NUMERICAL MODEL CALIBRATION OF A CANTILEVER BEAM USING THE

LOCAL CORRESPONDENCE PRINCIPLE

Andrew Martins Vieira

Graduation Project presented to the Naval

and Ocean Engineering Program of the Polytechnic

School, Federal University of Rio de Janeiro, as a

partial fullfilment to the requirments to obtain the

title of Engineer.

Advisor: Ulisses Admar Barbosa Vicente

Monteiro

Rio de Janeiro

March 2020

NUMERICAL MODEL CALIBRATION OF A CANTILEVER BEAM USING THE

LOCAL CORRESPONDENCE PRINCIPLE

Andrew Martins Vieira

GRADUATION PROJECT SUBMITTED TO THE FACULTY STAFF OF

THE NAVAL AND OCEAN ENGINEERING PROGRAM OF THE POLYTECHNIC

SCHOOL OF THE FEDERAL UNIVERSITY OF RIO DE JANEIRO AS A PARTIAL

FULLFILMENT TO THE REQUIREMENTS TO OBTAIN THE DEGREE OF

NAVAL AND OCEAN ENGINEER.

Examined by:

___________________________________________________

Prof. Ulisses A. Monteiro, D.Sc., DENO/UFRJ

(Advisor)

___________________________________________________

Prof. Luiz A. Vaz, D.Sc., DENO/COPPE/UFRJ

___________________________________________________

Eng. Ricardo Homero Ramírez Guetiérrez, D.Sc, COPPETEC/UFRJ

___________________________________________________

Eng. Claudio de Oliveira Mendonça, PETROBRAS

RIO DE JANEIRO, RJ - BRAZIL

MARCH 2020

iii

Vieira, Andrew Martins

Numerical Model Calibration of a Cantilever Beam using

the Local Correspondence Principle/Andrew Martins Vieira. - Rio

de Janeiro: UFRJ/Polytechnic School, 2020.

XII, 70 p.: il.; 29,7 cm.

Advisor: Ulisses Admar Barbosa Vicente Monteiro

Graduation Project – UFRJ Polytechnic School/Naval and

Ocean Engineering Program, 2020

Bibliographic References: p. 69-70.

1. Calibration. 2. Vibration. 3. Numerical Model Reduction.

4. Modal Analysis. I. Barbosa Vicente Monteiro, Ulisses Admar.

II. Federal University of Rio de Janeiro, Polytechnic School, Naval

and Ocean Engineering Program. III. Numerical Model Calibration

of a Cantilever Beam using the Local Correspondence Principle.

iv

DEDICATION

À minha mãe, Andrea.

v

ACKNOWLEDGMENT

Agradeço fundamentalmente à minha família, Andrea, Alain pai e filho, Alexia,

Graça, por ter me proporcionado a oportunidade de chegar a este momento e ao suporte

incodicional durante toda a minha vida para realizar minhas aspirações e desejos.

A Ulisses Barbosa, Cláudio Mendonça e Ricardo Homero por toda orientação e

aprendizado obtidos durante esse projeto, cujas contribuições foram fundamentais.

Aos meus amigos Gabriela Bloise, Ana Operti, Pedro Kaskus, Laura Coutinho,

Victor Oliveira, Guilherme Valsa, Eduardo Nascimento, Ícaro Reis por compartilharem

comigo essa jornada com todo o suporte emocional e acadêmico que foi necessário.

Ao prof. Richard Schachter por ter acreditado e valorizado meu trabalho e

potencial.

Agradeço por último a oportunidade divina que me foi dada de percorrer esse

trecho da estrada da vida, que apesar de altos e baixos, tenho a esperança que servirá de

aprendizado para o espírito e expansão da consciência.

vi

Abstract of Undergraduate Project presented to POLI/UFRJ as a partial

fulfillment of the requirements for the degree of Engineer.

Numerical Model Calibration of a Cantilever Beam using the Local Correspondence

Principle

Andrew Martins Vieira

March/2020

Advisor: Ulisses Admar Barbosa Vicente Monteiro

Course: Naval and Ocean Engineering

It is not economically viable, nor technically interesting the installation of

accelerometers in all locations of interest in a given machine or structure. To estimate

vibration amplitudes in non-instrumented locations there is the need to employ

calibrated numerical models. In this project it has been applied a method for calibration

of a clamped beam numerical model using Operational Modal Analysis, Subspace

Reduction techniques in physical domain (Guyan) and modal domain (SEREP) and the

Local Correspondence Principle. The results have shown an improvement in the

correlation of all 5 modes analyzed, with MAC values surpassing 0,97 and a

significative improvement in the DR between the fitted numerical and experimental

models, leaving behind as a recommendation for future work, the application of the

proposed method in structures and industrial equipment.

Keywords: Calibration, Vibration, Numerical Model Reduction, Modal Analysis.

vii

viii

TABLE OF CONTENTS

TABLE OF CONTENTS ..................................................................................... x

LIST OF SYMBOLS ............................................................................................ x

Latin symbols .................................................................................................... x

Greek Symbols ................................................................................................. xi

Acronyms ......................................................................................................... xi

Superscripts.....................................................................................................xii

Subscripts........................................................................................................xii

1 INTRODUCTION .................................................................................. 1

1.1 Project Objective ..................................................................................... 2

1.2 Project Outline ........................................................................................ 3

2 THEORETICAL BACKGROUND ....................................................... 4

2.1 Modal Analysis ....................................................................................... 4

2.2 Experimental Modal Analysis................................................................. 5

2.3 Operational Modal Analysis ................................................................... 7

2.3.1 Estimation of Power Spectral Density (PSD) Functions ................. 7

2.3.2 Enhanced Frequency Domain Decomposition (EFDD) Method..... 8

2.4 Validation Techniques .......................................................................... 13

2.4.1 Modal Assurance Criterion (MAC) ............................................... 13

2.4.2 Coordinate Modal Assurance Criterion (COMAC) ...................... 13

2.4.3 Relative Difference Between Modes (DR).................................... 14

2.5 Post-Processing of Modal Parameter Estimate ..................................... 14

2.5.1 Normalization by mass .................................................................. 15

2.6 GUYAN Reduction Method ................................................................. 17

2.7 System Equivalent Reduction Expansion Process (SEREP) ................ 20

2.8 Local Correspondence Principle ........................................................... 22

3 METHODOLOGY ............................................................................... 26

3.1 FEM Numerical Model ......................................................................... 26

3.2 Experiment and OMA ........................................................................... 26

3.3 FEM Model Preliminary Optimization ................................................. 26

3.4 Modal Parameters Post-Processing and Model Reduction ................... 27

3.5 Adjustment of Numerical Mode Shapes ............................................... 28

ix

3.6 Virtual Sensing ..................................................................................... 29

3.7 Proposed Method Flowchart ................................................................. 30

4 CASE STUDY ...................................................................................... 33

4.1 FEM Numerical Model ......................................................................... 33

4.1.1 Geometry and Material Properties................................................. 33

4.1.2 Mesh and Element Type ................................................................ 34

4.1.3 Boundary Conditions ..................................................................... 35

4.1.4 Solution settings ............................................................................ 35

4.1.5 Mesh test ........................................................................................ 35

4.2 Experimental Setup ............................................................................... 36

4.2.1 Instruments used ............................................................................ 36

4.2.2 Experiment assembly..................................................................... 38

4.2.3 Experiment output ......................................................................... 40

4.2.4 Signal Processing – Operational Modal Analysis ......................... 41

4.3 FEM numerical model optimization ..................................................... 43

4.3.1 Boundary Condition ...................................................................... 44

4.3.2 Spring stiffness optimization ......................................................... 44

4.4 Complex to Real operation to the experimental mode shapes .............. 49

4.5 Model Reduction ................................................................................... 50

4.5.1 GUYAN Method ........................................................................... 50

4.5.2 SEREP method .............................................................................. 51

4.5.3 Mode shapes comparison between GUYAN, GUYAN-SEREP and

Full numerical models and Experimental ...................................... 53

4.6 Adjustment of Numerical Mode Shapes by Experimental Mode Shapes

based on the Local Correspondence Principle ...................................... 59

5 CONCLUSIONS AND RECOMMENDATIONS ............................... 66

6 REFERENCES ..................................................................................... 69

x

LIST OF SYMBOLS

Latin symbols

x system’s vibration response;

𝑞 modal coordinates;

𝑴 mass matrix in physical domain;

𝒎 modal mass matrix;

𝑪 damping coefficient matrix in physical domain;

𝑲 stiffness matrix in physical domain;

�̈� acceleration response vector;

�̇� velocity response vector;

𝑋 displacement response vector;

𝐹 external forces vector;

𝑡 time (s);

𝑓 natural frequency (Hz);

𝑯 frequency response function matrix;

𝑮 Power Spectral Density Matrix;

𝑪𝒗 Covariance matrix of responses;

𝑬 Expectation Operator;

𝐸 Young’s Modulus (N/mm²);

𝑼 Matrix of singular vectors;

𝑺 Singular Value diagonal matrix;

𝐴 Auto-correlation function;

𝑻 Transformation Matrix;

𝑷 Transformation matrix between modified and unmodified modes;

𝑹 Condensation Matrix;

xi

𝑉 Volume (m³);

U Displacement;

ROT Rotation;

Greek Symbols

𝜙 mode shape;

𝚽 modal matrix;

𝛿 logarithmic decrement;

𝜉 damping coefficient;

𝚿 unscaled modal matrix;

𝜶 scaling factor diagonal matrix;

𝜌 specific mass (kg/m³);

𝜈 poisson’s ratio;

λ eigenvalue;

Acronyms

FEM Finite Element Method;

DOF Degree-Of-Freedom;

SDOF Single-Degree-Of-Freedom;

EMA Experimental Modal Analysis;

OMA Operational Modal Analysis;

SEREP System Equivalent Reduction Expansion Process;

FRF Frequency Response Function;

IRF Impulse Response Function;

EFDD Enhanced Frequency Domain Decomposition;

FDD Frequency Domain Decomposition;

PSD Power Spectral Density;

xii

SVD Single Value Decomposition;

MAC Modal Assurance Criterion;

COMAC Coordinate Modal Assurance Criterion;

DR Difference Relative;

SIMO Single-Input Multiple-Output;

BC Boundary Condition.

Superscripts

-1 Matrix inverse;

T Transpose;

H Hermitian Operator;

+ Moore-Penrose pseudo inverse;

^ Estimate.

Subscripts

xx physical coordinates;

qq modal coordinates;

d damped;

FE numerical finite element;

exp experimental;

m master DOFs;

s slave DOFs;

r reduced matrix;

a active DOFs;

d deleted DOFs;

fit fitting DOFs;

obs observational DOFs;

adj adjusted mode shapes.

1

1 INTRODUCTION

Any structure as any mechanical system, even if not perceptible, is moving. This

is happening because the system is always interacting, hence exchanging energy, with

the surrounding environment. If these movements are really small (oscillations) and

around an equilibrium point, they are called vibrations.

Linear systems arise when studying the mechanics of structures under small

deformations and small rotations/displacements (vibrations). The behavior of any non-

linear problem around an equilibrium position can also be modeled by a linear system.

(Ogno, M. G. L.)[1]. Hence, studying the linear dynamics of a system is at the very

heart of most structural studies and will be specially considered hereafter.

Any type of mechanical system is submitted to different types of shipments

during its lifetime. These loads may have diverse natures: external or internal,

deterministic or random, controlled or not. The specifications of intensity, duration and

periodicity of these forces, when interacting with system’s properties, define the

dynamic behavior of the same.

Winds, hurricanes, waves, maritime currents, the unbalance itself and

misalignment of a mechanical system’s components are factors capable of influencing

the dynamic behavior of a system.

Controlling the characteristics of these excitation forces is a complex and

sometimes impossible task. On the other hand, these forces can lead to behaviors that

generate undesirable conditions, such as: fatigue, high stresses and vibration levels,

noises, resonance, among others. These operating conditions negatively alter the system

performance, causing operational problems and damage to its components. The dynamic

behavior of structures and equipment can be evaluated through modal analysis.

(Machado e Silva, L. B.)[2]

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. (Brincker, R.,

Ventura, C. R.)[3]

In any significant effort related to structural dynamic modeling, analytical and

experimental models are always employed. The analytical models are critical to the

2

early design process as well as the further prediction of model characteristics including

dynamic stress/strain, response analysis, fatigue and failure of critical components.

One way to estimate vibration levels in the structure regions of interest is by

using finite element method (FEM) models, Qu [4] and Chen et al [5]. However, these

analytical models are based upon assumptions as to the actual structural characteristics

and, as such, need to be validated by experimental results. These experimental models

are critical to the success of any structural dynamic analysis and contain elements that

cannot be obtained analytically. (Avitabile, P.)[6]

According to Friswell et al [7], errors between the model and the actual structure

can be minimized by performing calibration using experimental data, but one more

difficulty is found, the number of degrees of freedom of the finite element model is

much higher than number of degrees of freedom that can be monitored, therefore

calibration cannot be performed directly. As noted by Avitabile, P. [6], the concept of

model reduction and model expansion play a significant role in this important aspect of

modeling especially in the efficient comparison of the large analytical set of DOF

(Degree-Of-Freedom) to the relatively small set of experimental DOF. In addition, these

reduction and expansion processes play a significant role in the correlation and updating

of analytical models.

Many numerical modeling optimization approaches require the measured vector

to be available at the full set of finite element DOF. Likewise, model updating at the set

of tested DOF requires the large model to be reduced to the number of measured DOFs

but without distortion of the reduced model.

1.1 Project Objective

Considering the restrictions in testing large structures such as the unviability in

applying controlled and measured excitation forces and the impossibility of reaching

certain points of interest for measurement, the project thesis objective is defined as

follows:

“Develop a method for calibrating numerical models of structures which direct

measurement of all locations of interest is not feasible, using operational modal

analysis, subspace reduction techniques and the local correspondence principle.”

3

1.2 Project Outline

The research of this Project is divided into 4 parts or chapters.

In Chapter 2, the theoretical background of all tools, methods and techniques

applied in this project are reviewed. Particularly, Experimental Modal Analysis (EMA)

and Operational Modal Analysis (OMA) are addressed where special consideration for

OMA will be given by engaging OMA frequency domain techniques. In addition, the

subspace reduction methods GUYAN and SEREP will be investigated. Also, the Local

Correspondence Principle and its application will be introduced with the aim to adjust

numerical mode shapes by experiment. Finally, in that Chapter, the validation methods

MAC, COMAC and DR will be presented.

In Chapter 3, a prescriptive method for the behavior prediction in structures is

proposed in line with the Project Objective. Briefly, the method explains how to adjust

the numerical mode shapes (FEM) by the experimental mode shapes obtained from

OMA using subspace GUYAN and SEREP reduction techniques and the Local

Correspondence Principle and how to use that for virtual sensing.

In Chapter 4, a case study with an aluminum beam is presented where the

proposed method is applied together with the validation procedure over the results.

In Chapter 5, based on the method and results presented in the previous chapters

a number of conclusions and recommendations for future work will be delineated.

4

2 THEORETICAL BACKGROUND

2.1 Modal Analysis

Modal Analysis is a process used to determine the mechanical properties of a

system, namely: Natural frequencies, damping ratios and vibration modes. With such

information, it’s possible to represent the dynamic behavior of a structure.

As noted by Ogno, Marco G. L. [1], free vibration fully characterizes the

structure’s dynamical properties. For that reason, in a linear or linearized system, the

response (𝒙(𝑡)) can be expressed as the linear combination of its number 𝑛0 of modes

𝜙𝑖 and their time dependent amplitude, or modal coordinates 𝒒𝒊(𝑡) as given in Eq. (2.1):

𝑥(𝑡) = ∑ 𝑞𝒊(𝑡)𝜙𝑖

𝑛0

𝑖=1

𝑛0 → ∞ (2.1)

A real structure is continuous, thus having infinite degrees of freedom. However, it’s

only possible to detected or describe a certain finite number of modes. This limitation

yields the idea of modal approximation where the Eq. (2.1) is truncated for a certain

number of modes 𝑁. The number of modes taken determines the accuracy of the

approximation.

𝑥(𝑡) ≅ ∑ 𝑞𝒊(𝑡)𝜙𝑖

𝑁

𝑖=1

𝑁 ∈ ℕ (2.2)

To practically estimate the described modal properties of a system a ‘Modal

Analysis’ is performed. (Machado e Silva, L. B.)[2] Modal Analysis has a wide range of

applications, for example: Vibration Control, Operational Stability of Turbomachinery,

Identification of Excitation Forces, Reduction of Dynamic Models and the focus of this

project: Calibration of Finite Element Method (FEM) Models. These days, the main

techniques are subdivided in Experimental Modal Analysis (EMA) and Operational

Modal Analysis (OMA).

5

2.2 Experimental Modal Analysis

(Machado e Silva, L. B.)[2] The Experimental Modal Analysis uses both inputs

and outputs to obtain modal parameters of a given system. The structure is excited by

known forces in specific known locations and the vibrational responses are measured in

other locations by distinct output sensors. In order to identify the modal parameters in

EMA, the Frequency Response Function (FRF) or Impulse Response Function is

needed. A brief definition of the FRF is given below.

(Qu, Z. Q.)[4] The dynamic Eq. of motion for a vibrating structure considering

damping effects is defined by:

𝑴�̈�(𝑡) + 𝑪�̇�(𝑡) + 𝑲𝑋(𝑡) = 𝐹(𝑡) (2.3)

Where �̈�(𝑡), �̇�(𝑡) and 𝑋(𝑡) are the acceleration response vector, velocity

response vector, and displacement response vector, respectively. 𝐹(𝑡) is the vector of

applied forces. 𝑴, 𝑪 and 𝑲 are the mass, damping and stiffness matrices respectively.

Suppose that all the components of vectors 𝑋(𝑡) and 𝐹(𝑡) are Fourier

transformable and their transformations are 𝑋(𝜔) and 𝐹(𝜔), respectively, and that

�̇�(𝑡) = 𝑋(𝑡) = 0 for 𝑡 = 0. The Fourier transformation of Eq. (2.3) is given by:

(𝑲 + 𝑗𝜔𝑪 − 𝜔2𝑴)𝑋(𝜔) = 𝐹(𝜔) (2.4)

where 𝜔 is the circular frequency of exciting forces. 𝑗 = √−1. The vector 𝑋(𝜔) is

called the frequency response vector, which can be expressed as:

𝑋(𝜔) = 𝑯(𝜔)𝐹(𝜔) (2.5)

Thus, the matrix 𝑯(𝜔), for Multiple-Degrees-of-Freedom (MDOF), is defined as:

𝑯(𝜔) = (𝑲 + 𝑗𝜔𝑪 − 𝜔2𝑴)−1 (2.6)

(Ogno, Marco G. L.)[1] The element of the matrix 𝑯(𝜔) is referred as the

frequency response function (FRF). The FRF describes the input-output relationship

6

between two points on a structure as a function of frequency. The FRF matrix 𝑯(𝜔)

represents the dynamics of the structure between all pairs of input and output DOFs,

where its columns correspond to inputs and rows correspond to outputs. In order to

estimate the modal domain of the structure the FRF matrix must be calculated through

modal testing.

(Ogno, Marco G. L.)[1] Once the modal test is completed and the whole FRF

matrix is known, the modal parameter of the system can be estimated by means of curve

fitting over the estimated set of FRFs. The modal domain can also be identified by

curve fitting in time domain on an equivalent set of Impulse Responses Functions

(IRFs).

Figure 2.1 - Simplified flowchart of Experimental Modal Analysis

In many cases, it can be hard or even impossible to obtain the excitation forces

acting on a system (inputs), this poses a limitation upon EMA methods since exciting a

complex structure can be cost prohibitive, not to mention the chance of locally

damaging the structure.

With such limitation in mind, the Output-Only Modal Analysis emerges as an

alternative. (Machado e Silva, L. B.)[2] In this type of analysis, only the responses of

7

the system are needed to estimate its modal parameters. This analysis is relevant when

excitation forces are unknown when it comes to magnitude and frequency. However,

this type of analysis also presents limitation like natural modes not normalized by mass

and the absence of excitation of some modes. One type of Output-Only Modal Analysis

is the Operational Modal Analysis (OMA).

2.3 Operational Modal Analysis

(Machado e Silva, L. B.)[2] The Operational Modal Analysis (OMA) consists of

using the responses of a system in operational conditions or under environmental

excitation to obtain its modal parameters. One of the advantages is that it can be applied

without spoiling the functioning of the system. The limitations present in EMA are

common in large structures such as buildings, bridges and offshore structures, so OMA

is adequate and usually preferred in those situations. OMA techniques are divided, in

general, in two groups: Time Domain and Frequency Domain.

The focus of this project will be the Frequency Domain techniques, by the

following reasons: They are easier to implement compared to time domain techniques;

They are effective for what’s intended; The main goal of the project isn’t to compare

and study the OMA methods specifically.

The frequency domain technique that will be reviewed is the EFDD (Enhanced

Frequency Domain Decomposition).

2.3.1 Estimation of Power Spectral Density (PSD) Functions

(Machado e Silva, L. B.)[2] The Power Spectral Density functions represent the

energy distribution in the frequency domain of a given time series. These functions

contain data about the modal characteristics present in the signal.

The variety of PSD estimators can be distinguished, mainly, in two categories:

direct and indirect. In the direct category, the estimators are based in the Period Chart

method, while in the indirect category they are based in the Correlation Chart method.

8

2.3.2 Enhanced Frequency Domain Decomposition (EFDD) Method

Frequency Domain Decomposition (FDD) Method

(Ogno, Marco G. L.)[1] The general concept behind the FDD technique is to

perform a decomposition of the system response into a set of independent Single

Degree-of-Freedom (SDOF) systems, one for each mode.

The FDD method uses a matrix factorization tool called Single Value

Decomposition (SVD) that is applied to the matrix 𝑮𝑥𝑥(𝑓) composed by the calculated

PSDs for all sensors. This process allows concentrating all spectral data in one chart

which is the single value chart of the PSD function matrix. The singular values are

obtained from the auto-spectral density of the SDOF systems, and the corresponding

singular vectors are estimates of the mode shapes.

The FDD assumptions are:

1. Uncorrelatedness, in both time and space, white noise inputs;

2. A linear or linearly behaving system;

3. A stationery and time invariant system;

4. A lightly damped system.

(Ogno, Marco G. L.)[1] The lightly damped assumption is directly related to the

fact that in case of highly damped system the modal frequencies peaks are damped out,

and the distinction between noise and structural modes is not clear anymore.

The theory behind the FDD methods is shortly described below:

As mentioned in Subchapter 2.1, a response of a linear or linearized system can

be described by:

𝑥(𝑡) ≅ ∑ 𝑞𝒊(𝑡)𝜙𝑖

𝑁

𝑖=1

𝑁 ∈ ℕ (2.7)

Where:

• 𝑞𝒊(𝑡): Vector containing the modal coordinates;

9

• 𝜙𝑖: eigenmodes or mode shapes of the Multiple Degree-of-Freedom

system;

The covariance matrix of the responses is described by the following relation:

𝑪𝒗𝑥𝑥(𝜏) = 𝑬[𝑥(𝑡 + 𝜏)𝑥(𝑡)𝑇] (2.8)

Where:

• 𝑬: Expectation Operator;

By expressing Eq. (2.8) in the modal coordinates form of Eq. (2.7) the

covariance matrix of the responses 𝑪𝒗𝑥𝑥 can be expressed by the matrix of the modal

coordinates 𝑪𝒗𝑞𝑞 pre-multiplied by 𝚽 and post-multiplied by 𝚽𝑇 as shown in Eq. (2.9).

𝑪𝒗𝑥𝑥(𝜏) = 𝑬[𝚽𝑞(𝑡 + 𝜏)𝑞(𝑡)𝑇𝚽𝑇] = 𝚽𝑪𝒗𝑞𝑞𝚽𝑇 (2.9)

Transforming Eq. (2.9) into the frequency domain by applying the Fourier

Transform, the covariance matrix (in the time domain) becomes the Power Spectral

Density (PSD) matrix 𝑮𝑥𝑥(𝑓) (in the frequency domain). Therefore, if and only if, the

inputs are uncorrelated and distributed all over the structure, than the PSD of the modal

coordinates 𝑮𝑞𝑞(𝑓)becomes diagonal and in case of white noise also constant 𝑮𝑞𝑞(𝑓) ∝

𝑰 and hence the following relation holds:

𝑮𝑥𝑥(𝑓) = 𝚽(𝑓)𝑮𝑞𝑞(𝑓)𝚽𝑇 ∝ 𝚽(𝑓)𝑰𝚽(𝑓)𝑇 (2.10)

Also, it’s known that the Power Spectral Matrix is Hermitian and some

complexity is expected in the vibration modes, thus the Hermitian operator is applied

instead of the Transpose operator, so the Eq. (2.10) becomes:

𝑮𝑥𝑥(𝑓) = 𝚽(𝑓)[𝑔𝑛2(𝑓)]𝚽(𝑓)𝐻 (2.11)

Where 𝑔𝑛2(𝑓) are the diagonal elements of the PSD of modal coordinates

𝑮𝑞𝑞(𝑓).

10

All the necessary modal information is contained in the PSD of the output

𝑮𝑥𝑥(𝑓), but in order to identify eigenvalues and eigenvectors, a Singular Value

Decomposition (SVD) is performed and Eq. (2.11) becomes:

𝑮𝑥𝑥(𝑓) = 𝑼(𝑓)𝑺𝑖(𝑓)𝑼(𝑓)𝐻 (2.12)

Where 𝑼(𝑓) is the matrix of the singular vectors and 𝑺𝑖(𝑓) is the diagonal

matrix containing all the singular values for each degree of freedom.

It must be paid attention that the SVD method described in Eq. (2.12) doesn’t

fully relate to the theoretical PSD decomposition, thus the method results only in an

estimated solution. This holds due to the assumptions of white noise excitation and non-

correlation of modal coordinates.

Enhanced Frequency Domain Decomposition (EFDD) Method

(Ogno, Marco G. L.)[1] The EFDD method is an extension of the FDD method

in order to estimate the

modal damping of the system under observation. In fact the EFDD begins as soon the

SVD of the PSD matrix is completed and the SV peaks are identified.

The advantage of this method with respect to the classic FDD method, is that

by using these SDOF modal domains, their modal damping ratios can be estimated.

In the vicinity of the natural frequency, singular vectors having a high MAC

(Section 2.4.1) value are obtained, enabling the establishment of a Single-Degree-Of-

Freedom (SDOF) spectral density function, for a specific mode, which is transformed to

the time domain yielding an auto-correlation function of the SDOF system.

(Machado e Silva, L. B.)[2] The auto-correlation function verifies the level of

dependency or correlation between a signal in instant (𝑡) and another signal in instant

(𝑡 + 𝜏).

11

Figure 2.2 - Representation of a time series signal

The auto-correlation fuction also contains all the modal information present in a

dynamic system that’s possible to extract by identification methods in time domain. It is

defined as:

𝐴𝑥𝑥(𝜏) = 𝐸[𝑥(𝑡)𝑥(𝑡 + 𝜏)] (2.13)

It can also be expressed in a practical way by the following Eq.:

𝐴𝑥𝑥(𝜏) =

1

𝑇∫ 𝑥(𝑡)𝑥(𝑡 + 𝜏)

𝑇

0

(2.14)

Figure 2.3 illustrates a graphical representation typical of an auto-correlation

function.

Figure 2.3 - Representation of an auto-correlation function

12

(Machado e Silva, L. B.)[2] From this auto-correlation function, the damped

natural frequency is obtained by determining the number of zero-crossing as a function

of time using a simple least-squares fit. The undamped natural frequency 𝑓 as a function

of the damped natural frequency 𝑓𝑑 is given by Eq. (2.15):

𝑓 =

𝑓𝑑

√1 − 𝜉2 (2.15)

(Machado e Silva, L. B.)[2] The damping ratio is obtained from the logarithmic

decrement of the auto-correlation function again using a simple least-squares fit

(Jacobsen et al. [8]). This technique calculates the damping ratio of a SDOF using the

identified peak values (𝑣0, 𝑣1, … , 𝑣𝑛) in the free vibration chart in time domain of the

system (Figure 2.4). In the EFDD method this is represented by the auto-correlation

function as described above.

Figure 2.4 - Logarithmic decrement method

First, the logarithmic decrement (𝛿) is calculated by Eq. (2.16):

𝛿 = ln (

𝑣𝑛

|𝑣𝑛+1|) =

1

𝑚 ln (

𝑣𝑛

|𝑣𝑛+𝑚|) (2.16)

Where 𝑣𝑛 and 𝑣𝑛+𝑚 correspond to two peak values in the auto-correlation

function apart by ‘m’ cicles.

13

(Machado e Silva, L. B.)[2] Since the auto-correlation function calculated

doesn’t represent exactly an exponential decrement function, the logarithmic decrement

curve (𝛿) has to be calculated and then, with a least-square-fit its value leading to

related damping ratio, given by Eq. (2.17):

𝜉 =

𝛿

√4𝜋2 + 𝛿2 (2.17)

2.4 Validation Techniques

2.4.1 Modal Assurance Criterion (MAC)

(Mendonça, C. O., et al.) [18] MAC is basically a linear and quadratic regression

correlation coefficient that measures the consistency between two vectors. MAC values

range from 0 to 1, where 0 indicates inconsistency or orthogonality between vectors and

1 indicates perfect consistency (differing only by a scale factor).

Considering the estimation, by two different methods, of the i-th vibration

modes, {𝜙𝑖1} e {𝜙𝑖

2}, the MAC between these modes is given by:

MAC({𝜙𝑖

1}, {𝜙𝑖2}) =

|{𝜙𝑖1}𝐻{𝜙𝑖

2}|2

({𝜙𝑖1}𝐻{𝜙𝑖

1})({𝜙𝑖2}𝐻{𝜙𝑖

2}) (2.18)

When MAC is calculated between vibration modes estimated by only one

method, Eq. (2.18) is changed to:

MAC({𝜙𝑖}, {𝜙𝑗}) =

|{𝜙𝑖}𝐻{𝜙𝑗}|2

({𝜙𝑖}𝐻{𝜙𝑖}) ({𝜙𝑗}𝐻

{𝜙𝑗}) (2.19)

2.4.2 Coordinate Modal Assurance Criterion (COMAC)

According to Friswell et. al. [7] and Ewins [9], MAC is an important tool in

mode correlation, but may pose a challenge in correlation of modes that are closely

spaced in frequency or when the selected locations for measurement or modeling are

14

insufficient. In this sense, a variant of MAC, called a co-ordinate MAC or COMAC, can

be used for error finding. COMAC values reflect the discrepancy between the compared

modal forms and can be calculated as follows:

𝐶𝑂𝑀𝐴𝐶(𝑖) = (∑ (𝚽𝐴)𝑖,𝑗(𝚽𝐵)𝑖,𝑗

𝐿𝑗=1 )

2

(∑ (𝚽𝐴)𝑖,𝑗2𝐿

𝑗=1 )(∑ (𝚽𝐵)𝑖,𝑗2𝐿

𝑗=1 ) (2.20)

Where 𝐿 and 𝑖 represent respectively the number of modes being compared and

the coordinate being evaluated and 𝚽𝐴 e 𝚽𝐵 represent the modal matrices being

correlated. COMAC values close to 1 indicate that all mode coordinates associated with

degree of freedom i are equal, values below 0.9 indicate discrepancy in the evaluated

degree of freedom.

2.4.3 Relative Difference Between Modes (DR)

(Mendonça, C. O., et al.) [18] The relative difference assesses the level of

variances in amplitudes of each degree of freedom between the modes being compared

and is calculated as follows:

𝐷𝑅(𝑖, 𝑗) = |

(𝚽𝐴)𝑖,𝑗 − (𝚽𝐵)𝑖,𝑗

(𝚽𝐴)𝑖,𝑗| (2.21)

Where, Φ𝐴 e Φ𝐵 represent the modal matrices being compared and indexes i and

j represent the degree of freedom and the mode being evaluated respectively. Values

close to 0 indicate that the amplitudes of the degrees of freedom analyzed are alike.

2.5 Post-Processing of Modal Parameter Estimate

(Rainieri, Fabbrocino)[10] Most of the OMA methods provide their results in the

form of complex natural frequencies and complex mode shapes. Complex modes are

often obtained from modal tests due to measurement noise (poor signal-to-noise ratio).

However, the degree of complexity is usually moderate. Taking into account that OMA

provides only un-scaled mode shapes, there is the need for simple approaches to scaling

and complex-to-real conversion of the estimated mode shape vectors.

15

The need for complex-to-real conversion of the estimated mode shapes stems

from one of the typical applications of modal data, the comparison between the

experimental values of the modal properties and those obtained from numerical models.

In fact, the latter are usually obtained from undamped models and, consequently, the

mode shapes are real-valued.

Whenever normal modes are expected from the modal test, the simplest

approach to carry out the complex-to-real conversion consists in analyzing the phase of

each mode shape component and setting it equal to 0º or 180º depending on its initial

value. If the phase angle lies in the first or in the fourth quadrant it is set equal to 0º ; it

is set equal to 180º if it lies in the second or in the third quadrant. To be rigorous, this

approach should be applied only in the case of nearly normal modes, when the phase

angles differ no more than ±10° from 0° to 180°. However, it is frequently extended to

all phase angles (Ewins)[9].

2.5.1 Normalization by mass

According to Aenlle, Brincker [11] If we consider an analytical or finite element

model of a structure with no damping, the eigenvalue solution of Eq. (2.3) with no

damping, the equation of motion of the structure subjected to a force F(t) is given by the

following:

𝑴𝐹𝐸𝚽𝐹𝐸𝜔𝐹𝐸2 = 𝑲𝐹𝐸𝚽𝐹𝐸 (2.22)

Where 𝚽𝐹𝐸 and 𝜔𝐹𝐸2 are the mass normalized mode shapes and the natural

frequencies, respectively, and the subscript ‘FE’ indicates the finite element model.

The unscaled 𝚿𝐹𝐸 and the scaled or mass normalized 𝚽𝐹𝐸 mode shapes are

related by the expression:

𝚽𝐹𝐸 = 𝚿𝐹𝐸𝜶𝐹𝐸 (2.23)

Where 𝜶𝐹𝐸 is the scaling factor diagonal matrix which can be expressed as

follows:

16

𝜶𝐹𝐸 = [√𝚿𝐹𝐸

𝑇 𝑴𝐹𝐸𝚿𝐹𝐸]

−1

(2.24)

From Eq. (2.23) it is derived that the scaling factor 𝜶𝐹𝐸 and the modal mass

𝒎𝐹𝐸 are related by the following:

𝒎𝐹𝐸 = [𝜶𝐹𝐸2 ]−1 (2.25)

Considering another discrete model, ‘the experimental model’ that represents the

reality, i.e. the dynamic behavior of the real structure is described by the stiffness matrix

𝑲𝑒𝑥𝑝 and the mass matrix 𝑴𝑒𝑥𝑝, where the subscript ‘exp’ indicated experimental. If

the experimental model can be considered as a dynamic modification of the analytical

one, and the modification is given by the mass Δ𝑴 and stiffness Δ𝑲 matrices, i.e.:

𝑴𝑒𝑥𝑝 = 𝑴𝐹𝐸 + Δ𝑴 𝑎𝑛𝑑 𝑲𝑒𝑥𝑝 = 𝑲𝐹𝐸 + Δ𝑲 (2.26)

The eigenvalue Eq. of the experimental mode is given by the following:

𝑴𝑒𝑥𝑝𝚽𝑒𝑥𝑝𝜔𝑒𝑥𝑝2 = 𝑲𝑒𝑥𝑝𝚽𝑒𝑥𝑝 (2.27)

Where 𝜔𝑒𝑥𝑝2 and 𝚽𝑒𝑥𝑝 are the natural frequency and modal matrix, respectively.

According to the structural modification theory (SDM), the mode shapes of the

experimental model 𝚽𝑒𝑥𝑝 can be expressed as follows:

𝚽𝑒𝑥𝑝 = 𝚽𝐹𝐸𝑷 (2.28)

Which means that the mode shapes in the experimental (modified) model are

expressed as a linear combination of the mode shapes of the unmodified structure

(FEM) using transformation matrix 𝑷. In case of unscaled mode shapes, Eq. (2.28)

becomes

𝚿𝑒𝑥𝑝 = 𝚿𝐹𝐸𝑷 (2.29)

17

Where

𝑩 = 𝜶𝐹𝐸𝑷𝜶𝑒𝑥𝑝−1 (2.30)

And 𝜶𝐹𝐸 and 𝜶𝑒𝑥𝑝 are diagonal matrices containing the scaling factors of the FE

and experimental models respectively.

The scaling factor of the experimental mode shapes is given by the following:

𝜶𝑒𝑥𝑝 = [√𝚿𝑒𝑥𝑝𝑇 𝑴𝑒𝑥𝑝𝚿𝑒𝑥𝑝]

−1

= [√𝚿𝑒𝑥𝑝𝑇 𝑴𝐹𝐸𝚿𝑒𝑥𝑝 + 𝚿𝑒𝑥𝑝

𝑇 Δ𝑴𝚿𝑒𝑥𝑝]−1

(2.31)

From Eq. (2.31) it is derived that, if the matrix Δ𝑴 is small, i.e., a reasonably

good estimation of the mass matrix can be achieved, the approximation 𝑴𝑒𝑥𝑝 ≅ 𝑴𝐹𝐸

can be taken and the scaling factor of the experimental mode shapes can be estimated

from:

�̂�𝑒𝑥𝑝 ≅ [√𝚿𝑒𝑥𝑝𝑇 𝑴𝐹𝐸𝚿𝑒𝑥𝑝]

−1

(2.32)

Therefore, the modal mass can be estimated as follows:

�̂�𝐹𝐸 ≅ 𝚿𝑒𝑥𝑝𝑇 𝑴𝐹𝐸𝚿𝑒𝑥𝑝 (2.33)

Finally, the approximated mass normalized experimental mode shapes are

obtained by:

�̂�𝑒𝑥𝑝 = 𝚿𝑒𝑥𝑝�̂�𝑒𝑥𝑝 (2.34)

2.6 GUYAN Reduction Method

(Mendonça, C. O., et al.) [18] Guyan's reduction or condensation method was developed

for static problems, but its application can be extended to the dynamic analysis of

structures and equipment, Qu [1], therefore, it has the following:

𝑲𝑋 = 𝐹 (2.35)

18

Where 𝑲, 𝑋 and 𝐹 represent the stiffness matrix, displacement vector and force vector

of the complete model, respectively. Since total degrees of freedom can be categorized

as master and slave, then Eq. (2.18) can be rearranged as follows:

[𝑲𝑚𝑚 𝑲𝑚𝑠

𝑲𝑠𝑚 𝑲𝑠𝑠] {

𝑋𝑚

𝑋𝑠} = {

𝐹𝑚

𝐹𝑠} (2.36)

In Eq. (2.36), the subscripts m and s indicate master and slave respectively. Expanding

the matrix multiplication on the left side of Eq. (2.36), we have:

𝑲𝑚𝑚𝑋𝑚 + 𝑲𝑚𝑠𝑋𝑠 = 𝐹𝑚 (2.37)

𝑲𝑠𝑚𝑋𝑚 + 𝑲𝑠𝑠𝑋𝑠 = 𝐹𝑠 (2.38)

In Eqs. (2.37) and (2.38), it is observed that the displacements of the slave degrees of

freedom have two parts: the first, due to the interaction with the master degrees of

freedom (coupled displacement), and the second, due to the external forces acting on

them (relative displacements). Thus, manipulating Eqs. (2.37) and (2.38), we have:

𝑲𝑟𝑋𝑚 = 𝐹𝑟 (2.39)

Eq. (2.39) is the static equilibrium Eq. corresponding to the master degrees of freedom,

where 𝑲𝑟 and 𝐹𝑟 are known as the reduced model stiffness matrix and equivalent force

vector, respectively, and are defined as:

𝑲𝑟 = 𝑲𝑚𝑚 − 𝑲𝑚𝑠𝑲𝑠𝑠−1𝑲𝑠𝑚 (2.40)

𝐹𝑟 = 𝐹𝑚 − 𝑲𝑚𝑠𝑲𝑠𝑠−1𝐹𝑠 (2.41)

To determine a relationship between the master and slave degrees of freedom, Guyan's

method assumes that 𝐹𝑠 = 0, hence from Eq. (2.38) we have:

𝑋𝑠 = 𝑹𝑋𝑚 where, 𝑹 = −𝑲𝑠𝑠−1𝑲𝑠𝑚 (2.42)

19

The matrix R is known as Guyan Condensation Matrix. Note that this matrix is load

independent because the external forces in the slave degrees of freedom were ignored.

Thus, the displacement vector of Eq. (2.36) can be expressed as follows:

𝑋 = {

𝑋𝑚

𝑋𝑠} = 𝑻𝑋𝑚 with: 𝑻 = [

𝑰𝑹

] (2.43)

In Eq. (2.43) the matrix 𝑻 is known as the Coordinate Transformation Matrix. It is

worth remembering that Guyan's reduction method was developed for static problems,

but it can be used in dynamic analysis. Therefore, the motion Eq. of the complete model

without damping is considered:

𝑴�̈�(𝑡) + 𝑲𝑋(𝑡) = 𝐹(𝑡) (2.44)

Where 𝑴 and 𝑲 are the mass and stiffness matrices of the full model, �̈� and 𝑋

are the acceleration and mass vectors of all degrees of freedom and 𝐹 is the vector of

external forces acting on the different degrees of freedom of the model.

Similarly, to static analysis, Eq. (2.44) can be expressed in terms of master and

slave degrees of freedom:

[𝑴𝑚𝑚 𝑴𝑚𝑠

𝑴𝑠𝑚 𝑴𝑠𝑠] {

𝑋𝑚(𝑡)̈

𝑋�̈�(𝑡)} + [

𝑲𝑚𝑚 𝑲𝑚𝑠

𝑲𝑠𝑚 𝑲𝑠𝑠] {

𝑋𝑚(𝑡)𝑋𝑠(𝑡)

} = {𝐹𝑚(𝑡)𝐹𝑠(𝑡)

} (2.45)

From Eq. (2.4.11) one can obtain:

𝑴𝑠𝑚𝑋�̈�(𝑡) + 𝑴𝑠𝑠𝑋�̈�(𝑡) + 𝑲𝑠𝑚𝑋𝑚(𝑡) + 𝑲𝑠𝑠𝑋𝑠(𝑡) = 0 (2.46)

In Eq. (2.46), it was assumed that 𝐹𝑠(𝑡) = 0, similar to the static problem. Also,

assuming that �̈�(𝑡) = 0 and 𝑋(𝑡) = 0, the relationship between primary and secondary

degrees of freedom is equal to Eq. (2.42). Thus, since the transformation matrix 𝑻 is

independent of time, deriving twice from Eq. (2.43), we have:

�̈�(𝑡) = 𝑻𝑋�̈�(𝑡) (2.47)

20

Substituting Eqs. (2.46) and (2.43) in Eq. (2.45) and pre-multiplying by matrix 𝑻

transpose, results:

𝑴𝑟𝑋�̈�(𝑡) + 𝑲𝑟𝑋𝑀(𝑡) = 𝐹𝑟(𝑡) (2.48)

Eq. (2.48) is the Eq. of motion of the reduced model, where 𝑴𝑟 and 𝑲𝑟 are the

reduced mass and stiffness matrices, respectively, and 𝐹𝑟 is the equivalent force vector,

and are calculated as follows:

𝑲𝑟 = 𝑻𝑇𝑲𝑻, 𝑴𝑟 = 𝑻𝑇𝑴𝑻, 𝐹𝑟(𝑡) = 𝑻𝑇𝐹(𝑡) (2.49)

It should be noted that vibration responses in slave degrees of freedom are

difficult to predict if the force vector acting on them is not zero. Therefore, in the

reduction process, it is recommended to maintain as master all degrees of freedom

whose vibration response is of interest.

2.7 System Equivalent Reduction Expansion Process (SEREP)

(Mendonça, C. O., et al.) [18] The SEREP method was developed for the dynamic

condensation of models, Maia et al [12] and Qu [4] using the modal approach.

Therefore, the solution of Eq. (2.44) can be as follows:

𝑋(𝑡) = 𝚽𝑞(𝑡) (2.50)

Where, 𝚽 e 𝑞(𝑡) are the modal matrix and modal coordinates, respectively, of

the complete model. However, obtaining the modal matrix is practically impossible in

large models. In this sense, the modal truncation technique is often used. Thus, if p

modes of the complete model are used in modal superposition, Eq. (2.50) can be

rewritten as:

𝑋(𝑡) = 𝚽𝑝𝑞𝑝(𝑡) (2.51)

Similarly, to Guyan's method, Eq. (2.51) can be rearranged as follows:

21

𝑋(𝑡) = {

𝑋𝑚(𝑡)

𝑋𝑠(𝑡)} = {

𝚽𝑚𝑝

𝚽𝑠𝑝} 𝑞𝑝(𝑡) (2.52)

From Eq. (2.52) one can obtain:

𝑋𝑚(𝑡) = 𝚽𝑚𝑝𝑞𝑝(𝑡) (2.53)

𝑋𝑠(𝑡) = 𝚽𝑠𝑝𝑞𝑝(𝑡) (2.54)

Eq. (2.53) is a description of the responses for the master degrees of freedom in

terms of the modal matrix of the master themselves. It can also be noted that 𝚽𝑚𝑝 is

generally not a square matrix and depends directly on the degrees of freedom and modes

considered. Thus, SEREP considers that the number of master degrees of freedom is

greater than the number of modes considered (m > p).

As m > p, this means that we have more equations than unknowns. Therefore,

Eq. (2.53) can be placed in the normal form (compatibility of degrees of freedom),

projecting this Eq. as:

𝑌𝑝(𝑡) = 𝚽𝑚𝑝𝑇 𝑋𝑚(𝑡) (2.55)

Merging Eq. (2.53) into Eq. (2.55) yields:

𝑌𝑝(𝑡) = 𝚽𝑚𝑝𝑇 𝚽𝑚𝑝�̂�𝑝(𝑡) (2.56)

Where, �̂�𝑝(𝑡) is an approximate solution of 𝑞𝑝(𝑡), and can be calculated by

manipulating Eq. (2.56):

�̂�𝑝(𝑡) = (𝚽𝑚𝑝𝑇 𝚽𝑚𝑝)

−1𝑌𝑝(𝑡) (2.57)

Merging Eq. (2.55) into Eq. (2.57):

�̂�𝑝(𝑡) = 𝚽𝑚𝑝+ 𝑋𝑚(𝑡) (2.58)

Where, 𝚽𝑚𝑝+ is the generalized inverse of 𝚽𝑚𝑝 and is defined as:

22

𝚽𝑚𝑝+ = (𝚽𝑚𝑝

𝑇 𝚽𝑚𝑝)−1

𝚽𝑚𝑝𝑇 (2.59)

Merging Eq. (2.58) into Eq. (2.54):

𝑋𝑠(𝑡) = 𝑹𝑋𝑚(𝑡) (2.60)

Where 𝑹 is the SEREP dynamic condensation matrix and is calculated as:

𝑹 = 𝚽𝑠𝑝𝚽𝑚𝑝+ (2.61)

Furthermore, the transformation matrix 𝑇 can be calculated by substituting Eq.

(2.58) in Eq. (2.52):

𝑻 = 𝚽𝑠𝑝𝚽𝑚𝑝

+ = [𝚽𝑚𝑝 𝚽𝑚𝑝

+

𝚽𝑠𝑝 𝚽𝑚𝑝+ ] (2.62)

Thus, the reduced stiffness and mass matrices can be calculated using Eq. (2.49).

(Maia, Silva)[13] The main advantages of the SEREP process are the following

ones:

1. The reduced model has exactly the same frequencies and mode shapes as the full

system for the selected modes of interest; and

2. The quality of the results is insensitive to the selection of the full DOFs that are

kept in the reduced model.

2.8 Local Correspondence Principle

(Brincker et al.)[14] It is known from the structural modification theory,

Sestieriand D'Ambrogio [15], that considering the complete mode shape matrix 𝑨 of a

system 𝑨 and the similar complete mode shape matrix 𝑩 of a system 𝑩 – that is system

𝑨 subjected to arbitrary changes of the stiffness and the mass matrices – then there’s a

linear relationship between the two mode shape matrices:

23

𝑨 = 𝑩𝑷 (2.63)

Where the linear relation is defined by the transformation matrix 𝑷. Considering

only one mode shape 𝑎 in the mode shape matrix 𝑨 yields the relation:

𝑎 = 𝑩𝑝 (2.64)

Where the vector 𝑝 is the vector that describes how vector 𝑎 is created as a linear

combination of the mode shape vectors in 𝑩.

Now, it will be considered the mode shape 𝑎 as an experimentally obtained

mode shape trying to express it as a linear combination of FE mode shapes – the mode

shapes in the modal matrix 𝑩.

Assuming the case which the mode shape matrix 𝑩 is incomplete, i.e. is not

including all mode shapes of the system, then Eq. (2.36) only holds approximately:

𝑎 ≅ 𝑩𝒂𝑝 (2.65)

and 𝑩𝒂 is the FE mode shape matrix reduced to the number of active DOFs in

the experiment. The reduced mode shape matrix 𝑩𝒂 is found by taking the full mode

shape matrix of the FE model:

𝑩 = [

𝑩𝒂

𝑩𝒅] (2.66)

and then removing all unwanted DOFs to use in Eq. (2.65), the so-called deleted

DOFs gathered in the partition matrix 𝑩𝒅.

If the changes of stiffness and mass between the two systems 𝑨 and 𝑩 are small,

and if the set of mode shapes including the incomplete mode shape matrix 𝑩 is chosen

well, then it is assumed that the approximation given by Eq. (2.65) is good.

The linear combination vector 𝑝 is found from Eq. (2.65) by solving the Eq. with

respect to the vector estimate:

�̂� = 𝑩𝒂+𝑎 (2.67)

24

where 𝑩𝒂+ is the pseudo-inverse of 𝑩𝒂. If the number of DOFs in the vector 𝑎 is

larger than the number of modes in the mode shape matrix 𝑩, then an over determined

problem is in consideration, and then the estimate:

�̂� = 𝑩𝒂+�̂� (2.68)

is a smoothed version of the experimental mode shape vector 𝒂. Once the

smoothed estimate has been obtained, the smoothed estimate can be used for expansion,

because in Eq. (2.65) any DOF can be included in the estimate just by expanding the

mode shape matrix 𝑩𝒂 to include the considered DOFs. The smoothed estimate can be

expanded to full size simply by including all deleted DOFs.

�̂� = 𝑩�̂� (2.69)

The approach is based on the assumption of a fixed FE subspace. It is clear that

the quality of the smoothing and the subsequent expansion of an experimental mode

shape depend totally on the choice of this subspace. In order to have an estimate of the

optimal FE subspace, a ranked list of FE mode shapes to be included in the FE subspace

is obtained through the Local Correspondence Principle and a criterion for the number

of mode shapes to be included from the ranked list must be established.

The Local Correspondence Principle can be summarized as:

For any perturbation of the mass or stiffness matrix, any perturbed mode shape

can be expressed approximately as a linear combination of a limited set of unperturbed

mode shapes. The limited set of mode shapes only need to consist of the corresponding

unperturbed mode shape and a limited number of unperturbed mode shapes around (in

terms of frequency) the unperturbed mode.

Using the LC principle as introduced above, a ranked list of mode shapes to be

included in the smoothing set is obtained simply by including the unperturbed mode

shapes according to their distance to the considered mode in terms of frequency. In

practical problems the frequencies of the experimental modes and those of FE model

might be quite different, and therefore the first mode shape in the mentioned ranked list

25

must be found as the FE shape that has the largest MAC value with the considered

experimental mode shape – denoted as the primary FE mode shape.

A criterion to determine the optimal number of mode shapes to include is

needed. First, the experimental DOFs are divided into a fitting set and an observation

set. As a result, a considered experimental mode shape 𝑎 is then known in the set of

fitting DOFs defining 𝑎𝑓𝑖𝑡, and in the set of observation DOFs defining 𝑎𝑜𝑏𝑠, the

number of DOFs in the fitting set is N.

With the ranked list of mode shapes, it’s possible to determine a number of

mode shape cluster matrices 𝑩𝑓𝑖𝑡,𝑛up to the number of modes M considered, where

𝑩𝑓𝑖𝑡,1 includes only one FE mode shape (the primary mentioned above), 𝑩𝑓𝑖𝑡,2 includes

two FE mode shapes (the two closest to the experimental mode considered in terms of

frequency) and so on.

For the considered experimental mode in the fitting DOFs 𝑎𝑓𝑖𝑡and for the n-th

cluster of FE modes 𝑩𝑓𝑖𝑡,𝑛 yields an expression equivalent to Eq. (2.67):

�̂�𝑛 = 𝑩𝑓𝑖𝑡,𝑛+ 𝑎𝑓𝑖𝑡 (2.70)

And the experimental mode shape can be estimated according to Eq. (2.69)

using the full set of DOFs:

�̂�𝑛 = {

�̂�𝑓𝑖𝑡,𝑛

�̂�𝑜𝑏𝑠,𝑛} = [

𝑩𝑓𝑖𝑡,𝑛

𝑩𝑜𝑏𝑠,𝑛] �̂�𝑛 (2.71)

Where 𝑩𝑜𝑏𝑠,𝑛 is the mode shape cluster matrix defined over the observation set

of DOFs.

This division of the experimental DOFs into a fitting set of DOFs and an

observation set of DOFs is performed to be able to observe the occurrence of overfitting

as this would not be possible using all DOFs known in the experiment. When the

number of modes m approaches the number of DOFs in the fitting set M, then the errors

on the fitting DOFs approach zero, but DOFs in between (the observation DOFs) get

large errors. Thus, a suitable measure of the quality of the fit is given by:

𝐹𝑖𝑡𝑜𝑏𝑠(𝑛) =

|�̂�𝑜𝑏𝑠,𝑛𝐻 𝑎𝑜𝑏𝑠|

2

�̂�𝑜𝑏𝑠,𝑛𝐻 �̂�𝑜𝑏𝑠,𝑛

(2.72)

26

3 METHODOLOGY

The method proposed in this project in order to fulfill the objective stated in

Subchapter 1.1 is detailed in this Chapter.

3.1 FEM Numerical Model

For a given a structure of interest, a preliminary finite element numerical model

shall be developed and then the first estimate for the modal solution (natural frequencies

and mode shapes) must be obtained by the FEM. This is done first to have a preliminary

understanding of the modal parameters of the structure to ideally execute an experiment

in the real structure, for instance, to avoid positioning sensors on the nodes of the

structure.

3.2 Experiment and OMA

Then, the experiment on the real structure in operational conditions must be

executed, as it has been noted, the focus of this method are structures where it’s not

viable or feasible to measure the forces acting on the structure during an experiment,

therefore only the output data, response of interest is collected.

With the signal output from the experiment, an Operational Modal Analysis

(OMA) shall be performed, as mentioned in Subchapter 2.3, the OMA method applied

can be either in the frequency domain such as FDD, EFDD and CFDD or in the time

domain like SSI-UPC. The OMA will return the modal parameters of the structure in

operational conditions: natural frequencies, damping ratios (depending on the method)

and mode shapes (usually complex).

3.3 FEM Model Preliminary Optimization

After the OMA analysis, differences between the preliminary numerical model

results and the experimental results may arise due to the impossibility of perfectly

modeling the structure’s manufacturing defects and boundary conditions. Thus, a first

adjustment in the FEM numerical model properties is needed to approximate the

solution to the experimental result, for instance, by adjusting the assumed boundary

conditions in the model. After this first adjustment, the mass and stiffness matrices of

the numerical model are obtained.

27

3.4 Modal Parameters Post-Processing and Model Reduction

As revised in Subchapter 2.3, OMA usually returns complex frequencies and

complex mode shapes due to measurement noise. Since there is a need of comparing

these experimental complex mode shapes to the modes obtained numerically from an

undamped model, a complex-to-real operation to the experimental mode shapes as

described in Subchapter 2.6 must be done.

Then, with the full mass and stiffness matrices of the FEM model, a partial

reduction is done by the GUYAN method, in this procedure the selected DOFs for the

reduction must include the DOFs where the sensors of the experiment were positioned

in the experiment. In addition, the number of remaining DOFs for reduction must be

chosen in order to avoid large errors in this partial reduction process, since the selection

of DOFs affects the results given by the GUYAN method. The idea is to reduce the

model to a number of DOFs only until the solution is not significantly affected but still

preserving a small computational effort.

With the reduced mass and stiffness matrices obtained from GUYAN as

described, the solution from the eigenvalue problem yields the reduced modal matrix.

Then the SEREP method is applied to completely reduce the model to the sensor’s

DOFs using the partially reduced modal, mass and stiffness matrices obtained from

GUYAN. As noted in Subchapter 2.7 by Silva, Maia [13] the reduced model by SEREP

has exactly the same frequencies and mode shapes as the original system for the

selected modes of interest. That is the reason explaining the GUYAN-SEREP hybrid

reduction. The idea is to reduce the model in the physical domain using GUYAN to a

degree that the results are almost unaffected and further computational effort is small to

then apply SEREP. In Section 4.5.3 it will be shown that a complete reduction using

only GUYAN gives results with large errors, while using only SEREP might not be

computationally efficient or possible since there would be a need for solving the full

model to obtain the modal matrix.

The solution of the eigenvalue problem from the reduced mass and stiffness

matrices from SEREP give the reduced mode shapes with DOFs matching the

experimental mode shapes. These reduced numerical mode shapes are intrinsically

normalized by mass, so only normalization by the unit is needed. However, as noted,

the experimental mode shapes from OMA come without normalization by the mass and

unit; therefore, this operation must be applied as described in Section 2.5.1.

28

With both numerical and experimental mode shapes properly normalized, an

adjustment or calibration of the numerical modes by the experimental modes is

performed based on the Local Correspondence Principle as revised in Subchapter 2.8.

3.5 Adjustment of Numerical Mode Shapes

Subchapter 2.8 (Brincker et al) [14] describes the Local Correspondence

Principle and its application by assuming the experimental modes are the ‘perturbed’,

while the numerical modes are ‘unperturbed’ so the experimental modes are adjusted by

the numerical ones. In this project it has assumed the other way around, i.e. that the

experimental modes carry all the real information of the system in operational

conditions. Therefore, the goal is to numerically model the structure to match these

operational conditions. It´s impossible to model a structure exactly as the real one since

the manufacturing errors cannot be modeled, thus the mass matrix will always differ to

a degree from model to real. Also, the boundary conditions may not be exactly modeled

like the real conditions. With this concept in mind, Eq. (2.43) becomes:

�̂�𝑛 = {

�̂�𝑓𝑖𝑡,𝑛

�̂�𝑜𝑏𝑠,𝑛} = [

𝑩𝑓𝑖𝑡,𝑛

𝑩𝑜𝑏𝑠,𝑛] �̂�𝑛 (3.1)

Where, the term �̂�𝑛 is defined as the adjusted numerical mode instead of the

experimental:

To apply the Local Correspondence Principle as described in Subchapter 2.8, the

DOFs in the mode shape being adjusted must be separated by fitting DOFs and

observational DOFs. When the DOFs are split in a way that the observational group

contain only 1 DOF, the MAC criterion for the adjusted observational set cannot be

verified since it will always be equal to 1, because there’s only 1 value, therefore the set

of observational DOFs must have optimally 2 DOFs.

For each mode shape to be adjusted there’s a need to identify the optimum

number of modes to be used in the fitting process. In addition to that, the optimum

combination of fitting DOFs is addressed as well. This is done by the following

algorithm:

1. list the full set of possible combinations of fitting DOFs. The number of

combinations is given by:

29

𝐶𝑏𝑠

𝑛 = (𝑛𝑠

) =𝑛!

𝑠! (𝑛 − 𝑠)! (3.2)

Where 𝑛 is the total number of DOFs in the mode shape, and 𝑠 = 𝑛 − 2.

2. For each mode to be adjusted and each of the sets of fitting DOFs, the

adjustment is tested by one mode, then two modes and so on using Eq.s

(2.42) and (2.45). The order that these modes are chosen is the closest

modes to the mode being adjusted in terms of frequency.

3. The ideal number of modes for fitting is determined by looking for which

the MAC of the adjusted observational DOFs reaches a maximum.

Therefore, each set of fitting DOFs will have an optimum number of

modes to be used.

4. Finally, the set of fitting DOFs and its respective optimum number of

modes that returns the best fit is obtained by applying MAC to the adjusted

mode shapes against the experimental mode shapes.

3.6 Virtual Sensing

Using the calibrated numerical mode shapes and the response output from the

experiment, the modal coordinates can be estimated by:

�̂�(𝑡) = 𝚽𝑎𝑑𝑗+ 𝑥(𝑡) (3.3)

Where �̂�(𝑡) are the estimate modal coordinates in time domain, 𝚽𝑎𝑑𝑗+ is the

pseudo-inverse of the reduced, adjusted numerical modal matrix (mode shapes) and

𝑥(𝑡) is the output response from the experiment in time domain for the DOFs where the

sensors were positioned.

Finally, multiplying the expanded modal matrix to the unmeasured DOFs of

interest by the estimated modal coordinates yield a prediction of the response in the

unmeasured DOFs of interest:

30

�̂�𝑖(𝑡) = 𝚽𝑖�̂�(𝑡) (3.4)

3.7 Proposed Method Flowchart

31

32

33

4 CASE STUDY

The method proposed in Chapter 3 was applied to a case study comprising an

impact test experiment in a clamped beam made of aluminum as described below.

4.1 FEM Numerical Model

A FEM numerical model was developed at first to have a preliminary notion of

the modal parameters of the beam, mainly to avoid positioning the sensors over the

nodes where the displacement will be zero, spoiling the measurement. One important

assumption for the model is that the system is undamped, this assumption is pertinent by

the reason that the proposed method is not interested in a system’s response in the

resonance regions, where it would be crucial to consider damping effects.

4.1.1 Geometry and Material Properties

1. The geometrical dimensions of the beam (L = 2,145 m, H = 0,0254 m, T =

0,00635 m) were assigned;

2. The volume of the beam was calculated as:

𝑉𝑏𝑒𝑎𝑚 = 𝐿 × 𝐻 × 𝑇 = 3,459 × 10−4 𝑚³

3. The total mass of the beam was obtained from a precise weighing-

machine:

𝑀𝑏𝑒𝑎𝑚 = 0,922 𝑘𝑔

4. The specific mass of the beam 𝜌𝑏𝑒𝑎𝑚 was defined as:

𝜌𝑏𝑒𝑎𝑚 =𝑀𝑏𝑒𝑎𝑚

𝑉𝑏𝑒𝑎𝑚= 2664,99 𝑘𝑔/𝑚³

5. The mass of each accelerometer is defined as 𝑀𝑎𝑐𝑐 = 50 𝑔 as shown in

Table 4.2.1;

34

6. The Young’s Modulus 𝐸 and Poisson Ratio 𝜈 of the aluminum 5052 were

assigned from the obtained charted values [16]:

𝐸𝑎𝑙−5052 = 70 𝐺𝑃𝑎 𝜈𝑎𝑙−5052 = 0,33

4.1.2 Mesh and Element Type

1. The keypoints were assigned to match the sensors coordinates and the

clamped point, as shown in Table 4.1:

Table 4.1 – FEM model keypoints position in relation to the beam clamped point.

Keypoints Coordinates [m]

#1 -0,835

#2 0,0

#3 0,11

#4 0,41

#5 0,72

#6 1,01

#7 1,31

2. The element type of the geometry was determined as beam element;

3. A line segment between each keypoint was assigned and further

subdivided to build the grid. The optimized subdivision of these segments

was determined afterwards by a grid test;

4. The element type of the sensors was determined as punctual mass type;

5. The sensors’ mass was assigned to the nodes in each of the last 5 keypoints

in Table 4.1 to match their coordinates in the experiment;

35

4.1.3 Boundary Conditions

The displacements (U) in x and y-direction and rotation (ROT) in x and z

direction of all nodes were set to zero (UX = 0, UY = 0, ROTX = 0, ROTZ = 0), since

in the experiment there’s only displacement in the z-direction and rotation in the y-

direction.

In addition, the clamped point in the experiment (z = 0) the displacement in z

and rotation in y were set to zero as well (UZ = 0, ROTY = 0) to model the clamped

condition.

4.1.4 Solution settings

The idea that motivates this project is that there are structural problems that

generate mass, damping and stiffness matrices with sizes that do not allow a feasible

direct solution, by a computational effort standpoint. Thus, by means of a case study,

the solution command is set only to evaluate the suitability of the numerical model and

to compare the sets of boundary conditions mentioned in the last section. A suitable

numerical model is needed to further validate the proposed method. The settings in the

script for the solution are:

1. Analysis type - Modal Analysis;

2. Lumped Mass - 0;

3. Pre-stress - 0;

4. The extraction method applied is the Block Lanczos;

5. The maximum number of eigenvalues to be extracted has been set to 30;

6. The range of eigenvalues to be extracted has been set as 0-1250 Hz;

7. Mode shapes normalized by the Mass Matrix;

4.1.5 Mesh test

A grid test was performed to determine an optimized grid size, i.e. a size where

it can be considered the solution converged. The convergence criteria applied is a

relative average relative difference between frequencies of successive grid steps reaches

a value below 0,1%. The gird step size was determined by the number of subdivisions

for each of the 6 line segments, these subdivisions were determined as a constant

multiplied by the length of the segment, so the variable is the constant which started

with 6 and doubled each iteration. Only the first 10 eigenvalues were used for this test.

36

The test was performed to the model with clamped boundary condition. Table 4.2 shows

the comparison between number of subdivisions for the line segments and its respective

eigenvalue solutions.

Table 4.2 – Grid test: Eigenvalues (f [Hz]) vs. Number of line segment

subdivisions (N)

f N 6 12 25 50 100 200

#1 2,4341 2,4323 2,4315 2,4313 2,4313 2,4313

#2 7,5634 7,5461 7,5413 7,5402 7,5399 7,5399

#3 15,490 15,272 15,181 15,164 15,160 15,158

#4 44,833 43,038 42,313 42,178 42,142 42,132

#5 51,493 48,281 47,473 47,297 47,253 47,242

#6 93,526 85,841 82,940 82,399 82,256 82,220

#7 170,49 140,96 134,14 132,69 132,33 132,24

#8 196,71 166,46 154,03 152,03 151,51 151,37

#9 307,37 241,4 218,66 214,79 213,77 213,51

#10 436,42 295,22 266,67 260,82 259,40 258,96

Avg. Rel.

diff. - 10,67% 3,88% 0,42% 0,20% 0,03%

Based on the criteria and analyzing Table 4.2 a grid size resultant of the subdivision of

the line segments in 100 has been determined as an optimized grid size for the geometry

of the study case, since further smaller subdivisions won’t give results with an average

relative difference greater than 0,1%. This subdivision resulted in a grid size of 215

nodes with elements of about 10 mm in length each.

4.2 Experimental Setup

4.2.1 Instruments used

The instruments used in the experiment are listed in Table 4.3:

37

Table 4.3 – List of instruments employed in the aluminum beam experiment.

Experiment Instruments

Acquisition System/Analogic-Digital

Converter

Chassis NI-9172 with modules NI-9233

USB, National Instruments®, 32

Channels, 50 kHz, 24 bits resolution,

Control Software Labview® 8.0

Personal Computer Laptop Toshiba, Intel Core i7-

2640M@2.8GHz, 8GB RAM memory

Vibration Transducer

Piezo-Electric accelerometer, PCB – IMI

624B11, 100 mV/g, 50g, 2,4-5000Hz, 100

m waterproof, armored housing integral

cable

Impact Hammer PCB, 086D50 model, 0,23mV/N

(Minette, R. S.) [17] For signal acquisition, A National Instruments’® system

was used with software Labview® 8.0. The system has a analogic-digital converter of

24 bits with sampling rate from 2 kHz to 50kHz for each channel, up to 32 channels

(NI-9178 model with boards USB NI-9233). The system also has an analogic filter anti-

aliasing.

The accelerometer used was a piezo-electric type with 100 mV/g sensibility,

acquisition range from 2 Hz to 10 kHz, waterproof from manufacturer PCB-IMI, model

624B11.

38

Figure 4.1 - Measuring Instruments: a) A/D NI® board b) Waterproof piezo-electric

accelerometer

4.2.2 Experiment assembly

The experiment has been performed at LCDAV-CENPES: Dynamic Behavior

and Vibration Analysis Laboratory of Petrobras’ Development and Research Center.

The system’s type assembled is SIMO (Single-Input Multiple-Output): The

beam was clamped in its base and all excitation impacts were applied near the clamped

point, as shown is Figure 4.2.a. Five accelerometers were attached and distributed

almost evenly along the beam from the clamped point as it is also shown in Figure 4.2.a.

39

Figure 4.2.a - Experiment Assembly LCDAV/CENPES

The beam’s material is aluminum 5052, with a rectangular section of 25,4 x 6,35

mm (1” x ¼”). It has a full length of 2,145 m, however, the length of the segment which

the accelerometers were positioned is 1,31 m from the clamped point to the top (Figure

4.2.b). The position of the accelerometers are described in Table 4.5, the coordinate

system has its vertical zero point in the clamped point. A total of 10 impacts were

applied near the clamped point.

Table 4.5 – Accelerometers positions in relation to the beam clamped point

Accelerometer Position [m] (relative to the clamped point)

#1 0,11

#2 0,41

#3 0,72

#4 1,01

#5 1,31

40

Figure 4.2.b – Schematic of beam experiment (illustrative image)

4.2.3 Experiment output

The collected output signal data in time domain is shown for each channel in

Figure 4.3. Every black circle marking represents an independent impact (15 in total).

The output of the experiment was then processed in a Modal Analysis commercial

software to obtain the modal parameters of the system.

Figure 4.3 - Clamped Aluminum beam experiment output signal in time domain

41

4.2.4 Signal Processing – Operational Modal Analysis

In the modal analysis software, the geometry has been built, the experimental

data has been imported with a sampling rate set at 25 kHz to match the experiment

sampling rate.

The settings to prepare the data for analysis are described below:

• Detrending – Enabled

• Decimation – 0 – 1250 kHz

• Filtering – Disabled

• Projection Channels – Disabled

• Spectral Density Estimation Resolution – 2048

• Harmonic Detection – Disabled

It was verified that the mentioned decimation (0-1250 kHz) and spectral

resolution (2048) defined were ideal to estimate the first handful of frequencies and

mode shapes.

Then, with the data preparation ready, the task proceeded to apply the estimation

method. The Enhanced Frequency Domain Decomposition (EFDD) (as described in

Section 2.3.2) was applied. A partial Single Value Decomposition (SVD) diagram

resulted from the analysis is shown in Figure 4.4.

42

Figure 4.4 - Single Value Decomposition (SVD) plot (0 to 325Hz) of the

clamped beam experiment

The first mode had to be identified manually. This can be explained by the fact

that the impacts were applied near the clamped point, therefore provoking higher modes

as it can be seen in Figure 4.5. The consequence is that lower frequency modes will

show lower energy in the spectrum and might be harder to identify. The first 8 natural

frequencies obtained from the analysis are shown in Table 4.6, together with the

damping ratio estimate.

Table 4.6 – Natural frequencies obtained from the beam experiment using OMA-EFDD.

Natural Frequency [Hz] Damping Ratio [%]

2,394 9,336

14,819 2,503

41,223 2,085

81,189 0,651

136,96 2,682

43

200,928 0,652

279,589 1,089

386,259 0,898

For each of the first five modes, the complex mode shapes were obtained. They

will be further processed as indicated in the proposed method described in Chapter 3.

Figure 4.5 – Beam vibration modes: a) 1st mode b) n-th mode

4.3 FEM numerical model optimization

In the preliminary model the clamped boundary condition (no displacement, no

rotation) was assigned to the experiment clamped point in all coordinate-axis (x, y, z).

However, this condition might not represent the real boundary condition, as it was

verified the beam shows visually very low amplitude vibration in the unexcited segment

of the beam, characterizing an imperfect clamped condition. Thus, in the experiment

clamped point (z = 0), it was assigned a zero-displacement condition (UZ = 0) and the

no rotation condition (ROTY = 0) was substituted by a spring with stiffness optimized

to approximate the eigenvalue solution to the frequencies obtained from the EFDD. This

boundary condition modification will couple the eigen solution from the DOFs of both

44

segments of the beam. In a perfect clamped condition, the eigen solution from both

segments is uncoupled. Further, the application of both models will be compared.

4.3.1 Boundary Condition

In the optimized FEM model, the no-rotation in y (ROTY = 0) was replaced by a

spring allowing rotation in that direction. To do that, a node had to be created in an

arbitrary position out of the structure and the spring was inserted connecting this

arbitrary node with the node from the support. The stiffness was determined by

optimization through the relative difference between the numerical solution and the

experimental solution, the result of this optimization will be addressed further.

4.3.2 Spring stiffness optimization

The stiffness of the spring that better models the real boundary condition is

unknown, however it is clear that the stiffness in the region is very high, therefore, it

was assigned initially a very high stiffness value for the spring and after, the value was

decremented until the numerical solution best approximated the experimental results.

The criterion applied was when the relative difference between the numerical and

experimental frequencies started diverging by another decrement in the spring stiffness.

Table 4.7 shows the comparison between solutions for several values of spring stiffness.

45

Table 4.7 – Spring stiffness optimization results.

Spring Stiffness [Nm/rad]

Freq.

[Hz]

Exp.

EFDD

Clamped

B.C.

1

× 104

5

× 104

4

× 103 3 × 103

2,5

× 103

2,4

× 103

#1 2,394 2,4313 2,4178 2,4044 2,3978 2,3869 2,3782 2,376

#3 14,819 15,160 15,079 15,004 14,969 14,913 14,870 14,86

#4 41,223 42,142 41,905 41,660 41,536 41,328 41,162 41,12

#6 81,189 82,256 81,894 81,572 81,424 81,194 81,025 80,985

#8 136,960 151,51 149,92 148,72 148,24 147,56 147,12 147,02

#9 200,928 213,77 212,56 211,56 211,12 210,47 210,02 209,91

#11 279,589 293,19 291,74 290,73 290,33 289,78 289,43 289,35

Avg.

Rel.

Error

- 4,18% 3,55% 3,01% 2,77% 2,475% 2,402% 2,405%

It has been concluded after analyzing Table 4.7 that the average relative error

stops decreasing at a spring stiffness of 2,5× 103 Nm/rad, thus this was the value

assigned to the numerical model.

Table 4.8 shows the relative error between individual experimental vs. numerical

with a spring of stiffness of 3 × 103 Nm/rad. It can be seen that for the lower

frequencies, the relative error is relatively small, while for the higher frequencies the

error increases significantly.

46

Table 4.8 – Comparison between natural frequencies obtained from Experiment and

Numerical FEM after B.C. optimization.

Vibration Modes Natural Frequency

(Experimental)

Natural Frequency

(Numerical FEM) Relative Error

#1 2,394 2,3782 0,66%

#3 14,819 14,870 0,34%

#4 41,223 41,162 0,15%

#6 81,189 81,025 0,20%

#8 136,960 147,12 7,42%

#9 200,928 210,02 4,53%

#11 279,589 289,43 3,52%

After the calibration of the spring B.C., the mass and stiffness matrices together

with the node numbering vector and ordering of the model were extracted in MMF

format in a .txt file for further processing. The first 8 modes shapes obtained from FEM

solution are shown is figures 4.6 to 4.13 below, it was note that modes #2, #5 and #7

were not identified by OMA-EFDD due to the positioning of the accelerometers.

Figure 4.6 – FEM solution mode shape #1 (f = 2,3782 Hz)

47

Figure 4.7 - FEM solution mode shape #2 (f = 7,2879 Hz)

Figure 4.8 - FEM solution mode shape #3 (f = 14,87 Hz)

Figure 4.9 - FEM solution mode shape #4 (f = 41,162 Hz)

48

Figure 4.10 - FEM solution mode shape #5 (f = 46,065 Hz)

Figure 4.11 - FEM solution mode shape #6 (f = 82,025 Hz)

49

Figure 4.12 - FEM solution mode shape #7 (f = 127,9 Hz)

Figure 4.13 - FEM solution mode shape #8 (f = 147,12 Hz)

4.4 Complex to Real operation to the experimental mode shapes

Since OMA methods result in mode shapes with some complexity, as it is the

case here, a complex to real operation as described in Subchapter 2.5 was applied to the

experimental mode shapes obtained. These complex vectors were extracted in

magnitude/phase form, so the operation is very straightforward: If the phase angle lies

in the 1st or 4th quadrant (between -90° and 90°) the new real vector will be the

magnitude of the complex vector with positive sign. In opposition, if it lies in the 2nd or

3rd quadrant the new real vector will be the magnitude of the complex vector with

negative sign.

50

4.5 Model Reduction

4.5.1 GUYAN Method

The GUYAN method has been revised in Subchapter 2.6. In order to apply the

method, it is needed to choose which nodes will be the master nodes. The precision of

the method is proportional to the number of master nodes chosen. The goal of first

applying the GUYAN method is to reduce the model to a degree where it´s possible to

solve the eigenvalue problem to obtain the modal domain so the SEREP method can be

applied. If the GUYAN method is applied to get a complete reduction to the few nodes

of interest, substantial errors may be obtained since the number and selection of Master

DOFs have influence over the solution. Thus, the number of master degrees of freedom

chosen for GUYAN was optimized to obtain eigenvalues close to the eigenvalues

obtained from the solution of the full model. In all cases the nodes related to the sensors

were included. The stop criterion was when the average relative difference between two

subsequent solutions reaches a value below 0,1%. Table 4.9 shows the number of

master degrees of freedom and the related frequency values.

51

Table 4.9 – Analysis of the influence of the number of Master degrees-of-freedom in the

GUYAN natural frequency results.

Number of Master DOF

Vibration

Modes 5 11 16 26 57 91

#1 2,3782 2,3781 2,3781 2,3781 2,3781 2,3781

#2 - 7,1375 6,8915 6,8862 6,8854 6,8854

#3 14,7364 14,877 14,8719 14,8717 14,8706 14,8706

#4 37,5596 41,2862 41,2083 41,1765 41,1593 41,1589

#5 55,0133 48,2337 47,3733 46,2462 46,0064 46,0065

#6 91,0793 81,24 81,1551 81,1136 81,0306 81,0271

#7 - 151,266 144,1323 132,8619 127,8839 127,8832

#8 - - 157,484 148,5908 147,1813 147,1224

#9 - 228,4923 213,8304 213,1786 210,1728 210,0402

#10 - - - 269,6793 253,2336 253,2478

Avg. Rel.

Diff. - 6,80% 2,09% 1,82% 1,29% 0,01%

Examining Table 4.9 it has been concluded that 57 master degrees of freedom is

the minimum number which the solution doesn’t diverge over 0,1% since the average

difference between frequencies with 91 Master DOF and 57 Master DOF observed was

only 0,01%. The solution by GUYAN for 5 Master DOF was used for comparison with

the hybrid reduction GUYAN-SEREP.

4.5.2 SEREP method

The SEREP method has been revised in Subchapter 2.7. The partial reduction by

GUYAN generated reduced modal, mass and stiffness matrices that will be inputs for

the SEREP method. In addition, it is necessary to choose the modes for which the

SEREP method will be applied. It has been verified by comparing the eigenvalues from

the solution after GUYAN and full model that the 2nd, 5th and 7th natural frequencies

(ordered from lowest to highest) are not among the first five obtained from the

experiment. These frequencies are related to mode shapes from the beam segment that

was not measured during the experiment, thus the selected modes for the SEREP

52

procedure were the 1st, 3rd, 4th, 6th and 8th modes to match the modes obtained from the

experiment.

In addition, the displacement (UZ) DOFs related to the position of the sensors in

the experiment were chosen as active DOFs for the reduction. Completely reduced mass

and stiffness matrices were finally obtained and the eigenvalue problem was solved.

The natural frequencies obtained were compared to the ones obtained from GUYAN,

GUYAN-SEREP, full model and experimental as it is shown in Table 4.10, while the

relative error between eigenvalues obtained from each method and experimental is

shown in Table 4.11.

Table 4.10 – Comparison (values) between natural frequencies resulted from

experiment vs. numerical (full model, reduced by GUYAN and reduced by GUYAN-

SEREP)

Natural Frequency [Hz]

Vibration

Modes EFDD

Numerical

(Full Model)

GUYAN-

SEREP GUYAN

#1 2,394 2,3781 2,3780 2,3781

#3 14,819 14,8706 14,8705 14,7364

#4 41,223 41,1586 41,1592 37,5596

#6 81,189 81,0256 81,0306 91,0793

#8 136,96 147,1153 147,1812 -

Table 4.11 - Comparison (relative error) between natural frequencies resulted from

experiment vs. numerical (full model, reduced by GUYAN and reduced by GUYAN-

SEREP)

Relative Error

Vibration Modes Exp. x Full Exp. x GUYAN-

SEREP Exp. x GUYAN

#1 0,66% 0,67% 0,66%

#3 0,35% 0,35% 0,56%

#4 0,16% 0,15% 8,89%

#6 0,01% 0,20% 10,86%

#8 6,90% 6,94% -

53

Analyzing Tables 4.10 and 4.11, it is understood that the solution of GUYAN-

SEREP reduced model resulted in approximately the same frequencies of the solution of

the full numerical model and that only for the frequency #8 the relative difference

compared to the experimental frequency was over 1%. On the other hand, it became

clear that the solution of the reduced model by only GUYAN (5 Master DOF) gave

results with large errors, especially for higher modes (#4 and #6), not to mention that it

wasn’t able to solve for mode #8 (solved for mode #5, though).

4.5.3 Mode shapes comparison between GUYAN, GUYAN-SEREP and Full

numerical models and Experimental

The mode shapes from the numerical model are intrinsically mass normalized,

so they had to be normalized by the unit only. Nevertheless, the mode shapes resulted

from OMA analysis come without normalization, therefore normalization by the mass

was performed as described in Section 2.5.1, followed by normalization by the unit.

With all the mode shapes properly normalized, they were all plotted as shown in

Figures 4.14 to 4.15.

Figure 4.14 – Mode shape comparison between GUYAN, GUYAN-SEREP and

Experimental for mode 1

54

Figure 4.15 - Mode shape comparison between GUYAN, GUYAN-SEREP and

Experimental for mode 3

Figure 4.16 - Mode shape comparison between GUYAN, GUYAN-SEREP and

Experimental for mode 4

55

Figure 4.17 - Mode shape comparison between GUYAN, GUYAN-SEREP and

Experimental for mode 6

Figure 4.18 - Mode shape comparison between GUYAN, GUYAN-SEREP and

Experimental for mode 8

Looking at Figures 4.14 to 4.18, it is clear that the GUYAN method reduction

for 5 Master DOF gives mode shapes with large errors for higher modes. In contrast,

GUYAN-SEREP almost gives the same result as the full model mode shapes, so close

56

that the full model mode shapes can’t be seen in the plots because they’re ‘under’ the

GUYAN-SEREP lines.

The MAC method (Section 2.4.1) and DR method (Section 2.4.3) were applied

to compare the mode shapes from the reduced model GUYAN-SEREP against the full

model to validate the reduction procedure. The MAC plot for this comparison is shown

in Figure 4.19 and the MAC values in Figure 4.20, while the DR values are displayed in

bar chart in Figure 4.21.

Figure 4.19 – MAC bar chart comparing the reduced model by GUYAN-SEREP

modes against the Full model modes

57

Figure 4.20 - MAC value chart comparing the reduced model by GUYAN-

SEREP modes against the Full model modes

Figure 4.21 – DR bar chart comparing the reduced model by GUYAN-SEREP

modes against the Full model modes

By analyzing the MAC values, it is clear the correlation between the mode

shapes from GUYAN-SEREP vs. the full model is excellent. The lowest value being

between 8th modes, but all extremely close of value 1. In addition, the DR bar chart

shows that all values are very close to 0, thus validating the reduction procedure, these

58

results are in line with the plots of the mode shapes as it could be seen that the modes

are almost the same.

Finally, the MAC and DR were applied to compare the mode shapes from the

reduced model against the experimental mode shapes. The MAC plot for this

comparison is shown in Figure 4.22, the MAC values are shown in Figure 4.23 and the

DR plot is shown in Figure 4.24.

Figure 4.22 - MAC bar chart comparing the reduced model by GUYAN-SEREP

modes against the Experimental modes

59

Figure 4.23 - MAC value chart comparing the reduced model by GUYAN-

SEREP modes against the Experimental modes

Figure 4.24 - DR bar chart comparing the reduced model by GUYAN-SEREP

modes against the Experimental modes

Analyzing Figure 4.23, it can be seen that there is already a fair correlation

between GUYAN-SEREP and experimental modes, with the lowest MAC value being

between the 8th modes with 0,937. In addition, examining the DR plot it’s clear that

there are some DOFs in some modes where the DR value is moderate to high, so the

need for calibration is glaring. An adjustment of the numerical modes was performed to

improve the correlation and relative difference.

4.6 Adjustment of Numerical Mode Shapes by Experimental Mode Shapes

based on the Local Correspondence Principle

The procedure for adjustment of the numerical mode shapes described in

Subchapter 3.5 was applied. For the 5 DOFs in this case study, there’s a total

combination of 10 sets of fitting DOFs. The result of the adjustment procedure is

represented here by plotting the fitted, unfitted and experimental mode shapes as

displayed in Figures 4.25 to 4.29.

60

Figure 4.25 - Mode shape comparison between fitted, unfitted reduced numerical

models and Experimental for mode 1

Figure 4.26 - Mode shape comparison between fitted, unfitted reduced numerical

models and Experimental for mode 3

61

Figure 4.27 - Mode shape comparison between fitted, unfitted reduced numerical

models and Experimental for mode 4

Figure 4.28 - Mode shape comparison between fitted, unfitted reduced numerical

models and Experimental for mode 6

62

Figure 4.29 - Mode shape comparison between fitted, unfitted reduced numerical

models and Experimental for mode 8

Table 4.12 and Figure 4.30 shows the MAC values between the fitted mode

shapes against the experimental mode shapes, while Figure 4.31 shows the DR plot and

Table 4.13 displays the COMAC (reviewed in Section 2.4.2) values.

Table 4.12 – MAC: Fitted reduced numerical model modes vs. Experimental modes

Fitted Mode Shapes

1 3 4 6 8

Exper

imen

tal

Mode

Shap

es

1 0,9985 0,0691 0,0572 0,0617 0,0981

3 0,0666 0,9996 0,0828 0,0939 0,0369

4 0,0512 0,1089 0,9900 0,0808 0,0500

6 0,0364 0,0450 0,0377 0,9874 0,3166

8 0,0721 0,0319 0,0180 0,1766 0,9730

63

Figure 4.30 - MAC bar chart comparing the fitted reduced model by GUYAN-

SEREP modes against the Experimental modes

Figure 4.31 - DR bar chart comparing the fitted reduced model by GUYAN-

SEREP modes against the Experimental modes

64

Table 4.13 – COMAC (Fitted Reduced numerical model)

Accelerometer #1 #2 #3 #4 #5

COMAC 0,9880 0,9944 0,9806 0,9897 0,9853

A bar chart comparing MAC values between fitted and unfitted mode shapes

against the experimental mode shapes is shown in Figure 4.32 to show the

improvement.

Figure 4.32 - Bar chart displaying the improvement in MAC from the unfitted

reduced numerical model to the fitted against the experimental.

Analyzing Figure 4.32 and comparing the DR plots from the unfitted model in

contrast with the fitted, it is clear that the adjustment procedure proves to be effective in

approximating the numerical mode shapes to the experimental mode shapes. All MAC

values were above 0,97, while the vast majority of DOFs in each mode had their DR

reduced.

Evaluating the COMAC in Table 4.13, it is confirmed that the mode coordinates

associated with the DOFs are the same, since the values are all extremely close to 1.

Table 4.14 shows, for each mode, the optimum combination of fitting DOFs and

the optimum number of modes used in the adjustment.

65

Table 4.14 – Optimum DOF set and number of modes in the adjustment of each mode

shape

Mode Optimum fitting DOF set Optimum number of modes

#1 2 3 5 3

#3 3 4 5 3

#4 1 4 5 2

#6 1 2 5 3

#8 3 4 5 3

Therefore, the numerical FEM model, reduced by GUYAN-SEREP and adjusted

by Local Correspondence Principle gives a good representation of the studied system.

66

5 CONCLUSIONS AND RECOMMENDATIONS

The first outcome that can be outlined is that a preliminary FEM numerical

model of the structure to be studied has been found to be extremely useful in supporting

the experimental setup as displayed in Subchapter 4.3. As it has been shown in Table

4.7, the average relative error between the first 8 natural frequencies in the experiment

and the preliminary FEM model was 4,18%. The analysis of the mode shapes from the

preliminary model supported an optimized positioning of the sensors in the experiment.

Furthermore, the application of the OMA-EFDD modal analysis (Subchapter

4.2) has shown to be effective in determining the modal parameters of the system,

confirming its advantages over EMA. However, it is concluded from the results that the

experiment setup was limited since the first mode presented little energy in the PSD

spectrum leading to a harder, manual identification of the mode, therefore the impacts

should have been performed in a way to properly excite the mode. Likewise, it’s

concluded that there should have been vibration measurement on the other side of the

beam to identify the mode shapes related to larger amplitudes on that side in order to

have the full experimental modal parameters of the system.

When it comes to the comparison between the FEM model and the real structure

(Subchapter 4.3), it has become evident the difference or modification between one

another, even though relatively small in this case. The attempt to update the FEM model

through optimizing the boundary condition proved to be able to reduce such difference

considerably, but not completely. The average relative error between the first 8 natural

frequencies dropped from 4,18% to 2,402%.

The hybrid Guyan-SEREP reduction procedure (Subchapter 4.5) demonstrated

that it succeeds in combining the advantages of both techniques. Guyan provides a

reduction with smaller computational effort and the errors in natural frequency and

mode shapes are minimized when the number and selection of Master DOFs are

properly selected, therefore being excellent as a partial step reduction. On top of that,

SEREP achieves the goal of fully reducing the system keeping exactly the same natural

frequencies and mode shapes from the original system. The MAC between the reduced

model mode shapes from this hybrid method compared to the full model were all above

0,99999999 (or 10−8 shy of 1) and all DR values were in the order of 10−4. These

results conclude the efficiency of the reduction procedure.

67

Regarding the calibration procedure (Subchapter 4.6), it has been confirmed that

the application of the local correspondence principle is adequate to adjust the numerical

mode shapes. The correlation from the unfitted to the fitted mode shapes has

considerably improved as shown is figures 4.22, 4.23, 4.30 and Table 4.12, while the

DR between each DOF/mode greatly decreased is the majority of cases as displayed in

figures 4.24 and 4.31. In addition, it has been observed the possibility of overfitting: For

the adjustment of mode shape #4, the maximum number of modes that could be applied

to the adjustment was 3, however, the optimum number ended up being only 2,

therefore, for that case, 3 modes led to an overfitting. Even though it has been registered

a boost in both MAC and DR, it’s obvious the adjustment has room for improvement,

for instance the #8 mode still registered DR values over 0,2 for 3 out of 5 DOFs while

the MAC went up to only 0,973, when we should expect DR values under 0,1 and MAC

values over 0,99 ideally. This could be reached using more modes in the adjustment, but

for that, more sensors would have to be setup in the experiment, an optimization in the

algorithm applying the local correspondence principle (Subchapter 3.5) can’t be ruled

out either.

Recommendations for Future Work

The first recommendation stems from the experimental results from OMA-

EFDD. It’s recommended to use the information given by the preliminary FEM model

as best as possible. Such preliminary model can be used to identify not only the ideal

positions of the sensors, but how many would be needed to acquire an optimized

number of modes to adequately represent the system. By evaluating the modal mass

matrix obtained through this preliminary model, it’s possible to even identify the modes

which have the most contributions in order to not miss them in a further vibration

measurement experiment. It’s important to remember that the response can be

approximated by a linear combination of modes, the higher the number of modes, the

better the approximation. However, only a handful of modes will likely compose the

largest contributions to the mass matrix, so this identification can be crucial for virtual

sensing. Therefore, such analysis is recommended for future work.

Another line of research regards the FEM numerical modelling itself; it has been

explored to a limited extent the influence of the boundary conditions in the FEM model

solution and how it leads to modifications from the real structure. A simplified attempt

68

to update the B.C. was done, but a considerable difference in the numerical solution was

registered. Therefore, a more thorough study about this B.C. influence and methods to

optimize/update it is suggested for future work. The quality of the FEM model affects

the final calibration, so it’s important to improve the knowledge over this procedure.

Moreover, a study of the impact in the adjustment procedure of number of

DOFs/modes employed in the local correspondence principle could be performed as

well. In this project only 5 DOFs/modes were used so this study would have been

limited here, but it became clear that 5 DOFs/modes left considerable room for

improvement in the adjustment.

Finally, the adequate application of the adjusted numerical mode shapes for

virtual sensing was out of the scope of this project. Nevertheless, the application of

calibrated mode shapes in virtual sensing is the ultimate goal, highlighting the

importance of studying this procedure. Requirements that may arise during the attempt

for virtual sensing will determine what’s needed to be obtained from the whole method

proposed in this project, for instance, it could be concluded that the selection and

number of modes wasn’t sufficient to satisfactorily predict the vibration response of

interest. Thus, such study is highly fundamental and suggested.

69

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