Structural Dynamics in Engineering Design E-Book

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Veröffentlichungsjahr: 2024

Beschreibung

STRUCTURAL DYNAMICS IN ENGINEERING DESIGN

World-class authors describe and illustrate how structural dynamics is applied to the engineering design process

Structural Dynamics in Engineering Design covers the fundamentals of structural dynamics and its application to the engineering design process, providing all of the necessary information to implement an optimal design process. Each of its seven chapters is written by an expert in the field and provides the reader with the structural dynamic theoretical background and its more practical aspects for the implementation of an advanced design capability.

The first three chapters are dedicated to the underlying theory of the three main processes: the fundamentals of vibration theory, the basis of experimental dynamics and the main numerical analysis tools (including reference to the finite element method). Having laid the foundation of the design philosophy, the following three chapters present the reader with the three disciplines of identification, nonlinear analysis and validation/updating. The final chapter presents some applications of the approach to real and complex engineering cases.

Key features:

• Takes a multi-disciplinary approach and contains critical information on theory, testing and numerical analysis for structural dynamics.
• Includes a chapter on industrial applications (including aircraft design and ground vibration testing), which illustrates the design process and explains how structural dynamics is applied at different stages.

The book is a must-have for researchers and practitioners in mechanical and aerospace engineering (in particular test engineers, CAE analysts and structural dynamicists), as well as graduate students in mechanical and aerospace engineering departments.

Leseprobe

STRUCTURAL DYNAMICS IN ENGINEERING DESIGN

Nuno M. M. MaiaDario Di MaioAlex CarrellaFrancesco MaruloChaoping ZangJonathan E. CooperKeith WordenTiago A. N. Silva

This edition first published 2024

© 2024 John Wiley & Sons Ltd

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The right of Nuno Maia, Dario Di Maio and Alex Carrella to be identified as the authors of the editorial material in this work has been asserted in accordance with law

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Library of Congress Cataloging-in-Publication Data

Hardback: 9781118770702

ePDF: 9781118770672

epub: 9781118770689

oBook: 9781118770696

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Cover design by Wiley

Set in 12/14pt TimesNewRomanPSMT by Integra Software Services Pvt. Ltd, Pondicherry, India

to all who use structural dynamics to understand and engineer the world around us

Contents

Cover

Title Page

Copyright Page

Dedication

Acronyms

Preface

List of Authors

About the Companion Website

Chapter 1 Theoretical Background

1.1 Introduction

1.2 Fundamental Concepts

1.2.1 Types of Signals

1.2.2 Degrees-of-Freedom/Discretisation

1.2.3 Elements of a Vibrating System

The Spring Element

The Damper Element

The Mass Element

The Torsional Vibration Model

1.2.4 The Simple Harmonic Motion

1.3 Establishing the Dynamic Equilibrium Equations

1.3.1 The Dynamic Equilibrium Equations based on Vectorial Mechanics

1.3.2 The Dynamic Equilibrium Equations based on Analytical Mechanics

The Principle of Conservation of Energy

The Generalisation of the Principle of Conservation of Energy

The Principle of Virtual Work

Hamilton’s Principle

Lagrange Equations

1.4 The Single Degree-of-Freedom System

1.4.1 The Dynamic Equilibrium Equation

1.4.2 Equivalent Systems

1.4.3 Undamped Free Vibration Response

1.4.4 Viscously Damped Free Vibration Response

1.4.5 Forced Vibration

Response due to a Harmonic Force of Constant Amplitude

Response due to a Harmonic Force of Variable Amplitude

Response due to an Imposed Harmonic Displacement

Transmissibility of Motion

Transmissibility of Forces

Response due to a Harmonic Force, with Hysteretic Damping

Response to a Periodic Excitation

Response to a Non-Periodic Excitation

Response to a Random Excitation

1.5 Discrete Systems with Multiple Degrees-of-Freedom

1.5.1 The Dynamic Equilibrium Equation with Viscous Damping

1.5.2 Undamped Vibration Response

Natural Frequencies and Mode Shapes

Free Response

Transformation of Coordinates

Forced Response

Orthogonality Properties and Normalisation of the Mode Shapes

Undamped Free Response using the Orthogonality Properties

1.5.3 Viscously Damped Vibration Response

Natural Frequencies and Mode Shapes

Free Response

Forced Response

1.5.4 Hysteretically Damped Vibration Response

Natural Frequencies and Mode Shapes

Forced Response

1.5.5 Graphical Representation of an

FRF

1.6 Continuous Systems

1.6.1 Free Vibration of Uniform Bars

Solving the Equilibrium Equation

Natural Frequencies and Mode Shapes

Free Response

1.6.2 Free Vibration of Uniform Beams

Solving the Equilibrium Equation

Natural Frequencies and Mode Shapes

Free Response

Bibliography

Chapter 2 Vibration Testing and Analysis

2.1 Introduction

2.2 Test Set-up

2.3 Fundamentals of Data Acquisition

2.4 Understanding and Analysing Measured Data

2.4.1 Validating Experimental Measurements: the Coherence Function

2.4.2 Time-Response to Harmonic, Stepped-Sine, Excitation

2.4.3 Other Types of Excitation

Sine Sweep

Multi-Sine Sweep

Chirp

Random, Pseudo-Random and Periodic Random

2.4.4 A Different Representation: the Lissajous Curves

2.4.5 Hammer Testing

versus

Shaker Testing

2.5 Tips and Tricks for Dynamical Data Analysis

2.5.1 Time Domain Analysis

Acceleration

versus

Time

Average Acceleration in a Time Interval

versus

Time

Root-Mean-Square Acceleration in a Time Interval

versus

Time

Trim Mean Acceleration

versus

Time

Quasi-steady Three-dimensional Histogram

Examples of Analysis in Time Domain

2.5.2 Frequency Domain Analysis

Power Spectral Density

versus

Frequency

Cumulative RMS Acceleration

versus

Frequency

Root-Mean-Square Acceleration

versus

Frequency

One Third Octave Band RMS Acceleration

versus

Frequency

Power Spectral Density

versus

Frequency and Time (Spectrogram)

Examples of Analysis in the Frequency Domain

References

Further Reading

Chapter 3 Numerical Methods

3.1 Introduction

3.2 Approximation Methods in Time Domain

3.2.1 Central Difference Method

3.2.2 Runge-Kutta’s Method

3.2.3 Houbolt’s Method

3.2.4 Wilson’s Method

3.2.5 The Newmark-

β

Method

3.2.6 Numerical Case Comparison:

SDOF

System

3.3 Approximation Methods for Natural Frequencies

3.3.1 Dunkerley’s Method

3.3.2 Rayleigh’s Method

3.3.3 Ritz’ Method

3.3.4 Holzer’s Method

3.4 Matrix Methods

3.4.1 Bisection Method

3.4.2 Sturm Sequences Method

3.4.3

MATLAB ROOTS

Method

3.4.4 Cholesky’s Decomposition

3.4.5 Matrix Iteration Method

3.4.6 Jacobi’s Method

3.4.7 Singular Value Decomposition

3.4.8 Principal Component Analysis

3.5 The Finite Element Method

3.5.1 Basic Idea of the

FEM

3.5.2 General Procedure for Finite Element Analysis

3.5.3 Bars and Trusses

3.5.4 Beams

Bibliography

Chapter 4 Linear System Identification

4.1 Introduction

4.1.1 Outline of the Chapter

4.1.2 Assumptions about the Measured Data

4.1.3 Categories of System Identification Methods

Phase Separation

versus

Phase Resonance Methods

Phase Separation Methods

4.1.4 Mathematical Models

Frequency Domain Models

Relationship between the Time and Frequency Domains

Time Domain Models

4.1.5 Example: the Wing-Pylon Model

Initial Interpretation of the

FRFs

Considerations on the Impulse Response Functions

4.2 System Identification Methods

4.2.1 Time Domain or Frequency Domain?

4.2.2 Single Degree of Freedom – Frequency Domain

Peak Picking / Half-Power Points

Circle Fitting

Inverse Fit Method

4.2.3 Single Degree of Freedom – Time Domain

Logarithmic Decrement

4.2.4 Effect of Multiple Modes – When can

SDOF

Methods not be used?

4.3

MDOF

Frequency Domain Methods

4.3.1 Nonlinear

FRF

Curve-Fit

4.3.2 Rational Fraction Polynomial Method

Pole Identification

4.3.3 Stability Plots – How Many Modes are there?

4.3.4 Mode Shape Estimation – Least-Squares Frequency Domain

4.3.5

PolyMAX

4.4

MDOF

Time Domain Methods

4.4.1 Extended Logarithmic Decrement Method

Natural Frequency Estimation

Damping and Amplitude Estimates

4.4.2 Least-Squares Complex Exponential

Step 1 - Estimation of

AR

Coefficients, Natural Frequencies and Damping Ratios

Step 2 – Determination of the System Poles using Stability Plots

Step 3 – Determination of the Mode Shapes

4.4.3 Polyreference Method

4.4.4 Eigensystem Realisation Algorithm

4.4.5 Reverse Data Fitting

4.5 Ambient Excitation – Operational Modal Analysis

4.5.1 Ambient Analysis – Frequency Domain

Frequency Domain Decomposition Method

Enhanced Frequency Domain Decomposition Method

4.5.2 Ambient Analysis – Time Domain

Random Decrement Method

Generation of Impulse Response Functions from

PSDs

and

CSDs

Generation of Impulse Response Functions from Auto and Cross Correlations

Eigensystem Realisation Algorithm using Data Correlations

4.6 Phase Resonance (Normal Modes / Force Appropriation) Testing

4.6.1 Square

FRF

Matrices

Asher’s Method

Modified Asher’s Method

Traill-Nash Method

4.6.2 Rectangular

FRF

Matrices

Extended Asher’s Method

Multivariate Mode Indicator Function

Normal Mode Purity Function

Application to Wing-Pylon Data

4.6.3 Rank Reduction Force Appropriation Methods

Modified Multivariate Mode Indicator Function

SVD

Multipoint Excitation Method

4.7 Overall Approach for Linear System Identification

References

Chapter 5 Nonlinearity in Engineering Dynamics

5.1 The Significance of Nonlinearity

5.1.1 Nonlinearity in Fundamental Physics

5.1.2 Nonlinearity in Epidemiology

5.1.3 Nonlinearity in Meteorology

5.1.4 Nonlinearity in Structural Dynamics

5.2 Solution of Nonlinear Equations of Motion

5.2.1 *Exact Solutions

5.2.2 Approximate Solutions: Perturbation Theory

5.2.3 Numerical Solutions: Simulation

The Euler Method

Runge-Kutta’s Methods

Simulating Nonlinear Systems

5.2.4 Qualitative Solutions: The Phase Plane

5.3 Signatures of Nonlinearity

5.3.1 Definition of Linearity: the Principle of Superposition

5.3.2 Harmonic Distortion

5.3.3 Homogeneity and

FRF

Distortion

5.3.4 Reciprocity

5.3.5 The Coherence Function

5.3.6 Nonlinearity in the Measurement Chain

5.4 Common Types of Nonlinearity

5.4.1 Cubic Stiffness

5.4.2 Bilinear Stiffness or Damping

5.4.3 Nonlinear Damping

5.4.4 Coulomb Friction

5.4.5 Piecewise Linear Stiffness

5.5 Linearisation: Effective

FRFs

for Nonlinear Systems

5.5.1 Harmonic Balance

5.5.2 Harmonic Generation in Nonlinear Systems

5.5.3 Sum and Difference Frequencies

5.5.4 Harmonic Balance revisited

5.5.5 Nonlinear Damping

5.5.6 Two Systems of Particular Interest

Quadratic Stiffness

Bilinear Stiffness

5.5.7 *Statistical Linearisation

Theory

Application to Duffing’s Equation

5.6 Chaos

References

Chapter 6 Updating of Numerical Models

6.1 Introduction

6.2 Model Matching

6.2.1 Model Reduction

Guyan’s Reduction

Dynamic Reduction

Improved Reduction System

System Equivalent Reduction Expansion Process

Modal Truncation

Component Mode Synthesis

Sum of Weighted Accelerations Technique

Reduction of Damped Models

6.2.2 Expansion of Measured Data

Kidder’s Method

Expansion using Analytical Modes

Expansion of Frequency Response Functions

Modified Kidder’s Method

A Complete Matrix of Frequency Response Functions

6.3 Model Correlation

6.3.1 Modal Domain

6.3.2 Frequency Domain

6.3.3 A Brief Note on Model Validation

6.4 Deterministic Model Updating

6.4.1 Direct Optimum Matrix Updating Method

6.4.2

FRF

-based Direct Updating Method

6.4.3 Sensitivity-based Model Updating

Eigensensitivity Approach

FRF

Sensitivity-based Approach

6.4.4 On the Localisation of Modelling Errors

6.5 Stochastic Model Updating

6.5.1 Fundamentals of Probability and Statistics

Random Variables

Statistical Hypothesis Tests

6.5.2 Updating the Parameter Covariance Matrix of a Model

The Perturbation Method

Small Perturbation about the Mean

An Equivalent Formulation

6.5.3 Selection of Parameters for Stochastic Model Updating

Towards Updating Parameters Selection

Selection of Parameters using Orthogonal Projections

6.5.4 A non-Probabilistic Approach to Model Updating

6.5.5 On the Bayesian Approach to Model Updating

References

Chapter 7 - Industrial Case Studies

7.1 General Introduction

7.2 An Engineering Application: the Ground Vibration Test

7.2.1 Definition of the Objective(s)

7.2.2 Perform Pre-Test Analysis

7.2.3 Testing Activities

7.2.4 Test Plan

7.2.5 Data Verification

7.2.6 Modal Identification

7.2.7 Model Validation and Updating

7.3 Presentation of the Test Cases

7.3.1 Introduction to the

SLDV

Measurement System

7.3.2 Response Model using the

SLDV

Measurement Method

7.4 Case study 1: Experimental Model Validation of a Composite Fan Blade

7.4.1 Modal Test of a Composite Blade

7.4.2 Modal Test under Free-Free Boundary Conditions

7.4.3 Fixed-Free Modal Test

7.4.4 Normal Mode Shapes and Correlation with

FE

Modes

7.4.5 Modal Test for

MAC

Correlation

7.5 Case Study 2: Experimental Model Validation of a Tower Rotor Test Rig

7.5.1 Structural Analysis of the Tower

7.5.2 Experimental Model Validation

7.6 Case Study 3: Nonlinear Behaviour of Bolted Flanges

7.6.1 Rapid Validation of Fine Mesh

FEM

Axisymmetric Casings and Assemblies

7.6.2 Sector Test Planning for Large Axisymmetric

FMFEMs

7.6.3 Test Plan for the Characterisation of Nonlinear Vibrations in Bolted Flanges

7.6.4 Test Set-ups for Characterisation of Bolted Flanges Nonlinear Vibrations

7.6.5 Test Method

7.6.6 Experimental Results

Full Assembly of the Aero-engine Casing

Experimental Results of the Sector Flange

7.6.7 Validation of the

FE

Model

Linear and Nonlinear Models

Validated Numerical Results

7.7 Case study 4: Experimental Nonlinear Modal Analysis of the Tail Drive Shaft System of a Helicopter

7.7.1 The Identification Process of Nonlinear Stiffness Parameters of the Tail Drive Shaft System

7.7.2 Identification Method of the Nonlinear Stiffness Parameters

7.7.3 Validation of the Linear Model of the Tail Drive Shaft System

7.7.4 Nonlinear Identification of the Tail Drive Shaft System

7.8 Final Notes

References

Appendices

Appendix A

Appendix B

Appendix C

Index

End User License Agreement

List of Tables

CHAPTER 02

Table 2.1 Effect of time...

Table 2.2 Characteristics...

Table 2.3 Time domain...

Table 2.4 Statistical data...

Table 2.5 Frequency domain...

CHAPTER 03

Table 3.1 The computed result...

Table 3.2 Comparison of the...

Table 3.3 ROOTS result.

Table 3.4 Natural...

Table 3.5 Example of...

Table 3.6 Example of...

Table 3.7 Comparison...

Table 3.8 Comparison...

Table 3.9 Natural frequencies...

CHAPTER 04

Table 4.1 Preferred...

Table 4.2 Phase Separation...

Table 4.3 Peak picking...

Table 4.4 Modal Estimates...

Table 4.5 Parameter...

Table 4.6 Typical Stability...

Table 4.7 Estimated natural...

Table 4.8 Parameters of...

Table 4.9 Amplitudes...

Table 4.10 First seven...

Table 4.11 Identified...

Table 4.12 Phase Resonance...

Table 4.13 Computed MMIF...

CHAPTER 06

Table 6.1 Parameter and...

Table 6.2 Parameters and...

Table 6.3 Initial and...

CHAPTER 07

Table 7.1 List of actions...

Table 7.1 Measurement...

Table 7.2 Natural...

Table 7.3 Natural...

Table 7.4 Lock points...

Table 7.5 Natural...

Table 7.6 Damping...

Table 7.7 Paired mode...

Table 7.8 Paired mode...

Table 7.9 Paired mode...

Table 7.10 Comparison...

Table 7.11 Modal analysis...

Table 7.12 Frequency...

List of Illustrations

CHAPTER 01

Figure 1.1 Classification...

Figure 1.2 Examples of...

Figure 1.3 Example of...

Figure 1.4 Elements of a...

Figure 1.5 The spring, damper...

Figure 1.6 The torsional...

Figure 1.7 Example of a...

Figure 1.8 Free body diagram...

Figure 1.9 SDOF system...

Figure 1.10 SDOF system...

Figure 1.11 Over-damped...

Figure 1.12 Under-damped...

Figure 1.13 Complete response...

Figure 1.14 Variation of...

Figure 1.15 Response of...

Figure 1.16 Evolution of...

Figure 1.17 Evolution of...

Figure 1.18 Relation between...

Figure 1.19 Model of an SDOF...

Figure 1.20 Variation of the...

Figure 1.21 Model of an SDOF...

Figure 1.22 Transmissibility...

Figure 1.23 Example of a...

Figure 1.24 New transmissibility...

Figure 1.25 System transmitting...

Figure 1.26 Variation of...

Figure 1.27 Example of a...

Figure 1.28 Discretisation...

Figure 1.29 Non-periodic...

Figure 1.30 Impulses that...

Figure 1.31 Unitary...

Figure 1.32 Unitary...

Figure 1.33 Impulse...

Figure 1.34 Simple...

Figure 1.35 Response of...

Figure 1.36 Response of...

Figure 1.37 Dividing a...

Figure 1.38 Example of...

Figure 1.39 Example of...

Figure 1.40 Division of...

Figure 1.41 Division of...

Figure 1.42 Example of...

Figure 1.43 Example of...

Figure 1.44 Example of...

Figure 1.45 Example of...

Figure 1.46 Example of...

Figure 1.47 Example of...

Figure 1.48 Real and...

Figure 1.49 Examples of...

Figure 1.51 Free body...

Figure 1.50 Uniform bar.

Figure 1.52 Uniform beam...

Figure 1.53 Free body diagram...

CHAPTER 02

Figure 2.1 Basic definitions...

Figure 2.2 More complete...

Figure 2.3 Examples of...

Figure 2.4 Examples of...

Figure 2.5 Examples of...

Figure 2.6 Digitising...

Figure 2.7 Analog signal...

Figure 2.8 Digital signal...

Figure 2.9 Effect of...

Figure 2.10 Windowing effect...

Figure 2.11 Example of six...

Figure 2.12 FFT Analysis of...

Figure 2.14 FFT Analysis of...

Figure 2.13 FFT Analysis of...

Figure 2.15 (a) Single Degree...

Figure 2.16 Impulse excitation...

Figure 2.17(a) Double impulse.

Figure 2.17(b) Double impulse...

Figure 2.17(c) Double impulse...

Figure 2.18(a) Slow rate sine...

Figure 2.18(b) Fast rate sine...

Figure 2.19 Four degrees of...

Figure 2.20 Time-histories...

Figure 2.21 Estimation of...

Figure 2.22 Time-histories...

Figure 2.23 Coherence function ...

Figure 2.24 (a) Cantilever beam...

Figure 2.25 High frequency...

Figure 2.26 Low frequency...

Figure 2.27 Time-histories...

Figure 2.28 Linear sine...

Figure 2.29 Time-histories...

Figure 2.30 Time-histories...

Figure 2.31 Lissajous plots...

Figure 2.32 Time-histories...

Figure 2.33 Simulated...

Figure 2.34 Moving average...

5 (a) Measured Sound...

Figure 2.36 Acceleration...

Figure 2.37 Cumulative...

CHAPTER 03

Figure 3.1 SDOF system.

Figure 3.2 Time response.

Figure 3.3 Central difference...

Figure 3.4 Displacement...

Figure 3.5 Comparison...

Figure 3.6 Equally spaced...

Figure 3.7 Response of...

Figure 3.8 Linear acceleration...

Figure 3.9 Response of the...

Figure 3.10 Acceleration...

Figure 3.11 Response of...

Figure 3.12 Time response...

Figure 3.13 Example of an...

Figure 3.14 3 DOF system.

Figure 3.15 Approximate...

Figure 3.16 Semi-definite...

Figure 3.17 3 DOF free-free...

Figure 3.18 Resultant force...

Figure 3.19 Mode shapes.

Figure 3.20 The computed...

Figure 3.21 Mode shapes.

Figure 3.22 Mode shapes.

Figure 3.23 Mode shapes.

Figure 3.24 Time signals .

Figure 3.25 Mode shape...

Figure 3.26 Size of the...

Figure 3.27 Reconstructed...

Figure 3.28 Mode shape...

Figure 3.29 The size of...

Figure 3.30 Reconstructed...

Figure 3.31 Example of a...

Figure 3.32 Sping element.

Figure 3.33 Assembly of...

Figure 3.34 Uniform bar...

Figure 3.35 Coordinate...

Figure 3.36 Lumped mass...

Figure 3.37 Finite element...

Figure 3.38 Element assembly...

Figure 3.39 Beam element.

Figure 3.41 Clamped beam...

Figure 3.42 2 finite...

Figure 3.43 Complete...

Figure 3.44 Element...

Figure 3.45 Global...

CHAPTER 04

Figure 4.1 Typical Modal...

Figure 4.2 Types of...

Figure 4.3 Phase Separation...

Figure 4.4 Differences...

Figure 4.5(a) Stiffness...

Figure 4.5(b) Effect...

Figure 4.6 Summed FRFs...

Figure 4.7 Sum of FRFs...

Figure 4.8 Relationship...

Figure 4.9 Schematic...

Figure 4.10 Wing-Pylon...

Figure 4.11 Test Stations...

Figure 4.13(a) Measured...

Figure 4.13(b) Measured...

Figure 4.14 Sample FRF...

Figure 4.15 Summed FRFs...

Figure 4.16 Reciprocity...

Figure 4.17 Sample FRFs...

Figure 4.18 Raw Impulse...

Figure 4.19 Truncated...

Figure 4.20 Schematic...

Figure 4.21 Sample measured...

Figure 4.22 Estimated mode...

Figure 4.23(a) Mobility...

Figure 4.23(b) Damping...

Figure 4.23(c) Angle, slope...

Figure 4.24 Circle...

Figure 4.25 Real and...

Figure 4.26 Regenerated...

Figure 4.27 SDOF free...

Figure 4.28 Straight line...

Figure 4.29 FRFs (Bode and...

Figure 4.30(a) Time histories...

Figure 4.30(b) Time histories...

Figure 4.30(c) Close modes...

Figure 4.31 Curve-fitting...

Figure 4.32 Stability Plot...

Figure 4.33 Comparison of...

Figure 4.34 Stability plot...

Figure 4.35 Transient...

Figure 4.36 Selection...

Figure 4.37 Interpolated...

Figure 4.38 Addition and...

Figure 4.39 Typical impulse...

Figure 4.40(a) Estimated...

Figure 4.40(b) Estimated...

Figure 4.41(a) Least-squares error.

Figure 4.41(b) Singular values plots.

Figure 4.42(a) ERA Stability...

Figure 4.42(b) ERA –...

Figure 4.43(a) Forwards...

Figure 4.43(b) Reversed...

Figure 4.44(a) Reversed...

Figure 4.44(b) Reversed...

Figure 4.45(a) ERA. All...

Figure 4.45(b) Reversed...

Figure 4.46 Basis of Operational...

Figure 4.47 Ambient time...

Figure 4.48 Auto-power...

Figure 4.49 Cross Power...

Figure 4.50 Cross Power...

Figure 4.51 Trigger level...

Figure 4.52 Time Histories...

Figure 4.53 Summation of...

Figure 4.54 Triggered time...

Figure 4.55 Zero Crossing...

Figure 4.56 Trigger points...

Figure 4.57 Time histories...

Figure 4.58 Averaged IRFs...

Figure 4.59 Random Decrement...

Figure 4.60 Impulse Response...

Figure 4.61 Cross-Correlation...

Figure 4.62 Power Spectra of...

Figure 4.63 Asher determinant...

Figure 4.64 Modified Asher...

Figure 4.65 Traill-Nash...

Figure 4.66 Extended Asher...

Figure 4.67 Sum of FRFs...

Figure 4.68 Phase Scatter...

Figure 4.69 SVD approach...

CHAPTER 05

Figure 5.1 Feynman...

Figure 5.2 Feynman...

Figure 5.3 Feynman...

Figure 5.4 Feynman...

Figure 5.5 Feynman...

Figure 5.6 Schematic...

Figure 5.7 Deflection...

Figure 5.8 An encastré...

Figure 5.9 Predicted FRF...

Figure 5.10 Hamming FRF...

Figure 5.11 Predicted FRF...

Figure 5.12 Hamming FRF...

Figure 5.13 Predicted FRF...

Figure 5.14 Predicted FRF...

Figure 5.15 Hamming FRF...

Figure 5.16 Simulated...

Figure 5.17 Simulated...

Figure 5.18 Simulated...

Figure 5.19 Simulated...

Figure 5.20 Simulated...

Figure 5.21 Simulated...

Figure 5.22 Simulated...

Figure 5.23 Potential...

Figure 5.24 Potential...

Figure 5.25 Potential...

Figure 5.26 Phase portrait...

Figure 5.27 Phase portrait...

Figure 5.28 Schematic...

Figure 5.29 Schematic...

Figure 5.30 Asymmetric...

Figure 5.31 Output responses...

Figure 5.32 Measured force...

Figure 5.33 Example of...

Figure 5.34 Example of...

Figure 5.35 Schematic...

Figure 5.36 Coherence...

Figure 5.37 Coherence...

Figure 5.38 Coherences...

Figure 5.39 The main types...

Figure 5.40 Characteristic...

Figure 5.41 Amplitudes of...

Figure 5.42 Amplitudes of...

Figure 5.43 Amplitudes of...

Figure 5.44 Amplitudes of...

Figure 5.45 Potential energy...

Figure 5.46 Potential energy...

Figure 5.47 Harmonic balance...

Figure 5.48 Bilinear stiffness...

Figure 5.49 Bilinear stiffness...

Figure 5.50 Harmonic balance...

Figure 5.51 FRFs from equivalent...

Figure 5.52 FRF from equivalent...

Figure 5.53 Segment of the...

Figure 5.54 Poincaré’s...

Figure 5.55 Reconstruction of...

Figure 5.56 Comparison of...

Figure 5.57 Poincaré’s...

Figure 5.58 Samples from...

Figure 5.59 Strange attractor...

Figure 5.60 Attractor for...

CHAPTER 06

Figure 6.1 A clamped-clamped...

Figure 6.2 Comparison of...

Figure 6.3 Comparison of...

Figure 6.4 Change in the...

Figure 6.5 Comparison of...

Figure 6.6 Change in the...

Figure 6.7 Three degree...

Figure 6.8 Convergence plots...

Figure 6.9 Example of a...

Figure 6.10 Example of...

Figure 6.11 Simulation...

Figure 6.12 Scatter plot...

Figure 6.13 Scatter plot...

Figure 6.14 Convergence...

Figure 6.15 Scatter plot...

Figure 6.16 Convergence...

Figure 6.17 Scatter plot...

Figure 6.18 Convergence...

Figure 6.20 Projection of...

Figure 6.19 Decomposition...

Figure 6.21 Decomposition...

Figure 6.22 Projection of...

Figure 6.23 Scatter plot...

Figure 6.24 Convergence...

Figure 6.25 Projection of...

Figure 6.28 Cosine distance:...

Figure 6.26 Cosine distance:...

Figure 6.27 Cosine distance:...

Figure 6.29 Pin-jointed...

Figure 6.30 Cosine distance...

Figure 6.31 Cosine distance...

Figure 6.32 Identified...

Figure 6.33 Scatter plot...

Figure 6.34 Scatter plot...

CHAPTER 07

Figure 7.1 Example of...

Figure 7.2 Example of...

Figure 7.3 Example of...

Figure 7.4 Comparison...

Figure 7.5 MAC between...

Figure 7.6 Schematic of...

Figure 7.7 Blade set-up...

Figure 7.8 FRFs from modal...

Figure 7.9 Comparison of...

Figure 7.10 Comparison of...

Figure 7.11 First on the...

Figure 7.12 Measured (red)...

Figure 7.13 AutoMAC.

Figure 7.14 Test rig...

Figure 7.15 Fixed-Free...

Figure 7.16 ODSs from...

Figure 7.17 FE model on...

Figure 7.18 MAC correlation...

Figure 7.19 MAC between...

Figure 7.20 Rotor Test...

Figure 7.21 Tower strut...

Figure 7.22 Driving point...

Figure 7.23 The locations...

Figure 7.24 Suspension of...

Figure 7.25 Driving Point...

Figure 7.26 FRF and Coherence...

Figure 7.27 Low- and...

Figure 7.28 FRFs with side...

Figure 7.29 FRF and poles...

Figure 7.30 An example of the...

Figure 7.31 AutoMAC matrix and...

Figure 7.32a (a) Paired mode...

Figure 7.32b (b) Paired mode...

Figure 7.33 MAC matrix...

Figure 7.34 MAC matrix...

Figure 7.35 Paired mode...

Figure 7.36 FE model (a),...

Figure 7.37 Predicted...

Figure 7.38 Example of...

Figure 7.39 FRFs (zoom)...

Figure 7.40 Mode shape...

Figure 7.41 FRFs (zoom)...

Figure 7.42 Mode shape...

Figure 7.43 FRF at the...

Figure 7.44 Selected mode...

Figure 7.45 Position 1.

Figure 7.46 Position 2.

Figure 7.47 Position 3.

Figure 7.48 Position 4.

Figure 7.49 Position 5.

Figure 7.50 Position 6.

Figure 7.51 Excitation...

Figure 7.52 Example of...

Figure 7.53 Natural frequency...

Figure 7.54 Damping Loss...

Figure 7.55 Damping factors...

Figure 7.56 (a) Linear...

Figure 7.57 The frequency...

Figure 7.58 Change in damping...

Figure 7.59 Three steps for...

Figure 7.60 Evaluation of...

Figure 7.61 Modal test of...

Figure 7.62 Mobility FRF...

Figure 7.63 The simplified...

Figure 7.64 Schematic diagram...

Figure 7.65 Mode shapes...

Figure 7.66 Mode shapes...

Figure 7.67 Accelerance FRFs...

Figure 7.68 Displacement FRFs...

Figure 7.69 Natural frequency...

Figure 7.70 Curve-Fitting of...

Figure 7.71 Natural frequency...

Figure 7.72 Nonlinear...

Guide

Cover

Title Page

Copyright Page

Dedication

Table of Contents

Acronyms

Preface

List of Authors

About the Companion Website

Begin Reading

Appendix A

Appendix B

Appendix C

Index

End User License Agreement

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Acronyms

A

ADC

Analog to Digital Conversion

AR

Autoregressive

ARMA

Autoregressive Moving Average

ARMAX

Autoregressive Moving Average with eXhogneous inputs

ASMAC

Alternated Search Modal Assurance Criterion

C

CB

Craig-Bampton method

CDF

Cumulative Distribution Function

CMIF

Complex Mode Indicator Function

CMS

Component Mode Synthesis

CMU

Computational Model Updating

CoV

Coefficient of Variation

COMAC

Coordinate Modal Assurance Criterion

CSAC

Cross Signature Assurance Criterion

CSC

Cross Signature Correlation

CSD

Cross Power Spectral Density

CSF

Cross Signature Scale Factor

D

DAC

Digital to Analog Conversion

DIC

Digital Image Correlation

DIRS

Dynamic Improved Reduction System

DOF

Degree of Freedom

E

eCDF

empirical Cumulative Distribution Function

EFDD

Enhanced Frequency Domain Decomposition

EI

Effective Independence method

EJ

Engineering Judgement

EMA

Experimental Modal Analysis

ERA

Eigensystem Realisation Algorithm

ERA/DC

Eigensystem Realisation Algorithm using Data Correlations

F

FAAC

Frequency Amplitude Assurance Criterion

FDAC

Frequency Domain Assurance Criterion

FDD

Frequency Domain Decomposition

FE

Finite Element

FEA

Finite Element Analysis

FEM

Finite Element Method

FFT

Fast Fourier Transform

FI

Fisher Information matrix

FMFEM

Fine Mesh Finite Element Method

FRAC

Frequency Response Assurance Criterion

FRF

Frequency Response Function

FRSF

Frequency Response Scale Factor

FS-SLDV

Fast Scan Scanning Laser Doppler Velocimeter

G

GAC

Global Amplitude Criterion

GFEM

Global Finite Element Model

GSC

Global Shape Criterion

GVT

Ground Vibration Test

I

IIRS

Iterated Dynamic Improved Reduction System

IRF

Impulse Response Function

IRS

Improved Reduction System

K

KMO

Kaiser-Meyer-Olkin criterion

KS-test

Kolmogorov-Smirnov goodness-of-fit test

L

LAC

Local Amplitude Criterion

LDV

Laser Doppler Velocimeter

LHS

Latin Hypercube Sampling

LSCE

Least-Squares Complex Exponential

LSFD

Least Squares Frequency Domain

M

MAC

Modal Assurance Criterion

MBA

Modal-Based Assembly

MCMC

Markov Chain Monte-Carlo

MCS

Monte-Carlo Simulations

MDOF

Multiple Degree of Freedom

MEMS

Micro-Electro-Mechanical Systems

MIMO

Multiple Input Multiple Output

MISO

Multiple Input Single Output

MMIF

Multivariate Mode Indicator Function

ModMMIF

Modified Multivariate Mode Indicator Function

MPC

Modal Phase Collinearity index

MSF

Modal Scale Factor

N

NMPF

Normal Mode Purity Function

O

ODS

Operational Deflection Shape

OMA

Operational Modal Analysis

P

PA

Horn’s Parallel Analysis

PCA

Principal Component Analysis

PDF

Probability Density Function

PID

Proportional Integral Derivative

PSD

Power Spectral Density

Q

QTH

Quasi-Steady Three-Dimensional Histogram

R

RFM

Response Function Method

RK

Runge-Kutta

RK4

Runge-Kutta of 4

th

order

RMS

Root Mean Square

RSS

Root Sum of Squares

RVAC

Response Vector Assurance Criterion

S

svs

Singular Values

SDE

Stochastic Differential Equation

SDOF

Single Degree of Freedom

SEREP

System Equivalent Reduction Expansion Process

SHM

Structural Health Monitoring

SLDV

Scanning Laser Doppler Velocimeter

SLE

Simultaneous Linear Equations

SIMO

Single Input Multiple Output

SISO

Single Input Single Output

SNR

Signal-to-Noise-Ratio

SPL

Sound Pressure Level

SVD

Singular Value Decomposition

SWAT

Sum of Weighted Accelerations Technique

T

TMF

Trim Mean Filtered

TR

Transmissibility

W

WEM

Whole Engine Models

Preface

The very first idea for this book came, some years ago, from Alex Carrella, who at the time was a young postdoctoral researcher, working within a University Technology Centre, integrating a group focused on applied research for a specific industry. In that case it was about vibration of helicopters. The partnership between academia and industry meant that an academic had to use the engineering pragmatism to solve some pressing issues, while practising engineers embrace the more rigorous and lengthier yet innovative practice of academia. Needless to say, the result is a fast transfer of technology to the industry and a much- needed flow of funds to academia to advance knowledge, as resources are of primary importance. For instance, in the process of preparing, carrying out and post-processing the data of a Ground Vibration Test (GVT) there were many questions to be answered, all within the science of structural dynamics, but related to different disciplines, each of them in a different book (or several books on the subject). A pragmatic approach would have been to have one tome with all that was needed enabling the counterpart in the industry to have a book on one’s desk where he/she could dig a little deeper and have a more theoretical notion on a specific subject. Hence the idea of creating a volume to be kept on the desk of practising engineers and ‘applied-researchers’ for having a reference for most topics related to structural dynamics.

However, to create a book on the subject of structural dynamics particularly interesting to the industry is quite an ambitious objective to achieve, as the industry seeks the necessary knowledge to make things happen in a relatively fast way, the so-called “know-how”, whereas academics explore the theoretical foundations to explain the physical phenomena, what one may call the “know-why”. To find the right balance between these two perspectives is not an easy task. Although most of the co-authors of this textbook are scholars, they have the notion of the industrial environment and of the needs of those involved in the daily practice, sometimes due to some industrial experience, or because of close participation in research projects involving various types of companies.

Structural Dynamics is a vast world and no book can encompass the wide variety of themes. Each subject can become a book on its own. Therefore, a judicious choice had to be made and it was decided that the book would have 7 chapters, where Chapter 1 underlines the main fundamental aspects of vibration theory, from the very simple single degree of freedom system to the more general multiple degree of freedom, pointing out relevant aspects that are used in practice; Chapter 2 addresses the main practical problems that may be found in testing a structure, analysing the results and how to tackle the encountered issues in order to solve them; Chapter 3 presents the most important numerical tools that are commonly used and provides the necessary insight on how the various methods work; Chapter 4 describes in detail methods of analysing the results from dynamic tests and how to identify the dynamic properties, so to build a reliable mathematical model that represents the behaviour of a structure when in real operational conditions; Chapter 5 gives a comprehensive and solid background on the nonlinear behaviour of a system, as often the nonlinear aspects cannot be ignore by the analyst engineer; Chapter 6 describes the updating of numerical models, to improve their performance and provide better and accurate estimates of the real behaviour of the structure, either from a deterministic or from a stochastic point of view; in all these first six chapters simple examples are given, to illustrate the application of the various subjects. Finally, Chapter 7 provides some real industrial applications, with emphasis on aeronautical structures.

It is our believe that this book will be useful not only for industry, but also for students doing their master or doctorate studies. Sections identified by an asterisk mean that they may be skipped in a first reading.

Acknowledgements should be addressed to Prof. Hugo Policarpo, who helped producing most of the graphs of Chapter 1, as well as solving some text processing issues; to the work of Dr. Julian Londono-Monsalve in generating the experimental FRFs used in Chapter 4; to the “Aircraft Research Association Limited” and Dr. delli Carri for the test case 2 in Chapter 7; to Rolls-Royces plc. for the test articles of test cases 1 and 3; and to Dr. C. Schwingshackl for the FE model validations of test case 3 in Chapter 7.

The Editors,

Nuno M. M. Maia

Dario Di Maio

Alex Carrella

List of Authors

Nuno M. M. MaiaUniversity of Lisbon, Portugal

Francesco MaruloUniversity of Naples Federico II, Italy

Chaoping ZangNanjing University of Aeronautics and Astronautics, P. R. China

Jonathan E. CooperUniversity of Bristol, U.K.

Keith WordenUniversity of Sheffield, U.K.

Tiago A. N. SilvaUniversidade Nova de Lisboa, Portugal

Dario Di MaioUniversity of Twente, The Netherlands

Alex CarrellaVibration and Acoustic Consultant, Belgium

About the Companion Website

This book is accompanied by a companion website which includes a number of resources created by author for students and instructors that you will find helpful.

www.wiley.com\go\carrella\Structural Dynamics in Engineering Design

The student website includes the Figures PDF of chapter 7.