3. Gear Teeth - salemcompany.com · What is the pitch diameters ratio of a 19 T. pinion with a 59 T. gear, with tooth ... a 40 tooth gear with an 8.000" pitch diameter has a diametral

Page 27

27

I f you’re asked to divide two numbers like this 216928 ÷ 391167 . . . what’s the first thing you

reach for? An electronic calculator, of course.But did you know that calculators were in usebefore such a thing as electronics even existed?

In 1876 there was a centennial exhibit in Phila-delphia, Pennsylvania. On display was a workingmechanical calculator called “Grant’s DifferenceEngine”. It was 5 feet × 8 feet in size, andweighed almost a ton, but it could mechanicallydo accurate arithmetic computations. Three yearsbefore, in 1873, a young man from Maine hadinvented it while a student at Harvard. Aftermonths of painstaking effort he had succeeded inbuilding a working model for display.

The young man’s name was George Grant, and hecan be called, not the father of computers, but thefather of the American gear industry. How did heget involved with gears?

His calculators required many small, but preci-sion gears, and in 1876 few knew how precisiongears should be designed and made. So GeorgeGrant himself worked out much of the theory ofmodern gear forms, profiles and geometry.

He set up shop in Charlestown, Boston, Mass-achusetts to cut the first gears he needed, andlater started four separate gear companies. Threedescendants survive today - Grant Gear Co.,Boston Gear Co., and Philadelphia Gear Co.

Grant wrote a book, “A Treatise on GearWheels” which was widely read and whichhelped to begin a process of standardizing gearteeth - so that gears made by different manufac-turers, any place in the country, can mesh to-gether properly, and so that tooling to make gearscan be made in standard sizes at reduced cost.

Gears in Mesh

Imagine two spur gears in mesh.

For purposes of illustration, let’s say one gear has40 T., and the other has 20 T. Gear ratio is 2 : 1.

Speed Ratio

The speed ratio between the two shafts will beexactly 2 : 1. It is not 2.001 : 1 or 1.999 : 1, butexactly 2.000 : 1.

3. Gear Teeth

George Grant

Page 28 3. Gear Teeth

28

If it is not quite exact - say 2.001 : 1 - then thespeed ratio will also be 2.001 : 1.

The speed ratio will be exactly the same as theroll diameter ratio, with the larger roll having theslower speed.

Pitch Circle

Call the circumference of these imaginary rollsthe pitch circle

Imagine we put both gears and rolls on the shaftstogether, side by side.

The outside diameter of the rolls (pitch circles)will come approximately, but not exactly, halfway up the teeth.

Center Distance

The distance between the center of each shaft iscalled center distance, and is measured ininches.

Imagine we now slide the gears off the shafts,and put in their place two rolls, or plain cylin-ders.

Imagine that the rolls press against each other. Ifone roll rotates, it will press on the other andcause it to rotate also.

Imagine also that the rolls are covered withsomething like hard rubber so that they don’t slipor slide against each other at all. They only rollagainst each other.

Let’s say we want the shafts to turn at the sameexact 2 : 1 speed ratio as they did with the gears.To get that exact 2 : 1 ratio one roll has to beexactly two times as big as the other.

3. Gear Teeth Page 29

29

Every set of gears in mesh has a set of imaginarypitch circles. Pitch circle diameters (pitchdiameters) have the same ratio as the gear ratio.Pitch circles touch each other and roll withoutslipping.

Pitch Diameter

Pitch circle diameteris called the gear’spitch diameter(P.D.).

Study Questions A.

1. Imagine a 21 T. pinion in mesh with a 63 T. gear. What is the ratio of P.D. gear to P.D.pinion?

2. Which goes slower, the 21 T. pinion or the 63 T. gear?

3. What is the pitch diameters ratio of a 19 T. pinion with a 59 T. gear, with tooth size6 D.P.?

On a gear set in mesh, the ratio between the twopitch diameters is the same as the gear ratio.

=P.D. #1 Number of Teeth #1P.D. #2 Number of Teeth #2


30

Pitch Radius

For a given center distance and speed ratio, thereis only one pair of pitch circles (and P.D.’s) thatwill work.

Half of one pitch diameter plus half of the otherpitch diameter equals the center distance. (Half adiameter is called a radius.)

4. What RPM does the pinion turn?

5. If a miniature man sits exactly on the pitch circle of the gear shown inquestion 4, he will go round and round at a speed of 1649.3 feet perminute, or 18.7 miles per hour.

How fast would he go if he sat on the pinion pitch circle?


31

If you know the gear ratio and center distance,you can figure out pitch diameters of both gearand pinion.

For example, if gear ratio is 2 : 1, the gear pitchradius will be 2/3rds of the center distance.Pinion pitch radius will be 1/3rd of center dis-tance.

If gear ratio is 4 : 1, pinion radius will be 1/5th ofcenter distance, gear radius will be 4/5ths ofcenter distance.

If ratio is 59 T. to 19 T. (59 : 19), pinion radiuswill be 19/78ths of center distance. Gear radiuswill be 59/78ths of center distance.

(59 + 19 = 78)

With a 59 : 19 ratio and center distance of10.000", pinion P.D. is:

P.D. Pinion = 2 × × 10 = 4.872"

Pitch diameter of the gear is:

P.D. Gear = 2 × × 10 = 15.128"

To check ourselves, P.D. gear plus P.D. piniondivided by two must equal center distance.

1978

5978

+ = 10.000" — OK

As a further check we know that

= Gear ratio

= 3.105 = 3.105

OK

15.1282

4.8722

P.D. gearP.D. pinion

15.1284.872

5919


32

Circularpitch

Parts of Gear Teeth

Addendum

Height of a tooth above the pitch circle is calledaddendum.

Dedendum

Depth of tooth below the pitch line is calleddedendum.

Addendum

Dedendum

Clearance

Study Questions B.

1. Gears are in mesh, one with 22 T., the other with 66 T. What is theratio of pitch diameters?

2. If a gear ratio is 4 : 1 and center distance is 5.000", what are pitchdiameters of gear and pinion?

3. If meshing gears have 61 T., and 14 T. and run on a center distanceof 9.625", what are the two pitch diameters?

Clearance

Dedendum is slightly larger than addendum. Thedifference allows some clearance when the gearsmesh.

Circular Pitch

Distance along the pitch circle from the center ofone tooth to the center of the next is circularpitch (C.P.).


33

Wholedepth

Circular pitch, addendum and dedendum aremeasured in inches.

Pressure Angle

Angle or slant of the side of the tooth at the pitchcircle is pressure angle (P.A.).

Pressure angle

Tooth Thickness

Tooth thickness is the thickness measured alongthe pitch line. It’s equal to half the circular pitch,less backlash allowance.

Whole Depth

Whole depth is addendum plus dedendum.

Tooth thickness

Study Questions C.

Write answers on the blank lines.

1. Write in the names of these tooth parts:

2. Write in the correct names:


34

Diametral Pitch

In the early days, gear tooth sizes were notstandardized. Gears would be made in pairs, andeach gear would run only with the one other gearin the world that matched.

The first standardization was to make circularpitches in round numbers . . . 1/2", or 3/4" or 1",for example.

Some of these “circular pitch” gears are still inuse today on old machinery.

If you have a 1/2" C.P. gear with 20 T., the pitchcircle circumference is 1/2" × 20 = 10.000".Pitch diameter is 10.000" ÷ Pi (π) = 10 ÷ 3.1416= 3.183". Pitch circumference comes out to anice round whole number, but pitch diametercomes out to an awkward decimal number. Sinceengineers use pitch diameter more than circum-ference, however, it would be nicer if P.D. cameout a whole round number.

In the early 1900’s a new standard that makesP.D.’s come out whole became popular. It wascalled diametral pitch and is still used today.

T.P.D.

T.D.P.

Diametral pitch is defined as the number of teethper inch of pitch diameter. Abbreviation is D.P.

D.P. = P.D. =

For example, a 40 tooth gear with an 8.000" pitchdiameter has a diametral pitch of 5 D.P. (40 ÷ 8= 5 D.P.)

Or, a 30 tooth, 3 D.P. gear has a pitch diameter of10.000" (30 ÷ 3 = 10).

Or, a gear with a 10.000" P.D. and 6 D.P. musthave 60 teeth.

Divide number of teeth by D.P. to get pitchdiameter, or divide number of teeth by P.D. to getdiametral pitch.

It is much easier to calculate pitch diameters withD.P. tooth sizes. That’s why D.P. is used insteadof circular pitch.

Study Questions D.

Write answers in the spaces allowed.

1. What diametral pitch are these gear teeth? 11.000" P.D., 33 teeth.

2. How many teeth does an 8 D.P., 3.000" P.D. gear have?


35

3. What is the pitch diameter of these gears?

a. 50 T., 5 D.P.

b. 48 T., 6 D.P.

c. 21 T., 2 D.P.

d. 17 T., 3 D.P.

e. 17 T., 12 D.P.

Tooth Parts

Standards have been worked out for toothproportions.

Addendum

Addendum =

For example, addendum of a 3 D.P. tooth is 1/3= 0.333"

1D.P.

1.157D.P.

Dedendum

Dedendum =

For example, dedendum of a 3 D.P. tooth is1.157/3 = 0.3857"


36

Whole Depth

Whole depth of tooth equals addendum plusdedendum.

Whole Depth =

For example, whole depth of a 4 D.P. tooth is2.157/4 = 0.5393"

C.P. to D.P.

Diametral pitch is related to circular pitch likethis:

D.P. = C.P. = (Pi = π)

For example, the circular pitch of a 6 D.P. tooth isPi/6 = 3.1416/6 = 0.5236"

Theoretical or nominal tooth thickness is half thecircular pitch. (Actual tooth thickness is a littleless in practice.)

So, for example a 6 D.P. tooth has tooth thicknessof Pi/6 × 1/2 = 3.1416/6 × 1/2 = 0.2618"

Pressure Angle

The first standard for pressure angle was 14 1/2°.

2.157D.P.

πC.P.

πD.P.

In the early 1900’s a new standard was graduallyintroduced using 20° pressure angle.

Today both standards are used, 20° and 14 1/2°.

You can see that a 20° tooth is more tapered thana 14 1/2° tooth.

All tooth sizes - circular pitch, addendum andwhole depth are the same for 14 1/2° and 20°teeth. Only the shape of the tooth is different.

Gears with different pressure angles will not runtogether properly.

20° pressure angle teeth will not run with 14 1/2°teeth.

Spur gears with the same pressure angle, and thesame D.P. will run together properly, no matterwhat the number of teeth. A 20 T. gear will runwith a 90 T. gear, or a 15 T. gear, or any othernumber of teeth, as long as the pressure angle anddiametral pitch are the same.


37

Study Questions E.

Write your answers in the spaces allowed.

1. What is the addendum of these teeth?

a. 2 D.P.

b. 1 D.P.

c. 12 D.P.

d. 16 D.P.

e. 1 D.P.

2. Which tooth is bigger in height and thicker, a 2 D.P. tooth or a 5 D.P tooth?

3. What is the O.D. of this gear?

30 T., 3 D.P., 14 1/2° P.A.

Explanation . . . Outside diameter can be found by addingtwice the addendum to the pitch diameter.

12


38

4. What is the O.D. of this spur gear?

20 T., 2 D.P., 20° P.A.

There is a shortcut formula to find O.D. Add two to the number ofteeth in the gear, and divide by the D.P.

For example, to find the O.D. of a 28 T., 3 D.P. gear you wouldstart with 28 T., add two, making 30 T., and divide by 3 D.P. . . .answer is 10.000".

5. What’s the O.D. of this gear?

90 T., 16 D.P., 14 1/2° P.A.

6. What is the whole depth of an 8 D.P. tooth?

7. What is the circular pitch of these gear teeth?

a. 8 D.P.

b. 16 D.P.

c. 3 D.P.

8. Which tooth is thicker at the pitch line, 14 1/2° P.A. or 20°P.A.?


39

9. Will a 15 T., 6 D.P., 20° P.A. gear mesh properly with a 60 T., 6 D.P., 14 1/2° P.A. gear?

Stub Teeth

We’ve learned that the addendum for a standardgear tooth is:

Addendum =

Occasionally you will find other gear systemswith different addendum and dedendum propor-tions.

1D.P.

One system is called American Standard 20°Stub. It’s the same as 20° full depth except theteeth are shorter and stubbier.

American Standard 20° stub teeth have theseproportions:

Addendum = Dedendum =

Whole Depth =

Another stub tooth standard is a Fellows stub,developed by the Fellows Gear Shaper Co., oncelocated in Athol, Massachusetts. In this system,diametral pitch is given as a fraction.

For example: 10/12 D.P. or 6/8 D.P.

0.8D.P.

1.0D.P.

1.8D.P.

Tooth thickness and circular pitch are based on thetop number.

6/8 DP has circular pitch equal to 6 DP or .

Depth and addendum is based on the bottomnumber.

6/8 DP has an addendum equal to 8 DP., or 1/8 .

Fellows stub teeth have a 20° pressure angle.

Common Fellows stub teeth are these pitches:

4/5 D.P., 6/8 D.P., 8/10 D.P., 10/12 D.P.,

12/14 D.P.

There is an old system called Nuttall stub, basedon circular pitch. In Nuttall stub the addendum is

equal to .C.P.4

π6


40

Study Questions F.

Write answers in the spaces provided.

1. What is the addendum of these gear teeth?

a. 6 D.P., 20° full depth.

b. 10 D.P., 20° stub.

c. 10/12 D.P. stub.

d. 2 D.P., 20° stub.

2. What is the P.D. and O.D. of these gears?

a. 24 T., 6 D.P., 14 1/2° full depth.

b. 24 T., 6 D.P., 20° full depth.

c. 24 T., 6 D.P., 20ο stub.

d. 30 T., 4/5 D.P., stub.

Metric Gears

Metric length measurements are:

1 meter (m) = 39.37 inches

1 centimeter (cm) = 1/100th of a meter = 0.3937"

1 millimeter (mm) = 1/1000th meter = 0.0394"

1 inch = 25.4 mm

1 Micron = 1/1,000,000th of a meter = 0.000039"

In countries using the metric system gears aremade to a metric standard. Tooth sizes are notmeasured by diametral pitch, but by module.


41

P.D. in millimetersnumber of teeth

Module

The module of a gear is defined as the number ofmillimeters of pitch diameter per tooth.

Module =

Module is the reciprocal of diametral pitch, usingmillimeters instead of inches.

In the D.P. system, the larger the D.P., the smallerthe tooth. In the module system, the larger themodule the larger the tooth.

Module teeth can be 14 1/2° or 20° pressureangle. In Europe 15° is a standard pressureangle.

Study Questions G.

1. a. What is the P.D. in millimeters of a 60T., 3 Module gear?

b. What is its P.D. in inches? (25.4 mm = 1 inch).

c. What is the D.P. of the above gear?

2. Which is larger, a 3 module tooth or a 6 module tooth?

Which is larger, a 3 D.P. tooth or a 6 D.P. tooth?

3. What is the P.D. of a 4 mod., 32 T. gear?

4. How many mm in an inch?

5. A gear has a P.D. of 60 mm and has 20 teeth. What is the module of the gear?


42

Module and Diametral Pitch

Diametral pitch and module are related like this:

Module =

Module Addendum

Addendum for module gears is equal to themodule. For example, a 3 module gear has anaddendum equal to 3 mm, 6 module has a 6 mmaddendum.

25.4D.P.

16

Module Clearance

Clearance usually equals times the module.

Module Whole Depth

Whole depth equals 2.166 times module.

Whole depth (mm) = Module × 2.166

Module gears can be either 14 1/2° or 20°pressure angle, or sometimes 15°.

Study Questions H.

1. What is the addendum of an 8 module tooth?

2. What is the O.D. of a 40 T., 4 module gear?

3. What is the module of a 6 D.P. tooth?

4. What is the D.P. of a 5 module tooth?


43

Fine Pitch Gears

Sometimes gears with teeth smaller than 20 D.P.are called fine pitch gears. Those bigger than16 D.P. can be called coarse pitch.

Many fine pitch gears are made with the sametooth proportions as coarse pitch.

Addendum =

Whole Depth =

You may find, however, that some fine pitchgears are made with longer tooth proportions.

Addendum =

Whole Depth =

Pressure angle for fine pitch gears is usually14 1/2° or 20°.

1D.P.

2.157D.P.

1.350D.P.

3.027D.P.

Splines

Splines are teeth cut in a shaft and a bore, andused to prevent the bore and part from spinningon the shaft.

Splines are stronger than keyways and allow youto more easily slide a wheel, pulley or other partback and forth on the shaft.

SAE Splines

One spline standard is the SAE flat sided spline.

Six and ten tooth SAE splines are common.

Diameters range from 3/4" to 6".

SAE splines are called out on a drawing like this,for example:

SAE 1 – 6B spline

6 = number of teeth.B = means slide under no load.(A = means permanent fit, no sliding.)(C = means slide under load.)

38


44

SAE splines are standardized, so that any twoparts with the same spline specification should fittogether. Parts do not have to be made in pairs.

Involute Splines

Another spline standard is involute spline. Thesesplines have curved profiles, same as gear teeth,except the splines are much stubbier than gearteeth.

Involute splines are described using diametralpitch (DP), same as gears. They have a standardpressure angle of 30°.

13 T., D.P., 30° P.A.

This spline has 13 teeth; diametral pitch is 16 D.P.

Therefore pitch diameter is = 0.8125" P.D.

Addendum is ".

Therefore O.D. is + + = 0.875" O.D.

They have an addendum and tooth depth onlyhalf as deep as standard gear teeth. Involutespline descriptions are similar to Fellows stubgear teeth, and look like this, for example:

1632

1316

132

13 1 116 32 32

30°

Study Questions I.

1. What is the O.D. of these splines?

a. SAE 1 – 10B

b. 14 T., D.P., 30o P.A.

c. 21 T., D.P., 30o P.A.

d. 13 T., D.P., Inv. Spline

2. How many teeth are in these splines?

a. SAE 1 – 6B Spline

18

1224

1632

816

18


45

1632

1020

b. Involute spline 1" O.D., DP.

c. 16 T., D.P., 30ο PA.

Backlash

If everything were perfect the thickness of a toothon the pitch line would be exactly equal to thewidth of the space on the pitch line of the matinggear.

In practice teeth are made a little smaller, so thatthere is room for oil and dirt particles, and toallow for small inaccuracies in the gears. Theslight space left between gear teeth is calledbacklash.

In normal operation, the exact amount of back-lash is not critical because the gears are onlypushing against each other in one direction. Insome cases where the gears reverse direction, orthe load reverses, amount of backlash becomesimportant.

Teeth press here

Amount of backlash has been standardized bygear manufacturers. It varies from about 0.003"on 16 D.P. gears to about 0.010" on 3 D.P. gears.

To feel the amount of backlash in a gear set, holdone gear from turning, and by hand twist theother back and forth. You can feel a slight tap orrattle as the teeth move back and forth.

In almost every case it’s better to have too muchbacklash rather than too little.

Usually a lot of backlash will not affect gearoperation at all, unless the load reverses. Theload will keep the gears pressing against eachother in one direction.

If there is too little backlash the gear teeth will bejammed together and tend to bind, get hot, makenoise and wear out quickly.


46

Study Questions J.

1. If gears bind together when turned by hand,would you want to reduce or increase back-lash?

2. If gears bind together when turned, should youbring centers closer, or pull centers fartherapart?

Section Summary

In this section you found that every set of spurgears has a set of pitch circles that roll to-

gether and have the same exact linear speed. Yousaw that the ratio of pitch diameters is the sameas the ratio of gear teeth. You learned that centerdistance is half of one P.D. plus half of themating P.D. You saw what diametral pitch is,and module teeth in the metric system. Toothproportions are spelled out for addendum,dedendum, and whole depth. A formula foroutside diameter is given. Teeth can have 14 1/2°or 20° pressure angle.

Stub teeth are shorter than full depth teeth. Somefine pitch teeth are longer than coarse pitchteeth. Splines can be SAE straight sided, madefor sliding under load or no load, or made for nosliding. Involute splines have teeth similar toshort gear teeth, with 30° pressure angle. Back-lash is play or looseness between meshing gearteeth. It is often not harmful, but prevents bind-ing.

The next section will cover worm gearing. Youwill find that worm gear speed ratios are calcu-lated the same as spur gears - only names aredifferent. You will learn what multiple threadworms are, what helix angle and lead mean, andwhat R.H. and L.H. mean.

Proceed to the next section.

Documents

3. Gear Teeth - salemcompany.com · What is the pitch diameters ratio of a 19 T. pinion with a 59 T. gear, with tooth ... a 40 tooth gear with an 8.000" pitch diameter has a diametral