Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 369

where x is a position vector locating the differential element of volume or mass with
respect to the origin. The power of x occurring in the integrand indicates the order
of the moment of the mass—it is to the zero power in the definition of the mass of
the object itself—and it is to the first order in the definition of the mass center. The
mass moment of inertia, which is the second moment of mass, arises as a convenient
definition that occurs in the development of the conservation of angular momentum
for an object as an integral of the moment of the linear momentum of an element of
mass, d m , over the object. The expression for the angular momentum H of an object
O as the integral over its volume of the cross product of the position vector x and
the linear momentum r x d v ¼ x d m of an element of mass d m , is given by

¼ O

r _ H x x d v : (4.22, repeated)

If the object is instantaneously rotating about an axis with an angular velocity o

and the rotational velocity x at the mass element, d m is given by _

x _ ¼ o x :

Substitution of this expression for the velocity into (4.22, repeated), an alternate
expression for H is obtained,

ð

H ¼ x ðo x Þr d v : (A.121)

O

The integrand in this new representation for H can be expressed differently using
the vector identity r

ð p q Þ¼ð r q Þ p ð r p Þ q proved in Example A.8.4, thus

x ðo x Þ¼ð x x Þoð x oÞ x :

Incorporating this expression into the integrand in (A.121) it is easily seen that H
has the representation

ð
O ½ð

H ¼ o x x Þ 1 ð x x Þr d v : (A.122a)

This result is written more simply as

H ¼ I o; (A.122b)

where the definition of the mass moment of inertia tensor,

ð
O ½ð

I x x Þ 1 ð x x Þr d v ; (A.123)