Biomedical Engineering Reference
In-Depth Information
1
1
þ
2
t
fð
t
2
X
I
¼
þ
t
Þ
x
1
tx
2
3
t
g;
1
1
t
2
X
II
¼
2
t
f
tx
1
þð
þ
t
Þ
x
2
þ
2
t
g;
X
III
¼
x
3
1
1
þ
then, substituting these expressions into the previous equations for the velocities,
the spatial representation of this motion is obtained:
1
1
v
1
¼
2
t
f
x
1
þ
x
2
þ
3
þ
t
g;
v
2
¼
2
t
f
x
1
þ
x
2
þ
2
t
g;
v
3
¼
0
:
1
þ
1
þ
It is known from the first calculation in this example that this motion is one of
zero acceleration,
0 . This may be verified by calculating the
acceleration of the spatial representation of the motion above using the material
time derivative (
2.29
), thus
€
x
1
¼ €
x
2
¼ €
x
3
¼
a
1
¼
@
v
1
@
v
1
@
v
1
v
2
@
v
1
v
3
@
v
1
t
þ
x
1
þ
x
2
þ
@
@
@
x
3
2
1
1
¼
2
f
x
1
þ
x
2
þ
3
þ
t
gþ
2
t
þ
2
f
2
ð
x
1
þ
x
2
Þþ
5
g¼
0
;
1
þ
ð
þ
2
t
Þ
ð
þ
2
t
Þ
1
1
a
2
¼
@
v
2
@
v
1
@
v
2
v
2
@
v
2
v
3
@
v
2
t
þ
x
1
þ
x
2
þ
@
@
@
x
3
2
1
1
¼
2
f
x
1
þ
x
2
þ
2
t
g
2
t
þ
2
f
2
ð
x
1
þ
x
2
Þþ
5
g¼
0
:
þ
1
ð
1
þ
2
t
Þ
ð
1
þ
2
t
Þ
The tensor of velocity gradients
L
for the motion (
2.12
)isobtainedby
substituting the spatial representation for the motion obtained above into
(
2.31
); thus
2
4
3
5
:
110
110
000
1
L ¼
1
þ
2
t
The rate-of-deformation tensor
D
for this motion is equal to
L
. The spin tensor
W
is zero for this motion.
Problems
2.2.1. For the first six motions of the form (
2.10
) given in Problem 2.1.1, namely
2.1.1(a) through 2.1.1(f), determine the velocity and acceleration in the
material (Lagrangian) representation, the velocity and acceleration in the
spatial (Eulerian) representation, and the three tensors
L
,
D
, and
W
. Discuss
briefly how these algebraic calculations relate to the geometry of the motion.
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