Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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34
Mechanical
shock
Rectangular
pulse
Initial
peak
saw
tooth pulse
Shock
response
spectra domains
35
Terminal
peak
saw
tooth pulse
Arbitrary
triangular pulse
36
Mechanical shock
Trapezoidal
pulse
Figure
2.7.
Trapezoidal
shock pulse
where
Shock
response
spectra
domains
37
For
an
isosceles trapezoid,
we set 6
r
= 0
0
- 6
d
. If the
rise
and
decay each have
a
duration equal
to 10% of the
total duration
of the
trapezoid,
we
have
Absolute
acceleration shock response spectrum
In
the
most usual
cases
where
the
excitation
is
defined
by an
absolute
acceleration
of the
support
or by a
force applied directly
to the
mass,
the
response
of
the
system
can be
characterized
by the
absolute acceleration
of the
mass (which
could
be
measured using
an
accelerometer fixed
to
this mass):
the
response spectrum
is
then called
the
absolute acceleration shock response spectrum. This spectrum
can
be
useful
when absolute acceleration
is the
parameter
easiest
to
compare with
a
characteristic
value (study
of the
effects
of a
shock
on a
man, comparison with
the
specification
of an
electronics component etc).
Relative
displacement shock spectrum
In
similar
cases,
we
often
calculate
the
relative displacement
of the
mass with
respect
to the
base
of the
system, displacement which
is
proportional
to the
stress
created
in the
spring (since
the
system
is
regarded
as
linear).
In
practice,
one in
general
expresses
in
ordinates
the
quantity
co
0
z
sup
called
the
equivalent static
acceleration.
This product
has the
dimension
of an
acceleration,
but
does
not
represent
the
acceleration
of the
mass, except when damping
is
zero; this term
is
then
strictly equal
to the
absolute
acceleration
of the
mass. However, when damping
is
close
to the
current values observed
in
mechanics,
and in
particular when
2
4
=
0.05,
one can
assimilate
as a first
approximation
co
0
z
sup
to the
absolute
acceleration
y
sup
of the
mass
m
[LAL 75].
2.3. Definitions
Response
spectrum
A
curve representative
of the
variations
of the
largest response
of a
linear one-
degree-of-freedom
system subjected
to a
mechanical excitation, plotted against
its
natural
frequency f
0
=
for
a
given value
of its
damping
ratio.
38
Mechanical shock
Very
often
in
practice,
it is the
stress (and thus
the
relative displacement) which
seems
the
most interesting parameter,
the
spectrum being primarily used
to
study
the
behaviour
of a
structure,
to
compare
the
severity
of
several
shocks
(the
stress
created
is
a
good indicator),
to
write
test
specifications
(it is
also
a
good comparison
between
the
real environment
and the
test environment)
or to
dimension
a
suspension (relative displacement
and
stress
are
then
useful).
The
quantity
co
0
z
su
is
termed pseudo-acceleration.
In the
same way,
one
terms
pseudo-velocity
the
product
o)
0
z
sup
.
2
The
spectrum giving
co
0
z
sup
versus
the
natural
frequency is
named
the
relative
displacement
shock spectrum.
In
each
of
these
two
important categories,
the
response spectrum
can be
defined
in
various ways according
to how the
largest response
at a
given
frequency is
characterized.
Primary
positive shock response spectrum
or
initial positive shock response
spectrum
The
highest positive response observed during
the
shock.
Primary
(or
initial)
negative shock response spectrum
The
highest negative
response
observed during
the
shock.
Secondary
(or
residual
shock)
response spectrum
The
largest response observed
after
the end of the
shock. Here also,
the
spectrum
can
be
positive
or
negative.
Positive
(or
maximum positive) shock response spectrum
The
largest
positive
response
due to the
shock, without reference
to the
duration
of
the
shock.
It is
thus about
the
envelope
of the
positive primary
and
residual
spectra.
Shock
response spectra domains
39
Negative
(or
maximum negative) shock response spectrum
The
largest negative response
due to the
shock, without reference
to the
duration
of
the
shock.
It is in a
similar
way the
envelope
of the
negative primary
and
residual
spectra.
Example
Maximax
shock response spectrum
Envelope
of the
absolute values
of the
positive
and
negative spectra.
Which
spectrum
is the
best?
The
damage
is
supposed proportional
to the
largest
value
of the
response, i.e.
to the
amplitude
of the
spectrum
at the frequency
considered,
and it is of
little importance
for the
system whether this maximum
z
m
takes place during
or
after
the
shock.
The
most interesting
spectra
are
thus
the
positive
and
negative
spectra,
which
are
most
frequently
used
in
practice,
with
the
maximax
spectrum.
The
distinction between positive
and
negative
spectra
must
be
made
each time
the
system,
if
disymmetrical, behaves differently,
for
example under
different
tension
and
compression.
It is,
however,
useful
to
know these various
definitions
so as to be
able
to
correctly interpret
the
curves published.
Figure
2.8. Shock response spectra
of
a
rectangular shock pulse
40
Mechanical
shock
2.4. Standardized response spectra
For
a
given
shock,
the
spectra
plotted
for
various values
of the
duration
and the
amplitude
are
homothetical.
It is
thus interesting,
for
simple shocks
to
have
a
standardized
or
reduced spectrum plotted
in
dimensionless co-ordinates,
while
plotting
on the
abscissa
the
product
f
0
t
(instead
of f
0
) or co
0
t and on the
ordinate
the
spectrum/shock pulse amplitude ratio
co
0
z
m
/x
m
,
which,
in
practice, amounts
to
tracing
the
spectrum
of a
shock
of
duration equal
to 1 s and
amplitude
1
m/s
2
.
Figure
2.9. Standardized
SRS
of
a
half-sine
pulse
These standardized
spectra
can be
used
for two
purposes:
-
plotting
of the
spectrum
of a
shock
of the
same
form,
but of
arbitrary amplitude
and
duration;
-
investigating
the
characteristics
of a
simple shock
of
which
the
spectrum
envelope
is a
given spectrum (resulting
from
measurements
from the
real
environment).
The
following
figures
give
the
spectra
of
reduced shocks
for
various pulse forms,
unit
amplitude
and
unit duration,
for
several values
of
damping.
To
obtain
the
spectrum
of a
particular shock
of
arbitrary amplitude
x
m
and
duration
T
(different
from
1)
from
these spectra,
it is
enough
to
regraduate
the
scales
as
follows:
-
for
the
amplitude;
by
multiply
the
reduced values
by x
m
;
- for the
abscissae, replace each value
<}>
(= f
0
T
) by f
0
=
Shock
response
spectra
domains
41
We
will
see
later
on how
these
spectra
can be
used
for the
calculation
of
test
specifications.
Half-sine
pulse
Figure
2.11.
Standardized primary
and
residual relative displacement
SRS
of
a
half-sine
pulse
Figure 2.10.
Standardized
positive
and
negative relative displacement
SRS
of
a
half-sine
pulse
42
Mechanical
shock
Figure
2.12.
Standardized
positive
and
negative absolute acceleration
SRS
of
a
half-sine
pulse
Versed-sine
pulse
Figure
2.13.
Standardized
positive
and
negative relative displacement
SRS
of
a
versed-sine pulse
Shock
response spectra
domains
43
Terminal
peak
saw
tooth pulse
Figure 2.14.
Standardized
primary
and
residual relative displacement
SRS
of
a
versed-sine pulse
Figure
2.15.
Standardized
positive
and
negative relative displacement
SRS
of
a
TPS
pulse