Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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164
Mechanical shock
The
surface
of the
piston subjected
to the
pressure
is
different
according
to
whether
it is on the
higher
face
or the
lower
face,
as
long
as it is
supported
in the
higher position
on the
Teflon seat [THO 64].
Initially,
the
moving piston,
rod and
table rose
by
filling
the
lower cylinder
(reference
pressure).
The
higher chamber
is
then
inflated
to a
pressure
of
approximately
five
times
the
reference pressure. When
the
force
exerted
on the
higher
face
of the
piston exceeds
the
force
induced
by the
pressure
of
reference,
the
piston
releases.
The
useful
surface
area
of the
higher
face
increases quickly
and the
piston
is
subjected
in a
very short time
to a
significant force exerted towards
the
bottom.
It
involves
the
table which compresses
the
programmers (elastomers, lead
cones etc) placed
on the top of the
body
of the
jack.
This machine
is
assembled
on
four
rubber bladders
filled
with
air to
uncouple
it
from the
floor
of the
building.
The
body
of the
machine
is
used
as
solid mass
of
reaction.
The
interest behind this lies
in its
performance
and its
compactness.
6.5. Specific
test
facilities
When
the
impact velocity
of
standard machines
is
insufficient,
one can use
other
means
to
obtain
the
desired velocity:
-
Drop
testers,
equipped
for
example with
two
vertical
(or
inclined) guide cables
[LAL
75], [WHI], [WHI 63].
The
drop height
can
reach
a few
tens
of
metres.
It is
wise
to
make sure that
the
guidance
is
correct
and in
particular, that
friction is
negligible.
It is
also desirable
to
measure
the
impact velocity (photo-electric cells
or
any
other
device).
- Gas
guns, which initially
use the
expansion
of a gas
(often
air) under pressure
in
a
tank
to
propel
a
projectile carrying
the
test item towards
a
target equipped with
a
programmer fixed
at the
extremity
of a gun on a
solid reaction mass [LAZ 67],
[LAL 75], [WHI], [WHI
63] and
[YAR 65].
One finds the
impact mode
to be as
above.
It is
necessary that
the
shock created
at the
time
of the
velocity setting
in the
gun
is of low
amplitude with regard
to the
specified shock carried
out at the
time
of
the
impact. Another operating mode consists
of
using
the
phase
of the
velocity
setting
to
program
the
specified shock,
the
projectile then being braked
at the end of
the
gun by a
pneumatic device, with
a
small acceleration with
respect
to the
principal shock.
A
major
disadvantage
of
guns
is
related
to the
difficulty
of
handling
cables instrumentation, which must
be
wound
or
unreeled
in the
gun,
in
order
to
follow
the
movement
of the
projectile.
-Inclined-plane
impact
testers
[LAZ 67], [VIG 61b]. These were especially
conceived
to
simulate shocks undergone during
too
severe handling operations
or in
Standard shock machines
165
trains. They
are
made
up
primarily
ofa
carriage
on
which
the
test item
is fixed,
travelling
on an
inclined rail
and
coming
to run up
against
a
wooden barrier.
Figure
6.15.
Inclined
plane impact tester
(CONBUR
tester)
The
shape
of the
shock
can be
modified
y
using elastomeric "bumpers'
or
springs.
Tests
of
this type
are
often
named
'CONBUR
tests'.
6.6. Programmers
We
will describe only
the
most-frequently used programmers used
to
carry
out
half-sine,
terminal peak
saw
tooth
and
trapezoid shock pulses.
6.6.1.
Half-sine
pulse
These shocks
are
obtained using
an
elastic material interposed between
the
table
and
the
solid mass reaction.
Shock
duration
The
shock duration
is
calculated
by
supposing that
the
table
and the
programmer,
for
this length
of
time, constitute
a
linear mass-spring system with only
one-degree-
of-freedom.
The
differential equation
of the
movement
can be
written
where
m =
mass
of the
moving assembly (table
+
fixture
+
test item)
k
=
stiffness
constant
of the
programmer
i.e.
166
Mechanical
shock
2
n
The
solution
of
this equation
is a
sinusoid
of
period
T = . It is
valid only
co
0
during
the
elastomeric material compression
and its
relaxation,
so
long
as
there
is
contact
between
the
table
and the
programmer, i.e. during
a
half-period.
If t is the
shock duration,
we
thus have
This
expression
shows that, theoretically,
the
duration
can be
regarded
as a
function
alone
of the
mass
m and of the
stiffness
of the
target.
It is in
particular
independent
of the
impact velocity.
The
mass
m and the
duration
i
being known,
we
deduce
from it the
stiffness
constant
k of the
target:
Maximum
deformation
of
the
programmer
If
Vj
is the
impact velocity
of the
table
and x
m
the
maximum deformation
of the
programmer during
the
shock,
it
becomes,
by
equalizing
the
kinetic loss
of
energy
and
the
deformation energy during
the
compression
of the
programmer
yielding
Shock
amplitude
k
From
[6.25],
one
has,
in
absolute terms,
m x
m
= k x
m
,
yielding
x
m
= x
m
—
m
and,
according
to
[6.30]
Standard shock machines
167
where
the
impact velocity
YJ
is
equal
to
with
g =
acceleration
of
gravity
(1 g =
9.81 m/s
2
)
H
=
drop height
This
relation, established theoretically
for
perfect rebound, remains usable
in
practice
as
long
as the
rebound velocity remains higher than approximately
50% of
the
impact velocity.
Having
determined
k from m and T, it is
enough
to act on the
impact velocity, i.e.
on
the
drop height,
to
obtain
the
required shock amplitude.
Characteristics
of
the
target
For
a
cylindrical programmer,
we
have
where
S and L are
respectively
the
cross-section
and the
height
of the
programmer
and
where
E is
Young's modulus
of
material
in
compression.
Depending
on the
materials available, i.e. possible values
of E, one
chooses
the
values
of L and S
which lead
to a
realizable programmer
(by
avoiding
too
large
a
height
to
diameter ratio
to
eliminate
the
risks
from
buckling). When
the
table
has a
large surface,
it is
possible
to
place
four
programmers
to
distribute
the
effort.
The
cross-section
of
each programmer
is
then calculated starting
from the
value
of S
determined above
and
divided
by 4.
The
elasticity modulus which intervenes here
is the
dynamic modulus, which
is
in
general larger than
the
static modulus. This divergence
is
mainly
a
function
of the
type
of
elastomeric material used, although other factors such
as the
configuration,
the
deformation
and the
load
can
have
an
effect
The
ratio dynamic modulus
E
d
to
static modulus
E
s
ranges
in
general between
1 and 2. It can
exceed
2 in
certain
cases
[LAZ 67].
The
greatest
values
of
this ratio
are
observed with most damped
materials.
For
materials such
as
rubber
and
Neoprene,
it is
close
to
unity.
168
Mechanical shock
Figure
6.16.
High
frequencies at
impact
Figure 6.17.
Impact module with conical
impact face
(open
module)
If
the
surface
of
impact
is
plane,
a
wave created
at the
time
of the
impact
is
propagated
in the
cylinder
and
makes
several
up and
down excursions. From
it at the
beginning
of the
signal
the
appearance
of a
high frequency oscillation which distorts
the
desired half-sine pulse results.
To
avoid this phenomenon,
the front
face
of the
programmer
is
designed
to be
slightly
conical,
in
order
to
insert
the
load material gradually (open module).
The
shock thus created
is
between
a
half-sine
and a
versed-sine
pulse.
Propagation
time
of
the
shock wave
So
that
the
target
can be
regarded
a
simple spring
and not as a
system with
distributed constants,
it is
necessary that
the
propagation time
of the
shock wave
through
the
target
is
weak with
respect
to the
duration
T of the
shock.
If a is the
velocity
of the
sound
in
material constituting
the
target
and h its
height, this
condition
is
written:
i.e., since
a
(E
=
Young's modulus,
p=
density)
and
Standard shock machines
169
If
the
mass
of the
target
is
equal
to M
c
= hSp (S =
cross-section),
it is
necessary then that
i.e.
Rebound
The
coefficient
of
restitution
is a
function
of the
material.
The
smallest rebounds
are
obtained with
the
elastic materials that
are
most strongly damped.
The
metal springs have small damping
and
thus produce significant rates
of
rebound,
often
about 75%.
The
elastomers vary greatly, with
the
rate
of
rebound
which
can be
located
as
being between
0 and 75% of the
drop height.
The
coefficient
of
restitution
is
also
a
function
of the
configuration
and of the
deformation
of
elastic material.
The
targets which
are
made
up of
very
soft
material,
presenting great deformations, lead
in
general
to
significant rebounds, whereas
the
elastomeric materials, which
are
stiff
and
thin,
are
calculated
to
become deformed
only
by a few
hundredths
of
millimetre,
and
produce only very little rebound [LAZ
67].
A
not
very substantial rebound
can
mean that
the
material
of the
programmer
reacts
during
the
impact like
a
viscoelastic material,
the
table taking
a
rebound
velocity higher than
the
relaxation velocity
of the
material [BRO 63].
To
create
a
perfectly half-sine shock pulse with this type
of
programmer,
one
needs
a
perfect rebound, with
a
rebound velocity equal
to the
impact
velocity.
It is
necessary thus that damping
is
zero.
The
shock pulse obtained under these
conditions
is
symmetrical. When
the
rate
of
rebound
decreases,
the
return
of
acceleration
to
zero (relaxation)
is
faster than
the
rise
of
acceleration.
170
Mechanical
shock
Figure 6.18.
Distortsion
of
the
half-sine
pulse related
to the
damping
of
the
material
A
good
empirical rule
is to
limit
the
maximum dynamic deformation
of the
programmer
from 10 to 15% of its
initial
thickness.
If
this
limit
is
exceeded,
the
shape
obtained
risks non-linear
tendencies.
Standard
shock
machines
171
Example
Realization
of a
half-sine shock
300
m/s
2
,
10 ms. It is
supposed that
the
mass
of
the
moving assembly (table
+
fixture
+
test
item)
is
equal
to 600 kg.
The
elastomeric programmers
often
have
a
coefficient
of
restitution
k
(
VR
= -k
Vj)
of
about
50 %.
From
[6.28]
The
impact velocity
is
calculated
from
[6.31]:
2
V:
_
3
which
leads
to the
drop height
H = « 47 10 m.
During
the
impact,
the
2g
elastomeric
target will
be
deformed
to a
height equal
to
[6.30]:
The
velocity change during
the
shock
is
equal
to
2 2 ~>
AV
= - x
m
t =
—
300 10 ^ »
1.91 m/s.
It is
checked that
AV = 2 v
{
.
7i 7i
with
L
being
the
height
of the
target,
its
diameter
D is
calculated
from
ES
u
.
L
f\ ")
If
the
target
is an
elastomer
of
hardness
60
Shore, then
E » 4 10 N/m
yielding,
if
L = 0.1 m, D «
0.137
m. It
remains
to
check that
the
stress
in
material does
not
exceed
the
acceptable value.
172
Mechanical
shock
NOTES:
1.
The
relations [6.27]
and
[6.31] were established
by
supposing that
the
material
of
the
target
is
perfectly
elastic
and
that
the
rebound
is
perfect.
If
it is not
the
case, these relations give only
one
approximation
of x
m
and T (or k and
vi).
In
difficult
cases,
it is
undoubtedly quicker
to
carry
out a
first test,
to
measure
the
values
of x
m
and i
obtained, then
to
correct
k and
Vj
using
i.e.,
according
to the
drop
height
Index
1
corresponds
to the first
shock carried out, index
2
with
the
required
shock. These relationship remain usable
as
long
as
there
is a
certain rebound
and as
long
as the
shock remains symmetrical.
It is
unfortunately
difficult
to
maintain
the
same shape
of the
shock when
one
tries
to
modify
its
amplitude
and its
duration.
Thus,
when
a
rubber target
is
deformed
by
more than
30%
approximately
its
length
at
rest,
its
characteristic force-displacement becomes non-linear, which leads
to a
distortion
of
the
profile
of
the
shock [BRO
63]
[WHI]
[WHI63].
^dv Sp
2. For a
confined
material (liquid
for
example),
we
have
k =
V
(E
dv
=
bulk dynamic modulus,
V =
volume
of the
liquid contained
and
Sp
=
effective
area
of
the
piston
compressing
the
liquid).
The
manufacturers provide cylindrical modules made
up of an
elastomer
sandwiched between
two
metal
plates.
The
programmer
is
composed
of a
stacked
modules
of
various stiffnesses
(Figure
6.19).
It
is
enough
for a
relatively
low
number
of
different modules
to
cover
a
broad
range
of
shock durations
by
combinations
of
these elements [BRE 67], [BRO 66a],
[BRO 66b]
and
[GRA 66].
Standard
shock machines
Figure
6.19.
Distribution
of
the
modules
(half-sine
shock pulse)
The
modules
are in
general distributed between
the
bottom
of the
table
and the
top of the
solid mass
of
reaction
to
regularly distribute
the
load
at the
time
of the
shock
in the
lower part
of the
table.
One
thus avoids exciting
its
bending mode
at
lower
frequency and
amplifying
the
vibrations
due to
resonance
of the
table.
The
programmers
for
very short duration shock
are
made
up of a
high-strength
thermoplastic material
and
with
a
large numbers
of
modules.
The
selected plastic
is
highly
resilient
and
very hard.
It is
used
within
its
yield stress
and can
thus
be
useful
almost indefinitely. Reproducibility
is
very good.
The
programmer
is
composed
of a
cylinder
of
this material stuck
on a
plane
circular plate screwed
to the
lower part
of the
table
of the
shock machine.
6.6.2.
Terminal
peak
saw
tooth shock pulse
Programmers
using crushable materials
We
showed that,
at the
time
of a
shock
by
impact without rebound,
the
deflection
varies according
to
time according
to the law
which
can be
written, since x(t)
= xm
—
and
F(t)
= -m
x(t),
es
173