Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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254
Mechanical
shock
Gradually,
the
peaks
of
maximum amplitude come before
it
decreases
at a
regular
rate.
The
positive
and
negative peaks
are
almost symmetrical. Therefore,
it is
well
adapted
to a
reproduction
on
shaker.
9.4.1.2.
D.O. Smallwood
expression
D.O. Smallwood [SMA
85]
defined ZERD function
by the
relation:
where
It
is
this definition which
we
will
use
hereafter.
where
This
function
resembles
a
damped sinusoid
and has the
advantage
of
leading
to
zero velocity
and
displacement
at the end of the
shock.
Figure 9.17.
ZERD
waveform
of
D.K.
Fisher
and
M.R.
Posehn
(example)
Control
of a
shaker using
a
shock
response
spectrum
255
9.4.2.
Velocity
and
displacement
By
integration
we
obtain
the
velocity:
It
can be
interesting
to
consider
the
envelope
of
this signal.
If we
pose:
If
n is
small,
T
also
is
small,
so
that C(t)
~ Q t and
q(t)
= Q t cos [a t +
9(t)].
and
the
displacement
256
Mechanical
shock
Figure
9.18. ZERD
waveform
of
D.O. Smallwood
(example)
Acceleration
becomes:
and for n
small
The
maximum
of the
envelope
t
e"
11
takes place
for t = .
Since
is
nQ
r|Q
the
time-constant,
the
function
reaches
its
maximum
in
time
1 / rj Q. The
value
of
this
maximum
is , the
maximum
of cos
[Q
t +
9(t)] being equal
to 1
Qne
A
is
thus
the
amplitude
of
a(t).
1
1
1
Control
of a
shaker
using
a
shock
response
spectrum
257
9.4.3.
Comparison
of
the
ZERD
waveform
with
a
standard decaying sinusoid
This comparison
can be
carried
out
through
the
envelopes,
Figure 9.19.
Comparison
of
the
ZERD
and
decaying
sinusoid
waveform
envelopes
The
plotting
of
these
two
curves shows that they have:
-
the
same amplitude
A;
- the
same slope (-1)
in a
semi-logarithmic scale
for
a t
large;
- a
different
decrement with
9.4.4. Reduced
response
spectra
9.4.4.1.
Influence
of
the
damping
n
of
the
signal
The
response spectra
of
this shock
are
plotted
in
Figure 9.20 with
a Q
factor
equal
to 10;
they
correspond
to
signals
according
to the
times
defined
for n =
0.01,
0.05
and 0.1
respectively.
and
where
decreasing less quickly
258
Mechanical
shock
Figure
9.20. ZERD
waveform
-
influence
of
damping
n on the SRS
The
peak
of
these spectra
becomes
increasingly narrow
as n
decreases.
9.4.4.2.
Influence
of
the Q
factor
The
response
spectra
of
Figure 9.21
are
plotted
from a
ZERD waveform
(n =
0.05)
for Q = 50, 10 and 5,
respectively.
Figure
9.21. ZERD
waveform
-
influence
of
the
Q
factor
on the SRS
Control
of a
shaker using
a
shock
response
spectrum
259
9.5. WAVSIN waveform
9.5.1.
Definition
R.C.
Yang ([SMA 74a], [SMA 75], [SMA 85], [YAN
70] and
[YAN 72])
proposed
(initially
for the
simulation
of the
earthquakes)
a
signal
of the
form:
where
N is an
integer (which,
we
will
see it,
must
be odd and
higher than
1). The
first
term
of
a(t)
is a
window
of
half-sine
form
of
half-period
i. The
second
describes
N
half-cycles
of a
sinusoid
of
greater
frequency
(f), modulated
by the
preceding window.
where
260
Mechanical shock
9.5.2.
Velocity
and
displacement
The
function
a(t)
can
also
be
written:
Figure
9.22.
Example
of
WAVSIN
waveform
Example
Figure 9.22 shows
an
acceleration signal WAVSIN plotted
for
Velocity
At
the end of the
shock
Control
of a
shaker
using
a
shock
response
spectrum
261
Whatever
the
value
of N
Figure
9.23.
Velocity
corresponding
to the
Figure
9.24. Displacement corresponding
to
waveform
in
Figure 9.22
the
waveform
in
Figure 9.22
Displacement
For
.
.
262
Mechanical shock
If
N is
odd(N
= 2n + l):
For
the
displacement
to be
zero
at the end of the
shock,
it is
thus necessary that
N
is
an odd
integer.
The
advantages
of
this waveform are:
- The
residual velocity
and
displacement associated with each elementary signal
a(t)
are
zero.
-
With
N
being odd,
the two
functions
sine intervening
in
a(t)
are
maximum
for
The
maximum
of
a(t)
is
thus
a
m
.
- The
components have
a
finite
duration, which makes
it
possible
to
avoid
the
problems
involved
in a
possible truncation
of the
signals (the case
of
decaying
sinusoids
for
example).
9.5.3.
Response
of
a
one-degree-of-freedom
system
Let
us set f
0
as the
natural
frequency of
this system
and Q as its
quality factor.
To
simplify
the
writing,
let us
express a(t)
in
dimensionless coordinates,
in the
form:
where
Control
of a
shaker using
a
shock response spectrum
263
9.5.3.1.
Relative response displacement
where:
yielding