Biomedical Engineering Reference
In-Depth Information
properties of the material of the cushion. To provide further reduction in
the injury risk, an active, controlled shock isolator can be used instead of
the cushion.
The technique of Chapter 4 will be used to analyze the limiting per-
formance that can be provided by an active isolator integrated into the
injury reduction system. The object to be protected will be represented by
a two-degree-of-freedom system.
The governing equations of the dynamical system comprising the seat
pan (identified as the base) and the seated person (occupant) are
m
1
X
1
C( X
1
+
−˙
x
2
)
+
K(X
1
−
x
2
)
=
m
1
g,
(5.1)
m
2
x
2
−
C( X
1
−
x
2
)
−
K(X
1
−
x
2
)
=
F(t)
+
m
2
g,
(5.2)
z
¨
=−
v(t),
(5.3)
where the coordinate
z
measures the displacement of the seat pan, while
X
1
and
x
2
measure the displacements of the upper and lower torsos of the seated
person model, respectively, relative to a fixed (inertial) reference frame;
m
1
and
m
2
represent the masses of the upper and lower torsos, respectively;
g
is the acceleration due to gravity;
C
and
K
are the damping and stiffness
coefficients of the vertebral column;
v(t)
is the seat pan deceleration pulse;
and
F
is the control force acting between the seat pan and the lower torso
that is exerted on the lower torso. The coordinate axes
z
,
X
1
,and
x
2
are
directed vertically downward. The origins of the coordinates are chosen so
that the spring, modeling the elastic properties of the spine, is unstrained
for
X
1
=
x
2
.
Introduce the new variables
m
1
g
K
F
x
1
=
X
1
−
,
F
=
+
(m
1
+
m
2
)g,
(5.4)
in terms of which the system of Eqs. (5.1) - (5.3) becomes
m
1
x
1
¨
+
C(
x
1
˙
−˙
x
2
)
+
K(x
1
−
x
2
)
=
0
,
(5.5)
m
2
x
2
−
C(x
1
−
x
2
)
−
K(x
1
−
x
2
)
=
F(t),
(5.6)
z
=−
v(t).
(5.7)
x
2
corre-
sponds to the spring statically compressed by the weight of the upper torso,
including the neck and head (represented by the mass
m
1
). The force
F
is
the sum of the control force
The origin of the coordinate
x
1
is chosen so that the state of
x
1
=
F
and the total weight of the object. Along
F
, the force
F
will be referred to as the control force.
with the force
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