Introduction to Operational Modal Analysis - Rune Brincker - ebook

Introduction to Operational Modal Analysis ebook

Rune Brincker

419,99 zł


Comprehensively covers the basic principles and practice ofOperational Modal Analysis (OMA). * Covers all important aspects that are needed to understand whyOMA is a practical tool for modal testing * Covers advanced topics, including closely spaced modes, modeshape scaling, mode shape expansion and estimation of stress andstrain in operational responses * Discusses practical applications of Operational ModalAnalysis * Includes examples supported by MATLAB® applications * Accompanied by a website hosting a MATLAB® toolbox forOperational Modal Analysis

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Table of Contents


Title Page



Chapter 1: Introduction

1.1 Why Conduct Vibration Test of Structures?

1.2 Techniques Available for Vibration Testing of Structures

1.3 Forced Vibration Testing Methods

1.4 Vibration Testing of Civil Engineering Structures

1.5 Parameter Estimation Techniques

1.6 Brief History of OMA

1.7 Modal Parameter Estimation Techniques

1.8 Perceived Limitations of OMA

1.9 Operating Deflection Shapes

1.10 Practical Considerations of OMA

1.11 About the Book Structure


Chapter 2: Random Variables and Signals

2.1 Probability

2.2 Correlation

2.3 The Gaussian Distribution


Chapter 3: Matrices and Regression

3.1 Vector and Matrix Notation

3.2 Vector and Matrix Algebra

3.3 Least Squares Regression


Chapter 4: Transforms

4.1 Continuous Time Fourier Transforms

4.2 Discrete Time Fourier Transforms

4.3 The Laplace Transform

4.4 The




Chapter 5: Classical Dynamics

5.1 Single Degree of Freedom System

5.2 Multiple Degree of Freedom Systems

5.3 Special Topics


Chapter 6: Random Vibrations

6.1 General Inputs

6.2 White Noise Inputs

6.3 Uncorrelated Modal Coordinates


Chapter 7: Measurement Technology

7.1 Test Planning

7.2 Specifying Dynamic Measurements

7.3 Sensors and Data Acquisition

7.4 Data Quality Assessment

7.5 Chapter Summary – Good Testing Practice


Chapter 8: Signal Processing

8.1 Basic Preprocessing

8.2 Signal Classification

8.3 Filtering

8.4 Correlation Function Estimation

8.5 Spectral Density Estimation


Chapter 9: Time Domain Identification

9.1 Common Challenges in Time Domain Identification

9.2 AR Models and Poly Reference (PR)

9.3 ARMA Models

9.4 Ibrahim Time Domain (ITD)

9.5 The Eigensystem Realization Algorithm (ERA)

9.6 Stochastic Subspace Identification (SSI)


Chapter 10: Frequency-Domain Identification

10.1 Common Challenges in Frequency-Domain Identification

10.2 Classical Frequency-Domain Approach (Basic Frequency Domain)

10.3 Frequency-Domain Decomposition (FDD)

10.4 ARMA Models in Frequency Domain


Chapter 11: Applications

11.1 Some Practical Issues

11.2 Main Areas of Application

11.3 Case Studies


Chapter 12: Advanced Subjects

12.1 Closely Spaced Modes

12.2 Uncertainty Estimation

12.3 Mode Shape Expansion

12.4 Modal Indicators and Automated Identification

12.5 Modal Filtering

12.6 Mode Shape Scaling

12.7 Force Estimation

12.8 Estimation of Stress and Strain


Appendix A: Nomenclature and Key Equations

Appendix B: Operational Modal Testing of the Heritage Court Tower

B.1 Introduction

B.2 Description of the Building

B.3 Operational Modal Testing

B.4 Vibration Measurements

B.5 Analysis of the HCT Cases


Appendix C: Dynamics in Short

C.1 Basic Equations

C.2 Basic Form of the Transfer and Impulse Response Functions

C.3 Free Decays

C.4 Classical Form of the Transfer and Impulse Response Functions

C.5 Complete Analytical Solution

C.6 Eigenvector Scaling

C.7 Closing Remarks



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Table of Contents


Begin Reading

List of Illustrations

Chapter 1: Introduction

Figure 1.1 Illustration of the concept of OMA. The nonwhite noise loads are modeled as the output from a filter loaded by a white noise load

Figure 1.2 Example of peak-picking technique for identification of natural frequencies of vertical vibration of a five-span bridge

Figure 1.3 Example of peak-picking technique

Chapter 2: Random Variables and Signals

Figure 2.1 (a) The PDF for the corresponding random variable . (b) The considered signal

Figure 2.2 (a) Density function for the random variable . (b) Distribution function for the random variable

Figure 2.3

Figure 2.4 Each interval on the axis for the random variable defines a number of associated time intervals

Figure 2.5 Joint density function for the random variables and . The probability that both variables are inside the infinitesimal area is equal to the infinitesimal volume

Figure 2.6 Two cases of distributed variables where the left case (a) has little or no correlation and the case to the right (b) has a clear correlation, thus in the case to the right it is natural to introduce the linear model and then try to minimize the error

Figure 2.7 Autocorrelation is simply introduced as the correlation between the considered variable at time and time

Figure 2.8 The autocorrelation function is symmetric, the initial value is equal to the variance and for large values of the time lag the function approaches zero

Figure 2.9

Chapter 3: Matrices and Regression

Figure 3.1

Figure 3.2

Chapter 4: Transforms

Figure 4.1 General periodic function with period to be expressed in terms of an infinite series of harmonics

Figure 4.2 Graphical representation of the Fourier coefficients at the discrete frequencies

Figure 4.3 Schematic of the real and imaginary parts of the Fourier coefficients. The real part is an even function of frequency whereas the imaginary part is an odd function of frequency forcing the imaginary part to be zero at DC and at the Nyquist frequency

Figure 4.4

Figure 4.5

Figure 4.6 The


-transform can be considered a result of letting the frequency axis (that normally goes from to in Fourier transform theory) go from 0 to (a) and then map the frequency axis onto the unit circle, (corresponding to ) , (b). This automatically introduces the periodic spectral density that is a result of the discrete time function; the stable solutions are mapped into the unit circle and the unstable solutions are mapped outside of the unit circle

Figure 4.7

Figure 4.8

Chapter 5: Classical Dynamics

Figure 5.1 Mechanical representation of a SDOF system and corresponding free-body-diagram

Figure 5.2

Figure 5.3

Figure 5.4 (a) Shows a Dirac delta impulse applied at and (c) shows the corresponding response given by the shifted IRF . (b) Shows the impulse of a continuously acting force in the infinitesimal time interval from to and (d) shows the corresponding scaled impulse response

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10 In case of repeated poles, the corresponding mode shapes are not necessarily orthogonal, but one of the vectors might be chosen for the basis – in this case – and the other one – in this case – is then rotated in the subspace defined by until it is perpendicular to

Figure 5.11 In the case of closely spaced modes, the change of mode shapes from the unperturbed set to the perturbed set of mode shapes the change is mainly a rotation of the unperturbed set in the subspace spanned by , the rotation angle is defined as the angle given by Eq. (5.181)

Figure 5.12

Chapter 6: Random Vibrations

Figure 6.1 (a) Illustrates Parseval's theorem stating that the area under the auto spectral density equals the variance of the signal. (b) Illustrates that for a bandpass-filtered signal the area of the spectral density inside the band defines the variance of the filtered signal. The auto spectral density illustrates the distribution of energy as a function of frequency

Figure 6.2

Figure 6.3

Chapter 7: Measurement Technology

Figure 7.1

Example of a 2D study

: Measurement plan for a straight bridge where the longitudinal displacements are neglected, thus only the transverse horizontal and vertical displacements and the torsions are to be estimated. Since the locations of the node points are uncertain, two 2D sensors are used in the reference set and then three sensors are used in the roving set. In this case, the minimum number of sensors needed to perform the measurements is seven and the number of data sets is equal to the number of cross sections where the mode shape is to be estimated

Figure 7.2

Example of a 2D study

: Measurement plan for a simple rectangular “box type” building where the vertical displacements are neglected, thus only the two horizontal displacements and the torsion of the building is to be estimated. In this case, no modes have node points at the top corners, thus only one 2D reference sensor can be used and three sensors can be used in the roving set. In this case, the minimum number of sensors needed to perform the measurements is five and the number of data sets is equal to the number of floors where the mode shape is to be estimated

Figure 7.3 Minimum and maximum vertical seismic noise power spectral density (PSD) according to Peterson [[5]] in dB relative to . The top plot is the Peterson high noise model (HNM) and bottom plot is the Peterson low noise model (LNM). The values were computed as 20 log of acceleration relative to

Figure 7.4 Singular value plots in dB of the response spectral matrix of a 30 cm-by-30 cm thin steel plate with 16 sensors in a 4 × 4 grid. (a) The result when a randomly moving load is used – in this case the back end of a pencil is being moved randomly over the entire surface of the plate. (b) The results when a stationary single input loading is being used instead. The distinct separation of the singular value lines in (a) indicates that the spectral matrix has rank of 5–6 over the whole frequency band, whereas in the (b) the rank is limited to one in nearly the whole frequency band. It is obvious that the case shown on the (b) has significantly less system information compared to the case shown on the (a)

Figure 7.5 Simplified model of a sensor

Figure 7.18 (a) Calculated sensor self-noise spectrum using Eq. (7.71). (b) Calculated signal spectrum using Eq. (7.73), both in dB (rel. to ). From Brincker and Larsen [9].

Figure 7.19 Plot of the singular values of the spectral matrix in dB (rel. to ) of four data channels measuring the same signal. Two sensors with high noise level (piezoelectric accelerometers) are visible in the data, the noise floor of the two other sensors (geophones) is not visible since the noise level of these sensors falls below the noise floor of the measurement amplifier, which is indicated by the lowest singular value

Figure 7.6 Typical form of the FRF of a sensor. Since the FRF is a complex quantity, it is common practice to present it in two companion plots: one plot of the magnitude of the FRF (a) and one plot of the phase between the imaginary and real components of the FRF (b). As illustrated the frequency response of a typical sensor might be disturbed by a cutoff filter close to DC and by the natural frequency of the sensor in the upper frequency region

Figure 7.7 When a bar of piezoelectric material is loaded by an axial force, a negative charge is built up at one end and a positive charge at the other end. (a) A single crystal is deformed. (b) When two crystals are connected in series the charges from the two crystals are canceled out at the joint

Figure 7.8 Typical FRF of the 4508 accelerometer from Brüel and Kjær

Figure 7.9 (a) Definition of base and coil displacements. (b) Theoretical transfer function for the SM6 geophone sensor element, from Brincker et al. [14]

Figure 7.10 Measurement principle for a FBA where the spring in Figure 7.5 is replaced by an active servo system consisting of a displacement sensor, an actuator often realized by a coil around a permanent magnet and finally a servo amplifier creating the control current to the actuator as proportional to the measured displacement of the seismic mass

Figure 7.11 FRF of the Kinemetrics EPI sensor. The natural frequency of the sensor is at 224 Hz, but due to the high damping of the sensor, no peak is visible in the magnitude and a large deviation of the phase is present. Information about the sensor provided by courtesy of Kinemetrics

Figure 7.12 An analog antialiasing filter must always be applied between the sensor and the ADC in order to effectively remove the energy of the signal outside of the Nyquist band. The left plot shows the signal from the sensor that might include signal components outside of the Nyquist band. Applying the low-pass filter shown in the middle will cut of these components, and finally the signal that reaches that ADC will have only components in the Nyquist band

Figure 7.13 Matching problem of sensors and ADC. A nonmatching maximum input and noise floor might significantly reduce the measurement range of the combined system consisting of the sensor and ADC. The ideal case is where the two measurement elements match reasonably so that the system noise floor and system dynamic range are approximately equal to the same properties of the individual elements

Figure 7.14 Noise cancelation using twisted pairs. (a) Shows a wire without twisting in which the full influence of the electromagnetic radiation is picked up by the wire acting as a loop antenna. (b) Shows a pair with a single twist. If the noise is the same in the two loops, then the twist creates two noise signals of opposite polarity in the signal path and the noise is canceled out

Figure 7.15 Ground loops occur when several groundings are active in the same measurement setup. The problem can be solved by disconnecting the shielding to the sensor casing (the two connections marked “x”) or by isolating the sensor casing from the ground

Figure 7.16 A simple way to perform a traceable calibration of a vibration sensor is to use a broad input from a shaker and measure the response with a displacement laser (a) or a reference sensor (b)

Figure 7.17 (a) Spectral density function of a signal from a piezoelectric accelerometer that is influenced by noise, in dB (rel. to ). (b) Coherence between two simultaneous measurements using the same type of piezoelectric accelerometer in the same location

Figure 7.20 Classical pendulum with a sensor attached to the mass

Figure 7.21 A tower primarily acting as a clamped beam with a sensor at the top. In this case, the tilt will significantly increase the measured acceleration if modes are present in the low frequency band

Chapter 8: Signal Processing

Figure 8.1 For detrending, but also for subsequent signal processing it might be practical to segment the data as shown in this Figure Each data segment is captured by overlapping and window tapering such that the sum of all data segments is equal to the original signal

Figure 8.2 Time-frequency plot of channel one of the first data set of the Heritage Court Tower building test showing a relatively stationary loading case

Figure 8.3 (a) Normalized probability density function of the response of a structural mode, and (b) Probability density function of harmonic component with unit amplitude

Figure 8.4 Plot of the singular values of the spectral matrix of the response of a plate loaded by random input and a harmonic around 90 Hz. The harmonic results in the first three peaks (the harmonic itself and its two over harmonics) and the random input results in the remaining peaks representing random response. As it appears the harmonics are visible in several singular values whereas the random response does not show this characteristic. From Brincker et al. [5]

Figure 8.5 Frequency response functions of the three commonly used filters: Top plots: High-pass filter with cut-off frequency , middle plots: Low-pass filter with cut-off frequency , and bottom plots: Band-pass filter with cut-off frequencies and . Left plots have natural axes and right plots have double logarithmic axes making easier to see the slope (or steepness) of the filters around cut-off frequencies

Figure 8.6 The relative impulse response function of the mixed filter given by Eq. (8.15) for different values of

Figure 8.7 Integration in the time domain corresponds to division in the frequency domain by . However for frequencies close to DC the pole at DC must be removed and replaced by a high-pass filter

Figure 8.8

Figure 8.9

Figure 8.10 Unbiased estimation of correlation functions by the Welch method. After data segments have been defined, they are zero padded to double length and Fourier transformed, the spectral density is calculated according to Eq. (8.52) and smoothed by averaging the spectral density estimates from the different data segments. The resulting spectral estimate is transformed back to time domain by inverse FFT and the bias is removed dividing by the triangular window. The outer most part of the estimate is discarded in order not to amplify the noise by division with the basic lag window where it is approaching zero

Figure 8.11 Estimation of a random decrement signature by defining a triggering condition (in this case the level triggering condition ) capturing the data segments around the triggering points and averaging them to form the signature

Figure 8.12

Figure 8.13

Figure 8.14

Chapter 9: Time Domain Identification

Figure 9.1

Figure 9.2 First three mode shapes of the HCT building found by AR/PR identification and merging the mode shapes components from the four data sets corresponding to the physical modes shown in Table 9.1

Figure 9.3 First three mode shapes of the HCT building found by ITD identification and merging the mode shapes components from the four data sets corresponding to the physical modes shown in Table 9.2

Figure 9.4

Figure 9.5

Figure 9.6

Chapter 10: Frequency-Domain Identification

Figure 10.1

Figure 10.2 Minimizing the bias from forcing the singular vector to geometrical orthogonality. When the left mode is dominating then the vector is dominating the modal contribution and the corresponding SVD estimate is close to the vector even though the two mode shapes are not geometrically orthogonal. Therefore, any mode shape in FDD is estimated as the first singular vector at any frequency line where the corresponding singular value is dominating

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Chapter 11: Applications

Figure 11.1 FE model of HCT building (courtesy of Ventura et al. [6]). (a) Complete Model. (b) Details of Model

Figure 11.2 First six mode shapes of updated FE model of HCT building

Figure 11.3 Comparison of reduced FE mode shapes of updated model and experimental determined mode shapes of HCT building

Figure 11.4 Comparison of 3D plots of MAC matrices for first six mode shapes of HCT building. (a) MAC matrix before updating. (b) MAC matrix after updating

Figure 11.5 South view of City Tower building

Figure 11.6 Plan and elevation views of City Tower

Figure 11.7 Typical sensor locations (see arrows marked 1–4)

Figure 11.8 Average of the singular values of the spectral density matrices for all test setups of the City Tower building with all 32 stories completed using four projection channels

Figure 11.9 First three mode shapes of City Tower building with 32 stories completed obtained from the FDD technique

Figure 11.10 Frequency trends for the tower during its construction phase

Figure 11.11 Overall view of the Vasco da Gama cable-stayed bridge

Figure 11.12 Measurement locations on the Vasco da Gama cable-stayed bridge

Figure 11.13 FDD plot for data sets 4 and 6 – sections 4 and 10, and 6 and 10

Figure 11.14 Stabilization diagram for data set 6 – measurement sections 6 and 10

Figure 11.15 MAC matrix plot for mode shapes estimated by FDD and SSI methods

Figure 11.16 Most relevant mode shapes identified by the FDD (left) and the SSI (right) methods

Figure 11.17 Investigated container vessel

Figure 11.18 Measurement model at ship yard: acquisition of six data sets using four References

Figure 11.19 Singular values of spectral density matrices of all data sets up to 3.5 Hz from ship yard measurement

Figure 11.20 Stabilization diagram for the SSI technique

Figure 11.21 Mode shapes 1 and 3 identified using operational modal analysis at the ship yard

Figure 11.22 Mode shapes 2, 4, 5, and 6 identified using operational modal analysis at the ship yard

Chapter 12: Advanced Subjects

Figure 12.1 Measures of the best fit for a plate structure. (a) MAC value calculated on the fitting set of DOF's. (b) MAC value calculated on the observation set of DOF's. As it appears, the measure in the left plot is an increasing function whereas the measure in the right plot has a clear optimum

Figure 12.2 (a) Generalized stabilization diagram depicting an identification condition on the vertical axis together with a modal quantity on the horizontal axis. (b) A clustering plot where two modal quantities are depicted in the same plot for different identification conditions

Figure 12.3 Example of modal coherence and modal domain. (a) SVD plot of the spectral density of the system with two closely spaced modes. (b) Modal coherence function . (c) Modal domain of the two modes

Appendix B: Operational Modal Testing of the Heritage Court Tower

Figure B.1 North building face from Robson street

Figure B.2 East building face from Hamilton street

Figure B.3 South building face from Hamilton street

Figure B.4 Data acquisition and analysis computers

Figure B.5 Photo of accelerometer setup

Figure B.6 Typical accelerometer locations and directions on every second floor (all dimensions are in meters)

Figure B.7 Ground floor accelerometer locations and directions

Figure B.8 FDD plot showing the SVD diagram found as an average of the SVD diagrams of the four data sets

Figure B.9 Mode shapes estimated by the FDD peak picking technique

Figure B.10 SSI stabilization diagram for data set 1

Figure B.11 Mode definition by combining the four selected models from the four data sets named measurement 1, 2, 3, and 4

Figure B.12 MAC matrix 3D plot

List of Tables

Chapter 1: Introduction

Table 1.1 General characteristics of structural response

Table 1.2 General characteristics of structural response

Table 1.3 Organization of this book

Chapter 4: Transforms

Table 4.1 Laplace transform table

Table 4.2


-transform table

Chapter 6: Random Vibrations

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Chapter 7: Measurement Technology

Table 7.1

Table 7.2 Uncertainty on the most important parameters of FE and modal models used for simulation studies

Table 7.3 Minimum time series length's for different damping ratios

Table 7.4 Recommended number of reference sensors

Table 7.5 Minimum excitation levels

Table 7.6 Classification of typical problems in relation to number of independent inputs

Table 7.7 Characteristics of commonly used sensors

Table 7.8 Advantages and disadvantages of commonly used sensors

Table 7.9

Chapter 8: Signal Processing

Table 8.1 Commonly used RD triggering conditions and their effects

Chapter 9: Time Domain Identification

Table 9.1

Table 9.2

Table 9.3

Table 9.4

Chapter 10: Frequency-Domain Identification

Table 10.1

Table 10.2

Table 10.3

Table 10.4

Chapter 11: Applications

Table 11.1 Comparison of first six natural frequencies of the HCT building before and after model updating

Table 11.2 Frequencies (Hz) for City Tower from five stages in its construction

Table 11.3 Identified natural frequencies and modal damping ratios

Table 11.4 Natural frequencies


(Hz) and damping ratios


(%) identified by classical and operational modal analysis at the ship yard

Appendix B: Operational Modal Testing of the Heritage Court Tower

Table B.1 Operational modal testing Test Setup Locations and Directions for April 28, 1998 Test

Table B.2 MAC matrix table

Table B.3 Summary of modal results for the two techniques


Rune Brincker

Technical University of Denmark

Carlos E. Ventura

University of British Columbia, Canada



This edition published in 2015

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Library of Congress Cataloging-in-Publication Data

Brincker, Rune.

Introduction to operational modal analysis / Rune Brincker, Technical University of Denmark, Denmark, Carlos Ventura, University of British Columbia, Canada.

pages cm

Includes bibliographical references and index.

ISBN 978-1-119-96315-8 (cloth)

1. Modal analysis. 2. Structural analysis (Engineering) I. Ventura, Carlos. II. Title.

TA654.15.B75 2015




After many years of working on various aspects of operational modal analysis (OMA), conducting vibration tests, analyzing test data from a variety of structures, and giving courses to promote the use and understanding of OMA, we decided to write a book on this topic during a meeting at SVS in Aalborg in the summer 2003. Two years later, in the summer of 2005, we secluded ourselves for one month at the Ventura's family coffee farm in Guatemala where we prepared the first outline of the book and started on the writing process. At that time, we focused our efforts on some key areas, mainly testing, classical dynamics, and signal processing. However, due to a number of circumstances related to work commitments and personal and family affairs, our serious writing process was delayed until the summer of 2011 and continued on until the summer of 2014 when we finished this first edition of the book. It has been a long and demanding effort, but in the end we have prepared a book that reflects our personal views of OMA, both in terms of theory and practice.

It would be preposterous to say that we are the only specialists in the field qualified to write one of the first books on OMA, but both of us have extensive experience with this technology and we recognized that as a team, we were well qualified to write the first book that dealt in a formal manner with the theory behind OMA. We noticed that in the early 1990s people started to pay more attention to OMA and realized the advantages of OMA techniques to determine modal properties of structures. We also noticed that the theory behind OMA was not well understood and that many people were hesitant to use these techniques because of a lack of a clear understanding of why these work so well. So, in 2003 we decided to make an effort to “demystify OMA,” and since then the need for such a book has just grown. When we look at our final product, it seems like we made a good decision to work on this book, and we are confident that it will help people working in the field of OMA.

This book is written to be used as a textbook by students, mainly graduate students and PhD students working in research areas where OMA is applied, but it can also be used by scientists and professionals as a reference book for the most important techniques presently being used to analyze vibration data obtained by using OMA testing techniques.

Some people might argue that the classical experimental modal analysis (EMA) and OMA are the same thing, and, therefore, there is no need for a special theory for OMA. But we have compelling reasons to disagree based on our understanding of the fundamental theory of OMA. Our opinion is that actually EMA and OMA are quite different; they have a different history, they use a different technology, they have a different theoretical background, and finally, their applications are different. OMA is indeed a special field that needs to be introduced properly, its mathematical basis and background need to be adequately explained, and good testing practices need to be introduced in order to obtain good data and meaningful results. This is why this book is needed.

The theory in this book is presented heuristically rather than rigorously, thus many mathematical details are omitted for the sake of brevity and conciseness. The aim is not to cover the whole subject in great detail but rather to present a consistent overview of the theories needed to understand the topic and to point out the what these theories have in common and how these theories can be implemented in practice.

This book is rich in mathematical equations that are needed for formulating the theory of OMA, but extensive derivations of equations and formulas are avoided. Our aim was to present each equation or formula in its simplest and clearest formulation. This is also a book rich in simple and clear explanations that will help the reader understand the background for the formulas and how to use them in an effective way in order to perform OMA.

During the writing process, we have been privileged to receive excellent advice from colleagues from around the world who also work on OMA techniques. Without all this advice, we would not have been able to complete the book in a manner that makes us proud of our efforts. We would like to thank all these colleagues for spending their time giving us feedback. We would like to thank Dr. Spilios Fassois and Dr. Nuno Maia for giving us feedback on Chapters 2 and 3, Dr. Anders Brandt and Dr. Henrik Karstoft for their feedback on Chapters 4 and 8, Dr. Manuel Lopez-Aenlle for his feedback on Chapters 5 and 12, Dr. Frede Christensen for his feedback on Chapter 5, Dr. George James and Dr. Lingmi Zhang for their feedback on Chapter 6, Dr. Lingmi Zhang for also giving us feedback on Chapters 9 and 10, Dr. James Brownjohn for his feedback on Chapter 7, and Dr. Bart Peeters for his feedback on Chapter 10. The many useful comments from our PhD students and coworkers, Martin Juul, Anela Bajric, Jannick B. Hansen, Peter Olsen, Anders Skafte, and Mads K. Hovgaard are very much appreciated. We would also like to thank Dr. Palle Andersen of SVS for providing insightful comments on the use of time-domain and frequency-domain techniques as implemented on the ARTeMIS program. The case studies presented in Chapter 11 are based on papers published by the authors and other colleagues. Special thanks are extended to Dr. Alvaro Cunha, Dr. Elsa Caetano, and Dr. Sven-Eric Rosenov for sharing with us the data sets for two of the case studies described in this chapter.

We like to thank our colleagues and friends who have encouraged us to write this book. And last, but not least, we would like to thank our wives, Henriette and Lucrecia for their unconditional support all these years and for having infinite patience with us while we struggled with the preparation of the various drafts of each chapter, and for their willingness to allow us to spend time working on this book rather than attending to family affairs – without their support and unconditional love, this book would have never been a reality.

June 2015

Rune Brincker

Carlos E. Ventura

Chapter 1Introduction

Torture numbers and they'll confess to anything

– Gregg Easterbrook

The engineering field that studies the modal properties of systems under ambient vibrations or normal operating conditions is called Operational Modal Analysis (OMA) and provides useful methods for modal analysis of many areas of structural engineering. Identification of modal properties of a structural system is the process of correlating the dynamic characteristics of a mathematical model with the physical properties of the system derived from experimental measurements.

It is fair to say that processing of data in OMA is challenging; one can even say that this is close to torturing the data, and it is also fair to say that fiddling around long enough with the data might lead to some strange or erroneous results that might look like reasonable results. One of the aims of this book is to help people who use OMA techniques avoid ending up in this situation, and instead obtain results that are valid and reasonable.

In OMA, measurement data obtained from the operational responses are used to estimate the parameters of models that describe the system behavior. To fully understand this process, one should have knowledge of classical structural mechanics, matrix analysis, random vibration concepts, application-specific simplifying assumptions, and practical aspects related to vibration measurement, data acquisition, and signal processing.

OMA testing techniques have now become quite attractive, due to their relatively low cost and speed of implementation and the recent improvements in recording equipment and computational methods. Table 1.1 provides a quick summary of the typical applications of OMA and how these compare with classical modal testing, also denoted experimental modal analysis (EMA), which is based on controlled input that is measured and used in the identification process.

Table 1.1 General characteristics of structural response

Mechanical engineering

Civil engineering


Artificial excitation

Artificial excitation

Impact hammerShakers (hydraulic, electromechanical, etc.)Controlled blastsWell-defined measured input

Shakers, mainly hydraulicDrop weightsPull back testsEccentric shakers and excitersWell defined, measured, or unmeasured inputsControlled blasts


Artificial excitation

Natural excitation

Scratching deviceAir flowAcoustic emissionsUnknown signal, random in time and space

WindWavesTrafficUnknown signal, random in time and space, with some spatial correlation

Source: Adapted from American National Standard: “Vibration of Buildings – guidelines for the measurement of vibrations and their effects on buildings,” ANSI S2.47-1990 (ASA 95-1990).

The fundamental idea of OMA testing techniques is that the structure to be tested is being excited by some type of excitation that has approximately white noise characteristics, that is, it has energy distributed over a wide frequency range that covers the frequency range of the modal characteristics of the structure. However, it does not matter much if the actual loads do not have exact white noise characteristics, since what is really important is that all the modes of interest are adequately excited so that their contributions can be captured by the measurements.

Referring to Figure 1.1, the concept of nonwhite, but broadband loading can be explained as follows. The loading is colored, thus does not necessarily have an ideal flat spectrum, but the colored loads can be considered as the output from an imaginary (loading) filter that is loaded by white noise.

Figure 1.1 Illustration of the concept of OMA. The nonwhite noise loads are modeled as the output from a filter loaded by a white noise load

It can be proved that the concept of including an additional filter describing the coloring of the loads does not change the physical modes of the system, see Ibrahim et al. [1] and Sections 7.2.7 and 8.3.7. The coloring filter concept shows that in general what we are estimating in OMA is the modal model for “the whole system” including both the structural system and the loading filter.

When interpreting the modal results, this has to be kept in mind, because, some modes might be present due to the loading conditions and some might come from the structural system. We should also note that in practice we often estimate a much larger number of modes than the expected physical number of modes of the considered system.

This means that we need to find ways to justify which modes belong to the structural system, which modes might describe the coloring of the loading, and finally which modes are just noise modes that might not have any physical meaning. These kinds of considerations are important in OMA, and will be further illustrated later in this book.

We can conclude these first remarks by saying that OMA is the process of characterizing the dynamic properties of an elastic structure by identifying its natural modes of vibration from the operating responses. Each mode is associated with a specific natural frequency and damping factor, and these two parameters can be identified from vibration data from practically any point on the structure. In addition, each mode has a characteristic “mode shape,” which defines the spatial distribution of movement over the entire structure.

1.1 Why Conduct Vibration Test of Structures?

Vibration measurements are made for a variety of reasons. They could be used to determine the natural frequencies of a structure, to verify analytical models of the structure, to determine its dynamic response under various environmental conditions, or to monitor the condition of a structure under various loading conditions. As structural analysis techniques continually evolve and become increasingly sophisticated, awareness grows of potential shortcomings in their representation of the structural behavior. This is prevalent in the field of structural dynamics. The justification and technology exists for vibration testing and analysis of structures.

Large civil engineering structures are usually too complex for accurate dynamic analysis by hand. It is typical to use matrix algebra based solution methods, using the finite element method of structural modeling and analysis, on digital computers. All linear models have dynamic properties, which can be compared with testing and analysis techniques such as OMA.

1.2 Techniques Available for Vibration Testing of Structures

Let us discuss in some detail the two main types of modal testing: the EMA that uses controlled input forces and the OMA that uses the operational forces.

Both forced vibration and in-operation methods have been used in the past and are capable of determining the dynamic characteristics of structures. Forced vibration methods can be significantly more complex than in-operation vibration tests, and are generally more expensive than in-operation vibration tests, especially for large and massive structures. The main advantage of forced vibration over in-operation vibration is that in the former the level of excitation and induced vibration can be carefully controlled, while for the latter one has to rely on the forces of nature and uncontrolled artificial forces (i.e., vehicle traffic in bridges) to excite the structure, sometimes at very low levels of vibration. The sensitivity of sensors used for in-operation vibration measurements is generally much higher than those required for forced vibration tests.

By definition, any source of controlled excitation being applied to any structure in order to induce vibrations constitutes a forced vibration test. In-operation tests that rely on ambient excitation are used to test structures such as bridges, nuclear power plants, offshore platforms, and buildings. While ambient tests do not require traffic shutdowns or interruptions of normal operations, the amount of data collected is significant and it can be a complex task to analyze this data thoroughly.

The techniques for data analysis are different. The theory for forced vibration tests of large structures is well developed and is almost a natural extension of the techniques used in forced vibration tests of mechanical systems. In contrast, the theory for ambient vibration tests still requires further development.

1.3 Forced Vibration Testing Methods

Forced vibration tests or EMA methods are generally used to determine the dynamic characteristics of small and medium size structures. In rare occasions, these methods are used on very large structures because of the complexity associated with providing significant levels of excitation to a large, massive structure. In these tests, controlled forces are applied to a structure to induce vibrations. By measuring the structure's response to these known forces, one can determine the structure's dynamic properties. The measured excitation and acceleration response time histories are used to compute frequency response functions (FRFs) between a measured point and the point of input. These FRFs can be used to determine the natural frequencies, mode shapes, and damping values of the structure using well-established methods of analysis. One can apply controlled excitation forces to a structure using several different methods. Forced vibrations encompass any motion in the structure induced artificially above the ambient level. Methods of inducing motion in structures include:

Mechanical shakers


Eccentric mass

Hydraulic, including large shaking tables in laboratories

Transient loads

Pull-back and release, initial displacement

Impact, initial velocity

Man-excited motions

Induced ground motion

Underground explosions

Blasts with conventional explosives above the ground

The three most popular methods for testing structures are shaker, impact, and pull back or quick-release tests. A brief description of these methods follows:

Shaker tests:

Shakers are used to apply forces to structures in a controlled manner to excite them dynamically. A shaker must produce sufficiently large forces, to effectively excite a large structure in a frequency range of interest. For very large structures, such as long-span bridges or tall buildings, the frequencies of interest are commonly less than 1 Hz. While it is possible to produce considerable forces with relatively small shakers at high frequencies, such as those used to test mechanical systems, it is difficult to produce forces large enough to excite a large structure at low frequencies. Although it is possible to construct massive, low frequency shakers, these are expensive to build, transport, and mount. In such cases, alternative methods to excite the structure are desirable.

Impact tests:

Impact testing is another method of forced vibration testing. Mechanical engineers commonly use impact tests to identify the dynamic characteristics of machine components and small assemblies. The test article is generally instrumented with accelerometers, and struck with a hammer instrumented with a force transducer. While impact testing is commonly used to evaluate small structures, a number of problems may occur when this method is used to test larger structures. To excite lower modes of a large structure sufficiently, the mass of the impact hammer needs to be quite large. Not only is it difficult to build and use large impact hammers with force transducers, but the impact produced by a large hammer could also cause considerable local damage to the test structure.

Pull back tests:

Pull back or quick-release testing has been used in some occasions for testing of large structures. This method generally involves inducing a prescribed temporary deformation to a structure and quickly releasing it, causing the structure to vibrate freely. Hydraulic rams, cables, bulldozers, tugboats, or chain blocks have been used to apply loads that produce a static displacement of the structure. The goal of this technique is to quickly release the load and record the free vibrations of the structure as it tends to return to its position of static equilibrium. The results from quick release tests can be used to determine natural frequencies, mode shapes, and damping values for the structure's principal modes.

1.4 Vibration Testing of Civil Engineering Structures

What makes testing of large civil engineering structures different than testing mechanical systems? As we have just discussed, the obvious answer to this question is that the forces are larger and the frequencies are lower in large structures. But there is more than that. First, in general, analytical models of existing large structures are based on geometric properties taken from design or construction drawings and material properties obtained from small specimens obtained from the structure. A series of assumptions are also made to account for the surrounding medium and its interaction with the structure (such as soil-structure interaction in the case of buildings and bridges, and soil–water–structure interaction in the case of dams, wharves, and bridges) and the composite behavior of structural elements. This, in general, is not the case for mechanical systems. And second, in the field of mechanical engineering, there are a number of integrated systems that can handle very efficiently the experimental testing, system identification, and model refinement. These integrated systems are very sophisticated as they combine the results of several decades of research in the field. Due to their relatively small size, most mechanical specimens can be tested in laboratories under controlled conditions. There is no such advantage for the verification of dynamic models of large civil engineering structures.

During normal operating conditions, a building is subjected to ambient vibrations generated by wind, occupants, ventilation equipment, and so on. As we have argued earlier, a key assumption of the analysis of these ambient vibrations is that the inputs causing motion have “nearly” white noise characteristics in the frequency range of interest. This assumption implies that the input loads are not driving the system at any particular frequency and therefore any identified frequency associated with significant strong response reflects structural modal response. However, in reality, some of the ambient disturbances, such as, for instance, an adjacent machine operating at a particular frequency may drive the structure at that frequency. In this case, the deformed shapes of the structure at such driving frequencies are called operational modes or operational deflection shapes. This means that a crucial requirement of methods to analyze ambient vibration data is the ability to distinguish the natural structural modes from any imposed operational modes.

The integrated systems, developed for mechanical engineering applications are not practical and economical to test large civil engineering structures. Bridges form vital links in transportation networks and therefore a traffic shutdown required to conduct a forced vibration test would be costly. Controlled forced vibration tests of buildings may disturb the occupants and may have to be conducted after working hours, thus increasing the cost of the testing. Therefore, routine dynamic tests of bridges and buildings must be based on ambient methods, which do not interfere with the normal operation of the structure.

1.5 Parameter Estimation Techniques

The methods that have been developed for analyzing data from forced and in-operation vibration tests range from linear deterministic models to nonlinear stochastic models. The applications range from improving mathematical models of systems to damage detection, to identifying the input of a system for controlling its response. Parameter estimation methods using dynamic signals can be classified as

time-domain methods

frequency-domain methods

joint frequency–time domain methods

The theory behind the first two methods is described in more detail in this book.

1.6 Brief History of OMA

Although very significant advances in OMA testing techniques have occurred since the early 1990s, there is a wealth of information about different uses of OMA since the 1930s. Even ancient history shows evidence of the use of the OMA concepts to better understand why and how structures vibrate.

Pythagoras is usually assumed to be the first Greek philosopher to study the origin of musical sound. He is supposed to have discovered that of two stretched strings fastened at the ends the higher note is emitted by the shorter one. He also noted that if one has twice the length the other, the shorter will emit a note an octave above the other. Galileo is considered the founder of modern Physics and in his book “Discourses Concerning Two New Sciences” in 1638: At the very end of the “First Day,” Galileo has a very remarkable discussion of the vibration of bodies. He describes the phenomenon of sympathetic vibrations or resonance by which vibrations of one body can produce similar vibrations in another distant body. He also did an interesting comparison between the vibrations of strings and pendulums in order to understand the reason why sounds of certain frequencies appear to the ear to combine pleasantly whereas others are discordant.

Daniel Bernoulli's publication of the Berlin Academy in 1755 showed that it is possible for a string to vibrate in such a way that a multitude of simple harmonic oscillations are present at the same time and that each contributes independently to the resultant vibration, the displacement at any point of the string at any instant being the algebraic sum of the displacements for each simple harmonic at each node. This is what is called the Principle of “Coexistence,” which is what we know today as the Superposition Principle. Today, we also refer to this as the method of Modal Superposition. Joseph Fourier's publication “Analytical Theory of Heat” in 1822 presents the development of his well-known theorem on this type of expansion. Isaac Newton in the second book of his “Principia” in 1687 made the first serious attempt to present a theory of wave propagation. John Strutt, 3rd Baron Rayleigh (1842–1919) through his investigations of sound and vibration provided the basis for modern structural dynamics and how mass, stiffness and damping are interrelated and determine the dynamic characteristics of a structural system.

The first studies on shocks and vibrations affecting civil engineering structures in the twentieth century were carried out at the beginning of the 1930s to improve the behavior of buildings during earthquakes. M.A. Biot introduced the concept of the shock spectrum to characterize the response of buildings to earthquakes and to compare their severity. G. Housner, refined the concept by defining it as the shock response spectrum (SRS) to clearly identify that it characterizes the response of a linear one-degree-of-freedom system subjected to a prescribed ground shaking. After the 1933 Long Beach earthquake in California, in 1935, D.S. Carder conducted tests of ambient vibrations in more than 200 buildings and applied rudimentary OMA techniques to determine the natural modes of vibrations of these buildings. The results of this investigation were used in the design codes to estimate natural frequencies of new buildings. The seminal work of M. Trifunac in 1972 showed that the analysis of ambient and forced vibrations led to the same results for practical engineering purposes.

The development of OMA techniques since the mid-1990s can be followed by reading the proceedings of the annual International Modal Analysis Conference ( and, most recently, those from the International Operational Modal Analysis Conference (

1.7 Modal Parameter Estimation Techniques

In contrast to EMA, OMA testing does not require any controlled excitation. Instead, the response of the structure to “ambient” excitation sources such as wind, traffic on or beneath the structure, and microtremors is recorded. Many existing textbooks provide an extensive overview of input–output modal parameter estimation methods. See for instance, Heylen et al. [2] and Ewins [3]. In the operational case, ignoring the need to measure the input is justified by the assumption that the input does not contain any specific information, or expressed in other words, the input is approximately white noise. As with EMA, the measured time signals can be processed in the time domain or in the frequency domain. Since the forcing function is unknown, frequency response functions between force and response signals cannot be calculated. Instead, the analysis relies on correlation functions and spectral density functions estimated from the operational responses.

Further, since OMA sensors and cables can be expensive, a limited number of sensors are used and some of these sensors are roved over the structure to obtain several data sets. In order to be able to assemble mode shapes using the parts of the mode shape estimated by the different data sets, some of the response signals are declared as reference signals. The reference sensors are kept in the same place when recording all data sets while the remaining sensors are moved progressively over the structure.

A fast method to estimate modal parameters from OMA tests is the rather simple “peak-picking” frequency-domain technique. This technique has been used extensively over the years for all kinds of applications. The basic idea of the peak-picking technique is that when a structure is subjected to ambient excitations, it will have strong responses near its natural frequencies. These frequencies can be identified from the peaks in the power spectral densities (PSD) computed for the time histories recorded at the measurement points. This concept is illustrated in the example presented in Figure 1.2. The significant peaks of the PSDs for the OMA measurements conducted on a five-span bridge can be associated with the natural frequencies of vertical vibrations of this bridge.

Figure 1.2 Example of peak-picking technique for identification of natural frequencies of vertical vibration of a five-span bridge

(Source: courtesy of Felber [4])

The method has been widely used for many years. One practical implementation of this method was developed by Felber [4] and will be used here as an example of the application of early OMA techniques. In this implementation, the natural frequencies are determined as the peaks of the averaged normalized power spectral densities (ANPSDs). It is then assumed that the coherence function computed for two simultaneously recorded response signals has values close to one at the natural frequencies; see Bendat and Piersol [5]. This also helps to decide which frequencies can be considered as natural frequencies for the structure. The components of the mode shapes are determined by the values of the transfer functions obtained at the natural frequencies. It should be noted that in the context of ambient testing, transfer function does not mean the ratio of response over force, but rather the ratio of the response measured at a location of the structure with respect to the reference sensor. Every transfer function yields a mode shape component at a natural frequency. In this method it is assumed, however, that the dynamic response at resonance is only determined, or controlled by one mode. The validity of this assumption increases when modes are well separated and when the modal damping is low (see more about this in Section 10.2).

Felber implemented a novel procedure based on this idea to expedite the modal identification of ambient vibration data, and this seminal work was the motivation for the development of interactive techniques for fast and efficient implementation of the peak-picking technique. Three programs were used for the process, one to generate the ANPSDs for the identification of natural frequencies, another to compute the transfer functions between sensors (program ULTRA) and the third to visualize and animate the mode shapes (program VISUAL). All three programs were used for preliminary analysis in the field and as well as for further analysis in the office. Figure 1.3 shows a flowchart of the implementation of the methodology developed by Felber.

Figure 1.3