Biomedical Engineering Reference
In-Depth Information
T
x
¼
Q
ð
t
Þ
X
þ
h
ð
t
Þ;
Q
ð
t
Þ
Q
ð
t
Þ
¼
1
for all
X
O
ð
Þ:
0
(2.5)
A motion of the form (
2.2
) is said to be a planar motion if the particles always
remain in the same plane. In this case (
2.2
) becomes
x
1
¼ w
1
ð
X
I
;
X
II
;
t
Þ;
x
2
¼ w
2
ð
X
I
;
X
II
;
t
Þ;
x
3
¼
X
III
:
(2.6)
Another subset of the motion is a deformation of an object from one configura-
tion to another, say from the configuration at
t
¼
0 to the configuration at
t
¼
t
*. In
this case the motion (
2.2
) becomes a deformation
x
¼ Cð
X
Þ
for all
X
O
ð
0
Þ;
(2.7)
where
t
Þ
Cð
X
Þ¼wð
X
;
for all
X
O
ð
0
Þ:
(2.8)
A 3D motion picture or 3D video of the motion of an object may be represented
by a subset of the motion (
2.2
) because a discrete number of images (frames) per
second are employed,
x
¼ wð
X
;
n
=zÞ
for all
X
O
ð
0
Þ;
n
¼
0
;
1
;
2
; ...;
(2.9)
z
where
is the number of images (frames) per second.
Example 2.1.1
Consider the special case of a planar motion given by
x
1
¼
A
ð
t
Þ
X
I
þ
C
ð
t
Þ
X
II
þ
E
ð
t
Þ;
x
2
¼
D
ð
t
Þ
X
I
þ
B
ð
t
Þ
X
II
þ
F
ð
t
Þ;
x
3
¼
X
III
;
(2.10)
where
A
(
t
),
B
(
t
),
C
(
t
),
D
(
t
),
E
(
t
),
F
(
t
) are arbitrary functions of time. Further
specialize this motion by the selections
A
ð
t
Þ¼
1
þ
t
;
C
ð
t
Þ¼
t
;
E
ð
t
Þ¼
3
t
;
B
ð
t
Þ¼
1
þ
t
;
D
ð
t
Þ¼
t
;
F
ð
t
Þ¼
2
t
;
(2.11)
for
A
(
t
),
B
(
t
),
C
(
t
),
D
(
t
),
E
(
t
), and
F
(
t
). With these selections the motion becomes
x
1
¼ð
þ
t
Þ
X
I
þ
tX
II
þ
3
t
;
x
2
¼
tX
I
þð
þ
t
Þ
X
II
þ
2
t
;
x
3
¼
X
III
:
1
1
(2.12)
The problem is to find the positions of the unit square whose corners are at the
material points (
X
I
,
X
II
)
¼
(0, 0), (
X
I
,
X
II
)
¼
(1, 0), (
X
I
,
X
II
)
¼
(1, 1), (
X
I
,
X
II
)
¼
(0, 1) at times
t
¼
1 and
t
¼
2.
Solution
: For convenience let the spatial (
x
1
,
x
2
,
x
3
) and material (
X
I
,
X
II
,
X
III
)
coordinate systems coincide and then consider the effect of the motion (
2.12
) on the
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