanent magnet in the polygonal prism permanent magnet array, a distribution model of the projection of the permanent magnet array on three coordinate axes as shown in figure 2 can be obtained,
m M x ＝λ xx λ xy M h
m M y ＝λ yx λ yy M h
m M z ＝ m M zs + m M zr
m M zs ＝λ zxs λ zys M v
m M zr ＝λ zxr λ zyr M v
wherein the method comprises the steps of m M x Is a series expression of magnetic field components projected along a z-axis by a horizontal x-direction magnetized permanent magnet in a magnetic field distribution model, m M y Is a series expression of magnetic field components projected along a z-axis by a horizontal y-direction magnetized permanent magnet in a magnetic field distribution model, m M z Is a series expression of magnetic field components projected along a z-axis by a horizontal z-direction magnetized permanent magnet in a magnetic field distribution model. M is M h ＝B h /μ 0 ，M v ＝B v /μ 0 Superscript m The representation coordinate system is a magnet coordinate system. Mu (mu) 0 Is vacuum magnetic permeability. Lambda (lambda) ij Is the projection of rectangular permanent magnet magnetized in the i direction on the j axis, lambda ijs Lambda is the projection of a square permanent magnet magnetized in the i direction on the j axis ijr Is the projection of the triangular permanent magnet magnetized in the i direction on the j axis. m M zs And (3) with m M zr Sum of (1) is m M z 。 m M zs Is a series expression of a vertical magnetization square column permanent magnet magnetic field distribution model, m M zr is a series expression of a vertical magnetization triangle column permanent magnet magnetic field distribution model.
Fig. 12 is a square wave projection of an x-direction magnetized permanent magnet on an x-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention. Fig. 13 is a square wave projection of a y-direction magnetized permanent magnet on an x-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention. Fig. 14 is a square wave projection of a z-magnetized square permanent magnet on an x-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention. Fig. 16 is a square wave projection of an x-direction magnetized permanent magnet on a y-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention. Fig. 17 is a square wave projection of a y-direction magnetized permanent magnet on a y-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention. Fig. 18 is a square wave projection of a z-magnetized square permanent magnet on a y-axis in a polygonal prism permanent magnet array according to a second embodiment of the present invention.
The projection of the rectangular permanent magnet and the square permanent magnet on the coordinate axis is simple to calculate, and a detailed deduction process is provided in the related article. In the embodiment, the projection of the triangular permanent magnet on the coordinate axis and the distribution model of the magnetic induction intensity in the corresponding air gap magnetic field are deduced by double points.
Fig. 5 is a schematic diagram of discretization of a triangular permanent magnet, including an x-axis projection point 3, a y-axis projection point 4, and a right trapezoid 5. Wherein the x-axis projection points 3 include a first x-axis projection point 31, a second x-axis projection point 32, a third x-axis projection point 33, and a fourth x-axis projection point 34; the y-axis projection points 4 include a first y-axis projection point 41, a second y-axis projection point 42, a third y-axis projection point 43, and a fourth y-axis projection point 44; the right trapezoid 5 comprises-a first right trapezoid 51, a second right trapezoid 52, a third right trapezoid 53 and a fourth right trapezoid 54; the x-axis magnetic pitch punctuation is 35 and the y-axis magnetic pitch punctuation is 45.
In the second embodiment, each triangle is equally divided into N right trapezoid shapes along the x-axis direction, so the right trapezoid height 68 has a size of g/N. According to symmetry, the projection points of the two vertexes of the right-angle side of the kth right trapezoid on the x coordinate axis are a first x-axis projection point 31, a second x-axis projection point 32, a third x-axis projection point 33 and a fourth x-axis projection point 34, and the projection points of the two vertexes of the hypotenuse of the kth right trapezoid on the y coordinate axis are a first y-axis projection point 41, a second y-axis projection point 42, a third y-axis projection point 43 and a fourth y-axis projection point 44, and the corresponding coordinates are as follows.
Considering that the surface area of the right trapezoid is the length of the upper bottom edge plus the length of the lower bottom edge multiplied by the height divided by two, and when the number of equally divided N is sufficiently large, the right trapezoid can be equivalently rectangular. Therefore, half of the length of the upper bottom edge and the length of the lower bottom edge are taken as the side length of the rectangle, and the height is consistent with the height of the right trapezoid. Taking the first right trapezoid 51 as an example, the y-axis coordinate of the midpoint of the hypotenuse is as follows,
after the rectangular trapezoid permanent magnet 51 is regarded as a rectangular permanent magnet, square wave projection of the nth rectangular permanent magnet on the x axis, which is obtained by discretizing all triangular permanent magnets in the chamfer area of the z-direction magnetized permanent magnet, is shown in fig. 15, square wave projection on the y axis is shown in fig. 19, and a series expression of a magnetic field distribution model of the rectangular permanent magnet can be obtained after calculation by adopting a fourier series methodAs will be described below,
wherein k and l are harmonic numbers in the x-axis positive direction and the y-axis positive direction of the magnet coordinate system, respectively, the expressions of the intermediate variables D (kn) and D (ln) are as follows,
when the chamfer region is not discretized, the equation containing the independent variable x is used for representing each bevel edge, and the series expression of the magnetic field distribution obtained by adopting the Fourier series method is adoptedAs will be described below,
wherein the intermediate variable T k ，T l And h x The expression of (c) is as follows,and calculating results of magnetic field components projected on the coordinate axis z by all triangular permanent magnets in the chamfer area which is not subjected to discretization by adopting a Fourier series method.
It can be seen that although the projection of the triangle area on each coordinate axis can also be represented by a fourier series method, the intermediate variable T l There is an inclusion argument x-coupled term h x I.e. the inner nesting margin of the sine functionThe sum of the sine function and the chord function is multiplied. This coupling term results in an increase in the calculation time and also results in an abnormally complex process of calculating the air-gap magnetic field by deviating the magnetic field boundary, even in an equation that uses the triple volume fraction to calculate the electromagnetic force, which tends to be insoluble.
The expression of the projection of the triangular permanent magnet on the coordinate axis can be obtained rapidly and simply based on the discrete method. Similarly, the fourier series method can be used to obtain the projection expression of other rectangular permanent magnets and square permanent magnets on the coordinate axis. And superposing the projections of the two to obtain the expression of projection of the permanent magnets with different magnetization directions and different shapes on the coordinate axis of the whole permanent magnet array, as follows.
Wherein the intermediate variables A (k), B (k) and C (k) are expressed as follows,
the space can be divided into three parts, namely an air domain above the permanent magnets, a permanent magnet domain and an air domain below the permanent magnets, according to the arrangement mode of the permanent magnet array. By introducing magnetic scale potential and solving bias guide for the magnetic field at the boundary, the expression of each directional component of the air gap magnetic field in the air field above the permanent magnet can be obtained,
wherein the intermediate variable K 1 P, Q and K 1 As will be described below,
z t and z b The coordinates of the projection of the upper surface and the lower surface of the permanent magnet array on the z axis are respectively, and z is the coordinate of the projection point of any point above the permanent magnet array on the z axis.
In order to compare the reliability of the air-gap field modeling in the second embodiment of the present invention, points uniformly distributed on four parallel curves in a certain plane above the permanent magnet array are taken, and the vertical component of the magnetic field at each point is calculated by adopting a finite element method and a discrete method according to the present invention, and the result is shown in fig. 6. The scattered points in the graph are the results of finite element method calculation, and the curves are the results of discrete method calculation. The method has the advantages that the matching effect of the two methods is good, and the reliability of the discretization modeling method is also laterally proved.
The above is only an expression of the air gap field of the permanent magnet array in the second embodiment of the present invention, where the heights of the vertical magnetized permanent magnet and the horizontal magnetized permanent magnet are the same, and the upper surfaces are flush, so that the assembly is convenient to complete in the processing process.
When the vertical magnetization permanent magnet and the horizontal magnetization permanent magnet are different in height or are installed in a mode that the upper surface and the lower surface are not parallel, the vertical magnetization permanent magnet is mainly divided into a case that the horizontal magnetization permanent magnet is positioned below as shown in fig. 7 and a case that the horizontal magnetization permanent magnet is positioned above as shown in fig. 8. For the two cases, only the z-direction coordinate in the air-gap field expression is changed according to the actual situation, and the air-gap field of the vertical magnetized permanent magnet and the air-gap field of the horizontal magnetized permanent magnet are added to obtain the permanent magnet array air-gap field expressions with different heights and different installation height drops.
Example III
As shown in fig. 9, which is a top view of the permanent magnet array in the third embodiment, compared with the second embodiment, the permanent magnet array in the third embodiment is replaced by a dodecagonal permanent magnet from an octagonal permanent magnet, and the shape of the horizontally magnetized permanent magnet remains unchanged. The vertically magnetized permanent magnets in this embodiment can be seen as a superposition of a large square permanent magnet with a small square permanent magnet and two triangular permanent magnets.
Example IV
As shown in fig. 10, which is a top view of the permanent magnet array in the fourth embodiment, compared with the second embodiment, the shape of the permanent magnet array in the fourth embodiment, which is magnetized horizontally, is replaced by an octagonal prism, and the shape of the permanent magnet array in the fourth embodiment, which is magnetized vertically, is kept unchanged. In this embodiment, the distribution model of the horizontal magnetization permanent magnet may refer to the calculation mode of the vertical magnetization permanent magnet model in the second embodiment, and it should be noted that the structural size also changes.
Example five
As shown in fig. 11, which is a top view of the permanent magnet array in the fifth embodiment, in comparison with the previous embodiments, in the fifth embodiment, the shape of the vertical magnetized permanent magnet is the same as that of the third embodiment, and the shape of the horizontal magnetized permanent magnet is the same as that of the fourth embodiment. The modeling of the permanent magnet in this embodiment can be referred to the modeling method in the previous embodiment, but the calculation of the expression is also more complicated, and appropriate processing is required according to the requirement of real-time control.
In the above embodiments, the height, the installation gap, the installation height difference, the magnetization intensity and the number of sides of the polygonal prism of the vertical magnetization permanent magnet and the horizontal magnetization permanent magnet can be adjusted, and parameterized representation can be performed in a discrete method, so that the optimization is convenient to follow according to actual requirements.
The invention provides a permanent magnet array of a low-harmonic long-stroke magnetic levitation planar motor and an air gap field discrete modeling method thereof. A polygonal prism permanent magnet is used to form a low-harmonic permanent magnet array. The method is characterized in that the triangular prism permanent magnet is equally divided into a plurality of right trapezoid prism permanent magnets along right-angle sides, the right trapezoid prism permanent magnets are equivalent to rectangular prism permanent magnets, and a plurality of rectangular prism permanent magnets are overlapped to obtain the air-gap field distribution of the topological structure permanent magnet array. The different structural dimensions of the permanent magnet are parameterized, which is beneficial to optimizing calculation in the design process. The permanent magnet array topological structure provided by the invention can reduce electromagnetic force fluctuation under the condition that the electromagnetic force amplitude is almost unchanged, and meanwhile, the lifted discrete modeling method can be used for rapidly calculating the electromagnetic force amplitude, so that the application in the real-time control process is facilitated.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.

Claims (8)

1. A low-harmonic permanent magnet array of a long-stroke magnetic levitation planar motor is characterized in that the array is a p-row q-column array formed by alternately and periodically arranging a first permanent magnet unit and a second permanent magnet unit, wherein p is more than or equal to 2, and q is more than or equal to 2;
the first permanent magnet unit comprises vertical magnetized permanent magnets arranged according to a Halbach arrangement mode and horizontal magnetized permanent magnets distributed on the upper, lower, left and right of the vertical magnetized permanent magnets; the vertical magnetization permanent magnet is a polygonal prism with 4 alpha sides, which is formed by chamfering four right-angle parts of a square column body, wherein alpha is more than or equal to 2; the horizontal magnetization permanent magnet is a rectangular cylinder or a 4 alpha-side polygonal prism formed by chamfering four right-angle parts of the rectangular cylinder;
the second permanent magnet unit and the first permanent magnet unit have the same structure and opposite polarization directions.
2. A long-stroke magnetically levitated planar motor low-harmonic permanent magnet array according to claim 1, characterized in that the first permanent magnet unit comprises first vertical magnetized permanent magnets (11) arranged according to Halbach arrangement, and first positive upper magnetized permanent magnets (14), first negative lower magnetized permanent magnets (12), first negative left magnetized permanent magnets (15) and first positive right magnetized permanent magnets (13) distributed on upper, lower, left and right horizontal magnetization of the first vertical magnetized permanent magnets (11).
3. A long-stroke magnetically levitated planar motor low-harmonic permanent magnet array according to claim 2, characterized in that the second permanent magnet unit comprises second positive vertical magnetized permanent magnets (21) arranged according to Halbach arrangement, and second negative downward magnetized permanent magnets (22), second positive upward magnetized permanent magnets (24), second positive right magnetized permanent magnets (23) and second negative left magnetized permanent magnets (25) distributed on upper, lower, left and right horizontal magnetization of the second positive vertical magnetized permanent magnets (21).
4. A method for modeling a low harmonic permanent magnet array of a long-stroke magnetic levitation planar motor according to any one of claims 1-3, comprising the steps of:
(S1) discretizing the permanent magnet subjected to chamfering into a superposition of a rectangular cylinder without chamfering and a triangular permanent magnet in a chamfering area, discretizing the triangular permanent magnet into enough right-angle trapezoidal permanent magnets, and equivalent the right-angle trapezoidal permanent magnets into rectangular permanent magnets;
(S2) superposing magnetic field component square wave functions projected by all equivalent rectangular permanent magnets in the permanent magnet array along the x axis to obtain a progression expression A of the magnetic field distribution model on the x axis, superposing magnetic field component square wave functions projected by the permanent magnets along the y axis to obtain a progression expression B of the magnetic field distribution model on the y axis, and adding the A and the B to obtain the progression expression of the magnetic field distribution model
(S3) obtaining the series expression of the vertical magnetization square column magnetic field distribution model of all the chamfering areas in the permanent magnet array by adopting a Fourier series method m M zs And the series expression of the rectangular column permanent magnet magnetic field distribution model of all horizontal x-direction magnetization m M x Series expression of rectangular cylinder permanent magnet magnetic field distribution model with all horizontal y-magnetization m M y ；
(S4) will m M zs 、 m M x 、 m M y And performing deflection guide on the air domain above the permanent magnet and the magnetic field boundary of the permanent magnet domain to obtain an expression of the magnetic induction intensity in the air gap magnetic field.
5. The modeling method of claim 4, wherein a center of a gap position of the upper surface of the permanent magnet array without the permanent magnets is used as an origin, a right direction in a paper surface is used as an x-axis forward direction, a vertical x-axis direction in the paper surface is used as a y-axis forward direction, a vertical paper surface is used as a z-axis forward direction, and axes of the permanent magnets are parallel to the z-axis, so that a magnet coordinate system is established.
6. The modeling method of claim 4, wherein in the step (S1), triangular permanent magnets are equally spaced apart along the x-axis to form a sufficient number of rectangular trapezoidal permanent magnets, and when the number of the rectangular trapezoidal permanent magnets to be segmented is more than one hundred, the rectangular trapezoidal permanent magnets are equivalent to rectangular permanent magnets.
7. The modeling method of claim 4, wherein in the step (S1), the rectangular column includes square column permanent magnets magnetized vertically and rectangular column permanent magnets magnetized horizontally.
8. The modeling method of claim 7, wherein when the permanent magnet magnetized vertically adopts an octagonal prism obtained by chamfering four right-angle portions of a square column, and the permanent magnet magnetized horizontally adopts a rectangular column, the expression of the magnetic induction intensity in the air-gap field obtained in step (S4) is as follows:
wherein the intermediate variable K 1 P and Q are as follows,
in the method, in the process of the invention,for the component of the magnetic induction in the air-gap field in the x-axis,/->For the component of the magnetic induction in the air-gap field on the y-axis,/->Mu, the component of the magnetic induction intensity in the air gap field in the z-axis 0 Is vacuum permeability, mu r K is the harmonic order of the positive direction of the x axis, l is the harmonic order of the positive direction of the y axis and B is the relative magnetic permeability h To magnetize the permanent magnet horizontally, B v For the magnetization intensity of the vertical magnetization permanent magnet, e is the installation gap between the horizontal magnetization permanent magnet and the vertical magnetization permanent magnet, C is 1/2 of the width of the vertical magnetization permanent magnet, f is 1/2 of the length of the horizontal magnetization permanent magnet, A (l) is calculated by replacing k in A (k) with l, B (l) is calculated by replacing k in B (k) with l, and C (l) is calculated by replacing k in C (k) with l; τ is the magnetic pitch of the permanent magnet array, g is the rectangular side length of the chamfer part, N is the number of N right-angle trapezoidal permanent magnets which are obtained by equally dividing the triangular permanent magnets into N right-angle trapezoidal permanent magnets along the x axis, namely the number of the scattered right-angle trapezoidal permanent magnets, and N is the nth right-angle trapezoidal permanent magnetRight angle trapezoidal permanent magnet; z t And z b Coordinates of projections of the upper surface and the lower surface of the permanent magnet array on the z axis; m x、 m and y and z are the coordinates of projection points of any point above the permanent magnet array on the x axis, the y axis and the z axis respectively. />
CN202311338692.6A 2023-10-16 2023-10-16 Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method Pending CN117477889A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202311338692.6A CN117477889A (en) 2023-10-16 2023-10-16 Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202311338692.6A CN117477889A (en) 2023-10-16 2023-10-16 Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method

Publications (1)

Publication Number Publication Date
CN117477889A true CN117477889A (en) 2024-01-30

Family

ID=89635541

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202311338692.6A Pending CN117477889A (en) 2023-10-16 2023-10-16 Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method

Country Status (1)

Country Link
CN (1) CN117477889A (en)

Similar Documents

Publication Publication Date Title
Jin et al. Robust multi-objective optimization of a 3-pole active magnetic bearing based on combined curves with climbing algorithm
CN103208867A (en) Magnet unit, magnet array, magnetic levitation planar motor and lithographic device using magnetic levitation planar motor
CN102862395B (en) System and method for automatically generating three-dimensional model
CN101527484A (en) Gas-magnet mixing suspended planar motor with easily expanded horizontal stroke
CN102097982A (en) Permanent-magnet synchronous magnetic suspension planar motor
CN113609599B (en) Wall surface distance effective unit calculation method for aircraft turbulence flow-around simulation
Toan et al. Using modification of visibility-graph in solving the problem of finding shortest path for robot
Zhao et al. Modeling and characteristic analysis of air-cored linear synchronous motors with racetrack coils for electrodynamic suspension train
CN117477889A (en) Low-harmonic permanent magnet array of long-stroke magnetic levitation planar motor and modeling method
Fu et al. Cascaded iterative learning motion control of precision maglev planar motor with experimental investigation
Luo et al. Layout problem of multi-component systems arising for improving maintainability
CN103580444A (en) Linear motor
CN107819391B (en) Permanent magnet array and planar motor
Wang et al. Magnet array of planar motor using permanent magnets with different magnetisation intensity and height
WO2023226272A1 (en) Magnetic suspension planar motor workbench with double-layer winding coarse-fine drive
Hu et al. A 6-DOF Extended Unified Wrench Model for Maglev Planar Motor
CN207321085U (en) A kind of novel permanent magnetic array and planar motor
CN102663163B (en) Topological optimization design method based on geometrical background meshes and with consideration of manufacturing constraints of drawing die
CN112234779B (en) Motor production method and system based on variable-thickness Halbach permanent magnet array
Liu et al. Magnetic Field Modeling of Linear Halbach Array for Wallclimbing Robot Based on Radial Basis Function Neural Network
Zhang et al. Modeling and analysis of a magnetically levitated synchronous permanent magnet planar motor with concentric structure winding
Fang et al. Halbach Square Array Structure for Optimizing Magnetic Adsorption
Wu et al. Sequential Structure and Control Co-design of Lightweight Precision Stages with Active control of flexible modes
CN113688557B (en) Particle position dynamics-based soft body real-time cutting simulation method
Zheng et al. Optimisation method of magnetic levitation actuator for rotary table

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination