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8/9/2019 Mechanisms With Lower Pairs
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Chapter 9
Mechanisms with
Lower Pairs
10/29/2013
Dr. Mohammad Abuhiba, PE1
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9.1. Introduction
When the two elements of a pair have a
surface contact and a relative motion takes
place, the surface of one element slides
over the surface of the other, the pair
formed is known aslower pair
.
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9.2. Pantograph
A pantograph is aninstrument used to
reproduce to an enlarged or
a reduced scale and as
exactly as possible the pathdescribed by a given point.
Bars BA & BC are extended
to O & E respectively, suchthat:
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9.2. Pantograph
For all relative positions of
the bars, triangles OAD &
OBE are similar and points
O, D and E are in one
straight line.
Point E traces out same
path as described by D
From similar triangles OADand OBE,
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
O = a point on circumference
of a circle of diameter OP
OA = any chord
B = a point on OA, such thatOAOB = constant
Locus of a point B will be a
straight line perpendicular to
diameter OP Draw BQ perpendicular to OP
Triangles OAP & OQB are
similar
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
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OP is constant
If OAOB is constant,
then OQ will beconstant.
Point B moves along
straight path BQ
which is
perpendicular to OP.
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
eaucellier mechanism
Pin atA is constrained to
move alongcircumference of a circlewith fixed diameter OP,by means of link O1A.
AC = CB = BD = DA; OC= OD ; and OO1= O1A
Product OAOB remainsconstant, when link O1Arotates.
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
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OC & BC are of constantlength
OBOA remains constant
B traces a straight path
perpendicular to OP
eaucellier mechanism
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
arts mechanism
FC = DE& CD = EF
O,A,B divide links FC, CD, EF in the same ratio
BOCE is a trapezium and OA & OB are respectivelyparallel toFD & CE.
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9.4. Exact Straight Line Motion
Mechanisms Made up of Turning Pairs
arts mechanism
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9.5. Exact Straight Line Motion Consisting of
One Sliding Pair -
Scott Russells Mechanism
OA =AP =AQ
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9.6. Approximate Straight Line Motion
Mechanisms - atts
me h nism
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9.6. Approximate Straight Line Motion
Mechanisms -
Tchebicheffs
mechanism
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OA = O1B
P, mid of AB traces out an
approximately straight line parallel to
OO1 P is exactly above O or O1 in the
extreme positions (when BA lies
along OA or when BA lies along BO1)
P will lie on a straight line parallel to
OO1, in the two extreme positionsand in the mid position, if the
lengths of the links are in
proportionsAB:OO1:OA = 1:2:2.5
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9.8. Steering Gear Mechanism
Used for changing direction of two or more of the
wheel axles with reference to the chassis.
In automobiles, front wheels are placed over the
front axles, which are pivoted at points A and B
(Fig. 9.15).
These points are fixed to the chassis.
Back wheels are placed over the back axle, at
the two ends of the differential tube.
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9.8. Steering Gear Mechanism
When the vehicle takes a turn, the front wheels along
with the respective axles turn about the respective
pivoted points.
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9.8. Steering Gear Mechanism
To avoid skidding (slipping of wheels sideways),
the two front wheels must turn about the same
instantaneous center I which lies on the axis of
the back wheels.
If the instantaneous center of the two front
wheels do not coincide with the instantaneous
center of the back wheels, the skidding on thefront or back wheels will definitely take place.
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9.8. Steering Gear Mechanism
The condition for correct steering is that all
the four wheels must turn about the same
instantaneous center.The axis of the inner wheel makes a larger
turning angle than the angle subtended by
the axis of outer wheel.
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9.8. Steering Gear Mechanism
a = Wheel track
b = Wheel base
c = Distance between pivotsA and B
From triangle IBP,
From triangle IAP,
Fundamental equation for correct steering
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9.9. Davis Steering Gear
Fig. 9.16
Exact steering
gear mechanism
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9.9. Davis Steering Gear
Slotted links AM & BH are attached to front
wheel axle, which turn on pivots A & B
respectively.Rod CD is constrained to move in direction of its
length, by sliding members at P & Q.
These constraints are connected to slotted link
AM & BH by a sliding and a turning pair at each
end.
Steering is affected by moving CD to right or left.
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9.9. Davis Steering Gear
a = Vertical distance betweenAB & CD
b = Wheel base
d = Horizontal distance betweenAC & BD
c = Distance between pivotsA & B of front axle
x = Distance moved byAC toAC = CC = DD
a = Angle of inclination of links AC & BD, tovertical
From triangleA AC,
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9.9. Davis Steering Gear
From triangleAAC,
From triangle BBD,
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9.9. Davis Steering Gear
For correct steering,
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Example 9.1
In a Davis steering gear, the distance
between the pivots of the front axle is 1.2m
and the wheel base is 2.7m. Find the
inclination of the track arm to the
longitudinal axis of the car, when it is
moving along a straight path.
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9.10. Ackerman Steering Gear
The difference between Ackerman andDavis steering gears are :
1. Whole mechanism of Ackerman steering
gear is on back of front wheels; whereas inDavis steering gear, it is in front of wheels.
2. Ackerman steering gear consists of turning
pairs, whereas Davis steering gear consists
of sliding members.
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9.10. Ackerman Steering Gear
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9.10. Ackerman Steering Gear
MechanismABCD is a four bar crank chainBC =AD&AB CD
The following are positions for correct steering:
1. When vehicle moves along a straight path, links AB & CD
are parallel and shorter links BC & AD are equally inclined
to longitudinal axis of vehicle.
2. When vehicle is steering to left, position of gear is shown by
dotted lines in Fig. 9.17. In this position, lines of front wheel
axle intersect on back wheel axle at I, for correct steering.
To satisfy the fundamental equation for correct
steering, links AD & DC are suitably proportioned.
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9 11 Universal or Hookes Joint
Used to connect two shafts, whichare intersecting at a small angle
End of each shaft is forked to U-typeand each fork provides two bearingsfor arms of a cross.
Arms of cross are perpendicular toeach other.
Motion is transmitted from driving
shaft to driven shaft through a cross. Inclination of the two shafts may be
constant, but in actual practice itvaries, when the motion istransmitted.
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9.12. Ratio of Shafts Velocities
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9.13. Max Min Speeds of Driven Shaft
w1will be max for a given value of awhen denominator
of above equationis min. This will happen, when
w1
is min when denominator of above equation is max.
This will happen when
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9.13. Max Min Speeds of Driven Shaft
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9.14. Condition for Equal Speeds
of the Driving and Driven Shafts
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9.15. Angular Acceleration of the
Driven Shaft
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For angular acceleration to be maximum
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9.16. Max Fluctuation of Speed
Max fluctuation of speed of driven shaft approximately
varies as square of angle between the two shafts
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9.17
Double Hookes
Joint
In order to have a constant velocity ratio of driving and driven
shafts, an intermediate shaft with a Hookes joint at each
end is used. This joint gives a velocity ratio equal to unity, if
1. Axes of driving & driven shafts are in same plane, and
2. Driving & driven shafts make equal angles with the
intermediate shaft.
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Example. 9.2
Two shafts with an included angle of 160 are
connected by a Hookes joint. The driving shaft
runs at a uniform speed of 1500 rpm. The driven
shaft carries a flywheel of mass 12 kg and 100
mm radius of gyration. Find the max angular
acceleration of the driven shaft and the max
torque required.
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Example 9.3
The angle between the axes of two shafts
connected by Hookesjoint is 18. Determine the
angle turned through by the driving shaft when
the velocity ratio is maximum and unity.
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Example 9.4
Two shafts are connected by a Hookesjoint. The
driving shaft revolves uniformly at 500 rpm. If the
total permissible variation in speed of the driven
shaft is not to exceed 6% of the mean speed,
find the greatest permissible angle between the
center lines of the shafts.
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Example 9.5
Two shafts are connected by a universal joint.
The driving shaft rotates at a uniform speed of
1200 rpm. Determine the greatest permissible
angle between the shaft axes so that the total
fluctuation of speed does not exceed 100 rpm.
Also calculate the maximum and minimum
speeds of the driven shaft.
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Example 9.6
The driving shaft of a Hookes joint runs at a uniform
speed of 240 rpm and the angle between the shafts
is 20. The driven shaft with attached masses has a
mass of 55 kg at a radius of gyration of 150 mm.
1. If a steady torque of 200 N.m resists rotation of
the driven shaft, find the torque required at the
driving shaft, when q = 45.
2. At what value of awill the total fluctuation of
speed of the driven shaft be limited to 24 rpm ?
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Example 9.7
A double universal joint is used to connect two
shafts in the same plane. The intermediate shaft
is inclined at an angle of 20 to the driving shaft
as well as the driven shaft. Find the maximum
and minimum speed of the intermediate shaft
and the driven shaft if the driving shaft has a
constant speed of 500 rpm.
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