Tài liệu FLUID-STRUCTURE INTERACTIONSSLENDER STRUCTURES AND AXIAL FLOW VOLUME 1 ppt

I FLUID-STRUCTURE INTERACTIONS SLENDER STRUCTURES AND AXIAL FLOW VOLUME FLUID-STRUCTURE INTERACTIONS SLENDER STRUCTURES AND AXIAL FLOW VOLUME MICHAEL P PAIDOUSSIS Department of Mechanical Engineering, McGill University, Montreal, Que'bec, Canada W ACADEMIC PRESS SAN DIEGO LONDON NEW YORK BOSTON SYDNEY TOKYO TORONTO This book is printed on acid-free paper Copyright 1998 by ACADEMIC PRESS All Rights Reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Academic Press 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http:llwww.apnet.com Academic Press Limited 24-28 Oval Road, London NW 7DX, UK http:llwww.hbuk.co.uWap/ ISBN 0-12-544360-9 A catalogue record for this book is available from the British Library Library of Congress Catalog Card Number: 98-86469 Typeset by Laser Words, Madras, India Printed in Great Britain by WBC Book Manufacturers, Bridgend, Mid-Glamorgan 98 99 00 01 02 03 WB Preface Artwork Acknowledgnierits xi xiv Introduction 1.1 General overview 1.2 Classification of flow-induced vibrations 1.3 Scope and contents of volume 1.4 Contents of volume Concepts Definitions and Methods 2.1 Discrete and distributed parameter systems 2.1.1 The equations of motion 2.1.2 Brief review of discrete systems 2.1.3 The Galerkin method via a simple example 2.1.4 Galerkin’s method for a nonconservative system 2.1.5 Self-adjoint and positive definite continuous systems 2.1.6 Diagonalization, and forced vibrations of continuous systems 2.2 The fluid mechanics of fluid-structure interactions 2.2.1 General character and equations of fluid flow 2.2.2 Loading on coaxial shells filled with quiescent fluid 2.2.3 Loading on coaxial shells filled with quiescent viscous fluid 2.3 Linear and nonlinear dynamics Pipes Conveying Fluid: Linear Dynamics I 3.1 Introduction 3.2 The fundamentals 3.2.1 Pipes with supported ends 3.2.2 Cantilevered pipes 3.2.3 On the various bifurcations 3.3 The equations of motion 3.3.1 Preamble 3.3.2 Newtonian derivation 3.3.3 Hamiltonian derivation 3.3.4 A comment on frictional forces 3.3.5 Nondimensional equation of motion 3.3.6 Methods of solution 6 12 16 17 18 23 23 36 46 51 59 59 60 60 63 67 69 69 71 76 82 83 84 V vi CONTENTS 3.4 Pipes with supported ends 3.4.1 Main theoretical results 3.4.2 Pressurization, tensioning and gravity effects 3.4.3 Pipes on an elastic foundation 3.4.4 Experiments 3.5 Cantilevered pipes 3.5.1 Main thcoretical rcsults 3.5.2 The effect of gravity 3.5.3 The effect of dissipation 3.5.4 The S-shaped discontinuities 3.5.5 On destabilization by damping 3.5.6 Experiments 3.5.7 The effect of an elastic foundation 3.5.8 Effects of tension and refined fluid mechanics modelling 3.6 Systems with added springs, supports, masses and other modifications 3.6.1 Pipes supported at = 1/L < 3.6.2 Cantilevered pipes with additional spring supports 3.6.3 Pipes with additional point masses 3.6.4 Pipes with additional dashpots 3.6.5 Fluid follower forces 3.6.6 Pipes with attached plates 3.6.7 Concluding remarks 3.7 Long pipes and wave propagation 3.7.1 Wave propagation 3.7.2 Infinitely long pipe on elastic foundation 3.7.3 Periodically supported pipes 3.8 Articulated pipes 3.8.1 The basic dynamics 3.8.2 N-Degree-of-freedom pipes 3.8.3 Modified systems 3.8.4 Spatial systems Pipes Conveying Fluid: Linear Dynamics I1 4.1 Introduction 4.2 Nonuniform pipes 4.2.1 The equation of motion 4.2.2 Analysis and results 4.2.3 Experiments 4.2.4 Other work on submerged pipes 4.3 Aspirating pipes and ocean mining 4.3.1 Background 4.3.2 Analysis of the ocean mining system 4.3.3 Recent developments 4.4 Short pipes and refined flow modelling 4.4.1 Equations of motion 4.4.2 Method of analysis 88 88 98 102 103 111 111 115 118 123 130 133 149 150 153 153 157 164 167 168 170 172 173 173 174 178 183 184 186 190 194 196 196 196 196 203 208 211 213 213 214 217 220 221 224 CONTENTS 4.5 4.6 4.7 4.8 vii 4.4.3 The inviscid fluid-dynamic force 4.4.4 The fluid-dynamic force by the integral Fourier-transform method 4.4.5 Refined and plug-flow fluid-dynamic forces and specification of the outflow model 4.4.6 Stability of clamped-clamped pipes 4.4.7 Stability of cantilevered pipes 4.4.8 Comparison with experiment 4.4.9 Concluding remarks on short pipes and refined-flow models 4.4.10 Long pipes and refined flow theory 4.4.11 Pipes conveying compressible fluid Pipes with harmonically perturbed flow 4.5.1 Simple parametric resonances 4.5.2 Combination resonances 4.5.3 Experiments 4.5.4 Parametric resonances by analytical methods 4.5.5 Articulated and modified systems 4.5.6 Two-phase and stochastically perturbed flows Forced vibration 4.6.1 The dynamics of forced vibration 4.6.2 Analytical methods for forced vibration Applications 4.7.1 The Coriolis mass-flow meter 4.7.2 Hydroelastic ichthyoid propulsion 4.7.3 Vibration attenuation 4.7.4 Stability of deep-water risers 4.7.5 High-precision piping vibration codes 4.7.6 Vibration conveyance and vibration-induced flow 4.7.7 Miscellaneous applications Concluding remarks 225 Pipes Conveying Fluid: Nonlinear and Chaotic Dynamics 5.1 Introductory comments 5.2 The nonlinear equations of motion 5.2.1 Preliminaries 5.2.2 Hamilton's principle and energy expressions 5.2.3 The equation of motion of a cantilevered pipe 5.2.4 The equation of motion for a pipe fixed at both ends 5.2.5 Boundary conditions 5.2.6 Dissipative terms 5.2.7 Dimensionless equations 5.2.8 Comparison with other equations for cantilevers 5.2.9 Comparison with other equations for pipes with fixed ends 5.2.10 Concluding remarks 5.3 Equations for articulated systems 5.4 Methods of solution and analysis 228 229 232 236 238 240 241 241 242 243 250 253 258 258 261 261 261 265 267 268 269 270 271 273 274 275 276 277 277 278 279 281 283 285 287 287 288 290 294 295 296 299 INDEX Modal matrix 10, 12 Mode exchange, switching, veering 113, 123, 185 Mode localization 183 Mode shapes 94, 113, 135-7, 140, 167 Model dynamical problem 1, 59 Modem methods of nonlinear dynamics 301, 487-501 Modified inextensible theory for curved pipes 416 curved cantilevered pipes 452-7 curved supported pipes 446-52 Modulated waves (MW) 335 see also Quasiperiodic motions Moment-curvature relation 503 Momentum, rate of change 74, 197 Movement-induced excitation (MIE) Multiple scales method 300, 395, 400 Multistep methods 300 ~ Navier-Stokes equations 23, 25 Negative pressurization 219 Negative stiffness 66, 134, 317 Newtonian derivation linear equations of motion 71-6, 197-203, 221-4, 417-25 nonlinear equations of motion 502-5 Newtonian methods Newton’s second law 72 Non-autonomous systems 491 Nonconservative hydrodynamic forces 468 Nonconservative systems 64-5, 67, 118, 121, 124 educational models 275 Galerkin’s method 16-7, 265-7 see also Destabilization; Cantilevered pipes Nondegenerate hi furcations 498 Nondimensional See also Dimensionless Nondimensional equations of motion articulated pipes 187, 299 curved pipes 426-8 linear for pipes 83-4, 85, 162, 174, 203, 223 nonlinear for pipes 288-90, 299 Nongyroscopic conservative system 96 Nongyroscopic nonconservative system 124 Nonhomogeneous equation of motion 84, 86, 261, 265 Nonhyperbolic fixed point 493 Nonlinear control 13 Nonlinear dynamics 51 -8, 277-348 applications to pipe problem 506- 15 basic methods 483-501 curved pipes 457-9 modem methods of 301, 487-501 Nonlinear equations of motion 278-96, 296-9, 317, 357, 373, 397, 407 Hamiltonian derivation 278-87 567 methods of solution and analysis 299-302, 483-501 Newtonian derivation 502-5 Nonlinear harmonic oscillator 491 Nonlinear inertial terms 288-90, 293, 300, 373 Nonlinear motions three-dimensional (3-D) articulated cantilevered pipes 323 -4 impulsively excited 15-6 limit-cycle motion, cantilevered pipe 333-6 pipe with added end-mass 379-83 pipe-spring system 344-5, 389-92 two-dimensional (2-D) articulated cantilevered pipes 319-23, 324-7 limit-cycle motion, cantilevered pipe 328-33 pipe with added end-mass 370-9 pipe with end-mass defect 383-7 pipe-spring system 336-44 up-standing cantilever 345-8 Nonlinear parametric resonance 394-412 Nonlinear restoring and damping functions 54, 289 Nonlinear spring 357-63 Nonlinear tension effects 293-5, 302 Non-Newtonian fluids 23 Nonself-adjoint system 17 Non-slender pipes 220-40 applicabilitykomparison of various versions of theory 232-8, 240 cantilevered pipe stability 236 clamped-clamped pipe stability 232 effect of slenderness 234-5, 239 equations of motion 223, 478 experiments 238 general analysis 224 integral Fourier-transform method 228 nondimensional parameters 223 outflow models 229 plug-flow models for EBPF and TPF theories 225 refined flow modelling for TRF theory 220, 226, 229 Timoshenko theory 220-1, 478 differences in equations of motion 478 eigenfunctions of Timoshenko beam 480 Nontrivial equilibria 309 Nonuniform pipes comparison of theory to experiment 209 effect of immersion in liquid 206 effect of internavexternal tapering 204, 206 effect of slenderness 207 equation of motion 196-203 experimental observations 209 nondimensional equation 203 INDEX Normal coordinates 11 Normal form method 489-91 applied to pipe problem 507, 15 examples 490 for Hopf bifurcation 301, 325, 342 for parametrically perturbed Hopf bifurcation 398, 408 Numerical time-difference methods 300 Numerical tools 302 Nutating oscillations 379 Ocean mining 213-20 system analysis 214-7, 219 Ocean Thermal Energy Conversion (OTEC) plants 219 One-degree-of-freedom linear system 51 One-degree-of-freedom system 54, 131 Open systems 9, 79 Orbital stability 483 Ordinary differential equations (ODES) 7, 333, 398, 495 Orthogonality 10, 14 biorthogonality 12, 22 weighted 10, 12, 17 Oscillation-induced flow 274, 412- 13 Oscillatory instability See Flutter Oscillatory Reynolds number 26, 35 Outflow models 152-3,230-2 Out-of-plane motion of curved pipes 415, 426, 428-30,434-6,439,451,529 cantilevered semi-circular pipe 455 -6 clamped-clamped semi-circular pipe 439, 445-6,449 Out-of-plane oscillation, pipes with slanted end-nozzle 330 Pdidoussis flutter 69, 92-4, 97 Parametric excitation 213, 242, 250 Parametric resonances 213, 242-61, 394-412 analytical methods 258, 395, 397-8, 402-3, 407 - articulated pipes 258, 406 Bolotin’s method 243 cantilevered pipes 246, 402 combination resonances 250 comparison theorylexperiment 255 -7 experiments 253-8, 395-6, 410-2 Floquet analysis 250 nonlinear articulated systems 406 conservative systems 394 nonconservative systems 402 periodically supported pipes 259 pipes with spring supports 259 pipes with supported ends 245, 394 primary resonance 243 -4 principal resonance 243 secondary resonance 243-4 simple parametric resonances 243 suppression of flutter 256 theoretical resonance maps, cantilevered pipes 248-9, 252, 256-7 theoretical resonance maps, clamped-clamped pipes 246-7, 25 1, 255 up-standing cantilever 389 with two-phase flow 261 Partial differential equations (PDEs) 7-8, 333, 495-501 Penalty function technique 332 Pendular oscillations 379 Period doubling articulated cantilevered pipes 322, 392 bifurcation 35 1, 486 constrained pipes 348 -66 experimental 35 parametrically perturbed systems 400, 402 pipes with end-mass defect 385 pipes with supported ends 400, 402 route to chaos 358, 387 up-standing pipes 387 Periodic excitation See Parametric excitation; Parametric resonances Periodic motions 62, 326, 360, 375 see also Flutter Periodic solution 486, 501 Periodically supported pipes 178-83 Perturbation equations 338, 347 Perturbation method 300 Phase angle 126, 491 Phase difference 109, 129, 132 Phase-plane diagrams, plots articulated systems 327, 393-4 cantilevered, chaotic systems 353, 358-9, 365, 369, 375, 382, 388-90, 393-4 curved pipes 458 general 54-7 parametrically excited systems 401-2 pipes with supported ends 313 Phase-plane trajectories See Phase-plane diagrams Phase portrait of averaged system 405 Phase velocity 173 Pinned-pinned curved pipes 415, 448 Pinned-pinned pipes 62, 89, 94, 101, 104, 110, 245, 305, 307, 401 see also Pipes with supported ends Pipe conveying fluid See Articulated pipes; Aspirating pipes; Cantilevered pipes; Curved pipes; Harmonically perturbed flow in pipes; Pipes with supported ends Pipe flows 33 INDEX Pipe-spring systems 157-64, 357, 336-45, 389-91 Pipe strings 271, 316 382 Pipe-whip 276 Pipes supported at intermediate points 153 see also Pipes with added springs Pipes with added point-masses 164-7 368-87 chaotic dynamics 368-87 stabilizatioddestabilization 166 Pipes with added springs at intermediate points 160 at one end 157 chaotic dynamics 389-92 nonlinear dynamics three-dimensional (3-D) motion 344-5 two-dimensional (2-D) motion 336-44 rotational and translational 162 stabilizatioddestabilization 158-60 Pipes with additional dashpots 167 Pipes with attached plates 170 ichthyoid propulsion 269 Pipes with axially sliding downstream end 75, 92, 314-5 Pipes with fixed ends See Pipes with supported ends; Curved pipes Pipes with follower-jet attachments 168 Pipes with supported ends experiments buckling (divergence) 103 deflection-induced tensioning 108, 110 effects of tensioning, pressurization 107 frequency-flow relation 105, 110 nonlinear effects 315, 317 parametric resonances 253-4 zero-frequency condition 108, 110, 315 linear dynamics Argand diagrams 89, 91, 95 basic dynamics 60 88 characteristic curves 97 coupled-mode flutter 92 critical flow velocities 89-90 effect of Coriolis forces 95 effect of damping 95 effect of slenderness 235 ‘effective’ critical flow velocity 102 gravity effects 101 gyroscopic restabilization 97 Hamiltonian Hopf bifurcation 93 impossible inventions 99 non-classical normal modes 95 Paidoussis-type flutter 92-93 parametric resonances 245-7, 250 periodically supported pipes 178 pipes on elastic foundation 102 post-divergence dynamics 91 pressurization effects 98 refined-flow modelling 226, 229 569 short pipes 229-30, 232 tensioning effects 98 nonlinear dynamics axially sliding downstream end 14 chaotic dynamics 394-402 comparison of nonlinear equations of motion 294 coupled-mode flutter nonexistence 303 - 16 effect of amplitude on frequency 302 equations of motion 289 flutter of the Hamiltonian system 312 Holmes’ finite dimensional analysis 304 Holmes’ infinite dimensional analysis 308 impulsively excited 3-D motions 315 parametric resonances 245, 250, 394 post-divergence dynamics 303 see also Pipes with added springs; Parametric resonances; Curved pipes Pitchfork bifurcations 53-7, 62, 67 and parametric resonances 403 articulated cantilevered pipes 321 -7 cantilevered pipes with a spring 339, 346 constrained pipes 358 pipes with added end-mass 377 supported pipes 277, 304, 306, 308-9, 515 symmetry breaking 308-9 through bifurcation theory and unfolding parameters 494 transcritical 486 up-standing cantilever 345 Planar oscillations 328-33 336-44, 348-79, 383-90 See also Curved pipes; In-plane motion of curved pipes; In-plane oscillation, pipes with slanted end-nozzle Plug-flow approximation 74, 225 -6, 274, 279 Euler-Bernoulli theory (EBPF) 220, 233, 239 fluid-dynamic forces 229-32 nonlinear equations of motion 279 short pipes 220, 225 Timoshenko theory (TPF) 220, 233, 235-40 Plunger pump 253 PoincarC maps 350-1, 356, 381-2, 385 PoincarC return map 385-6 Point end-mass 368-87 Poisson bracket 489 Poisson ratio 75, 202, 221, 285 Polyethylene cantilevered pipes 146-7 Positive definite continuous systems 17 Positive definite matrix 10 Positive definite system 10, 17, 310-1, 486 Positive semi-definite system 17 Post-divergence amplitude 315, 317 Post-divergence dynamics 91 -5, 303-14 Post-divergence dynamics in pulsating flow 400 - Post-divergence flutter 91 -4, 340 570 INDEX Post-divergence restabilization 97 Potential energy 81, 281 -6, 312 Potential flow theory 25 Potential well 368, 387 Power spectral densities (PSDs) 147, 363 Power spectrum (PS) 350-2, 356, 381 -4, 389 Prandtl’s mixing-length hypothesis 33 Pressure drop, effect on dynamics 138, 145 Pressurization 75, 98-102 Primary resonances 243-4, 398 Principal coordinates 11 Principal parametric resonance 243,256 Principal primary resonance 245-6 Probability density functions (PDFs) 350-6 Pseudovectors 517 Pump-induced pulsations 316 Pump analogy 65 Quantitative feedback theory (QFT) 41 Quarter-circular clamped-axially-sliding pipe 462 Quasiperiodic motions 250, 326-7, 384, 410 articulated cantilevered pipe 325 -7 constrained pipes 366 due to parametric resonances 405 pipes with added end-mass 374-83 pipes with a spring 383-92 pipes with supported ends 313-4 see also Combination resonance Quasiperiodicity 366, 374, 379 Quenching 41 Radial-flow turbine analogy 65 Rate of change of momentum 74, 197 Rational analytic methods 300 Rayleigh-Ritz method 8, 301 Receptance 262-4 Refined-flow modelling for pipes 220-42 application to long pipes 241 application to short pipes 232-40 Refined fluid-mechanics model 226-32 Relaminarization 385 Research applications-oriented curiosity-driven flow-induced vibrations classifications -4 practical experiences slender systems Resonance boundary See Parametric resonances Resonance oscillations See Parametric resonances Restabilization 92, 13 Return map 385 Reynolds general transport equation 78 Reynolds number (Re) 24-6, 32-3, 130 Reynolds stress tensor 34 Reynolds stresses 29, 32 Riemann-Hugoniot catastrophe 308 Rigid-body motion 40, 44 Ritz-Galerkin method 8, 301 Room-temperature vulcanizing (RTV) silicone rubber 471 Rotary pipe motions 334 Rotating planar oscillations 379 Rotatory inertia 222, 233, 478 Route to chaos intermittency 385 parametric resonance 400,402, 41 1-12 period-doubling 358, 387, 400, 402 quasiperiodicity 366, 374, 379 Runge-Kutta method 300, 357,408 Saddle 54, 307, 347 Saddle-node bifurcation 404 Scleronomic constraints Secondary bifurcation 372-8 Secondary parametric resonance 243-4, 249, 253 Secular term, resonancehonresonance term 490 Self-adjoint system 17 Self-excited oscillations 59 Semi-definite matrix 10 Separatrices 57 Severed pipes 276 Shape optimization 276 Shells, manufacturing methods 473 Short pipes equations of motion 221-4, 478-9 experimental methods 473 linear dynamics 220-42 method of analysis 224-5 see also Non-slender pipes Single-mode amplified oscillations 68 see also Flutter Singular systems analysis 518-9 Simple parametric resonances 243-50 Simply-supported pipes See Pinned-pinned pipes; Pipes with supported ends Slender body approximation 26, 60, 73 Slender-body theory 26-8 Slender systems Small-deflection approximation 72 Solar wind 275 Spatial correlations 30 Spatial systems, articulated pipes 194-5 Spring, nonlinear 357-63 Sprinkler system Feynman’s riddle 217 fluttering sprinkler 276 S-shaped discontinuities cantilevered pipes 123-30 early attempts to understanding 123 INDEX peculiar dynamics, associated with 159, 165, 167, 329-33, 334 recent work 126 Stability 1, 66, 68, 97, 339 articulated pipes 188-95, 321 aspirating pipes 13-20 cantilevered pipes 111-22, 133-53, 204- 12, 236 - concept 483-4 curved pipes 438-57, 459-62 deep-water risers 271 -3 nonuniform pipes 204- 12 pipes with added springs, masses, etc 153-72 pipes with supported ends 88-102, 103-111, 232-6 short pipes 232-40 see also Asymptotic stability; BifurcatiodBifurcation theory; Lyapunov stability; Critical flow velocities Stability boundaries 341 Stability diagram See Critical flow velocities Stable in the large 54 Stable trajectory 483 Standing waves (SW) 334-6 Static equilibrium of curved pipes 436-7 Static instability 51, 515 see also Buckling; Divergence Steady-state configurations of curved pipes 442 Stiffness 41, 62, 66 Stiffness coefficient 36 Stochastically perturbed flows 261 Stokes number 26, 43 Straight cantilevered pipes See Cantilevered pipes Straight pipes with supported ends See Pipes with supported ends Strain energy 282, 284, 286 Stress couples 424 Stress softening 138 Structural damping model See Hysteretic damping Submerged pipes 196, 211, 213, 241 partial immersion 212 Symbolic manipulation computational software 301 -2 Symmetry breaking 308, 325, 333, 345 Syblenls circulatory 64, 67 continuous 7, 12, 18 decoupled 11, 22 discrete 6-22 distributed parameter 6, 22 first-order 1 gyroscopic conservative 62-3, 92, 96 57 nonconservative 16-7, 64-5, 118, 121, 124, 275 positive definite 10, 17, 310- positive semi-definite 10, 17 self-adjoint 17 Takens-Bogdanov point 392 Taylor’s hypothesis 30 Temporal correlations 30 Tension effects, cantilevered pipes 150-3 Tension-gravity effects, pipes with supported ends 98, 101, 316 Tensioning 75, 98- 102 Three-dimensional (3-D) nonlinear motions See Nonlinear motions Three-dimensional (3-D) potential flow 226-7 Time scale of turbulence 30 Timoshenko beam 225, 416, 478 eigenfunctions 480- theory 220- 1,478 Timoshenko equations of motion 478-82 Timoshenko plug-flow theory (TPF) 220, 233, 235-40 Timoshenko refined-flow (TRF) theory 221, 233-5, 237, 239-41 Topological features 347 Torsional divergencehtter 171 Tower 392 Transcritical bifurcation 486 Transversely sliding downstream end 460 Travelling bands 276 Travelling waves (TW) 334-6 Trilinear spring See Nonlinear spring Truncated cones 204 Truncation factor 206 Turbulence 275 energy 32-3 energy spectrum frequency spectrum intensity 1-2 K - E model 34 kinetic energy 32, 34 length scale 30 modelling 33-5 scales 32 Turbulent flow 24, 29-35, 73 Turbulent phase of intermittent oscillation 385 Twist 419, 523 Twisted pipes 416 Two-dimensional (2-D) nonlinear motions See Nonlinear motions Two-equation models for turbulence 33 Two-phase flow 45, 213, 261 Unfolding parameters 325-6, 342-5, 493-5, 500 double degeneracy 494 572 Unfolding parameters (cont.) Hopf bifurcation 493 pitchfork bifurcation 494 Unstable fixed point 54 Unstable in the small 54 Up-standing cantilever linear dynamics 116-7, 143-4 nonlinear dynamics 345-8, 387-9 van der Pol oscillator 325 Variational techniques Variational vector function 360 Velocity potential 38, 226 Vena contractu 151 Vibration attenuation by flow 270-2 Vibration codes for piping 273 Vibration conveyance 274 Vibration-induced flow 274, 412 Vibration-suppression system 272 Virtual displacements 40 Virtual mass See Added mass Virtual work 40 Viscoelastic damping 161, 287 see also Kelvin-Voigt damping Viscoelastic-hysteretic model 209 Viscosity Boussinesq eddy viscosity concept 33 dynamic 23 INDEX eddy 29, 33-4 kinematic 23 molecular 29 Viscous damping, effect of 120-3, 312-4 Viscous damping coefficient 72, 527 dimensionless viscous damping coefficient 84 Viscous flow 442 Volume 2, contents Vorticity 46 Water-aspirating pipes 213 Wave propagation 173-83 energy-trapping modes 182 frequency dispersion 174 in infinitely long pipes 174 in periodically supported pipes 178 in pipe-strings 173 mode localization 183 phase velocity 173 pipes on elastic foundations 174 propagatiodstop bands 181 Wavenumber 31-2, 173, 227 Weak solution 11 Weighted orthogonality 10, 12, 17 Young’s modulus 72, 76 Zero-frequency condition 108, 110, 315 RNAL Journal of Fluids and Structures Editor Michael Pai'doussis McGill University, Canada JFS is a high quality outlet for original full-length papers, brief communications, review articles and special issues in any aspect of fluid-structure interaction With excellent web coverage through IDEAL, the Academic Press Online Journal Library, publication of supplementary data and images through this media is also considered Research areas Flow-induced excita Solid-fluid interactions Response of rnechanica structures to flow a Unsteady fluid dynamic Aeroelasticity and www.academicpress.com/jfs F hid-Structure Interactions is the first of two volumes which concentrates on the dynamics of slender bodies within, or containing axial flow Pai'doussis has provided an authoritative and cohesive text with far-reaching applications, including both the fundamentals and mechanisms leading to flow-induced vibrations The text discusses the causes of vibration as well as the symptoms, and also provides long-term solutions For example, discovering the source of the vibrations, rather than suggesting a buffer t o reduce the effect of the vibrations The broad comprehensive treatment of this important area, and the complete bibliography of main works, provide the tools with which the reader can solve similar problems t o those illustrated Fluid-Structure Interactions include: The Non-linear behaviour of Fluid-Structure interactions The possible existence of chaotic oscillations The use of this area as a model t o demonstrate new mathematical techniques This book will prove invaluable t o researchers, pr and students in fluid-structure interactions, flow-induced vibrations, and dynamics and vibrations Michael Pai'doussis is Thomas Workman Professor of Mechanical Engineering at McGill University, Montreal, Canada ACADEMIC PRESS HARCOURT BRACE & COMPANY,PUBLISHER! S A N DIEGO LONDON BOSTON NEW YORK TOKYO TORONTO PRIKIEDI GREAT BRITAIN N ISBN 0-12-544360-9 ... 88 98 10 2 10 3 11 1 11 1 11 5 11 8 12 3 13 0 13 3 14 9 15 0 15 3 15 3 15 7 16 4 16 7 16 8 17 0 17 2 17 3 17 3 17 4 17 8 18 3 18 4 18 6 19 0 19 4 19 6 19 6 19 6 19 6 203 208 211 213 213 214 217 220 2 21 224 CONTENTS 4.5 4.6 4.7... 2.0 18 4 17 .16 6 - 2. 016 6 16 .936 52 I25 2. 016 4 16 . 912 1. 826 2. 016 3 16 .906 1. 754 2. 016 3 16 .904 1. 738 R2(El/m L4)-’I2 R ( E l / m L4)-‘I2 16 2 .1. 4 SLENDER STRUCTURES AND AXIAL FLOW Galerkin''s method... and refined flow modelling 4.4 .1 Equations of motion 4.4.2 Method of analysis 88 88 98 10 2 10 3 11 1 11 1 11 5 11 8