gear measurement

95

Measureoverpins

Involute curve

7. Measurement

Track and Field

In 1905, encouraged by President Theodore Roosevelt, the Intercollegiate Athletic Associa-

tion of the United States was founded in NewYork City. Their original purpose was to reformcollege football.

In 1910, they changed their name to the NationalCollege Athletic Association, and in 1921 thefirst National Collegiate Track and Field Champi-onships were held.

The NCAA Track and Field Rule Book definesthe rules, events, equipment, personnel and eventhe construction of the track facilities. It statesthat there must be a visible starting line, whichshould measure 2 inches (5.08 cm) wide. Forraces not run in lanes, the starting line should becurved.

This means that if there are more runners thanthere are lanes, the start line must be curved. Thestart is then called a “waterfall start,” and thecurved start line is the involute of a circle.

Measuring Gear Teeth

Gear teeth can be measured using special toothcalipers, or by meshing with special precisionmaster gears, or with profile testers.

There are two ways of measuring gear tooththickness, however, that require no specialequipment.

You can hold round pins or balls in the spacebetween gear teeth, and measure over the pins todetermine tooth thickness.

You can measure across two or more teeth, fromthe involute curve of one tooth, to the involutecurve of another.

7. Measurement

96

Measure pitch lineto back

Rack teeth too thin.

Rack teeth too thick

Measure over pin

Measuring Thickness of Rack Teeth

Imagine a rack mounted on a base plate. It isimportant that the distance from the pitch line tothe back of the rack is correct. It is importantthat the teeth are the correct thickness. If toothick, the pinion will mesh too tightly, or be tootight to mesh at all. If too loose, there will be toomuch backlash between rack and pinion.

To measure the thickness of rack teeth, we canput a round pin or ball in the tooth space, andmeasure from the top of the pin to the back of therack.

7. Measurement Page 97

97

Pin radius,PinRad

Pressureangle, PA

Pitchline

Backing to pitch line,BkgPL

Space thicknesson pitch line

cos PAtan PA

CircP – TThkPL2(tan PA)

sin PAcos PA

1sin PA


CircP – TThkPL(cos PA)

CircP – TThkPL.968148

CircP – TThkPL.939693

π + 4(tan PA)2DP (cos PA)

Measurement over a pin works out like this.

Space thickness on pitch line = Circular pitch – Tooth thickness on pitch line [abbreviated CircP – TThkPL]

Measure over pin =

PinRad (1 + sin PA + ) –

+ BkgPL

From trigonometry we know that:

(sin PA)2 + (cos PA)2 = 1

tan PA =

Measure over pin =

PinRad (1 + ) –

+ BkgPL

Approximate estimate for pin diameter is:

Maximum pin dia. ~

Ideal pin diameter is about ~

For 14½° involute teeth, equations work out likethis:

Ideal pin diameter =

Measure over pin = 2.4970 ( Pin dia.) –1.9334 ( CircP – TThkPL) + BackingPL

For 20° involute teeth, equations work out likethis:

Ideal pin diameter =

Measure over pin = 1.9619 ( Pin dia.) –1.3737 ( CircP – TThkPL) + BackingPL

For example, picture 10 DP, 20° PA rack cut withno backlash allowance. Dimension from pitchline of rack to the back of the rack is .500".

An ideal measuring pin diameter would be about.167" diameter.

7. Measurement

98

π10

.1571.939693

π2 x 10

1sin PA

1.342020

CircP – TThkPL(.939693)


.3142 – .15712(.363970)

Assuming we have on hand a .1700" diameterpin, here are equations for pin measurement.

Ideal pin diameter is about ~

CircP = = .3142

TThkPL = .1571

Ideal pin diameter = = .1672

Measure over pin = 1.9619 (.1700 ) –1.3737 ( .1571) + .5000 =

.6177" measure over pinto back of rack.

Using general equations:

Pin radius = .0850"

Tooth thickness on pitch line = = .1571"

sin 20° = .342020cos 20° = .939693tan 20° = .363970

Backing to pitch line of rack = .5000"

Measure over pin =

Pin Rad (1 + ) –

+ BkgPL

Measure over pin =

.0850 (1 + ) –

+ .5000"

= .6177" measure over pin to back of rack

Study Questions A.

1. Imagine a rack, 6 DP, 14½º pressure angle, cut into a 1.000" squaresteel bar with no backlash.

What would a measurement over a .2800" diameter pin be?


99

2. a. Imagine a rack, 6 DP, 14½º pressure angle, cut into a 1.000" squaresteel bar with no backlash.

What would a measurement over a .2500" diameter pin be?

b. How far above the top of the rack tooth does the pin project?

Measuring Gear Teeth Over Pins

Gear tooth thickness can be measured with pinsor balls placed in tooth spaces.

A computer program, Involute Calculations, forcalculating measurement over pins or balls forspur gears and helical gears is given onwww.salemcompany.com, and is on a CDavailable from Salem Co.

The program allows you to calculate the pinmeasurement for any inch size or metric size gearand any pin diameter. Use it. It will save muchtedious hand calculator work.

The derivations and calculations below areincluded as examples, to show the properties ofinvolute gear teeth, and how involute equationscan be manipulated.

Picture a round pin or steel ball resting in a spacebetween two teeth.Measure

overpins

Base circle

Referencepitchcircle

Base circleradius

a

bc

d

e

inv e

7. Measurement

100

Remember - whenmeasuring angles inradians, the size ofthe angle, in radians,equalsthis length divided

bythis length.

angle

inv e

a

b

c

d

e

a + b + c

half the tooth thickness (measuredon reference pitch circle in radians) ref pitch radius

ref tooth thicknessref PD

PA at reference pitchradius, in radians

ball diameter(ref PD) (cos ref PA)

ball radiusbase radius

π(number of teeth in gear)

half of ref circular pitchref pitch radius

base radiuscos e

base radiuscos e

We can see that angle a + angle b + angle c –angle d = inv e. Knowing inv e, we can calculateangle e, using iteration.

We can calculate angles a, b, c, and d. Addangles a plus b plus c, and subtract angle d. Theresult is the involute of angle e.

a + b + c – d = inv e

Knowing inv e, we can calculate angle e.

Knowing angle e and the base circle radius, wecan calculate the length from the center of thegear to the center of the ball.

From the diagram we can see that

Angle a =

=

We can also see that:

angle b = involute ( )

And:

angle c = in radians

=

And also:

angle d =

=

Then we can calculate involute e:

inv e = angle a + angle b + angle c – angle d

Use iteration to calculate angle e from inv e.

Then calculate radius to ball center:

radius to ball center =

If there are an even number of teeth in the gear,

Measurement over pins =

(radius to pin center) ( 2 )+ pin diameter =

( ) ( 2 ) + pin diameter


101

We can write the above equations above like this:

inv e = + inv PA + –

Pitch diameter (PD) is the reference pitch diam-eter. Tooth thickness, circular pitch, and pressureangle are measured at the reference pitch diam-eter (PD). T is the number of teeth in the gear.

PD =

Base dia = PD x cos PA

Use iteration to calculate angle e from Inv e.

Radius to center of pin =

If the gear has an even number of teeth,

Measurement over pins =

( ) + pin diameter

For example, imagine measuring a spur gear with30 teeth, 6 DP, 20º pressure angle, no backlashallowance, using pins or balls with a diameter of.280".

Angle a =

Tooth thickness = x = = .2618"

PD = = 5.000"

Angle a = = .052360

Angle b = inv(pressure angle) = inv(20º)

20º = x 2π = .349066 radians

inv(20º) = tan 20º – 20º, in radians

angle b = inv(20º) = .014904

angle c =

cos 20º = .939693

angle c = =

= .059594

Angle d = = .104720

Then:

Inv e = a + b + c – d

= .052360 + .014904 + .059594 – .104720

Inv e = .022138

Calculate angle e from the value of inv e:

e(est) = (2.8 × inv e) .33

e(est) = (2.8 × .022138) .33

e(est) = .399446 radians

inv e(est) = .022694

new e(est) = .396190

inv new e(est) = .022119

new e(est) = .396300

inv new e(est) = .022138

Therefore e = .396300 radians

cos e = .922496

tooththickness

PDpin diabase dia

πT

TDP

base radiuscos e

base diametercos e

Tooth thicknessPitch diameter

πDP

π6 x 2

306

.26185.000

12

20360

pin diameterPD (cos PA)

.28005.0000 (.939693)

pin diameterPD (cos PA)

πT

7. Measurement

102

base diametercos e

5.0000 x .939693.922496

π8

244

tooth thicknessPD

.3846996

pin diabase dia

.4205.63816

πT

π24

base diametercos e

Angle a

Base circle

Angle b =Involute(Angle a)

5.63816.921452

.33

For an even number of teeth, measurement over pins =

( ) + ball diameter

= + .2800 = 5.3732

Measurement over .2800" pins = 5.3732"

Example:

Picture a spur gear with 24 teeth, 4 DP, 20º PA,cut with a backlash allowance of .008".

Using a pin diameter of .420", what will measure-ment over pins be?

20º = .349066 radianscos 20º = .939693inv 20º = .014904

Tooth thickness = – .008" = .384699

PD = = 6.000"

Base diameter = 6.0000 x cos 20º = 5.63816

Angle a = =

= = .064117

angle b = inv PA = inv 20º = .014904

angle c = = = .074492

angle d = = = .130900

inv e = a + b + c – d

= .064117 + .014904 + .074492 – .130900

inv e = .022613

e(est) = (2.8 × inv e)

e(est) = .402254

inv e(est) = .023199

e(est) = .398872

inv e(est) = .022592

e(est) = .398994

inv e(est) = .022613

e = .398994 radians

cos e = .921452

For an even number of teeth, measurement over pins =

( ) + ball diameter =

+ .4200 = 6.5388

Measure over .4200" dia. pins = 6.5388"

You can use the work sheets attached for thesecalculations. A sheet with the above calculationsis included.


103

If we use a pin or ball that's too small, it will fallinto the tooth space and rest on the root diameterof the teeth, or it will not project above the gearoutside diameter, and we won't be able to fit amicrometer or caliper tip between the teeth to geta measurement.

If it's too big, it will rest on the tips of the teeth,not on the involute flanks.

There are two standards for pins to use for anysize spur or helical tooth.

Pin diameter =

Pin diameter =

If you use these pin or ball sizes, you can findtables that will give exact measurements for spurgears of any number of teeth, for 20º or 14½ºteeth.

If you don't have a pin or ball of this exact size,you can use something close, and calculate themeasurement from the formulas in this book, oruse the computer program, Involute Calculations,on our web site, www.salemcompany.com, oravailable on CD from Salem Company.

Ball or pin is too big.

Ball or pin is too small.

Ball or pin is okay.

1.680DP

1.728DP

Study Questions B.

1. What should measurement over pins be for a 56 tooth spur gear,10 DP, 20º pressure angle, zero backlash allowance, .1728" pindiameter. Use tables or Involute Calculations computer program.

4 7. Measurement

104

Measureoverpins

Even number of teeth

Center toOD of pin

Center toOD of pin

2. What should measurement over pins be for this gear:

60 teeth, 6 DP, 20º pressure angle spur gearpin diameter = .2800"Zero backlash allowance.

When measuring over pins or balls, there is adifference between a gear with an even numberof teeth (evenly divisible by 2), and one with anodd number of teeth.

We have learned how to calculate a measurementfrom the center of a gear to the outside diameterof a pin or ball.

With an even number of teeth, measurement overpins equals twice the dimension from center ofgear to OD of pin.

For even numbers of teeth:

Measure over pins =(Measure to center of pin x 2) + Pin diameter

With odd numbers of teeth, we have to correct forthe fact that the pins are not exactly oppositeeach other.

For odd numbers of teeth:

Measure over pins = (Measure to pin center )(cos ( ))(2)

+ Pin diameter

For example:

35 teeth, 8 DP, 14½° PA, no backlash

.2160" pin diameter.

Use tables or SalemCompany website.

See worksheet.

Measurement over pins is 4.6773"

360º4 × (No. of Teeth)

90°No. teeth


105

Study Questions C.

1. What should measurement over pins be for this spur gear?

75 Teeth, 10 DP, 20º PA, .004" backlash allowance, pin diameter .1680"

2. What is measurement over pins for a spur gear with 21 teeth, 4 DP, 20ºpressure angle, .4380" diameter pins. Allowance for backlash is .008".

3. Imagine a spur gear, 40 teeth, 5 DP, 14½º PA. Using .3400" diameter pins,you measure 8.429" over the pins. About how much backlash allowance iscut into the gear teeth?

4. A spur gear has 24 teeth, 8 DP, 20º PA.What should it measure over .2160" diameter pins, with no backlash allowance?

7. Measurement

106

If we know the pin measurement of a gear, here'show to calculate tooth thickness.

Rearrange tooth thickness equation like this:

Step 1:

For even number of teeth,

cos e =

For an odd number of teeth,

cos e = ( )

Step 2:

angle e = arccos (cos e)

Step 3:

inv e = tan e – e

Step 4:

tooth thickness at ref PD =

PD ( inv e – inv PA – + )

= ( inv e – inv PA – + )

PD is reference pitch diameterPA is reference pressure angleT is number of teeth in the gearDP is reference diametral pitch

For example, consider this spur gear, 30 teeth, 6DP, 20º PA. Measure over .2800" diameter pinsis 5.3732".

What is tooth thickness?

Step 1:

cos e =

cos e = = .922498

Step 2:

e = .396294 radians

Step 3:

inv e = .022138 inv 20º = .014904

Step 4:

= .059594

= .104720

tooth thickness on ref pitch circle =

5.0000 (.022138 –.014904 –.059594 +.104720) =

.2618" tooth thickness

ball dia.PD (cos PA)

πT

TDP

πT

ball dia.(DP)T (cos PA)

(Number of teeth)(cos PA)DP (Measure over pin – pin

dia.)

(Number of teeth)(cos PA)DP (Measure over pin – pin

dia.)

(30) (.939693)6(5.3732 – .2800)

ball diaPD (cos PA)

π30

(Number of teeth)(cos PA)DP (Measure over pin – pin dia.)

cos 90°T


107

Consider this spur gear, 24 teeth, 4 DP, 20º PA.Measure over .4200" diameter pins is 6.5388".


Step 1:

cos e =

cos e = = .921448

Step 2:

e = .399005 radians

Step 3:

inv e = .022616 inv 20º = .014904

Step 4:

tooth thickness at ref PD =

PD ( inv e – inv PA – + )

= .074492 = .130900

tooth thickness on ref pitch circle =

6.0000 (.022616 –.014904 –.074492 +.130900) = .3847" tooth thickness

Tooth thickness with no backlash allowance =.3927".

Allowance for backlash = .3927 – .3847 = .0080"

Study Questions D.

1. Consider a spur gear with 46 teeth, 10 DP, 14½° PA.Measure over .1728" diameter pins is 4.8344".


(24) (.939693)4(6.5388 – .4200)

(Number of teeth)(cos PA)DP (Measure over pin – pin dia.)

ball dia.PD (cos PA)

ball dia.(DP)T (cos PA)

πT

πT

7. Measurement

108

2. Consider this spur gear, 45 teeth, 10 DP, 14½° PA.Measure over .1728" diameter pins is 4.8043".



109

Summary

In this section we learned two methods to measure the thickness of gear teeth. One is by

measuring over pins or balls placed in the toothspaces. The other is by measuring from toothflank to tooth flank across two or more teeth.

Rack tooth thickness is determined by measur-ing over pins or balls. The exact calculation isstraightforward for both 20° and 14½º teeth. Becareful that the micrometer spindle used tomeasure over the pins does not touch the rackitself, only the pin.

Pin measurements for spur gears have beenprinted in tables for standard pin sizes. A com-puter program, Involute Calculations, for bothspur and helical gears for any pin size can be runfrom Salem Company's web site,www.salemcompany.com, or run from a CD diskavailable from Salem Company. Use rack toothapproximations to estimate tooth thicknesschange for a given change in pin measurement.

For 14½º teeth:Tooth thickness change ~ (½) (Change in measure from pin to gear center)

For 20º teeth:Tooth thickness change ~ (¾) (Change in measure from pin to gear center)

Be sure the pin touches the involute flanks of thegear teeth, not the bottom or tip of the tooth.

Measurements over pins for spur and helicalgears can be calculated using involute functions.Following the derivations will help us understandthe properties of involute curves on gear teeth.

Work sheets are furnished to allow calculationsusing a scientific type calculator, or calculationscan be made on the computer program, InvoluteCalculations.

In the next section we will learn how to use span measurements to find tooth thickness.

A span measurement is a measure over two ormore teeth, from the involute curved flank of onetooth to the involute curved flank of another.Span measurements can be made with microme-ters or vernier calipers.

A span measurement is actually a measurementalong a line tangent to the base circle. It issometimes called a base tangent measurement.

Tables of span measurements for spur gears areavailable, for 14½° and 20° teeth.

A computer program for span measurements ofboth spur and helical gear teeth can be run fromSalem Company's web site, salemcompany.com.

A change in span measurement of .0010" equals achange in backlash allowance of .0010", wherethe backlash is measured at the base circlediameter.

Circular pitch, diametral pitch and pressure angleof helical gears can be measured in two direc-tions:

1.Normal (at right angles) to the tooth

2.Circumferential (around the circumference ofthe gear). Sometimes called transverse.

Span measure calculations can be made usinginvolute functions, and are fairly simple if youuse a scientific type calculator. Work sheets forcalculating span measurements for both spur andhelical gears are included.

Derivation of the equations for span measure-ments for spur and helical gears is shown andwill be helpful in learning more about involutecurves in gear teeth.

7. Measurement

110

gear measurement

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