Noise and Vibration Analysis: Signal Analysis and Experimental Procedures

Cover
Title Page
Copyright
Contents
About the Author
Preface
Acknowledgments
List of Abbreviations
Annotation
Chapter 1 Introduction
1.1 Noise and Vibration
1.2 Noise and Vibration Analysis
1.3 Application Areas
1.4 Analysis of Noise and Vibrations
1.4.1 Experimental Analysis
1.5 Standards
1.6 Becoming a Noise and Vibration Analysis Expert
1.6.1 The Virtue of Simulation
1.6.2 Learning Tools and the Format of this Book
Chapter 2 Dynamic Signals and Systems
2.1 Introduction
2.2 Periodic Signals
2.2.1 Sine Waves
2.2.2 Complex Sines
2.2.3 Interacting Sines
2.2.4 Orthogonality of Sines
2.3 Random Signals
2.4 Transient Signals
2.5 RMS Value and Power
2.6 Linear Systems
2.6.1 The Laplace Transform
2.6.2 The Transfer Function
2.6.3 The Impulse Response
2.6.4 Convolution
2.7 The Continuous Fourier Transform
2.7.1 Characteristics of the Fourier Transform
2.7.2 The Frequency Response
2.7.3 Relationship Between the Laplace and Frequency Domains
2.7.4 Transient Versus Steady‐State Response
Chapter Summary
2.9 Problems
References
Chapter 3 Time Data Analysis
3.1 Introduction to Discrete Signals
3.1.1 Discrete Convolution
3.2 The Sampling Theorem
3.2.1 Aliasing
3.2.2 Discrete Representation of Analog Signals
3.2.3 Interpolation and Resampling
3.3 Filters
3.3.1 Analog Filters
3.3.2 Digital Filters
3.3.3 Smoothing Filters
3.3.4 Acoustic Octave Filters
3.3.5 Analog RMS Integration
3.3.6 Frequency Weighting Filters
3.4 Time Series Analysis
3.4.1 Min‐ and Max‐Analysis
3.4.2 Time Data Integration
3.4.3 Time Data Differentiation
3.4.4 FFT‐Based Processing
Chapter Summary
3.6 Problems
References
Chapter 4 Statistics and Random Processes
4.1 Introduction to the Use of Statistics
4.1.1 Ensemble and Time Averages
4.1.2 Stationarity and Ergodicity
4.2 Random Theory
4.2.1 Expected Value
4.2.2 Errors in Estimates
4.2.3 Probability Distribution
4.2.4 Probability Density
4.2.5 Histogram
4.2.6 Sample Probability Density Estimate
4.2.7 Average Value and Variance
4.2.8 Central Moments
4.2.9 Skewness
4.2.10 Kurtosis
4.2.11 Crest Factor
4.2.12 Correlation Functions
4.2.13 The Gaussian Probability Distribution
4.3 Statistical Methods
4.3.1 Hypothesis Tests
4.3.2 Test of Normality
4.3.3 Test of Stationarity
4.3.3.1 Frame Statistics
4.3.3.2 The Reverse Arrangements Test
4.3.3.3 The Runs Test
4.4 Quality Assessment of Measured Signals
Chapter Summary
4.6 Problems
References
Chapter 5 Fundamental Mechanics
5.1 Newton's Laws
5.2 The Single Degree‐of‐Freedom System (SDOF)
5.2.1 The Transfer Function
5.2.2 The Impulse Response
5.2.3 The Frequency Response
5.2.4 The Q‐Factor
5.2.5 SDOF Forced Response
5.3 Alternative Quantities for Describing Motion
5.4 Frequency Response Plot Formats
5.4.1 Magnitude and Phase
5.4.2 Real and Imaginary Parts
5.4.3 The Nyquist Plot – Imaginary Versus Real Part
5.5 Determining Natural Frequency and Damping Ratio
5.5.1 Peak in the Magnitude of FRF
5.5.2 Peak in the Imaginary Part of FRF
5.5.3 Resonance Bandwidth (3 dB Bandwidth)
5.5.4 Circle in the Nyquist Plot
5.6 Rotating Mass
5.7 Some Comments on Damping
5.7.1 Hysteretic Damping
5.8 Models Based on SDOF Approximations
5.8.1 Vibration Isolation
5.8.2 Resonance Frequency and Stiffness Approximations
5.9 The Two Degree of Freedom System (2DOF)
5.10 The Tuned Damper
Chapter Summary
5.12 Problems
References
Chapter 6 Modal Analysis Theory
6.1 Waves on a String
6.2 Matrix Formulations
6.2.1 Degree of Freedom
6.3 Eigenvalues and Eigenvectors
6.3.1 Undamped System
6.3.2 Mode Shape Orthogonality
6.3.3 Modal Coordinates
6.3.4 Proportional Damping
6.3.5 General Damping
6.4 Frequency Response of MDOF Systems
6.4.1 Frequency Response from [M], [C], [K]
6.4.2 Frequency Response from Modal Parameters
6.4.3 Frequency Response from [M], [K], and ζ – Modal Damping
6.4.4 Mode Shape Scaling
6.4.5 The Effect of Node Lines on FRFs
6.4.6 Antiresonance
6.4.7 Impulse Response of MDOF Systems
6.5 Free Decays
Chapter Summary
6.7 Problems
References
Chapter 7 Transducers for Noise and Vibration Analysis
7.1 The Piezoelectric Effect
7.2 The Charge Amplifier
7.3 Transducers with Built‐In Impedance Converters, “IEPE”
7.3.1 Low‐Frequency Characteristics
7.3.2 High‐Frequency Characteristics
7.3.3 Transducer Electronic Data Sheet, TEDS
7.4 The Piezoelectric Accelerometer
7.4.1 Frequency Characteristics
7.4.2 Mounting Accelerometers
7.4.3 Electrical Noise
7.4.4 Choosing an Accelerometer
7.5 The Piezoelectric Force Transducer
7.6 The Impedance Head
7.7 The Impulse Hammer
7.8 Accelerometer Calibration
7.9 Measurement Microphones
7.10 Microphone Calibration
7.11 The Geophone
7.12 MEMS‐based Sensors
7.13 Shakers for Structure Excitation
7.14 Some Comments on Measurement Procedures
7.15 Problems
References
Chapter 8 Frequency Analysis Theory
8.1 Periodic Signals – The Fourier Series
8.2 Spectra of Periodic Signals
8.2.1 Frequency and Time
8.3 Random Processes
8.3.1 Spectra of Random Processes
8.4 Transient Signals
8.5 Interpretation of Spectra
Chapter Summary
8.7 Problems
References
Chapter 9 Experimental Frequency Analysis
9.1 Frequency Analysis Principles
9.1.1 Nonparametric Frequency Analysis
9.2 Octave and Third‐Octave Band Spectra
9.2.1 Time Constants
9.2.2 Real‐time Versus Serial Measurements
9.3 The Discrete Fourier Transform (DFT)
9.3.1 The Fast Fourier Transform, FFT
9.3.2 The DFT in Short
9.3.3 The Basis of the DFT
9.3.4 Periodicity of the DFT
9.3.5 Properties of the DFT
9.3.6 Relation Between DFT and Continuous Spectrum
9.3.7 Leakage
9.3.8 The Picket‐Fence Effect
9.3.9 Time Windows for Periodic Signals
9.3.9.1 Amplitude Correction of Window Effects
9.3.9.2 Power Correction of Window Effects
9.3.9.3 Comparison of Common Windows
9.3.9.4 Frequency Resolution
9.3.10 Time Windows for Random Signals
9.3.11 Oversampling in FFT Analysis
9.3.12 Circular Convolution and Aliasing
9.3.13 Zero Padding
9.3.14 Frequency Domain Processing
9.3.15 Zoom FFT
Chapter Summary
9.5 Problems
References
Chapter 10 Spectrum and Correlation Estimates Using the DFT
10.1 Averaging
10.2 Spectrum Estimators for Periodic Signals
10.2.1 The Autopower Spectrum
10.2.2 Linear Spectrum
10.2.3 Phase Spectrum
10.3 Estimators for PSD and CSD
10.3.1 The Periodogram
10.3.2 Welch's Method
10.3.3 Window Correction for Welch Estimates
10.3.4 Bias Error in Welch Estimates
10.3.5 Random Error in Welch Estimates
10.3.6 The Smoothed Periodogram Estimator
10.3.7 Bias Error in Smoothed Periodogram Estimates
10.3.8 Random Error in Smoothed Periodogram Estimates
10.4 Estimators for Correlation Functions
10.4.1 Correlation Estimator by Long FFT
10.4.2 Correlation Estimator by Welch's Method
10.4.3 Variance of the Correlation Estimator
10.4.4 Effect of Measurement Noise on Correlation Function Estimates
10.5 Estimators for Transient Signals
10.5.1 Windows for Transient Signals
10.6 A Signal Processing Framework for Spectrum and Correlation Estimation
10.7 Spectrum Estimation in Practice
10.7.1 Linear Spectrum Versus PSD
10.7.2 Example of a Spectrum of a Periodic Signal
10.7.3 Practical PSD Estimation
10.7.4 Spectrum of Mixed Property Signal
10.7.5 Calculating RMS Values in Practice
10.7.6 RMS from Linear Spectrum of Periodic Signal
10.7.7 RMS from PSD
10.7.8 Weighted RMS Values
10.7.9 Integration and Differentiation in the Frequency Domain
10.8 Multichannel Spectral and Correlation Analysis
10.8.1 Matrix Notation for MIMO Spectral Analysis
10.8.2 Arranging Spectral Matrices in MATLAB/Octave
10.8.3 Multichannel Correlation Functions
Chapter Summary
10.10 Problems
References
Chapter 11 Measurement and Analysis Systems
11.1 Principal Design
11.2 Hardware for Noise and Vibration Analysis
11.2.1 Signal Conditioning
11.2.2 Analog‐to‐Digital Conversion, ADC
11.2.2.1 Quantization and Dynamic Range
11.2.2.2 Setting the Measurement Range
11.2.2.3 Sampling Accuracy
11.2.2.4 Anti‐alias Filters
11.2.2.5 Sigma–Delta ADCs
11.2.3 Practical Issues
11.2.4 Hardware Specifications
11.2.4.1 Absolute Amplitude Accuracy
11.2.4.2 Anti‐alias Protection
11.2.4.3 Simultaneous Sampling
11.2.4.4 Cross‐Channel Match
11.2.4.5 Dynamic Range
11.2.4.6 Cross‐Channel Talk
11.2.5 Transient (Shock) Recording
11.3 FFT Analysis Software
11.3.1 Block Processing
11.3.2 Data Scaling
11.3.3 Triggering
11.3.4 Averaging
11.3.5 FFT Setup Parameters
Chapter Summary
11.5 Problems
Problems
References
Chapter 12 Rotating Machinery Analysis
12.1 Vibrations in Rotating Machines
12.2 Understanding Time–Frequency Analysis
12.3 Rotational Speed Signals (Tachometer Signals)
12.4 RPM Maps
12.4.1 The Waterfall Plot
12.4.2 The Color Map Plot
12.5 Smearing
12.6 Order Tracks
12.7 Synchronous Sampling
12.7.1 DFT Parameters after Resampling
12.8 Averaging Rotation‐Speed‐Dependent Signals
12.9 Adding Change in RMS with Time
12.10 Parametric Methods
Chapter Summary
12.12 Problems
References
Chapter 13 Single‐input Frequency Response Measurements
13.1 Linear Systems
13.2 Determining Frequency Response Experimentally
13.2.1 Method 1 – The H1 Estimator
13.2.2 Method 2 – The H2 Estimator
13.2.3 Method 3 – The Hc Estimator
13.3 Important Relationships for Linear Systems
13.4 The Coherence Function
13.5 Errors in Determining the Frequency Response
13.5.1 Bias Error in FRF Estimates
13.5.2 Random Error in FRF Estimates
13.5.3 Bias and Random Error Trade‐offs
13.6 Coherent Output Power
13.7 The Coherence Function in Practice
13.7.1 Nonrandom Excitation
13.8 Impact Excitation
13.8.1 The Force Signal
13.8.2 The Response Signal and Exponential Window
13.8.3 Impact Testing Software
13.8.4 Compensating for the Influence of the Exponential Window
13.8.5 Sources of Error
13.8.6 Improving Impact Testing by Alternative Processing
13.9 Shaker Excitation
13.9.1 Signal‐to‐noise Ratio Comparison
13.9.2 Pure Random Noise
13.9.3 Burst Random Noise
13.9.4 Pseudo‐random Noise
13.9.5 Periodic Chirp
13.9.6 Stepped‐sine Excitation
13.10 Examples of FRF Estimation – No Extraneous Noise
13.10.1 Pure Random Excitation
13.10.2 Burst Random Excitation
13.10.3 Periodic Excitation
13.11 Example of FRF Estimation – With Output Noise
13.12 Examples of FRF Estimation – With Input and Output Noise
13.12.1 Sources of Error during Shaker Excitation
13.12.2 Checking the Shaker Attachment
13.12.3 Other Sources of Error
Chapter Summary
13.14 Problems
References
Chapter 14 Multiple‐Input Frequency Response Measurement
14.1 Multiple‐Input Systems
14.1.1 The 2‐Input/1‐Output System
14.1.2 The 2‐Input/1‐Output System – Matrix Notation
14.1.3 The H1 Estimator for MIMO
14.1.4 Multiple Coherence
14.1.5 Computation Considerations for Multiple‐Input System
14.1.6 The Hv Estimator
14.1.7 Other MIMO FRF Estimators
14.2 Conditioned Input Signals
14.2.1 Conditioned Output Signals
14.2.2 Partial Coherence
14.2.3 Ordering Signals Prior to Conditioning
14.2.4 Partial Coherent Output Power Spectra
14.2.5 Backtracking the H‐Systems
14.2.6 General Conditioned Systems
14.3 Bias and Random Errors for Multiple‐Input Systems
14.4 Excitation Signals for MIMO Analysis
14.4.1 Pure Random Noise
14.4.2 Burst Random Noise
14.4.3 Periodic Random Noise
14.4.4 The Multiphase Stepped‐Sine Method (MPSS)
14.5 Data Synthesis and Simulation Examples
14.5.1 Burst Random – Output Noise
14.5.2 Burst and Periodic Random – Input Noise
14.5.3 Periodic Random – Input and Output Noise
14.6 Real MIMO Data Case
Chapter Summary
14.8 Problems
References
Chapter 15 Orthogonalization of Signals
15.1 Principal Components
15.1.1 Principal Components Used to Find Number of Sources
15.1.2 Data Reduction
15.2 Virtual Signals
15.2.1 Virtual Input Coherence
15.2.2 Virtual Input/Output Coherence
15.2.3 Virtual Coherent Output Power
15.3 Noise Source Identification (NSI)
15.3.1 Multiple Source Example
15.3.2 Automotive Example
Chapter Summary
15.5 Problems
References
Chapter 16 Experimental Modal Analysis
16.1 Introduction to Experimental Modal Analysis
16.1.1 Main Steps in EMA
16.2 Experimental Setup
16.2.1 Points and DOFs
16.2.2 Selecting Measurement DOFs
16.2.3 Measurement System
16.2.4 Sensor Considerations
16.2.5 Data Acquisition Strategies
16.2.6 Suspension
16.2.7 Measurement Checks
16.2.8 Calibration
16.2.9 Data Acquisition
16.2.10 Mode Indicator Functions
16.2.11 Data Quality Assessment
16.2.12 Checklist
16.3 Introduction to Modal Parameter Extraction
16.4 SDOF Parameter Extraction
16.4.1 The Least Squares Local Method
16.4.2 The Least Squares Global Method
16.4.3 The Least Squares (Local) Polynomial Method
16.5 The Unified Matrix Polynomial Approach, UMPA
16.5.1 Mathematical Framework
16.5.2 Choosing Model Order
16.5.3 Matrix Coefficient Normalization
16.5.4 Data Compression
16.6 Time Versus Frequency Domain Parameter Extraction for EMA
16.7 Time Domain Parameter Extraction Methods
16.7.1 Converting Bandpass Filtered FRFs into IRFs
16.7.2 The Ibrahim Time Domain Method
16.7.3 The Multiple‐Reference Ibrahim Time Domain Method (MITD)
16.7.4 Prony's Method
16.7.5 The Least Squares Complex Exponential Method
16.7.6 Polyreference Time Domain
16.7.7 The Modified Multiple‐Reference Ibrahim Time Domain Method (MMITD)
16.8 Frequency Domain Parameter Extraction Methods
16.8.1 The Least Squares Complex Frequency Domain Method
16.8.2 The Frequency Domain Direct Parameter Identification Method (FDPI)
16.8.3 The Frequency Z‐Domain Direct Parameter Method, FDPIz
16.8.4 The Complex Mode Indicator Function, CMIF Method
16.9 Methods for Mode Shape Estimation and Scaling
16.9.1 Least Squares Frequency Domain – Single Reference Case
16.9.2 Least Squares Frequency Domain – Multiple Reference Case
16.9.3 Least Squares Frequency Domain – Multiple Reference Without MPFs
16.9.4 Least Squares Time Domain
16.9.5 Scaling Modal Model When Poles and Mode Shapes Are Known
16.10 Evaluating the Extracted Parameters
16.10.1 Synthesized FRFs
16.10.2 The MAC Matrix
Chapter Summary
16.12 Problems
References
Chapter 17 Operational Modal Analysis (OMA)
17.1 Principles for OMA
17.2 Data Acquisition Principles
17.3 OMA Modal Parameter Extraction for OMA
17.3.1 Spectral Functions for OMA Parameter Extraction
17.3.2 Correlation Functions for OMA Parameter Extraction
17.3.3 Half Spectra
17.3.4 Time versus Frequency Domain Parameter Extraction for OMA
17.3.5 Modal Parameter Estimation Methods for OMA
17.3.6 Least Squares Frequency Domain, OMA Versions
17.4 Scaling OMA Modal Models
17.4.1 Scaling an OMA Model Using the Mass Matrix
17.4.2 The OMAH Method
Chapter Summary
17.6 Problems
References
Chapter 18 Advanced Analysis Methods
18.1 Shock Response Spectrum
18.2 The Hilbert Transform
18.2.1 Computation of the Hilbert Transform
18.2.2 Envelope Detection by the Hilbert Transform
18.2.3 Relating Real and Imaginary Parts of Frequency Response Functions
18.3 Cepstrum Analysis
18.3.1 Power Cepstrum
18.3.2 Complex Cepstrum
18.3.3 The Real Cepstrum
18.3.4 Inverse Cepstrum
18.4 The Envelope Spectrum
18.5 Creating Random Signals with Known Spectral Density
18.6 Identifying Harmonics in Noise
18.6.1 The Three‐Parameter Sine Fit Method
18.6.2 Periodogram Ratio Detection, PRD
18.7 Harmonic Removal
18.7.1 Frequency Domain Editing, FDE
18.7.2 Cepstrum‐Based Harmonic Removal Methods
Chapter Summary
18.9 Problems
References
Chapter 19 Practical Vibration Measurements and Analysis
19.1 Introduction to a Plexiglas Plate
19.2 Forced Response Simulation
19.2.1 Frequency Domain Forced Response for Periodic Inputs
19.2.2 Frequency Domain Forced Response for Random Inputs
19.2.3 Time Domain Computation of Forced Response for Any Inputs
19.2.3.1 Time Domain Response by Frequency Domain Computation
19.2.3.2 Time Domain Response by Digital Filters
19.2.4 Plexiglas Plate Forced Response Example
19.3 Spectra of Periodic Signals
19.4 Spectra of Random Signals
19.5 Data with Random and Periodic Content
19.5.1 Car Idling Sound
19.5.2 Container Ship Measurement
19.6 Operational Deflection Shapes – ODS
19.6.1 Plexiglas Plate ODS Example – Single Reference
19.6.2 Plexiglas Plate ODS Example – Multiple‐Reference
19.7 Impact Excitation and FRF Estimation
19.8 Plexiglas EMA Example
19.8.1 FRF Quality Assessment
19.8.2 EMA Modal Parameter Extraction, MPE
19.9 Methods for EMA Modal Parameter Estimation, MPE
19.9.1 Time Domain Variable Settings
19.9.2 High‐Order Methods for EMA MPE
19.9.3 Low‐Order Methods for EMA MPE
19.9.4 The Complex Mode Indicator Function, CMIF
19.9.5 Calculating Scaled Mode Shapes
19.10 Conclusions of EMA MPE
19.11 OMA Examples
19.11.1 OMA Using Synthesized Data for Plexiglas Plate
19.11.2 OMA on Measured Data of Plexiglas Plate
19.11.3 OMA of a Suspension Bridge
19.11.4 OMA on Container Ship
References
Appendix A Complex Numbers
Appendix B Logarithmic Diagrams
Appendix C Decibels
Appendix D Some Elementary Matrix Algebra
Appendix E Eigenvalues and the SVD
E.1 Eigenvalues and Complex Matrices
E.2 The Singular Value Decomposition (SVD)
Appendix F Organizations and Resources
Appendix G Checklist for Experimental Modal Analysis Testing
Bibliography
Index
EULA