Lalanne C. Mechanical Vibrations and Shocks: Mechanical Shock Volume II
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Mechanical
Shock
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Mechanical
Vibration
&
Shock
Mechanical
Shock
Volume
II
Christian Lalanne
HPS
HERMES
PENTON SCIENCE
First published
in
1999
by
Hermes Science Publications, Paris
First published
in
English
in
2002
by
Hermes Penton
Ltd
Apart
from any
fair
dealing
for the
purposes
of
research
or
private study,
or
criticism
or
review,
as
permitted under
the
Copyright, Designs
and
Patents
Act
1988, this publication
may
only
be
reproduced,
stored
or
transmitted,
in any
form
or by any
means, with
the
prior permission
in
writing
of the
publishers,
or in the
case
of
reprographic reproduction
in
accordance with
the
terms
and
licences issued
by
the
CLA. Enquiries concerning reproduction outside these terms should
be
sent
to the
publishers
at
the
undermentioned address:
Hermes Penton Science
120
Pentonville Road
London
N1 9JN
©
Hermes Science Publications, 1999
©
English language edition Hermes Penton Ltd,
2002
The right of
Christian Lalanne
to be
identified
as the
author
of
this work
has
been
asserted
by him in
accordance with
the
Copyright, Designs
and
Patents
Act
1988.
British Library Cataloguing
in
Publication Data
A
CIP
record
for
this
book
is
available
from the
British Library.
ISBN
1
9039
9604
X
Printed
and
bound
in
Great Britain
by
Biddies
Ltd, Guildford
and
King's
Lynn
www.biddies,co.uk
Contents
Introduction
xi
List
of
symbols
xiii
1
Shock analysis
1
1.1.
Definitions
1
1.1.1.
Shock
1
1.1.2. Transient signal
2
1.1.3.
Jerk
2
1.1.4. Bump
2
1.1.5. Simple
(or
perfect) shock
3
1.1.6. Half-sine shock
3
1.1.7.
Terminal peak
saw
tooth shock (TPS)
or
final
peak
saw
tooth shock (FPS)
3
1.1.8. Initial peak
saw
tooth shock (IPS)
3
1.1.9.
Rectangular shock
3
1.1.10.
Trapezoidal
shock
3
1.1.11.
Versed-sine
(or
haversine) shock
3
1.1.12. Decaying sinusoidal pulse
4
1.2. Analysis
in
the
tune domain
4
1.3. Fourier transform
4
1.3.1. Definition
4
1.3.2.
Reduced Fourier transform
6
1.3.3. Fourier transforms
of
simple shocks
7
1.3.3.1. Half-sine pulse
7
1.3.3.2. Versed-sine pulse
8
1.3.3.3. Terminal peak
saw
tooth pulse (TPS)
9
1.3.3.4. Initial peak
saw
tooth pulse (IPS)
10
1.3.3.5. Arbitrary triangular pulse
12
1.3.3.6. Rectangular pulse
14
vi
Mechanical shock
1.3.3.7. Trapezoidal pulse
15
1.3.4.
Importance
of the
Fourier
transform
17
1.4
Practical calculations
of the
Fourier transform
18
1.4.1.
General
18
1.4.2. Case: signal
not yet
digitized
18
1.4.3.
Case: signal already digitized
20
2
Shock
response
spectra
domains
23
2.1.
Main
principles
23
2.2. Response
of a
linear
one-degree-of-freedom
system
26
2.2.1.
Shock defined
by a
force
26
2.2.2.
Shock defined
by an
acceleration
27
2.2.3.
Generalization
28
2.2.4. Response
of a
one-degree-of-freedom system
to
simple shocks
33
2.3. Definitions
37
2.4. Standardized response
spectra
40
2.5.
Difference
between shock response spectrum (SRS)
and
extreme response spectrum (ERS)
47
2.6. Algorithms
for
calculation
of the
shock response spectrum
47
2.7. Subroutine
for the
calculation
of the
shock response spectrum
48
2.8. Choice
of the
digitization
frequency of the
signal
52
2.9. Example
of use of
shock response spectra
53
2.10.
Use of
shock response spectra
for the
study
of
systems
wi th se ver al degress of fr ee dom
3
Characteristics
of
shock
response
spectra
59
3.1. Shock response
spectra
domains
59
3.2. Characteristics
of
shock response spectra
at low frequencies 60
3.2.1.
General characteristics
60
3.2.2. Shocks with velocity changed
from
zero
TT60
3.2.3.
Shocks with
AV = 0 and AD = 0 at end of
pulse
69
3.2.4.
Shocks with
AV = 0 and AD = 0 at end of
pulse
72
3.2.5.
Notes
on
residual spectrum
74
3.3. Characteristics
of
shock response spectra
at
high
frequencies 75
3.4. Damping influence
77
3.5. Choice
of
damping
78
3.6. Choice
of frequency
range
80
3.7. Charts
81
3.8. Relation
of
shock response spectrum
to
Fourier spectrum
81
3.8.1. Primary shock response spectrum
and
Fourier
transform
81
3.8.2.
Residual shock response spectrum
and
Fourier transform
83
3.8.3.
Comparison
of the
relative severity
of
several shocks using
their Fourier
spectra
and
their shock response spectra
86
3.9. Characteristics
of
shocks
of
pyrotechnic origin
88
Contents
vii
3.10. Care
to be
taken
in the
calculation
of
spectra
90
3.10.1.
Influence
of
background noise
of the
measuring equipment..
90
3.10.2.
Influence
of
zero
shift
92
4
Development
of
shock
test
specifications
95
4.1.
General
95
4.2. Simplification
of the
measured signal
96
4.3.
Use of
shock response
spectra
98
4.3.1. Synthesis
of
spectra
98
4.3.2. Nature
of the
specification
99
4.3.3.
Choice
of
shape
100
4.3.4. Amplitude
101
4.3.5.
Duration
101
4.3.6.
Difficulties
105
4.4.
Other methods
107
4.4.1.
Use of a
swept sine
107
4.4.2. Simulation
of
shock response spectra using
a
fast
swept sine
108
4.4.3. Simulation
by
modulated random noise
112
4.4.4.
Simulation
of a
shock using random vibration
113
4.4.5. Least favourable response technique
114
4.4.6. Restitution
of a
shock response spectrum
by
a
series
of
modulated sine pulses
115
4.5. Interest behind simulation
of
shocks
on a
shaker using
a
shock spectrum
117
5.
Kinematics
of
simple shocks
121
5.1.
General
121
5.2. Half-sine pulse
121
5.2.1. Definition
121
5.2.2.
Shock motion study
122
5.2.2.1. General expressions
122
5.2.2.2. Impulse mode
124
5.2.2.3.
Impact mode
126
5.3. Versed-sine pulse
136
5.3.1. Definition
136
5.3.2.
Shock motion study
137
5.4. Rectangular pulse
139
5.4.1. Definition
139
5.4.2.
Shock motion study
139
5.5. Terminal peak
saw
tooth pulse
142
5.5.1.
Definition
142
5.5.2.
Shock motion study
143
5.6. Initial peak
saw
tooth
pulse
145
5.6.1.
Definition
145
viii Mechanical shock
5.6.2. Shock motion study
145
6
Standard shock machines
149
6.1. Main types
149
6.2. Impact shock machines
151
6.3. High impact shock machines
160
6.3.1. Lightweight high impact shock machine
160
6.3.2.
Medium weight high impact shock machine
162
6.4. Pneumatic machines
163
6.5. Specific
test
facilities
164
6.6. Programmers
165
6.6.1.
Half-sine pulse
165
6.6.2.
Terminal peak
saw
tooth shock pulse
173
6.6.3.
Rectangular pulse
-
trapezoidal
pulse
180
6.6.4.
Universal shock programmer
181
6.6.4.1. Generating
a
half-sine shock pulse
182
6.6.4.2.
Generating
a
terminal peak
saw
tooth
shock pulse
182
6.6.4.3.
Trapezoidal
shock pulse
183
6.6.4.4. Limitations
183
7
Generation
of
shocks using shakers
189
7.1. Principle behind
the
generation
of a
simple shape
signal
versus time
189
7.2. Main advantages
of the
generation
of
shock using shakers
190
7.3. Limitations
of
electrodynamic shakers
191
7.3.1. Mechanical limitations
191
7.3.2. Electronic limitations
193
7.4.
The use of the
electrohydraulic shakers
193
7.5. Pre-and
post-shocks
193
7.5.1.
Requirements
193
7.5.2.
Pre-
or
post-shock
195
7.5.3.
Kinematics
of the
movement
for
symmetrical pre-
and
post-shock
198
7.5.3.1.
Half-sine pulse
198
7.5.3.2.
TPS
pulse
206
7.5.3.3.
Rectangular pulse
207
7.5.3.4.
IPS
pulse
208
7.5.4.
Kinematics
of the
movement
for a
pre-shock
or
a
post-shock
alone
208
7.5.5. Abacuses
212
7.5.6.
Influence
of the
shape
of
pre-
and
post-pulses
213
7.5.7. Optimized pre-
and
post-shocks
216
7.6. Incidence
of
pre-
and
post-shocks
on the
quality
of
simulation
220
7.6.1.
General
220
Contents
ix
7.6.2.
Influence
of the
pre-
and
post-shocks
on the
time
history response
of a
one-degree-of-freedom system
220
7.6.3.
Incidence
on the
shock response spectra
223
8
Simulation
of
pyroshocks
227
8.1.
Simulations using pyrotechnic facilities
227
8.2. Simulation using metal
to
metal impact
230
8.3. Simulation using electrodynamic shakers
231
8.4. Simulation using conventional shock machines
232
9
Control
of a
shaker using
a
shock
response
spectrum
235
9.1. Principle
of
control
by a
shock response spectrum
235
9.1.1. Problems
235
9.1.2. Method
of
parallel
9.1.3. Current numerical methods
237
9.2. Decaying sinusoid
239
9.2.1. Definition
239
9.2.2.
Response spectrum
239
9.2.3. Velocity
and
displacement
242
9.2.4.
Constitution
of the
total signal
243
9.2.5. Methods
of
signal compensation
244
9.2.6.
Iterations
250
9.3. D.L. Kern
and
C.D. Hayes'
function
251
9.3.1. Definition
251
9.3.2.
Velocity
and
displacement
252
9.4. ZERD
function
253
9.4.1. Definition
253
9.4.1.1. D.K. Fisher
and
M.R. Posehn expression
253
9.4.1.2. D.O. Smallwood expression
254
9.4.2.
Velocity
and
displacement
255
9.4.3.
Comparison
of the
ZERD waveform with
a
standard decaying sinusoid
257
9.4.4. Reduced response spectra
257
9.4.4.1. Influence
of the
damping
n of the
signal
257
9.4.4.2. Influence
of the Q
factor
258
9.5. WAVSIN waveform
259
9.5.1.
Definition
259
9.5.2.
Velocity
and
displacement
260
9.5.3.
Response
of a
one-degree-of-freedom system
262
9.5.3.1. Relative response displacement
263
9.5.3.2. Absolute response acceleration
265
9.5.4. Response spectrum
265
9.5.5. Time history synthesis
from
shock spectrum
266
9.6.
SHOC waveform
267
9.6.1. Definition
267
filterss
236