Biomedical Engineering Reference
In-Depth Information
Unlike the control law of Eq. (4.153), this control law does not contain an
impulsive component.
The time history of the displacement of body 1 corresponding to this
control is
⎧
⎨
V
2
t
2
4
D
3
2
D
3
V
Vt
−
if 0
≤
t
≤
,
x
1
(t)
=
(4.155)
if
t >
2
D
3
V
⎩
D
3
.
Subtract Eq. (4.155) from Eq. (4.150) to obtain the time history of the
motion of body 2 relative to body 1,
⎧
⎨
V
2
−
2
D
2
D
3
4
D
3
2
D
3
V
t
2
≤
≤
if 0
t
,
D
2
t
2
2
2
D
3
V
V
D
2
,
x
2
(t)
−
x
1
(t)
=
(4.156)
Vt
−
−
D
3
if
<t
≤
⎩
V
2
2
D
2
−
V
D
2
.
if
t
>
D
3
From this relation, the relative displacement of bodies 1 and 2 monotonically
increases from 0 to the minimum peak value
J
min
=
V
2
/(
2
D
2
)
−
D
3
.
1
4.2.3
Shock Isolation of a Multibody Object
Consider a generalization of the model defined in Section 4.2.2. Let body
2 (the object to be isolated) consist of a rigid housing and a finite number
of point masses that can move relative to the housing. These point masses
can be connected to each other and to the housing by elastic and damp-
ing members that may have linear or nonlinear characteristics. Figure 4.11
depicts the simplest of such models, in which the object is represented by
only two bodies, a housing and one point mass. The forces acting on the
point masses depend only on the displacements of these masses relative
to the housing. Let
m
be the mass of the housing,
μ
i
the magnitude of
the
i
th mass,
x
2
the displacement of the housing relative to the fixed ref-
erence frame,
y
i
the displacement of the
i
th mass relative to the housing,
y
[
y
1
, ..., y
n
]the
n
-vector of displacements of the masses relative to
the housing,
n
the number of the point masses, and
f
i
(
y
,
=
y
)
the total force
acting on the
i
th mass. The motion of this model is governed by the system
˙
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