NOISE AND VIBRATION ANALYSIS -- Contents -- About the Author -- Preface -- Acknowledgements -- List of Abbreviations -- Notation -- 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 -- 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 -- 2.8 Chapter Summary -- 2.9 Problems -- References -- 3 Time Data Analysis -- 3.1 Introduction to Discrete Signals -- 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 -- 3.5 Chapter Summary -- 3.6 Problems -- References -- 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 -- 4.4 Quality Assessment of Measured Signals -- 4.5 Chapter Summary -- 4.6 Problems -- References -- 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 vs. Real Part -- 5.5 Determining Natural Frequency and Damping -- 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 -- 5.11 Chapter Summary -- 5.12 Problems -- References -- 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 Time Domain Simulation of Forced Response -- 6.6 Chapter Summary -- 6.7 Problems -- References -- 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 Shakers for Structure Excitation -- 7.12 Some Comments on Measurement Procedures -- 7.13 Problems -- References -- 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 -- 8.6 Chapter Summary -- 8.7 Problems -- References -- 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 -- 9.3.10 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 Zoom FFT -- 9.4 Chapter Summary -- 9.5 Problems -- References -- 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 Estimator for Correlation Functions -- 10.5 Estimators for Transient Signals -- 10.5.1 Windows for Transient Signals -- 10.6 Spectrum Estimation in Practice -- 10.6.1 Linear Spectrum Versus PSD -- 10.6.2 Example of a Spectrum of a Periodic Signal -- 10.6.3 Practical PSD Estimation -- 10.6.4 Spectrum of Mixed Property Signal -- 10.6.5 Calculating RMS Values in Practice -- 10.6.6 RMS From Linear Spectrum of Periodic Signal -- 10.6.7 RMS from PSD -- 10.6.8 Weighted RMS Values -- 10.6.9 Integration and Differentiation in the Frequency Domain -- 10.7 Multi-channel Spectral Analysis -- 10.7.1 Matrix Notation for MIMO Spectral Analysis -- 10.7.2 Arranging Spectral Matrices in MATLAB/Octave -- 10.8 Chapter Summary -- 10.9 Problems -- References.

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 -- 11.2.3 Practical Issues -- 11.2.4 Hardware Specifications -- 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 -- 11.4 Chapter Summary -- 11.5 Problems -- References -- 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 -- 12.11 Chapter Summary -- 12.12 Problems -- References -- 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 Non-random 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.