Biomedical Engineering Reference
In-Depth Information
h
4
24
10
03
I
Area
Q
T
Q
¼
:
Æ
The parallel axis theorem for the moment of inertia matrix
I
is derived by
considering the mass moment of inertia of the object
O
about two parallel axes,
I
ee
about
e
and
I
e
0
e
0
about
e
0
.
I
e
0
e
0
is given by
I
e
0
e
0
¼ e
0
I
0
e
0
;
(A.135)
where the moment of inertia matrix
I
0
is given by
ð
O
fð
I
0
¼
x
0
x
0
Þ
x
0
x
0
Þgrð
x
0
;
d
v
0
:
1
ð
t
Þ
(A.136)
Let
d
be a vector perpendicular to both
e
and
e
0
and equal in magnitude to the
perpendicular distance between
e
and
e
0
, thus
x
0
¼
0, and
e
0
·
d
x
þ
d
,
e
·
d
¼
¼
0.
Substituting
x
0
¼
d
in
I
0
, it follows that
x
þ
ð
O
fð
I
0
¼
x
x
þ
d
d
þ
2
x
d
Þ
1
grð
x
;
t
Þ
d
v
ð
O
fð
x
x
þ
d
d
þ
d
x
þ
x
d
Þgrð
x
;
t
Þ
d
v
(A.137)
or if (A.137) is rewritten so that the constant vector
d
is outside the integral signs,
0
@
1
A
rðx; tÞ
d
v
I
0
¼ 1
ð
ð
O
rðx; tÞ
d
v þ 1
2
d
ð
O
fðx xÞrðx; tÞ
d
v þ 1ðd dÞ
x
O
0
@
1
A
d
ð
O
fðx xÞrðx;
ð
O
rðx;
ð
ð
t
Þ
d
v
ðd dÞ
t
Þ
d
v
d
xrðx;
t
Þ
d
v
xrðx;
t
Þ
d
v
O
O
then recalling the definitions (A.119) of the mass
M
O
of
O
and (A.120) of the center
of mass
x
cm
of the object
O
, this result simplifies to
I
0
¼
I
þfð
d
d
Þ
1
ð
d
d
Þg
M
O
þ
2
M
O
1
ð
x
cm
d
Þ
M
O
ð
d
x
cm
þ
x
cm
Þ:
(A.138)
d
Thus, when the origin of coordinates is taken at the center of the mass, it follows
that
x
cm
¼
0 and
I
0
¼
I
cm
þfð
d
d
Þ
1
ð
d
d
Þg
M
O
:
(A.139)
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