J.
Peraire,
S.
Widnall
16.07
Dynamics
Fall
2008
Version
2.0
Lecture
L25
 3D
Rigid
Body
Kinematics
In
this
lecture,
we
consider
the
motion
of
a
3D
rigid
body.
We
shall
see
that
in
the
general
threedimensional
case,
the
angular
velocity
of
the
body
can
change
in
magnitude
as
well
as
in
direction,
and,
as
a
consequence,
the
motion
is
considerably
more
complicated
than
that
in
two
dimensions.
Rotation
About
a
Fixed
Point
We
consider
first
the
simplified
situation
in
which
the
3D
body
moves
in
such
a
way
that
there
is
always
a
point,
O
,
which
is
fixed.
It
is
clear
that,
in
this
case,
the
path
of
any
point
in
the
rigid
body
which
is
at
a
distance
r
from
O
will
be
on
a
sphere
of
radius
r
that
is
centered
at
O
.
We
point
out
that
the
fixed
point
O
is
not
necessarily
a
point
in
rigid
body
(the
second
example
in
this
notes
illustrates
this
point).
Euler’s
theorem
states
that
the
general
displacement
of
a
rigid
body,
with
one
fixed
point
is
a
rotation
about
some
axis.
This
means
that
any
two
rotations
of
arbitrary
magnitude
about
di
ff
erent
axes
can
always
be
combined
into
a
single
rotation
about
some
axis.
At
first
sight,
it
seems
that
we
should
be
able
to
express
a
rotation
as
a
vector
which
has
a
direction
along
the
axis
of
rotation
and
a
magnitude
that
is
equal
to
the
angle
of
rotation.
Unfortunately,
if
we
consider
two
such
rotation
vectors,
θ
1
and
θ
2
,
not
only
would
the
combined
rotation
θ
be
di
ff
erent
from
θ
1
+
θ
2
,
but
in
general
θ
1
+
θ
2
=
θ
2
+
θ
1
.
This
situation
is
illustrated
in
the
figure
below,
in
which
we
consider
a
3D
rigid
body
undergoing
two
90
o
rotations
about
the
x
and
y
axis.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
It
is
clear
that
the
result
of
applying
the
rotation
in
x
first
and
then
in
y
is
di
ff
erent
from
the
result
obtained
by
rotating
first
in
y
and
then
in
x
.
Therefore,
it
is
clear
that
finite
rotations
cannot
be
treated
as
vectors,
since
they
do
not
satisfy
simple
vector
operations
such
as
the
parallelogram
vector
addition
law.
This
result
can
also
be
understood
by
considering
the
rotation
of
axes
by
a
coordinate
transformation.
Consider
a
transformation
(
x
) = [
T
1
](
x
),
and
a
subsequent
coordinate
transformation
(
x
) = [
T
2
](
x
).
The
(
x
)
coordinate
obtained
by
the
transformation
(
x
) = [
T
1
][
T
2
](
x
)
will
not
be
the
same
as
the
coordinates
obtained
by
the
transformation
(
x
) = [
T
2
][
T
1
](
x
),
in
other
words,
order
matters.
Angular
Velocity
About
a
Fixed
Point
On
the
other
hand,
if
we
consider
infinitesimal
rotations
only,
it
is
not
di
ﬃ
cult
to
verify
that
they
do
indeed
behave
as
vectors.
This
is
illustrated
in
the
figure
below,
which
considers
the
e
ff
ect
of
two
combined
infinitesimal
rotations,
d
θ
1
and
d
θ
2
,
on
point
A
.
(
figure
reproduced
from
J.L.
Meriam
and
K.L.
Kraige,
Dynamics,
5th
edition,
Wiley)
As
a
result
of
d
θ
1
,
point
A
has
a
displacement
d
θ
1
×
r
,
and,
as
a
result
of
d
θ
2
,
point
A
has
a
displacement
d
θ
2
×
r
.
The
total
displacement
of
point
A
can
then
be
obtained
as
d
θ
×
r
,
where
d
θ
=
d
θ
1
+
d
θ
2
.
Therefore,
it
follows
that
angular
velocities
ω
1
=
θ
˙
1
and
ω
2
=
θ
˙
2
can
be
added
vectorially
to
give
ω
=
ω
1
+
ω
2
.
This
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 MAtuka
 Rigid Body, Rotation, Angular velocity, Euclidean geometry

Click to edit the document details