Structural Dynamics and Vibration in Practice: An Engineering Handbook
Chapter 1 Basic Concepts
1.1 Statics, dynamics and structural dynamics
1.2 Coordinates, displacement, velocity and acceleration
1.3 Simple harmonic motion
1.3.1 Time history representation
1.3.2 Complex exponential representation
1.4 Mass, stiffness and damping
1.4.1 Mass and inertia
1.4.2 Stiffness
1.4.3 Stiffness and flexibility matrices
1.4.4 Damping
1.5 Energy methods in structural dynamics
1.5.1 Raleigh?s energy method
1.5.2 The principle of virtual work
1.5.3 Lagrange?s equations
1.6 Linear and non-linear systems
1.7 Systems of units
1.7.1 Absolute and gravitational systems
1.7.2 Conversion between systems
1.7.3 The SI system
Chapter 2 The Linear Single Degree of Freedom System: Classical Methods
2.1 Setting up the differential equation of motion
2.1.1 Single degree of freedom system with force input
2.1.2 Single degree of freedom system with base motion input
2.2 Free response of single-DOF systems by direct solution of the equation of motion
2.3 Forced response of the system by direct solution of the equation of motion
Chapter 3 The Linear Single Degree of Freedom System: Response in the Time Domain
3.1 Exact analytical methods
3.1.1 The Laplace transform method
3.1.2 The convolution or Duhamel integral
3.1.3 Listings of standard responses
3.2 ?Semi-analytical? methods
3.2.1 Impulse response method
3.2.2 Straight-line approximation to input function
3.2.3 Superposition of standard responses
3.3 Step-by-step numerical methods using approximate derivatives
3.3.1 Euler method
3.3.2 Modified Euler method
3.3.3 Central difference method
3.3.4 The Runge?Kutta method
3.3.5 Discussion of the simpler finite difference methods
3.4 Dynamic factors
3.4.1 Dynamic factor for a square step input
3.5 Response spectra
3.5.1 Response spectrum for a rectangular pulse
3.5.2 Response spectrum for a sloping step
Chapter 4 The Linear Single Degree of Freedom System: Response in the Frequency Domain
4.1 Response of a single degree of freedom system with applied force
4.1.1 Response expressed as amplitude and phase
4.1.2 Complex response functions
4.1.3 Frequency response functions
4.2 Single-DOF system excited by base motion
4.2.1 Base excitation, relative response
4.2.2 Base excitation: absolute response
4.3 Force transmissibility
4.4 Excitation by a rotating unbalance
4.4.1 Displacement response
4.4.2 Force transmitted to supports
Chapter 5 Damping
5.1 Viscous and hysteretic damping models
5.2 Damping as an energy loss
5.2.1 Energy loss per cycle – viscous model
5.2.2 Energy loss per cycle – hysteretic model
5.2.3 Graphical representation of energy loss
5.2.4 Specific damping capacity
5.3 Tests on damping materials
5.4 Quantifying linear damping
5.4.1 Quality factor, Q
5.4.2 Logarithmic decrement
5.4.3 Number of cycles to half amplitude
5.4.4 Summary table for linear damping
5.5 Heat dissipated by damping
5.6 Non-linear damping
5.6.1 Coulomb damping
5.6.2 Square law damping
5.7 Equivalent linear dampers
5.7.1 Viscous equivalent for coulomb damping
5.7.2 Viscous equivalent for square law damping
5.7.3 Limit cycle oscillations with square-law damping
5.8 Variation of damping and natural frequency in structures with amplitude and time
Chapter 6 Introduction to Multi-degree-of-freedom Systems
6.1 Setting up the equations of motion for simple, undamped, multi-DOF systems
6.1.1 Equations of motion from Newton?s second law and d?Alembert?s principle
6.1.2 Equations of motion from the stiffness matrix
6.1.3 Equations of motion from Lagrange?s equations
6.2 Matrix methods for multi-DOF systems
6.2.1 Mass and stiffness matrices: global coordinates
6.2.2 Modal coordinates
6.2.3 Transformation from global to modal coordinates
6.3 Undamped normal modes
6.3.1 Introducing eigenvalues and eigenvectors
6.4 Damping in multi-DOF systems
6.4.1 The damping matrix
6.4.2 Damped and undamped modes
Chapter 7 Eigenvalues and Eigenvectors
7.1 The eigenvalue problem in standard form
7.1.1 The modal matrix
7.2 Some basic methods for calculating real eigenvalues and eigenvectors
7.2.1 Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination
7.2.2 Matrix iteration
7.2.3 Jacobi diagonalization
7.3 Choleski factorization
7.4 More advanced methods for extracting real eigenvalues and eigenvectors
7.5 Complex (damped) eigenvalues and eigenvectors
Chapter 8 Vibration of Structures
8.1 A historical view of structural dynamics methods
8.2 Continuous systems
8.2.1 Vibration of uniform beams in bending
8.2.2 The Rayleigh?Ritz method: classical and modern
8.3 Component mode methods
8.3.1 Component mode synthesis
8.3.2 The branch mode method
8.4 The finite element method
8.5 Symmetrical structures
Chapter 9 Fourier Transformation and Related Topics
9.1 The Fourier series and its developments
9.1.1 Fourier series
9.1.2 Fourier coefficients in magnitude and phase form
9.1.3 The Fourier series in complex notation
9.1.4 The Fourier integral and fourier transforms
9.2 The discrete Fourier transform
9.2.1 Derivation of the discrete fourier transform
9.2.2 Proprietary DFT codes
9.2.3 The fast fourier transform
9.4 Response of systems to periodic vibration
9.4.1 Response of a single-DOF system to a periodic input force Chapter
10 Random Vibration
10.1 Stationarity, ergodicity, expected and average values
10.2 Amplitude probability distribution and density functions
10.2.1 The Gaussian or normal distribution
10.3 The power spectrum
10.3.1 Power spectrum of a periodic waveform
10.3.2 The power spectrum of a random waveform
10.4 Response of a system to a single random input
10.4.1 The frequency response function
10.4.2 Response power spectrum in terms of the input power spectrum
10.4.3 Response of a single-DOF system to a broadband random input
10.4.4 Response of a multi-DOF system to a single broad-band random input
10.5 Correlation functions and cross-power spectral density functions
10.5.1 Statistical correlation
10.5.2 The autocorrelation function
10.5.3 The cross-correlation function
10.5.4 Relationships between correlation functions and power spectral density functions
10.6 The response of structures to random inputs
10.6.1 The response of a structure to multiple random inputs
10.6.2 Measuring the dynamic properties of a structure
10.7.1 Computing spectral density functions
10.7.2 Computing correlation functions
10.7.3 Leakage and data windows
10.7.4 Accuracy of spectral estimates from random data
10.8 Fatigue due to random vibration
10.8.1 The Rayleigh distribution
10.8.2 The S?N diagram
Chapter 11 Vibration Reduction
11.1 Vibration isolation
11.1.1 Isolation from high environmental vibration
11.1.2 Reducing the transmission of vibration forces
11.2 The dynamic absorber
11.2.1 The centrifugal pendulum dynamic absorber
11.3 The damped vibration absorber
11.3.1 The springless vibration absorber
Chapter 12 Introduction to Self-Excited Systems
12.1 Friction-induced vibration
12.1.1 Small-amplitude behaviour
12.1.2 Large-amplitude behaviour
12.1.3 Friction-induced vibration in aircraft landing gear
12.2 Flutter
12.2.1 The bending-torsion flutter of a wing
12.2.2 Flutter equations
12.2.3 An aircraft flutter clearance program in practice
12.3 Landing gear shimmy
Chapter 13 Vibration testing
13.1 Modal testing
13.1.1 Theoretical basis
13.1.2 Modal testing applied to an aircraft
13.2 Environmental vibration testing
13.2.1 Vibration inputs
13.2.2 Functional tests and endurance tests
13.2.3 Test control strategies
13.3 Vibration fatigue testing in real time
13.4 Vibration testing equipment
13.4.1 Accelerometers
13.4.2 Force transducers
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Chapter 1 Basic Concepts
1.1 Statics, dynamics and structural dynamics
1.2 Coordinates, displacement, velocity and acceleration
1.3 Simple harmonic motion
1.3.1 Time history representation
1.3.2 Complex exponential representation
1.4 Mass, stiffness and damping
1.4.1 Mass and inertia
1.4.2 Stiffness
1.4.3 Stiffness and flexibility matrices
1.4.4 Damping
1.5 Energy methods in structural dynamics
1.5.1 Raleigh?s energy method
1.5.2 The principle of virtual work
1.5.3 Lagrange?s equations
1.6 Linear and non-linear systems
1.7 Systems of units
1.7.1 Absolute and gravitational systems
1.7.2 Conversion between systems
1.7.3 The SI system
Chapter 2 The Linear Single Degree of Freedom System: Classical Methods
2.1 Setting up the differential equation of motion
2.1.1 Single degree of freedom system with force input
2.1.2 Single degree of freedom system with base motion input
2.2 Free response of single-DOF systems by direct solution of the equation of motion
2.3 Forced response of the system by direct solution of the equation of motion
Chapter 3 The Linear Single Degree of Freedom System: Response in the Time Domain
3.1 Exact analytical methods
3.1.1 The Laplace transform method
3.1.2 The convolution or Duhamel integral
3.1.3 Listings of standard responses
3.2 ?Semi-analytical? methods
3.2.1 Impulse response method
3.2.2 Straight-line approximation to input function
3.2.3 Superposition of standard responses
3.3 Step-by-step numerical methods using approximate derivatives
3.3.1 Euler method
3.3.2 Modified Euler method
3.3.3 Central difference method
3.3.4 The Runge?Kutta method
3.3.5 Discussion of the simpler finite difference methods
3.4 Dynamic factors
3.4.1 Dynamic factor for a square step input
3.5 Response spectra
3.5.1 Response spectrum for a rectangular pulse
3.5.2 Response spectrum for a sloping step
Chapter 4 The Linear Single Degree of Freedom System: Response in the Frequency Domain
4.1 Response of a single degree of freedom system with applied force
4.1.1 Response expressed as amplitude and phase
4.1.2 Complex response functions
4.1.3 Frequency response functions
4.2 Single-DOF system excited by base motion
4.2.1 Base excitation, relative response
4.2.2 Base excitation: absolute response
4.3 Force transmissibility
4.4 Excitation by a rotating unbalance
4.4.1 Displacement response
4.4.2 Force transmitted to supports
Chapter 5 Damping
5.1 Viscous and hysteretic damping models
5.2 Damping as an energy loss
5.2.1 Energy loss per cycle – viscous model
5.2.2 Energy loss per cycle – hysteretic model
5.2.3 Graphical representation of energy loss
5.2.4 Specific damping capacity
5.3 Tests on damping materials
5.4 Quantifying linear damping
5.4.1 Quality factor, Q
5.4.2 Logarithmic decrement
5.4.3 Number of cycles to half amplitude
5.4.4 Summary table for linear damping
5.5 Heat dissipated by damping
5.6 Non-linear damping
5.6.1 Coulomb damping
5.6.2 Square law damping
5.7 Equivalent linear dampers
5.7.1 Viscous equivalent for coulomb damping
5.7.2 Viscous equivalent for square law damping
5.7.3 Limit cycle oscillations with square-law damping
5.8 Variation of damping and natural frequency in structures with amplitude and time
Chapter 6 Introduction to Multi-degree-of-freedom Systems
6.1 Setting up the equations of motion for simple, undamped, multi-DOF systems
6.1.1 Equations of motion from Newton?s second law and d?Alembert?s principle
6.1.2 Equations of motion from the stiffness matrix
6.1.3 Equations of motion from Lagrange?s equations
6.2 Matrix methods for multi-DOF systems
6.2.1 Mass and stiffness matrices: global coordinates
6.2.2 Modal coordinates
6.2.3 Transformation from global to modal coordinates
6.3 Undamped normal modes
6.3.1 Introducing eigenvalues and eigenvectors
6.4 Damping in multi-DOF systems
6.4.1 The damping matrix
6.4.2 Damped and undamped modes
Chapter 7 Eigenvalues and Eigenvectors
7.1 The eigenvalue problem in standard form
7.1.1 The modal matrix
7.2 Some basic methods for calculating real eigenvalues and eigenvectors
7.2.1 Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination
7.2.2 Matrix iteration
7.2.3 Jacobi diagonalization
7.3 Choleski factorization
7.4 More advanced methods for extracting real eigenvalues and eigenvectors
7.5 Complex (damped) eigenvalues and eigenvectors
Chapter 8 Vibration of Structures
8.1 A historical view of structural dynamics methods
8.2 Continuous systems
8.2.1 Vibration of uniform beams in bending
8.2.2 The Rayleigh?Ritz method: classical and modern
8.3 Component mode methods
8.3.1 Component mode synthesis
8.3.2 The branch mode method
8.4 The finite element method
8.5 Symmetrical structures
Chapter 9 Fourier Transformation and Related Topics
9.1 The Fourier series and its developments
9.1.1 Fourier series
9.1.2 Fourier coefficients in magnitude and phase form
9.1.3 The Fourier series in complex notation
9.1.4 The Fourier integral and fourier transforms
9.2 The discrete Fourier transform
9.2.1 Derivation of the discrete fourier transform
9.2.2 Proprietary DFT codes
9.2.3 The fast fourier transform
9.4 Response of systems to periodic vibration
9.4.1 Response of a single-DOF system to a periodic input force Chapter
10 Random Vibration
10.1 Stationarity, ergodicity, expected and average values
10.2 Amplitude probability distribution and density functions
10.2.1 The Gaussian or normal distribution
10.3 The power spectrum
10.3.1 Power spectrum of a periodic waveform
10.3.2 The power spectrum of a random waveform
10.4 Response of a system to a single random input
10.4.1 The frequency response function
10.4.2 Response power spectrum in terms of the input power spectrum
10.4.3 Response of a single-DOF system to a broadband random input
10.4.4 Response of a multi-DOF system to a single broad-band random input
10.5 Correlation functions and cross-power spectral density functions
10.5.1 Statistical correlation
10.5.2 The autocorrelation function
10.5.3 The cross-correlation function
10.5.4 Relationships between correlation functions and power spectral density functions
10.6 The response of structures to random inputs
10.6.1 The response of a structure to multiple random inputs
10.6.2 Measuring the dynamic properties of a structure
10.7.1 Computing spectral density functions
10.7.2 Computing correlation functions
10.7.3 Leakage and data windows
10.7.4 Accuracy of spectral estimates from random data
10.8 Fatigue due to random vibration
10.8.1 The Rayleigh distribution
10.8.2 The S?N diagram
Chapter 11 Vibration Reduction
11.1 Vibration isolation
11.1.1 Isolation from high environmental vibration
11.1.2 Reducing the transmission of vibration forces
11.2 The dynamic absorber
11.2.1 The centrifugal pendulum dynamic absorber
11.3 The damped vibration absorber
11.3.1 The springless vibration absorber
Chapter 12 Introduction to Self-Excited Systems
12.1 Friction-induced vibration
12.1.1 Small-amplitude behaviour
12.1.2 Large-amplitude behaviour
12.1.3 Friction-induced vibration in aircraft landing gear
12.2 Flutter
12.2.1 The bending-torsion flutter of a wing
12.2.2 Flutter equations
12.2.3 An aircraft flutter clearance program in practice
12.3 Landing gear shimmy
Chapter 13 Vibration testing
13.1 Modal testing
13.1.1 Theoretical basis
13.1.2 Modal testing applied to an aircraft
13.2 Environmental vibration testing
13.2.1 Vibration inputs
13.2.2 Functional tests and endurance tests
13.2.3 Test control strategies
13.3 Vibration fatigue testing in real time
13.4 Vibration testing equipment
13.4.1 Accelerometers
13.4.2 Force transducers
link
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