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`
`""FLUIDSTRUCTURE .
`fINTERACTIONS
`
`gVMumel
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 1
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 1
`
`

`

`
`
`
`FLUID—STRUCTURE INTERACTIONS
`
`SLENDER STRUCTURES AND AXIAL FLOW
`
`VOLUME 1
`
`MICHAEL P. PAIDOUS SIS
`Deparzmem of Mechanical Engineering,
`MCGEH University,
`Montreal, Québec. Canada
`
`I
`
`‘
`
`ACADEMIC PRESS
`SAN DIEGO LONDON
`NEW YORK
`BOSTON
`SYDNEY TOKYO TORONTO
`
`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 2
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 2
`
`

`

`,
`Ircffrtrfe .....
`Arrww'k Ackttrm
`_
`1 lntroductmn
`|.i General
`1.2 Citlssiiic
`_
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`{mm a:
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`Es
`
`2 Concepts, De
`2.| Discrcle
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`1
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`2-1-4
`C
`2'5 3
`2.1.6
`'
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`2.2.1
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`2.2.2
`L
`2.2.3
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`This book is printed an IlCidifi'BC paper.
`Cupyrighl © 1998 by ACADEMIC PRESS
`AH Highn- Hus-rived.
`No part of this publication may he rcprotiuccd or transmitted in any form (tr by any means.
`ciccm‘ntic or mechanical. including photocopy, recording. or any ini'urmtltinn slnragc and rctrievul
`system. withuttl pcrmissinn in writing l'rmn the publisher.
`Acudmnic Press
`525 B Srrccl, Suite 1900, San Diego, California tJ‘QIDI-4405, USA
`httinf/WWW.apnutt‘nrn
`d
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`24—23 Oval Road. Londun NWI 7DX. UK
`hltp:wawwiihuk.cu.ukr’:in/
`
`ISBN 0427544360—9
`A catalogue record for this book is uvaiiuble from the British Library
`Library of Congress Catalog Card Number: 98—86469
`Typeset hy Laser Words, Madras. indin
`Printed in Great Britain by WBC Bonk Manufacturers, Bridgentl, Mid-Gimnorgun
`
`L
`(
`98 )9 00 D] 02 03 WE J 8 7 6 5 4 3 2]
`
`3 Pipes Crmvcy
`3. l
`Inlrndnct
`3.2 The fund
`3.2.I
`Pi
`3.2.2 C
`3.2.3 0
`3.3 The aqua
`3.3.]
`Pr
`3.3.2 N
`3.3.3 H
`3.3.4 A
`3.3.5 N
`3.3.6 M
`
`i
`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 3
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 3
`
`

`

`|
`
`i
`I
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`" l
`
`Contents
`
`Preface ................................................
`Artwork Acknowledgments ....................................
`
`1 Introduction
`1.1 General overview ....................................
`1.2 Classification of How-induced vibrations .....................
`1.3
`Scope and contents of volume I ...........................
`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 neoconservative system ...........
`2.1.5
`Selfiadjotnt and positive definite continuous systems ........
`2.1.6 Diagonalization. and l'orcecl vibrations ol‘ continuous
`systems ......................................
`2.2 The fluid mechanics of fluid—structure interactions ...............
`2.2.] General character and equations of fluid ilow .............
`2.2.2 Loading on coaxial shells tilled with quiescent fluid .........
`2.2.3 Loading on coaxial shells filled with quiescent
`viseous lluid ...................................
`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 Cantilcvcred pipes ...............................
`3.2.3 On the various bifurcations .........................
`3.3 The equations 01' 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 ..............................
`
`.
`x1
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`46
`51
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`59
`59
`60
`60
`63
`67
`69
`69
`7]
`76
`82
`83
`84
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`”MIA HALL LIBRARY.
`“mm.“ V
`
`1lr
`
`I
`I
`
`it“ by any means.
`in storage and retrieval
`
`mrgtm
`
`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 4
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 4
`
`

`

`
`
`vi
`
`CONTENTS
`
`3.4
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`1
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`1
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`4.4
`
`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 theoretical results
`...........................
`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
`153
`modifications .......................................
`153
`3.6.1
`Pipes supported at g 2 UL < 1
`......................
`157
`3.6.2 Cantilevered pipes with additional spring supports ..........
`164
`3.6.3
`Pipes with additionai point masses ....................
`167
`3.6.4
`Pipes with additional dashpots .......................
`168
`3.6.5
`Fluid follower forces .............................
`170
`3.6.6 Pipes with attached plates ..........................
`172
`3.6.7 Concluding remarks ..............................
`173
`3.7 Long pipes and wave propagation ..........................
`173
`3.7.1 Wave propagation ...............................
`174
`3.7.2
`Infinitely long pipe on elastic foundation ................
`178
`3.7.3 Periodically supported pipes ........................
`183
`3.8 Articulated pipes .....................................
`184
`3.8.1 The basic dynamics ..............................
`1 86
`3.8.2 N-Degree—01L1'reedom pipes .........................
`190
`3.8.3 Modified systems ...............................
`194
`3.8.4 Spattai systems .................................
`196
`4 Pipes Conveying Fluid: Linear Dynamics II
`196
`Introduction ........................................
`4.1
`196
`4.2 Nonanifortn pipes ....................................
`196
`4.2.1 The equation of motion ...........................
`203
`4.2.2 Analysis and results ..............................
`208
`4.2.3 Experiments ...................................
`21 |
`4.2.4 Other work on submerged pipes ......................
`213
`4.3 Aspirating pipes and ocean mining .........................
`213
`4.3.1 Background ...................................
`
`214
`4.3.2 Analysis of the ocean mining system ..................
`217
`4.3.3 Recent developments .............................
`
`220
`Short pipes and refined 110w modelling ......................
`
`4.4.1
`Equations of motion ..............................
`22?
`4.4.2 Method of analysis ..............................
`2'24
`
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`88
`88
`98
`102
`103
`I
`1 1
`1 11
`1 15
`l. 18
`123
`130
`133
`149
`150
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`4.5
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`4.6
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`4-7
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`4-3
`5 Pip
`5“
`5'2
`
`PGS v. WESTERNGECO (1PR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 5
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 5
`
`

`

`|i
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`CONTENTS
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`vii
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`225
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`4.4.3 The inviscid fluid—dynamic force
`4.4.4 The fluid-dynamic force by the integral Fourier—transform
`method 228
`4.4.5 Relined and plug-110w fluid-dynamic forces and specification
`oftheoutflowmodei...._..__..................,.
`4.4.6 Stability of clamped—clamped pipes
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`Stability of cantilevered pipes
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`4.4.8 Comparison with experiment
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`4.4.9 Concluding remarks on short pipes and refined-[low
`models
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`4.4.10 Long pipes and refined [low theory .
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`4.4.1] Pipes conveying compressible fluid ,
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`4.5 Pipes with harmonically perturbed llow .
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`4.5.1
`Simple parametric resonances
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`4.5.2 Combination resonances ...........................
`4.5.3 Experiments............................_......
`4.5.4 Parametric resonances by analytical methods
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`4.5.5 Articulated and modified systems
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`4.6 Forcedvibration.....................................
`4.6.1 The dynamics of forced vibration .
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`4.7 Applications..,.....................................
`4.7.1 The Coriolis mass—110w meter
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`4.7.2 Hydroelastic ichthyoid propulsion .....................
`4.7.3 Vibration attenuation .............................
`4.7.4 Stability of deepewater risers ........................
`4.7.5 High-precision piping vibration codes ..................
`4.7.6 Vibration conveyance and vibrationainduced llow ..........
`4.7.7 Miscellaneous applications .........................
`4.8 Concluding remarks ...................................
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`24]
`24]
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`26]
`261
`26]
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`270
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`273
`274
`275
`276
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`.....
`.....
`.....
`.....
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`.....
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`.....
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`88
`88
`98
`102
`103
`111
`Ill
`115
`] 18
`123
`130
`133
`I49
`ISO
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`I53
`153
`157
`164
`167
`168
`170
`172
`I73
`173
`174
`173
`I83
`184
`I86
`:90
`194
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`196
`[96
`195
`196
`203
`208
`2' l
`313
`213
`214
`217
`:20
`22]
`224
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`'
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`i
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`5 Pipes Conveying Fluid: Nonlinear and Chaotic Dynamics
`5.1
`Introductory comments .................................
`5.2 The nonlinear equations of motion .........................
`5.2.]
`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 lined at both ends ........
`52.5 Boundary conditions .............................
`5.2.6 Dissipative terms ................................
`5.2.7 Dimensionless equations ...........................
`5.2.8 Comparison with other equations for cantilevers
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`5.2.9 Comparison with other equations for pipes with fixed ends
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`5.2.10 Concluding remarks ..............................
`5‘3 Equations for articulated systems ..........................
`5'4 Methods 01' solution and analysis ..........................
`
`2'77
`277
`278
`279
`281
`283
`285
`287
`287
`288
`290
`294
`295
`296
`299
`
`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 6
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 6
`
`

`

`I
`
`1
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`I
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`1
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`V
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`J
`H
`
`viii
`
`CONTENTS
`
`5.5 Pipes with supported cnds ...............................
`5.5.1 The effect of amplitude on frequency ..................
`5.5.2 The post-divergence dynamics .......................
`5.5.3 Pipes with an axially sliding downstream end .............
`5.5.4
`Impulsively excited 37D motions .....................
`5.6 Articulated cantilevered pipes ............................
`5.6.1 Cantilever with constrained end ......................
`5.6.2 Unconstrained cantilevers ..........................
`56.3 Concluding comment .............................
`5.7 Cantilevered pipes ....................................
`5.7.1
`2-D iimit~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 dcgeneracies ......................
`5.8.5 Chaos in the articulated system ......................
`5.9 Nonlinear parametric resonances ..........................
`5.9.1
`Pipes with supported ends ..........................
`5.9.2 Cantilevered pipes ...............................
`5.10 Osciilation—induced [low ................................
`5.1 1 Concluding remarks ...................................
`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 incxtensible pipe ..............
`6.2.6 Equations of motion of an extensible pipe ...............
`6.3 Finite element analysis .................................
`6.3.1 Analysis for inextcnsibie pipes .......................
`6.3.2 Analysis for extensible 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 Curved cantilevered pipes ...............................
`65.] Modified inextensiblc and extensible theories .............
`6.5.2 Nonlinear and chaotic dynamics ......................
`
`302
`302
`303
`314
`315
`316
`316
`317
`327
`328
`328
`333
`336
`348
`348
`348
`366
`368
`387
`392
`394
`394
`402
`4i2
`413
`415
`415
`417
`417
`421
`426
`426
`428
`429
`430
`431
`436
`437
`438
`440
`446
`448
`451
`452
`452
`457
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`6.6 Cunt;
`6.6.1:
`6.6.2;
`
`Appendices
`
`A Fi rSt-prin
`COHVeyi II;
`
`B Analytical
`
`C Deslabiliz
`
`D Experimet
`DJ Male:
`13.2 Short
`0.3 Fiexu
`D4 Mean
`
`E The Time:
`E.l The c
`13.2 The e
`E3 The it
`
`F Some of ti
`
`Fl
`
`Lyapu
`F.1.|
`1:12
`F.1.3
`F.2 Ccntn
`E3 Norm.
`R4 Then
`E5 Bil‘urt
`E6 Partia
`F.6.1
`F.6.2
`F63
`
`G Newtoniar
`
`Conveying
`G.l Cantil
`6.2 Pipef
`
`H Nonlinear
`H.1 Ccntt“;
`H2 Norm;
`H21]
`1-1.2.23'
`
`1 P
`
`GS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 7
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 7
`
`

`

` ix
`
`CONTENTS
`
`
`
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`6.6 Curved pipes with an axially sliding end
`6.6.1 Transversely sliding downstream end
`6.6.2 Axiolly sliding downstream end .
`.
`Appendices
`A First~prineiples Derivation of the Equation of Motion of a Pipe
`Conveying Fluid
`B Analytical Evaluation of b", (:5, and d"
`C Destabilization by Damping: T. Brooke Benjamin’s Work
`D Experimental Methods for Elastomcr Pipes
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`D.] Materiais, equipment and procedures .
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`D2 Short pipes, shells and cylinders .
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`D3 Flexural rigidity and damping constants .
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`D4 Measurement of frequencies and damping
`E The 'l‘imoshenko Equations of Motion and Associated Analysis
`E.1Thecquntionsol'thion................................
`E.2 The eigenfunctions of a Timoshenko beam .
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`E3 Tltcintegmlslk”.....................................
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`F Some of the Basie Methods of Nonlinear Dynamics
`F] Lyapunovmethod....................................
`F.l.| The concept of Lyapunov stability ....................
`F.l.2 Linearization ..................................
`F.I.3 Lyapunovdirectmethod.............l.............
`F.2 Centremunifoldreduetion...............................
`F.3 Normall'orms.......................................
`13.4 Themethodofaveraging...............................
`ES Bifurcation theory and unfolding parameters .
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`E6 Partial differential equations
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`Fifil The method of averaging revisited
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`G Newtonian Derivation of the Nonlinear Equations of Motion of a l’ipe
`Conveying Fluid
`G.1C‘antileveredpipc
`G-ZPipctixedatbmhends...................._...,........
`H :?"llc';e[::fibynfl_mics Theory Applied to :1 Pipe Conveying Fluid
`>
`manifold .....................................
`H’ZNor""‘"i“’““
`H'z‘lDynlulillicinstabiliiy..............................
`H.2.28tat1eiiistahtlity......._...........,.....l......
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`459
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`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 8
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`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 8
`
`

`

`2
`
`CONTENTS
`
`I The Fractal Dimension from the Experimental Pipe-vibration Signal
`J Detailed Analysis for the Derivation of the Equations of Motion 0f
`Chapterfi
`1]
`Relationship between (x0, y”, Zn) and (x. y, z) ..................
`J.2 The expressions for curvature and twist ......................
`J.3 Derivation of the fluid—aeeeleration vector ....................
`1.4 The equations of motion for the pipe
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`K Matrices for the Analysis of an Extensible Curved Pipe
`Conveying Fluid
`References ..............................................
`Index .................................................
`
`516
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`522
`522
`523
`523
`524
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`529
`53 1
`558
`
`A word about
`question to aris
`induced vibrrttic
`Flow-induced
`but also in tote
`the point
`that I?
`they are unnoyi
`leading to the Cu:
`ntlelearreaetor
`virtually all suel
`free thereafter —
`redesigned and 5
`a book emphasi:
`induced vibr'ntt‘o
`sense of the won
`eases, the ureter
`
`cause of the orig
`time—worn hatter
`
`supports. usually
`to re—emerge um
`space: moreover.
`may actually he
`Another ausw
`of the material a
`work in the area
`
`aiming to come)
`that is fun to rem
`of the undoubted
`
`complete hibliog
`which the reader
`A second post
`By glancing thn;
`axial-flow-rclatec‘
`instability of eylj
`cross~llow—relate(
`are already well (‘
`fundamentals are
`understood; (iii) 1
`axial—flow problel
`
`
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 9
`
`PGS v. WESTERNGECO (IPR2014-00687)
`WESTERNGECO Exhibit 2046, pg. 9
`
`