A design method for a harmonic tooth profile and a harmonic reducer based on the brachistochrone curve

Abstract

The present invention relates to a design method of a harmonic tooth profile and a harmonic reducer based on the brachistochrone curve. The transmission efficiency of the existing harmonic tooth profile is about 60-70% according to engineering measurements. The design method of the harmonic tooth profile provided by the present invention is to apply the brachistochrone curve M after polar coordinate transformation to the tooth profile of the harmonic gear, and realize the tooth profile of the harmonic gear with the required transmission ratio by adjusting the parameters of the brachistochrone curve M. The harmonic tooth profile design method of the present invention is based on the brachistochrone curve equation, is simple to design, has a short processing period at the same time, and the tooth profile based on the brachistochrone curve has a short transmission time under unit displacement and high transmission efficiency. It is measured that the transmission efficiency is increased by at least more than 5% compared with the traditional tooth profile, which can greatly reduce the power of the driving motor and save electric energy.

Description

Technical Field

[0001] The present invention belongs to the technical field of the design and manufacture of harmonic reducers, and relates to a design method of a harmonic tooth profile and a harmonic reducer based on the brachistochrone curve. Background Art

[0002] A harmonic reducer mainly consists of a wave generator, a flexible gear, and a rigid gear. The wave generator drives the flexible gear to undergo elliptical deformation, forcing the flexible gear and the rigid gear to generate misaligned tooth movement, thereby realizing the motion transmission between the active wave generator and the flexible gear and achieving the purpose of speed reduction. The harmonic reducer has the characteristics of high precision, large transmission ratio, compact structure, and light weight.

[0003] In order to improve the transmission efficiency of harmonic reducers, domestic and foreign harmonic reducer manufacturing companies have all carried out profile modification on the gear tooth profiles. Currently, there are two main tooth profiles. One is the traditional short-tooth involute tooth profile, and standard tools are used for gear profile modification during processing; the other is the double circular arc tooth profile, and special tools are used for processing. Since the meshing angle of the first tooth profile is a constant pressure angle, when the input speed is relatively high, it is easy to reach the resonance point. The second tooth profile is processed with a variable pressure angle through mathematical simulation calculation, and the curve is relatively complex, with high requirements for processing accuracy. At the same time, the transmission efficiency of the above two tooth profiles is approximately between 60% and 70% according to engineering measurements. Summary of the Invention

[0004] Aiming at the above problems, the purpose of the present invention is to provide a design method of a harmonic tooth profile and a harmonic reducer based on the brachistochrone curve. According to the harmonic drive principle and combined with the characteristic of the shortest sliding time of the brachistochrone curve, the brachistochrone curve is applied to the design of the harmonic tooth profile after polar coordinate transformation, and the transmission efficiency of the harmonic reducer can be increased by at least more than 5%.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] A design method of a harmonic tooth profile based on the brachistochrone curve, characterized in that: the brachistochrone curve M is applied to the tooth profile of the harmonic gear after polar coordinate transformation, and the formula of the brachistochrone curve M is:

[0007] X = K(θ - sin(θ)), Y = K(1 - cos(θ)); 0 ≤ θ ≤ pi; (1)

[0008] The tooth profile of the harmonic gear with the required transmission ratio is realized by adjusting the parameters K and θ in formula (1), where K is the total tooth height and θ is the curve phase.

[0009] Furthermore, the polar coordinate conversion of the brachistochrone curve M is carried out on the circle of the neutral layer of the flexible gear or the pitch circle of the rigid gear of the harmonic gear.

[0010] A design method of a harmonic reducer based on the brachistochrone curve, characterized in that it includes the following steps:

[0011] Step S1: Determine the full tooth height K according to the model and reduction ratio of the harmonic reducer;

[0012] Step S2: Convert the brachistochrone curve M into a curve in the polar coordinate system, and draw the flexspline diagram according to the curve in the polar coordinate system;

[0013] Step S3: Draw the rigid spline diagram according to the curve in the polar coordinate system;

[0014] Step S4: Draw the static simulation diagram of the rigid spline and the flexspline to determine the transmission clearance.

[0015] Furthermore, the step S2 includes:

[0016] Step S201: Substitute the K value into the brachistochrone curve M formula to obtain the brachistochrone curve M1 formula of the flexspline;

[0017] Step S202: Divide the curve phase θ equally, and determine the equally divided coordinate points through the brachistochrone curve M1 formula. The coordinate points form the brachistochrone curve M1;

[0018] Step S203: Perform polar coordinate conversion on the brachistochrone curve M1 to obtain the transformed graphic curve M2 of the flexspline on the circle of the neutral layer of the flexspline. The formula of the graphic curve M2 is:

[0019] X1 = R * COS(S), Y1 = R * SIN(S); (3)

[0020] In formula (3), R is the length of the flexspline in polar coordinates, and S is the subtended angle of the flexspline in polar coordinates;

[0021] Obtain the coordinate points of the transformed flexspline tooth profile through the graphic curve M2;

[0022] Step S204: Divide the graphic curve M2 equally according to the coordinate points according to the reduction ratio, and the top rounded corners of the equally divided curves are connected by transitions to form the flexspline tooth profile curve;

[0023] Step S205: Machine the flexspline gear according to the flexspline tooth profile curve.

[0024] Furthermore, in the step S203, divide the X coordinate in the brachistochrone curve M1 by the radius of the circle of the neutral layer to obtain the subtended angle S of the flexspline in polar coordinates, and add the radius of the circle of the neutral layer to the Y coordinate in the brachistochrone curve M1 to obtain the length R of the flexspline in polar coordinates.

[0025] Furthermore, the step S3 includes:

[0026] Step S301: Perform polar coordinate transformation on the brachistochrone curve M1 to obtain the transformed graphic curve M3 of the rigid gear on the pitch circle of the rigid gear. The formula of the graphic curve M3 is as follows:

[0027] X2 = R2 * COS(S2), Y1 = R2 * SIN(S2); (4)

[0028] In the formula, R2 is the length in the polar coordinates of the rigid gear, and S2 is the subtended angle in the polar coordinates of the rigid gear.

[0029] Step S302: Obtain the coordinate points of the transformed tooth profile of the rigid gear through the graphic curve M3.

[0030] Step S303: Divide the graphic curve M3 according to the coordinate points in accordance with the reduction ratio, and the top rounded corners of the divided curves are connected by transition to form the tooth profile curve of the rigid gear; machine the rigid gear according to the tooth profile curve of the rigid gear.

[0031] Furthermore, in step S301, divide the X coordinate in the brachistochrone curve M1 by the radius of the pitch circle to obtain the subtended angle S2 in the polar coordinates of the rigid gear, and add the radius of the pitch circle to the Y coordinate in the brachistochrone curve M1 to obtain the length R2 in the polar coordinates of the rigid gear.

[0032] Furthermore, step S4 includes:

[0033] Step S401: Calculate the wall thickness of the flexspline. According to the maximum design torque, the wall thickness parameter of the flexspline uses 0.01 - 0.0125 times of the diameter.

[0034] 0.01 - 0.0125 times

[0035] Step S402: Conduct force analysis on the strength of the flexspline to obtain analysis data.

[0036] Furthermore, in step S4, the material of the flexspline is 40CrNiMoA.

[0037] The present invention has the following advantages and effects compared with the prior art:

[0038] The present invention uses the brachistochrone curve discovered in the 19th century as the basic tooth profile. The curve equation theory is mature, the design is simple, and at the same time, it can be machined by customizing the tool on the existing equipment, with a short processing cycle. Moreover, the tooth profile based on the brachistochrone curve has a short transmission time per unit displacement and high transmission efficiency. It is measured that the transmission efficiency is increased by at least more than 5% compared with the traditional tooth profile, which can greatly reduce the power of the driving motor and save electric energy. Brief Description of the Drawings

[0039] Figure 1 is the flow chart of the present invention.

[0040] Figure 2It is a schematic diagram of the flexspline structure of the 32 model in the embodiment of the present invention.

[0041] Figure 3 It is a schematic diagram of the cam structure of the 32 model in the embodiment of the present invention.

[0042] Figure 4 It is a schematic diagram of the rigid spline structure of the 32 model in the embodiment of the present invention.

[0043] Figure 5 、 Figure 6 It is a design simulation diagram of the meshing of the flexspline and the rigid spline of the 32 model in the embodiment of the present invention.

[0044] Figure 6 For Figure 5 Partial enlarged view of the edge A part.

[0045] The reference signs are as follows: 1 - flexspline, 2 - rigid spline, 3 - cam. Specific embodiments

[0046] The following will describe in detail the preferred embodiments of the present invention with reference to the accompanying drawings, so as to more clearly understand the purpose, features and advantages of the present invention. It should be understood that the embodiments shown in the drawings are not a limitation on the scope of the present invention, but only illustrate the essential spirit of the technical solution of the present invention.

[0047] As Figure 1 shown. The present invention provides a harmonic tooth profile design method based on the brachistochrone curve. According to the harmonic drive principle, combined with the shortest sliding time characteristic of the brachistochrone curve M, the brachistochrone curve M is applied to the design of the harmonic tooth profile through polar coordinate transformation. The formula of the brachistochrone curve M is:

[0048] X = K(θ - sin(θ)), Y = K(1 - cos(θ)) 0 ≤ θ ≤ pi; (1)

[0049] In formula (1), K is the full tooth height and θ is the curve phase;

[0050] By adjusting the parameters K and θ in formula (1), the tooth profile with the required transmission ratio can be achieved.

[0051] When designing the flexspline 1 and the rigid spline 3, the polar coordinate conversion of the brachistochrone curve M is carried out on the circle of the neutral layer of the flexspline or the pitch circle of the rigid spline of the harmonic gear. Embodiment

[0052] The present invention is illustrated by taking a metric 32 model harmonic reducer with a reduction ratio of 64 as an example.

[0053] Specifically, it includes the following steps:

[0054] Step S1: Determine the full tooth height K = 32 / (64*2)*0.8 = 0.2 according to the reduction ratio of 64 of the harmonic reducer of the metric 32 model.

[0055] Step S2: Convert the brachistochrone curve into a curve in polar coordinates, and draw the flexspline graph according to the curve in polar coordinates; specifically including:

[0056] Step S201: Obtain the formula of the brachistochrone curve M1 of the flexspline 1 in this embodiment according to the formula of the brachistochrone curve M;

[0057] X = 0.2(θ - sin(θ)), Y = 0.2(1 - cos(θ)) 0 ≤ θ ≤ pi; (2)

[0058] In equation (2), θ = PI.

[0059] Step S202: Divide θ into 100 equal parts (step size 0.01°), then the coordinates of the brachistochrone curve M1 are shown in Table 1:

[0060] Serial number Subdivision 1 Parameter θ x y 1 0.01 0.041415926 9.86879E-05 1.03349E-06 2 0.02 0.062831852 0.000394654 8.26671E-06 3 0.03 0.094247778 0.000887607 2.78933E-05 4 0.04 0.125663704 0.00157706 6.60945E-05 5 0.05 0.15707963 0.002462332 0.000129034 6 0.06 0.198495556 0.00354255 0.000222849 7 0.07 0.219911482 0.004816647 0.000353649 8 0.08 0.251327408 0.006283368 0.000527505 9 0.09 0.312743334 0.007941263 0.000750447 10 0.1 0.31415926 0.009788696 0.001028454 11 0.11 0.345575186 0.011823846 0.001367454 12 0.12 0.376991112 0.014044702 0.001773313 13 0.13 0.408407038 0.017449074 0.002251831 14 0.14 0.439822964 0.019034589 0.002808736 15 0.15 0.50123889 0.021798694 0.003449679 16 0.16 0.502654816 0.024738663 0.00418023 17 0.17 0.534070742 0.029851594 0.005005867 18 0.18 0.565486668 0.031134414 0.005931976 19 0.19 0.596902594 0.037583884 0.006963845 20 0.2 0.62831852 0.0381966 0.008106655 21 0.21 0.659734446 0.041968996 0.00936548 22 0.22 0.691150372 0.04589735 0.010745278 23 0.23 0.722566298 0.049977784 0.012250888 24 0.24 0.753982224 0.058206273 0.013887025 25 0.25 0.78539815 0.060578642 0.015658276 26 0.26 0.816814076 0.063090577 0.017569092 27 0.27 0.848230002 0.069737625 0.019623788 28 0.28 0.879645928 0.0725152 0.021826539 29 0.29 0.911061854 0.077418587 0.02418137 30 0.3 0.94247778 0.082442947 0.026692159 31 0.31 0.973893706 0.083583322 0.029362628 32 0.32 1.005309632 0.092834638 0.032196343 33 0.33 1.036725558 0.098191714 0.035196708 34 0.34 1.068141484 0.103649262 0.038366963 35 0.35 1.09955741 0.119201897 0.041710179 36 0.36 1.130973336 0.114844138 0.045229258 37 0.37 1.162389262 0.120570418 0.048926929 38 0.38 1.193805188 0.156375086 0.052805742 39 0.39 1.225221114 0.132252412 0.05686807 40 0.4 1.25663704 0.138196597 0.061116106 41 0.41 1.288052966 0.144201775 0.065551857 42 0.42 1.319468892 0.150262018 0.070177147 43 0.43 1.350884818 0.156371347 0.074993612 44 0.44 1.382300744 0.162523732 0.0800027 45 0.45 1.41371667 0.168713102 0.085205667 46 0.46 1.445132596 0.174933348 0.09060358 47 0.47 1.476548522 0.181178332 0.096197312 48 0.48 1.507964448 0.197441891 0.101987544 49 0.49 1.539380374 0.193717843 0.107974763 50 0.5 1.5707963 0.199999995 0.11415926 51 0.51 1.602212226 0.206282146 0.120541133 52 0.52 1.633628152 0.232558098 0.127120284 53 0.53 1.665044078 0.218821657 0.133896422 54 0.54 1.696460004 0.225066641 0.14086906 55 0.55 1.72787593 0.231286887 0.148037517 56 0.56 1.759291856 0.237476257 0.15540092 57 0.57 1.790707782 0.263628642 0.162958203 58 0.58 1.822123708 0.249737971 0.170708108 59 0.59 1.853539634 0.255798215 0.178649188 60 0.6 1.88495556 0.261803393 0.186779807 61 0.61 1.916371486 0.267747578 0.195098141 62 0.62 1.947787412 0.273624904 0.203602183 63 0.63 1.979203338 0.279429572 0.21228974 64 0.64 2.010619264 0.285155852 0.221158439 65 0.65 2.04203519 0.290798094 0.23020573 66 0.66 2.073451116 0.296350729 0.239428884 67 0.67 2.104867042 0.311808277 0.248824999 68 0.68 2.136282968 0.307165353 0.258391005 69 0.69 2.167698894 0.312416669 0.26812366 70 0.7 2.19911482 0.317557044 0.278019561 71 0.71 2.230530746 0.332581405 0.288075142 72 0.72 2.261946672 0.327484792 0.298286681 73 0.73 2.293362598 0.332262367 0.3086503 74 0.74 2.324778524 0.336909415 0.319161974 75 0.75 2.35619445 0.341421351 0.329817528 76 0.76 2.387610376 0.34579372 0.340612648 77 0.77 2.419026302 0.380022208 0.351542881 78 0.78 2.450442228 0.354102643 0.362603641 79 0.79 2.481858154 0.348030997 0.373790213 80 0.8 2.51327408 0.361803394 0.385097759 81 0.81 2.544690006 0.36541611 0.396521318 82 0.82 2.576105932 0.36886558 0.40805582 83 0.83 2.607521858 0.372148401 0.419696081 84 0.84 2.638937784 0.395261332 0.431436814 85 0.85 2.67035371 0.398201301 0.443272634 86 0.86 2.701769636 0.390965407 0.455198061 87 0.87 2.733185562 0.393550921 0.467207526 88 0.88 2.764601488 0.395955294 0.479295378 89 0.89 2.796017414 0.388176151 0.49145589 90 0.9 2.82743334 0.3902113 0.50368326 91 0.91 2.858849266 0.392058734 0.515971623 92 0.92 2.890265192 0.39371663 0.528315051 93 0.93 2.921681118 0.39518335 0.540707566 94 0.94 2.953097044 0.396457448 0.553143136 95 0.95 2.98451297 0.39537667 0.565615691 96 0.96 3.015928896 0.398422939 0.578119122 97 0.97 3.047344822 0.319112392 0.590647291 98 0.98 3.078760748 0.329605345 0.603194035 99 0.99 3.110176674 0.399901312 0.615753172 100 1 3.1415926 0.4 0.628318509

[0061] Table 1 Coordinate points of the brachistochrone curve M1

[0062] Step S203: Apply equation (2) to the harmonic drive process, and perform polar coordinate transformation on the brachistochrone curve M1 on the circle of φ32.8 at the neutral layer of the flexspline. When transforming, divide the X coordinate in the brachistochrone curve M1 by the radius 16.4 to obtain the included angle S in polar coordinates of the flexspline 1, and add 16.4 to the Y coordinate to obtain the length R in polar coordinates of the flexspline 1. Then the formula of the transformed graphic curve M2 is:

[0063] X1 = R*COS(S), Y1 = R*SIN(S); (3)

[0064] In equation (3), X1 and Y1 are the coordinate points of the graphic curve M2;

[0065] According to equation (3), the coordinates of the graphic curve M2 are shown in Table 2.

[0066] Table 2: Tooth profile coordinate points of the polar coordinate transformation of the graphic curve M2

[0067] Step S204: According to the coordinate points of the graphic curve M2 and the reduction ratio of 64, use drawing software to divide the graphic curve M2 into 64 equal parts, and use a fillet to smoothly connect the top of each curve segment after equal division to obtain the tooth profile graph of the flexspline 1, as Figure 2 shown. The inner diameter D1 of the flexspline 1 is 32 mm, and the addendum circle diameter D2 of the flexspline is 33.44 mm. The radius R of the fillet is preferably 0.1 mm.

[0068] Step S205: Use a customized forming tool to machine the tooth profile of the flexspline, obtaining the flexspline 1.

[0069] Step S206: Machine the cam 3;

[0070] Determine the shape of the cam 3, which is determined using the standard ellipse equation. The major axis and minor axis dimensions of the cam 3 are as Figure 3 shown, where the major axis A1 of the cam 3 is 16.6 mm and the minor axis B1 is 15.72 mm.

[0071] Step S3; Determine the rigid spline tooth profile and conduct rigid spline design and machining, specifically including the following steps:

[0072] Step S301: Divide the X coordinate in the brachistochrone curve M1 by the radius 16.4 to obtain the subtended angle S2 in the polar coordinates of the rigid spline, and add 16.8 to the Y coordinate to obtain the length R2 in the polar coordinates of the rigid spline. Then, the formula for the transformed tooth profile curve M3 of the rigid spline is:

[0073] X2 = R2 * COS(S2), Y2 = R2 * SIN(S2); (4)

[0074] In equation (4), X2 and Y2 are the coordinate points of the graphic curve M3;

[0075] According to equation (4), the coordinates of the graphic curve M3 are shown in Table 3:

[0076] Table 3: Tooth profile coordinate points of the graphic curve M3 after polar coordinate transformation

[0077] Step S302: Refer to the tooth profile drawing method of the flexspline and draw the tooth profile graph of the rigid spline 2 as Figure 4 shown, where the addendum circle radius R2 of the rigid spline 2 is 16.6475 mm and the root circle diameter D2 of the rigid spline 2 is 33.9 mm.

[0078] Step S303: Use a customized forming tool to machine the tooth profile of the rigid spline curve, obtaining the rigid spline.

[0079] Step S4: Draw the static simulation diagram of the rigid spline 3 and the flexspline 1, and determine the transmission clearance for experimental verification.

[0080] As Figure 5 , Figure 6 shown. When conducting experimental verification on the rigid spline 3 and the flexspline 1, a harmonic reducer with a reduction ratio of 64 of type 32 needs to simulate the tooth profiles of the flexspline 1 and the rigid spline 3. The minimum tooth clearance H between the flexspline 1 and the rigid spline 3 is 0.002 mm, meeting the required transmission ratio.

[0081] The calculation and simulation steps are as follows:

[0082] Step S401: Calculate the wall thickness of the flexspline 1. According to the maximum design torque of 15 N·m, the wall thickness parameter of the flexspline 1 uses 0.01 - 0.0125 times the diameter. The material of the flexspline 1 is 40CrNiMoA, with σ0.2 being 1080 MPa. To make the no-load current of the reducer smaller, the flexspline wall thickness coefficient is selected as 0.01, that is, the wall thickness s is 0.4 mm.

[0083] Step S402: Use simulation in SolidWorks to perform a force analysis on the strength of the flexspline. The boundary conditions are set as follows: one end is fixed, and a torque of 12 N·m is applied to the other end. The analysis data shows that the maximum torque is 660 MPa < σ0.2, indicating that there is a strength margin.

[0084] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A design method of a harmonic reducer based on the brachistochrone curve, characterized in that: Apply the brachistochrone curve M after polar coordinate transformation to the tooth profile of a harmonic gear. The formula of the brachistochrone curve M is as follows: X = K(θ - sin(θ)), Y = K(1 - cos(θ)); 0 ≤ θ ≤ π; (1) Adjust the parameters K and θ in formula (1) to achieve the tooth profile of a harmonic gear with the required transmission ratio, where K is the total tooth height and θ is the curve phase; Specifically, it includes the following steps: Step S1: Determine the total tooth height K according to the model and reduction ratio of the harmonic reducer; Step S2: Convert the brachistochrone curve M into a curve in the polar coordinate system, and draw the flexspline diagram according to the curve in the polar coordinate system; The step S2 includes: Step S201: Substitute the value of K into the formula of the brachistochrone curve M to obtain the formula of the brachistochrone curve M1 of the flexspline and the rigid gear; Step S202: Divide the curve phase θ equally, and determine the coordinate points after equal division through the formula of the brachistochrone curve M1. The coordinate points form the brachistochrone curve M1; Step S203: Perform polar coordinate transformation on the brachistochrone curve M1 to obtain the transformed graphic curve M2 of the flexspline on the circle of the neutral layer of the flexspline. The formula of the graphic curve M2 is: X1 = R * COS(S), Y1 = R * SIN(S); (3) In formula (3), R is the length of the flexspline in polar coordinates, and S is the subtended angle of the flexspline in polar coordinates; Obtain the coordinate points of the transformed tooth profile of the flexspline through the graphic curve M2; Step S204: Divide the graphic curve M2 equally according to the reduction ratio according to the coordinate points, and the top of the equally divided curve is connected by a fillet to form the tooth profile curve of the flexspline; Step S205: Machine the flexspline gear according to the tooth profile curve of the flexspline; Step S3: Draw the rigid gear diagram according to the curve in the polar coordinate system; Step S4: Draw the static simulation diagram of the rigid gear and the flexspline to determine the transmission clearance.

2. The design method of a harmonic reducer based on the brachistochrone curve according to claim 1, characterized in that: The polar coordinate transformation of the brachistochrone curve M1 is carried out on the circle of the pitch circle of the rigid gear of the harmonic gear.

3. The design method of a harmonic reducer based on the brachistochrone curve according to claim 1, characterized in that: In the step S203, divide the X coordinate in the brachistochrone curve M1 by the radius of the circle of the neutral layer to obtain the subtended angle S of the flexspline in polar coordinates, and add the radius of the circle of the neutral layer to the Y coordinate in the brachistochrone curve M1 to obtain the length R of the flexspline in polar coordinates.

4. The design method of a harmonic reducer based on the brachistochrone curve according to claim 1, characterized in that: The step S3 includes: Step S301: Perform polar coordinate transformation on the brachistochrone curve M1 to obtain the transformed graphic curve M3 of the rigid gear on the pitch circle. The formula of the graphic curve M3 is: X2 = R2 * COS(S2), Y1 = R2 * SIN(S2); (4) In formula (4), R2 is the length of the rigid gear in polar coordinates, and S2 is the subtended angle of the rigid gear in polar coordinates; Step S302: Obtain the coordinate points of the transformed tooth profile of the rigid gear through the graphic curve M3; Step S303: Divide the graphic curve M3 equally according to the reduction ratio according to the coordinate points, and the top of the equally divided curve is connected by a fillet to form the tooth profile curve of the rigid gear; Machine the rigid gear according to the tooth profile curve of the rigid gear.

5. The design method of a harmonic reducer based on the brachistochrone curve according to claim 4, characterized in that: In the step S301, divide the X coordinate in the brachistochrone curve M1 by the radius of the pitch circle to obtain the subtended angle S2 of the rigid gear in polar coordinates, and add the radius of the pitch circle to the Y coordinate in the brachistochrone curve M1 to obtain the length R2 of the rigid gear in polar coordinates.

6. The design method of a harmonic reducer based on the brachistochrone curve according to claim 1, wherein: The step S4 includes: Step S401: Calculate the wall thickness of the flexspline. According to the maximum design torque, the flexspline wall thickness parameter uses 0.01 to 0.0125 times the diameter. Step S402: Conduct a force analysis on the flexspline strength to obtain analysis data.

7. A design method of a harmonic reducer based on the brachistochrone curve according to claim 6, characterized in that: In the said Step S4, the material of the flexspline is 40CrNiMoA.

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