# Structural Dynamics and Vibration in Practice

## An Engineering Handbook

**By**

- Douglas Thorby

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 featuresâ¢ Worked example based makes it a thoroughly practical resourceâ¢ Aimed at those studying to enter, and already working in industry;â¢ Presents an applied practice and testing based approach while remaining grounded in the theory of the topicâ¢ Makes the topic as easy to read as possible, omitting no steps in the development of the subject;â¢ Includes the use of computer based modelling techniques and finite elementsâ¢ Covers theory, modelling testing and control in practiceWritten 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. It 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.

**Audience**

Reference text/supplementary text for senior undergraduate and graduate students of aeronautical engineering plus mechanical, civil and related engineering disciplines; Graduates starting structural dynamics work

Paperback, 420 Pages

Published: January 2008

Imprint: Butterworth Heinemann

ISBN: 978-0-7506-8002-8

## Contents

- 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 Raleighâs energy method 1.5.2 The principle of virtual work 1.5.3 Lagrangeâs 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: Responsein 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 âSemi-analyticalâ 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 method3.3.4 The RungeâKutta 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: Responsein 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 response4.2.2 Base excitation: absolute response4.3 Force transmissibility4.4 Excitation by a rotating unbalance4.4.1 Displacement response 4.4.2 Force transmitted to supportsChapter 5 Damping 5.1 Viscous and hysteretic damping models 5.2 Damping as an energy loss 5.2.1 Energy loss per cycle â viscous model 5.2.2 Energy loss per cycle â hysteretic model 5.2.3 Graphical representation of energy loss 5.2.4 Specific damping capacity5.3 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 damping5.8 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 Newtonâs second law and dâAlembertâs principle 6.1.2 Equations of motion from the stiffness matrix 6.1.3 Equations of motion from Lagrangeâs 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 6.4.3 Damping inserted from measurements 6.4.4 Proportional damping 6.5 Response of multi-DOF systems by normal mode summation 6.6 Response of multi-DOF systems by direct integration 6.6.1 Fourth-order RungeâKutta method for multi-DOF systemsChapter 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 RayleighâRitz method: classical and modern 8.3 Component mode methods 8.3.1 Component mode synthesis 8.3.2 The branch mode method8.4 The finite element method8.4.1 An overview 8.5 Symmetrical structures Chapter 9 Fourier Transformation and Related Topics 9.1 The Fourier series and its developments 9.1.1 Fourier series9.1.2 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 codes9.2.3 The fast fourier transform 9.3 Aliasing 9.4 Response of systems to periodic vibration9.4.1 Response of a single-DOF system to a periodic input force Chapter 10 Random Vibration10.1 Stationarity, ergodicity, expected and average values10.2 Amplitude probability distribution and density functions 10.2.1 The Gaussian or normal distribution 10.3 The power spectrum10.3.1 Power spectrum of a periodic waveform10.3.2 The power spectrum of a random waveform10.4 Response of a system to a single random input 10.4.1 The frequency response function10.4.2 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 input10.5 Correlation functions and cross-power spectral density functions10.5.1 Statistical correlation10.5.2 The autocorrelation function10.5.3 The cross-correlation function10.5.4 Relationships between correlation functions and power spectral density functions 10.6 The response of structures to random inputs10.6.1 The response of a structure to multiple random inputs 10.6.2 Measuring the dynamic properties of a structure 10.7 Computing power spectra and correlation functions using the discretefourier transform 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 vibration10.8.1 The Rayleigh distribution10.8.2 The SâN diagramChapter 11 Vibration Reduction 11.1 Vibration isolation 11.1.1 Isolation from high environmental vibration 11.1.2 Reducing the transmission of vibration forces11.2 The dynamic absorber11.2.1 The centrifugal pendulum dynamic absorber11.3 The damped vibration absorber11.3.1 The springless vibration absorberChapter 12 Introduction to Self-Excited Systems12.1 Friction-induced vibration12.1.1 Small-amplitude behaviour12.1.2 Large-amplitude behaviour12.1.3 Friction-induced vibration in aircraft landing gear12.2 Flutter12.2.1 The bending-torsion flutter of a wing12.2.2 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 strategies13.3 Vibration fatigue testing in real time 13.4 Vibration testing equipment 13.4.1 Accelerometers 13.4.2 Force transducers 13.4.3 Exciters Appendix A A Short Table of Laplace Transforms Appendix B Calculation of Flexibility Influence CoefficientsAppendix C Acoustic Spectra