Biomedical Engineering Reference
In-Depth Information
and, substituting for
F
using (
2.42
),
r
O
u
¼r
u
þ½r
u
½r
u
thus, to neglect terms of second order,
r
O
u
r
u
:
(2.46)
For infinitesimal motions, the movement of boundaries due to motion is
neglected because the small movement is equivalent to the difference in the use
of material and spatial coordinates, which is insignificant. Therefore in all the
following considerations of infinitesimal motions the coordinates
x
will be used
without reference to their material or spatial character, because the result is correct
independent of their character. The second important conclusion is that, for infini-
tesimal motions,
F
has the representation
T
F
¼
1
þ½r
u
ð
x
;
t
Þ
:
(2.47)
In the special case when the infinitesimal motion is a rigid object rotation,
u
]
T
. The requirement that
Q
be orthogonal,
Q
T
F
¼
Q
and
Q
¼
1
þ
[
∇
Q
¼
T
T
u
]
T
)
u
]
T
)
T
u
]
T
Q
Q
¼
1
,
Q
Q
¼
(
1
þ
[
∇
(
1
þ
[
∇
¼
1
þ
[
∇
u
)
T
þ
(
∇
u
)
þ
(
∇
(
∇
u
)
¼
1
,meansthat
T
ðr
u
Þ
þr
u
¼
;
0
(2.48)
u
)
T
since (
u
) represents terms of the second order terms that are
neglected. Defining the symmetric and skew symmetric parts of
∇
(
∇
∇
u
as
E
and
Y
,
T
T
E
¼ð
1
=
2
Þððr
u
Þ
þr
u
Þ;
Y
¼ð
1
=
2
Þððr
u
Þ
r
u
Þ;
(2.49)
it is seen from (
2.48
) that
E
must be zero when the infinitesimal motion is a rigid
object rotation. It may also be seen that the orthogonal rotation
Q
characterizing the
infinitesimal rigid object rotation is given by
Q
¼
1
þ
Y
;
(2.50)
Y
T
, and
YY
T
is a second
order term, since it is a square of the coefficients (
2.37
) which are neglected
compared to the values of
Y
.
Returning to the total infinitesimal motion, the definitions (
2.49
)of
E
and
Y
may
be used to rewrite (
2.42
)as
where
Y
, defined by (
2.49
), is skew symmetric,
Y
¼
F
¼
1
þ
E
þ
Y
:
(2.51)
It has been established that
F
represents only the rotational and deformational
motion because the translational portion of the motion, being independent of the
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