Biomedical Engineering Reference
In-Depth Information
α
that was discussed previously (these angles do not need to be equal). To quantify the
angular velocity, the difference in velocity at points A and B must first be determined.
Since this element is a differential element, the velocity difference between A and B is
/
β
Δ
v
y
.
(This velocity acts in the y-direction even though the original line segment was in the
x-direction.) Similarly, the difference in velocity from point A to point C would be
v
x
.
This makes the assumption that the x-velocity at A is defined as v
x
and the y-velocity at A
is defined as v
y
for both differences. Thus, the change in displacement for point A, B and
C would be 0,
Δ
v
y
1
@
v
y
@
v
x
1
@
v
x
@
t, respectively. To formulate
these displacements (denoted as s), a change in velocity is multiplied by a change in time
(i.e.,
x
Δ
x
2
v
y
Δ
t and
2
y
Δ
y
2
v
x
Δ
@
v
y
@
Δ
s
5Δ
v
Δ
t). Also, the change in velocity is defined by the velocity at B v
y
1
Δ
x
x
minus the velocity at A(v
y
). The partial derivative term for the velocity at B arises from the
original
0 for a differential element. The dis-
placement of C follows the same formulation but is negative because the point moves in
the negative x-axis direction. Now take the limit as time approaches zero of the change in
the angle (
Δ
v
y
, but the assumption is made that
Δ
x
-
α
β
or
) (from
Equation 2.22
to obtain
Equation 2.23
), to obtain the elements
angular velocity
0
1
t
0
1
v
y
1
@
v
y
@
@
v
y
@
@
A
Δ
x
Δ
x
2
v
y
Δ
x
x
Δ
x
Δ
t
@
A
5
@
0
Δα
BB
0
=
AB
0
Δ
1
Δ
v
y
@
lim
Δ
lim
Δ
5
lim
Δ
5
lim
Δ
t
D
t
-
Δ
t
Δ
t
t
Δ
x
x
t
0
t
0
t
0
-
-
-
0
@
1
A
Δ
t
0
@
1
A
52
@
1
@
v
x
@
@
v
x
@
2
y
Δ
2
Δ
v
x
y
v
x
x
y
Δ
y
Δ
t
0
Δβ
CC
0
=
AC
0
1
Δ
v
x
@
lim
Δ
lim
Δ
5
lim
Δ
5
lim
Δ
t
2
t
D
Δ
Δ
t
Δ
t
Δ
y
y
t
-
t
-
0
t
-
0
t
-
0
ð
2
:
23
Þ
Therefore, angular velocity about
the z axis will equal
the average of these two
quantities
1
2
@
v
y
@
x
2
@
v
x
@
ω
z
5
ð
2
:
24
Þ
y
A similar analysis conducted about the other two axes would yield
y
2
@
v
y
@
1
2
@
v
z
@
1
2
@
v
x
@
z
2
@
v
z
@
ω
x
5
;
and
ω
y
5
ð
2
:
25
Þ
z
x
Combining terms for the angular velocity rotation vector becomes
-
-
-
y
2
@
v
y
@
@
v
y
@
1
2
@
v
z
@
@
v
x
@
z
2
@
v
z
@
x
2
@
v
x
@
1
-
2
curl
-
ω
5
1
1
5
ð
2
:
26
Þ
z
x
y
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