CN109630652B - Three-arc harmonic gear slotting cutter and tooth profile design method thereof - Google Patents

Three-arc harmonic gear slotting cutter and tooth profile design method thereof Download PDF

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CN109630652B
CN109630652B CN201910017231.6A CN201910017231A CN109630652B CN 109630652 B CN109630652 B CN 109630652B CN 201910017231 A CN201910017231 A CN 201910017231A CN 109630652 B CN109630652 B CN 109630652B
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CN109630652A (en
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周青华
杨勇
王家序
时志奇
黄彦彦
周广武
蒲伟
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Sichuan University
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/17—Toothed wheels
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H49/00—Other gearings
    • F16H49/001—Wave gearings, e.g. harmonic drive transmissions
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/0826—Novikov-Wildhaber profile
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F30/17—Mechanical parametric or variational design
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    • G06—COMPUTING; CALCULATING OR COUNTING
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Abstract

The invention discloses a three-arc harmonic gear slotting cutter, belonging to the technical field of harmonic reducer gear machining cutters, wherein the basic tooth profile of cutter teeth consists of three arc sections which are sequentially tangent; the radius of the upper arc section is rho1The radius of the middle arc section is rho2And the radius rho of the lower circular arc segment3(ii) a The tooth profile design method comprises the steps of designing a flexible gear three-arc tooth profile and a rigid gear three-arc tooth profile which meet conditions, and obtaining an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile by adopting a gear meshing kinematics method according to the obtained rigid gear three-arc tooth profile; obtaining an upper arc section Ac1 and a lower arc section Ac3 and related parameters thereof by adopting arc fitting; obtaining a middle arc segment Ac2 and related parameters thereof by a plane analytic geometry method; and finally, determining auxiliary parameters according to the process requirements of the harmonic gear to be processed so as to achieve the purpose of processing the working tooth profile of the three-arc harmonic gear.

Description

Three-arc harmonic gear slotting cutter and tooth profile design method thereof
Technical Field
The invention belongs to the technical field of harmonic reducer gear machining cutters, and particularly relates to a three-arc harmonic gear slotting cutter and a tooth profile design method thereof.
Background
Harmonic gear transmission is a transmission that is acted upon by a wave generator to control the elastic deformation of a flexible gear to transmit force and motion. The transmission gear pair has a series of advantages of large transmission ratio, light weight, small volume, large number of meshing gear pairs, high transmission stability and precision and the like, and becomes a key part of equipment transmission systems in the fields of industrial robots, aerospace, machine tools, medical instruments and the like.
When the us scholars MUSSER patented the harmonic gear drive in 1959, a linear tooth profile with a large pressure angle was adopted, which can satisfy the requirement of a fixed transmission ratio, but does not consider the tangential displacement on the neutral line of the flexspline, the movement of the symmetrical line of the teeth of the flexspline caused by the curvature change when the neutral line is deformed, and the like, and the tooth profile has some disadvantages in application.
Because the tooth forms of the harmonic gear transmission flexible gear and the harmonic gear can obviously influence the performance of the harmonic reducer, the research on the tooth form of the harmonic gear tooth obtains great attention for further improving the performance of the harmonic transmission, and the research current situations are as follows:
the involute tooth profile is most widely used due to the advantages of easy processing, mature manufacturing and the like, but when the tooth profile is meshed, a conjugate area is distributed in a smaller interval at two sides of a long shaft of a wave generator, and sharp point meshing is mostly generated due to loaded deformed meshing teeth (Shenqinsha, Yeqingtai. harmonic gear transmission theory and design [ M ] Beijing: mechanical industry publisher, 1985: 1-33, 127-;
a double-arc tooth profile having an S-shape was proposed and patented by ISHIKAWA (reference ISHIKAWA S. tooth of profile of strain wave: US Patent, No.4823638[ P ].1989.), and ISHIKAWA was improved in 1995 on an existing flexspline tooth profile (reference ISHIKAWA S. flexible contact type gear drive of non-profile-shifted two-circular-arc composite profile: US Patent, No.5458023[ P ]. 1995.). The tooth profile not only realizes continuous contact of meshing areas, but also improves the stress condition of tooth roots and the meshing quality of transmission and improves the bearing capacity and the torsional rigidity because of the simultaneous multiple pairs of meshing teeth (reference document: great strength, Yang Jia Jun, Wang Xuefu. analysis of the motion characteristics of double-arc tooth profile harmonic gear transmission [ J ]. university of China, 2000, 28 (01): 12-14.), additionally, the meshing tooth grooves are wider;
the common tangent tooth profile connecting the convex circular arc tooth profile and the concave circular arc tooth profile of the flexible gear in the common tangent type double circular arc tooth profile is replaced by a section of circular arc, so that a three-circular arc tooth profile can be obtained, the tooth profile has the advantages of a double circular arc tooth profile, and the defect that a conjugate tooth profile is not easily formed by a straight-line tooth profile part when the tooth profile angle is small is avoided. And theoretically, the three-arc tooth profile can obtain a larger theoretical conjugate interval (reference document: Chenxian, static Chenchen faithful. design of the three-arc tooth profile of the continuous conjugate cup-shaped or hat-shaped harmonic gear: China, CN201710436032.X), and can improve the meshing performance of the transmission of the harmonic gear, so that the development of a machining tool for machining the corresponding three-arc harmonic gear is necessary.
Disclosure of Invention
In view of the above, in order to solve the above problems in the prior art, the present invention provides a three-arc harmonic gear shaper cutter to achieve the purpose of processing a working tooth profile of a three-arc harmonic gear according to a processing requirement, and also provides a tooth profile design method applied to the three-arc harmonic gear shaper cutter to achieve the purpose of accurately calculating each parameter of the shaper cutter.
The technical scheme adopted by the invention is as follows: the utility model provides a three circular arc harmonic gear pinion cutter, includes handle of a knife and locates the cutter body on this handle of a knife, the profile face of cutter body is equipped with the sword tooth, and the tooth top anterior angle and the tooth top relief angle of sword tooth are gamma and alpha respectivelytThe basic tooth profile of the cutter tooth is formed by three arc sections which are sequentially tangent, and the three arc sections are an upper arc section, a middle arc section and a lower arc section respectively; the radius of the upper arc section is rho1The radius of the middle arc section is rho2And the radius rho of the lower circular arc segment3(ii) a An upper tangent point is formed between the upper arc section and the middle arc section, and the included angle of the upper tangent point is delta1A lower tangent point is formed between the middle arc section and the lower arc section and the included angle of the lower tangent point is delta2(ii) a The circle center offset of the upper arc section is xaAnd the amount of circle center shift is yaThe circle center offset of the lower arc section is xfAnd the amount of circle center shift is yf(ii) a The rho1、ρ2、ρ3、δ1、δ2、xa、ya、xfAnd yfThe parameter value of (A) is determined by the working tooth profile of the processed three-arc harmonic gear through calculation and meshing.
Further, the basic tooth profile of the cutter tooth further includes a tip fillet, a root fillet, a tip, a root, a tip clearance, and a root clearance, the tip fillet having a radius raRadius of root fillet is rfThe height of the tooth top is haThe height of the tooth root is hfA tip clearance of caA root space of cf. Said ra、rf、ha、hf、caAnd cfThe parameter values of (a) are determined by the process requirements of the three-arc harmonic gear to be machined.
Further, the normal tooth profile of the cutter tooth consists of a tooth top fillet, a tooth root fillet, the upper arc section, the middle arc section and the lower arc section.
The invention also provides a tooth profile design method of the three-arc harmonic gear slotting cutter, which comprises the following steps:
(1) designing a flexible gear three-arc tooth profile and a rigid gear three-arc tooth profile which meet the requirements according to the design requirements of 'double conjugation' and 'secondary conjugation' met in the meshing process of a flexible gear and a rigid gear of a harmonic gear;
(2) according to the obtained three-arc tooth profile of the rigid wheel, a gear meshing kinematics method is adopted, and an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile which are respectively conjugated with the three-section arc tooth profiles of the upper arc, the middle arc and the lower arc in the three-arc tooth profile of the rigid wheel in the gear shaper are obtained by solving a meshing equation of the gear shaper for cutting the rigid wheel;
(3) arc fitting is adopted for the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C to respectively obtain an upper arc section Ac1 and a lower arc section Ac3 so as to determine the radius rho of the upper arc section1Radius rho of lower arc segment3The circle center offset x of the upper arc segmentaThe circle center displacement amount y of the upper arc sectionaThe circle center offset x of the lower arc segmentfThe circle center displacement amount y of the lower arc sectionf；
(4) Through planeAnalyzing the geometric method, and obtaining a middle arc section Ac2 corresponding to the theoretical conjugate tooth profile of the section B to determine the radius rho of the middle arc section2Included angle delta of upper tangent point1Angle delta with lower tangent point2；
(5) Determining auxiliary parameters according to the technological requirements of the harmonic gear to be processed, wherein the auxiliary parameters comprise: radius of tooth top fillet raRoot fillet radius rfTooth top height haRoot height hfTop clearance caRoot space cf。
Further, the step (2) of calculating the theoretical conjugate tooth profile at the a section, the theoretical conjugate tooth profile at the B section and the theoretical conjugate tooth profile at the C section is as follows:
(2.1) establishing a coordinate system S for transforming the coordinate of a rigid wheel in the pinion cutter machining harmonic gear and fixedly connecting with the rigid wheel1(O1,x1,y1) To a coordinate system S fixedly connected with the pinion cutter2(O2,x2,y2) Is transformed into a matrix M21Comprises the following steps:
Figure BDA0001939506530000041
wherein r is1Is the reference circle radius of the machined rigid wheel, r2The reference circle radius of the original section of the pinion cutter;
(2.2) establishing a coordinate System S1To S2Bottom vector transformation matrix W21The following were used:
Figure BDA0001939506530000042
(2.3) the two surfaces must satisfy the following equation of mesh at the point of contact, according to the velocity vector of the relative motion perpendicular to the normal vector of the profiles at the point of contact of the mutually enveloping profiles:
ni·vi (12)＝0 (i＝1,2)
in the formula, niAnd vi (12)Respectively expressed in a coordinate system SiThe common vector and the relative velocity vector of the middle two conjugate curved surfaces at the contact point;
(2.4) in a coordinate system S2Substituting into the meshing equation in step (2.3) and converting it into:
Figure BDA0001939506530000051
(2.5) defining the matrix phi, order
Figure BDA0001939506530000052
Then there is
Figure BDA0001939506530000053
I.e. n1 TΦr1＝0
Wherein:
Figure BDA0001939506530000054
Figure BDA0001939506530000055
substitution into
Figure BDA0001939506530000056
In the method, the following steps are obtained:
Figure BDA0001939506530000057
and (2.6) calculating the conjugate tooth profile of the slotting tool by inputting respective tooth profile equations and normal line equations and calling the same phi matrix for the circular arc tooth profiles of all the sections of the rigid wheel represented by different functions.
Further, the calculation method in the step (2.6) is specifically as follows:
(2.6.1) adopting a numerical discrete method, representing the three-arc tooth profile of the rigid wheel with the arc length u as a parameter by using s discrete points, and for any point j, ordering: u. ofjIs the parameter value of the point;
(2.6.2) mixing ujSubstituting the sagittal diameter and normal vector of the point on the flexspline tooth profile corresponding to (j ═ 1,2, …, s) into the equation n1 TΦr1When the point generates conjugate motion, the rotation angle of the rigid wheel relative to the initial position is obtained
Figure BDA0001939506530000061
Its value is recorded as
Figure BDA0001939506530000062
(2.6.3) according to each
Figure BDA0001939506530000063
Determining correspondence M12The value of each element in the transformation matrix M is obtained12And then substituted into the following equation:
Figure BDA0001939506530000064
and (2.6.4) obtaining an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile which are respectively conjugated with the upper arc, the middle arc and the lower arc of the three-arc tooth profile of the rigid wheel through coordinate transformation.
Further, in the step (3), arc fitting is carried out on the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C in the gear shaper cutter by adopting a least square method.
The invention has the beneficial effects that:
1. by adopting the three-arc harmonic gear slotting cutter disclosed by the invention, the cutter tooth basic tooth profile of the slotting cutter is composed of three arc sections which are sequentially tangent, and the parameters of each arc section in the cutter tooth are determined by calculating and meshing the working tooth profile of the processed three-arc harmonic gear, so that the three-arc tooth profile harmonic gear processed by adopting the three-arc harmonic gear slotting cutter can effectively increase the number of meshing teeth in the transmission process, and the meshing backlash between the flexible gear tooth profile and the rigid gear tooth profile is more uniformly distributed in the whole meshing interval, thereby being beneficial to improving the bearing capacity and the transmission precision of the harmonic gear; the harmonic gear transmission device can overcome the defect that most conjugate areas of the harmonic gear transmission device with involute tooth profiles are distributed in a smaller range near the long shaft area of a wave generator, and can also overcome the defect that a straight-line section tooth profile part connecting a convex circular arc tooth profile and a concave circular arc tooth profile of a flexible gear is difficult to form conjugate tooth profiles when the tooth profile angle is smaller.
2. By adopting the tooth profile design method of the three-arc harmonic gear shaper cutter, which is disclosed by the invention, the theoretical conjugate tooth profile at the A section, the theoretical conjugate tooth profile at the B section and the theoretical conjugate tooth profile at the C section are obtained by calculation by using a gear meshing kinematics method, then the theoretical conjugate tooth profile at the A section and the theoretical conjugate tooth profile at the C section are both fitted by adopting arcs, so as to obtain an upper arc section Ac1, a lower arc section Ac3 and corresponding parameter values, and then a middle arc section Ac2 is obtained by calculation through a plane analytic geometry method, the three-arc harmonic gear shaper cutter obtained by the tooth profile design method can process the accurate three-arc tooth profile harmonic gear by accurately obtaining the corresponding parameter value of the basic tooth profile in the shaper cutter, and the meshing tooth number can be effectively increased in transmission, so that the meshing backlash between the flexible gear tooth profile and the rigid gear tooth profile is more uniformly distributed in the whole meshing interval, and the bearing capacity and the transmission precision of the harmonic gear are favorably improved.
Drawings
FIG. 1 is a schematic view of the overall structure of a three-arc harmonic gear slotting cutter provided by the invention;
FIG. 2 is a schematic front view of a three-arc harmonic gear slotting cutter provided by the present invention;
FIG. 3 is a schematic sectional view taken along line A-A of FIG. 2;
FIG. 4 is a basic tooth profile of the cutter teeth of the three-arc harmonic gear shaper cutter provided by the present invention;
fig. 5 is a schematic diagram of coordinate transformation of rotary motion in the same direction in the tooth profile designing method of the three-arc harmonic gear shaper cutter provided by the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
As shown in fig. 1-4, the present embodiment discloses a three-arc harmonic gear slotting cutter, which includes a cutter handle 1 and a cutter body 2 disposed on the cutter handle 1, wherein the cutter teeth 3 are disposed on the profile surface of the cutter body 2, and the tooth top rake angle and the tooth top relief angle of the cutter teeth 3 are γ and α, respectivelytIn the present embodiment, the structure design of the three-arc harmonic gear slotting cutter is similar to that of a common involute straight gear slotting cutter, and the main difference is the basic tooth profile part of the cutter body 2.
The basic tooth profile of the cutter teeth 3 is formed by three arc sections which are sequentially tangent, the three arc sections are an upper arc section, a middle arc section and a lower arc section, and the middle arc section is a middle connecting arc; the radius of the upper arc section is rho1The radius of the middle arc section is rho2And the radius rho of the lower circular arc segment3(ii) a An upper tangent point is formed between the upper arc section and the middle arc section, an included angle between a tangent line of the upper tangent point and the Y axis is an included angle of the upper tangent point, and the included angle of the upper tangent point is delta1A lower tangent point is formed between the middle arc segment and the lower arc segment, the included angle between the tangent of the lower tangent point and the Y axis is the included angle of the lower tangent point, and the included angle of the lower tangent point is delta2(ii) a The circle center offset of the upper arc section is xaAnd the amount of circle center shift is yaThe center of the lower arc section is deviatedThe displacement is xfAnd the amount of circle center shift is yf(ii) a The rho1、ρ2、ρ3、δ1、δ2、xa、ya、xfAnd yfThe parameter value of (A) is determined by the working tooth profile of the processed three-arc harmonic gear through calculation and meshing. As shown in fig. 4, the basic tooth profile of the cutter tooth 3 provided in the present embodiment is composed of five sections, i.e., an upper convex circular arc tooth profile, a lower concave circular arc tooth profile, a middle-joining concave circular arc tooth profile, a tip fillet, and a root fillet.
The basic tooth profile of the cutter tooth 3 further includes an addendum fillet, a dedendum fillet, an addendum, a dedendum, a tip clearance, and a root clearance, the addendum fillet having a radius raRadius of root fillet is rfThe height of the tooth top is haThe height of the tooth root is hfA tip clearance of caA root space of cf. Said ra、rf、ha、hf、caAnd cfThe parameter values of the three-arc harmonic gear are auxiliary parameters which are determined by the process requirements of the processed three-arc harmonic gear.
The angle delta of the upper tangent point1Angle delta with lower tangent point2According to different requirements in the actual tooth profile design, the included angle delta of the tangent point1Angle delta with lower tangent point2The parameter values of the two can be adjusted correspondingly.
The normal tooth profile of the cutter tooth 3 comprises an addendum fillet, a dedendum fillet, an upper arc section, a middle arc section and a lower arc section, preferably, the upper arc section is of a convex circular shape, the middle arc section is of a concave circular shape, and the lower arc section is of a concave circular shape.
Aiming at the three-arc harmonic gear slotting cutter disclosed by the embodiment, the tooth profile design method suitable for machining the three-arc harmonic gear slotting cutter is also provided, and comprises the following steps:
(1) designing a flexible gear three-arc tooth profile and a rigid gear three-arc tooth profile which meet the requirements according to the design requirements of 'double conjugation' and 'secondary conjugation' met in the meshing process of a flexible gear and a rigid gear of a harmonic gear; wherein, the flexible gear and the rigid gear are the basic characteristics of the harmonic gear, and the details are not repeated here.
(2) According to the obtained three-arc tooth profile of the rigid wheel, a gear meshing kinematics method is adopted, and the meshing equation of the rigid wheel cut by the slotting tool is solved to obtain an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile in the slotting tool, wherein the A-section theoretical conjugate tooth profile, the B-section theoretical conjugate tooth profile and the C-section theoretical conjugate tooth profile are respectively conjugated with the upper arc, the middle arc and the lower arc of the three-arc tooth profile of the rigid wheel.
(3) Arc fitting is adopted for the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C to respectively obtain an upper arc section Ac1 and a lower arc section Ac3 so as to determine the radius rho of the upper arc section1Radius rho of lower arc segment3The circle center offset x of the upper arc segmentaThe circle center displacement amount y of the upper arc sectionaThe circle center offset x of the lower arc segmentfThe circle center displacement amount y of the lower arc sectionf；
Preferably, in the step, arc fitting is carried out on both the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C in the gear shaper cutter by adopting a least square method; the least square method (also called the least square method) is a mathematical optimization technique, which can easily find unknown data by minimizing the square sum of errors and finding the optimal function matching of the data, and minimizes the square sum of errors between the found data and actual data, and the least square method is mainly used for curve fitting.
(4) Obtaining a middle arc section Ac2 corresponding to the theoretical conjugate tooth profile of the section B by a plane analytic geometry method to determine the radius rho of the middle arc section2Included angle delta of upper tangent point1Angle delta with lower tangent point2(ii) a Among these, the planar analytic geometry is a branch of geometry for conducting a graphic study by means of an analytic expression, and the analytic geometry usually uses a two-dimensional planar rectangular coordinate system to study various general planar curves such as straight lines, circles, conic curves, cycloid curves, and star-shaped lines, and uses a three-dimensional spatial rectangular coordinate system to study planes, a,Spheres and other general space curved surfaces, and simultaneously researching the equations of the space curved surfaces and defining the concepts and parameters of some graphs.
(5) Determining auxiliary parameters according to the technological requirements of the harmonic gear to be processed, wherein the auxiliary parameters comprise: radius of tooth top fillet raRoot fillet radius rfTooth top height haRoot height hfTop clearance caRoot space cf。
The method for calculating the theoretical conjugate tooth profile at the section A, the theoretical conjugate tooth profile at the section B and the theoretical conjugate tooth profile at the section C in the step (2) comprises the following steps:
(2.1) establishing coordinate transformation of a rigid gear in the harmonic gear machined by the slotting cutter, and as shown in figure 5, establishing a coordinate system S fixedly connected with the rigid gear1(O1,x1,y1) To a coordinate system S fixedly connected with the pinion cutter2(O2,x2,y2) Is transformed into a matrix M21Comprises the following steps:
Figure BDA0001939506530000101
wherein r is1Is the reference circle radius of the machined rigid wheel, r2The reference circle radius of the original section of the pinion cutter;
(2.2) establishing a coordinate System S1To S2Bottom vector transformation matrix W21The following were used:
Figure BDA0001939506530000111
(2.3) according to the fact that the velocity vector of the relative motion is perpendicular to the normal vector of the tooth profile at the contact point of the mutually enveloping tooth profiles (reference: litterland phi. l. gear meshing principle (second edition) [ M ] luxian, shanghai: shanghai scientific press, 1984:33-80.), the two curved surfaces must satisfy the following meshing equation at the contact point:
ni·vi (12)＝0 (i＝1,2)
in the above formula, niAnd vi (12)Respectively expressed in a coordinate system SiThe common vector and the relative velocity vector of the middle two conjugate curved surfaces at the contact point;
(2.4) in a coordinate system S2Substituting into the meshing equation in step (2.3) and converting it into:
Figure BDA0001939506530000112
(2.5) defining the matrix phi, order
Figure BDA0001939506530000113
Then there is
Figure BDA0001939506530000114
I.e. n1 TΦr1＝0
Wherein:
Figure BDA0001939506530000115
Figure BDA0001939506530000121
substitution into
Figure BDA0001939506530000122
In the method, the following steps are obtained:
Figure BDA0001939506530000123
(2.6) when the rigid wheel tooth profile of the three-arc harmonic gear is expressed by a piecewise function with the arc length as a parameter, the uniqueness and the continuity of the mathematical description of the tooth profile can be ensured. The improved kinematics method is to encapsulate the motion parameters describing the complex motion rule of the rigid wheel in a phi matrix, and as the phi matrix does not contain the geometric parameters of conjugate surfaces, the phi matrix has uniqueness no matter what form the two conjugate surfaces are as long as the motion rule and the coordinate system are selected to be certain. Therefore, the conjugate tooth profile of the slotting tool can be calculated by inputting the respective circular tooth profile equations and normal line equations and calling the same phi matrix for the circular tooth profiles of the sections of the rigid wheel represented by different functions.
The calculation method of the step (2.6) is specifically as follows:
(2.6.1) adopting a numerical discrete method, representing the three-arc tooth profile of the rigid wheel with the arc length u as a parameter by using s discrete points, and for any point j, ordering: u. ofjIs the parameter value of the point;
(2.6.2) mixing ujSubstituting the sagittal diameter and normal vector of the point on the flexspline tooth profile corresponding to (j ═ 1,2, …, s) into the equation n1 TΦr1When the point generates conjugate motion, the rotation angle of the rigid wheel relative to the initial position is obtained
Figure BDA0001939506530000124
Its value is recorded as
Figure BDA0001939506530000125
(2.6.3) according to each
Figure BDA0001939506530000126
Determining correspondence M12The value of each element in the transformation matrix M is obtained12And then substituted into the following equation:
Figure BDA0001939506530000127
and (2.6.4) obtaining an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile which are respectively conjugated with the upper arc, the middle arc and the lower arc of the three-arc tooth profile of the rigid wheel through coordinate transformation.
The application of the tooth profile design method of the three-arc harmonic gear slotting cutter disclosed above includes but is not limited to machining three-arc tooth profiles in a taper shank-shaped slotting cutter, a disc-shaped slotting cutter or a bowl-shaped slotting cutter, and the taper shank-shaped slotting cutter, the disc-shaped slotting cutter or the bowl-shaped slotting cutter can machine three-arc harmonic gears meeting the requirements of users.
The three-arc harmonic gear processed by the tooth profile design method of the three-arc harmonic gear slotting cutter can effectively increase the number of meshing teeth in transmission, so that meshing backlash between a flexible gear tooth profile and a rigid gear tooth profile is more uniformly distributed in the whole meshing interval, and the bearing capacity and the transmission precision of the harmonic gear are improved.
The invention is not limited to the above alternative embodiments, and any other various forms of products can be obtained by anyone in the light of the present invention, but any changes in shape or structure thereof, which fall within the scope of the present invention as defined in the claims, fall within the scope of the present invention.

Claims (5)

1. A tooth profile design method of a three-arc harmonic gear shaper cutter is characterized by comprising the following steps:
(1) designing a flexible gear three-arc tooth profile and a rigid gear three-arc tooth profile which meet the requirements according to the design requirements of 'double conjugation' and 'secondary conjugation' met in the meshing process of a flexible gear and a rigid gear of a harmonic gear;
(2) according to the obtained three-arc tooth profile of the rigid wheel, a gear meshing kinematics method is adopted, and an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile which are respectively conjugated with the three-section arc tooth profiles of the upper arc, the middle arc and the lower arc in the three-arc tooth profile of the rigid wheel in the gear shaper are obtained by solving a meshing equation of the gear shaper for cutting the rigid wheel;
the method for calculating the theoretical conjugate tooth profile at the A section, the theoretical conjugate tooth profile at the B section and the theoretical conjugate tooth profile at the C section comprises the following steps:
(2.1) establishing a coordinate system S for transforming the coordinate of a rigid wheel in the pinion cutter machining harmonic gear and fixedly connecting with the rigid wheel1(O1,x1,y1) To a coordinate system S fixedly connected with the pinion cutter2(O2,x2,y2) Is transformed into a matrix M21Comprises the following steps:
Figure FDA0003076364610000011
wherein r is1Is the reference circle radius of the machined rigid wheel, r2Is the reference circle radius of the original section of the pinion cutter,
Figure FDA0003076364610000012
the rotation angle of the rigid wheel relative to the initial position when conjugate motion occurs;
(2.2) establishing a coordinate System S1To S2Bottom vector transformation matrix W21The following were used:
Figure FDA0003076364610000021
(2.3) the two surfaces must satisfy the following equation of mesh at the point of contact, according to the velocity vector of the relative motion perpendicular to the normal vector of the profiles at the point of contact of the mutually enveloping profiles:
ni·vi (12)＝0(i＝1，2)
in the formula, niAnd vi (12)Respectively expressed in a coordinate system SiThe common vector and the relative velocity vector of the middle two conjugate curved surfaces at the contact point;
(2.4) in a coordinate system S2Substituting into the meshing equation in step (2.3) and converting it into:
Figure FDA0003076364610000022
(2.5) defining a matrix phi, packaging motion parameters for describing the complex motion rule of the rigid wheel in the phi matrix, and enabling the motion parameters to be in a form of a complete matrix phi
Figure FDA0003076364610000023
Then there is
Figure FDA0003076364610000024
I.e. n1 TΦr1＝0
Wherein:
Figure FDA0003076364610000031
Figure FDA0003076364610000032
substitution into
Figure FDA0003076364610000033
In the method, the following steps are obtained:
Figure FDA0003076364610000034
(2.6) for the circular arc tooth profiles of all the sections of the rigid wheel represented by different functions, the conjugate tooth profile of the slotting tool can be calculated only by inputting respective tooth profile equations and normal line equations and calling the same phi matrix;
(3) arc fitting is adopted for the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C to respectively obtain an upper arc section Ac1 and a lower arc section Ac3 so as to determine the radius rho of the upper arc section1Radius rho of lower arc segment3The circle center offset x of the upper arc segmentaThe circle center displacement amount y of the upper arc sectionaThe circle center offset x of the lower arc segmentfThe circle center displacement amount y of the lower arc sectionf；
(4) Obtaining a middle arc section Ac2 corresponding to the theoretical conjugate tooth profile of the section B by a plane analytic geometry method to determine the radius rho of the middle arc section2Included angle delta of upper tangent point1Angle delta with lower tangent point2；
(5) Determining auxiliary parameters according to the technological requirements of the harmonic gear to be processed, wherein the auxiliary parameters comprise: radius of tooth top fillet raRoot fillet radius rfHeight h of tooth topaThe height h of the tooth rootfTop clearance caRoot space cf；
The three-arc harmonic gear slotting cutter,comprises a knife handle and a knife body arranged on the knife handle, the profile surface of the knife body is provided with knife teeth, and the tooth crest front angle and the tooth crest rear angle of the knife teeth are respectively gamma and alphatThe basic tooth profile of the cutter tooth is formed by three arc sections which are sequentially tangent, and the three arc sections are an upper arc section, a middle arc section and a lower arc section respectively; the radius of the upper arc section of the three-arc harmonic gear slotting cutter is rho1The radius of the middle arc section is rho2And the radius rho of the lower circular arc segment3(ii) a An upper tangent point is formed between the upper arc section and the middle arc section of the three-arc harmonic gear slotting cutter, and the included angle of the upper tangent point is delta1A lower tangent point is formed between the middle arc section and the lower arc section and the included angle of the lower tangent point is delta2(ii) a The circle center offset of the arc section on the three-arc harmonic gear slotting cutter is xaAnd the amount of circle center shift is yaThe circle center offset of the lower arc section is xfAnd the amount of circle center shift is yf(ii) a Rho of the three-arc harmonic gear slotting cutter1、ρ2、ρ3、δ1、δ2、xa、ya、xfAnd yfThe parameter value of (A) is determined by the working tooth profile of the processed three-arc harmonic gear through calculation and meshing.
2. The method for designing the tooth profile of the three-arc harmonic gear shaper cutter according to claim 1, wherein the calculation method of the step (2.6) is as follows:
(2.6.1) adopting a numerical discrete method, representing the three-arc tooth profile of the rigid wheel with the arc length u as a parameter by using s discrete points, and for any point j, ordering: u. ofjIs the parameter value of the point;
(2.6.2) mixing ujSubstituting the sagittal diameter and normal vector of the point on the flexspline tooth profile corresponding to (j ═ 1,2, …, s) into the equation n1 TΦr1When the point generates conjugate motion, the rotation angle of the rigid wheel relative to the initial position is obtained
Figure FDA0003076364610000041
Its value is recorded as
Figure FDA0003076364610000042
(2.6.3) according to each
Figure FDA0003076364610000043
Determining correspondence M12The value of each element in the transformation matrix M is obtained12And then substituted into the following equation:
Figure FDA0003076364610000044
and (2.6.4) obtaining an A-section theoretical conjugate tooth profile, a B-section theoretical conjugate tooth profile and a C-section theoretical conjugate tooth profile which are respectively conjugated with the upper arc, the middle arc and the lower arc of the three-arc tooth profile of the rigid wheel through coordinate transformation.
3. The method for designing the tooth profile of the slotting tool of the three-arc harmonic gear according to claim 1, wherein in the step (3), arc fitting is carried out on the theoretical conjugate tooth profile at the section A and the theoretical conjugate tooth profile at the section C in the slotting tool by a least square method.
4. The method of claim 1 wherein the basic tooth profile of the cutter teeth of the triple-arc harmonic gear shaper cutter further comprises tip fillet, root fillet, addendum, dedendum, tip relief, and root relief, the radius of the tip fillet of the cutter teeth of the triple-arc harmonic gear shaper cutter being raThe radius of the root fillet is rfThe height of the tooth top is haThe height of the tooth root is hfA tip clearance of caA root space of cfR of the cutter teeth of the three-arc harmonic gear slotting cuttera、rf、ha、hf、caAnd cfThe parameter values of (a) are determined by the process requirements of the three-arc harmonic gear to be machined.
5. The method of claim 1 wherein the normal tooth profile of the cutter teeth of the triple-arc harmonic gear shaper cutter consists of a tip fillet, a root fillet, an upper arc segment, a middle arc segment and a lower arc segment.
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