Nonlinearity in Structural Dynamics Detection Identification and Modelling 1st Edition by K Worden, GR Tomlinson ISBN 0750303565 9780750303569

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ISBN 10: 0750303565 
ISBN 13: 9780750303569
Author: K Worden, GR Tomlinson

Nonlinearity in Structural Dynamics Detection Identification and Modelling 1st Table of contents:

1 Linear systems
1.1 Continuous-time models: time domain
1.2 Continuous-time models: frequency domain
1.3 Impulse response
1.4 Discrete-time models: time domain
1.5 Classification of difference equations
1.5.1 Auto-regressive (AR) models
1.5.2 Moving-average (MA) models
1.5.3 Auto-regressive moving-average (ARMA) models
1.6 Discrete-time models: frequency domain
1.7 Multi-degree-of-freedom (MDOF) systems
1.8 Modal analysis
1.8.1 Free, undamped motion
1.8.2 Free, damped motion
1.8.3 Forced, damped motion
2 From linear to nonlinear
2.1 Introduction
2.2 Symptoms of nonlinearity
2.2.1 Definition of linearity—the principle of superposition
2.2.2 Harmonic distortion
2.2.3 Homogeneity and FRF distortion
2.2.4 Reciprocity
2.3 Common types of nonlinearity
2.3.1 Cubic stiffness
2.3.2 Bilinear stiffness or damping
2.3.3 Piecewise linear stiffness
2.3.4 Nonlinear damping
2.3.5 Coulomb friction
2.4 Nonlinearity in the measurement chain
2.4.1 Misalignment
2.4.2 Vibration exciter problems
2.5 Two classical means of indicating nonlinearity
2.5.1 Use of FRF inspections—Nyquist plot distortions
2.5.2 Coherence function
2.6 Use of different types of excitation
2.6.1 Steady-state sine excitation
2.6.2 Impact excitation
2.6.3 Chirp excitation
2.6.4 Random excitation
2.6.5 Conclusions
2.7 FRF estimators
2.8 Equivalent linearization
2.8.1 Theory
2.8.2 Application to Duffing’s equation
2.8.3 Experimental approach
3 FRFs of nonlinear systems
3.1 Introduction
3.2 Harmonic balance
3.3 Harmonic generation in nonlinear systems
3.4 Sum and difference frequencies
3.5 Harmonic balance revisited
3.6 Nonlinear damping
3.7 Two systems of particular interest
3.7.1 Quadratic stiffness
3.7.2 Bilinear stiffness
3.8 Application of harmonic balance to an aircraft component ground vibration test
3.9 Alternative FRF representations
3.9.1 Nyquist plot: linear system
3.9.2 Nyquist plot: velocity-squared damping
3.9.3 Nyquist plot: Coulomb friction
3.9.4 Carpet plots
3.10 Inverse FRFs
3.11 MDOF systems
3.12 Decay envelopes
3.12.1 The method of slowly varying amplitude and phase
3.12.2 Linear damping
3.12.3 Coulomb friction
3.13 Summary
4 The Hilbert transform—a practical approach
4.1 Introduction
4.2 Basis of the method
4.2.1 A relationship between real and imaginary parts of the FRF
4.2.2 A relationship between modulus and phase
4.3 Computation
4.3.1 The direct method
4.3.2 Correction methods for truncated data
4.3.3 Fourier method 1
4.3.4 Fourier method 2
4.3.5 Case study of the application of Fourier method 2
4.4 Detection of nonlinearity
4.4.1 Hardening cubic stiffness
4.4.2 Softening cubic stiffness
4.4.3 Quadratic damping
4.4.4 Coulomb friction
4.5 Choice of excitation
4.6 Indicator functions
4.6.1 NPR: non-causal power ratio
4.6.2 Corehence
4.6.3 Spectral moments
4.7 Measurement of apparent damping
4.8 Identification of nonlinear systems
4.8.1 FREEVIB
4.8.2 FORCEVIB
4.9 Principal component analysis (PCA)
5 The Hilbert transform—a complex analytical approach
5.1 Introduction
5.2 Hilbert transforms from complex analysis
5.3 Titchmarsh’s theorem
5.4 Correcting for bad asymptotic behaviour
5.4.1 Simple examples
5.4.2 An example of engineering interest
5.5 Fourier transform conventions
5.6 Hysteretic damping models
5.7 The Hilbert transform of a simple pole
5.8 Hilbert transforms without truncation errors
5.9 Summary
6 System identification—discrete time
6.1 Introduction
6.2 Linear discrete-time models
6.3 Simple least-squares methods
6.3.1 Parameter estimation
6.3.2 Parameter uncertainty
6.3.3 Structure detection
6.4 The effect of noise
6.5 Recursive least squares
6.6 Analysis of a time-varying linear system
6.7 Practical matters
6.7.1 Choice of input signal
6.7.2 Choice of output signal
6.7.3 Comments on sampling
6.7.4 The importance of scaling
6.8 NARMAX modelling
6.9 Model validity
6.9.1 One-step-ahead predictions
6.9.2 Model predicted output
6.9.3 Correlation tests
6.9.4 Chi-squared test
6.9.5 General remarks
6.10 Correlation-based indicator functions
6.11 Analysis of a simulated fluid loading system
6.12 Analysis of a real fluid loading system
6.13 Identification using neural networks
6.13.1 Introduction
6.13.2 A linear system
6.13.3 A nonlinear system
7 System identification—continuous time
7.1 Introduction
7.2 The Masri–Caughey method for SDOF systems
7.2.1 Basic theory
7.2.2 Interpolation procedures
7.2.3 Some examples
7.3 The Masri-Caughey method for MDOF systems
7.3.1 Basic theory
7.3.2 Some examples
7.4 Direct parameter estimation for SDOF systems
7.4.1 Basic theory
7.4.2 Display without interpolation
7.4.3 Simple test geometries
7.4.4 Identification of an impacting beam
7.4.5 Application to measured shock absorber data
7.5 Direct parameter estimation for MDOF systems
7.5.1 Basic theory
7.5.2 Experiment: linear system
7.5.3 Experiment: nonlinear system
7.6 System identification using optimization
7.6.1 Application of genetic algorithms to piecewise linear and hysteretic system identification
7.6.2 Identification of a shock absorber model using gradient descent
8 The Volterra series and higher-order frequency response functions
8.1 The Volterra series
8.2 An illustrative case study: characterization of a shock absorber
8.3 Harmonic probing of the Volterra series
8.4 Validation and interpretation of the higher-order FRFs
8.5 An application to wave forces
8.6 FRFs and Hilbert transforms: sine excitation
8.6.1 The FRF
8.6.2 Hilbert transform
8.7 FRFs and Hilbert transforms: random excitation
8.7.1 Volterra system response to a white Gaussian input
8.7.2 Random excitation of a classical Duffing oscillator
8.8 Validity of the Volterra series
8.9 Harmonic probing for a MDOF system
8.10 Higher-order modal analysis: hypercurve fitting
8.10.1 Random excitation
8.10.2 Sine excitation
8.11 Higher-order FRFs from neural network models
8.11.1 The Wray-Green method
8.11.2 Harmonic probing of NARX models: the multi-layer perceptron
8.11.3 Radial basis function networks
8.11.4 Scaling the HFRFs
8.11.5 Illustration of the theory
8.12 The multi-input Volterra series
8.12.1 HFRFs for a continuous-time MIMO system
8.12.2 HFRFs for a discrete-time MIMO system
9 Experimental case studies
9.1 An encastré beam rig
9.1.1 Theoretical analysis
9.1.2 Experimental analysis
9.2 An automotive shock absorber
9.2.1 Experimental set-up
9.2.2 Results
9.2.3 Polynomial modelling
9.2.4 Conclusions
9.3 A bilinear beam rig
9.3.1 Design of the bilinear beam
9.3.2 Frequency-domain characteristics of the bilinear beam
9.3.3 Time-domain characteristics of the bilinear beam
9.3.4 Internal resonance
9.3.5 A neural network NARX model
9.4 Conclusions
A A rapid introduction to probability theory
A.1 Basic definitions
A.2 Random variables and distributions
A.3 Expected values
A.4 The Gaussian distribution
B Discontinuities in the Duffing oscillator FRF
C Useful theorems for the Hilbert transform
C.1 Real part sufficiency
C.2 Energy conservation
C.3 Commutation with differentiation
C.4 Orthogonality
C.5 Action as a filter
C.6 Low-pass transparency
D Frequency domain representations of δ(t) and ϵ(t)
E Advanced least-squares techniques
E.1 Orthogonal least squares
E.2 Singular value decomposition
E.3 Comparison of LS methods
E.3.1 Normal equations
E.3.2 Orthogonal least squares
E.3.3 Singular value decomposition
E.3.4 Recursive least squares
F Neural networks
F.1 Biological neural networks
F.1.1 The biological neuron
F.1.2 Memory
F.1.3 Learning
F.2 The McCulloch-Pitts neuron
F.2.1 Boolean functions
F.2.2 The MCP model neuron
F.3 Perceptrons
F.3.1 The perceptron learning rule
F.3.2 Limitations of perceptrons
F.4 Multi-layer perceptrons
F.5 Problems with MLPs and (partial) solutions
F.5.1 Existence of solutions
F.5.2 Convergence to solutions
F.5.3 Uniqueness of solutions
F.5.4 Optimal training schedules
F.6 Radial basis functions
G Gradient descent and back-propagation
G.1 Minimization of a function of one variable
G.1.1 Oscillation
G.1.2 Local minima
G.2 Minimizing a function of several variables
G.3 Training a neural network
H Properties of Chebyshev polynomials
H.1 Definitions and orthogonality relations
H.2 Recurrence relations and Clenshaw’s algorithm
H.3 Chebyshev coefficients for a class of simple functions
H.4 Least-squares analysis and Chebyshev series
I Integration and differentiation of measured time data
I.1 Time-domain integration
I.1.1 Low-frequency problems
I.1.2 High-frequency problems
I.2 Frequency characteristics of integration formulae
I.3 Frequency-domain integration
I.4 Differentiation of measured time data
I.5 Time-domain differentiation
I.6 Frequency-domain differentiation
J Volterra kernels from perturbation analysis
K Further results on random vibration
K.1 Random vibration of an asymmetric Duffing oscillator
K.2 Random vibrations of a simple MDOF system
K.2.1 The MDOF system
K.2.2 The pole structure of the composite FRF
K.2.3 Validation

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K Worden,GR Tomlinson,Nonlinearity,Structural Dynamics,Modelling