Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 370

Fig. A.2 A diagram for the calculation of the mass moment of inertia of an object about the axis
characterized by the unit vector e ; x is the vector from the origin O of coordinates to the element of
mass dm , x ( xe ) e is the perpendicular distance from the axis e to the element of mass dm

has been introduced; note that it is a symmetric tensor. The rotational kinetic energy
of the spinning object is then given by

K rot ¼ð = Þo I o:

1 2 (A.124)

A second perspective on the mass moment of inertia tensor, without the angular
momentum motivation, is the following: Let e represent the unit vector passing
through the origin of coordinates, then x ( xe ) e is the perpendicular distance from
the e axis to the differential element of volume or mass at x (Fig. A.2 ). The second
or mass moment of inertia of the object O about the axis e , a scalar, is denoted by I ee
and given by

ð
O ð

I ee ¼ x ð x e Þ e Þð x ð x e Þ e Þrð x ; t Þ d v : (A.125a)

This expression for I ee may be changed in algebraic form by noting first that

2 and

ð x ð x e Þ e Þð x ð x e Þ e Þ¼ x x ð x e Þ

thus, from A(108),

2 3

ð
O fðx xÞ1 ðx xÞgrðx;

4 5 e:

I ee ¼ e t Þ d v (A.125b)

If the notation (A.123) for the mass moment of inertia tensor I is introduced,then
the representation for (A.125b) simplifies to

I ee ¼ e I e : (A.125c)