Biomedical Engineering Reference
In-Depth Information
it is rigidly attached to the base to that transmitted to the isolated object.
This ratio is an analogue of the ratio
a/w
in Eq. (2.39), which measures the
isolation effectiveness in terms of the ratio of the rattlespace allotted to the
object to the deceleration path of the base. For the case under consideration,
the rattlespace is defined by the expression corresponding to
t > τ
1
on the
right-hand side of Eq. (2.54),
f
0
u
0
1
.
f
0
τ
2
2
M
μ
M
−
R
=
(2.56)
This relation can be solved for
μf
0
/(Mu
0
)
to obtain
f
0
μ
u
0
M
=
2
MR
f
0
τ
2
.
1
+
(2.57)
If the entire system (the base and the object) is decelerated to a complete
stop by the impact, then
(M
+
m)V
f
0
τ
=
.
(2.58)
This relation follows from Newton's second law for the system's center of
mass,
m) X
(M
+
=
F,
(2.59)
where
X
is the coordinate of the center of mass defined by
mx
+
Mz
M
X
=
.
(2.60)
+
m
If the base and the object move with the same velocity
V
before the impact
occurs, Eq. (2.59) is subject to the initial conditions
X(
0
)
X(
0
)
=
0
,
=
V.
(2.61)
Then the solution of this equation for
F
defined by Eq. (2.50) yields
⎨
f
0
t
2
2
(M
+
m)
Vt
−
if 0
≤
t
≤
τ,
X(t)
=
(2.62)
V
t
+
⎩
f
0
τ
2
2
(M
+
m)
f
0
τ
M
+
m
−
if
t > τ.
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