Table Of ContentCarl Sisemore · Vít Babuška
The Science and
Engineering
of Mechanical
Shock
The Science and Engineering of Mechanical Shock
Carl Sisemore • Vít Babuška
The Science and Engineering
of Mechanical Shock
123
CarlSisemore VítBabuška
Albuquerque,NM,USA Albuquerque,NM,USA
Thisbookdescribesobjectivetechnicalmaterial.Anysubjectiveviewsoropinionsthatmight
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To
Sara,Seth,Hosea,Josiah,andHanna;
PavelandTomáš;
andthenextgenerationofengineers.
Preface
Mechanical shock is often the misunderstood and neglected half of system envi-
ronmental testing and analysis. Engineers design their systems to pass vibration
exposure and hope that the systems pass shock testing. It has been said that the
vibration laboratory is where components go for testing and the shock laboratory
is where they go to die. The primary reason for this is that mechanical shock is
generally not taught in either the undergraduate or graduate curriculum nor is it
particularly straightforward. There are relatively few books dedicated to the study
ofmechanicalshock,andmostofthoseareengineeringdiscipline-specific.
Whilesomeundergraduatetextsonvibrationsmayincludeachapteronmechan-
ical shock, thorough treatments of the topic are lacking. Nevertheless, shock is a
mechanicalenvironmentthataffectsalmosteverydevice—fromshipstospacecraft
and from cell phones to shipping packages. Civil engineers worry about blast-
inducedshocksandearthquakes.Mechanicalengineersareconcernedwithimpact-
induced shocks, while naval engineers must consider the effects of underwater
explosiveshocksonships.Satellitesmustberobusttopyroshocks.
Thegoalofthisbookistobringtogetherengineeringinformationaboutmechan-
icalshockfromvariousdisciplinesintoacohesivetext.Thescienceofmechanical
shock involves understanding of the basic physics of the event: how the dynamics
of the system interplay with the high-speed transient nature of the excitation and
thenumericalmethodsforquantifyingtheshockeventanditsdamagepotentialto
various structures, systems, and components. The book includes information from
naval and military applications, earthquake engineering, aerospace, and spacecraft
inonereference.
While the book is intended to provide basic information about the field of
mechanicalshock,thebookandthetopicarenotthemselvessimplistic.Itisassumed
that the reader has a firm understanding of mechanical vibrations and differential
equations. The book presents a historical context to the field of mechanical shock
including numerous practical examples and anecdotes. Mathematical derivations
of the fundamental principles and data processing methods are included and
elucidated.Developmentofthesingledegree-of-freedomoscillatorytheoryandthe
theoryoftheshockresponsespectraarecoveredindetailinChaps.3and4.There
vii
viii Preface
isasection onnonspectral techniques foranalyzing shock testdata.Various types
of shock testmachines and their capabilities are covered inChap.9.Inaddition, a
lengthydiscussiononthemethodsforderivingshocktestspecificationsisincluded
inChap.11.Theanalysisofmulti-degree-of-freedomsystemswithshockresponse
spectra is discussed in Chap.8. The energy spectrum, which is another type of
responsespectrumthatisusedprimarilyintheearthquakeengineeringcommunity,
is discussed at the end of the book, in Chap.12. Throughout the book, example
problems illustrate the main concepts. Problems are included at the end of the
chapterstofurtherthereader’sunderstanding.
Bothauthorshaveworkedinthefieldofshockandvibrationanalysisandtesting
for many years. Our experience has been predominately with naval, aerospace,
and spacecraft systems, although there has also been some brief work with
earthquakesurvivability.Throughoutourcareers,wehavefoundthataconsiderable
amount of information about mechanical shock is available if one is willing to
search. Unfortunately, the information is widely dispersed, and not every source
is necessarily a good source. The finer points of the science are often buried in
“blackbox”routinesandanalysiscodes.Acombined,handyreference,groundedin
structuraldynamictheorywithwideapplicability,hasnotbeenreadilyavailable.It
isintothisarenathattheauthorsplacethiswork.Itisourhopethatthereaderwill
findthetextusefulandinteresting.Afterall,mechanicalshockisafunbusiness.
Albuquerque,NM,USA CarlSisemore
Albuquerque,NM,USA VítBabuška
May2019
Contents
1 Introduction................................................................. 1
1.1 IntroductiontoMechanicalShock ................................. 2
1.2 HistoryofShockEngineering...................................... 4
1.2.1 NavalShock............................................... 5
1.2.2 CivilEngineeringShock.................................. 8
1.2.3 AircraftShock............................................. 10
1.2.4 SpaceVehicleShock...................................... 10
1.3 EffectsofShockonSystems ....................................... 13
1.3.1 StructuralFailure.......................................... 13
1.3.2 FunctionalFailure......................................... 14
1.4 TheNeedforDynamicAnalysis................................... 15
1.5 Summary ............................................................ 18
Problems ..................................................................... 18
References.................................................................... 19
2 CommonMechanicalShockEnvironments ............................. 21
2.1 ImpactShock........................................................ 21
2.1.1 TransportationShock ..................................... 24
2.1.2 RailShock................................................. 27
2.2 DropShock.......................................................... 28
2.3 Pyroshock ........................................................... 31
2.3.1 ExplosiveBoltTheoryofOperation ..................... 34
2.4 BallisticShock...................................................... 34
2.4.1 RecoilTheory ............................................. 36
2.4.2 Shot-to-ShotDispersion .................................. 37
2.5 SeismicShock....................................................... 38
2.6 UnderwaterShock .................................................. 40
2.7 Summary ............................................................ 43
Problems ..................................................................... 43
References.................................................................... 44
ix
x Contents
3 SingleDegree-of-FreedomSystems....................................... 45
3.1 SDOFGoverningEquations........................................ 46
3.2 SolutionoftheDifferentialEquation .............................. 52
3.3 FreeVibration....................................................... 52
3.3.1 FreeVibrationofUndampedSDOFSystems............ 53
3.3.2 FreeVibrationofDampedSDOFSystems .............. 53
3.4 ForcedResponse.................................................... 58
3.4.1 BasicShockExcitation ................................... 58
3.4.2 GeneralShockExcitation................................. 68
3.5 Summary ............................................................ 82
Problems ..................................................................... 83
References.................................................................... 84
4 ShockEnvironmentCharacterizationUsingShockResponse
Spectra....................................................................... 87
4.1 HistoryoftheShockResponseSpectra............................ 89
4.1.1 ShockSpectraCalculations............................... 91
4.1.2 Maxi-MaxSRS............................................ 95
4.1.3 PrimarySRS............................................... 98
4.1.4 ResidualSRS.............................................. 99
4.1.5 UsesofthePositiveandNegativeSRS .................. 101
4.1.6 RelationshiptoFourierSpectra........................... 102
4.2 TypesofShockSpectra............................................. 104
4.2.1 AbsoluteAccelerationSRS............................... 105
4.2.2 RelativeDisplacementSRS............................... 107
4.2.3 Pseudo-VelocitySRS ..................................... 109
4.2.4 RelativeVelocitySRS..................................... 109
4.3 CodinganSRSFunction ........................................... 111
4.4 Pseudo-VelocitySRSonTripartitePaper.......................... 112
4.5 Non-uniquenessoftheSRS ........................................ 113
4.6 Summary ............................................................ 115
Problems ..................................................................... 116
References.................................................................... 117
5 ClassicalShockTheory .................................................... 119
5.1 ClassicalShocksHaveSimilarSRS ............................... 119
5.2 InterpretationofClassicalShockSRS............................. 124
5.2.1 SlopesindBperOctave .................................. 125
5.2.2 WhytheLow-FrequencySlopeis6dB/Octave.......... 126
5.2.3 EstimatingVelocityfromtheSRS........................ 131
5.2.4 EffectofDampingontheSRS ........................... 133
5.3 ShockBandwidth ................................................... 135
5.4 PositiveandNegative,PrimaryandResidual...................... 138
5.5 Summary ............................................................ 141
Problems ..................................................................... 141
References.................................................................... 142
Contents xi
6 OscillatoryandComplexShockTheory ................................. 143
6.1 OscillatoryShockWaveform....................................... 144
6.2 ShockBandwidth ................................................... 148
6.3 EffectsofShockLength............................................ 150
6.4 UnderstandingtheSRSInflectionPoints.......................... 152
6.5 Two-SidedShockPulse............................................. 155
6.6 ComplexShocks.................................................... 158
6.7 Summary ............................................................ 162
Problems ..................................................................... 162
7 DesignforShockwithSDOFSpectra.................................... 165
7.1 UseoftheSRSforDesign.......................................... 166
7.2 EquivalentSRS ..................................................... 168
7.3 RelationshipofPseudo-VelocitytoStress......................... 170
7.3.1 Stress–Pseudo-VelocityDerivationforanAxially
LoadedRod................................................ 172
7.3.2 Stress–Pseudo-VelocityDerivationforaBeam
inBending................................................. 176
7.3.3 Examples .................................................. 181
7.4 ExtremeLoading.................................................... 185
7.5 PlasticDesignofFoundations...................................... 188
7.6 FatigueLoading..................................................... 189
7.7 ShockSpectrumandStrainEnergy................................ 190
7.7.1 Example:MachineryFoundation......................... 192
7.7.2 Example:SmallComponentAttachmentScrews........ 193
7.8 ShockDesignGuidance ............................................ 194
7.9 Summary ............................................................ 196
Problems ..................................................................... 196
References.................................................................... 197
8 Multi-Degree-of-FreedomSystems ....................................... 199
8.1 IntroductiontoMDOFModels..................................... 201
8.1.1 EquationsofMotioninAbsoluteCoordinates........... 201
8.1.2 EquationsofMotioninRelativeCoordinates............ 201
8.1.3 EquationsofMotioninModalCoordinates ............. 202
8.1.4 Damping................................................... 204
8.1.5 TheOutputEquation...................................... 206
8.2 MDOFShockResponseSpectra................................... 207
8.2.1 RelationshiptotheSDOFEquationofMotion.......... 208
8.2.2 DampinginMDOFSystems ............................. 208
8.2.3 TheRoleoftheOutputEquation......................... 209
8.2.4 ModalResponseCombination............................ 209
8.2.5 Multi-AxisShocks ........................................ 213
8.2.6 Examples .................................................. 214
8.3 Summary ............................................................ 226
Problems ..................................................................... 227
References.................................................................... 227
