NBSIR 73-272

On Characterizing IVIaster Involute

Profiles

Edgar G. Erber B. Nelson Norden

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

August 28, 1973

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
r
NBSIR 73-272

ON CHARACTERIZING MASTER INVOLUTE

PROFILES

Edgar G. Erber

B. Nelson Norden

Institute for Basic Standards

National Bureau of Standards Washington, D. C. 20234

August 28, 1973

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

IMATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

CONTENTS

Page

Abstract , 1

I. Introduction 1

II. Procedures in Involute Calibrations 2

III. Example of Calibration 7

IV. Sources of Error 9

V. Summary 11

ILLUSTRATIONS

Figure Page

1. Illustration of Basic Gear Elements 12

2. Illustration of the Generation of the Involute Profile . . 13

3. Illustration of Measurement of Involute Profile 14

4. Involute Master 15

5. Establishing Reference Axis on Measuring Machine (a) ... 16

6. Establishing Reference Axis on Measuring Machine (b) ... 17

7. Checking Involute Master for Roundness 18

8. Installation of Master Involute into Measuring Machine . . 19

9. Check for Parallelism between Involute Axis and Reference

Axis of Measuring Machine 20

10. Establishing Base Radius Distance on Involute Master ... 21

11. Bringing Probe into Contact with Involute Master 22

12. Probe used only as Nulling Device 23

13. Rotation of Involute Master 24

14. Rotation of Involute Master 25

15. Rotation of Involute Master 26

16. X-Axis of Machine Moved to Tram Surface 27

17. Reduction of Involute Data 28

18. Plot of Involute Profile Errors from Nominal 29
£
ON CHARACTERIZING MASTER

INVOLUTE PROFILES

by

Edgar G. Erber and B. Nelson Norden

Abstract

The involute profile plays an important role in gear metrology.

It is necessary to quantify that profile on involute masters so

that an accurate form for gear teeth may be transferred to the gear

teeth from the master profile. The outlining of the procedures for

calibration of a master involute profile on a two coordinate axis

measuring machine is the function of this paper.

I. INTRODUCTION

The curve called an involute plays a vital role in gear metrology. The involute fulfills the requirements of gear teeth for transmitting smooth (uniform) angular motion.''' It is a familiar curve

since it can be described by the end of a taut string as it is wound

upon or unwound from a fixed cylindrical surface. Figure 1 shows

the basic elements of a gear with the corresponding definitions of

each term.

For metrological purposes, an alternative definition of the involute profile is useful. Figure 2 illustrates the principle of

"''"The Involute Curve and Involute Gearing," Published by The Fellows

Gear Shaper Company, 1969.
involute generation from a base circle. The base circle diameter is

equivalent to the base circle diameter of a gear. To construct the

involute profile from this base circle:

(a) the circumference of the circle is divided into a number

of segments

(b) from each division point a tangent is constructed

(c) the length of each corresponding arc is set off along each

tangent , and

(d) the resultant curve through each terminal point is drawn.

From Figure 2 we see:

Arc BC = (Radius of Base Circle X (6)

where 9 = angle COB (in radians) and is normally called the angle of

roll.

Since Arc BC = AB then

AB = (Radius of Base Circle) x (6)

This simple equation permits master involute profiles to be

generated or calibrated by comparison of the measured AB to that calculated by the above equation.

II. PROCEDURES

The calibration procedure is derived directly from the fact that

a line normal to the involute profile is tangent to the base circle.

Figure 3 illustrates the measurement process from these basic principle

If an indicator measuring lateral displacement, is mounted at a height

above the center of the base circle equal to the base circle radius
and the involute profile is rotated through an angle B about an axis

concentric with the base circle center, then

A (indicator change) = (radius of base circle) X (B)

.

If the process is repeated for an incremental B, a complete profile

for the actual curve is obtained.

In our non -perfect real world, the method described above has

to be modified somewhat and the following discussion is the process

realized at NBS for measurement of the most common form of involute

master. Figure 4 is a photograph of a typical involute master which

is used to qualify involute profile measuring instruments. The true

involute surface is the surface characterized. The instrument used

for measuring is a two -axis (x,y) measuring machine where the third

axis (Z ) is a precision spindle. The manual vertical movement of

the spindle housing will be considered the Z axis in the measurement

process

.

(1) Initially a reference axis for the base circle must be

established on the measuring machine before a meaningful measurement

can be made. A precision 1.4 inch diameter test cylinder (arbor),

which has been produced with tight tolerances on roundness (10 micro

inches) and taper (5 microinches/inch) , is used to define that reference

axis (Figures 5 and 6). The cylinder (arbor) is mounted at one end on

a rotary table and on a precision adjustable cone point at the other

end. By successive adjustments of the adjustable cone point, the vertical and horizontal planes of the reference axis are aligned to within a value limited only by the quality of the arbor itself (approximately
0.000020 inch). To insure that measurements are made normal to the

reference axis, the spindle is rotated through a small angle in the

x-y plane (Figure 6) until a maximum reading is indicated on the meter.

At this position, the operator is assued that the "sensitive" axis of

the probe is normal to the reference axis. The spindle of the measuring

machine is then locked into position.

(2) The coordinate system which is used is as follows:

The involute master is then examined for wear and/or burrs and

roundness tracings of the centers for mounting are recorded to assure

that the quality is sufficient for measurement (Figure 7).

To provide a suitable reference from which to quantify these

master profiles

(1) the internal mounting centers on the involute master and

the external mounting centers used to hold the involute should be

made to a deviation (TIR) from roundness not in excess of thirty micro

inches , and

Y-Axis of

Measuring

Machine

Z-Axis

Master Involute

Profile

X-Axis Normal

to Paper

4
(2) the surface finish of the cylindrical reference surface

shall be approximately seven microinches AA or better (arithmetic

average).

If one considers the fact that the external mounting centers for

holding the involute master are probably not round, then one can obtain

a better appreciation of how difficult it is to define a reference

axis for the involute master. If the number of lobes on both the

internal and external centers are the same; then the condition is

reached where the high areas of the external mounting centers may

fit into the low areas of the internal centers on the involute master

or vice versa. For the opposite case, the high areas of the external

centers will contact the high areas of the internal centers. If the

number of lobes in both centers is not the same, then various positions

of the reference involute master will result when rotated.

(3) After one is convinced the involute master is of sufficient

quality to warrant measurement, the test cylinder is withdrawn and the

involute master is placed within the mounting centers (Figure 8). One

can ascertain the degree of parallelism between the face and the reference axis by tramming the surface along the y-axis (Figure 9).

(4) The steps to establish the correct base radius is composed of:

(a) establishing a reference zero (i.e. locating the X-Y axis of the involute mounting axis) by moving the spindle housing in the vertical Z direction while the probe is in contact with the involute mounting cylinder

(Figure 10) , and (b) raising the spindle housing on the measuring machine

in the positive Z direction the required distance for the base radius by
using steel gage blocks (Figure 10).

(5) The rotary table is then rotated CCW (Figure 11) and the

probe brought into contact with the machined part of the involute master.

Note: most involute profiles have a six degree run-out and measurement

should not be started until the master has been rotated approximately

six degrees beyond the initial point where the probe contacts the surface. Only after this rotation does the true involute profile begin.

(6) The probe is only used as a nulling device (Figure 12) and

displacements are obtained directly from the y-axis scale of the measuring machine which has a least count of 20 microinches with 10 micro

inch interpolation capability. The first reading is, of course, the

reference reading.

(7) The slide of the measuring machine is then moved back to permit the involute master to be rotated by the rotary table through an

appropriate angle (usually four degrees to give approximately ten data

points). The y-slide is then brought forward to again bring the probe

into contact with the surface at a new point. The process is repeated

until the end of the true involute profile is reached (Figure 13, 14, 15)

(8) The involute profile may be checked for non-parallelism to the

reference axis. After steps 5-7 have been performed, the x-slide of

the measuring machine can be translated some small amount (^.1 inch)

and the above procedures repeated (Figure 16).

6
III. EXAMPLE OF CALIBRATION

An example of an involute calibration follows: the data is from

a test on an involute master having a base radius of 0.500 inch. Three

sets of readings were recorded with the x-axis displaced 0.100 inch

between each set of readings. By using the initial center reading5 as

headings

Left

r- ' Center Rif^ht

Rotary Table Y- Axis Rotary Table Y-Axis Rotary Table Y-Axis

316° 8' 13" 5. 53065 316° 8' 13" 5.53059 316° 8

r -, oM

J

I 5

.

53052

321° 8' 13" 5. 5 7434 321° 8' 13" 5.57430 321° 8 13" 5. 5 7422

326° 8' 13" 5. 61800 326° 8' 13" 5.61795 326° 8 13" 5. 61787

331° 8' 13" 5. 66162 331° 8' 13" 5.66157 331° 8 13" 5. 66149

336° 8' 13" 5. 70527 336° 8' 13" 5.70522 336° 8 5. 70515

341° 8' 13" 5. 74889 341° 8' 13" 5.74886 341° 8 13" 5. 74878

346° 8' 13" 5. 79255 346° 8' 13" 5. 79251 346° 8 • 13" 5

.

79242

351° 8' 13" 5. 83619 351° 8' 13" 5.83614 351° 8 13" 5. 83606

356° 8' 13" 5. 87984 356° 8' 13" 5.87979 356° 8 ' 13" 5. 87971

the reference reading, the plane of the involute face is not parallel to

the reference axis (since 5.53059 i=- 5.53065 ^ 5.53052) and one must determine whether this affects measurement of the true involute profile.

Reference

7
The probe is set up to measure A^^ (because of maximizing on the

test cylinder to read normal displacements to the reference axis) and

we see

(Measured component a) = (A-j^) X (cos a)

when the plane of the involute face is tilted at angle a to the reference axis. From the previous table of readings, we find an average ry

of 2.1 minutes and since the cosine of this angle is very close to

unity, this introduces no appreciable error in the measurement process.

The deviations of the involute surface from a true involute are

computed by comparing the measured values to the calculated nominal

values. In this Case Aj which represents the linear distance imparted

when the involute profile is rotated through an angle 6, is equal to

the base radius multiplied by 9, or for 6=5 degrees

A= (0.500 inch) X (5°) X (jf^o

)

A = 0.043633 inch.

From this, the profile errors may be calculated (Figure 17) and plotted

(Figure 18) as the angle of roll, the abscissa, and the deviations from

nominal the ordinate. For example at 9 = 5 degrees, the theoretical

value of the involute should be

Reference Reading + A = New Value

5.53059 + 0.043633 = 5.574223 inches.

And since our reading at 321° 8' 13" was 5.57430, we find the true profile is 77 microinches greater than theory predicts.

8
IV. SOURCES OF ERROR

In any measurement process there are errors which affect the ultimate accuracy of the calibration. It is useful to classify these errors

into (1) random observational errors, and (2) what can be defined as

model ambiguity . The definition of the latter term is that error which

arises because the object being measured has not been perfectly made,

i.e. the difference between the embodiment of the concept in the material

object to that embodiment in the model. For example, if one measures

the diameter of a cylinder which is badly out -of -round , then that

measured diameter is undefined to the extent that the object (out -of

round cylinder) differs from the model (perfectly round cylinder).

Model Ambiguity Factors :

(1) Because the character of the involute axis will be determined

by the quality of the centering devices, the estimated uncertainty from

this model ambiguity will be on the same order as the roundness of the

centers.

(2) The amount of surface roughness also contributes to the total

uncertainty. It is very difficult to locate and relocate a reference

surface if there is a large degree of waviness on the surface.

(3) The actual involute master will have the involute profile

generated with a base radius which is not exactly the value which is

stated for that particular master. For example, if the base radius

is actually 0.500050 inch then over a 40° angle of roll, the actual

A is 35 microinches greater than that measured A on the assumption

9
that the base radius is exactly 0.500000 inch.*

Measurement Errors

(1) The stylus probe tip can be a potential source of error. If

the probe is a knife-edge, care must be taken to insure the edge is on

the vertical tangent to the base circle. If the involute profile is

incorrect or has a small high spot, the probe will contact the highest

point on the profile, thus resulting in an erroneous profile. A spheri

cal probe is preferable but again reasonable care must be taken to

position on the vertical tangent to the base circle.

(2) The sources of observational errors involve the setting of

the rotary table used to rotate the involute profile and the ability

to repeat a reading on the axis of the measuring machine. The observed

standard deviation of repeated readings on the measuring machine is

approximately 10 microinches. The rotary table is accurate to one

second which corresponds to a 2.4 microinch uncertainty when the base

radius is 0.500 inch.

These random errors combined with estimations of error due to mode

ambiguities indicate that it is difficult to quantify an involute profile to better than + 0.000050 inch in most cases.

*The defining of the base radius of an involute master to be 0.500000

inch inherently means that any errors in the form are to be referenced

to that base radius. This does not mean that the actual base radius

needs to be known, but that this is one of the model ambiguities in the

measurement process.

10
V. SUMMARY

The procedures and problems Inherent in the calibration of master

involute profiles have been discussed. A reasonably well -equipped

metrology lab can apply the same basic principles for the calibration

of their involute masters.

11
INVOLUTE

TOOTH BASE CIRCLE

OF PINION

PITCH CIRCLE OF

PINION AND GEAR

FIGURE 1. Illustration of Basic Gear Elements

Definitions:

Involute Teeth of gears are those in which the active portion of

the profile is the involute of a circle.

Base Circle is the circle frora which the involute tooth profiles

are derived.

Pitch Circle is the imaginary circle which rolls without slipping

with a pitch circle of a mating gear.

Pitch Point is the point of tangency of two pitch circles and is

on the line of centers.

12
INVOLUTE CURVE

BASE CIRCLE

FIGURE 2. Illustration of the generation of the involue profile,

The tangent to point B is extended and the length of

line AB is equal to the arc BC on the base circle.

Arc BC = (Radius)^ ^. ^ x (0). ^ , , , Base Circle included angle.

13
ROTATED

POSITION

FIGURE 3. Illustration of the measurement of an Involute

profile. A reference reading is first obtained

and then the involute master is rotated through

an angle, 3, which produces

A = (Base Radius) x (6)

14
True Involute

FIGURE 4. Involute Master

15
FIGURE 5. Establishing Reference Axis on Measuring Machine

16
FIGURE 6. Establishing Reference Axis on Measuring Machine

17
18
FIGURE 8. Installation of Master Involute into Measuring Machine

19
FIGURE 9. Check for Parallelism between Involute Axis

and Reference Axis of Measuring Machine

20
FIGURE 10. Establishing Base Radius Distance on Involute Master

21
22
FIGURE 12. Probe used only as Nulling Device

23
24
FIGURE 14. Rotation of Involute Master

25
FIGURE 15. Rotation of Involute Master

26
FIGURE 16. X-Axis of Machine Moved to Tram Surface

27
REDUCTION OF INVOLUTE DATA

A = (Radius) (6)= (0.500) (5) (Tr^) = 0. 043633 in.

Angle of Roll

Degrees

Observed Reading

Inches

Theory

Inches

Obs-Theory

10-6lnches

0 5.53059 5.530590 0

5 5.57430 5.574223 +77

10 5 61795 5 617856 +94

15 5.66157 5.661489 +81

20 5.70522 5.705122 +98

25 5. 74886 5.748755 +105

30 5.79251 5.792388 +122

35 5.83614 5.836021 +119

40 5.87979 5.879654 +136 .

FIGURE 17

28
S3H0NI0a0ll/M

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FORM NBS-114A 11-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 73-272

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

ON CHARACTERIZING MASTER INVOLUTE PROFILES

5. Publication Date August 28, 1973

6. Performing Organization Code

7. AUTHOR(S)

Edgar G. Erber and B. Nelson Norden

8. Performing Organization

NBSIR 73-272

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

10. Project/Task/Worlc Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

SAME

13. Type of Report & Period CoveredFinal

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The involute profile plays an important role in gear metrology. It is necessary

to quantify that profile on involute masters so that an accurate form for gear

teeth may be transferred from the master profile. The procedures for calibration

of a master involute profile on a two coordinate axis measuring machine is the

function of this paper.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Calibration; gear; involute.

18, AVAILABILITY STATEMENT

I I UNLIMITED. EFOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

29

22. Price

USCOMM-C3C 66244-P7I


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