CN108645323A - Involute spur gear wheel total profile deviation assessment method under the influence of installation error - Google Patents

Involute spur gear wheel total profile deviation assessment method under the influence of installation error Download PDF

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CN108645323A
CN108645323A CN201810646085.9A CN201810646085A CN108645323A CN 108645323 A CN108645323 A CN 108645323A CN 201810646085 A CN201810646085 A CN 201810646085A CN 108645323 A CN108645323 A CN 108645323A
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coordinate system
gear
flank profil
measurement point
axis
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CN108645323B (en
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汤洁
魏建杰
石照耀
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Beijing University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B5/00—Measuring arrangements characterised by the use of mechanical techniques
    • G01B5/20—Measuring arrangements characterised by the use of mechanical techniques for measuring contours or curvatures
    • G01B5/202—Measuring arrangements characterised by the use of mechanical techniques for measuring contours or curvatures of gears

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Abstract

The invention discloses the involute spur gear wheel total profile deviation assessment methods under the influence of installation error, initially set up coordinate system of machine O0‑X0Y0Z0, tested gear is measured under coordinate system of machine, two Section Points, gear upper end millet cake and the flank profil measurement point up and down that position includes tested gear mandrel is measured, obtains the flank profil measurement point under the influence of installation error；According to two Section Points of mandrel under coordinate system of machine, make Least Square Circle fitting, calculates revolution paraxial equation, establish coordinate system O1‑X1Y1Z1, flank profil measurement point is reverted into initial position, corrects the error of rotary axle of flank profil measurement point；Least square plane fitting is made to tested gear upper surface measurement point, is Z with tested gear upper surface normal vector direction2Axis direction establishes coordinate system O2‑X2Y2Z2, correct the heeling error of flank profil measurement point；The rotation angle for rotating to measurement point around the rolling circle center of circle by calculating design flank profil obtains total profile deviation evaluation numerical value, to correct the influence that installation error evaluates total profile deviation.

Description

Involute spur gear wheel total profile deviation assessment method under the influence of installation error
Technical field
The present invention relates to Precision Inspection and instrument field, the gear-profile under the influence of specifically a kind of installation error is inclined The assessment method of difference.
Background technology
Gear is most important and most widely used transmission mechanism in machine driving.Gear is widely used in automobile The fields such as manufacture, aerospace, wind power equipment and equipment manufacturing.The accuracy of gear, quality directly affect the efficiency of equipment, make an uproar Sound, kinematic accuracy and service life.With the development of science and technology gear is as key components and parts, above-mentioned every profession and trade is to design of gears The accuracy of manufacture proposes requirements at the higher level.Since gear testing technology complements each other with gear machining technology, the inspection to gear Survey is just particularly important.
It is known as gear individual event deviation as the measurement of measurement item using total profile deviation, spiral deviation and circular pitch deviation to measure.Tooth Wheel individual event deflection measurements reflect the difference of machine tooling ability and institute's workpieces processing and design size.
Total profile deviation refers to the amount of practical tooth off-design flank profil, the telegoniometer in end face and perpendicular to involute profile Value.Total profile deviation is mainly by the foozle and installation error of cutter during Gear Processing and machine tool drive chain short cycle Caused by the composite factors such as error.Standard ISO 1328-1：Total profile deviation is defined as tooth profile total deviation, profile geometry by 2013 Deviation and flank profil dip deviation.
The measurement method of total profile deviation generally includes coordinate method and generating, the instrument measured to gear-profile deviation Have very much, including gear measuring center, three coordinate measuring machine, laser gear measurement machine, gear meshingtester etc., middle gear Measuring center is most widely used.
When being measured to gear-profile deviation using gear measuring center, due to gear measuring center it is upper and lower it is top not To just and measure when the centre of gyration and gear geometric center it is misaligned, to introduce installation error.
The presence of installation error makes the actual measurement location deviation theory of gear measure position, if not to practical measurement Data are modified, and will reduce the measurement accuracy of gear measuring center, or even cause gear rating mistake.In order to reduce peace Influence of the error to gear measurement is filled, the value of installation error should be adjusted to as small as possible, but this requires a great deal of time simultaneously And installation error cannot completely eliminate.Therefore, quantifier elimination is surveyed with important to the gear-profile deviation under the influence of installation error Meaning.
Installation error is divided into three parts by the present invention：1) gear error of rotary axle is since gear measuring center pushes up up and down The sharp not site error between just caused practical rotating shaft and theoretical rotating shaft；2) Gear tipping error is in gear geometry Error of tilt between mandrel and practical rotating shaft；3) gear geometric eccentricity error is the geometric center of gear measured section and returns Turn the site error between center.Traditional installation error treating method often only considers one of which installation error, processing Process is typically the influence for analyzing installation error to total profile deviation evaluation result, and the compensation of installation error is carried out at result.
The present invention is directed to be measured to the gear with installation error, the gear measurement number under the influence of installation error is obtained According to by the assessment method of invention by gear measurement data correction to ideal mounting position and according to standard ISO 1328-1： 2013 pairs of gear-profile deviations are evaluated, and the gear-profile amount of deflection under ideal position and rating are obtained.
Invention content
To solve the problems, such as to propose in background technology, the present invention proposes the gear teeth under the influence of a kind of new installation error The assessment method of wide deviation.It is inclined that installation error is divided into gear error of rotary axle, Gear tipping error and gear geometry by this method Heart error is established machine by determination, the determination of gear upper surface law vector and the reverse in the basic circle center of circle to rotating shaft and is sat Transformation relation between mark system and workpiece coordinate system, to which the flank profil measurement point under the influence of installation error is adapted to ideal installation Position, according to standard ISO 1328-1：2013 pairs of total profile deviations are evaluated, and the total profile deviation obtained under ideal mounting position is commented Determine result and rating.
The technical solution adopted by the present invention is the involute spur gear wheel total profile deviation evaluation side under the influence of installation error The realization process of method, this method is as follows：
1) coordinate system of machine O is established0-X0Y0Z0, tested gear is measured under coordinate system of machine, measures position packet Include two Section Points, gear upper end millet cake and the flank profil measurement point up and down of tested gear mandrel；
2) according to two Section Points for being tested gear mandrel under coordinate system of machine, make Least Square Circle fitting, obtain two The central coordinate of circle in section is Z with two sections circle center line connecting direction1Axis direction is positive direction in orientation, by coordinate system O0-X0Y0Z0 Successively around X0Axis and Y0Axis rotation makes Z0Axis and Z1Axis is parallel, in plane X0O0Y0It is interior that Z is made by the translation of coordinate system0Axis and Z1Axis It overlaps, obtains coordinate system O1-X1Y1Z1.Establish coordinate system O0-X0Y0Z0With coordinate system O1-X1Y1Z1Between transformation relation, obtain Coordinate system O1-X1Y1Z1Under gear upper end millet cake and flank profil measurement point, and flank profil measurement point is reverted into initial position；
3) according to O1-X1Y1Z1Coordinate system lower gear upper surface measurement point makees plane fitting, obtains gear upper end facial plane side Journey and its normal vector are Z with the normal vector direction2Axis direction, for just in orientation, with Z1The intersection point O of axis and measured section2For original Point.By coordinate system O1-X1Y1Z1Successively around X1Axis and Y1Axis rotation makes Z1Axis and Z2Axis is parallel, makes O by the translation of coordinate system1Point With O2Point overlaps, and obtains coordinate system O2-X2Y2Z2；
4) coordinate system O is established1-X1Y1Z1With coordinate system O2-X2Y2Z2Between transformation relation.Obtain coordinate system O2-X2Y2Z2 Under flank profil measurement point, due to the presence of installation error so that flank profil actual measurement location deviates measured section, therefore in coordinate It is O2-X2Y2Z2Under flank profil measurement point position is modified；
5) according to coordinate system O2-X2Y2Z2Under revised flank profil measurement point establish least square target equation, pass through newton Solution by iterative method is tested the basic circle center of circle of gear, in X2O2Y2Translational coordination system O in plane2-X2Y2Z2, make Z2Axis is cut by tested The face basic circle center of circle, establishes workpiece coordinate system Ow-XwYwZw, obtain coordinate system O2-X2Y2Z2With coordinate system Ow-XwYwZwBetween transformation Relationship obtains the flank profil measurement point under workpiece coordinate system；
6) in workpiece coordinate system Ow-XwYwZwIt is lower according to standard ISO 1328-1：The total profile deviation of 2013 pairs of gears is commented Fixed, the rotation angle for rotating to measurement point around the rolling circle center of circle by calculating design flank profil obtains total profile deviation numerical value, to Correct the influence that installation error evaluates total profile deviation.
Ideal mounting position is the workpiece coordinate system position corrected after installation error.Two sections up and down of tested gear mandrel Millet cake or two Section Points for top circle.
Advantage of the invention is that：
Gear with installation error is measured, the flank profil measurement data under the influence of installation error is obtained, passes through hair Measurement data is adapted to ideal mounting position and according to standard ISO 1328-1 by bright assessment method：2013 pairs of gear-profiles are inclined Difference is evaluated, and the gear-profile amount of deflection under ideal position and rating is obtained, to have modified installation error to tooth The influence of wide deviation evaluation.
Description of the drawings
Fig. 1 coordinate system of machine establishes schematic diagram.
Fig. 2 number of gear teeth number schematic diagrames.
The amendment schematic diagram of Fig. 3 flank profil measurement points.
Fig. 4 non-linear least square target equations establish schematic diagram.
Fig. 5 workpiece coordinate systems establish schematic diagram.
Fig. 6 involute cylindrical gear total profile deviation Evaluation model schematic diagrames.
Fig. 7 is the implementing procedure figure of this method.
Specific implementation mode
To make the derivation of the present invention be more clear, the present invention is done furtherly below in conjunction with attached drawing and derivation formula It is bright.
Establish coordinate system of machine O as shown in Figure 10-X0Y0Z0, with the rotating shaft of gear measuring center and rotating platform Intersection point is origin, and X, Y, the tri- axis rail directions Z are change in coordinate axis direction.In coordinate system of machine O0-X0Y0Z0Under gear is surveyed Amount measures two Section Points, gear upper end millet cake and flank profil measurement point up and down that position includes tested gear mandrel.
Mandrel section and gear upper surface are measured when gear measuring center rotating shaft registration is zero, by this position Initial position when as gear measurement ensures that gear measuring center rotating shaft is motionless, only operates X at this time0、Y0And Z0Axis changes. Mandrel Section Point is denoted as { P0 Aj_k(X0 Aj_k, Y0 Aj_k, Z0 Aj_k)}；Gear upper end millet cake is denoted as { P0 Uj(X0 Uj, Y0 Uj, Z0 Uj)}；
Flank profil point measurement needs rotating shaft and X0、Y0And Z0Axis is four-axle linked to be measured, and flank profil point is denoted asWith
Wherein, L and R it is non-not Biao Shi the left and right flank of tooth, i indicate i-th of tooth, j indicate j-th point, k represents k-th of section, U Gear upper surface is represented, A represents mandrel surface.
The number of teeth sorts as shown in Fig. 2, in the case where measuring initial position, with X0The right flank of tooth in lower section first (including with X0Axis phase Hand over) where tooth be the 1st tooth, in a clockwise direction be incremented by.
To two groups of mandrel surface point { P0 Aj_k(X0 Aj_k, Y0 Aj_k, Z0 Aj_k) Least Square Circle fitting, fitting circle are carried out respectively Heart coordinate is denoted as (A respectively1 0,B1 0,Z1 0) and (A2 0,B2 0,Z2 0).Obtaining revolution paraxial equation is：
Obtain revolution paraxial equation and plane X0O0Y0The coordinate of intersection point be
As shown in Figure 1, being Z with rotating shaft direction1Axis direction, with Z1Axis and plane X0O0Y0Intersection point O1For origin, By by coordinate system O0-X0Y0Z0Translation makes O0Point and O1Point overlaps, by successively around X0Axis and Y0Axis rotates so that Z0Axis and Z1Axis It overlaps, to obtain coordinate system O1-X1Y1Z1。
The direction vector of rotating shaft isTake is just, to obtain successively counterclockwise Around X0Axis and Y0The angle of axis rotation is respectively γ0 xAnd γ0 y。
Obtain coordinate system O0-X0Y0Z0And O1-X1Y1Z1Between transformational relation be：
Above-mentioned transformation is to correct error of rotary axle.At this point, in coordinate system O1-X1Y1Z1It is lower by flank profil measurement point around Z1Axis is inverse Hour hands turn roundAngle obtains the initial position co-ordinates of measurement point, is denoted as { P at this time1 Lij(X1 Lij, Y1 Lij, Z1 Lij) and { P1 Rij (X1 Rij, Y1 Rij, Z1 Rij)}.Spin matrix is：
To gear upper end millet cake { P1 Uj(X1 Uj, Y1 Uj, Z1 Uj) least square plane fitting is carried out, obtain gear upper surface method Vector is denoted as
Assuming that in coordinate system O1-X1Y1Z1The height value in tested gear section is downThe then friendship of rotating shaft and measured section It puts and is
By gear upper surface law vector andPoint determines that linear equation is：
As shown in Figure 1, withPoint is origin, and above-mentioned rectilinear direction is Z2Axis direction, by by coordinate system O1-X1Y1Z1It is flat Shifting makes O1Point and O2Point overlaps, by successively around X1Axis and Y1Axis rotates so that Z1Axis and Z2Overlapping of axles, to obtain coordinate system O2- X2Y2Z2。
It is obtained around X according to fit Plane law vector1Axis and Y1The angle of axis rotation is respectively γ1 xAnd γ1 y。
Obtain coordinate system O1-X1Y1Z1And O2-X2Y2Z2Between transformational relation be：
Above-mentioned transformation is corrected gear heeling error.At this point, in coordinate system O2-X2Y2Z2Under to flank profil measure carry out such as Flank profil measurement point is adapted to measured section X by amendment shown in Fig. 32O2Y2On, ensure to be tested flank profil point in the same plane, repair Point after just is denoted as { P2 Lij(X2 Lij, Y2 Lij, 0) } and { P2 Rij(X2 Rij, Y2 Rij, 0) }.
In view of the basic circle center of circle of gear measured section and O2The gear geometric eccentricity error that point does not introduce again, it is therefore desirable to Determine rolling circle central coordinate of circle.According to involute basic circle center of circle reverse schematic diagram as shown in Figure 4, exist on involute profile Multiple measurement points, for any one measurement point PK(x2 K,y2 K), according to principle of forming involute:
θK=ξK-αK
Wherein：
Ideally, all measurement points are on same involute, therefore, the involute starting point of excessively each measurement point It is identical, i.e., excessively each measurement point PK(XK,YK) involute starting point and circle center line connecting and X2Axle clamp angle δKIt is identical.
Wherein：
δK=arctan ((y2 K-b)/(x2 K-a))-θK(right flank profil)
δK=arctan ((x2 K-a)/(y2 K-b))-θK(left flank profil)
Under actual conditions, due to the presence of measurement error and flank profil itself error, flank profil measurement point is not opened gradually in same On line, therefore least square target equation is established according to residual sum of squares (RSS) minimum principle:
Wherein：
N is the number of measurement point.
Flank profil is measured into the above-mentioned target equation of point data substitution and is solved according to Newton iteration method, the basic circle center of circle is obtained Coordinate is denoted as Ow 0(aw 0,bw 0)
Translational coordination system O2-X2Y2Z2, makeWith the obtained center of circleIt overlaps, obtains work as shown in Figure 5 Part coordinate system Ow-XwYwZw.Establish coordinate system O2-X2Y2Z2With workpiece coordinate system Ow-XwYwZwBetween transformation relation be：
Above-mentioned transformation is corrected gear geometric eccentricity error.At this point, obtaining flank profil measurement point under workpiece coordinate system and being {Pw Lij(Xw Lij, Yw Lij, 0) } and { Pw Rij(Xw Rij, Yw Rij, 0) }.
Gear-profile deviation is evaluated under workpiece coordinate system, establishes design involute equation as shown in FIG. 6：
ξ is the exhibition angle of involute.α is pressure angle.θ is the angle of spread.
Left tooth Profile Design involute is with YwThe intersection point of axis and basic circle is starting point, and equation is：
According to standard ISO 1328-1：2013 calculate flank profil control diameter dcfAnd its length Lcf, tooth top shape diameter dFaAnd Its length LFa, tooth tip diameter daAnd its length La, evaluation range Lα。
Flank profil measurement point is screened, arbitrary profile point P is calculatedw Rij(Xw Rij, Yw Rij, Zw C) corresponding involute angle of spread ξRij。
According to angle of spread ξRij, the coordinate value for calculating design involute is (xw D_Rij,yw D_Rij)。
Design involute is rotated around workpiece coordinate system origin, makes each survey of the design involute by practical involute Point is measured, rotation angle γ is calculatedRij。
According to the definition of total profile deviation it is found that tooth profile total deviation is：
Fα=rb·(max(γRij)-min(γRij))
Calculate deviation discrete data：
To discrete discrepancy data carry out least squares line fitting, obtain being fitted average flank profil line equation be：
yEi=kEi·xEi+bEi
kEiFor the slope of most average flank profil line.
Calculating flank profil dip deviation is：
fHα=kEi·(La-Lcf)
Evaluation range LαInterior calculating profile geometry deviation is：
ffα=| max (γRij·rb-yEi)-min(γRij·rb-yEi)|
Algorithm is evaluated by Matlab to the total profile deviation under the influence of above-mentioned installation error to emulate.It is 5mm with modulus, The number of teeth 18, for the involute spur gear wheel of reference diameter angle 90mm, according to world standards for gears ISO 1328-1：2013 2 class precisions of middle regulation choose emulation data-oriented：
The installation error studied according to the present invention gives every error value：
It is by the measurement point under coordinate system of machine of the MATLAB generations with installation error：
Algorithm according to the present invention carries out total profile deviation evaluation to the above-mentioned flank profil measurement data with installation error, obtains Simulation figure and total profile deviation numerical value.
If without the amendment of installation error, total profile deviation evaluation directly is carried out to above-mentioned flank profil measurement data, is imitated True figure and total profile deviation numerical value.
Obtain given total profile deviation numerical value and the simulation result table of comparisons.
It can be seen that for modulus be 5mm by the above-mentioned table of comparisons, the number of teeth 18, reference diameter is 2 grades of involutes of 90mm Spur gear wheel, the total profile deviation assessment method under the influence of installation error is to the flank profil measurement with installation error through the invention Data are evaluated, and obtained evaluation result and given value are almost the same, and gear rating is identical；Without installation error It corrects the evaluation result directly evaluated to the flank profil measurement data with installation error and differs larger with given value, tooth It is 5 grades to take turns rating, is far longer than given grade.Prove that the total profile deviation assessment method under the influence of installation error of the present invention is repaiied The just influence of installation error, algorithm is correctly feasible.

Claims (6)

1. the involute spur gear wheel total profile deviation assessment method under the influence of installation error, it is characterised in that：The reality of this method Existing process is as follows：
1) coordinate system of machine O is established0-X0Y0Z0, tested gear is measured under coordinate system of machine, it includes tested to measure position Two Section Points, gear upper end millet cake and the flank profil measurement point up and down of gear mandrel；
2) according to two Section Points for being tested gear mandrel under coordinate system of machine, make Least Square Circle fitting, obtain two sections Central coordinate of circle, be Z with the circle center line connecting direction of two sections1Axis direction is positive direction in orientation, by coordinate system O0-X0Y0Z0According to It is secondary around X0Axis and Y0Axis rotation makes Z0Axis and Z1Axis is parallel, in plane X0O0Y0It is interior that Z is made by the translation of coordinate system0Axis and Z1Axis weight It closes, obtains coordinate system O1-X1Y1Z1；Establish coordinate system O0-X0Y0Z0With coordinate system O1-X1Y1Z1Between transformation relation, sat Mark system O1-X1Y1Z1Under gear upper end millet cake and flank profil measurement point, and flank profil measurement point is reverted into initial position；
3) according to O1-X1Y1Z1Coordinate system lower gear upper surface measurement point makees plane fitting, obtain gear upper surface plane equation and Its normal vector is Z with the normal vector direction2Axis direction, for just in orientation, with Z1The intersection point O of axis and measured section2For origin；It will Coordinate system O1-X1Y1Z1Successively around X1Axis and Y1Axis rotation makes Z1Axis and Z2Axis is parallel, makes O by the translation of coordinate system1Point and O2Point It overlaps, obtains coordinate system O2-X2Y2Z2；
4) coordinate system O is established1-X1Y1Z1With coordinate system O2-X2Y2Z2Between transformation relation；Obtain coordinate system O2-X2Y2Z2Under Flank profil measurement point, due to the presence of installation error so that flank profil actual measurement location deviates measured section, therefore in coordinate system O2-X2Y2Z2Under flank profil measurement point position is modified；
5) according to coordinate system O2-X2Y2Z2Under revised flank profil measurement point establish least square target equation, pass through Newton iteration Method solves the basic circle center of circle for being tested gear, in X2O2Y2Translational coordination system O in plane2-X2Y2Z2, make Z2Axis is by being tested gear Workpiece coordinate system O is established in the basic circle center of circlew-XwYwZw, obtain coordinate system O2-X2Y2Z2With coordinate system Ow-XwYwZwBetween transformation close System, obtains the flank profil measurement point under workpiece coordinate system；
6) in workpiece coordinate system Ow-XwYwZwUnder the total profile deviation of gear is evaluated, by calculating design flank profil around gear base The rotation angle that the round heart rotates to measurement point obtains total profile deviation numerical value, is evaluated to total profile deviation to correcting installation error It influences.
2. the involute spur gear wheel total profile deviation assessment method under the influence of installation error according to claim 1, It is characterized in that：Ideal mounting position is the workpiece coordinate system position corrected after installation error.
3. the involute spur gear wheel total profile deviation assessment method under the influence of installation error according to claim 1, It is characterized in that：Two Section Points up and down or two Section Points for top circle of tested gear mandrel.
4. the involute spur gear wheel total profile deviation assessment method under the influence of installation error according to claim 1, It is characterized in that：Installation error is divided into gear error of rotary axle, Gear tipping error and gear geometric eccentricity error.
5. the involute spur gear wheel total profile deviation assessment method under the influence of installation error according to claim 1, It is characterized in that：According to standard ISO 1328-1：2013 pairs of gear-profile deviations are evaluated.
6. the involute spur gear wheel total profile deviation assessment method under the influence of installation error according to claim 1, It is characterized in that：The eccentricity of tested gear is by solving in coordinate system O2-X2Y2Z2The least square target equation of lower foundation obtains It arrives.
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