Biomedical Engineering Reference
In-Depth Information
by a viscous elastic system, this force-velocity curve should have been linear, the loss of
force being always proportional to the velocity. The slope of the curve would then represent
the coefficient of viscosity.” Essentially the same experiment was repeated for rectus eye
muscle by Close and Luff in 1974 with similar results.
The classic force-velocity experiment was performed to test the viscoelastic model for
muscle as described previously in Section 13.5.4. Under these conditions, it was first
assumed that the inertial force exerted by the load during isotonic shortening could be
ignored. The second assumption was that if mass was not reduced enough by the lever ratio
(enough to be ignored), then taking measurements at maximum velocity provided a
measurement at a time when acceleration is zero, and, therefore, inertial force equals zero.
If these two assumptions are valid, then the experiment would provide data free of the
effect of inertial force as the gravity force is varied. Both assumptions are incorrect. The first
assumption is wrong, since the inertial force is never minimal (minimal would be zero) and
therefore has to be taken into account. The second assumption is wrong, since, given an
inertial mass not equal to zero, the maximum velocity depends on the forces that act prior
to the time of maximum velocity. The force-velocity relationship is carefully reexamined
with the inertial force included in the analysis in this section.
The dynamic characteristics for the linear muscle model are described with a force-
velocity curve calculated via the lever system presented in Figure 13.20 and according to
the isotonic experiment. For the rigid lever, the displacements
x
1
and
x
3
are directly propor-
tional to the angle y
1
and to each other, such that
¼
x
1
¼
x
1
3
y
1
ð
13
:
42
Þ
d
d
3
The equation describing the torques acting on the lever is given by
3
‥
1
¼
d
1
K
se
x
2
x
1
2
Mgd
3
þ
Md
ð
Þ
d
1
B
2
ð
x
2
x
1
Þ
ð
13
:
43
Þ
The equation describing the forces at node 2, inside the muscle, is given by
F
¼
K
lt
x
þ
B
x
2
þ
B
ð
x
2
x
Þ
K
se
x
ð
x
Þ
ð
13
:
44
Þ
2
1
2
1
2
1
Equation (13.43) is rewritten by removing y
1
using Eq. (13.42), hence
2
x
Mg
d
d
1
þ
M
d
3
3
d
1
‥
1
¼
K
se
x
2
x
1
ð
Þ
B
2
ð
x
2
x
1
Þ
ð
13
:
45
Þ
Ideally, to calculate the force-velocity curve for the lever system,
x
1
(
t
) is found first. Then
‥
1
x
1
V
max
¼
x
1
ð
T
Þ
ðÞ
and
ðÞ
are found from
x
1
(
t
). Finally, the velocity is found from
, where
time
is the time it takes for the muscle to shorten to the stop, according to the experi-
mental conditions of Close and Luff. While this velocity may not be maximum velocity
for all data points, the symbol
T
V
max
is used to denote the velocities in the force-velocity
curve for ease in presentation. Note that this definition of velocity differs from the
Fenn and Marsh definition of velocity. Fenn and Marsh denoted maximum velocity as
V
‥
1
¼
x
:
It should be noted that this is a third-order system and the solution for
ð
T
Þ
, where time
T
is found when
x
ð
T
Þ¼
0
max
1
) is not
trivial and involves an exponential approximation (for an example of an exponential
x
1
(
t
