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Elsevier Ltd., 2008. 419 p. — ISBN:978-0-7506-8002-8.This straightforward text, primer and reference introduces the theoretical, testing and control aspects of structural dynamics and vibration, as practised in industry today.

Written by an expert engineer of over 40 years experience, the book comprehensively opens up the dynamic behavior of structures and provides engineers and students with a comprehensive practice based understanding of the key aspects of this key engineering topic.

Key features

Worked example based makes it a thoroughly practical resource

Aimed at those studying to enter, and already working in industry;

Presents an applied practice and testing based approach while remaining grounded in the theory of the topic

Makes the topic as easy to read as possible, omitting no steps in the development of the subject;

ncludes the use of computer based modelling techniques and finite elements

Covers theory, modelling testing and control in practice

Written with the needs of engineers of a wide range of backgrounds in mind, this book will be a key resource for those studying structural dynamics and vibration at undergraduate level for the first time in aeronautical, mechanical, civil and automotive engineering. It will be ideal for laboratory classes and as a primer for readers returning to the subject, or coming to it fresh at graduate level.

t is a guide for students to keep and for practicing engineers to refer to: its worked example approach ensures that engineers will turn to Thorby for advice in many engineering situations.

Presents students and practitioners in all branches of engineering with a unique structural dynamics resource and primer, covering practical approaches to vibration engineering while remaining grounded in the theory of the topic

Written by a leading industry expert, with a worked example lead approach for clarity and ease of understanding

Makes the topic as easy to read as possible, omitting no steps in the development of the subject; covers computer based techniques and finite elementsContents

Preface

Acknowledgements

**Basic Concepts**

Statics, dynamics and structural dynamics

Coordinates, displacement, velocity and acceleration

Simple harmonic motion

Time history representation

Complex exponential representation

Mass, stiffness and damping

Mass and inertia

Stiffness

Stiffness and flexibility matrices

Damping

Energy methods in structural dynamics

Rayleigh’s energy method

The principle of virtual work

Lagrange’s equations

Linear and non-linear systems

Systems of units

Absolute and gravitational systems

Conversion between systems

The SI system

References

**The Linear Single Degree of Freedom System: Classical Methods**

Setting up the differential equation of motion

Single degree of freedom system with force input

Single degree of freedom system with base motion input

Free response of single-DOF systems by direct solution of the equation of motion

Forced response of the system by direct solution of the equation of motion

**The Linear Single Degree of Freedom System: Response in the Time Domain**

Exact analytical methods

The Laplace transform method

The convolution or Duhamel integral

Listings of standard responses

‘Semi-analytical’ methods

Impulse response method

Straight-line approximation to input function

Superposition of standard responses

Step-by-step numerical methods using approximate derivatives

Euler method

Modified Euler method

Central difference method

The Runge–Kutta method

Discussion of the simpler finite difference methods

Dynamic factors

Dynamic factor for a square step input

Response spectra

Response spectrum for a rectangular pulse

Response spectrum for a sloping step

References

**The Linear Single Degree of Freedom System: Response in the Frequency Domain**

Response of a single degree of freedom system with applied force

Response expressed as amplitude and phase

Complex response functions

Frequency response functions

Single-DOF system excited by base motion

Base excitation, relative response

Base excitation: absolute response

Force transmissibility

Excitation by a rotating unbalance

Displacement response

Force transmitted to supports

References

**Damping**

Viscous and hysteretic damping models

Damping as an energy loss

Energy loss per cycle – viscous model

Energy loss per cycle – hysteretic model

Graphical representation of energy loss

Specific damping capacity

Tests on damping materials

Quantifying linear damping

Quality factor, Q

Logarithmic decrement

Number of cycles to half amplitude

Summary table for linear damping

Heat dissipated by damping

Non-linear damping

Coulomb damping

Square law damping

Equivalent linear dampers

Viscous equivalent for coulomb damping

Viscous equivalent for square law damping

Limit cycle oscillations with square-law damping

Variation of damping and natural frequency in structures with amplitude and time

**Introduction to Multi-degree-of-freedom Systems**

Setting up the equations of motion for simple, undamped, multi-DOF systems

Equations of motion from Newton’s second law and d’Alembert’s principle

Equations of motion from the stiffness matrix

Equations of motion from Lagrange’s equations

Matrix methods for multi-DOF systems

Mass and stiffness matrices: global coordinates

Modal coordinates

Transformation from global to modal coordinates

Undamped normal modes

Introducing eigenvalues and eigenvectors

Damping in multi-DOF systems

The damping matrix

Damped and undamped modes

Damping inserted from measurements

Proportional damping

Response of multi-DOF systems by normal mode summation

Response of multi-DOF systems by direct integration

Fourth-order Runge–Kutta method for multi-DOF systems

** Eigenvalues and Eigenvectors**

The eigenvalue problem in standard form

The modal matrix

Some basic methods for calculating real eigenvalues and eigenvectors

Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination

Matrix iteration

Jacobi diagonalization

Choleski factorization

More advanced methods for extracting real eigenvalues and eigenvectors

Complex (damped) eigenvalues and eigenvectors

References

**Vibration of Structures**

A historical view of structural dynamics methods

Continuous systems

Vibration of uniform beams in bending

The Rayleigh–Ritz method: classical and modern

Component mode methods

Component mode synthesis

The branch mode method

The finite element method

An overview

Equations of motion for individual elements

Symmetrical structures

References

**Fourier Transformation and Related Topics**

The Fourier series and its developments

Fourier series

Fourier coefficients in magnitude and phase form

The Fourier series in complex notation

The Fourier integral and Fourier transforms

The discrete Fourier transform

Derivation of the discrete Fourier transform

Proprietary DFT codes

The fast Fourier transform

Aliasing

Response of systems to periodic vibration

Response of a single-DOF system to a periodic input force

References

**Random Vibration**

Stationarity, ergodicity, expected and average values

Amplitude probability distribution and density functions

The Gaussian or normal distribution

The power spectrum

Power spectrum of a periodic waveform

The power spectrum of a random waveform

Response of a system to a single random input

The frequency response function

Response power spectrum in terms of the input power spectrum

Response of a single-DOF system to a broadband random input

Response of a multi-DOF system to a single broad-band random input

Correlation functions and cross-power spectral density functions

Statistical correlation

The autocorrelation function

The cross-correlation function

Relationships between correlation functions and power spectral density functions

The response of structures to random inputs

The response of a structure to multiple random inputs

Measuring the dynamic properties of a structure

Computing power spectra and correlation functions using the discrete Fourier transform

Computing spectral density functions

Computing correlation functions

Leakage and data windows

Accuracy of spectral estimates from random data

Fatigue due to random vibration

The Rayleigh distribution

The S–N diagram

References

**Vibration Reduction**

Vibration isolation

Isolation from high environmental vibration

Reducing the transmission of vibration forces

The dynamic absorber

The centrifugal pendulum dynamic absorber

The damped vibration absorber

The springless vibration absorber

References

**Introduction to Self-Excited Systems**

Friction-induced vibration

Small-amplitude behavior

Large-amplitude behavior

Friction-induced vibration in aircraft landing gear

Flutter

The bending-torsion flutter of a wing

Flutter equations

An aircraft flutter clearance program in practice

Landing gear shimmy

References

**Vibration testing**

Modal testing

Theoretical basis

Modal testing applied to an aircraft

Environmental vibration testing

Vibration inputs

Functional tests and endurance tests

Test control strategies

Vibration fatigue testing in real time

Vibration testing equipment

Accelerometers

Force transducers

Exciters

References

Appendix A A Short Table of Laplace Transforms

Appendix B Calculation of Flexibility Influence Coefficients

Appendix C Acoustic Spectra

Index

Written by an expert engineer of over 40 years experience, the book comprehensively opens up the dynamic behavior of structures and provides engineers and students with a comprehensive practice based understanding of the key aspects of this key engineering topic.

Key features

Worked example based makes it a thoroughly practical resource

Aimed at those studying to enter, and already working in industry;

Presents an applied practice and testing based approach while remaining grounded in the theory of the topic

Makes the topic as easy to read as possible, omitting no steps in the development of the subject;

ncludes the use of computer based modelling techniques and finite elements

Covers theory, modelling testing and control in practice

Written with the needs of engineers of a wide range of backgrounds in mind, this book will be a key resource for those studying structural dynamics and vibration at undergraduate level for the first time in aeronautical, mechanical, civil and automotive engineering. It will be ideal for laboratory classes and as a primer for readers returning to the subject, or coming to it fresh at graduate level.

t is a guide for students to keep and for practicing engineers to refer to: its worked example approach ensures that engineers will turn to Thorby for advice in many engineering situations.

Presents students and practitioners in all branches of engineering with a unique structural dynamics resource and primer, covering practical approaches to vibration engineering while remaining grounded in the theory of the topic

Written by a leading industry expert, with a worked example lead approach for clarity and ease of understanding

Makes the topic as easy to read as possible, omitting no steps in the development of the subject; covers computer based techniques and finite elementsContents

Preface

Acknowledgements

Statics, dynamics and structural dynamics

Coordinates, displacement, velocity and acceleration

Simple harmonic motion

Time history representation

Complex exponential representation

Mass, stiffness and damping

Mass and inertia

Stiffness

Stiffness and flexibility matrices

Damping

Energy methods in structural dynamics

Rayleigh’s energy method

The principle of virtual work

Lagrange’s equations

Linear and non-linear systems

Systems of units

Absolute and gravitational systems

Conversion between systems

The SI system

References

Setting up the differential equation of motion

Single degree of freedom system with force input

Single degree of freedom system with base motion input

Free response of single-DOF systems by direct solution of the equation of motion

Forced response of the system by direct solution of the equation of motion

Exact analytical methods

The Laplace transform method

The convolution or Duhamel integral

Listings of standard responses

‘Semi-analytical’ methods

Impulse response method

Straight-line approximation to input function

Superposition of standard responses

Step-by-step numerical methods using approximate derivatives

Euler method

Modified Euler method

Central difference method

The Runge–Kutta method

Discussion of the simpler finite difference methods

Dynamic factors

Dynamic factor for a square step input

Response spectra

Response spectrum for a rectangular pulse

Response spectrum for a sloping step

References

Response of a single degree of freedom system with applied force

Response expressed as amplitude and phase

Complex response functions

Frequency response functions

Single-DOF system excited by base motion

Base excitation, relative response

Base excitation: absolute response

Force transmissibility

Excitation by a rotating unbalance

Displacement response

Force transmitted to supports

References

Viscous and hysteretic damping models

Damping as an energy loss

Energy loss per cycle – viscous model

Energy loss per cycle – hysteretic model

Graphical representation of energy loss

Specific damping capacity

Tests on damping materials

Quantifying linear damping

Quality factor, Q

Logarithmic decrement

Number of cycles to half amplitude

Summary table for linear damping

Heat dissipated by damping

Non-linear damping

Coulomb damping

Square law damping

Equivalent linear dampers

Viscous equivalent for coulomb damping

Viscous equivalent for square law damping

Limit cycle oscillations with square-law damping

Variation of damping and natural frequency in structures with amplitude and time

Setting up the equations of motion for simple, undamped, multi-DOF systems

Equations of motion from Newton’s second law and d’Alembert’s principle

Equations of motion from the stiffness matrix

Equations of motion from Lagrange’s equations

Matrix methods for multi-DOF systems

Mass and stiffness matrices: global coordinates

Modal coordinates

Transformation from global to modal coordinates

Undamped normal modes

Introducing eigenvalues and eigenvectors

Damping in multi-DOF systems

The damping matrix

Damped and undamped modes

Damping inserted from measurements

Proportional damping

Response of multi-DOF systems by normal mode summation

Response of multi-DOF systems by direct integration

Fourth-order Runge–Kutta method for multi-DOF systems

The eigenvalue problem in standard form

The modal matrix

Some basic methods for calculating real eigenvalues and eigenvectors

Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination

Matrix iteration

Jacobi diagonalization

Choleski factorization

More advanced methods for extracting real eigenvalues and eigenvectors

Complex (damped) eigenvalues and eigenvectors

References

A historical view of structural dynamics methods

Continuous systems

Vibration of uniform beams in bending

The Rayleigh–Ritz method: classical and modern

Component mode methods

Component mode synthesis

The branch mode method

The finite element method

An overview

Equations of motion for individual elements

Symmetrical structures

References

The Fourier series and its developments

Fourier series

Fourier coefficients in magnitude and phase form

The Fourier series in complex notation

The Fourier integral and Fourier transforms

The discrete Fourier transform

Derivation of the discrete Fourier transform

Proprietary DFT codes

The fast Fourier transform

Aliasing

Response of systems to periodic vibration

Response of a single-DOF system to a periodic input force

References

Stationarity, ergodicity, expected and average values

Amplitude probability distribution and density functions

The Gaussian or normal distribution

The power spectrum

Power spectrum of a periodic waveform

The power spectrum of a random waveform

Response of a system to a single random input

The frequency response function

Response power spectrum in terms of the input power spectrum

Response of a single-DOF system to a broadband random input

Response of a multi-DOF system to a single broad-band random input

Correlation functions and cross-power spectral density functions

Statistical correlation

The autocorrelation function

The cross-correlation function

Relationships between correlation functions and power spectral density functions

The response of structures to random inputs

The response of a structure to multiple random inputs

Measuring the dynamic properties of a structure

Computing power spectra and correlation functions using the discrete Fourier transform

Computing spectral density functions

Computing correlation functions

Leakage and data windows

Accuracy of spectral estimates from random data

Fatigue due to random vibration

The Rayleigh distribution

The S–N diagram

References

Vibration isolation

Isolation from high environmental vibration

Reducing the transmission of vibration forces

The dynamic absorber

The centrifugal pendulum dynamic absorber

The damped vibration absorber

The springless vibration absorber

References

Friction-induced vibration

Small-amplitude behavior

Large-amplitude behavior

Friction-induced vibration in aircraft landing gear

Flutter

The bending-torsion flutter of a wing

Flutter equations

An aircraft flutter clearance program in practice

Landing gear shimmy

References

Modal testing

Theoretical basis

Modal testing applied to an aircraft

Environmental vibration testing

Vibration inputs

Functional tests and endurance tests

Test control strategies

Vibration fatigue testing in real time

Vibration testing equipment

Accelerometers

Force transducers

Exciters

References

Appendix A A Short Table of Laplace Transforms

Appendix B Calculation of Flexibility Influence Coefficients

Appendix C Acoustic Spectra

Index

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