CN109027186B - Discretization envelope design method for gear ratio rack of steering gear - Google Patents

Discretization envelope design method for gear ratio rack of steering gear Download PDF

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CN109027186B
CN109027186B CN201810879278.9A CN201810879278A CN109027186B CN 109027186 B CN109027186 B CN 109027186B CN 201810879278 A CN201810879278 A CN 201810879278A CN 109027186 B CN109027186 B CN 109027186B
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rack
gear
tooth
points
gear ratio
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CN109027186A (en
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韩星会
代银
华林
徐曼
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Wuhan University of Technology WUT
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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
    • F16H55/26—Racks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation

Abstract

The invention relates to a discretization envelope design method of a gear ratio rack of a steering gear, which comprises the following steps: s1, establishing a bevel gear end face model formed by dense discrete points to obtain an envelope diagram of a gear ratio rack tooth profile; s2, drawing equidistant straight line clusters parallel to the rack in the range of the rack tooth height, and solving the intersection point of the straight line clusters and the envelope line; s3, screening out an intersection point in the effective length range of the rack, namely a point in the tooth groove of the rack; s4, extracting rack tooth groove boundary points; s5, reordering the rack tooth profile points by an interval point taking method and connecting the points to obtain the single longitudinal section tooth profile of the rack; and S6, establishing a plurality of helical gear section models parallel to the end face, repeatedly enveloping and extracting to obtain tooth profiles of all longitudinal sections of the rack, and fitting the tooth profiles to obtain the gear ratio rack. The invention combines the generating simulation method and the gear element method, envelopes the rack tooth profile by the discretization idea, and provides a new numerical extraction algorithm to obtain the gear ratio rack tooth profile point, thereby having the advantages of high efficiency and accuracy.

Description

Discretization envelope design method for gear ratio rack of steering gear
Technical Field
The invention relates to the field of involute helical surface gear and rack pair design, in particular to a discretization envelope design method of a gear ratio rack of a steering gear.
Background
For a traditional steering gear with a fixed transmission ratio, after the transmission ratio of the steering gear is reduced, although the steering becomes lighter and more labor-saving, the steering operation time is prolonged, and the steering sensitivity of an automobile is reduced, so that the contradiction between lightness and flexibility exists. The speed ratio steering gear can well adjust the contradiction between light and flexible, so the speed ratio steering gear is widely applied to automobiles.
The existing design methods of the gear ratio rack comprise a conjugate principle analysis method, a generating simulation cutting method and a gear element method. The solution of the curved surface equation is not unique when the contact ratio is more than 1 by the conjugate principle analytic method. The generating simulation method obtains a gear ratio rack model by utilizing the Boolean operation function of three-dimensional software, although the design process is simple, the software needs to be developed secondarily, and the following fitting of a curved surface is required, so that the design error is large. The algorithm of the gear element method is complicated.
Disclosure of Invention
The invention aims to solve the technical problem of providing a discretization envelope design method of a gear ratio rack of a steering gear, which has the advantages of no principle error, high efficiency and accuracy.
The technical scheme adopted by the invention for solving the technical problems is as follows: the discretization envelope design method of the gear ratio rack of the steering gear comprises the following steps:
s1, establishing a bevel gear end face model formed by dense discrete points, simulating the tooth division generating motion of gear shaping processing according to a gear ratio function, and solving an instantaneous pose to obtain an envelope graph of a gear ratio rack tooth profile;
s2, drawing equidistant straight line clusters parallel to the rack in the range of the rack tooth height, and calculating the intersection point with the envelope line by using a distance screening method;
s3, screening out an intersection point in the effective length range of the rack, namely a point in the tooth groove of the rack;
s4, extracting rack tooth groove boundary points, namely, rack tooth profile points;
s5, reordering the rack tooth profile points by an interval point taking method and connecting the points to obtain the single longitudinal section tooth profile of the rack;
and S6, establishing a plurality of helical gear section models parallel to the end face, repeating the steps S2, S3, S4 and S5 to obtain tooth profiles of all longitudinal sections of the rack, and fitting the tooth profiles to obtain the gear ratio rack.
In the foregoing solution, the step S1 specifically includes: is established by nrHelical gear end face tooth profile model r formed by discrete points1(x1，y1) As a gear shaping tool, the gear shaping tool simulates the tooth dividing generating movement in gear shaping of a gear ratio rack, solves the gear ratio rack and pinion movement relation based on a gear ratio curve and expresses the gear ratio rack and pinion movement relation as a coordinate transformation matrix Mt1Calculating n in the course of machining by means of coordinate transformationtThe pose of the end face model of the bevel gear at each moment is obtained, so that the tooth profile envelope graph r of the gear ratio rack is obtainedt(xt,yt)＝Mt1·r1(x1,y1) And the range of the envelope figure is larger than the effective length of the rack,
Figure GDA0002518199490000021
wherein
Figure GDA0002518199490000022
Angle of rotation of helical gear at time t, StFor the corresponding displacement of the gear ratio rack normally engaged at time t, the calculation formula is
Figure GDA0002518199490000023
Figure GDA0002518199490000024
As a function of the speed ratio,
Figure GDA0002518199490000025
the angle of the helical gear rotation at any moment.
In the foregoing solution, the step S2 specifically includes: according to the gear ratio gear rack meshing principle, the range of the y value of the rack tooth profile point needing to be extracted is determined to be [ rf+c,ra+c]，ra、rfRespectively the gear tooth top circle radius, the gear root circle radius and c is a top clearance; drawing an equidistant straight line cluster r parallel to the x axis in the rack tooth profile point interval of the enveloping graph in the step S13(x3,y3)，y3∈[rf+c,ra+c]Defining high dispersion accuracy rho of gear teeth3The value is the distance between the straight lines in the equidistant straight line cluster and the number of the straight lines
Figure GDA0002518199490000026
Figure GDA0002518199490000026
0＜ρ3＜0.1(ra-rf) (ii) a The method of screening tolerance zone points is adopted to calculate the fuzzy intersection point of the equidistant straight line cluster and the envelope line
Figure GDA0002518199490000027
Storing the data points in the intersection matrix, and screening the tolerance band points by using coordinate value y ∈ [ y [ ]3i-,y3i+](i＝0，1…n)，y3iIs the y value of each straight line in the equidistant straight line cluster,
Figure GDA0002518199490000028
in the foregoing solution, the step S3 specifically includes: determining effective rack length
Figure GDA0002518199490000031
Figure GDA0002518199490000032
The tooth pitch of the safety gear pitch line at the end part of the rack,
Figure GDA0002518199490000033
total number of gear rotations of
Figure GDA0002518199490000034
z1The number of teeth of the pinion gear is,
Figure GDA0002518199490000035
is the maximum rotation angle of the gear rotation,
Figure GDA0002518199490000036
is the minimum rotation angle of the gear rotation; screening the intersection points obtained in the step S2, extracting the intersection points positioned in the effective length interval of the rack, namely the intersection points in the tooth grooves of the rack, and judging the intersection points to be the points
Figure GDA0002518199490000037
And storing the rack tooth space points with the same y value in the same matrix, wherein the y value of each point in the matrix is the same, and thus obtaining n tooth space point matrixes of the gear ratio rack, wherein x is the abscissa of the intersection point, and y is the ordinate of the intersection point.
In the above scheme, n is described in step S1r、ntDetermined by the following equation: defining gear dispersion accuracy ρ1The value is the micro-segment distance after the total length of the gear outline is dispersed, the micro-segment distance after the circumference of the gear tooth top circle is dispersed is adopted for carrying out substitution calculation, and the gear envelope step precision rho is defined2The value is a minute degree after the total degree of rotation of the gear is dispersed,
Figure GDA0002518199490000038
gear singleThe number of discrete points on the contour line of the tooth is distributed in the way that
Figure GDA0002518199490000039
nr1、nr2Number of discrete points, n, of the involutes from side to side, respectivelyr3、nr4Number of discrete points, n, of left and right transition lines, respectivelyr5、nr6The number of discrete points of the addendum line and the dedendum line, respectively.
In the above scheme, in step S4, the specific extraction algorithm is as follows: firstly, the tooth space point matrixes stored in the step S3 are arranged in ascending order according to the x coordinate values of the points, namely the first row of the matrixes, and then the distance l between every two adjacent tooth space points in the tooth space point matrixes after the arrangement is sequentially obtained as | x |i-xi+1Selecting the smallest tooth thickness s of the tooth crest line from the different tooth thicknesses of the gear ratio rackkaLet l0＝skaWhen l is greater than l0Then, the two points are explained as tooth profile points and are kept in the original matrix in ascending order; and sequentially screening all the matrixes in the S3 to obtain n speed ratio rack tooth profile point matrixes.
In the foregoing scheme, in step S5, the specific method includes: vertically splicing the tooth profile point matrixes in the step S4 into a large matrix according to the ascending order of the y coordinate value, and setting the gear ratio rack to have z2Tooth then has 2z2The strip tooth profile has n straight lines in the equidistant straight line cluster, so that 2z is extracted2n number of tooth profile points, resulting in a 2z2A matrix of n rows and 2 columns; taking the number of rows i + j.2z from the matrix2(i＝1，2…2z2(ii) a j is 0, 1 … n-1), the tooth profile point matrixes on the same tooth profile are obtained by storing the values of i respectively, and the tooth profile points are connected to obtain the gear ratio rack tooth profile meshed with the end face of the bevel gear.
In the foregoing solution, the step S6 specifically includes: axially dispersing the helical gear to establish n within the tooth width1Equidistant helical gear section model r parallel to end facen1(xn1,yn1)＝Mn1·r1(x1,y1) (ii) a Defining gear axial dispersion accuracy rho4The value of which is the spacing between equidistant sections, then
Figure GDA0002518199490000041
b is the tooth width, wherein the transformation matrix is
Figure GDA0002518199490000042
β, repeating steps S1, S2, S3, S4 and S5 to obtain tooth profiles of a plurality of longitudinal sections of the gear ratio rack, and connecting the tooth profiles to model the gear ratio rack.
The discretization envelope design method of the gear ratio rack of the steering gear has the following beneficial effects:
the method is based on the gear shaping machining principle, combines a generating simulation method and a gear element method, obtains a gear ratio rack tooth profile envelope graph by solving poses of a plurality of helical gear sections at different moments, and provides a new numerical extraction algorithm to screen intersection points of a equidistant linear cluster and the envelope graph to obtain a gear ratio tooth profile point, and finally obtains a gear ratio rack model.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a model of an end face of a helical gear according to an embodiment of the present invention;
FIG. 2 is a graphical representation of a transmission ratio curve designed according to an embodiment of the present invention;
FIG. 3 is a schematic illustration of a rack tooth profile envelope in an embodiment of the present invention;
FIG. 4 is a schematic diagram of equidistant linear clusters in an embodiment of the invention;
FIG. 5 is a schematic view of points within rack slots in an embodiment of the present invention;
FIG. 6 is a schematic representation of a gear ratio rack tooth profile point extracted in an embodiment of the present invention;
FIG. 7 is a schematic representation of a variable speed ratio rack tooth profile in an embodiment of the present invention;
FIG. 8 is a tooth profile line schematic of a plurality of longitudinal sections of a variable speed ratio rack in an embodiment of the present invention;
FIG. 9 is a multi-section assembly view of the variable ratio rack and pinion of the embodiment of the present invention;
FIG. 10 is a schematic representation of a variable ratio rack in an embodiment of the present invention.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
The invention discloses a discretization envelope design method of a gear ratio rack of a steering gear, which comprises the following steps as shown in figures 1-9:
s1, determining the precision of the face model of the helical gear and the precision of the envelope graph according to the design precision of the rack, and establishing a matrix n in MatlabrHelical gear end face tooth profile model r formed by discrete points1(x1，y1) As a gear shaping tool, as shown in fig. 2, a split-tooth generating motion in a gear-ratio rack gear shaping process is simulated. Solving the gear ratio rack and pinion motion relation based on the gear ratio curve and expressing the relation as a coordinate transformation matrix Mt1Calculating n in the course of machining by means of coordinate transformationtThe pose of an end face model of the bevel gear at each moment, wherein a coordinate transformation matrix is
Figure GDA0002518199490000051
In the matrix
Figure GDA0002518199490000052
Angle of rotation of helical gear at time t, StFor the corresponding displacement of the gear ratio rack normally engaged at time t, the calculation formula is
Figure GDA0002518199490000053
Figure GDA0002518199490000054
As a function of the speed ratio,
Figure GDA0002518199490000055
the angle of rotation of the bevel gear at any moment is shown in fig. 2. According to the formula rt(xt,yt)＝Mt1·r1(x1,y1) And obtaining a tooth profile envelope graph of the gear ratio rack, as shown in FIG. 3, and enabling the envelope graph range to be larger than the effective length of the rack so as to ensure the envelope integrity of each tooth of the rack. Wherein n isr、ntDetermined by the following equation. Defining gear dispersion accuracy ρ1The value is the micro-segment distance after the total length of the gear outline is dispersed, the micro-segment distance after the circumference of the gear tooth top circle is dispersed is adopted for carrying out substitution calculation, and the gear envelope step precision rho is defined2The value is a minute degree after the total degree of the rotation of the gear is dispersed.
Figure GDA0002518199490000056
The distribution mode of the number of discrete points of the contour line of a single tooth of the gear is
Figure GDA0002518199490000057
nr1、nr2Number of discrete points, n, of the involutes from side to side, respectivelyr3、nr4Number of discrete points, n, of left and right transition lines, respectivelyr5、nr6The number of discrete points of the addendum line and the dedendum line, respectively.
S2, determining the range of the y value of the rack tooth profile point needing to be extracted as r according to the gear ratio gear rack meshing principlef+c,ra+c]，ra、rfThe radius of the gear tooth top circle, the radius of the gear root circle and c is a top clearance. Drawing an equidistant straight line cluster r parallel to the x axis in the interval of the rack tooth profile points of the envelope graph in S13(x3,y3)，y3∈[rf+c,ra+c]Defining high discrete accuracy ρ of gear teeth as shown in FIG. 43The value is the distance between the straight lines in the equidistant straight line cluster and the number of the straight lines
Figure GDA0002518199490000061
Figure GDA0002518199490000061
0＜ρ3＜0.1(ra-rf). The method of screening tolerance zone points is adopted to calculate the fuzzy intersection point of the equidistant straight line cluster and the envelope line
Figure GDA0002518199490000062
The screening method of tolerance band points stored in the intersection point matrix comprises screening envelope graph data points with the criterion of coordinate value y ∈ [ y [3i-,y3i+](i＝0，1…n)，y3iIs the y value of each straight line in the equidistant straight line cluster,
Figure GDA0002518199490000063
s3, determining the effective length of the rack
Figure GDA0002518199490000064
For the safe tooth pitch at the end part of the rack,
Figure GDA0002518199490000065
total number of gear rotations of
Figure GDA0002518199490000066
z1The number of teeth of the pinion gear is,
Figure GDA0002518199490000067
is the maximum rotation angle of the gear rotation,
Figure GDA0002518199490000068
is the minimum angle of rotation of the gear. The intersection points obtained in step S2 are screened to extract intersection points located within the effective length interval of the rack, that is, points within the tooth grooves of the rack, as shown in fig. 5, where the criterion is
Figure GDA0002518199490000069
And storing the rack tooth space points with the same y value in the same matrix, wherein the y value of each point in the matrix is the same, and thus obtaining n tooth space point matrixes of the gear ratio rack, wherein x is the abscissa of the intersection point, and y is the ordinate of the intersection point.
And S4, extracting a boundary point of the rack tooth groove, namely the gear ratio rack tooth profile point. The distance between the tooth profile points with the same height on both sides of the tooth is the tooth thickness s at the positionkBecause the precision of the envelope map is higher, the distance between two adjacent non-tooth-profile points with the same y value is far less than that of the adjacent non-tooth-profile pointssk. The specific extraction algorithm based on this principle is to arrange the tooth space point matrices stored in step S3 in ascending order according to the x coordinate values of the points, i.e., the first row of the matrix, and then sequentially find the distance l between every two adjacent tooth space points in the sorted tooth space point matrix as | x |i-xi+1Selecting the smallest tooth thickness s of the tooth crest line from the different tooth thicknesses of the gear ratio rackkaLet l0＝skaWhen l is greater than l0Then, the two points are explained as tooth profile points, and are kept in the original matrix in ascending order. All the matrixes in the step S3 are sequentially screened to obtain n matrixes of gear ratio rack tooth profile points, and the tooth profile points are shown in FIG. 6.
And S5, extracting points belonging to the same tooth profile curve by adopting an interval point taking method for the mixed tooth profile points of the rack. The specific method comprises the following steps: vertically splicing the tooth profile point matrix in the S4 into a large matrix according to the ascending order of the y coordinate value, and setting the gear ratio rack to have z2Tooth then has 2z2The strip tooth profile has n straight lines in the equidistant straight line cluster, so that 2z is extracted2n number of tooth profile points, resulting in a 2z2A matrix of n rows and 2 columns; taking the number of rows i + j.2z from the matrix2(i＝1，2…2z2(ii) a j is 0, 1 … n-1), the tooth profile point matrixes on the same tooth profile are obtained by respectively storing the values of i, and the tooth profile point matrixes are connected after three-dimensional software is introduced to obtain a gear ratio rack tooth profile meshed with the end face of the helical gear, as shown in fig. 7;
s6, dispersing the helical gear along the axial direction, and establishing n within the range of the tooth width1Equidistant helical gear section model r parallel to end facen1(xn1,yn1)＝Mn1·r1(x1,y1). Defining gear axial dispersion accuracy rho4The value of which is the spacing between equidistant sections, then
Figure GDA0002518199490000071
b is the tooth width, where n1Determining values by design accuracy, wherein the matrix is transformed
Figure GDA0002518199490000072
β is the angle of the difference between the corresponding position on the section and the end face, the steps S1, S2, S3, S4 and S5 are repeated to obtain the tooth profile of a plurality of longitudinal sections of the gear ratio rack, as shown in FIG. 8, the tooth profiles of a plurality of sections of the gear and the gear ratio rack after being dispersed along the axial direction are assembled together, the corresponding relation of the corresponding sections of the gear and the gear rack can be seen, the principle of the discretization envelope is intuitively understood, as shown in FIG. 9, the tooth profiles of a plurality of longitudinal sections of the gear ratio rack are connected together to model to obtain a three-dimensional model of the gear ratio rack, as shown in FIG..
The tooth surface of the rack of the gear-ratio rack-and-pinion steering gear is designed according to the method by taking the helical gear with determined rack design requirements and known basic parameters as an example. The basic parameters of the standard bevel gear are shown in table 1.
TABLE 1 helical gear basic parameters
Figure GDA0002518199490000073
Establishing n in MatlabrHelical gear end face tooth profile model r formed by 3600 discrete points1(x1，y1) As a gear shaping tool, as shown in fig. 1.
Designing a function of the transmission ratio according to the steering demand of the vehicle
Figure GDA0002518199490000085
As shown in fig. 2; solving the gear ratio rack and pinion motion relation based on the gear ratio curve and expressing the relationship as a coordinate transformation matrix
Figure GDA0002518199490000081
In the matrix
Figure GDA0002518199490000082
Angle of rotation of helical gear at time t, StFor the corresponding displacement of the gear ratio rack normally engaged at time t, the calculation formula is
Figure GDA0002518199490000083
In simulating gear ratio rack-gear shapingGenerating motion by teeth division, and calculating n in the processing process in a coordinate transformation modetPose of the end face model of the helical gear at 1301 moments, thus according to the formula rt(xt,yt)＝Mt1·r1(x1,y1) And obtaining a tooth profile envelope graph of the gear ratio rack, wherein the envelope graph range is larger than the effective length of the rack, and the envelope graph consists of 4383600 data points, as shown in FIG. 3.
According to the gear tooth height and the meshing principle and considering the influence of the tip clearance, the gear tooth height interval is determined to be [5.9577, 10.7077 ]]Determining the number n of straight lines of the straight line cluster to be 24 and the distance rho according to the design precision3Equal distance straight line cluster r parallel to the x axis is drawn in the rack tooth height interval3(x3,y3)，y3∈[5.9577,10.7077]As shown in fig. 4, when the coordinate value is 0.1, the coordinate value y ∈ [ y ] is programmed to be selected3i-0.01,y3i+0.01](i ═ 0, 1 … n), i.e., the intersection points of equidistant linear clusters and the envelope curve
Figure GDA0002518199490000084
According to the design requirement of the rack, because the gear and the rack can not be meshed to the last tooth when being meshed, safety teeth and safety tooth moments S are required to be arranged at two ends of the rack 12, 7.4577, pi/6, 7.8097, and finally determining the effective length interval [ S ] of the rack]Is [ -82.7996, 82.7996]And screening out the intersection points in the effective length interval of the rack, namely the points in the tooth grooves of the rack, as shown in figure 5, and respectively storing the points in the tooth grooves according to the intersected straight lines.
Adopting a sortrows function to arrange the tooth socket points respectively stored in the S3 in an ascending order according to the x coordinate value, namely the first row of the matrix; the boundary point of the tooth space of the rack is the tooth profile point, and the distance between the tooth profile points with the same height at the two sides of the tooth is the tooth thickness s at the positionkAnd because the precision of the envelope graph is higher, the distance between two adjacent non-tooth-profile points is far less than skTherefore, the distance l ═ x between every two adjacent tooth space points in the sorted set of tooth space points is sequentially obtainedi-xi+1I, selecting the smallest of the different tooth thicknesses of the gear ratio rackTooth thickness s of tooth top lineka0.8, let l0＝skaWhen l is greater than a set value l, 0.80Then, the two points are described as tooth profile points, and all the tooth profile points of the gear ratio rack are sequentially screened out, as shown in fig. 6.
All the tooth profile points are stored in a matrix according to the extraction sequence, the gear ratio rack has 20 teeth and 40 tooth profile lines, and 960 tooth profile points are generated because the equidistant straight line cluster has 24 straight lines; programming and extracting a 1 st point in matlab, then extracting a next point every 40 points, extracting 24 points in total, and storing the points together to obtain all points of a tooth profile; the above point-taking process is repeated from the 1 st point to the 40 th point in order, and points on the same tooth profile are stored and introduced into Pro/E to obtain a gear ratio rack tooth profile, as shown in FIG. 7.
Calculating a transformation matrix between a helical gear section model and an end face model
Figure GDA0002518199490000091
β is the angle of the phase difference between the corresponding positions on the cross section and the end face, n is established according to the design accuracy 110 helical gear section models r parallel to end facen1(xn1,yn1)＝Mn1·r1(x1,y1) Steps S1, S2, S3, S4 and S5 are repeated to obtain tooth profiles of a plurality of longitudinal sections of the transmission gear ratio rack, as shown in fig. 8. The gear after being dispersed along the axial direction and a plurality of section tooth profiles of the gear ratio rack are assembled together, the corresponding relation of each section of the gear rack can be seen, and the principle of the discretization envelope can be intuitively understood, as shown in fig. 9. The tooth profiles of a plurality of longitudinal sections of the gear ratio rack are connected and modeled to obtain a three-dimensional model of the gear ratio rack, as shown in fig. 10.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (7)

1. A discretization envelope design method of a gear ratio rack of a steering gear is characterized by comprising the following steps:
s1, establishing a bevel gear end face model formed by dense discrete points, simulating the tooth division generating motion of gear shaping processing according to a gear ratio function, and solving an instantaneous pose to obtain an envelope graph of a gear ratio rack tooth profile;
s2, drawing equidistant straight line clusters parallel to the rack in the range of the rack tooth height, and calculating the intersection point with the envelope line by using a distance screening method;
s3, screening out an intersection point in the effective length range of the rack, namely a point in the tooth groove of the rack;
s4, extracting rack tooth groove boundary points, namely, rack tooth profile points;
s5, reordering the rack tooth profile points by an interval point taking method and connecting the points to obtain the single longitudinal section tooth profile of the rack;
s6, establishing a plurality of helical gear section models parallel to the end face, repeating the steps S2, S3, S4 and S5 to obtain tooth profiles of all longitudinal sections of the rack, and fitting the tooth profiles to obtain the gear ratio rack;
the step S1 specifically includes: is established by nrHelical gear end face tooth profile model r formed by discrete points1(x1，y1) As a gear shaping tool, the gear shaping tool simulates the tooth dividing generating movement in gear shaping of a gear ratio rack, solves the gear ratio rack and pinion movement relation based on a gear ratio curve and expresses the gear ratio rack and pinion movement relation as a coordinate transformation matrix Mt1Calculating n in the course of machining by means of coordinate transformationtThe pose of the end face model of the bevel gear at each moment is obtained, so that the tooth profile envelope graph r of the gear ratio rack is obtainedt(xt,yt)＝Mt1·r1(x1,y1) And the range of the envelope figure is larger than the effective length of the rack,
Figure FDA0002518199480000011
wherein
Figure FDA0002518199480000012
Angle of rotation of helical gear at time t, StFor the corresponding displacement of the gear ratio rack normally engaged at time t, the calculation formula is
Figure FDA0002518199480000013
Figure FDA0002518199480000014
As a function of the speed ratio,
Figure FDA0002518199480000015
the angle of the helical gear rotation at any moment.
2. The discretized envelope design method for a steering gear ratio rack according to claim 1, wherein said step S2 is embodied as: according to the gear ratio gear rack meshing principle, the range of the y value of the rack tooth profile point needing to be extracted is determined to be [ rf+c,ra+c]，ra、rfRespectively the gear tooth top circle radius, the gear root circle radius and c is a top clearance; drawing an equidistant straight line cluster r parallel to the x axis in the rack tooth profile point interval of the enveloping graph in the step S13(x3,y3)，y3∈[rf+c,ra+c]Defining high dispersion accuracy rho of gear teeth3The value is the distance between the straight lines in the equidistant straight line cluster and the number of the straight lines
Figure FDA0002518199480000021
The method of screening tolerance zone points is adopted to calculate the fuzzy intersection point of the equidistant straight line cluster and the envelope line
Figure FDA0002518199480000022
Storing the data points in the intersection matrix, and screening the tolerance band points by using coordinate value y ∈ [ y [ ]3i-,y3i+](i＝0，1…n)，y3iIs the y value of each straight line in the equidistant straight line cluster,
Figure FDA0002518199480000023
3. the discretized envelope design method for a steering gear ratio rack according to claim 2, wherein said step S3 is embodied as: determining effective rack length
Figure FDA0002518199480000024
Figure FDA0002518199480000025
The tooth pitch of the safety gear pitch line at the end part of the rack,
Figure FDA0002518199480000026
total number of gear rotations of
Figure FDA0002518199480000027
z1The number of teeth of the pinion gear is,
Figure FDA0002518199480000028
is the maximum rotation angle of the gear rotation,
Figure FDA0002518199480000029
is the minimum rotation angle of the gear rotation; screening the intersection points obtained in the step S2, extracting the intersection points positioned in the effective length interval of the rack, namely the intersection points in the tooth grooves of the rack, and judging the intersection points to be the points
Figure FDA00025181994800000210
And storing the rack tooth space points with the same y value in the same matrix, wherein the y value of each point in the matrix is the same, and thus obtaining n tooth space point matrixes of the gear ratio rack, wherein x is the abscissa of the intersection point, and y is the ordinate of the intersection point.
4. The discretized envelope design method for a steering gear ratio rack according to claim 1Method, characterized in that n is described in step S1r、ntDetermined by the following equation: defining gear dispersion accuracy ρ1The value is the micro-segment distance after the total length of the gear outline is dispersed, the micro-segment distance after the circumference of the gear tooth top circle is dispersed is adopted for carrying out substitution calculation, and the gear envelope step precision rho is defined2The value is a minute degree after the total degree of rotation of the gear is dispersed,
Figure FDA00025181994800000211
the distribution mode of the number of discrete points of the contour line of a single tooth of the gear is
Figure FDA00025181994800000212
nr1、nr2Number of discrete points, n, of the involutes from side to side, respectivelyr3、nr4Number of discrete points, n, of left and right transition lines, respectivelyr5、nr6The number of discrete points of the addendum line and the dedendum line, respectively.
5. The discretized envelope design method for a steering gear ratio rack according to claim 1, wherein in step S4, the specific extraction algorithm is: firstly, the tooth space point matrixes stored in the step S3 are arranged in ascending order according to the x coordinate values of the points, namely the first row of the matrixes, and then the distance l between every two adjacent tooth space points in the tooth space point matrixes after the arrangement is sequentially obtained as | x |i-xi+1Selecting the smallest tooth thickness s of the tooth crest line from the different tooth thicknesses of the gear ratio rackkaLet l0＝skaWhen l is greater than l0Then, the two points are explained as tooth profile points and are kept in the original matrix in ascending order; and sequentially screening all the matrixes in the S3 to obtain n speed ratio rack tooth profile point matrixes.
6. The discretized envelope design method for a steering gear ratio rack according to claim 1, wherein in step S5, the specific method comprises: vertically splicing the tooth profile point matrixes in the step S4 into a large matrix according to the ascending order of the y coordinate value, and setting the gear ratio rack to have z2Tooth then has 2z2The strip tooth profile has n straight lines in the equidistant straight line cluster, so that 2z is extracted2n number of tooth profile points, resulting in a 2z2A matrix of n rows and 2 columns; taking the number of rows i + j.2z from the matrix2(i＝1，2…2z2(ii) a j is 0, 1 … n-1), the tooth profile point matrixes on the same tooth profile are obtained by storing the values of i respectively, and the tooth profile points are connected to obtain the gear ratio rack tooth profile meshed with the end face of the bevel gear.
7. The discretized envelope design method for a steering gear ratio rack according to claim 1, wherein said step S6 is embodied as: axially dispersing the helical gear to establish n within the tooth width1Equidistant helical gear section model r parallel to end facen1(xn1,yn1)＝Mn1·r1(x1,y1) (ii) a Defining gear axial dispersion accuracy rho4The value of which is the spacing between equidistant sections, then
Figure FDA0002518199480000031
b is the tooth width, wherein the transformation matrix is
Figure FDA0002518199480000032
β, repeating steps S1, S2, S3, S4 and S5 to obtain tooth profiles of a plurality of longitudinal sections of the gear ratio rack, and connecting the tooth profiles to model the gear ratio rack.
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