Development of Scanning Measurement of Tooth FlankForm of ...

Development of Scanning Measurement of Tooth Flank Form of Generated Face Mill

55

Development of Scanning Measurement of ToothFlankForm of Generated Face Mill Hypoid Gear Pair

with Reference to the Conjugate MatingTooth Flank form Using 2 Axes Sensor

RYOHEI TAKEDA and MASAHARU KOMORI*Department of Mechanical Engineering and Science, Kyoto University, Yoshidahonmachi, Sakyo-ku

Kyoto-shi, Kyoto 606-8501, Japan*e-mail: [email protected]

[Received: 31.01.2011 ; Revised: 14.02.2011 ; Accepted: 22.02.2011]

AbstractThe three diamensional (3D) scanning measurement method of tooth flank form of generated face millgear pair is proposed, which is based on the conjugate mating tooth flank form. In this method,deviation sensor is placed at the position which has certain offset from the gear axis and synchronizedto the gear rotation. This method realizes the 3D tooth flank form measurement and multi pointsscanning measurement by using 2 axes sensors. Based upon this proposal of the measurement method,tooth flank form measuring machine was developed and the experimental measurements were carriedout. It is confirmed that this method is capable of high speed and high resolution measurement of finewaviness and it has high repeatability. It is confirmed that the proposed method, is capable of 3Dscanning tooth form measurement of generated face mill hypoid gear pairs with high repeatability athigh speed.

© Metrology Society of India, All rights reserved 2011.

1. Indroduction

Hypoid gears have an advantage in strength andsmooth rotation over spiral bevel gears and thus theyare widely used in rear wheel drive and 4 wheel drivevehicles. Recently engine noise and road noise ofvehicles have been improved and therefore betternoise and vibration quality is demanded for hypoidgears. For hypoid gears, slight form deviations on thetooth surfaces affect their noise and vibrationcharacteristics. For that reason quality control of toothsurfaces by means of measurement of tooth flank formbecomes important.

For hypoid gear, a lot of studies have been carriedout on theoretical tooth geometry [1-2]. Most of themdeal with non generated face mill hypoid gears, whichare most widely used [1]. After CNC controlled beveland hypoid gear cutting machines and grindingmachines were developed, such studies were relatedto the CNC control [3-6].

On the other hand, many studies about gear toothflank form measurement were reported [7-8]. Therewere also some studies about the analysis of hypoidgear cutting error from the tooth form measurement[9-10], calculation of correction of gear cuttingmachine setting parameter based on the toothmeasurement data or curve fit of the real tooth surface

MAPAN - Journal of Metrology Society of India, Vol. 26, No. 1, 2011; pp. 55-67ORIGINAL ARTICLE

Ryohei Takeda and Masaharu Komori

56

using setting parameter of gear cutting machine[11-15]. However, in those researches, coordinatemeasuring machine (CMM) were used. In that case,only 5 points were measured in profile direction. Noiseand vibration of gears are caused by small formdeviation of the tooth flank. Therefore, it is necessaryto measure multiple points on the tooth surface. Inaddition, to realize it in a short time, scanningmeasurement is desirable. But high speed scanningmeasurement of complex 3D surface like hypoid gearsusing 3 axes deviation sensor causes deterioration ofmeasuring accuracy and increase of cost of sensor.

In the earlier studies, non generated face millhypoid gears are taken, which can be produced bysimple cutting method. However, recently demand forgenerated face mill hypoid gear is increasing. Thesedays, multi purpose 4 wheel drive vehicle is verypopular. In case of 4 wheel drive system based onfront drive vehicle, hypoid gear of low ratio is used inthe transfer gear box because in many cases it isdifficult to apply non generated cutting such asFormate® or Helixform® to cut such wheels. In thatcase, generated hypoid gear must be used. In generatedhypoid gear, wheel tooth flank form becomes muchcomplicated and it’s quality control is much moredifficult than non generated hypoid gear. For thatreason, tooth flank form measuring technology ofgenerated hypoid gears is required. However, there isno report about the research of the measurement ofgenerated hypoid gear tooth, especially multi pointscanning measurement of it.

In this report, scanning measurement method oftooth flank form of generated face mill hypoid gearpair is developed. To solve the problems of high speedscanning measurement using 3 axes deviation sensor,focusing on the geometrical characteristics of hypoidgear surface, new measurement method is proposed,in which position of gear and deviation sensor ischosen so that direction of normal vector of toothsurface becomes close to the sensing direction ofsensor at the measuring point. This method does notneed 3 axes deviation sensor. Using this method,technique of scanning measurement of 3 dimensionalsurface form by 2 axes deviation sensor is developed.

In the earlier studies about hypoid gearmeasurement, reference surface of pinion and wheelwere created based on the envelope of gear cutter. Inthat case, the measurement result, which is the

deviation from the reference surface, does not indicatenoise and vibration characteristics of measured geardirectly, and it is not suitable to evaluate gearperformance. But in the method of using conjugatesurface of gear pair as the reference surfaces [16], thosereference surfaces are conjugate to each other and donot have any transmission error. Therefore it isadvantageous for the stand point of quality control ofgear performance. The research of this concept ofconjugate reference surface was done for the nongenerated face mill hypoid gear, but not for generatedhypoid gear. In this research, it is aimed to apply theconcept of conjugate reference surface to generatedhypoid gear and realize the measuring method, whichis suitable for the evaluation of gear performance.

The analysis theory of tooth geometry of generatedhypoid gear is shown based on the concept ofconjugate tooth surface. Using this analysis, themethod to make the directions of normal vector oftooth surface almost parallel to the sensing directionof sensor at the measuring point, is presented. Usingthis result, multipoint measuring method is proposed.Based on such studies a tooth flank form measuringmachine is developed and experimental measurementsare performed and the effectiveness of the proposedmethod is verified.

2. Analysis Method of Conjugate Surface ofGenerated Hypoid Gear

2.1. Purpose of Tooth Form Measurement and SuitableReference Surface

The purposes of hypoid gear tooth flank formmeasurement are (i) correction of gear cutting machinesetting and (ii) quality control by means of prediction oftransmission error and tooth contact pattern.

On the other hand, there are 2 types of concept aboutreference surface of hypoid gear pair. One is type (A),which is to define the reference surface of both wheel andpinion as the surface derived from the envelope of gearcutter. The other one is type (B), which is to define thereference surface of wheel as the surface derived from theenvelope of gear cutter and that of mating pinion as theconjugate surface of wheel [16].

To achieve purpose (i), to make correction of gearcutting machine setting, it is required to identify thedeviation of real tooth flank form from the theoretical cuttingform, and the reference surface type (A) is suitable for it.


57

On the other hand, for the purpose (ii), to predicttransmission error and tooth contact pattern and controlits quality, it is suitable to use reference surface of conjugatesurface (type (B)), which represent ideal tooth meshcondition, and identify the deviation of real tooth flankform of gear pair from the ideal condition. It is similar toevaluation of spur and helical gear, for which involutehelicoid is used as the reference surface. The involutehelicoids of mating gear pair are conjugate to each otherand measured result of them indicate the deviation fromthe conjugate condition. Thus, it is convenient to predicttransmission error and tooth contact pattern of the gearpair. In case of type (A), reference surfaces of gear pair arenot conjugate to each other and it is not possible to estimatetransmission error or tooth contact pattern directly fromthe measured result. However, in case of type (B), referencesurfaces of gear pair are conjugate to each other and it iseasy to estimate transmission error or tooth contact pattern.In addition, hypoid gears are commonly finished by heattreatment and lapping processes and during theseprocesses, tooth surfaces are distorted largely from thecutting form. Therefore, it is not effective to measure thefinished tooth surfaces using reference surfaces of type (A).Therefore, in case of quality control of performance bytooth flank form measurement, reference surface of type(B) is adopted.

2.2. Parameters for the Theoretical Surface of GeneratedFace Mill Hypoid Gear

Both sides of the tooth of the generated face mill wheelare cut by face mill cutter similar to that of the nongenerated wheel as shown in Fig. 1 using the same cuttingmachine used for hypoid pinion. Recent CNC controlledhyoid pinion cutting machines do not have mechanicaldevices such as eccentric angle and cradle which oldmechanical machines have, but the relative motion of thecutter and work piece are duplicating that of the oldmechanical machines. Figure 2 shows the Gleason No.106 generator. In this machine, cutter is attached on thetilting mechanism, "Cutter spindle rotation angle", whichadjust the cutter axis tilt angle θi and it is mounted on theswivel mechanism, "Swivel angle", which adjust thedirection of the tilting axis θi . In addition, it is mountedon the table to adjust eccentricity of the cutter axis fromthe machine centre, "Eccentric angle", which create radiusof the orbital motion of the cutter axis Sr, and on therotational table, which rotate around the machine centreand create orbital motion of the cutter, "Cradle". Workpiece spindle is located on the vertical slide, which adjustthe vertical position of the spindle, "Blank offset EM", andit is mounted on the work piece axial adjustmentmechanism, "Machine centre to back ΔA ". Since it is set

Fig. 1. Definition of cutter coordinate


58

on the rotational table which adjust the root angle of thework piece, "Machine root angle y1", and on the in feedslide "Sliding base", which adjust the in feed position ofthe work piece ∆B. Figure 3 shows the relationshipbetween the cutter and the work piece, using followingcoordinate systems;

Of , Xf , Yf , Zf : Cutter coordinate system.Ob, Xb , Yb, Zb : Swivel angle coordinate system.Ocl , Xcl, Ycl , Zcl : Cradle coordinate system.O1, X1 , Y1, Z1 : Work piece coordinate system.Oh , Xh , Yh , Zh : Adjusted work piece coordinate system.

The work spindle and the cradle then rotate with the

relationship defined by Eq. (1), where 1ω and arethe angular velocities of the work spindle and the cradle.

(1)

Using these 8 machine parameters, Em,∆B, Y1 , ∆A,θi , θi , Sr and If, and cutter dimension Rcl and α1 , thegenerated wheel tooth flank form can be defined.

2.3. Calculation of the Theoretical Tooth Surface of the Generated Face Mill Method

The generated work tooth and the cutting edge contactat point M are shown in Fig. 4 having common normal

vector N and move in the different directions with the

velocity of wV and cV . At the contact point M , thenormal component of the velocity of the 2 surfaces mustbe equal. This condition can be described as follows;

− ⋅ =w c( ) 0V V N (2)

where

= ×C CV Mω (3)

Fig. 2. Gleason no.106 hypoid gear generator

Fig. 3. Relationship of cutter and work gear Fig. 4. Contact point of cutter and work gear tooth

ω c

ω=

f C 1/I ω


59

= × −w 1 1( )V M Oω (4)

Point M on the cutting edge is determined by the

rotation angle αc of cutter axis zf , the distance along thegenerating cone Sp (Fig. 1) and rotation angle of cradle α1,which moves the cutter axis around the cradle axis Ocl.

(Fig. 3). The normal vector of the point M can be

described by αcandα1 where α1 and α3 are related as;

=f 1 3/I α α (5)

and are described as follows.

α α (6)

α α= C 3( , )N N (7)

SubstitutingEqs. (3) and (4) into Eq. (2), we obtained;

× − × − ⋅ =C 1 1 0( ( ))M M O Nω ω (8)

Then substituting Eqs. (6) and (7) into Eq. (8), Sp isdetermined by α1 and α3;

=p p C 3( , )S S α α (9)

The parameter Sp can be eliminated from Eq. (6) and

is describe as follows;

= C 3( , )M M α α (10)

Then by coordinate transform, the contact point inthe work gear coordinate P is also described as follows;

= C 3,( )P P α α (11)

2.4. Calculation of Conjugate Pinion Surface

Now, the wheel tooth surface of the generated facemill hypoid gear is defined as the envelope of the cuttingedge controlled by the cutting machine setting. Pinion

tooth surface is defined as the conjugate surface to thewheel. Figure 5 shows the wheel and the pinioncoordinate systems and the contact of their surfaces asfollows;

αg : Wheel rotation angleαp : Pinion rotation angleOg-Xg , Yg , Zg : Wheel coordinate system G, of which 3rd

axis is the wheel axis and 2nd axis is the offset direction.Op -Xp , Yp , Zp : Pinion coordinate system P, of which3rdaxis is the pinion axis and 2nd axis is the offset direction.

When wheel rotates, the coordinate of arbitrary pointand its normal vector on wheel surface are changed.Suppose that initial position and normal vector of certainpoint Q on the coordinate system G is as Eqs.(12) and(13).

( ) ( )= =g0 g0 c 3 g0 g0 g0, , ,T

P P x y zα α (12)

( )= =,g0 0 3 xg0, yg0, zg0cT

N N N N Ng α α (13)

When wheel is rotated by αg, position vector gr and

normal vector , the point Q can be described asfollows;

g

Op

EOp g

Zp

Xp

Yp

Yg

Zg

Xg

r g

r p

p

g

p

ωg

v

vg

- g

pv

v gpv -

gω

r p

rg

ωppv =

= ×

×

Σ

α

α

The coordinate system of gear

The coordinate system of pinion

O

ω

Q

Fig. 5. Definition of gear and pinion coordinate

=

p C 3( , , )M M S

( )

gN

M


60

( )= + + =g g g g g g g, ,T

jr x i y z k x y z (14)

( )=g xg yg zg, ,T

N N N N (15)

where,

α α= −g g g0 g g0cos sinx x y

α α= +yg g xg0 g yg0sin cosN N N

=zg zg0N N

, ,ji k : x, y, z unit vector of xe , ye and ze axis.

If rotation vector of the wheel is gω , pinion is pω ,

shaft angle of wheel and pinion is pgO , and shaft position

vector is (absolute value E= | pgO |), velocity vector ofpoint Q is written as follows (Fig. 5);

( )=g g0, 0,T

ω ω (16)

= ×g g gV rω (17)

If the rotation ratio of the wheel and the pinion is= p g/γ ω ω , angular velocity vector pω , position vector

pr and velocity vector of contact point pV of pinion canbe described in the wheel coordinate G as follows;

= − + − ΣΣp p p( ) ( )sin cosi kω ω ω (18)

= + = + − +p pg g g g g( )O jEr r x i y z k

( )= −g g g,, TEx y z (19)

= ×p p pV rω (20)

The contact points satisfy the following condition;

− ⋅ =p g g( ) 0V V N (21)

In such cases, the following relation is obtained;

γ γ

α

α

− ⋅ Σ + + ⋅ Σ +

+ ⋅ Σ ⋅

+ − Σ ⋅ + ⋅ ⋅ Σ − ⋅ ⋅

Σ

+ Σ ⋅ + ⋅ ⋅ Σ − ⋅ ⋅

Σ ⋅ =

0gx0g0gy0g

zg0

xg0 g0 yg0 g0 zg0

g

yg0 g0 xg0 g0 z 0

g

( 1/ ) ( 1/ )

(

)

(

) 0

cos cossin

cos sinsin cos

cos sinsin sin

g

N

N E

E N N N

E N N N

Nx y

z y

z x (22)

From Eq. (22), the wheel rotation angle αg is obtained,when point Q on the wheel is in contact with pinion surface .Pinion rotation angle αv is then given by;

γα α= ⋅p g (23)

By coordinate transformation, the position vector andthe normal vector of the contact point of pinion surfacecan be obtained on pinion coordinate system P.

2.5. Example of Calculation

Based on the above mentioned theory of the toothgeometry, analysis program of the generated hypoid gearis developed. Table 1 shows the gear dimension used forthe analysis. Figure 6 (a) shows the calculated image ofthe wheel tooth surface form and Fig. 6(b) shows themating conjugate pinion surface form. As shown in Fig.6, 3D tooth form of the wheel and the mating pinioncan be obtained.

3. Scanning Method of the 3D Tooth Form by 2Axes Deviation Sensor with the Offset Position

3.1. Measuring Machine Structure

α α= +g g g0 g g0sin cosy x y

=g g0z z

α α= −xg g xg0 g yg0cos sinN N N


61

The normal CMM machines utilize 3 linear axes anddo not use rotational axes at the measurement in manycases. However, the purpose of this research is to realizemulti points scanning measurement, and to scan wholetooth surface without interference it is desirable to rotategears during the measurement. From this stand point, thestructure of gear measuring machine with precisionrotational table is adopted. Fig. 7 shows the view of thedeveloped measuring machine of hypoid gear tooth flankform. A measuring gear is held on the main spindle axis(T axis), using upper and lower centre or special mandrelto be attached on the spindle. The probe and the sensorhead can be moved by orthogonal 3 axes (X, Y and Z).Those moving axis are guided by the linear slide and thedriven by the ball screws and motors. Each axis isequipped with linear scale and the actual position ismonitored.

3.2. Proposal of the Measuring Method of the 3 D Tooth Flank Form by 2 Axes Sensor with Offset Measuring Position

Hypoid gear tooth surface has 3D geometry and the

Table 1Gear dimension

Pinion Gear

Number of teeth 19 24Module - 4.0Face width - 24.0Pinion offset 5.0 -Shaft angle 90Pitch angle 39.27 50.58Spiral angle (Hand) 40.15 (RH) 34.40 (LH)Cutting method Generated Generated

Face Mill Face Mill

Fig. 6. Calculated tooth surface of the generated facemill hypoid gear

(b) Conjugate pinion tooth surface

(a) Gear tooth surface

Fig. 7. Structure of the measureing machine of hypoidgear tooth flank form


62

directions of normal vectors of the surface vary in thewide range. For this reason, normally, it is consideredthat 3D sensor is required. But increase of sensor axismakes it difficult to keep high precision and causesincrease of cost. The measuring method of 3D surface by2 axes sensor with offset measuring position is proposed.

The normal vectors of the points on the tooth surfacevary widely in the fixed coordinate of the gear, but if thegear is rotated on the spindle in accordance with theposition of the probe, it is presumable that the variationrange of the normal vector can be limited in the coordinateof sensor.

Now we assume that the probe position is offset fromthe axis of the gear as shown in Fig. 8 (a). The probe andthe measuring surface make contact in the X-Z offset planeduring the measurement. In other words, contact positionis located in the offset plane and the gear is rotated so thatthe corresponding point on the gear surface comes to theoffset plane. By this manner, it is possible to minimize theX component of normal vector of the tooth surface in thecoordinate of sensor.

An example of calculation result of normal vector bythe theoretical formula discussed in the previous sectionis shown in Fig. 9 using the gear dimension of Table. 1.Three different probe offset positions are taken. Thisexample shows normal vector on tooth trace measuringline, which is shown in Fig. 8 (b). When the offset is set to8mm, the variation of X component of normal vector staysaround 0, and the variation range is within ± few degrees.From this example, it can be said that if the probe offset ischosen properly, the variation of X component of normalvector can be minimized. The example of the profilemeasurement is shown in Fig. 10. In this case also, if theprobe offset is chosen properly, the variation of Xcomponent of normal vector can be minimized.

Now the X component of the normal vector of themeasuring point can be small enough. But depending onthe position of the tooth surface, it is not always 0. Thereforethe output of the deviation sensor is converted to thenominal direction. The method is discussed in the nextsection. As a result, the deviation sensor with 2 axes,which are corresponding to Y and Z axes of the machinecoordinate can measure tooth flank form and 3 axes sensor

Probe head of sensor

Yp

O

=Constant

Y

X

Gear tooth surface

Normal vector

Moving direction of Probe headon tooth trace measurement

Gear rotationto target point

θt

Offset plane

R

Z

Profile Toe

Heel

Tooth trace

Profile

Tooth trace

Toe

Heel

Fig. 8. Measuring method using a 2 axis sensor with offset

(a) Probe position with offset

(b) Profile and tooth trace measurement

-2 0

-1 5

-1 0

- 5

0

5

1 0

1 5

2 0

25 30 35 40 45 5 0

4 mm8 mm12 mm

Offset position

Incl

inat

ion

angl

e of n

orm

al v

ecto

r fro

m Y

axi

s on

X-Y

pla

ne (d

eg)

20

15

10

5

0

-5

-10

-15

-20

25 30 35 40 45 50

X Coordinate (mm)

Fig. 9. Normal vector on tooth trace measuring line underdifferent offset condition of the sensor


63

is not required. Therefore, 3D measurement can berealized by the more accurate and less expensive 2 axessensor. In this example, although the gear with right handspiral angle is shown, in the case of the gear with lefthand spiral angle, the same result is obtained by offsettingthe probe to the left side.

3.3. Calculation Method of the Tooth Flank Form Deviation

The tooth flank form deviation is defined as thedistance between the nominal surface (reference surface)and the measured surface in the direction of the normalvector. The nominal surface consists of the coordinatedata and normal vector of the target points. In the case ofthe developed measuring machine, spindle rotation angledata is also included. The probe is controlled so that itcontacts with theoretical point designated by the nominaldata on the X-Z offset plane. As described in the previoussection, X component of the normal vector of tooth surfaceis small but not zero. In other words, to measure toothflank form using 2 axes sensor on the Y-Z plane, thedirection of the normal vector of the surface and sensingdirection of the 2 axes sensor is not perfectly identical.Then the output of the sensor must be converted to thetooth flank form deviation in the normal direction of thesurface. In the case of Fig. 11, the probe is deviated fromthe target point on the nominal surface, A. The amount ofthe probe deviation is obtained as the output from thesensor. From the probe deviation and the normal vector,

the deviation in the normal direction AB is calculated asthe tooth surface deviation E. It is obtained as follows;

( ) ( )( )( )

= − + − + −

+ + −

x 0 y y 0

z z 0

E n X X Y Y

n Z d Z

n d (24)

where, (X, Y, Z) : Output data of the linear scale (X0, Y0, Z0) : Target position of the probe (nominal

coordinate) (nx, ny, nz) : Normal vector of the target point (dy, dx,) : Output data of 2 axes sensor

The target position and its normal vector are obtained byrotating the pinion or wheel with θt as shown in Fig. 8.

3.4. Developed Measuring Machine

Figure 12 shows the picture of the developedmeasuring machine of the hypoid gear tooth flank form.Figure 13 shows the sectional view of the 2D deviationsensor used in the measuring machine. The probe is heldby 2 sets of the parallel flat springs, and the linearity ofthe output data is controlled less than ± 0.001/0.2mm ,which means that the senor is replaced by 0.2mm and

-20

-15

-10

-5

0

5

10

15

20

15 16 17 18 19 20

4 mm8 mm

12 mm

Offset position

Z Coordinate (mm)

Incl

inat

ion

angl

e of

nor

mal

vec

tor

from

Y a

xis

on X

-Y p

lane

(deg

)

20

15

10

5

0

-5

-10

-15

-20

15 16 17 18 19 20

Fig. 10. Normal vector on profile measuring line underdifferent offset condition of the sensor

Fig. 11. Definition of tooth flank form deviationconsidering sensing direction of a 2 axis sensor

C

C ：Contact point of probe

A

B

Nominal surface

Measured surface

X

Z

Y

A : Point on the nominal surfaceB : Point on the measured surfaceAB: ( E )

AB

Deviation

Output

of

Deviat

ion se

nsor

Normal vector


64

the deviation of output data is within ± 0.001. Thedeveloped machine measures the gear tooth flank formby scanning the target line. The data storage system iscreated so that it can measure more than 100 points ineach target line. As shown in Fig. 8 (b), profile measurementis performed from root to tip and that of tooth trace isperformed from toe to heel by scanning. It is same manneras profile and helix measurement of involute gear.

4. Measurement Experiment of the Tooth Flank Form

4.1. Experimental Measurement

Measurement experiment was conducted usingthe gear set with dimension shown in Table. 1. It wasconfirmed that the measurement can be performedwithout any problem. The measurement along a profileline was performed in a short time (about 10 seconds).

The measurement results thus obtained areshown in Fig. 14 of concave tooth flank of the wheel.Toe, mean and heel measurement are shown fromupper to lower. The result of tooth trace measurementat the middle of tooth height is also shown. Themeasurement results of the convex tooth of matingpinion are shown in middle. The results show thatthe small form deviation of the tooth surface can bemeasured by the multipoint scanning. In case ofhypoid gear measurement by normal CMM, onlyabout 5 points are measured and such a small formdeviation cannot be recognized. From this result, it isconfirmed that the small form deviation of the toothsurface can be measured by this method.

4.2 Estimation of the Meshing Condition Utilizing the Measurement Data

Because the reference surface of the mating piniontooth surface is conjugate to that of wheel, abscissasaxis of the graph of the wheel and the pinionmeasurement chart mean common contact line duringgear mesh. In other words, by observing thosemeasurement results on the same abscissa, deviation

Fig. 12. Developed measuring machine of hypoid gear

Fig. 13. Two axis deviation sensor


65

of the pinion and the wheel tooth surfaces at apotentially contacting point can be estimated. Theright figure in Fig. 14 shows the condition in whichthe pinion tooth trace data is made upside down, andpinion profile data is made upside down and isreversed horizontally, and they are overlapped to thewheel measurement data. Relative deviations from theconjugate condition are displayed. Therefore, theoccurance of the contact of the tooth can be estimatedfrom these results. For example, overlapped figure(right figure in Fig. 14) indicate that tooth contactoccurs in the middle area of tooth on the tooth traceline.

Measurement of 29 lines of profile and a line oftooth trace were performed for the wheel and thepinion. Measurement data of the wheel and the pinionsurfaces were combined and relative deviation wascalculated for each point. The composite tooth flank

form deviation of the pinion and the gear is thenobtained as shown in the Fig. 15. The ordinate lineshows composite deviation. From the composite toothflank form deviation, outline of tooth contact isobserved. In other words, it can be said that if thishypoid gear set is assembled in ideal position, toothcontact will occur in the middle area between toe andheel. In the profile direction, gear root area (pinion tiparea) will contact strongly, but gear tip area (pinionroot area) might not contact depending on load.

By realizing scanning measurement, it becamepossible to find out small form deviation of the toothsurface and also possible to measure many lines onthe surface within the practical time. In addition, bythe adoption of conjugate reference surface, it becamepossible to estimate the tooth contact pattern in detail.

4.3 Measurement Ability of the Small Form Deviation

Fig. 14. Result of tooth form measurement by proposed method


66

and the Repeatability of the Measurement

In Fig. 16, the result of 10 times measurements ofthe same profile line and tooth trace line of the wheelis shown. The gear used in this experiment was notprocessed by lapping and therefore has rather largewaviness on tooth flank. The measuring result showsthe small waviness on the flank well. In addition, tenmeasured data is almost identical and the differenceamong ten measured data is less than 1 μm. Therefore

the results are highly repeatable.

The measurements were performed at the highspeed of 10 s for a line. Even in such a condition, highresolution to small form deviation and highrepeatability were accomplished. These result showsthat the developed measuring machine realized highaccuracy.

5. Conclusion

Generated face mill hypoid gear is used in thetransfer gear box of the front drive based 4 wheel drivepassenger car. In this research, tooth flank formmeasuring method of the generated face mill hypoidgear was developed for the purpose of control of gearperformance such as estimation of the tooth contactpattern.

The measuring method of the tooth flank formwas proposed. The measurement of 3D tooth flankform using 2 axes deviation sensor was realized byoffsetting the sensor from the measuring gear axis andmoving it’s position according to the gear rotation.Using this method, multi points scanning of the toothsurface was proposed. Based upon these proposals,the gear tooth flank form measuring machine wasdeveloped. By experimental measurements, using thedeveloped machine, it was confirmed that themeasurement of small form deviation of the tooth flankwas possible with high repeatability at a high speedscanning measurement. In addition, using theconjugate reference surface, it was demonstrated thatthe estimation of the tooth contact condition waspossible directly from the tooth flank formmeasurement data. From these result, it was confirmedthat the proposed scanning measurement method ofthe tooth flank form of the generated face mill hypoidgear is effective for the quality control of the gear set.

Acknowledgement

This research was conducted in collaboration withMazda Motor Corporation and Osaka Seimitsu KikaiCo. Ltd., Japan.

References

[1] F.L. Litvin, et al., Methods of Synthesis andAnalysis for Hypoid Gear-Drives of ̀ Formate'

Fig. 15. Composite tooth flank form deviation of pinionand gear

-12

-10

-8

-6

-4

-2

0

2

4

6

8

-6

-4

-2

0

2

4

6

8

10

12

14

RootTip

(b) Tooth trace measurement result

Toe

Heel

（a）Profile measurement result

Toot

h fla

nk d

evia

tion

(μm

)

-12-10-8-6-4-202468

-6-4-202468

101214

Toot

h fla

nk d

eviat

ion

(μm

)

Fig. 16. Result of 10 times measurement of the sameprofile and trace line on gear tooth flank


67

and ̀ Helixform', Part 1, 2, and 3, ASME J. Mech.Des., 103 (1981) 83-113.

[2] F.L. Litvin , et al., Design and Stress Analysis ofLow-Noise Adjusted Bearing Contact SpiralBevel Gears, ASME Journal of MechanicalDesign, 124 (2002) 524-532.

[3] H. J. Stadtfeld, Hand book of Bevel and HypoidGears, Rochester Institute of Technology,(1993).

[4] H.J. Stadtfeld, et al., The Ultimate Motion Graph,ASME Journal of Mechanical Design, 122 (2000)317-322.

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