Biomedical Engineering Reference
In-Depth Information
⎨
V
U
1
V
U
≤
3
4
V
U
,
V
t
+
for
−
t<
−
2
U
1
V
2
3
4
V
U
≤
t<
0
,
−
V
t
+
for
−
x(t)
˙
=
1
V
t
(3.152)
⎩
V
2
2
U
t<
V
−
for 0
≤
2
U
,
V
2
U
.
≥
0
for
t
These functions are presented in Figs. 3.10 and 3.11 in a normalized form.
The curve of Fig. 3.10 plots
Ux/V
2
versus
Ut/V
and the curve of Fig. 3.11
shows
x/V
versus
Ut/V
.
From Eqs. (3.151) and (3.152) and the curves of Figs. 3.10 and 3.11 it
can be observed that the object moves in the direction of the anticipated
shock pulse during the time interval
˙
−
V/U
≤
t<
−
V/(
2
U)
;attheinstant
=−
t
V/(
2
U)
, the velocity vanishes and the displacement has a maximum
Ux
V
2
0.06
0.03
0.00
-0.03
-0.06
-1.0
-0.5
0.0
0.5
1.0
Ut/V
FIGURE 3.10
Minimization of the peak displacement magnitude: normalized time his-
tory of the relative displacement of the object.
x
V
0.50
0.25
0.00
-0.25
-0.50
-1.0
-0.5
0.0
0.5
1.0
Ut/V
FIGURE 3.11
Minimization of the peak displacement magnitude: normalized time his-
tory of the relative velocity of the object.
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