Biomedical Engineering Reference
In-Depth Information
isolation is ineffective if the deceleration or acceleration path of the base
substantially exceeds the rattlespace.
The most important characteristics of the system's behavior associated
with shock isolation are the force acting on the object to be protected
and the time history of the motion of the object relative to the base. The
motion of the object relative to the base is governed by the differential
equation
m
y
¨
=
u
−
mv(t),
y
=
x
−
z,
(2.40)
where the coordinate
y
characterizes the displacement of the object relative
to the base. To obtain Eq. (2.40), subtract the second relation of Eq. (2.24)
multiplied by
m
from the first equation. In accordance with Eq. (2.26), the
variables
y
and
y
should be subjected to zero initial conditions
˙
y(
0
)
=
0
,
y(
0
)
˙
=
0
.
(2.41)
The solution of Eq. (2.40) subject to the initial conditions of Eq. (2.41)
can be expressed by the integral
ξ)
u(ξ)
m
v(ξ)
dξ.
t
y(t)
=
(t
−
−
(2.42)
0
Substituting
u/m
of Eq. (2.30) and
v
of Eq. (2.27) into Eq. (2.42) gives
⎧
⎨
w)t
2
2
(a
−
V
a
,
if 0
≤
t
≤
wt
2
2
V
2
2
a
V
a
V
w
,
y
=
(2.43)
Vt
−
−
if
<t
≤
⎩
a
w
−
1
V
2
2
a
if
t
>
V
w
.
0to
(V
2
/
2
a)
The coordinate
y
monotonically increases from 0 at
t
=
(a/w
V/w
and, hence, the rattlespace is expressed by Eq. (2.33).
The integral of Eq. (2.42) can be expressed as the difference of two
integrals,
−
1
)
at
t
=
t
t
ξ)
u(ξ)
m
=
−
−
−
y(t)
(t
dξ
(t
ξ) v(ξ) dξ.
(2.44)
0
0
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