Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 372

The matrix of tensor components of the moment of inertia tensor I in a three-
dimensional space is given by,

2 3

I 11 I 12 I 13

4 5 ;

I ¼ I 12 I 22 I 23 (A.131)

I 13 I 23 I 33

where the components are given by

ð
O ð

ð
O ð

x 2 þ x 3 Þrð x 1 þ x 3 Þrð I 11 ¼ x ; t Þ d v ; I 22 ¼ x ; t Þ d v ;

ð
O ð

ð
O ð

x 2 þ x 1 Þrð I 33 ¼ x ; t Þ d v ; I 12 ¼ x 1 x 2 Þrð x ; t Þ d v

ð
O ð

ð
O ð

I 13 ¼ x 1 x 2 Þrð x ; t Þ

d v

; I 23 ¼ x 2 x 3 Þrð x ; t Þ

d v

:

(A.130)

Example A.9.1

Determine the mass moment of inertia of a rectangular prism of homogeneous
material of density and side lengths a , b , and c about one corner. Select the

coordinate systems so that its origin is at one corner and let a , b , c represent the
distances along the x 1 , x 2 , x 3 axes, respectively. Construct the matrix of tensor
components referred to this coordinate system.

r

Solution: The integrations (A.132) yield the following results:

ð
O ð

ð
O ð

x 2 þ x 3 Þrð x 2 þ x 3 Þ

I 11 ¼ x ; t Þ d v ¼ r d x 1 d x 2 d x 3

ð

b;c
0 ; 0 ð

d x 2 d x 3 ¼ r

abc
3 ð

x 2 þ x 3 Þ b 2 c 2 ¼ a r þ Þ;

I 22 ¼ r

abc
3 ð

I 33 ¼ r

abc
3 ð

a 2 c 2 a 2 b 2 þ Þ; þ Þ;

ð
O ð

c ð

a:b

d x 1 d x 2 ¼ r

abc
4

I 12 ¼ x 1 x 2 Þrð x ; t Þ d v ¼r

0 ð

x 1 x 2 Þ ð ab Þ

0 ;

I 13 ¼ r

abc
4

I 23 ¼ r

abc
4

ð ac Þ; ð bc Þ;