Gear Geometry & Profile Theory

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Gear Geometry &Profile Theory.Spur & Helical Gears

Gear geometry plays important role in power transmission. Every Gear designer

needs to go through the gear profile and geometry study. The purpose of this

article is to provide primary introduction to the beginners to have strong baseof Gear profile.

2013

ASHUTOSH KUMAR

ME-DESIGN

2/17/2013


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Tooth profileA profile is one side of a tooth in a cross section between the outside circle and

the root circle. Usually a profile is the curve of intersection of a tooth surface and

a plane or surface normal to the pitch surface, such as the transverse, normal, or

axial plane.

The fillet curve (root fillet) is the concave portion of the tooth profile where it

joins the bottom of the tooth space.

The attainment of a no fluctuating velocity ratio is dependent on the profile of

the teeth. Friction and wear between two gears is also dependent on the tooth

profile. There are a great many tooth profiles that will give a constant velocity

ratio, and in many cases, given an arbitrary tooth shape, it is possible to develop

a tooth profile for the mating gear that will give a constant velocity ratio.

However, two constant velocity tooth profiles have been by far the most

commonly used in modern times. They are the cycloid and the involute. The

cycloid was more common until the late 1800s; since then the involute has

largely superseded it, particularly in drive train applications. The cycloid is in

some ways the more interesting and flexible shape; however the involute has

two advantages: it is easier to manufacture, and it permits the center to center

spacing of the gears to vary over some range without ruining the constancy of

the velocity ratio. Cycloidal gears only work properly if the center spacing is

exactly right. Cycloidal gears are still used in mechanical clocks.


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An undercut is a

curve lies inside

juncture with the

finishing operatio

Without undercu

tangent

Prametric EquatiX = rb(sin co

Y = rb(cos si

- Involute Roll a

Involute gear ProInvolute gear pro

between a pair o

the gears causes

tooth surfaces.

ondition in generated gear teeth when any pa

f a line drawn tangent to the working profile at

fillet. Undercut may be deliberately introduce

ns. With undercut the fillet curve intersects th

the fillet curve and the working profile have a

n of Involute Curve:

)

)

gle, rb Radius of base circle.

file:

ile is the involutes of a circle. In involute gear

gear teeth occurs at a single instantaneous po

he location of this contact point to move acros

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t of the fillet

its point of

to facilitate

working profile.

common

esign, contact

int . Rotation of

s the respective


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Line of Action:The path traced by the contact point between any pair of gear is known as the

Line of Action (also called Pressure Line or Line of Contact). It is the line alongwhich the gear force act.

Involute Tooth:Tooth having involute curve is known as Involute tooth. A property of the

involute tooth form is that if the gears are meshed properly, the line of action is

straight and passes through the Pitch Point of the gears. When this is true, the

gears obey the Fundamental Law of Gearing: The angular velocity ratio between

two gears of a gear set must remain constant throughout the mesh. This

property results in smooth transmission of power without speed or torquevariations as pairs of teeth go into or come out of mesh.

Conjugate action:If a pair of gear produce constant angular velocity ratio and also maintains it

while meshing then such a action is called Conjugate action.

Conjugate curve:The curve which produces conjugate action is known as Conjugate curve.

Involute Polar angle():It may be defined as the angle between the radius vector to a point, P, on an

involute curve and a radial line to the intersection, A, of the curve with the base

circle. Refer figure shown below.

This angle can also be represented as involute or inv .

Pressure angle():The Pressure Angle is the acute angle between the line of action and a normal to

the line connecting the gear centers. The pressure angle of the gear is a function

of the involute tooth shape and pairs of gears must have the same pressure

angle in order for the teeth to mesh properly.

The pressure angle shown below resembles to the pressure angle at point P.


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So we have,

r = rb.sec

Length of involute (L):In the above shown figure Length of involute for A-B can be calculated as shown

below:

L = rb . (tan2/2)

Radius of Curvature (R):Radius of Curvature (R ) at point P can be calculated as below:

R = rb . tan

Involute roll angle:Expressed as , the involute roll angle is the angle whose arc on the base circle of

radius unity equals the tangent of the pressure angle() at a selected point on

the involute.


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General nomenclat

Rotational frequMeasured in rota

Angular frequencMeasured in radi

ra

Number of teethHow many teeth

re

ncy, n

ion over time, such as RPM.

y,

ns per second.

/second

Z

a gear has, an integer.

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Gear, wheelThe larger of two interacting gears or a gear on its own.

PinionThe smaller of two interacting gears.

Path of contactPath followed by the point of contact between two meshing gear teeth.

Line of action, pressure lineLine along which the force between two meshing gear teeth is directed. It has

the same direction as the force vector. In general, the line of action changes

from moment to moment during the period of engagement of a pair of teeth.For involute gears, however, the tooth-to-tooth force is always directed along

the same linethat is, the line of action is constant. This implies that for involute

gears the path of contact is also a straight line, coincident with the line of

actionas is indeed the case.

Pitch point, pPoint where the line of action crosses a line joining the two gear axes.

Pitch circle, pitch lineCircle centered on and perpendicular to the axis, and passing through the pitch

point. A predefined diametral position on the gear where the circular tooth

thickness, pressure angle and helix angles are defined.

Pitch diameter, dA predefined diametral position on the gear where the circular tooth thickness,

pressure angle and helix angles are defined. The standard pitch diameter is a

basic dimension and cannot be measured, but is a location where other

measurements are made. Its value is based on the number of teeth, the normal

module (or normal diametral pitch), and the helix angle. It is calculated as:


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Module, mA scaling factor used in metric gears with units in millimeters whose effect is to

enlarge the gear tooth size as the module increases and reduce the size as the

module decreases. Module can be defined in the normal (mn), the transverse

(mt), or the axial planes (ma) depending on the design approach employed and

the type of gear being designed. Module is typically an input value into the gear

design and is seldom calculated.

Operating pitch diametersDiameters determined from the number of teeth and the center distance at

which gears operate. Example for pinion:

2

1

Pitch surfaceIn cylindrical gears, cylinder formed by projecting a pitch circle in the axial

direction. More generally, the surface formed by the sum of all the pitch circles

as one moves along the axis.

Angle of actionAngle with vertex at the gear center, one leg on the point where mating teeth

first make contact, the other leg on the point where they disengage.

Arc of actionSegment of a pitch circle subtended by the angle of action.

Pressure angle, The complement of the angle between the direction that the teeth exert force

on each other, and the line joining the centers of the two gears. For involute

gears, the teeth always exert force along the line of action, which, for involute

gears, is a straight line; and thus, for involute gears, the pressure angle is

constant.

Outside diameter, DODiameter of the gear, measured from the tops of the teeth.


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Root diameterDiameter of the

Addendum, aRadial distance f

Dedendum, bRadial distance f

Whole depth,The distance fro

dedendum or to

ClearanceDistance betwee

Working depthDepth of engage

Circular pitch, pDistance from o

on the same gear

Diametral pitch,Ratio of the num

per inch or teeth

Base circleIn involute gears

radius of the bas

gear, measured at the base of the tooth.

rom the pitch surface to the outermost point o

rom the depth of the tooth trough to the pitch

the top of the tooth to the root; it is equal to

working depth plus clearance.

the root circle of a gear and the addendum cir

ent of two gears, that is, the sum of their oper

e face of a tooth to the corresponding face of a

, measured along the pitch circle.

ber of teeth to the pitch diameter. Could be mea

per centimeter.

, where the tooth profile is the involute of the b

circle is somewhat smaller than that of the pit

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the tooth.

surface.

addendum plus

le of its mate.

ting addendums.

adjacent tooth

sured in teeth

se circle. The

h circle.


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Base pitch, normIn involute gears

an adjacent tooth

InterferenceContact between

InterchangeableA set of gears, a

Helical gear nomencla

Helix angle,Angle between a

case of a spur ge

Normal circular pCircular pitch in

Transverse circulCircular pitch in

pitch".

Pn = Pcos

al pitch,

, distance from one face of a tooth to the corres

on the same gear, measured along the base circ

teeth other than at the intended parts of their su

et

y of which will mate properly with any other.

ure

tangent to the helix and the gear axis. It is zero

ar, albeit it can consider as the hypotenuse angl

itch, Pn

the plane normal to the teeth.

r pitch, P

the plane of rotation of the gear. Sometimes jus

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onding face of

le.

rfaces.

in the limiting

as well.

t called "circular


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Tooth contact nomenclature Point of contact

Any point at which two tooth profiles touch each other.

Line of contactA line or curve along which two tooth surfaces are tangent to each other.

Path of actionThe locus of successive contact points between a pair of gear teeth, during the

phase of engagement. For conjugate gear teeth, the path of action passes through

the pitch point. It is the trace of the surface of action in the plane of rotation.


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Line of actionThe path of action for involute gears. It is the straight line passing through the

pitch point and tangent to both base circles.

Surface of actionThe imaginary surface in which contact occurs between two engaging tooth

surfaces. It is the summation of the paths of action in all sections of the engaging

teeth.

Plane of actionThe surface of action for involute, parallel axis gears with either spur or helical

teeth. It is tangent to the base cylinders.


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Zone of action (contact zone)For involute, parallel-axis gears with either spur or helical teeth, is the

rectangular area in the plane of action bounded by the length of action and the

effective face width.

Path of contactThe curve on either tooth surface along which theoretical single point contact

occurs during the engagement of gears with crowned tooth surfaces or gears that

normally engage with only single point contact.

Length of actionThe distance on the line of action through which the point of contact moves

during the action of the tooth profile.

Arc of action, QtThe arc of the pitch circle through which a tooth profile moves from the

beginning to the end of contact with a mating profile.

Arc of approach, QaThe arc of the pitch circle through which a tooth profile moves from its

beginning of contact until the point of contact arrives at the pitch point.


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Arc of recess, QrThe arc of the pi

the pitch point u

Contact ratio, mcThe number of a

beginning to the

of the average n

comes and goes

Transverse contaThe contact ratio

angular pitch. Fo

length of action

Face contact ratiThe contact ratio

pitch.

Total contact ratiThe sum of the t

ch circle through which a tooth profile moves f

til contact ends.

,

ngular pitches through which a tooth surface rot

end of contact. In a simple way, it can be defin

mber of teeth in contact during the period in w

ut of contact with the mating gear.

ct ratio, mp,

in a transverse plane. It is the ratio of the angle

r involute gears it is most directly obtained as t

o the base pitch.

, mF,

in an axial plane, or the ratio of the face width

o, mt,

ansverse contact ratio and the face contact ratio

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rom contact at

ates from the

d as a measure

ich a tooth

of action to the

e ratio of the

to the axial

.


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Limit diameterDiameter on a g

minimum for int

referred to as thethe end of active

Start of active prIntersection of th

Face advanceDistance on a pit

position at whic

to the position w

ar at which the line of action intersects the max

rnal pinion) addendum circle of the mating gea

start of active profile, the start of contact, the eprofile.

file (SAP)

e limit diameter and the involute profile.

ch circle through which a helical or spiral tooth

contact begins at one end of the tooth trace on

here contact ceases at the other end.

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imum (or

r. This is also

d of contact, or

moves from the

the pitch surface


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Tooth thickness Circular thickness

Length of arc between the two sides of a gear tooth, on the specified datumcircle.

Transverse circular thicknessCircular thickness in the transverse plane.

Normal circular thicknessCircular thickness in the normal plane. In a helical gear it may be considered as

the length of arc along a normal helix.

Axial thicknessIn helical gears and worms, tooth thickness in an axial cross section at the

standard pitch diameter.

Base circular thicknessIn involute teeth, length of arc on the base circle between the two involute curves

forming the profile of a tooth.


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Normal chordal thicknessLength of the chord that subtends a circular thickness arc in the plane normal to

the pitch helix. Any convenient measuring diameter may be selected, not

necessarily the standard pitch diameter.

Chordal addendum (chordal height)Height from the top of the tooth to the chord subtending the circular thickness

arc. Any convenient measuring diameter may be selected, not necessarily the

standard pitch diameter.

Profile shiftDisplacement of the basic rack datum line from the reference cylinder, made

non-dimensional by dividing by the normal module. It is used to specify the

tooth thickness, often for zero backlash.

Rack shiftDisplacement of the tool datum line from the reference cylinder, made non-

dimensional by dividing by the normal module. It is used to specify the tooththickness.


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Measurement over pinsMeasurement of the distance taken over a pin positioned in a tooth space and a

reference surface. The reference surface may be the reference axis of the gear, a

datum surface or either one or two pins positioned in the tooth space or spacesopposite the first. This measurement is used to determine tooth thickness.

Span measurementMeasurement of the distance across several teeth in a normal plane. As long asthe measuring device has parallel measuring surfaces that contact on an

unmodified portion of the involute, the measurement will be along a line tangent

to the base cylinder. It is used to determine tooth thickness.


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Modified addendum teethTeeth of engaging gears, one or both of which have non-standard addendum.

Full-depth teethTeeth in which the working depth equals 2.000 divided by the normal diametral

pitch.

Stub teethTeeth in which the working depth is less than 2.000 divided by the normal

diametral pitch.

Equal addendum teethTeeth in which two engaging gears have equal addendums.

Long and short-addendum teethTeeth in which the addendums of two engaging gears are unequal.


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Pitch nomenclature


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PitchThe distance between a point on one tooth and the corresponding point on an

adjacent tooth. It is a dimension measured along a line or curve in the transverse,

normal, or axial directions. The use of the single word pitch without qualification

may be ambiguous, and for this reason it is preferable to use specificdesignations such as transverse circular pitch, normal base pitch, axial pitch.

Circular pitch, pArc distance along a specified pitch circle or pitch line between corresponding

profiles of adjacent teeth.

Transverse circular pitch, ptCircular pitch in the transverse plane.

Normal circular pitch, pn, peCircular pitch in the normal plane, and also the length of the arc along the normal

pitch helix between helical teeth or threads.

Axial pitch, pxLinear pitch in an axial plane and in a pitch surface. In helical gears and worms,

axial pitch has the same value at all diameters. In gearing of other types, axial

pitch may be confined to the pitch surface and may be a circular measurement.

The term axial pitch is preferred to the term linear pitch. The axial pitch of ahelical worm and the circular pitch of its worm gear are the same.

Normal base pitch, pN, pbnAn involute helical gear is the base pitch in the normal plane. It is the normal

distance between parallel helical involute surfaces on the plane of action in the

normal plane, or is the length of arc on the normal base helix. It is a constant

distance in any helical involute gear.


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Transverse base pitch, pb, pbtIn an involute gear, the pitch on the base circle or along the line of action.

Corresponding sides of involute gear teeth are parallel curves, and the base pitch

is the constant and fundamental distance between them along a common normalin a transverse plane.

Diametral pitch (transverse), PdRatio of the number of teeth to the standard pitch diameter in inches.

Normal diametral pitch, PndValue of diametral pitch in a normal plane of a helical gear .

Angular pitch, N, Angle subtended by the circular pitch, usually expressed in radians.

degrees


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Gear Geometry & Profile Theory

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