Download - 2 Limit Position
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Minimum Transmission Angle at 2=00
Maximum Transmission Angle at 2=1800
The limits of link lengths are theoretical. However, additional considerations
must be given regarding the quality of the transmission of motion between
the links for any practical design. One common consideration is the
transmission angle of the mechanism.
The ideal value of trans-
mission angle is 90. In
this instance, the line of
action of the interactiveforce between links 3and 4 matches the line of action of
motion of the kinematic pair between
the two links. This ideal value cannot
be maintained in a moving mechanism.However, it is generally acceptable if
transmission angles fall within the
range, 45 < < 135
Values outside this range result in inefficient transmission of motion.
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Maximum Transmission Angle at 2=2700
Minimum Transmission Angle at 2=900
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Figure 2.18 Configuration of a four-bar mechanism with a small transmission angle.
The figure shows the configuration of a
four-bar mechanism having a small
transmission angle, outside theacceptable range.
Driving torque M12 is applied to link 2. As shown in Figure (b), the direction of
force transmitted from link 3 on link 4 results in a small torqueM14 about point
O4 , but a high bearing force at the same point. For transmission angles close to
0 or 180, there is a tendency for a mechanism to bind.
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Applying the cosine law for the triangle
formed by points O2O4B,
Using the cosine law again for the triangle
formed by points DO4B,
22122212 cos24 rrrrrBO
cos2 432
4
2
3
2
4rrrrrBO
43
22
4
2
31
2cos 4
rr
rrr BO
The transmission angle for a particularangular displacement of link 2.
For a four-bar mechanism, extreme values
of the transmission angle occur when
max
min
0
2
0
2 180,0 and
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Mechanics of MachinesCleghorn
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Figure 2.19 Transmission angle of a four-bar mechanism.
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Mechanics of MachinesCleghorn
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Figure 2.12 Limit positions of a slider crank mechanism: (a) mechanism, (b) geometry of first limit position, (c) geometry of second limit position.
Limit Position
Limit Position
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Limit PositionLimit Position
122 180
32
11
1sin
rr
r
23
11
2sin
rr
r
2
1
2
321 rrrs
2
1
2
232
rrrs
21
2
13
2
1
2
3221 rrrrrrsssstroke
The time ratio as the time for the
slider to move in one direction
between the limit positions, divided
by the time it takes to move in the
opposite direction between the
same limit positions. The time ratiois dimensionless.
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Mechanics of MachinesCleghorn
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The time ratio of the mechanism is
The average velocity of the slider,2
2
2
1
2
t
t
TR
2
2
21
1
,4
ss
t
strokev
leftavgTo the left,
To the right, 2
2
21,4 22
ss
t
strokev
leftavg
Between the limit positions, The time taken for the slider to move
To the left To the right
2
2
1
t
2
2
2
2
t
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Mechanics of MachinesCleghorn
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Example : For the slider crank mechanism with dimensions and crank angular speed;
r1=2.0 cm, r2=3.5cm and r3=10cm ;the angular velocity, 20.00 rads-1 CCW. Determine
the average slider velocity to the left and right.
01
32
11
152.8
105.3
2sinsin
rr
r
01
23
11
292.17
5.310
2sinsin
rr
r
rad31.34.18952.892.17180180 00122
cmrrrrrrsssstroke 17.7212132123221
1
2
2
21
1
,4 35.43
2031.3
17.7
cms
ss
t
strokev
leftavg
1
2
2
21
1
,4 14.482
cmssst
strokevleftavg
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1321
2
32
2
1
2
4cos2 rrrrrrr
141
2
4
2
1
2
32cos2 rrrrrr
From triangle (b)
321
2
4
2
32
2
111
2cos
rrr
rrrr
41
2
4
2
32
2
111
2cos
rr
rrrr
The amplitude of motion of the rocker may be expressed as, 214
(b)(c)
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22312
23
2
1
2
4cos2 rrrrrrr
241
2
4
2
1
2
23cos2 rrrrrr
From triangle (c)
231
2
4
2
23
2
11
22cos rrr
rrrr
41
2
4
2
23
2
11
2 2cos rr
rrrr
The amplitude of motion of the rocker maybe expressed as, 214
(c)
2
2
4
,4
CCWavg
To the left,To the right,
2
2
4
,4 2
CWavg
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Figure 2.15 Limit position of a quick-return mechanism.
1
21cosrr
421 OO
rr 22 2PO
rr Where,
It shows a quick-return mechanism in
one of its two limit positions. We willassume that link 2 is driven at a
constant rate in the counterclockwise
direction. From Figure points O2, P2,
and O4 form a right triangle. Therefore
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Figure 2.15 Limit position of a quick-return mechanism.
Due to symmetry about the vertical axis
,the rotation of link 2 required to move link
6 to the right, from the second limit
position to the first, is 2.
The time required to execute this motion is,
2
1
21
2
1
cos22
rr
t
the time required to move link 6 to the left, from
the first limit position to the second, is
2
1
21
2
2
cos2222
rr
t
The time ratio of the motions of link 6 to the
right and to the left is
2
1
t
tTR