Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
Подождите немного. Документ загружается.
x
Mechanical shock
9.6.2.
Velocity
and
displacement
270
9.6.3.
Response spectrum
271
9.6.3.1.
Influence
of
damping
n of the
signal
271
9.6.3.2. Influence
of the Q
factor
on the
spectrum
272
9.6.4.
Time history synthesis
from
shock spectrum
273
9.7. Comparison
of the
WAVSIN, SHOC waveforms
and
decaying sinusoid
274
9.8.
Use of a
fast
swept sine
274
9.9. Problems encountered during
the
synthesis
of the
waveforms
278
9.10. Criticism
of
control
by a
shock response spectrum
280
9.11.
Possible
improvements
282
9.11.1.
IBS
proposal
283
9.11.2.
Specification
of a
complementary parameter
284
9.11.2.1.
Rms
duration
of the
shock
284
9.11.2.2.
Rms
value
of the
signal
286
9.11.2.3.
Rms
value
in the frequency
domain
287
9.11.2.4. Histogram
of the
peaks
of the
signal
288
9.11.2.5.
Use of the
fatigue
damage spectrum
288
9.11.3. Remarks
on the
characteristics
of
response spectrum
288
9.12. Estimate
of the
feasibility
of a
shock specified
by its SRS 289
9.12.1 C.D. Robbins
and
E.P. Vaughan's method
289
9.12.2.
Evaluation
of the
necessary force, power
and
stroke
291
Appendix.
Similitude
in
mechanics
297
A1.
Conservation
of
materials
297
A2.
Conservation
of
acceleration
and
stress
299
Mechanical
shock
tests:
a
brief historical background
301
Bibliography
303
Index
315
Synopsis
of
five volume
series
319
Introduction
Transported
or
on-board equipment
is
very
frequently
subjected
to
mechanical
shocks
in the
course
of its
useful
lifetime
(material handling, transportation,
etc.).
This
kind
of
environment, although
of
extremely short duration
(from
a fraction of a
millisecond
to a few
dozen milliseconds)
is
often
severe
and
cannot
be
neglected.
The
initial
work
into shocks
was
carried
out in the
1930s
on
earthquakes
and
their
effect
on
buildings. This resulted
in the
notion
of the
shock
response
spectrum.
Testing
on
equipment started during
World
War II.
Methods continued
to
evolve
with
the
increase
in
power
of
exciters, making
it
possible
to
create
synthetic shocks,
and
again
in the
1970s, with
the
development
of
computerization, enabling
tests
to
be
directly conducted
on the
exciter employing
a
shock response spectrum.
After
a
brief recapitulation
of the
shock shapes most widely used
in
tests
and of
the
possibilities
of
Fourier analysis
for
studies taking account
of the
environment
(Chapter
1),
Chapter
2
presents
the
shock response spectrum
with
its
numerous
definitions
and
calculation methods.
Chapter
3
describes
all the
properties
of the
spectrum, showing that important
characteristics
of the
original signal
can be
drawn
from it,
such
as its
amplitude
or
the
velocity change
associated
with
the
movement during
the
shock.
The
shock
response
spectrum
is the
ideal
tool
for
drafting specifications.
Chapter
4
details
the
process
which makes
it
possible
to
transform
a set of
shocks
recorded
in
the
real environment into
a
specification
of the
same severity,
and
presents
a few
other methods that have been proposed
in the
literature.
Knowledge
of the
kinematics
of
movement during
a
shock
is
essential
to the
understanding
of the
mechanism
of
shock machines
and
programmers. Chapter
5
gives
the
expressions
for
velocity
and
displacement according
to
time
for
classic
shocks, depending
on
whether they occur
in
impact
or
impulse mode.
xii
Mechanical shock
Chapter
6
describes
the
principle
of
shock machines currently most widely used
in
laboratories
and
their associated programmers.
To
reduce costs
by
restricting
the
number
of
changes
in
test
facilities, specifications expressed
in the
form
of a
simple
shock (half-sine, rectangle,
saw
tooth with
a
final
peak)
can
occasionally
be
tested
using
an
electrodynamic exciter. Chapter
7
sets
out the
problems encountered,
stressing
the
limitations
of
such means, together with
the
consequences
of
modification,
that have
to be
made
to the
shock profile,
on the
quality
of the
simulation.
Pyrotechnic devices
or
equipment
(cords,
valves, etc.)
are
very
frequently
used
in
satellite launchers
due to the
very high degree
of
accuracy that they provide
in
operating sequences. Shocks induced
in
structures
by
explosive charges
are
extremely
severe,
with very specific characteristics. Their simulation
in the
laboratory requires specific means,
as
described
in
Chapter
8.
Determining
a
simple shape shock
of the
same severity
as a set of
shocks,
on the
basis
of
their response spectrum,
is
often
a
delicate operation. Thanks
to
progress
in
computerization
and
control facilities, this
difficulty
can
occasionally
be
overcome
by
expressing
the
specification
in the
form
of a
response spectrum
and by
controlling
the
exciter directly
from
that spectrum.
In
practical terms,
as the
exciter
can
only
be
driven with
a
signal that
is a
function
of
time,
the
software
of the
control
rack determines
a
time signal with
the
same spectrum
as the
specification displayed.
Chapter
9
describes
the
principles
of the
composition
of the
equivalent shock, gives
the
shapes
of the
basic signals most
often
used, with their properties,
and
emphasizes
the
problems that
can be
encountered, both
in the
constitution
of the
signal
and
with respect
to the
quality
of the
simulation obtained.
Containers must protect
the
equipment carried
in
them
from
various
forms
of
disturbance related
to
handling
and
possible accidents. Tests designed
to
qualify
or
certify
containers include shocks
mat are
sometimes
difficult,
not to say
impossible,
to
produce, given
the
combined weight
of the
container
and its
content.
One
relatively
widely used possibility consists
of
performing shocks
on
scale models,
with
scale
factors
of the
order
of 4 or 5, for
example. This same technique
can be
applied, although less
frequently, to
certain vibration tests.
At the end of
this
volume,
the
Appendix summarizes
the
laws
of
similarity
adopted
to
define
the
models
and to
interpret
the
test
results.
List
of
symbols
The
list below gives
the
most
frequent
definition
of the
main symbols used
in
this
book. Some
of the
symbols
can
have another meaning locally which will
be
defined
in the
text
to
avoid
any
confusion.
a
max
Maximum value
of
a(t)
a(
t)
Component
of
shock
x( t)
A
c
Amplitude
of
compensation
signal
A(0)
Indicial admittance
b
parameter
b of
Basquin's
relation
N o
b
= C
c
Viscous damping constant
C
Basquin's
law
constant
(N
o
b
= C)
d(t) Displacement associated
with
a(t)
D
Diameter
of
programmer
D(f
0
)
Fatigue damage
e
Neper's
number
E
Young's modulus
or
energy
of a
shock
ERS
Extreme
response
spectrum
E(t) Function characteristic
of
swept sine
f
Frequency
of
excitation
f
0
Natural frequency
F(t) External force applied
to
system
P
rms
Rms
value
of
force
F
m
Maximum value
of
F(t)
g
Acceleration
due to
gravity
h
Interval (f/f
0
)
or
thickness
of the
target
h(t)
Impulse
response
H
Drop
height
H
R
Height
of
rebound
H(
)
Transfer
function
i v=f
IPS
Initial
peak
saw
tooth
3(Q)
Imaginary
part
of
X(O)
k
Stiffness
or
coefficient
of
uncertainty
K
Constant
of
proportionality
of
stress
and
deformation
^nns
Rms
value
of
#(t)
i
m
Maximum
of
l(t)
t(t)
Generalized
excitation
(displacement)
t(t\
First
derivative
of
^(t)
xiv
Mechanical
shock
"i(t)
Second derivative
of
t(i)
L
Length
L(Q) Fourier transform
of
^(t)
m
Mass
n
Number
of
cycles undergone
by
test-bar
or
material
N
Number
of
cycles
to
failure
p
Laplace variable
or
percentage
of
amplitude
of
shock
q
0
Value
of
q(0)
for 0=0
q
0
Value
of
q(0)
for 0=0
q(0)
Reduced response
q(
0)
First derivative
of q( 0)
q( 0)
Second derivative
of q( 0)
Q
Q
factor (quality factor)
Q( p)
Laplace transform
of q (0)
R
e
Yield stress
R
m
Ultimate tensile strength
R(Q)
Fourier
transform
of the
system response
9?(Q)
Real part
of
X(Q)
s
Standard deviation
S
Area
SRS
Shock
response
spectrum
s( )
Power spectral density
t
Time
t
d
Decay time
to
zero
of
shock
tj
Fall duration
t
r
Rise time
of
shock
t
R
Duration
of
rebound
T
Vibration duration
T
0
Natural period
TPS
Terminal
peak
saw
tooth
u(t) Generalized response
u(t) First derivative
of
u(t)
u(t) Second derivative
of
u(t)
v
f
Velocity
at end of
shock
Vj
Impact velocity
V
R
Velocity
of
rebound
v(t)
Velocity x(t)
or
velocity
associated
with a(t)
v( )
Fourier transform
of
v(t)
x
m
Maximum value
of
x(t)
x(t)
Absolute displacement
of
the
base
of a
one-degree-of-
freedom
system
x(t) Absolute velocity
of the
base
of
a
one-degree-of-£reedom
system
x(t) Absolute acceleration
of the
base
of a
one-degree-of-
freedom
system
\
ms
Rms
value
of
x(t)
x
m
Maximum value
of
x(t)
X
m
Amplitude
of
Fourier
transform
X(Q)
X(Q) Fourier transform
of
x(t)
y(t) Absolute response
of
displacement
of
mass
of a
one-degree-of-freedom
system
y(t) Absolute response velocity
of
the
mass
of a
one-degree-
of-freedom
system
y(t) Absolute response
acceleration
of
mass
of a
one-degree-of-freedom
system
z
m
Maximum value
of
z(t)
z
s
Maximum static relative
displacement
z
sup
Largest value
of z( t)
z(t) Relative
response
displacement
of
mass
of a
one-degree-of-freedom
system with
respect
to its
base
z(t) Relative response velocity
z(t) Relative response
acceleration
a
Coefficient
of
restitution
5(t) Dirac delta
function
AV
Velocity change
(ft
Dimensionless product
f
0
T
(j)(Q)
Phase
TI
Damping
factor
of
damped
sinusoid
TJ
C
Relative damping
of
compensation signal
X.(
)
Reduced excitation
A(p) Laplace
transform
of
A,(
)
K
3.14159265...
6
Reduced time (co
0
t)
0
d
Reduced decay time
List
of
symbols
xv
6
m
Reduced rise time
6
0
Value
of 0 for t = t
p
Density
a
Stress
cr
cr
Crushing stress
a
m
Maximum stress
T
Shock duration
T!
Pre-shock duration
t
2
Post-shock duration
i
nns
Rms
duration
of a
shock
co
c
Pulsation
of
compensation
signal
co
0
Natural pulsation
(2 n f
0
)
Q
Pulsation
of
excitation
(27Cf)
£
Damping
factor
This page intentionally left blank
Chapter
1
Shock
analysis
1.1.
Definitions
1.1.1.
Shock
Shock
is
defined
as a
vibratory excitation with duration between once
and
twice
the
natural period
of the
excited mechanical system.
Figure
1.1.
Example
of
shock
Shock occurs when
a
force,
a
position,
a
velocity
or an
acceleration
is
abruptly
modified
and
creates
a
transient
state
in the
system considered.
2
Mechanical shock
The
modification
is
normally regarded
as
abrupt
if it
occurs
in a
time period
which
is
short compared
to the
natural period concerned (AFNOR definition)
[NOR].
1.1.2.
Transient
signal
This
is a
vibratory signal
of
short duration
(of a
fraction
of
second
to a few
tens
of
seconds),
the
mechanical shock,
for
example,
in the
air-braking phase
on
aircraft
etc.
Figure
1.2.
Example
of
transient signal
1.1.3.
Jerk
A
jerk
is
defined
as the
derivative
of
acceleration with respect
to
time. This
parameter thus characterizes
the
rate
of
variation
of
acceleration with time.
1.1.4. Bump
A
bump
is a
simple shock which
is
generally repeated many times when testing
(AFNOR) [NOR].
Example
The
GAM
EG 13
(first part
-
Booklet
43 -
Shocks) standard proposes
a
test
characterized
by a
half-sine:
10 g, 16 ms,
3000 bumps (shocks)
per
axis,
3
bumps
a
second [GAM86].
Shock
analysis
3
1.1.5.
Simple
(orperfect)
shock
Shock whose signal
can be
represented exactly
in
simple mathematical terms,
for
example half-sine, triangular
or
rectangular shock.
1.1.6.
Half-sine
shock
Simple
shock
for
which
the
acceleration-time curve
has the
form
of a
half-period
(part positive
or
negative)
of a
sinusoid.
1.1.7.
Terminal
peak
saw
tooth shock
(TPS)
or
final peak
saw
tooth shock
(FPS)
Simple shock
for
which
the
acceleration-time curve
has the
shape
of a
triangle
where acceleration increases linearly
up to a
maximum value
and
then instantly
decreases
to
zero.
1.1.8.
Initial peak
saw
tooth shock
(IPS)
Simple shock
for
which
the
acceleration-time curve
has the
shape
of a
triangle
where acceleration instantaneously increases
up to a
maximum,
and
then decreases
to
zero.
1.1.9.
Rectangular shock
Simple
shock
for
which
the
acceleration-time curve increases instantaneously
up
to
a
given value, remains constant throughout
the
signal
and
decreases
instantaneously
to
zero.
1.1.10.
Trapezoidal
shock
Simple shock
for
which
the
acceleration-time curve grows linearly
up to a
given
value,
remains constant during
a
certain time
after
which
it
decreases linearly
to
zero.