MACHINERY''''S HANDBOOK 27th CD Episode 3 potx

If 56 is used as an approximate number possibly after one or more trial solutions to find an mate number and resulting gear ratio coinciding with available gears: approxi-The tooth numbe

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

0.6923 27/39 27/39 0.7234 … 34/47 0.6930 19/37 + 7/39 19/37 + 7/39 0.7240 3/27 + 19/31 34/41 − 4/38

Part

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

Approximate Indexing for Small Angles.—To find approximate indexing movements

for small angles, such as the remainder from the method discussed in Angular Indexing

starting on page 1990, on a dividing head with a 40:1 worm-gear ratio, divide 540 by thenumber of minutes in the angle, and then divide the number of holes in each of the availableindexing circles by this quotient The result that is closest to a whole number is the bestapproximation of the angle for a simple indexing movement and is the number of holes to

0.9630 30/43 + 13/49 1/51 + 50/53 0.9910 22/23 + 1/29 21/46 + 31/58 0.9640 21/39 + 20/47 52/57 + 3/58 0.9920 39/41 + 2/49 40/53 + 14/59

Part

of a Turn

B&S, Becker, Hendey, K&T,

& Rockford

Cincinnati and LeBlond

be moved in the corresponding circle of holes If the angle is greater than 9 degrees, thewhole number will be greater than the number of holes in the circle, indicating that one ormore full turns of the crank are required Dividing by the number of holes in the indicatedcircle of holes will reduce the required indexing movement to the number of full turns, andthe remainder will be the number of holes to be moved for the fractional turn If the angle isless than about 11 minutes, it cannot be indexed by simple indexing with standard B & Splates (the corresponding angle for standard plates on a Cincinnati head is about 8 minutes,and for Cincinnati high number plates, 2.7 minutes See Tables 5, 6a, and 6b for indexingmovements with Cincinnati standard and high number plates).

Example:An angle of 7° 25′ is to be indexed Expressed in minutes, it is 445′ and 540divided by 445 equals 1.213483 The indexing circles available on standard B & S platesare 15, 16, 17, 18, 19, 20, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, and 49 Each of thesenumbers is divided by 1.213483 and the closest to a whole number is found to be 17 ÷1.213483 = 14.00926 The best approximation for a simple indexing movement to obtain

7° 25′ is 14 holes on the 17-hole circle

Differential Indexing.—This method is the same, in principle, as compound indexing

(see Compound Indexing on page 1984), but differs from the latter in that the index plate isrotated by suitable gearing that connects it to the spiral-head spindle This rotation or dif-ferential motion of the index plate takes place when the crank is turned, the plate movingeither in the same direction as the crank or opposite to it, as may be required The result is

that the actual movement of the crank, at every indexing, is either greater or less than its

movement with relation to the index plate The differential method makes it possible toobtain almost any division by using only one circle of holes for that division and turning theindex crank in one direction, as with plain indexing

The gears to use for turning the index plate the required amount (when gears are required)are shown by Tables 4a and 4b, Simple and Differential Indexing with Browne & Sharpe Indexing Plates, which shows what divisions can be obtained by plain indexing, and when

it is necessary to use gears and the differential system For example, if 50 divisions arerequired, the 20-hole index circle is used and the crank is moved 16 holes, but no gears arerequired For 51 divisions, a 24-tooth gear is placed on the wormshaft and a 48-tooth gear

on the spindle These two gears are connected by two idler gears having 24 and 44 teeth,respectively

To illustrate the principle of differential indexing, suppose a dividing head is to be gearedfor 271 divisions Table 4b calls for a gear on the wormshaft having 56 teeth, a spindle gearwith 72 teeth, and a 24-tooth idler to rotate the index plate in the same direction as thecrank The sector arms should be set to give the crank a movement of 3 holes in the 21-holecircle If the spindle and the index plate were not connected through gearing, 280 divisionswould be obtained by successively moving the crank 3 holes in the 21-hole circle, but thegears cause the index plate to turn in the same direction as the crank at such a rate that,when 271 indexings have been made, the work is turned one complete revolution There-fore, we have 271 divisions instead of 280, the number being reduced because the totalmovement of the crank, for each indexing, is equal to the movement relative to the index

plate, plus the movement of the plate itself when, as here, the crank and plate rotate in the

same direction

If they were rotated in opposite directions, the crank would have a total movement equal

to the amount it turned relative to the plate, minus the plate's movement Sometimes it is

necessary to use compound gearing to move the index plate the required amount for eachturn of the crank The differential method cannot be used in connection with helical or spi-ral milling because the spiral head is then geared to the leadscrew of the machine

Finding Ratio of Gearing for Differential Indexing.—To find the ratio of gearing for

differential indexing, first select some approximate number A of divisions either greater or less than the required number N For example, if the required number N is 67, the approxi-

mate number A might be 70 Then, if 40 turns of the index crank are required for 1 tion of the spindle, the gearing ratio R = (A − N) × 40/A If the approximate number A is less than N, the formula is the same as above except that A − N is replaced by N − A Example:Find the gearing ratio and indexing movement for 67 divisions.

revolu-The fraction 12⁄7 is raised to obtain a numerator and a denominator to match gears that areavailable For example, 12⁄7 = 48⁄28

Various combinations of gearing and index circles are possible for a given number ofdivisions The index numbers and gear combinations in the accompanying Tables 4a and4b apply to a given series of index circles and gear-tooth numbers The approximate num-

ber A on which any combination is based may be determined by dividing 40 by the fraction

representing the indexing movement For example, the approximate number used for 109divisions equals 40 ÷ 6⁄16, or 40 × 16⁄6 = 106 2⁄3 If this approximate number is inserted inthe preceding formula, it will be found that the gear ratio is 7⁄8, as shown in the table

Second Method of Determining Gear Ratio: In illustrating a somewhat different method

of obtaining the gear ratio, 67 divisions will again be used If 70 is selected as the mate number, then 40⁄70 = 4⁄7 or 12⁄21 turn of the index crank will be required If thecrank is indexed four-sevenths of a turn, sixty-seven times, it will make 4⁄7 × 67 = 382⁄7 rev-olutions This number is 15⁄7 turns less than the 40 required for one revolution of the work(indicating that the gearing should be arranged to rotate the index plate in the same direc-tion as the index crank to increase the indexing movement) Hence the gear ratio 15⁄7 = 12⁄7

approxi-To Find the Indexing Movement.—The indexing movement is represented by the

frac-tion 40/A For example, if 70 is the approximate number A used in calculating the gear ratio

for 67 divisions, then, to find the required movement of the index crank, reduce 40⁄70 toany fraction of equal value and having as denominator any number equal to the number ofholes available in an index circle

Use of Idler Gears.—In differential indexing, idler gears are used to rotate the index plate

in the same direction as the index crank, thus increasing the resulting indexing movement,

or to rotate the index plate in the opposite direction, thus reducing the resulting indexing

movement

Example 1:If the approximate number A is greater than the required number of divisions

N, simple gearing will require one idler, and compound gearing, no idler Index plate and

crank rotate in the same direction

Example 2:If the approximate number A is less than the required number of divisions N,

simple gearing requires two idlers, and compound gearing, one idler Index plate and crankrotate in opposite directions

When Compound Gearing Is Required.—It is sometimes necessary, as shown in the

table, to use a train of four gears to obtain the required ratio with the gear-tooth numbersthat are available

Example:Find the gear combination and indexing movement for 99 divisions, assuming

that an approximate number A of 100 is used

The final numbers here represent available gear sizes The gears having 32 and 28 teeth arethe drivers (gear on spindle and first gear on stud), and gears having 40 and 56 teeth are

If A 70, gearing ratio (70–67) 40

70

7 - gear on spindle (driver)gear on worm (driven) -

To illustrate, 40

70471221number of holes indexednumber of holes in index circle

×

driven (second gear on stud and gear on wormshaft) The indexing movement is sented by the fraction 40⁄100, which is reduced to 8⁄20, the 20-hole index circle being used here.

repre-Example:Determine the gear combination to use for indexing 53 divisions If 56 is used

as an approximate number (possibly after one or more trial solutions to find an mate number and resulting gear ratio coinciding with available gears):

approxi-The tooth numbers above the line here represent gear on spindle and first gear on stud approxi-The tooth numbers below the line represent second gear on stud and gear on wormshaft.

To Check the Number of Divisions Obtained with a Given Gear Ratio and Index

Movement.—Invert the fraction representing the indexing movement Let C = this

inverted fraction and R = gearing ratio.

Example 1:If simple gearing with one idler, or compound gearing with no idler, is used: number of divisions N = 40C − RC.

For instance, if the gear ratio is 12⁄7, there is simple gearing and one idler, and the indexingmovement is 12⁄21, making the inverted fraction C, 21⁄12; find the number of divisions N.

Example 2:If simple gearing with two idlers, or compound gearing with one idler, is used: number of divisions N = 40C + RC.

For instance, if the gear ratio is 7⁄8, two idlers are used with simple gearing, and the ing movement is 6 holes in the 16-hole circle, then number of divisions:

1×7 - 72×40

24×56 -

Indexing movement 40

5657

5×7

7×7 - 35 holes49-hole circle

6 -

×

8166 -

80 70 60

90 100 110 120

150 160 170

Table 4b (Continued) Simple and Differential Indexing

Browne & Sharpe Indexing Plates

Gear on Worm

No 1 Hole

Gear on Spindle

Idlers

First Gear

on Stud

Second Gear on Stud

No 1 Hole

No 2 Hole b

Table 4b (Continued) Simple and Differential Indexing

Browne & Sharpe Indexing Plates

Gear on Worm

No 1 Hole

Gear on Spindle

Idlers

First Gear

on Stud

Second Gear on Stud

No 1 Hole

No 2 Hole b

Table 4b (Continued) Simple and Differential Indexing

Browne & Sharpe Indexing Plates

Gear on Worm

No 1 Hole

Gear on Spindle

Idlers

First Gear

on Stud

Second Gear on Stud

No 1 Hole

No 2 Hole b

a See Note on page2011

b On B & S numbers 1, 1 1⁄2 , and 2 machines, number 2 hole is in the machine table On numbers 3 and

4 machines, number 2 hole is in the head

Table 4b (Continued) Simple and Differential Indexing

Browne & Sharpe Indexing Plates

Gear on Worm

No 1 Hole

Gear on Spindle

Idlers

First Gear

on Stud

Second Gear on Stud

No 1 Hole

No 2 Hole b

Table 6b (Continued) Indexing Movements for High Numbers

Cincinnati Milling Machine

Indexing Tables.—Indexing tables are usually circular, with a flat, T-slotted table, 12 to

24 in in diameter, to which workpieces can be clamped The flat table surface may be izontal, universal, or angularly adjustable The table can be turned continuously through

hor-360° about an axis normal to the surface Rotation is through a worm drive with a ated scale, and a means of angular readout is provided Indexed locations to 0.25° withaccuracy of ±0.1 second can be obtained from mechanical means, or greater accuracy from

gradu-an autocollimator or sine-gradu-angle attachment built into the base, or under numerical control.Provision is made for locking the table at any angular position while a machining operation

is being performed

Power for rotation of the table during machining can be transmitted, as with a dividinghead, for cutting a continuous, spiral scroll, for instance The indexing table is usuallymore rigid and can be used with larger workpieces than the dividing head

Block or Multiple Indexing for Gear Cutting.—With the block system of indexing,

numbers of teeth are indexed at one time, instead of cutting the teeth consecutively, and thegear is revolved several times before all the teeth are finished For example, when cutting agear having 25 teeth, the indexing mechanism is geared to index four teeth at once (seeTable 7) and the first time around, six widely separated tooth spaces are cut The secondtime around, the cutter is one tooth behind the spaces originally milled On the third index-ing, the cutter has dropped back another tooth, and the gear in question is thus finished byindexing it through four cycles

The various combinations of change gears to use for block or multiple indexing are given

in the accompanying Table 7 The advantage claimed for block indexing is that the heatgenerated by the cutter (especially when cutting cast iron gears of coarse pitch) is distrib-uted more evenly about the rim and is dissipated to a greater extent, thus avoiding distor-tion due to local heating and permitting higher speeds and feeds to be used

Table 7 gives values for use with Brown & Sharpe automatic gear cutting machines, butthe gears for any other machine equipped with a similar indexing mechanism can be calcu-lated easily Assume, for example, that a gear cutter requires the following change gearsfor indexing a certain number of teeth: driving gears having 20 and 30 teeth, respectively,and driven gears having 50 and 60 teeth

Then if it is desired to cut, for instance, every fifth tooth, multiply the fractions 20⁄60 and

30⁄50 by 5 Then 20⁄60 × 30⁄50 × 5⁄1 = 1⁄1 In this instance, the blank could be divided sothat every fifth space was cut, by using gears of equal size The number of teeth in the gearand the number of teeth indexed in each block must not have a common factor

Table 7 Block or Multiple Indexing for Gear Cutting

Table 8 Indexing Movements for 60-Tooth Worm-Wheel Dividing Head

Linear Indexing for Rack Cutting.—When racks are cut on a milling machine, two

gen-eral methods of linear indexing are used One is by using the graduated dial on the screw and the other is by using an indexing attachment The accompanying Table 9 showsthe indexing movements when the first method is employed This table applies to millingmachines having feed-screws with the usual lead of 1⁄4 inch and 250 dial graduations eachequivalent to 0.001 inch of table movement

feed-Multiply decimal part of turn (obtained by above formula) by 250, to obtain dial reading

for fractional part of indexing movement, assuming that dial has 250 graduations

Table 9 Linear Indexing Movements for Cutting Rack Teeth on a Milling Machine

These movements are for table feed-screws having the usual lead of 1 ⁄ 4 inch

Note: The linear pitch of the rack equals the circular pitch of gear or pinion which is to

mesh with the rack The table gives both standard diametral pitches and their equivalentlinear or circular pitches

Example:Find indexing movement for cutting rack to mesh with a pinion of 10 diametral

pitch

Indexing movement equals 1 whole turn of feed-screw plus 64.2 thousandths or divisions

on feed-screw dial The feed-screw may be turned this fractional amount by setting dialback to its zero position for each indexing (without backward movement of feed-screw),

or, if preferred, 64.2 (in this example) may be added to each successive dial position asshown below

Dial reading for second position = 64.2 × 2 = 128.4 (complete movement = 1 turn × 64.2additional divisions by turning feed-screw until dial reading is 128.4)

Third dial position = 64.2 × 3 = 192.6 (complete movement = 1 turn + 64.2 additionaldivisions by turning until dial reading is 192.6)

Fourth position = 64.2 × 4 − 250 = 6.8 (1 turn + 64.2 additional divisions by turning screw until dial reading is 6.8 divisions past the zero mark); or, to simplify operation, setdial back to zero for fourth indexing (without moving feed-screw) and then repeat settingsfor the three previous indexings or whatever number can be made before making a com-plete turn of the dial

feed-Pitch of Rack Teeth Indexing, Movement Pitch of Rack Teeth Indexing, Movement Diametral

No of 0.001 Inch Divisions Diametral Pitch

Linear or Circular

No of Whole Turns

No of 0.001 Inch Divisions

=

Counter Milling.—Changing the direction of a linear milling operation by a specific

angle requires a linear offset before changing the angle of cut This compensates for theradius of the milling cutters, as illustrated in Figs 1a and 1b

For inside cuts the offset is subtracted from the point at which the cutting directionchanges (Fig 1a), and for outside cuts the offset is added to the point at which the cuttingdirection changes (Fig 1b) The formula for the offset is

where x = offset distance; r = radius of the milling cutter; and, M = the multiplication factor (M = tan θ⁄2) The value of M for certain angles can be found in Table 10

Table 10 Offset Multiplication Factors

Multiply factor M by the tool radius r to determine the offset dimension

Fig 1a Inside Milling Fig 1b Outside Milling

x

radius

Inside angle

;;

r

Cutter path

Outside angle

x= rM

TABLE OF CONTENTS

2026

GEARS AND GEARING

2029 Definitions of Gear Terms

2033 Sizes and Shape of Gear Teeth

2033 Nomenclature of Gear Teeth

2033 Properties of the Involute Curve

2034 Diametral and Circular Pitch

Systems

2035 Formulas for Spur Gear

2036 Gear Tooth Forms and Parts

2038 Gear Tooth Parts

2039 Tooth Proportions

2040 Fine Pitch Tooth Parts

2040 American Spur Gear Standards

2041 Formulas for Tooth Parts

2041 Fellows Stub Tooth

2041 Basic Gear Dimensions

2042 Formulas for Outside and Root

Diameters

2045 Tooth Thickness Allowance

2045 Circular Pitch for Given Center

2052 Involute Gear Milling Cutter

2052 Circular Pitch in Gears

2052 Increasing Pinion Diameter

2054 Finishing Gear Milling Cutters

2055 Increase in Dedendum

2055 Dimensions Required

2056 Tooth Proportions for Pinions

2058 Minimum Number of Teeth

2058 Gear to Mesh with Enlarged

2061 True Involute Form Diameter

2062 Profile Checker Settings

2067 Determining Amount of Backlash

2068 Helical and Herringbone Gearing

2069 Bevel and Hypoid Gears

2070 Providing Backlash

2070 Excess Depth of Cut

2070 Control of Backlash Allowances

2071 Measurement of Backlash

2072 Control of Backlash

2072 Allowance and Tolerance

2073 Angular Backlash in Gears

2073 Inspection of Gears

2073 Pressure for Fine-Pitch Gears

2074 Internal Gearing

2074 Internal Spur Gears

2074 Methods of Cutting Internal Gears

2074 Formed Cutters for Internal Gears

2074 Arc Thickness of Gear Tooth

2074 Arc Thickness of Pinion Tooth

2074 Relative Sizes of Internal Gear

2075 Rules for Internal Gears

2076 British Standard for Spur and Helical Gears

2077 Addendum Modification

HYPOID AND BEVEL GEARING

2080 Hypoid Gears

2081 Bevel Gearing

2081 Types of Bevel Gears

2083 Applications of Bevel and Hypoid Gears

2083 Design of Bevel Gear Blanks

2084 Mountings for Bevel Gears

2084 Cutting Bevel Gear Teeth

2085 Nomenclature for Bevel Gears

2085 Formulas for Dimensions

2089 Numbers of Formed Cutters

2091 Selecting Formed Cutters

2092 Offset of Cutter

2093 Adjusting the Gear Blank

2094 Steels Used for Bevel Gear

2095 Circular Thickness, Chordal Thickness

GEARS, SPLINES, AND CAMS

TABLE OF CONTENTS

2027

GEARS, SPLINES, AND CAMS WORM GEARING

2095 Standard Design for Fine-pitch

2096 Formulas for Wormgears

2098 Materials for Worm Gearing

2098 Single-thread Worms

2098 Multi-thread Worms

HELICAL GEARING

2099 Helical Gear Calculations

2099 Rules and Formulas

2099 Determining Direction of Thrust

2100 Determining Helix Angles

2100 Pitch of Cutter to be Used

2103 Shafts at Right Angles, Center

2109 Factors for Selecting Cutters

2109 Outside and Pitch Diameters

2109 Milling the Helical Teeth

2110 Fine-Pitch Helical Gears

2111 Center Distance with no Backlash

2112 Change-gears for Hobbing

2114 Helical Gear Hobbing

OTHER GEAR TYPES

2115 Planetary Bevel Gears

2116 Ratios of Epicyclic Gearing

2119 Ratchet Gearing

2119 Types of Ratchet Gearing

2120 Shape of Ratchet Wheel Teeth

2120 Pitch of Ratchet Wheel Teeth

2121 Module System Gear Design

2121 German Standard Tooth Form

2122 Tooth Dimensions

2123 Rules for Module System

2124 Equivalent Diametral Pitches,

Circular Pitches

CHECKING GEAR SIZES

2125 Checking Externall Spur Gear Sizes

2126 Measurement Over Wires

2130 Checking Internal Spur Gear

2139 Measurements for Checking Helical Gears using Wires

2140 Checking Spur Gear Size

2142 Formula for Chordal Dimension

GEAR MATERIALS

2144 Gearing Material

2144 Classification of Gear Steels

2144 Use of Casehardening Steels

2144 Use of “Thru-Hardening” Steels

2144 Heat-Treatment for Machining

2145 Making Pinion Harder

2145 Forged and Rolled Carbon Steels

2147 Steels for Industrial Gearing

2147 Bronze and Brass Gear Castings

2149 Materials for Worm Gearing

TABLE OF CONTENTS

2028

GEARS, SPLINES, AND CAMS

(Continued)

SPLINES AND SERRATIONS

2162 Types and Classes of Fits

2162 Classes of Tolerances

2163 Maximum Tolerances

2164 Fillets and Chamfers

2165 Spline Variations

2165 Effect of Spline Variations

2165 Effective and Actual Dimensions

2166 Space Width and Tooth Thickness

Limits

2166 Effective and Actual Dimensions

2167 Combinations of Spline Types

2167 Interchangeability

2167 Drawing Data

2169 Spline Data and Reference

Dimensions

2169 Estimating Key and Spline Sizes

2170 Formulas for Torque Capacity

2171 Spline Application Factors

2171 Load Distribution Factors

2172 Fatigue-Life Factors

2172 Wear Life Factors

2172 Allowable Shear Stresses

2173 Crowned Splines for Large

Misalignments

2174 Fretting Damage to Splines

2174 Inspection Methods

2175 Inspection with Gages

2175 Measurements with Pins

2176 Metric Module Splines

2180 Tooth Thickness Total Tolerance

2181 Selected Fit Classes Data

2182 Straight Splines

2182 British Standard Striaght Splines

2183 Splines Fittings

2184 Standard Splined Fittings

2185 Dimensions of Standard Splines

2186 Polygon-type Shaft Connections

CAMS AND CAM DESIGN

2188 Classes of Cams

2188 Cam Follower Systems

2189 Displacement Diagrams

2194 Cam Profile Determination

2195 Modified Constant Velocity Cam

2197 Pressure Angle and Radius of Curvature

2198 Cam Size for a Radial Follower

2200 Cam Size for Swinging Roller Follower

2201 Formulas for Calculating Pressure Angles

2203 Radius of Curvature

2205 Cam Forces, Contact Stresses, and Materials

2210 Calculation of Contact Stresses

2211 Layout of Cylinder Cams

2211 Shape of Rolls for Cylinder Cams

2212 Cam Milling

2213 Cutting Uniform Motion Cams

Axial thickness is the distance parallel to the axis between two pitch line elements of the

same tooth

Backlash is the shortest distance between the non-driving surfaces of adjacent teeth when

the working flanks are in contact

Base circle is the circle from which the involute tooth curve is generated or developed Base helix angle is the angle at the base cylinder of an involute gear that the tooth makes

with the gear axis

Base pitch is the circular pitch taken on the circumference of the base circles, or the

dis-tance along the line of action between two successive and corresponding involute tooth

profiles The normal base pitch is the base pitch in the normal plane and the axial base pitch is the base pitch in the axial plane.

Base tooth thickness is the distance on the base circle in the plane of rotation between

invo-lutes of the same pitch

Bottom land is the surface of the gear between the flanks of adjacent teeth.

Center distance is the shortest distance between the non-intersecting axes of mating gears,

or between the parallel axes of spur gears and parallel helical gears, or the crossed axes ofcrossed helical gears or worm gears

Central plane is the plane perpendicular to the gear axis in a worm gear, which contains the

common perpendicular of the gear and the worm axes In the usual arrangement with theaxes at right angles, it contains the worm axis

Chordal addendum is the radial distance from the circular thickness chord to the top of the

tooth, or the height from the top of the tooth to the chord subtending the circular thicknessarc

Chordal thickness is the length of the chord subtended by the circular thickness arc The

dimension obtained when a gear tooth caliper is used to measure the tooth thickness at thepitch circle

Circular pitch is the distance on the circumference of the pitch circle, in the plane of

rota-tion, between corresponding points of adjacent teeth The length of the arc of the pitchcircle between the centers or other corresponding points of adjacent teeth

Circular thickness is the thickness of the tooth on the pitch circle in the plane of rotation, or

the length of arc between the two sides of a gear tooth measured on the pitch circle

Clearance is the radial distance between the top of a tooth and the bottom of a mating tooth

space, or the amount by which the dedendum in a given gear exceeds the addendum of itsmating gear

Contact diameter is the smallest diameter on a gear tooth with which the mating gear

makes contact

Contact ratio is the ratio of the arc of action in the plane of rotation to the circular pitch, and

is sometimes thought of as the average number of teeth in contact This ratio is obtainedmost directly as the ratio of the length of action to the base pitch

Contact ratio – face is the ratio of the face advance to the circular pitch in helical gears Contact ratio – total is the ratio of the sum of the arc of action and the face advance to the

circular pitch

Contact stress is the maximum compressive stress within the contact area between mating

gear tooth profiles Also called the Hertz stress

Cycloid is the curve formed by the path of a point on a circle as it rolls along a straight line When such a circle rolls along the outside of another circle the curve is called an epicyc- loid, and when it rolls along the inside of another circle it is called a hypocycloid These

curves are used in defining the former American Standard composite Tooth Form

Dedendum is the radial or perpendicular distance between the pitch circle and the bottom

of the tooth space

Diametral pitch is the ratio of the number of teeth to the number of inches in the pitch

diameter in the plane of rotation, or the number of gear teeth to each inch of pitch ter Normal diametral pitch is the diametral pitch as calculated in the normal plane, or thediametral pitch divided by the cosine of the helix angle

diame-Efficiency is the torque ratio of a gear set divided by its gear ratio.

Equivalent pitch radius is the radius of curvature of the pitch surface at the pitch point in a

plane normal to the pitch line element

Face advance is the distance on the pitch circle that a gear tooth travels from the time pitch

point contact is made at one end of the tooth until pitch point contact is made at the otherend

Fillet radius is the radius of the concave portion of the tooth profile where it joins the

bot-tom of the tooth space

Fillet stress is the maximum tensile stress in the gear tooth fillet.

Flank of tooth is the surface between the pitch circle and the bottom land, including the

gear tooth fillet

Gear ratio is the ratio between the numbers of teeth in mating gears.

Helical overlap is the effective face width of a helical gear divided by the gear axial pitch Helix angle is the angle that a helical gear tooth makes with the gear axis at the pitch circle,

unless specified otherwise

Hertz stress, see Contact stress.

Highest point of single tooth contact (HPSTC) is the largest diameter on a spur gear at

which a single tooth is in contact with the mating gear

Interference is the contact between mating teeth at some point other than along the line of

action

Internal diameter is the diameter of a circle that coincides with the tops of the teeth of an

internal gear

Internal gear is a gear with teeth on the inner cylindrical surface.

Involute is the curve generally used as the profile of gear teeth The curve is the path of a

point on a straight line as it rolls along a convex base curve, usually a circle

Land The top land is the top surface of a gear tooth and the bottom land is the surface of the

gear between the fillets of adjacent teeth

Lead is the axial advance of the helix in one complete turn, or the distance along its own

axis on one revolution if the gear were free to move axially

Length of action is the distance on an involute line of action through which the point of

contact moves during the action of the tooth profile

Line of action is the portion of the common tangent to the base cylinders along which

con-tact between mating involute teeth occurs

Lowest point of single tooth contact (LPSTC) is the smallest diameter on a spur gear at

which a single tooth is in contact with its mating gear Gear set contact stress is mined with a load placed on the pinion at this point

deter-Module is the ratio of the pitch diameter to the number of teeth, normally the ratio of pitch

diameter in mm to the number of teeth Module in the inch system is the ratio of the pitchdiameter in inches to the number of teeth

Normal plane is a plane normal to the tooth surfaces at a point of contact and perpendicular

to the pitch plane

Number of teeth is the number of teeth contained in a gear.

Outside diameter is the diameter of the circle that contains the tops of the teeth of external

gears

Pitch is the distance between similar, equally-spaced tooth surfaces in a given direction

along a given curve or line

Pitch circle is the circle through the pitch point having its center at the gear axis Pitch diameter is the diameter of the pitch circle The operating pitch diameter is the pitch

diameter at which the gear operates

Pitch plane is the plane parallel to the axial plane and tangent to the pitch surfaces in any

pair of gears In a single gear, the pitch plane may be any plane tangent to the pitch faces

sur-Pitch point is the intersection between the axes of the line of centers and the line of action Plane of rotation is any plane perpendicular to a gear axis.

Pressure angle is the angle between a tooth profile and a radial line at its pitch point In

involute teeth, the pressure angle is often described as the angle between the line of action

and the line tangent to the pitch circle Standard pressure angles are established in

con-nection with standard tooth proportions A given pair of involute profiles will transmitsmooth motion at the same velocity ratio when the center distance is changed Changes incenter distance in gear design and gear manufacturing operations may cause changes inpitch diameter, pitch and pressure angle in the same gears under different conditions

Unless otherwise specified, the pressure angle is the standard pressure angle at the dard pitch diameter The operating pressure angle is determined by the center distance

stan-at which a pair of gears operstan-ate In oblique teeth such as helical and spiral designs, thepressure angle is specified in the transverse, normal or axial planes

Principle reference planes are pitch plane, axial plane and transverse plane, all

intersect-ing at a point and mutually perpendicular

Rack: A rack is a gear with teeth spaced along a straight line, suitable for straight line

motion A basic rack is a rack that is adopted as the basis of a system of interchangeablegears Standard gear tooth dimensions are often illustrated on an outline of a basic rack

Roll angle is the angle subtended at the center of a base circle from the origin of an involute

to the point of tangency of a point on a straight line from any point on the same involute.The radian measure of this angle is the tangent of the pressure angle of the point on theinvolute

Root diameter is the diameter of the circle that contains the roots or bottoms of the tooth

spaces

Tangent plane is a plane tangent to the tooth surfaces at a point or line of contact Tip relief is an arbitrary modification of a tooth profile where a small amount of material is

removed from the involute face of the tooth surface near the tip of the gear tooth

Tooth face is the surface between the pitch line element and the tooth tip.

Tooth surface is the total tooth area including the flank of the tooth and the tooth face Total face width is the dimensional width of a gear blank and may exceed the effective face

width as with a double-helical gear where the total face width includes any distance arating the right-hand and left-hand helical gear teeth

sep-Transverse plane is a plane that is perpendicular to the axial plane and to the pitch plane In

gears with parallel axes, the transverse plane and the plane of rotation coincide

Trochoid is the curve formed by the path of a point on the extension of a radius of a circle

as it rolls along a curve or line A trochoid is also the curve formed by the path of a point

on a perpendicular to a straight line as the straight line rolls along the convex side of a

base curve By the first definition, a trochoid is derived from the cycloid, by the second definition it is derived from the involute.

True involute form diameter is the smallest diameter on the tooth at which the point of

tan-gency of the involute tooth profile exists Usually this position is the point of tantan-gency ofthe involute tooth profile and the fillet curve, and is often referred to as the TIF diameter

Undercut is a condition in generated gear teeth when any part of the fillet curve lies inside

a line drawn at a tangent to the working profile at its lowest point Undercut may be duced deliberately to facilitate shaving operations, as in pre-shaving

intro-Whole depth is the total depth of a tooth space, equal to the addendum plus the dedendum

and equal to the working depth plus clearance

Working depth is the depth of engagement of two gears, or the sum of their addendums.

The standard working distance is the depth to which a tooth extends into the tooth space

of a mating gear when the center distance is standard

Definitions of gear terms are given in AGMA Standards 112.05, 115.01, and 116.01 tled “Terms, Definitions, Symbols and Abbreviations,” “Reference Information—BasicGear Geometry,” and “Glossary—Terms Used in Gearing,” respectively; obtainable fromAmerican Gear Manufacturers Assn., 500 Montgomery St., St., Alexandria, VA 22314

enti-American National Standard and Former enti-American Standard Gear Tooth Forms

ANSI B6.1-1968, (R1974) and ASA B6.1-1932

Basic Rack of the 20-Degree and 25-Degree Full-Depth Involute Systems

Basic Rack of the 14 1 ⁄ 2 -Degree Full-Depth Involute System

Basic Rack of the 20-Degree Stub Involute System

Approximation of Basic Rack for the 14 1 ⁄ 2 -Degree Composite System

Gears for Given Center Distance and Ratio.—When it is necessary to use a pair of

gears of given ratio at a specified center distance C1, it may be found that no gears of dard diametral pitch will satisfy the center distance requirement Gears of standard diame-

stan-tral pitch P may need to be redesigned to operate at other than their standard pitch diameter

D and standard pressure angle φ The diametral pitch P1 at which these gears will operate is

(1)

where N p =number of teeth in pinion

N G =number of teeth in gear

and their operating pressure angle φ1 is

(2)

Thus although the pair of gears are cut to a diametral pitch P and a pressure angle φ, they

operate as standard gears of diametral pitch P1 and pressure angle φ1 The pitch P and

pres-sure angle φ should be chosen so that φ1 lies between about 18 and 25 degrees

The operating pitch diameters of the pinion D p1 and of the gear D G1 are

The base diameters of the pinion D PB1 and of the gear D GB1 are

The basic tooth thickness, t1, at the operating pitch diameter for both pinion and gear is

(5)

The root diameters of the pinion D PR1 and gear D GR1 and the corresponding outside

diam-eters D PO1 and D GO1 are not standard because each gear is to be cut with a cutter that is not

standard for the operating pitch diameters D P1 and D G1

The root diameters are

where b c is the hob or cutter addendum for the pinion and gear

The tooth thicknesses of the pinion t P2 and the gear t G2 are

-=

φ1

P1P

=

t P2 N P P

The outside diameter of the pinion D PO and the gear D GO are

Example:Design gears of 8 diametral pitch, 20-degree pressure angle, and 28 and 88

teeth to operate at 7.50-inch center distance The gears are to be cut with a hob of inch addendum

t G2 N G P

The tooth thickness on the pitch circle can be determined very accurately by means ofmeasurement over wires which are located in tooth spaces that are diametrically opposite

or as nearly diametrically opposite as possible Where measurement over wires is not sible, the circular or arc tooth thickness can be used in determining the chordal thicknesswhich is the dimension measured with a gear tooth caliper

fea-Circular Thickness of Tooth when Outside Diameter has been Enlarged.—When the

outside diameter of a small pinion is not standard but is enlarged to avoid undercut and toimprove tooth action, the teeth are located farther out radially relative to the standard pitchdiameter and consequently the circular tooth thickness at the standard pitch diameter is

increased To find this increased arc thickness the following formula is used, where t = tooth thickness; e = amount outside diameter is increased over standard; φ = pressure

angle; and p = circular pitch at the standard pitch diameter.

Example:The outside diameter of a pinion having 10 teeth of 5 diametral pitch and a

pressure angle of 141⁄2 degrees is to be increased by 0.2746 inch The circular pitch lent to 5 diametral pitch is 0.6283 inch Find the arc tooth thickness at the standard pitchdiameter

equiva-Circular Thickness of Tooth when Outside Diameter has been Reduced.—If the

out-side diameter of a gear is reduced, as is frequently done to maintain the standard center tance when the outside diameter of the mating pinion is increased, the circular thickness ofthe gear teeth at the standard pitch diameter will be reduced.This decreased circular thick-

dis-ness can be found by the following formula where t = circular thickdis-ness at the standard pitch diameter; e = amount outside diameter is reduced under standard; φ = pressure angle;

and p = circular pitch.

Example:The outside diameter of a gear having a pressure angle of 141⁄2 degrees is to bereduced by 0.2746 inch or an amount equal to the increase in diameter of its mating pinion.The circular pitch is 0.6283 inch Determine the circular tooth thickness at the standardpitch diameter

Chordal Thickness of Tooth when Outside Diameter is Standard.—T o f i n d t h e

chordal or straight line thickness of a gear tooth the following formula can be used where t c

= chordal thickness; D = pitch diameter; and N = number of teeth.

Example:A pinion has 15 teeth of 3 diametral pitch; the pitch diameter is equal to 15 ÷ 3

or 5 inches Find the chordal thickness at the standard pitch diameter

sin 5sin6° 5×0.10453 0.5226 inch

Chordal Thicknesses and Chordal Addenda of Milled, Full-depth

Gear Teeth and of Gear Milling Cutters

T =chordal thickness of gear tooth and cutter tooth at pitch line;

H =chordal addendum for full-depth gear tooth;

A =chordal addendum of cutter = (2.157 ÷ diametral pitch) − H