Environmental Engineering Reference
In-Depth Information
This equation gives
L
in the body-fixed frame. In this frame
L
does not change
with time, but this does not mean that there will be no torque since the frame is
non-inertial. In the lab frame, which is inertial, the body rotates about the x
3
axis
and
L
will rotate with the body, giving rise to a torque (see Figure 10.10). While the
L
3
component clearly doesn't change with time, the components of
L
in the x
1
−
x
2
plane form a time-dependent vector
L
12
with magnitude L
12
:
(Mb
2
ω)
2
1
4
2
2
1
4
1
8
(Mb
2
ω)
2
.
L
12
=
L
1
+
L
2
=
+
=
Since
L
12
sweeps out a circle in the x
1
−
x
2
plane with constant angular speed ω
we can write
d
L
12
d
t
=−
L
12
ω
r
12
,
where
r
12
is a radial unit vector in the x
1
−
x
2
plane. Now that we have gone to the
effort of shifting to an inertial frame, we are in a position to calculate the torque:
d
L
d
t
d
L
12
d
t
1
√
8
Mb
2
ω
2
r
12
.
τ
=
=
=−
(10.45)
Although we managed to solve the last example, we had to figure out that
L
12
precesses in a circle in the lab frame. It would be useful if we had a more general,
algebraic, way of solving this problem and the formalism that we shall develop in
the next section will allow us to do just that. Moreoever, it will permit us to finally
move away from the special case of rotation about a fixed axis.
10.5 EULER'S EQUATIONS
To depart from the special case of fixed-axis rotation and deal with the general
rotational motion of a rigid body our starting point is Eq. (4.9), used in conjunction
with Eq. (10.14):
d
L
d
t
=
d
d
t
(
I
ω
)
=
τ
.
(10.46)
This equation is valid in the lab frame (i.e. our generic inertial frame) or in a
non-rotating, accelerating frame provided we work relative to the centre of mass.
But it is not generally valid in the body fixed frame, which is a rotating frame.
Thus to solve for the motion of the body we might consider starting with a set of
axes that are fixed in the lab frame. However, as we have previously stressed, to do
that necessitates the use of a time-dependent moment of inertia matrix. As a result,
it is usually more convenient to work within a body-fixed frame of reference and
modify the equations of motion accordingly, i.e. we can no longer use Eq. (10.46)
directly but must transform it into the body-fixed frame.
Using the general rule for transforming the time-derivative of a vector between
the lab and a rotating frame (Eq. (8.16)), we have
