Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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204
Mechanical shock
Generation
of
shocks
using shakers
205
[7.43]
[7.44]
The
expressions
of the
largest velocity during
the
movement
and
those
of the
maximum
and
residual displacements
are
brought together
in
Table 7.1.
Table
7.1.
Half-sine
~
maximum velocity
and
displacement
-
residual displacement
Similar
expressions
can be
established
for the
other shock shapes (TPS,
rectangle,
IPS
pulses).
The
results appear
in
Tables
7.2 to
7.4.
Half-sine
Maximum
velocity
Symmetrical
pre-
and
post-shocks
Maximum
displacement
Residual
displacement
Half-sine
Triangles
Rectangles
206
Mechanical
shock
7.5.3.2.
TPS
pulse
Table 7.2.
TPS
pulse
-
maximum velocity
and
displacement
—
residual displacement
1.
TJ is the
total
duration
of the
pre-shock
(or
post-shock
if
they
are
equal).
12 i
s tne
duration
of
the
first part
of the
pre-shock when
it is
composed
of two
straight-line segments
(or of the
last part
of the
post-shock).
13 is the
total duration
of the
post-shock when
it is
different
from Tj.
TPS
Symmetrical
pre-
and
post-shocks
Half-sine
Triangles
Rectangles
Durations
of
pre-
and
post-
shocks
Maximum
velocity
Maximum
displacement
Residual
displacement
Generation
of
shocks
using
shakers
7.5.3.3.
Rectangular pulse
Table 7.3.
Rectangular
pulse
—
maximum velocity
and
displacement
-
residual displacement
Rectangle
Half-sine
Symmetrical
pre-
and
post-shocks
Triangles
Rectangles
Durations
of
pre-and
post-
shocks
Maximum
velocity
Maximum
displacement
Residual
displacement
207
208
Mechanical
shock
7.5.3.4.
IPS
pulse
Table
7.4.
IPS
pulse
-
maximum velocity
and
displacement
-
residual displacement
7.5.4.
Kinematics
of
the
movement
for a
pre-shock
or a
post-shock alone
In
the
case
of a
pre-shock
or a
post-shock alone,
the
maximum velocity, equal
to
the
velocity change
AV
related
to the
shock, takes place
at the
time
of
transition
between
the
compensation signal
and the
shock itself.
The
displacement starts
from
zero
and
reaches
its
largest value
at the end of the
movement (without changing
sign). Like
the
velocity,
it is
negative with
a
pre-shock
and
positive with
a
post-
shock. Tables
7.5-7.8
bring together
the
expressions
for
this displacement according
to the
shape
of the
principal shock
and of
that
of the
compensation signal, with
the
same
notations
and
conventions
as
those
in the
preceding paragraphs.
IPS
Half-sine
Symmetrical
pre and
post-shocks
Triangles
Rectangles
Durations
of
pre-
and
post-
shocks
Maximum
velocity
Maximum
displacement
Residual
displacement
Generation
of
shocks
using
shakers
209
Table 7.5.
Half-sine
withpre-
or
post-shock
only
-
maximum velocity
and
displacement
—
residual displacement
Pre-shock
or
post-shock only
Half-sine
Half-sine
Triangle
Rectangle
Duration
of
pre-shock
or
post-shock
Maximum
velocity
Pre-shock
Post-shock:
Residual
displacement
210
Mechanical
shock
Table 7.6.
TPS
with
pre-
or
post-shock
only
-
maximum velocity
and
displacement
-
residual displacement
TPS
Half-sine
Pre-shock
or
post-shock
only
Triangle
Rectangle
Pre-shock:
Pre-shock:
Duration
of
pre-shock
or
post-shock
Post-shock:
Post-shock:
Maximum
velocity
Pre-shock:
Post-shock:
Pre-shock: Pre-shock: Pre-shock:
Residual
displacement
Post-shock:
Post-shock:
Post-shock:
Generation
of
shocks
using shakers
211
Table 7.7.
Rectangular
pulse
withpre-
or
post-shock
only
—
maximum velocity
and
displacement
—
residual displacement
Rectangle
Half-sine
Pre-shock
or
post-shock
only
Triangle
Rectangle
Duration
of e
pre-shock
or
post-shock
Maximum
velocity
Pre-shock:
Post-shock:
Residual
displacement
212
Mechanical
shock
Table 7.8.
IPS
pulse
with
pre-
or
post-shock
only
-
maximum velocity
and
displacement
—
residual displacement
7.5.5.
Abacuses
For
a
given shock
and for
given pre-
and
post-shocks shapes,
we can
calculate,
by
integration
of the
expressions
of the
acceleration,
the
velocity
and the
displacement
as a
function
of
time,
as
well
as the
maximum values
of
these
parameters,
in
order
to
compare them
with
the
characteristics
of the
facilities.
This work
was
carried
out for
pre-
and
post-shocks
-
respectively half-sine,
triangular
and
rectangular [LAL
83] in
order
to
establish
abacuses allowing quick
Pre-shock
or
post-shock only
IPS
Half-sine
Triangle
Rectangle
Pre-shock:
Pre-shock:
Duration
of
pre-shock
or
post-shock
Post-shock:
Post-shock:
Maximum
velocity
Pre-shock: Post-shock:
Pre-shock: Pre-shock:
Pre-shock:
Residual
displacement
Post-shock:
Post-shock:
Post-shock:
evaluation
of the
possibility
of
realization
of a
specified shock
on a
given
test
facility
(characterized
by its
limits
of
velocity
and of
displacement). These abacuses
are
made
up of
straight line segments
on
logarithmic scales (Figure 7.16):
-AA',
corresponding
to the
limitation
of
velocity:
the
condition
v
m
< V
L
(V
L
=
acceptable maximum velocity
on the
facility
considered) results
in a
relationship
of the
form
x
m
t <
constant
(independent
of p);
- CC, DD,
etc, greater slope corresponding
to the
limitation
in
displacement
for
various values
of p (p =
0.05, 0.10, 0.25, 0.50
and
1.00).
A
particular shock will
be
thus realizable
on
shaker only
if the
point
of co-
ordinates
T, x
m
(duration
and
amplitude
of the
shock considered)
is
located under
these lines, this
useful
domain increasing when
p
increases.
Figure
7.16.
Abacus
of
the
realization
domain
of
a
shock
7.5.6.
Influence
of
the
shape
of
pre-
and
post-pulses
The
analysis
of the
velocity
and the
displacement varying with time associated
with
some simple shape shocks shows that [LAL 83]:
- For all the
shocks having
a
vertical axis
of
symmetry,
the
residual displacement
is
zero.
- For a
shock
of
given amplitude, duration
and
shape,
the
maximum
displacement
during movement
is
most important with half-sine pre-
and
post-
shocks.
It is
weaker
in the
case
of the
triangles, then rectangles.