Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
Подождите немного. Документ загружается.
214
Mechanical
shock
-
Triangular pre-post-shocks lead
to the
greatest duration
of
signal,
the
rectangle
giving
the
smallest duration. Under these
criteria
of
displacement
and
duration,
it is
thus
preferable
to use
rectangular
or
triangular pre-
and
post-shocks.
The
rectangle
has
however
the
disadvantage
of
having slope discontinuities which make
its
reproduction
difficult
and
which
in
addition
can
excite resonances
at
high
frequencies.
It
seems, however, interesting
to try to
approach this
form
[MIL 64].
- The
maximum displacement
decreases,
as one
might
expect,
when
p
increases.
It
seems, however, hazardous
to
retain values higher than 0.10 (although possible
with
certain control systems),
the
total shock communicated
to the
specimen being
then
too
deformed compared with
the
specification, which results
in
response
spectra appreciably
different
from
those
of the
pure shocks [FRA 77].
Example
Half-sine
shock with half-sine symmetrical pre-
and
post-shocks.
Electrodynamic
shaker
Figure
7.17.
Half-sine
pulse
with
half-sine
symmetrical
pre-
and
post-shocks
Generation
of
shocks
using shakers
215
Figure
7.18. Acceleration, velocity
and
displacement during
the
shock
of
Figure
7.17
Figure
7.19. Abacus
for a
half-sine
shock with
half-sine
pre-
and
post-shocks
216
Mechanical
shock
Figure 7.20.
Comparison between maximum displacements obtained with
the
typical
shape
of
pre-
and
post-shocks
7.5.7.
Optimized
pre-
and
post-shocks
At
the
time
of the
realization
of a
shock
on
shaker,
the
displacement starts
from
the
equilibrium position, passes through
a
maximum, then returns
to the
initial
position.
One
uses
in
fact
only half
of the
available stroke.
For
better
use of the
capacities
of the
machine,
we saw
(Figure 7.1) that
it is
possible
to
shift
the
zero
position
of the
table.
Another
method
was
developed [FAN
81] in
order
to
fulfill
the
following
objectives:
- to
take into account
the
tolerances
on the
shape
of the
signal
allowed
by the
standards (R.T. Fandrich refers
to
standard MIL-STD 8IOC);
- to
best
use the
possibilities
of
the
shaker.
The
solution suggested consists
of
defining:
1.
A
pre-shock made
up of the first two
terms
of the
development
in a
Fourier
series
of a
rectangular pulse (with coefficients modified
after
a
parametric analysis),
of the
form:
Generation
of
shocks
using
shakers
217
Figure 7.21.
Optimizedpre-shock
The
rectangular shape
is
preferred
for the
reasons already mentioned,
the
choice
of
only
the first two
terms
of the
development
in
series being intended
to
avoid
the
disadvantages related
to
slope discontinuities.
The
pre-shock consists
of one
period
of
this signal, each half-period having
a
different
amplitude:
-
positive arch:
where
x
m
is the
amplitude
of the
shock
to be
realized
(in
m/s
2
)
and f is the
fundamental
frequency of the
signal, estimated
from the
relationship
[7.45]
where
g =
9.81 m/s
2
. This expression
is
calculated
by
setting
the
maximum
displacement during
the
pre-shock lower than
the
possible maximum displacement
on
the
shaker (for example 1.27 cm). This maximum displacement takes
place
at the
end of the
first
arch, comparable
at
first
approximation with
a
rectangle.
-
negative
arch:
duration,
the
maximum displacement
is
equal
to
yielding,
if f
is
its
218
Mechanical shock
Figure
7.22.
Velocity
during
the
optimizedpre-shock
and the
shock
Figure 7.23.
Acceleration, velocity
and
displacement during
the
pre-shock
if
The
total duration
of the
pre-shock
is
thus equal
to
m
The
factor
of
0.05 corresponds
to the
tolerance limit
of the
quoted standard
before
the
principal shock (5%).
The
constant 0.24
is the
reduced amplitude
of the
first
arch,
the
real amplitude
for a
shock
of
maximum value
x
m
being equal
to
0.24
(0.046
x
m
).
The
second arch
has
unit amplitude.
The
table being, before
the
test,
in
equilibrium
in a
median position,
the
objective
of
this pre-shock
is two
fold:
- to
give
to the
velocity, just before
the
principal shock,
a
value close
to one of
the
two
limits
of the
shaker,
so
that during
the
shock,
the
velocity
can use all the
range
of
variation permitted
by the
machine (Figure 7.22);
Generation
of
shocks
using
shakers
219
- to
place,
in the
same way,
the
table
as
close
as
possible
to one of the
thrusts
so
that
the
moving element
can
move during
the
shock
in all the
space between
the two
thrusts (limitation
in
displacement equal, according
to the
machines,
to
2.54
or
5.08 cm).
Figure
7.24.
Acceleration, velocity
and
displacement during thepre-shock
and the
shock
(half-sine)
2. A
post-shock
composed
of one
period
of a
signal
of the
shape
K
t
y
sin(2
7i fj t)
where
the
constants
K, y and fj are
evaluated
in
order
to
cancel
the
acceleration,
the
velocity
and the
displacement
at the end of the
movement
of the
table.
Figure
7.25.
Overall movement
for
a
half-sine
shock
220
Mechanical
shock
The
frequency and the
exponent
are
selected
in
order
to
respect
the
ratio
of the
velocity
to the
displacement
at the end of the
principal shock.
The
amplitude
of the
post-shock
is
adjusted
to
obtain
the
desired velocity change.
Figure
7.25 shows
the
total signal obtained
in the
case
of a
principal shock
half-
sine
30 g, 11 ms.
This methodology
has
been improved
to
provide
a
more general solution [LAX
01].
7.6.
Incidence
of
pre-
and
post-shocks
on the
quality
of
simulation
7.6.1.
General
The
specification
of
shock
is in
general expressed
in the
form
of a
signal varying
with
time (half-sine, triangle etc).
We saw the
need
for an
addition
of
pre- and/or
post-shock
to
cancel
the
velocity
at the end of the
shock when
it is
carried
out on an
electrodynamic shaker.
There
is no
difference
in
principle between
the
realization
of a
shock
by
impact
after
free or
accelerated
fall
and the
realization
of a
shock
on a
shaker.
On a
shock
machine
of the
impact type,
the
test
item
and the
table have zero velocity
at the
beginning
of the
test.
The free or
accelerated
fall
corresponds
to the
pre-shock phase.
The
rebound,
if it
exists,
corresponds
to the
post-shock.
The
practical
difference
between
the two
methods lies
in the
characteristics
of
shape, duration
and
amplitude
of
pre-
and
post-shocks.
In the
case
of
impact,
the
duration
of
these signals
is in
general longer than
in the
case
of the
shocks
on the
shaker,
so
that
the
influence
on the
response appears
for
systems
of
lower natural
frequency.
7.6.2.
Influence
of
the
pre-
and
post-shocks
on the
time
history
response
of
a
one-
degree-of-freedom
system
To
highlight
the
problems,
we
treated
the
case
of a
specification which
can be
realized
on a
shaker
or on a
drop table, applied
to a
material
protected
by a
suspension
with
a 5 Hz
natural
frequency and
with
a Q
factor equal
to 10.
Nominal
shock
-
half-sine pulse
ms.
Shock
on
shaker
-
identical pre-shock
and
post-shock;
-
half-sine shape;
-
amplitu
-
duration such that:
Shock
by
impact
-
free
fall
-
shock with rebound
to 5
-
velocity
of
impact:
-
velocity
of
rebou
-
drop height
Generation of shocks using shakers 221
222
Mechanical
shock
The
duration
of the
fall
is tj
where
Figure
7.26 shows
the
response
a>
0
z(t)
of a
one-degree-of-freedom system
(f
0
=5Hz,
£ =
0.05):
-
for
z
0
= z
0
= 0
(conditions
of
the
response spectrum);
- in the
case
of
a
shock with impact;
- in the
case
of a
shock
on
shaker.
We
observed
in
this example
the
differences between
the
theoretical response
at
5 Hz and the
responses actually obtained
on the
shaker
and
shock machine.
According
to the
test
facility
used,
the
shock applied
can
under-test
or
over-test
the
Duration
of the
rebound
Figure 7.26.
Influence
of
the
realization mode
of
a
half-sine
shock
on the
response
of
a
one-degree-of-freedom
system
Generation
of
shocks
using
shakers
223
material.
For the
estimate
of
shock severity
one
must
take
account
of the
whole
of
the
signal
of
acceleration.
7.6.3. Incidence
on the
shock
response
spectra
In
Figure 7.27,
for £ =
0.05
, is the
response spectrum
of:
-the nominal shock, calculated under
the
usual conditions
of the
spectra
(z
0
= z
0
= 0);
- the
realizable shock
on
shaker,
with
its
pre-
and
post-shocks,
- the
realizable shock
by
impact, taking
of
account
of the
fall
and
rebound
phases.
Figure 7.27.
Influence
of
the
realization
mode
of
a
half-sine
shock
on the SRS
One
notes
in
this example that for:
-
f<}
< 10 Hz, the
spectrum
of the
shock
by
impact
is
lower than
the
nominal
spectrum,
but
higher than
the
spectrum
of the
shock
on the
shaker,
- 10 Hz < f
0
< 30 Hz, the
spectrum
of the
shock
on the
shaker
is
much
overestimated,
- f
0
> 30 Hz, all the
spectra
are
superimposed.
This
result appears logical when
we
remember that
the
slope
of the
shock
spectrum
at the
origin
is, for
zero damping, proportional
to the
velocity change
associated
with
the
shock.
The
compensation signal added
to
bring back
to
zero
the
velocity change thus makes
the
slope
of the
spectrum
at the
origin zero.
In
addition,
the
response
spectrum
of the
compensated signal
can be
larger than
the
spectrum
of
the
theoretical signal close
to the frequency
corresponding
to the
inverse
of the
duration
of the
compensation signal.
It is
thus advisable
to
make sure that
the