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Kinematics of Rigid Bodies
Applications: Analysis of cams, gears, shafts, linkages, connecting rods, etc.
Deﬁnition:
A
rigid body
is a special case of a system of particles wherein the distances between
all particles remain unchanged.
For a system of particles with general threedimensional motion prescribed for each particle,
there are up to
3
N
possible degrees of freedom for a system of
N
particles. With the restriction
on distance between two points
A
and
B
so that

~
r
A/B

=
constant, the only possible change with
respect time for
~
r
A/B
is due to rotation. Since the rotational degrees of freedom can also be
represented by a vector, the general description of rigid body motion can be reduced to
(i) 3 translations and 3 rotations for threedimensional problems and
(ii) 2 translations and 1 rotation for twodimensional (plane) problems.
Motion of an Arbitrary Point on a Rigid Body
If the translation components of a rigid body is
given at a reference point
B
and the rotation vector is
also known for the rigid body, then the motion at any
arbitrary point
A
on the rigid body can be determined
completely.
Consider ﬁrst the relationship
~
r
A
=
~
r
B
+
~
r
A/B
,
in which
~
r
B
is the known motion of the reference
point
B
while
~
r
A
is the motion of the arbitrary point
A
to be determined. The relative displacement
~
r
A/B
has the additional restriction of

~
r
A/B

=
constant for
a rigid body.
A
B
O
~r
B
A
/
To obtain the velocity at point
A
, take derivative of the position vector
~
r
A
with respect to time
to yield
d~r
A
dt
=
B
dt
+
A/B
dt
or
~v
A
=
B
+
A/B
where the relative velocity
A/B
is deﬁned as
A/B
=
A/B
dt
=
~ω
×
~
r
A/B
for a rigid body because

~
r
A/B

=
constant and the change occurs only due to rotation. Hence, for
a rigid body
A
=
B
+
×
~
r
A/B
.
The above equation indicates that once
B
and
are known, the velocity everywhere on the
rigid body can be calculated easily.
To obtain the acceleration vector, perform another time derivative to yield
d~v
A
dt
=
B
dt
+
d~ω
dt
×
~
r
A/B
+
×
A/B
dt
or
~a
A
=
B
+
~α
×
~
r
A/B
+
×
(
×
~
r
A/B
)
.
–6/1–
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View Full Document Example – Plane Rigid Body Kinematics
A rigid bar
AP
is connected to sliders
A
and
B
as shown in the ﬁgure.
Slider
A
is restricted to move only in the horizontal
direction while slider
B
can move only in
the vertical direction. If at the instant shown,
slider
A
is moving to the left with a velocity
of 4 mm/sec, ﬁnd
~v
B
,
~ω
and
P
.
P
300 mm
200 mm
B
45
o
A
Solution:
Since points
A
,
B
and
P
are all on the same rigid body, we can write
B
=
A
+
×
~
r
B/A
and
P
=
A
+
×
~
r
P/A
.
Of course, we can also use
P
=
B
+
×
~
r
P/B
if the velocity at
B
is known. The angular velocity,
, is the same at any location along the rigid
bar.
With
A
given as
{
4
,
0
,
0
}
T
, let
B
=
0
v
B
0
,~
ω
=
0
0
ω
and
~
r
B/A
=

200
√
2
/
2
200
√
2
/
2
0
.
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.
 Spring '08
 DocWong

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