Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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244
Mechanical shock
frequencies and
weak
at the
high
frequencies. It
can,
for
example, decrease
in a
linear
way from 0.3 to
0.01
between
the two
ends
of the
spectrum.
NOTE:
If we
have acceleration signals which lead
to the
specified
spectrum,
we
could
use the
Prony method
to
estimate
the frequencies and
damping
factors [GAR
86].
1
-n
being chosen,
we can
calculate, using
the
relation
[9.3],
for Q = —
given
(damping chosen
for
plotting
of the
shock spectrum
of
reference),
the
ratio
R of the
peak
of the
spectrum
to the
amplitude
of the
decaying
sinusoid.
This value
of R
makes
it
possible
to
determine
the
amplitude
a
max
of the
decaying sinusoid
at the
particular frequency.
Knowing that
the
amplitude
amax
of the
first
peak
of the
decaying sinusoid
is
related
to the
constant
A by the
relation
[9.2]:
we
determine
the
value
of A for
each elementary sinusoid.
For n
small
( <
0.08
), we
have
9.2.5.
Methods
of
signal compensation
Compensation
can be
carried
out in
several ways.
1.
By
truncating
the
total signal until
it is
realizable
on the
shaker. This
correction
can,
however, lead
to an
important degradation
of the
corresponding
spectrum [SMA
73].
2. By
adding
to the
total signal
(sum
of all the
elementary signals)
a
highly
damped decaying
sinusoid,
shifted
in
time, defined
to
compensate
for the
velocity
and
the
displacement [SMA
74b]
[SMA
75]
[SMA
85].
3. By
adding
to
each component
two
exponential compensation functions, with
a
phase
in the
sinusoid
[NEL
74]
[SMA
75]
Control
of a
shaker using
a
shock
response spectrum
245
Compensation
using
a
decaying
sinusoid
In
order
to
calculate
the
characteristics
of the
compensating pulse,
the
complete
acceleration signal used
to
simulate
the
specified spectrum
can be
written
in the
form:
where
0i
is
the
delay applied
to the ith
elementary signal.
A
c
, w
c
, n
c
and 9 are the
characteristics
of the
compensating signal (decaying sinusoid).
The
calculation
of
these constants
is
carried
out by
cancelling
the
expressions
of the
velocity
x and of
the
displacement
x
obtained
by
integration
of x.
246
Mechanical
shock
The
cancellation
of the
velocity
and
displacement
for t
equal
to
infinity
leads
to:
yielding
Figure 9.8.
Acceleration
pulse
compensated
Figure 9.9.
Velocity
associated with
the
by
a
decaying sinusoid signal compensated
by a
decaying sinusoid
Control
of a
shaker using
a
shock
response
spectrum
247
Figure
9.10.
Displacement associated with
the
signal compensated
by a
decaying sinusoid
The
constants
A
c
and 0
characterizing
the
compensating sinusoid
are
thus
a
function
of the
other parameters (Q
c
, n
c
).
The frequency of the
compensating
Compensation
using
two
exponential
signals
This method, suggested
by
Nelson
and
Prasthofer
[NEL
74]
[SMA 85], consists
of
adding
to the
decaying sinusoid
two
exponential signals
and a
phase
shift
9.
Each
elementary waveform
is of the
form:
The
exponential terms
are
defined
in
order
to
compensate
the
velocity
and the
displacement (which must
be
zero
at the
beginning
and the end of the
shock).
The
phase
0 is
given
to
cancel acceleration
at t = 0.
With this method, each individual
component
is
thus compensated.
The
choice
of the
parameters
a, b and c is
somewhat arbitrary.
The
idea being
to
create
a
signal resembling
to a
damped
decaying sinusoid,
one
chooses:
(
<O
waveform
f
c
= —
should
be
about
a
half
or
third
of the
smallest
frequency of the
V
27iJ
points
selected
to
represent
the
specified spectrum.
The
damping
n
c
is
selected
to be
between
0.5 and 1
[SMA 74b].
The
constants
A
c
and 0 can
then
be
determined.
248
Mechanical shock
where
The
acceleration, velocity
and
displacement
are
zero
at the
beginning
and the end
of the
shock
if:
With
these notations,
the
velocity
and the
displacement
are
given
by:
Control
of a
shaker
using
a
shock response spectrum
249
Figure 9.11 shows
as an
example
a
signal compensated according
to
this
method.
It is
noted that
the
first
negative peak
is
larger than
the
first
positive peak.
The
waveform
resembles overall
a
decaying sinusoid.
Figure
9.11.
Waveform
compensated
by two
exponential signals
Figures
9.12
and
9.13 give
the
corresponding velocity
and
displacement.
Figure
9.12.
Velocity
after
compensation
by
Figure
9.13.
Displacement
after
two
exponential signals compensation
by two
exponential signals
250
Mechanical shock
NOTE:
We
also
find the
equivalent
form
where
k1 and k2
have
the
same
definition
as
previously.
9.2.6. Iterations
Once
all
these coefficients
are
determined,
one
calculates
the
spectrum
of the
signal thus made
up. All
this work
was
carried
out up to now by
supposing
the
influence
of
each decaying sine
of frequency fi on the
other points
of the
spectrum
to be
negligible. This assumption
is
actually
too
simplifying,
and the
spectrum
obtained
does
not
match
the
specified spectrum.
It is
thus necessary
to
proceed
to
successive iterations
to
refine
the
values
of the
amplitudes
A i of the
components
of
x(t):
The
iterations
can be
carried
out by
correcting
the
amplitudes
of the
elementary
waveforms
by a
simple rule
of
three. More elaborate relations were proposed, such
as
[BOI 81], [CRI 78]:
where
-
S
c
^
n
'(fj)
is the
amplitude
of the
spectrum calculated using
the
values
Ai of the
nth
iteration
at the frequency fi;
-
S(fi)
is the
value
of the
reference spectrum
to the frequency fi;
-
S
c
(fi)
is the
amplitude
of the
spectrum calculated with
the
values
Ai of the
nth
iteration
at the frequency fi,
except
for the
coefficient
Ai
which
is
added with
an
increment
AAi
(0.05 gives satisfactory results);
with
Control
of a
shaker using
a
shock response spectrum
251
- p is a
weighting factor (0.5 gives
an
acceptable velocity
of
convergence).
If
the
procedure converges (and
it is
fortunately
the
case
in
general),
the
reconstituted spectrum
and the
specified spectrum
are
very close
at the frequencies
fi
selected.
The
agreement
can be
worse between
the frequencies fi.
These
intermediate values
can be
modified
by
changing
the
sign
of the
constants
Ai
(taken
to
be
initially positive)
[NEL
74].
One
often
retains
the
empirical
rule
which consists
of
alternating
the
signs
of the
components.
If
the
levels
are too
high between
the frequencies fi, one
must decrease
the
values
of ni. If, on the
contrary, they
are too
weak,
it is
necessary
to
increase
the
values
of ni or to add
components between
the fi frequencies.
The
calculated spectrum must match
the
specification
as
well
as
possible,
as
close
to the
peaks
of the
spectrum
as to the
troughs.
One
acts sometimes
on the
frequency
f
c
to
readjust
to
zero
the
residual velocity
V
R
and
displacement
d
R
(to
increase
f
c
led to a
reduction
in V
R
and d
R
in
general [SMA 74a]).
9.3. D.L. Kern
and
C.D. Hayes' function
9.3.1.
Definition
To
reproduce
a
given spectrum
frequency
after
frequency
(and
not
simultaneously
a
whole spectrum), D.L. Kern
and
C.D. Hayes [KER
84]
proposed
to
use a
shock
of the
form:
where
f= frequency
n
=
damping
A
=
amplitude
of the
shock
e =
Neper number
The
signal resembles
the
response
of
one-degree-of-freedom system
to a
Dirac
impulse function.
It can
seem interesting
a
priori
to
examine
the
potential
use of
this
function
for the
synthesis
of the
spectra.
The first
parameters
to be
considered
are
residual velocity
and
displacement.
252
Mechanical shock
9.3.2.
Velocity
and
displacement
Velocity
By
integration
we
obtain:
The
velocity,
different
from
zero
for t = 0, can be
cancelled
by
adding
the
term
Displacement
The
integration
of
v(t) gives:
One
cannot cancel
the
velocity
and the
displacement
at the
same time
for t = 0
and
t
large.
As
an
example,
the
curves
of
Figures 9.14, 9.15
and
9.16 show
the
acceleration,
the
velocity
and the
displacement
for f = 1 and n =
0.05.
but
then,
the
residual velocity
is
zero
no
longer.
Control
of a
shaker
using
a
shock
response
spectrum
253
Figure
9.15.
Velocity
Figure
9.16.
Displacement
9.4. ZERD
function
9.4.1.
Definition
9.4.1.1.
D.K. Fisher
and
M.R.
Posehn
expression
The
use of a
decaying sinusoid
to
compose
a
shock
of
given
SRS has the
disadvantage
of
requiring
the
addition
of a
compensation
waveform
intended
to
reduce
to
zero
the
velocity
and the
displacement
at the end of the
shock. This signal
modifies
the
response
spectrum
at the low frequencies
and,
in
certain cases,
can
harm
the
quality
of
simulation.
D.K.
Fisher
and
M.R. Posehn [FIS
77]
proposed using
a
waveform
which they
named ZERD (Zero Residual Displacement) defined
by the
expression:
Figure
9.14.
Example
of
D.L.Kern
and
C.D.
Hayes'
waveform