Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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194
Mechanical shock
-
pre-
and
post-shocks, possibly
of
equal durations.
Figure
7.2.
Possibilities
for
pre-
and
post-shocks positioning
Another parameter
is the
shape
of
these pre-
and
post-shocks,
the
most used
shapes being
the
triangle,
the
half-sine
and the
rectangle.
Figure
7.3.
Shapes
of
pre-
and
post-shock pulses
Due
to
discontinuities
at the
ends
of the
pulse,
the
rectangular compensation
is
seldom satisfactory [SMA 85].
One
often
prefers
a
versed-sine applied
to all the
signal (Hann window)
which
has the
advantages
of
being zero
and
smoothed
at the
ends (first zero derivative)
and to
present symmetrical pre-
and
post-shocks.
In
all the
cases,
the
amplitude
of
pre-
and
post-shocks
must remain small
with
respect
to
that
of the
principal
shock
(preferably lower than approximately 10%),
in
order
not to
deform
too
much
the
temporal signal
and
consequently,
the
shock
spectrum.
For a
given shape
of
pre-
and
post-shocks,
this choice thus imposes
the
duration.
Generation
of
shocks using shakers
195
7.5.2.
Pre-
or
post-shock
As
an
example, treated below
is the
case
of a
terminal peak
saw
tooth shock
pulse
(amplitude unit, duration equal
to 1)
with
rectangular pre- and/or post-shocks
(ratio
p of the
absolute values
of the
pre-
and
post-shocks
amplitude
and of the
principal
shock amplitude equal
to
0.1).
Figure 7.4.
Terminal
peak
saw
tooth with rectangular
pre-
and
post-shocks
Figure
7.4
shows
the
signal
as a
function
of
time.
The
selected parameter
is the
duration
i
l
of
the
pre-shock.
Figure 7.5.
Influence
of
pre-shock duration
on
velocity during
the
shock
It
is
important
to
check that
the
velocity
is
always zero
at the
beginning
and the end
of the
shock (Figure 7.5). Between these
two
limits,
the
velocity remains positive
196
Mechanical
shock
when
there
is
only
one
post-shock
(TJ = 0) and
negative
for a
pre-shock alone
(T!
=
5.05).
Figure
7.6.
Influence
of
pre-shock duration
on
displacement during
the
shock
Figure
7.6
shows
the
displacement corresponding
to
this movement
for the
same
values
of the
duration
tj of the
pre-shock between
TJ = 0 and TJ =
5.05
s.
Figure
7.7
shows that
the
residual displacement
at the end of the
shock
is
zero
for
T!
» 2.4 s. The
largest displacement during shock, envelope
of the
residual
displacement
and of the
maximum displacement,
is
given according
to TJ in
Figure
7.8
(absolute values). This displacement
has a
minimum
at t1 « 2s.
Figure 7.7.
Influence
of
the
pre-shock
duration
on the
residual displacement
Figure 7.8.
Influence
of
the
pre-shock
duration
on the
maximum displacement
Generation
of
shocks
using
shakers
197
Figure 7.9.
TPS
pulse with pre-shock alone
(1) and
post-shock alone
(2)
If
we
compare
the
kinematics
of the
movements
now
corresponding
to the
realization
of a TPS
shock with only
one
pre-shock
(1) and
only
one
post-shock (2),
we
note,
from
Figures
7.9 to
7.11, that [YOU 64]:
- the
peak amplitude
of
the
velocity
is (in
absolute value) identical;
- in
(2),
the
acceleration peak takes place when
the
velocity
is
very large.
It is
thus
necessary
to be
able
to
provide
the
maximum force when
the
velocity
is
significant
[MIL 64];
- in (1) to the
contrary,
the
velocity
is at a
maximum when acceleration
is
zero.
Figure
7.10.
Velocity
curve with pre-shock
alone
(1) and
post-shock alone
(2)
Figure
7.11.
Displacement curve with pre-
shock alone
(I) and
post-shock alone
(2)
Solution
(1), which requires
a
less
powerful
power
amplifier,
thus seems
preferable
to
(2).
The use of
symmetrical pre-
and
post-shocks
is
however
better,
because
of a
certain number
of
additional advantages [MAG 72]:
198
Mechanical
shock
- for the
same duration
x of the
specified shock
and for the
same value
of
maximum
velocity,
the
possible maximum level
of
acceleration
is
twice
as
big;
- the
maximum force
is
provided
at the
moment when acceleration
is
maximum,
i.e. when
the
velocity
is
zero
(one will
be
able
to
thus have
the
maximum current).
The
solution with symmetrical pre-post-shocks
requires
minimal electric power.
Figure 7.12.
Kinematics
of
the
movement with pre-shock alone (I), symmetrical
pre-
and
post-shocks
(2) and
post-shock alone
(3)
7.5.3. Kinematics
of
the
movement
for
symmetrical
pre-
and
post-shock
7.5.3.1.
Half-sine
pulse
Half-sine
pulse with
half-sine
pre-
and
post-shocks
Duration
of
pre-
and
post-shocks
[LAL 83]:
- the final
displacement
is
minimal.
If the
specified shock
is
symmetrical (with
respect
to the
vertical line
this
residual displacement
is
zero [YOU 64];
Generation
of
shocks
using
shakers
199
Figure
7.13.
Half-sine
-with
half-sine
symmetrical
pre-
and
post-shocks
The
following relations give
the
expressions
of the
acceleration,
the
velocity
and
the
displacement
as a
function
of
time
in
each interval
of
definition
of the
signal.
Velocity
Displacement
200
Mechanical
shock
Half-sine
pulse with triangular pre-
and
post-shocks
Figure 7.14.
Half-sine
with triangular symmetrical
pre-
and
post-shocks
Duration
of
pre-
and
post-shocks
Generation
of
shocks using shakers
201
(by
supposing that
the
slope
of the
segment joining
the top of the
triangle
to the
foot
of
the
half-sine
is
equal
to the
slope
of the
half-sine
at its
origin).
Rise time
of
pre-shock
202
Mechanical shock
Generation
of
shocks using shakers
203
Half-sine
pulse
with rectangular
pre-
and
post-shocks
Duration
of
pre-
and
post-shocks
Figure
7.15.
Half-sine
with rectangular symmetrical
pre-
and
post-shocks