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
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.
DEDICATION
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.
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.
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
3.6 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
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;
𝑋̈ acceleration response vector; 𝑋̇ 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;
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;
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 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 non-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
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.”
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.
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).
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:
𝑴𝑋̈(𝑡) + 𝑪𝑋̇(𝑡) + 𝑲𝑋(𝑡) = 𝐹(𝑡) (2.3)
Where 𝑋̈(𝑡), 𝑋̇(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 𝑋̇(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)
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
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.
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:
• 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).
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).
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.
(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.
(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 {𝜙𝑖2}, the MAC between these modes is given by:
MAC({𝜙𝑖1}, {𝜙𝑖2}) = |{𝜙𝑖 1}𝐻{𝜙 𝑖2}| 2 ({𝜙𝑖1}𝐻{𝜙 𝑖1})({𝜙𝑖2}𝐻{𝜙𝑖2}) (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{𝜙 𝑖}) ({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
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 𝐿 𝑗=1 ) 2 (∑𝐿𝑗=1(𝚽𝐴)𝑖,𝑗2)(∑𝑗=1𝐿 (𝚽𝐵)𝑖,𝑗2) (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.
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 = 𝑲
𝐹𝐸𝚽𝐹𝐸 (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:
𝜶𝐹𝐸 = [√𝚿𝐹𝐸𝑇 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 = 𝑲
𝑒𝑥𝑝𝚽𝑒𝑥𝑝 (2.27)
Where 𝜔𝑒𝑥𝑝2 and 𝚽
𝑒𝑥𝑝 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
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+ 𝚿𝑒𝑥𝑝𝑇 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:
𝜶̂𝑒𝑥𝑝≅ [√𝚿𝑒𝑥𝑝𝑇 𝑴𝐹𝐸𝚿𝑒𝑥𝑝] −1
(2.32)
Therefore, the modal mass can be estimated as follows:
𝒎̂𝐹𝐸 ≅ 𝚿𝑒𝑥𝑝𝑇 𝑴
𝐹𝐸𝚿𝑒𝑥𝑝 (2.33)
Finally, the approximated mass normalized experimental mode shapes are obtained by:
𝚽̂𝑒𝑥𝑝= 𝚿𝑒𝑥𝑝𝜶̂𝑒𝑥𝑝 (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:
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:
[𝑲𝑲𝑚𝑚 𝑲𝑚𝑠 𝑠𝑚 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−1𝑲
𝑠𝑚 (2.40)
𝐹𝑟 = 𝐹𝑚− 𝑲𝑚𝑠𝑲𝑠𝑠−1𝐹
𝑠 (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:
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 = [ 𝑰
𝑹] (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:
𝑴𝑋̈(𝑡) + 𝑲𝑋(𝑡) = 𝐹(𝑡) (2.44)
Where 𝑴 and 𝑲 are the mass and stiffness matrices of the full model, 𝑋̈ 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)̈ 𝑋𝑠̈ (𝑡)} + [ 𝑲𝑚𝑚 𝑲𝑚𝑠 𝑲𝑠𝑚 𝑲𝑠𝑠] { 𝑋𝑚(𝑡) 𝑋𝑠(𝑡)} = { 𝐹𝑚(𝑡) 𝐹𝑠(𝑡)} (2.45)
From Eq. (2.4.11) one can obtain:
𝑴𝑠𝑚𝑋𝑚̈ (𝑡) + 𝑴𝑠𝑠𝑋𝑠̈ (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 𝑋̈(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:
Substituting Eqs. (2.46) and (2.43) in Eq. (2.45) and pre-multiplying by matrix 𝑻 transpose, results:
𝑴𝑟𝑋𝑚̈ (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)
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 𝚽𝑚𝑝𝑞̂𝑝(𝑡) (2.56)
Where, 𝑞̂𝑝(𝑡) is an approximate solution of 𝑞𝑝(𝑡), and can be calculated by manipulating Eq. (2.56):
𝑞̂𝑝(𝑡) = (𝚽𝑚𝑝𝑇 𝚽 𝑚𝑝)
−1
𝑌𝑝(𝑡) (2.57)
Merging Eq. (2.55) into Eq. (2.57):
𝚽𝑚𝑝+ = (𝚽𝑚𝑝𝑇 𝚽𝑚𝑝)−1𝚽𝑚𝑝𝑇 (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:
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:
where 𝑩𝒂+ is the pseudo-inverse of 𝑩
𝒂. 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:
𝑎̂ = 𝑩𝒂+𝑝̂ (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.
𝑎̂ = 𝑩𝑝̂ (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
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):
𝑝̂𝑛 = 𝑩𝑓𝑖𝑡,𝑛+ 𝑎𝑓𝑖𝑡 (2.70)
And the experimental mode shape can be estimated according to Eq. (2.69) using the full set of DOFs:
𝑎̂𝑛 = {𝑎̂𝑓𝑖𝑡,𝑛 𝑎̂𝑜𝑏𝑠,𝑛} = [
𝑩𝑓𝑖𝑡,𝑛
𝑩𝑜𝑏𝑠,𝑛] 𝑝̂𝑛 (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) =
|𝑎̂𝑜𝑏𝑠,𝑛𝐻 𝑎𝑜𝑏𝑠|2
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.
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
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:
𝑎̂𝑛 = {𝑎̂𝑓𝑖𝑡,𝑛 𝑎̂𝑜𝑏𝑠,𝑛} = [
𝑩𝑓𝑖𝑡,𝑛
𝑩𝑜𝑏𝑠,𝑛] 𝑝̂𝑛 (3.1)
Where, the term 𝑎̂𝑛 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:
𝐶𝑏𝑠𝑛 = ( 𝑛 𝑠) =
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:
𝑞̂(𝑡) = 𝚽𝑎𝑑𝑗+ 𝑥(𝑡) (3.3)
Where 𝑞̂(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:
𝑥̂𝑖(𝑡) = 𝚽𝑖𝑞̂(𝑡) (3.4) 3.7 Proposed Method Flowchart
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;
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;
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
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
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.
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.
