CN113361031B - Three-dimensional tooth shape design method for flexible gear of harmonic reducer - Google Patents
Three-dimensional tooth shape design method for flexible gear of harmonic reducer Download PDFInfo
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Abstract
The invention discloses a third harmonic reducer flexible gearThe tooth profile design method comprises the steps of establishing tooth profile parameter equations of a flexible gear and a rigid gear; calculating the minimum meshing backlash CX of the flexible gear tooth profile and the rigid gear tooth profile under different polar angles; calculating circumferential displacement v of the flexible gear generated by fit clearance between wave generator components and fit clearance between the wave generator and the inner wall of the flexible gear δ (ii) a Calculating circumferential displacement of flexible gear generated by elastic contact deformation of flexible bearing roller and inner and outer ring channels
Calculating circumferential displacement v of neutral layer curve of flexible gear caused by stretching action p (ii) a Calculating the circumferential displacement change v of each section of the flexible gear ring relative to the main section KP (ii) a Calculating the minimum meshing side clearance v of different sections of the flexible gear ring at different conjugate angles Ti (ii) a Calculating the minimum meshing side clearance CX of the section by adopting a numerical iteration method, and solving the displacement h of the tooth profiles of different sections of the flexible gear ring; obtaining a three-dimensional tooth profile displacement curve of the flexible gear; the design method fully considers the circumferential tensile displacement of the flexible gear ring under the load working condition, and the problem of avoiding the meshing interference of the harmonic reducer is solved.
Description
Technical Field
The invention relates to the technical field of harmonic reducers, particularly discloses a three-dimensional tooth shape design method of a flexible gear of a harmonic reducer, and particularly relates to a three-dimensional tooth shape design method of a flexible gear of a harmonic reducer, which is used for changing the thickness of a gear ring rim of the flexible gear.
Background
The harmonic reducer is mainly composed of a flexible gear, a rigid gear and a wave generator, is a speed reducer which realizes gear tooth meshing motion by means of periodic controllable elastic deformation of the flexible gear under the action of the wave generator and completes motion and power transmission, and has a series of advantages of large transmission ratio, wide transmission ratio range, high precision and the like. The high-sensitivity servo system can be used for precise transmission of a high-sensitivity servo system and can also be used for power transmission for transmitting large loads, and good transmission performance can be realized.
Because of the structural characteristics of the harmonic reducer, the tooth form of the rigid gear has the same shape along the tooth direction and is a spur gear, after the flexible gear is assembled with the wave generator, the gear ring part is in an approximate elliptical shape under the action of the wave generator, the cup bottom part of the flexible gear is fixedly connected with the output shaft without deformation and keeps a circular shape, and the phenomenon of taper deformation of the flexible gear from the cup mouth to the cup bottom is formed. In order to avoid the assembling interference and the meshing interference between the rigid gear tooth profile and the flexible gear tooth profile, the displacement h needs to be applied to the tooth profiles of different sections of the flexible gear tooth ring, namely, the thickness of the wheel rim of the flexible gear tooth ring is changed.
Disclosure of Invention
In view of this, the invention provides a three-dimensional tooth profile design method for a flexible gear of a harmonic reducer, which aims to solve the problems that the backlash-free contact design method for a flexible gear tooth profile and a rigid gear tooth profile in the assembly state of the conventional harmonic reducer has the phenomenon that meshing interference is easily caused when a flexible gear and a rigid gear are engaged during bearing and meshing, and the operation reliability and the service life of the harmonic reducer are affected.
In order to achieve the purpose, the invention provides the following technical scheme:
a three-dimensional tooth shape design method of a harmonic reducer flexible gear comprises the following steps:
s1: establishing a flexbile gear tooth profile parameter equation with the tooth profile arc length s as a variable;
s2: establishing a rigid wheel tooth profile parameter equation with the tooth profile arc length s as a variable;
s3: according to the motion rule of the transmission gear tooth of the harmonic reducer, calculating the minimum meshing backlash CX of the tooth profile of the flexible gear and the tooth profile of the rigid gear under different polar angles in an assembly state;
s4: calculating circumferential displacement v of the flexible gear generated by fit clearance between wave generator components (mainly clearance between the roller and the inner and outer rings) and fit clearance between the wave generator and the inner wall of the flexible gear δ ；
S5: calculate due toCircumferential displacement of flexible gear caused by elastic contact deformation of flexible bearing roller and inner and outer ring channels
S6: after the gear surface engaging force of the flexible gear is calculated, the circumferential displacement v of the neutral layer curve caused by the stretching action p ；
S7: calculating the circumferential displacement change v of each section of the flexible gear ring relative to the main section due to the torsion influence of the flexible gear cup body KP ；
S8: calculating the minimum meshing side clearance v of different sections of the flexible gear ring at different conjugate angles in consideration of the factors Ti ；
S9: judging and calculating the size of the minimum meshing side clearance CX of the section by adopting a numerical iterative calculation method, and solving the displacement h of the tooth profiles of different sections of the flexible gear ring;
s10: and (4) performing arc fitting on the displacement h of different sections obtained by calculation in the step (S9) by adopting a least square method to obtain a displacement curve suitable for the flexible gear hobbing.
Further, in step S1, a coordinate system S is established at the neutral layer curve of the flexible gear by the flexible gear tooth profile f (x f oy f ) The tooth profile parameter equation with the arc length s of the tooth profile from the tooth profile vertex as a parameter is
Further, in the step S2, a coordinate system S is established by taking the rigid wheel tooth profile as an origin at the rigid wheel revolution center c (x c oy c ) The tooth profile parameter equation with the arc length s of the tooth profile from the root of the tooth as a parameter is
Further, the backlash in step S3 refers to the shortest distance between the engaging tooth surfaces that may come into contact after the harmonic gear is assembled, and generallyExpressed by a circumferential side gap AB between a flexible gear working tooth surface and a rigid gear working tooth surface under the condition of no load, taking the periodicity and the symmetry of the meshing process of harmonic gears into consideration, and adopting a flexible gear rotating angle beta to be in the range of 0, pi/2]The circumferential backlash reflects the engagement backlash throughout the transmission. The calculation principle is as follows: in a rigid-wheel coordinate system S c (x c oy c ) In the middle, the coordinate value of the flexbile gear tooth profile arbitrary point A is (x) jA ,y jA ) At this time, the radius r corresponding to the point A j
Using the origin o as the center of a circle, r j The circular rigid wheel tooth profile with radius is intersected at the point B in a rigid wheel coordinate system S c (x c oy c ) In the formula, the coordinates of the point B are (x) jB ,y jB ). Therefore, A, B has a circumferential backlash of two points
Calculating the corresponding circumferential backlash l on the tooth profile of the flexible gear in the motion state AB At this time, the side clearance l AB Is to calculate the engagement backlash CX of the corresponding gear pair in the section
CX＝min(l AB )。
Further, the fitting clearance δ between the wave generator parts in step S4 mainly includes the clearance δ of the flexible bearing 1 And the fit clearance delta between the contact surfaces of the flexible gear and the bearing outer ring 2
δ＝K(δ 1 +δ 2 )
In the formula: k is the clearance increase coefficient caused by the abrasion of harmonic gear parts;
eliminating the displacement v caused by the gap delta δ Comprises the following steps:
in the formula:
is the extreme angle of the flexible gear deformation end.
Further, the circumferential displacement of the flexspline caused by the elastic contact deformation of the flexspline bearing rollers and the inner and outer race grooves in step S5 is
In the formula: delta. For the preparation of a coating R0 The flexibility of the wave generator is mainly the radial displacement w caused by the elastic contact deformation of the roller of the flexible gear bearing and the inner and outer ring channels for the maximum contact deformation of the flexible gear bearing under load R The displacement depends on the contact force of the flexible bearing roller and the distribution of the force along the circumferential direction of the wave generator, and is in accordance with
Further, after the flexible gear is subjected to the tooth surface engaging force in the step S6, the neutral layer curve can cause circumferential displacement v due to the stretching action p Is composed of
In the formula: t is the load torque, E is the modulus of elasticity of the flexspline material, b w The equivalent tooth width of the flexible gear engaged is shown, and t is the wall thickness of the flexible gear ring.
Further, in step S7, after the flexible gear cup is twisted due to the load, the analysis of the relative position of the gear teeth is not a point displacement caused by the twisting, but a rotation angle λ of the flexible gear bus caused by the twisting KP I.e. by
λ KP ＝Tr 0 /GI P
In the formula: g is shear modulus; I.C. A P Is the polar moment of inertia, I P ＝2πr 0 3 t，r 0 The flexible gear cup body is arranged at the inner edge of the same sectionRadius of the circumferential twist.
The circumferential displacement change v of each section of the flexible gear ring gear relative to the main section is influenced by the torsion of the flexible gear KP Is composed of
v KP ＝T(l 0 -l)/2πGr 0 2 t
In the formula: l 0 The axial distance between the main section and the bottom of the flexible gear cup is shown, and l is the axial distance between the section to be calculated and the bottom of the flexible gear cup.
Further, in the step S8, in the three-dimensional tooth profile design, the meshing side clearance amount v which is required to be reserved at different conjugate angles of different sections Ti Is composed of
Further, the method for determining and calculating the size of the cross-section minimum engagement side clearance CX in step S9 is as follows: if CX>v Ti +Δ 1 (Δ 1 A value smaller than the machining accuracy of the flexible gear teeth), h = h + Δ 2 (Δ 2 Is the incremental step size of the iteration); if CX<v Ti Then h = h- Δ 2 (ii) a If v is Ti <CX<v Ti +Δ 1 Then the iteration ends. And (4) iterating in the above way, obtaining the magnitude of the flexible gear tooth displacement of the section, wherein in order to improve the calculation efficiency, the initial flexible gear tooth displacement h of the subsequent section is the displacement h obtained by calculation of the previous section.
The beneficial effect of this scheme lies in:
compared with the existing research results, the three-dimensional tooth profile design method for the flexible gear of the harmonic gear, disclosed by the invention, considers the influence of factors such as contact deformation caused by fit clearance and load of different parts during harmonic gear assembly, torsion and circumferential stretching of a flexible gear cup body and the like, calculates the minimum meshing backlash which needs to be reserved for different sections of a flexible gear ring under the condition of ensuring that harmonic transmission does not generate meshing interference, and obtains the tooth profile displacement of different sections of different flexible gear rings. The invention obtains the deflection curve of the flexible gear three-dimensional tooth shape design, ensures that the harmonic gear transmission does not have meshing interference under rated load based on the deflection curve, reduces the tooth surface abrasion of the harmonic gear and improves the bearing capacity of the harmonic gear.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
Drawings
FIG. 1 is a two-dimensional tooth profile diagram of a flexible gear in a harmonic reducer according to the invention;
FIG. 2 is a two-dimensional tooth profile of a rigid wheel in the harmonic reducer according to the present invention;
FIG. 3 is a simplified diagram of analysis of backlash in a harmonic reducer according to the present invention;
FIG. 4 is a front, main, and rear cross-sectional definition of a harmonic reducer according to the present invention;
FIG. 5 is a flow chart of the calculation of the three-dimensional tooth profile displacement in the harmonic reducer according to the present invention;
FIG. 6 is a schematic diagram of the front section movement of the harmonic reducer of the present invention;
FIG. 7 is a schematic diagram of the motion of the main section in the harmonic reducer of the present invention;
FIG. 8 is a schematic diagram of the motion of the rear section of the harmonic reducer of the present invention;
FIG. 9 is a curve of the compliance gear shift after the harmonic reducer of the present invention is fitted.
Detailed Description
The following is further detailed by way of specific embodiments:
the invention aims to solve the technical problem of providing a three-dimensional tooth profile design method for a flexible gear of a harmonic reducer; the method is established on the premise that two-dimensional tooth profile parameters of a harmonic reducer are available, in order to avoid the transmission meshing interference phenomenon of the harmonic reducer, the circumferential stretching amount of a flexible gear ring gear under a load working condition is considered, the displacement h of different sections of the flexible gear ring gear is calculated, and a three-dimensional displacement curve of a flexible gear is obtained.
The technical scheme of the invention is explained in detail by taking a double-arc harmonic gear as an example and combining the accompanying drawings as follows:
Step 2 and FIG. 2 are schematic diagrams of two-dimensional tooth profiles of harmonic gear rigid gear, and a coordinate system S established by taking the revolution center of the rigid gear as an origin c (x c oy c ) In the method, a tooth profile parameter equation taking the tooth profile arc length s from the root bottom point as a parameter is
Step 3, the meshing backlash refers to the shortest distance between meshing tooth surfaces which are possibly contacted after the harmonic gear is assembled, and is usually represented by a circumferential backlash AB between a flexible gear working tooth surface and a rigid gear working tooth surface under the no-load condition, as shown in fig. 3, in consideration of the periodicity and symmetry of the harmonic gear meshing process, a flexible gear rotating angle beta epsilon [0, pi/2 ]]The circumferential backlash reflects the engagement backlash throughout the transmission. The calculation principle is as follows: in a rigid-wheel coordinate system S c (x c oy c ) In the middle, the coordinate value of the flexbile gear tooth profile arbitrary point A is (x) jA ,y jA ) At this time, the radius r corresponding to the point A j
Using the origin o as the center of a circle, r j The circular rigid wheel tooth profile with radius is intersected at the point B in a rigid wheel coordinate system S c (x c oy c ) In the formula, the coordinates of the point B are (x) jB ,y jB ). Therefore, A, B has a circumferential backlash of two points
Calculating the corresponding circumferential backlash l on the tooth profile of the flexible gear in the motion state AB At this time, the side clearance l AB Is to calculate the engagement backlash CX of the corresponding gear pair in the section
CX＝min(l AB )
δ＝K(δ 1 +δ 2 )
In the formula: k is the clearance increase factor due to harmonic gear part wear.
Eliminating the displacement v caused by the gap delta δ Comprises the following steps:
in the formula:
is the extreme angle of the flexible gear deformation end.
Maximum contact deflection delta of a flexspline bearing under load R0 According to the document "Load Analysis of Flexible Ball Bearing in a Harmonic Reducer", xiong Y, zhu Y, yan K. Journal of mechanical design,2020, 142 (2): 022302. Therefore, the circumferential displacement due to the contact deformation of the compliant bearing is
6, after the flexible gear is subjected to the tooth surface engaging force, the neutral layer curve can cause circumferential displacement v due to the stretching action p According to the document "M.H. Itanov. Harmonic gearing [ M]Beijing: national defense industry Press, 1987. "calculated
In the formula: t is the load torque, E is the modulus of elasticity of the flexspline material, b w The equivalent tooth width of the flexible gear engaged is shown, and t is the wall thickness of the flexible gear ring.
Step 7, after the flexible gear cup body is twisted due to the load, the relative position analysis of the gear teeth is not the displacement of the point caused by the twisting, because the relative position analysis of the gear teeth is the same in the same section along the circumferential direction, but the rotation angle lambda of the flexible gear bus is caused KP 。
λ KP ＝Tr 0 /GI P
In the formula: g is shear modulus; i is P Is the polar moment of inertia, I P ＝2πr 0 3 t，r 0 The radius of the flexible gear cup body which is twisted in the same section along the circumferential direction.
The circumferential displacement change v of each section of the flexible gear ring gear relative to the main section is influenced by the torsion of the flexible gear KP Is composed of
v KP ＝T(l 0 -l)/2πGr 0 2 t
In the formula: l 0 The axial distance between the main section and the bottom of the flexible gear cup is shown, and l is the axial distance between the section to be calculated and the bottom of the flexible gear cup.
Step 8, in the three-dimensional tooth profile design, the meshing side clearance amount of different sections required to be reserved at different conjugate anglesv Ti Is composed of
And 9, step 5, a flow chart for calculating the tooth profile displacement h of each section of the flexible gear ring gear is shown in fig. 5. Judging and calculating the size of the minimum meshing side clearance CX of the cross section if CX > v Ti +Δ 1 (Δ 1 A value smaller than the machining accuracy of the flexible gear teeth), h = h + Δ 2 (Δ 2 Is the incremental step size of the iteration); if CX < v Ti Then h = h- Δ 2 (ii) a If v is Ti ＜CX＜v Ti +Δ 1 Then the iteration ends. Iteration is carried out in this way, the magnitude of the flexible gear tooth displacement of the section is obtained, and in order to improve the calculation efficiency, the initial flexible gear tooth displacement h of the subsequent section is the displacement h obtained by calculation of the previous section; FIG. 4 is a front, main, and rear cross-sectional definition of a harmonic reducer according to the present invention; FIGS. 6-8 are motion simulation diagrams of the front, main, and rear sections after deflection;
and step 10, performing least square arc fitting on the displacement h of the different sections obtained by calculation in the step 9 to obtain a displacement curve applicable to the flexible gear hobbing. FIG. 9 is a graph of compliance gear deflection.
The foregoing is merely an example of the present invention and common general knowledge of known specific structures and features of the embodiments is not described herein in any greater detail. It should be noted that, for those skilled in the art, without departing from the structure of the present invention, several changes and modifications can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicability of the present invention.
Claims (9)
1. A three-dimensional tooth form design method of a harmonic reducer flexible gear is characterized by comprising the following steps: the method comprises the following steps:
s1: establishing a flexbile gear tooth profile parameter equation with the tooth profile arc length s as a variable;
s2: establishing a rigid wheel tooth profile parameter equation with the tooth profile arc length s as a variable;
s3: calculating the minimum meshing backlash CX of the tooth profile of the flexible gear and the tooth profile of the rigid gear under different polar angles in an assembly state according to the motion law of the transmission gear of the harmonic reducer;
s4: calculating circumferential displacement v of the flexible gear generated by the fit clearance between the roller and the inner and outer rings between wave generator components and the fit clearance between the wave generator and the inner wall of the flexible gear δ ；
S5: calculating circumferential displacement of the flexspline due to elastic contact deformation of the flexible bearing roller and the inner and outer race grooves
S6: after the gear surface engaging force of the flexible gear is calculated, the circumferential displacement v of the neutral layer curve caused by the stretching action p ；
S7: calculating the circumferential displacement change v of each section of the flexible gear ring relative to the main section due to the torsion influence of the flexible gear cup body KP ；
S8: calculating the minimum meshing side clearance v of different sections of the flexible gear ring at different conjugate angles in consideration of the factors Ti ；
S9: judging and calculating the size of the minimum meshing side clearance CX of the section by adopting a numerical iterative calculation method, and solving the displacement h of the tooth profiles of different sections of the flexible gear ring; the method for judging and calculating the size of the minimum meshing side clearance CX of the cross section comprises the following steps: if CX>v Ti +Δ 1 ，Δ 1 If the machining accuracy is smaller than the machining accuracy of the flexible gear teeth, h = h + Δ 2 ，Δ 2 Is the incremental step size of the iteration; if CX<v Ti Then h = h- Δ 2 (ii) a If v is Ti <CX<v Ti +Δ 1 If so, the iteration is ended; iteration is carried out in this way, the flexible gear tooth profile displacement of the section is obtained, and in order to improve the calculation efficiency, the flexible gear tooth initial displacement h of the subsequent section is the displacement h obtained by calculation of the previous section;
s10: and (5) performing least square circular arc fitting on the displacement h of different sections obtained by calculation in the step (S9) to obtain a displacement curve suitable for the flexible gear hobbing.
2. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 1, characterized in that: in the step S1, a coordinate system S is established at the neutral layer curve of the flexible gear by the flexible gear tooth profile f (x f oy f ) The tooth profile parameter equation with the arc length s of the tooth profile from the tooth profile vertex as a parameter is
3. The three-dimensional tooth form design method of the harmonic reducer flexible gear according to claim 2, characterized in that: in the step S2, a coordinate system S is established by taking the rigid wheel tooth profile as an origin at the rigid wheel revolution center c (x c oy c ) The tooth profile parameter equation with the arc length s of the tooth profile from the root of the tooth as a parameter is
4. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 3, characterized in that: the meshing backlash in the step S3 is the shortest distance between meshing tooth surfaces that may be contacted after the harmonic gear is assembled, and is usually expressed by a circumferential backlash AB between a flexible gear working tooth surface and a rigid gear working tooth surface under a no-load condition, and in consideration of periodicity and symmetry of a harmonic gear meshing process, a flexible gear rotating angle β e [0, pi/2 ]]The circumferential backlash reflects the meshing backlash in the whole transmission process; the calculation principle is as follows: in a rigid-wheel coordinate system S c (x c oy c ) In the middle, the coordinate value of the flexbile gear tooth profile arbitrary point A is (x) jA ,y jA ) At this time, the radius r corresponding to the point A j
Using the origin o as the center of a circle, r j The circular rigid wheel tooth profile with radius is intersected at the point B in a rigid wheel coordinate system S c (x c oy c ) In the formula, the coordinates of the point B are (x) jB ,y jB ) Therefore, the circumferential backlash at A, B is
Calculating the corresponding circumferential backlash l on the tooth profile of the flexible gear in the motion state AB At this time, the side clearance l AB Is to calculate the engagement backlash CX of the corresponding gear pair in the section
CX＝min(l AB )。
5. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 4, characterized in that: in step S4, the fit clearance delta between the wave generator components mainly comprises the clearance delta of the flexible bearing 1 And the fit clearance delta between the contact surfaces of the flexible gear and the bearing outer ring 2
δ＝K(δ 1 +δ 2 )
In the formula: k is the clearance increase coefficient caused by the abrasion of harmonic gear parts;
eliminating the displacement v caused by the gap delta δ Comprises the following steps:
in the formula:
is the extreme angle of the flexible gear deformation end.
6. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 5, characterized in that: in step S5, the flexible gear circumferential displacement generated by the elastic contact deformation of the flexible gear bearing roller and the inner and outer ring channels is
In the formula: delta R0 The flexibility of the wave generator is mainly the radial displacement w caused by the elastic contact deformation of the roller of the flexible gear bearing and the inner and outer ring channels for the maximum contact deformation of the flexible gear bearing under load R The displacement depends on the contact force of the flexible bearing roller and the distribution of the force along the circumferential direction of the wave generator, and is in accordance with
7. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 6, characterized in that: after the flexible gear is subjected to the tooth surface engaging force in the step S6, the neutral layer curve can cause circumferential displacement v due to the stretching action p Is composed of
In the formula: t is the load torque, E is the modulus of elasticity of the flexspline material, b w The equivalent tooth width of the flexible gear engaged is shown, and t is the wall thickness of the flexible gear ring.
8. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 7, characterized in that: in the step S7, after the flexible gear cup body is twisted due to the load action, the analysis of the relative position of the gear teeth is not the point displacement caused by the twisting, but the corner lambda of the flexible gear bus is caused KP I.e. by
λ KP ＝Tr 0 /GI P
In the formula: g is shear modulus; i is P Is the polar moment of inertia, I P ＝2πr 0 3 t，r 0 The radius of the flexible gear cup body twisted along the circumferential direction in the same section;
the circumferential displacement change v of each section of the flexible gear ring gear relative to the main section is influenced by the torsion of the flexible gear KP Is composed of
v KP ＝T(l 0 -l)/2πGr 0 2 t
In the formula: l 0 The axial distance between the main section and the bottom of the flexible gear cup is shown, and l is the axial distance between the section to be calculated and the bottom of the flexible gear cup.
9. The three-dimensional tooth form design method of the harmonic reducer flexspline of claim 8, characterized in that: in the step S8, in the three-dimensional tooth profile design, the meshing side clearance amount v which is required to be reserved at different conjugate angles of different sections Ti Is composed of
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