Biomedical Engineering Reference
In-Depth Information
For the particular case of
F(t)
specified by Eq. (2.50), we have
μ
M
f
0
.
max
t
|
u(t)
|=
(2.52)
The peak magnitude of the force transmitted to the object can be reduced
by means of shock isolation. To do this, it is necessary that
<f
0
μ/M
during the entire time of motion. Assume that the shock isolator responds
to the rectangular shock pulse of Eq. (2.50) with the rectangular pulse
|
u
|
−
u
0
if 0
≤
t
≤
τ
1
,
u(t)
=
(2.53)
0 f
t > τ
1
,
where
u
0
is a positive constant satisfying the inequality
u
0
<f
0
μ/M
and
τ
1
is a positive time instant at which the relative velocity of the object,
y
,
vanishes. Substituting Eqs. (2.50) and (2.53) into Eqs. (2.48) or (2.49) gives
˙
⎧
⎨
⎩
f
0
M
−
t
2
1
2
u
0
μ
if 0
≤
t
≤
τ,
tτ
−
τ
2
2
f
0
M
u
0
2
μ
t
2
−
if
τ<t
≤
τ
1
,
y
=
(2.54)
f
0
u
0
1
f
0
τ
2
2
M
μ
M
−
if
t > τ
1
,
where
f
0
u
0
μ
M
τ.
τ
1
=
(2.55)
Since the isolation must reduce the load transmitted to the object, the
inequality
u
0
<f
0
μ/M
must hold, and hence
τ
1
> τ
.
Equation (2.54) for a dynamic disturbance is similar to Eq. (2.43) for
a kinematic disturbance. This similarity can be observed from the analogy
between the relations of Eqs. (2.40), (2.27), and (2.30) and those of Eqs.
(2.47), (2.50), and (2.53), respectively. This analogy implies that to obtain
Eq. (2.55) from Eq. (2.43) one should replace
a
,
w
,and
V
in the latter
equation by
f
0
/M
,
u
0
/μ
,and
f
0
τ/M
, respectively.
A relation similar to that of Eq. (2.39) will be derived to evaluate the
effectiveness of isolation for a dynamic disturbance. This effectiveness is
measured by the ratio
μf
0
/(Mu
0
)
, which, in accordance with Eqs. (2.52)
and (2.53), characterizes the ratio of the force transmitted to the object when
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