Biomedical Engineering Reference
In-Depth Information
3.2.5.3 Balance of Angular Momentum
The second basic law of mechanics is the principle of conservation of angular
momentum
d
0
dd
0
dt
M
0
¼
d
0
with
ð
3
:
121
Þ
stating that the sum of all external moments M
0
acting on a body and with respect
to the fixed point 0 are equal to the time rate of change of the angular momentum
vector d
0
, with respect to the same point. The resulting moment vector M
0
is thus
M
0
¼
M
0
þ
M
0
ð
3
:
122
Þ
with
M
0
¼
Z
V
M
0
¼
Z
A
x
kdV
and
x
t
n
dA
:
ð
3
:
123
Þ
In (
3.123
), M
0
is the vector of volume moments and M
0
the vector of surface
moments where x is the position vector pointing to the volume element dV in the
CCFG (cf. Fig.
3.20
). The vector of angular momentum is defined by
d
0
:
¼
Z
V
qx
vdV
:
ð
3
:
124
Þ
Substitution of (
3.122
)to(
3.124
)in(
3.121
) and using C
AUCHY
's lemma (
3.93
)
leads to the global balance of angular momentum
Z
x
kdV
þ
Z
Z
x
ð
n
S
Þ
dA
¼
d
dt
qx
vdV
:
ð
3
:
125
Þ
V
A
V
Similar to the balance of linear momentum, the surface integral in (
3.125
) must
be transformed into a volume integral using the G
AUSS
' integral theorem. With the
identities x
ð
n
S
Þ¼ð
n
S
Þ
x
¼
n
ð
S
x
Þ
and
r
r
ð
S
x
Þ¼ðr
r
S
Þ
x
þ
ð
3
Þ
S
x
ðr
r
S
Þþ
ð
3
Þ
S as well as the G
AUSS
' theorem, the surface
integral can be reformulated to
R
A
x
ð
n
S
Þ
dA
¼
R
V
r
r
ð
S
x
Þ
dV
¼
R
V
½
x
ðr
r
S
Þ
ð
3
Þ
S
dV. Furthermore, analogue to the differentiation done in
(
3.117
) and using
ð
dx
=
dt
Þ
v
v
v
¼
0 for the right-hand side of (
3.125
), it
follows
ð Þ
dV
¼
R
V
qx
vdV
:
Substitution of the previous terms in (
3.125
) and appropriate factoring (note
that the term in parentheses in (
3.126
) represents the balance of linear momentum
(
3.120
) and thus vanishes!) leads to
dt
R
V
qx
vdV
¼
R
V
q x
v
þ
x
d
