spur gear

1. GEAR

2. The slipping of a belt or a rope is a common phenomenon
in the transmission of motion or power between two shafts.
The effect of slip is to reduce the velocity ratio of the
Drive / System.
In precision machine, in which a definite velocity ratio is
important (as in watch mechanism) the only positive drive
is by means of Gears or toothed wheels.
Introduction

3. Transmission of motion or power between two shafts.
 Belt/Rope Drives - Large center distance of the shafts.
 Chain Drives - Medium center distance of the shafts.
 Gear Drives - Small center distance of the shafts.

4. GEAR
Introduction- Gear can be defined as the mechanical
element used for transmitting power and rotary motion from
one shaft to another by means of progressive engagement of
projections called teeth.
Smaller of the pair is called PINION and the larger of the
pair is called the GEAR.

5. ADVANTAGES OF GEAR DRIVES
 Compact as compared to belt and chain drives.
 Positive drives.
 wide range of speed ratios.(6:1 to 4900:1)
 High speed ratio than belt drives
 Used for shafts parallel, intersecting, non-parallel, non-
intersecting.
 Large power transmission
 Transmits power at higher speed.

6. LIMITATIONS OF GEAR DRIVES
 Costly.
 Can’t be used over a long distance.
 Requires precise alignment of shafts.
 Requires continuous lubrication.
 The manufacture of gears require special tools and
equipment.
 The error in cutting teeth may cause vibrations and noise
during operation

7. Classification of Toothed Wheels / Gears
1. According to the position of axes of the shafts.
The axes of the two shafts between which the motion is to be transmitted, may be
(a) Parallel, (b) Intersecting, and (c) Non-intersecting and non-parallel.
a). Parallel
• The two parallel and co-planar shafts connected by the gears is these gears are called spur
gears and the arrangement is known as spur gearing.
• These gears have teeth parallel to the axis of the wheel
• helical gearing, in which the teeth are inclined to the axis. The single and double helical
gears connecting parallel shafts are shown in Fig. (a) and (b) respectively
• The double helical gears are known as herringbone gears

8. b) Intersecting Axes
•The two non-parallel or intersecting, but coplanar shafts connected by gears are called bevel
gears and the arrangement is known as bevel gearing.
•The bevel gears, like helical gears, may also have their teeth inclined to the face of the bevel,
in which case they are known as helical bevel gears.
c) The two non-intersecting and non-parallel
•The two non-intersecting and non-parallel i.e. non-coplanar shafts are connected by gears
and are called skew bevel gears or spiral gears or Skewed Gears
•The arrangement is known as skew bevel gearing or spiral gearing.

9. 2. According to the peripheral velocity of the gears.
The gears, according to the peripheral velocity of the gears, may be
classified as :
(a)Low velocity: velocity less than 3 m/s
(b)Medium velocity :velocity between 3 and 15 m / s
(c) High velocity : velocity of gears is more than 15 m / s,

10. Classification of Toothed Wheels / Gears
3. According to the type of gearing.
The gears, according to the type of gearing, may be classified as :
(a) External gearing, (b) Internal gearing, and (c) Rack and pinion.

11. 4. According to the position of teeth on the gear surface.
The teeth on the gear surface may be
(a) Straight, (b) Inclined, and (c) Curved.
We have discussed earlier that the spur gears have straight teeth
whereas helical gears have their teeth inclined to the wheel rim. In
case of spiral gears, the teeth are curved over the rim surface

12. CLASSIFICATION OF GEARS.
1- Parallel axis Gears.- Spur Gears.
Helical Gears.
Herringbone Gears.
Rack and pinion.
Internal gears.
2- intersecting axis Gears- Bevel Gears.
3- Non intersecting and Worm Gears
perpendicular axis Gears
4- Non intersecting and Crossed
Non parallel axis Gears- Helical Gears

13. Parallel axis Gears SPUR GEAR

14. SPUR GEAR

15. Parallel axis Gears Helical gear
Parallel Helical gears Heringbone gears
(Double Helical gears)

16. Parallel axis Gears Rack and pinion

17. Intersecting axis Gears- Bevel gear

18. Non-intersecting and
perpendicular axis Gears-worm

19. Non-intersecting, Non-parallel-
CROSSED HELICAL GEAR

21. TERMINOLOGY IN SPUR GEAR

22. Terms used in Gears
1. Pitch circle. It is an imaginary circle which by pure rolling action, would give
the same motion as the actual gear.
2. Pitch circle diameter. It is the diameter of the pitch circle. The size of the gear is
usually specified by the pitch circle diameter. It is also called as pitch diameter.
3. Pitch point. It is a common point of contact between two pitch circles.
4. Pitch surface. It is the surface of the rolling discs which the meshing gears have
replaced at the pitch circle.
5. Pressure angle or angle of obliquity. It is the angle between the common normal
to two gear teeth at the point of contact and the common tangent at the pitch point.
It is usually denoted by φ. The standard pressure angles are 14 / 1 2° and 20°.
6. Addendum. It is the radial distance of a tooth from the pitch circle to the top of the
tooth.
7. Dedendum. It is the radial distance of a tooth from the pitch circle to the bottom of

23. 8. Addendum circle. It is the circle drawn through the top of the teeth and is
concentric with the pitch circle.
9. Dedendum circle. It is the circle drawn through the bottom of the teeth. It is also
called root circle.
Note : Root circle diameter = Pitch circle diameter × cos φ, where φ is the pressure
angle.
10. Circular pitch. It is the distance measured on the circumference of the pitch circle
from a point of one tooth to the corresponding point on the next tooth. It is usually
denoted by pc. Mathematically,
Circular pitch, pc = π D/T
where
D = Diameter of the pitch circle, and T = Number of teeth on the wheel.
A little consideration will show that the two gears will mesh together correctly, if the
two wheels have the same circular pitch.
Note : If D 1 and D2 are the diameters of the two meshing gears having the teeth T1
and T2 respectively; then for them to mesh correctly,

25. 13. Clearance. It is the radial distance from the top of the tooth to the bottom of the
tooth, in a meshing gear. A circle passing through the top of the meshing gear is
known as clearance circle.
14. Total depth. It is the radial distance between the addendum and the dedendum
circle of a gear. It is equal to the sum of the addendum and dedendum.
15. Working depth. It is radial distance from the addendum circle to the clearance
circle. It is equal to the sum of the addendum of the two meshing gears.
16. Tooth thickness. It is the width of the tooth measured along the pitch circle.
17. Tooth space. It is the width of space between the two adjacent teeth measured
along the pitch circle.
18. Backlash. It is the difference between the tooth space and the tooth thickness, as
measured on the pitch circle.
19. Face of the tooth. It is surface of the tooth above the pitch surface.
20. Top land. It is the surface of the top of the tooth.

26. 21. Flank of the tooth. It is the surface of the tooth below the pitch surface.
22. Face width. It is the width of the gear tooth measured parallel to its axis.
23. Profile. It is the curve formed by the face and flank of the tooth.
24. Fillet radius. It is the radius that connects the root circle to the profile of the tooth.
25. Path of contact. It is the path traced by the point of contact of two teeth from the
beginning to the end of engagement.
26. Length of the path of contact. It is the length of the common normal cut-off by the
addendum circles of the wheel and pinion.
27. Arc of contact. It is the path traced by a point on the pitch circle from the beginning to
the end of engagement of a given pair of teeth. The arc of contact consists of two parts,
i.e.
(a) Arc of approach. It is the portion of the path of contact from the beginning of the
engagement to the pitch point.
(b) Arc of recess. It is the portion of the path of contact from the pitch point to the end of
the engagement of a pair of teeth.
Note : The ratio of the length of arc of contact to the circular pitch is known as contact
ratio i.e. number of pairs of teeth in contact.

27. Condition for Constant Velocity Ratio of Gears–Law of Gearing
• Consider the portions of the two teeth, one on the wheel 1 (or
pinion) and the other on the wheel 2, as shown by thick line curves
in Fig. 28.7.
• Let the two teeth come in contact at point Q, and the wheels rotate
in the directions as shown in the figure.
• Let T T be the common tangent and MN be the common normal to
the curves at point of contact Q.
•

28. • From the centers O1 and O2, draw O1M and O2N perpendicular to MN.
• A little consideration will show that the point Q moves in the direction QC, when
considered as a point on wheel 1, and in the direction QD when considered as a
point on wheel 2.
• Let v 1 and v2 be the velocities of the point Q on the wheels 1 and 2 respectively.
If the teeth are to remain in contact, then the components of these velocities along
the common normal MN must be equal.

29. • We see that the angular velocity ratio is inversely proportional to the ratio of
the distance of P from the centers O 1 and O2, or the common normal to the
two surfaces at the point of contact Q intersects the line of centers at point P
which divides the center distance inversely as the ratio of angular
velocities.

30. • Therefore, in order to have a constant angular velocity ratio for all positions of the wheels, P
must be the fixed point (called pitch point) for the two wheels.
• In other words, the common normal at the point of contact between a pair of teeth must
always pass through the pitch point.
• This is fundamental condition which must be satisfied while designing the profiles for the
teeth of gear wheels. It is also known as law of gearing
Notes : 1. The above condition is fulfilled by teeth of involute form, provided that the
root circles from which the profiles are generated are tangential to the common normal.
2. If the shape of one tooth profile is arbitrary chosen and another tooth is designed to
satisfy the above condition, then the second tooth is said to be conjugate to the first.
The conjugate teeth are not in common use because of difficulty in manufacture and
cost of production.
3. If D 1 and D2 are pitch circle diameters of wheel 1 and 2 having teeth T1 and T2
respectively, then velocity ratio,

31. Forms of Teeth
We have discussed in (Note 2) that conjugate teeth are not in common use.
Therefore, in actual practice, following are the two types of teeth commonly
used.
1.Cycloidal teeth ; and 2. Involute teeth.
1. Cycloidal Teeth
A cycloid is the curve traced by a point on the circumference of a circle
which rolls without slipping on a fixed straight line.

32. Locus of a point on a straight line which rolls without slipping, on the
circumference of the circle. Example: An involute profile is generated by the
end of the string/tape being unwound from a cylinder or by a point on a line as
the line rolls on the circumference of a circle without slipping.
2. Involute Teeth
Tooth form
Involute gears in Action

33. Comparison Between Involute and Cycloidal Gears

34. STANDARD SYSTEMS OF GEAR TOOTH
• A tooth system is a standard which specifies the pressure angle
and the relations for addendum, dedendum, working depth,
tooth thickness and a fillet radius in terms of module.
Objective- To attain interchangeability of gears of all tooth
number but of the same pressure angle and module.
•1) 14.5° composite system.
•2) 14.5° full depth involute.
•3) 20° full depth involute.
•4) 22.5° full depth involute.
•5) 20° stub tooth involute.

35. GEAR MATERIAL
Desirable properties of gear material-
• Sufficient endurance strength in bending.
• Sufficient surface endurance strength.
• Low coefficient of friction.
• Low and consistent thermal distortion.

36. TYPES OF GEAR MATERIALS
• 1) Cast Iron.- Large size and complicated shapes
Advantages of cast irons-
•Cheap
•Good damping property
•Good surface endurance strength but low bending
endurance strength.
•Graphite presents in C.I. works as lubricant.
•Easily cast in complicated shapes
•Good machinability
Disadvantages of C.I.-
•Brittle, low bending strength.

37. TYPES OF GEAR MATERIALS
• 2) Steel-
•High bending endurance strength.
•High surface endurance strength.
•Ductile
•Expensive
•Poor damping property.
•Sensitive to thermal distortion during heat treatment.
•Steel pinion and C.I. gear.

38. TYPES OF GEAR MATERIALS
• 3) Non- Ferrous metals- copper, Zink, tin, aluminum,
bronze- worm gear.
4) Sintered Metals- low cost gears
• 5) Non-metals- Bakelite, Nylon
•Cheap, easy to manufacture,
•Light weight, tolerate errors in tooth profile.
•Do not have sufficient strength.

39. Design Considerations for a Gear Drive
In the design of a gear drive, the following data is usually given :
• 1. The power to be transmitted.
• 2. The speed of the driving gear,
• 3. The speed of the driven gear or the velocity ratio, and
• 4. The center distance.
The following requirements must be met in the design of a gear drive :
• (a) The gear teeth should have sufficient strength so that they will not fail
under static loading or dynamic loading during normal running conditions.
• (b) The gear teeth should have wear characteristics so that their life is
satisfactory.
• (c) The use of space and material should be economical.
• (d) The alignment of the gears and deflections of the shafts must be
considered because they effect on the performance of the gears.
• (e) The lubrication of the gears must be satisfactory

40. Force analysis of spur gears

41. Force analysis of spur gears
1) Tangential force Ft-Responsible for power transmission.
• Ft=Tangential force.
• P= Power transmitted, W
• V= Pitch line velocity,m/s
• Tp= torque acting on pinion.
• Tg= torque acting on Gear.
• dp= P.C.D. of pinion.
• dg= P.C.D. of Gear.
• np= pinion speed, rpm.
• ng= Gear speed, rpm.
• Tangential force acting on the driving gear opposes the rotation of the
driving gear whereas assists the rotation of driven gear.
2
/
2
/
V
P
Ft
g
g
p
p
d
T
d
T



1000
60
1000
60 


g
g
p
p n
d
n
d 


42. Force analysis of spur gears
2) Radial Force- (Fr) Not useful component.
• Tends to separate two gears.
• Acts along the radial line through the pitch point and towards the
centre.
3)Resultant force (F)=
tanφ
Ft
Fr 

cosφ
Ft
F
F
F
2
r
2
t 



43. GEAR TOOTH FAILURE
• Gear tooth failure
.
.
Bending failure or
breaking of tooth
Wear failure or surface
destruction
Pitting Scoring Corrosive
wear
Abrasive
Wear
Destructive
Pitting
Initial
Pitting

44. TYPES OF GEAR TOOTH FAILURE
1) Bending failure or tooth breakage.-
• Gear tooth is like a cantilever and subjected to repetitive bending
stresses.
• Tooth breakage happens when total load exceeds beam strength.
• Avoided by adjusting module, face width, providing fillet radius.
2) Wear failure-
Removes complete surface layer.
• i) Pitting- Fatigue failure due to repetitive contact stresses.
• A) Initial pitting- Because of high spots, surface irregularities,
errors in tooth profile. It is a temporary phenomena.
• Can be minimized by improving surface finish and proper
alignment of gears.

45. B) Destructive pitting- Due to repetitive contact stresses.
• Depends upon Hertz contact stress, surface endurance
strength, number of cycles.
• Can be avoided by improving surface endurance
strength.(Hardness).
• Spalling-It is another surface fatigue failure in which the
cracks originates below the gear tooth surface due to sub
surface stresses.
• Spalling can be avoided by providing sufficient case depth in
surface hardening.

46. TYPES OF GEAR TOOH FAILURE
ii Scoring- (scuffing or galling)
Metal to metal contact due to lubrication failure. Excessive frictional heat and
overheating of teeth. Welding and Tearing action. (stick-slip phenomena)
iii Abrasive Wear-
Damage caused by particles like dirt, dust in lubricant. Surface injury caused by
particles trapped between mating teeth. Can be avoided by proper filtration of
lubricant, use high viscosity oils, increasing surface hardness.
iv Corrosive wear-
Due to chemical action by improper lubricant, surrounding atmosphere.

48. BEAM STRENGTH OF SPUR GEAR TOOTH
Beam strength of the spur gear tooth is the maximum tangential load
the gear tooth can take without tooth failure.
Lewis Equation for Beam strength of spur Gear tooth
Each tooth is considered as a cantilever beam fixed at the base.
Radial force Fr- Induces a direct compressive stress of small
magnitude and can be neglected.
Tangential Force Ft- Induces a bending stress which tends to break
the tooth.

49. M = Max. bending moment at section BC = WT. h
WT = tangential load acting on gear tooth
h= Length of tooth
y = Half the thickness of the tooth
t = Thickness of tooth section at BC
b = Face width of the gear tooth
I = Moment of inertia about the center line of the tooth = b.t3 /12
Max. bending stress at pt. B

50. Max. bending stress at section BC
From Equation tangential force acting on gear tooth
The variables ‘t’ & ‘h’ depends upon the size of the gear
tooth and its profile.
Let t = x pc and h= kpc
pc = Circular pitch
x , k= Constants
σw =
M • y
I
=
(Ft × h) (t / 2)
bt3/12
=
Ft(6 ℎ)
bt2
WT =
σt. b. t2
6ℎ

52. LEWIS FORM FACTOR
The Lewis form factor y is the function of tooth shape and the
point of application of load. Lewis form factor depends upon
1) Number of teeth.
2) Tooth system.
3) Point of application of load.

53. Permissible Working Stress for Gear Teeth in the Lewis
Equation
• The permissible working stress (σw) in the Lewis equation depends
upon the material for which an allowable static stress (σo) may be
determined.
• The allowable static stress is the stress at the elastic limit of the
material. It is also called the basic stress.
• In order to account for the dynamic effects which become more
severe as the pitch line velocity increases, the value of σ w is
reduced.
• According to the Barth formula, the permissible working stress,

55. Note : The allowable static stress (σ o) for steel gears is approximately one-
third of the ultimate tensile strength (σ u) i.e. σ o = σ u / 3.

56. Dynamic Load on Gear Tooth
The tooth errors combined with the mass of the pinion and gear
result in inertia forces.
Additional forces arising out of the inertia effects are known as
dynamic loads.
Dynamic loads arises due to
• Inaccuracies in tooth profile.
• Errors in tooth profile.
• Run out of the gear.
• Inertia of the rotating masses.
• Deflection of the teeth.
• Stiffness of the rotating parts.
Depends upon tooth error and pitch line velocity.

60. Static Tooth Load

62. WEAR STRENGTH OF SPUR GEAR TOOTH
Wear Tooth Load
• The maximum load that gear teeth can carry, without premature wear,
depends upon the radii of curvature of the tooth profiles and on the elasticity
and surface fatigue limits of the materials.
• The maximum or the limiting load for satisfactory wear of gear teeth, is
obtained by using the following
Buckinghams equation for wear strength of a gear tooth-

63. WEAR STRENGTH OF SPUR GEAR TOOTH
• The load stress factor depends upon the maximum fatigue limit of
compressive stress, the pressure angle and the modulus of
elasticity of the materials of the gears.
• According to Buckingham, the load stress factor is given by the
following relation :

64. Notes : 1. The surface endurance limit for steel may be obtained from the
following equation :
σ es= (2.8 × B.H.N. – 70) N/mm2
2. The maximum limiting wear load (Ww) must be greater than the dynamic
load (WD).

65. Design Procedure for Spur Gears
In order to design spur gears, the following procedure may be followed :
1. First of all, the design tangential tooth load is obtained from the power
transmitted and the pitch line velocity by using the following relation :