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Kinematics and dynamics of rigid bodies 65
Y 2
G 1 X 1
Y 1
G 3 X 3
Y
Y 3
G 2 X 2
G
X
a 2
a 1 a 3
Fig. 2.32
Application of parallel axes theorem
A practical application of the parallel axes theorem given in (2.198) is provided
using the simplified representation of a tie rod as shown in Figure
2.32. The body can be considered an assembly of three components with
centres of mass at G 1 , G 2 and G 3 . The mass centre of the entire body is
located at G. The components have masses m 1 , m 2 and m 3 and moments of
inertia about the local z-axis at each mass centre I G1zz , I G2zz and I G3zz .
Applying the parallel axes theorem would in this case give a moment of
inertia I Gzz for the body using
I Gzz I G1zz m 1 a 1 2 I G2zz m 2 a 2 2 I G3zz m 3 a 3
2
(2.199)
2.12 Principal axes
The principal axes of any rigid body are those for which the products of
inertia are all zero resulting in an inertia matrix of the form
[] I
⎡I1
0 0⎤
⎢
0 I2
0
⎥
⎢ ⎥
⎣⎢
0 0 I3⎦⎥
(2.200)
where in this case I 1 , I 2 and I 3 are the principal moments of inertia. The
three planes formed by the principal axes are referred to as the principal
planes as shown in Figure 2.33. In this example the geometry chosen is a
solid cylinder to demonstrate the concept.
For the cylinder the principal axes are represented by the frame O 2 positioned
at the mass centre of the body. In this case each of the principal
planes is a plane of symmetry for the body. As can be seen for each element
of mass with positive co-ordinates there are other elements of mass,
reflected in each of the principal planes, with negative co-ordinates. The
result of this is that the products of inertia are all zero.
Returning to the consideration of the angular momentum of a body given in
(2.178) this can now be written as shown in equation (2.201) where I 1 , I 2
and I 3 are the principal moments of inertia for Body 2 taken about the origin
of frame O 2. The moment of inertia matrix is referred to the principal
axes, again frame O 2 and the products of inertia are zero:
{H 2 } 1/2 [I 2 ] 2/2 { 2 } 1/2