Modelling and Estimation of Damping in Non-linear Random Vibration · Modelling and Estimation of...

Modelling and Estimation of

Damping in Non-linear

Random Vibration

MEK

Department ofMechanicalEngineering

MaritimeEngineering

TechnicalUniversity ofDenmark

Finn Rüdinger

PhD thesis

October 2002

Modelling and Estimation of Damping inNon-linear Random Vibration

Finn Rudinger

TECHNICAL UNIVERSITY OF DENMARKDEPARTMENT OF MECHANICAL ENGINEERING

MARITIME ENGINEERINGOCTOBER 2002

Published in Denmark byTechnical University of Denmark

Copyright c© F. Rudinger 2002All rights reserved

Maritime EngineeringDepartment of Mechanical EngineeringTechnical University of DenmarkStudentertorvet, Building 101E, DK-2800 Kgs. Lyngby, DenmarkPhone +45 4525 1360, Telefax +45 4588 4325E-mail: [email protected]: http://www.mek.dtu.dk/

Publication Reference DataRudinger, F.

Modelling and Estimation of Damping in Non-linear RandomVibrationPhD ThesisTechnical University of Denmark, Maritime Engineering.October, 2002ISBN 87-89502-64-7Keywords: Random vibration, Power spectral density, System

identification, Vortex-induced vibrations

Preface

The Ph.d. project is part of the research programme ”Damping Mechanisms in the Dynamicsof Structures and Materials” funded by the Danish Technical Research Council. This pro-gramme is carried out in collaboration between a research group at the Technical Universityof Denmark, the Department of Structural Dynamics at Aalborg University and several ad-vanced consulting engineering companies. The Ph.d. project was initiated at the Departmentof Structural Mechanics and Materials and was completed at the Department of MechanicalEngineering, both at the Technical University of Denmark.

First of all, I would like to thank my advisor Professor Steen Krenk for his guidance throughthe project. Apart from being a great source of inspiration in the work itself, he has alsotaught me many things within the areas of applied mechanics and mathematics.

Meetings within the research programme have been held with regular intervals during theperiod of the Ph.d. project. The objective of these meetings is to exchange ideas betweenthe different groups and to discuss the results of each of the subprojects with respect tothe overall plan. I would therefore like to express my gratitude to the other participants ofthe research programme for interesting interaction at these meetings, especially the groupat Aalborg University headed by Professor Søren R. K. Nielsen.

In the spring of 1999 I was involved in an analysis of the vibrations of the pylons at theØresund high bridge during the construction of the bridge. Working with response recordsobtained by field measurements gave me an impression of the difference between the cleanrecords from digital simulation and ”real records” and served as an interesting supplementto the more theoretical work. I would like to express my gratitude to Senior Researcher,Ph.d., Guy L. Larose from the Danish Maritime Institute, who made my involvement in theproject possible.

In the autumn of 1999 I had the opportunity to spend one semester at the University ofNotre Dame, Indiana, with Professor Bill F. Spencer as advisor. During this stay I studiedexperimental structural dynamics and followed courses in control theory and linear systems.This experience was very rewarding and I am indebted to Professor Spencer and his coworkersfor making my visit both interesting and enjoyable.

During the last six month of 2001 I stayed at the European Laboratory for Structural As-sessment, Joint Research Centre of the European Commission at Ispra, Italy, as a Marie

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Curie exchange student. In this research centre I was introduced to the fascinating world offull scale testing, and had the opportunity to carry out characterization tests for a specifictype of damper along with more theoretical work, some of which is included in this thesis. Iam very grateful for having had the chance to experience this interesting area of structuraldynamics testing, and I wish to thank the staff at the Laboratory, especially my advisor,Ph.d. and Head of the Laboratory, Georges Magonette and his team.

Finn Rudinger

Kgs. Lyngby, Oct 2, 2002

The thesis is based on the following publications:

• Krenk, S. and Rudinger, F. (2000), System identification from non-linear stochasticresponse, in Melchers, R. E. and Stewart, M. G. (editors), Applications os Statisticsand Probability, pp. 805 - 812, Balkema, Rotterdam.

• Rudinger, F. and Krenk, S. (2001), Non-parametric system identification from non-linear stochastic response, Probabilistic Engineering Mechanics, 16:233 - 243.

• Rudinger, F. and Krenk, S. (2002), Stochastic analysis of self-induced vibrations, Mec-canica, 37:3 - 14.

• Krenk, S., Lin, Y. K. and Rudinger, F. (2002), Effective system properties and spectraldensity in random vibration with parametric excitation, Journal of Applied Mechanics,69:161 - 170.

• Rudinger, F. and Krenk, S. (2002), Non-linear stochastic oscillator models of vortex-induced vibrations, in Corotis, R. B., Schueller, G. I. and Shinozuka, M. (editors),Structural Safety and Reliability, p. no. 452, Balkema, Rotterdam.

• Rudinger, F. and Krenk, S. (2003), Spectral density of an oscillator with power lawdamping excited by white noise, Journal of Sound and Vibration (in press).

• Rudinger, F. and Krenk, S. (2003), Identification of non-linear oscillator with paramet-ric excitation, in 4th International Conference on Computational Stochastic Mechanics(in press).

• Rudinger, F. and Krenk, S. (2003), Spectral density of oscillator with bilinear stiffnessand white noise excitation, Probabilistic Engineering Mechanics (in press).

Executive Summary

In many cases in structural and mechanical engineering the excitation of the mechanicalsystem is of a stochastic nature. Examples include wind load on high rise buildings andbridges, wave loads on offshore structures and ships, the load on the suspension systemof a vehicle travelling a rough road and base excitation of structures in connection withearthquakes. If the excitation is modelled as a stochastic process the response will also be astochastic process and must be described in probabilistic terms. If the system is linear andtime invariant and the excitation is Gaussian procedures exist for complete characterizationof the response statistics. If the system is non-linear it is generally difficult to find exactsolutions, even in the simple case of single-degree-of-freedom systems.

When the dynamic response of a structure is analyzed, linear theory is often used. Nomechanical systems are completely linear, but linearization around the equilibrium positionis acceptable in many cases - at least within some range of deformation. In some cases theconfiguration of the system suggests that a non-linear model should be used, e.g. systemswith elastic stops, where the stiffness changes when a certain level of deformation is reached.If the system is linear, the response is Gaussian if the excitation is Gaussian, and even inthe case of a non-Gaussian excitation the linear properties of the structure will tend to makethe response more Gaussian. A non-Gaussian response therefore indicates the presence ofnon-linear effects. An example is the phenomenon of vortex-induced vibrations, where theresponse changes from Gaussian to sinusoidal in the lock-in regime.

In the present work the stochastic response of non-linear single-degree-of-freedom systems isconsidered. The thesis is divided into three parts. In the first part exact and approximatesolutions for the statistics of the response are investigated. In the second part system identi-fication procedures are developed, i.e. methods for estimation of the system from records ofthe stochastic response. In the last part of the thesis a model for vortex-induced vibrationsis proposed. A brief summary of the methods and main results in each of the three parts isgiven below.

In order to assess the probability of failure of a structure or mechanical device it is neces-sary to have information about the extreme value statistics of the system. When a systemapproches the failure domain, it will often have exceeded the limit of the linear range, and itmay therefore be important to include non-linear effects. A simple way to estimate such prob-abilities is by means of Monte Carlo simulation. In this method stochastic response records

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are generated numerically and analyzed, but since the events of interest are rare events, thisapproach will often be extremely time consuming. Another approach is to consider exactor approximate solutions for the response statistics, if these are available. Evaluation ofan extreme event requires knowledge of the probability density. However, the probabilitydensity of the stationary response does not contain information about the correlation timescale of the process. This information can be derived from the covariance function oftenrepresented by the Fourier transform known as the power spectral density. The power spec-tral density can be interpreted as the distribution of the energy in the frequency domain,and therefore contains information of the periods of the oscillations in the response. Thisfunction is therefore also required if the random fatigue is to be evaluated.

For non-linear single-degree-of-freedom systems some exact solutions are known for the prob-ability density of the state space variables, and a number of methods for obtaining approx-imate solutions have been developed. Solutions in terms of the power spectral density aremuch more scarce. In the present work a method proposed by Krenk and Roberts (1999)for obtaining approximate solutions for the power spectral density of the response of non-linear stochastic oscillators is investigated and extended to include systems with response-dependent excitation, where the effective damping of the system may be modified by therelation between the excitation and the response. The free undamped response at a givenenergy level is described by a set of modified phase plane variables conserving polar symme-try. Assuming local similarity with the free undamped response, an approximation to thecovariance function of the response at a given energy level is derived. The power spectraldensity at this energy level is obtained by applying the Fourier transform and the total spec-trum is evaluated by integration over all energy levels weighting each with the probabilitydensity of the energy. The probability density is approximated by the method of dissipationenergy balancing, if an exact solution does not exist. The quality of the method is evaluatedby considering a number of examples and comparing with results obtained by Monte Carlosimulation. The technique generally captures the broadening of the peak and the presenceof higher harmonics for systems with non-linear stiffness very accurately. For strong levels ofnon-linearity minor deviations are observed at resonance and in the quasi-static limit. Forsystems with negative effective damping in some energy ranges, the method is less accurateoutside the resonance region.

When the behaviour of structures and other mechanical systems is described by mathematicalmodels, these are only approximations that may or may not be accurate. For linear systemsthe stiffness is generally well-defined and can be obtained from knowledge of the geometryand material properties. The damping, on the other hand, is generally difficult to assessand cannot be evaluated theoretically. Normally, the damping is estimated from engineeringjudgement. In the case of systems with non-linear behaviour the stiffness may be equallydifficult to estimate. It is therefore of interest to be able to identify these properties, eitherfrom field measurements to verify an assumed behaviour, or in connection with laboratorytesting to establish a model for the behaviour. Most techniques in system identificationassumes knowledge of the input or excitation. However, in the cases of wind and wave loadon structures the excitation is not easily measured. This indicates a need for development ofmethods for identification of stiffness, damping and excitation of non-linear structures from

Executive Summary v

records of the stochastic response.

In this thesis a non-parametric identification procedure for stationary single-degree-of-freedomsystems with broad band excitation is proposed. The procedure is based on records of thestochastic response and applies to both systems with response-independent excitation and toa specific class of systems with response-dependent excitation. For a lightly damped systemthe change in energy over a typical period of oscillation is small and the kinetic energy at themean-level crossing will therefore give an estimate of the potential energy at the followingextreme. The scatter of these estimates is reduced by forming averages at various energylevels, and a non-parametric estimate of the stiffness is obtained by an iterative procedure.The damping is also estimated in a non-parametric way by considering the decay of thecovariance functions of the modified phase plane variables introduced in the first part of thethesis. The accessible information is the effective damping at various energy levels. Applyingthe information from the damping estimation a polynomial representation of the excitationintensity as a function of the energy level is obtained from a generic form of the probabilitydensity of the response. The accuracy of the estimation procedure is evaluated by applica-tion to simulated records. The stiffness estimation is shown to give excellent results. Thedamping estimation gives accurate results for systems with linear stiffness. For systems withnon-linear stiffness the damping is overestimated. The deviation reaches 15 - 20 % for thestrongest levels of non-linearity considered. The inaccuracy in the damping is reflected inthe estimation of the excitation as well.

The wind-field around a structural element will under certain conditions result in the shed-ding of vortices. These will give rise to loading in the across-wind direction and may lead toviolent vibration when the shedding frequency approaches the eigenfrequency of the struc-ture. In this case the response is approximately harmonic, while it is mainly Gaussian outsidethis regime. Models used in current design codes do not capture this transition.

In the last part of the thesis a non-linear stochastic oscillator model of vortex-induced re-sponse incorporating the structural damping as a parameter is proposed. It is shown howthis model captures the transition from a Gaussian response to a sinusoidal response in thelock-in regime. The probability density and spectral density is evaluated by the methodsdiscussed in the first part of the thesis, and the peak-factor is investigated numerically con-sidering stochastic records of the response. It is discussed how a model of this type can becalibrated by the identification procedure proposed in the second part of the thesis.

vi Executive Summary

Synopsis

Belastning af konstruktioner eller andre mekaniske systemer vil ofte være stokastisk af natur.Som eksempler kan nævnes vindlast pa høje bygninger og broer, bølgelast pa offshore kon-struktioner og skibe, belastning af ophænget af et køretøj, der kører over ujævnt terræn, samtjordskælvsbelastning af konstruktioner. Hvis belastningen modelleres som en stokastisk pro-ces, vil responset ligeledes være en stokastisk proces og ma beskrives statistisk. Hvis detbetragtede system er lineært og tidsinvariant og belastningsprocessen er Gaussisk, kan sys-temets respons karakteriseres fuldstændigt. Hvis systemet er ikke-linært, er det genereltvanskeligt at finde eksakte løsninger, selv i det simple tilfælde af et een-frihedsgrads-system.

Lineær teori benyttes ofte nar en konstruktions dynamiske respons analyseres. Ingen meka-niske systemer er helt lineære, men linearisering omkring ligevægtsstillingen vil ofte være ac-ceptabel som tilnærmelse - i det mindste indenfor et vist deformationsinterval. I visse tilfældevil systemets konfiguration antyde, at en ikke-lineær model bør benyttes. Som eksempel kannævnes systemer med elastiske stødpuder, der aktiveres, nar flytningen nar en vis værdi.Hvis systemet er lineært, og belastningen er en Gaussisk proces, vil responset ligeledes væreGaussisk, og selv hvis belastningen er ikke-Gaussisk vil konstruktionens lineære opførsel gøreresponset mere Gaussisk. Et ikke-Gaussisk respons er derfor tegn pa et system med ikke-lineær opførsel. Som eksempel kan nævnes belastning paført ved rytmisk hvirvelafløsning,hvor responset ændres fra Gaussisk til harmonisk, nar hvirvelafløsningsfrekvensen nærmersig systemets egenfrekvens.

I denne afhandling betragtes det stokastiske respons af et ikke-lineært een-frihedsgrads-system. Afhandlingen er inddelt i tre dele. I første del undersøges eksakte og tilnærmedeløsninger for det stokastiske respons. I anden del udvikles metoder til system identifikation,det vil sige metoder, der kan estimere systemets karakteristika pa baggrund af tidsserieraf det stokastiske respons. I sidste del af afhandlingen opstilles en model for svingningerinduceret af hvirvelafløsning. Et kort resume af de betragtede metoder og resultater i de tredele gives i det følgende.

For at kunne vurdere sandsynligheden for svigt af konstruktioner eller andre mekaniske sys-temer er det nødvendigt at kende systemernes ekstrem-værdi-fordelinger. Nar et systemnærmer sig grænsen for begyndende svigt, vil det ofte have overskredet det lineære omrade,og det kan derfor være nødvendigt at medtage ikke-lineære effekter. Monte Carlo simulering

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er en simpel metode, der kan benyttes til at estimere sandsynlighedsfordelinger for ikke-lineære systemer. Ved denne metode genereres stokastiske tidsserier numerisk, hvorefter deanalyseres, men eftersom hændelserne af interesse er sjældne hændelser, er denne metodeekstremt tidskrævende. Alternativt kan eksakte eller approximerede løsninger for responsetsstatistik benyttes, hvis sadanne eksisterer. Bestemmelse af sandsynligheden for en ekstremhændelse kræver kendskab til frekvensfunktionen. Frekvensfunktionen for det stationærerespons indeholder ingen information om korrelationstiden i responset. Denne informa-tion er indeholdt i kovariansfunktionen, der ofte er givet ved sin Fourier transformerede,som betegnes spectraltætheden. Denne funktion angiver fordelingen af energien i respon-set pa de forskellige frekvenser, og indeholder derfor information om perioderne i responset.Spectraltætheden ma derfor bestemmes, hvis den stokastiske udmattelse af systemet skalvurderes.

Nogle eksakte løsninger for frekvensfunktionen for systemets tilstandsvariable eksisterer foreen-frihedsgrads-systemer, og metoder til bestemmelse af approximerede løsninger er tilligeblevet udviklet. Løsninger for spektraltætheden er knap sa udbredt. I denne afhandlingundersøges en metode oprindeligt udviklet af Krenk og Roberts (1999) til bestemmelse afapproximerede løsninger for spektraltætheden af ikke-lineære stokastiske oscillatorers re-spons. Denne metode udvides til at omfatte systemer med respons-afhængig belastning,hvor den effektive dæmpning i visse tilfælde kan ændres af forholdet belastningen og respon-set. Det frie udæmpede respons ved et givet energi-niveau beskrives ved et sæt modificeredefaseplansvariable, der udviser polær symmetri i faseplanen. Ved at antage, at responsetlokalt kan betragtes som en perturbation af det frie udæmpede respons, kan en approxima-tiv løsning for kovariansfunktionen ved et givet energi-niveau bestemmes. Spektraltæthedenved dette energi-niveau bestemmes dernæst ved at anvende Fourier transformationen og denubetingede spektraltæthed fas ved integration over alle energi-niveauer, hvor hvert energi-niveau vægtes med frekvensfunktionen for energien. Denne frekvensfunktion bestemmesapproximativt ved en metode betegnet dissipations-energi-udligning, i de tilfælde hvor eneksakt løsning ikke kan bestemmes. Metodens anvendelighed undersøges gennem en rækkeeksempler hvor den approximative analytiske løsning sammenlignes med resultater fra MonteCarlo simulering. Metoden er generelt i stand til med stor præcision at fange forøgelsen afbredden af resonansspidsen samt tilstedeværelsen af superharmoniske komponenter i respon-set i form af resonans spidser ved overtonerne, for systemer med ikke-lineær stivhed. Forsystemer med stærkt ikke-lineær opførsel ses mindre afvigelser omkring resonans og i denkvasi-statiske grænse. For systemer med negativ dæmpning for visse energi-niveauer ermetoden mindre præcis udenfor resonans-omradet.

Nar konstruktioner og andre mekaniske systemers opførsel beskrives ved matematiske mod-eller, vil disse kun være mere eller mindre nøjagtige approximationer. For lineære systemerer stivheden generelt veldefineret og kan bestemmes ud fra kendskab til geometri og materialeegenskaber. Dæmpningen derimod er generelt svær at vurdere og kan ikke bestemmes teo-retisk. Normalt estimeres dæmpningen ud fra ingeniørmæssig erfaring. I tilfælde af systemermed ikke-lineær opførsel kan stivheden ligeledes være svær at bestemme. Det er derfor af in-teresse at kunne identificere disse egenskaber; enten ud fra malinger af den færdige konstruk-tions respons for at verificere en antaget opførsel, eller i forbindelse med laboratorieforsøg

Synopsis ix

for at opstille en model for systemet. De fleste metoder indenfor system identifikation byg-ger pa antagelsen om en kendt eller malelig belastning. Hvis et system udsættes for vind-eller bølgelast vil det dog ofte være svært at male belastningen. Disse observationer visernødvendigheden af at udvikle metoder til identifikation af systemers stivhed, dæmpning ogpatrykte belastning ud fra malinger af det pagældende systems stokastiske respons.

I denne afhandling opstilles en metode til ikke-parametrisk identifikation af stationære een-frihedsgrads-systemer med stokastisk belastning over et bredt frekvensomrade. Metodenbaserer sig pa tidsserier af det stokastiske respons og kan benyttes for systemer med respons-uafhængig belastning samt for en klasse af systemer hvor belastningen er respons-afhængig.For svagt dæmpede systemer er ændringen i den mekaniske energi over een svingning lilleog den kinetiske energi ved middel-værdi krydsninger vil derfor give et estimat af den po-tentielle energi ved det efterfølgende lokale maximum. Spredningen forbundet med disseestimater reduceres ved at midle indenfor givne energi-niveauer, og et ikke-parametriskestimat af stivheden bestemmes ved en iterativ fremgangsmade. Dæmpningen estimeresligeledes pa en ikke-parametrisk made ved at betragte den aftagende kovarians funktion afde modificerede faseplansvariable beskrevet i første del af afhandlingen. Den tilgængeligeinformation er størrelsen af den effektive dæmpning ved forskellige energi-niveauer. Vedat anvende estimatet af dæmpningen, kan et estimat af belastningsintensiteten som funk-tion af energi-niveauet i form af en Taylor-udvikling bestemmes ud fra et generelt udtrykfor frekvensfunktionen af responset. Nøjagtigheden af estimationsmetoden vurderes ved atanvende den pa stokastisk simulerede tidsserier af responset. Det demonstreres, at estima-tionen af stivheden generelt giver meget nøjagtig resultater. Dæmpningsestimationen givernøjagtige resultater for systemer med lineær stivhed. For systemer med ikke-lineær stivhedvil dæmpningen generelt blive overvurderet. Afvigelser pa op til 15 - 20 % observeres for destærkest ikke-linære tilfælde, der betragtes. Unøjagtigheden i estimationen af dæmpningenafspejler sig ligeledes i estimationen af belastningsintensiteten.

Strømningen af vinden omkring en konstruktionsmodel vil under visse betingelser resul-terer i rytmisk afløsning af hvirvler pa konstruktionens læside. Dette vil medføre belast-ninger pa konstruktioner i vindens tværretning og kan resultere i voldsomme svingningerhvis frekvensen af afløsningen af hvirvler nærmer sig konstruktionens egenfrekvens. I dettetilfælde bliver konstruktionens respons tilnærmelsesvis harmonisk, mens det hovedsageligter Gaussisk under andre forhold. De normer, der i dag benyttes ved dimensionering, byggerpa modeller, der ikke fanger denne overgang.

I den sidste del af afhandlingen opstilles en stokastisk oscillator model for svingninger gener-eret af rytmisk hvirvelafløsning, som inddrager konstruktionens dæmpning som parameter.Det demonstreres hvordan denne model fanger overgangen fra det Gaussiske til det har-moniske respons ved resonans. Frekvensfunktionen og spectraltætheden bestemmes vedmetoderne diskuteret i første del af afhandlingen. Forholdet mellem responsets maximumog dets standart afvigelse undersøges numerisk ved betragtning af simulerede tidsserier forresponset. Det diskuteres hvordan metoderne til system identifikation omtalt i anden delaf afhandlingen kan benyttes til kalibrering af en stokastisk oscillator model for hvirvel-inducerede svingninger.

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Contents

Executive Summary iii

Synopsis (in Danish) vii

Contents x

Symbols xvii

1 Introduction 1

1.1 Stochastic Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Probability Density 5

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Statistical linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Non-linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Systems with External Excitation . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Caughey’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Method of equivalent non-linearization . . . . . . . . . . . . . . . . . 10

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2.3 Systems with Parametric Excitation . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Simple class of exact solutions . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Method of dissipation energy balancing . . . . . . . . . . . . . . . . . 14

2.4 Probability Density Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Modified state space variables . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Probability density of energy . . . . . . . . . . . . . . . . . . . . . . . 17

3 Power Spectral Density 19

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Extension of statistical linearization . . . . . . . . . . . . . . . . . . . 20

3.1.2 Local similarity with free response . . . . . . . . . . . . . . . . . . . . 21

3.2 Covariance Function and Power Spectral Density . . . . . . . . . . . . . . . 21

3.3 Systems with Linear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 The linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.2 Systems with external excitation . . . . . . . . . . . . . . . . . . . . 23

3.3.3 Systems with parametric excitation . . . . . . . . . . . . . . . . . . . 24

3.4 Systems with Non-linear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Free undamped vibration . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.2 Effective damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.3 Preliminary approximation for spectral density . . . . . . . . . . . . . 28

3.4.4 Local solution for covariance function and spectral density . . . . . . 29

4 Simulation Procedure 31

4.1 White noise approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Generating the Response Process . . . . . . . . . . . . . . . . . . . . . . . . 33

Contents xiii

5 Probability Density and Spectral Density: Examples 35

5.1 Oscillator with power law viscous damper . . . . . . . . . . . . . . . . . . . 35

5.1.1 Non-dimensional formulation . . . . . . . . . . . . . . . . . . . . . . . 36

5.1.2 Equivalent non-linearization . . . . . . . . . . . . . . . . . . . . . . . 36

5.1.3 Statistical linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Linear Oscillator with Parametric Excitation . . . . . . . . . . . . . . . . . . 41

5.2.1 Probability density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Oscillator with stable limit cycle behaviour . . . . . . . . . . . . . . . . . . . 45

5.3.1 Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.2 Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3.4 Alternative procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Oscillator with Bilinear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 53


5.4.2 Free undamped vibration . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4.3 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 System Identification from Non-linear Stochastic Response 63

6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1.1 Stiffness estimation in the time domain . . . . . . . . . . . . . . . . . 63

6.1.2 Estimation of damping relative to excitation intensity . . . . . . . . . 64

6.1.3 Estimation of absolute damping in the time domain . . . . . . . . . . 65

xiv Contents

6.1.4 Frequency domain and FPK equation . . . . . . . . . . . . . . . . . . 65

6.1.5 Non-parametric identification methods . . . . . . . . . . . . . . . . . 66

6.2 Stiffness Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.1 Development of mechanical energy . . . . . . . . . . . . . . . . . . . 67

6.2.2 Standard regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.3 Iterative regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3 Damping and Excitation Estimation for Systems with External Excitation . 71


6.3.2 Estimation of absolute damping . . . . . . . . . . . . . . . . . . . . . 73

6.3.3 Estimation of excitation intensity . . . . . . . . . . . . . . . . . . . . 74

6.4 Damping and Excitation Estimation for Systems with Parametric Excitation 74

6.4.1 Estimation of absolute damping . . . . . . . . . . . . . . . . . . . . . 75

6.4.2 Estimation of stationary potential . . . . . . . . . . . . . . . . . . . . 75

6.4.3 Estimation of excitation amplitude function . . . . . . . . . . . . . . 76

7 System Identification: Examples 77

7.1 System with External Excitation . . . . . . . . . . . . . . . . . . . . . . . . 77

7.1.1 Stiffness estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


7.1.3 Covariance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.4 Estimation of damping and excitation intensity . . . . . . . . . . . . 83

7.2 System with Parametric Excitation . . . . . . . . . . . . . . . . . . . . . . . 85

7.2.1 Non-dimensional formulation . . . . . . . . . . . . . . . . . . . . . . . 86

7.2.2 Natural period of vibration . . . . . . . . . . . . . . . . . . . . . . . . 87


7.2.4 Characteristic parameters for the system . . . . . . . . . . . . . . . . 89

7.2.5 Stiffness estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2.6 Damping estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2.7 Excitation estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Contents xv

8 Modelling of Vortex-induced Vibrations 109

8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.2 Stochastic Oscillator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3 Effects of Turbulence on Aerodynamic Damping . . . . . . . . . . . . . . . . 113

8.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9 Conclusion 119

9.1 Probability Density and Spectral Density . . . . . . . . . . . . . . . . . . . . 119

9.2 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

9.3 Modelling of Vortex-induced Vibration . . . . . . . . . . . . . . . . . . . . . 121

9.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

References 125

List of PhD Theses Available from the Department 131

xvi Contents

Symbols

Roman Symbols

A Transformation matrix

A1 Initial value of covariance function

A2 Value of covariance function at first negative extreme

Aj Amplitude

B Unit Wiener process

C/C1/C2 Normalizing constant

Deq Non-dimensional equivalent damping function

E[ ] Mean value operator

G Potential energy

H Damping potential

Heq Equivalent damping potential

Iu Turbulence intensity in the longitudinal direction

Ix Turbulence intensity in the transverse direction

K Complete elliptic integral of the first kind

Rx Covariance function of displacement

Rx/Rz2 Covariance function of velocity

Rz1 Covariance function of first modified phase plane variable

S0 Intensity of additive white noise process

SW Power spectral density of broad band process

Sij/S Spectral intensity matrix of white noise process

Sx Power spectral density of displacement

Sy Power spectral density of non-dimensional displacement

Sc Scruton number

T Natural period of vibration

T Non-dimensional natural period

xvii

xviii Symbols

U Non-dimensional unit white noise process, mean wind speed

Ucr Critical wind speed

Vλ Coefficient of variation for energy

W Unit white noise process

W Vector with correlated white noise processes

W0 Additive white noise process

WI Vector with independent unit white noise processes

Wj Additive or parametric white noise process

X Displacement (stochastic process)

X0 Length scale

Y Non-dimensional displacement (stochastic process)

a Transition displacement for system with bilinear stiffness

a Internal force function in state space formulation

ak Taylor series coefficients

b Excitation coefficient function in state space formulation

bj Excitation coefficient function

bk Taylor series coefficients

c Twice the fraction of kinetic energy relative to mechanical energy, dampingcoefficient

cj Coefficient in Fourier series

ck Taylor series coefficients

d Non-dimensional damping function, structural length scale

deq Non-dimensional equivalent damping function

f Internal force function

g Stiffness function

h Damping function

heq Equivalent damping function

i Imaginary unit

kp Peak factor

m Effective mass

px,x Probability density of state space variables

pλ Probability density of mechanical energy

q Histogram of energy at mean level crossings

r Non-dimensional frequency

s Twice the fraction of potential energy relative to mechanical energy

s1/s2 Non-dimensional spectral intensity

sj Coefficient in Fourier series

Symbols xix

t Time

t Non-dimensional time

t0 Time scale

u State space vector

x Displacement

〈xmax〉T Expected maximum displacement within the time period T

y Non-dimensional displacement

z1 First modified phase plane variable

z2 Second modified phase plane variable (velocity)

zj Non-dimensional modified phase plane variables

Greek Symbols

Γ Gamma function

Λ Mechanical energy (stochastic process)

Ω Frequency ratio for system with bilinear stiffness

Ωj Damped eigenfrequency

α Damping law exponent, negative linear damping coefficient, cubic damping co-efficient

β Non-dimensional damping coefficient, cubic damping coefficient, non-linear stiff-ness parameter

γ Damping ratio, measure of damping non-linearity

γ Non-dimensional damping ratio

δ Dirac delta function, measure of parametric excitation intensity

ε Difference/error

ζ Damping ratio

〈ζ〉 Measure of damping magnitude

ζa Aerodynamic damping

ζe Effective damping ratio

ζeq Equivalent damping ratio

ζs Structural damping ratio

ηλ Effective damping

κ Ratio between effective stiffness and apparent stiffness

λ Mechanical energy

λ0 Reference energy level

ν Non-dimensional parameter

ξ Non-dimensional energy

xx Symbols

ρ Correlation coefficient, air density

σ Excitation amplitude function

σW Standard deviation of broad band process

σx Standard deviation of displacement

σx Standard deviation of velocity

σλ Standard deviation of energy

τ Time lag, non-dimensional time

ϕ Phase

ψ Stationary potential

ω Frequency, instantaneous angular frequency

〈ω〉 Mean frequency

ω0 Eigenfrequency for system with linear stiffness

ω1/ω2 Eigenfrequency for system with bilinear stiffness

ωλ Eigenfrequency for system with non-linear stiffness

Chapter 1

Introduction

1.1 Stochastic Structural Dynamics

The vibration sensitive structures, which are considered in structural dynamics, are oftenexcited by loads, which are unknown or unpredictable. Typical examples are loads generatedby wind, waves or seismic activity. One way to approach these types of problems is to modelthe loads as stochastic processes.

Loads generated by the wind are often characterized by having a broad spectrum, whichmeans that the energy in the excitation is distributed on a wide range of frequencies. Lightlydamped structures will have a response, which is concentrated in a narrow frequency range.The excitation can therefore as a reasonable approximation be modelled as an ideal whitenoise, i.e. an excitation with equal distribution of energy on all frequencies. In the case ofwave load this assumption is generally not valid, since the energy of the train of waves willbe concentrated around the frequency corresponding to the period of the waves. In suchcases the loads can be modelled by specifying the spectrum, and the excitation process isthus characterized in the frequency domain rather than in the time domain. If the excitationprocess is assumed to be a normal process, it is completely characterized by its spectrum.

Wind and wave loads are often relatively stationary, i.e. changes in the characteristics of theexcitation process will be slow compared to the time scales of the response of the structure.This is not the case for earthquake excitation, where the ground acceleration takes place ina short period of time. In this case the response will be transient and cannot be character-ized by a spectrum (distribution of energy on frequencies). However, the excitation can bemodelled as a stationary process multiplied by a deterministic function, which is zero untilthe ground motion starts, and then grows and dies out over a short period of time in thesame way as a typical earthquake. Another way to investigate the response to earthquakesis by considering historical ground acceleration records, but the stochastic nature of the phe-nomenon is not taken into account by this approach. In this thesis the excitation is assumed

1

2 Chapter 1. Introduction

to be stationary and broad band, and the theory is thus mainly applicable to wind-inducedvibrations.

If the systems considered are linear and the excitation process is a normal process a largebody of theory is developed for the evaluation of the response statistics. For non-linearsystems, on the other hand, closed form solutions are only available for a small numberof systems - in most cases single-degree-of-freedom systems with stationary white noiseexcitation (ideal broad band excitation). In this thesis exact and approximate solutionsfor the probability density and spectral density of the response of non-linear oscillatorswith stationary broad band excitation are investigated. Part of this theory is then used todevelop a system identification procedure. The case of vortex-induced vibrations of structuralmembers is considered as an area of structural dynamics where the theory may have practicalinterest. A number of textbooks addresses the theory of stochastic structural dynamics, seee.g. Lin (1967) or Soong and Grigoriu (1993). More advanced theory have been presentedby Lin and Cai (1995).

1.2 Objectives of the Work

The material presented in this thesis is to a large extend based on a formulation introducedby Krenk and Roberts (1999). In this paper the vibration of an oscillator is represented by aset of modified phase plane variables. Local similarity with the free undamped vibration forlightly damped systems is used to approximate the autocovariance function of the response,which leads to an approximate expression for the power spectral density. In the present workthis theory is further investigated and extended to include systems, where the excitation isa function of the response. One of the main objectives of the project is to use the ideasof local similarity to develop an identification procedure, where the stiffness, damping andexcitation of a non-linear stochastic oscillator can be estimated from records of the stochasticresponse. Finally, the methods developed will be applied to a typical problem in structuraldynamics, namely that of vortex-induced vibrations. Present models used in design codesdo not predict the response accurately and non-linear stochastic oscillator models offer apromising alternative to these models.

1.3 Notation

In texts on probabilistic theory, stochastic variables are often represented by capital let-ters. The same lower case letters are then used to represent the variables in deterministicstatements, e.g. PX < x is the probability of the stochastic variable X being smallerthan the deterministic value x. Some texts use the somewhat less concise notation, wherethe same symbol characterizes both the stochastic variable and the independent variable

1.4 Outline of Thesis 3

in deterministic statements concerning the stochastic variable, e.g. the probability densityfunction.

In the first part of this thesis (Chapters 2 - 5) it is important to distinguish between stochasticand deterministic variables, since stochastic variables sometimes are ”frozen” at a givenvalue. The mechanical energy is an example of such a variable. It is in general a stochasticprocess, but in some of the derivations a specific energy level is considered and the energyshould thus appear as a deterministic variable in the corresponding expressions. In this partof the thesis capital letters are therefore used for stochastic variables and lower case lettersare used in deterministic equations in order to avoid confusion. In the remaining part of thethesis (Chapters 6 - 8) less theory is developed and the simpler notation of using lower caseletters for the stochastic variables is employed.

1.4 Outline of Thesis

The thesis is divided into 9 chapters. Chapter 1 contains a rather general discussion ofstochastic structural dynamics, a formulation of the objectives of the work and a few com-ments on the notation. In Chapter 2 the probability density of stochastic oscillators isdiscussed. Exact and approximate solutions for stationary systems with broad band exci-tation are investigated. Approximate solutions for the power spectral density are derivedin Chapter 3. The accuracy and validity of the theory is evaluated by comparing the theo-retical solutions with results obtained by stochastic simulation. In chapter 4 the simulationtechnique used to obtain the stochastic records is discussed. Chapter 5 contains a num-ber of examples, where the theory for the probability density and power spectral densityis evaluated for specific systems. Stochastic records obtained by the simulation techniqueintroduced in chapter 4 are used to evaluate the quality of the approximate solutions.

In Chapter 6 the reverse problem of system identification is considered. A method for esti-mation of stiffness, damping and excitation from non-linear stochastic response is proposed.Two examples where the system identification technique is demonstrated are given in Chap-ter 7. Simulated records are used to evaluate the accuracy of the technique. In Chapter 8 anon-linear stochastic oscillator model for vortex-induce vibrations is proposed. The theorydeveloped in the previous chapters applies to this system. The thesis is concluded by asummary of the main results obtained in Chapter 9. Areas where further work is needed areidentified.

The thesis consists of three main parts. In the first part exact and approximate solutionsfor the probability density and spectral density of the stochastic response are considered.Chapters 2, 3 and 5 constitute this part. In the second part system identification proceduresare investigated. This part is covered by Chapters 6 and 7. The final part is given in Chapter8, where some of the techniques are used to model vortex-induced vibrations.

4 Chapter 1. Introduction

Chapter 2

Probability Density

In structural and mechanical engineering, problems involving unpredictable or stochasticvariables or processes are frequently encountered, and in these cases a probabilistic analysismay be the most rational way of approaching the problem. In many problems involvingthe dynamical behaviour of mechanical systems, the dominating source of uncertainty orunpredictability is the excitation. Examples of such cases are wave and wind load on offshorestructures, the motion of ships in random sea waves, wind load on bridges and high risebuildings, traffic load on bridges and earthquake excitation.

The performance of the structure will normally depend on the amplitude of the vibrations.In extreme cases excessive vibration may lead to failure of the structure, but even if thisextreme event does not occur, vibration at large amplitudes may lead to fatigue damage inthe structural components. Large vibrations visible to the eye will furthermore reduce thepublic confidence in the safety of the structure. If a bridge deck is experiencing vibrations ofsuch a magnitude, most people would refuse to cross the bridge. In a densely populated areathe effects of a bridge, which is non-operational for many hours due to excessive vibrations,may be a heavy economical loss in terms of lost working hours.

If the excitation is given in terms of a stochastic process, the response of the mechanicalsystem is also a stochastic process. In order to assess the probability of the occurance ofextreme events and evaluate possible fatigue damage in the structure, it is necessary tobe able to evaluate the response statistics with reasonable accuracy. In this chapter thestationary probability density of the response is addressed.

2.1 Background

In non-linear random vibration much effort has been directed towards the establishment ofsolutions in terms of the probability density of the response. If the excitation is a white

5

6 Chapter 2. Probability Density

noise, the state space vector is a Markov process, and the joint probability density of thestate space variables can be obtained as the solution to the corresponding Fokker-Planckequation. Analytical solutions to this partial differential equation are generally difficult tofind. A number of exact solutions have been reported by e.g. Caughey (1971), Caughey andMa (1982), Yong and Lin (1987), Lin and Cai (1988b) and Lin and Cai (1995). Lin andCai (1988a) discuss a method of generating systems with a given probability distribution.Most of the solutions focus on the stationary probability density of single-degree-of-freedomsystems. Stationary solutions are obtained by solving the reduced Fokker-Planck equation,where the time derivative of the probability density vanishes.

2.1.1 Statistical linearization

A classical way of obtaining an approximate solution to a non-linear random vibrationproblem is the method of statistical linearization (also known as equivalent linearizationor stochastic linearization), where the non-linear equation of motion is replaced by an equiv-alent linear one, which is easily solved. The difference between the two systems is a stochasticprocess, and the equivalent linear system is obtained by requiring that the mean value ofthis process is zero and by minimizing its variance. This can however be done in severalways, e.g. by using the statistics of the original system or the equivalent system, as dis-cussed by Crandall (2001). For strongly non-linear systems the response can be stronglynon-Guassian. In these cases the method of statistical linearization will give a reasonablyaccurate representation of the mean value and the variance of the probability density of theresponse. However, the Gaussian nature of the response of the equivalent linear system willimply that the probability density function cannot be accurately represented over the entirerange. In fact, extreme value statistics will normally be poorly approximated by the methodof statistical linearization. The method can be used for systems with any number of degreesof freedom and is therefore frequently used despite the shortcomings mentioned above. Athorough discussion of the method including a number of examples are given in the mono-graph by Roberts and Spanos (1990). For a brief account of the method, see e.g. Madsenet al. (1986).

2.1.2 Non-linear methods

An alternative approximate method is that of equivalent non-linearization introduced byCaughey (1986). In this method the original non-linear system is replaced by an equivalentsystem, which is also non-linear, but to which an exact solution exists. The equivalentsystem is also in this case determined by minimizing the error between the two systems. Forlightly damped systems the method of equivalent non-linearization will generally give a veryaccurate approximation to the probability density function. However, the method can onlybe used for single-degree-of-freedom systems and is from a computational point of view morecomplicated than the method of statistical linearization.

2.2 Systems with External Excitation 7

The method of equivalent non-linearization can be extended to include single-degree-of-freedom systems with parametric excitation, i.e. where the excitation is a function of theresponse. In this case the method is based on an exact stationary solution obtained by Linand Cai (1988b) for a class of systems termed the class of stationary potential. This classof systems is discussed in detail by Lin and Cai (1995). It was shown by Cai and Lin (1988)and Cai et al. (1992) that approximate solutions can be obtained by replacing a stochasticsystem by an equivalent system belonging to the class of stationary potential, for which asolution exists. For the class of stationary potential an energy balance equation must besatisfied, and the equivalent system is determined by replacing the individual terms in theequation of motion by mean values, that satisfy the energy balance equation. The methodis thus known as dissipation energy balancing.

Another important method, which is frequently used, is that of stochastic averaging originallyintroduced by Stratonovich (1963). In this method the state space variables are replacedby energy and phase through the van der Pol transformation. Replacing the coefficients inthe equation of motion by average values over one period of vibration, the energy processis uncoupled from the phase process. For an oscillator this means that the Fokker-Planckequation can be reduced from two dimensions to one dimension, see e.g. Roberts (1989)or Roberts and Spanos (1986). This is a great advantage, since a general solution to thestationary one-dimensional Fokker-Planck equation exists.

When systems with parametric excitation are considered distinction must be made betweensystems with an arbitrarily small but finite correlation time in the excitation and systemswith ideal white noise excitation. When the physical broad band excitation process is re-placed by an ideal white noise process, the system must be modified by including the so-calledWong-Zakai correction term, accounting for the effect of small but finite correlation in thephysical process, Wong and Zakai (1965). The Wong-Zakai term may change the effectivesystem properties such as stiffness and damping as discussed by Zhu and Lin (1991) andKrenk et al. (2002). It has been shown by Cai and Lin (1997a) that the method of dis-sipation energy balancing is equivalent to the quasi-conservative averaging method, if theWong-Zakai correction is accounted for.

In Section 2.2 the method of equivalent non-linearization is considered. In Section 2.3 themore general method of energy balancing will be discussed. The procedure follows that ofKrenk et al. (2002). The method is essentially the same as that originally introduced by Caiand Lin (1988), though the notation has been simplified.

2.2 Systems with External Excitation

If the excitation does not depend on the response, it is termed external. The normalizedequation of motion for an oscillator excited by external white noise can be expressed as

X + h(X, X)X + g(X) = W0(t) (2.2.1)


where X is the displacement and a dot indicates the derivative with respect to time. Thestochastic nature of the response is indicated by the use of a capital letter. The second termon the left hand side of the equation represents the forces due to damping, and h(X, X) istermed the damping function. The third term represents the internal forces caused by staticloading and g(X) is termed the stiffness function. The right hand side of the equation is theexternal white noise excitation, which is characterized by the following covariance function,

E[W0(t)W0(t + τ)] = 2πS0δ(τ) (2.2.2)

where E[ ] is the expectation operator. S0 is the intensity of the white noise and δ isthe Dirac delta function, indicating that a white noise process has infinite variance. Thepotential energy of the system is obtained by integration of the stiffness function,

G(X) =

∫ X

0

g(X)dX (2.2.3)

The lower integration limit can be chosen arbitrarily, since any constant can be added tothe potential energy by redefining the level of zero potential energy. For mathematicalconvenience the equilibrium position X = 0 is chosen as reference level. The mechanicalenergy is obtained as the sum of the potential and kinetic energy as

Λ(X, X) = G(X) + 12X2 (2.2.4)

It should be noticed that the energy is defined as energy per mass and thus has the dimensionlength2/time2. The equation of motion (2.2.1) is given in the form of a force balance.Multiplying this equation by X it takes the form of an energy balance equation, which canbe expressed as

dΛ

dt= W0(t)X − h(X, X)X2 (2.2.5)

where the term W0(t)X represents the energy supplied by the excitation and h(X, X)X2

represents the energy dissipated by the damping. From (2.2.5) it is observed that the energyis conserved for free undamped vibrations.

2.2.1 The Fokker-Planck equation

White noise is characterized by being uncorrelated for any finite time separation as seen by(2.2.2). Since the future excitation is independent of the excitation in the past, the statespace vector (X, X) will depend only on the present state and the future input. This meansthat the state space vector is a Markov process, and the probability density of the state spacevector can thus be identified as the solution to a partial differential equation known as theFokker-Planck equation (also referred to as the FPK equation or the forward Kolmogorov

2.2 Systems with External Excitation 9

equation). The derivation of the general form of this equation is given by e.g. Lin and Cai(1995). For the system governed by (2.2.5) the Fokker-Planck equation is given by

−x∂px,x

∂x+

∂

∂x(hxpx,x) + g

∂px,x

∂x+ πS0

∂2px,x

∂x2=

∂px,x

∂t(2.2.6)

where px,x(x, x; t) is the joint probability density of the stochastic variables (X, X). In orderto obtain a particular solution a set of initial conditions and boundary conditions are needed.The initial conditions are typically given in the form: px,x(x, x; t0) = δ(x − x0)δ(x − x0)for x(t0) = x0 and x(t0) = x0. Different types of boundary conditions can be specified,e.g. absorbing or reflecting boundary conditions, as discussed in detail by Lin and Cai (1995).If the process (X, X) is not constrained, natural boundary conditions apply: px,x(x, x; t) → 0for x, x → ±∞. This is the condition, which will be assumed in the following. If theright hand side of (2.2.6) is non-zero the equation is termed the non-stationary or non-homogeneous Fokker-Planck equation. If px,x is independent of the time the right hand side ofthe equation disappears and the equation is termed the stationary, reduced or homogeneousFokker-Planck equation. In this case initial conditions are not required.

2.2.2 Caughey’s solution

A special class of exact solutions can be obtained if it is assumed that the damping is afunction of the energy only. In this case the equation of motion reads

X + h(Λ)X + g(X) = W0(t) (2.2.7)

It is now assumed that the solution is given in the form px,x = px,x(λ), i.e. that the statespace variables enter the expression only through the mechanical energy λ. In this case thedifferential operators in (2.2.6) can be formally expressed as

∂

∂x= x

d

dλ,

∂

∂x= g(x)

d

dλ(2.2.8)

By this reformulation the first and the third term in (2.2.6) are seen to cancel. The homo-geneous Fokker-Planck equation takes the following form,

d

dλ

(hxp + πS0x

dp

dλ

)= 0 (2.2.9)

and the expression in the brackets must be constant. A particular solution to this equationis given by

px,x(x, x) = C exp

(−H(λ)

πS0

), H(λ) =

∫ λ

0

h(λ) dλ (2.2.10)

where H(λ) is termed the damping potential and C is a normalizing constant. The naturalboundary conditions require that the probability density vanishes for λ → ∞. This require-ment is fulfilled if H(λ) → ∞ for λ → ∞. Damping functions not satisfying this criterion


can easily be constructed, e.g. if h(λ) ∼ λ−2, and in such cases (2.2.10) is not a solution.However, most damping models representing energy dissipation mechanisms of structuralmaterials or joints do fulfill this requirement (e.g. linear viscous damping or dry friction).In the following it is therefore assumed that H(λ) → ∞ for λ → ∞ and that (2.2.10)consequently is a solution. This solution was originally obtained by Caughey (1971).

2.2.3 Method of equivalent non-linearization

The exact solution presented in the previous subsection can be used to obtain approximatesolutions for the more general case. It is now assumed that the equation of motion is given by(2.2.1). An equivalent equation of motion of the form (2.2.7), with an equivalent dampingfunction heq(λ), is then identified by minimizing the difference between the two systems.Subtraction of the equations of motion (2.2.1) and (2.2.7) yields

ε = h(X, X)X − heq(Λ)X (2.2.11)

where ε is the difference or error between the original system and the equivalent system. Aspecific energy level is now considered. The mean value of the square of the error for thisenergy level is given by

E[ε2|λ] = E[X2|λ]h2eq − 2E[hX2|λ]heq + E[h2X2|λ] (2.2.12)

where E[ |λ] is the conditional mean value. The expression is seen to be a parabola in heq(λ).The minimum of E[ε2|λ] is obtained for

heq(λ) =E[h(X, X)X2|λ]

E[X2|λ](2.2.13)

thereby identifying the equivalent damping function. Once the equivalent damping functionis determined an approximate solution is given by (2.2.10) with H(λ) replaced by Heq(λ) =∫ λ

0h(λ)dλ. For lightly damped systems the technique will yield a very accurate solution. The

method is known as equivalent non-linearization and was originally introduced by Caughey(1986). This method is illustrated in the example given in Section 5.1.

2.3 Systems with Parametric Excitation

The right hand side of (2.2.1) is independent of the response. This type of excitation is knownas external or additive excitation. However, there are cases where the excitation is a functionof the structural motion. If the excitation is generated by the velocity of a moving medium,e.g. air or water, the forces experienced by the structure will be related to the differencebetween the velocity of the medium and the velocity of the structure. Though the velocity

2.3 Systems with Parametric Excitation 11

of the structure is negligible in certain situations, this will not always be the case, and evena small dependence of the structural motion on the excitation may lead to large differencesin extreme value statistics. Another loading case, where the structural motion influences theexcitation, is that of vortex-induced across-wind vibrations of slender structural members.In this particular case the motion of the structure will modify the fluid flow and therebyindirectly the excitation. This case is considered in Chapter 8.

2.3.1 Equation of motion

If the excitation is a function of the response it is termed parametric or multiplicativeexcitation. The equation of motion can in this case be expressed as

X + f(X, X) = bj(X, X) Wj(t) (2.3.1)

where repeated index implies summation. f(X, X) is denoted the internal force function andrepresents the damping and stiffness of the system. For systems with parametric excitationthe damping and stiffness terms may not be readily identified since the excitation may leadto changes in both stiffness and damping, as will be discussed in the following. The righthand side of the equation represents the excitation. Wj(t) are white noise processes, eachof which are multiplied by an excitation coefficient function bj(X, X). The excitation termis external (additive) if bj(X, X) = bj is independent of X and X. If bj(X, X) depends oneither X or X (or both), the white noise excitation term is parametric (multiplicative). Thecovariance function of the white noise processes is given by

E[Wi(t)Wj(t + τ)] = 2πSij δ(τ) (2.3.2)

where Sij is the spectral intensity matrix, which is symmetric. The diagonal elements in Sij

represent the intensity of each of the excitation processes. The off-diagonal elements givethe correlation between the different excitation processes Wj(t). If these excitation processesare uncorrelated the off-diagonal elements in Sij are zero. The equation of motion (2.3.1)can be replaced by an equivalent equation given by

X + f(X, X) = σ(X, X) W (t) (2.3.3)

where W (t) is a unit white noise, i.e. E[W (t)W (t+τ)] = δ(τ), and σ(X, X) is the excitationamplitude function. Comparing the variance of the right hand side of (2.3.1) and (2.3.3) itis seen that the amplitude function is given by

σ(X, X)2 = 2πSij bi(X, X) bj(X, X) (2.3.4)

Again, the repeated indices on the right hand side imply summation. With this relationbetween the amplitude function σ(X, X) and the coefficient functions bj(X, X) the systems(2.3.1) and (2.3.3) are equivalent in the strong sense, i.e. the two systems share the samenon-homogeneous Fokker-Planck equation. If all excitation terms are external, σ(X, X) is


constant and (2.3.3) reduces to the special case (2.2.1). Equation (2.3.3) can be written asa first order evolution equation in the state space (X, X) as

d

dt

[X

X

]=

[X

−f(X, X)

]+

[0

σ(X, X)

]W (t) (2.3.5)

In equation (2.3.5) W (t) should be interpreted as a smooth but rapidly fluctuating function,an interpretation often associated with Stratonovich (1963). If W (t) is replaced by an idealwhite noise, (2.3.5) can be written on incremental form as the following Ito-type equation

d

[X

X

]=

X

−f(X, X) +1

4

∂σ(X, X)2

∂X

dt +

[0

σ(X, X)

]dB(t) (2.3.6)

dB(t) is here the increment of a unit Wiener process, the formal derivative of which is aunit white noise. In the formulation (2.3.6) X and X are evaluated at the beginning ofthe increment, and dB(t) is thus independent of X and X. The increment σ(X, X)dB(t)is therefore uncorrelated. In the formulation (2.3.5) W (t) and σ(X, X) may be correlated.The difference between the two formulations is the modification of the internal force functionf(X, X) by the term −1

4∂σ(X, X)2/∂X known as the Wong-Zakai correction, Wong and

Zakai (1965). This correction term vanishes in the case of additive white noise.

2.3.2 The Fokker-Planck equation

For a system governed by (2.3.1) the non-homogeneous Fokker-Planck equation is given by

∂

∂x(−xpx,x) +

∂

∂x

(fpx,x − 1

4

∂σ2

∂xpx,x +

1

2

∂

∂x

(σ2px,x

))=

∂px,x

∂t(2.3.7)

It is observed, that (2.3.7) reduces to (2.2.6) if the excitation amplitude σ is constant and ifthe internal force function is written as f(X, X) = h(X, X)X+g(X). Again, only stationarysolutions (solutions to the homogeneous Fokker-Planck equation) are considered. Even inthe stationary case it is difficult to obtain exact solutions to the Fokker-Planck equation.However, a class of exact solutions can be obtained by writing the probability density in theform of an exponential function as

px,x(x, x) = C exp(−ψ(λ)) (2.3.8)

The requirement of non-negative probability density is automatically fulfilled by this formu-lation. ψ(λ) is termed the stationary potential and is assumed to be a function of mechanicalenergy only. The argument H(λ)/πS0 in (2.2.10) is a special case of a stationary potential.C is a normalizing constant. Since px,x is a function of the energy, the partial derivativeswith respect to x and x are related by

x∂px,x

∂x= g(x)

∂px,x

∂x(2.3.9)

2.3 Systems with Parametric Excitation 13

where g(x) is the stiffness, which has not yet been identified. Using this identity the station-ary Fokker-Planck equation can be rewritten as

∂

∂x

[(f − g − 1

4

∂σ2

∂x

)px,x +

1

2

∂

∂x

(σ2px,x

)]= 0 (2.3.10)

This equation can be further simplified if the exponential representation for the probabilitydensity is introduced. With the expression (2.3.8) the stationary Fokker-Planck equationreduces to

∂

∂x

[(f − g +

1

4

∂σ2

∂x− 1

2xσ2dψ

dλ

)C exp(−ψ)

]= 0 (2.3.11)

A solution is obtained if the expression in the inner bracket is equal to zero. If a functionψ(λ) can be identified satisfying the equation

f(x, x) +1

4

∂σ(x, x)2

∂x− 1

2xσ(x, x)2 dψ(λ)

dλ= g(x) (2.3.12)

then (2.3.8) is a solution to the stationary Fokker-Planck equation. It is furthermore re-quired, that ψ(λ) → ∞ for λ → ∞ such that the probability density vanishes at infinity inaccordance with the natural boundary conditions. It is observed that the right hand side of(2.3.12) only depends on the displacement. For most systems defined by (2.3.1), it is notpossible to find a stationary potential satisfying an equation of the type (2.3.12). Systems,for which a solution of this type exists, are said to belong to the class of stationary potential,Lin and Cai (1988b).

2.3.3 Simple class of exact solutions

If both the damping and the excitation are functions of the mechanical energy only, a par-ticular class of exact solutions can be obtained. In this case the equation of motion is givenby

X + h(Λ)X + g(X) = σ(Λ)W (t) (2.3.13)

with Λ given by (2.2.4). The internal force function f(X, X) is here separated into a stiffnessterm g(X) and a damping term h(Λ)X, where the damping function h(Λ) is assumed to bea function of the mechanical energy. The amplitude function σ(Λ) is still obtained from(2.3.4), where the coefficient functions are assumed to be of the form bj(X, X) = bj(Λ).Equation (2.3.12) can in this case be expressed as

h(λ)x = −1

4

dσ2

dλx +

σ2

2

dψ

dλx (2.3.14)

From this equation the stationary potential is identified as

dψ

dλ=

1

σ2

(2h(λ) +

1

2

dσ2

dλ

)(2.3.15)


which is seen to depend only on the energy. An exact solution in terms of the probabilitydensity of the state space variables is then obtained from (2.3.8) as

px,x(x, x) = C exp

(−∫ λ

0

1

σ(λ)2

(2h(λ) +

1

2

dσ(λ)2

dλ

)dλ

)(2.3.16)

In the case of a constant amplitude function σ(Λ) = σ (external excitation), equation (2.3.16)reduces to the solution obtained by Caughey (1971). The equation (2.3.13) may seem torepresent a rather special case, but if a system is lightly damped, h(X, X) and σ(X, X)may be replaced by their average values at a given energy level following the procedure ofequivalent non-linearization. Equation (2.3.16) would then yield an approximate solution.

2.3.4 Method of dissipation energy balancing

The condition (2.3.12) is given in the form of a force balance equation. Even if this equationcannot be satisfied exactly, it can still be used to determine approximate solutions. In thegeneral case the separation of the internal force function f(X, X) into a part containingthe stiffness and a part containing the damping may not be apparent from the equation ofmotion. An equivalent stiffness can then be identified as the mean value of the left handside of (2.3.12) for a given displacement level as

g(x) = E

[f(x, X) +

1

4

∂σ2

∂X− σ2

2

∂ψ

∂X

∣∣∣∣ x

](2.3.17)

where the conditional mean value is defined as

E[ ∗ |x ] =

∫ ∞

−∞∗ px|x(x|x) dx (2.3.18)

x

x′

x = const

x

x′

λ = const

Figure 2.3.1: Conditional expectations: a) Stiffness function via E[ |x], b) Stationary poten-tial via E[ |λ].

2.4 Probability Density Relations 15

px|x is the marginal probability density of X for a given value of X. The integration isillustrated in Figure 2.3.1a. The last term in (2.3.17) can be integrated by parts, whichyields the following simplified expression for the equivalent stiffness,

g(x) = E

[f(x, X) − 1

4

∂σ2

∂X

∣∣∣∣ x

](2.3.19)

The stationary potential is thus eliminated from the equation. It is interesting to see, thatthe stiffness function is the conditional mean value of the internal force function includingthe Wong-Zakai correction term as it appears in the Ito equation (2.3.6). In certain casesthe expression (2.3.19) can be evaluated without knowledge of the probability density, i.e. ifthe terms in the argument are independent of X or proportional with X, in which casethe conditional mean value is zero. However, if this is not the case it may be necessary todetermine the probability density first. The force balance equation (2.3.12) takes the formof an energy balance equation when multiplied by x. It is thus rewritten as

f(x, x)x = g(x)x − 1

4

∂σ2

∂xx +

σ2

2

dψ

dλx2 (2.3.20)

where the left hand side represents the energy dissipated by the internal force function. Sat-isfying this equation in the mean for a given energy level, an equivalent stationary potentialcan be identified as

dψ

dλ=

E

[f(X, X)X +

1

4

∂σ2

∂XX

∣∣∣∣ λ

]

E[

12σ2X2

∣∣∣ λ] (2.3.21)

This result is equivalent to the result obtained by Cai and Lin (1988) by the method ofdissipation energy balancing. Once ψ′(λ) is determined from (2.3.21) an approximation tothe probability density can be obtained from (2.3.8). However, evaluation of the conditionalmean values in (2.3.21) may require knowledge of the probability density, which suggests aniterative procedure, where a parametric representation of the probability density is assumed,and the parameters gradually determined. g(x) determined by (2.3.19) and ψ′(λ) determinedby (2.3.21) identifies an equivalent system with the following equation of motion,

X +

(12σ2ψ′(Λ)X − 1

4

∂σ2

∂X

)+ g(X) = σW (t) (2.3.22)

The probability density obtained is an exact solution for an oscillator governed by thisequation. The examples given in Sections 5.2 and 5.3 demonstrate the method of dissipationenergy balancing.

2.4 Probability Density Relations

So far only the joint probability density of the state space variables has been considered.Another important function is the probability density of the energy. In order to derive the


x

x′

z1

z2

Figure 2.4.1: a) Traditional phase plane (x, x), b) Modified phase plane (z1, z2).

relations between this function and the probability density of the state space variables givenin the previous sections it proves advantageous to introduce a set of modified state spacevariables.

2.4.1 Modified state space variables

For a system with non-linear stiffness free undamped vibration does not describe a circlein the traditional phase plane, see Figure 2.4.1a. A set of modified phase plane variablesrestoring polar symmetry in the case of free undamped vibrations were introduced by Krenkand Roberts (1999). They are related to the potential and kinetic energy by

z1 = sign(x)√

2G(x) , z2 = sign(x)√

2(

12x2)

= x (2.4.1)

The second modified phase plane variable thus reduces to the velocity. As seen from (2.4.1),z1 is related to the potential energy in the same way as x is related to the kinetic energy.This rescaling of the x-axis in the phase plane leads to a representation where the energy isproportional to the square of the distance to the origin of the coordinate system since

z21 + z2

2 = 2λ (2.4.2)

and free undamped vibration at a given energy level thus describes a circle as shown inFigure 2.4.1b. A polar representation for z1 and z2 is introduced as

z1 =√

2λ sin ϕ , z2 =√

2λ cos ϕ (2.4.3)

where ϕ(t) is the phase angle. The instantaneous angular velocity is defined as

ω =dϕ

dt(2.4.4)

2.4 Probability Density Relations 17

For a system with linear stiffness the instantaneous angular velocity is a constant. For asystem with non-linear stiffness this is not the case, as indicated by the trajectories in Figure2.4.1b, where the marks on the circles define intervals of equal duration. The angular velocityis thus largest around x = 0, i.e. at the displacement extremes. This corresponds to a systemwith hardening stiffness. The instantaneous angular velocity is obtained by considering thetime derivative of z1 for fixed λ,

dz1

dt=

√2λ cos ϕ

dϕ

dt=

dx

dt

dϕ

dt(2.4.5)

which yields the following relation between ω and dz1/dx,

ω =1

x

dz1

dt=

dz1

dx=

|g(x)|√2G(x)

(2.4.6)

The instantaneous velocity is thus seen to be equal to the derivative of the modified phaseplane variable z1 with respect to the displacement x, when trajectories at constant energylevels are considered.

2.4.2 Probability density of energy

Equations (2.4.1) and (2.4.3) define a transformation from the state space variables (x, x)to the energy and phase (λ, ϕ). The relationship between the probability density of the twosets of variables is given by the Jacobi determinants∣∣∣∣∂(z1, z2)

∂(λ, ϕ)

∣∣∣∣ = 1 ,

∣∣∣∣∂(z1, z2)

∂(x, x)

∣∣∣∣ = ω (2.4.7)

where the first determinant is evaluated from (2.4.1) and the last one follows from (2.4.6).The probability densities of (x, x) and (λ, ϕ) are thus related by

pλ,ϕ(λ, ϕ) =1

ω(x)px,x(x, x) (2.4.8)

The marginal probability density of the energy can now be obtained by integrating over thephase. A particularly simple relation exists in the case where the probability density of thetraditional state space variables is a function of the energy only. In this case integration overthe phase yields

pλ(λ) = T (λ)px,x(λ) (2.4.9)

where the coefficient T (λ), given by

T (λ) =

∫ 2π

0

dϕ

ω(λ, ϕ)=

∫ 2π

0

dt (2.4.10)


is seen to be the natural period of vibration for a given energy level. By writing dt = dx/x,the classical expression is obtained as

T (λ) = 2

∫ xmax

xmin

dx√2(λ − G(x))

(2.4.11)

where xmin and xmax are the minimum and maximum displacement for free undampedvibration at energy level λ. In the case of systems governed by an equation of the type(2.2.7) and (2.3.13), the assumption px,x = px,x(λ) is exact and (2.4.9) therefore gives anexact relation. In the more general cases (2.2.1) and (2.3.1) where the probability density ofthe state space variables is not a function of the energy only, equation (2.4.9) is not satisfied.However, in these cases the solutions for px,x will be approximations, and (2.4.9) thus yieldsan approximate expression for the probability density of the energy. The accuracy of theapproximation for pλ is of the same order of magnitude as the approximation for px,x, sinceboth are based on the assumption of independence of the phase.

Chapter 3

Power Spectral Density

The probability density function does not contain all the statistical information of a stochas-tic process. If e.g. the special case of the exact solution given by (2.2.10) is considered, itcan be seen, that only the ratio between the damping potential H(λ) and the white noiseintensity S0 enters the equation. The absolute damping is, however, a measure of the cor-relation time scale of the response, and this information is not contained in the stationaryprobability density function.

Another function, which yields important information of the response statistics is the powerspectral density. It is defined as the Fourier transform of the covariance function, and givesthe distribution of the energy on the frequency components. This information is necessaryfor evaluation of random fatigue, since it determines the distribution of frequencies andthereby the number of oscillations performed. Furthermore, the width of the resonance peakof the power spectral density is a measure of the magnitude of the damping. The problemof obtaining solutions for the power spectral density has been given less attention than thatof obtaining the probability density of the response. Only very few exact solutions for non-linear systems or systems with parametric excitation have been reported, see e.g. Dimentberget al. (1995b); Dimentberg and Lin (2002).

3.1 Background

The power spectral density of a linear stochastic system is given by exact analytical expres-sions if the excitation is either a white noise or the output of a linear equation where theinput is a white noise, i.e. a white noise with a rational filter. Since any function in a givenrange can be approximated by a rational function to any degree of accuracy, these solutionsare quite general. They do, however, require that the excitation is a Gaussian process. Ifthis is the case, an exact solution can also be obtained, if the power spectral density of the

19

20 Chapter 3. Power Spectral Density

excitation process is given. The method of statistical linearization discussed in the previ-ous chapter therefore offers a relatively simple way of obtaining an approximate solution,and this method can be applied to systems with any number of degrees of freedom. Thisprocedure gives an accurate prediction of the mean frequency of the response, but only forsmall levels of non-linearity in the stiffness will the spectrum be accurately represented overa large part of the frequency range. For systems with a strongly non-linear behaviour inthe stiffness, this method will generally give a very poor representation of the spectrum, seee.g. Bouc (1994), Soize (1995) or Fogli et al. (1996).

Cai and Lin (1997b) have proposed a method based on an extension of cumulant-neglectclosure, normally applied to obtain approximations to higher order moments. By consideringthe response at two different times an approximation to the covariance function is obtained.Comparison with simulated data show good agreement. However, the method involves asubstantial amount of computation and relies on the convergence of the cumulant-neglectclosure scheme.

3.1.1 Extension of statistical linearization

An extension of the statistical linearization procedure was proposed by Miles (1989). In thismethod the natural frequency of the equivalent system is given as a function of the amplitude,which is a stochastic process. The spectrum is thereby evaluated as the linear spectrum fora given amplitude, and the total spectrum is obtained by integrating over all amplitudelevels weighting each by the probability density at that amplitude. The technique wasdemonstrated for a system with linear-cubic stiffness (Duffing oscillator) and in a followingpaper for an oscillator with bilinear stiffness, Miles (1993). In both cases the system wasassumed to be lightly damped, and the excitation was assumed to be ideal broad band.Methods of this type have been further investigated by Fogli et al. (1996) and Fogli andBressolette (1997) considering an oscillator with bilinear stiffness. Soize (1995) proposeda similar method, where the non-linear system is replaced by an equivalent linear systemwith stochastic stiffness and damping parameters. A joint probability density of the stiffnessand damping parameters is determined by assuming a parametric representation of theprobability density, and requiring that the first and second moments of the state spacevariables of the original system and the equivalent linear system are identical. The spectrumfor given values of the stiffness and damping is obtained as the linear spectrum. The totalspectrum is then obtained by a weighted average using the joint probability density of thestiffness and damping parameters, and the technique is demonstrated by considering thepower spectral density of the Duffing oscillator.

For a system with non-linear stiffness the eigenfrequency is a function of the amplitude. Sincethe stochastic response takes place at various levels of the amplitude, a shift of frequencywill be observed in the power spectral density, i.e. a broadening of the resonance peak.The techniques mentioned above, Fogli et al. (1996); Fogli and Bressolette (1997); Miles(1989, 1993); Soize (1995), generally capture the broadening of the peak at the fundamental

3.2 Covariance Function and Power Spectral Density 21

frequency well, but are not capable of capturing the resonance peaks at higher harmonics. Amore accurate approximation to the power spectral density is obtained by a method proposedby Bouc (1994), where the higher harmonics are accounted for via a perturbation technique.The method was demonstrated for the Duffing oscillator, and in a later paper, Bellizzi andBouc (1996), for an oscillator with bilinear stiffness.

3.1.2 Local similarity with free response

A method based on local similarity with the free undamped response was proposed byKrenk and Roberts (1999). The response is here described by a set of modified phase planevariables conserving polar symmetry, and the free undamped response at a given energylevel is expanded in a Fourier series. An approximation to the covariance function at a givenenergy level is obtained by assuming local similarity with the free undamped response. Theapproximation is based on the series expansion of the modified phase plane variables andis therefore also given in terms of an infinite series. An approximation to the spectrumat this energy level is then derived by applying the Fourier transform to the covariancefunction. The result is an infinite series, where the first term corresponds to the fundamentalfrequency, the second term to the first higher harmonic, and so forth. The total spectrum isobtained by integrating over all energy levels weighting each with the probability density ofthe energy. The method was originally demonstrated considering the Duffing oscillator withlinear-quadratic-cubic damping and both white and coloured noise excitation, and has beenextended to include systems with parametric white noise excitation, Krenk et al. (2002). Themethod has furthermore been applied to systems with hyperbolic stiffness, Krenk (1999),and bilinear stiffness, Rudinger and Krenk (2003c). It is this method, which will be discussedin the following.

3.2 Covariance Function and Power Spectral Density

Initially the covariance function and power spectral density are defined. A stationary processX(t) is considered. The covariance function of the process is given by

Rx(τ) = E[X(t)X(t + τ)] (3.2.1)

The covariance function is thus given as the covariance of the stochastic variables X(t1) andX(t2) and for a stationary process it is a function of the time difference τ = t2 − t1 only.The value Rx(0) gives the variance of X(t). The covariance function and the power spectraldensity form a Fourier transform pair,

Sx(ω) =1

2π

∫ ∞

−∞Rx(τ)e−iωτ dτ , Rx(τ) =

∫ ∞

−∞Sx(ω)eiωτ dω (3.2.2)


where Sx(ω) is the power spectral density and i is the imaginary unit. Rx(τ) is an evenfunction and Sx(ω) is therefore real and even. The Fourier transforms can thus be replacedby cosine transforms as

Sx =1

π

∫ ∞

0

Rx(τ) cos(ωτ) dτ , Rx(τ) = 2

∫ ∞

0

Sx(ω) cos(ωτ) dω (3.2.3)

The transformations (3.2.2) and (3.2.3) define the two-sided spectral density, as opposed tothe one-sided spectral density defined as 2Sx(ω), 0 ≤ ω < ∞. From (3.2.2b) it is seen that∫∞−∞ Sx(ω)dω = E[X2].

3.3 Systems with Linear Stiffness

The aim of the method discussed in this chapter is to establish solutions for the powerspectral density of the response of general non-linear oscillators with broad band excitation.However, since the method simplifies considerably for systems with linear stiffness, this caseis initially discussed in a separate section. It is furthermore instructive to consider the case oflinear stiffness, because it facilitates the understanding of the principles behind the method.

3.3.1 The linear system

Initially, the simple case of a linear system with external excitation is considered. In thiscase the equation of motion reduces to

X + 2ζω0X + ω20X = W0(t) (3.3.1)

where ζ is the damping ratio and ω0 is the eigenfrequency. W0(t) is a white noise process.The mean value of the mechanical energy is given by

E[Λ] = E[

12ω2

0X2 + 1

2X2]

=πS0

2ζω0

(3.3.2)

where S0 is the intensity of the white noise. An analytical solution for the power spectraldensity of the response is given by

Sx(ω) =S0

(ω20 − ω2)2 + (2ζω0ω)2

(3.3.3)

With the expression (3.3.2) for the mean value of the energy, the power spectral density canbe rewritten as

Sx(ω) =

∫ ∞

0

2ζω0λpλ(λ)

π

dλ

(ω20 − ω2)2 + (2ζω0ω)2

(3.3.4)

Writing the equation of motion in this form suggests a way of generalizing the formula toinclude systems with non-linear behaviour, as discussed in the following. The stochasticproperties of linear systems are discussed in detail by Nielsen (1997).

3.3 Systems with Linear Stiffness 23

3.3.2 Systems with external excitation

It is now assumed that the equation of motion has the following slightly more general form,

X + h(Λ)X + ω20X = W0(t) (3.3.5)

This is a special case of a system where Caugheys solution applies, see Section 2.2.2. Inthe linear case discussed in the previous subsection h(Λ) = 2ζω0, and it therefore seemsreasonable to assume that an approximation to the spectral density can be obtained from(3.3.4) by replacing the factor 2ζω0 with h(λ) as

Sx(ω) ∫ ∞

0

|h(λ)|λpλ(λ)

π

dλ

(ω20 − ω2)2 + (h(λ)ω)2

(3.3.6)

This equation can also be expressed as

Sx(ω) ∫ ∞

0

Sx(ω|λ)pλ(λ)dλ (3.3.7)

where the energy conditional spectral density Sx(ω|λ) is given by

Sx(ω|λ) =|h(λ)|λ

π

1

(ω20 − ω2)2 + (h(λ)ω)2

(3.3.8)

Using this formulation it is assumed that a spectral density can be associated with any givenenergy level. It is difficult to give this conditional spectral density a physical interpretation.It should rather be considered as a mathematically convenient way of decomposing thesystem, which will help generalize the procedure. The absolute value of the damping appearsin the expression since the damping may be negative in certain energy ranges, and the conceptof a negative spectral density does not make sense. It is observed that∫ ∞

−∞Sx(ω)dω =

∫ ∞

0

|h(λ)|λpλ(λ)

π

∫ ∞

−∞

dω

(ω20 − ω2)2 + (h(λ)ω)2

dλ =E[Λ]

ω20

(3.3.9)

since ∫ ∞

−∞

dω

(ω20 − ω2)2 + (h(λ)ω)2

=π

ω20|h(λ)| (3.3.10)

see e.g. Gradshteyn and Ryzhik (1980). For a system given by (3.3.5) it can be shown thatω2

0E[X2] = E[X2] and thereby that E[Λ]/ω20 = E[1

2ω2

0X2 + 1

2X2] = E[X2]. The formulation

(3.3.6) is thus consistent in the sense that the area under the spectral density function is equalto the variance of the displacement. Equation (3.3.6) is never the less an approximation.

In the more general case where the damping function is given in the form h(X, X) it isassumed that the equivalent damping function heq(Λ) (obtained by the method of equivalentnon-linearization discussed in Section 2.2.3) can be used in the expressions. In this case themethod can be considered as an extension of the method of equivalent non-linearization tothe frequency domain. A system of this type is analyzed in the example in Section 5.1.


3.3.3 Systems with parametric excitation

The last system with linear stiffness, which will be considered is a system governed by thefollowing equation,

X + h(Λ)X + ω20X = σ(Λ)W (t) (3.3.11)

where W (t) is a unit white noise and σ(Λ) is the excitation amplitude function introducedin (2.3.4). The system (3.3.11) is seen to be a special case of the class of systems discussedin Section 2.3.3. Multiplying (3.3.11) by the velocity X, it is rewritten as an energy balanceequation as

dΛ

dt= σ(Λ)W (t)X − h(Λ)X2 (3.3.12)

For linear systems the response after a given time t can be considered as a sum of twocontributions, one contribution from the initial conditions, i.e. the value of the state spacevariables at time t (the zero input response), and one contribution due to the excitation aftertime t (the zero-initial-condition response). Such a separation can generally not be made fornon-linear systems.

Time

Dis

plac

emen

t

Figure 3.3.1: Separation of response in an uncorrelated and a fully correlated part.

For linear systems the part of the response originating from the initial conditions is givenby the free damped vibration. For stochastic systems with external excitation this willcorrespond to the correlated part of the response, which for lightly damped systems willresemble the free undamped response with an exponentially decaying envelope. Therefore,

3.3 Systems with Linear Stiffness 25

it seems reasonable as an approximation to divide the response of a lightly damped systeminto a fully correlated part and a part due to the excitation, see e.g. the discussion by Krenket al. (2002).

The idea of this separation is illustrated in Figure 3.3.1. In the second half of the timerange considered the stochastic response is indicated by the dashed line and the free dampedresponse is indicated by the solid line. The difference between these two curves is thenassumed to be the part of the response due to the excitation.

The energy balance equation (3.3.12) is now considered for a given energy level. Taking theconditional mean value of this equation the following expression is obtained,

E

[dΛ

dt

∣∣∣∣λ]

= E[σ(Λ)W (t)X|λ] − E[h(Λ)X2|λ] (3.3.13)

where E[ ] indicates the mean value. In the case of an ideal white noise the excitation isindependent of the response. The first term in (3.3.13) therefore vanishes for the correlatedpart of the response, and the equation governing the correlated part of the response is reducedto

E

[dΛ

dt

∣∣∣∣λ]

cor

−E[h(Λ)X2|λ] (3.3.14)

where the subscript ’cor’ indicates that only the energy of the correlated part of the responseis considered in this equation. If the excitation has a small but finite correlation time theexcitation will be correlated with the response and the first term in (3.3.13) contributesto the correlated part of the response. The effect is taken into account by including theWong-Zakai term in the damping function and (3.3.14) is rewritten as

E

[dΛ

dt

∣∣∣∣λ]

cor

−E

[h(Λ)X2 − 1

4

dσ2

dΛX2

∣∣∣∣λ]

(3.3.15)

The effective damping of the system for a given energy level is defined as the ratio betweenthe time derivative of the correlated part of the energy and the energy itself. This relationcan be expressed as

E

[dΛ

dt

∣∣∣∣λ]

cor

= −ηλλ (3.3.16)

where ηλ is the effective damping. From (3.3.15) it is seen that the effective damping is givenby

ηλ = h(λ) − 1

4

dσ(λ)2

dλ(3.3.17)

where the relation E[X2|λ] = λ, which is valid for a system governed by (3.3.11), has beenapplied. It is observed from (3.3.17) that parametric excitation may influence the effective


damping through the Wong-Zakai term. The energy conditional spectral density is nowobtained from (3.3.8) by replacing the damping function h(λ) with the effective damping,

Sx(ω|λ) =|ηλ|λ

π

1

(ω20 − ω2)2 + (ηλω)2

(3.3.18)

If the damping and excitation are given in the form h(X, X) and σ(X, X) the method ofdissipation energy balancing discussed in Section 2.3.4 can be used to define an equivalentsystem. However, in this case the stiffness may be modified and the determination of theeffective damping becomes less apparent. In this case the theory given in the next sectionfor systems with non-linear stiffness should be applied. A system of this type is consideredin the example given in Section 5.2.

3.4 Systems with Non-linear Stiffness

In the expressions for the power spectral density given in the previous section, e.g. (3.3.8)and (3.3.18), the natural frequency ω0 appears in the denominator. For systems with non-linear stiffness one natural frequency cannot be defined, since the period of free undampedoscillations is a function of the energy level. The natural frequency is thus a function ofthe energy, which has the effect of broadening the peak of the spectrum, an effect whichfor systems with linear stiffness is associated with an increase in the damping. Since thefree undamped vibrations are not pure harmonics, a Fourier series expansion will includeharmonics at higher frequencies. This will have the effect of introducing peaks at higherharmonics in the power spectral density. The method discussed in the previous section fordetermination of approximate expressions for the power spectral density therefore becomesmore complicated, when generalized to systems with non-linear stiffness.

3.4.1 Free undamped vibration

First a description of the free undamped vibrations is addressed. The modified phase planevariables z1 and z2 introduced in Section 2.4.1 are reconsidered. These two variables cannow be expanded in a Fourier series as

z1(t)√2λ

= sin ϕ(t) =∞∑

j=1,3,...

sj(λ) sin

(2πjt

T (λ)

)

z2(t)√2λ

= cos ϕ(t) =∞∑

j=1,3,...

cj(λ) cos

(2πjt

T (λ)

) (3.4.1)

since both variables are periodic with period T (λ) given by (2.4.11). ϕ(t) is here thephase angle introduced in the transformation (2.4.3) from (z1, z2) to (λ, ϕ). Representa-tions of this type have also been considered by Dimentberg et al. (1995a) in connection with

3.4 Systems with Non-linear Stiffness 27

stochastic averaging. The coefficients sj(λ) and cj(λ) are obtained by multiplying (3.4.1) bysin(2πkt/T (λ)) and cos(2πkt/T (λ)), respectively, where k is an integer. By integrating withrespect to time from t = 0 to t = T (λ) and using orthogonality, sj and cj are obtained as

sj(λ) =

√2

T (λ)√

λ

∫ T (λ)

0

z1(t) sin

(2πjt

T (λ)

)dt

cj(λ) =

√2

T (λ)√

λ

∫ T (λ)

0

z2(t) cos

(2πjt

T (λ)

)dt

(3.4.2)

The coefficients of the sine and cosine expansions are related to the potential and kineticenergy. Taking the mean value of the square of the expansions (3.4.1), the following relationsare obtained,

〈2G(x)〉λ

= 2 〈sin2 ϕ〉 =∞∑

j=1,3,...

sj(λ)2 = s(λ)

〈x2〉λ

= 2 〈cos2 ϕ〉 =∞∑

j=1,3,...

cj(λ)2 = c(λ)

(3.4.3)

where 〈〉 indicates the average over one period of oscillation. s(λ) is thus twice the meanvalue of the potential energy relative to the total energy, and c(λ) is twice the mean valueof the kinetic energy relative to the total energy for free undamped vibration. It is observedthat s(λ) + c(λ) = 2. In the case of a system with linear stiffness s1(λ) = c1(λ) = 1, and allother coefficients vanish.

3.4.2 Effective damping

A system governed by the general equation of motion (2.3.1) is now considered, and theapproach discussed in Section 3.3.3 for evaluation of an effective damping at a given energylevel is generalized. The energy balance equation is in this case expressed as

dΛ

dt= σ(X, X)XW (t) −

(f(X, X) − g(X)

)X (3.4.4)

where g(X) is evaluated from (2.3.17). Again, the mean value for a given energy level isconsidered,

E

[dΛ

dt

∣∣∣∣λ]

= E[σ(X, X)XW (t)|λ] − E[f(X, X)X − g(X)X|λ] (3.4.5)

It can be shown that the term E[g(X)X|λ] is zero if the probability density of the statespace vector is given in the form px,x = px,x(λ), i.e. if the variable X and X only enter theexpression via the mechanical energy. In the present case it is assumed that an approximate


solution is given in the form px,x = C exp(−ψ(λ)), see (2.3.8), so within the accuracy of theapproximations already considered, E[g(X)X|λ] can be assumed to be zero. The energy ofthe correlated part of the response is thus governed by

E

[dΛ

dt

∣∣∣∣λ]

cor

−E

[f(X, X)X − 1

4

∂σ2

∂XX

∣∣∣∣λ]

(3.4.6)

similarly to (3.3.15), where the last term is the contribution from E[σ(X, X)XW (t)|λ] whenW (t) has an arbitrarily small but finite correlation time scale, i.e. when W (t) is broadbanded, but not an ideal white noise. The contribution from the excitation is seen to be theWong-Zakai correction term as it appears in the Ito-equation (2.3.6). Defining the effectivedamping as the ratio between the right hand side of (3.4.6) and the negative value of theenergy, as in (3.3.16), the effective damping is obtained as

ηλ =1

λE

[f(X, X)X − 1

4

∂σ2

∂XX

∣∣∣∣λ]

(3.4.7)

It is seen from this expression that parametric excitation may influence the effective damping.This effect is illustrated in the example given in Section 5.2.

3.4.3 Preliminary approximation for spectral density

Before arriving at a final approximate expression for the power spectral density, some ofthe features to be expected from such a solution will be discussed. From the Fourier seriesexpansions (3.4.1) it is seen that the free undamped response can be expanded in harmonicswith frequencies 2πj/T (λ), j = 1, 3, ..., and the spectrum at a given energy level shouldtherefore have peaks at each of these frequencies. It is therefore assumed that the spectraldensity can be expanded in series with terms of the type

Sx,j(ω|λ) =|ηλ|λ

π

1

(j2ω2λ − ω2)2 + (ηλω)2

, ωλ =2π

T (λ)(3.4.8)

where ωλ is the fundamental frequency at energy level λ. Each of the terms in (3.4.8) havea resonance peak at jωλ. The fundamental peak will normally be the most pronounced inthe spectrum, even for systems with very strong non-linearities in the stiffness. The energyconditional spectrum is therefore obtained by a weighted summation of the terms (3.4.8),i.e. as

Sx(ω|λ) =∞∑

j=1,3,...

wj(λ)Sx,j(ω|λ) (3.4.9)

where wj(λ) are weight factors. The unconditional spectral density Sx(ω) is then obtainedfrom (3.3.7). It is not yet clear how the weighting factors wj(λ) should be chosen. In thenext subsection a more detailed analysis will identify these factors.

3.4 Systems with Non-linear Stiffness 29

3.4.4 Local solution for covariance function and spectral density

A more formal derivation of the approximate solution for the power spectral density is givenin this section. The procedure identifies the weight functions introduced in (3.4.9). Thefirst step is to establish approximate solutions for the covariance functions of the state spacevariables.

The response is now considered at time t and at a later time t + τ . As in Sections 3.3.3and 3.4.2 it is assumed that the response can be divided into an uncorrelated and a fullycorrelated part. For a given energy at time t, the energy of the correlated part of the responseat time t + τ is approximated by

λt+τ λt exp(−ηλ,tτ) (3.4.10)

which can be considered as a local solution to (3.3.16). The subscript ’cor’ used in Sections3.3.3 and 3.4.2 will be omitted in this section in order to simplify the notation. Equation(3.4.10) is only accurate if the energy is approximately constant in the interval [t; t + τ ],which will be the case if τ is small compared to the time scale η−1

λ of the energy dissipation,i.e. if ηλτ 1.

The velocity is now considered at these two times. For a given energy level λ at time t, thevelocity at time t and the correlated part of the velocity at time t + τ can be expressed as

xt =√

2λ cos ϕt , xt+τ √

2λ exp(−12ηλ,tτ) cos ϕt+τ (3.4.11)

where the notation introduced in (2.4.3) has been used. The covariance function of thevelocity at the energy level λ is now defined as

Rx(τ |λ) = E[xtxt+τ |λ] 2λ exp(−12ηλτ) E[cos ϕt cos ϕt+τ |λ] (3.4.12)

where the last expression is based on the representation (3.4.11). The mean value can beevaluated from the series expansion in (3.4.1b). This yields the following series expansionfor the covariance function at a given energy level,

Rx(τ |λ) λ exp(−12ηλτ)

∞∑j=1,3,...

cj(λ)2 cos(jωλτ) (3.4.13)

The covariance function is thus given by an oscillatory expression multiplied by an exponen-tially decaying function. The complementary modified phase plane variable z1 introduced in(2.4.1a) can be represented in the same way. Evaluation of the covariance function of thisvariable yields

Rz1(τ |λ) λ exp(−12ηλτ)

∞∑j=1,3,...

sj(λ)2 cos(jωλτ) (3.4.14)

where the expansion (3.4.1a) has been used. The two expressions (3.4.13) and (3.4.14) areidentical except for the participation factors c2

j and s2j . An approximation to the covariance


function of the response process X(t) can be obtained by integrating (3.4.13) twice. However,for Rx(τ |λ) to be the conditional covariance function of X(t) the derivative must vanish atτ = 0. A consistent approximation for the covariance function of X(t) at energy level λ isgiven by, see Krenk and Roberts (1999) or Krenk et al. (2002),

Rx(τ |λ) λ exp(−12ηλτ)

∞∑j=1,3,...

(cj(λ)

jωλ

)2cos(Ωjτ) +

ζj√1 − ζ2

j

sin(Ωjτ)

(3.4.15)

where the damping ratio ζj and the damped natural frequency Ωj are given by

ζj =ηλ

2jωλ

, Ωj = j√

1 − ζ2j ωλ (3.4.16)

Each term in (3.4.15) represents the covariance function of a linear single-degree-of-freedomsystem with damping ratio ζj and natural frequency jωλ. For small values of the dampingratios ζj, the expression (3.4.15) will be of the same form as (3.4.13) and (3.4.14) for themodified phase plane variables z1 and z2 = x. The additional term in (3.4.15) ensures thatdRx/dτ = 0 for τ = 0. The energy conditional spectral density is obtained by taking theFourier transform of (3.4.15) as

Sx(ω|λ) =|ηλ|λ

π

∞∑j=1,3,...

cj(λ)2

(j2ω2λ − ω2)2 + η2

λω2

(3.4.17)

From this expression it is seen that a consistent choice of the weight factors in (3.4.9) is givenby cj(λ)2. The total spectrum is again obtained from (3.3.7) by integrating over all energylevels weighting each with the probability density of the energy. The expression (3.4.17)reduces to (3.3.18) in the case of a system with linear stiffness, and to the expression (3.3.8)if the system has linear stiffness and external excitation. The effects of the higher harmonicson the spectral density is illustrated in the example in Section 5.4.

Chapter 4

Simulation Procedure

In Chapter 5 the accuracy of the approximate solutions introduced in Chapters 2 and 3 willbe investigated by comparing with results obtained by stochastic simulation. The simulationtechnique is therefore discussed in this chapter.

An ideal white noise is a process with an infinite variance and a correlation time, which isinfinitely small, as seen by 2.2.2. It thus represents an idealization, which can never occurin the physical reality, and which cannot be represented by a digital record generated bya computer. The white noises applied as excitation processes are thus approximations toideal white noises and the first section in this chapter discusses how these broad bandedprocesses can be generated. It should be noticed that the theory presented in the previoustwo chapters assumes that the excitation is not an ideal white noise by including the Wong-Zakai correction, see e.g. equation (2.3.6).

Once the record of the approximation to the white noise is generated, the response processis obtained by numerical integration. This procedure is discussed in the second section.

4.1 White noise approximation

Consider a sequence of independent random numbers Wi, each of which has a Gaussiandistribution with zero mean and standard deviation σW . A stochastic process W (t) is gener-ated by linear interpolation of the random numbers Wi, each separated by a time increment∆t = ti+1 − ti, see Figure 4.1.1a. As can be derived from results obtained by Clough andPenzien (1975), the power spectral density of this process is given by

SW (ω) = S0

[sin(1

2ω∆t)

12ω∆t

]4

, S0 = σ2W

∆t

2π(4.1.1)

31

32 Chapter 4. Simulation Procedure

Wi−1

Wi

Wi+1

ti−1

ti

ti+1

0 0.5 10

1

S W /

S 0

ω ∆t / 2π

Figure 4.1.1: a) Piecewise linear process W (t), b) Spectral density of piecewise linear process.

The function is depicted in Figure 4.1.1b, and it is observed, that the spectrum is approx-imately constant SW (ω) S0 for ω∆t 1. Alternatively, one could assume that W (t)remains constant within each time increment, i.e. that W (t) = Wi for ti ≤ t < ti+1. Thisidea is sketched in Figure 4.1.2a. This way of sampling is sometimes referred to as zero orderhold. The spectral density of a process of this type can be expressed as

SW (ω) = S0

[sin(1

2ω∆t)

12ω∆t

]2

(4.1.2)

with S0 given by (4.1.1b). The spectral densities of these two types of processes are thusseen to correspond to different powers of the function sin(1

2ω∆t)/1

2ω∆t. The expression

(4.1.2) is shown in Figure 4.1.2b by the solid line, where it is compared to the expression(4.1.1) indicated by the dashed line. As can be seen by this figure, zero order hold leads toa better approximation of a white noise than the linear interpolation scheme. However, in

Wi−1

Wi

Wi+1

ti−1

ti

ti+1

0 0.5 10

1

S W /

S 0

ω ∆t / 2π

Figure 4.1.2: a) Piecewise constant process W (t), b) Spectral density of piecewise constantprocess (–) and piecewise linear process (- -).

4.2 Generating the Response Process 33

the examples in the following chapter, it will be the linear interpolation, which will be usedas an approximation to the white noise excitation.

If a structure or a system excited by the process W (t) (generated either by linear interpolationor by zero order hold) has a typical frequency 〈ω〉, such that 〈ω〉∆t 1, then the excitationwill be experienced as a close approximation to white noise by the structure or system. Atypical frequency could be the standard deviation of the spectral density of the response,

〈ω〉2 =

∫∞−∞ ω2 Sx(ω) dω∫∞−∞ Sx(ω) dω

=σ2

x

σ2x

(4.1.3)

which can be interpreted as a mean frequency. Equation (4.1.3) only gives a good measureif one mode of the response is dominant, i.e. if the response has one dominating frequency.For the simulations used in the examples considered in Chapter 5 (and later in Chapter 7) atleast 50 time steps are taken within a typical period, i.e. 50∆t ≤ 2π/〈ω〉. This correspondsto ∆t〈ω〉 < 0.13, and the value of the spectral density at this frequency is SW (〈ω〉) = 0.997S0

with SW (ω) given by (4.1.1). The approximation to an ideal white noise is considered to besufficiently good.

By the procedure described above a number of samples of independent unit white noises aregenerated and stored in the vector WI(t). A set of correlated white noises with spectralintensity matrix S is obtained by the linear transformation

W(t) = AWI(t) (4.1.4)

where A is a solution to the equation

AAT = 2πS (4.1.5)

The solution to this equation is not unique. A solution can be obtained by e.g. Choleskyfactorization.

4.2 Generating the Response Process

Once the white noise excitation processes have been generated a sample of the responseprocess is obtained by numerical integration of the equation of motion. This method issometimes referred to as Monte Carlo simulation, but will be termed stochastic simulationin this thesis. The numerical integration requires a smooth excitation, which is not the casefor the processes shown in Figures 4.1.1a and 4.1.2a. These functions are, however, smoothin the intervals ti < t < ti+1 and the integration can therefore be performed for each timestep at a time. The first order equation of motion (2.3.1) can be written as

u = a(u) + b(u)W(t) (4.2.1)

34 Chapter 4. Simulation Procedure

where the state space vector u and the function a(u) are given by

u =

[x

x

], a(u) =

[u2

−f(u1, u2)

](4.2.2)

and the functions b(u) and W(t) are given by

b(u) =

[0 . . . 0

b1(u1, u2) . . . bn(u1, u2)

], W(t) =

W1(t)

...

Wn(t)

(4.2.3)

f(x, x), bj(x, x) and Wj(t) are introduced in Section 2.3.1 and n is the number of termson the right hand side of (2.3.1). The integration procedure, which is used, is 4th orderRunge-Kutta integration. The algorithm of this integration scheme is given in Table 4.2.1,where it is assumed that the state space vector ui at time ti is known, and the state spacevector ui+1 at time ti+1 is evaluated.

u1 = a(ui)∆t + b(ui)W(ti)∆t

u2 = a(ui + 12u1)∆t + b(ui + 1

2u1)(

12W(ti) + 1

2W(ti+1))∆t

u3 = a(ui + 12u2)∆t + b(ui + 1

2u2)(

12W(ti) + 1

2W(ti+1))∆t

u4 = a(ui + u3)∆t + b(ui + u3)W(ti+1)∆t

ui+1 = 16u1 + 1

3u2 + 1

3u3 + 1

6u4

Table 4.2.1: Algorithm for 4th order Runge-Kutta integration.

Chapter 5

Probability Density and SpectralDensity: Examples

In this chapter the theory presented in Chapters 2 and 3 will be illustrated by four exam-ples. The first example is the relatively simple case of an oscillator with power law viscousdamping, linear stiffness and external excitation. The results given here have been presentedby Rudinger and Krenk (2003b). The second example is the case of a linear system withparametric excitation. This example illustrates the influence of parametric excitation onthe effective stiffness and damping of the system. The results are included in the paper byKrenk et al. (2002) and in a slightly modified version by Rudinger and Krenk (2000a). Inthe third example an oscillator with stable limit cycle is considered. The special feature ofthis example is that the damping is negative for certain energy levels. These results wereoriginally given by Rudinger and Krenk (2002b). Finally, an oscillator with bilinear stiffness,linear damping and external excitation is considered. This example illustrates the ability ofthe method presented in Chapter 3 to capture the higher harmonics in the frequency domain.The results were presented by Rudinger and Krenk (2003c).

5.1 Oscillator with power law viscous damper

A system with a relatively simple non-linear behaviour is an oscillator with linear stiffnessand power law viscous damping. If an oscillator of this type is excited by additive whitenoise the equation of motion can be expressed as

X + h(X)X + ω20X = W0(t) , h(X)X = c sign(X) |X|α (5.1.1)

where ω0 is the natural angular frequency and W0(t) is a white noise with intensity S0. h(X)is the damping function, which is characterized by the damping coefficient c, which has thedimension length1−α · timeα−2, and the damping law exponent α, which is a number in the

35

36 Chapter 5. Probability Density and Spectral Density: Examples

range α ∈ [0, 1]. For α = 1 the case of linear viscous damping is retrieved. Dry frictioncorresponds to α = 0. A damping function of this type describes the behaviour of theJarret Elastomeric Spring Dampers reasonably well, see e.g. Terenzi (1999) and Rudingerand Magonette (2002).

5.1.1 Non-dimensional formulation

Initially the problem is rewritten in non-dimensional form. A length scale and a time scaleare introduced as

X0 =

√2πS0

ω30

, t0 =1

ω0

(5.1.2)

The equation of motion is then multiplied by t20/X0, whereby the following non-dimensionalequation of motion is obtained,

Y + d(Y )Y + Y = U(τ) (5.1.3)

The excitation process U(τ) is a unit white noise, i.e. a white noise with intensity 1/2π. Thenon-dimensional time τ , displacement Y and velocity Y are given by

τ =t

t0, Y =

X

X0

, Y =dY

dτ(5.1.4)

A dot used in connection with Y thus indicates the derivative with respect to τ . The non-dimensional damping function d(Y ) is given by

d(Y )Y = β sign(Y ) |Y |α , β =c

ω0

(2πS0

ω0

)α−12

(5.1.5)

where β is a non-dimensional damping coefficient. The equation of motion is seen only todepend on the parameters α and β in this form.

5.1.2 Equivalent non-linearization

In the method of equivalent non-linearization, Caughey (1986), an equivalent non-linearsystem is introduced as

Y + deq(Λ)Y + Y = U(τ) , Λ = 12Y 2 + 1

2Y 2 (5.1.6)

where Λ is the mechanical energy non-dimensionalized by multiplication with t20/X20 from

(5.1.2). deq(Λ) is a non-dimensional equivalent damping function, which is assumed to be afunction of the mechanical energy only. The equivalent damping function is evaluated by

deq(λ) =E[d(Y )Y 2 |λ]

E[Y 2 |λ](5.1.7)

5.1 Oscillator with power law viscous damper 37

where E[ |λ] is the mean value for a given energy level. The mean value in (5.1.7) can beevaluated by considering free undamped vibration at energy level λ as discussed by Krenket al. (2002). This corresponds to assuming a uniform distribution of the phase. Freeundamped vibration at energy level λ is given by

y =√

2λ sin τ , y =√

2λ cos τ (5.1.8)

The equivalent damping function is now evaluated from (5.1.7) considering the harmonicmotion given by (5.1.8),

deq(λ) =

1

2π

∫ 2π

0

(β|y|α+1

∣∣λ) dτ

1

2π

∫ 2π

0

(y2∣∣λ) dτ

= aλ(α−1)/2 , a =β2(α+1)/2

√π

Γ(12α + 1)

Γ(12α + 3

2)

(5.1.9)

where Γ( ) is the gamma function. The probability of the mechanical energy of the equivalentsystem (5.1.6) is given by

pλ(λ) = C exp(−2Deq(λ)) , Deq(λ) =

∫ λ

0

deq(λ) dλ (5.1.10)

which is a special case of Caugheys solution (2.2.10) transformed from state space variablesto the energy by (2.4.9). C is here a normalizing constant and Deq(λ) is the equivalentdamping potential. With the equivalent damping function deq(λ) evaluated in (5.1.9), theprobability density of the energy reduces to

pλ(λ) = C exp

(− 4a

α + 1λ(α+1)/2

), C =

α + 1

2Γ(

2α+1

) ( 4a

α + 1

) 2α+1

(5.1.11)

where a is given in (5.1.9b). The spectral density at a given energy level is obtained from(3.3.8) as

Sy(r|λ) =deq(λ)λ

π

1

(1 − r2)2 + deq(λ)2r2, r =

ω

ω0

(5.1.12)

where r is a non-dimensional frequency. The total spectrum is now obtained from (3.3.7) byintegration over all energy levels as

Sy(r) =

∫ ∞

0

Sy(r|λ)pλ(λ)dλ =

∫ ∞

0

aλ(α+1)/2

π

pλ(λ)

(1 − r2)2 + a2λα−1r2dλ (5.1.13)

The integration in (5.1.13) is carried out numerically in the examples given in the following.

5.1.3 Statistical linearization

As an alternative to the method of equivalent non-linearization the method of statisticallinearization is now considered. The non-linear equation of motion is in this case replaced


by an equivalent one, which is easily solved. Based on the non-dimensional formulation(5.1.3) the equivalent linear system is expressed as

Y + 2ζeqY + Y = U(τ) (5.1.14)

where ζeq is the equivalent damping ratio. The probability density of the non-dimensionalvelocity Y and the non-dimensional energy Λ introduced in (5.1.6b) for the linear system(5.1.14) are given by

py(y) =

√2ζeq

πexp(−2ζeqy

2) , pλ(λ) = 4ζeq exp(−4ζeqλ) (5.1.15)

which are seen to be a zero mean normal distribution and an exponential distribution. ζeq

is obtained by

2ζeq = E

[∂

∂Y(d(Y )Y )

]=

∫ ∞

−∞py(y)

∂

∂y(d(y)y) dy (5.1.16)

Solving this equation for ζeq the following value is obtained,

ζeq =1

2

(αβΓ(1

2α)√

π

) 2α+1

(5.1.17)

The power spectral density of the non-dimensional linear system (5.1.14) is simply

2πSy(r) =1

(1 − r2)2 + (2ζeqr)2(5.1.18)

where r is the non-dimensional frequency introduced in (5.1.12b). Both the probabilitydensity (5.1.15) and the power spectral density (5.1.18) are seen only to depend on theparameters α and β in the combination ζeq in (5.1.17).

5.1.4 Numerical examples

In order to investigate the results derived, the analytical expressions for the probabilitydensity of the energy and for the spectral density of the response are compared to resultsobtained from stochastic records. The simulation of the stochastic records follows the pro-cedure described in Chapter 4. 50 time steps are taken per natural period.

In Figures 5.1.1 - 5.1.4 the probability density of the energy and the power spectral density ofthe response are given for various combinations of the parameter α, governing the magnitudeof non-linearity, and the damping level of the system, as defined by ζeq. The dashed linecorresponds to the analytical solution obtained by statistical linearization and the solid linecorresponds to the method of equivalent linearization discussed in Sections 2.2.3 and 3.3.2.The dots correspond to stochastic simulation of records with a length of 50,000 naturalperiods.

5.1 Oscillator with power law viscous damper 39

0 10 2010

−3

10−2

10−1

100

p λ

λ0 1 2

10−1

100

101

102

2πS y

r

Figure 5.1.1: Probability density and spectral density for α = 0.2 and ζeq = 0.1 (β = 0.355),– equivalent non-linearization, - - statistical linearization, • stochastic simulation.

0 10 2010

−3

10−2

10−1

100

p λ

λ0 1 2

10−1

100

101

102

2πS y

r


Figures 5.1.1 and 5.1.2 show the probability density and the spectral density for ζeq = 0.1.In the first case α = 0.2 and in the second case α = 0.1. In both cases the probability densityis approximated very accurately by the method of equivalent non-linearization. Statisticallinearization on the other hand gives a relatively poor estimate of the probability density,especially the tail. This is an effect of the Gaussian nature of the response of the equivalentlinear system leading to an exponential distribution of the energy, i.e. a straight line in thesemi-logarithmic plots in Figures 5.1.1a - 5.1.4a. As to the spectral density of the response,both methods seem to give accurate results, except at the resonance peak, where the methodof statistical linearization slightly underestimates the peak.

In Figures 5.1.3 and 5.1.4 the equivalent damping is increased to ζeq = 0.3, which can beseen by the significant broadening of the peak, when comparing with the spectral densitiesin Figures 5.1.1b and 5.1.2b. The damping law exponent is again chosen as α = 0.2 andα = 0.1, respectively. The probability density is seen to be accurately evaluated by the


0 5 1010

−3

10−2

10−1

100

p λ

λ0 1 2

10−1

100

101

2πS y

r


0 5 1010

−3

10−2

10−1

100

p λ

λ0 1 2

10−1

100

101

2πS y

r


method of equivalent non-linearization, while the statistical linearization procedure displaysthe same shortcomings as observed in Figures 5.1.1a and 5.1.2a. The accuracy of the statis-tical linearization with respect to the power spectral density is seen to be decreasing withincreasing damping. The peaks in the spectra shown in Figures 5.1.3b and 5.1.4b are thusless accurately represented by the statistical linearization than in the previous case in Figures5.1.1b and 5.1.2b. This is partly due to the broadening of the peak and thereby broadeningof the range containing the inaccuracy. The method of equivalent non-linearization givesmore accurate results. However, in the last case for ζeq = 0.3 and α = 0.1 this method seemsto overestimate the peak slightly.

5.2 Linear Oscillator with Parametric Excitation 41

5.2 Linear Oscillator with Parametric Excitation

If the excitation of a system is parametric, the internal force function should be modified bythe Wong-Zakai correction term when the effective stiffness and damping of the system areevaluated, as demonstrated in Sections 2.3.4 and 3.4.2. In this example a linear oscillatorwith parametric excitation is considered, and the influence of the effective system propertieson the power spectral density is shown. A system of this type has also been investigated byZhu and Lin (1991). The equation of motion is given by

X + ω0

(2ζ + W2(t)

)X + ω2

0

(1 + W1(t)

)X = W0(t) (5.2.1)

Here W0(t), W1(t) and W2(t) are white noise processes with spectral density matrix Sij. Itis assumed that S01 = S02 = 0, whereby the non-parametric excitation term is uncorrelatedwith the two parametric excitation terms. The excitation amplitude follows from (2.3.4) as

σ(X, X)2 = 2π(S00 + ω4

0X2S11 + 2ω3

0XXS12 + ω20X

2S22

)(5.2.2)

For a system of this type an exact solution for the probability density cannot be obtained,and the approximation method introduced in Section 2.3.4 must be applied.

5.2.1 Probability density

The effective stiffness of the system is evaluated from (2.3.19), which in this case reduces tothe following expression,

g(x) = (κω0)2x , κ2 = 1 − πω0S12 (5.2.3)

where κ2 is the ratio between the effective stiffness and the apparent stiffness of the system.It is seen, that correlation between the two parametric excitation processes W1(t) and W2(t)leads to a change of effective stiffness. The elastic potential is obtained by integration of thestiffness and the mechanical energy follows as

λ = 12x2 + 1

2(κω0)

2x2 (5.2.4)

In the present case, the modified phase plane variables introduced in (2.4.1) reduce to

(κω0)x =√

2λ sin ϕ , x =√

2λ cos ϕ (5.2.5)

by which the state vector (x, x) is represented by the energy and phase (λ, ϕ). The undampedfree response is harmonic, and the expectation for a given energy level therefore reduces toan average over the phase angle ϕ. The equation (2.3.21) for the derivative of the stationarypotential then takes the form

dψ

dλ=

1

2π

∫ 2π

0

(f(x, x)x +

1

4

∂σ(x, x)2

∂xx

∣∣∣∣λ)

dϕ

1

2π

∫ 2π

0

(12σ(x, x)2x2|λ) dϕ

(5.2.6)


with x and x given by (5.2.5). Evaluation of the integrals gives

dψ

dλ=

4ζω0 + 2πω20S22

2πS00 + (πω20κ

−2S11 + 3πω20S22) λ

(5.2.7)

This expression is rewritten by introducing a reference energy level λ0 and a non-dimensionalparameter ν,

λ0 =2S00

ω20κ

−2S11 + 3ω20S22

, ν =4ζ − πω0κ

−2S11 − πω0S22

πω0κ−2S11 + 3πω0S22

(5.2.8)

whereby it takes the simple form

dψ

dλ=

ν + 1

λ0 + λ(5.2.9)

After introducing the non-dimensional energy ξ = λ/λ0 integration of (5.2.9) leads to theprobability density function

pξ(ξ) =ν

(1 + ξ)ν+1, ν > 0 (5.2.10)

The distribution of ξ is seen to depend on only one variable ν. The probability density isdefined for ν > 0. However, the mean value is given by

E[ξ] =1

ν − 1, ν > 1 (5.2.11)

so for 0 < ν ≤ 1 the distribution does not have a mean value, which implies that the varianceof the displacement is infinite and the idea of a stationary process meaningless. The followinginvestigation therefore concentrates on systems satisfying the condition ν > 1.

5.2.2 Spectral density

The correlation between the two parametric excitation processes is quantified via the corre-lation coefficient

ρ =S12√S11S22

, −1 ≤ ρ ≤ 1 (5.2.12)

The effective damping is evaluated by (3.4.7) as

ηλ = 2ζeω0 , ζe = ζ − 12πω0S22 (5.2.13)

ζe is the effective damping ratio, which reduces to the parameter ζ for W2(t) ≡ 0. ForW2(t) = 0 the energy input of the excitation is biased, which leads to a reduction in theeffective damping. The energy conditional one-sided spectral density reduces to

Sx(ω|λ) =ηλλ

π

2

((κω0)2 − ω2)2 + η2λω

2(5.2.14)

5.2 Linear Oscillator with Parametric Excitation 43

where the effective eigenfrequency κω0 enters. The integration of (3.3.7) for the unconditionalone-sided spectral density can be carried out explicitly, whereby

Sx(r)ω0

σ2x

=2ζe/κ

π

1

(1 − r2)2 + (2ζe/κ)2 r2, r =

ω

κω0

(5.2.15)

The right hand side in this representation integrates to one. The variance of the displacementis given by

σ2x =

E[λ]

(κω0)2=

λ0

(κω0)2 (ν − 1)(5.2.16)

where the requirement ν > 1 appears again.

5.2.3 Numerical examples

Examples illustrating the effect of the intensities S12 and S22 on the effective stiffness anddamping are now considered. Table 5.2.1 gives the parameters of two different systemsunder two different loading situations. The systems have damping ratios ζ = 0.05 andζ = 0.1 and the excitation processes W1(t) and W2(t) have identical intensities, but are eitheruncorrelated or fully correlated. In each case the effective damping ratio ζe given by (5.2.13b)is reduced by about 20 %, independent of the correlation. The relative natural frequencyκ from (5.2.3b), on the other hand, is equal to unity for the uncorrelated excitation, butreduced for the correlated excitation. The shape parameter ν from (5.2.8b) is also reducedby correlation of the excitation processes. The probability densities corresponding to theparameter combinations given in Table 5.2.1 are shown in Figure 5.2.1. The solid linesrepresent the theoretical densities given by (5.2.10). Since the non-dimensional form of theprobability density only depends on the shape parameter ν, and ν 2 in all four cases,the four probability density functions are very similar. The theoretical results are comparedto results obtained by numerical simulation of 20,000 periods using the simulation methoddiscussed in Chapter 4. The probability density of the simulated records are shown by crossesand dots in Figure 5.2.1. The theoretical results are seen to agree very well with the resultsobtained by stochastic simulation.

ζ 0.0500 0.1000

ω0Sjj 0.0064 0.0120

ρ 0 1 0 1

ζe 0.040 0.040 0.081 0.081

κ 1.000 0.990 1.000 0.981

ν 2.000 1.985 2.153 2.122

Table 5.2.1: System parameters with S11 = S22.


10−1

100

101

10−3

10−2

10−1

100

101

p λ/λ 0

λ/λ0

ρ=0ρ=1

10−1

100

101

10−3

10−2

10−1

100

101

p λ/λ 0

λ/λ0

ρ=0ρ=1

Figure 5.2.1: Probability density pξ(ξ), a) ζ = 0.05, ω0S11 = ω0S22 = 0.0064, b) ζ = 0.1,ω0S11 = ω0S22 = 0.012.

0.8 1 1.20

5

10

S xω0 /

σ x2

ω/ω0

ρ=0ρ=1

0.8 1 1.20

1

2

3

4

5

S xω0 /

σ x2

ω/ω0

ρ=0ρ=1

Figure 5.2.2: Auto-spectral density Sx(ω)ω0/σ2x, a) ζ = 0.05, ω0S11 = ω0S22 = 0.0064, b)

ζ = 0.1, ω0S11 = ω0S22 = 0.012.

The spectral densities for the parameter combinations given in Table 5.2.1 are shown inFigure 5.2.2. The solid lines correspond to the theoretical expression (5.2.15a). The reductionof the natural frequency for positively correlated excitation processes (κ = 0.990 and κ =0.981) is clearly illustrated in the figure. The spectra are compared to results obtained bynumerical simulation of 400,000 periods. The spectra obtained from simulation are givenby the crosses and dots in Figure 5.2.2. The theoretical results agree very well with theresults obtained from stochastic simulation, thus confirming the theoretical prediction of thereduction of the natural frequency.

Figure 5.2.3 shows the same simulated spectra as in Figure 5.2.2, but here compared tothe theoretical predictions that would result from neglecting the parametric correction term(the Wong-Zakai term) in the definition (3.4.7) of the effective damping coefficient, leadingto ζe = ζ, irrespective of the parametric excitation. Comparison of Figures 5.2.2 and 5.2.3clearly illustrates the effect of the parametric correction term on the shape of the spectral

5.3 Oscillator with stable limit cycle behaviour 45

0.8 1 1.20

5

10

S xω0 /

σ x2

ω/ω0

ρ=0ρ=1

0.8 1 1.20

1

2

3

4

5

S xω0 /

σ x2

ω/ω0

ρ=0ρ=1

Figure 5.2.3: Auto-spectral density Sx(ω)ω0/σ2x, a) ζ = 0.05, ω0S11 = ω0S22 = 0.0064, b)

ζ = 0.1, ω0S11 = ω0S22 = 0.012.

density, and the accuracy of the definition of the effective damping coefficient ηλ by (3.4.7).

Finally, a few words concerning the stability of the system should be added. The additiveexcitation term will not influence the stability of the system. In the case where W1(t) ≡ 0an exact stability limit can be obtained following a procedure described by e.g. Lin and Cai(1995), Section 6.3. An equation governing the logarithm of the Euclidean norm of the statespace vector can be established. This equation is integrated from 0 to t. Letting t tend toinfinity, the following stability criterion is obtained,

πω0S22

ζ< 2 for S11 = 0 (5.2.17)

From (5.2.13b) it is seen, that this corresponds to requiring positive effective damping.The stability of this system has been discussed by Dimentberg (1982) for the special caseS11 = S22. The solution (5.2.15a) has also been obtained by Dimentberg and Lin (2002).In this paper the spectral density is obtained as the mean square of a measuring filter, thevalue of which can be determined from moment equations, and the procedure is thus exact.

5.3 Oscillator with stable limit cycle behaviour

If the response of an oscillator has a non-trivial solution in the case where all external andparametric excitation is removed from the system, the oscillator is said to have a stable limitcycle behaviour, see e.g. Lin and Cai (1995). Such a system is believed to give a reasonablyaccurate representation of vortex-induced vibration, Rudinger and Krenk (2002a). Theproblem of modelling vortex-induced vibrations of a structural element will be discussed inChapter 8. In the following a system with stable limit cycle behaviour is investigated. Thesystem is governed by an equation of the following form

X + ω0 (−α + βΛ + W1(t)) X + ω20X = W2(t) , α, β ≥ 0 (5.3.1)


where W1(t) and W2(t) are white noise processes and ω0 is the natural angular frequency. αand β are non-negative parameters of the model. The mechanical energy Λ is given by

Λ = 12X2 + G(X) = 1

2X2 + 1

2ω2

0X2 (5.3.2)

which means that the stiffness is linear. In the case where the excitation is removed (W1(t) ≡W2(t) ≡ 0) a stationary non-zero solution is given by harmonic motion at energy levelλ0 = α/β. This value will serve as a reference energy level in the following.

5.3.1 Probability Density

The internal force function f(X, X), introduced in the more general equation (2.3.1), of thesystem defined by (5.3.1) is identified as

f(X, X) = ω0 (−α + βΛ) X + ω20X (5.3.3)

and the coefficient functions of the white noise excitation processes are given by

b1(X, X) = −ω0X , b2(X, X) = 1 (5.3.4)

It is observed that the term containing W1(t) is a parametric excitation term while theterm W2(t) corresponds to external excitation. It is assumed that W1(t) and W2(t) areuncorrelated. The amplitude function introduced in (2.3.4) is thus expressed as

σ(X, X)2 = 2π(S11ω

20X

2 + S22

)(5.3.5)

where S11 and S22 are the intensities of the white noise processes W1(t) and W2(t). An exactstationary solution to the Fokker-Planck equation cannot be obtained in this case, and theexpression given by (2.3.8) with the stationary potential identified from (2.3.21) must beused as an approximation. Applying the van der Pol transformation the state space vector(x, x) is replaced by the energy and the phase (λ, ϕ),

xω0 =√

2λ cos (ω0t + ϕ) , x = −√

2λ sin (ω0t + ϕ) (5.3.6)

The energy conditional mean values given in (2.3.21) can be evaluated by

E[∗|λ] =

∫ 2π

0

∗ pϕ|λ(ϕ|λ)dϕ , pϕ|λ(ϕ|λ) =1

2π(5.3.7)

where the conditional probability density in (5.3.7b) shows that the energy and the phaseare uncorrelated and that the phase has a uniform distribution. This is only correct underthe assumption that the internal force function can be expressed in the form (2.3.12), whichis actually not the case for the system considered here. However, the solution we are lookingfor is not an exact solution to the system governed by (5.3.1), but an exact solution to an


equivalent system governed by (2.3.22) (in the general case). For this system (5.3.7) yieldsan exact relation. Equation (2.3.22) is thus reduced to

dψ

dλ=

1

2π

∫ 2π

0

(f(x, x)x + 1

4

∂σ(x, x)2

∂xx

∣∣∣∣λ)

dϕ

1

2π

∫ 2π

0

(12σ(x, x)2x2|λ) dϕ

(5.3.8)

With x and x given by (5.3.6), the above expression is readily evaluated as

dψ

dλ=

2(πω20S11 − ω0α) + 2ω0βλ

2πS22 + 3πω20S11λ

(5.3.9)

As in the examples considered in the previous sections a non-dimensional formulation isintroduced to simplify the expressions. A non dimensional energy variable is introduced as

ξ =λ

λ0

(5.3.10)

Equation (5.3.9) is then rewritten as

dψ

dξ=

23s1 − 1 + ξ

s2 + s1ξ, s1 =

3πω0S11

2α, s2 =

πβS22

ω0α2(5.3.11)

where s1 and s2 are non-dimensional intensities of the two processes W1(t) and W2(t). Inte-gration of (5.3.11) yields

ψ =ξ

s1

−(

s2

s21

− 2s1 − 3

3s1

)ln(s1ξ + s2) (5.3.12)

The probability density of ξ is thus expressed as

pξ(ξ) = C (s1ξ + s2)ν−1 exp

(− ξ

s1

), ν =

s2

s21

+1

s1

+ 13

(5.3.13)

where C is a normalizing constant. Equation (5.3.13) is an exact solution to a systemgoverned by the following equation

X +(ω2

0S11X2 + S22)(πω0S11 − α + βΛ)ω0X

S22 + 32ω2

0S11Λ− πω2

0S11X − ω0W1(t)X + ω20X = W2(t)

(5.3.14)

obtained from (2.3.22). The variable ξ = λ/λ0 thus has a stationary probability density givenby (5.3.13) as an exact solution, λ being the energy of the variable governed by (5.3.14).


5.3.2 Spectral Density

The effective damping of the system is evaluated by (3.4.7). As in the evaluation of thederivative of the stationary potential in (5.3.8), the mean values in (3.4.7) are evaluated byconsidering free undamped response at constant energy level λ,

ηλ =

∫ 2π

0

(f(x, x)x − 1

4

∂σ(x, x)2

∂xx

∣∣∣∣ λ

)dϕ = ω0α

(λ

λ0

− 1 − 23s1

)(5.3.15)

where the last term originates from the Wong-Zakai correction of the internal force function.The effective damping coefficient is seen to depend on the energy only through the non-dimensional energy variable ξ = λ/λ0. The spectral density at a given energy level isobtained from (3.3.18) as

Sx(r|ξ) =2λ0α

πω30

| ξ − 1 − 23s1 | ξ

(1 − r2)2 + (ξ − 1 − 23s1)2α2r2

, r =ω

ω0

(5.3.16)

where r is the frequency relative to the natural frequency. Sx is here the one-sided spectraldensity. The total spectral density is finally obtained from (3.3.7). In a non-dimensionalformulation the spectral density can be expressed as

ω30Sx(r)

λ0

=4

π

∫ ∞

0

| ζ(ξ) |(1 − r2)2 + 4ζ(ξ)2r2

ξpξ(ξ) dξ , 2ζ(ξ)ω0 = ηλ (5.3.17)

where ζ(ξ) is an equivalent damping ratio, which is a function of the energy level. Theintegral in (5.3.17) is evaluated numerically in the following examples.

5.3.3 Numerical Examples

In order to investigate the quality of the approximate expressions for the probability densityof the energy and the spectral density of the displacement, records of the stochastic responseare simulated, analyzed and the results compared with the theoretical expressions. Thesimulation method discussed in Chapter 4 is used to simulate records of a length of 10,000natural periods.

In Figures 5.3.1 - 5.3.3 theoretical results for the probability density are compared with scaledhistograms obtained from the simulated records for various parameter combinations. Twosystems are simulated. The original system represented by (5.3.1), to which the analyticalsolution is an approximation corresponds to the dots, while the crosses correspond to thesystem given by (5.3.14), to which the theoretical solution is exact.

It is observed that the parameter α has no effect on the probability density. As s1 ands2 tend to zero the distribution tends to a delta function at ξ = 1, which means that theoscillations will take place at the equilibrium energy level λ0. As s1 and s2 increase the


10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξ

10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξFigure 5.3.1: Probability density of non-dimensional energy, s1 = 0.1, s2 = 0, a) α = 0.02,b) α = 0.05.

10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξ

10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξ

Figure 5.3.2: Probability density of non-dimensional energy, s1 = 0.1, s2 = 0.1, a) α = 0.02,b) α = 0.05.

equilibrium energy λ0 becomes less dominant. As seen by the figures, the theoretical andnumerical results show excellent agreement.

The spectral density is also evaluated for the parameter combinations corresponding to thoseused in Figures 5.3.1 - 5.3.3 for the probability density. The results are shown in Figures5.3.4 - 5.3.6. In Figure 5.3.7 an additional parameter combination is considered.

For small values of s1 and s2 (Figures 5.3.4 and 5.3.5) the asymptotic behaviour (r → 0 andr → ∞) of the theoretical spectra (solid lines) deviate substantially from the results basedon Fast Fourier Transform of the simulated records (dots). The peak on the other hand ispredicted quite well. For large values of s1 and s2 (Figures 5.3.6 and 5.3.7) the agreementbetween theoretical and numerical results is satisfactory for the entire frequency range. Itis observed, that the parameter α, which did not influence the probability density functionsgiven in Figures 5.3.1 - 5.3.3, is a measure of the damping. Increasing values of α tend to


10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξ

10−1

100

101

10−3

10−2

10−1

100

101

ξ

p ξFigure 5.3.3: Probability density of non-dimensional energy, s1 = 0.1, s2 = 1, a) α = 0.02,b) α = 0.05.

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.4: Spectral density of displacement, s1 = 0.1, s2 = 0, a) α = 0.02, b) α = 0.05.

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.5: Spectral density of displacement, s1 = 0.1, s2 = 0.1, a) α = 0.02, b) α = 0.05.


10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.6: Spectral density of displacement, s1 = 0.1, s2 = 1, a) α = 0.02, b) α = 0.05.

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.7: Spectral density of displacement, s1 = 1, s2 = 1, a) α = 0.02, b) α = 0.05.

broaden the peak.

5.3.4 Alternative procedure

The technique used to obtain an approximate expression for the spectral density of theresponse has been demonstrated successfully by Krenk and Roberts (1999) and Krenk (1999).However, these investigations only consider cases where the damping is positive for all energylevels, and the question whether ηλ or the absolute value |ηλ| should appear in the expressionfor the spectral density at a given energy level does therefore not arise.

The expression for the spectral density given by (5.3.17) is consistent in the sense that∫∞0

Sx(ω)dω = σ2x, where σx is the standard deviation of the response. As seen by Figures

5.3.4 and 5.3.5, the estimate of the spectral density is less accurate for narrow probabilitydistributions of the energy, which is due to the fact that the energy levels where the damping


10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.8: Spectral density of displacement, s1 = 0.1, s2 = 0.1, a) α = 0.02, b) α = 0.05,using equation (5.3.18).

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

10−1

100

101

10−4

10−2

100

102

r

S x ω03 /

λ 0

Figure 5.3.9: Spectral density of displacement, s1 = 0.1, s2 = 0.1, a) α = 0.02, b) α = 0.05,using equation (5.3.20).

is negative constitute a relatively large part of the total probability mass in these cases, seeFigures 5.3.1 and 5.3.2. If on the other hand the energy is mainly distributed in the rangewhere the damping is positive, as in the case shown in Figure 5.3.3, the estimated spectrumwill in general show good agreement with the numerical results, see e.g. Figure 5.3.6.

If the absolute value sign in the expression (3.3.18) for the spectral density at a given energylevel is removed, the spectrum will become negative if the damping is negative for thatparticular energy. For a system with linear stiffness the energy conditional spectrum wouldin that case be given by

Sx(ω|λ) ηλλ

π

[2

(ω20 − ω2)2 + η2

λω2

](5.3.18)

Sx(ω|λ) is here the one-sided spectral density and has therefore been multiplied with a factorof 2 compared to (3.3.18). The above equation is now used to evaluate the spectrum, and

5.4 Oscillator with Bilinear Stiffness 53

the parameter combinations corresponding to the results shown in Figure 5.3.5 are nowreconsidered. With the new definition of the spectrum the results shown in Figure 5.3.8 areobtained.

As seen by the figure, the asymptotic behaviour is now accurately predicted by the theory,while the agreement is poor at the resonance peak. When this approach is used the areaunder the theoretical function (solid line) is smaller than the variance of the displacement.

Alternatively, one can take advantage of the fact, that the first approach yields a goodestimate in the resonance range, while the second approach gives accurate estimates ofthe asymptotic behaviour. For a linear system the phase difference ϕ between harmonicexcitation and harmonic response is given by

tan(ϕ) =2r

1 − r2ζ(ξ) (5.3.19)

If the system is linear the damping ratio ζ will not be a function of the energy. However, inthe present case the damping is a function of the energy level due to the non-linearity in thedamping function, and (5.3.19) will be used as an approximation to the phase difference ata given energy level. The phase difference is ϕ = 0 rad for r = 0, ϕ = 1

2π rad for r = 1 and

ϕ = π rad for r → ∞. The two expressions (5.3.16) and (5.3.18) can therefore be weightedthe following way,

Sx(ω|λ) cos2(ϕ)ηλλ

π

[2

(ω20 − ω2)2 + η2

λω2

]

+ sin2(ϕ)|ηλ|λ

π

[2

(ω20 − ω2)2 + η2

λω2

] (5.3.20)

where ϕ is given by (5.3.19). Using this approach the results shown in Figure 5.3.9 areobtained. As seen by the figure a very accurate estimate is obtained. It should however bementioned, that the area under the theoretical function is smaller than the variance of thedisplacement.

The spectral density of an oscillator with negative damping at certain energy levels (van derPol oscillator) has also been considered by Spanos et al. (2002). In this case the excitationwas additive but non-white, and an approximate solution was obtained by writing both inputand output in the form of series expansions, and relating the input and output coefficientsby solving a non-linear set of equations.

5.4 Oscillator with Bilinear Stiffness

A system with bilinear stiffness is considered. Such a system may occur in many practicalsituations, e.g. if linear springs are arranged in such a way, that some of them will be activated


g(x)

xa

a

1

1

ω12

ω22

Figure 5.4.1: Bilinear stiffness function.

only when a certain level of displacement is reached. An example could be a ship moored to adock wall or a piece of machinery vibrating within a clearance defined by elastic stops. Thisparticular problem has consequently been thoroughly investigated. The spectral density ofthe response of an oscillator of this type to stochastic excitation has been investigated bye.g. Miles (1993), Bellizzi and Bouc (1996), Fogli et al. (1996) and Fogli and Bressolette(1997).

In Figure 5.4.1 the bilinear stiffness function is shown. Mathematically the function isdescribed the following way

g(x) =

ω2

1x , |x| ≤ a

ω22x + a(ω2

1 − ω22)sign(x) , |x| > a

(5.4.1)

It is seen that g(−x) = −g(x). The initial stiffness is given by ω21. When the absolute value

of the displacement exceeds a, the stiffness changes to ω22. It is assumed that both ω1, ω2

and a are larger than zero. For ω1 = ω2, a → 0 or a → ∞ the linear system is retrieved.The potential energy is obtained by integration of (5.4.1) as

G(x) =

12ω2

1x2 , |x| ≤ a

12ω2

2x2 + a(ω2

1 − ω22)|x| + 1

2a2(ω2

2 − ω21) , |x| > a

(5.4.2)

The potential energy is thus described by two parabolas with continuity of the zeroth andfirst derivative of the intersection. Since g(x) is an odd function, G(x) is even.



The minimum and maximum displacement for free undamped vibration at energy level λ isgiven by G(xmin) = G(xmax) = λ. In the present case the potential energy is symmetric andconsequently xmin = −xmax. From (5.4.2) xmax is obtained as

xmax

a=

√2λ

ω1a, λ ≤ 1

2ω2

1a2

(1 − Ω2) + Ω

√Ω2 − 1 +

2λ

a2ω21

, λ > 12ω2

1a2

(5.4.3)

where Ω = ω1/ω2 is the frequency ratio. It is observed that√

Ω2 − 1 + 2λ/a2ω21 > Ω from

where it follows that xmax/a > 1 for λ > 12ω2

1a2. Due to symmetry in the potential energy,

the natural period (2.4.11) is reduced to

T (λ) =4√2

∫ xmax

0

dx√λ − G(x)

=

2π

ω1

, λ ≤ 12ω2

1a2

4

ω1

sin−1

(ω1a√2λ

)+

4

ω2

cos−1

(Ω√

Ω2 − 1 + 2λ/a2ω21

), λ > 1

2ω2

1a2

(5.4.4)

It is now assumed that the damping of the system is linear, and that the excitation isexternal. The equation of motion can in this case be expressed as

X + γX + g(X) = W0(t) (5.4.5)

γ is the damping coefficient and W0(t) is a white noise with intensity S0. The probabilitydensity of the energy of the system (5.4.5) follows from (2.2.10) and (2.4.9) as

pλ(λ) = C exp

(− γλ

πS0

)

×

2π

ω1

, λ ≤ 12ω2

1a2

4

ω1

sin−1

(ω1a√2λ

)+

4

ω2

cos−1

(Ω√

Ω2 − 1 + 2λ/a2ω21

), λ > 1

2ω2

1a2

(5.4.6)

where C is a normalizing constant. The probability density is conveniently investigatedconsidering a non-dimensional formulation. Non-dimensional variables are introduced as

ξ =λ

λ0

, λ0 = 12ω2

1a2 , β =

γλ0

πS0

=γω2

1a2

2πS0

(5.4.7)


λ0 is thus a reference energy level corresponding to the transition from the region with thestiffness ω2

1 to the region with the stiffness ω22. For free undamped vibration the condition

ξ ≤ 1 yields harmonic vibration. β is a non-dimensional measure of the reference energylevel to the input intensity. The natural period is now expressed as

T (ξ) =ω1T (λ)

2π=

1 , ξ ≤ 1

2

πsin−1

(1√ξ

)+

2Ω

πcos−1

(Ω√

Ω2 − 1 + ξ

), ξ > 1

(5.4.8)

The probability density of the non-dimensional energy variable ξ is given by

pξ(ξ) = λ0 pλ(λ/λ0)

= C exp(−βξ)

1 , ξ ≤ 1

2

πsin−1

(1√ξ

)+

2Ω

πcos−1

(Ω√

Ω2 − 1 + ξ

), ξ > 1

(5.4.9)

It is thus seen that the probability distribution of ξ only depends on the parameter β andon the frequency ratio Ω.

0 5 100

0.1

0.2

0.3

0.4

pξ

ξ0 5 10

0

0.1

0.2

pξ

ξ

Figure 5.4.2: Probability density of non-dimensional energy, a) Ω = 0.2, β = 0.2, b) Ω = 5,β = 0.2.

In order to verify the validity of the result comparison is made with records obtained bystochastic simulation. In Figure 5.4.2a the probability density of ξ is given for Ω = 0.2 andβ = 0.2. In Figure 5.4.2b the parameters are Ω = 5 and β = 0.2. The dots correspond tohistogram points obtained by processing a simulated record with a length of approximately50,000 mean periods. The results are seen to agree very well.


5.4.2 Free undamped vibration

Free undamped vibration at energy level λ is now considered. Only the range 0 ≤ t ≤ 14T (λ)

needs to be investigated due to the vertical and horizontal symmetry. For λ ≤ 12ω2

1a2 the

system is linear and the phase plane variables (x, x) are simply given by

x(t) = A1 sin(ω1t)

x(t) = ω1A1 cos(ω1t), A1 =

√2λ

ω1

(5.4.10)

where A1 is the amplitude. For λ < 12ω2

1a2 the solution is still valid until the time t1, where

x(t) crosses into the range where the stiffness changes from ω21 to ω2

2,

x(t) = A1 sin(ω1t)

x(t) = ω1A1 cos(ω1t), 0 ≤ t ≤ t1 =

1

ω1

sin−1

(ω1a√2λ

)(5.4.11)

For t > t1 the solution in terms of x(t) can be obtained by solving the following lineardifferential equation

x + ω22x = a(ω2

2 − ω21) (5.4.12)

The equation is non-homogeneous due to the non-zero equilibrium condition in the casewhere ω1 = ω2. The initial conditions are given by (5.4.11) at t = t1 since both x(t) andx(t) are continuous. The solution is given by

x(t) = a(1 − Ω2) + A2 cos(ω2(14T (λ) − t))

x(t) = ω2A2 sin(ω2(14T (λ) − t))

, t1 < t ≤ 14T (λ) (5.4.13)

where only one arbitrary constant A2 appears, since the solution must have a form wherethe velocity is zero at t = 1

4T (λ). The constant A2 is the amplitude of the harmonic part of

the motion. Continuity in x(t) at t = t1 yields

aΩ2 = A2 cos(ω2t2)

t2 = 14T (λ) − t1 =

1

ω2

cos−1

(Ω√

Ω2 − 1 + 2λ/a2ω21

)(5.4.14)

where t2 is the remaining part of the quarter period after the crossing of the level x = a. t2is determined from (5.4.4). Evaluation of A2 from (5.4.14a) gives the following value,

A2 = Ω√

a2Ω2 − a2 + A21 (5.4.15)

The same result is obtained if continuity in x(t) at time t = t1 is used to determine thisconstant. The modified phase plane variables are determined as

z1(t) =√

2λ − z2(t)2 =√

2λ − ω22A

22 sin2(ω2(

14T (λ) − t))

z2(t) = ω2A2 sin(ω2(14T (λ) − t))

(5.4.16)


Again, the solution is most easily investigated by introducing a non-dimensional formulation.The time t and z1(t) and z2(t) are made non-dimensional by the following rescaling,

t = ω1t , t1 = ω1t1 = sin−1

(1√ξ

), zj(t) =

zj(t)√2λ

(5.4.17)

where the non-dimensional energy variable ξ is introduced in (5.4.7a). z1(t) and z2(t) areexpressed as

z1(t) =

sin(t) , ξ ≤ 1 ∨ 0 ≤ t ≤ t1√√√√1 − Ω2 − 1 + ξ

ξsin2

(12πT (ξ) − t

Ω

), otherwise

z2(t) =

cos(t) , ξ ≤ 1 ∨ 0 ≤ t ≤ t1√Ω2 − 1 + ξ

ξsin

(12πT (ξ) − t

Ω

), otherwise

(5.4.18)

Finally the modified phase plane variables are expanded in a Fourier series. The coefficientsare obtained by (3.4.2). These equations can be given in the following non-dimensional form,

sj(ξ) =4

π T (ξ)

∫ 12

πT (ξ)

0

z1(t) sin

(jt

T (ξ)

)dt

cj(ξ) =4

π T (ξ)

∫ 12

πT (ξ)

0

z2(t) cos

(jt

T (ξ)

)dt

(5.4.19)

where the non-dimensional period T (ξ) is defined in (5.4.8). Due to symmetry the integrationcan be reduced to the first quarter of the period and the result multiplied by 4. The aboveformulation shows that the coefficients sj(ξ) and cj(ξ) only depend on ξ, Ω and j. In thefollowing the coefficients will be obtained by numerical integration.

The Fourier series expansion is now compared to the exact solution in order to estimatethe number of terms necessary for an accurate prediction of the response. In Figure 5.4.3the non-dimensional phase plane variables z1/

√2λ and z2/

√2λ are shown as function of the

non-dimensional time 4t/T (λ). Only one quarter of the period is considered. The solid lineis the analytical solution. The first modified phase plane variable z1/

√2λ starts at 0 while

the second z2/√

2λ starts at 1. In Figure 5.4.3a the parameters are ξ = 10 and Ω = 10. Thesolution for an expansion with one term is shown by the dashed line and the solution foran expansion with three terms (j = 1, 3, 5) is shown by the dashed/dotted line. Inclusionof only one term in the series expansion gives a rather inaccurate representation, whereasinclusion of three terms leads to an accurate representation of the function. In Figure 5.4.3the functions are shown for the parameters ξ = 10 and Ω = 0.2 and expansions with one termand three terms have been included as well. The accuracy of the truncated series expansionsis of the same order of magnitude as in the previous case.


0 0.5 10

0.5

1

1.5

z i / (2

λ)1/

2

4t / T(λ)0 0.5 1

0

0.5

1

1.5

z i / (2

λ)1/

2

4t / T(λ)

Figure 5.4.3: Modified state space variables, (–) exact, (- -) 1 term, (- · -) 3 terms. a) ξ = 10,Ω = 10, b) ξ = 10, Ω = 0.2.

5.4.3 Spectral density

The spectral density is obtained from the expression (3.4.17) and (3.3.7). In the present casethe spectral density is most conveniently expressed in the following non-dimensional form,

Sx(r)ω31

λ0

=γ

π

∫ ∞

0

c(ξ) ξ pξ(ξ)∞∑

j=1,3,...

2cj(ξ)2

(j2T (ξ)−2 − r2)2 + γ2c(ξ)2r2

dξ

γ =γ

ω1

, r =ω

ω1

(5.4.20)

In the following examples the integration in this equation is performed numerically, so theequation takes the form of a double summation. From the above expression and fromthe earlier expressions for pξ(ξ), cj(ξ) and T (ξ) it is seen that the spectral density in thisnon-dimensional form is a function of Ω, β and γ only. It is difficult to give a preciseinterpretation of the influence on the solution of these parameters, but a few primarilyqualitative observations can be made. If Ω = 1 the system is linear. γ is in this case twicethe damping ratio. The parameter β will not influence the shape of the spectrum but beinversely proportional to the mean energy of the response, which is proportional to the areaunder the spectrum. For Ω < 1 the system is characterized by hardening stiffness and forΩ > 1 by softening stiffness. In both cases γ will be a measure of the damping. Apart fromgoverning the magnitude of the response, β will also influence the non-linearity. For verylow or very high values of β the response is governed by the stiffness ω2

1 or ω22, respectively.

This will have the effect of reducing the non-linear characteristics of the response.

In Figures 5.4.4 - 5.4.6 a number of numerical examples are shown for different combinationsof the parameters Ω, β and γ. The dots indicate solutions obtained by stochastic simulation.In order to demonstrate the effect of the truncation of the series in (5.4.20) both the solutionwith one term and the solution with three terms (j = 1, 3, 5) are included, indicated by


0 1 2 3 4 510

−4

10−3

10−2

10−1

100

101

S x ω13 /

λ 0

r0 2 4 6 8

10−5

10−4

10−3

10−2

10−1

100

101

S x ω13 /

λ 0

r

Figure 5.4.4: Spectral density, a) Ω = 0.6, β = 1, b) Ω = 0.2, β = 1, γ = 0.05. (•) Stochasticsimulation, (- -) Theory (1 term), (–) Theory (3 terms).

0 1 2 3 410

−4

10−3

10−2

10−1

100

101

102

S x ω13 /

λ 0

r0 1 2 3 4

10−4

10−3

10−2

10−1

100

101

102

S x ω13 /

λ 0

r

Figure 5.4.5: Spectral density, a) Ω = 2, β = 0.5, b) Ω = 5, β = 0.5, γ = 0.05. (•) Stochasticsimulation, (- -) Theory (1 term), (–) Theory (3 terms).

a dashed line and a solid line, respectively. In Figure 5.4.6a the solution with two terms(j = 1, 3) is also included by the dashed/dotted line.

In Figure 5.4.4a the values are chosen as Ω = 0.6, β = 1 and γ = 0.05. The degree of non-linearity in this case is relatively small, as seen by the magnitude of the peak of the higherharmonic. It is clearly seen that inclusion of only one term in the solution leads to failureto capture the second peak, and that the additional terms have no influence (are negligiblysmall) near the fundamental frequency. The analytical results are seen to agree very well withthe results obtained by stochastic simulation, except for the fact that the theory predicts alocal peak at ω = ω1. This corresponds to oscillations taking place at small energy levels,where the response does not enter into the region of changed stiffness ω2

2. However, thislocal peak is not as pronounced in the simulated response. This observation was also madeby Fogli et al. (1996). A more detailed analysis shows, that this is not a result of the Fast


0 2 4 6 8 1010

−6

10−4

10−2

100

102

S x ω13 /

λ 0

r0.5 1 1.5

10−4

10−2

100

102

S x ω13 /

λ 0

r

Figure 5.4.6: Spectral density, Ω = 0.3, β = 1, γ = 0.01. (•) Stochastic simulation, (- -)Theory (1 term), (- · -) Theory (2 terms), (–) Theory (3 terms).

Fourier Transform procedure carried out numerically for the simulated record. Changing thenumber of FFT-points, the windowing, the magnitude of the time increment or increasingthe length of the record does not alter this trend.

In Figure 5.4.4b the level of non-linearity is increased by decreasing the frequency ratioto Ω = 0.3, and letting the other parameters remain unchanged. The bandwidth of thefundamental frequency is now larger (due to the frequency sweep), and both the magnitudeand bandwidth of the higher harmonics are increased. Again, the pronounced local peakpredicted by the theoretical result does not show in the spectrum of the simulated record.Furthermore, a small deviation is seen at zero frequency, a feature which is also observablein results given by Fogli et al. (1996), Krenk and Roberts (1999) and Krenk (1999). Adiscrepancy reported by Krenk and Roberts (1999) at high frequencies, seems to be due totruncation after the first two terms.

In the above example the system is characterized by hardening stiffness, i.e. ω2 > ω1. InFigure 5.4.5 systems with softening stiffness are considered. In Figure 5.4.5a the values arechosen as Ω = 2, β = 0.5 and γ = 0.05. As opposed to the cases shown in Figure 5.4.4,the peak is now located below ω = ω1 due to the fact that ω2 < ω1. The non-linearity isrelatively small, as seen by the magnitude of the higher harmonic. Again, a small deviationis seen at zero frequency.

In Figure 5.4.5b the non-linearity is increased by increasing the value of the frequency ratioto Ω = 5, while the other parameters remain unchanged. This is seen to broaden both thefundamental frequency peak and the peak of the first higher harmonic. The discrepancy atzero frequency is larger in this case compared to the case in Figure 5.4.5a, and the accuracya bit lower, though still satisfactory. The local peak at ω = ω1 observed in Figures 5.4.4aand 5.4.4b in the theoretical solution is less pronounced for the results shown in Figures5.4.5a and 5.4.5b.

Finally the parameter combination Ω = 0.3, β = 1 and γ = 0.01 is investigated. This rela-


tively strong non-linearity in combination with decreasing value of the damping parameter γresults in a spectrum, where it is possible to distinguish both the second and third harmonic.In Figure 5.4.6a both the solutions with two terms and three terms have been included, andthe effect of including the third term on the third harmonic is evident. In Figure 5.4.6b therange around the first peak is investigated. A larger number of FFT-points have been used.The deviation between the theoretical local peak at ω = ω1 and the results obtained fromthe simulated record is clearly observed.

Chapter 6

System Identification from Non-linearStochastic Response

In the previous section the systems investigated were assumed to be known, and the problemconsisted in determining the probabilistic characteristics of the response given a probabilisticdescription of the excitation process. In the present section the reverse procedure, namelythat of system identification, will be investigated. A large part of the theory introduced inChapters 2 and 3 is applied in this section as well, for the system identification proceduresdiscussed in the following. Some of the methods developed in this chapter have been pre-sented in earlier publications, see Krenk and Rudinger (2000); Rudinger and Krenk (2000b,2001, 2003a).

6.1 Background

The response of vibration sensitive structures to random excitation depends on stiffnessand damping properties of the structure along with the characteristics of the excitation.For a structure behaving according to linear theory, the stiffness is readily obtained fromknowledge of geometry and material properties. If the structure is non-linear the stiffnessmay not be easily evaluated. The damping will normally not be available from theory evenin the linear case, and may depend on the type of loading. In the case of field measurementsor wind-tunnel testing of structural elements, the excitation may not be measurable orreliably estimated. It is therefore of interest to be able to estimate the stiffness, dampingand excitation of the structure from records of the stochastic response.

6.1.1 Stiffness estimation in the time domain

Vibration sensitive structures are characterized by a low level of damping. This will re-sult in narrow band response to broad band excitation, and implies that the response can

63

64 Chapter 6. System Identification from Non-linear Stochastic Response

be regarded as a slow change of free undamped oscillations, see e.g. Krenk and Roberts(1999). The slow variation of the energy level of the response formed the basis of a stiffnessestimation procedure introduced by Roberts et al. (1992). In this work the elastic energyis estimated at the extremes of the displacement by interpolation of the kinetic energy atmean-level crossings. In a later paper, Roberts et al. (1994), this approach is replaced bya method where the elastic potential is obtained from the natural period estimated by thetime interval between two mean-level crossings. In both cases a number of correspondingvalues of displacement and potential energy are found, and the stiffness parameters are ob-tained by a least squares fit to these points. For non-trivial damping levels the data exhibita considerable statistical scatter, and thus an essential aspect of the grouping and averagingof data is to reduce scatter. In general, the conditional distribution of energy at the nextextreme or length of the next half period is not known. It is therefore important to avoid theintroduction of bias in the formation of mean values. An iterative scheme for grouping dataand forming averages while gradually reducing the systematic estimation error due to changein energy level was proposed by Krenk and Rudinger (2000). The result of the approach is anon-parametric estimate of the elastic potential as function of displacement represented bya number of points. This procedure is discussed in Section 6.2. The method was originallyinvestigated for systems with external excitation, but has been shown to work equally wellfor systems with parametric excitation, Rudinger and Krenk (2003a).

6.1.2 Estimation of damping relative to excitation intensity

The stiffness estimation procedures discussed above are based on local considerations relatingthe states of the system at various times. A global procedure for estimation of dampingrelative to input intensity was proposed by Roberts et al. (1992, 1994). The method isbased on the generic solution obtained by Caughey (1971) for the probability density of theenergy, and the accessible information is the equivalent damping relative to the excitationintensity, i.e. the average damping at a given energy level obtained from the original systemby equivalent non-linearization, Caughey (1986). A drawback of the method is that itrequires knowledge of the elastic potential and thus relies on the quality of the stiffnessestimation. A method was proposed by Krenk and Rudinger (2000), where the energy issampled at mean energy levels only. This has the advantage that no knowledge of the stiffnessis required, since the energy is purely kinetic at mean-level crossings. Furthermore, the biasintroduced in (2.4.9) in the form of the natural period when performing the transformationfrom state space variables to energy is removed by this way of sampling the energy. Theresult of this procedure is a non-parametric estimate of the equivalent damping as a functionof the energy relative to the excitation intensity.

Since the above procedure relies on the generic expression for the probability density of theenergy it can be used only for oscillators with external broad band excitation, which can beapproximated by white noise. However, by considering the special class of exact solutionsdiscussed in Section 2.3.3, the method can be extended to include this particular class ofsystems with parametric excitation, Rudinger and Krenk (2003a). In this case the result

6.1 Background 65

of the estimation procedure is the stationary potential, which contains both the damping,the excitation and the derivative of the excitation intensity as a function of the energy. Thedamping and excitation can however still be separated if an independent estimate of thedamping can be obtained.

6.1.3 Estimation of absolute damping in the time domain

In the previous subsections it was discussed how the damping could be estimated relative tothe excitation intensity for oscillators with external excitation, and how the method wouldresult in an estimate of a particular combination of the damping and excitation in the caseof an oscillator with parametric excitation. The estimation procedure is therefore completedif an estimate of the absolute damping is obtained. The estimation of the damping generallyleads to more inaccurate results than estimation of stiffness and ratio between damping andexcitation intensity.

A number of different methods for estimating the absolute damping of the system were sug-gested by Roberts et al. (1992, 1994). Two of these are based on estimates of the covariancefunction for the energy and the displacement, respectively, and they are thus local procedurescarried out in the time domain. A method was proposed by Rudinger and Krenk (2001) inwhich the covariance functions of the modified phase plane variables introduced in Section2.4.1 are estimated. From these functions it is possible to extract a non-parametric estimateof the effective damping. This method will be discussed in Sections 6.3.2 and 6.4.1. Themethod can also be used for systems with parametric excitation.

6.1.4 Frequency domain and FPK equation

A possible alternative to the time domain methods was proposed by Roberts et al. (1994),where a frequency domain representation is used. This method consists in determiningthe excitation intensity from a least squares fit to the spectral density of the displacementcorresponding to the linearized equation. In combination with some of the methods discussedin the previous subsections this yields a complete characterization of the system. The ideaof operating in the frequency domain has been further pursued by Roberts et al. (1995),where the system identification is based on a least squares fit to the spectral densities of thevarious terms in the equation of motion. However, the spectrum is not a good discriminatorof non-linear damping, and the probability density function is therefore used to separatethe linear and non-linear damping terms. In a later paper, Vasta and Roberts (1998), theentire system identification is carried out in the frequency domain. Apart from consideringthe second order spectrum (power spectral density) the method also makes use of the fourthorder spectrum. Methods based on higher order spectra have also been applied to systemswith non-Gaussian excitation, Roberts and Vasta (2000d), where the characteristics of theexcitation are identified as well.


Stochastic averaging procedures can be used to reduce the dimension of the Fokker-Planckequation by a factor of two. A procedure of this type was used by Roberts and Vasta(2000c) to obtain the one-dimensional Fokker-Planck equation for the energy of a stochasticoscillator. The drift and diffusion coefficients are estimated from the stochastic response andby comparing these to the theoretical coefficients appearing in the Fokker-Planck equationthe parameters of the non-linear system are identified. This method also enables estimationof the spectrum of the excitation, which may be non-white. In a later paper, Roberts andVasta (2000a), it was shown that more accurate estimates of the stiffness and damping canbe obtained if stochastic records for two different levels of excitation intensity are available.The method has been extended to include systems with parametric excitation, Roberts andVasta (2000b, 2002).

6.1.5 Non-parametric identification methods

In most of the methods discussed in the previous subsections, Roberts et al. (1992, 1994,1995), Vasta and Roberts (1998), Roberts and Vasta (2000a,b,c,d, 2002), the system identi-fication is parametric, which means that the non-linear system is represented by a numberof parameters, and that the identification consists in determining these parameters. If it isassumed that the stiffness is linear-cubic, the damping is linear-quadratic and the excita-tion is a white noise, this type of identification will yield five parameters: the two stiffnesscoefficients, the two damping coefficients and the intensity of the white noise excitation.However, the results of such procedures may give unreliable estimates if the properties can-not be modelled by the assumed expressions, e.g. if the stiffness of the system cannot beaccurately approximated by the linear-cubic model. A non-parametric approach has beenproposed by Iourtchenko and Dimentberg (2002). However, this method only addressessystems with linear stiffness.

The methods discussed in the following are mostly non-parametric in the sense, that noparametric models of the terms in the equation of motion are required. In the case ofstiffness and damping estimation the result of the identification procedure will simply be thetwo functions represented by a number of points. As to the excitation intensity, which is thelast step in the procedure, a parametric representation is required. In the examples presenteda polynomial expansion will be applied. In the case of a system with external white noiseexcitation the intensity is just a number and the entire procedure can be considered as anon-parametric approach.

6.2 Stiffness Estimation

The stiffness estimation procedure discussed in this section was originally introduced byKrenk and Rudinger (2000) for oscillators with external excitation, but has also been shownto give accurate results for systems with parametric excitation, Rudinger and Krenk (2003a).

6.2 Stiffness Estimation 67

x

t

vi

ai

vi+1

ai+1

Figure 6.2.1: Sample of stochastic response.

6.2.1 Development of mechanical energy

Consider a sample of the stochastic response, as shown in Figure 6.2.1. The velocity atthe i’th mean-level-crossing and the displacement at the subsequent extreme are denoted vi

and ai, respectively. At mean-level-crossings the energy is purely kinetic and at displace-ment extremes the energy is purely elastic. So the pair (vi, ai) corresponds to the energies(1

2v2

i , G(ai)). If the level of damping is small, the energy process is slowly varying, and 12v2

i

will approximately equal G(ai). It is therefore assumed that 12v2

i G(ai) in some averagedsense, and the important thing is to perform the averaging without introducing bias.

In Figure 6.2.2a the pairs (12v2

i , a2i ) obtained from simulation have been plotted as the dots.

The system considered is an oscillator with linear damping and linear-cubic stiffness. Since12v2

i G(ai) the points are centered around the solid line, which is the theoretical expressionG(x). x2 has been chosen as the independent variable, because in the linear case G(x) =12ω2

0x2 appears as a straight line, when the independent variable is defined this way. Thus

(Displacement)2

Ene

rgy

(Displacement)2

Ene

rgy

Figure 6.2.2: (12v2

i , a2i ) and G(x), a) strongly damped system, b) lightly damped system.


the straight dashed line shown in Figure 6.2.2a is the linear case corresponding to removalof the cubic term from g(x). By plotting G(x) versus x2 the level of non-linearity in thestiffness is easy to observe.

The points in Figure 6.2.2a show a considerable level of scatter around the theoretical line.In this figure a system with relatively strong damping has been considered (the dampingratio is around 5 % at mean energy level). In Figure 6.2.2b the damping has been reducedby a factor of 10. This reduces the scatter as observed from the two figures. As the dampingtends to zero all points in the diagram would tend to the solid line corresponding to aninfinitely slow change of energy level. The deviation between the points and the line is thusa measure of the energy drift over half a period.

From the results shown in Figure 6.2.2 it seems plausible to assume that the solid line, andthus an estimate of G(x), can be extracted from the cloud of points, if some averaging isperformed. In the following two subsections two different ways of performing these averagesare discussed.

6.2.2 Standard regression

The standard approach would be a regression analysis, in which the data are grouped withrespect to intervals along one of the axes. If the variable x is considered as the independentvariable the regression would be performed as indicated in Figure 6.2.3. The x-axis has herebeen divided into a number of intervals as shown in Figure 6.2.3a. If averaging is performedwithin each of the groups defined by the vertical lines the estimate shown in Figure 6.2.3bis obtained. It should be pointed out, that this estimate is based on around 34,000 datapoints. Only 1000 of these are shown in Figure 6.2.3a in order to avoid a too dense cloud.The function G(x) is shown by the solid line.

A bias is clearly introduced by this estimation approach as seen by Figure 6.2.3b. For highenergy levels the estimate of G(x) is too small and for low energy levels the estimate of

x2

G(x

)

x2

G(x

)

Figure 6.2.3: Estimate of G(x), regression along vertical axis.

6.2 Stiffness Estimation 69

x2

G(x

)

x2

G(x

)

Figure 6.2.4: Estimate of G(x), regression along horizontal axis.

G(x) is too high. Performing the averaging along the y-axis corresponds to dividing thedata-points into groups of equal energy at the displacement extremes ai (see Figure 6.2.1).Due to the energy drift the expected value of the energy at the previous mean-level crossingis closer to the mean value of the energy. If the energy is higher than the mean energy level,the value of G(x) will therefore be underestimated, and for energies below the mean energylevel, G(x) will be overestimated, when the regression is performed this way. The pointwhere the estimated values cross the theoretical function corresponds to the mean energylevel.

Alternatively, one could divide the y-axis into a number of intervals as shown in Figure 6.2.4.The estimate displays the opposite behaviour as that observed in Figure 6.2.3. For highenergy levels, G(x) is overestimated, and for low levels of energy, G(x) is underestimated,though the bias is not as pronounced in this region. The reason for the systematic error isthe same as in the previous case. The points are divided into groups of equal energies 1

2v2

i

at mean-level crossings and at the subsequent extreme the energy will on average be closerto the mean energy level. Again, the estimated values cross the theoretical elastic potentialfunction around the mean energy level.

6.2.3 Iterative regression

The two types of bias shown in the previous subsection were introduced because the groupingof data was performed either with respect to the value of the energy at the extreme or withrespect to the value of the energy at the mean-level crossing. It therefore seems reasonableto assume that the estimate will be unbiased if the grouping and averaging of the data pointsis performed with respect to the mean value of the energy at the mean-level crossing andsubsequent extreme. However, this approach cannot be readily applied since the potentialenergy at the extremes is originally unknown - the aim of the procedure is in fact to identifythis function. This leads to an iterative procedure, which will be discussed in the following.


x2

G(x

)

x2

G(x

)

Figure 6.2.5: Estimate of G(x), regression along lines of constant energy, 1st iteration.

x2

G(x

)

x2

G(x

)

Figure 6.2.6: Estimate of G(x), regression along lines of constant energy, 3rd iteration.

To each point (vi, ai) a mean energy level is defined as

λi = 12(1

2v2

i + G(ai)) (6.2.1)

The idea is now to divide the points (vi, ai) into groups of approximately equal mean energies.Since G(ai) is initially unknown this must be done by iteration.

Consider the data points shown in Figure 6.2.5a. In order to begin the iteration procedurethe stiffness is initially assumed to be linear and is estimated as

〈ω〉2 =σ2

x

σ2x

(6.2.2)

If the stiffness is non-linear (6.2.2) gives an estimate of the stiffness of an equivalent linearsystem obtained by statistical linearization. An initial estimate of the mean energy definedby (6.2.1) is thus given by

λi,1 = 12(1

2v2

i + 12〈ω〉2a2

i ) (6.2.3)

6.3 Damping and Excitation Estimation for Systems with External Excitation 71

The dashed lines in Figure 6.2.5a correspond to equal values of λi,1, and the lines thus definea number of ranges. Averaging of the points within each range leads to a non-parametricestimate of G(x) shown by the dots in Figure 6.2.5b. The real function G(x) is shown by thesolid line and the elastic energy is seen to be slightly overestimated for high and low energylevels. This is due to the fact that the grouping of data points was performed accordingto λi,1 given by (6.2.3) where G(x) is assumed to be linear. From Figure 6.2.5b a betterestimate G1(x) is obtained and the mean energy is now evaluated using this estimate ofG(x),

λi,2 = 12(1

2v2

i + G1(ai)) (6.2.4)

In the present case G1(x) has been obtained by linear interpolation of the estimated pointsshown in Figure 6.2.5b, but a polynomial fit to the points can also be used, Rudinger andKrenk (2001). The result after three iterations is shown in Figure 6.2.6a. The dashed linescorrespond to equal values of λi,3 in Figure 6.2.6a and the dots in Figure 6.2.6b show thenon-parametric estimate. A very satisfactory estimate of the elastic energy is obtained. Forpractical purposes the first estimate shown in Figure 6.2.5b may be sufficiently accurate.

6.3 Damping and Excitation Estimation for Systems

with External Excitation

The stiffness estimation procedure discussed in the previous section is quite general. Itcan be applied to systems with both external and parametric excitation, as will be shownin Chapter 7. The remaining part of the estimation procedure, namely that of dampingand excitation estimation, is conveniently treated in two steps. In this section the case ofexternal excitation is considered, and in the next section the more general case of a systemwith parametric excitation is considered. A system with external excitation can off coursebe treated as a special case of a system with parametric excitation. However, in the firstcase the unknowns are a function and a value (the damping function and the excitationintensity) and in the second case the unknowns are two functions (the damping functionand the excitation amplitude function). The case of external excitation therefore simplifiesconsiderably.

In this section, it is assumed that the system is governed by an equation of motion of thetype (2.2.1). The stiffness g(X) has been estimated by the method discussed in the previoussection. The remaining part of the procedure consists in estimating the damping functionh(X, X) and the excitation intensity S0.


The estimation of the damping relative to the excitation intensity for systems with externalexcitation is based on the expression (2.2.10) for the probability density of the state space


variables. This expression can be rewritten as

Heq(λ)

πS0

= − ln

(px,x(x, x)

C

)(6.3.1)

where Heq(λ) is the integral of the equivalent damping function given by (2.2.13). Theprocedure discussed in this subsection yields an estimate of the quantity on the left handside of this equation. The accessible information is thus the equivalent damping potentialrelative to the intensity of the white noise excitation. Equation (6.3.1) is not an exactrelation but an approximation corresponding to the method of equivalent non-linearization.Estimating Heq(λ)/πS0 from the expression (6.3.1) is not straight forward, since px,x is afunction of the two variables x and x. However, by the transformation (2.4.9), equation(6.3.1) can be written in a form involving only the mechanical energy as

Heq(λ)

πS0

= − ln

(pλ(λ)

T (λ)C

)(6.3.2)

where T (λ) is the natural period given by (2.4.11) and C is the normalizing constant forthe probability density px,x(x, x) of the state space variables. It should be observed that thepresence of the constant C in (6.3.2) is not in itself a problem, since it is known that the lefthand side is equal to zero for λ = 0.

Estimating the equivalent damping potential relative to the white noise intensity from (6.3.2)has the disadvantage, that it requires knowledge of the potential energy G(x). This functionmust be known or estimated accurately in order to evaluate the energy from the record ofthe stochastic response. Furthermore, G(x) is needed for evaluation of the natural periodT (λ). As will be shown in the examples in the next chapter, G(x) can be evaluated veryaccurately, so this is not a severe restriction for the use of (6.3.2) for estimation. It is,however, an advantage if Heq(λ)/πS0 can be evaluated without considering the potentialenergy, since this would require less computation and will eliminate any sensibility to scatterin the estimate of G(x).

If the energy is sampled only at mean level crossings a bias is introduced, which removesthe factor 1/T (λ) from the expression (6.3.2). At mean level crossings the energy is purelykinetic, and is known irrespective of the estimation of the potential energy. Equation (6.3.2)can therefore be modified as

Heq(λ)

πS0

= − ln (q(λ)) (6.3.3)

where q(λ) is a histogram of the kinetic energy sampled at mean level crossings and scaledsuch that q(0) = 1. Equation (6.3.3) thus yields a non-parametric estimate of Heq(λ)/πS0.

6.3 Damping and Excitation Estimation for Systems with External Excitation 73

6.3.2 Estimation of absolute damping

In the previous subsection the estimation procedure did not require knowledge of the poten-tial energy G(x). In this subsection, where estimation of the absolute damping is discussed,it is necessary to assume that G(x) is known or reliably estimated. Under this assumptionthe record (x(t), x(t)) can be transformed to the modified phase plane defined in (2.4.1)and denoted (z1, z2) = (sign(x)

√2G(x), x). If the covariance function of the modified phase

plane vector is interpreted as the mean value of the inner product of (z1, z2)t and (z1, z2)t+τ

it can be expressed as

R(τ |λ) = E[(z1, z2)t · (z1, z2)t+τ |λ] = Rz1(τ |λ) + Rz2(τ |λ) (6.3.4)

for a given energy level. When introducing the approximations for Rz1(τ |λ) and Rz2(τ |λ)given by (3.4.14) and (3.4.13), the following approximate expression for R(τ |λ) is obtained,

R(τ |λ) = λ exp(−12heq(λ)c(λ)τ)

∞∑j=1,3,...

(s2j + c2

j) cos(jωλτ) (6.3.5)

where the effective damping ηλ in (3.4.13) and (3.4.14) has been reduced to the equivalentdamping function heq(λ) multiplied by the factor c(λ) from (3.4.3b) representing the fractionof kinetic energy. This reduction is due to the assumption of external excitation. R(τ |λ) isnow considered for τ = 0 and τ = π/ωλ = 1

2T (λ) corresponding to half a period of oscillation

at energy level λ. Evaluating R(τ |λ) at these values of τ yields

R(0|λ) = 2λ = A1

R(12T (λ)|λ) −2λ exp(−1

4heq(λ)c(λ)T (λ)) = A2

(6.3.6)

The simplicity of the second expression is due to the fact that c(λ) + s(λ) = 2, as seen from(3.4.3), and shows the advantage of considering the phase plane vector rather than any ofthe two phase plane variables z1 and z2. The points A1 and A2 on the covariance function

A(τ)

A1

A2

τ = ½Tλ

τ

R =

Rz1

+ R

z2

Figure 6.3.1: Sketch of covariance function for lightly damped systems.


R(τ |λ) are sketched in Figure 6.3.1. From (6.3.6) it is seen, that an estimate of heq(λ) canbe obtained by

heq(λ) =4

c(λ)T (λ)ln

(−A1

A2

)(6.3.7)

R(12T (λ)|λ) is approximately equal to the value of R(τ |λ) at the first negative extreme as

shown in Figure 6.3.1, and 12T (λ) is then approximated by the value of τ at the first negative

extreme. In fact, the extremes of the function R(τ |λ) do not lie exactly on the envelope, butthis is a higher order effect, which is negligible for lightly damped systems. The functionR(τ |λ) can be estimated from the record by taking conditional average values of the modifiedphase plane record (z1, z2). The points A1 and A2 are then identified and (6.3.7) gives anestimate of the equivalent damping at a given energy level or for a given energy range. 1

2T (λ)

is estimated as the time separation between A1 and A2 and c(λ) can be estimated by takingtwice the mean value of the kinetic energy divided by the mechanical energy for all parts ofthe record at a given energy level. This leads to a non-parametric estimate of heq(λ) as afunction of the energy.

6.3.3 Estimation of excitation intensity

As the final stage of the method the intensity of the white noise S0 is estimated. This isa relatively straight forward procedure, since estimates of

∫ λ

0heq(λ)dλ/πS0 and heq(λ) have

been obtained in the previous two subsections. Initially the non-parametric estimate ofheq(λ) is fitted by a polynomial. Integration of this function yields an estimate of Heq(λ)also in the form of a polynomial. Equation (6.3.3) is now rewritten as

0 =Heq(λ)

πS0

+ ln(q(λ)) (6.3.8)

where q(λ) is given in the form of a histogram (as discussed in Section 6.3.1) and Heq(λ)is the polynomial mentioned above. S0 is thus the only unknown, and it is estimated byminimizing the square of the right hand side of (6.3.8). This completes the identificationof the system. The estimation procedure for a system with external excitation discussed inthis section is demonstrated in the example given in Section 7.1.

6.4 Damping and Excitation Estimation for Systems

with Parametric Excitation

A specific type of oscillator with parametric excitation is now considered, and it is assumedthat the equation of motion is given by (2.3.13). This means that the damping functionand the excitation amplitude function depend only on the mechanical energy, and this class

6.4 Damping and Excitation Estimation for Systems with Parametric Excitation 75

of systems is discussed in Section 2.3.3. This may seem like a rather special case, butmany lightly damped systems can be represented very accurately by a system belonging tothis class, by replacing the damping function h(x, x) and the excitation amplitude functionσ(x, x) with average values for oscillations at a given energy level. Many of the methodspresented in Section 6.3 are reconsidered, though in a slightly more general form.

6.4.1 Estimation of absolute damping

The estimation of the damping follows the procedure discussed in Section 6.3.2. Again, thecovariance function R(τ |λ) of the phase plane vector introduced in (6.3.4) is considered. Theequation (6.3.5) still holds as an approximation if the factor heq(λ)c(λ) is replaced by theeffective damping ηλ introduced in (3.4.7), and an estimate of the effective damping is thusgiven by

ηλ =4

T (λ)ln

(−A1

A2

)(6.4.1)

where A1 and A2 are defined in (6.3.6) and illustrated in Figure 6.3.1. This estimationprocedure is actually quite general and can be applied to a system governed by (2.3.1).

6.4.2 Estimation of stationary potential

The estimation of the stationary potential follows the procedure introduced in Section 6.3.1,and will just be briefly discussed. The probability density function is now assumed to begiven in the form (2.3.8), and the stationary potential can thus be expressed as

ψ(λ) = − ln

(px,x(x, x)

C

)(6.4.2)

It is thus seen that Heq(λ)/πS0 is equal to the stationary potential in the special casediscussed in Section 6.3.1. If the probability density of the mechanical energy is considered,(6.4.2) is rewritten as

ψ(λ) = − ln

(pλ(λ)

T (λ)C

)(6.4.3)

corresponding to (6.3.2). Again, the factor 1/T (λ) is removed by sampling the energy onlyat mean level crossings and (6.4.3) is expressed as

ψ(λ) = − ln (q(λ)) (6.4.4)

where q(λ) is a histogram of the energy at mean level crossings obtained from the stochasticrecord. Equation (6.4.4) yields a non-parametric estimate of the stationary potential ψ(λ).It should be mentioned, that the scaling of the histogram is of no importance, since anyconstant can be added to ψ(λ) without changing the solution. In the following estimationof the excitation only the derivative of the stationary potential is needed. In order for q(λ)to retain the meaning in Section 6.3.1, the histogram is scaled such that q(0) = 1.


6.4.3 Estimation of excitation amplitude function

The general relation between the stationary potential and the functions defining the system isgiven by (2.3.15), for the particular class of systems considered here. However, this expressioninvolves the damping function h(λ), which has not been estimated directly. A more usefulrelation is obtained by considering the effective damping ηλ. For a system governed by(2.3.13), equation (3.4.7) reduces to

ηλ =1

λE

[h(Λ)X2 − 1

4

dσ2

dΛX2

∣∣∣∣λ]

=

(h(λ) − 1

4

dσ2

dλ

)c(λ) (6.4.5)

where c(λ) is the relative fraction of kinetic energy defined in (3.4.3b). With h(λ) given by(6.4.5), equation (2.3.15) can be rewritten as

0 =dψ

dλ− 2

σ(λ)2

(ηλ

c(λ)+

1

2

dσ(λ)2

dλ

)(6.4.6)

and an estimate of σ(λ)2 is obtained by minimizing the square of the right hand side of thisequation. However, this requires a parametric representation of the functions ψ(λ), ηλ/c(λ)and σ(λ)2. Polynomial representations are used,

ψ(λ) =N∑

k=0

akλk ,

ηλ

c(λ)=

N∑k=0

bkλk , σ(λ)2 =

N∑k=0

ckλk (6.4.7)

The polynomial for ψ(λ) is obtained by fitting the non-parametric estimate obtained bythe procedure described in the previous section. The polynomial for ηλ/c(λ) is obtained byfitting the damping estimates derived by the procedure discussed in Section 6.4.1 divided byan estimate of c(λ), which is obtained by considering the mean value of the kinetic energydivided by the total energy for all points of the record with a given energy level. Theestimation of σ(λ) completes the identification. An example of this procedure is given inSection 7.2.

Chapter 7

System Identification: Examples

In order to demonstrate the system identification techniques proposed in the previous chaptertwo different examples are considered. In the first example the excitation is external, andthe estimation of the damping and excitation thus follows the procedure described in Section6.3. The results given here can be found in Krenk and Rudinger (2000), Rudinger and Krenk(2000b, 2001).

The second system, which is considered, has parametric excitation, and in this case theestimation of damping and excitation follows the procedure introduced in Section 6.4. Theresults given in this example have not been published previously, but a few preliminaryresults for a system of this type were presented by Rudinger and Krenk (2003a).

7.1 System with External Excitation

The first system, which will be investigated, is a non-linear system with external excitation.The equation of motion is given by

x + h(x, x)x + g(x) = W0(t) (7.1.1)

where W0(t) is a white noise with intensity S0. The non-linear damping function is of theform

h(x, x) = 2ζω0(1 + αx2) (7.1.2)

and the stiffness function is given by

g(x) =ω2

0

βsinh(βx) (7.1.3)

where α and β are non-negative parameters of the model. For α = 0 and β → 0 the linearsystem is retrieved. In this case ζ and ω0 are the damping ratio and natural frequency,

77

78 Chapter 7. System Identification: Examples

respectively. A system of this type was investigated in detail by Krenk (1999). The potentialenergy is evaluated by integration of (7.1.3) as

G(x) =ω2

0

β2(cosh(βx) − 1) (7.1.4)

The natural period of vibration is determined from (2.4.11) as

T (λ) =4K(k)√

ω20 + 1

2β2λ

, k2 =12β2λ

ω20 + 1

2β2λ

(7.1.5)

where K(k) is the complete elliptic integral of the first kind. The effective damping isobtained from (2.2.13) as

heq(λ) = 2ζω0(1 + 32αλ) (7.1.6)

by considering free undamped vibration at energy level λ. The probability density of thestate space variables is given by

px,x(x, x) = C exp

(−Heq(λ)

πS0

), Heq(λ) =

∫ λ

0

heq(λ)dλ = 2ζω0(λ+ 34αλ2) (7.1.7)

which is an exact solution to the equivalent non-linear system obtained by replacing h(x, x)with heq(λ) in (7.1.1). If the system is linear the mean energy level is given by

λ0 =πS0

2ζω0

(7.1.8)

For the non-linear systems this energy will be used as a reference energy level indicating theorder of magnitude of the response. The non-linearity is then characterized by the stiffnessnon-linearity parameter (β/ω0)

2λ0 and the damping non-linearity parameter αλ0. For thelinear case (α = β = 0) the damping ratio is given by ζ = πS0/2λ0ω0, λ0 being the meanenergy. For the non-linear system an equivalent damping is introduced as

ζeq =πS0

2ω0E[λ](7.1.9)

where E[λ] is the mean energy of the non-linear system. This parameter gives the magnitudeof the damping.

7.1.1 Stiffness estimation

The procedure described in Section 6.2 for estimation of the stiffness is now applied tosimulated records. The function, which is estimated, is the potential energy G(x), which forthe present system is given by (7.1.4).

7.1 System with External Excitation 79

0 0.5 1 1.5 2 2.50

1

2

3

4

5

½v i2 /<

λ>

ai2 /<a

i2>

0 0.5 1 1.5 2 2.50

1

2

3

4

5

G /<

λ>

x2 /<ai2>

Figure 7.1.1: Result of the estimation after 3 iterations, a) original points, b) estimate ofG(x).

Since the stiffness is non-linear, several iterations are needed. In Figure 7.1.1 the result afterthree iterations is shown. In Figure 7.1.1a the original points (ai, vi) are displayed. Thedashed lines in this figure show lines of equivalent mean energy λi,3 = 1

2(1

2v2

i +G2(ai)), whereG2(x) is the estimate of the elastic potential after the second iteration. The average pointswithin each of the groups defined by the dashed lines are shown in Figure 7.1.1b, and theseare seen to give an excellent estimate of the theoretical elastic potential shown by the solidline. The estimate is based on a record with 162,000 points (ai, vi) obtained by the stochasticsimulation method discussed in Chapter 4. Only 500 of these are shown in Figure 7.1.1a.The parameters are ζeq = 0.05, (β/ω0)

2λ0 = 16 and αλ0 = 0, which means that the systemhas linear damping and strongly non-linear stiffness.


The procedure discussed in Section 6.3.1 is now applied to estimate the equivalent dampingpotential relative to the white noise intensity of the excitation process from records of thestochastic response. Two different methods are considered. In the first method the energyis sampled at equal intervals in time and an estimate of pλ(λ) is obtained. The estimateof Heq(λ)/πS0 is then based on (6.3.2). Sampling the energy at every time step requiresknowledge of the potential energy G(x), and the use of (6.3.2) requires knowledge of thenatural period T (λ). The results of this procedure are shown in Figures 7.1.2a - 7.1.5a. InFigures 7.1.2a and 7.1.3a αλ0 = 0 and the damping is thus linear, which is seen by thestraight lines in the semi-logarithmic plots. In Figures 7.1.4a and 7.1.5a αλ0 = 0.8 and thedamping is non-linear, as seen by the curvature of the lines. In Figures 7.1.2a and 7.1.4a thestiffness is linear, while (β/ω0)

2λ0 = 8 for the cases shown in 7.1.3a and 7.1.5a correspondingto a relatively strong level of non-linearity in the stiffness.

In the second method the energy is sampled at mean level crossings and thereby a fixednumber of times per period. This eliminates the dependence of the natural period, and the


0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>0 1 2 3 4 5 6 7 8 9 10

10−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>

Figure 7.1.2: Estimation of damping relative to input intensity, ζeq = 0.05, (β/ω0)2λ0 = 0

and αλ0 = 0.

0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>0 1 2 3 4 5 6 7 8 9 10

10−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>


and αλ0 = 0.

estimate is obtained from (6.3.3). No information concerning the potential energy nor thenatural period is thus required. The results of this method are shown in Figures 7.1.2b- 7.1.5b for the same parameter combinations. As seen from the Figures 7.1.2 - 7.1.5 thesecond method yields estimates, which are as accurate as those obtained by the first method,which relies on an estimate of the potential energy and of the natural period of the system,and is based on more computation.

The results shown in Figures 7.1.2 - 7.1.4 are generally satisfactory though a small bias isobserved for large energy levels. When both stiffness and damping are non-linear this biasis larger, as seen from Figure 7.1.5.


0 1 2 3 4 5 6 710

−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>0 1 2 3 4 5 6 7

10−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>


and αλ0 = 0.8.

0 1 2 3 4 5 6 7 810

−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>0 1 2 3 4 5 6 7 8

10−5

10−4

10−3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>


and αλ0 = 0.8.

7.1.3 Covariance functions

The estimation of the absolute damping is based on the approximate expressions for thecovariance functions of the modified phase plane variables given by (3.4.13) and (3.4.14), asdiscussed in Section 6.3.2. The validity of these expressions is therefore investigated initially.

First the simple case of a linear system is considered. In this case the expressions (3.4.13)and (3.4.14) reduce to

Rz1(τ |λ) = Rz1(τ |λ) = λ exp(−ζω0τ) cos(ω0τ) (7.1.10)

and the first phase plane variable is proportional to the displacement: z1 = ω0x. Whencomparing the expression (7.1.10) with analytical solutions for linear single-degree-of-freedom


0 0.5 1 1.5−2

−1

0

1

2

Rz 1(τ

|λ)/

λ 0

ω0t/2π

0 0.5 1 1.5−2

−1

0

1

2

Rz 2(τ

|λ)/

λ 0

ω0t/2π

Figure 7.1.6: Covariance functions for z1 and z2 at four different energy levels, ζeq = 0.05,(β/ω0)

2λ0 = 0 and αλ0 = 0.

0 0.5 1 1.5−2

−1

0

1

2

Rz 1(τ

|λ)/

λ 0

ω0t/2π

0 0.5 1 1.5−2

−1

0

1

2

Rz 2(τ

|λ)/

λ 0

ω0t/2π

Figure 7.1.7: Covariance functions for z1 and z2 at four different energy levels, ζeq = 0.06,(β/ω0)

2λ0 = 8 and αλ0 = 0.

systems with external white noise excitation it is seen that (7.1.10) is only an approximation,valid for lightly damped systems.

In Figure 7.1.6 the covariance functions of the modified phase plane variables at four differentenergy levels are shown. The dashed lines are the expressions (7.1.10), and the solid lines arethe estimates based on stochastic records. The damping ratio is ζ = 0.05 and the estimatesare based on simulation of 50,000 natural periods. The approximate theoretical expression(7.1.10) is seen to agree very well with the simulation results.

A system with strongly non-linear behaviour ((β/ω0)2λ0 = 8) is considered next. The system

has approximately the same level of damping as the linear system considered in Figure 7.1.6(ζeq = 0.06) and a linear damping term (αλ0 = 0). In Figure 7.1.7 estimates of the covariancefunctions of z1 and z2 at four different energy levels are shown by the solid lines and comparedto the approximate theoretical expressions (3.4.13) and (3.4.14), given by the dashed lines.


The estimates shown in Figure 7.1.7 are based on simulation of approximately 80,000 periodsat mean energy level. It is observed, that the natural frequency changes with energy level,as expected for systems with non-linear stiffness. The system is furthermore characterizedby an average drift in the energy level towards the mean value. For a system with non-linear stiffness the averaging performed to estimate the covariances is an averaging of theprocess at various energy levels and thus at various frequencies. This averaging of oscillationswith different natural frequencies will appear as additional damping - the unsyncronizedoscillations will tend to cancel. This effect is clearly observed in Figure 7.1.7, where theestimated functions from the simulated record show a higher level of damping. However,only the initial part of the record (0 ≤ τ ≤ 1

2T (λ)) is needed for the damping estimation,

and in this range the estimated functions are still reasonably accurate, despite the strongnon-linearity in the stiffness.

7.1.4 Estimation of damping and excitation intensity

Finally, the absolute damping and the excitation intensity are estimated. The estimate of theequivalent damping function heq(λ) follows the procedure described in Section 6.3.2 and isbased on estimated covariance functions investigated in the previous subsection. In Figures7.1.8a - 7.1.10a the non-parametric estimate of heq(λ) is given for a linear system (ζeq = 0.05,(β/ω0)

2λ0 = 0, αλ0 = 0), a system with non-linear stiffness (ζeq = 0.06, (β/ω0)2λ0 = 8,

αλ0 = 0) and a system with non-linear damping (ζeq = 0.05, (β/ω0)2λ0 = 0, αλ0 = 0.8),

respectively. Ten different energy levels have been considered, and the equivalent damping isthus estimated at ten different points indicated by the dots shown in the figures. The solidline is the theoretical damping function given by (7.1.6). The estimates are based on recordswith 50,000 natural periods for the linear system and 80,000 periods at mean energy level forthe non-linear systems. In the two cases with linear stiffness (Figures 7.1.8a and 7.1.10a), theestimates are very accurate. In the case of non-linear stiffness (Figure 7.1.9a) the damping

0 1 2 30

0.5

1

1.5

h eq /

2ζω

0

λ/<λ>0 1 2 3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>

Figure 7.1.8: Estimation of damping and excitation intensity, ζeq = 0.05, (β/ω0)2λ0 = 0 and

αλ0 = 0.


0 1 2 30

0.5

1

1.5

h eq /

2ζω

0

λ/<λ>0 1 2 3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>

Figure 7.1.9: Estimation of damping and excitation intensity, ζeq = 0.06, (β/ω0)2λ0 = 8 and

αλ0 = 0.

0 1 2 30

1

2

3

h eq /

2ζω

0

λ/<λ>0 1 2 3

10−2

10−1

100

101

exp(−

Heq

/ πS

0)

λ/<λ>

Figure 7.1.10: Estimation of damping and excitation intensity, ζeq = 0.05, (β/ω0)2λ0 = 0

and αλ0 = 0.8.

is overestimated. This is the result of averaging oscillations at various frequencies whenestimating the covariance functions, as discussed in the previous subsection. The error is inthis case about 15 - 20 % for 1 < λ/〈λ〉 < 3, and smaller for the initial part of the λ-axis.For smaller levels of non-linearity in the stiffness this bias will be less pronounced.

The non-parametric estimates of the damping are now fitted by a polynomial indicated bythe dotted lines in Figures 7.1.8a - 7.1.10a. This polynomial fit is integrated to obtain anestimate of Heq(λ). S0 is then determined by minimizing the square of (6.3.4), as discussed inSection 6.3.3. This corresponds to fitting exp(−Heq(λ)/πS0) to histograms of the type shownin Figures 7.1.2 - 7.1.5. The results of this procedure are shown in Figures 7.1.8b - 7.1.10b.The dots represent the histogram values, and the solid lines the functions exp(−Heq(λ)/πS0)with Heq(λ) given by the polynomial and S0 determined to give the best fit in a least squaressense. The estimates of S0 obtained by the fits shown in the figures are S0,est/S0 = 0.98,1.14, 1.02, respectively. In the cases of linear stiffness the error is thus 2 % and in the case

7.2 System with Parametric Excitation 85

0 5 10 15 200

0.5

1

1.5

(λ/ω0)2 λ

0

S 0,es

t /S0

Figure 7.1.11: Estimation of excitation intensity as a function of the non-linear stiffnessparameter (linear damping).

of non-linear stiffness the error is 14 %.

Finally, the quality of the estimate of the excitation intensity is evaluated as a function ofthe non-linear stiffness parameter in the case of linear damping. The results are shown inFigure 7.1.11 by the dots. For low values of (β/ω0)

2λ0 the estimate is very accurate. Forhigher levels of non-linear behaviour in the stiffness the accuracy decreases.

7.2 System with Parametric Excitation

The second system, which will be investigated in terms of the system identification techniquesproposed, is a system with parametric excitation. The system has both non-linear stiffnessand damping and is governed by an equation of the type

¨x + h(λ) ˙x + g(x) = W1(t) +√

λW2(t) (7.2.1)

where λ is the mechanical energy, h(λ) is the damping function and g(x) is the stiffnessfunction. The right hand side represents the excitation and the system considered is seento belong to the class of systems discussed in Section 2.3.3 to which an exact solution existsin terms of the probability density of the state space variables. The tilde-notation is usedto indicate that the system is not given in non-dimensional form, which will be done in the

following. The coefficient functions are identified as b1(λ) = 1 and b2(λ) =√

λ. The firstterm on the right hand side of the equation of motion corresponds to an external excitationterm, while the second term corresponds to a parametric excitation term. The intensitiesW1(t) and W2(t) are denoted S11 and S22, and the processes are assumed to be uncorrelated,i.e. S12 = 0. The stiffness function g(x) is assumed to be given by

g(x) = ω20x(1 + βx2) (7.2.2)


corresponding to the stiffness of a Duffing oscillator. ω0 is the natural frequency for thelinear case (β = 0). β is a non-negative stiffness parameter. The damping function h(λ) isassumed to have the following form,

h(λ) = 2ζω0(1 + sign(ζ)αλ) (7.2.3)

where ζ is the damping ratio for the corresponding linear system (α = β = S22 = 0). αis a non-negative damping parameter. For ζ < 0, the first term in (7.2.3) is negative, andthe system has a non-trivial stationary solution in the absence of excitation (a stable limitcycle), see e.g. Lin and Cai (1995).

7.2.1 Non-dimensional formulation

In order to reduce the number of parameters describing the system, the equation of motionis expressed in a non-dimensional form. A time scale and a length scale are introduced as

t0 =1

ω0

, x0 =

√πS11

2|ζ|ω0

(7.2.4)

where x0 is the standard deviation of the linear system obtained for β = α = S22 = 0.Non-dimensional time, displacement and energy are defined as

t =t

t0, x =

x

x0

, λ =t20λ

x20

(7.2.5)

The equation of motion is now multiplied by t20/x0. It can then be expressed in the followingnon-dimensional form,

x + h(λ)x + g(x) = W1(t) +√

λW2(t) (7.2.6)

The non-dimensional coefficient functions are thus b1(λ) = 1 and b2(λ) =√

λ. The spectralintensities S11 and S22 of W1(t) and W2(t) are evaluated as

S11 =2|ζ|π

, S22 =S22

ω0

(7.2.7)

The non-dimensional stiffness function g(x) is identified as

g(x) = x(1 + βx2) , β =πS11β

2|ζ|ω30

(7.2.8)

where β is a non-dimensional parameter for the stiffness non-linearity. The non-dimensionaldamping function h(λ) is given by

h(λ) = 2ζ(1 + sign(ζ)αλ) , α =πS11α

2|ζ|ω0

(7.2.9)

where α is a non-dimensional parameter for the damping non-linearity. The original sixparameters ζ, ω0, α, β, S11 and S22 are thus reduced to the four non-dimensional parametersζ, α, β and S22, as one would expect according to the Buckingham pi-theorem, since theproblem involves the two dimensions length and time.


7.2.2 Natural period of vibration

The elastic energy is obtained by integration of the stiffness as

G(x) = 12x2(1 + 1

2βx2) (7.2.10)

Free undamped vibration at energy level λ is now considered. The maximum response isgiven by the equation G(xmax) = λ leading to the following expression,

βx2max = −1 +

√1 + 4βλ (7.2.11)

where xmax is the positive root and xmin = −xmax due to the symmetric stiffness. Thenatural period of vibration can be found from (2.4.11). In the present case of a linear-cubicstiffness, the natural period is expressed as

T (λ) =4√

1 + βx2max

K(m) , m =12βx2

max

1 + βx2max

(7.2.12)

where K(m) is the complete elliptic integral of the first kind, see e.g. Krenk and Roberts(1999).


The excitation amplitude function of the non-dimensional system introduced in (7.2.6) isobtained from (2.3.4), which in the present case yields

σ(λ)2 = 4|ζ| + 2πS22λ (7.2.13)

The equation for the derivative of the argument function ψ′(λ) given in (2.3.15) is expressedas

dψ

dλ=

2ζ + 12πS22 + 2|ζ|αλ

2|ζ| + πS22λ(7.2.14)

The integration of (7.2.14) is divided into two cases. For S22 = 0 ψ(λ) is evaluated as

ψ(λ) =2|ζ|αλ

πS22

− ln(2|ζ| + πS22λ)ν , ν =4ζ2α − 2πζS22 − 1

2π2S2

22

π2S222

(7.2.15)

In the case where S22 = 0 the expression for ψ(λ) reduces to

ψ(λ) = sign(ζ)λ + 12αλ2 (7.2.16)


10−1

100

101

10−3

10−2

10−1

100

101

p λ

λ10

010

110

210

−3

10−2

10−1

100

p λ

λ

Figure 7.2.1: Probability density. a) ζ = 0.05, α = 2, S22 = 0.1, • β = 0, + β = 10. b)ζ = −0.05, α = 0.1, β = 1, • S22 = 0, + S22 = 0.02.

The probability density for the state space variables (x, x) is now obtained from (2.3.8) ordirectly from (2.3.16) as

px,x(x, x) =

C1

(2|ζ|πS22

+ λ

)ν

exp

(−2|ζ|α

πS22

λ

), S22 = 0

C2 exp(−sign(ζ)λ − 12αλ2) , S22 = 0

ν = α

(2ζ

πS22

)2

− 2ζ

πS22

− 12

(7.2.17)

where C1 and C2 are normalizing constants. The solution is seen to depend only on theparameters α, sign(ζ) and 2ζ/πS22. In the case where S22 = 0 only α and sign(ζ) controlsthe solution. The probability density of the energy is obtained from (2.4.9). With theexpression (7.2.12) for the natural period and (7.2.17) for the joint probability density of thestate space variables, the probability density of the energy can be expressed as

pλ(λ) =

4 C1 K(

12− 1

21√

1+4βλ

)4√

1 + 4βλ

(2|ζ|πS22

+ λ

)ν

exp

(−2|ζ|α

πS22

λ

), S22 = 0

4 C2 K(

12− 1

21√

1+4βλ

)4√

1 + 4βλexp(−sign(ζ)λ − 1

2αλ2) , S22 = 0

(7.2.18)

where ν is given in (7.2.17b). For a given sign of ζ, this probability density is seen to dependonly on the three parameters: β, α and 2ζ/πS22.

In order to verify the solution obtained for the probability density of the non-dimensionalenergy variable λ it is compared to results obtained by stochastic simulation following theprocedure described in Chapter 4. In Figure 7.2.1a two cases are considered, one with


linear stiffness (β = 0) and the other with non-linear stiffness (β = 0). The solid line anddashed line correspond to expression (7.2.18), while the dots and crosses are obtained fromhistograms of the simulated records. Records with a length of 10, 000 · 2π are considered,corresponding to 10,000 periods in the case of linear stiffness. As seen by the figure thetheoretical results are in agreement with the results obtained from stochastic simulation. Theeffect of including a non-linear stiffness term is seen to be quite small, when the probabilitydensity of the energy is considered.

In Figure 7.2.1a the damping ratio is ζ = 0.05, but this parameter only enters the expressionfor the probability density through the value of sign(ζ) and the parameter combination2ζ/πS22. In Figure 7.2.1b results are shown for ζ = −0.05. In the case of a negative dampingratio the system is characterized by having a stable limit cycle, i.e. a non-zero equilibriumstate in the absence of excitation. In the absence of excitation the distribution of energy isa delta function at energy level λ = 1/α corresponding to free undamped vibration at thisenergy level.

In Figure 7.2.1b two cases are shown, one for external excitation (S22 = 0), and the otherfor parametric excitation (S22 = 0). Again, the solid and dashed lines correspond to thetheoretical expression (7.2.18), while the dots and crosses are obtained from simulated recordsof length 10, 000 · 2π. For S22 = 0 the solution tends to a delta function for α → 0. Thepresence of the parametric excitation is seen to have a strong influence on the probabilitydensity of the energy. The results from simulation are seen to verify the theoretical solution.

7.2.4 Characteristic parameters for the system

The non-dimensional system governed by (7.2.6) is completely characterized by the fourparameters ζ, β, α and S22, but these parameters are difficult to interpret directly. Thusthe value of 〈ω〉2 defined in (6.2.2) gives an indication of the degree of non-linearity in thestiffness, whereas β is a more indirect measure of the stiffness non-linearity. The followingparameters are introduced,

〈ζ〉 =ζ + |ζ|αE[λ]

〈ω〉 , γ = αE[λ] sign(ζ) , δ =S22E[λ]

S11

=πS22E[λ]

2|ζ| (7.2.19)

The parameter 〈ζ〉 is obtained by setting 2〈ζ〉〈ω〉 equal to the mean value of the dampingfunction h(λ). In the case of a linear system 〈ζ〉 reduces to the damping ratio ζ. This par-ameter is thus a measure of the level of damping. γ is the mean value of the ratio betweenthe non-linear and the linear term in the damping function and is therefore a measure of thedamping non-linearity. Finally, δ is the ratio between the variance of the parametric excita-tion term and the external excitation term, and therefore gives a measure of the intensity ofthe parametric excitation relative to the intensity of the external excitation. In the followingthe numerical examples considered will be described in terms of the parameters 〈ω〉2, 〈ζ〉, γand δ.


0 0.5 1 1.50

2

4

x2

G(x

)

0 0.5 1 1.50

2

4

x2

G(x

)

0 1 2 30

2

4

x2

G(x

)

0 1 2 30

2

4

x2

G(x

)

0 2 4 60

2

4

x2

G(x

)

0 2 4 60

2

4

x2

G(x

)

Figure 7.2.2: Estimation of G(x), 2nd iteration, γ = 0, δ = 0. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

7.2.5 Stiffness estimation

The stiffness estimation follows the procedure described in Section 6.2. A number of examplesof estimation of the stiffness are shown in Figures 7.2.2 - 7.2.7, where the results after the2nd iteration are depicted. The results correspond to six different combinations of theparameters γ and δ. In each figure the results for three different values of the stiffness non-linearity parameter 〈ω〉2 and two different values of the damping magnitude parameter 〈ζ〉are shown. The 1st, 2nd and 3rd row of the subfigures in each figure thus correspond to〈ω〉2 = 1, 2 and 4, respectively. The left and the right columns correspond to 〈ζ〉 = 0.01 and0.04, respectively. The highest value of the damping is identical to an equivalent logarithmicdecrement of 25 %. This is a relatively high level of damping if vibration sensitive structures


0 0.5 10

1

2

x2

G(x

)

0 0.5 10

1

2

x2

G(x

)

0 1 20

1

2

x2

G(x

)

0 1 20

1

2

x2

G(x

)

0 2 40

1

2

x2

G(x

)

0 2 40

1

2

x2

G(x

)

Figure 7.2.3: Estimation of G(x), 2nd iteration, γ = 0.5, δ = 0. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

are considered. The first row corresponds to the relatively simple case where the stiffness islinear.

In Figure 7.2.2 γ = δ = 0, which means that the damping is linear and the excitation isexternal. The results in this figure contain the special case of a purely linear system in thetwo subfigures in the first row.

In Figure 7.2.3 γ = 12

and δ = 0. The mean value of the non-linear damping term is thushalf the mean value of the linear damping term. The excitation is also in this case external.

In Figure 7.2.4 γ = 0 and δ = 12. The damping is thus linear and the variance of the

parametric excitation at mean energy level is equal to half the value of the variance of the


0 1 2 30

2

4

6

x2

G(x

)

0 1 2 30

2

4

6

x2

G(x

)

0 2 4 60

2

4

6

x2

G(x

)

0 2 4 60

2

4

6

x2

G(x

)

0 6 120

2

4

6

x2

G(x

)

0 6 120

2

4

6

x2

G(x

)

Figure 7.2.4: Estimation of G(x), 2nd iteration, γ = 0, δ = 0.5. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

external excitation term.

In Figure 7.2.5 γ = δ = 12. This case is thus a combination of the previous two cases,

i.e. non-linear damping and parametric excitation are considered simultaneously.

In the above cases only systems with positive damping over the entire energy range havebeen considered. This means that non of these systems have a stable limit cycle, see e.g. theexample in Section 5.3. If ζ and thereby γ is negative, a system with a stable limit cyclewill occur. Such a system is considered in Figure 7.2.6 where γ = −3

2and δ = 1

2. In

this particular case the parameter 〈ζ〉 is not a good discriminator of the magnitude of thedamping, since the damping may have large negative values for small energy levels and largepositive values for high energy levels without having a large value of damping at the mean


0 0.5 1 1.50

1

2

3

x2

G(x

)

0 0.5 1 1.50

1

2

3

x2

G(x

)

0 1 2 30

1

2

3

x2

G(x

)

0 1 2 30

1

2

3

x2

G(x

)

0 2 4 60

1

2

3

x2

G(x

)

0 2 4 60

1

2

3

x2

G(x

)

Figure 7.2.5: Estimation of G(x), 2nd iteration, γ = 0.5, δ = 0.5. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

energy level, where the negative and positive contributions will approximately balance oneanother. As will be seen in the following, when the estimation of the damping is considered,the energy drift is much higher in this case than in the previous cases.

In the last case considered, corresponding to the results shown in Figure 7.2.7, the parametersare γ = δ = 1. The linear and non-linear damping terms are thus of the same order ofmagnitude at the mean energy level. This is also the case for the intensities of the externaland parametric excitation terms.

The dots shown in Figures 7.2.2 - 7.2.7 correspond to the non-parametric estimate of theelastic potential after the second iteration. In all cases the function is estimated by 12points covering the range 0 < λ < 4 E[λ]. The solid line is the real function G(x). As seen


0 1 20

2

4

x2

G(x

)

0 1 20

2

4

x2

G(x

)

0 2 40

2

4

x2

G(x

)

0 2 40

2

4

x2

G(x

)

0 4 80

2

4

x2

G(x

)

0 4 80

2

4

x2

G(x

)

Figure 7.2.6: Estimation of G(x), 2nd iteration, γ = −1.5, δ = 0.5. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

by the results an excellent estimate is obtained in almost all cases. The only exceptionsare the estimates shown in the last two subfigures in the second column in Figure 7.2.6.In these two cases the estimates at high energy levels are significantly smaller than thetheoretical functions. The reason why these estimates are less accurate is partly given abovein connection with the discussion of systems with stable limit cycle. As mentioned earlier,the numerical values of the damping may in the case of a system with stable limit cycle bequite high at low and high energy levels. The inaccuracies observed in Figure 7.2.6 are dueto high levels of damping at high energy levels, which leads to a substantial level of scatterof the points (vi, ai) and thereby a less accurate estimate of the stiffness.

It should be mentioned that the method works quite well for systems with much higher


0 0.5 1 1.50

1

2

3

x2

G(x

)

0 0.5 1 1.50

1

2

3

x2

G(x

)

0 1 2 30

1

2

3

x2

G(x

)

0 1 2 30

1

2

3

x2

G(x

)

0 2 4 60

1

2

3

x2

G(x

)

0 2 4 60

1

2

3

x2

G(x

)

Figure 7.2.7: Estimation of G(x), 2nd iteration, γ = 1, δ = 1. left) 〈ζ〉 = 0.01, right)〈ζ〉 = 0.04. 1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

level of non-linearity in the stiffness, as shown in the previous example, see Figure 7.1.1.However, in the present case the stochastic records will be used for damping estimationand estimation of the excitation intensity as well, and these estimation techniques looseaccuracy, if the non-linearity in the stiffness is too strong. As will be seen in the following, thedamping estimation for 〈ω〉2 = 4 is significantly less accurate than the damping estimationfor 〈ω〉2 = 1. The results shown for the stiffness estimation are based on records with alength of 40, 000 T (E[λ]).


0 2 40

0.04

0.08

λ / <λ>

ηλ

0 2 40

0.1

0.2

0.3

λ / <λ>

ηλ

0 2 40

0.02

0.04

0.06

λ / <λ>

ηλ

0 2 40

0.1

0.2

λ / <λ>

ηλ

0 2 40

0.02

0.04

λ / <λ>

ηλ

0 2 40

0.05

0.1

0.15

λ / <λ>

ηλ

Figure 7.2.8: Estimation of ηλ, γ = 0, δ = 0. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1st row)〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

7.2.6 Damping estimation

The estimation of damping follows the procedure described in Section 6.4.1. The resultsshown in Figures 7.2.8 - 7.2.13 are based on the same records as the results shown in Fig-ures 7.2.2 - 7.2.7 for the stiffness estimation. The records considered have a duration of100, 000 T (E[λ]) (only the first 40 % of these records were considered for the stiffness estima-tion). The dots correspond to non-parametric estimates. The solid lines are the theoreticalvalues of the effective damping given by

ηλ =[2ζ + 2|ζ|αλ − 1

2πS22

]c(λ) (7.2.20)


0 2 40

0.05

0.1

0.15

λ / <λ>

ηλ

0 2 40

0.2

0.4

λ / <λ>

ηλ

0 2 40

0.05

0.1

λ / <λ>

ηλ

0 2 40

0.1

0.2

0.3

λ / <λ>

ηλ

0 2 40

0.02

0.04

0.06

λ / <λ>

ηλ

0 2 40

0.1

0.2

λ / <λ>

ηλ

Figure 7.2.9: Estimation of ηλ, γ = 0.5, δ = 0. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1st row)〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

where c(λ) is introduced in (2.3.14a) as twice the ratio between the mean value of the kineticenergy and the total energy for free undamped vibrations. This quantity is evaluated nu-merically. The last term in the bracket in (7.2.9) originates from the Wong-Zakai correction.The dashed lines in Figures 7.2.10 - 7.2.13 correspond to removal of this term in (7.2.10) andthereby illustrates the effect of the correction. In the case of additive excitation (Figures7.2.8 and 7.2.9) the correction does not occur. As observed from the values on the x-axis,16 points are estimated covering the range 0 < λ < 4〈λ〉.

As mentioned in connection with the stiffness estimation, the two subfigures in the firstrow of Figures 7.2.8 - 7.2.13 correspond to systems with linear stiffness. In these cases theestimates are very accurate. Considering the length of the records (100,000 periods) the level


0 2 40

0.02

0.04

0.06

λ / <λ>

ηλ

0 2 40

0.1

0.2

0.3

λ / <λ>

ηλ

0 2 40

0.02

0.04

λ / <λ>

ηλ

0 2 40

0.1

0.2

λ / <λ>

ηλ

0 2 40

0.02

0.04

λ / <λ>

ηλ

0 2 40

0.05

0.1

λ / <λ>

ηλ

Figure 7.2.10: Estimation of ηλ, γ = 0, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1st row)〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

of scatter observed in the figures is, however, relatively high. This is due to the fact that theestimates are based on the decrease of the envelope of the covariance function over half aperiod of vibration, which is a small value for lightly damped systems. The estimate is thusquite sensitive to statistical scatter in the estimate of the covariance function. Assessmentof the damping in structural engineering normally involves a high level of uncertainty. Inmany cases estimates with some scatter would therefore be acceptable for practical purposes,and in such cases records with only 10,000 or maybe as little as 1,000 periods of vibrationmight be enough. As seen from the results in Figures 7.2.10, 7.2.11 and 7.2.13 the estimatesclearly identify the effective damping with the inclusion of the Wong-Zakai correction termand thereby verify the expression for the effective damping given by (3.4.7).


0 2 40

0.05

0.1

0.15

λ / <λ>

ηλ

0 2 40

0.2

0.4

λ / <λ>

ηλ

0 2 40

0.04

0.08

λ / <λ>

ηλ

0 2 40

0.1

0.2

0.3

λ / <λ>

ηλ

0 2 40

0.02

0.04

λ / <λ>

ηλ

0 2 40

0.05

0.1

0.15

λ / <λ>

ηλ

Figure 7.2.11: Estimation of ηλ, γ = 0.5, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

In the rows 2 and 3 in Figures 7.2.8 - 7.2.13 the stiffness is non-linear, with stiffness non-linearity parameters 〈ω〉2 = 2 and 〈ω〉2 = 4, respectively, and the non-linearity is thusstrongest in the last rows. The non-linearity in the stiffness is seen to have a strong effecton the damping estimation. A minor bias is introduced in the results shown in the secondrow in the form of an overestimation of the damping. This bias is further pronounced inthe results given in the last row. the error arises because of the drift in the energy, whichtakes place within half a period of vibration. The approximation for the covariance functionat a given energy level (6.3.5) assumes that the energy stays approximately constant, and arequirement for accuracy in the damping estimation is therefore an energy envelope, whichis approximately constant over half a period of vibration. In reality the energy will fluctuatesubstantially within this period of time, even if the system is lightly damped. For systems


0 2 4−0.2

0

0.2

0.4

0.6

λ / <λ>

ηλ

0 2 4−0.5

0

0.5

1

1.5

λ / <λ>

ηλ

0 2 4−0.2

0

0.2

0.4

λ / <λ>

ηλ

0 2 4−0.5

0

0.5

1

λ / <λ>

ηλ

0 2 4−0.1

0

0.1

0.2

λ / <λ>

ηλ

0 2 4−0.2

0

0.2

0.4

0.6

λ / <λ>

ηλ

Figure 7.2.12: Estimation of ηλ, γ = −1.5, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

with linear stiffness this is of minor importance, since the harmonic function in the expressionfor the covariance function has the same frequency for all energy levels. If the stiffness isnon-linear, on the other hand, the frequency of the harmonic function will change withenergy level, and the drift in energy will result in an averaging of oscillations taking placewith different frequencies. The effect will be a smearing of the peak, and the result in termsof the final estimate is an overestimation of the damping. The effect is illustrated in Figure7.1.7 where the covariance functions for a system with non-linear stiffness are shown.

In the first and second column of Figures 7.2.8 - 7.2.13 the damping magnitude parameteris 〈ζ〉 = 0.01 and 〈ζ〉 = 0.04, respectively. In the results for 〈ζ〉 = 0.04 both the level ofscatter and the bias appear to be smaller than in the case where 〈ζ〉 = 0.01. A higher level


0 2 40

0.05

0.1

0.15

λ / <λ>

ηλ

0 2 40

0.2

0.4

0.6

λ / <λ>

ηλ

0 2 40

0.05

0.1

λ / <λ>

ηλ

0 2 40

0.2

0.4

λ / <λ>

ηλ

0 2 40

0.02

0.04

0.06

λ / <λ>

ηλ

0 2 40

0.1

0.2

λ / <λ>

ηλ

Figure 7.2.13: Estimation of ηλ, γ = 1, δ = 1. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1st row)〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

of damping will result in a higher decrease in the envelope of the covariance function. Theestimation of the damping is based on estimating this difference, i.e. the difference betweenA1 and A2 in (6.3.6). The larger this difference the more accurate the estimate, and a higherlevel of damping will therefore result in a reduction of scatter in the estimate. The reductionin the magnitude of the bias for the case 〈ζ〉 = 0.04 compared to the case 〈ζ〉 = 0.01 ismore difficult to understand. For 〈ζ〉 = 0.01 the fluctuations in the energy process should besmaller than in the case where 〈ζ〉 = 0.04 and since the bias is a result of changes in energylevel, one would intuitively assume, that the bias should be strongest for 〈ζ〉 = 0.04. Thereason for the opposite behaviour of the results is not presently known.

An interesting feature of the technique is that it does not seem to be influenced by non-


0 2 40

0.05

0.1

0.15

λ / <λ>

σ2

0 2 40

0.2

0.4

λ / <λ>

σ2

0 2 40

0.05

0.1

λ / <λ>

σ2

0 2 40

0.1

0.2

0.3

λ / <λ>

σ2

0 2 40

0.02

0.04

0.06

λ / <λ>

σ2

0 2 40

0.1

0.2

λ / <λ>

σ2

Figure 7.2.14: Estimation of σ(λ)2, γ = 0, δ = 0. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

linearity in the damping and appearance of parametric excitation, at least within the par-ameter range investigated here. This is most clearly observed by comparing the results inFigure 7.2.8, where γ = δ = 0, with the results in Figure 7.2.13, where γ = δ = 1. Theaccuracy of the damping estimation in these two cases is of the same order of magnitude.

The results shown in Figure 7.2.12 correspond to a system with a stable limit cycle. Here, itis interesting to observe that the method is capable of identifying the negative damping in therange of low energy levels. The λ-value of the estimated points in all figures is not the valueof the energy at time τ = 0, but the average energy in the interval 0 < τ < 1

2T (λ), i.e. the

mean energy in the interval between the points A1 and A2 in Figure 6.3.1. In Figure 7.2.12the λ-values of the estimated points are seen to be located closer to 〈λ〉 when comparing


0 2 40

0.04

0.08

λ / <λ>

σ2

0 2 40

0.1

0.2

0.3

λ / <λ>

σ2

0 2 40

0.02

0.04

0.06

λ / <λ>

σ2

0 2 40

0.1

0.2

λ / <λ>

σ2

0 2 40

0.02

0.04

λ / <λ>

σ2

0 2 40

0.05

0.1

0.15

λ / <λ>

σ2

Figure 7.2.15: Estimation of σ(λ)2, γ = 0.5, δ = 0. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

with the other figures. This is the result of high numerical values of damping for low andhigh energy levels, as discussed earlier in connection with the stiffness estimation.

7.2.7 Excitation estimation

The last step consists in determining the excitation amplitude function σ(λ) and follows theprocedure described in Section 6.4.3. Initially, estimates of the argument function ψ(λ) areobtained by the procedure described in Section 6.4.2. These estimates will not be shownhere, but a few things should be mentioned about the approach and the quality of the results.


0 2 40

0.1

0.2

0.3

λ / <λ>

σ2

0 2 40

0.5

1

λ / <λ>

σ2

0 2 40

0.1

0.2

0.3

λ / <λ>

σ2

0 2 40

0.4

0.8

λ / <λ>

σ2

0 2 40

0.1

0.2

λ / <λ>

σ2

0 2 40

0.2

0.4

0.6

λ / <λ>

σ2

Figure 7.2.16: Estimation of σ(λ)2, γ = 0, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

The function ψ(λ) is approximated by a 3rd order polynomial, i.e. N = 3 in (6.4.7a). Thecoefficients in the polynomial are obtained via (6.4.4) by a least squares fit, where q(λ) isobtained by an arbitrarily scaled histogram of the kinetic energy at mean level crossings.The estimates of ψ(λ) are generally very accurate.

However, the estimate of ψ(λ) is in itself not important. As seen by (6.4.6), which is theequation used for the estimate, it is the derivative of ψ(λ), which is important. This functionis generally more sensitive than ψ(λ) with respect to the number of histogram points andthe energy-range considered. Too many points will lead to too much scatter, and if too fewpoints are used the polynomial will take small local deviations into account. In the presentcase a range of 0 < λ < 6〈λ〉 is considered for the fit and 30 histogram points are used in


0 2 40

0.1

0.2

λ / <λ>

σ2

0 2 40

0.4

0.8

λ / <λ>

σ2

0 2 40

0.05

0.1

0.15

λ / <λ>

σ2

0 2 40

0.2

0.4

0.6

λ / <λ>

σ2

0 2 40

0.05

0.1

λ / <λ>

σ2

0 2 40

0.2

0.4

λ / <λ>

σ2

Figure 7.2.17: Estimation of σ(λ)2, γ = 0.5, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

most of the cases.

The results shown in Figures 7.2.14 - 7.2.19 are based on the same stochastic records asthe results shown for the stiffness and damping estimation. The polynomial estimate of thedamping, which is required in order to determine the estimate of σ(λ), as seen by (6.4.6), isobtained from the estimated points shown in Figures 7.2.7 - 7.2.13. It is the quantity ηλ/c(λ),which is fitted by the polynomial, where c(λ) can be estimated in a straight forward manner,as discussed in Section 6.4.3. 2nd order polynomials are used, i.e. N = 2 in (6.4.7b). As seenfrom the results shown in connection with the damping estimation a bias in the estimateis introduced when the stiffness is non-linear. Since the excitation estimation is based onthe estimate of the damping this inaccuracy must be expected to be reflected in the results


0 2 40

0.2

0.4

0.6

λ / <λ>

σ2

0 2 40

1

2

λ / <λ>

σ2

0 2 40

0.2

0.4

λ / <λ>

σ2

0 2 40

0.5

1

1.5

λ / <λ>

σ2

0 2 40

0.1

0.2

0.3

λ / <λ>

σ2

0 2 40

0.5

1

λ / <λ>

σ2

Figure 7.2.18: Estimation of σ(λ)2, γ = −1.5, δ = 0.5. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04.1st row) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

shown in Figures 7.2.14 - 7.2.19.

σ(λ)2 is represented by a 3rd order polynomial (N = 3 in (6.4.7c)) in the results shown inFigures 7.2.14 - 7.2.19, where the dashed line is the estimate and the solid line is the realfunction. Generally, the results are seen to be most accurate in the cases of linear stiffness,i.e. the results shown in the first row of the subfigures.

The results shown in Figures 7.2.14 - 7.2.15, 7.2.17 and 7.2.19 are quite similar in terms ofaccuracy. In the case of linear stiffness (〈ω〉2 = 1) the error is only a few percent for thesefour cases. For moderate non-linearity in the stiffness (〈ω〉2 = 2) it can roughly be statedthat the error is of the order of magnitude 5 - 10 %. The error increases to 10 - 20 % in the


0 2 40

0.1

0.2

λ / <λ>

σ2

0 2 40

0.4

0.8

λ / <λ>

σ2

0 2 40

0.05

0.1

0.15

λ / <λ>

σ2

0 2 40

0.2

0.4

0.6

λ / <λ>

σ2

0 2 40

0.05

0.1

λ / <λ>

σ2

0 2 40

0.2

0.4

λ / <λ>

σ2

Figure 7.2.19: Estimation of σ(λ)2, γ = 1, δ = 1. left) 〈ζ〉 = 0.01, right) 〈ζ〉 = 0.04. 1strow) 〈ω〉2 = 1, 2nd row) 〈ω〉2 = 2, 3rd row) 〈ω〉2 = 4.

case of strong non-linearity in the stiffness (〈ω〉2 = 4). The magnitude of the damping doesnot seem to have a strong influence on the quality of the estimate. Considering the accuracyof the damping estimation discussed earlier (and on which the excitation estimation is based)the results must be considered satisfactory.

The results shown for the case of linear damping (γ = 0) and parametric excitation (δ = 12)

in Figure 7.2.16 are less satisfactory. Here only the first case (〈ω〉2 = 1, 〈ζ〉 = 0.01) havean error within a few percent. The results for the cases of light damping (〈ζ〉 = 0.01) andnon-linear stiffness (〈ω〉2 = 2, 4) show very high errors of about 100 % for the highest energylevels. It is difficult to give a reasonable explanation for the lack of accuracy in this case, butthe combination of linear damping and parametric excitation may generally lead to strange


behaviour of the response or even instability in the case of strong intensities of the parametricexcitation term.

The results shown in Figure 7.2.18 correspond to a system with a stable limit cycle and theresults are also in this case less satisfactory. The estimates seem to be equally inaccurate forall six parameter combinations. Though the mean value of σ(λ)2 is predicted fairly accuratelythe inclination of the functions is underestimated. In this case there are, however, somereasons for the increasing error. Firstly, the relative error on the estimate of the dampingis relatively high in the range where ηλ crosses zero. Secondly, the large drift in energyleads to a reduction of the energy range for which the damping can be estimated, see Figure7.2.12. Finally, the damping is numerically quite large in this particular case for high and lowvalues of the energy, as discussed earlier. Since most of the theory used in the estimationprocedure is based on the assumption of a lightly damped system, this may also lead toreduced accuracy of the estimate.

Chapter 8

Modelling of Vortex-inducedVibrations

Fluid flow around a structural element leads to the formation of vortices in the wake of theflow. When the Reynolds number exceeds a certain level the flow becomes unstable resultingin the shedding of vortices, which will form the socalled von Karman vortex-street. Thismay happen to structural members under certain meteorological conditions. The alternatingshedding of vortices in the wake of the flow will result in periodic loading in the across-winddirection. For slender structures this may give rise to violent across-wind vibration if theshedding frequency is close to the eigenfrequency of the structure. Furthermore, the mo-tion of the structure can modify the shedding frequency, such that it syncronizes with theeigenfrequency of the structure, a phenomenon known as lock-in. In extreme cases, thevortex-induced vibrations may lead to failure of the structure, but even if this extreme eventdoes not occur, the vortex-induced vibrations can cause fatigue damage in the structural ma-terial. It is therefore important to be able to predict the vortex-induced response accurately.The shedding of vortices from a structural element is sketched in Fig. 8.0.1.

Ux

Figure 8.0.1: Sketch of vortex-shedding from structural element.

The model, which will be discussed in this chapter, was originally introduced by Rudingerand Krenk (2000c, 2002b) in a simplified version corresponding to the system consideredin Section 5.3. The presentation given in this chapter follows an extension of this model,Rudinger and Krenk (2002a).

109

110 Chapter 8. Modelling of Vortex-induced Vibrations

8.1 Background

Computation of a flow field around a moving elastic body is a very complicated task. Thegeneral case of a turbulent three-dimensional flow cannot be analyzed today, even withadvanced computer codes. Design codes are therefore normally based on simple oscillatormodels, where the structure is represented by a single-degree-of-freedom, and the pressuregenerated by the flow field is included in the excitation. The models used focus on eitherGaussian response or harmonic response. The spectral model, introduced by Vickery andClark (1972), is a stochastic model where the lift coefficient is described as a zero-meanGaussian process. Turbulence effects are included by assuming a linear relationship betweenthe turbulence intensity and the bandwidth of the spectrum of the lift coefficient, Vickeryand Basu (1983a,b). Since the model is linear, the response is also Gaussian. However,experimental results by van Koten (1984) indicate that the response is sinusoidal for windspeeds close to the critical wind speed.

The structural damping is often represented by the Scruton number as defined by

Sc =4πζsm

ρd2(8.1.1)

where ζs is the damping ratio of the structure, m is the effective mass of the structure, ρ isthe density of the air and d is a length scale of the structure, e.g. the diameter. Accordingto the spectral model the response magnitude is inversely proportional to the square rootof the structural damping. However, recent data analysis by Lollesgaard (2000) seems toindicate that the response magnitude is inversely proportional to the structural damping.Some of the results are shown in Fig. 8.1.1, where the maximum response over a period of60 s is depicted as a function of the Scruton number. The different data sets correspond todifferent turbulence intensities and to different turbulence length scales. The results indicateinverse proportionality between response magnitude and structural damping.

100

101

102

10−2

10−1

100

Sc

x max

/ d

100

101

102

10−2

10−1

100

Sc

x max

/ d

Figure 8.1.1: Response magnitude as function of the Scruton number, Lollesgaard (2000).

8.2 Stochastic Oscillator Model 111

The harmonically excited model, introduced by Ruscheweyh (1982), assumes harmonic ex-citation with amplitude inversely proportional to the structural damping. The model islinear and the response is therefore also harmonic and proportional to the amplitude of theexcitation. Thus, the model has the important properties of inverse proportionality betweenresponse magnitude and structural damping, and sinusoidal distribution of the response.However, the model does not include the effects of turbulence, which may reduce the re-sponse considerably, and the transition from a Gaussian to a sinusoidal response, as thewind speed approaches critical wind speed, is not captured by the model. The shortcomingsof the models used in codes have also been illustrated by Ruscheweyh and Sedlacek (1988),where results from three different design codes are seen to deviate substantially.

The above observations indicate a need for development of models capable of capturing thecharacteristics of the response more accurately. In the double oscillator model proposed byKrenk and Nielsen (1999), the moving mass of fluid is modelled as an additional degree offreedom. The model is consistent with respect to the energy exchange between the structureand the fluid, and has been extended to include effects of turbulence, Nielsen and Krenk(1998). A model of this type has also been applied by Lewandowski (2002) to study thevortex-induced vibrations of beams by a finite element approach. The double oscillator modelgives an accurate representation of the two response branches in the lock-in regime, andleads to probability distributions of the response, which are in agreement with experimentalresults. However, when turbulence is included, the model is not analytically tractable, andinvestigation relies on stochastic simulation. The model is therefore assumed to be toocomplicated to form the basis of design procedures used in codes.

In the following a stochastic oscillator model is considered. The excitation is assumed to bean ideal white noise, which enables determination of approximate analytical solutions by themethods discussed in Chapters 2 and 3. It is demonstrated that this model has the correctqualitative behaviour with respect to the response characteristics discussed in the above.

8.2 Stochastic Oscillator Model

The structure is assumed to behave as a stochastically excited non-linear single-degree-of-freedom system. The angular frequency of the structure is ω0 and the damping ratiocorresponding to linear structural damping is ζs. The equation of motion is assumed to begiven by

x + h(λ) x + ω20x = W0(t) , λ = 1

2x2 + 1

2ω2

0x2 (8.2.1)

where W0(t) is a white noise process with intensity S0 and λ is the mechanical energy. h(λ)is the energy dependent damping function given by

h(λ) = ω0

(2(ζs − ζa) + ζ2

s αλ)

(8.2.2)


α and ζa are positive parameters. The parameter ζa represents negative aerodynamic damp-ing. The non-linear structural damping term ζ2

s αλ ensures a self-limiting response whenζa > ζs. For ζa = α = 0 the system will be linear with the following standard deviation andmean energy level

σ2x =

E[λ]

ω20

=πS0

2ζsω30

(8.2.3)

so in this limit, where the response is Gaussian, the magnitude is inversely proportional tothe square root of the structural damping. In the limit S0 = 0 and ζa > ζs a stable solutionis given by harmonic motion at the following energy level

λ =2(ζa − ζs)

ζ2s α

(8.2.4)

corresponding to h(λ) = 0. In this limit the response has a sinusoidal distribution and themagnitude, which is proportional to

√λ, is inversely proportional to the structural damping

when ζa ζs. The probability density is given by, Caughey (1971),

pλ(λ) = C exp(− 1

πS0

∫ λ

0

h(e)de)

= C exp(− ω0 (2ζαλ + 1

2ζ2s α

2λ2)

παS0

)(8.2.5)

where ζ = ζs − ζa and C is a normalizing constant. It is observed, that the distribution ofαλ depends only on the parameters ζω0/αS0 and ζ2

s ω0/αS0.

0 1 2 3 4 510

−2

10−1

100

p E

[λ]

λ / E[λ]

ζ ω0 / α S

0 = 0.1

10−1

100

101

10−1

100

101

p E

[λ]

λ / E[λ]

ζ ω0 / α S

0 = −0.1

Figure 8.2.1: Energy probability density, ζ2s ω0/αS0 = 10−4 (−) , 10−2 (−−) , 100 (− · −).

In Fig. 8.2.1 the probability distribution is given for different combinations of the two par-ameters ζω0/αS0 and ζ2

s ω0/αS0. In Fig. 8.2.1a ζ is negative and in Fig. 8.2.1b ζ is positive.Figure 8.2.1a contains the special case of a Gaussian distribution of the displacement forζ2s ω0/αS0 → 0, which corresponds to an exponential distribution of the energy. In Fig. 8.2.1b

the linear part of the damping term is negative, and this class of solutions contains the spe-cial case of a concentrated distribution for the energy as ζ2

s ω0/αS0 → 0, which corresponds

8.3 Effects of Turbulence on Aerodynamic Damping 113

10−4

10−3

10−2

10−1

100

101

102

103

ζs

E[λ

α]½

10−4

10−3

10−2

10−1

10−2

10−1

100

ζs

V

λ

Figure 8.2.2: Mean value and coefficient of variation, ω0/αS0 = 1, ζa = 0 (−) , 0.01 (−−) ,0.1 (− · −).

to a sinusoidal distribution of the displacement. The coefficient of variation for the energyis defined as

Vλ =σλ

E[λ](8.2.6)

where σλ is the standard deviation of the energy. For a Gaussian response Vλ = 1 and fora sinusoidal response Vλ = 0. The value of Vλ is thus a measure of the type of response. InFig. 8.2.2a the mean value of

√λα is given as a function of ζs for ω0/αS0 and various values of

ζa. In Fig. 8.2.2b the coefficient of variation is given for the same values. It is observed thatwhen ζs > ζa the non-dimensional amplitude

√λα is approximately proportional to 1/

√ζs

and the coefficient of variation is close to unity indicating a predominantly Gaussian response.When ζa > ζs the non-dimensional amplitude is approximately proportional to 1/ζs and thecoefficient of variation is very small indicating a distribution of the displacement close tosinusoidal. This behaviour is in agreement with experimental observations as discussed inthe previous section.

8.3 Effects of Turbulence on Aerodynamic Damping

If the effects of turbulence on the aerodynamic damping are taken into account, the originalsystem (8.2.1) is modified the following way

x + ω0

(2(ζs − ζa) + ζ2

s αλ + W1(t))

x + ω20x = W2(t) (8.3.1)

where W1(t) and W2(t) are independent white noise processes. It is assumed that W1(t) canbe related to the along wind turbulence, since this term has the effect of destabilizing the non-zero equilibrium for ζa > ζs, and similarly W2(t) is related to the across-wind turbulence.


10−1

100

101

10−2

10−1

100

λ/E[λ]

p λE[λ

]

10−1

100

101

10−2

10−1

100

λ/E[λ]

p λE[λ

]Figure 8.4.1: Energy probability density, ζs = 0.05, ζ = −0.1, αS22/ω0 = 0.1, a) ω0S11 =0.01, b) ω0S11 = 0.1.

The intensities of the processes S11 and S22 will therefore be functions of the turbulenceintensities.

The procedure applied to obtain an approximate solution in terms of the probability densityof the energy was shown in Section 2.3.4 and will not be discussed here. An approximatesolution is given by

pλ(λ) = C(

32πω0S11αλ +

παS22

ω0

)ν

exp(− 2ζ2

s

3πω0S11

αλ)

ν =4ζ2

s αS22

3ω20S11

− 4ζ

3πω0S11

− 23

(8.3.2)

It is seen that the solution for the non-dimensional energy λα depends on ζ, ζs, S11ω0 andαS22/ω0. The effective damping of the system introduced in (3.3.17) is in the present caseevaluated as

ηλ = ω0ζ2s αλ + 2ζω0 − πω2

0S11 (8.3.3)

where the last term is due to the Wong-Zakai correction. An approximation to the powerspectral density is obtained by the method introduced in Section 3.3.3. Only one termappears in the conditional spectrum given by (3.3.18) due to the linearity in the stiffness.The integral in (3.3.7) is evaluated numerically in the following examples.

8.4 Numerical Examples

The simulation method discussed in Chapter 4 is now used to obtain samples of the stochasticresponse. The probabilistic density of the energy and the power spectral density of the re-sponse are evaluated from the records and compared to the approximate theoretical solution.Furthermore, the peak factor of the response will be investigated.

8.4 Numerical Examples 115

10−1

100

101

10−4

10−2

100

102

ω/ω0

S xω0 /

E[λ

]

10−1

100

101

10−4

10−2

100

102

ω/ω0

S xω0 /

E[λ

]Figure 8.4.2: Spectral density of displacement, ζs = 0.05, ζ = −0.1, αS22/ω0 = 0.1, a)ω0S11 = 0.01, b) ω0S11 = 0.1.

In Fig. 8.4.1 the probability density is shown for ζs = 0.05, ζ = −0.1 and αS22/ω0 = 0.1. InFig. 8.4.1a ω0S11 = 0.01 and in Fig. 8.4.1b ω0S11 = 0.1. The dots correspond to histogramsof simulated records. The solid lines represent the theoretical expression (8.3.2). The effectof the parametric excitation term W1(t) is a broadening of the probability mass. As seen byFig. 8.4.1 the results from stochastic simulation agree well with the approximate theoreticalexpression.

In Fig. 8.4.2 the spectra of the displacement are shown for the same parameter combinationsas those considered in Fig. 8.4.1. Again, the points correspond to analysis of the simulatedrecords (Fast Fourier Transform of the displacement record), while the solid lines representthe theoretical expression obtained from (3.3.18) and (3.3.7) with the effective damping givenby (3.3.17). The effect of increasing the intensity of the parametric white noise term W1(t)is seen to be a broadening of the spectral peak. The approximation is less accurate thanthe approximation to the energy probability density, though still satisfactory. It should bementioned that an improved expression for the power spectral density can be obtained bythe method discussed in Section 5.3.4 (see the results shown in Figure 5.3.9).

The peak factor is given by the ratio between the maximum value of the response and thestandard deviation during a given time T ,

kp =E[xmax]T

σx

(8.4.1)

The observation period T is normally large compared to a typical period of oscillation. In thepresent case an observation period of 1000 natural periods is considered (T = 1000 · 2π/ω0).If the response is purely harmonic the peak factor is kp =

√2 and if the response is Gaussian

the peak factor is approximately 3.87 for a time interval of 1000 natural periods. It is difficultto obtain analytical expressions for the peak factor for non-linear systems, as the ones givenby (8.2.1) and (8.3.1), and the peak factor is therefore estimated from records obtained bystochastic simulation.


10−4

10−3

10−2

10−1

1

2

3

4

5k p

ζs

10−4

10−3

10−2

10−1

1

2

3

4

5

k p

ζs

Figure 8.4.3: Peak factor as function of structural damping ratio, ω0S11 = 0, αS22/ω0 = 0.01,a) ζa = 0.001, b) ζa = 0.01.

In Fig. 8.4.3 the peak factor is shown as a function of the structural damping ratio forω0S11 = 0 and αS22/ω0 = 0.01. In Fig. 8.4.3a ζa = 0.001 and in Fig. 8.4.3b ζa = 0.01. Thedashed lines correspond to peak factors for purely sinusoidal response and for narrow bandGaussian response. The transition occurs in an interval around ζs = ζa. The behaviour ofthe peak factor as a function of structural damping is in agreement with experimental resultspresented by Vickery and Basu (1983a) and by Lollesgaard (2000).

8.5 Discussion

An important feature of the model is the representation of the excitation term as a whitenoise process, which enables the determination of approximate analytical solutions. Clearly,the excitation originating from vortex-shedding is not an ideal broad band process. However,the inclusion and proper scaling of the non-linear structural damping term in the equationof motion introduces a scaling of the response magnitude for critical wind speeds, where theaerodynamic damping is negative. This way the effect of narrow-band excitation is capturedby white noise excitation.

In the present case the equation of motion for vortex-induced response is assumed to begiven by (8.3.1). This equation depends on the structural damping ratio ζs, but informationconcerning the structural configuration, the wind speed relative to the critical wind speedand the turbulence intensities do not appear explicitly in the equation. This information isassumed to be hidden in the parameters ζa, α, S11 and S22.

As an alternative to the model discussed in this chapter one could assume a more generalmodel given by the following equation of motion,

x + hx + g(x) = σW (t) (8.5.1)

8.5 Discussion 117

where W (t) is a unit white noise. h is the damping function, g(x) is the stiffness and σ isthe excitation amplitude function. For systems with linear stiffness, such as those defined by(8.2.1a) and (8.3.1), g(x) = ω2

0x. The system (8.5.1) is thus more general, but includes thecase of linear stiffness as a special case. It is furthermore assumed that for a given structuralconfiguration, i.e. a circular cross section, the functions h and σ are given by

h = h(λ, ζs, U/Ucr, Iu, Ix) , σ = σ(λ, ζs, U/Ucr, Iu, Ix) (8.5.2)

where ζs again represents the structural damping ratio, U is the wind speed, Ucr is the criticalwind speed and Iu and Ix are the turbulence intensities in the longitudinal and transversedirections, respectively. λ is the mechanical energy given by

λ = 12x2 +

∫ x

0

g(x)dx (8.5.3)

which reduces to (8.2.1b) in the case of linear stiffness. For a given structure and given windconditions ζs, U/Ucr, Iu and Ix will be fixed, and the damping and excitation amplitudewill be functions of the mechanical energy only. The system will in this case belong to theclass of systems discussed in Section 2.3.3 and h(λ), g(x) and σ(λ) can be determined fromwind tunnel testing using the system identification technique described in Chapter 6 anddemonstrated in Section 7.2. The idea of proposing a stochastic model with undeterminedparameters and using system identification techniques to calibrate the model has also beenconsidered by Christensen and Roberts (1998).

In the light of the above discussion, the model proposed in this chapter should not beconsidered as an ultimate model, which (upon calibration) could be readily implemented indesign codes. The model should be seen as a way of demonstrating that the experimentallyverified behaviour can be captured by relatively simple non-linear stochastic oscillator modelswith broad band excitation. For practical purposes it would be more reasonable to assume amodel of the general type given by (8.5.1). This model can be directly calibrated as discussedin the previous paragraph. For a general discussion of wind load on structures and designprocedures, see e.g. Dyrbye and Hansen (1997).

Chapter 9

Conclusion

The main findings in the three parts of the thesis will be summarized below. A few commentsaddressing the future aspects of the problems considered in the thesis are also given.

9.1 Probability Density and Spectral Density

In the first part of the thesis the probability density and power spectral density of the sta-tionary response of non-linear stochastic oscillators is considered. Special classes of systemshaving exact solutions for the probability density are discussed, Caughey (1971, 1986). Itis shown how approximate analytical solutions can be obtained by determining equivalentsystems belonging to a class of systems for which an exact solution exists. For systems withexternal excitation the method is known as equivalent non-linearization, Caughey (1986),and for systems with parametric excitation the method is referred to as dissipation energybalancing, Cai and Lin (1988), and yields the same result as stochastic averaging.

The spectral density is evaluated following a method originally proposed by Krenk andRoberts (1999) for systems with additive stiffness and later extended to include systemswith parametric excitation, Krenk et al. (2002). An approximation to the spectral densityat a given energy level is obtained as the Fourier transform of a local approximation for thecovariance function. The total spectrum is obtained by integration over all energy levelswith the probability density of the energy as a weight function.

The theory is investigated considering a number of examples. Based on these numericalinvestigations some conclusions concerning the accuracy of the method can be drawn. Theprobability density, which is presented graphically in the examples, is the probability densityof the mechanical energy. The systems investigated are generally lightly damped, and forthese systems the approximate solutions for the probability density of the energy showexcellent agreement with the results obtained from stochastic simulation. Some deviations

119

120 Chapter 9. Conclusion

must be expected for systems with high levels of damping, but these cases are generally notas important in structural dynamics. Some of the damping levels considered in the examplesare actually relatively high compared to typical values for vibration sensitive structures.

As to the approximate solutions for the spectral density of the response the method generallycaptures the broadening of the resonance peak and the appearance of higher harmonics inthe case of systems with non-linear stiffness. However, there are a few cases, where theresults are not completely satisfactory. For systems with strongly non-linear stiffness themethod tends to underestimate the spectral density slightly in the quasi-static limit (at zerofrequency). In the case of a system with bilinear stiffness the theory predicts a local peakat the frequency corresponding to the first part of the bilinear stiffness function. This peakis less pronounced in the results based on stochastic simulation, as discussed in the examplein Section 5.4 and by Rudinger and Krenk (2003c).

Some systems are characterized by having a non-zero equilibrium position. This will occurif the damping is negative for small energy levels. These systems are said to have a stablelimit cycle, and the spectral densities for these types of systems are generally less accurate,especially if the probability of being in the energy range with negative damping is large. Theeffective damping appears in the expression for the spectral density at a given energy level.Since the spectral density should be positive for all energy levels the absolute value of thedamping is used in this expression. For systems with linear stiffness this is also consistentin the sense that the area under the spectral density function is equal to the variance of thedisplacement. This approach gives a reasonable representation of the peak of the spectraldensity, but the asymptotic behaviour - the behaviour for high and low frequencies - is notrepresented very well. However, if the real value of the damping (negative or positive) isused in the expression for the energy conditional spectral density, the asymptotic behaviourof the spectral density is captured with good accuracy, while the peak is underestimated.This suggests a weighting of these two ways of obtaining the result. Such a weighting isdiscussed in the examples in Section 5.3 and by Rudinger and Krenk (2002b), and is seen togive accurate results both for the asymptotic behaviour and in the region around the peak.

9.2 System Identification

In the second part of this thesis, some of the principles formulated by Krenk and Roberts(1999) are used to develop a system identification technique. The method developed hastwo features, which makes it unusual compared to other techniques used in system iden-tification. Firstly, the excitation is assumed to be unknown and must be estimated alongwith the system, and secondly, the estimates of the stiffness and damping functions arenon-parametric, i.e. no specific parametric representation of the stiffness and damping func-tions is required. The method was originally introduced by Rudinger and Krenk (2001)for systems with external excitation, and has later been extended to include systems withparametric excitation, Rudinger and Krenk (2003a). The method is applicable to lightlydamped non-linear oscillators with stationary broad band excitation.

9.3 Modelling of Vortex-induced Vibration 121

The identification procedure takes place in three steps. Initially, the stiffness is estimated bycomparing the energy at mean-level crossings and following local extremes. The procedureleads to a non-parametric estimate of the potential energy function. In the second step, thedamping is estimated from the decay of the autocovariance functions of a set of modifiedphase plane variables at different energy levels. The procedure yields a non-parametricestimate of the effective damping as a function of the energy level. Finally, the excitationintensity is estimated from a generic expression for the probability density of the energy.In this case it is necessary to use a polynomial representation. The result of this part ofthe procedure is an estimate of the excitation intensity as a function of the energy level.The procedure is slightly different in the case of a system with external excitation, since theintensity of the excitation in this case is a value and not a function of the energy.

The stiffness estimation generally gives excellent results, even for systems with very highlevels of non-linearity in the stiffness. The estimate does not seem to be influenced by non-linearity in the damping or by the presence of a parametric excitation term, and can be basedon relatively short records. For damping ratios above 5 % the accuracy of this identificationtechnique will start to decrease.

The damping estimates generally show a higher level of scatter than the stiffness estimates.This behaviour is to be expected, since the stiffness is obtained by considering the oscillations(the motion governed by the fast time scale) and the damping is obtained from the slow decayin correlation (the motion governed by the slow time scale). This tendency is also presentin identification techniques for linear systems. The estimates are very accurate for systemswith linear stiffness, but the damping tends to be overestimated in the case of systems withnon-linear stiffness. For strong non-linearities the deviations are in the range 15 - 20 %.Non-linearity in the damping and the presence of parametric excitation seem to have verylittle influence on the quality of the estimate.

The estimation of the excitation is based on the estimate of the damping. If the dampingestimate is biased, which is the case for systems with non-linear stiffness as discussed in theprevious paragraph, this inaccuracy will also influence the excitation estimate. In this casethe estimates are also very accurate for systems where the stiffness is linear. As the non-linearity in the stiffness increases the accuracy of the estimates decreases, but the deviationsare generally within 30 %. One exception is the case of a system with linear damping andparametric excitation, where high levels of error are observed (up to 100 %). The reason forthe inaccuracy in this particular case is not understood, but may be due to an approach tothe stability limit, which will be exceeded for sufficiently high intensities of the parametricexcitation term.

9.3 Modelling of Vortex-induced Vibration

In the third part of the thesis a model of vortex-induced vibrations is considered. The across-wind motion of the structural element is modelled as a stochastic oscillator with non-linear


damping and parametric excitation. The model was originally proposed by Rudinger andKrenk (2002a).

Experimental observations suggests that the probability density of the vortex-induced re-sponse undergoes a transition from a Gaussian response to a sinusoidal response, as thewind velocity approaches the critical velocity. The amplitude of the response is inverselyproportional to the structural damping in the lock-in regime and inversely proportional tothe square root of the structural damping outside this regime. It is demonstrated how thesequalities can be incorporated in a non-linear stochastic oscillator model by introducing adamping function, which depends on the structural damping. The probability density andspectral density are evaluated using the methods introduced in he first part of the thesis.Furthermore, the peak factor is investigated by stochastic simulation verifying the transitionfrom a harmonic response to a Gaussian response with increasing structural damping.

The basic idea of this last part of the thesis is not to establish a model, which can be readilyused. The objective is rather to point out that non-linear stochastic oscillators can bedesigned in such a way as to capture the characteristics of the vortex-induced response. As aconcluding remark to part three it is discussed how the identification procedure introducedin the second part could be used to calibrate the model.

9.4 Future work

The systems, which have been considered in the present work are non-linear single-degree-of-freedom systems with stationary broad band excitations. Though some situations do occurin both structural and mechanical engineering, where models of this type can be used as anapproximation, there are also many cases where more complicated models must be used.

Often several structural modes are activated by the excitation and it is thus necessary tomodel the structure as a system with several degrees of freedom. The assumption of broadband excitation requires that the bandwidth of the excitation spectrum is significantly largerthan the bandwidth of the response spectrum. In the case of wave load the energy of theexcitation is concentrated around the frequency corresponding to the period of the incomingwave, and the assumption of a broad band excitation will often be too crude. It will thereforebe necessary to model the load as a stochastic process with correlation.

In the case of wind or wave loads the assumption of stationarity is often reasonable. Thechange of the probabilistic characteristics of the excitation process is slow compared tothe typical time scales of the response. When the excitation is caused by earthquakes theassumption of stationarity does not hold, and the concept of a transient response must betaken into account.

Finally, the damping is assumed to be viscous, i.e. to depend only on the state of the systemand not on the history of the response. If memory is to be incorporated in the damping termof the system, additional state variables must be included in the model.

9.4 Future work 123

Both in terms of establishing solutions for the probability density and the power spectraldensity of the response and developing system identification procedures it must be concluded,that future work should address systems with several degrees of freedom, systems withcorrelated excitation, cases of transient loading and thereby transient response and systemswith hysteretic damping. Extension of the theory presented in this thesis to such cases is byno means trivial.

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PhD ThesesDepartment of Naval Architecture and Offshore Engineering

Technical University of Denmark · Kgs. Lyngby

1961 Strøm-Tejsen, J.Damage Stability Calculations on the Computer DASK.

1963 Silovic, V.A Five Hole Spherical Pilot Tube for three Dimensional Wake Measurements.

1964 Chomchuenchit, V.Determination of the Weight Distribution of Ship Models.

1965 Chislett, M.S.A Planar Motion Mechanism.

1965 Nicordhanon, P.A Phase Changer in the HyA Planar Motion Mechanism and Calculation of PhaseAngle.

1966 Jensen, B.Anvendelse af statistiske metoder til kontrol af forskellige eksisterende tilnærmelses-formler og udarbejdelse af nye til bestemmelse af skibes tonnage og stabilitet.

1968 Aage, C.Eksperimentel og beregningsmæssig bestemmelse af vindkræfter pa skibe.

1972 Prytz, K.Datamatorienterede studier af planende bades fremdrivningsforhold.

1977 Hee, J.M.Store sideportes indflydelse pa langskibs styrke.

1977 Madsen, N.F.Vibrations in Ships.

1978 Andersen, P.Bølgeinducerede bevægelser og belastninger for skib pa lægt vand.

1978 Romeling, J.U.Buling af afstivede pladepaneler.

1978 Sørensen, H.H.Sammenkobling af rotations-symmetriske og generelle tre-dimensionale konstruk-tioner i elementmetode-beregninger.

1980 Fabian, O.Elastic-Plastic Collapse of Long Tubes under Combined Bending and Pressure Load.

131

132 List of PhD Theses Available from the Department

1980 Petersen, M.J.Ship Collisions.

1981 Gong, J.A Rational Approach to Automatic Design of Ship Sections.

1982 Nielsen, K.Bølgeenergimaskiner.

1984 Nielsen, N.J.R.Structural Optimization of Ship Structures.

1984 Liebst, J.Torsion of Container Ships.

1985 Gjersøe-Fog, N.Mathematical Definition of Ship Hull Surfaces using B-splines.

1985 Jensen, P.S.Stationære skibsbølger.

1986 Nedergaard, H.Collapse of Offshore Platforms.

1986 Yan, J.-Q.3-D Analysis of Pipelines during Laying.

1987 Holt-Madsen, A.A Quadratic Theory for the Fatigue Life Estimation of Offshore Structures.

1989 Andersen, S.V.Numerical Treatment of the Design-Analysis Problem of Ship Propellers using VortexLattice Methods.

1989 Rasmussen, J.Structural Design of Sandwich Structures.

1990 Baatrup, J.Structural Analysis of Marine Structures.

1990 Wedel-Heinen, J.Vibration Analysis of Imperfect Elements in Marine Structures.

1991 Almlund, J.Life Cycle Model for Offshore Installations for Use in Prospect Evaluation.

1991 Back-Pedersen, A.Analysis of Slender Marine Structures.

List of PhD Theses Available from the Department 133

1992 Bendiksen, E.Hull Girder Collapse.

1992 Petersen, J.B.Non-Linear Strip Theories for Ship Response in Waves.

1992 Schalck, S.Ship Design Using B-spline Patches.

1993 Kierkegaard, H.Ship Collisions with Icebergs.

1994 Pedersen, B.A Free-Surface Analysis of a Two-Dimensional Moving Surface-Piercing Body.

1994 Hansen, P.F.Reliability Analysis of a Midship Section.

1994 Michelsen, J.A Free-Form Geometric Modelling Approach with Ship Design Applications.

1995 Hansen, A.M.Reliability Methods for the Longitudinal Strength of Ships.

1995 Branner, K.Capacity and Lifetime of Foam Core Sandwich Structures.

1995 Schack, C.Skrogudvikling af hurtiggaende færger med henblik pa sødygtighed og lav modstand.

1997 Simonsen, B.C.Mechanics of Ship Grounding.

1997 Olesen, N.A.Turbulent Flow past Ship Hulls.

1997 Riber, H.J.Response Analysis of Dynamically Loaded Composite Panels.

1998 Andersen, M.R.Fatigue Crack Initiation and Growth in Ship Structures.

1998 Nielsen, L.P.Structural Capacity of the Hull Girder.

1999 Zhang, S.The Mechanics of Ship Collisions.

1999 Birk-Sørensen, M.Simulation of Welding Distortions of Ship Sections.

134 List of PhD Theses Available from the Department

1999 Jensen, K.Analysis and Documentation of Ancient Ships.

2000 Wang, Z.Hydroelastic Analysis of High-Speed Ships.

2000 Petersen, T.Wave Load Prediction—a Design Tool.

2000 Banke, L.Flexible Pipe End Fitting.

2000 Simonsen, C.D.Rudder, Propeller and Hull Interaction by RANS.

2000 Clausen, H.B.Plate Forming by Line Heating.

2000 Krishnaswamy, P.Flow Modelling for Partially Cavitating Hydrofoils.

2000 Andersen, L.F.Residual Stresses and Deformations in Steel Structures.

2000 Friis-Hansen, A.Bayesian Networks as a Decision Support Tool in Marine Applications.

PhD ThesesMaritime Engineering · Department of Mechanical Engineering

Technical University of Denmark · Kgs. Lyngby

2001 Lutzen, M.Ship Collision Damage.

2001 Olsen, A.S.Optimisation of Propellers Using the Vortex-Lattice Method.

2002 Rudinger, F.Modelling and Estimation of Damping in Non-linear Random Vibration.

Studentertorvet, Building 101EDK-2800 Kgs. LyngbyDenmarkPhone + 45 4525 1360Fax + 45 4588 [email protected]

Maritime Engineering

Department ofMechanical Engineering

Technical Universityof Denmark

ISBN 87-89502-64-7

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