Short Notes: Gears and Gear Trains | Short Notes for Mechanical Engineering PDF Download

 Page 1


Gears and Gear Trains 
Gears and Gear Trains
• Gears is a positive drive which transmits exact velocity ratio from one shaft to 
another shaft, gear drive is used when the distance between driver and 
follower is very less, when two or more gears are made to mesh with each 
other to transmit power from one shaft to another, the combination is called 
gear trains.
Gears
• Gears are used to transmit motion from one shaft to another or between a 
shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or 
connector and transmit the motion by direct contact.
Gear Terminology
• Circular Pitch (p): It is a distance measured along the circumference of the 
pitch circle from a point on one tooth to the corresponding point on the 
adjacent tooth.
P =
7 t d 
T
where, p = circular pitch 
d = pitch diameter 
T = number of teeth
Page 2


Gears and Gear Trains 
Gears and Gear Trains
• Gears is a positive drive which transmits exact velocity ratio from one shaft to 
another shaft, gear drive is used when the distance between driver and 
follower is very less, when two or more gears are made to mesh with each 
other to transmit power from one shaft to another, the combination is called 
gear trains.
Gears
• Gears are used to transmit motion from one shaft to another or between a 
shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or 
connector and transmit the motion by direct contact.
Gear Terminology
• Circular Pitch (p): It is a distance measured along the circumference of the 
pitch circle from a point on one tooth to the corresponding point on the 
adjacent tooth.
P =
7 t d 
T
where, p = circular pitch 
d = pitch diameter 
T = number of teeth
Face width
• Diametrical Pitch (P)
It is the number of teeth per unit length of the pitch circle diameter in inches.
p - t -
d
• Module (m)
It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch.
d
m — —
T
=¦ p= -m
• Gear Ratio (G)
It is the ratio of the number of
where, T = number of teeth on 
t = number of teeth on the
• Velocity Ratio
The velocity ratio is defined as the ratio of the angular velocity of the follower to 
the angular velocity of the driver gear.
ra = £ i= L ± = Zl
o y j .Vj d i 7 * .
• Addendum Circle: A circle bounding the ends of the teeth, in a normal section 
of the gear.
• Dedendum Circle: The circle bounding the spaces between the teeth, in a 
normal section of the gear.
• Addendum: The radial distance between the pitch circle and the addendum 
circle.
• Dedendum: The radial distance between the pitch circle and the root circle,
Standard Proportions for 20° Full Depth System
The following standard proportions are used in 20° full depth system
teeth on the gear to that on the pinion.
r
the gear 
pinion
• Standard Proportions 
° Addendum ha = m 
° Clearance C = 0.25 m
Page 3


Gears and Gear Trains 
Gears and Gear Trains
• Gears is a positive drive which transmits exact velocity ratio from one shaft to 
another shaft, gear drive is used when the distance between driver and 
follower is very less, when two or more gears are made to mesh with each 
other to transmit power from one shaft to another, the combination is called 
gear trains.
Gears
• Gears are used to transmit motion from one shaft to another or between a 
shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or 
connector and transmit the motion by direct contact.
Gear Terminology
• Circular Pitch (p): It is a distance measured along the circumference of the 
pitch circle from a point on one tooth to the corresponding point on the 
adjacent tooth.
P =
7 t d 
T
where, p = circular pitch 
d = pitch diameter 
T = number of teeth
Face width
• Diametrical Pitch (P)
It is the number of teeth per unit length of the pitch circle diameter in inches.
p - t -
d
• Module (m)
It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch.
d
m — —
T
=¦ p= -m
• Gear Ratio (G)
It is the ratio of the number of
where, T = number of teeth on 
t = number of teeth on the
• Velocity Ratio
The velocity ratio is defined as the ratio of the angular velocity of the follower to 
the angular velocity of the driver gear.
ra = £ i= L ± = Zl
o y j .Vj d i 7 * .
• Addendum Circle: A circle bounding the ends of the teeth, in a normal section 
of the gear.
• Dedendum Circle: The circle bounding the spaces between the teeth, in a 
normal section of the gear.
• Addendum: The radial distance between the pitch circle and the addendum 
circle.
• Dedendum: The radial distance between the pitch circle and the root circle,
Standard Proportions for 20° Full Depth System
The following standard proportions are used in 20° full depth system
teeth on the gear to that on the pinion.
r
the gear 
pinion
• Standard Proportions 
° Addendum ha = m 
° Clearance C = 0.25 m
o
Whole depth h = 2.25 m 
° Tooth space = 1.5700 m 
o Dedendum hf = 1.25 m 
o Working depth h^ = 2m 
o Tooth thickness (s) = 1.5708 m 
° Fillet radius = 0.4 m
• Form of Teeth: The form of teeth in a gear system can be as follows.
• Conjugate Teeth: Two curves of any arbitrary shape that fulfill the law of 
gearing can be used as the profiles of teeth. Such a gear The profiles of tooths 
which are generally used given as
• Cycloidal Profile Teeth: As cycloid is the locus of a point on the circumference 
of a circle that rolls without slipping on a fixed straight line In this type, the 
faces of the teeth are epicycloids and the flanks the hypocycloids.
• Epicycloid Teeth: An epicycloid is the locus of point on the circumference of a 
circle that rolls without slipping on a fixed straight line.
Arc PK = arc KJG
• Hypocycloid Teeth
A hypocycloid is the locus of a point on the circumference of a circle that rolls 
without slipping inside the circumference of another circle
Arc AC = Arc CD
• Involute Profile Teeth
An involute is defined as the locus of a point on a straight line which rolls without 
slipping on the circumference of a circle. From figure at the start the tracing point 
is at A as the rolls on the circumference of the circle, the path ABC traced out by 
the point A is the involute.
D is regarded as the instantaneous centre of rotation of 8. The motion of 8 is 
perpendicular to BD. Since, BD is tangent to the base circle, the normal to the 
involute is a tangent to the base circle.
Page 4


Gears and Gear Trains 
Gears and Gear Trains
• Gears is a positive drive which transmits exact velocity ratio from one shaft to 
another shaft, gear drive is used when the distance between driver and 
follower is very less, when two or more gears are made to mesh with each 
other to transmit power from one shaft to another, the combination is called 
gear trains.
Gears
• Gears are used to transmit motion from one shaft to another or between a 
shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or 
connector and transmit the motion by direct contact.
Gear Terminology
• Circular Pitch (p): It is a distance measured along the circumference of the 
pitch circle from a point on one tooth to the corresponding point on the 
adjacent tooth.
P =
7 t d 
T
where, p = circular pitch 
d = pitch diameter 
T = number of teeth
Face width
• Diametrical Pitch (P)
It is the number of teeth per unit length of the pitch circle diameter in inches.
p - t -
d
• Module (m)
It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch.
d
m — —
T
=¦ p= -m
• Gear Ratio (G)
It is the ratio of the number of
where, T = number of teeth on 
t = number of teeth on the
• Velocity Ratio
The velocity ratio is defined as the ratio of the angular velocity of the follower to 
the angular velocity of the driver gear.
ra = £ i= L ± = Zl
o y j .Vj d i 7 * .
• Addendum Circle: A circle bounding the ends of the teeth, in a normal section 
of the gear.
• Dedendum Circle: The circle bounding the spaces between the teeth, in a 
normal section of the gear.
• Addendum: The radial distance between the pitch circle and the addendum 
circle.
• Dedendum: The radial distance between the pitch circle and the root circle,
Standard Proportions for 20° Full Depth System
The following standard proportions are used in 20° full depth system
teeth on the gear to that on the pinion.
r
the gear 
pinion
• Standard Proportions 
° Addendum ha = m 
° Clearance C = 0.25 m
o
Whole depth h = 2.25 m 
° Tooth space = 1.5700 m 
o Dedendum hf = 1.25 m 
o Working depth h^ = 2m 
o Tooth thickness (s) = 1.5708 m 
° Fillet radius = 0.4 m
• Form of Teeth: The form of teeth in a gear system can be as follows.
• Conjugate Teeth: Two curves of any arbitrary shape that fulfill the law of 
gearing can be used as the profiles of teeth. Such a gear The profiles of tooths 
which are generally used given as
• Cycloidal Profile Teeth: As cycloid is the locus of a point on the circumference 
of a circle that rolls without slipping on a fixed straight line In this type, the 
faces of the teeth are epicycloids and the flanks the hypocycloids.
• Epicycloid Teeth: An epicycloid is the locus of point on the circumference of a 
circle that rolls without slipping on a fixed straight line.
Arc PK = arc KJG
• Hypocycloid Teeth
A hypocycloid is the locus of a point on the circumference of a circle that rolls 
without slipping inside the circumference of another circle
Arc AC = Arc CD
• Involute Profile Teeth
An involute is defined as the locus of a point on a straight line which rolls without 
slipping on the circumference of a circle. From figure at the start the tracing point 
is at A as the rolls on the circumference of the circle, the path ABC traced out by 
the point A is the involute.
D is regarded as the instantaneous centre of rotation of 8. The motion of 8 is 
perpendicular to BD. Since, BD is tangent to the base circle, the normal to the 
involute is a tangent to the base circle.
Comparison between Cycloidal Teeth and Involute Teeth
• Cyclodial Teeth
° Pressure angle maximum at beginning of engagement, then reduce to 
zero at pitch point and again maximum at end of engagement, 
o Less smooth running of gears.
° Involve double curve, epicycloid and hypocycloid for the teeth 
° Costier manufacturing
o Exact centre distance is reuired to transmit a constant velocity ratio.
° Phenomenon of interference does not occur at all.
° Spreading flank teeth and strong.
° Convex flank always has contact with a concave face resulting in less 
wear.
• Involute Teeth
° Pressure angle is constant throughtout the engagement of teeth. 
o Smooth running of gears.
° Involve single curve 
° Cheaper manufacturing
° A little variation in centre distance does not effect the velocity ratio 
° Interference can occur if the condition of minimum number of teeth on a 
gear is not followed.
° Raidal flank teeth and weaker compare to cycloidal.
° Two convex surfaces are in contact and thus there is more wear.
• Methods of Gear Manufacturing
° Casting, blanking, machining.
° For power transmitting gears (made of steel), the following methods 
generally used are milling, rack generation, hobbing, fellow gear shaper 
method.
o For good accuracy and surface finish following finishing process are 
used grinding, lapping, sharing, barnishing.
The followings are the parameters used in manufacturing of gears as
• Path of Contact (CD)
According to the figure, the path of the contact is given by the relation
CD=CP+PD 
= (iCF-PF) + {DE-PE)
yjR ' — R : cos: o — i?sin o,4-
R
r * — r cos* o — r s m o ,
Page 5


Gears and Gear Trains 
Gears and Gear Trains
• Gears is a positive drive which transmits exact velocity ratio from one shaft to 
another shaft, gear drive is used when the distance between driver and 
follower is very less, when two or more gears are made to mesh with each 
other to transmit power from one shaft to another, the combination is called 
gear trains.
Gears
• Gears are used to transmit motion from one shaft to another or between a 
shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or 
connector and transmit the motion by direct contact.
Gear Terminology
• Circular Pitch (p): It is a distance measured along the circumference of the 
pitch circle from a point on one tooth to the corresponding point on the 
adjacent tooth.
P =
7 t d 
T
where, p = circular pitch 
d = pitch diameter 
T = number of teeth
Face width
• Diametrical Pitch (P)
It is the number of teeth per unit length of the pitch circle diameter in inches.
p - t -
d
• Module (m)
It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch.
d
m — —
T
=¦ p= -m
• Gear Ratio (G)
It is the ratio of the number of
where, T = number of teeth on 
t = number of teeth on the
• Velocity Ratio
The velocity ratio is defined as the ratio of the angular velocity of the follower to 
the angular velocity of the driver gear.
ra = £ i= L ± = Zl
o y j .Vj d i 7 * .
• Addendum Circle: A circle bounding the ends of the teeth, in a normal section 
of the gear.
• Dedendum Circle: The circle bounding the spaces between the teeth, in a 
normal section of the gear.
• Addendum: The radial distance between the pitch circle and the addendum 
circle.
• Dedendum: The radial distance between the pitch circle and the root circle,
Standard Proportions for 20° Full Depth System
The following standard proportions are used in 20° full depth system
teeth on the gear to that on the pinion.
r
the gear 
pinion
• Standard Proportions 
° Addendum ha = m 
° Clearance C = 0.25 m
o
Whole depth h = 2.25 m 
° Tooth space = 1.5700 m 
o Dedendum hf = 1.25 m 
o Working depth h^ = 2m 
o Tooth thickness (s) = 1.5708 m 
° Fillet radius = 0.4 m
• Form of Teeth: The form of teeth in a gear system can be as follows.
• Conjugate Teeth: Two curves of any arbitrary shape that fulfill the law of 
gearing can be used as the profiles of teeth. Such a gear The profiles of tooths 
which are generally used given as
• Cycloidal Profile Teeth: As cycloid is the locus of a point on the circumference 
of a circle that rolls without slipping on a fixed straight line In this type, the 
faces of the teeth are epicycloids and the flanks the hypocycloids.
• Epicycloid Teeth: An epicycloid is the locus of point on the circumference of a 
circle that rolls without slipping on a fixed straight line.
Arc PK = arc KJG
• Hypocycloid Teeth
A hypocycloid is the locus of a point on the circumference of a circle that rolls 
without slipping inside the circumference of another circle
Arc AC = Arc CD
• Involute Profile Teeth
An involute is defined as the locus of a point on a straight line which rolls without 
slipping on the circumference of a circle. From figure at the start the tracing point 
is at A as the rolls on the circumference of the circle, the path ABC traced out by 
the point A is the involute.
D is regarded as the instantaneous centre of rotation of 8. The motion of 8 is 
perpendicular to BD. Since, BD is tangent to the base circle, the normal to the 
involute is a tangent to the base circle.
Comparison between Cycloidal Teeth and Involute Teeth
• Cyclodial Teeth
° Pressure angle maximum at beginning of engagement, then reduce to 
zero at pitch point and again maximum at end of engagement, 
o Less smooth running of gears.
° Involve double curve, epicycloid and hypocycloid for the teeth 
° Costier manufacturing
o Exact centre distance is reuired to transmit a constant velocity ratio.
° Phenomenon of interference does not occur at all.
° Spreading flank teeth and strong.
° Convex flank always has contact with a concave face resulting in less 
wear.
• Involute Teeth
° Pressure angle is constant throughtout the engagement of teeth. 
o Smooth running of gears.
° Involve single curve 
° Cheaper manufacturing
° A little variation in centre distance does not effect the velocity ratio 
° Interference can occur if the condition of minimum number of teeth on a 
gear is not followed.
° Raidal flank teeth and weaker compare to cycloidal.
° Two convex surfaces are in contact and thus there is more wear.
• Methods of Gear Manufacturing
° Casting, blanking, machining.
° For power transmitting gears (made of steel), the following methods 
generally used are milling, rack generation, hobbing, fellow gear shaper 
method.
o For good accuracy and surface finish following finishing process are 
used grinding, lapping, sharing, barnishing.
The followings are the parameters used in manufacturing of gears as
• Path of Contact (CD)
According to the figure, the path of the contact is given by the relation
CD=CP+PD 
= (iCF-PF) + {DE-PE)
yjR ' — R : cos: o — i?sin o,4-
R
r * — r cos* o — r s m o ,
where, r = pitch circle radius of pinion 
R = pitch circle radius of wheel 
ra = addendum circle radius of pinion 
Ra = Addendum circle radius of wheel
• Arc of Contact
From the figure, the arc of the contact as given by
path of contact (CD)
Arc of contact = ¦
cos 6
A rc of a p p roa ch = 
A rc of recess =
C P 
cos 6
PD
cos < f >
Number of teeth within the arc
CD 1
cos < t > p
Angle of action 5P = arc of contact
360
x ----- -
2 pr
(p = circular pitch)
• Interference in Involute Gear
° Meshing of two non-conjugate (non-involute) teeth is known as 
interference because the two teeth do not slide properly and thus rough 
action and binding occurs.
° For equal addenda of the wheel and pinion, the addendum radius of the 
wheel decides whether the interference will occur or not.
• Minimum Number of Teeth
° Maximum value of the addendum radius of wheel to avoid interference 
can be upto BE
(BE)Z =(BF)2 + (FE)2
B E
r
L + 2
~ R R
— sin'o