Escola de Engenharia Rui Jorge Freitas Rodrigues Guimarães Development of a signal ...

Escola de Engenharia

Rui Jorge Freitas Rodrigues Guimarães Development of a signal processing tool for the post-processing of structural dynamics results obtained through the application of the finite element method.

Tese de Mestrado Mestrado Integrado em Engenharia Mecânica Trabalho efetuado sob a orientação do Doutor José Felipe Bizarro de Meireles Co-orientador Doutor Paulo Jorge da Rocha Soares Antunes

Dezembro 2014

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Agradecimentos A realização desta dissertação de mestrado só foi possível com os importantes contributos de pessoas, que, através de apoio científico ou incentivos pessoais, me levaram a concretizar este projecto, e sem os quais não o teria conseguido. Aos meus orientadores, Doutor José Felipe Bizarro de Meireles e Doutor Paulo Jorge da Rocha Soares Antunes, dirijo uma palavra de agradecimento, por todo o tempo que disponibilizaram a este projecto e pela forma como me souberam orientar. Ao Doutor Gustavo Rodrigues Dias e Doutor Júlio César Machado Viana, que me permitiram usufruir, por diversos momentos, dos recursos da empresa em prol da realização deste trabalho, e todos os colaboradores da Critical Materials S.A. que de alguma forma me auxiliaram na realização deste trabalho, envio também uma palavra de agradecimento. Gostaria de agradecer à minha família e amigos que nunca deixaram de acreditar no meu trabalho. Por fim, uma especial palavra de apreço e agradecimento a quem teve a generosidade (e muitas vezes paciência) de ser o meu maior suporte, Maria João.

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Abstract The increasing usage of the finite element methods (FEM) applied to structural dynamic analysis problems is resulting in a growing interaction with auxiliary tools, such as signal processing tools, able to extend the FEM efficiency. The incorporation of such tools in finite element analysis (FEA) software is a reality. Therefore, this work aimed the development of a signal processing tool for the post-processing of structural dynamics results obtained through the application of the finite element method. The tool was developed as a plug-in, in order to be fully embedded into Abaqus® Non-linear

FEA platform, and provides those signal processing algorithm, considered as being the most relevants in the structural dynamic analysis field. In order to demonstrate the potentialities of the developed tool three case-studies were developed where is proved the advantages of such tool when applied to structural dynamics analysis problems. The name decided for the tool was SPA-Signal Processing for Abaqus®.

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Resumo A crescente utilização do método dos elementos finitos (FEM) aplicado à resolução de problemas relacionados com a dinâmica estrutural tem resultado numa crescente interacção com ferramentas auxiliares, tais como ferramentas para o processamento de sinal, capazes de aumentar a eficiência do próprio FEM. A inclusão destas ferramentas em software’s dirigidos à análise de elementos finitos (FEA) é uma realidade. Assim sendo, pretendeu-se, com este trabalho, o desenvolvimento de uma ferramenta para pós-processamento de resultados obtidos em análise de dinâmica estrutural recorrendo ao método dos elementos finitos. A ferramenta foi desenvolvida sob a forma de plug-in, de modo a ser embebida na plataform Abaqus® Non-linear FEA, e comtempla os algoritmos de processamento de sinal considerados mais relevantes no domínio da dinâmica estrutural. Com o objetivo de demonstrar as potencialidades da ferramenta desenvolvida foram conduzidos três casos de estudo onde se prova o interesse da ferramenta na aplicação a problemas de dinâmica estrutural. O nome definido para a ferramenta foi SPA-Signal Processing for Abaqus®.

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Contents Agradecimentos

iii

Abstract

v

Resumo

vii

Contents

ix

List of figures

xi

List of tables

xvii

List of symbols

xix

1 Introduction

1

1.1

Motivation

2

1.2

Outline of the thesis

3

2 State of the art 2.1

5

Dynamic analysis

5

2.1.1

Single degree of freedom

2.1.2

Multi degree of freedom system

2.2

6

Signal processing

23 25

2.2.1

Signal acquisition

26

2.2.2

Signal types

28

2.3

Theoretical introduction to the problem

31

2.3.1

Signal processing of discrete signals

32

2.3.2

Structural dynamics analysis

35

2.3.3

Finite element analysis

49

3 Signal Processing for Abaqus®

53

3.1

General view

54

3.2

Modules

57

3.2.1

Transformations

58

3.2.2

Statistical operations

62

3.2.3

Signal operations

66

3.2.4

Advanced operations

68 ix

x

Contents

4 Case-studies 4.1

Oberst beam method for material characterisation

71 71

4.1.1

Introduction

71

4.1.2

Objective

72

4.1.3

Definitions and acronyms

72

4.1.4

Contextualization

72

4.1.5

Procedure

73

4.1.6

FE modelling

73

4.1.7

Results

76

4.1.8

Results discussion

80

4.2

Study about Lamb waves’ sensitivity in the damage identification

81

4.2.1

Introduction

81

4.2.2

Objective

82

4.2.3

Definitions and acronyms

83

4.2.4

Contextualization

83

4.2.5

FE modelling

84

4.2.6

Results discussion

87

4.3

A PRODDIA® deployment

93

4.3.1

Introduction

93

4.3.2

Objective

93

4.3.3

Definitions and acronyms

94

4.3.4

Contextualization

94

4.3.5

FE modelling

95

4.3.6

Mesh convergence studies

99

4.3.7

Sensor Positioning

104

4.3.8

Results discussion

105

5 Conclusions and future work

107

Bibliography

109

Annex A Transforms methods

113

Annex B SPA requirements

117

Annex C SPA technical details

129

List of figures Figure 2.1 Dynamic loading. (Source: Clough and Penzien (1993))

6

Figure 2.2 Damped SDOF system disturbed by a force, 𝑓(𝑡).

6

Figure 2.3 Damped SDOF system disturbed by a base motion, 𝑥𝑏(𝑡).

7

Figure 2.4 Free vibration of an undercritically-damped SDOF system. (Source: Braun et al. (2002))

9

Figure 2.5 Vector diagram of forces.

11

Figure 2.6 FRF magnitude and phase angle varying with the damping ratio, for a given undercritically-damped SDOF system.

12

Figure 2.7 Arbitrary periodic loading. (Source: Clough and Penzien (1993))

12

Figure 2.8 Arbitrary impulsive loading.

13

Figure 2.9 Impulse response function for an arbitrary undercritically-damped SDOF system.

14

Figure 2.10 Derivation of the Duhamel's integral.

15

Figure 2.11 3-D plots representing the transfer function magnitude, 𝐻 (left), and phase angle, 𝜃 (right), in the 𝑠-plane, highlighting the frequency response function, represented by the dashed line. 17 Figure 2.12 FRF amplitude, highlighting the resonance, for an arbitrary undercritically-damped SDOF system. (Source: Clough and Penzien (1993)) 19 Figure 2.13 Schematic representation of Convolution, Cross-correlation and Autocorrelation operations. (Source: wikipedia.org)

21

Figure 2.14 Damped MDOF system.

23

Figure 2.15 Exemplificative response of a MDOF system to a transient loading in the time domain (left) – acceleration – and, the respective frequency par, in the Fourier domain (right) – accelerance. 25 Figure 2.16 Typical examples of signals: (A) Continuous (analogue) signal; (B) Discrete signal (digital), sampled at every ∆ seconds. 27 Figure 2.17 Schematic diagram of a general data acquisition system. A/D, analogue-to-digital. 28 Figure 2.18 Signal types. (Source: Randall (1987))

28

Figure 2.19 Sinusoidal signal with a period 𝑇 [s].

29

Figure 2.20 Hum noise signal. (Source: Shin and Hammond (2008))

29

xi

xii

List of figures

Figure 2.21 White noise signal.

30

Figure 2.22 Typical example of transient signals.

30

Figure 2.23 Typical example of non-stationary signals. (Source: Shin and Hammond (2008))

31

Figure 2.24 Magnitude spectrum of a FRF highlighting the resonance peaks, and the half-power (3 dB) points, for each mode. (Source: Brüel & Kjær (1999)) 35 Figure 2.25 Impact of modifications on the structural dynamic behaviour of a cantilever beam. (Source: Silva and Maia (1999)) 36 Figure 2.26 Tacoma Narrows Bridge roadway, vibrating (left) and finally collapsing (right), due to excitation forces produced by wind (traveling at 64𝑘𝑚ℎ). (Source: wikipedia.org) 37 Figure 2.27 Frequency response of an MDOF (A) system, and the highlighting of the respective SDOF modal contributions (B) – Note: The plots are zooming the first three modes in range). (Source: Agilent Technologies (2000)) 39 Figure 2.28 System descriptors, time and frequency domain. (Source: Bilošová (2011))

41

Figure 2.29 Basic experimental rig for modal testing. (Source: Brüel & Kjær (1999))

42

Figure 2.30 “Free” conditions. (Source: Agilent Technologies (2000))

43

Figure 2.31 Grounded condition. (Source: SEM Modal Analysis Technical Division (2011))

43

Figure 2.32 Complex mechanical structures in operation: (A) Offshore petroleum platform; (B) Wind turbine. (Source: Barakah Offshore Petroleum (2014) and Vestas Wind Systems A/S (2014)) 43 Figure 2.33 Civil structures under monitoring: (A) Bridge; (B) Stadium suspended roof. (Source: SC Solutions (2013) and Magalhães et al. (2006)) 44 Figure 2.34 Shake excitation: (A) shaker mass influence; (B) typical experimental setups. (Source: Agilent Technologies (2000))

45

Figure 2.35 Impact hammer excitation: (A) Instrumented impact hammer; (B) typical curves of excitations with different “tips”. (Source: National Instruments Corporation (2014) and Agilent Technologies (2000)) 45 Figure 2.36 Influence of mass loading on measured response. (Source: Agilent Technologies (2000)) 46 Figure 2.37 2-D and 3-D representation of a MAC matrix values, for the modes shape correlation. (Source: Braun et al. (2002)) 47 Figure 2.38 Several sets of (x, y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the non-linearity and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). 49 Figure 2.39 Abaqus® Visualization module main areas.

50

List of figures

xiii

Figure 3.1 Accessing to SPA plug-in on Abaqus (visualization module).

54

Figure 3.2 SPA-Signal Processing for Abaqus logo.

54

Figure 3.3 SPA plug-in’s GUI main dialog, pointing Main category, Modules and Data tabs.

55

Figure 3.4 Domain transformation operations with two acquired signals.

57

Figure 3.5 General view of Transformation category, presenting the typical aspect of From File, From XYData and From Model tabs. 58 Figure 3.6 Defining input variables for FFT procedure.

59

Figure 3.7 FFT resultant spectrums: Magnitude and Phase angle

60

Figure 3.8 Input and output definition to obtain the FRF response (compliance).

60

Figure 3.9 Compliance resultant spectrums, Magnitude and Phase angle.

61

Figure 3.10 General view of Statistical Operations category, presenting the typical aspect of From File, From XYData and From Model tabs. 62 Figure 3.11 Inputs definition for both “source data”, Data 1 and Data 2, to run MAC procedure. 63 Figure 3.12 MAC correlation output matrix result.

64

Figure 3.13 Inputs definition to obtain the FRAC correlation between two systems responses obtained from experimental and analytical tests. 64 Figure 3.14 FRAC correlation output.

65

Figure 3.15 General view of Signal Operations category, presenting the typical aspect of From File, From XYData and From Model tabs. 66 Figure 3.16 Acceleration signal.

67

Figure 3.17 Defining inputs to perform the correlation.

67

Figure 3.18 Autocorrelation of signal presented in Figure 3.16.

68

Figure 3.19 General view of Advanced Operations category, presenting the typical aspect of From File, From XYData and From Model tabs. 68 Figure 3.20 Selection of the magnitude part resultant for the FFT procedure in section 3.2.1.1. 69 Figure 3.21 Results obtained from the extraction of modal damping (%) values from a compliance response (magnitude part). 70 Figure 4.1 Experimental setup. Cantilever Oberst Beam. (Source: Antunes, et al. (2009))

73

Figure 4.2 Oberst beam general dimensions.

74

Figure 4.3 Oberst beam FE model.

75

Figure 4.4 Input force from Hammer (see Figure 4.3).

76

xiv

List of figures

Figure 4.5 SPA procedure to obtain the compliance spectrums (see example in section 3.2.1.1): input (“Hammer”) – Concentrated load node set; output (“Sensor”) – Sensor node set. 77 Figure 4.6 Transient signals in time domain: Input (“hammer”) and output (“sensor”).

78

Figure 4.7 Highlight of three identified modes from the resultant compliance magnitude part spectrum analysis (inputs in Figure 4.5).

78

Figure 4.8 Compliance spectrum – LabView®. (Source: Antunes et al. (2009))

79

Figure 4.9 Fraction of critical damping (𝜉) with frequency– LabView®. (Source: Antunes et al. (2009)) 79 Figure 4.10 Fraction of critical damping [%]: experimental Vs analytical results.

80

Figure 4.10 Phase velocity dispersion curves for Lamb waves in steel plate, showing shapes of fundamental modes. (Source: Braun et al. (2002)) 82 Figure 4.12 Test scenario.

83

Figure 4.13 Test scenario general dimensions.

84

Figure 4.14 Mesh details, highlighting the “damage” area (only applicable for the damage model).

86

Figure 4.15 15 kHz (sine wave) tone burst 3.5 cycles Hann windowed pulse

87

Figure 4.16 Electrical potential boundary conditions.

87

Figure 4.17 FFT (magnitude part) spectrum obtained from the transformed time domain velocity from a central point of the PZT top layer, obtained using SPA FFT module. 88 Figure 4.18 Lamb wave propagation state (elapsed time 0.0002s) in the CFRP plate.

89

Figure 4.19 Wavelet propagation, from PZT point to “Sensor” point. Boundary reflection effect is also emphasised. 89 Figure 4.20 Baseline and damage models in a given, step time frame.

90

Figure 4.21 Comparison of signals (normal to surface velocity): baseline Vs damage models. 90 Figure 4.22 Picking point to extract data using the SPA Correlation module.

91

Figure 4.23 Comparison of cross-correlated signals (normal to surface velocity): baseline Vs damage models, obtained using the SPA Correlation module.

92

Figure 4.24 Experimental setup to test CSHM1 structure.

94

Figure 4.25 CSHM1 component general dimensions and composite plies layup scheme.

95

Figure 4.26 CSHM1 composite plies stacking order.

96

Figure 4.27 CSHM1 final layup: complete, reinforcement, and overlap areas.

96

Figure 4.28 Example of complete and reinforcement layups (discretised in element).

97

List of figures

Figure 4.29 Loads and boundary conditions.

xv

98

Figure 4.30 Characterization of linear and quadratic shell elements: Linear – S3 (3-node triangular general-purpose shell, finite membrane strains) and S4R (4-node generalpurpose shell, reduced integration with hourglass control, finite membrane strains); Quadratic – STRI65 (6-node triangular thin shell, using five degrees of freedom per node) S8R (8-node doubly curved thick shell, reduced integration). 99 Figure 4.31 Meshes – (i) Mesh A: used to define Mesh A-S4R and Mesh A-S8R; (ii) Mesh B – used to define Mesh B-S8R. 100 Figure 4.32 Points (“sensors”) locations from where modal displacement, U2, was extracted. 101 Figure 4.33 Inputs selection to run the SPA’s MAC analysis (Mesh S4R Vs Mesh B-S8R).

101

Figure 4.34 SPA’s MAC analysis output results (Mesh S4R Vs Mesh B-S8R).

102

Figure 4.35 MAC output matrix results: (i) Mesh A-S4R Vs Mesh B-S8R; (ii) Mesh A-S8R Vs Mesh B-S8R. 103 Figure 4.36 CSHM1first four eigenmodes (study “1st moment”).

104

Figure 4.37 CSHM1resultant Sensor Positioning field (study “1st moment”). Locations for four sensors. 105

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List of tables Table 2.1 Sensor and physical phenomenon target.

27

Table 2.2 Physical quantities possible to be measured from a system.

40

Table 3.1 Requirement table template.

53

Table 3.2 Modules domains.

56

Table 4.1 Initial material properties.

74

Table 4.2 Modal damping values: analytical.

78

Table 4.3 Modal damping values: experimental.

80

Table 4.4 Initial material properties.

85

Table 4.5 CSHM1 composite plies layup.

96

Table 4.6 Initial material properties.

97

Table 4.7 Mass values for the experimental and analytical models.

98

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xviii

List of symbols 𝑎0 , 𝑎𝑛

Fourier coefficients constants

𝐴

Generic complex number (magnitude)

𝐴1 , 𝐴2

Constants

𝑏0 , 𝑏𝑛

Fourier coefficients constants

𝑐 𝑘𝑔𝑠

−1

Damping coefficient

[𝑐] 𝑘𝑔𝑠

−1

Damping coefficient matrix

𝑒 𝐸 𝑁𝑚

Exponential −2

𝑓 𝐻𝑧

Young’s modulus Natural cyclic frequency

𝑓(𝑡) 𝑁

Applied loading in time domain

𝐹(𝑖𝜔)

Load vector in frequency domain (Fourier transform)

𝐹(𝑠)

Load vector in the Laplace domain

𝑓𝐼 , 𝑓𝐷 , 𝑓𝑆 𝑁

Inertial, damping, and spring forces, respectively

𝑓𝐼 , 𝑓𝐷 , 𝑓𝑆 𝑁

Inertial, damping, and spring forces, respectively

𝐺 𝑁𝑚−2

Shear modulus

𝑔(𝑡)

Generic function in time domain

ℎ(𝑡)

Impulse response function in time domain

𝐻(𝑖𝜔)

Frequency response function in frequency

𝑖 √−1

Imaginary number

𝑗

Arbitrary coordinate

𝐼0 𝑁𝑠 𝑘 𝑁𝑚

Impulse-momentum −1

[𝑘] 𝑁𝑚−1 𝑚 𝑘𝑔 [𝑚] 𝑘𝑔

Spring stiffness constants, frequency domain index, arbitrary coordinate Spring stiffness constants matrix Mass, time index Mass matrix

𝑛

Integer, constant, time domain index

𝑁

Number of time increments, number of degree of freedom, number of resonance peaks

𝑅𝑥𝑥 (𝜏)

Autocorrelation function

xix

xx

List of symbols

𝑅𝑥𝑦 (𝜏)

Cross-correlation function

𝑟

Mode number, Pearson’s correlation coefficient

𝑠

Laplace constant

𝑆𝑋𝑋 (𝑖𝜔) 𝑔2 𝐻𝑧 −1 Power spectral density function 𝑆𝑋𝑌 (𝑖𝜔) 𝑔2 𝐻𝑧 −1 Cross-spectral density function 𝑡, 𝑡0 𝑠

Time

𝑇, 𝑇𝑝 𝑠

Period of vibration

𝑣0 𝑚𝑠 −1

Initial velocity

𝑥0 𝑚

Initial displacement

𝑥(𝑡) 𝑚

Displacement function in time domain, generic time domain signal

𝑥𝑏 (𝑡) 𝑚

Base motion function in time domain

𝑥𝑐 (𝑡), 𝑥𝑝 (𝑡), 𝑚

Particular and complementary solutions, respectively.

𝑦(𝑡)

Generic time domain signal

𝛽

Frequency ratio

𝛿(𝑡)

Dirac’s delta function in time domain

𝜃 𝑟𝑎𝑑

Angle

𝜆

Eigenvalue

𝜉, 𝜉𝑛 , 𝜉𝑟

Damping ratios

𝜏 𝑠

Time delay

𝜙

Mass-normalised Eigenvector

Φ

Modal matrix

𝜓

Eigenvector (mode shape vector)

Ψ

Mode shapes matrix

𝜌

Vector magnitude

𝜔 𝑟𝑎𝑑𝑠

−1

Circular frequency, eigenvalue

𝜔𝑛 𝑟𝑎𝑑𝑠

−1

Undamped natural circular frequency

𝜔𝑑 𝑟𝑎𝑑𝑠

−1

Damped natural circular frequency

Operators ( )∗

Complex conjugate

( ̇)

First order derivate

( ̈)

Second order derivate

| |

Vector norm

(∗)

Convolution

List of symbols

xxi

(⋆)

Correlation

∆

Increment

𝑑

Derivate

𝑑𝑒𝑡

Matrix determinant

𝐹{}, 𝐹 −1 {}

Fourier transform, inverse Fourier transform

𝐿{}, 𝐿−1 {}

Laplace transform, inverse Laplace transform

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List of acronyms Abaqus®

Abaqus® Non-linear FEA package

API

Application programming interface

CAD

Computer Aid Design

CSD

Cross-spectral density

DAQ

Data acquisition system

DOF

Degree of freedom

DFT

Discrete Fourier transform

FEA

Finite element analysis

FE

Finite elements

FEM

Finite element method

FFT

Fast Fourier transform

FRAC

Frequency response assurance criteria

FRF

Frequency response function

GUI

Graphical user interface

IDFT

Inverse discrete Fourier transform

IEEE

Professional association for the advancement of technology

IFFT

Inverse fast Fourier transform

IRF

Impulse response function

MAC

Modal assurance criteria

MDOF

Multiple degree of freedom

ODB

Output database

PSD

Power spectral density

RMSD

Root mean square deviation

SDA

Structural dynamic analysis

SDOF

Single degree of freedom

SHM

Structural heath monitoring

SPA

Signal Processing for Abaqus

TF

Transfer function

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Chapter 1 1 Introduction Nowadays, Structural Dynamics Analysis (SDA) is largely associated with computational mechanics, which has in the Finite Element Method (FEM) the most significant analysis tool. The application of computational mechanical procedures, such as FEM, to design complex structures, like bridges, cars, planes, satellites, space crafts, dams, tool machines, etc., is considered fundamental in modern engineering fields. FEM enables the access to information from analytical models, which is used to predict the dynamic behaviour of the analysed structure or component. In the SDA field, post-processing operations on dynamic signals are crucial for evaluating, more in detail, particular aspects such as resonance frequencies, punctual accelerations, vibration spectra, etc. In this way, is crucial to have tools that can be used to improve and optimize such signal processing operations, specially, tools embedded in the analysis framework. As stated by the IEEE Signal Processing Society, in their most recent constitution document (IEEE Signal Processing Society, 2012), “signal processing is the enabling technology for the

generation, transformation, extraction, and interpretation of information.”, and for Oppenheim et al. (1998), signal processing is concerned with the representation, transformation, and manipulation of signals and the information they contain. SDA, in particular, has benefited, and keep collecting profits, from the continuum improvement of the signal processing techniques. These technics are used by dynamicist analysts to improve the FEM capabilities, extending the method beyond the project phase. Examples of such extended application, are given by Viana et al. (2011), Kluska et al. (2012), Epameinondas et al. (2009), in works involving Finite Element (FE) models to assess damage in mechanical structures in the context of Structural Health Monitoring Techniques (SHM). The continuum increase of computational power together with the enhancement of the theories behind FEM development, led Khennane (2013) to write that FEM will continually increase its capability of recreate more accurate models from real structures, enabling the access to relevant and more precise information about the structural dynamic behaviour of real structures or components. By other side, the complexity of FE models tends to increase. This means that, with 1

2

Chapter 1. Introduction

the continuous increase in the complexity of modelled structures, large amounts of data will be available for post-processing operations. FE analysts, normally, use third-party software to execute dynamic signal processing operations compromising the interactivity with the FE model. Time spent in pre-process data files for posterior upload to external software’s is high and tends to increase the probability of analysis errors, compromising the ability to execute “on-the-fly” operations. Moreover, dedicated numerical computational tools for signal processing are very expensive and, most of the time, are not able to interact, directly, with the FE model.

1.1 Motivation Current structural dynamics analysis demands quick and reliable solutions for complex problems. Dynamic analysts have been using FEM to solve highly complex dynamic problems. They been comparing experimental and analytical data, using signal processing techniques, in order to predict and compare structural dynamic behaviour. In this field, FEM proves to be a truly effective and reliable method to solve dynamic problems, and for some authors (Clough and Penzien, 1993), has become the standard tool for structural dynamicists. Massive use of FEM led to a huge improvement on the FEM tools, which has in commercial finite-element codes some of the best applications to solve problems related with SDA. Commercial finite-element software became so advanced that even nonlinearity (material or geometrical), contact, structural interaction with fluids, metal forming, crash simulation, and several other problems, can be modelled and solved with extreme efficiency. The increasing reliability of FEM methods led many experts to relay on finite-element software to aid in the structural design, taking some authors (Khennane, 2013) to affirm that, nowadays, all but simple structural analysis, in structural design are carried out using FEM. The trend on the analysis of SDA is to include more dynamical insight on current problems, which generates temporal and frequency dependent information of the relevant physical variables. These resultant variables play the most important role on this procedure, since the main aim of the FEM it is to get information from the virtual model and compare/correlate them to the signals acquired from the real structure (Antunes et al., 2012). Although numerical methods can give reliable solutions for many complex problems, still necessary to use experimental tests to validate many of these analytical solutions (Meireles, 2007). Even so, FEM allows to test methodologies and theories, involving structural dynamic behaviour,

Chapter 1. Introduction

3

without the mandatory experimental trials avoiding the premature spend of time and money on experimental setups. Given the actual reliability of FEM software, dynamicists can generate several FE models, with different inputs (geometries, material constitutive modelling, loads, boundary conditions, etc.), collect outputs, and use signal processing techniques to obtain some conclusions before stepping to the experimental domain. Such use of FEM creates large amount of data to be handled, that is usually exported from the original FE software in order to be post-processed by a third-party software. However, to avoid deceiving conclusions or mismatch of experimental Vs analytical data, it is mandatory to guarantee a consistent manipulation of outputted results. This work intends to design and obtain a signal processing tool able to post-process Finite Element Analysis (FEA) in an embedded form, increasing the efficiency of dynamic analysis in the SDA field. This tool aims to be used by advanced FE analysts that use Abaqus® Non-linear FEA package (from now on, designated simply as Abaqus®), as platform for FEA. A fully embedded solution, capable of performing the most relevant signal processing operations, mathematical and statistical operations is the final goal of the present work.

1.2 Outline of the thesis In the chapter 2 are review some fundamentals on the Dynamic analysis theory. Signal processing is also briefly covered in this section, providing the basis for understanding, acquisition, processing and in interpretation of results. This chapter ends with the theoretical introduction to the problem, in which important fields of the SDA field, such modal analysis and modal testing are reviewed. Also, a quick reference to actual FEA software, responsible for the increasing efficiency of the FEM analysis, is made. Chapter 3 presents the proposed signal processing tool, together with a quick presentation of the developed tool. In chapter 4 are presented a set of case-studies that demonstrate some of the potentialities of the developed tool. Conclusions and future work are outlined in chapter 5.

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Chapter 2 2 State of the art This chapter covers theoretical and practical aspects associated with structural dynamic analysis. A brief review on the fundamentals of dynamic analysis theory is provided in section 2.1. By other hand, section 2.2 presents a high-level coverage on the signal processing subject. In section 2.3 is presented the theoretical introduction to the problem. Here, some practical applications of signal processing in the SDA field are reported. A review on the signal processing algorithms (those incorporated in the proposed signal processing tool) is made. A particular field, inside SDA, namely, modal analysis, is emphasised due to its importance in the dynamic analysis field. Finally, a section covering aspects associated with finite element analysis application and available software solutions is provided.

2.1 Dynamic analysis Authors like Girard and Roy (2008), Clough and Penzien (1993), and other, refer structural dynamics analysis as the study of structures subjected to a mechanical environment, which depends on time and leading to a movement. Structural dynamic problems, in opposition to its counterpart static-load, both load and responses varies in time. Therefore, is not possible to define internal forces (moments and shears) and resultant displacements based, simply, on the external load. When a force 𝑓(𝑡), as depicted in Figure 2.1, is dynamically applied, the introduced accelerations produces inertial forces, with opposite direction, that must be accounted, in addition with the external force, in order to equilibrate the internal forces at any time (𝑡). In this time-varying characteristic, the addition of inertia and presence of energy-loss mechanisms (damping), generates a considerably more complicated solution when compared with its static counterpart.

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Chapter 2. State of the art

Figure 2.1 Dynamic loading. (Source: Clough and Penzien (1993))

2.1.1 Single degree of freedom Vibrating systems can be classified by the number of degrees of freedom (DOF) of motion. The number of degrees of freedom is the number of independent coordinates needed to describe motion completely. Simplest vibrating systems are the single degree of freedom (SDOF) – some authors might referrer as 1-DOF – oscillators. For some classical examples, covered in several (SDA) textbooks, SDOF provide a good physical understanding of the behaviour of a vibration structure.

2.1.1.1 Forced vibration The simplest SDOF vibrating system, possessing energy-loss mechanism, is the mass-springdamper system, depicted in Figure 2.2. The presence of energy-loss mechanisms in SDOF is introduced using a viscous damping1 element, which is indicated by a dashpot, simply defined as damper. In SDOF mass-spring-damper system, the damping force is proportional to the velocity of the mass, but opposite to its motion.

Figure 2.2 Damped SDOF system disturbed by a force, 𝑓(𝑡).

Damped SDOF system contains a point mass 𝑚 attached to a rigid support through a linear massless spring with a stiffness 𝑘 (units: N m-1) and a viscous damper 𝑐 (units: kg s-1). In

Among the available damping models, viscous damping is most commonly used, and is the one that will be used. To get a proper coverage on damping models, see Braun et al. (2002). 1

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practice, the system can be disturbed either by a force, 𝑓(𝑡), acting on the mass or by a forced movement of the support, 𝑥𝑏 (𝑡) (Figure 2.3), usually referred as “base motion”. When a damped SDOF system, is disturbed by an external force, 𝑓(𝑡), the vertical displacement, 𝑥(𝑡), of the mass, measured from the equilibrium position, completely describe its motion, and according to Newton's second law, is governed by equation (2.1), 𝑓𝐼 (𝑡) + 𝑓𝐷 (𝑡) + 𝑓𝑆 (𝑡) = 𝑓(𝑡)

(2.1)

where 𝑓𝐼 (𝑡) = 𝑚𝑥̈ (𝑡) the inertial force, product of the mass and acceleration (𝑥̈ = 𝑑 2 𝑥 ⁄𝑑𝑡 2 ), 𝑓𝐷 (𝑡) = 𝑐𝑥̇ (𝑡) the damper force, product of the damping constant c and the velocity (𝑥̇ = 𝑑𝑥⁄𝑑𝑡), and 𝑓𝑆 (𝑡) = 𝑘𝑥(𝑡) the spring force (Clough and Penzien, 1993). So, motion of the damped SDOF system can be described by the linear differential equation: 𝑚𝑥̈ (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = 𝑓(𝑡)

(2.2)

When the damped SDOF system is disturbed by a base motion, 𝑥𝑏 (𝑡), as represented in Figure 2.3,

Figure 2.3 Damped SDOF system disturbed by a base motion, 𝑥𝑏 (𝑡).

the equilibrium of forces is written as 𝑓𝐼 (𝑡) + 𝑓𝐷 (𝑡) + 𝑓𝑆 (𝑡) = 0

(2.3)

where the damping and spring forces are represented as in equation (2.2), and the inertial force has the contribution of both displacements (mass and base motion), 𝑥𝑇 (𝑡) = 𝑥(𝑡) + 𝑥𝑏 (𝑡): 𝑓𝐼 (𝑡) = 𝑚𝑥̈ 𝑇 (𝑡) = 𝑚𝑥̈ (𝑡) + 𝑚𝑥̈ 𝑏 (𝑡)

(2.4)

so, the motion of the damped SDOF system can be described by the following linear differential equation:

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Chapter 2. State of the art

𝑚𝑥̈ (𝑡) + 𝑚𝑥̈ 𝑏 (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = 0

(2.5)

which can be rearranged to conveniently isolate the dynamic input given by the base motion: 𝑚𝑥̈ (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = −𝑚𝑥̈ 𝑏 (𝑡)

(2.6)

In this equation, the structural displacements originated by the base acceleration, can be interpreted as an disturbance provided by an external force 𝑝′ (𝑡) = −𝑚𝑥̈ 𝑏 (𝑡).

2.1.1.2 Free vibration To describe a vibrating system, besides the equation of motion, is necessary to define the initial and boundary conditions. When the initial state of a damped vibrating system provides potential or kinetic energy, due to initial displacements or initial velocities, the resultant vibration occurs without the application of external forces. This phenomenon is defined as “free vibration”. Equation (2.7) describes the free vibration of a SDOF, mass-spring-damper, system: 𝑚𝑥̈ (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = 0

(2.7)

The free vibration response, obtained as the solution of equation (2.7) may be expressed in the following form: 𝑥(𝑡) = 𝐴𝑒 𝜆𝑡

(2.8)

where 𝐴 is an arbitrary complex constant. Considering a simple harmonic motion to describe the vibrating motion (as the one described by equation (2.8), for instance), equation (2.7) can be transformed into characteristic equation (see Braun et al. (2002) for further highlighting): 𝑚𝜆2 + 𝑐𝜆 + 𝑘 = 0

(2.9)

The roots or eigenvalues of the characteristic equation (2.9) are: 𝜆1,2 = −

𝑐 𝑘 𝑐 2 ± 𝑖√ − ( ) 2𝑚 𝑚 2𝑚

(2.10)

which can be, conveniently, represented in terms of undamped natural frequency, 𝜔𝑛 = √𝑘⁄𝑚 , of the equivalent undamped vibrating system (units: 𝑟𝑎𝑑𝑠/𝑠), and damping ratio (fraction of critical damping), 𝜉 = 𝑐⁄2𝑚𝜔𝑛 :

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𝜆1,2 = −𝜉𝜔𝑛 ± 𝑖𝜔𝑑

(2.11)

where 𝜔𝑑 = 𝜔𝑛 √1 − 𝜉 2 is considered the damped natural frequency of the vibrating system, with radians per second as units. Each root has two parts: the real part or decay rate, which defines damping in the system; and the imaginary part, or oscillatory rate, which defines the damped natural frequency, 𝜔𝑑 . This free vibration response, with decaying amplitude, is called “undercritically-damped vibration” (Clough and Penzien, 1993). Making use of equation (2.8) and the eigenvalues values, 𝜆, given by equation (2.11), the free vibration response for an undercritically-damped system becomes: 𝑥(𝑡) = 𝑒 −𝜉𝜔𝑛𝑡 (𝐴1 cos 𝜔𝑑 𝑡 + 𝐴2 sin 𝜔𝑑 𝑡)

(2.12)

where 𝐴1 and 𝐴2 are determined by the initial conditions, 𝑥0 and 𝑣0 , initial displacement and velocity, respectively. Transforming equation (2.12) into: 𝑥(𝑡) = 𝑒 −𝜉𝜔𝑛𝑡 (𝑥0 cos 𝜔𝑑 𝑡 +

𝑣0 + 𝜉𝜔𝑛 𝑥0 sin 𝜔𝑑 𝑡) 𝜔𝑑

(2.13)

In Figure 2.4 the dotted curves indicate the decay in the amplitude of free vibration, which is controlled by the factor 𝜉, and the filled line indicate the oscillatory rate, which is controlled by 𝜔𝑑 .

Figure 2.4 Free vibration of an undercritically-damped SDOF system. (Source: Braun et al. (2002))

When 𝜉 = 1, the system decays without oscillation, and the motion is called “critically damped vibration”. This means that, in practice, normal oscillatory systems have the damping ratio defined between: 0 < 𝜉 < 1 (see Clough and Penzien (1993) to have a full coverage on damped systems subject).

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2.1.1.3 Response to arbitrary loading Randall (1987) wrote that the formulation of the response of a linear physical system get simpler when it is formulated in terms of the response itself. In practice, a forced vibration often consists in a two parts motion: transient response, which disappears after a period of time, and a steadystate response, which remains after the transient response has disappeared. Thus, the response will present the following solution: 𝑥(𝑡) = 𝑥𝑐 (𝑡) + 𝑥𝑝 (𝑡)

(2.14)

where 𝑥𝑐 (𝑡) is the complementary solution, corresponding to the homogeneous equation (2.13), and 𝑥𝑝 (𝑡) is the particular solution, referring to the steady-state response. While determining the response due to loads that are of longer duration, the complementary solution is often ignored and emphasis is placed only on the particular solution. Since the solution to the homogeneous equation is the same for any load case, and is already depicted above, only particular solutions are represented next.

Harmonic loading When a given undercritically-damped SDOF system, is subjected to a harmonically varying load 𝑓(𝑡), described by a sine-wave with an amplitude 𝜌𝑓 , the equation (2.2) is transformed into: 𝑚𝑥̈ (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = 𝜌𝑓 sin 𝜔𝑡

(2.15)

Since the excitation force is harmonically varying, the steady-state displacement of the mass is assumed to be sinusoidal with the same frequency, 𝜔, and a magnitude 𝜌𝑥𝑝 : 𝑥𝑝 (𝑡) = 𝜌𝑥𝑝 sin(𝜔𝑡 − 𝜃)

(2.16)

Note that this response includes a phase lag, 𝜃, caused by the damping in the system. Differentiating twice to derive the velocity and acceleration, respectively, and substituting on equation (2.15), is possible to solve the equation by considering the contribution of each structural component as a vector with magnitude and phase components:

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11

Figure 2.5 Vector diagram of forces.2

Using the structural elements from Figure 2.5 is possible to relate the applied force in order to get the system response: 𝜌𝑓 = √(𝑘𝜌𝑥𝑝 − 𝑚𝜔 2 𝜌𝑥𝑝 )2 + (𝑐𝜔𝜌𝑥𝑝 )2

(2.17)

Which lead to: 𝜌𝑥𝑝 𝜌𝑓

=

1 √(𝑘 − 𝑚𝜔 2 )2 + (𝑐𝜔)2

(2.18)

and: 𝑐𝜔 ) 𝑘 − 𝑚𝜔 2

𝜃 = tan−1 (

(2.19)

This complex expression, with magnitude, 𝜌𝑥𝑝 ⁄𝜌𝑓 , and phase angle, 𝜃, is called FRF-Frequency Response Function, and is referred again, for a detailed review, further on this chapter. The amplitude of the steady-state response depends on the excitation frequency, 𝜔, and damping ratio, 𝜉. Figure 2.6 shows that the amplitude of the steady-state response approaches a maximum value as the excitation is near the undamped natural frequency, 𝜔𝑛 . Moreover, the smaller the damping ratio, the higher the peak.

2

Based on the original scheme from Braun et al. (2002).

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Chapter 2. State of the art

Figure 2.6 FRF magnitude and phase angle varying with the damping ratio, for a given undercriticallydamped SDOF system.

Periodic loading In Figure 2.7, a general periodic loading function 𝑓(𝑡) repeats itself in a fixed time period 𝑇, such that 𝑓(𝑡 + 𝑇) = 𝑓(𝑡) for all values of 𝑡.

Figure 2.7 Arbitrary periodic loading. (Source: Clough and Penzien (1993))

This arbitrary periodic loading function can be expanded in an infinite series of sinusoidal functions, called the Fourier series: ∞

𝑓(𝑡) = 𝑎0 + ∑(𝑎𝑛 cos 𝑛𝜔𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑡)

(2.20)

𝑛=1

where 𝜔 = 2𝜋⁄𝑇 and the Fourier coefficients 𝑎0 , 𝑎𝑛 and 𝑏𝑛 , are calculated by: 𝑇

1 𝑎0 = ∫ 𝑓(𝑡) 𝑑𝑡 𝑇

(2.21)

0

𝑇

2 𝑎𝑛 = ∫ 𝑓(𝑡) cos 𝑛𝜔𝑡 𝑑𝑡 , 𝑇

𝑛 = 1, 2, 3, …

(2.22)

𝑛 = 1, 2, 3, …

(2.23)

0

𝑇

2 𝑏𝑛 = ∫ 𝑓(𝑡) cos 𝑛𝜔𝑡 𝑑𝑡 , 𝑇 0

Chapter 2. State of the art

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For this case, the particular solution of equation (2.2) to a periodic force can be interpreted as a sequence of 𝑛 harmonic solutions, given by equation (2.16). I.e., the particular solution for equation: 𝑚𝑥̈ (𝑡) + 𝑐𝑥̇ (𝑡) + 𝑘𝑥(𝑡) = 𝑎𝑛 cos 𝑛𝜔𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑡

(2.24)

can be given by: ∞

𝑎0 𝑥𝑝 (𝑡) = + ∑ 𝜌𝑛 sin(𝑛𝜔𝑡 − 𝜃𝑛 ) , 2𝑘

𝑛 = 1, 2, 3, …

(2.25)

𝑛=1

where 𝜌𝑛 = |𝑥𝑝 |𝑛 , representing the magnitude of the vector 𝑥𝑝 , which can be found in equation (2.18), and 𝜃𝑛 represents the phase angle, for each 𝑛 (for a full cover on this subject see Clough and Penzien (1993)).

Impulsive loading The time history of an impulsive load in general is difficult to measure. However, its temporal effect can be quantified. An impulsive load consists in a single impulse of arbitrary form, as illustrated in Figure 2.8, where the duration 𝑡0 is arbitrary small, such that 𝑡0 = 0+ .

Figure 2.8 Arbitrary impulsive loading.

Once 𝑓(𝑡) = 0 for 𝑡 > 𝑡0 the impulse-excited motion is equivalent to the free vibration problem, and is solved using equation (2.7) for 𝑡 > 𝑡0 , with initial conditions, 𝑥(𝑡0 ) and 𝑥̇ (𝑡0 ),deduced from the impulse-momentum principle deduced from Newton's second law, 𝑡0

𝐼0 = ∫ 𝑓(𝑡)𝑑𝑡

(2.26)

0

respectively, 𝑥(𝑡0 ) = 𝑥0 ,

𝑥̇ (𝑡0 ) = 𝑣0 +

𝐼0 𝑚

(2.27)

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Therefore, for an undercritically-damped system, initially at rest, the impulse response is: 𝑥(𝑡) =

𝐼0 −𝜉𝜔 𝑡 𝑛 sin 𝜔 𝑡 𝑒 𝑑 𝑚𝜔𝑑

(2.28)

(Note: See Braun et al. (2002) to further highlights on this matter.) For the special case, when the considered impulsive load, 𝑓(𝑡), results into a unitary impulse-momentum, 𝐼0 , the equation (2.28) returns the IRF-Impulse Response Function, ℎ(𝑡): ℎ(𝑡) =

1 −𝜉𝜔 𝑡 𝑛 sin 𝜔 𝑡 𝑒 𝑑 𝑚𝜔𝑑

(2.29)

The IRF represents the unique characteristic of the system/structure, therefore very useful in the dynamical structural analysis field and can be traduced by: 𝑓(𝑡) → [𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑦𝑠𝑡𝑒𝑚, ℎ(𝑡)] → 𝑥(𝑡) In this case, the 𝑓(𝑡) function, can be considered the Dirac's delta function, 𝛿(𝑡), with the following properties: ∞

∫ 𝛿(𝑡 − 𝜏) 𝑑𝑡 = 1

(2.30)

−∞ ∞

∫ 𝑔(𝑡)𝛿(𝑡 − 𝜏) 𝑑𝑡 = 𝑔(𝑡)

(2.31)

−∞

such that:

Figure 2.9 Impulse response function for an arbitrary undercritically-damped SDOF system.

where 𝜏 represents a time delay. (Note: See Braun et al. (2002) to get a full cover on Dirac’s delta function and its properties.)

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2.1.1.4 Domain transformation Since in most practical cases, the interest is on finding the response of the system to an external excitation. The deductive approach used to define the impulse response function derived in equation (2.29) can be used to obtain the response of any system perturbed by an external excitation. For a given loading, 𝑓(𝑡), as represented in Figure 2.10, and considering a portion of the excitation load, 𝑓(𝜏), acting at time 𝑡 = 𝜏, and for a very short time, 𝑑𝜏, as being very shortduration impulse, 𝑓(𝜏) 𝑑𝜏, on the structure, is possible to use the equation (2.29) to obtain the response. Notice that, this equation is exact for impulse with duration 𝑡 = 0+ , resulting in approximate results for impulses with finite duration.

Figure 2.10 Derivation of the Duhamel's integral.3

The 𝑥𝑝 (𝑡) part of the solution, which does not have any arbitrary constants, can be obtained using the convolution (Duhamel's) integral: 𝑡

𝑥𝑝 (𝑡) = (𝑓 ∗ ℎ)(𝑡) = ∫ 𝑓(𝜏)ℎ(𝑡 − 𝜏) 𝑑𝜏

(2.32)

0

From where results the impulse response 𝑑𝑥(𝑡), derived from equation (2.29), replacing the 𝑡 variable for (𝑡 − 𝜏): 𝑑𝑥(𝑡) =

𝑓(𝜏) 𝑑𝜏 −𝜉𝜔 (𝑡−𝜏) 𝑛 𝑒 sin 𝜔𝑑 (𝑡 − 𝜏) , 𝑚𝜔𝑑

𝑡≥𝜏

(2.33)

which is used to obtain the system response: Adaptation from the original, of Clough and Penzien (1993) textbook used to undamped systems, to the undercritically-damped system example. 3

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Chapter 2. State of the art 𝑡

1 𝑥𝑝 (𝑡) = ∫ 𝑓(𝜏) 𝑒 −𝜉𝜔𝑛(𝑡−𝜏) sin 𝜔𝑑 (𝑡 − 𝜏) 𝑑𝜏 , 𝑚𝜔𝑑

𝑡≥0

(2.34)

0

knowing that ℎ(𝑡 − 𝜏) is represented by: ℎ(𝑡 − 𝜏) =

1 −𝜉𝜔 (𝑡−𝜏) 𝑛 𝑒 sin 𝜔𝑑 (𝑡 − 𝜏) 𝑚𝜔𝑑

(2.35)

However, solving the integral from equation (2.32), in time domain, can be a difficult task, especially when the force function 𝑓(𝑡) is not a simple function. To overcome this fact, numerical and transform methods, Laplace and Fourier, here represented by 𝐿{} and 𝐹{}, respectively, are used in such cases (Note: in following operations, the upper case letters are used for representation in the frequency domain, and lower case letters for representation of time domain). Using domain transformation methods is possible to transform the above integral into an algebraic product which is easier to compute. Then, the solution in the time domain is obtained by performing the inverse transform (frequency to time domain transformation).

Laplace transform Applying the Laplace transform to the equation (2.2) results on equation, (𝑚𝑠 2 + 𝑐𝑠 + 𝑘)𝑋(𝑠) = 𝐹(𝑠)

(2.36)

where 𝑋(𝑠) and 𝐹(𝑠) are the Laplace transform (see Annex A1 to get a quick review on Laplace transforms) of 𝐿{𝑥(𝑡)} and 𝐿{𝑓(𝑡)}, respectively, in terms of the Laplace variable 𝑠. Simplifying equation (2.36), and reorganizing it in order to get the response, is obtained: 𝑋(𝑠) = 𝐻(𝑠)𝐹(𝑠)

(2.37)

where 𝐻(𝑠), represents the TF-Transfer Function, more precisely the “compliance” and is given by: 𝐻(𝑠) =

𝑋(𝑠) 1 = 2 𝐹(𝑠) (𝑚𝑠 + 𝑐𝑠 + 𝑘)

(2.38)

For the undercritically-damped case, the two complex roots (or “poles”) for the complementary solution of equation (2.36) are given by: 𝑠1,2 = −

𝑐 𝑘 𝑐 2 ± 𝑖√ − ( ) 2𝑚 𝑚 2𝑚

(2.39)

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Thus, in this case, the system response, 𝐻(𝑠), in the Laplace domain, can be written, evidencing the 𝜔𝑑 term, as: 𝐻(𝑠) =

−𝑖 1 1 [ − ] 2𝑚𝜔𝑑 (𝑠 − 𝑠1 ) (𝑠 − 𝑠2 )

(2.40)

Using the inverse function is possible to get the 𝐻(𝑠) in the time domain, represented by: ℎ(𝑡) =

−𝑖 −1 1 1 𝐿 { − }, (𝑠 − 𝑠1 ) (𝑠 − 𝑠2 ) 2𝑚𝜔𝑑

𝑡>0

(2.41)

Recalling that 𝐿−1 {1⁄(𝑠 − 𝑎)} = 𝑒 𝑎𝑡 , ℎ(𝑡) becomes: ℎ(𝑡) =

1 −𝜉𝜔 𝑡 𝑛 sin 𝜔 𝑡 , 𝑒 𝑑 𝑚𝜔𝑑

𝑡>0

(2.42)

This response, deduced using the Laplace transformation, is the same obtained in equation (2.35), with 𝑡 = (𝑡 − 𝜏), proving the claimed equivalency of the results, using transform and time domain (to further understanding on this subject see Randall (1987) and Braun et al. (2002) textbooks). Figure 2.11 shows a 3-D representation of the magnitude and phase of the TF, 𝐻(𝑠), for an undercritically-damped system depicting the location of the poles in the complex 𝑠-plane, highlighting the intersection with the imaginary axis, 𝑖𝜔. This intersection represents the FRF, 𝐻(𝑖𝜔).

Figure 2.11 3-D plots representing the transfer function magnitude, |𝐻| (left), and phase angle, 𝜃 (right), in the 𝑠-plane, highlighting the frequency response function, represented by the dashed line.

Fourier transform Fourier transforms are considered as special cases of the Laplace transforms, when 𝑠 = 𝑖𝜔. It is because of its ability to use the power of modern computers that the use of Fourier transform has clearly overcame the Laplace transform for determining the system response.

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Due to the importance of Fourier transform, for practical applications, besides the extended review presented in the Annex A2, it is present next, in this section, the formulas for the (direct) transform: ∞

𝑋(𝑖𝜔) = 𝐹{𝑥(𝑡)} ∫ 𝑥(𝑡)𝑒 −𝑖𝜔𝑡 𝑑𝑡

(2.43)

−∞

which being an even, “two-sided”, function of frequency is usually represented in the format of “one-sided” spectrum using the following conditions (Randall, 1987): 𝑋(𝑖𝜔) = 2𝑋(𝑖𝜔)

𝑖𝜔 > 0

= 𝑋𝑘 (𝑖𝜔)

𝑖𝜔 = 0

=0

𝑖𝜔 < 0

The inverse in given by: ∞

𝑥(𝑡) = 𝐹 −1 {𝑋(𝑖𝜔)} = ∫ 𝑋(𝑖𝜔)𝑒 𝑖𝜔𝑡 𝑑𝑓

(2.44)

−∞

Note that for practical considerations, from this point, the Fourier transform will be represented using the 𝜔 variable instead of 𝑖𝜔. Solving the equation (2.32) using Fourier transform: 𝐹{𝑥(𝑡)} = 𝐹{(𝑓 ∗ ℎ)(𝑡)} = 𝐹(𝜔) × 𝐻(𝜔)

(2.45)

(Note: To see other properties of the Fourier transform see Annex A2) Returns a system response, 𝐻(𝜔), given by: 𝐻(𝜔) =

𝑋(𝜔) 1 = 𝐹(𝜔) 𝑚(𝜔𝑛2 + 2𝑖𝜉𝜔𝜔𝑛 − 𝜔 2 )

(2.46)

It is interesting to note that FRF is the same as the steady-state response of the system to a harmonic exponential excitation, of unit amplitude, and excitation frequency 𝜔 (i.e. 𝑒 𝑖𝜔𝑡 ). This response, is given by: 𝑥𝑠𝑠 (𝜔) = 𝐻(𝜔)𝑒 𝑖𝜔𝑡 = |𝐻(𝜔)|𝑒 𝑖(𝜔𝑡−𝜃) =

1 1 𝑒 𝑖(𝜔𝑡−𝜃) , 𝑘 √(1 − 𝛽 2 )2 + (2𝜉𝛽)2

2𝜉𝛽 𝜃 = tan−1 ( ) 1 − 𝛽2

(2.47)

where 𝛽 = 𝜔⁄𝜔𝑛 . Using the inverse of Fourier transform is possible to obtain the system response in time domain, 𝑥(𝑡), that is equivalent to the ℎ(𝑡), from equation (2.35), with 𝑡 = (𝑡 − 𝜏):

(2

Chapter 2. State of the art

19 ∞

ℎ(𝑡) = 𝐹

−1 {𝐻(𝜔)}

= ∫ −∞

1 𝑒 𝑖𝜔𝑡 𝑑𝜔 2 2 ) 𝑚(𝜔𝑛 + 2𝑖𝜉𝜔𝜔𝑛 − 𝜔

1 −𝜉𝜔 𝑡 𝑛 sin 𝜔 𝑡 = 𝑒 𝑑 𝑚𝜔𝑑

(2.48)

The equivalence is demonstrated using an analytical approach that use the so-called Cauchy relation (or contour integral) to perform the improper integral (see Braun et al. (2002) textbook to further information).

Half-power bandwidth The reader might notice that the response given by equation (2.47) is a recover of the response given by equations (2.18) and (2.19). Let take the change to introduce a new concept regarding the analysis of the FRF amplitude curve, as the one schematised in Figure 2.12 , used to obtain the damping ratio, 𝜉.

Figure 2.12 FRF amplitude, highlighting the resonance, for an arbitrary undercritically-damped SDOF system. (Source: Clough and Penzien (1993))

There are several methods, based on different properties of the curve in Figure 2.12, which can be used to deduce the damping ratio. However, the most popular method is the “half-power bandwidth” method (or band-width method, or even -3db method) whereby the damping ratio is

20

Chapter 2. State of the art

determined, at a resonance4 peak, from the frequency at which the response amplitude 𝜌 is reduced to the level 𝜌𝑚𝑎𝑥 ⁄√2. Deduced from Figure 2.12, is possible to obtain the following relation, for small values of damping (Clough and Penzien 1993): 𝛽2 − 𝛽1 𝛽2 + 𝛽1

𝜉=

(2.49)

Since the 𝛽 = 𝜔⁄𝜔𝑛 , and 𝜔 = 2𝜋𝑓, is possible to obtain the damping ratio in order to the frequency, 𝑓, which is usually more convenient for practical cases: 𝜉=

𝑓2 − 𝑓1 𝑓2 + 𝑓1

(2.50)

Power spectral density Whenever a vibrating system is excited by random force(s), the response is also random. One of the methods available to obtain the system response is the direct use of the basic excitation/response relation for PSD-Power Spectral Density. Whenever the response is stationary, its PSD, represented by 𝑆𝑋𝑋 (𝜔) can be predicted in terms of the excitation PSD, represented by 𝑆𝐹𝐹 (𝜔), and relevant complex frequency response H, using the basic relation: 𝑆𝑋𝑋 (𝜔) = |𝐻 2 (𝜔)|𝑆𝐹𝐹 (𝜔)

(2.51)

Considering an arbitrary waveform, 𝑥(𝑡), to represent a stationary random process, which is represented only over a finite interval, − 2⁄𝑠 < 𝑡 < 2⁄𝑠, its PSD (𝑆𝐹𝐹 (𝜔)) can be deduced from: ∞

|𝑥(𝑡)2 | = ∫ 𝑆𝑋𝑋 (𝜔) 𝑑𝜔

(2.52)

−∞

where the function: 2

2⁄𝑠

𝑆𝑋𝑋 (𝜔) ≡ lim

𝑠→∞

|∫−2⁄𝑠 𝑥(𝑡) 𝑒 −𝑖𝜔𝑡 𝑑𝑡| 2𝜋𝑠

(2.53)

Resonance is the condition that results the maximum steady-state response amplitude, assumed to occur when the frequency of the applied loading equals the undamped natural vibration frequency. Knowing that for low values of damping the maximum steady-state response amplitude occurs at a frequency ratio slightly less than unity. 4

Chapter 2. State of the art

21

Clough and Penzien (1993), as demonstrated in his textbook, state that if the stationary process being considered is ergodic5 the 𝑆𝑋𝑋 (𝜔) function can be obtained, from the autocorrelation function 𝑅𝑥𝑥 (𝜏), using the Fourier transform: ∞

1 𝑆𝑋𝑋 (𝜔) = ∫ 𝑅𝑥𝑥 (𝜏)𝑒 −𝑖𝜔𝜏 𝑑𝜏 2𝜋

(2.54)

−∞

∞

𝑅𝑥𝑥 (𝜏) = ∫ 𝑆𝑥𝑥 (𝜔)𝑒 𝑖𝜔𝜏 𝑑𝜔

(2.55)

−∞

(Note: To properly check how the presented equations are deducted see the following textbooks: Clough and Penzien (1993); Braun et al. (2002)) Autocorrelation function, 𝑅𝑥𝑥 (𝜏), in its turn, it is a special case of the cross-correlation function, 𝑅𝑥𝑦 (𝜏) where two signals, 𝑥(𝑡) and 𝑦(𝑡), are correlated using the following equation: 𝑇

1 𝑅𝑥𝑦 (𝜏) = (𝑥 ⋆ 𝑦)(𝜏) = ∫ 𝑥(𝑡)𝑦(𝑡 + 𝜏) 𝑑𝑡 𝑇

(2.56)

0

Figure 2.13 shows schematic representations of autocorrelation, 𝑅𝑥𝑥 (𝜏), crosscorrelation, 𝑅𝑥𝑦 (𝜏), at the same time, highlighting the “similarities” with convolution function, previously referred.

Figure 2.13 Schematic representation of Convolution, Cross-correlation and Autocorrelation operations. (Source: wikipedia.org)

Transforming 𝑅𝑥𝑦 (𝜏) using the Fourier transform, similarly to what is done for 𝑅𝑥𝑥 (𝜏) in equation (2.54) will result in an important physical quantity in the field of dynamic analysis named Cross-Spectral Density (CSD), represented by

A process is defined as ergodic when any average obtained with respect to time 𝑡 along any member 𝑟 of the ensemble is exactly equal to the corresponding average across the ensemble at an arbitrary time 𝑡 (Clough and Penzien 1993). 5

22

Chapter 2. State of the art

𝑆𝑋𝑌 (𝜔) = 𝐹{𝑅𝑥𝑥 (𝜏)} ∞

= ∫ 𝑅𝑥𝑦 (𝜏)𝑒 −𝑖𝜔𝜏 𝑑𝜏 −∞ ∞

∞

= ∫ ∫ 𝑥(𝑡)𝑦(𝑡 + 𝜏)𝑒 −𝑖𝜔𝜏 𝑑𝜏𝑑𝑡

(let 𝑡1 = 𝑡 + 𝜏)

−∞ −∞ ∞

∞

= ∫ 𝑥(𝑡)𝑒 𝑖𝜔𝑡 𝑑𝑡 ∫ 𝑦(𝑡1 )𝑒 −𝑖𝜔𝑡1 𝑑𝑡1 −∞

−∞

= 𝑋 ∗ (𝜔)𝑌(𝜔)

(2.57)

Where 𝑋 ∗ (𝜔) is the complex conjugate (𝑋(−𝑖𝜔) = 𝑋 ∗ (𝑖𝜔)). Note that the function 𝑆𝑋𝑌 (𝜔) result in the two-sided cross spectrum. Therefore, the same approach used for the Fourier transform, in equation (2.43), is used to obtain the one-sided spectrum: 𝐺𝑋𝑌 (𝜔) = 2𝑆𝑋𝑌 (𝜔)

𝜔>0

= 𝑆𝑋𝑌 (𝜔)

𝜔=0

=0

𝜔0

= 𝑋(𝑘)

𝑓=0

=0

𝑓0

1 |𝑋(𝑘)|2 = ∙ , ∆𝑓 𝑁2

𝑘 = 0, 1, … , 𝑁 − 1

𝑓=0

𝐺𝑋𝑋 (𝑘) = 2

=0

(2.71)

𝑓0

1 𝑋 ∗ (𝑘)𝑌(𝑘) , 𝑘 = 0, 1, … , 𝑁 − 1 ∆𝑓 𝑁2

𝑓=0

𝐺𝑋𝑌 (𝑘) = 2 =

=0

(2.72)

𝑓