Environmental Engineering Reference
In-Depth Information
These are known as Euler's equations
3
. Notice that they are coupled, non-linear,
differential equations of motion, which give the time dependence of
ω
under the
influence of a net external torque
.By
coupled
, we mean that each equation
contains variables that appear in the other two equations and by
non-linear
we
mean that the equations involve products of the components of
τ
. You should
compare this structure with that of Newton's Second Law, which yields three linear,
uncoupled differential equations in a Cartesian basis. This additional complexity
means that we cannot hope to provide a general solution to Euler's equations, rather
we shall use them to examine some interesting special cases.
ω
10.6 THE FREE ROTATION OF A SYMMETRIC TOP
10.6.1 The body-fixed frame
Consider a rigid body tossed into the air: the body experiences a net external
force due to gravity, but if the gravitational field is uniform there will be no net
torque about the centre of mass and the gravitational field will not induce any
rotation of the body. In this section we investigate this kind of torque-free rotation.
We simplify matters by focussing on an object that is a solid of revolution about
one axis (often called a symmetric top). To be entitled to use Euler's equations
(Eq. (10.53)), we must be certain that Eq. (10.46) is valid and since we would like
to consider the possibility that the centre of mass is accelerating (e.g. as it is for
an object tossed through the air) that means we must always compute the angular
momentum and torque about the centre of mass.
e
3
e
2
e
1
Figure 10.11
A free symmetric top with body-fixed axes.
We need first to identify the principal axes and here the symmetry of the top
helps us since the axis of symmetry is automatically a principal axis; we will label
this as
e
3
. The other two principal axes are labelled
e
1
and
e
2
(see Figure 10.11)
and symmetry with respect to rotations about
e
3
means that any pair of orthogonal
unit vectors lying in the plane perpendicular to
e
3
will suffice. The symmetry also
dictates that the corresponding principal moments of inertia are the same, i.e.
I
1
=
I
2
=
I.
(10.54)
3
After Leonhard Paul Euler (1707 - 1783).
