A Design and Calculation Method for Low Torsional Stiffness of Permanent Magnet Couplings

Abstract

The present invention discloses a design and calculation method for low torsional stiffness of a permanent magnet coupling. Through the design of a low torsional stiffness structure, this method can effectively reduce the transmission of vibration from the driving end to the load end, improve the service life of the shafting components and the reliability of transmission, and has high practical application value. In addition, according to the characteristics of the magnetic circuit formation of the tubular permanent magnet coupling, an equivalent magnetic circuit of the tubular permanent magnet coupling is constructed based on the equivalent magnetic circuit method. By solving the magnetic circuit, the torque and stiffness analytical formulas of the tubular permanent magnet coupling are derived. Without the need for time-consuming and complex finite element calculations, the torque and torsional stiffness of the tubular permanent magnet coupling in the starting stage can be quantitatively analyzed, which can provide important technical support for the initial design, torsional stiffness prediction, and vibration reduction and isolation effect evaluation of the tubular permanent magnet coupling. It has the advantages of simple operation, strong adaptability, and short calculation time in practical engineering applications, and is a calculation method with engineering practical application value.

Description

Technical Field

[0001] The present invention belongs to the technical field of permanent magnet flexible vibration isolation and drive, and relates to a structural design of a low dynamic torsional stiffness of a cylindrical permanent magnet coupling and an analytical calculation method for torsional stiffness. Background Art

[0002] Torsional vibration is a special vibration form of the rotating machinery shafting, which widely exists in transmission systems such as automobiles, ships, aerospace, and wind power generation. It is one of the important reasons for the generation of noise, fatigue, and damage of shafting components. As a common type of part in the transmission system, the coupling plays an important role in controlling the vibration transmission between the driving end and the driven end. However, the traditional coupling belongs to the contact type transmission, and inevitably has problems such as wear, fatigue damage, and direct vibration transmission. In recent years, with the development of permanent magnet materials, permanent magnet drive devices represented by cylindrical permanent magnet couplings have developed rapidly. With the non-contact transmission characteristics, it can effectively reduce the vibration transmission from the driving end to the load end. However, the magnetic force generated by the permanent magnet has strong non-linearity, and a small offset between the two magnets will cause a large change in the acting force. The traditional cylindrical permanent magnet coupling with a full distribution of magnetic steel configuration has a large torsional stiffness at the starting stage of the shafting, which will lead to insufficient absorption capacity for the torsional vibration and impact load of the shafting, exacerbate the vibration transmission from the driving end to the load end, and further accelerate the fatigue and damage of the shafting components, affecting the life and reliability of the transmission shafting. Therefore, the problem of large torsional stiffness under the load starting condition is one of the important reasons why the vibration isolation effect and buffer vibration absorption capacity of the traditional configuration cylindrical permanent magnet coupling cannot meet the design expectations. In addition, the current research on the transmission characteristics of cylindrical permanent magnet couplings mainly relies on finite element simulation analysis, which takes a long time in the calculation process, there are cases where the returned solution does not converge, and the calculation accuracy is greatly affected by factors such as mesh division, boundary conditions, and simulation environment settings, making it difficult to iterate the structural parameters in the design stage of cylindrical permanent magnet couplings, and the calculation time cost is relatively high. Therefore, it is of great significance to propose a structural design of a low dynamic torsional stiffness under the load starting condition of a cylindrical permanent magnet coupling and a rapid analytical prediction method for torsional stiffness to guide the initial design.

[0003] Regarding the research on the transmission characteristics of the tubular permanent magnet coupling and the problem of structural optimization design, Li Guangjun, Guo Heyi, Xu Lei and others, in the article "Magnetic Field Simulation Analysis and Structural Optimization Design of Cylindrical Permanent Magnet Couplings", fully considered the actual operating conditions for the long-term safe and reliable operation of the cylindrical permanent magnet couplings used in stirring devices, established a three-dimensional magnetic field simulation model of the cylindrical permanent magnet coupling, analyzed the influence laws of various structural parameters on the magnetic torque, and carried out structural optimization design for the problem of permanent magnet demagnetization failure. However, the analysis process relied on finite element analysis and did not give a specific analytical calculation method. Moreover, the object of structural optimization only focused on reducing the influence of temperature rise on the safe and reliable operation of the tubular permanent magnet coupling, and did not conduct structural optimization on the torsional stiffness of the tubular permanent magnet coupling. Regarding the problem of torsional stiffness optimization design of the tubular permanent magnet coupling, Zheng Yisheng, Lu Guanglin and Zhang Xinandeng and others, in the patent "A Quasi-Zero Stiffness Coupling" (CN201610032023.X), proposed a quasi-zero stiffness coupling composed of rubber and permanent magnets. By selecting the geometric parameters of the permanent magnets of the inner and outer magnetic rings, the negative stiffness generated by the magnetic spring is close to the positive stiffness of the rubber bearing, so that the total stiffness reaches the quasi-zero state. This method can theoretically effectively reduce the torsional stiffness of the tubular permanent magnet coupling under the load start condition. However, the combination of rubber and permanent magnets increases the structural complexity, and the design and selection of positive and negative stiffness are more difficult, and no specific selection method is given in the article.

[0004] Therefore, it is of great significance to propose a low dynamic torsional stiffness structural design and torsional stiffness analytical calculation method for the tubular permanent magnet coupling for the design and research of the tubular permanent magnet coupling. Summary of the Invention

[0005] In order to make up for the defects of the existing technology, the present invention invented a low dynamic torsional stiffness structural design and torsional stiffness analytical calculation method for the tubular permanent magnet coupling. Its purpose is to optimize the structure of the existing tubular permanent magnet coupling to achieve the purpose of reducing the torsional stiffness in the starting stage and enhancing the ability of the tubular permanent magnet coupling to absorb torsional vibration. Combined with the torque analytical calculation model established based on the equivalent magnetic circuit method, the torsional stiffness in the starting condition stage is calculated and evaluated, providing important technical support for the initial structural design, torsional stiffness calculation, buffer vibration absorption, and vibration isolation and shock reduction effect evaluation of the tubular permanent magnet coupling. This method has good practicability, operability and convenience in practical engineering applications.

[0006] The technical solution of the present invention:

[0007] A low torsional stiffness design and calculation method for a tubular permanent magnet coupling. The structural design method is characterized by using the area difference between the surfaces of the outer rotor permanent magnet 2 and the inner rotor permanent magnet 3 that participate in magnetic field coupling. When the inner rotor permanent magnet 3 moves within a small deflection angle range, the excess magnetic flux from the outer rotor permanent magnet 2 forms an auxiliary magnetic circuit through the convergence effect of the high magnetic permeability partition 4. Therefore, the magnetic flux of the main magnetic circuit passing through the inner rotor permanent magnet 3 remains almost unchanged. According to Lenz's law, the torque acting on the inner rotor permanent magnet 2 remains almost unchanged under a small deflection angle, thus realizing the "flexible" design of the tubular permanent magnet coupling during the starting stage and achieving the purpose of reducing torsional stiffness. The analytical calculation method for torsional stiffness is characterized in that, first, an equivalent magnetic circuit of any adjacent two pairs of magnetic poles is constructed based on the equivalent magnetic circuit method; second, according to the series-parallel relationship of the magnetic circuit, the magnetic resistance and magnetic flux at each place are calculated; then, based on the electromagnetic field theory, the total magnetic energy at the air gap between adjacent two pairs of magnetic poles is calculated, and combined with the principle of virtual work, the torque analytical formula of the tubular permanent magnet coupling is obtained, and the torsional stiffness analytical formula is obtained after derivation; finally, the numerical value of the deflection angle is substituted to quantitatively evaluate the torque and torsional stiffness of the tubular permanent magnet coupling under the starting condition. The specific steps of this calculation method are as follows:

[0008] Step 1: Determine the key structural parameters of the tubular permanent magnet coupling

[0009] Assume that the thicknesses of the outer rotor back iron 1 and the inner rotor back iron 5 are sufficient and the leakage magnetic flux is negligible; first, determine the key structural parameters of the tubular permanent magnet coupling; among them, the thicknesses of both the outer rotor permanent magnet 2 and the inner rotor permanent magnet 3 are h1, the lengths are both l, and the coercive forces are both H c , the number of pole pairs is p; the outer radius of the outer rotor permanent magnet 2 is R1, the outer radius of the inner rotor permanent magnet 3 is R2, the thicknesses of both the outer rotor back iron 1 and the inner rotor back iron 5 are h2, the central angle of the high magnetic permeability partition 4 is α0, and the air gap thickness between the inner and outer rotors is a g , and the outer rotor permanent magnet 2 and the inner rotor permanent magnet 3 are evenly distributed along the circumference, with the N poles and S poles evenly alternating. During the load-starting stage, the relative deflection angle between the inner and outer rotors along the axis is θ;

[0010] Step 2: Construct an equivalent magnetic circuit of any adjacent two pairs of magnetic poles based on the equivalent magnetic circuit method

[0011] According to the structural symmetry, select any adjacent two pairs of magnetic poles to construct the equivalent magnetic circuit of the tubular permanent magnet coupling and make the following assumptions: ① The magnetic circuit is divided into several segments, each segment having the same cross-sectional area and the same magnetic medium; ② The magnetic field strength in each segment of the magnetic circuit is the same everywhere, and the direction is consistent with the magnetic circuit path; ③ On each cross-section of the magnetic circuit, the magnetic flux is evenly distributed; ④ Neglect the leakage magnetic flux at the air gap and other places, assume the magnetic circuit is an ideal magnetic circuit, do not consider saturation, and set the magnetic permeability of the ferromagnetic material as a constant value;

[0012] Step 3: Calculate the magnetic resistance and magnetic flux at each place in the magnetic circuit according to the series-parallel relationship of each magnetic resistance

[0013] First, calculate the reluctance and magnetic flux of the outer rotor permanent magnet 2. The formula is as follows:

[0014]

[0015] In the formula, R pm1 is the reluctance of the outer rotor permanent magnet, H -1 ; μ0 is the permeability of free space, and μ0 = 4π×10 -7 H / m; μ pm is the relative permeability of the permanent magnet, and μ pm = 1.05; Φ pm1 is the magnetic flux of the outer rotor permanent magnet, Wb;

[0016] Secondly, calculate the reluctance and magnetic flux of the inner rotor permanent magnet 3. The formula is as follows:

[0017]

[0018] In the formula, R pm2 is the reluctance of the inner rotor permanent magnet, H -1 ; Φ pm2 is the magnetic flux of the inner rotor permanent magnet, Wb;

[0019] Then, calculate the reluctance of the high-permeability separator 4. The formula is as follows:

[0020]

[0021] In the formula, R y is the reluctance of the high-permeability separator, H -1 ; μ iron is the relative permeability of the separator, and μ iron = 4000;

[0022] Finally, calculate the reluctance and magnetic flux at the air gap. The formula is as follows:

[0023]

[0024] In the formula, A(θ) is the equivalent magnetic flux area of the air gap between the inner and outer rotors, m 2 ; R g is the air gap reluctance, H -1 ; μ air is the relative permeability of air, and μ air = 1; R / / is the parallel reluctance of R y and R pm2 , H -1 ; Φ g is the magnetic flux at the air gap between adjacent pairs of magnetic poles, Wb; K fK is the back ferromagnetic resistance effect coefficient, which is generally 1 to 1.2, and is taken as 1.1 here; I is the magnetic flux leakage coefficient, without considering the influence of magnetic flux leakage, and is taken as 1 here;

[0025] Step 4: Based on the electromagnetic field theory, calculate the total magnetic energy at the air gap between two adjacent pairs of magnetic poles, and combine it with the principle of virtual work to obtain the torque analytical formula of the cylindrical permanent magnetic coupling after correction. After differentiation, obtain the torsional stiffness analytical formula

[0026] First, the total magnetic energy at the air gap between two adjacent pairs of magnetic poles is calculated according to the electromagnetic field theory. The formula is as follows:

[0027]

[0028] Where W g is the total magnetic energy in the air gap between two adjacent pairs of magnetic poles, J; Λ g is the magnetic permeability at the air gap between two adjacent pairs of magnetic poles, H;

[0029] Then, combined with the principle of virtual work, the torque of the cylindrical permanent magnet coupling is calculated as follows:

[0030]

[0031] Where, T(θ) is the torque of the cylindrical permanent magnet coupling, N·m; dW g , dθ are W g , the differential corresponding to θ;

[0032] Finally, the torque analytical formula of the cylindrical permanent magnet coupling is derived to calculate the torsional stiffness. The formula is as follows:

[0033]

[0034] Where K(θ) is the torsional stiffness of the cylindrical permanent magnetic coupling, N·m / deg; dT(θ) is the differential of T(θ);

[0035] Step 5: Substitute the deflection angle value to calculate the torque and torsional stiffness of the cylindrical permanent magnet coupling.

[0036] Substituting the relative deflection angle θ of the inner and outer rotors along the axis during the load-starting stage of the cylindrical permanent magnet coupling into the analytical formulas of T(θ) and K(θ), the torque T and torsional stiffness K of the cylindrical permanent magnet coupling under the starting condition can be obtained. The calculation is now complete.

[0037] The beneficial effects of the present invention are as follows: This calculation method does not require time-consuming and complex finite element calculations. Instead, based on the characteristics of the magnetic circuit formation of the tubular permanent magnet coupling, an equivalent magnetic circuit is constructed, and the torque and torsional stiffness are analytically calculated according to the principle of virtual work, thereby realizing the quantitative analysis of the torsional stiffness of the tubular permanent magnet coupling during the starting stage and the effective evaluation of the vibration reduction and isolation effect. It has the advantages of simple operation and short calculation time in practical engineering applications. In addition, through the structural design of the tubular permanent magnet coupling with low dynamic torsional stiffness, its ability to absorb torsional vibration can be enhanced, the transmission of vibration from the driving end to the load end can be effectively reduced, the service life of the shafting components and the transmission reliability can be improved, and it has high practical application value. Description of the Drawings

[0038] Figure 1 is a schematic structural diagram of a low torsional stiffness tubular permanent magnet coupling. In the figure: 1 - outer rotor back iron, 2 - outer rotor permanent magnet, 3 - inner rotor permanent magnet, 4 - high magnetic permeability partition, 5 - inner rotor back iron; h1 - permanent magnet thickness, h2 - back iron thickness, R1 - outer diameter of the outer rotor permanent magnet, R2 - outer diameter of the inner rotor permanent magnet, a g - air gap thickness, p - number of pole pairs, θ - relative deflection angle of the inner and outer rotors along the axis, α0 - central angle of the high magnetic permeability partition; main magnetic circuit - magnetic flux loop formed by the permanent magnet, auxiliary magnetic circuit - magnetic flux loop formed by the permanent magnet and the high magnetic permeability partition;

[0039] Figure 2 is a flow chart of the stiffness analytical calculation method for a low torsional stiffness tubular permanent magnet coupling;

[0040] Figure 3 is an equivalent magnetic circuit of any two adjacent pairs of poles of a low torsional stiffness tubular permanent magnet coupling, where (a) is the equivalent magnetic circuit formed by two adjacent pairs of poles, and (b) is the simplified magnetic circuit; in the figure: R pm1 is the magnetic resistance of the outer rotor permanent magnet, Φ pm1 is the magnetic flux of the outer rotor permanent magnet, R pm2 is the magnetic resistance of the inner rotor permanent magnet, Φ pm2 is the magnetic flux of the inner rotor permanent magnet, R i1 is the magnetic resistance of the outer rotor back iron, R i2 is the magnetic resistance of the inner rotor back iron, R y is the magnetic resistance of the high magnetic permeability partition, R g is the magnetic resistance of the air gap, Φ g is the magnetic flux at the air gap between two adjacent pairs of poles, R / / is R y and R pm2 is the parallel magnetic resistance of, K f is the back iron magnetic resistance effect coefficient. Detailed Embodiments

[0041] The following further elaborates on the implementation cases of the present invention in combination with the attached drawings and technical solutions.

[0042] In this implementation case, a cylindrical permanent magnet coupling with a rated power of 1.26 kW and 18 pairs of pole pairs is selected for the torsional stiffness analysis and calculation during the starting condition stage.

[0043] Figure 1 It is a structural schematic diagram of a low torsional stiffness cylindrical permanent magnet coupling. Among them, the thickness h1 of the outer rotor permanent magnet 2 and the inner rotor permanent magnet 3 of this cylindrical permanent magnet coupling is 8 mm, the length l is 50 mm, and the coercive force is H c = 907000 A / m, the number of pole pairs p = 18 pairs, the outer radius R1 of the outer rotor permanent magnet 2 is 52 mm, the outer radius R2 of the inner rotor permanent magnet 3 is 41 mm, the thickness h2 of the outer rotor back iron 1 and the inner rotor back iron 5 is 8 mm, the central angle α0 of the high magnetic permeability separator 4 is 4 deg, and the air gap thickness a g = 3 mm, and the relative deflection angle θ between the inner and outer rotors along the axis during the load-starting stage is 1 deg.

[0044] Figure 2 It is a flow chart of the stiffness analysis and calculation method for a low torsional stiffness cylindrical permanent magnet coupling. The specific calculation steps are as follows:

[0045] The first step is to determine the key structural parameters of the cylindrical permanent magnet coupling

[0046] First, determine the key structural parameters of the cylindrical permanent magnet coupling. Among them, the thickness h1 of the outer rotor permanent magnet 2 and the inner rotor permanent magnet 3 of this cylindrical permanent magnet coupling is 8 mm, the length l is 50 mm, and the coercive force is H c = 907000 A / m, the number of pole pairs p = 18 pairs, the outer radius R1 of the outer rotor permanent magnet 2 is 52 mm, the outer radius R2 of the inner rotor permanent magnet 3 is 41 mm, the thickness h2 of the outer rotor back iron 1 and the inner rotor back iron 5 is 8 mm, the central angle α0 of the high magnetic permeability separator 4 is 4 deg, and the air gap thickness a g = 3 mm, and the relative deflection angle θ between the inner and outer rotors along the axis during the load-starting stage is 1 deg;

[0047] The second step is to construct the equivalent magnetic circuit of any two adjacent pairs of magnetic poles based on the equivalent magnetic circuit method

[0048] Figure 3 It is an equivalent magnetic circuit of any two adjacent pairs of magnetic poles of a low torsional stiffness cylindrical permanent magnet coupling. Among them, (a) is the equivalent magnetic circuit formed by two adjacent pairs of magnetic poles, and (b) is the simplified magnetic circuit;

[0049] The third step is to calculate the magnetic resistance and magnetic flux at each place in the magnetic circuit according to the series-parallel relationship of each magnetic resistance

[0050] Substituting the known key parameters into equations (1) to (9), the magnetic reluctance R of the outer rotor permanent magnet can be obtained. pm1 = 7.254×10 6 H -1 、The magnetic flux Φ of the outer rotor permanent magnet pm1 = 1.029×10 -3 Wb, the magnetic reluctance R of the inner rotor permanent magnet pm2 = 1.178×10 7 H -1 、The magnetic flux Φ of the inner rotor permanent magnet pm2 = 6.335×10 -4 Wb, the magnetic reluctance R of the high-permeability separator y = 9.521×10 3 H -1 、R y And R pm2 The parallel magnetic reluctance R of / / = 9.513×10 3 H -1 、The analytical formula for the equivalent magnetic flux area A(θ) of the air gap between the inner and outer rotors is:

[0051]

[0052] The analytical formula for the air gap magnetic reluctance R g is:

[0053]

[0054] The magnetic flux Φ g at the air gap between adjacent pairs of magnetic poles is given by the analytical formula:

[0055]

[0056] Step 4: According to the electromagnetic field theory, calculate the total magnetic energy at the air gap between adjacent pairs of magnetic poles, and combined with the principle of virtual work, the torque analytical formula of the tubular permanent magnet coupling is corrected, and the torsional stiffness analytical formula is obtained after differentiation

[0057] First, substituting Φ g and R g into equation (10), the analytical formula for the total magnetic energy W g at the air gap between adjacent pairs of magnetic poles is:

[0058]

[0059] For the convenience of subsequent differentiation, the analytical formula of W g is polynomially fitted in the range of 0 to 2 degrees in Matlab, and the fitted analytical formula of W g is:

[0060] W g=-2.073×10 -5 θ 4 -2.914×10 -5 θ 3 +0.001764θ 2 -7.076×10 -6 θ+0.06302;

[0061] Then, combined with the principle of virtual work, the torque analytical formula of the cylindrical permanent magnet coupling is calculated by formula (11):

[0062]

[0063] Finally, the torque analytical formula of the cylindrical permanent magnet coupling is derived, and the torsional stiffness analytical formula is obtained as follows:

[0064]

[0065] Step 5: Substitute the deflection angle value to calculate the torque and torsional stiffness of the cylindrical permanent magnet coupling.

[0066] Substituting the relative deflection angle θ=1deg of the inner and outer rotors along the axis during the load starting stage of the cylindrical permanent magnet coupling into the analytical expressions of T(θ) and K(θ), we can obtain the torque T=3.46N·m and the torsional stiffness K=3.20N·m / deg during the starting stage of the cylindrical permanent magnet coupling. The calculation is now completed.

[0067] This method fully considers the problem of insufficient torsional vibration absorption capacity of the cylindrical permanent magnet coupling due to the large dynamic torsional stiffness under load starting conditions in actual applications. Through the design of low torsional stiffness structure, the transmission of vibration from the drive end to the load end can be effectively reduced, the service life of the shaft system components and the reliability of the transmission can be improved, and it has high practical application value. In addition, according to the magnetic circuit formation characteristics of the cylindrical permanent magnet coupling, the equivalent magnetic circuit of the cylindrical permanent magnet coupling is constructed based on the equivalent magnetic circuit method. The torque and stiffness analytical expressions of the cylindrical permanent magnet coupling are derived by solving the magnetic circuit. Without the need for time-consuming and complex finite element calculations, the torque and torsional stiffness of the cylindrical permanent magnet coupling in the starting stage can be quantitatively analyzed. It can provide important technical support for the initial design of the cylindrical permanent magnet coupling, torsional stiffness prediction and vibration reduction and isolation effect evaluation. In actual engineering applications, it has the advantages of simple operation, strong adaptability and short calculation time. It is a calculation method with practical engineering application value.

Claims

1. A design and calculation method for a permanent magnet coupling with low torsional stiffness, characterized in that, The steps are as follows: First step: Determine the key structural parameters of the tubular permanent magnet coupling: Assume that the thickness of the outer rotor back iron (1) and the inner rotor back iron (5) is sufficient and the leakage magnetic flux is negligible; determine the key structural parameters of the tubular permanent magnet coupling; among them, the thickness of the outer rotor permanent magnet (2) and the inner rotor permanent magnet (3) is h1, the length is l, and the coercive force is H c , the number of pole pairs is p; the outer radius of the outer rotor permanent magnet (2) is R1, the outer radius of the inner rotor permanent magnet (3) is R2, the thickness of the outer rotor back iron (1) and the inner rotor back iron (5) is h2, the central angle of the high magnetic permeability separator (4) is α0, and the air gap thickness between the inner and outer rotors is a g , and the outer rotor permanent magnet (2) and the inner rotor permanent magnet (3) are evenly distributed along the circumference, the N poles and S poles are evenly and alternately arranged, and the relative deflection angle of the inner and outer rotors along the axis is θ during the load-starting stage; Second step: Construct the equivalent magnetic circuit of any two adjacent pairs of magnetic poles based on the equivalent magnetic circuit method: According to the structural symmetry, select any two adjacent pairs of magnetic poles to construct the equivalent magnetic circuit of the tubular permanent magnet coupling, and make the following assumptions: ① The magnetic circuit is divided into several segments, each segment has the same cross-sectional area and the same magnetic medium; ② The magnetic field intensity in each segment of the magnetic circuit is the same everywhere, and the direction is consistent with the magnetic circuit path; ③ The magnetic flux is evenly distributed on each cross-section of the magnetic circuit; ④ Ignore the magnetic leakage at the air gap, etc., assume that the magnetic circuit is an ideal magnetic circuit, do not consider saturation, and set the magnetic permeability of the ferromagnetic material as a constant value; Third step: Calculate the magnetic resistance and magnetic flux at each part of the magnetic circuit according to the series-parallel relationship of each magnetic resistance: First, calculate the magnetic resistance and magnetic flux of the outer rotor permanent magnet (2), and the formula is as follows: wherein, R pm1 is the reluctance of the outer rotor permanent magnet, H -1 ; μ0 is the permeability of free space, and μ0 = 4π × 10 -7 H / m; μ pm is the relative permeability of the permanent magnet, and μ pm = 1.05; Φ pm1 is the magnetic flux of the outer rotor permanent magnet, Wb; Secondly, calculate the magnetic resistance and magnetic flux of the inner rotor permanent magnet (3), and the formula is as follows: where, R pm2 is the reluctance of the inner rotor permanent magnet, H -1 ; Φ pm2 is the magnetic flux of the inner rotor permanent magnet, Wb; Then, calculate the magnetic resistance of the high magnetic permeability separator (4), and the formula is as follows: where, R y is the reluctance of the high-permeability partition, H -1 ; μ iron is the relative permeability of the partition, and μ iron = 4000; Finally, calculate the magnetic resistance and magnetic flux at the air gap, and the formula is as follows: where \(A(\theta)\) is the equivalent magnetic flux area of the air gap between the inner and outer rotors, \(m\) 2 ; \(R\) g is the air gap reluctance, \(H\) -1 ; \(\mu\) air is the relative magnetic permeability of air, and \(\mu\) air = 1; \(R\) / / is the parallel reluctance of \(R\) y and \(R\) pm2 , \(H\) -1 ; \(\varPhi\) g is the magnetic flux at the air gap between adjacent pairs of magnetic poles, \(Wb\); \(K\) f is the back iron reluctance effect coefficient, generally taken as 1 - 1.2, here taken as 1.1; \(K\) I is the leakage magnetic coefficient, without considering the influence of leakage magnetic, here taken as 1; Fourth step: According to the electromagnetic field theory, calculate the total magnetic energy at the air gap between two adjacent pairs of magnetic poles, and combine the principle of virtual work to obtain the torque analytical formula of the tubular permanent magnet coupling after correction, and obtain the torsional stiffness analytical formula after derivation. First, calculate the total magnetic energy at the air gap between two adjacent pairs of magnetic poles according to the electromagnetic field theory, and the formula is as follows: Where W g is the total magnetic energy at the air gaps between two adjacent pairs of magnetic poles, in J; Λ g is the magnetic conductance at the air gaps between two adjacent pairs of magnetic poles, in H; Then, combine the principle of virtual work to calculate the torque of the tubular permanent magnet coupling, and the formula is as follows: where \(T(\theta)\) is the torque of the tubular permanent magnet coupling, in N·m; \(dW\) g , \(d\theta\) are the differentials corresponding to \(W\) g , \(\theta\) respectively; finally, the derivative of the torque analytical formula of the tubular permanent magnet coupling is taken to calculate the torsional stiffness, and the formula is as follows: In the formula, K(θ) is the torsional stiffness of the tubular permanent magnet coupling, N·m / deg; dT(θ) is the differential of T(θ); Fifth step: Substitute the value of the deflection angle to calculate the torque and torsional stiffness of the tubular permanent magnet coupling: Substitute the relative deflection angle θ of the inner and outer rotors along the axis during the load startup stage of the tubular permanent magnet coupling into the analytical formulas of T(θ) and K(θ) respectively, and obtain the torque T and torsional stiffness K of the tubular permanent magnet coupling under the startup working condition stage. So far, the calculation is completed.

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