Dr.T.V.Govindaraju,SSEC

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ME44 KINEMATICS OF MACHINES
Dr.T.V.GovindarajuPrincipal, Shirdi Sai
Engineering College Anekal, Bangalore-562
106email govindaraju_tv_at_yahoo.com
Dr.T.V.Govindaraju,SSEC
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CHAPTER 4 SPUR GEARS
Introduction of Spur Gears Classification of
Gears Application of Gear Drives Spur Gear
Terminology Definitions Some Important formulae
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CHAPTER 4 SPUR GEARS
Law of gearing Constant velocity ratio Conjugate
action Tooth profiles Advantages of involute and
cycloidal tooth profile Properties of involute
profile Systems of gear tooth Involumetry
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Path of contact
Consider a pinion driving wheel as shown in
figure. When the pinion rotates in clockwise, the
contact between a pair of involute teeth begins
at K (on the near the base circle of pinion or
the outer end of the tooth face on the wheel) and
ends at L (outer end of the tooth face on the
pinion or on the flank near the base circle of
wheel).
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Path of contact
MN is the common normal at the point of contacts
and the common tangent to the base circles. The
point K is the intersection of the addendum
circle of wheel and the common tangent. The point
L is the intersection of the addendum circle of
pinion and common tangent.
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Path of contact
http//en.wikipedia.org/wiki/ImageInvolute_wheel.
gif
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Path of contact
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Path of contact
The length of path of contact is the length of
common normal cut-off by the addendum circles of
the wheel and the pinion. Thus the length of part
of contact is KL which is the sum of the parts of
path of contacts KP and PL. Contact length KP is
called as path of approach and contact length PL
is called as path of recess.
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Path of contact
ra O1L Radius of addendum circle of
pinion, and R A O2K Radius of addendum
circle of wheel
r O1P Radius of pitch circle of pinion,
and R O2P Radius of pitch circle of wheel.
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Path of contact
Radius of the base circle of pinion O1M
O1P cos? r cos? and radius of the
base circle of wheel O2N O2P cos ?
R cos?
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Path of contact
From right angle triangle O2KN
Path of approach KP
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Path of contact
Similarly from right angle triangle O1ML
Path of recess PL
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Path of contact
Length of path of contact KL
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Arc of contact
Arc of contact 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. In Figure,
the arc of contact is EPF or GPH.
Dr.T.V.Govindaraju,SSEC
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Arc of contact
Considering the arc of contact GPH. The arc GP is
known as arc of approach and the arc PH is called
arc of recess. The angles subtended by these arcs
at O1 are called angle of approach and angle of
recess respectively.
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Arc of contact
Length of arc of approach arc GP
Length of arc of recess arc PH
Length of arc contact arc GPH arc GP arc PH
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Contact Ratio (or Number of Pairs of Teeth in
Contact)
The contact ratio or the number of pairs of teeth
in contact is defined as the ratio of the length
of the arc of contact to the circular pitch.
Mathematically,
Where
and m Module.
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Number of Pairs of Teeth in Contact
Continuous motion transfer requires two pairs of
teeth in contact at the ends of the path of
contact, though there is only one pair in contact
in the middle of the path, as in Figure.
R
r
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Number of Pairs of Teeth in Contact
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Number of Pairs of Teeth in Contact
The average number of teeth in contact is an
important parameter - if it is too low due to the
use of inappropriate profile shifts or to an
excessive centre distance.The manufacturing
inaccuracies may lead to loss of kinematic
continuity - that is to impact, vibration and
noise.
R
r
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Number of Pairs of Teeth in Contact
The average number of teeth in contact is also a
guide to load sharing between teeth it is termed
the contact ratio
R
r
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Length of path of contact for Rack and Pinion
The rack is part of toothed wheel of infinite
diameter. The base circle diameter and profile of
the involute teeth are straight lines.
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Length of path of contact for Rack and Pinion
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Length of path of contact for Rack and Pinion

• Let
• r Pitch circle radius of the pinionO1P
• Pressure angle
• ra. Addendum radius of the pinion
• a Addendum of rack
• EFLength of path of contact

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Length of path of contact for Rack and Pinion
EF Path of approach EP Path of recess PF
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Length of path of contact for Rack and Pinion
From triangle O1NF
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Summary
Length of path of contact in spur gear Arc of
contact in spur gear Contact ratio in spur
gear Number of pair of teeth in contact in spur
gears Length of path of contact in Rack and
pinion
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Exercise 1
Two spur wheels have 24 and 30 teeth with a
standard addendum of 1 module. The pressure angle
is 20?. Calculate the path of contact and arc of
contact.
Solution Data t 24 T30 addendum 1m and
?20?
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Exercise 1- Continued
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Exercise 1- Continued
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Exercise 1- Continued
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Exercise 2
Two gears in mesh have a module of 8 mm and a
pressure angle of 20?. The larger gear has 57
teeth while the pinion has 23 teeth. If the
addenda on pinion and gear wheel are equal to one
module (1m), find 1. The number of pairs of teeth
in contact and 2. The angle of action of the
pinion and the gear wheel.
Solution Data t 23 T57 addendum 1m 8mm
and ?20?
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Exercise 2 - continued
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Exercise 2 - continued
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Exercise 2 - continued
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Exercise 2 - continued
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Exercise 2 - continued
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Exercise 3
The following data refers to two mating involute
gears of 20? pressure angle. Number of teeth on
pinion is 20. Gear ratio 2, speed of pinion is
250 rpm, module 12 mm. If the addendum on each
wheel is such that the path of approach and the
path of recess on each side are half of the
maximum permissible length, find the maximum
velocity of sliding during approach and recess
and the length of arc of contact.
Solution Data t 20 G2 m 12mm
n250rpm and ?20?
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Exercise 3 - continued
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Exercise 3 - continued
From the given condition
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Exercise 3 - continued
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Exercise 3 - continued
Path of approach KP
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Exercise 3 - continued
Path of recess PL
Path of contact Path of approach Path of
recess 20.52 41.04
61.56 mm
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Exercise 3 - continued
Path of recess PL
Path of contact Path of approach Path of
recess 20.52 41.04
61.56 mm
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Exercise 3 - continued
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Exercise 3 - continued
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