Chapter 3
Velocity and Acceleration Analysis
3.1 Introduction
The motion of a rigid body (
RB
) is deFned when the position vector, velocity and
acceleration of all points of the rigid body are deFned as functions of time with
respect to a Fxed reference frame with the origin at
O
0
.
Let
ı
0
,
j
0
, and
k
0
, be the constant unit vectors of a Fxed orthogonal Cartesian
reference frame
x
0
y
0
z
0
and
ı
,
j
and
k
be the unit vectors of a body Fxed (mobile or
rotating) orthogonal Cartesian reference frame
xyz
(±ig. 3.1). The unit vectors
ı
0
,
j
0
,
and
k
0
of the primary reference frame are constant with respect to time.
r
1
r
O
x
ı
y
j
z
k
(
RB
)
ω
ı
0
j
0
k
0
x
0
y
0
z
0
O
0
O
M
α
Fig. 3.1
±ixed orthogonal Cartesian reference frame with the unit vectors [
ı
0
,
j
0
,
k
0
]; body Fxed
(or rotating) reference frame with the unit vectors
[
ı
,
j
,
k
]
; the point
M
is an arbitrary point,
M
∈
(
RB
)
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3 Velocity and Acceleration Analysis
A reference frame that moves with the rigid body is a
bodyfxed
(or rotating)
reference frame. The unit vectors
ı
,
j
, and
k
of the bodyFxed reference frame are
not constant, because they rotate with the bodyFxed reference frame. The location
of the point
O
is arbitrary.
The position vector of a point
M
,
M
∈
(
RB
)
, with respect to the Fxed reference
frame
x
0
y
0
z
0
is denoted by
r
1
=
r
O
0
M
and with respect to the rotating reference
frame
Oxyz
is denoted by
r
=
r
OM
. The location of the origin
O
of the rotating
reference frame with respect to the Fxed point
O
0
is deFned by the position vector
r
O
=
r
O
0
O
. Then, the relation between the vectors
r
1
,
r
and
r
0
is given by
r
1
=
r
O
+
r
=
r
O
+
x
ı
+
y
j
+
z
k
,
(3.1)
where
x
,
y
, and
z
represent the projections of the vector
r
=
r
OM
on the rotating
reference frame
r
=
x
ı
+
y
j
+
z
k
.
The magnitude of the vector
r
=
r
OM
is a constant as the distance between the
points
O
and
M
is constant,
O
∈
(
RB
)
, and
M
∈
(
RB
)
. Thus, the
x
,
y
and
z
compo
nents of the vector
r
with respect to the rotating reference frame are constant. The
unit vectors
ı
,
j
, and
k
are timedependent vector functions. The vectors
ı
,
j
and
k
are the unit vector of an orthogonal Cartesian reference frame, thus one can write
ı
·
ı
=
1
,
j
·
j
=
1
,
k
·
k
=
1
,
(3.2)
ı
·
j
=
0
,
j
·
k
=
0
,
k
·
ı
=
0
.
(3.3)
3.2 Velocity Field for a Rigid Body
The velocity of an arbitrary point
M
of the rigid body with respect to the Fxed
reference frame
x
0
y
0
z
0
, is the derivative with respect to time of the position vector
r
1
v
=
d
r
1
dt
=
d
r
O
0
M
=
d
r
O
+
d
r
=
v
O
+
x
d
ı
+
y
d
j
+
z
d
k
+
dx
ı
+
dy
j
+
dz
k
,
(3.4)
where
v
O
=
˙
r
O
represent the velocity of the origin of the rotating reference frame
O
1
x
1
y
1
z
1
with respect to the Fxed reference frame
Oxyz
. Because all the points in
the rigid body maintain their relative position, their velocity relative to the rotating
reference frame
xyz
is zero, i.e., ˙
x
=
˙
y
=
˙
z
=
0.
The velocity of point
M
is
v
=
v
O
+
x
d
ı
+
y
d
j
+
z
d
k
=
v
O
+
x
i
+
y
j
+
z
˙
k
.
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 Spring '09
 D.A
 Acceleration, Special Relativity, Velocity, reference frame, acceleration analysis

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