Biomedical Engineering Reference
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(g) Rigid rotation (ccw).
A
(
t
)
¼
cos (
p
t
/2),
B
(
t
)
¼
cos(
p
t
/2),
C
(
t
)
¼
sin(
p
t
/
2),
D
(
t
)
¼
sin(
p
t
/2),
E
(
t
)
¼
0,
F
(
t
)
¼
0 and values of
t
¼
0, 1, 2, 3, 4.
2.1.2. Sketch the shape and position of the square with corners at (
1,
1), (1,
1),
0, 1, 2, 3, 4. The square is subjected to the
motion in (
2.10
) with the values of
A
(
t
),
B
(
t
),
C
(
t
),
D
(
t
),
E
(
t
), and
F
(
t
) being
those given in 2.1(f), the rigid rotation (clockwise) motion.
2.1.3. For the six motions of the form (
2.10
) given in Problem 2.1.1, namely 2.1.1
(a) through 2.1.1(f), compute the deformation gradient tensor
F
, its Jacobian
J
, and its inverse
F
1
. Discuss briefly the significance of each of the tensors
computed. In particular, explain the form or value of the deformation
gradient tensor
F
in terms of the motion.
2.1.4. Using the planar homogeneous deformation (
2.10
), with the values of
A
,
B
,
C
,
D
,
E
, and
F
calculated in Example 2.1.2, show that deformation (
2.10
)
predicts the final positions of the three markers when the initial marker
locations
(1, 1), and (
1, 1) at times
t
¼
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
ð
II
Þ¼
(0, 0),
ð
II
Þ¼
(1, 0), and
ð
II
Þ¼
(0, 1) are substituted into it.
2.1.5. In Example 2.1.2 an experimental technique in widespread use in the mea-
surement of the planar homogeneous motion of the deformable object was
described and a system of equations was set and solved to determine the
time-dependent parameters appearing in the equations describing the planar
homogeneous motion. Consider the same problem when the problem is not
planar, but three-dimensional. How many markers are necessary in three
dimensions and how must the markers be arranged so that they provide the
information necessary to determine the time-dependent parameters
appearing in the equations describing the three-dimensional homogeneous
motion? Explain the process.
2.2 Rates of Change and the Spatial Representation
of Motion
The velocity
v
and acceleration
a
of a particle
X
are defined by
X
fixed
;
X
fixed
;
2
¼
@w
@
¼
@
w
v
¼ _
x
a
¼ €
x
(2.24)
t
@
t
2
where
X
is held fixed because it is the velocity or acceleration of that particular
particle that is being determined. The
spatial description of motion
(as opposed
to the
material description of motion
represented by (
2.2
)) is obtained by solving
(
2.2
) for
X
,
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