Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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294
Mechanical
shock
This yields
an
estimate
of the
maximum stroke:
Relations
[9.101], [9.102]
and
[9.103] show that,
if n is the
slope
of the
SRS,
the
This
power,
the
stroke
and the
force
are
respectively
of the
form
p
max
1 1
is a
function
increasing
for n >
—
and
decreasing
for n < —.
2 2
F
m
=k
3
(where
kj,k
2
and k
3
are
constants).
Thus,
in
logarithmic scales:
- the
necessary power
is
given
by the
highest point
of
contact between
the SRS
and
the
line
of
slope
!£;
- the
maximum stroke corresponds
to the
highest point
of
contact
of the SRS
with
a
line
of
slope
2;
- the
force
at the
highest point
of
contact between
the SRS and a
line
of
zero
slope.
Figure
9.37. Calculation
of
the
force,
the
power
and the
stroke necessary
to
carry
out a
shock
of
given
SRS
The
coordinates
of
these points
of
contact
are
used
in
relations [9.101], [9.102]
and
[9.103]
to
calculate
the
maximum force,
the
necessary power
and the
stroke.
Control
of a
shaker
using
a
shock response spectrum
295
If
the
damping
£
sp
of the
test
equipment
is not
known,
one can use the
approximate
value
of
0.05.
NOTES.
1.
In the
case
of
a
specification
defined
by a
velocity SRS,
one
will consider
in
the
same
way the
points
of
intersection
of
the
spectrum with [DES 83]:
- a
line
of
slope -1/2
for
maximum power;
- a
line
of
slope
1 for the
stroke;
- a
line
of
slope
-1 for
the
force.
Table 9.2.
Slopes
of
the
straight
lines
allowing
evaluation
of
the
possibility
of
carrying
out
a
shock
of
a
given
SRS on a
shaker
Quantity
Power
Stroke
Force
Slope
Relative
displacement
SRS
(or
absolute
acceleration)
1
2
2
0
Velocity
SRS
1
2
1
-1
2.
If
the
dynamic behaviour
of
the
material
is
known
and
if
it is
known that,
in
the
frequency domain studied,
the
material does
not
have
any
resonance,
one
will
be
able
to
take £,
sp
= 0.5 for
transfer
function H(Q) equal
to 1.
3.
This
method provides
only
one
order
of
magnitude
of
the
force,
the
power
and
the
stroke necessary
to
carry
out a
shock
specified
from an SRS on
shaker because:
- the
relations used
are
approximations;
-
according
to the
shape
of the
elementary waveform used
to
build
the
drive
waveform,
the
actually generated
waveform
will require conditions slightly
different
from
those estimated
by
this test.
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Appendix
Similitude
in
mechanics
In
certain cases where
the
material
is too
heavy
or
bulky
to be
tested using
the
usual
test facilities,
it is
possible
to
carry
out
tests
on a
specimen
on a
reduced scale.
This method
isoften
used,
for
example
in
certification
tests
applied
to
containers
of
transport
of
nuclear matter which comprise,
for
example,
of a
test
of free
fall
of 9m
on
a
concrete flagstone (type
B
containers).
The
determination
of the
model
can be
made according
to two
different
assumptions.
Al.
Conservation
of
materials
With
this assumption, which most
frequently
used,
the
materials
of the
object
on
scale
1 and the
model
on a
reduced
scale
are the
same.
The
stresses
and the
velocities
(in
particular
the
velocity
of
sound
in
each material)
are
retained.
Let
us set L as a
length
on
scale
1 and i the
length corresponding
to the
reduced
scale.
The
scale ratio [BAK 73], [BRI 31], [FOC 53], [LAN 56], [MUR 50],
[PAS
67] and
[SED
72] is:
L
i
The
velocity being retained, i.e.
V =
—
= v = -, the
changing duration
is
given
by
T t
298
Mechanical shock
In
a
similar way, analysis
of the
dimensions leads
to the
following
relations:
Table
A.I. Scale factor
of
various
parameters
Surface
Volume
Density
Mass
Frequency
Force
Energy
Power
Pressure
(stress)
s
= )?S
w = tf W
Unchanged
m = X M
3>
cp
= —
X
f=X
2
F
e=X
3
E
p = X
2
P
Unchanged
Several requirements exist
at the
time
of the
definition
of the
model
on a
reduced
scale, such
as:
- the
clearance limits
on the
scale,
the
manufacturing tolerances,
the
states
of
surface;
-
for
the
screwed parts,
the
account taken,
if
possible,
of the
number
of
screws
(dimensions
on the
scale).
If the
ratio
of the
scale leads
to too
small screws,
one can
replace them
by
screws
of
bigger size,
of
lower number, taking account
of the
total
area
of the
cross-section
of the
whole
of the
screws;
- in the
case
of
stuck parts, taking account
of
the
bonding strength
in
similitude;
acceleration
becomes
Appendix
299
- in the
case
of
measurements taken
on the
model
by
similarity, taking into
account
the
similitude
of the frequency
bandwidth
of the
measuring equipment. This
is
not
always easy
to do, in
particular
in the
case
of
impact tests where
the
shock
to
be
measured already
has on
scale
1
high
frequency
contents;
- the
effects
of
gravity (when they
are not
negligible);
- as far as
possible
the
choice
of
sensors
in
similitude (dimensions, mass etc).
It
is
advisable
to
also consider
the
scale
factor
with
respect
to
their
functional
characteristics (acceleration
and the frequency are 1 / X
times
on a
reduced scale).
Difficulties
can
arise whose importance need
to be
evaluated according
to the
case studied.
For
example,
the
gradient
of the
stresses
in the
part
is not
taken into
account.
If G is
this
gradient
on
scale
1 and if £ is the
stress,
we
have
on a
reduced
scale
i.e.
A2.
Conservation
of
acceleration
and
stress
We
always have
Acceleration being preserved,
it
becomes, since
yielding
Table
A.2
summarizes
the
main relations.
300
Mechanical
shock
Table
A.2.
Scale
factors
Velocity
Surface
Volume
Density
Mass
Frequency
Force
Energy
Power
Pressure
(stress)
v
=
X
3
/
2
V
s = A
2
S
w = ?i
3
W
D
d = —
K
m = X M
$
(p
~ /—
4\
f=X
2
F
e = A? E
p =
X
5
/
2
P
Unchanged
Mechanical shock
tests:
a
brief historical background
The
very
first
tests
were performed
by the US
Navy around 1917 [PUS
77]
[WEL 46], Progress, which
was
slow
up to
World
War II,
accelerated
as from the
1940s.
The
following
list gives
a few of the
major
dates.
1932 First publication
on the
shock response spectrum
for the
study
of
earthquakes.
1939 First high impact shock machine (pendular drop-hammer)
for the
simulation
of the
effects
of
submarine explosions
on
on-board equipment
[CLE
72]
[OLI 47].
1941
Development
of a
10-foot
free
fall
test
method [DEV 47].
1945 Drafting
of the first
specifications
for
aircraft
equipment [KEN
51]
under
various environmental conditions (A.F. Specification 410065).
1947 Environmental measurements
on
land vehicles
for
drafting
specifications
[|PRI 47].
1947
Use of an air gun to
simulate shocks
on
electronic components (Naval
Ordnance Laboratory) [DEV 47].
1948 Free
fall
machine
on
sand, with monitoring
of the
amplitude
and
duration
of
the
shock
(US Air
Force) [BRO 61].
1955
Use of
exciters
for
shock simulation (reproduction
of
simple shapeshocks)
[WEL 61].
302
Mechanical shock
1964 Taking into account shocks
of
pyrotechnic origin, demonstrating
the
difficulty
of
simulating them with classic facilities. Development
of
special
facilities [BLA 64].
1966 Initial research into shock simulation
on
exciters
driven
from a
shock
response
spectrum [GAL 66]. These methods were
only
fully
developed
in
the
mid-1970s.
1984
The
shock response spectrum becomes
the
benchmark
in the
MIL-STD
810
D
standard
for the
definition
of
specifications.
Bibliography
[AER
66]
Aerospace systems pyrotechnic shock data (Ground
test
and flight),
Final
Report,
Contract
NAS 5,
15208, June 1966, March 1970.
[BAC
82]
BACA
T.J., Characterization
of
conservatism
in
mechanical shock testing
of
structures,
PhD
Dissertation, Department
of
Civil
Engineering, Stanford
University,
September 1982,
or
Sandia Report, Sand
82-2186,
1982.
[BAC
83]
BACA
T.J., 'Evaluation
and
control
of
conservatism
in
drop
table
shock
tests',
The
Shock
and
Vibration Bulletin,
n° 53,
Part
1, May
1983,
p.
166/170,
or
Sandia
Report,
82 -
1548C, 1982.
[BAC
84]
BACA
T.J., 'Alternative shock characterizations
for
consistent shock test
specification',
The
Shock
and
Vibration Bulletin,
n° 54,
Part
2,
June 1984,
p.
109/130.
[BAC
86]
BACA
T.J.
and
BLACKER
T.D., 'Relative conservatism
of
drop table
and
shaker
shock
tests',
The
Shock
and
Vibration Bulletin,
n° 56,
Part
1,
August 1986,
p.289/298.
[BAC
89]
BACA
T.J.
and
BELL
R.G., 'Identification
of
error sources
in
shock
data',
IES
Proceedings, 1989,
p.
1/7.
[BAI
79] BAI M. and
THATCHER
W.,
'High
G
pyrotechnic shock simulation using metam-
to-metal impact',
The
Shock
and
Vibration Bulletin,
49,
Part
1,
September
1979, p.97/100.
[BAK
73]
BAKER
W.E.,
WESTENE
P.S.
and
DODGE
F.T., Similarity methods
in
engineering
dynamics
-
Theory
and
practice
of
scale modeling, Spartan Books, Hayden
Book Company, Inc., Rochelle Park,
New
Jersey, 1973.
[BAR
73]
BARNOSKJ
R.L.
and
MAURER
J.R.,
Transient
characteristics
of
single systems
of
modulated random
noise',
Journal
of
Applied Mechanics, March 1973,
p.
73/77.
[BAR
74]
BARTHMAER
J.P.,
'Shock
testing under minicomputer control',
IES
Proceedings, 1974,
p.
207/215.
[BEL
88]
BELL
R.G.
and
DAVIEN.T.,
'Shock
response spectrum anomalies which occur
due
to
imperfections
in the
data',
Sandia Report, SAND-88-1486C, 1988.