A magnetically solid magnetic reducer and its multi-objective optimization method

Abstract

The present invention belongs to the technical field of permanent magnet speed regulation transmission, and discloses a magnetic solid magnetic reducer and a multi-objective optimization method thereof. The magnetic solid magnetic reducer has inner and outer back irons of a dovetail groove structure, which can prevent the permanent magnet from being dislocated and falling off during high-speed rotation, improve space utilization, and ensure the high efficiency and stability of its transmission; the magnetic adjustment ring is made of laminated silicon steel sheets to reduce eddy current loss, and a connecting bridge and a screw are used to jointly ensure its structural strength. The multi-objective optimization method involved performs analytical calculations on the magnetic field of each sub-region of the magnetic reducer, combines the Maxwell stress tensor method to solve the output torque, torque density, and permanent magnet volume as the fitness function of the NSGA-II multi-objective optimization algorithm, and then through a series of iterative steps, finally obtains a set of optimal parameters, so that the magnetic reducer has better comprehensive performance. It not only improves the calculation efficiency, but also maintains the accuracy of the results and the diversity of the solutions, and has good application prospects in practical engineering.

Description

Technical Field

[0001] The present invention belongs to the technical field of magnetic transmission, and specifically relates to a magnetically fixed magnetic reducer and a multi-objective optimization method thereof. Background Art

[0002] Reducers are essential transmission devices in national industrial production. As core components of mechanical transmission systems, they play a key role in industrial equipment, robotics, new energy vehicles, large ships, and other fields. Their primary function is to match the power source and load requirements by reducing speed and increasing torque, while also improving system control accuracy and energy efficiency. Traditional reducers utilize mechanical gear contact transmission and employ processes such as elastic pads for vibration reduction and isolation. While these reducers offer low cost and mature technology, they still suffer from limited vibration reduction, short lifespan, and the need for regular maintenance. Furthermore, their lifespan and reliability cannot be guaranteed in extreme operating conditions such as those associated with long-term operation on large ocean vessels. With breakthroughs in high-performance permanent magnet materials, permanent magnet transmission devices are finding applications in a variety of industrial applications. Magnetic reducers, with their unique contactless transmission mode, are attracting increasing attention from engineering experts. These reducers can significantly improve the system's vibration and noise reduction performance and lifespan. Conventional magnetic reducers use an integrally formed magnetic tuning ring, and most permanent magnets are installed using surface-mount or embedded solutions. This has problems such as structural instability and low efficiency. It cannot meet the extreme working conditions of long time, high speed and heavy load in working environments such as large marine vessels, which seriously hinders the application of magnetic reducers in large marine environments and greatly restricts the development of magnetic transmission technology. In addition, the existing magnetic reducers mainly use finite element simulation methods in the iterative optimization process, which requires design domains with different structural parameters, adds physical fields and multiple boundary condition constraints to different geometric regions, and refines more grids to approximate the actual situation and optimize the structural parameters based on this. This method has high hardware requirements and makes the finite element calculation process complicated, the calculation speed is slow, and even non-convergence occurs, which becomes a key obstacle in the magnetic reducer optimization process. Therefore, the invention of a magnetic-solid magnetic reducer with a compact structure, firm permanent magnet installation, stable structure and high transmission efficiency and its multi-objective optimization method has irreplaceable value and significance for improving the vibration reduction and isolation capability of large marine ship transmission systems, increasing service life, and achieving large-scale promotion of magnetic reducers in this field.

[0003] Regarding magnetic reducers, Sumika Otosaka, in her patent "Magnetic Gear Device" (CN202380017279.1), proposed a magnetic gear consisting of an inner rotor, a pole piece module, and an outer rotor. While this approach reduces the material of the magnets and the magnetic pole pieces, resulting in greater torque transmission, it still suffers from drawbacks such as unstable permanent magnet fixation, large deformation of the magnetic ring, and low efficiency, making it unsuitable for the development needs of modern large marine vessels. Regarding structural parameter optimization methods for magnetic reducers, Tianbo She et al., in their paper "An Optimization Method for Coaxial Magnetic Field Modulated Magnetic Gears Based on Reluctance Network Analysis," utilized an equivalent reluctance analysis method, taking into account the central angles of the inner and outer permanent magnets and the magnetic pole pieces, and aiming to maximize output torque. They analyzed a structural parameter optimization method for magnetic reducers, typically magnetic gears, with the goal of maximizing output torque. While the proposed method achieves high computational accuracy and a compact structure, it suffers from a lengthy algorithm, high computational cost, a single objective function, and lacks consideration of the structural parameter optimization of other components, making it inappropriate for the design, analysis, and multi-parameter, multi-objective optimization of complete magnetic reducers.

[0004] Therefore, proposing a magnetic solid magnetic reducer and its multi-objective optimization method is of great significance for the design and characteristic research of low-noise vibration reduction, long life, and high-performance speed transmission basic components in the field of large marine ships. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention invents a magnetically fixed magnetic reducer. Its purpose is to achieve speed regulation between the input and output shafts, using the non-contact magnetic force between the outer permanent magnet, the magnetic tuning ring, and the inner permanent magnet as a medium. This reduces vibration and noise in the transmission system of large marine vessels, improving the operating performance and reliability of the transmission system.

[0006] The technical solution of the present invention:

[0007] A magnetically fixed magnetic reducer, first, insert the inner permanent magnet into the dovetail groove of the inner back iron, and then install the inner rotor end covers on both sides; then, connect the front end of the input shaft and the end section of the input shaft to the inner rotor covers on both sides respectively, and install the flat key into the keyway of the end section of the input shaft to complete the installation of the input shaft; then, insert the two sections of outer permanent magnets into the dovetail groove of the outer back iron, and then install the sleeve ring on one side, the outer back iron bearing is installed on the outer back iron through the shaft shoulder, and the outer rotor end cover is fixed to the end of the outer back iron close to the shaft shoulder to complete the installation of the outer rotor; then, the magnetic adjustment ring end cover and the magnetic adjustment ring pressure cover are respectively installed on both sides of the magnetic adjustment ring through screws, and anti-loosening nuts are screwed on both ends of the screw for axial limitation, and then the magnetic adjustment ring end cover is connected to the pressure cover to complete the installation of the magnetic adjustment mechanism; the pressure cover retaining ring is embedded in the pressure cover, the input shaft front section bearing is installed on the input shaft front section, and the input shaft front section on the input shaft is fixed through the input shaft front section bearing The outer cover is connected to the outer cover, and the magnetic adjustment ring gland bearing is installed on the shaft sleeve, and is limited by the shaft sleeve retaining ring. The shaft sleeve is connected to the end section of the input shaft through a flat key to complete the assembly of the input shaft and the magnetic adjustment structure; the outer rotor is matched with the outer shell through the outer back iron bearing, the shaft sleeve bearing is installed on the shaft shoulder of the shaft sleeve and matched with the collar, the sealing cover is connected to the collar, the output shaft is connected to the sealing cover, the outer shell cover is connected to the outer shell, the collar bearing is installed on the collar, and finally, the end cover is connected to the outer shell cover; the motor is used to drive the input shaft to rotate, and the magnetic field between the outer permanent magnet and the inner permanent magnet is modulated and interacted by the magnetic adjustment ring to realize power transmission and drive the output shaft to rotate; the permanent magnet of the present invention is firmly installed, the magnetic adjustment ring is smaller, the reliability is high, the service life is long, the structure is compact, and the vibration and noise are low, and it has high practicality and engineering application promotion value in the transmission system of large marine ships.

[0008] A magnetic fixed magnetic reducer, including a ring bearing 1, an outer shell cover 2, a ring 3, an inner rotor end cover 4, an outer back iron 5, an inner permanent magnet 6, an inner back iron 7, an outer back iron bearing 8, a pressure cover 9, an outer end cover 10, an input shaft front section bearing 11, an input shaft front section 12, a pressure cover retaining ring 13, a magnetic adjustment ring end cover 14, an outer rotor end cover 15, an outer permanent magnet 16, a magnetic adjustment ring 17, an outer shell 18, an input shaft end section 19, a shaft sleeve retaining ring 20, a magnetic adjustment ring pressure cover 21, The magnetic ring pressure cover bearing 22, the shaft sleeve bearing 23, the sealing cover 24, the flat key 25, the shaft sleeve 26, the output shaft 27, the screw 28 and the end cover 29; the cross section of the inner permanent magnet 6 is a "T"-shaped structure, with "V"-shaped notches formed on both sides; the inner back iron 7 is a cylindrical structure, and its outer surface is provided with periodic dovetail grooves with chamfers, and the dovetail grooves present a "V"-shaped structure, which cooperates with the "V"-shaped notches of the inner permanent magnet 6; the cross section of the outer permanent magnet 16 is It has a "T"-shaped structure with "V"-shaped notches on both sides; the outer back iron 5 is a torus structure with periodic chamfered dovetail grooves on its inner surface. The dovetail grooves are in a "V"-shaped structure, which cooperates with the "V"-shaped notches of the outer permanent magnet 16; the magnetic adjustment ring 17 is composed of a plurality of magnetic adjustment pole blocks and magnetic resistance blocks alternatingly, both of which are in the shape of a fan ring. The magnetic adjustment pole blocks are made of high magnetic permeability materials, and the magnetic resistance blocks are made of low magnetic permeability materials. Each magnetic adjustment pole block There is a small hole in the center, and the magnetic pole pieces are connected to form a whole through a connecting bridge close to the center of the circle; the front section 12 of the input shaft has a "C"-shaped keyway at one end, the middle shoulder area is thickened, and a flange is attached to the other end, with a stop and a through hole on the flange; the end section 19 of the input shaft has a keyway at one end and a flange is attached to the other end, with a stop and a through hole on the flange; the shaft sleeve 26 is a hollow stepped cylinder with a keyway on the inner wall and a shoulder at each end.

[0009] The inner permanent magnet 6 is inserted into the dovetail groove of the inner back iron 7, and the two inner rotor end covers 4 are respectively installed on both sides of the inner back iron 7; the front section 12 of the input shaft is installed on the inner rotor end cover 4 on one side through the flange thereon, and the input shaft end section 19 is installed on the inner rotor end cover 4 on the other side through the flange thereon; the flat key 25 is installed in the keyway of the input shaft end section 19 to complete the installation of the input shaft; the outer permanent magnet 16 is divided into two sections, which are inserted into the dovetail grooves therein from both ends of the outer back iron 5, the collar 3 is installed at one end of the outer back iron 5, the outer back iron bearing 8 is installed at the other end of the outer back iron 5 through the shaft shoulder, and the outer rotor end cover 15 is fixed on the end of the outer back iron 5 close to the shaft shoulder to complete the installation of the outer rotor; the screw 28 passes through the small hole on the magnetic adjustment ring 17, and its two ends are respectively connected to the magnetic adjustment ring end cover 14 and the magnetic adjustment ring pressure cover 21, and the two ends of the screw 28 are axially limited, and the magnetic adjustment ring end cover 14 is fixed to the outer shell 18 through the pressure cover 9 to complete the adjustment Installation of the magnetic structure; the gland retaining ring 13 is embedded in the gland 9, the input shaft front section 12 on the input shaft is installed on the gland 9 through the input shaft front section bearing 11, the outer end cover 10 is connected to the gland 9, the magnetic adjustment ring gland bearing 22 is installed on the shaft sleeve 26, and is limited by the shaft sleeve retaining ring 20. The shaft sleeve 26 is installed on the magnetic adjustment ring gland 21 through the magnetic adjustment ring gland bearing 22, and the shaft sleeve 26 is connected to the input shaft end section 19 through the flat key 25 to complete the assembly of the input shaft and the magnetic adjustment structure; the outer back iron 5 on the outer rotor is matched with the outer shell 18 through the outer back iron bearing 8, the shaft sleeve bearing 23 is installed on the shaft shoulder of the shaft sleeve 26, and cooperates with the collar 3, the sealing cover 24 is connected to the collar 3, the output shaft 27 is connected to the sealing cover 24, the outer shell cover 2 is connected to the outer shell 18, the collar bearing 1 is installed on the collar 3, and cooperates with the outer shell cover 2 at the same time. Finally, the end cover 29 is connected to the outer shell cover 2 to complete the installation of the whole machine.

[0010] A multi-objective optimization method for a magnetic solid magnetic reducer, characterized by the following steps: the first step is to determine the key design parameters of the magnetic solid magnetic reducer; the key design parameters include the thickness of the inner back iron , outer back iron thickness , thickness of inner permanent magnet , thickness of outer permanent magnet , inner air gap thickness , External air gap thickness , adjust the thickness of the magnet pole , center hole diameter and axial length ; The second step is to conduct magnetic field analysis modeling, divide the sub-regions, list the partial differential equations to solve the radial and tangential magnetic induction intensities in region IV; list the partial differential equations for the regions in the magnetic solid magnetic reducer, obtain the expression of the magnetic vector potential in region IV, and then obtain the expression of the magnetic induction intensity in region IV based on the relationship between the magnetic vector potential and the magnetic induction intensity; since the Fourier coefficients required to solve the magnetic induction intensity have a linear relationship between different regions, it is necessary to solve different regions in turn; in the structure of the magnetic solid magnetic reducer, the air region between the inner permanent magnet (6) and the magnetic tuning ring (17) is the inner air gap, and the outer permanent magnet (16 ) and the magnetic adjustment ring (17) is the outer air gap; since the structure of the magnetic solid magnetic reducer is axially symmetrical, the magnetic solid magnetic reducer is divided into seven regions I to VII according to the boundaries of different material compositions, the inner permanent magnet (6) portion corresponds to region I, the inner air gap portion corresponds to region II, the magnetic adjustment ring (17) portion corresponds to region III, the outer air gap portion corresponds to region IV, the outer permanent magnet (16) portion corresponds to region V, the outer back iron (5) portion corresponds to region VI, and the inner back iron (7) portion corresponds to region VII, corresponding to seven circular rings respectively; the inner and outer radii of the circular ring in region I are 、 The inner and outer radii of the ring in region II are 、 The inner and outer radii of the ring in region III are 、 , the inner and outer radii of the ring in region IV are 、 The inner and outer radii of the ring in region V are 、 The inner and outer radii of the ring in area VI are 、 According to the Maxwell stress tensor formula, the magnitude of the outer rotor torque is related to the magnetic induction intensity of the adjacent area VI, so the magnetic induction intensity of area VI is solved; the third step is to solve the Fourier coefficient linear equations about the magnetic vector potential; the fourth step is to calculate the output torque, torque density, and permanent magnet volume; the fifth step is to initialize the population; in the multi-objective optimization algorithm NSGA-II, let the number of independent variables be ; Set the number of chromosomes to ; Population step length of offspring Is the length of , the values ​​are A vector of ; randomly generated Individuals, the values ​​of the decision variables contained in the individuals are within the upper and lower bounds; the fitness function is solved for the values ​​in each individual using the above algorithm, and the results are saved in the attributes of the individual; the sixth step is to perform iterative optimization calculations of the NSGA-II algorithm; after determining the calculation model and completing the initialization of the population, a set of optimal solutions, namely the Pareto frontier solution set, will be obtained through iterative calculations.

[0011] The specific implementation process of the second step is as follows: According to the magnetic vector potential and magnetic induction intensity In region IV, the following relationships exist: . .in, 、 Respectively represent the radial and tangential magnetic induction in region IV, is the polar diameter, is the polar angle, represents the magnetic vector potential in region IV; in solving 、 Before that, we need to solve the magnetic vector potential in region IV. , applying the partial differential equation to region IV: Combined with the boundary conditions, the separation of variables method is used to solve the magnetic vector potential in region IV: .in, 、 、 、 for The Fourier coefficients of for The constant term of the Fourier expansion of : . Where, is a positive integer, x and y are independent variables; from formulas (1), (2), and (4), the radial and tangential magnetic induction intensities of region IV are obtained: 、 The expression: . Where, The Fourier coefficients of 、 、 、 is an unknown number; if we want to get 、 The specific value of needs to be solved according to the boundary conditions of region IV, region III and region V to obtain the Fourier coefficient 、 、 、 ,have: . . . Where, is the number of iron pole pairs of the magnetic ring, is the central angle of the magnetoresistive block, 、 is the magnetic vector potential of the jth magnetoresistive block in region III The Fourier coefficients of for The constant term of the Fourier expansion of 、 is the magnetic vector potential in region V The Fourier coefficients of Representation function The derivative of is the outer rotor rotation angle; . . . Where, For the The azimuth angle corresponding to the magnetoresistive block is represents the remanence of the permanent magnet, is the number of pole pairs of the external permanent magnet; .

[0012] . In the formula, m, n, j are all positive integers; .because 、 、 、 The expressions (9) to (12) contain the Fourier coefficients of regions III and V. 、 、 、 , so it is necessary to solve regions III and V, list the partial differential equations and combine them with the boundary conditions to obtain the Fourier coefficients of region V: . .

[0013] Similarly, solve for region III and obtain the magnetic vector potential of the j-th magnetoresistive block in region III: The Fourier coefficients of :

[0014] . Where, 、 、 、 is the magnetic vector potential in region II Fourier coefficient; From Equations (22) and (23), we can know that the magnetic vector potential of the j-th magnetoresistive block in region III is The Fourier coefficients of The expression contains the Fourier coefficients of region II 、 、 、 , so it is necessary to solve region II and obtain the magnetic vector potential of region II The Fourier coefficients of : . . . Where, 、 for The Fourier coefficients of is the inner rotor rotation angle, Representation function The derivative of , and . .in, is the number of pole pairs of the inner permanent magnet; from equations (24) to (27), we can know that The Fourier coefficients of 、 The expression contains the Fourier coefficients of region I 、 , it is necessary to solve the region I and obtain the magnetic vector potential of region I The Fourier coefficients of : . At this point, the radial and tangential magnetic induction intensities of region IV are obtained. 、 The expression and solution method of ; According to formulas (22) and (23), the Fourier coefficients of region III are completely represented by the Fourier coefficients of regions II and IV, and the Fourier coefficients are obtained. 、 、 、 、 、 、 、 、 、 、 、 A set of linear equations.

[0015] The specific implementation process of the third step is as follows:

[0016] About Fourier coefficients 、 、 、 、 、 、 、 、 、 、 、 In a set of linear equations, since the Fourier coefficients are linearly related, to get the coefficients 、 、 、 , it is necessary to establish a linear equation system to solve;

[0017] According to formulas (9) to (12), (20), (21), (24) to (27), the linear equations are established: .in, is the Fourier coefficient vector to be solved, with a size of , is the number of truncated terms in the infinite series: .

[0018] Vector x consists of 12 The vector composition is: . P is the coefficient matrix of vector x, and its size is :

[0019] .

[0020] in, is an N-order unit matrix, For size The diagonal matrix whose value is the coefficient matrix of vector x; t is the constant vector in the linear equation system, and its size is : S1~S8 are the calculation items in formulas (9)~(12), (20), (21), (24)~(27) that are independent of n; solve for the Fourier coefficient vector Then, take 、 、 、 , obtained from formulas (7) and (8) 、 The value of .

[0021] The specific implementation process of the fourth step is as follows:

[0022] Solve for the outer rotor torque based on the Maxwell stress tensor:

[0023] .

[0024] in, is the axial length of the magnetic solid magnetic reducer, is the vacuum permeability;

[0025] .

[0026] Take the outer rotor torque The maximum value among them is taken as the output torque , the calculation formula of torque density Des:

[0027] .

[0028] Permanent magnet volume V m Calculation formula:

[0029] .

[0030] Pick 、 、 As the fitness function of the multi-objective optimization algorithm NSGA-II, ~ is the decision variable, and the function expression is:

[0031] .

[0032] in, ; 、 are the upper and lower bounds of the decision variables respectively; is the constraint violation function.

[0033] The specific implementation process of the sixth step is as follows:

[0034] 1) Perform fast non-dominated sorting on the population;

[0035] Set the constraint violation function:

[0036] .

[0037] in, represents the inequality constraint of the e-th individual in formula (25); 、 Represent the upper and lower bounds of the constraints respectively;

[0038] When sorting, classify and discuss based on the constraint violation, set individuals with smaller constraint violation values ​​as high priority, and if individual inv1 dominates another individual inv2, then add inv2 to the domination set of inv1. The number of individuals that simultaneously dominate inv2 On the contrary, if individual inv2 dominates individual inv1, then inv1 is added to the dominance set of inv2 The number of individuals that simultaneously dominate inv1 ; In the traversal process, if the number of individuals e is dominated =0, then put it into the first level of the Pareto frontier set middle;

[0039] Let ne=1, set Q is an empty set, traverse the Pareto frontier of the neth layer, visit all points inv3 dominated by the individual, and the number of individuals dominating inv3 -1, if at this time =0, then , that is, Q is placed in the ne+1th layer Pareto frontier; after the neth layer Pareto frontier traversal is completed, if the set Q is an empty set, it means that the Pareto frontier has been stratified, and we get 、 、 ...Exit the loop at this time, otherwise , ne is updated to ne+1, and the individuals on the next layer of Pareto frontier are traversed;

[0040] 2) Calculate the crowding degree of individuals in the population;

[0041] The congestion calculation formula is as follows:

[0042] .

[0043] in, represents the crowding distance of the e-th point, 、 Represents the fitness function value of the two points before and after the individual e, 、 Represents the maximum and minimum values ​​of the fitness function of the Pareto front solution set of the neth layer; 3) forms a new generation population; prioritizes individuals with high Pareto front levels and large crowding distances for crossover and mutation operations of decision variables, calculates the fitness function of the newly obtained offspring population, merges it with the parent population, performs non-dominated sorting, adopts an elite selection strategy, and prioritizes individuals with high Pareto front levels and large crowding distances as the new generation population; repeats processes 1) to 3) until the maximum number of iterations is reached, and finally obtains a set of optimal Pareto front solution sets as the multi-objective optimization results; at this point, the multi-objective optimization of the field-magnetic solid-state magnetic reducer has been completed.

[0044] The beneficial effect of the present invention is to propose a magnetically fixed magnetic reducer, which uses the non-contact magnetic force of the outer permanent magnet, the magnetic adjustment ring and the inner permanent magnet as the medium to achieve speed regulation transmission between the input shaft and the output shaft, thereby reducing the vibration and noise of the transmission system of large marine ships and improving the operating performance and reliability of the transmission system. The inner and outer back irons with dovetail groove structures can not only prevent the permanent magnets from being misaligned and falling off during high-speed rotation, but also greatly improve the spatial utilization of the permanent magnets, allowing as many permanent magnets as possible to be installed in a limited space, ensuring efficient and stable transmission; the magnetic adjustment ring made of laminated silicon steel sheets can significantly reduce eddy current losses, and the structural strength of the magnetic adjustment ring is ensured by the use of a connecting bridge and a screw. In addition, the present invention considers the structural parameters of the magnetic reducer and performs multi-objective optimization design to achieve the purpose of improving output torque, torque density, and reducing the amount of permanent magnets used. Insufficient output torque will affect the direct function of the magnetic reducer, resulting in the device being unable to drive the load; too low torque density will lead to bloated equipment and prolonged dynamic response time; too large a permanent magnet volume will lead to increased cost, increased eddy current loss, and increased operating temperature. A magnetic reducer with the three advantages of high output torque, high torque density, and low permanent magnet volume can achieve a larger rated load and better dynamic performance response. It has low manufacturing cost, small size, light weight, low operating temperature, high operating efficiency, and can be applied to space-constrained scenarios, such as the cabin of large ocean vessels. The multi-objective optimization method for the magnetic reducer proposed in the present invention can effectively avoid complex finite element simulation calculations, greatly facilitating the optimization of the structural parameters of the magnetic reducer. In addition, the calculation method fully considers the diversity of multi-objective optimization solutions and can more accurately match engineering requirements. The method is highly applicable in magnetic reducer performance calculation and structural design optimization, and is an optimization design method with engineering universality. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is a cross-sectional view of a magnetically solid magnetic reducer;

[0046] Figure 2 This is a schematic diagram of the overall structure of the magnetic solid magnetic reducer;

[0047] Figure 3 Schematic diagram of the internal permanent magnet structure;

[0048] Figure 4 Schematic diagram of the inner back iron structure;

[0049] Figure 5 Schematic diagram of the external permanent magnet structure;

[0050] Figure 6 Schematic diagram of the outer back iron structure;

[0051] Figure 7 Schematic diagram of the shaft sleeve structure;

[0052] Figure 8 Schematic diagram of the input shaft structure;

[0053] Figure 9 Schematic diagram of the outer rotor structure and assembly;

[0054] Figure 10 It is the assembly diagram of the magnetic tuning structure;

[0055] Figure 11 This is a schematic diagram of the assembly of the input shaft and the magnetic modulation structure;

[0056] Figure 12 It is a schematic diagram of the axial section of a magnetic solid magnetic reducer;

[0057] Figure 13 Schematic diagram of parameter analysis model of magnetic solid magnetic reducer;

[0058] Figure 14 This is a flow chart of a multi-objective optimization method for a magnetically solid magnetic reducer;

[0059] Figure 15 This is a comparison chart of the theoretical value and simulation value of the radial magnetic flux density in the outer air gap;

[0060] Figure 16 This is a comparison chart between the theoretical value and simulation value of the outer air gap tangential flux density;

[0061] Figure 17 is a diagram showing the relationship between the outer rotor rotation angle and the outer rotor rotation angle;

[0062] Figure 18 is the Pareto frontier solution set obtained after multi-objective optimization;

[0063] Figure 1 middle:

[0064] 1- collar bearing, 2- outer shell cover, 3- collar, 4- inner rotor end cover, 5- outer back iron, 6- inner permanent magnet, 7- inner back iron, 8- outer back iron bearing, 9- gland, 10- outer end cover, 11- input shaft front section bearing, 12- input shaft front section, 13- gland retaining ring, 14- magnetic adjustment ring end cover, 15- outer rotor end cover, 16- outer permanent magnet, 17- magnetic adjustment ring, 18- outer shell, 19- input shaft end section, 20- shaft sleeve retaining ring, 21- magnetic adjustment ring gland, 22- magnetic adjustment ring gland bearing, 23- shaft sleeve bearing, 24- sealing cover, 25- flat key, 26- shaft sleeve, 27- output shaft, 28- screw, 29- end cover. DETAILED DESCRIPTION

[0065] The embodiments of the present invention are further described below in conjunction with the accompanying drawings and technical solutions.

[0066] This embodiment selects a magnetic reducer with 4 pairs of inner permanent magnets, 11 pairs of outer permanent magnets, 15 pole pairs of magnetic tuning rings, and a reduction ratio of 2.75 for assembly and multi-objective optimization. The overall structure is as follows: Figure 2 shown.

[0067] Among them, the inner diameter of the ring bearing 1 is 120mm, the outer diameter is 150mm, the outer diameter of the outer cover 2 is 226mm, the inner diameter is 150mm, the inner diameter of the ring 3 is 72mm, the outer diameter is 157mm, and the length is 66.5mm. The outer diameter of the inner rotor end cover 4 is 66mm, the stepped hole diameter is 41.4mm, the inner diameter is 8mm, the outer diameter of the outer back iron 5 is 157mm, the inner diameter is 125mm, and the length is 10mm. The inner permanent magnet 6 uses 4 pairs of fan-shaped prismatic magnetic blocks with an inner radius of 23mm and an outer radius of 33mm. The inner back The outer diameter of iron 7 is 46mm, the inner diameter is 7mm, and the length is 60mm. The outer diameter of the outer back iron bearing 8 is 190mm, the inner diameter is 150mm, the inner diameter of the pressure cover 9 is 72mm, the outer diameter is 226mm, the diameter of the input shaft front section 12 is 30mm, the length is 133mm, the outer diameter of the outer end cover 10 is 108mm, the inner diameter is 52mm, the outer diameter of the input shaft front section bearing 11 and the shaft sleeve bearing 23 are both 72mm, the inner diameter is 50mm, the outer diameter of the pressure cover retaining ring 13 is 75mm, the inner diameter is 68mm, the magnetic ring end cover 1 is 1. 4 has an outer diameter of 142mm and an inner diameter of 72mm. The outer diameter of the outer rotor end cover 15 is 160mm and the inner diameter is 103mm. The outer permanent magnet 16 uses 11 pairs of sector-shaped prismatic magnets with an inner radius of 51.5mm and an outer radius of 62.5mm. The inner diameter of the magnetic adjustment ring 17 is 78mm, the outer diameter is 96mm, and the length is 120mm. The inner diameter of the housing 18 is 180mm, the outer diameter is 226mm, and the length is 132mm. The diameter of the input shaft end section 19 is 30mm and the length is 59mm. The inner diameter of the shaft sleeve retaining ring 20 is 4 2.5mm, outer diameter is 49mm, the inner diameter of the magnetic ring gland 21 is 58mm, the outer diameter is 96mm, and the length is 20.5mm. The inner diameter of the magnetic ring gland bearing 22 is 45mm, the outer diameter is 58mm, the inner diameter of the sealing cover 24 is 45mm, the outer diameter is 117mm, the width of the flat key 25 is 8mm, the height is 7mm, and the length is 22mm. The inner diameter of the shaft sleeve 26 is 30mm, the outer diameter is 55mm, and the length is 54.5mm. The diameter of the output shaft 27 is 30mm and the length is 101mm. The screw 28 uses 15 The tooth bar has a length of 140 mm, an inner diameter of the end cover 29 is 122 mm, and an outer diameter is 190 mm.

[0068] Internal permanent magnet 6, such as Figure 3 As shown, its cross section is a "T"-shaped structure, with "V"-shaped notches formed on both sides; the inner back iron 7, as shown Figure 4 As shown, it is a cylindrical structure, and its outer surface is provided with periodic dovetail grooves with chamfers. The dovetail grooves present a "V"-shaped structure, which cooperates with the "V"-shaped notch of the inner permanent magnet 6; the outer permanent magnet 16, as shown Figure 5 As shown, its cross section is a "T"-shaped structure with "V"-shaped notches on both sides. Each magnetic pole is divided into two permanent magnets; the outer back iron 5, as shown Figure 6 As shown, it is a torus structure, and its inner surface is provided with periodic dovetail grooves with chamfers. The dovetail grooves present a "V"-shaped structure, which cooperates with the "V"-shaped notch of the outer permanent magnet 16; the shaft sleeve 26, as shown Figure 7 As shown, it is a hollow stepped cylinder with a keyway on the inner wall and a shoulder at each end; the magnetic ring 17, as shown Figure 12 As shown, the main body is composed of several adjusting magnet pole pieces, each of which is provided with a small hole in the center of the adjusting magnet pole piece, and the adjusting magnet pole pieces are connected to form a whole by a connecting bridge close to the center side; the front section 12 of the input shaft has a "C"-shaped keyway at one end, a thickened shoulder area in the middle, and a flange plate attached to the other end, which is provided with a stopper and a through hole; the rear section 19 of the input shaft has a keyway at one end and a flange plate attached to the other end, which is provided with a stopper and a through hole.

[0069] The installation steps of the magnetic solid magnetic reducer are as follows:

[0070] First, insert the inner permanent magnet 6 into the dovetail groove of the inner back iron 7. The two inner rotor end covers 4 are respectively installed on both sides of the inner back iron 7. The input shaft front section 12 is installed on the inner rotor end cover 4 on one side, and the input shaft end section 19 is installed on the inner rotor end cover 4 on the other side. The flat key 25 is installed in the keyway of the input shaft end section 19 to complete the installation of the input shaft. Figure 8 shown.

[0071] Next, insert the two sections of external permanent magnets 16 into the dovetail grooves of the outer back iron 5, install the collar 3 on one end of the outer back iron 5, install the outer back iron bearing 8 on the other end of the outer back iron 5 through the shaft shoulder, and fix the outer rotor end cover 15 on the end of the outer back iron 5 close to the shaft shoulder to complete the installation of the outer rotor. Figure 9 shown.

[0072] Then, the two ends of the magnetic adjustment ring 17 are connected to the magnetic adjustment ring end cover 14 and the magnetic adjustment ring pressure cover 21 respectively through the screw 28. The two ends of the screw 28 are axially limited. The magnetic adjustment ring end cover 14 is fixed to the housing 18 through the pressure cover 9 to complete the installation of the magnetic adjustment structure. Figure 10 shown.

[0073] The gland retaining ring 13 is embedded in the gland 9, the input shaft front section bearing 11 is installed on the input shaft front section 12, the input shaft front section 12 on the input shaft is installed on the gland 9 through the input shaft front section bearing 11, the outer end cover 10 is connected to the gland 9, the magnetic adjustment ring gland bearing 22 is installed on the shaft sleeve 26, and is limited by the shaft sleeve retaining ring 20. The shaft sleeve 26 is installed on the magnetic adjustment ring gland 21 through the magnetic adjustment ring gland bearing 22, and the shaft sleeve 26 is connected to the input shaft end section 19 through the flat key 25, completing the assembly of the input shaft and the magnetic adjustment structure, as shown in FIG. Figure 11 shown.

[0074] The outer back iron 5 on the outer rotor cooperates with the outer shell 18 through the outer back iron bearing 8. The shaft sleeve bearing 23 is installed on the shaft shoulder of the shaft sleeve 26 and cooperates with the collar 3. The sealing cover 24 is connected to the collar 3. The output shaft 27 is connected to the sealing cover 24. The outer shell cover 2 is connected to the outer shell 18. The collar bearing 1 is installed on the collar 3 and cooperates with the outer shell cover 2. Finally, the end cover 29 is connected to the outer shell cover 2 to complete the installation of the entire machine.

[0075] At this point, a magnetically solid magnetic reducer has been installed.

[0076] A multi-objective optimization method for a magnetically solid magnetic reducer Figure 14 The specific steps are as follows:

[0077] The first step is to determine the design parameters of the magnetic reducer to be optimized;

[0078] The above-mentioned magnetic-solid magnetic reducer with 4 pairs of inner permanent magnets, 11 pairs of outer permanent magnets, 15 pole pairs of the magnetic tuning ring, and a reduction ratio of 2.75 was selected for multi-objective optimization. The parameters to be optimized are shown in Table 1:

[0079] Table 1 Structural parameters to be optimized

[0080]

[0081] The second step is to conduct magnetic field analysis and modeling, divide the sub-regions, and list partial differential equations to solve the radial and tangential magnetic induction intensities in region IV. The specific process is to list partial differential equations for the sub-regions in the magnetic reducer, obtain the expression of the magnetic vector potential in region IV, and then obtain the expression of the magnetic induction intensity in region IV based on the relationship between the magnetic vector potential and the magnetic induction intensity. Since the Fourier coefficients required to solve the magnetic induction intensity have a linear relationship between different regions, it is necessary to solve different regions in turn.

[0082] The magnetic reducer is divided into seven regions I to VII, and the region radius is ~ ,like Figure 13 As shown in Table 1, the radius parameter table of the sub-area of ​​the magnetic-solid magnetic reducer is calculated, as shown in Table 2:

[0083] Table 2 Radius parameter table

[0084]

[0085] Set the inner rotor rotation angle =0, outer rotor rotation angle =0, that is, the inner and outer rotors remain stationary, calculate the radial and tangential magnetic induction in the outer air gap 、 , Matlab program is written based on formulas (9)~(12), (20), (21), (24)~(27), and the calculated theoretical value results are compared with the finite element simulation results. Figure 15 、 Figure 16 The results show that the theoretical values ​​are consistent with the simulation values, and subsequent calculations can be performed.

[0086] Step 3: output torque, torque density and permanent magnet volume;

[0087] Set the inner rotor rotation angle = , outer rotor rotation angle =0, that is, the outer rotor remains stationary and the inner rotor rotates. The outer rotor torque T is calculated by formula (37), and the result is as follows: Figure 17 The results show that the theoretical value and the simulation value are consistent with each other, and subsequent calculations can be performed. , the torque density is obtained from formula (39): , the volume of the permanent magnet is obtained from formula (40): , and this calculation method is used as the fitness function of the multi-objective optimization algorithm NSGA-II for iterative calculation.

[0088] According to formula (41), the mathematical model of the optimization problem is determined as:

[0089] .

[0090] Among them, the design domain of key design parameters is shown in Table 3:

[0091] Table 3 Design domain of geometric parameters

[0092]

[0093] Step 4: Initialize the population;

[0094] In the genetic optimization algorithm NSGA-II, set the number of independent variables =9; Set the number of chromosomes to =100; population step size of offspring A vector of length 9, all of which are randomly generated = 100 individuals, and the values ​​of the decision variables contained in the individuals are all within the upper and lower bounds. The fitness function is solved using the above algorithm for the values ​​in each individual, and the result is saved in the individual's attributes.

[0095] Step 5: Perform iterative optimization of the NSGA-II algorithm;

[0096] Set the maximum number of iterations maxit=50 and perform the iterative steps:

[0097] Set the constraint violation function:

[0098] .

[0099] Repeat the process 1) to 3) to obtain a set of optimal Pareto frontier solutions as the multi-objective optimization results, such as Figure 18 As shown. Take a suitable point in the optimal Pareto front to calculate the optimized structural parameters, and the results are shown in Table 4:

[0100] Table 4 Optimized geometric parameters and performance indicators

[0101]

[0102] It can be concluded that after optimization, the output torque increased by 47.7%, the torque density increased by 356%, and the volume of the permanent magnet decreased by 69.3%, achieving the optimization effect.

[0103] At this point, the multi-objective optimization of the magnetic reducer has been completed.

[0104] This novel structure of the magnetic fixed magnetic reducer uses the non-contact magnetic force between the outer permanent magnet, the magnetic adjustment ring and the inner permanent magnet as the medium to achieve speed regulation transmission between the input shaft and the output shaft. The inner and outer rotor isolation assembly scheme adopted by the present invention enables the device to achieve mechanical contactless speed regulation transmission, which not only effectively avoids mechanical friction and loss, but also greatly improves reliability and life. In addition, due to the lack of direct mechanical contact, the vibration and noise generated during the speed regulation transmission are low, making it particularly suitable for the field of large marine vessels. In addition, the present invention achieves the fixation of the permanent magnets through dovetail grooves on the inner and outer rotors, which is easy to assemble and the permanent magnets will not be subjected to force displacement, misalignment and falling off under high-speed rotation conditions. The magnetic adjustment ring adopts an internal screw and connecting bridge structure, which improves the reliability and life of the magnetic reducer. In addition, the magnetic adjustment ring is made of laminated silicon steel sheets, which can reduce eddy current loss and improve the transmission efficiency of the magnetic reducer to a certain extent.

[0105] This method uses partial differential equations and the subdomain method to analytically calculate the magnetic field distribution in each subregion of the magnetic reducer. It then combines the Maxwell stress tensor method to solve for output torque, torque density, and permanent magnet volume. This calculation method is then used as the fitness function for the multi-objective optimization algorithm NSGA-II. This method fully considers the structural parameters of the magnetic reducer, establishes a mathematical model, and performs iterative calculations, ultimately establishing a multi-objective optimization method that conforms to the design rules for magnetic reducer parameters. Compared to finite element simulation, this method avoids complex software operations, significantly improves the computational efficiency of the theoretical model while maintaining computational accuracy, and greatly facilitates the optimization of magnetic reducer structural parameters. Furthermore, this calculation method fully considers the diversity of multi-objective optimization solutions, allowing for more accurate matching of specific engineering requirements. This method has strong applicability in magnetic reducer performance calculations and structural design optimization. It is a simple, low-cost, and highly accurate optimization design method with engineering applicability.

Claims

1. A magnetically fixed magnetic reducer, characterized in that: The magnetic fixed magnetic speed reducer comprises a sleeve bearing (1), an outer shell cover (2), a sleeve ring (3), an inner rotor end cover (4), an outer back iron (5), an inner permanent magnet (6), an inner back iron (7), an outer back iron bearing (8), a pressure cover (9), an outer end cover (10), an input shaft front section bearing (11), an input shaft front section (12), a pressure cover retaining ring (13), a magnetic adjustment ring end cover (14), an outer rotor end cover (15), an outer permanent magnet (16), a magnetic adjustment ring (17), an outer shell (18), an input shaft end section (19), a shaft sleeve retaining ring (20), a magnetic adjustment ring pressure cover (21), a magnetic adjustment ring pressure cover bearing (22), a shaft sleeve bearing (23), a sealing cover (24), a flat key (25), a shaft sleeve (26), an output shaft (27), a screw (28) and an end cover (29); The inner permanent magnet (6) is inserted into the dovetail groove of the inner back iron (7), and the two inner rotor end covers (4) are respectively installed on both sides of the inner back iron (7); the input shaft front section (12) is installed on the inner rotor end cover (4) on one side through the flange plate thereon, and the input shaft end section (19) is installed on the inner rotor end cover (4) on the other side through the flange plate thereon; the flat key (25) is installed in the keyway of the input shaft end section (19) to complete the installation of the input shaft; the outer permanent magnet (16) is divided into two sections, which are inserted into the dovetail groove inside the outer back iron (5) from both ends thereof, and the sleeve The ring (3) is installed at one end of the outer back iron (5), the outer back iron bearing (8) is installed at the other end of the outer back iron (5) through the shaft shoulder, and the outer rotor end cover (15) is fixed on the end of the outer back iron (5) close to the shaft shoulder, thereby completing the installation of the outer rotor; the screw (28) passes through the small hole on the magnetic adjustment ring (17), and its two ends are respectively connected to the magnetic adjustment ring end cover (14) and the magnetic adjustment ring pressure cover (21), and the two ends of the screw (28) are axially limited, and the magnetic adjustment ring end cover (14) is fixed to the housing (18) through the pressure cover (9), thereby completing the installation of the magnetic adjustment structure; The pressure cover retaining ring (13) is embedded in the pressure cover (9), the input shaft front section (12) on the input shaft is installed on the pressure cover (9) through the input shaft front section bearing (11), the outer end cover (10) is connected to the pressure cover (9), the magnetic adjustment ring pressure cover bearing (22) is installed on the shaft sleeve (26), and is limited by the shaft sleeve retaining ring (20), the shaft sleeve (26) is installed on the magnetic adjustment ring pressure cover (21) through the magnetic adjustment ring pressure cover bearing (22), and the shaft sleeve (26) is connected to the input shaft end section (19) through the flat key (25), completing the input shaft and Assembly of the magnetic adjustment structure; the outer back iron (5) on the outer rotor is matched with the outer shell (18) through the outer back iron bearing (8), the shaft sleeve bearing (23) is installed on the shaft shoulder of the shaft sleeve (26) and matched with the collar (3), the sealing cover (24) is connected to the collar (3), the output shaft (27) is connected to the sealing cover (24), the outer shell cover (2) is connected to the outer shell (18), the collar bearing (1) is installed on the collar (3) and matched with the outer shell cover (2), and finally, the end cover (29) is connected to the outer shell cover (2) to complete the installation of the whole machine.

2. The magnetically solid magnetic reducer according to claim 1, characterized in that: The cross section of the inner permanent magnet (6) is a "T"-shaped structure, with "V"-shaped notches formed on both sides; The inner back iron (7) is a cylindrical structure, and its outer surface is provided with periodic dovetail grooves with chamfers, and the dovetail grooves present a "V"-shaped structure, and the "V"-shaped structure cooperates with the "V"-shaped notch of the inner permanent magnet (6); The cross section of the outer permanent magnet (16) is a "T"-shaped structure, with "V"-shaped notches on both sides; The outer back iron (5) is a toroidal structure, and its inner surface is provided with periodic dovetail grooves with chamfers, and the dovetail grooves present a "V"-shaped structure, and the "V"-shaped structure cooperates with the "V"-shaped notch of the outer permanent magnet (16); The magnetic adjustment ring (17) is composed of a plurality of magnetic adjustment pole blocks and magnetic resistance blocks alternately, both of which are in the shape of a fan ring, the magnetic adjustment pole blocks are made of high magnetic permeability materials, and the magnetic resistance blocks are made of low magnetic permeability materials. A small hole is provided in the center of each magnetic adjustment pole block, and the magnetic adjustment pole blocks are connected to form a whole through a connecting bridge near the center side. The front section of the input shaft (12) has a "C"-shaped keyway at one end, a thickened shoulder area in the middle, and a flange at the other end, with a stop and a through hole provided on the flange; The input shaft end section (19) has a keyway at one end and a flange at the other end, with a stopper and a through hole provided on the flange; The shaft sleeve (26) is a hollow stepped cylinder with a keyway provided on the inner wall and a shaft shoulder provided at each end.

3. The multi-objective optimization method for a magnetically solid magnetic reducer according to claim 1, characterized in that: Here are the steps: The first step is to determine the key design parameters of the magnetic solid magnetic reducer; Key design parameters include inner back iron thickness , outer back iron thickness , thickness of inner permanent magnet , thickness of outer permanent magnet , inner air gap thickness , External air gap thickness , adjust the thickness of the magnet pole , center hole diameter and axial length ; The second step is to conduct magnetic field analysis modeling, divide the sub-regions, and list the partial differential equations to solve the median diameter and tangential magnetic induction intensity in region IV; The partial differential equations for the regions in the magneto-solid magnetic reducer are listed to obtain the expression for the magnetic vector potential in region IV. The expression for the magnetic induction intensity in region IV is then derived based on the relationship between the magnetic vector potential and the magnetic induction intensity. Because the Fourier coefficients required to solve the magnetic induction intensity have a linear relationship between different regions, the equations for each region need to be solved sequentially. In the structure of the magneto-solid magnetic reducer, the air region between the inner permanent magnet (6) and the magnetic adjustment ring (17) is the inner air gap, and the air region between the outer permanent magnet (16) and the magnetic adjustment ring (17) is the outer air gap. Since the structure of the magneto-solid magnetic reducer is axially symmetrical, the magneto-solid magnetic reducer is divided into seven regions I to VII according to the boundaries of different material compositions. The inner permanent magnet (6) portion corresponds to region I, the inner air gap portion corresponds to region II, the magnetic adjustment ring (17) portion corresponds to region III, the outer air gap portion corresponds to region IV, the outer permanent magnet (16) portion corresponds to region V, the outer back iron (5) portion corresponds to region VI, and the inner back iron (7) portion corresponds to region VII, corresponding to seven circular rings respectively. The inner and outer radii of the circular ring in region I are respectively 、 The inner and outer radii of the ring in region II are 、 The inner and outer radii of the ring in region III are 、 , the inner and outer radii of the ring in region IV are 、 The inner and outer radii of the ring in region V are 、 The inner and outer radii of the ring in area VI are 、 According to Maxwell's stress tensor formula, the magnitude of the outer rotor torque is related to the magnetic induction intensity of the adjacent area VI, so the magnetic induction intensity of area VI is solved. The third step is to solve the linear equations of Fourier coefficients of magnetic vector potential; Step 4: Calculate the output torque, torque density, and permanent magnet volume; Step 5: Initialize the population; In the multi-objective optimization algorithm NSGA-II, let the number of independent variables be ; Set the number of chromosomes to ; Population step length of offspring Is the length of , the values ​​are A vector of ; randomly generated Individuals, the values ​​of the decision variables contained in the individuals are all within the upper and lower bounds; the fitness function is solved for the values ​​in each individual using the above algorithm, and the results are saved in the attributes of the individual; Step 6: Perform iterative optimization calculation of NSGA-II algorithm; After determining the calculation model and completing the initialization of the population, a set of optimal solutions, namely the Pareto frontier solution set, will be obtained through iterative calculation.

4. The multi-objective optimization method of the magnetically solid magnetic reducer according to claim 3, characterized in that: The specific implementation process of the second step is as follows: According to the magnetic vector potential and magnetic induction intensity In region IV, the following relationships exist: ， ， in, 、 Respectively represent the radial and tangential magnetic induction in region IV, is the polar diameter, is the polar angle, represents the magnetic vector potential in region IV; In solving 、 Before that, we need to solve the magnetic vector potential in region IV. , applying the partial differential equation to region IV: ， Combined with the boundary conditions, the separation of variables method is used to solve the magnetic vector potential in region IV: ， in, 、 、 、 for The Fourier coefficients of for The constant term of the Fourier expansion of : ， ， Where, is a positive integer, x and y are independent variables; According to formulas (1), (2), and (4), the radial and tangential magnetic induction intensities of region IV are obtained: 、 The expression: ， ， Where, The Fourier coefficients of 、 、 、 is an unknown number; if we want to get 、 The specific value of needs to be solved according to the boundary conditions of region IV, region III and region V to obtain the Fourier coefficient 、 、 、 ,have: ， ， ， ， Where, is the number of iron pole pairs of the magnetic ring, is the central angle of the magnetoresistive block, 、 is the magnetic vector potential of the jth magnetoresistive block in region III The Fourier coefficients of for The constant term of the Fourier expansion of 、 is the magnetic vector potential in region V The Fourier coefficients of Representation function The derivative of is the outer rotor rotation angle; ， ， ， ， Where, For the The azimuth angle corresponding to the magnetoresistive block is represents the remanence of the permanent magnet, is the number of pole pairs of the external permanent magnet; ， ， Where m, n, and j are all positive integers; ， because 、 、 、 The expressions (9) to (12) contain the Fourier coefficients of regions III and V. 、 、 、 , so it is necessary to solve regions III and V, list the partial differential equations and combine them with the boundary conditions to obtain the Fourier coefficients of region V: ， ， Similarly, solve for region III and obtain the magnetic vector potential of the j-th magnetoresistive block in region III: The Fourier coefficients of : ， ， Where, 、 、 、 is the magnetic vector potential in region II Fourier coefficients of ; From Equations (22) and (23), we can know that the magnetic vector potential of the j-th magnetoresistive block in region III is The Fourier coefficients of The expression contains the Fourier coefficients of region II 、 、 、 , so it is necessary to solve region II and obtain the magnetic vector potential of region II The Fourier coefficients of : ， ， ， ， Where, 、 for The Fourier coefficients of is the inner rotor rotation angle, Representation function The derivative of , and ， ， in, is the number of pole pairs of the inner permanent magnet; From formulas (24) to (27), we can see that The Fourier coefficients of 、 The expression contains the Fourier coefficients of region I 、 , it is necessary to solve the region I and obtain the magnetic vector potential of region I The Fourier coefficients of : ， ， So far, the radial and tangential magnetic induction intensities of region IV are obtained. 、 The expression and solution method of ; According to formulas (22) and (23), the Fourier coefficients of region III are completely represented by the Fourier coefficients of regions II and IV, and the Fourier coefficients are obtained. 、 、 、 、 、 、 、 、 、 、 、 A set of linear equations.

5. The multi-objective optimization method for a magnetically solid magnetic reducer according to claim 4, characterized in that: The specific implementation process of the third step is as follows: About Fourier coefficients 、 、 、 、 、 、 、 、 、 、 、 In a set of linear equations, since the Fourier coefficients are linearly related, to get the coefficients 、 、 、 , it is necessary to establish a linear equation system to solve; According to formulas (9) to (12), (20), (21), (24) to (27), the linear equations are established: ， in, is the Fourier coefficient vector to be solved, with a size of , is the number of truncated terms in the infinite series: ， Vector x consists of 12 The vector composition is: ， P is the coefficient matrix of vector x, and its size is : ， in, is an N-order unit matrix, For size The diagonal matrix whose value is the coefficient matrix of vector x; t is the constant vector in the linear equation system, and its size is : ， S1~S8 are the calculation items in formulas (9)~(12), (20), (21), (24)~(27) that are not related to n; Solving for the Fourier coefficient vector Then, take 、 、 、 , obtained from formulas (7) and (8) 、 The value of .

6. The multi-objective optimization method of the magnetically solid magnetic reducer according to claim 5, characterized in that: The specific implementation process of the fourth step is as follows: Solve for the outer rotor torque based on the Maxwell stress tensor: ， in, is the axial length of the magnetic solid magnetic reducer, is the vacuum permeability; ， Take the outer rotor torque The maximum value among them is taken as the output torque , the calculation formula of torque density Des: ， Permanent magnet volume V m Calculation formula: ， Pick 、 、 As the fitness function of the multi-objective optimization algorithm NSGA-II, ~ is the decision variable, and the function expression is: ， in, ; 、 are the upper and lower bounds of the decision variables respectively; is the constraint violation function.

7. The multi-objective optimization method of the magnetically solid magnetic reducer according to claim 6, characterized in that: The specific implementation process of the sixth step is as follows: 1) Perform fast non-dominated sorting on the population; Set the constraint violation function: ， in, represents the inequality constraint of the e-th individual in formula (25); 、 Represent the upper and lower bounds of the constraints respectively; When sorting, classify and discuss based on the constraint violation, set individuals with smaller constraint violation values ​​as high priority, and if individual inv1 dominates another individual inv2, then add inv2 to the domination set of inv1. The number of individuals that simultaneously dominate inv2 On the contrary, if individual inv2 dominates individual inv1, then inv1 is added to the dominance set of inv2 The number of individuals that simultaneously dominate inv1 ; In the traversal process, if the number of individuals e is dominated =0, then put it into the first level of the Pareto frontier set middle; Let ne=1, set Q is an empty set, traverse the Pareto frontier of the neth layer, visit all points inv3 dominated by the individual, and the number of individuals dominating inv3 -1, if at this time =0, then , that is, Q is placed in the ne+1th layer Pareto frontier; after the neth layer Pareto frontier traversal is completed, if the set Q is an empty set, it means that the Pareto frontier has been stratified, and we get 、 、 ...Exit the loop at this time, otherwise , ne is updated to ne+1, and the individuals on the next layer of Pareto frontier are traversed; 2) Calculate the crowding degree of individuals in the population; The congestion calculation formula is as follows: ， in, represents the crowding distance of the e-th point, 、 Represents the fitness function value of the two points before and after the individual e, 、 Represents the maximum and minimum values ​​of the fitness function of the Pareto frontier solution set of the neth layer; 3) Forming a new generation of population; Prioritize individuals with high Pareto front rank and large crowding distance to perform crossover and mutation operations on decision variables, calculate the fitness function of the newly obtained offspring population, merge it with the parent population, perform non-dominated sorting, and adopt an elite selection strategy to prioritize individuals with high Pareto front rank and large crowding distance as the new generation population; Repeat process 1) to 3) until the maximum number of iterations is reached, and finally a set of optimal Pareto frontier solutions is obtained as the multi-objective optimization result; At this point, the multi-objective optimization of the field-solid magnetic reducer has been completed.

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