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CN108470102B - Design method and processing method of pinion tooth surface for pre-control of meshing performance - Google Patents

Design method and processing method of pinion tooth surface for pre-control of meshing performance Download PDF

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CN108470102B
CN108470102B CN201810237558.XA CN201810237558A CN108470102B CN 108470102 B CN108470102 B CN 108470102B CN 201810237558 A CN201810237558 A CN 201810237558A CN 108470102 B CN108470102 B CN 108470102B
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tooth surface
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numerical control
small wheel
tooth
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CN108470102A (en
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彭先龙
徐琪超
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Xian University of Science and Technology
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Abstract

本发明公开了一种面向啮合性能预控的小轮齿面设计方法及加工方法，所述小轮与直刃刀具单参数包络法近似加工的面齿轮啮合，所述小轮齿面的设计方法包括几何传动误差的预置、小轮齿面的反求和重新构造小轮齿面，所述几何传动误差为2至8阶的多项式函数并预置于小轮齿面的反求过程中，利用反求齿面和接触轨迹与椭圆的长度重新构造小轮齿面。进一步阐述了小轮齿面的加工方法，所述加工方法包括选用数控机床和成型刀具、初始数控规律的建立和数控规律的修正，通过数控规律的迭代修正，使得数控加工齿面逼近小轮齿面并加工小轮，本方法能够有效地预控小轮与面齿轮啮合的几何啮合性能。

Figure 201810237558

The invention discloses a pinion tooth surface design method and a processing method for pre-control of meshing performance. The method includes the presetting of the geometric transmission error, the inversion of the pinion tooth surface and the reconstruction of the pinion tooth surface, and the geometric transmission error is a polynomial function of order 2 to 8 and is preset in the inversion process of the pinion tooth surface , reconstruct the pinion tooth surface using the reversed tooth surface and the contact trajectory and the length of the ellipse. The processing method of pinion tooth surface is further expounded. The processing method includes the selection of CNC machine tools and forming tools, the establishment of the initial CNC law and the correction of the CNC law, and the iterative correction of the CNC law to make the CNC machined tooth surface approach the pinion teeth The method can effectively pre-control the geometric meshing performance of the meshing between the pinion and the face gear.

Figure 201810237558

Description

Small wheel tooth surface design method and machining method for meshing performance pre-control
The technical field is as follows:
the invention relates to the field of gear transmission, in particular to a method for designing and processing a small wheel tooth surface facing meshing performance pre-control.
Background art:
bevel gears in the transmission of cylindrical gears (small wheels) in mesh with bevel gears are called asThe face gear is machined based on the traditional method, so that the cutter universality is poor, special manufacturing equipment needs to be developed, the popularization and the application of the face gear are extremely not facilitated, and most of the face gear is only suitable for gears with orthogonal straight teeth and oblique teeth surfaces. Various face gears can be machined on the existing bevel gear machine by using the straight-edge cutter and by using a special fixture, as shown in fig. 1, the method is explained in detail in "machining equipment and machining method for manufacturing various face gears by using the straight-edge cutter", and is incorporated by reference. The face gear processed by the straight-edge cutter can adopt a double-parameter enveloping method, the face gear processed by the method is the same as the face gear processed by the traditional method, but the processing efficiency is not high, the face gear processed by the straight-edge cutter based on the single-parameter enveloping method has high processing efficiency due to line contact, but certain tooth surface deviation can be brought compared with the traditional processing method, and the deviated tooth surface is called as the approximate tooth surface sigma of the face gear2pThe tooth surface machined by the traditional method is called theoretical tooth surface sigma2Approximate tooth surface ∑2pWith theoretical tooth flank ∑2Inscribed in curve Cp, as shown in FIG. 2, and ∑2pApproximately Σ along Cp2However, the adverse effect of the tooth surface deviation on the meshing performance is more prominent, and particularly, the length of the major axis of the contact ellipse in the geometric meshing performance is shortened, which inevitably reduces the transmission strength.
However, in gear transmission, good meshing performance is far more important than machining accuracy, and the tooth surfaces of various gears are usually modified and designed to deviate from the theoretical tooth surfaces, and the prior patent technology discloses a series of tooth surface modification or modification design methods, such as: US6205879B1, US5580298, CN103577713A, CN103440356A, CN 104832623A. In addition, the prior literature also provides various tooth surface modifications or modified designs for the meshing performance of face gears or other gear transmissions machined by conventional methods, such as: journal of Mechanical Design, 2016, 138 (4): 043302-13; journal of the university of west ampere, 2017, 51 (07): 98-104; aeronautics, 2014, 29 (07): 1752-; chinese mechanical engineering, 2012, 23 (08): 992-996. Obviously, the straight edge cutter and the single parameter envelope method are used for efficiently machining the face gear, and necessary measures must be taken to eliminate the adverse effect of the face gear tooth surface deviation on the meshing performance.
In order to solve the technical problems, a method for designing the tooth surface of the small wheel facing to meshing performance pre-control is provided, and a method for machining the tooth surface is further explained.
The invention content is as follows:
in order to eliminate the adverse effect on the meshing performance caused by the deviation of the approximate tooth surface of the face gear processed by the straight-edge cutter single-parameter envelope method, the invention provides a small wheel tooth surface design method and a small wheel tooth surface processing method for meshing performance pre-control.
In order to achieve the purpose, the invention adopts the technical scheme that:
a method for designing a small wheel tooth surface facing meshing performance pre-control is characterized by comprising the following steps: presetting geometric transmission errors, reversely solving the tooth surfaces of the small wheels, and reconstructing the tooth surfaces of the small wheels;
the presetting of the geometric transmission error comprises the following steps of S101-S104:
s101 construction geometric transmission error polynomial
Figure GSB0000194301860000021
C is mentioned0，c1，c2，…，cσIs a coefficient to be determined, sigma is the order of the geometric transmission error polynomial, and a positive integer of 2-8 is taken, epsilon1Is a small wheel corner;
s102, determining the shape of a geometric transmission error curve according to the amplitude of the expected geometric transmission error in the meshing period and the slope of the intersection point of adjacent geometric transmission errors;
s103, determining the order sigma of the geometric transmission error polynomial according to the shape of the geometric transmission error curve, and selecting the geometric transmission error delta epsilon on the geometric transmission error curve2And known values of their 1 st derivative, the total number of these known values being σ +1, and determining Δ ε2And sigma +1 epsilon corresponding to known value of 1-order derivative1The value of (d);
s104 will be2And known value of its 1 st derivative, epsilon1Substituting the numerical value into a geometric transmission error polynomial to construct a matrix equation and solveSolving the coefficient to be determined c0，c1，c2，…，cσAnd is substituted back into the geometric transmission error polynomial to obtain delta epsilon2Angle epsilon with small wheel1The functional relationship of (a).
The reverse process of the small wheel tooth surface is as follows: when the small wheel has approximate tooth surface sigma2pWhen the face gear is engaged, according to the angle of rotation1＝(ε2N2-Δε2N2)/N1Rotating face gear and small gear, face gear approximate tooth surface sigma2pThe envelope of the tooth flank family on the small wheel is the inverse flank Σ of said small wheel12pOf said epsilon2Is the angle of rotation of the face gear, N1、N2The number of teeth of the small wheel and the face gear, respectively, the approximate tooth flank sigma of said face gear2pThe cutting tool is processed by a straight-edge cutting tool single-parameter enveloping method;
the reconstructing the pinion tooth surface includes steps S105 to S111:
s105, determining approximate tooth surface sigma of face gear2pWith its theoretical tooth flank sigma2And with a series of discrete points FiRepresents the internal common tangent line Cp, i is 1, 2, …, n is an odd number which is more than or equal to 3;
s106 point FiIs the tooth surface sigma2p、∑12pUsing differential geometry to solve the tooth flank sigma2p、∑12pConjugation Point FiA principal curvature, a principal direction;
s107 utilizing conjugate point FiThe main curvature and the main direction of the contact point determine the major axis direction of the contact ellipse at the point;
s108 at conjugate point FiOn the major axis of the contact ellipse, FiTwo ends a are respectively taken at two sides ofi1、ai2And make the length | Fiai1|＝|Fiai2A is the length of the desired preset contact ellipse major axis;
s109 over-contact end point a of ellipse long axisi1、ai2Are respectively made parallel to the conjugate point FiSquare of2p、∑12pStraight line of common normal line and tooth surface sigma12pIntersect at the point bi1、bi2；
S110 direction sigma12pRespectively extend inside ai1bi1、ai2bi2To Pi1、Pi2And make the length | bi1Pi1|＝|bi2Pi2|＝0.00635mm；
S111 over conjugate point FiAnd point Pi1、Pi2The second order and the smooth continuous curve with more than second order replace the conjugate point FiSquare of2p、∑12pContact line L of21Making n smooth continuous curves with second order and above, and forming small wheel tooth surface sigma by using said n smooth continuous curves1d。
The method for processing the small wheel tooth surface reconstructed by the small wheel tooth surface design method facing the meshing performance precontrol is characterized by comprising the following steps of: selecting a machining tool, selecting a numerical control machine tool, establishing an initial numerical control rule and correcting the numerical control rule;
the processing cutter adopts a forming cutter, and the working curved surface of the cutter used for cutting or grinding the tooth socket of the small wheel is formed by a standard involute tooth surface sigma1The sectional line of the normal surface of the pinion tooth socket rotates for a circle around the axis of the cutter;
the numerical control machine tool selected by the numerical control machine tool comprises 5 numerical control motion shafts, wherein three of the numerical control motion shafts are numerical control translational motion shafts X, Y, Z, the other two numerical control rotational motion shafts A, B are numerical control rotational motion shafts, a shaft A is a workpiece spindle, and a high-speed free rotation cutter spindle C is arranged on a shaft B;
the establishment of the initial numerical control rule comprises the following steps of S201 to S204:
s201, establishing that the cutter is used for processing the workpiece with standard involute tooth surface sigma1The abstract machining model of the small wheel;
s204, establishing the standard involute tooth surface sigma based on the principle that the position vector and normal vector of the cutter relative to the small wheel are equal in abstract machining and numerical control machining1The numerical control processing mathematical model of the small wheel;
s205 determining the turning angle of the small wheelψ1Expanding the motion law of each numerical control axis into psi as the motion law of each numerical control axis of independent variable1And obtaining initial numerical control law polynomial of each numerical control axis by the Taylor series at the position of 0:
Figure GSB0000194301860000031
Figure GSB0000194301860000032
Figure GSB0000194301860000033
Figure GSB0000194301860000034
Figure GSB0000194301860000035
g1～g7、g8～g14、g15～g21、g22～g28、g29～g35coefficients of numerical control regular polynomials from 0 th order to 6 th order of the numerical control axis X, Y, Z, A, B, respectively;
s204, the numerical control rule polynomial is substituted back into a numerical control machining mathematical model of the small wheel to obtain the numerical control tooth surface of the small wheel
Figure GSB0000194301860000036
The correction of the numerical control rule is used for correcting each order coefficient of the initial numerical control rule polynomial so as to enable the numerical control tooth surface of the small wheel
Figure GSB0000194301860000037
Approach to the small wheel tooth surface ∑1dAnd machining the small wheel, comprising steps S205-S208:
s205 the standard involute tooth surface sigma of the small wheel1Numerical control tooth surface
Figure GSB0000194301860000038
Tooth flank sigma1dDividing the grid into h-1 equal parts along the tooth width direction to obtain h longitudinal grid lines, dividing the grid into w-1 equal parts along the tooth height direction to obtain w vertical grid lines, wherein the intersection points of the longitudinal grid lines and the vertical grid lines are tooth surface grid points, the number of the grid points is λ 1, 2, 3, … and w × d from the tooth top to the tooth bottom;
s206 calculating small wheel tooth surface sigma1dStandard involute tooth surface sigma for small relative wheel1Normal deviation vector of (2): e1d＝n1dλ·(R1dλ-R1λ)，n1dλ、R1dλRespectively, the small wheel tooth surface ∑1dUnit normal vector, position vector, R, of the lambda-th grid point1λIs a standard involute tooth surface sigma of a small wheel1A position vector of the λ -th grid point;
s207, correcting each order coefficient of the numerical control law polynomial for the xi time, and the method comprises the following steps:
the method comprises the following steps: in solving the xi correction, the numerical control tooth surface of the small wheel
Figure GSB0000194301860000041
Standard involute tooth surface sigma for small relative wheel1Normal deviation vector of (2):
Figure GSB0000194301860000042
respectively, a small wheel digital control tooth surface
Figure GSB0000194301860000043
Unit normal vector and position vector of the lambda-th grid point in the xi correction;
step two: solving the correction matrix omega of the xi order(ξ)The elements are
Figure GSB0000194301860000044
And is
Figure GSB0000194301860000045
Figure GSB0000194301860000046
Respectively, a small wheel digital control tooth surface
Figure GSB0000194301860000047
In the xi correction, the polynomial coefficient g of the numerical control lawκWith disturbance quantity Δ gκThen, the unit normal vector, position vector, κ ═ 1, 2, 3 …, 35 for the λ -th grid point;
step three: solving the correction matrix equation to obtain the coefficients of each order of the zeta-corrected numerical control law polynomial
Figure GSB0000194301860000048
Comprises the following steps:
Figure GSB0000194301860000049
when xi is 1, taking each order coefficient of the initial numerical control law polynomial to solve the correction matrix equation, repeating the first step to the third step, and iteratively solving the correction matrix equation until xi is 1
Figure GSB00001943018600000410
When approaching to 0 vector, the small wheel numerical control tooth surface
Figure GSB00001943018600000411
Approach to the small wheel tooth surface ∑1dObtaining final coefficients of each order of the numerical control law polynomial;
s208, installing the cutter and the small wheel, compiling a numerical control program by utilizing coefficients of various orders of the final numerical control law polynomial, debugging a machine tool and machining the small wheel.
Compared with the prior art, the invention has the following advantages:
1. the meshing performance of the approximate tooth surface of the face gear processed by the controllable straight-edge cutter single-parameter envelope method;
2. the contact path, geometric transmission error and contact ellipse length of the meshing performance can be quantitatively expressed, and corresponding small wheel tooth surfaces can be designed and manufactured, so that the meshing performance is pre-controlled with high flexibility;
3. the processing efficiency of small gears and face gears is considered, the general number of teeth of the face gears is large, the efficiency of processing the face gears by the straight-edge cutter single-parameter envelope method is high, the number of teeth of the small gears is small, although the tooth surfaces of the small gears are complex, the tooth profile direction of the small gears is mainly formed by involute with certain displacement, and the efficiency is high by adopting a forming cutter to process.
Drawings
FIG. 1 is a schematic diagram of the machining of various face gears on a bevel gear machine tool using a straight edge tool and a special fixture.
In the figure: p-straight edge cutter, 2-various face gears, 3-cutter spindle box, 4-workpiece spindle box, 5-special cutter fixture, and 6-special workpiece fixture.
FIG. 2 is a graph of an approximate tooth surface sigma of a face gear processed by a straight-edge tool single parameter envelope method2pWith its theoretical tooth flank sigma2Schematic deviation of (a).
FIG. 3 is a diagram of a high order geometric transmission error curve.
FIG. 4 is a reverse schematic view of the flank of a small wheel.
FIG. 5 standard involute flank sigma of small wheel1Inverse of tooth flank sigma12pSchematic representation of (a).
FIG. 6 reconstruction of the pinion tooth flank ∑1dSchematic representation of (a).
FIG. 7 pinion flank Σ1dAnd a smooth continuous curve diagram of the second order and above on the tooth surface.
FIG. 8 is a view for machining the flank of a small wheel1dSchematic diagram of the numerical control machine tool.
FIG. 9 Small wheel flank ∑1dThe grid points on and their numbering scheme.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
the technical problem solved by the invention is as follows: aiming at the problem that the meshing performance of the approximate tooth surfaces of various face gears processed by a straight-edge cutter single-parameter envelope method is not very ideal, particularly the problem that the length of a contact ellipse is too short, a small wheel tooth surface design method for meshing performance pre-control is provided, and a processing method of the tooth surfaces is further explained.
Example (b): a design method and a processing method of a small gear tooth surface in oblique offset helical tooth surface gear transmission are provided.
The method for designing the tooth surface of the small wheel comprises the following steps: presetting geometric transmission errors, reversely solving small gear tooth surfaces and reconstructing small gear tooth surfaces, (1) presetting the geometric transmission errors as shown in steps S101 to S104:
s101 construction geometric transmission error polynomial
Figure GSB0000194301860000051
C is mentioned0，c1，c2，…，cσIs a coefficient to be determined, sigma is the order of the geometric transmission error polynomial, and a positive integer of 2-8 is taken, epsilon1Is a small wheel corner;
s102, determining the shape of a geometric transmission error curve according to the amplitude of the expected geometric transmission error in the meshing period and the slope of the intersection point of adjacent geometric transmission errors, wherein FIG. 3 is a 4-order transmission error curve which reflects the amplitude at the starting point, the amplitude and the slope at the end point of the meshing period and determines the shape of the error curve;
s103, determining the order sigma of the geometric transmission error polynomial according to the shape of the geometric transmission error curve, and selecting the geometric transmission error delta epsilon on the geometric transmission error curve2And known values of their 1 st derivative, the total number of these known values being σ +1, and determining Δ ε2And sigma +1 epsilon corresponding to known value of 1-order derivative1The values of (A) are known as 3 Deltaε in FIG. 32And 2 known values of its 1 st derivative, which correspond to ∈1Is also determined, e.g. epsilon11、ε12、ε13Shown;
s104 will be2And known value of its 1 st derivative, epsilon1Substituting the numerical value into a geometric transmission error polynomial to construct a matrix equation,solving the coefficient c to be solved0，c1，c2，…，cσAnd is substituted back into the geometric transmission error polynomial to obtain delta epsilon2Angle epsilon with small wheel1According to fig. 3, the following equation can be determined:
Figure GSB0000194301860000052
and further writing the obtained product into a matrix equation form, and solving each item coefficient in the geometric error polynomial.
(2) Reverse calculation of small wheel tooth surface: as shown in fig. 4, when the small wheel has an approximate tooth surface Σ2pWhen the face gear is engaged, according to the angle of rotation1＝(ε2N2-Δε2N2)/N1Rotating face gear and small gear, face gear approximate tooth surface sigma2pThe envelope of the tooth flank family on the small wheel is the inverse flank Σ of the small wheel12pThe relationship between the back-off tooth surface and the small wheel standard involute tooth surface is shown in FIG. 5, epsilon2Is the angle of rotation of the face gear, N1、N2The number of teeth of the small gear and the face gear respectively;
approximate tooth surface sigma of face gear2pThe gear is processed by a straight-edge cutter single-parameter enveloping method, and the specific description can refer to processing equipment and processing method for manufacturing various face gears by the straight-edge cutter.
(3) The reconfiguration of the pinion tooth surface, as shown in steps S105 to S111:
s105, determining the approximate tooth surface sigma of the bevel offset bevel tooth surface gear2pWith its theoretical tooth flank sigma2And with a series of discrete points FiDenotes the internal common tangent line Cp, i is 1, 2, …, n is an odd number equal to or more than 3, and FIG. 2 shows the approximate tooth surface sigma of the oblique offset helical tooth surface gear2pWith its theoretical tooth flank sigma2The internal common tangent Cp of (a);
s106 point FiIs the tooth surface sigma2p、∑12pUsing differential geometry to solve the tooth flank sigma2p、∑12pConjugation Point FiPrincipal curvature of, principalDirection;
s107 utilizing conjugate point FiThe principal curvature and principal direction of (F) determine the major axis direction of the contact ellipse at that point, and the vector η in fig. 6 represents the conjugate point FiThe direction of the major axis of the contact ellipse;
s108 As shown in FIG. 6, at the conjugate point FiOn the major axis of the contact ellipse, FiTwo ends a are respectively taken at two sides ofi1、ai2And make the length | Fiai1|＝|Fiai2A is the length of the desired preset contact ellipse major axis;
s109 As shown in FIG. 6, the end a of the major axis of the over-contact ellipsei1、ai2Are respectively made parallel to the conjugate point FiSquare of2p、∑12pStraight line of common normal line and tooth surface sigma12pIntersect at the point bi1、bi2；
S110 shows in FIG. 6, the direction ∑12pRespectively extend inside ai1bi1、ai2bi2To Pi1、Pi2And make the length | bi1Pi1|＝|bi2Pi2|＝0.00635mm；
S111 is shown in FIG. 6 as passing through conjugate point FiAnd point Pi1、Pi2Second order and above smooth continuous Curve Curve replaces conjugate point FiSquare of2p、∑12pContact line L of21The smooth continuous curve has n number, and the tooth surface sigma of the small wheel is formed by the smooth continuous curve with the second order and above as shown in FIG. 71d。
Small wheel tooth flank sigma reconstructed by the small wheel tooth flank design method1dThe processing method comprises the following steps: selecting a machining tool, selecting a numerical control machine tool, establishing an initial numerical control rule and correcting the numerical control rule;
(1) selecting a machining cutter:
the working tool is a profiled tool, as shown in fig. 8, which is used for cutting or grinding the working surface of the small wheel tooth socket and consists of a tool with standard involute tooth flanks sigma1Of small wheel gulletThe sectional line of the normal surface is formed by rotating a circle around the axis of the cutter;
(2) selecting a numerical control machine tool:
the numerical control machine tool selected by the numerical control machine tool comprises 5 numerical control motion shafts, as shown in 8, the machine tool comprises 5 numerical control motion shafts, three numerical control translational motion shafts X, Y, Z and two numerical control rotary motion shafts A, B, wherein the shaft A is a workpiece spindle, a small wheel is fixedly connected with the workpiece spindle A through an installation shaft, a high-speed freely-rotating tool spindle C is arranged, a tool is fixedly connected with the tool spindle C through a tool holder, a high-speed freely-rotating tool spindle C is arranged on the shaft B, and the tooth surface sigma of the small wheel is machined through the compound motion of the numerical control motion shafts X, Y, Z, A, B1d。
(3) The establishment of the initial numerical control rule comprises the following steps of S201 to S204:
s201 establishing a forming tool to process a tooth surface sigma with a standard involute1The abstract machining model of the small wheel;
s202, establishing a standard involute tooth surface sigma based on the principle that the position vector and normal vector of the cutter relative to the small wheel are equal in abstract machining and numerical control machining1The numerical control processing mathematical model of the small wheel;
s203, determining the rotation angle phi1Expanding the motion law of each numerical control axis into psi as the motion law of each numerical control axis of independent variable1And obtaining initial numerical control law polynomial of each numerical control axis by the Taylor series at the position of 0:
Figure GSB0000194301860000071
Figure GSB0000194301860000072
Figure GSB0000194301860000073
Figure GSB0000194301860000074
Figure GSB0000194301860000075
g1～g7、g8～g14、g15～g21、g22～g28、g29～g35coefficients of numerical control regular polynomials from 0 th order to 6 th order of the numerical control axis X, Y, Z, A, B, respectively;
s204, the numerical control rule polynomial of the numerical control shaft k is back substituted into a numerical control machining mathematical model of the small wheel to obtain the numerical control tooth surface of the small wheel
Figure GSB0000194301860000076
(4) And (3) correcting the numerical control rule:
the correction of the numerical control law is used for correcting each order coefficient of the initial numerical control law polynomial so as to ensure that the small wheel numerically-controlled machining tooth surface
Figure GSB0000194301860000077
Approach to the small wheel tooth surface ∑1dAnd machining the small wheel, comprising steps S205-S208:
s205 As shown in FIG. 9, the standard involute tooth surface sigma of the small wheel is processed1Numerical control tooth surface
Figure GSB0000194301860000078
Tooth flank sigma1dDividing the grid into h-1 equal parts along the tooth width direction to obtain h longitudinal grid lines, dividing the grid into w-1 equal parts along the tooth height direction to obtain w vertical grid lines, wherein the intersection points of the longitudinal grid lines and the vertical grid lines are tooth surface grid points, the number of the grid points is λ 1, 2, 3, … and h × w;
s206 calculating small wheel tooth surface sigma1dStandard involute tooth surface sigma for small relative wheel1Normal deviation vector of (2): e1d＝n1dλ·(R1dλ-R1λ)，n1dλ、R1dλAre respectivelySmall wheel tooth surface sigma1dUnit normal vector, position vector, R, of the lambda-th grid point1λIs a standard involute tooth surface sigma of a small wheel1A position vector of the λ -th grid point;
s207, correcting each order coefficient of the numerical control law polynomial for the xi time, and the method comprises the following steps:
the method comprises the following steps: in solving the xi correction, the numerical control tooth surface of the small wheel
Figure GSB0000194301860000081
Standard involute tooth surface sigma for small relative wheel1Normal deviation vector of (2):
Figure GSB0000194301860000082
respectively, a small wheel digital control tooth surface
Figure GSB0000194301860000083
Unit normal vector and position vector of the lambda-th grid point in the xi correction;
step two: solving the correction matrix omega of the xi order(ξ)The elements are
Figure GSB0000194301860000084
And is
Figure GSB0000194301860000085
Figure GSB0000194301860000086
Respectively, a small wheel digital control tooth surface
Figure GSB0000194301860000087
In the xi correction, a certain order coefficient g of a numerical control law polynomialκWith disturbance quantity Δ gκThen, the unit normal vector, position vector, κ ═ 1, 2, 3 …, 35 for the λ -th grid point;
step three: solving the correction matrix equation to obtain the coefficients of each order of the zeta-corrected numerical control law polynomial
Figure GSB0000194301860000088
Comprises the following steps:
Figure GSB0000194301860000089
when xi is 1, taking each order coefficient of the initial numerical control law polynomial to solve the correction matrix equation, repeating the first step to the third step, and iteratively solving the correction matrix equation until xi is 1
Figure GSB00001943018600000810
When approaching to 0 vector, the small wheel numerical control tooth surface
Figure GSB00001943018600000811
Approach to the small wheel tooth surface ∑1dObtaining final coefficients of each order of the numerical control law polynomial;
s208, installing a cutter and a small wheel, compiling a numerical control program by utilizing each order coefficient of the final numerical control law polynomial, debugging a machine tool and machining the small wheel.
All the steps are not only applicable to the design and manufacture of the small gear tooth surface in the oblique offset oblique tooth surface gear transmission, but also applicable to the design and processing of the small gear tooth surface in other types of surface gear transmissions, for example, in fig. 4, when gamma is 90 degrees and q is 0 and a surface gear is meshed with a straight-tooth small gear, the straight-tooth small gear tooth surface in the orthogonal straight-tooth surface gear transmission can be designed and processed, when gamma is 90 degrees and q is 0 and the surface gear is meshed with the straight-tooth small gear, the straight-tooth small gear tooth surface in the oblique offset oblique tooth surface gear transmission can be designed and processed, and the same, the tooth surfaces of the small bevel gears can be designed and processed, when the face gear is meshed with the small bevel gears, the tooth surfaces of the small bevel gears can be designed and processed, wherein the design and processing of the tooth surfaces of the small bevel gears can be regarded as the design and processing of the tooth surfaces of two small bevel gears, and the rotation directions of the teeth of the two small bevel gears are that one is left-handed and the other is right-handed.
Based on the above steps, it is possible to design and machine the corresponding flanks of the small wheels, sigma, as long as the desired preset geometric meshing performance can be given1dTooth surface similar to the face gear2pWill exhibit a preset geometric engagement performance.
While the preferred embodiments of the present invention have been described with reference to the accompanying drawings, it is to be understood that all changes in the foregoing embodiments, and equivalents thereof, may be resorted to by those skilled in the art, without departing from the scope of the invention as defined by the appended claims.

Claims (2)

1.一种面向啮合性能预控的小轮齿面设计方法，其特征在于包括以下步骤：几何传动误差的预置、小轮齿面的反求、重新构造小轮齿面；1. A method for designing a pinion tooth surface for pre-control of meshing performance, characterized in that it comprises the following steps: presetting of geometrical transmission error, inversion of pinion tooth surface, and reconstruction of pinion tooth surface; 所述几何传动误差的预置包括步骤S101～S104：The preset of the geometric transmission error includes steps S101 to S104: S101构造几何传动误差多项式
Figure FSB0000194301850000011
所述c0，c1，c2，…，cσ是待确定系数，σ是几何传动误差多项式的阶数，取2～8之间的正整数，ε1是小轮转角；S101 Construction of geometric transmission error polynomial
Figure FSB0000194301850000011
The c 0 , c 1 , c 2 , . S102根据啮合周期内期望的几何传动误差的幅值、相邻几何传动误差交点处的斜率确定几何传动误差曲线的形状；S102 determines the shape of the geometrical transmission error curve according to the amplitude of the expected geometrical transmission error in the meshing period and the slope at the intersection of adjacent geometrical transmission errors; S103根据几何传动误差曲线形状确定几何传动误差多项式的阶数σ，在几何传动误差曲线上选择几何传动误差Δε2及其1阶导数的已知值，这些已知值的总个数为σ+1，并确定Δε2及其1阶导数已知值对应的σ+1个ε1的数值；S103 Determine the order σ of the geometrical transmission error polynomial according to the shape of the geometrical transmission error curve, and select the known values of the geometrical transmission error Δε 2 and its first-order derivative on the geometrical transmission error curve, and the total number of these known values is σ+ 1, and determine the value of σ+1 ε 1 corresponding to the known value of Δε 2 and its first derivative; S104将Δε2及其1阶导数的已知值、ε1的数值代入几何传动误差多项式中，构建矩阵方程，求解待确定系数c0，c1，c2，…，cσ，并回代入几何传动误差多项式中，获得Δε2与小轮转角ε1的函数关系；S104 Substitute the known value of Δε 2 and its first-order derivative, and the value of ε 1 into the geometric transmission error polynomial, construct a matrix equation, solve the coefficients c 0 , c 1 , c 2 , ..., c σ to be determined, and substitute them into In the geometric transmission error polynomial, the functional relationship between Δε 2 and the small wheel angle ε 1 is obtained; 所述小轮齿面的反求过程是：当小轮与具有近似齿面∑2p的面齿轮啮合时，按转角关系ε1＝(ε2N2-Δε2N2)/N1转动面齿轮和小轮，面齿轮近似齿面∑2p在小轮上的齿面族的包络是小轮的反求齿面∑12p，所述ε2是面齿轮的转角，所述N1、N2分别是小轮和面齿轮的齿数，所述面齿轮的近似齿面∑2p由直刃刀具单参数包络法加工而成；The inverse process of the tooth surface of the pinion is: when the pinion meshes with the surface gear having an approximate tooth surface Σ 2p , the rotating surface is ε 1 =(ε 2 N 2 -Δε 2 N 2 )/N 1 according to the rotation angle relationship Gear and pinion, the approximate tooth surface Σ 2p of the face gear The envelope of the tooth surface family on the pinion is the inverse tooth surface of the pinion Σ 12p , the ε 2 is the rotation angle of the face gear, the N 1 , N 2 is the number of teeth of the pinion and the face gear, respectively, and the approximate tooth surface ∑ 2p of the face gear is processed by the single-parameter envelope method of the straight-edged tool; 所述重新构造小轮齿面包括步骤S105～S111：The reconstructing the pinion tooth surface includes steps S105-S111: S105确定面齿轮近似齿面∑2p与其理论齿面∑2的内公切线Cp，并用一系列的离散点Fi表示所述内公切线Cp，i＝1，2，…，n，n为≥3的奇数；S105 Determine the internal common tangent Cp of the approximate tooth surface Σ 2p of the face gear and its theoretical tooth surface Σ 2 , and use a series of discrete points F i to represent the internal common tangent Cp, where i=1, 2, ..., n, n is ≥ an odd number of 3; S106点Fi是齿面∑2p、∑12p的共轭点，利用微分几何，求解齿面∑2p、∑12p共轭点Fi处的主曲率、主方向；S106 point F i is the conjugate point of tooth surface ∑ 2p , ∑ 12p , using differential geometry to solve the principal curvature and main direction of tooth surface ∑ 2p , ∑ 12p conjugate point F i ; S107利用共轭点Fi处的主曲率、主方向确定该点处的接触椭圆长轴方向；S107 utilizes the principal curvature and principal direction at the conjugate point F i to determine the long axis direction of the contact ellipse at this point; S108在共轭点Fi处的接触椭圆长轴上、Fi的两侧分别取两个端点ai1、ai2，并使长度|Fiai1|＝|Fiai2|＝a，a是期望的预设的接触椭圆长轴的长度；S108 On the long axis of the contact ellipse at the conjugate point F i , two end points a i1 and a i2 are taken on both sides of F i respectively, and the length |F i a i1 |=|F i a i2 |=a, a is the expected length of the long axis of the preset contact ellipse; S109过接触椭圆长轴的端点ai1、ai2分别作平行于共轭点Fi处∑2p、∑12p公法线的直线分别与齿面∑12p相交于点bi1、bi2；S109 make straight lines parallel to the common normal of ∑ 2p and ∑ 12p at the conjugate point F i through the end points a i1 and a i2 of the long axis of the contact ellipse, respectively, and intersect the tooth surface ∑ 12p at points b i1 and b i2 ; S110向∑12p内部分别延长ai1bi1、ai2bi2至Pi1、Pi2，并使长度|bi1Pi1|＝|bi2Pi2|＝0.00635mm；S110 respectively extends a i1 b i1 and a i2 b i2 to P i1 and P i2 to the inside of ∑ 12p , and makes the length |b i1 P i1 |=|b i2 P i2 |=0.00635mm; S111以过共轭点Fi和点Pi1、Pi2的二阶及二阶以上的光滑连续曲线替代共轭点Fi处∑2p、∑12p的接触线L21，作n条二阶及二阶以上的光滑连续曲线，由这n条光滑连续曲线构成小轮齿面∑1d。S111 replaces the contact line L 21 of ∑ 2p and ∑ 12p at the conjugate point F i with a smooth continuous curve of the second order and above through the conjugate point F i and the points P i1 and P i2 , making n second order sums For smooth continuous curves above the second order, the n smooth continuous curves constitute the pinion tooth surface ∑ 1d . 2.利用如权利要求1所述的一种面向啮合性能预控的小轮齿面设计方法所重新构造的小轮齿面的加工方法，其特征在于该加工方法包括以下步骤：加工刀具选用、数控机床选用、初始数控规律的建立、以及数控规律的修正；2. Utilize the processing method of the pinion tooth surface reconfigured by a pinion tooth surface design method oriented to pre-control of meshing performance as claimed in claim 1, it is characterized in that this processing method comprises the following steps: processing tool selection, Selection of CNC machine tools, establishment of initial CNC laws, and correction of CNC laws; 所述加工刀具选用为成型刀具，该刀具用于切削或磨削小轮齿槽的工作曲面是由具有标准渐开线齿面∑1的小轮齿槽的法面截线绕刀具轴线旋转一周而形成的；The machining tool is selected as a forming tool, and the working surface of the tool for cutting or grinding the pinion tooth slot is made by the normal surface section of the pinion tooth slot with a standard involute tooth surface ∑ 1 and rotates around the tool axis once. formed; 所述数控机床选用的数控机床包括5个数控运动轴，其中三个为数控平移运动轴X、Y、Z，另外两个为数控旋转运动轴A、B，其中轴A为工件主轴，轴B上安装高速自由旋转刀具主轴C；The CNC machine tool selected by the CNC machine tool includes 5 CNC motion axes, three of which are CNC translational motion axes X, Y, and Z, and the other two are CNC rotary motion axes A and B, wherein axis A is the workpiece spindle, and axis B is the main axis of the workpiece. A high-speed free-rotating tool spindle C is installed on it; 所述初始数控规律的建立包括步骤S201～S204：The establishment of the initial numerical control law includes steps S201-S204: S201建立所述刀具加工具有标准渐开线齿面∑1的小轮的抽象加工模型；S201 establishes an abstract machining model of the tool machining a small wheel with a standard involute tooth surface ∑ 1 ; S202基于在抽象加工与数控加工中，刀具相对小轮的位置矢量和法向矢量相等的原则，建立具有标准渐开线齿面∑1的小轮的数控加工数学模型；S202 Based on the principle that the position vector and normal vector of the tool relative to the pinion are equal in the abstract machining and NC machining, the NC machining mathematical model of the pinion with the standard involute tooth surface ∑ 1 is established; S203确定以加工中小轮转角ψ1为独立变量的各数控轴的运动规律，将各数控轴的运动规律展开为ψ1＝0处的泰勒级数得各数控轴初始数控规律多项式：S203 determines the motion law of each NC axis with the small wheel rotation angle ψ 1 as an independent variable in the process, and expands the motion law of each NC axis into a Taylor series at ψ 1 =0 to obtain the initial NC law polynomial of each NC axis:
Figure FSB0000194301850000021
Figure FSB0000194301850000021
Figure FSB0000194301850000022
Figure FSB0000194301850000022
Figure FSB0000194301850000023
Figure FSB0000194301850000023
Figure FSB0000194301850000024
Figure FSB0000194301850000024
Figure FSB0000194301850000025
Figure FSB0000194301850000025
 g1～g7、g8～g14、g15～g21、g22～g28、g29～g35分别为数控轴X、Y、Z、A、B的数控规律多项式从0阶到6阶的系数；g 1 ～g 7 , g 8 ～g 14 , g 15 ～g 21 , g 22 ～g 28 , g 29 ～g 35 are the numerical control law polynomials of numerical control axes X, Y, Z, A, B respectively from 0th order to 6th order coefficient; S204将上述数控规律多项式回代入小轮的数控加工数学模型中，得小轮数控齿面
Figure FSB0000194301850000026
S204 Substitute the above numerical control law polynomial into the numerical control machining mathematical model of the small wheel to obtain the numerical control tooth surface of the small wheel
Figure FSB0000194301850000026
 所述数控规律的修正用于修正上述初始数控规律多项式的各阶系数，使小轮数控齿面
Figure FSB0000194301850000027
趋近于小轮齿面∑1d，并加工小轮，包括步骤S205～S208：The correction of the numerical control law is used to correct the coefficients of each order of the polynomial of the above-mentioned initial numerical control law, so as to make the numerical control tooth surface of the small wheel.
Figure FSB0000194301850000027
Approaching the pinion tooth surface ∑ 1d and processing the pinion, including steps S205-S208: S205将小轮标准渐开线齿面∑1、数控齿面
Figure FSB0000194301850000028
齿面∑1d沿齿宽方向分为h-1等份，得h条纵向网格线，沿齿高方向分为w-1等份，得w条竖向网格线，纵向、竖向网格线的交点是齿面网格点，自小轮一个端面到另一个端面，自齿顶到齿根，这些网格点的编号是λ＝1，2，3，…，h×w；S205 converts the standard involute tooth surface of the small wheel ∑ 1 and the numerical control tooth surface
Figure FSB0000194301850000028
The tooth surface ∑ 1d is divided into h-1 equal parts along the tooth width direction, and h longitudinal grid lines are obtained, and it is divided into w-1 equal parts along the tooth height direction, and w vertical grid lines are obtained. The intersection points of the grid lines are the grid points of the tooth surface, from one end face of the small wheel to the other end face, from the tooth top to the tooth root, the number of these grid points is λ=1, 2, 3, ..., h×w; S206计算小轮齿面∑1d相对小轮标准渐开线齿面∑1的法向偏差向量：E1d＝n1dλ·(R1dλ-R1λ)，n1dλ、R1dλ分别是小轮齿面∑1d第λ个网格点的单位法向矢量、位置矢量，R1λ是小轮标准渐开线齿面∑1第λ个网格点的位置矢量；S206 Calculate the normal deviation vector of the pinion tooth surface Σ 1d relative to the pinion standard involute tooth surface Σ 1 : E 1d =n 1dλ ·(R 1dλ -R 1λ ), n 1dλ and R 1dλ are the pinion tooth surfaces respectively ∑ 1d Unit normal vector and position vector of the λth grid point, R 1λ is the position vector of the λth grid point of the standard involute tooth surface of the small wheel ∑ 1 ; S207第ξ次修正数控规律多项式各阶系数，包括以下步骤：S207 Revise the coefficients of each order of the numerical control law polynomial for the ξth time, including the following steps: 步骤一：求解第ξ次修正中，小轮数控齿面
Figure FSB0000194301850000031
相对小轮标准渐开线齿面∑1的法向偏差向量：
Figure FSB0000194301850000032
分别是小轮数控齿面
Figure FSB0000194301850000033
第ξ次修正中第λ个网格点的单位法向矢量、位置矢量；Step 1: Solve in the ξth correction, the NC tooth surface of the small wheel
Figure FSB0000194301850000031
The normal deviation vector relative to the standard involute tooth surface of the pinion ∑ 1 :
Figure FSB0000194301850000032
Respectively, the small wheel CNC tooth surface
Figure FSB0000194301850000033
Unit normal vector and position vector of the λth grid point in the ξth revision; 步骤二：求解第ξ次的修正矩阵Ω(ξ)，其元素为
Figure FSB0000194301850000034
且
Figure FSB0000194301850000035
Figure FSB0000194301850000036
分别是小轮数控齿面
Figure FSB0000194301850000037
第ξ次修正中，数控规律多项式系数有扰动量Δgκ时，第λ个网格点的单位法向矢量、位置矢量，κ＝1，2，3…，35；Step 2: Solve the ξth correction matrix Ω (ξ) , whose elements are
Figure FSB0000194301850000034
and
Figure FSB0000194301850000035
Figure FSB0000194301850000036
Respectively, the small wheel CNC tooth surface
Figure FSB0000194301850000037
In the ξth revision, when the numerical control law polynomial coefficient has a disturbance Δg κ , the unit normal vector and position vector of the λth grid point, κ=1, 2, 3..., 35; 步骤三：求解修正矩阵方程，得第ξ次修正后的数控规律多项式各阶系数
Figure FSB0000194301850000038
为：Step 3: Solve the corrected matrix equation to obtain the coefficients of each order of the numerical control law polynomial after the ξth correction
Figure FSB0000194301850000038
for:
Figure FSB0000194301850000039
Figure FSB0000194301850000039
 当ξ＝1时，取初始数控规律多项式的各阶系数求解上述修正矩阵方程，重复步骤一到步骤三，迭代求解上述修正矩阵方程，直至
Figure FSB00001943018500000310
趋近于0向量时结束，则小轮数控齿面
Figure FSB00001943018500000311
趋近于小轮齿面∑1d，获得最终的数控规律多项式的各阶系数；When ξ=1, take the coefficients of each order of the initial numerical control law polynomial to solve the above correction matrix equation, repeat steps 1 to 3, and iteratively solve the above correction matrix equation until
Figure FSB00001943018500000310
When the vector approaches 0, it ends, then the NC tooth surface of the small wheel
Figure FSB00001943018500000311
Approaching the pinion tooth surface ∑ 1d , the coefficients of each order of the final numerical control law polynomial are obtained; S208安装所述刀具、小轮、利用最终数控规律多项式的各阶系数编写数控程序、调试机床、加工小轮。S208 installs the tool, the small wheel, uses the coefficients of each order of the final numerical control law polynomial to write a numerical control program, debugs the machine tool, and processes the small wheel.
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