Stochastic Dynamics

Soren Nielsen, Zili Zhang

Description

In chapter 1, the basic assumptions of the random vibration theory are emphasized. In chapters 2 and 3, pertinent results of stochastic variables and stochastic processes have been indicated. Chapter 4 deals with the stochastic response analysis of single degrees-of-freedom, multi-degrees-of-freedom and continuous linear structural systems. In principle, an introductory course on linear structural dynamics is presupposes. However, in order to make this textbook self-contained, short reviews of the most important results of linear deterministic vibration theory have been included in the start of the relevant sub-sections. Chapter 5 outlines the reliability theory for dynamically excited building structures, i.e., reliability theory for narrowbanded response processes. Finally, Chapter 6 gives an introduction to Monte Carlo simulation methods, which become increasingly important and useful as the computers become more and more powerful.

Sujets

• Stochastic process
• Mathematics

Informations

• Publié par: Aarhus University Press
• Date de parution: 01 juin 2017
• Nombre de lectures: 1
• EAN13: 9788771843477
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

STOCHASTICDYNAMICSSTOCHASTICDYNAMICS
SørenR.K.Nielsen
ZiliZhang
x(t)
b(t)(limitstatefunction)
ΔNΔN0
ΔN1
N
N1
N S0 0
St
t
t t+ΔtO
Lt
AarhusUniversityPressStochastic Dynamics
© SørenR.K.Nielsen, ZiliZhang and Aarhus University Press 2017
Cover: NetheEllinge Nielsen, Trefold
Publishing editor: SimonOlling Rebsdorf
Printed in Denmark 2017
eISBN9788771843477
AARHUSUNIVERSITYPRESS
Finlandsgade 29
DK-8200 Aarhus N
Denmark
www.unipress.dk
Weblinks wereactive when the book wasprinted. They mayno longer beactive.
INTERNATIONALDISTRIBUTION
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www.gazellebooks.com USA
www.isddistribution.comContents
1 INTRODUCTION 9
2 STOCHASTICVARIABLES 13
2.1 Introductiontostochasticvariables . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Combinedstochasticvariables . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Conditionaldistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 STOCHASTICPROCESSES 27
3.1 Introductiontostochasticprocesses . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Homogeneousprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Stochasticintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Stochasticdifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Stochasticdifferentialequations . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 Ergodicsamplingandergodicprocesses . . . . . . . . . . . . . . . . . . . . . 74
4 STOCHASTICVIBRATIONTHEORYFORLINEARSYSTEMS 85
4.1 Single-degree-of-freedomsystems . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Multi-degree-of-freedomsystems. . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Continuoussystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5 RELIABILITYTHEORYOFDYNAMICALLYEXCITEDSTRUCTURES 135
5.1 First-passagefailureproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Calculationofout-crossingfrequency . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Approximationstoprobabilityoffailure . . . . . . . . . . . . . . . . . . . . . 153
6 MONTECARLOSIMULATIONTECHNIQUE 171
6.1 Equivalentwhitenoiseprocesses . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2 Simulationmethodsbasedonstochasticfinitedifferenceequations . . . . . . . 182
6.3 Almostperiodicprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.4 Simulationofthefirst-passagetimeprobabilitydensityfunction . . . . . . . . 200
APPENDICES 205
INDEX 212
— 5 —PREFACE
Thepresenttextbookhasbeenwrittenbasedonpreviouslecturenotesforacourseonstochastic
vibrationtheorygivenintheautumnsemesteratAarhusUniversityforM.Sc.
studentsinstructuralengineering.
Inchapter1,thebasicassumptionsoftherandomvibrationtheoryareemphasized. Inchapters
2 and 3, pertinent results of stochastic variables and stochastic processes have been
indicated.
Chapter4dealswiththestochasticresponseanalysisofsingle-degree-of-freedom,multi-degreeof-freedom and continuous linear structural systems. In principle, an introductory course on
linear structural dynamics is presupposed. However, in order to make this textbook
selfcontained, short reviews of the most important results of linear deterministic vibration theory
have been included in the start of the relevant sub-sections. Chapter 5 outlines the reliability
theory for dynamically excited civil engineering structures, i.e., reliability theory for
narrowbandedresponseprocesses. Finally,Chapter6givesanintroductiontoMonteCarlosimulation
techniques, which become increasingly important and useful as the computers become more
andmorepowerful.
AarhusUniversity,December2016
SørenR. K.Nielsen
ZiliZhang
— 7 —Chapter1
INTRODUCTION
A structural system may be considered as an ensemble of mass particles, each of which is
exposedtoexternalloading,andtointernalforcesfromneighbouringmassparticles. Iftheinitial
conditions,andtheexternalandinternalforcesareperfectlydescribedforallmassparticles,the
motionof thesystemcan inprinciplebe determinedfromNewton’s2ndlawofmotion,leaving
noroomfor anyindeterminism. Thisistheprincipleofcausalityordeterminism,postulatedin
classicalphysics.
In practice, neither the initial conditions, nor the external and internal forces can be perfectly
determined for the mass particles. Usually, the design loads of the structure are some future
extreme loadings, which cannot be specified in space and time in the sense presumed above.
Even if this was the case, the internal forces cannot be observed or specified. Instead, these
internalforces have tobe theoreticallydeterminedby a mathematicalmodel. In thecontinuum
mechanics approach, the internal forces are expressed in terms of stresses. Significant
modelling errors may stem from the constitutive equation which relates the stress components to
the motions of the mass particles, and also from certain kinematical approximationsrendering
analytical solutions possible (e.g., classical beam-, plate-, and shell theories). These errors or
approximations introduce uncertainty into the predicted motions of the mass particles. The
accuracyofthepredictiondependsonthelevelofuncertaintywithwhichtheinitialvalues,and
theexternalandinternalforcescanbespecified.
Generally,uncertaintiesare classifiedaseitheraleatoricorepistemic.
Aleatoricuncertaintiesareuncertaintiesthatcannotberemovedorreducedbyanymeans.
Typical examples are external dynamic natural loads from winds, waves and earthquakes, but also
somemanmadeloadssuchastrafficloads.
Epistemicuncertaintiescan beremovedorreduced toa certain extent.
Anexampleisthemeasurementuncertainties inactive feedback vibrationcontrolproblems,whichcan be reduced by
bettermeasurementequipmentsorfilteringprocedures.
Anotherexampleisthestructuralmodelling uncertainty, which can be reduced by better numerical or analytical models. Epistemic
uncertainties can only be reduced to a certain extent, after which they should be considered as
aleatoric.
— 9 —10 Chapter1–INTRODUCTION
Bothaleatoricandepistemicuncertaintiesare quantifiedbystochasticmodels.
Thesubjectofstochasticmechanicsisthequantificationofuncertaintyofthestructuralresponse
basedonthequantifieduncertaintyoftheinitialvaluesandtheexternalandinternalforces.
The
′word"stochastic"(fromgreekstochasmos=presumerable),withtheusualmeaningthatsomethingishappeningbychance,isinawaymisleading,becausenothinginnatureisconsideredto
happenbychance. Stochasticmechanicsshouldmerelybeinterpretedasatoolforquantification
of uncertainty of some quantities in the mechanical problem. If these quantities are observed
or controlled, the uncertainty is removedor reduced. This wouldnot be possibleif an inherent
indeterminismispresentinnature.
As stated, a mathematical model must be adapted for the determination of internal forces.
The quantification of the errors associated with this choice is one of the unsolved problems in
stochasticmechanics. The frequently mentionedsuggestionto evaluatethe performance of the
modelagainstamoreelaboratedmodelprovidesinsightintotheaccuracyoftheselectedmodel,
but does not carry the uncertainty of the chosen model into an operable scheme, where the
modelling uncertainty can be weighted along with the uncertainties associated with the initial
valuesandtheexternalloadings.
Even for a well-defined mathematical model for the internal forces, additional uncertainty can
be introduced if the parameters of the model cannot be properly calibrated. As an example,
the beam in Fig. 1-1 is assumed to be modelled using Bernoulli-Euler beam theory. However,
the bending stiffness EI(x) in this modelmay stillbe uncertain. Thisis the case for reinforced
concretemembersunderdynamicloading,wherecrackedanduncrackedsectionsmayalternate
inanuncontrolledwayalongthebeam. Incontrarytothemorefundamentalmodellingproblem
fortheinternalforces,parameteruncertaintycaneasilybetreatedwithintherealmofstochastic
mechanics.
Instochasticmechanics,theuncertaintiesoftheexternalloadingsandpossibleparameterfields
of the mathematical model for the internal forces are usually modelled as stochastic variables
or stochastic processes. The stochastic structure of these stochastic variables and
stochastic processes is estimated from available measurements, statistical inference and engineering
U(x,t)
F(x,t)
EI(x)
x
Fig.1–1Bernoulli-Eulerbeamwithuncertainbendingstiffness.Chapter1–INTRODUCTION 11
Uncertain Statistical Stochasticanalysisofphysical Measurements model
quantity measurements
Fig.1–2Stochasticmodellingofuncertainphysicalquantity.
judgement. However, this statistical calibration process is not considered a subject of the
stochasticresponseanalysis,whichassumesthatsuchacalibrationhasalreadybeenperformed.
The various steps in a stochastic modelling problem have been illustrated in Fig. 1-2. Then,
the stocha