Biomedical Engineering Reference
In-Depth Information
The solution to these six equations is
X ð 1 Þ
II x ð 2 Þ
ðtÞX ð 1 Þ
II x ð 3 Þ
ðtÞX ð 2 Þ
II x ð 1 Þ
ðtÞþX ð 2 Þ
II x ð 3 Þ
ðtÞþX ð 3 Þ
II x ð 1 Þ
ðtÞX ð 3 Þ
II x ð 2 Þ
ðtÞ
1
1
1
1
1
1
A
ð
t
Þ¼
;
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 2 Þ
I
x ð 1 Þ
2
X ð 3 Þ
I
x ð 1 Þ
2
X ð 1 Þ
I
x ð 2 Þ
2
X ð 3 Þ
I
x ð 2 Þ
2
X ð 1 Þ
I
x ð 3 Þ
2
X ð 2 Þ
I
x ð 3 Þ
2
ð
t
Þ
ð
t
Þ
ð
t
Þþ
ð
t
Þþ
ð
t
Þ
ð
t
Þ
B
ð
t
Þ¼
;
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 1 I x ð 2 Þ
X ð 2 I x ð 1 Þ
X ð 1 I x ð 3 Þ
X ð 2 I x ð 3 Þ
X ð 3 I x ð 1 Þ
X ð 3 I x ð 2 Þ
CðtÞ¼
ð
t
Þþ
ð
t
Þþ
ð
t
Þ
ð
t
Þ
ð
t
Þþ
ð
t
Þ
1
1
1
1
1
1
;
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 1 Þ
II
x ð 2 Þ
2
X ð 2 Þ
II
x ð 1 Þ
2
X ð 1 Þ
II
x ð 3 Þ
2
X ð 2 Þ
II
x ð 3 Þ
2
X ð 3 Þ
II
x ð 1 Þ
2
X ð 3 Þ
II
x ð 2 Þ
2
ð
t
Þ
ð
t
Þ
ð
t
Þþ
ð
t
Þþ
ð
t
Þ
ð
t
Þ
D
ð
t
Þ¼
;
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 1 Þ
II X ð 2 I x ð 3 Þ
X ð 1 Þ
II X ð 3 I x ð 2 Þ
X ð 1 I X ð 2 Þ
II x ð 3 Þ
ð
t
Þ
ð
t
Þ
ð
t
Þ
1
1
1
E
ð
t
Þ¼
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 2 Þ
II X ð 3 Þ
x ð 1 Þ
1
X ð 2 Þ
II X ð 3 Þ
x ð 2 Þ
1
X ð 2 Þ
I
X ð 3 Þ
II
x ð 1 Þ
1
ð
t
Þþ
ð
t
Þ
ð
t
Þ
I
I
þ
;
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 1 Þ
II X ð 3 Þ
x ð 2 Þ
2
X ð 2 Þ
II X ð 3 Þ
x ð 1 Þ
2
X ð 2 Þ
I
X ð 1 Þ
II
x ð 3 Þ
2
ð
t
Þþ
ð
t
Þþ
ð
t
Þ
I
I
F
ð
t
Þ¼
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
X ð 2 Þ
II X ð 3 Þ
X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
þ
þ
I
I
I
I
I
I
X ð 2 Þ
II X ð 1 I x ð 3 Þ
X ð 3 Þ
II X ð 2 I x ð 1 Þ
X ð 1 I X ð 3 Þ
II x ð 2 Þ
ð
t
Þ
ð
t
Þþ
ð
t
Þ
2
2
2
þ
:
X ð 1 Þ
II X ð 2 Þ
X ð 1 Þ
II X ð 3 Þ
X ð 2 Þ
II X ð 1 Þ
þ X ð 2 Þ
II X ð 3 Þ
þ X ð 3 Þ
II X ð 1 Þ
X ð 3 Þ
II X ð 2 Þ
I
I
I
I
I
I
Example 2.1.3
Consider again the experimental technique described in Example 2.1.2, but in this
case a deformation rather than a motion, Fig. 2.4 . Suppose that the initial locations
of the markers are recorded relative to the fixed laboratory frame of reference as
ð
X ð 1 Þ
I
, X ð 1 Þ
X ð 2 Þ
I
, X ð 2 Þ
X ð 3 Þ
I
, X ð 3 Þ
(0, 1). The deformed
locations of the three markers relative to the same fixed laboratory frame of
reference are
II Þ¼
(0, 0),
ð
II Þ¼
(1, 0), and
ð
II Þ¼
x ð 1 Þ
1
, x ð 1 2 Þ¼
x ð 2 Þ
1
, x ð 2 2 Þ¼
x ð 3 Þ
1
, x ð 3 2 Þ¼
(2.5,
3.75). From these data the constant coefficients A , B , C , D , E , and F of the
homogeneous planar deformation ( 2.10 ) are determined.
Solution : The solution for the motion coefficients A ( t ), B ( t ), C ( t ), D ( t ), E ( t ), and F ( t )
obtained in Example 2.1.2 may be used in the solution to this problem. One simply
assigns the time-dependent positions in the formulas for A ( t ), B ( t ), C ( t ), D ( t ), E ( t ),
and F ( t ) to be fixed rather than time dependent by setting
ð
(1, 2),
ð
(2, 3.25), and
ð
x ð 1 Þ
1
, x ð 1 Þ
2
x ð 1 Þ
1
ð
ð
t
Þ
ð
t
ÞÞ ¼ ð
,
x ð 1 Þ
2
,( x ð 2 Þ
1
, x ð 2 Þ
2
x ð 2 1 , x ð 2 Þ
x ð 3 Þ
1
, x ð 3 Þ
2
x ð 3 1 , x ð 3 Þ
Þ
ð
t
Þ
ð
t
ÞÞ¼ð
Þ
, and
ð
ð
t
Þ
ð
t
ÞÞ¼ð
Þ
. The coefficients
2
2
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