Structural dynamics and vibration in practice : an engineering handbook

Contents Preface xiii Acknowledgements xv Chapter 1 Basic Concepts 1 1.1 Statics, dynamics and structural dynamics 1 1.2 Coordinates, displacement, velocity and acceleration 1 1.3 Simple harmonic motion 2 1.3.1 Time history representation 3 1.3.2 Complex exponential representation 5 1.4 Mass, stiffness and damping 7 1.4.1 Mass and inertia 7 1.4.2 Stiffness 10 1.4.3 Stiffness and flexibility matrices 12 1.4.4 Damping 14 1.5 Energy methods in structural dynamics 16 1.5.1 Rayleigh’s energy method 17 1.5.2 The principle of virtual work 19 1.5.3 Lagrange’s equations 21 1.6 Linear and non-linear systems 23 1.7 Systems of units 23 1.7.1 Absolute and gravitational systems 24 1.7.2 Conversion between systems 26 1.7.3 The SI system 27 References 28 Chapter 2 The Linear Single Degree of Freedom System: Classical Methods 29 2.1 Setting up the differential equation of motion 29 2.1.1 Single degree of freedom system with force input 29 2.1.2 Single degree of freedom system with base motion input 33 2.2 Free response of single-DOF systems by direct solution of the equation of motion 34 2.3 Forced response of the system by direct solution of the equation of motion 38 vii Chapter 3 The Linear Single Degree of Freedom System: Response in the Time Domain 45 3.1 Exact analytical methods 46 3.1.1 The Laplace transform method 46 3.1.2 The convolution or Duhamel integral 50 3.1.3 Listings of standard responses 53 3.2 ‘Semi-analytical’ methods 55 3.2.1 Impulse response method 56 3.2.2 Straight-line approximation to input function 56 3.2.3 Superposition of standard responses 56 3.3 Step-by-step numerical methods using approximate derivatives 59 3.3.1 Euler method 60 3.3.2 Modified Euler method 62 3.3.3 Central difference method 62 3.3.4 The Runge–Kutta method 65 3.3.5 Discussion of the simpler finite difference methods 69 3.4 Dynamic factors 70 3.4.1 Dynamic factor for a square step input 70 3.5 Response spectra 72 3.5.1 Response spectrum for a rectangular pulse 72 3.5.2 Response spectrum for a sloping step 74 References 76 Chapter 4 The Linear Single Degree of Freedom System: Response in the Frequency Domain 77 4.1 Response of a single degree of freedom system with applied force 77 4.1.1 Response expressed as amplitude and phase 77 4.1.2 Complex response functions 81 4.1.3 Frequency response functions 83 4.2 Single-DOF system excited by base motion 86 4.2.1 Base excitation, relative response 87 4.2.2 Base excitation: absolute response 91 4.3 Force transmissibility 93 4.4 Excitation by a rotating unbalance 94 4.4.1 Displacement response 95 4.4.2 Force transmitted to supports 96 References 97 Chapter 5 Damping 99 5.1 Viscous and hysteretic damping models 99 5.2 Damping as an energy loss 103 5.2.1 Energy loss per cycle – viscous model 103 5.2.2 Energy loss per cycle – hysteretic model 104 5.2.3 Graphical representation of energy loss 105 5.2.4 Specific damping capacity 106 5.3 Tests on damping materials 108 viii Contents 5.4 Quantifying linear damping 108 5.4.1 Quality factor, Q 108 5.4.2 Logarithmic decrement 109 5.4.3 Number of cycles to half amplitude 110 5.4.4 Summary table for linear damping 111 5.5 Heat dissipated by damping 112 5.6 Non-linear damping 112 5.6.1 Coulomb damping 113 5.6.2 Square law damping 113 5.7 Equivalent linear dampers 114 5.7.1 Viscous equivalent for coulomb damping 115 5.7.2 Viscous equivalent for square law damping 116 5.7.3 Limit cycle oscillations with square-law damping 117 5.8 Variation of damping and natural frequency in structures with amplitude and time 117 Chapter 6 Introduction to Multi-degree-of-freedom Systems 119 6.1 Setting up the equations of motion for simple, undamped, multi-DOF systems 119 6.1.1 Equations of motion from Newton’s second law and d’Alembert’s principle 120 6.1.2 Equations of motion from the stiffness matrix 120 6.1.3 Equations of motion from Lagrange’s equations 121 6.2 Matrix methods for multi-DOF systems 122 6.2.1 Mass and stiffness matrices: global coordinates 122 6.2.2 Modal coordinates 126 6.2.3 Transformation from global to modal coordinates 127 6.3 Undamped normal modes 132 6.3.1 Introducing eigenvalues and eigenvectors 132 6.4 Damping in multi-DOF systems 142 6.4.1 The damping matrix 142 6.4.2 Damped and undamped modes 143 6.4.3 Damping inserted from measurements 144 6.4.4 Proportional damping 145 6.5 Response of multi-DOF systems by normal mode summation 147 6.6 Response of multi-DOF systems by direct integration 155 6.6.1 Fourth-order Runge–Kutta method for multi-DOF systems 156 Chapter 7 Eigenvalues and Eigenvectors 159 7.1 The eigenvalue problem in standard form 159 7.1.1 The modal matrix 161 7.2 Some basic methods for calculating real eigenvalues and eigenvectors 162 7.2.1 Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination 162 7.2.2 Matrix iteration 165 7.2.3 Jacobi diagonalization 168 Contents ix 7.3 Choleski factorization 177 7.4 More advanced methods for extracting real eigenvalues and eigenvectors 178 7.5 Complex (damped) eigenvalues and eigenvectors 179 References 180 Chapter 8 Vibration of Structures 181 8.1 A historical view of structural dynamics methods 181 8.2 Continuous systems 182 8.2.1 Vibration of uniform beams in bending 182 8.2.2 The Rayleigh–Ritz method: classical and modern 189 8.3 Component mode methods 194 8.3.1 Component mode synthesis 195 8.3.2 The branch mode method 208 8.4 The finite element method 213 8.4.1 An overview 213 8.4.2 Equations of motion for individual elements 221 8.5 Symmetrical structures 234 References 235 Chapter 9 Fourier Transformation and Related Topics 237 9.1 The Fourier series and its developments 237 9.1.1 Fourier series 237 9.1.2 Fourier coefficients in magnitude and phase form 243 9.1.3 The Fourier series in complex notation 245 9.1.4 The Fourier integral and Fourier transforms 246 9.2 The discrete Fourier transform 247 9.2.1 Derivation of the discrete Fourier transform 248 9.2.2 Proprietary DFT codes 255 9.2.3 The fast Fourier transform 256 9.3 Aliasing 256 9.4 Response of systems to periodic vibration 260 9.4.1 Response of a single-DOF system to a periodic input force 261 References 265 Chapter 10 Random Vibration 267 10.1 Stationarity, ergodicity, expected and average values 267 10.2 Amplitude probability distribution and density functions 270 10.2.1 The Gaussian or normal distribution 274 10.3 The power spectrum 279 10.3.1 Power spectrum of a periodic waveform 279 10.3.2 The power spectrum of a random waveform 281 10.4 Response of a system to a single random input 286 10.4.1 The frequency response function 286 10.4.2 Response power spectrum in terms of the input power spectrum 287 x Contents 10.4.3 Response of a single-DOF system to a broadband random input 288 10.4.4 Response of a multi-DOF system to a single broad-band random input 296 10.5 Correlation functions and cross-power spectral density functions 299 10.5.1 Statistical correlation 299 10.5.2 The autocorrelation function 300 10.5.3 The cross-correlation function 302 10.5.4 Relationships between correlation functions and power spectral density functions 303 10.6 The response of structures to random inputs 305 10.6.1 The response of a structure to multiple random inputs 305 10.6.2 Measuring the dynamic properties of a structure 307 10.7 Computing power spectra and correlation functions using the discrete Fourier transform 310 10.7.1 Computing spectral density functions 312 10.7.2 Computing correlation functions 314 10.7.3 Leakage and data windows 317 10.7.4 Accuracy of spectral estimates from random data 318 10.8 Fatigue due to random vibration 320 10.8.1 The Rayleigh distribution 321 10.8.2 The S–N diagram 322 References 324 Chapter 11 Vibration Reduction 325 11.1 Vibration isolation 326 11.1.1 Isolation from high environmental vibration 326 11.1.2 Reducing the transmission of vibration forces 332 11.2 The dynamic absorber 332 11.2.1 The centrifugal pendulum dynamic absorber 336 11.3 The damped vibration absorber 338 11.3.1 The springless vibration absorber 342 References 345 Chapter 12 Introduction to Self-Excited Systems 347 12.1 Friction-induced vibration 347 12.1.1 Small-amplitude behavior 347 12.1.2 Large-amplitude behavior 349 12.1.3 Friction-induced vibration in aircraft landing gear 350 12.2 Flutter 353 12.2.1 The bending-torsion flutter of a wing 354 12.2.2 Flutter equations 358 12.2.3 An aircraft flutter clearance program in practice 360 12.3 Landing gear shimmy 362 References 366 Contents xi Chapter 13 Vibration testing 367 13.1 Modal testing 368 13.1.1 Theoretical basis 368 13.1.2 Modal testing applied to an aircraft 369 13.2 Environmental vibration testing 373 13.2.1 Vibration inputs 373 13.2.2 Functional tests and endurance tests 374 13.2.3 Test control strategies 375 13.3 Vibration fatigue testing in real time 376 13.4 Vibration testing equipment 377 13.4.1 Accelerometers 377 13.4.2 Force transducers 378 13.4.3 Exciters 378 References 385 Appendix A A Short Table of Laplace Transforms 387 Appendix B Calculation of Flexibility Influence Coefficients 389 Appendix C Acoustic Spectra 393 Index 397