Page 324 - Mechanics of Asphalt Microstructure and Micromechanics

316 Ch a p t e r N i n e

Where m p is the mass for a volume element 1 1 1 pixels and s is a number of
pixels between two adjacent scanned slices. For example, the moment inertia tensor for
2
the particle shown in Figure 9.23a can be obtained in mass•pixel unit as:
⎡ 3 473 0 607 −0 580⎤
.
.
.
⎢ ⎥
.
.
.
I = 0 607 1 881 −0 925 x 10 9
⎥
⎢
⎢ 2 686 ⎥
.
⎣ −0 580 −0.9925 . ⎦
The crucial property of a tensor is that its components should transform under a
rotation of coordinate axes in such a way as to keep its geometrical or physical meaning
invariant. In a particular case of rotation of axes, the moment inertia tensor becomes a
diagonal form where the products of inertia are null. This occurs when the three or-
thogonal axes of reference frame are parallel to the three “principal axes” of the object.
In other words, a rigid body is dynamically balanced when its angular momentum is
parallel to its angular velocity:
L = ω ⋅ω (9-37)
I = λ

Where l is some (scalar) number. For this to be true, the angular velocity, w, must
point along a principal axis of the moment of inertia tensor. The corresponding value of
I is called a principal moment of inertia. The principal moment of inertia of a rotating
body is defined by finding values of I such that:
⎡
⎡ L ⎤ I ⎡ I I ⎤ ω ⎤ ⎡ ω ⎤
⎢ x ⎥ ⎢ 11 12 13 ⎥ ⎢ x ⎥ ⎢ x ⎥
⎢
L = ⎢ L ⎥ = ⎢ I I I 23 ⎥ ⎢ω ⎥ = λ ω ⎥ (9-38)
⎢ y ⎥ I ⎢ 21 I 22 I ⎥ ⎢ y ⎥ ⎢ ω y ⎥
⎣
z ⎦
⎣ ⎢ L ⎥ ⎣ 31 3 32 33⎦ ⎢ ω z ⎦ ⎥ ⎣ ⎣ ⎢ z ⎦ ⎥
Which is an eigenvalue problem. The principal moments of inertia are the eigenval-
ues of the moment of inertia tensor and the corresponding eigenvectors are the direc-
tion cosines of the principal axis. There will be three eigenvalues, which will be called
I 1 , I 2 , and I 3 in order of decreasing magnitude and three corresponding principal axes
defined as w 1 , w 2 , and w 3 . The principal axes, w 1 , w 2 , and w 3 , are unit vectors and are

z
z y
y

x
x
(a) (b)
FIGURE 9.23 (a) Digitally reconstructed limestone particle (b) Simulated ellipsoid that has the
same mass momentums and principal directions as those of the real particle.