Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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64
Mechanical
shock
The
pseudovelocity
G)
0
z
sup
tends towards
AV
when
the
damping tends towards
zero.
If
damping
is
different
from
zero,
the
pseudovelocity tends towards
a
constant
value
lower than
AV.
2
The
residual positive
SRS of the
relative displacements (o>
0
z
sup
)
decreases
at
low
frequencies
with
a
slope equal
to 1,
i.e.,
on a
logarithmic scale,
with
a
slope
of
6dB/octave(£
= 0).
The
impulse absolute response
of a
linear one-degree-of-freedom system
is
given
by
(Volume
1,
relation
3.85):
where
If
damping
is
zero,
The
'input' impulse
can be
represented
in the
form
as
long
as
The
response which results
The
maximum
of the
displacement takes place during
the
residual response,
for
yielding
the
shock
response
spectrum
Characteristics
of
shock
response spectra
65
and
Figure 3.4.
TPS
shock
pulse
A
curve defined
by a
relation
of the
form
y = a f is
represented
by a
line
of
slope
n on a
logarithmic
grid:
The
slope
can be
expressed
by a
number
N of
dB/octave
according
to
The
undamped shock response spectrum plotted
on a
log-log grid thus
has a
slope
at the
origin equal
to 1, i. e. 6
dB/octave.
Terminal peak
saw
tooth
pulse
10 ms, 100
m/s
2
66
Mechanical shock
Figure 3.5.
Residual positive
SRS
(relative
displacements)
of
a
TPS
shock pulse
The
primary positive
SRS o>
0
z
su
_ always
has a
slope equal
to 2 (12
dB/octave)
(example Figure 3.6) [SMA 85].
Figure 3.6.
Primary positive
SRS
of
a
half-sine
shock pulse
The
relative displacement
z
sup
tends towards
a
constant value
z
0
= x
m
equal
to
the
absolute displacement
of the
support during
the
application
of the
shock pulse
(Figure
3.7).
At low
resonance
frequencies, the
equipment
is not
directly sensitive
to
accelerations,
but to
displacement:
Characteristics
of
shock response spectra
67
Figure 3.7.
Behaviour
of
a
resonator
at
very
low
resonance
frequency
The
system works
as
soft
suspension
which
attenuates accelerations
with
large
displacements [SNO 68].
This property
can be
demonstrated
by
considering
the
relative displacement
response
of a
linear one-degree-of-freedom system given
by
Duhamel's equation
(Volume
1,
Chapter
2):
After
integration
by
parts
we
obtain
68
Mechanical
shock
The
mass
m of an
infinitely flexible oscillator
and
therefore
of
infinite
natural
period
(f
0
= 0),
does
not
move
in the
absolute reference axes.
The
spectrum
of the
relative
displacement thus
has as an
asymptotic value
the
maximum value
of the
absolute
displacement
of the
base.
Example
Figure
3.8
shows
the
primary positive
SRS
z
sup
(f
0
)
of a
shock
of
half-sine
shape
100
m/s
2
,10
ms
plotted
for £ = 0
between 0.01
and 100 Hz.
Figure 3.8.
Primary
positive
SRS
of
a
half-sine
(relative displacements)
The
maximum displacement
x
m
under shock calculated
from the
expression
x(t)
for the
acceleration pulse
is
equal
to:
The
SRS
tends towards this value when
Characteristics
of
shock response spectra
69
For
shocks
of
simple shape,
the
instant
of
time
t at
which
the
first peak
of the
response takes place tends towards
,
tends towards zero [FUN 57].
The
primary positive spectrum
of
pseudovelocities has,
a
slope
of 6
dB/octave
at
the low
frequencies.
Figure 3.9.
Primary
positive
SRS
of
a
TPS
pulse
(four
coordinate
grid)
Example
3.2.3.
Shocks
for AV = 0 and AD * 0 at end
of
pulse
In
this
case,
for £, = 0 :
- The
Fourier transform
of
the
velocity
for f = 0,
V(0),
is
equal
to
Since acceleration
is the first
derivative
of
velocity,
the
residual spectrum
is
equal
to co
0
AD for low
values
of
co
0
.
The
undamped residual shock response
spectrum thus
has a
slope equal
to 2
(i.e.
12
dB/octave)
in
this range.
70
Mechanical
shock
Example
Shock
consisted
by one
sinusoid period
of
amplitude
100
m/s
2
and
duration
10
Figure
3.10.
Residual positive
SRS
of
a
'sine
1
period'
shockpulse
- The
primary relative displacement (positive
or
negative, according
to the
form
of
the
shock)
z
sup
tends towards
a
constant value equal
to x
m
,
absolute
displacement
corresponding
to the
acceleration pulse
x(t)
defining
the
shock:
Characteristics
of
shock
response
spectra
71
Example
Let us
consider
a
terminal peak
saw
tooth pulse
of
amplitude
100
m/s
2
and
duration
10 ms
with
a
symmetrical rectangular
pre-
and
post-shock
of
amplitude
10
m/s
2
.
The
shock
has a
maximum displacement given
by
(Chapter
7):
At
the end of the
shock, there
is no
change
in
velocity,
but the
residual
displacement
is
equal
to
Using
the
numerical
data
of
this example,
we
obtain
x
m
=
-4.428
mm
We
find
this value
of x
m
on the
primary negative spectrum
of
this shock
(Figure
3.11).
In
addition,
Figure 3.11.
Primary negative
SRS
(displacements)
of
a
TPS
pulse
with
rectangular
pre-
and
post-shocks
^residua]
=
-0-9576
10"
4
mm
72
Mechanical shock
3.2.4. Shocks
with
AV = 0 and AD = 0 at end
of
pulse
For
oscillatory type shocks,
we
note
the
existence
of the
following
regions [SMA
85]
(Figure 3.12):
-just
below
the
principal
frequency of the
shock,
the
spectrum
has,
on a
logarithmic
scale,
a
slope characterized
by the
primary response (about
3);
-
when
the frequency of
spectrum
decreases,
its
slope tends towards
a
smaller
value
of 2;
-
when
the
natural
frequency
decreases
further,
one
observes
a
slope equal
to 1
(6
dB/octave) (residual spectrum).
In a
general
way,
all the
shocks, whatever their
form,
have
a
spectrum
of
slope
of 1 on a
logarithmic scale
if the frequency is
rather
small.
The
primary negative
SRS o
0
z
sup
has a
slope
of 12
dB/octave;
the
relative
displacement
z
sup
tends towards
the
absolute displacement
x
m
associated with
the
shock movement
x(t).
Figure
3.12.
Shock response spectrum (relative displacements)
of
a
ZERD
pulse
Characteristics
of
shock response spectra
73
Examples
Figure
3.13.
Primary negative
SRS
of
a
half-sine
pulse
with
half-sine
pre-
and
post-shocks
Figure
3.14.
Primary negative
SRS
(displacements)
of
a
half-sine
pulse
•with
half-sine
pre-
and
post-shocks