A method for modeling dynamics of a three-orthogonal reluctance translational magnetic bearing

Abstract

This invention discloses a dynamic modeling method for a three-orthogonal magnetic reluctance translational magnetic bearing. Based on the equivalent magnetic circuit method, magnetic circuit analysis is performed. Utilizing Ohm's law for magnetic circuits, a single-pole electromagnetic force model of the three-orthogonal magnetic bearing is obtained. Using the lateral edge of a right-angled regular triangular pyramid as a reference, the magnetic bearing's three magnetic poles are evenly distributed circumferentially to construct an orthogonal model magnetic circuit. By analyzing the geometric characteristics of the three-orthogonal magnetic bearing, a coordinate transformation matrix (direction cosine matrix) for the electromagnetic force coordinate system and the position coordinate system is constructed. When the magnetically levitated mover is subjected to external disturbance, the current flowing through the magnetic bearing coil winding is changed according to the change in mover displacement, thereby achieving mover control. The dynamic modeling method for the three-orthogonal magnetic reluctance translational magnetic bearing described in this invention can realize three-degree-of-freedom translation and preset position levitation of the platform on which the magnetic bearing is mounted, and has broad application prospects in the field of attitude control technology for novel spacecraft.

Description

Technical Field

[0001] This invention relates to the field of magnetic bearing dynamics modeling technology, and in particular to a method for modeling the dynamics of a three-orthogonal magnetic reluctance translational magnetic bearing. Background Technology

[0002] Common reluctance magnetic bearings are combinations of uniaxial and dual radial bearings. The advantage of this combination is that the three degrees of freedom (DOF) are independent of each other, with minimal coupling and easy control. However, this combination also has disadvantages: placing the axial and radial bearings occupies a significant amount of space, and the relatively fixed structure cannot adapt to different task requirements. If a highly integrated multi-DOF system is adopted, the coupling between the various DOFs is high, making control more difficult or even uncontrollable.

[0003] The present invention relates to a dynamic modeling method for a three-orthogonal magnetic reluctance translational magnetic bearing, which can realize three-degree-of-freedom translation and preset position suspension of the platform on which the magnetic bearing is mounted, and is of great significance for further simplifying control and thus improving response speed and accuracy. Summary of the Invention

[0004] The purpose of this invention is to provide a dynamic modeling method for a three-orthogonal magnetic reluctance translational magnetic bearing, which can accurately establish a dynamic model of the three-orthogonal magnetic reluctance translational magnetic bearing to solve the problems of three-degree-of-freedom translational motion and suspension of the mover in the three-orthogonal magnetic bearing, and provide theoretical support for simplifying the control of magnetic levitation bearings and improving control accuracy.

[0005] To achieve the above objectives, this invention provides the following technical solution: Based on the equivalent magnetic circuit method, magnetic circuit analysis is performed. Using Ohm's law for magnetic circuits, a single-pole electromagnetic force model of a three-orthogonal magnetic bearing is obtained. Using the lateral edge of a right-angled regular triangular pyramid as a reference, the magnetic bearing is constructed with three sets of magnetic poles evenly distributed around its circumference, forming an orthogonal magnetic circuit model. By analyzing the geometric characteristics of the three-orthogonal magnetic bearing, a coordinate transformation matrix (direction cosine matrix) for the electromagnetic force coordinate system and the position coordinate system is constructed. When the magnetically levitated mover is subjected to external disturbance, the current flowing through the magnetic bearing coil winding is changed according to the change in the mover's displacement, thereby achieving control of the mover. The specific steps of the dynamic modeling method are as follows:

[0006] (1) Modeling of electromagnetic force of a single magnetic pole of a three-orthogonal magnetic bearing based on the equivalent magnetic circuit method

[0007] The three-orthogonal magnetic reluctance translational magnetic bearing consists of three sets of identical magnetic poles. The magnetic circuit of a single set of magnetic poles mainly includes the radial inner and outer magnetic poles formed by the inner and outer iron cores and windings, the mover, and the air gap between the mover and the radial inner and outer magnetic poles. The magnetic flux loops formed by the inner and outer magnetic poles all start from the N pole of the coil winding, pass through the air gap between the iron core and the mover, then through the air gap between the mover and the S pole, and return to the S pole to form a closed loop. Ignoring the effects of magnetic leakage of the magnet, the self-induced magnetic field of the coil, and edge effects, the equivalent magnetic circuit of a single magnetic bearing is established according to the equivalent magnetic circuit method. From the equivalent magnetic circuit, the total magnetic reluctance of the magnetic circuit of a single set of magnetic poles can be obtained as follows:

[0008]

[0009] Where R 11 R 12 R 21 R 22 R1 and R2 represent the air gap magnetic resistance between the inner and outer magnetic poles and the mover, respectively, and R1 and R2 represent the total magnetic resistance of the two-channel magnetic circuit of a single set of magnetic poles.

[0010] The reluctance of a single air gap is further expressed as:

[0011] R = δ / (μ0A) i (2)

[0012] Where δ is the air gap length between the mover and the magnetic pole face, μ0 is the permeability in vacuum, and A i Indicates the area of ​​each magnetic pole;

[0013] From the above equation, the magnetic flux of a single set of magnetic poles in two channels can be obtained:

[0014]

[0015] In the formula, I0 is the current in the coil when the mover is in the equilibrium position, I s This represents the control current, and N represents the number of coil turns.

[0016] The forces F1 and F2 acting on the mover in the two channels of a single magnetic pole can be expressed as:

[0017]

[0018] Substituting equations (1), (2), and (3) into the above equation yields the electromagnetic attraction force on the mover in a single magnetic pole two-channel configuration:

[0019]

[0020] From the above equation, the electromagnetic force experienced by the mover when it is in the preset equilibrium position is:

[0021]

[0022] As can be seen from the above formula, the electromagnetic force on the mover is proportional to the square of the current flowing through the coil winding;

[0023] When the mover is in the equilibrium position, the displacement detected by the position sensor is S, and the displacement from the equilibrium position is s. Then, the electromagnetic attraction forces exerted on the mover by the two ends of the single set of magnetic poles are as follows:

[0024]

[0025]

[0026] The total electromagnetic force exerted on the mover by a single set of magnetic poles is:

[0027]

[0028] Performing a Taylor expansion on the above equation and discarding higher-order terms, we obtain the formula for the electromagnetic attraction force after linearization of the mover's single degree of freedom:

[0029] F = -K s S+K i I s (10)

[0030] In the formula, K i Called current stiffness K s This is called displacement stiffness.

[0031] (2) Creation of the magnetic pole configuration of the three orthogonal magnetic bearings based on a right-angled regular triangular pyramid

[0032] The three sets of magnetic poles of the triorthogonal magnetic reluctance translational magnetic bearing are uniformly distributed along the circumference in the same radial plane, meaning that the angle between any two adjacent magnetic poles is 120 degrees. The three sets of magnetic poles are simultaneously mounted obliquely. The axis of each set of magnetic poles is defined as the direction of its electromagnetic force. The axes of the three sets of magnetic poles form a right-angled regular triangular pyramid (a regular triangular pyramid with isosceles right-angled triangles on the sides), and its base is the radial plane containing the magnetic poles. Based on the method of equal volume and the properties of equilateral triangles, we can obtain:

[0033] Calculate the volume using the side view as a reference:

[0034]

[0035] Find the volume given that the base is an equilateral triangle:

[0036]

[0037] It is easy to know from V1 = V2:

[0038]

[0039] From the geometric characteristics of a right-angled regular triangular pyramid, we can obtain:

[0040]

[0041] Solving equations (13) and (14), we get:

[0042] θ = 35.26° (15)

[0043] In the formula, l is the length of the lateral edge of the triangular pyramid, h is the height from the radial plane to the vertex of the triangular pyramid, and θ is the angle between the lateral edge of the triangular pyramid and the radial plane;

[0044] Based on the above formula, the electromagnetic force of the three magnetic poles can be decomposed in space. The electromagnetic force generated by the three magnetic poles a, b, and c can be decomposed into two orthogonal components F′ in the radial plane. x F′ y And a component F′ passing through the vertex of the triangular pyramid and perpendicular to the radial plane. z From the formula, we can obtain:

[0045]

[0046] The electromagnetic resultant force generated by each set of magnetic poles passes through the center of the sphere and is perpendicular to each other. By changing the magnitude of the current flowing through the coil winding, the radial two-degree-of-freedom translational motion and the axial one-degree-of-freedom translational motion of the mover can be controlled.

[0047] (3) Construction of the cosine array of electromagnetic force direction of the three orthogonal magnetic bearings

[0048] An electromagnetic force coordinate system is established with the vertex of the triangular pyramid as the origin O′. The three coordinate axes point along the three lateral edges α, β, and γ of the triangular pyramid towards the base of the pyramid. The coordinate system satisfies the Cartesian right-hand rule.

[0049] Establish a position coordinate system with the center O of the base (radial plane) of the triangular pyramid as the origin. The x-axis is the projection of the α-axis onto the radial plane, the z-axis is along OO′, and the y-axis lies in the radial plane, satisfying the right-hand rule.

[0050] To solve for the transformation matrix P between the electromagnetic force coordinate system O′-αβγ and the position coordinate system O-xyz, first rotate the position coordinate system O-xyz about the y-axis by θ to obtain the transition system Ox′yz′, whose x′ axis coincides with the α-axis of the electromagnetic force coordinate system O′-αβγ. Then rotate the coordinate system Ox′yz′ about the x′ axis by ω. It is easy to see that the position coordinate system has now been rotated to the electromagnetic force coordinate system.

[0051] Based on the above analysis, the elementary transformation matrices for the first and second rotations are as follows:

[0052]

[0053]

[0054] According to the transitivity of the direction cosine matrix, we have:

[0055]

[0056] Based on the spatial geometric relationship of a regular triangular pyramid, it is easy to find that ω = 135°;

[0057] (4) Modeling of three-degree-of-freedom translational decoupling magnetic reluctance of three orthogonal magnetic bearings

[0058] The resultant electromagnetic attraction force F corresponding to the displacement offset x F y F z The transformation matrix P and the electromagnetic attraction F provided by the reluctance magnetic bearing are used to... α F β F γ By combining these, we obtain a three-degree-of-freedom translational dynamics model of the mover;

[0059] According to Newton's second law, we get:

[0060]

[0061] Based on coordinate transformation, the electromagnetic attraction force F corresponding to the displacement deviation is compared with the electromagnetic attraction force provided by the reluctance bearing through the transformation matrix P. Connecting these points, we can obtain:

[0062]

[0063]

[0064] Substituting the values ​​of θ and ω, we get:

[0065]

[0066] Substitute P -1 Simplifying, we get:

[0067]

[0068] According to the formula for electromagnetic attraction after linearization of the single degree of freedom of the mover, we can obtain:

[0069]

[0070] The single-degree-of-freedom dynamic equation of the mover obtained by combining Newton's second law:

[0071]

[0072] Linearizing the resultant electromagnetic attraction forces in the x, y, and z directions of the position coordinate system yields a three-degree-of-freedom translational dynamic model of the mover:

[0073]

[0074] In the formula, S x S y S z It is the displacement measured by the displacement sensor; I α I β I γ The current is controlled by three sets of magnetic reluctance bearings.

[0075] Compared with the prior art, the present invention has the following beneficial effects: it provides a dynamic modeling method for a three-orthogonal magnetic reluctance translational magnetic bearing, which can accurately establish the dynamic model of the three-orthogonal magnetic reluctance translational magnetic bearing, solves the problem of three-degree-of-freedom translation and suspension of the platform on which the three-orthogonal magnetic bearing is mounted, and provides theoretical support for simplifying the control of magnetic levitation bearings and improving the control accuracy. Attached Figure Description

[0076] Figure 1 This is a flowchart of a dynamic modeling method for a three-orthogonal magnetic reluctance translational magnetic bearing.

[0077] Figure 2 This is a structural diagram of a three-orthogonal magnetic reluctance translational magnetic bearing system.

[0078] Figure 3 This is the equivalent magnetic circuit diagram of a single magnetic pole of a magnetic bearing.

[0079] Figure 4 This is a schematic diagram showing the relative positions of the electromagnetic force coordinate system and the position coordinate system.

[0080] Figure 5 This is a schematic diagram of the first coordinate system rotation.

[0081] Figure 6 This is a schematic diagram of the second coordinate system rotation. Detailed Implementation

[0082] The present invention will be further described in detail below with reference to specific embodiments. These descriptions are for explanation purposes only and not for limitation. Based on the equivalent magnetic circuit method, magnetic circuit analysis is performed. Using Ohm's law for magnetic circuits, a single-pole electromagnetic force model of a three-orthogonal magnetic bearing is obtained. Using the lateral edge of a right-angled regular triangular pyramid as a reference, the magnetic bearing is constructed with three sets of magnetic poles evenly distributed around its circumference, forming an orthogonal model magnetic circuit. By analyzing the geometric characteristics of the three-orthogonal magnetic bearing, a coordinate transformation matrix (direction cosine matrix) for the electromagnetic force coordinate system and the position coordinate system is constructed. When the magnetically levitated mover is subjected to external disturbance, the current flowing through the magnetic bearing coil winding is changed according to the change in the mover's displacement, thereby achieving control of the mover. The specific steps of the dynamic modeling method are as follows:

[0083] (1) Modeling of electromagnetic force of a single magnetic pole of a three-orthogonal magnetic bearing based on the equivalent magnetic circuit method

[0084] The three-orthogonal magnetic reluctance translational magnetic bearing consists of three sets of identical magnetic poles. The magnetic circuit of a single set of magnetic poles mainly includes the radial inner and outer magnetic poles formed by the inner and outer iron cores and windings, the mover, and the air gap between the mover and the radial inner and outer magnetic poles. The magnetic flux loops formed by the inner and outer magnetic poles all start from the N pole of the coil winding, pass through the air gap between the iron core and the mover, then through the air gap between the mover and the S pole, and return to the S pole to form a closed loop. Ignoring the effects of magnetic leakage of the magnet, the self-induced magnetic field of the coil, and edge effects, the equivalent magnetic circuit of a single magnetic bearing is established according to the equivalent magnetic circuit method. From the equivalent magnetic circuit, the total magnetic reluctance of the magnetic circuit of a single set of magnetic poles can be obtained as follows:

[0085]

[0086] Where R 11 R 12 R 21 R 22 R1 and R2 represent the air gap magnetic resistance between the inner and outer magnetic poles and the mover, respectively, and R1 and R2 represent the total magnetic resistance of the two-channel magnetic circuit of a single set of magnetic poles.

[0087] The reluctance of a single air gap is further expressed as:

[0088] R = δ / (μ0A) i (2)

[0089] Where δ is the air gap length between the mover and the magnetic pole face, μ0 is the permeability in vacuum, and A i Indicates the area of ​​each magnetic pole;

[0090] From the above equation, the magnetic flux of a single set of magnetic poles in two channels can be obtained:

[0091]

[0092] In the formula, I0 is the current in the coil when the mover is in the equilibrium position, I s This represents the control current, and N represents the number of coil turns.

[0093] The forces F1 and F2 acting on the mover in the two channels of a single magnetic pole can be expressed as:

[0094]

[0095] Substituting equations (1), (2), and (3) into the above equation yields the electromagnetic attraction force on the mover in a single magnetic pole two-channel configuration:

[0096]

[0097] From the above equation, the electromagnetic force experienced by the mover when it is in the preset equilibrium position is:

[0098]

[0099] As can be seen from the above formula, the electromagnetic force on the mover is proportional to the square of the current flowing through the coil winding;

[0100] When the mover is in the equilibrium position, the displacement detected by the position sensor is S, and the displacement from the equilibrium position is s. Then, the electromagnetic attraction forces exerted on the mover by the two ends of the single set of magnetic poles are as follows:

[0101]

[0102]

[0103] The total electromagnetic force exerted on the mover by a single set of magnetic poles is:

[0104]

[0105] Performing a Taylor expansion on the above equation and discarding higher-order terms, we obtain the formula for the electromagnetic attraction force after linearization of the mover's single degree of freedom:

[0106] F = -K s S+K i I s (10)

[0107] In the formula, K i Called current stiffness K s This is called displacement stiffness.

[0108] (2) Creation of the magnetic pole configuration of the three orthogonal magnetic bearings based on a right-angled regular triangular pyramid

[0109] The three sets of magnetic poles of the triorthogonal magnetic reluctance translational magnetic bearing are uniformly distributed along the circumference in the same radial plane, meaning that the angle between any two adjacent magnetic poles is 120 degrees. The three sets of magnetic poles are simultaneously mounted obliquely. The axis of each set of magnetic poles is defined as the direction of its electromagnetic force. The axes of the three sets of magnetic poles form a right-angled regular triangular pyramid (a regular triangular pyramid with isosceles right-angled triangles on the sides), and its base is the radial plane containing the magnetic poles. Based on the method of equal volume and the properties of equilateral triangles, we can obtain:

[0110] Calculate the volume using the side view as a reference:

[0111]

[0112] Find the volume given that the base is an equilateral triangle:

[0113]

[0114] It is easy to know from V1 = V2:

[0115]

[0116] From the geometric characteristics of a right-angled regular triangular pyramid, we can obtain:

[0117]

[0118] Solving equations (13) and (14), we get:

[0119] θ = 35.26° (15)

[0120] In the formula, l is the length of the lateral edge of the triangular pyramid, h is the height from the radial plane to the vertex of the triangular pyramid, and θ is the angle between the lateral edge of the triangular pyramid and the radial plane;

[0121] Based on the above formula, the electromagnetic force of the three magnetic poles can be decomposed in space. The electromagnetic force generated by the three magnetic poles a, b, and c can be decomposed into two orthogonal components F in the radial plane. x ′、F y ′ and a component F passing through the vertex of the triangular pyramid and perpendicular to the radial plane. z From the formula, we can obtain:

[0122]

[0123] The electromagnetic resultant force generated by each set of magnetic poles passes through the center of the sphere and is perpendicular to each other. By changing the magnitude of the current flowing through the coil winding, the radial two-degree-of-freedom translational motion and the axial one-degree-of-freedom translational motion of the mover can be controlled.

[0124] (3) Construction of the cosine array of electromagnetic force direction of the three orthogonal magnetic bearings

[0125] An electromagnetic force coordinate system is established with the vertex of the triangular pyramid as the origin O′. The three coordinate axes point along the three lateral edges α, β, and γ of the triangular pyramid towards the base of the pyramid. The coordinate system satisfies the Cartesian right-hand rule.

[0126] Establish a position coordinate system with the center O of the base (radial plane) of the triangular pyramid as the origin. The x-axis is the projection of the α-axis onto the radial plane, the z-axis is along OO′, and the y-axis lies in the radial plane, satisfying the right-hand rule.

[0127] To solve for the transformation matrix P between the electromagnetic force coordinate system O′-αβγ and the position coordinate system O-xyz, first rotate the position coordinate system O-xyz about the y-axis by θ to obtain the transition system Ox′yz′, whose x′ axis coincides with the α-axis of the electromagnetic force coordinate system O′-αβγ. Then rotate the coordinate system Ox′yz′ about the x′ axis by ω. It is easy to see that the position coordinate system has now been rotated to the electromagnetic force coordinate system.

[0128] Based on the above analysis, the elementary transformation matrices for the first and second rotations are as follows:

[0129]

[0130]

[0131] According to the transitivity of the direction cosine matrix, we have:

[0132]

[0133] Based on the spatial geometric relationship of a regular triangular pyramid, it is easy to find that ω = 135°;

[0134] (4) Modeling of three-degree-of-freedom translational decoupling magnetic reluctance of three orthogonal magnetic bearings

[0135] The resultant electromagnetic attraction force F corresponding to the displacement offset x F y F z The transformation matrix P and the electromagnetic attraction F provided by the reluctance magnetic bearing are used to... α F β F γ By combining these, we obtain a three-degree-of-freedom translational dynamics model of the mover;

[0136] According to Newton's second law, we get:

[0137]

[0138] Based on coordinate transformation, the electromagnetic attraction force F corresponding to the displacement deviation is compared with the electromagnetic attraction force provided by the reluctance bearing through the transformation matrix P. Connecting these points, we can obtain:

[0139]

[0140]

[0141] Substituting the values ​​of θ and ω, we get:

[0142]

[0143] Substitute P -1 Simplifying, we get:

[0144]

[0145] According to the formula for electromagnetic attraction after linearization of the single degree of freedom of the mover, we can obtain:

[0146]

[0147] The single-degree-of-freedom dynamic equation of the mover obtained by combining Newton's second law:

[0148]

[0149] Linearizing the resultant electromagnetic attraction forces in the x, y, and z directions of the position coordinate system yields a three-degree-of-freedom translational dynamic model of the mover:

[0150]

[0151] In the formula, S x S y S z It is the displacement measured by the displacement sensor; I α I β I γ The current is controlled by three sets of magnetic reluctance bearings.

[0152] The scope of protection of this invention is defined by the claims.

Claims

1. A method for dynamic modeling of a three-orthogonal magnetic reluctance translational magnetic bearing, characterized in that: Magnetic circuit analysis is performed based on the equivalent magnetic circuit method. Using Ohm's law for magnetic circuits, a single-pole electromagnetic force model of a three-orthogonal magnetic bearing is obtained. Using the lateral edge of a right-angled regular triangular pyramid as a reference, the magnetic bearing is constructed with three sets of magnetic poles evenly distributed around its circumference, forming an orthogonal magnetic circuit model. By analyzing the geometric characteristics of the three-orthogonal magnetic bearing, a coordinate transformation matrix (direction cosine matrix) for the electromagnetic force coordinate system and the position coordinate system is constructed. When the magnetically levitated mover is subjected to external disturbance, the current flowing through the magnetic bearing coil winding is changed according to the change in mover displacement, thereby controlling the mover. The specific steps of the dynamic modeling method are as follows: (1) Modeling of electromagnetic force of a single magnetic pole of a three-orthogonal magnetic bearing based on the equivalent magnetic circuit method The three-orthogonal magnetic reluctance translational magnetic bearing consists of three sets of identical magnetic poles. The magnetic circuit of a single set of magnetic poles mainly includes the radial inner and outer magnetic poles formed by the inner and outer iron cores and windings, the mover, and the air gap between the mover and the radial inner and outer magnetic poles. The magnetic flux loops formed by the inner and outer magnetic poles all start from the N pole of the coil winding, pass through the air gap between the iron core and the mover, then through the air gap between the mover and the S pole, and return to the S pole to form a closed loop. Ignoring the effects of magnetic leakage of the magnet, the self-induced magnetic field of the coil, and edge effects, the equivalent magnetic circuit of a single magnetic bearing is established according to the equivalent magnetic circuit method. From the equivalent magnetic circuit, the total magnetic reluctance of the magnetic circuit of a single set of magnetic poles can be obtained as follows: in , , , These represent the magnetic reluctance of the air gap between the inner and outer magnetic poles and the mover, respectively. , These represent the total magnetic reluctance of a single-pole two-channel magnetic circuit; The reluctance of a single air gap is further expressed as: in δ It is the air gap length between the mover and the magnetic pole face. Permeability in vacuum Indicates the area of ​​each magnetic pole; From the above equation, the magnetic flux of a single set of magnetic poles in two channels can be obtained: In the formula This is the current in the coil when the mover is in the equilibrium position. Represents control current. N Represents the number of coil turns; Force on the mover in a single magnetic pole two-channel system , Represented as: Substituting equations (1), (2), and (3) into the above equation yields the electromagnetic attraction force on the mover in a single magnetic pole two-channel configuration: From the above equation, the electromagnetic force experienced by the mover when it is in the preset equilibrium position is: As can be seen from the above formula, the electromagnetic force on the mover is proportional to the square of the current flowing through the coil winding; The displacement detected by the position sensor when the mover is in the equilibrium position is S The detection quantity of the offset equilibrium position is s Then the electromagnetic attraction forces exerted on the mover at both ends of the single set of magnetic poles are as follows: The total electromagnetic force exerted on the mover by a single set of magnetic poles is: Performing a Taylor expansion on the above equation and discarding higher-order terms, we obtain the formula for the electromagnetic attraction force after linearization of the mover's single degree of freedom: In the formula, Called current stiffness , This is called displacement stiffness. ; (2) Creation of the magnetic pole configuration of the three orthogonal magnetic bearings based on a right-angled regular triangular pyramid The three sets of magnetic poles of the three orthogonal magnetic reluctance translational magnetic bearing are uniformly distributed along the circumference in the same radial plane, that is, the angle between any two adjacent magnetic poles is 120 degrees. The three sets of magnetic poles are simultaneously mounted obliquely. The axis of each set of magnetic poles is defined as the direction of its electromagnetic force. The axes of the three sets of magnetic poles form a right-angled regular triangular pyramid in space, that is, a regular triangular pyramid with isosceles right-angled triangles on the sides. Its base is the radial plane where the magnetic poles are located. According to the method of equal volume and the properties of equilateral triangles, we can obtain: Calculate the volume using the side view as a reference: Find the volume given that the base is an equilateral triangle: according to It is easy to know: From the geometric characteristics of a right-angled regular triangular pyramid, we can obtain: Solving equations (13) and (14), we get: In the formula, l Let be the length of the lateral edge of the triangular pyramid. h The height from the radial plane to the vertex of the triangular pyramid. The angle between the lateral edge of the triangular pyramid and the radial plane; Based on the above formula, the electromagnetic force of the three sets of magnetic poles can be decomposed in space. a , b, c The generated electromagnetic force can be decomposed into two orthogonal components in the radial plane. , And a component passing through the vertex of the triangular pyramid and perpendicular to the radial plane. From the formula, we can obtain: The electromagnetic resultant force generated by each set of magnetic poles passes through the center of the sphere and is perpendicular to each other. By changing the magnitude of the current flowing through the coil winding, the radial two-degree-of-freedom translational motion and the axial one-degree-of-freedom translational motion of the mover can be controlled. (3) Construction of the cosine array of electromagnetic force direction of the three orthogonal magnetic bearings Using the vertex of the triangular pyramid as the origin Establish an electromagnetic force coordinate system, with the three coordinate axes along the three lateral edges of a triangular pyramid. α, β, γ The coordinate system points towards the base of the triangular pyramid and satisfies the Cartesian right-hand rule. Center of the radial plane of the base of the triangular pyramid Establish a position coordinate system for the origin. x The axis is α The projection of the axis onto the radial plane. z shaft edge , y The axis lies in the radial plane and satisfies the right-hand rule. Solving for the electromagnetic force coordinate system With position coordinate system Transformation matrix between P First, establish the position coordinate system Around y Axis rotation Obtain the transition system ,That Axis and electromagnetic force coordinate system of α Align the axes, then set the coordinate system Around Axis rotation Then, it is easy to see that the position coordinate system has been rotated to the electromagnetic force coordinate system; Based on the above analysis, the elementary transformation matrices for the first and second rotations are as follows: According to the transitivity of the direction cosine matrix, we have: It is easy to obtain from the spatial geometric relations of a regular triangular pyramid. ; (4) Modeling of three-degree-of-freedom translational decoupling magnetic reluctance of three orthogonal magnetic bearings The resultant force of the electromagnetic attraction corresponding to the displacement offset , , By transforming the matrix P Electromagnetic attraction provided by magnetic reluctance bearings , , By combining these, we obtain a three-degree-of-freedom translational dynamics model of the mover; According to Newton's second law, we get: The electromagnetic attraction force corresponding to the displacement deviation is determined by coordinate transformation. F By transforming the matrix P Electromagnetic attraction provided by magnetic reluctance bearings Connecting these points, we can obtain: Substitution and The value is obtained by solving: Substitution Simplifying, we get: According to the formula for electromagnetic attraction after linearization of the single degree of freedom of the mover, we can obtain: The single-degree-of-freedom dynamic equation of the mover obtained by combining Newton's second law: For position coordinate system x, y, z Linearizing the resultant electromagnetic attraction forces in the three directions yields a three-degree-of-freedom translational dynamic model of the mover: In the formula, 、 、 It is the displacement measured by the displacement sensor; , , The current is controlled by three sets of magnetic reluctance bearings.

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