Formula Sheet: Gears | Theory of Machines (TOM) - Mechanical Engineering PDF Download

 Page 1


Gears
1. Gear Terminology
• Pitch circle diameter (d): Diameter of the pitch circle.
• Number of teeth (N)
• Module (m): Ratio of pitch diameter to number of teeth,
m =
d
N
• Circular pitch (p): Distance between corresponding points of adjacent teeth on pitch
circle,
p =
pd
N
=pm
• Pressure angle (? ): Angle between the line of action and the tangent to the pitch
circle (usually 20
? )
• Addendum (a): Radial distance from pitch circle to tooth tip,
a =m
• Dedendum (b): Radial distance from pitch circle to root,
b = 1.25m
• Whole depth (h): Sum of addendum and dedendum,
h =a+b = 2.25m
2. Gear Ratio and Velocity Ratio
Gear ratio (i) =
N
2
N
1
=
d
2
d
1
=
?
1
?
2
where,
• N
1
,N
2
= Number of teeth of driver and driven gears
• d
1
,d
2
= Pitch diameters
• ?
1
,?
2
= Angular velocities
1
Page 2


Gears
1. Gear Terminology
• Pitch circle diameter (d): Diameter of the pitch circle.
• Number of teeth (N)
• Module (m): Ratio of pitch diameter to number of teeth,
m =
d
N
• Circular pitch (p): Distance between corresponding points of adjacent teeth on pitch
circle,
p =
pd
N
=pm
• Pressure angle (? ): Angle between the line of action and the tangent to the pitch
circle (usually 20
? )
• Addendum (a): Radial distance from pitch circle to tooth tip,
a =m
• Dedendum (b): Radial distance from pitch circle to root,
b = 1.25m
• Whole depth (h): Sum of addendum and dedendum,
h =a+b = 2.25m
2. Gear Ratio and Velocity Ratio
Gear ratio (i) =
N
2
N
1
=
d
2
d
1
=
?
1
?
2
where,
• N
1
,N
2
= Number of teeth of driver and driven gears
• d
1
,d
2
= Pitch diameters
• ?
1
,?
2
= Angular velocities
1
3. Velocity of Sliding (v
s
)
v
s
=?rtan? where,
• ? = Angular velocity
• r = Radius of pitch circle
• ? = Pressure angle
4. Law of Gearing
The angular velocity ratio of two gears in mesh remains constant if the common normal at
the point of contact passes through the pitch point.
5. Contact Ratio (? )
Average number of teeth in contact,
? =
Length of path of contact
Base pitch
=
b
p
b
where,
p
b
=pcos? and b = length of path of contact.
6. Beam Strength of Gear Tooth
s =
F
t
P
d
bY
where,
• s = Bending stress at tooth root
• F
t
= Tangential force at pitch circle
• P
d
= Diametral pitch =
N
d
• b = Face width
• Y = Lewis form factor (depends on tooth shape and number)
2
Page 3


Gears
1. Gear Terminology
• Pitch circle diameter (d): Diameter of the pitch circle.
• Number of teeth (N)
• Module (m): Ratio of pitch diameter to number of teeth,
m =
d
N
• Circular pitch (p): Distance between corresponding points of adjacent teeth on pitch
circle,
p =
pd
N
=pm
• Pressure angle (? ): Angle between the line of action and the tangent to the pitch
circle (usually 20
? )
• Addendum (a): Radial distance from pitch circle to tooth tip,
a =m
• Dedendum (b): Radial distance from pitch circle to root,
b = 1.25m
• Whole depth (h): Sum of addendum and dedendum,
h =a+b = 2.25m
2. Gear Ratio and Velocity Ratio
Gear ratio (i) =
N
2
N
1
=
d
2
d
1
=
?
1
?
2
where,
• N
1
,N
2
= Number of teeth of driver and driven gears
• d
1
,d
2
= Pitch diameters
• ?
1
,?
2
= Angular velocities
1
3. Velocity of Sliding (v
s
)
v
s
=?rtan? where,
• ? = Angular velocity
• r = Radius of pitch circle
• ? = Pressure angle
4. Law of Gearing
The angular velocity ratio of two gears in mesh remains constant if the common normal at
the point of contact passes through the pitch point.
5. Contact Ratio (? )
Average number of teeth in contact,
? =
Length of path of contact
Base pitch
=
b
p
b
where,
p
b
=pcos? and b = length of path of contact.
6. Beam Strength of Gear Tooth
s =
F
t
P
d
bY
where,
• s = Bending stress at tooth root
• F
t
= Tangential force at pitch circle
• P
d
= Diametral pitch =
N
d
• b = Face width
• Y = Lewis form factor (depends on tooth shape and number)
2
7. Power Transmitted by Gear
P =
2pNT
60
where,
• P = Power (W)
• N = Rotational speed (rpm)
• T = Torque (Nm)
8. Efficiency of Gear Drive
? =
Output Power
Input Power
=
T
2
?
2
T
1
?
1
Typically, gear efficiency ranges from 94% to 98%.
3