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
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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