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I
FLUID-STRUCTURE INTERACTIONS
SLENDER STRUCTURES AND AXIAL FLOW
VOLUME 1
FLUID-STRUCTURE INTERACTIONS
SLENDER STRUCTURES AND AXIAL FLOW
VOLUME 1
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 0 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 1 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 9 8 7 6 5 4 3 2 1
Preface ................................................ Artwork Acknowledgnierits ....................................
1 Introduction
1.1 General overview ....................................
1.2 Classification of flow-induced vibrations .....................
1.3 Scope and contents of volume 1 ........................... 1.4 Contents of volume 2 ..................................
2 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 ............................
3 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 ..............................
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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 ......
Systems with added springs, supports, masses and other
modifications ....................................... 3.6.1 Pipes supported at 6 = 1/L < 1 ...................... 3.6.2 Cantilevered pipes with additional spring supports ..........
3.6.3 Pipes with additional point masses .................... 3.6.4 Pipes with additional dashpots .......................
3.6.7 Concluding remarks ..............................
3.6
3.6.5 Fluid follower forces ............................. 3.6.6 Pipes with attached plates ..........................
3.7 Long pipes and wave propagation .......................... 3.7.1 Wave propagation ...............................
3.7.2 Infinitely long pipe on elastic foundation ................
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 .................................
3.7.3 Periodically supported pipes ........................
4 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 ..............................
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CONTENTS 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 ........................
4.4.10 Long pipes and refined flow theory ....................
4.4.11 Pipes conveying compressible fluid .................... 4.5 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 .............
4.6 Forced vibration .....................................
4.6.1 The dynamics of forced vibration .....................
4.6.2 Analytical methods for forced vibration ................. 4.7 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 ..................
4.7.7 Miscellaneous applications .........................
4.8 Concluding remarks ...................................
5.1 Introductory comments .................................
5.2 The nonlinear equations of motion .........................
Hamilton's principle and energy expressions .............. 5.2.3 The equation of motion of a cantilevered pipe ............
5.2.5 Boundary conditions ............................. 5.2.6 Dissipative terms ................................ 5.2.7 Dimensionless equations ...........................
5.2.8 Comparison with other equations for cantilevers ...........
5.2.10 Concluding remarks .............................. 5.3 Equations for articulated systems .......................... 5.4 Methods of solution and analysis ..........................
4.4.9 Concluding remarks on short pipes and refined-flow
models ......................................
4.7.6 Vibration conveyance and vibration-induced flow ..........
5 Pipes Conveying Fluid: Nonlinear and Chaotic Dynamics
5.2.1 Preliminaries ..................................
5.2.2
5.2.4 The equation of motion for a pipe fixed at both ends ........
5.2.9 Comparison with other equations for pipes with fixed ends ....
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... CONTENTS
VI11
5.5 Pipes with supported ends ............................... 5.5.1 The effect of amplitude on frequency .................. 5.5.2 The post-divergence dynamics .......................
Pipes with an axially sliding downstream end ............. 5.5.4 Impulsively excited 3-D motions .....................
5.6 Articulated cantilevered pipes ............................ 5.6.1 Cantilever with constrained end ...................... 5.6.2 Unconstrained cantilevers .......................... 5.6.3 Concluding comment ............................. 5.7 Cantilevered pipes .................................... 5.7.1 2-D limit-cycle motions ........................... 5.7.2 3-D limit-cycle motions ........................... 5.7.3 Dynamics under double degeneracy conditions ............ 5.7.4 Concluding comment ............................. 5.8 Chaotic dynamics .................................... 5.8.1 Loosely constrained pipes .......................... 5.8.2 Magnetically buckled pipes .........................
5.8.3 Pipe with added mass at the free end ..................
5.8.4 Chaos near double degeneracies ...................... 5.8.5 Chaos in the articulated system ...................... 5.9 Nonlinear parametric resonances ..........................
Pipes with supported ends .......................... 5.9.2 Cantilevered pipes ............................... 5.10 Oscillation-induced flow ................................ 5.11 Concluding remarks ...................................
5.5.3
5.9.1
6 Curved Pipes Conveying Fluid
6.1 Introduction ........................................
6.2 Formulation of the problem .............................. 6.2.1 Kinematics of the system .......................... 6.2.2 The equations of motion ........................... 6.2.3 The boundary conditions ...........................
6.2.4 Nondimensional equations .......................... 6.2.5 Equations of motion of an inextensible pipe ..............
6.2.6 Equations of motion of an extensible pipe ...............
6.3 Finite element analysis .................................
Analysis for inextensible pipes .......................
6.4 Curved pipes with supported ends ......................... 6.4.1 Conventional inextensible theory ..................... 6.4.2 Extensible theory ...............................
6.4.3 Modified inextensible theory ........................
6.4.4 More intricate pipe shapes and other work ............... 6.4.5 Concluding remarks ..............................
6.5.1 Modified inextensible and extensible theories ............. 6.5.2 Nonlinear and chaotic dynamics ......................
6.3.1
6.3.2 Analysis for extensible pipes ........................
6.5 Curved cantilevered pipes ...............................
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CONTENTS ix
6.6 Curved pipes with an axially sliding end .....................
6.6.1 Transversely sliding downstream end ..................
6.6.2 Axially sliding downstream end ......................
Appendices
A First-principles Derivation of the Equation of Motion of a Pipe
Conveying Fluid
B Analytical Evaluation of b. ... and d.
C Destabilization by Damping: T . Brooke Benjamin’s Work
D Experimental Methods for Elastomer Pipes
D.l
D.2
Materials. equipment and procedures ........................
Short pipes. shells and cylinders ........................... D.3 Flexural rigidity and damping constants ...................... D.4 Measurement of frequencies and damping ....................
E.l The equations of motion ................................
The eigenfunctions of a Timoshenko beam ....................
E.3 The integrals Zkn .....................................
F.l Lyapunov method ....................................
F.l . 1 The concept of Lyapunov stability .................... F . 1.2 Linearization ..................................
F . 1.3 Lyapunov direct method ...........................
F.2 Centre manifold reduction ...............................
F.3 Normal forms .......................................
F.4 The method of averaging ...............................
F.5 Bifurcation theory and unfolding parameters ...................
F.6 Partial differential equations .............................
F.6.1 The method of averaging revisited ....................
F.6.2 The Lyapunov-Schmidt reduction ....................
The method of alternate problems .....................
E The Timoshenko Equations of Motion and Associated Analysis
E.2
F Some of the Basic Methods of Nonlinear Dynamics
F.6.3
G Newtonian Derivation of the Nonlinear Equations of Motion of a Pipe
Conveying Fluid
G.l Cantilevered pipe ....................................
G.2 Pipe fixed at both ends .................................
H Nonlinear Dynamics Theory Applied to a Pipe Conveying Fluid
H.l Centre manifold .....................................
H.2 Normal form .......................................
H.2.2 Static instability ................................
H.2.1 Dynamic instability ..............................
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X CONTENTS
I The Fractal Dimension from the Experimental Pipe-vibration Signal 516
J Detailed Analysis for the Derivation of the Equations of Motion of
Relationship between (XO, yo, ZO) and (x, y, z) . . . . . . . . . . . . . . . . . . The expressions for curvature and twist . . . . . . . . . . . . . . . . . . . . . . Derivation of the fluid-acceleration vector . . . . . . . . . . . . . . . . . . . . The equations of motion for the pipe . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6 522
J.l 522
J.2 523
5.3 523
5.4 524
K Matrices for the Analysis of an Extensible Curved Pipe
Conveying Fluid 529
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , . . . . . . . . . . 53 1
Index ................................................. 558
Preface
A word about la raison d’2tre of this book could be useful, especially since the first
question to arise in the prospective reader’s mind might be: why another book on powinduced vibration?
Flow-induced vibrations have been with us since time immemorial, certainly in nature,
but also in artefacts; an example of the latter is the Aeolian harp, which also makes
the point that these vibrations are not always a nuisance. However, in most instances
they are annoying or damaging to equipment and personnel and hence dangerous, e.g.
leading to the collapse of tall chimneys and bridges, the destruction of heat-exchanger and
nuclear-reactor intemals, pulmonary insufficiency, or the severing of offshore risers. In
virtually all such cases, the problem is ‘solved’, and the repaired system remains troublefree thereafter - albeit, sometimes, only after a first and even a second iteration of the
redesigned and supposedly ‘cured’ system failed also. This gives a hint of the reasons why
a book emphasizing (i) thefundamentals and (ii) the mechanisnis givitig rise to thepowinduced vibration might be useful to researchers, designers, operators and, in the broadest
sense of the word, students of systems involving fluid-structure interactions. For, in many
cases, the aforementioned problems were ‘solved’ without truly understanding either the
cause of the original problem or the reasons why the cure worked, or both. Some of the
time-worn battery of ‘cures’, e.g. making the structure stiffer via stiffeners or additional
supports, usually work, but often essentially ‘sweep the problem under thc carpet’, for it
to re-emerge under different operating conditions or in a different part of the parameter
space; moreover, as we shall see in this book, for a limited class of systems, such measures
may actually be counterproductive.
Another answer to the original question ‘Why yet another book?’ lies in the choice
of the material and the style of its presentation. Although the discussion and citation of
work in the area is as complete as practicable, the style is not encyclopaedic; it is sparse,
aiming to convey the main ideas in a physical and comprehensible manner, and in a way
that isfun to read. Thus, the objectives of the book are (i) to convey an understanding
of the undoubtedly fascinating (even for the layman) phenomena discussed, (ii) to give a
complete bibliography of all important work in the field, and (iii) to provide some tools
which the reader can use to solve other similar problems.
A second possible question worth discussing is ‘Why the relatively narrow focus?’
By glancing through the contents, it is immediately obvious that the book deals with
axial-flow-related problems, while vortex-induced motions of bluff bodies, fluidelastic
instability of cylinder arrays in cross-flow, ovalling oscillations of chimneys, indeed all
cross-flow-related topics, are excluded. Reasons for this are that (i) some of these topics
are already well covered in other books and review articles; (ii) in at least some cases, the
fundamentals are still under development, the mechanisms involved being incompletely
understood; (iii) the cross-flow literature is so vast, that any attempt to cover it, as well as
axial-flow problems, would by necessity squeeze the latter into one chapter or two, at most.
xi
xii PREFACE
After extensive consultations with colleagues around the world, it became clear that there
was a great need for a monograph dealing exclusively with axial-flow-induced vibrations
and instabilities. This specialization translates also into a more cohesive treatment of the
material to be covered. The combination of axial flow and slender structures implies, in
many cases, the absence or, at most, limited presence of separated flows. This renders
analytical modelling and interpretation of experimental observation far easier than in
systems involving bluff bodies and cross-flow; it permits a better understanding of the
physics and makes a more elegant presentation of the material possible. Furthermore,
because the understanding of the basics in this area is now well-founded, this book
should remain useful for some time to come.
In a real sense, this book is an anthology of much of the author’s research endeavours
over the past 35 years, at the University of Cambridge, Atomic Energy of Canada in
Chalk River and, mainly, McGill University - with a brief but important interlude at
Cornell University. Inevitably and appropriately, however, vastly more than the author’s
own work is drawn upon.
The book has been written for engineers and applied mechanicians; the physical systems
discussed and the manner in which they are treated may also be of interest to applied
mathematicians. It should appeal especially to researchers, but it has been written for
practising professionals (e.g. designers and operators) and researchers alike. The material
presented should be easily comprehensible to those with some graduate-level understanding of dynamics and fluid mechanics. Nevertheless, a real attempt has been made to
meet the needs of those with a Bachelor’s-level background. In this regard, mathematics
is treated as a useful tool, but not as an end in itself.
This book is not an undergraduate text, although it could be one for a graduate-level
course. However, it is not written in rext-book format, but rather in a style to be enjoyed
by a wider readership.
I should like to express my gratitude to my colleagues, Professor. B.G. Newman for
his help with Section 2.2.1, Professors S.J. Price and A.K. Misra for their input mainly
on Chapters 3 and 6, respectively, Dr H. Alighanbari for input on several chapters and
Appendix F, and Professor D.R. Axelrad for his help in translating difficult papers in
Gernian.
I am especially grateful and deeply indebted to Dr Christian Semler for some special
calculations, many suggestions and long discussions, for checking and rechecking every
part of the book, and particularly for his contributions to Chapter 5 and for Appendix F,
of which he is the main creator. Also, many thanks go to Bill Mark for his willing help
with some superb computer graphics and for input on Appendix D, and to David Sumner
for help with an experiment for Section 4.3.
I am also grateful to many colleagues outside McGill for their help: Drs D.J. Maul1 and
A. Dowling of Cambridge, J.M.T. Thompson of University College London, S.S. Chen
of Argonne, E.H. Dowel1 of Duke, C.D. Mote Jr of Berkeley, F.C. Moon of Cornell,
J.P. Cusumano of Penn State, A.K. Bajaj of Purdue, N.S. Namachchivaya of the University of Illinois, S. Hayama and S. Kaneko of the University of Tokyo, Y. Sugiyama of
Osaka Prefecture, M. Yoshizawa of Keio, the late Y. Nakamura of Kyushu and many
others, too numerous to name.
My gratitude to my secretary, Mary Fiorilli, is unbounded, for without her virtuosity
and dedication this book would not have materialized.