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AME 352
INSTANT CENTERS
P.E. Nikravesh
1
Instant Center of Velocities
Instant center of velocities is a simple graphical method for performing velocity analysis on
mechanisms.
The method provides visual understanding on how velocity vectors are related.
In
this lesson we review the fundamentals of instant centers.
A Single Link and The Ground
Assume that the link shown is pinned to the ground at
O
.
The link has a known angular
velocity
±
(assume CCW).
The magnitude of the velocity of any point on the link, such as
A
,
B
,
or
C
, can be computed as
V
=
R
The direction of the velocity vector is determined by rotating the position vector
R
90
±
in the
direction of
.
Using the “rotated vector” notation, we have
V
=
±
R
(v.1)
This equation represents both the magnitude and the direction of
the velocity vector.
Note that the position vector has a constant
length.
This problem can also be stated as: For a link that is pinned
to the ground at
O
, the velocity of a point such as
A
is given.
Determine the angular velocity of the link and the velocity of
point
B
(or
C
).
The magnitude of the angular velocity is obtained by
measuring the magnitude of vector
R
AO
and then the above
equation is used to obtain the angular velocity.
After finding
,
the velocity of
B
is determined.
1, i
I
(i)
A
B
C
O
R
C,O
V
C
B,O
R
B
V
A,O
R
A
V
The pin joint that connects link
i
to the ground can be viewed as two coinciding points: point
O
i
on the link and point
O
1
on the ground (the ground is always given index
1
).
Point
O
i
has the
same velocity as point
O
1
.
Obviously, since the velocity of
O
1
is zero, the velocity of
O
i
is zero
as well.
These two points that are on two different bodies but coincide have identical
velocities—they form an
instant center
between the two bodies.
This instant center is denoted as
I
1,
i
(or
I
i
,1
).
In this example, the instant center between link
i
and the ground is an actual pin joint.
As we
will see next, an instant center may be an imaginary pin joint.
A side note: In the example shown
below, the given velocity is incorrect!
The
velocity of
A
must be perpendicular to
R
AO
.
O
Another incorrect problem statement is
shown below.
Velocity of
A
suggests a CW
rotation but velocity of
B
suggests the opposite.
O
C
V
C
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INSTANT CENTERS
P.E. Nikravesh
2
In the example shown here, link
i
is not pinned to the ground.
Assume that the velocities of two or more points on the link are
given.
The following observations can be made:
a) The normal axes to the velocity vectors, each passing through
its corresponding point, intersect at one point, denoted as
I
1,
i
.
b) The distances of these points from
I
1,
i
yield
V
A
R
A
,
I
1,
i
=
V
B
R
B
,
I
1,
i
=
V
C
R
C
,
I
1,
i
±
=
²
c) All velocity vectors give the same sense of direction for the
angular velocity (CCW in this illustration).
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This note was uploaded on 09/08/2010 for the course AME 352 taught by Professor Nikravesh during the Fall '08 term at University of Arizona Tucson.
 Fall '08
 NIKRAVESH

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