Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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174
Mechanical shock
To
generate
a
terminal peak
saw
tooth shock pulse,
any
target made
up of an
inelastic material (crushable material)
with
a
curve dynamic deflection-load which
follows
a
cubic
law is
thus appropriate [WHI].
To
obtain
a
perfect
TPS
shock pulse,
it
is
necessary that
and,
by
integration
Knowing
that
|F = m
x(t)
= a
cr
S(t),
it
becomes
where
S(t)
=
surface
of the
programmer
in
contact with
the
table
at
time
t.
a
cr
=
crush stress
of
material constituting
the
target.
The law
S(t)
is
thus relatively complicated.
If we set SQ = —, we can
write:
°cr
and
Standard shock machines
175
Example
Mass
of the
unit table
+
fixture
+
test
item:
400 kg
Maximum acceleration:
500
m/s
2
Shock
duration:
10 ms
a
cr
= 760
kg/cm
2
= 760 10
4
kg/cm
2
yielding
and
Figure 6.20 shows
the
variations
of
S(x) with
x.
Figure 6.20.
Evolution
of
the
impact area according
to the
crushed length
i.e.
176
Mechanical shock
It is
supposed that
S =
A,
x
(A,
=
constant)
and we
have
at
time
t
yielding
Let
us set
This gives
which
has as a
solution
x(t)=
x
m
sinh£2t.
Differentiating
twice,
we
have
successively
x(t)= Qx
m
coshQt
and
x(t)
= Q
x
m
sinhQt.
With this
assumption,
the
rise curve
is not
perfectly linear.
In
practice,
the
approximation
is
sufficient.
So
that
F(t)or
x(t)
presents
a
constant slope,
it is
thus enough that
the
cross-section
of the
programmer increases linearly according
to the
distance
to its
top
(point
of
impact),
i.e.
to
define
a
cone.
For
t = 0, x = 0 and x =
Vj
= AV . For t = T,
X(T)
= x
m
,
yielding
These relations make
it
possible
in
theory
to
determine
the
characteristics
of the
target.
The
calculations
are
however complex, with
Q
being related
to X.
Although
it is
possible
to
determine
by
calculation
the
required load deformation
characteristic, according
to a
particular
law of
acceleration,
it is
very
difficult
to use
this
information
in
practice.
The
difficulty
rests
in the
determination
of the
form
of
the
programmer
and the
characteristic
of the
dynamic crushing
to
produce
a
given
Standard shock machines
177
shock.
For
each machine
and
each shock,
it is
necessary
to
carry
out
preliminary
tests
to
check that
the
programmer
is
well calculated.
The
programmers
are
destroyed with each
test.
It is
thus
a
relatively expensive method.
One
prefers
to
use,
if
possible,
a
universal
programmer (Section
6.6.4).
The
material generally used
is
lead
or
honeycomb.
The
cones
can be
calculated
as
follows:
-
crushed length:
yielding
the
height
of the
cone
h > 1.2 x
m
(to
allow
material
to
become deformed
to
the
necessary height);
-
force maximum:
yielding
the
cross-section
S
m
of the
cone
at
height
x
m
:
When
all the
kinetic energy
of the
table
is
dissipated
by
crushing
of
lead,
acceleration
decreases
to
zero.
The
shock machine must have
a
very rigid solid mass
of
reaction,
so
that
the
time
of
decay
to
zero
is not too
long
and
satisfies
the
specification.
The
speed
of
this decay
to
zero
is a
function
of the
mass
of
reaction
and
of the
mass
of the
table:
if the
solid mass
of
reaction
has a not
negligible
elasticity, this time, already non-zero because
of the
imperfections inherent
in the
programmer,
can
become
too
long
and
unacceptable.
For
lead,
the
order
of
magnitude
of a
cr
is 760
kg/cm
2
(7.6
10
7
N/m
2
=
76
MPa).
The
range
of
possible durations lies between
2 and 20 ms
approximately.
Penetration
of
a
steel punch
in a
lead block
Another method
of
generating
a
terminal peak
saw
tooth shock pulse consists
of
using
the
penetration
of a
punch
of
required
form
in a
deformable material such
as
lead.
The
punch
is fixed
under
the
table
of the
machine,
the
block
of
lead
on the
solid reaction
mass.
The
velocity setting
of the
table
is
obtained,
for
example,
by
178
Mechanical shock
free
fall
[BOC 70], [BRO 66a]
and
[ROS 70].
The
duration
and the
amplitude
of the
shock
are
functions
of the
impact velocity
and the
point angle
of the
cone.
Figure 6.21.
Realization
of
a
TPS
shock
by
punching
of
a
lead block
Figure 6.22.
Penetration
of
the
steel punch
in a
lead block
The
force which tends
to
slow down
the
table during
the
penetration
of the
conical punch
in the
lead
is
proportional
to the
greatest
section S(x) which
is
penetrated,
at
distance
x from the
point.
If
cp
is the
point angle
of the
cone
yielding,
in a
simplified
way,
if m is the
total mass
of the
moving assembly,
by
equalizing
the
inertia
and
braking forces
in
lead
with
a
being
a
constant
function
of the
crush stress
of
lead
(by
supposing that
only
this parameter intervenes
and
that
the
other phenomena such
as
steel-lead
friction
are
negligible).
Let us set a
If
v is the
carriage velocity
at the
time
t and
Vj
the
impact velocity,
this
relation
can
be
written
Standard
shock machines
179
yielding
The
constant
of
integration
b is
calculated starting
from the
initial conditions:
for
x = 0, v = Vi
yielding
Let
us
write
[6.45]
in the
form
it
becomes
by
integration:
If
we set
and
we
obtain
Acceleration then results
from
[6.44]:
We
have
in
addition
v = vi yi - y . The
velocity
of the
table
is
cancelled when
all
its
kinetic energy
is
dissipated
by the
plastic deformation
of
lead. Then,
y = 1 and
180
Mechanical shock
Knowing
that
v
i
=
<J2
g H (H =
drop height),
the
shock duration
is not
very sensitive
to the
drop height.
From
these expressions,
we
can
establish
the
relations:
and
In
addition,
As an
indication, this method allows
us to
carry
out
shocks
of a few
hundreds
to a
few
thousands
of
grams, with durations
from 4 to 10 ms
approximately (for
a
mass
m
equal
to 25
kg).
6.6.3. Rectangular pulse
-
trapezoidal
pulse
This test
is
carried
out by
impact.
A
cylindrical programmer consists
of a
material which
is
crushed with
constant
force
(lead,
honeycomb)
or
using
the
universal programmer.
In the first
case,
the
characteristics
of the
programmer
can be
calculated
as
follows:
Standard
shock
machines
181
- the
cross-section
is
given according
to the
shock amplitude
to be
realized using
the
relation
yielding
-starting
from the
dynamics
of the
impact without rebound,
the
length
of
crushing
is
equal
to
and
that
of the
programmer must
be at
least equal
to 1.4 x
m
, in
order
to
allow
a
correct crushing
of the
matter with constant
force.
One
can say
that
the
shock amplitude
is
controlled
by the
cross-section
of the
programmer,
the
crush stress
of
material
and the
mass
of the
total carriage mass.
The
duration
is
affected only
by the
impact velocity.
For
this pulse shape also,
it is
possible
to use the
penetration
of a
rigid punch
in a
crushable material such
as
lead.
The two
methods produce relatively disturbed
signals, because
of
impact between
two
plane surfaces. They
are
adapted only
for
shocks
of
short duration, because
of the
limits
of
deformation.
A
long duration
indeed requires
a
plastic deformation over
a big
length;
but it is
difficult
to
maintain
constant
the
force
of
resistance
on
such
a
stroke.
The
honeycombs lend themselves
better
to the
realization
of a
long duration [GRA 66].
One
could also
use the
shearing
of a
lead plate.
6.6.4.
Universal
shock programmer
The MTS
Monterey programmer known
as
universal
can be
used
to
produce
half-sine,
TPS and
trapezoidal shock pulses
after
various adjustments.
This programmer consists
of a
cylinder
fixed
under
the
table
of the
machine,
filled
with
a gas
under pressure
and in the
lower part
of a
piston,
a rod and a
head
(Figure
6.23).
182
Mechanical
shock
6.6.4.1. Generating
a
half-sine
shock pulse
The
chamber
is put
under
sufficient
pressure
so
that, during
the
shock,
the
piston
cannot move (Figure
6.23).
The
shock pulse
is
thus formatted only
by the
compression
of the
stacking
of
elastomeric cylinders (modular programmers)
placed
under
the
piston head.
One is
thus brought back
to the
case
of
Section 6.6.1.
Figure
6.23.
Universal
programmer
MTS
(half-sine
and
rectangle
pulse
configuration)
Figure
6.24.
Universal
programmer
MTS
(TPS
pulse
configuration)
6.6.4.2.
Generating
a
terminal
peak
saw
tooth shock pulse
The gas
pressure (nitrogen)
in the
cylinder
is
selected
so
that,
after
compression
of
elastomer during duration
T, the
piston, assembled
in the
cylinder
as
indicated
in
Figure
6.24, suddenly
is
released
for a
force corresponding
to the
required maximum
acceleration
x
m
.
The
pressure which
was
exerted before separation over
the
whole area
of the
piston applies only
after
separation
to one
area equal
to
that
of the
rod, producing
a
negligible resistant force.
Standard shock machines
183
Acceleration
thus
passes very quickly
from x
m
to
zero.
The
rise phase
is not
perfectly
linear,
but
corresponds rather
to an arc of
versed-sine (since
if the
pressure
were
sufficiently
strong,
one
would obtain
a
versed-sine
by
compression
of the
elastomer alone).
Figure
6.25. Realization
of
a
TPS
shock pulse
6.6.4.3.
Trapezoidal
shock pulse
The
assembly here
is the
same
as
that
of the
half-sine pulse (Figure
6.23).
At the
time
of the
impact, there
is:
-
compression
of the
elastomer
until
the
force exerted
on the
piston balances
the
compressive
force
produced
by
nitrogen. This phase gives
the first
part (rise)
of the
trapezoid;
- up and
down displacement
of the
piston
in the
part
of the
cylinder
of
smaller
diameter, approximately with
constant
force (since volume varies little). This phase
corresponds
to the
horizontal part
of the
trapezoid;
-
relaxation
of
elastomer: decay
to
zero acceleration.
The
rise and
decay parts
are not
perfectly linear
for the
same reason
as in the
case
of the TPS
pulse.
6.6.4.4. Limitations
Limitations
of
the
shock machines
The
limitations
are
often
represented graphically
by
straight lines plotted
in
logarithmic
scales
delimiting
the
domain
of
realizable shocks (amplitude, duration).
The
shock machine
is
limited
by
[IMP]:
- the
allowable maximum force
on the
table.
To
carry
out a
shock
of
amplitude
x
m
, the
force
generated
on the
table, given
by