Kinematics Roadmap

1
Kinematics Roadmap
2
Instant Centers

• Instant Center A point at which there is no
  relative velocity between two links of a
  mechanism at that instant
• Does not work for acceleration

3
Kennedys Theorem

• The three instantaneous centers of three bodies
  moving relative to one another must lie along a
  straight line

Relates an angular velocity ratio to a ratio of
distances between instant centers 1. If the
relative IC lies between the absolute ICs the
angular velocity ratio is negative 2. If the
relative IC lies outside the other two, the
angular velocity ratio is positive
4
Instant Centers for Mechanisms
N of instant centers n of links
Sliders one IC is at the link and the slider and
the other IC has infinite locations in the
direction perpendicular to the sliding direction
5
Example 6-bar
Method Close two triangles with one line!
6
Velocity Analysis
Formula Method 1. Use w ratio to find ws 2.
Use V iwR Graphical Method 1. The magnitude of
velocity is proportional to the radius from the
point of reference to the point of interest
7
Instant Center Formula Method
w2 30 rad/s. Find VB, w3, VP, w4
B
(3
,
4
)
P
3
A
4
(2
,
3
)
w
2
2
(1
,
4
)
(1
,
2
)
1
8
Instant Center Graphical Method

• Need 2 scales
• One for dimensions- One for velocities

w2 CCW Find w3, w4, VB, VP
Vb
B
Vp
(3
,
4
)
P
3
VA
A
4
(2
,
3
)
(2,4)
VA
w
2
2
(1
,
4
)
(1
,
2
)
If you dont know w3, use this construction!
1
ICs - Given w2, find VA using (1,2) - Find w3
using VA and (1,3) - Find VP using w3 and (1,3) -
Find VB using w3 and (1,3) - Find w4 using VB and
(1,4)
VB
Ov
VP/B
VB/A
VA
VP
VP/A
(1,3)
9
Example 6-bar
1,4
5,6
1,6
Method Close two triangles with one line!
2,3
3,5
3,4
1,2
Solution Need ICs 1,2 1,6 2,6 to
determine w6 1,3 1,2 2,3 to determine w3
1,3 to determine V4

• Questions Given w2 2 rad/s and the
  scale 1 cm 1 cm,
• Determine V4
• Determine w6

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Example 6-bar
1,4
5,6
1,6
Solution 1,2 2,3 1.5 cm1,3 2,3 8
cm1,3 3,4 4.9 cm w3 (2)(1.5)/8 0.375
rad/s (CCW) V4 (4.9)(0.375) 1.84 cm/s ?
2,3
3,5
3,4
1,2
11
Example 6-bar
1,4
5,6
1,6
2,3
3,5
3,4
1,2
3,6
12
Example 6-bar
1,4
5,6
1,6
Solution 1,2 2,6 0.8 cm1,6 2,6 1.2
cm w6 (2)(0.8)/2.2 0.73 rad/s (CW)
2,6
2,3
3,5
3,4
1,2
3,6