Operational Modal Analysis in the Presence of Harmonic Excitations

Operational Modal Analysis in the

Presence of Harmonic Excitations

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 10 januari 2005 om

10:30 uur

door Prasenjit MOHANTY,

Bachelor in Technology, Indian Institute of Technology,

Kharagpur, India,

Prof. dr. ir. D.J. Rixen, M.Sc. Samenstelling promotiecommissie:

Prof. dr. ir. D.J. Rixen, Technische Universiteit Delft, promotor Prof. Dr.-Ing. M. Link, Universit¨at Kassel, Duitsland

Prof. dr. ir. A. de Boer, Universiteit Twente

Prof. dr. ir. M. Steinbuch, Technische Universiteit Eindhoven Prof. dr. ir. P.M.J. Van den Hof, Technische Universiteit Delft Prof. dr. ir. J.A. Mulder, Technische Universiteit Delft

iii

ACKNOWLEDGEMENTS

Since November 2000, I have enjoyed being a member of Engineering Mechanics group of the Mechanical Engineering and Marine Technology department, Delft University of Technology. I am very grateful to all my colleagues for creating such a wonderful atmosphere for research. I particularly appreciated the timely help of Jos van Driel and Jan Sterk in setting up for the experiments. I want to thank Nadine Conza, Ester Vlaanderen, Corinne du Burck, Cornelis van’t Hof, Dr. Paul van Woerkom, Prof. Leo Ernst and Dr. Arend L. Schwab for their continuous support during my stay in Delft.

I want to express my sincere gratitude to Prof. Daniel J. Rixen for his continuous innovating and challenging supervision. It was the luckiest moment in my career to have him as my supervisor as he is such a wonderful person as well as a wonderful professor.

I also want to thank Dr. Thomas G. Carne of Sandia National Laboratories, Albuquerque for his valuable suggestion during my research.

Delft, January 2005 Prasenjit Mohanty

vii

Samenvatting

De stabiliteit, betrouwbaarheid en levensduur van een constructie wordt voor een belangrijk deel bepaald door het dynamisch gedrag van de constructie. Daarom is het in veel geavanceerde technische toepassingen essentieel om: de dynamica van nieuwe ontwerpen door simulaties te kunnen voorspellen, de dy-namische kenmerken van prototypen te kunnen identificeren en bestaande syste-men te kunnen controleren. Operationele Modale Analyse (OMA) is een prak-tische methode om de dynamische eigenschappen van bestaande constructies tijdens bedrijf te onderzoeken. Het staat namelijk toe eigenparameters van con-structies te identificeren onder willekeurige excitatie, dit terwijl de krachtinput niet kan worden gemeten zoals vaak het geval is tijdens bedrijf.

Nochtans hebben de OMA methode beperkingen wanneer deze wordt toegepast op praktische gevallen. ´E´en van die beperking van OMA is dat de opgelegde ex-citatie aan het systeem een stochastische witte ruis moet zijn. Dit betekent dat, als er naast de witte ruis ook harmonische belastingen aanwezig zijn, de stan-daard OMA procedure niet kan worden toegepast. Bij de identificatie kunnen de extra harmonische responsies soms als virtuele modes worden beschouwd, maar wanneer de harmonische excitatie frequenties dicht bij eigenfrequenties van de constructies liggen is de OMA methode onbruikbaar geworden.

In dit proefschrift worden drie nieuwe procedures gepresenteerd om de toepas-baarheid van de standaard OMA methode uit te breiden. Indien de onbekende harmonische krachten kunnen worden samengevoegd tot een combinatie van sto-chastische witte ruis en harmonische excitatie kunnen, gebaseerd op het principe van de ”Natural Excitation Technique” (NExT), de drie procedures worden toegepast om de modale parameters van structuren te bepalen. Een dergelijke combinatie stochastische en harmonische excitatie is een vrij algemeen verschi-jnsel: bij voorbeeld de excitatie van de wind op de wieken van een windmolen en het akoestische lawaai in een auto kunnen worden beschouwd als de som van een harmonische en een stochastische excitatie.

In dit proefschrift worden drie identificatietechnieken besproken die een aan-passing zijn van de standaard methode zodanig dat bij de identificatie rekening kan worden gehouden met harmonische responsiecomponenten die een bekende frequentie hebben. In het bijzonder is de efficiency bekeken van de nieuwe pro-cedures welke de eigenfrequenties, de modale demping en de trillingsvormen bepalen.

De belangrijkste en originele bijdragen van dit proefschrift zijn:

• In hoofdstuk 3 wordt een gewijzigd Least Square Complexe Exponenti¨ele

(LSCE) methode gepresenteerd, welke de toepassing van OMA bij har-monische excitatie behandelt met als ingangssignaal de auto- en kruiscor-relatie van de responsies.

• In overeenstemming met de gewijzigde LSCE wordt in hoofdstuk 4 een

methode is gebaseerd op de ”Single Station Time Domain” (SSTD) meth-ode en is in het bijzonder geschikt voor de OMA methmeth-ode in de aan-wezigheid van harmonische excitaties.

• In hoofdstuk 5 wordt een gewijzigde ”Eigensystem Realization Algorithm”

(ERA) methode voorgesteld om effici¨ent de OMA methode te kunnen toepassen bij de aanwezigheid van harmonische excitaties. In tegenstelling tot de eerste twee methodes worden bij deze derde methode de modale pa-rameters niet direct ge¨ıdentificeerd maar eerst wordt de lineaire systeem-matrix bepaald waaruit de eigenparameters dan kunnen worden berekend.

• In hoofdstuk 6 wordt een methode voorgesteld om de trillingsvormen te

bepalen via de OMA methode in de aanwezigheid van harmonische exci-taties. De modale parameters zoals bepaald met de methode van Hoofd-stuk 3, 4 of 5 zijn een startpunt voor deze berekeningen.

De voorbeelden die in dit proefschrift worden gepresenteerd (een scharnierend ondersteunde balk en een vrije plaat) hebben aangetoond dat de voorgestelde methodes de nauwkeurigheid van de identificatie aanzienlijk verbeteren in vergelijk-ing met standaardtechnieken. Het toepassen van dezelfde technieken op meer complexe constructies zal waarschijnlijk dezelfde verbetering laten zien. De toepassing van de gepresenteerde methode en technieken op bestaande indus-tri¨ele constructies viel echter buiten de inhoud van dit proefschrift.

ix

Summary

The dynamic behavior of structures is a primordial factor determining its struc-tural stability, reliability and lifespan. Therefore, in many advanced engineering applications, it has become essential to predict the dynamics of new designs through simulations, to identify the dynamic characteristics of prototypes and finally to monitor systems in operation. Operational Modal Analysis (OMA) provides an interesting manner to investigate the dynamic properties the ture while in operation. Indeed it allows identifying eigenparameters of struc-tures excited randomly and for which the force inputs cannot be measured, as is often the case in operation.

However OMA methods have limitations when applied to practical cases. One limiting constraint of OMA is that the non-measured excitation to the sys-tem in operation must a stochastic white noise. This implies that if harmonic excitations are present in addition to random forces, the standard OMA pro-cedures cannot be applied in a straightforward way. The harmonic response components can sometimes be considered as virtual modes in the identification, but when the harmonic excitation frequencies are close to eigenfrequencies the standard OMA approaches may break down.

In this thesis, three new procedures are proposed to extend the applicability of the standard OMA methods. Based on the principle of Natural Excitation Technique (NExT), all the procedures can be used to extract modal parameters of structures in operation condition when the unknown input forces can be assimilated to a combination of stochastic white noise and harmonic excitation. Such combination of random and harmonic excitations are quite common in real life: wind excitations and acoustic noise for instance can be considered as stochastic and harmonic excitation forces can arise due for instance to the presence of rotating components.

This thesis discusses three identification algorithms adapted from standard methods in order to explicitly take into account harmonic response components of known frequency. In particular we have investigated the efficiency of the new procedures to properly identify eigenfrequencies, modal damping and mode shapes.

The major and original contributions of this thesis are:

• In chapter 3, a modified Least Square Complex Exponential (LSCE) method

is proposed, which handles as input the auto- and cross-correlation of the response signals to perform OMA in the presence of harmonic excitations.

• Similar to the modified LSCE, in chapter 4, a modified Ibrahim Time

Domain (ITD) method, based on the Single Station Time Domain (SSTD) method is proposed which also performs OMA in the presence of harmonic excitations.

• In chapter 5, a modified Eigensystem Realization Algorithm (ERA) method

is presented to perform efficiently OMA in the presence of harmonic exci-tations. Unlike the first two methods, this third approach does not directly

identify modal parameters but it first identifies the linear system matrix from which eigenparameters are then found.

• In chapter 6, a method has been proposed to identify mode shapes by

OMA in the presence of harmonic excitations. In this chapter modal parameters computed by the methods presented in the previous chapters are used as starting point to derive mode shapes.

The examples used in this thesis work (a simply supported beam and a free plate) have shown that the proposed methods significantly improve the accu-racy of the identification compared to standard OMA techniques and therefore applying them to complex structures is expected to yield similar improvements. Nevertheless application to industrial case studies was beyond the scope of the present thesis.

Contents

Summary in Dutch vii

Summary in English ix

1 Introduction 1

1.1 Research context . . . 2

1.2 Scientific challenge and thesis topic . . . 7

1.3 Outline and structure of the thesis . . . 8

2 Operational Modal Analysis 11 2.1 Introduction . . . 12

2.2 OMA: an overview . . . 13

2.3 The Natural Excitation Technique (NExT) . . . 14

2.3.1 Introduction . . . 14

2.3.2 Modal analysis of linear structures . . . 14

2.3.3 Derivation of correlation functions as impulse response functions . . . 17

2.4 OMA in the presence of harmonic excitation . . . 20

2.5 Conclusion . . . 21

3 Modified LSCE method for harmonics 23 3.1 Introduction . . . 24

3.2 Least Square Complex Exponential method . . . 24

3.3 Modification of LSCE method to Include Harmonic Components 26 3.4 Experimental example . . . 27

3.4.1 Pure stationary white noise excitation . . . 28

3.4.2 Single sine harmonic at 23.225Hz . . . 29

3.4.3 Two square shaped harmonics . . . 30

3.4.4 Three additional harmonics . . . 32

3.4.5 Robustness of the modified LSCE . . . 33

3.5 Conclusion . . . 35

3.6 Possible further developments . . . 37 xi

4 Modified ITD method for harmonics 41

4.1 Introduction . . . 42

4.2 A variant of the ITD . . . 42

4.3 A SIMO-SSTD method . . . 44

4.3.1 Modified SIMO-SSTD for harmonics . . . 45

4.4 Experimental examples . . . 47

4.4.1 Pure stationary white noise excitation . . . 47

4.4.2 Two additional harmonics . . . 48

4.4.3 Three additional harmonics . . . 50

4.4.4 Robustness of the modified SIMO-SSTD . . . 51

4.5 Conclusion . . . 52

5 Modified ERA method for harmonics 57 5.1 Introduction . . . 58

5.2 OMA and modified ERA . . . 58

5.2.1 Basics of the ERA algorithm . . . 58

5.2.2 Modified ERA for harmonics . . . 61

5.3 Experiment and Signal Processing . . . 64

5.3.1 Pure stationary white noise excitation . . . 65

5.3.2 Two additional harmonics . . . 66

5.3.3 Three additional harmonics . . . 68

5.3.4 Robustness of the modified ERA method . . . 69

5.4 Conclusion . . . 72

6 Mode Shapes in the Presence of Harmonics 75 6.1 Introduction . . . 75

6.2 Operational modal analysis in the presence of harmonics . . . 76

6.2.1 Extraction of mode shapes . . . 76

6.3 Steel plate test case . . . 78

6.3.1 Pure noise excitation . . . 79

6.3.2 Three additional harmonics with the standard LSCE . . . 81

6.3.3 Three additional harmonics with modified LSCE . . . 82

6.3.4 Discussion of results . . . 83

6.4 Conclusion . . . 84

7 Conclusions 89 7.1 Summary and main contribution . . . 89

List of Figures

1.1 Examples of structures in operating condition. bridge (a),

aero-plane (b), offshore oil platform (c), rocket (d) . . . 3

1.2 Organization of the text. . . 9

2.1 Peak picking method for modal analysis . . . 13

2.2 Sketch of the NExT method for doing OMA . . . 15

3.1 (a)Experimental setup for the beam structure. (b) slider arrange-ment at one end . . . 28

3.2 PSD of acceleration at position 1 for pure random excitation and stability diagram for the polyreference LSCE method. . . 29

3.3 PSD of acceleration at position 1 for a combined single harmonic and random excitation, and stabilization diagram of the polyref-erence LSCE method . . . 31

3.4 PSD of acceleration at position 1 for a combined single harmonic and random excitation, and stabilization diagram of the modified single reference LSCE . . . 32

3.5 PSD of the acceleration at position 1 for random loads and two additional periodic square excitations, and stabilization diagram from the polyreference LSCE method . . . 34

3.6 PSD of the acceleration at position 1 for random loads and two additional periodic square excitations, and stabilization diagram from the modified single reference LSCE method . . . 35

3.7 PSD of the acceleration at position 1 for random loads and three additional harmonic excitations, and stabilization diagram from the polyreference LSCE method . . . 37

3.8 PSD of the acceleration at position 1 for random loads and three additional harmonic excitations, and stabilization diagram from the modified single reference LSCE method . . . 38

3.9 Identified eigenfrequency (a) and damping ratio (b) for the second bending mode as a function of ωharmonic−ωbending ωbending . . . 40

3.10 Identified modal frequency (a) and damping ratio (b) as a func-tion of the harmonic frequency specified in the modified LSCE. . 40

4.1 Experimental Setup of a Plate . . . 48 4.2 Stability diagram for white noise excitation only computed by

the SIMO-SSTD method . . . 49 4.3 Stability diagram for white noise excitation only computed by

LSCE method . . . 50 4.4 PSD of the acceleration at position 1 for random loads and two

additional harmonic excitations, and stabilization diagram for the SIMO-SSTD (4.10) and for the SIMO-SSTD modified to explic-itly include harmonic components (4.18) . . . 52 4.5 PSD of the acceleration at position 1 for random loads and three

additional harmonic excitations, and stabilization diagram ob-tained from the SIMO-SSTD and the modified SIMO-SSTD. . . 54 4.6 Eigenfrequency vs. assumed harmonic frequency in the modified

SIMO-SSTD (harmonic excitations at 274Hz and 276Hz) . . . 55 4.7 Damping vs. assumed harmonic frequencies in the modified

SIMO-SSTD (harmonic excitations at 274Hz and 276Hz). . . 55 5.1 Stability diagram for white noise excitation only computed by

the ERA method . . . 65 5.2 Stability diagram for white noise loading and additional harmonic

excitations at 23.30 Hz and 23.80 Hz (standard and modified ERA methods). . . 67 5.3 Stability diagram for white noise loading and additional harmonic

excitations at 23.30 Hz, 23.80 Hz and 24.30Hz (standard and modified ERA methods). . . 69 5.4 Identified eigenfrequency for the second bending mode as a

func-tion of the Frequency Resolufunc-tion(FR), F R =ωharmonic−ωbending

ωbending . 71

5.5 Eigenfrequency and damping ratio identified by the modified ERA method when the assumed harmonic frequency is different from the true harmonic frequency (23.25 Hz) . . . 72 6.1 Experimental setup for a steel-plate placed in free-free condition,

excited by a shaker . . . 78 6.2 Stability diagram for the steel-plate while it is excited by the

white-noise . . . 80 6.3 Mode shapes identified in the presence of pure stationary noise

and taken as reference . . . 86 6.4 PSD of the acceleration at position 1 for random loads and three

additional periodic excitations, stabilization diagram from the standard LSCE method and the modified LSCE method . . . 87 6.5 Identified harmonic responses corresponding to harmonic frequency

144 Hz and 148 Hz. The harmonic response at 146 Hz is very similar. . . 87 6.6 First mode shape computed by the modified LSCE method for

List of Tables

3.1 Frequencies and associated damping (white noise only) . . . 30 3.2 Frequencies and associated damping identified by the

polyref-erence LSCE and the modified single refpolyref-erence LSCE methods (random and single harmonic excitation). . . 33 3.3 Frequencies and associated damping for both polyreference LSCE

and single reference LSCE method (two additional periodic exci-tations with square waveform) . . . 36 3.4 Frequencies and associated damping from the polyreference LSCE

and the modified single reference LSCE methods (three additional harmonic excitations). . . 39 4.1 Frequencies and associated dampings (white noise only) . . . 51 4.2 Frequencies and associated damping identified by the SIMO-SSTD

and the modified SIMO-SSTD. . . 53 4.3 Frequencies and associated damping identified by the SIMO-SSTD

and the modified SIMO-SSTD. . . 54 5.1 Frequencies and associated dampings identified by ERA (white

noise excitation only) . . . 66 5.2 Frequencies and associated damping identified by the standard

ERA and the modified ERA. System excited by stationary white noise and harmonic loads at 23.30Hz and 23.80Hz. . . 68 5.3 Frequencies and associated damping identified by the standard

ERA and the modified ERA. System excited by stationary white noise and harmonic loads at 23.30Hz, 23.80Hz and 24.30Hz. . . . 70 6.1 Identified frequencies and damping by white-noise excitation . . 80 6.2 Frequencies and associated damping identified by the standard

LSCE method in the presence of harmonic excitations at 144Hz, 146Hz and 148Hz . . . 81 6.3 Poles used for identification of the eigenmodes and MAC values

between the reference modes and the modes identified by the standard LSCE procedure applied to the signals with harmonics. 82

6.4 Frequencies and associated damping identified by the modified LSCE method in the presence of harmonic excitations at 144Hz, 146Hz and 148Hz . . . 83 6.5 Poles used for identification of the eigenmodes and MAC values

between the reference modes and the modes identified by the modified LSCE procedure applied to the signals with harmonics. 84

Chapter 1

Introduction

In this general introduction we will outline the major issues investigated within the present research. The research context is described in Sec. 1.1. The objective of the thesis, as well as the main contributions, are discussed in Sec. 1.2. The structure of the manuscript and the content of its chapters are discussed in Sec. 1.3

1.1

Research context

Dynamic design of a system aims at obtaining the desired dynamic character-istics of the system by specifying the right shape, size, configuration, material and manufacturing steps of its various parts. Desired dynamic characteristics include low vibration and noise levels meaning for instance tuning its natural frequencies to be well separated from the excitation spectrum. Other dynamic properties that are often required for systems include high dynamic stability and prescribed mode shapes or vibration patterns. Ultimately, the designer will have as an objective to develop quieter and more comfortable products and increase reliability and efficiency (ex: noise and vibrations in cars, reliability of long span bridges, efficiency of loudspeakers, accuracy of wafer-steppers, avoid flutter of aircrafts). While fulfilling the above mentioned requirements, the designer has to look for solutions within specific economical constraints and in a highly evo-lutive global market. Thus there is a need for lower design and development time calling for advanced analysis approaches such as optimum dynamic design. Sound dynamic design of real-life complex systems, like those mentioned above, cannot be ensured only on the basis of numerical and analytical investi-gations. Experimental Modal Analysis (EMA) of prototypes provides essential information on dynamical properties of the system during the process of de-signing, which is impossible to verify only by computer simulations. Sometimes EMA is carried out only on the part of the system, which has been redesigned and thus for which dynamic properties have to be identified [61]. This process is quite standard in industry and, combined with advanced numerical simulation techniques, helps to develop new products in a short time for the design cycle.

Modal analysis methods are based on the assumption that dynamic response of a system can be totally represented by its eigenparameters. Response at any point on the system is considered as the superposition of responses from the individual modes of the system at which the system vibrates naturally.

During the last few years, EMA technology has evolved in a very efficient and reliable method in the quest for accurate determination of dynamic prop-erties of the structures. It has become a standard procedure in many engineer-ing fields such as mechatronical design, transport technology, aeronautics and aerospace, civil engineering, biomechanics and biomedical engineering, vibro-acoustics, wind mills and microsystems. For all these applications, predicting and analyzing the dynamical behavior is essential.

In this modern era, structural design in fields like aircrafts, satellites, cars and buildings is characterized by the trend to become lighter, smaller, cheaper and mulitifunctional while satisfying stringent safety and reliability criteria. Availability of efficient optimization tools on one hand and ever increasing con-sumer expectation on the other have lead to tremendous changes in the way structures look today. The severe requirement on mechanical components have resulted in unwanted vibrations. The multifunctionality of products has called for the need to include multiphysical effects in the analysis.

The demand for better products and higher reliability of structures has in-creased the necessity to properly account for the dynamics during design,

man-1.1. RESEARCH CONTEXT 3

Figure 1.1: Examples of structures in operating condition. bridge (a), aeroplane (b), offshore oil platform (c), rocket (d)

ufacturing and operation of new concepts. Most commonly used finite element (FE) analysis provides robust modelling capability to analyze a structure for its dynamic and static properties thanks to the advent of modern high-performance computer. However it is often seen that dynamic properties computed by the FE analysis differ from the actual dynamic properties of the structure. The difference between simulations and reality can be traced back to the following reasons:

• FE analysis is based on a discretization of the reality, meaning that the

displacement fields are approximated by predefined shape functions within each element. Further assumptions such as mass lumping are also used in practice.

• Eventhough bulk properties of materials such as mass density and

elastic-ity coefficients are often well known, damping characteristics of material is difficult to properly account for in simulation. Evenmore structural damping which is related not only to material properties but also to dissi-pation in joints or in interaction with the surroundings can not be properly included in the models.

• The geometry of a specific project can significantly differ from what is

indicated on drawings and used for the FE model.

while doing numerical modal analysis which by essence is based on lin-earized models.

To fill the gap between reality and modelling, experimental techniques are necessary in order to confront measurement with simulated results. Updating techniques can then be applied to tune the models such that they represent real-ity more accurately. Note that complete dynamic analysis has become common practice thanks to

• advances in measurement techniques, sensors and signal processing, • increase of computer power and decrease of computer costs which allows

computing detailed dynamical behavior based on complex and large mod-els.

• the increase of computational power has also significantly broadened the

scope of experimental analysis by allowing to measure and process many signals in real time. Obviously this is essential for health monitoring and control of systems.

Determining the linear dynamic behavior of systems from the test is called Experimental Modal Analysis (EMA). EMA procedures are based on the as-sumption that the vibration response of a linear time-invariant dynamical sys-tem can be expressed as the linear combination of a set of simple harmonic motions called natural modes of vibration. It identifies dynamic properties of the structure in terms of natural frequencies, damping ratios and mode shapes. Natural modes of vibration are inherent to the dynamic system and are de-termined completely by its physical properties (mass, stiffness and damping). Response at any point on the structure can be considered as the sum of all modal responses of each of its individual modes. Based on eigenparameters identified from the modal analysis, a mathematical model of the dynamic behavior of the structure can be constructed. Such procedures are known as model synthesis and results in a modal model of the structure. A modal model of the structures plays an important role in dynamical behavior prediction for the structure.

In EMA, structures are usually excited with one or more excitation sources and responses of the structure are measured at one or more locations. Modal parameters can be identified from the frequency response function (FRF) of the response signal to the input signal or from the impulse response of the system. In such a frequency domain approach, eigenmodes and eigenfrequencies are iden-tified by fitting the measurement to a theoretical expression of the response as a superposition of modes. Another way consists in analyzing time signals such as impulse response and identify modal parameters by fitting a mode superposition expression on the time data. Such a procedure is known as time-domain modal identification.

In practice, within their operational environment, structures are excited nat-urally due to external or internal excitations. Think of buildings excited by the wind, bridges under the action of traffic or engine vibration on aircrafts or cars

1.1. RESEARCH CONTEXT 5 (see Figure 1.1). Such excitations can be considered as white-noise in nature. Even though the input force to the system can generally not be measured in such cases, it is still possible to apply modal analysis techniques to evaluate some of the modal parameters of the structures. This modal analysis procedure is called Operational Modal Analysis (OMA). OMA is also sometimes denoted as ambient vibration analysis or output-only analysis.

Different methods have been developed to do the modal parameters estima-tion using output-only data. The concepts underlying these methods are often similar, but they differ in the aspects of data reduction, mathematical modelling of equations or signal processing. Few methods work directly on the raw mea-sured data while others require pre-processing of the meamea-sured response. OMA methods can mostly be subdivided in the following categories:

• Methods performing auto- and cross-correlation of the time-signals and

handling the results as impulse response functions (the Natural Excitation Technique or NExT [28–30]. Time domain identification methods are then used to extract modal parameters from impulse responses of the system. To mention a few: Least Square Complex Exponential (LSCE) [9], Polyreference Complex Exponential method (PRCE) [62, 63], Ibrahim Time Domain Method (ITD) [25–27], Single Station Time Domain method [64] and Eigensystem Realization Algorithm (ERA) [31, 32].

• Methods based on discrete-time data models. Examples of such methods

are the Auto Regressive Moving Average Vector (ARMAV) [16–21] and the Auto Regressive Vector Time Series Modelling [48].

• Stochastic subspace methods such as the Canonical Variate and Balance

Realization Techniques [4, 49].

• The Maximum Likelihood frequency domain method, which works on

auto-and cross-spectrum functions of the response signals [23, 50, 55]. In OMA procedures, like in classical EMA, natural frequencies, modes and damping ratios are evaluated for a structure. However due to the absence of input force to the system, modal participation factors can not be computed. Re-cently however [53] a procedure was propose for deriving scaling factors based on sensitivity analysis for the perturbed system. Typically the modal partici-pation factors are found by performing OMA on the system with known added masses.

In practice modal analysis is a succession of three steps:

1. Planning and executing the experiments: in the initial phase, experiments must be planned based on information known at forehand such as fre-quency range, flexibility distribution, accessibility of measurement points, vibration nodes [1, 3] ... An important issue is to choose the boundary condition to apply on the test object. Two types of boundary condition can be used: free and fixed boundaries. Usually free surface boundary

condition can be achieved by hanging the structure from soft springs or putting the structure on a very soft surface like foam. Fixed boundary can be achieved by attaching one end of the structure to a heavy, highly stiff external object. Note that for bulky structures or structures that have to be tested in situ the boundary conditions are mostly imposed. Proper locations and orientations of the actuators and sensor must then be defined based on preliminary information one may have from previous tests or from models. Data acquisition parameters like measurement time, sampling frequencies must also be chosen properly depending on the ex-perimental goals. In classical EMA, excitation is provided either by an impact hammer or by a shaker. A hammer produces input in a broad frequencies range whereas shakers are more often used to apply specific excitation inputs in terms of prescribed amplitude and frequencies. In OMA, excitation is inherently provided by uncontrolled sources like wind, waves or acoustic noise.

2. Processing of the data and identification of modal parameters: During the measurement, signals are measured in time domain. Output data are then formatted according to the specific identification utilized. For frequency domain identification methods FRF curves are derived using for instance Fast Fourier Transforms (FFT) and windowing [3, 36]. For time domain identification approaches, the time signal might be filtered or used as such in the identification algorithms. Often the time-signal is transformed into an impulse response before performing the identification. This can be done by FFT and inverse FFT of the raw signal.

3. Validation of the modal models: In this process, modal parameters ex-tracted from the measurements are visualized and analyzed. So-called “numerical” modes have to be discarded.

Once the modal model has been found, it can be used in several different ways. For an engineer it is a matter of interest to investigate different proper-ties of a structure using a good modal model of the structure by experimental analysis. Some of the applications where the modal model is used are:

• Troubleshooting: this is a very general application of modal analysis.

Structure’s vibrational properties like natural frequencies, damping ratios and mode shapes measured from the experiments are used to understand the dynamics behavior of the structure to find out the root cause of prob-lems such as resonance and noise often encountered in real life [6, 61].

• Model updating: a mathematical model of a structure (e.g. using FE)

may not represent the correct dynamical behavior of the structure due to the fact that material properties, geometry, joint properties and bound-ary conditions for the model may not represent the real system given its complexity and uncertainties arising during the production. Experimen-tal modal analysis provides a tool to evaluate the modal parameters in practice and helps in updating the FE model to make it more realistic [2].

1.2. SCIENTIFIC CHALLENGE AND THESIS TOPIC 7

• Structural modification and sensitivity analysis: structural modification

consists in predicting the dynamical behavior of the structure assuming that structural modifications are brought on a nominal configuration. It allows to evaluate the effect of changes on the dynamics without having to actually modify the structure. This is particularly useful at the early stage of a new design to optimize its dynamic characteristics starting from a pre-existing system. The sensitivity analysis provides the rate of change in dynamical behavior for specified small system perturbations [14, 60]. It thus provides the information necessary to guide the designer in his decisions during the redesign phase.

• Structural health monitoring and damage detection: structural integrity

can be assessed by comparing modal parameters from the current state of a structure with the modal parameters of the structures in nominal conditions [4, 11, 51, 52]. Modal parameters can give indications on the severity of the damage as well as on its location (using model updating techniques).

• Performance evaluation: modal parameters and mode shapes can be

valu-able information to evaluate the dynamic performance of systems. For instance the quality and performance of a tennis racquet (described in terms of so-called sweet spots) are closely related to its eigenmodes (see e.g. [39])

• Force identification: this is an inverse problem compared to the response

prediction from the known force input [10, 22]. Force can be identified by measuring only the response of the structure, using a modal or FE model.

• Controller synthesis: a last common utilization of modal models is to

synthesize controllers. For instance in house appliance or in CD players, the control loop is typically designed based on the characteristic of the system such as measured in EMA.

As described here above, there are many important applications, where modal parameters provide essential information to solve complex engineering problems. So it is essential to perform the modal parameter estimation of a structure as accurately as possible.

1.2

Scientific challenge and thesis topic

This thesis addresses an essential issue in Operational Modal Analysis, namely accounting for non-random excitations that often appears in addition to white noise. Indeed one major disadvantage of output-only identification is that the input force to the system must be a stationary white noise while in practice periodic excitations inherent to the system are also present.

Some examples of structures were standard OMA techniques are well-suited are bridges, buildings, aeroplanes, spacecrafts, oil platforms or ships. In these

examples, the excitation forces during operation are caused by wind, waves and other broadband excitations such as road roughness or jet propulsion noise. In those cases, the input forces during operation can not be properly measured. On the other hand exciting those structures with external excitations (e.g. shakers) like in classical EMA is hardly feasible due to the fact that the structures are very large and heavy or because they can not be accessed easily. Especially when the dynamic properties are to be measured in operation (think about the vibration modes of a wing during flight) applying external excitation is often difficult, expensive and can even cause safety hazards. In this case, operational modal analysis is a suitable method to estimate modal properties of these kind of structures as input force to these systems are very difficult or often impossible to measure.

In many other structures such as wind mills, milling machines, turbines or cars, periodic exciting forces are present in addition to the white-noise equiva-lent excitations. Harmonically exciting forces can arise from moving components like the blades of an helicopter, the rotation frequency of turbines, teeth con-tact frequencies in gears or milling processes or car engine orders. In theory, for such problems, operational modal analysis techniques can still be used to extract modal properties. Indeed considering that harmonic excitation gener-ates harmonic frequencies in the response signals, the harmonic responses will be identified during the OMA identification as non-damped modes with very low damping (theoretically null), which distinguish them from true eigenmodes. However such a procedure fails to identify eigenmodes when the harmonic fre-quencies are close to the eigenfrefre-quencies of the structures or when the strength of the signal arising from harmonic excitations is very high compared to the response associated with the eigenfrequencies of the structures excited through the noise input.

Addressing the above drawback of OMA is primordial if OMA techniques are to be used in a wide range of practical applications. To tackle this important engineering and scientific challenge, three major novel methods are proposed in this thesis work to enable OMA in the presence of harmonically exciting forces in addition to stochastic force in a structure:

• a new method based on the Least Square Complex Exponential method

is proposed (Chapter 3 ).

• a new method related to the Ibrahim Time Domain method is described

(Chapter 4 ).

• a novel extension to the Eigensystem Realization Algorithm is outlined

in Chapter 5.

1.3

Outline and structure of the thesis

In this section a brief overview of the content of the different chapters is given in a Chapter-by-Chapter outline. (see also Figure 1.2).

1.3. OUTLINE AND STRUCTURE OF THE THESIS 9

Figure 1.2: Organization of the text.

After the present chapter describing the general context and the scientific challenges of our research, chapter 2 gives a short overview of OMA approaches and in particular explains the procedure known as the Natural Excitation Tech-nique (NExT). The NExT approach to OMA will be considered for the algo-rithmic developments exposed in the subsequent chapters.

Chapter 3 discusses OMA identification procedures based on the Least Square Complex Exponential method (LSCE). First we recall the theory and procedure of the standard LSCE algorithm. Then we propose a modified version of the LSCE that explicitly accounts for harmonic excitations during the identifica-tion. The effectiveness and robustness of the method is illustrated on a beam structure.

In chapter 4 a second method is developed relying on the principle of the Ibrahim Time Domain (ITD) method and on one of its variants, the Single Station Time Domain technique (SSTD). First, the SSTD method is generalized for multiple correlation signals considering several response channels. Then a novel technique is described that modifies the SSTD-like algorithm to account for harmonic excitations in OMA. A plate like structure is used as a testing

structure to show the performance of the modified ITD method in the presence of harmonic excitations.

In chapter 5 a third approach to OMA identification is described, based on the popular Eigensystem Realization Algorithm (ERA). The extension proposed for the ERA allows to properly account for harmonic excitations of known fre-quencies. Once more the beam structure as used in chapter 3 is used to show the performance of the method.

In chapter 6 a procedure to evaluate mode-shapes is discussed. Assuming that the modal parameters have been obtained by one of the methods described in the previous chapters, we can use the standard modeshape identification procedure in the time-domain based on the expression of responses in terms of complex modal contributions. The mode identification post-processing is illustrated on the plate structure in the presence of strong harmonic signals. Modal Assurance Criteria (MAC) values are considered in order to compare the modes identified with or without harmonic inputs.

Finally in chapter 7 we conclude the thesis by a highlighting the main results of our research and by indicating relevant and challenging research direction for the future.

Chapter 2

Operational Modal Analysis

2.1

Introduction

Dynamic analysis of structures is an essential engineering tool since it allows to construct a mathematical model of the structure based on measured modal para-meters. The experimental results can also be compared to numerical models and used for model updating. Reliable mathematical models obtained experimen-tally or by updating numerical models are primordial for designers and engineers since they can represent the dynamical behavior of the structure under various dynamic loadings (earthquake, winds, explosions) and therefore allow analyzing issues such as reliability, integrity or lifetime prediction of systems.

Experimental analysis can be performed in a traditional way, namely in a laboratory environment. In standard Experimental Modal Analysis (EMA) externally applied input forces are measured together with dynamic responses of the system at different locations. The forces applied by the excitations should be large enough to produce measurable response at the measuring points, but yet small enough for the structure to be considered as a linear system. EMA techniques for non-linear systems exist but are not yet mature and thus rarely applied in practice.

Exciting large engineering structures is cumbersome due to the excitation force levels required and do to accessibility issues. Simply exciting at one point of the structure may not be enough. For these reasons, when measuring large structures, multiple shakers at different locations need to be used, implying high costs and long experimentation time.

Some structures experience vibration due to natural excitations. Sometimes it becomes essential to test the dynamic properties of a structural system while it is functioning in its operational environment in terms of its internal stress state, its coupling with the environment or its thermal condition. In opera-tional condition, it is often impossible to apply standard EMA since on one hand applying excitations in a controlled way might be intricate and dangerous and, on the other hand, the presence of unmeasurable internal and ambient forces jeopardizes the application of identification techniques. When so-called

natural excitations are present during operation of a structural system, ambient

vibration testing or operational modal analysis (OMA) needs to be considered. The main advantages of OMA are:

• Testing is cheap and fast since no excitation equipment is needed. • Measurements are done in the actual operation conditions for the

struc-ture. Identified modal parameters represents the dynamic behavior of the structure such as used in reality.

• Testing does not interfere with the operation of the structure meaning for

instance that a power generator need not to be laid still to measure its turbine, or a bridge needs not to be closed to traffic when analyzed. In this chapter we will briefly outline common OMA techniques and discuss in particular the Natural Excitation Technique (NExT) that will be considered throughout this thesis.

2.2. OMA: AN OVERVIEW 13

2.2

Operational modal analysis techniques:

an overview

Peak-Picking techniques

A simple way of doing modal analysis from the response-only data is based on a Peak-Picking technique applied to the auto and cross-spectrum of the re-sponse signal [5]. This is a Single-Degree-Of-Freedom (SDOF) method, which involves selecting peaks in the spectra corresponding to resonance frequencies.

Peak-Picking can be a reliable approach when the modes of the system are well

separate and when the damping associated with each mode is low so that the in-fluence each mode on the dynamical behavior can be considered independently. An advanced version of Peak-Picking method is known as the Complex Mode

Indicator Function (CIMF) [57]. This method is based on singular value

de-composition (SVD) of the matrix of power spectra. The steps involved in the

Figure 2.1: Peak picking method for modal analysis

Peak-Picking method are:

• Individual peak regions are visually chosen and marked from the

cross-spectra between responses and a reference signal. In the vicinity of each peak, the maximum is found and associated to the eigenfrequency of the mode.

• Two points are identified on the cross-spectra curves, one below the rth_{eigenfrequency}

ωr and one above, both having an amplitude Cωr/

√

2 where Cωr is the

amplitude at the natural frequency ωr. These two points are called

half-power points.

• The damping ratio ζ_rof the associated mode is then computed by

ζ = ω2a− ω2b 4ω2 r ∼ = ∆ω 2ωr (2.1)

Other methods for OMA

Several other methods based on discrete-time data models have been proposed. The Maximum Likelihood Estimation(MLE) technique is a frequency domain method [23,54,56]. Other methods include the Auto Regressive Moving Average (ARMAV) [16–21], the Auto Regressive Vector Time Series Modelling [48] and Stochastic Subspace methods such as the Canonical Variate and the Balance Realization Techniques [4,49]. Describing those methods is beyond the scope of the present manuscript. The reader is referred to the references for a detailed description of those approaches.

2.3

The Natural Excitation Technique (NExT)

2.3.1

Introduction

The Natural Excitation Technique (NExT) [28–30] is a method using the mea-sured time response to white noise excitation to construct an impulse-like re-sponse signal of the structure. The NExT approach is based on the property that auto- and cross-correlation functions between response signals can be used as the impulse response functions of the system when the system is excited by a white noise force. This property will be outlined in this section. Figure 2.2 illustrates this concept underlying the NExT procedure. This fundamen-tal property pertains to linear systems that can be represented by the modes of the associated conservative system, namely systems that are lightly damped and which eigenfrequencies are well-separate [35]. Note that for linear systems where modal coupling through damping can not be neglected and where com-plex eigenmodes should be considered, the NExT method can, in theory, not be used since no proof exists to relate signal correlations to impulse-like responses. Using the fundamental property on which the NExT method is based, auto-and cross-correlation functions of the responses to white noise can be hauto-andled as being impulse response functions and the modal parameters of the system can be identified using standard time domain identification methods like Least Square Complex Exponential (LSCE) [9], the Ibrahim Time Domain(ITD) method [25– 27] and the Eigensystem Realization Algorithm (ERA) [32]. The time-domain identification techniques will be discussed in later chapters.

2.3.2

Modal analysis of linear structures

Considering a Multiple-Degree-Of-Freedom (MDOF) system, the linear(ized) equation of motion assuming viscous damping can be can be written as:

[M ] {¨x (t)} + [C] { ˙x (t)} + [K] {x (t)} = {f } (2.2) where

2.3. THE NATURAL EXCITATION TECHNIQUE (NEXT) 15

Figure 2.2: Sketch of the NExT method for doing OMA [M ] mass matrix [C] damping matrix [K] stiffness matrix {f } force vector {x} displacement vector { ˙x} velocity vector {¨x} acceleration vector

The solution of the dynamic equation can be written as a combination of the modal basis [Φ] of the conservative system,

{x(t)} =

N

X

r=1

{φ_r} qr= [Φ] {q (t)} (2.3)

where {q} is the vector of modal coordinates and where [Φ] is a matrix collecting

N eigenmodes of the system and satisfies the eigenvalue problem

([K] − ω2

r[M ]) {φr} = 0 (2.4)

The dynamic equation (2.2) then writes

[M ] [Φ] {¨q (t)} + [C] [Φ] { ˙q (t)} + [K] [Φ] {q (t)} = {f } (2.5) Pre-multiplying this equation by the mode shape matrix [Φ]T to project the dynamic equilibrium on the modal basis, we obtain

The eigenmodes [Φ] of the conservative system satisfy orthogonality with respect to the mass and stiffness matrices, namely

[Φ]T[M ] [Φ] =£r_m r ¤ and [Φ]T[K] [Φ] =£r_k r ¤ (2.7) where [r_m

r] and [rkr] are the modal mass and stiffness diagonal matrices respectively. Note that, according to (2.4), the modal mass and stiffness are related by kr/mr= ω2r.

Assuming that the eigenfrequencies of the system are well separated and that they are associated with low damping ratios, the real modes of the associated conservative system can be assumed to be identical to the complex modes of the damped system and we can assume that the eigenmodes [Φ] can be used to diagonalize the damping matrix [C] namely [35, 36]

[Φ]T[C] [Φ] =£r_c r

¤

(2.8) where [r_c

r] is the modal damping matrices. Finally the dynamic equations in the modal basis are obtained from (2.6):

£_r mr ¤ (N ×N ) {¨q (t)} (N ×1) +£r_c r ¤ (N ×N ) { ˙q (t)} (N ×1) +£r_k r ¤ (N ×N ) {q (t)} (N ×1) = [Φ]T{f } (N ×1) (2.9) Eq. (2.9) are known as the normal equations and can be written for a par-ticular mode r as

mrq¨r(t) + cr˙qr(t) + krqr(t) = {φr}T{f (t)} (2.10)

where {φ_r} is rth _{mode shape vector. Dividing the equation (2.10) by the}

rth_{modal mass m}

r, we get:

¨

qr(t) + 2ζrωrqr(t) + ω2rqr(t) = 1

mr{φr}T{f (t)} (2.11)

where ζ_rand ωrare the damping ratio and natural frequencies associated with

rth_mode.

The solution of Eq. (2.11), assuming a general force {f } is obtained from the convolution or Duhamel integral [58]:

qr(t) =

Z t

−∞

{φr}T{f (τ )} gr(t − τ ) dτ (2.12)

where gr(t) is the impulse response of a SDOF system

gr(t) = 1 mrωdr e−ζrωrt_sin¡_ωd rt ¢ (2.13) and ωd r= ωr ¡

1 − ζ2_r¢1/2is the damped natural frequency.

Now the response vector {x (t)} can be formulated as the summation of modal responses by using Eq. (2.12) and Eq. (2.3) as:

2.3. THE NATURAL EXCITATION TECHNIQUE (NEXT) 17 {x (t)} = N X r=1 {φ_r} {φ_r}T Z t −∞ {f (τ )} gr(t − τ ) dτ (2.14)

Eq. (2.14) can now be written for a single response, xik(t) at a point i, due

to single input force, fk(t), at point k.

xik(t) = N X r=1 φriφrk Z t −∞ fk(τ ) gr(t − τ ) dτ (2.15)

The impulse response function between input k and output i is obtained by assuming f (τ ) in Eq. (2.15) to be a unit Dirac delta function at τ = 0. So the response at the point i due to an input impulse at k, can be expressed as a summation of modal impulse responses:

xik(t) = N X r=1 φriφrk mrωdr e−ζrωrt_sin¡_ωd rt ¢ (2.16)

2.3.3

Derivation of correlation functions as impulse

re-sponse functions

In this section we derive the fundamental property used in the NExT method such as outlined in the milestone publications [28–30].

Let us call Rijk(T ) the cross-correlation function between two responses xik

and xjk at i and j due to a white-noise input at a location k. By definition

Rijk(T ) is the expected value of the product between xik and xjk evaluated

with a time delay T , namely [59]

Rijk(T ) = E [xik(t + T ) xjk(t)] (2.17)

where E is the expectation operator. For measured signals, the correlations can be computed as Rijk(T ) = lim T →∞ 1 T Z _{T /2} −T /2 xik(t)xjk(t − T )dt ' 1 n n X α=1 xik(tα+ T ) xjk(tα) (2.18) Substituting (2.15) into (2.17), Rijk(T )=E Ã_N X r=1 N X s=1 φ_riφ_rkφ_sjφ_sk Z t −∞ Z t+T −∞ gr(t+T− σ) gs(t − τ ) fk(σ) fk(τ ) dσdτ ! (2.19) Let us note that the expectation of the product between a random function a(t) and a non-random function b(t) satisfies [15]

Using this property in (2.19) where fk(t) is a random function and x (t) is a

non-random time function, one obtains

Rijk(T ) = N X r=1 N X s=1 φ_riφ_rkφ_sjφ_sk Z t −∞ Z t+T −∞ gr(t+T− σ) gs(t − τ ) E [fk(σ) fk(τ )] dσdτ (2.21) Using the definition of the autocorrelation function (2.17) and assuming f (t) to be white noise we have [59]

Rf f k(τ − σ) = E [fk(τ ) fk(σ)] = αkδ (τ − σ) (2.22)

where αk is a constant and δ (t) Dirac delta function.

Substituting (2.22) in (2.21) we get: Rijk(T ) = N X r=1 N X s=1 αkφriφrkφsjφsk Z t −∞ gr(t + T − σ) gs(t − σ) dσ (2.23)

Equation (2.23) can be further simplified by making a change of the variable for the integration variable: let λ = t − σ, then dλ = −dσ and the limits of integration for λ are +∞ and 0. Equation (2.23) then becomes:

Rijk(T ) = N X r=1 N X s=1 αkφriφrkφsjφsk Z _∞ 0 gr(λ + T ) gs(λ) dλ (2.24)

Using the definition of grfrom Eq. (2.13) we can separate the terms

depend-ing on λ and the ones dependdepend-ing on T as follows

gr(λ + T ) = 1 mrωdr e−ζrωr(λ+T )_sin¡_ωd r(λ + T ) ¢ (2.25) = 1 mrωdr e−ζrωr(λ+T )¡_{sin ω}d

rλ cos ωdrT + sin ωdrT cos ωdrλ

¢ = £e−ζrωrT_cos¡_ωd rT ¢¤ e−ζrωrλsin¡ωd rλ ¢ mrωdr +£e−ζrωrT_sin¡_ωd rT ¢¤ e−ζ_rωrλ_cos¡_ωd rλ ¢ mrωdr

Substituting (2.25) in (2.24) and using the definition (2.13) for gs(λ), on finds

Rijk(T ) = N X r=1 £ Grijke−ζrωrTcos ¡ ωdrT ¢ + Hijkr e−ζrωrTsin ¡ ωdrT ¢¤ (2.26)

2.3. THE NATURAL EXCITATION TECHNIQUE (NEXT) 19 where Gr

ijk and Hijkr are independent of T and are given by

½ Gr ijk Hr ijk ¾ = N X s=1 αkφriφrkφsjφsk mrωdrmsωds Z _∞ 0 e(−ζrωr−ζsωs)λ_{. sin}¡_ωd sλ ¢½_sin¡_ωd rλ ¢ cos¡ωd rλ ¢ ¾ dλ (2.27)

Equation (2.26) is the key result of this derivation. Examining Eq. (2.26), we can see that the cross-correlation function is indeed a linear combination of decaying sinusoids similar to the impulse response function of the original sys-tem (see Equation (2.16)); thus, the cross-correlation functions can be handled as impulse response functions and processed with time-domain identification techniques to estimate the modal parameters of the system. Note that in this approach the amplitude of the random inputs are not measured so that the con-stant αk in (2.27) is unknown. As a consequence the modal masses mr cannot

be estimated in (2.27) and one will say that the modes so-obtained are unscaled. It should not come as a surprise since the actual transfer function can not be evaluated since the inputs are not measured. Some perturbation techniques have however been proposed to obtain the scaling of the modes based on several analysis of the system with added mass [53]. Applying sensitivity analysis, the modal masses can be evaluated.

In (2.27) Gr

ijk and Hijkr can be further simplified by evaluating the definite

integral, leading to Grijk= N X s=1 αkφriφrkφsjφsk mrmsωdr · Irs J2 rs+ Irs2 ¸ (2.28) Hr ijk = N X s=1 αkφriφrkφsjφsk mrmsωdr · Jrs J2 rs+ Irs2 ¸ (2.29) where Irs = 2ωdr(ζrωr+ ζsωs) and Jrs= ³ ωd2 s − ωd 2 r ´ + (ζ_rωr+ ζsωs)2. Let

us then define γrs such that:

tan (γ_rs) = Irs/Jrs (2.30)

Equation (2.28) and (2.29) then write            Grijk= φri mrωdr N X s=1 βrsjk ¡ Jrs2 + Irs2 ¢−1/2 sin (γrs) Hr ijk= φri mrωdr N X s=1 βrsjk ¡ J2 rs+ Irs2 ¢−1/2 cos (γrs) (2.31) where we define βrs_jk=αkφrkφsjφsk ms (2.32)

Substituting (2.31) into Eq. (2.26), and summing over k, namely over all M input locations to find the cross-correlation function due to all inputs, we find:

Rij(T ) = N X r=1 φ_ri mrωdr N X s=1 M X k=1 βrs_jk¡J2 rs+ Irs2 ¢−1/2 e−ζrωrT_sin¡_ωd rT + γrs ¢ (2.33) The inner summations on s and k are merely a summation of constants times the sine function, with variable phase but fixed frequency. Equation (2.33) can therefore be rewritten as a single sine function with a new phase angle (Θr) and

new constant multipliers (Arj):

Rij(T ) = N X r=1 φ_riArj mrωdr e−ζrωrT_sin¡_ωd rT + Θr ¢ (2.34) This last result shows that the cross-correlation function Rij(T ) between

the response signal at j and a reference signal at i at an interval of time T can be expressed as a sum of N decaying sinusoids. Each decaying sinusoid is characterized by a natural frequency ωr and damping ratio ζr of the structure.

2.4

OMA in the presence of harmonic excitation

Operational Modal Analysis is based on the assumption that the input to the structure is stationary white noise. In practice however, structural vibration ob-served in operation cannot always be considered as pure white noise excitation. In many practical cases, vibrations are induced by a combination of white noise and harmonic excitations. Harmonic excitations in addition to random inputs can occur due to rotating components or fluctuating forces in electric actuators for instance. In these particular cases, OMA procedures are, strictly speaking, not applicable. However, one can view the harmonic response as the vibration of the structure according to a virtual eigenmode having zero damping and consider that only white-noise excitation is present. In this way, harmonic excitation re-sponses are taken into account during OMA as non-damped modes in addition to the genuine natural eigenmodes of the structure. Unfortunately in that case, the identification methods fail to identify modal parameters properly especially for modes with eigenfrequencies close to the harmonic frequencies. Special nu-merical filters can also be applied to filter-out harmonic components from the measured response. Unfortunately in practice, filters are not perfect and if the harmonic frequency is close to eigenfrequencies, the filtering will pollute the measured response so that the identified modal parameters are perturbed.

In [7], a procedure based on statistical properties of the outputs [34] was proposed in order to distinguish between harmonic response and narrow-band stochastic response for output-only modal testing. The authors in [7] define an indicator to identify harmonics from the signal when the Frequency Domain De-composition is applied, but the example presented assumes harmonic frequencies well separate from the eigenfrequencies.

2.5. CONCLUSION 21

2.5

Conclusion

This chapter briefly outlines the concepts of response-only modal testing. In particular we have described the theoretical basis for the Natural Excitation Technique (Sec. 2.3) working directly in the time domain and considering auto-and cross-correlation of the response signals. One of the major limitation of OMA has been described namely the fact that in the presence of harmonic excitation the procedures can, in many cases, no longer be applied. In the following chapters this issue will be further investigated.

Chapter 3

Modified LSCE method to

include known harmonic

excitations

In this chapter a modified version of Least Square Complex Exponential (LSCE) method is proposed to perform operational modal analysis in the presence of harmonic excitations. In the section (Sec. 3.2), a brief theoretical background is presented for the LSCE. The auto- and cross-correlation functions of the system as used in the Natural Excitation Technique (NExT, see section 2.3) are used as impulse response functions of the system to estimate the modal parameters. In the section (Sec. 3.3), the standard LSCE algorithm is extended to incorpo-rate harmonic excitations. Section (Sec. 3.4), contains a brief description of the test object. In section (Sec. 3.4.1), parameter values used as reference for the experimental tests are obtained from standard OMA procedures. Then in sections (Sec. 3.4.2, 3.4.3, 3.4.4) system parameters obtained in the presence of harmonic excitations are presented and discussed. In the section (Sec. 3.4.5), the robustness of the modified method is tested with respect to the difference be-tween excitation and eigen frequencies and with respect to the accuracy of the harmonic frequencies specified during the analysis.

This work has been published in [40, 41, 44, 47]

3.1

Introduction

Least Square Complex Exponential (LSCE) method is a time domain method which is applied to the impulse response function (IRF) of a system to compute modal parameters. In this method impulse response functions of the systems are modelled in a non-parametric way on the basis of Prony’s equation.

3.2

Least Square Complex Exponential method

Assuming that a system is excited by a stationary white noise, it has been shown in the Natural Excitation Technique (NExT)(Sec. 2.3) that the correla-tion funccorrela-tion Rij(T ) between the response signals i and j at a time interval of T

is similar to the response of the structure at i due to an impulse on j. Assuming damping to be small, this is expressed by the relation (2.34)

Rij(T ) = N X r=1 φ_riArj mrωdr e−ζrωrT_sin¡_ωd rT + Θr ¢ (3.1) where φ_ri is the ith_{component of the eigenmode number r of the conservative}

system, Arjis a constant associated to the jthresponse signal taken as reference,

mris the rthmodal mass, ζrand ωrare respectively the rthmodal damping ratio

and non-damped eigenfrequency, ωd r = ωr

q

1 − ζ2_r and Θr is the phase angle

associated with the rth_{modal response. Hence the correlation between signals is}

a superposition of decaying oscillations having damping and frequencies equal to the damping and frequencies of the structural mode. As a consequence, modal parameter identification techniques like the Least Square Complex Exponential method (LSCE) [9] can be used to extract the modal parameters from the correlation functions between measured responses to the noise input. In terms of the complex modes of the structure, the correlation function (3.1) can be written as (see for instance [13])

Rij(k∆t) = N X r=1 esrk∆t_C rj+ N X r=1 es∗ rk∆tC∗ rj (3.2) where sr= −ωrζr+ iωr q¡ 1 − ζ2r ¢

and where Crj is a constant associated with

the rth_{mode for the j}th_{response signal, which is the reference signal. ∆t is the}

sampling time step and the superscript∗ _{denotes the complex conjugate. Note}

that in conventional modal analysis, these constant multipliers are modal par-ticipation factors. Numbering all complex modes and eigenvalues in sequence,

3.2. LEAST SQUARE COMPLEX EXPONENTIAL METHOD 25 equation (3.2) can be written as

Rij(k∆t) = 2N X r=1 C0 rijesrk∆t (3.3)

As sr appears in complex conjugate forms in this expression, there exists a

polynomial of order 2N (known as Prony’s equation) of which expsr∆tare roots:

β0+ β1Vr1+ β2Vr2+ ... + β2N −1Vr2N −1+ Vr2N = 0 (3.4)

where Vr = esr∆t and where β2N = 1. {β} is the coefficient matrix of the polynomial. To determine the values of βi, let us multiply the impulse response

(3.3) for sample k by the coefficient βkand sum up these values for k = 0, . . . 2N

[12, 24, 36]: 2N X k=0 βkRij(k∆t) = 2N X k=0 Ã βk 2N X r=1 Crij0 Vrk ! = 2N X r=1 Ã Crij0 2N X k=0 βkVrk ! = 0 (3.5) Hence the coefficients {β} satisfy a linear equation which coefficients are the impulse response (or correlation functions) at 2N + 1 successive time samples. In order to determine those coefficients, relation (3.5) is written at least for 2N times, starting at successive time samples. In other words, many Prony’s equa-tions can be written to build up a linear system that determines the coefficients

{β}:

β₀Rn

ij+ β1Rn+1ij + ... + β2N −1Rijn+2N −1= −Rn+2Nij (3.6)

where we note Rk

ij = Rij(k∆t) and where n = 1...L. One build therefore a linear

system of L equations based on the Hankel matrix to determine the coefficients

{β}:

[R]ij{β} = −{R0}ij (3.7)

The procedure explained above uses a single correlation function. Since {β} are global quantities related to modal parameters, the same parameters can be de-rived from different correlation functions between any two response signals and from the auto-correlation functions. Hence, let us consider p response stations and adopt q of them as reference locations. Grouping the associated equations (3.7) for all correlation functions in one system yields

     [R]11 [R]12 .. . [R]qp     {β} = −          {R0_} 11 {R0_} 12 .. . {R0_} qp          (3.8)

A solution in a least square sense can be found for {β} from (3.8) using pseudo-inverse techniques. Observing that [R]ij = [R]ji, the least square problem will