2. COTENTS
• Classification of Gears,
• Spur Gear: Definition, Terminology,
• Fundamental Law of Toothed Gearing,
• Involute and Cycloidal Profile,
• Path of Contact, Arc of Contact,
• Conjugate Action, Contact Ratio,
• Minimum Number of Teeth,
• Interference and Under Cutting,
• Force Analysis and Friction in Gears.
3. GEARS
FUNCTIONS
• Reduce speed
• Increase torque
• Move power from one point to another
• Change direction of power
• Split power
CLASSIFICATION OF GEARS
1. Parallel Axis
i. Spur Gear ii. Helical Gear. iii. Rack iv. Internal Gear
2. Intersecting Axis
i. Bevel Gear ii. Spiral Bevel Gear iii. Miter Gear
3. Non Parallel & Non Intersecting
i. Worm ii. Worm Wheel iii. Screw Gear
14. SPUR GEAR
•Teeth are parallel to the axis of the
gear
•Advantages
Simple in construction
Easy to Manufacture
Highest Efficiency
Excellent Precision rating
Wide Velocity Range
Cost
•Disadvantages
Sudden Loading
High Impact Stresses
Excessive Noise at High Speed
16. Pitch Circle: The circles remain tangent throughout
entire engagement
• Circular Pitch: It is a distance between a point on a tooth at the
pitch circle to corresponding point on the next adjacent tooth
P=(p*D)/N
• Diametral Pitch, (Pd) – Number of teeth per inch of pitch diameter
• Addendum: The radial distance between the Pitch Circle and the top
of the teeth.
• Dedendum: The radial distance between the bottom of the tooth to
pitch circle.
• Arc of Action: Is the arc of the Pitch Circle between the beginning
and the end of the engagement of a given pair of teeth.
• Arc of Approach: Is the arc of the Pitch Circle between the first
point of contact of the gear teeth and the Pitch Point.
• Arc of Recession: That arc of the Pitch Circle between the Pitch
Point and the last point of contact of the gear teeth.
• Backlash: Play between mating teeth.
17. • Base Circle: The circle from which is generated the
involute curve upon which the tooth profile is based.
• Center Distance: The distance between centers of two gears.
•Circular Thickness: The thickness of the tooth measured along an arc
following the Pitch Circle
•Clearance: The distance between the top of a tooth and the bottom
of the space into which it fits on the meshing gear.
•Contact Ratio: The ratio of the length of the Arc of Action to the
Circular Pitch.
Face: The working surface of a gear tooth, located between the pitch
diameter and the top of the tooth.
Face Width: The width of the tooth measured parallel to the gear
axis.
Flank: The working surface of a gear tooth, located between the pitch
diameter and the bottom of the teeth
Gear: The larger of two meshed gears. If both gears are the same size,
they are both called "gears".
18. • Land: The top surface of the tooth.
Module: Millimeter of Pitch Diameter to Teeth.
•Line of Action: That line along which the point of contact between gear
teeth travels, between the first point of contact and the last.
•Pinion: The smaller of two meshed gears.
•Pitch Circle: The circle, the radius of which is equal to the distance from the
center of the gear to the pitch point.
•Diametral pitch: Teeth per millimeter of pitch diameter.
•Pitch Point: The point of tangency of the pitch circles of two meshing gears,
where the Line of Centers crosses the pitch circles.
•Pressure Angle: Angle between the Line of Action and a line perpendicular
to the Line of Centers.
•Root Circle: The circle that passes through the bottom of the tooth spaces.
•Working Depth: The depth to which a tooth extends into the space between
teeth on the mating gear.
20. FUNDAMENTAL LAW OF TOOTHED GEARING
Let,
ω1 = angular velocity of body 1
ω2 = angular velocity of body 2
Vc = linear velocity of point C
Vd = linear velocity of point D
∴ Vc = ω1 × AC and Vd = ω2× BD
Component of Vc along n-n = Vc x cos α
Component of Vd along n-n = Vd x cos β
Relative motion along n-n = Vc x cos α - Vd x cos β
< CAE = α and < DBF = β
25. Advantages of Cycloidal gear
Disadvantages of Cycloidal gear
1. Cycloidal gear do not have interference
2. Cycloidal tooth is generally stronger than an involute tooth.
3. Beacause of spreading flanks, they have high strength and
compact drives.
4.Cycloidal teeth have longer life since the contact is mostly
rolling which results in low wear
1. For a pair of cycloidal gears, there is only one theoretically
correct center distance for which a constant angular velocity
ration is possible
2. Manufacturing is difficult and hence costlier
26. APPLICATIONS
1. Cycloidal gear are extensively used in watches, clocks shown
in figure and in instruments where strength and interference
are prime considerations.
2. Cast bull gears of paper mill machinery
3. Crusher drives in sugar mill
28. ADVANTAGES OF INVOLUTE
1. Variation of Center distance does not affect the velocity ratio
2. Pressure angle remains constant throughout the engagement
which results in smooth running.
3. Straight teeth of basic rack for involute admit simple tools.
Hence, manufacturing becomes simple and cheap
30. Velocity of Sliding
• If the curved surfaces of two teeth of the gears 1 & 2
are to remain in contact, one can have a sliding
motion relative to the other along the common
tangent t-t at C or D.
Component of Vc along t-t = Vc x sin α
Component of Vd along t-t = Vd x sin β
31. Velocity of Sliding = Vc x sin α - Vd x sin β
= ω1 x AC (EC/AC) – ω2 x BD (FD / BD)
= ω1x EC – ω2 x FD
= ω1( EP + PC) – ω2 (FP – PD)
= ω1 EP+ ω1 PC – ω2 FP+ ω2 PC
= (ω1 + ω2) PC + ω1 EP – ω2 FP (ω1 EP = ω2 FP)
= (ω1 + ω2) PC
= Sum of angular velocities x distance between
the pitch point and the point of contact.
Velocity of Sliding
35. Path of Contact
Let,
The pinion 1 is driver and rotating clockwise.
The wheel 2 is driven and rotating CCW.
EF is their common tangent to the base circles.
Start of engagement at C
End of engagement at D
CD is the Path of Contact
36. Let, r = pitch circle radius of pinion
R = pitch circle radius of wheel
ra = addendum circle radius of pinion
Ra = addendum circle radius of wheel
Path of Contact = Path of approach + Path of recess
CD = CP + PD
CD = (CF – PF) + (DE – PE)
= [ √(Ra)2 - R2 cos2 φ - R sin φ ] + [√(ra)2 - r2 cos2 φ - r sin φ ]
= √(Ra)2 - R2 cos2 φ + √(ra)2 - r2 cos2 φ - (R + r) sin φ
Path of Contact
37. Arc of Contact
Arc of contact is the distance travelled by a point on either
pitch circle of the two wheels during the period of contact of
a pair of teeth.
38. Arc of contact, P′ P″ = Arc of approach P′P + Arc of recess PP″
Let the time to traverse the arc of approach is ta , then
Arc of approach, P′P = tangential velocity of P’ x Time of approach
= ω1 r × Time of approach (ta)
Arc FK is equal to the path FP as the point P is on the generator FP that rolls
on the base circle FHK to generate the involute PK.
Similarly, arc FH = Path FC.
Arc of Contact
39. Arc of recess, PP” = tangential velocity of P x Time of recess
= ω1 r × time of recess (tr)
Arc of Contact
40. Contact Ratio
The arc of contact is the length of
the pitch circle traversed by a point
on it during the mating of a pair of
teeth.
If n = 1.6,
one pair of teeth always in contact,
whereas two pair of teeth are in
contact for 60% of time.
41. Numerical – 1
• Two gears in mesh have a module of 8 mm and a pressure angle of 20o . The
larger gear has 57 while the pinion has 23 teeth. If the addenda on pinion and
gear wheels are equal to one module, find –
(i) the number of pairs of teeth in contact
(ii) the angle of action of pinion and the gear wheel
(iii) the ratio of the sliding to rolling velocity at
(a) the beginning of contact
(b) the pitch point
(c) the end of contact
42. • Given data: φ = 20o, T = 57, m = 8 mm,
t = 23, addendum = 1
R = = = 228 mm
Ra= R + m = 228 + 8 = 236 mm
r = = = 92 mm
ra = r + m = 92 + 8 = 100 mm
Numerical – 1
2
mT
2
578x
2
mt
2
238x
46. • Two 20o gears have a module pitch of 4 mm. The number of
teeth on gears 1 and 2 are 40 and 24 respectively. If the gear 2
rotates at 600 rpm, determine the velocity of sliding when the
contact is at tip of the tooth of gear 2. Take addendum equal to
one module.
Also, maximum velocity of sliding.
• Given Data :
φ = 20o, T = 40, NP = 600 mm,
t = 24, m = 4 mm,
Addendum = 1m = 4 mm
Numerical – 2
47. • Let 1 be the gear wheel and 2 the pinion.
R = = = 80 mm
Ra= R + m = 80 + 4 = 84 mm
r = = = 48 mm
ra = r + m = 48 + 4 = 52 mm
Ng = NP x = 360 rpm
Numerical – 2
2
mT
2
404x
2
mt
2
244x
T
t
48. • Let pinion (gear 2) be the driver.
The tip of the driving wheel is in contact with a tooth of the driven wheel at the end of
engagement.
Path of recess
= 9.458 mm
Velocity of sliding = (ωp + ωg ) x Path of recess
= 2 ∏ ( NP + Ng) x 9.458
= 2 ∏ (600 + 360) x 9.458
= 57049 mm / min = 950.8 mm/s
Numerical – 2
49. • In case the gear wheel is the driver, the tip of the pinion will be in contact with
the flank of a tooth of the gear wheel at the beginning of contact.
Path of approach
= 9.458 mm
Velocity of sliding = 950.8 mm/s
Thus, it is immaterial whether the driver is the gear wheel or the pinion, the
velocity of sliding is the same when the contact is at the tip of the pinion.
Numerical – 2
50. • Maximum velocity of sliding will depend upon the larger path considering
any of the wheels to the driver.
• Consider pinion to be the driver
Path of recess = 9.458 mm
Path of approach
= 10.117 mm
This is also the path of recess if the wheel becomes the driver.
Maximum velocity of sliding = (ωp + ωg ) x Maximum Path
= 61024 mm / min = 1017.1 mm / s
Numerical – 2
51. • Calculate (i) length of path of contact (ii) Arc
of Contact & (iii) The contact ratio when a
pinion having 23 teeth drives a gear having
teeth 57. The profile of the gears is involute
with pressure angle 20 o , module 8 mm and
addendum is one modume.
Numerical – 3
52. • Path of Contact = 39.78 mm
• Arc of Contact = 42.33 mm
• Contact Ratio = 1.68 = 2 (say)
Numerical – 3
53. • Power transmission through a pair of teeth is along the line of action or the normal
to the two involutes at the point of contact.
• The common normal is also a common tangent to the two base circles and passes
through the picth point.
• At any instant, the portion of tooth profiles which are in contact must be involute so
that the line of action does not deviate.
• If any of the two surfaces is not involute, the two surfaces would not touch each
other tangentially and the transmission of power would not be proper.
• Mating of two non conjugate (Non-involute) teeth is known as interference.
Interference in Involute Gears
54. • Thus, for equal addenda of the wheel and pinion, the addendum radius of
the wheel decides whether the interference will occur or not.
Interference in Involute Gears
55. • The maximum value of the addendum radius of the wheel to avoid interference can be upto BE.
(BE)2 = (BF)2 + (FE)2
(BE)2 = (BF)2 + (FP + PE)2
(BE)2 = (R cos φ)2 + (R sin φ + r sin φ )2
= R2 cos2 φ + R sin2 φ + r2 sin2 φ + 2rR sin2 φ
= R2 (cos2 φ + sin2 φ) + sin2 φ (r2 + 2rR)
= R2 + sin2 φ (r2 + 2rR)
= R2 [ 1 + 1/R2 (r2 + 2rR) sin2 φ ]
= R2 [ 1 + (r2/ R2 + 2r/R) sin2 φ ]
BE =
Minimum Number of Teeth on Wheel
]sin22)(r/RRr/1 R
56. • Therefore, maximum value of the addendum of the wheel can be –
= BE – Pitch circle radius
aw max = - R
aw max = R [ - 1 ]
Let the adopted value of the addendum in some case be aw times
the module of teeth.
Minimum Number of Teeth on Wheel
Φsin2)(r/RRr/1R 2
Φsin2)(r/RRr/1 2
57. • Let t = No. of teeth on pinion, T = No. of teeth on wheel
m = D / T = d / t
m = 2R / T = 2r / t
R = m T / 2 and r = m t / 2
Where G = T / t = Gear ratio, T =
Minimum Number of Teeth on Wheel
58. • For equal No. of teeth on the pinion and the wheel, G = 1 and
T min =
For a pressure angle of 20o, i.e. φ = 20o
T min =
= 12.31 or 13
Minimum Number of Teeth on Wheel
59. • The maximum value of the addendum radius of the pinion to avoid interference can be upto AF.
(AF)2 = (AE)2 + (FE)2
(AF)2 = (AE)2 + (FP + PE)2
(AF)2 = (r cos φ)2 + (R sin φ + r sin φ )2
= r2 cos2 φ + R2 sin2 φ + r2 sin2 φ + 2rR sin2 φ
= r2 (cos2 φ + sin2 φ) + sin2 φ (R2 + 2rR)
= r2 + sin2 φ (R2 + 2rR)
= r2 [ 1 + 1/r2 (R2 + 2rR) sin2 φ ]
= r2 [ 1 + (R2/ r2 + 2R/r) sin2 φ ]
BE =
Minimum Number of Teeth on Pinion
]2sin2)(R/rrR/1 r
60. • Therefore, maximum value of the addendum of the pinion can be –
= AF – Pitch circle radius
ap max = - r
ap max = r [ - 1 ]
Let the adopted value of the addendum in some case be ap times
the module of teeth.
ap max =
Minimum Number of Teeth on pinion
Φsin22)(R/rrR/1r
Φsin22)(R/rrR/1
1Φ2sin)2(1
2
GG
mt
61. • P is the pitch point and PE is the line of action.
• Engagement of the rack tooth with the pinion
tooth occurs at C.
• To avoid interference, the max addendum of the
rack can be increased in such a way that C
coincides with E.
• Thus, addendum of the rack must be less than
GE.
Interference Between Rack and Pinion
62. • Let the adopted value of the addendum of
the rack can be ar m.
where ar is the addendum coefficient by
which the standard value of the addendum
has been multiplied.
In Δ PGE, GE = PE sin φ
GE = (r sin φ) sin φ
GE = r sin2 φ
GE = sin2 φ
Interference Between Rack and Pinion
2
mt
63. • To avoid interference,
GE ≥ ar m
sin2 φ ≥ ar m
t ≥
where ar = 1, i.e. For standard addendum
t min = 2 / sin2 φ
For 20o pressure angle, i.e. φ = 20o
t min = 17.1 = 18
Interference Between Rack and Pinion
2
mt
2sin
2ar
64. • Path of contact = CP + DP
= +
Interference Between Rack and Pinion
cos
.ofrackadd
66. Force analysis of Spur Gear
1. Tangential force (Ft):
• It transmits torque or power
• It is tangent to pitch circle at pitch point.
𝐹𝑡 =
𝑃
𝑉
=
2𝜋𝑁𝑝𝑇𝑝
60
𝑥
60
𝜋𝐷𝑝𝑁𝑝
𝐹𝑡 =
2𝑇𝑝
𝐷𝑝
2. Radial force (Fr):
• Radial force always tends to separate the two gears.
• It acts along the radial line through the pitch point and directed
towards center.
• Fr = Ft x tan ∅
3. Resultant Force (F) : F = 𝐹𝑡 2 + 𝐹𝑟 2
68. • Let, Ft = Tangential force transmitted by meeting gear in N
FN = Normal force acting at point of contact in N
=
𝐹𝑡
𝐶𝑜𝑠 ∅
∅ = pressure angle
ω 1 = angular velocity of pinion in Rad/second
ω 2 = angular velocity of gear in Rad/second
Vs = velocity of sliding
μ = coefficient of friction between the material of mating gear
Friction between Gear Teeth
69. • If the pinion is driver and gear is driven then,
In ∆ O2AP and ∆ O2BS,
O2A = O2B
O2P = O2S
< O2AP = < O2BS = 90o
∴ AP = BS
AR = arc AE and BS = arc BE
Arc AE – Arc BE = AR – BS
Arc AB = AR – BS
Arc AB = RP = Path of recess
Friction between Gear Teeth
70. • ∴ Velocity of sliding = (ω 1 + ω 2) x (Dist. Of pitch point from point of contact)
∴ Max. velocity of sliding = (ω 1 + ω 2) x RP
Min. velocity of sliding = (ω 1 + ω 2) x 0
Avg. velocity of sliding = (ω 1 + ω 2) x
𝑅𝑃
2
= (ω 1 + ω 2) x
𝐴𝑟𝑐 AB
2
=
(ω 1 + ω 2) 𝑥 𝑂2
𝐴 𝑥 𝜃2
2
=
(ω 1 + ω 2) 𝑥 𝑟2
𝑐𝑜𝑠∅ 𝑥 𝜃2
2
Friction between Gear Teeth
71. • Frictional workdone during recess
= μ FN x Average velocity x time
(W.D)R = μ
𝐹𝑡
cos ∅
x
(ω 1 + ω 2) 𝑥 𝑟2
𝑐𝑜𝑠∅ 𝑥 𝜃2
2
x
𝜃2
𝜔2
=
1
2
x (1 +
𝜔1
𝜔2
) x 𝑟2 x (𝜃2)2 x μ 𝐹𝑡
It may be noted that it is independent of ∅
∴Frictional W. D. during approach
(W.D)A =
1
2
x (1 +
𝜔2
𝜔1
) x 𝑟1 x (𝜃1)2 x μ 𝐹𝑡
Friction between Gear Teeth
72. ∴ The total W. D. against friction for complete period of action is,
W.D. = (W.D.)A + (W.D.)R
W.D. =
1
2
x (1 +
𝜔2
𝜔1
) x 𝑟1 x (𝜃1)2 x μ 𝐹𝑡 +
1
2
x (1 +
𝜔1
𝜔2
) x 𝑟2 x (𝜃2)2 x μ 𝐹𝑡
=
μ 𝐹𝑡
2
[(1 +
𝜔2
𝜔1
) x 𝑟1 x (𝜃1)2 + (1 +
𝜔1
𝜔2
) x 𝑟2 x (𝜃2)2]
=
μ 𝐹𝑡
2
[(1 +
𝑟1
𝑟2
) x 𝑟1 x (𝜃1)2 + (1 +
𝑟2
𝑟1
) x 𝑟2 x (𝜃2)2]
=
μ 𝐹𝑡
2
[(
𝑟1+𝑟2
𝑟2
) x 𝑟1 x (𝜃1)2 + (
𝑟1+𝑟2
𝑟1
) x 𝑟2 x (𝜃2)2]
=
μ 𝐹𝑡
2
[(
𝑟1+𝑟2
𝑟1 𝑟2
) x 𝑟1
2 x (𝜃1)2 + (
𝑟1+𝑟2
𝑟1
) x 𝑟2
2 x (𝜃2)2]
=
μ 𝐹𝑡
2
x (
𝑟1+𝑟2
𝑟1 𝑟2
) x [ 𝑟1
2 x (𝜃1)2 + 𝑟2
2 x (𝜃2)2]
Friction between Gear Teeth
73. • We know that, 𝑟1 x 𝜃1 = l1 = Path of recess
𝑟2 x 𝜃2 = l2 = Path of approach
l = total path of contact
l = l1 + l2
K =
μ 𝐹𝑡
2
x (
𝑟1+𝑟2
𝑟1 𝑟2
) = constant
W.D. = K [ l1
2 + l2
2]
W.D. = K [ l1
2 + (l-l1)2]
For max. W.D. & min frictional work difference the above equation with r. t. l1 &
equate to zero.
𝑑
𝑑𝑙1
[K [ l1
2 + (l-l1)2] = 0
2l1 = l
It means, for total arc of action the work loss in friction will be minimum when the
length of path of recess is half of the total path of contact.
Friction between Gear Teeth
