Stability of Axially Moving Materials PDF
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Solid Mechanics and Its Applications Nikolay Banichuk Alexander Barsuk Juha Jeronen Tero Tuovinen Pekka Neittaanmäki Stability of Axially Moving Materials Solid Mechanics and Its Applications Volume 259 Founding Editor G. M. L. Gladwell, University of Waterloo, Waterloo, ON, Canada Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof mechanicsasitrelatestosolids.Thescopeoftheseriescoverstheentirespectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods,beams,shellsandmembranes;structuralcontrolandstability;soils,rocksand geomechanics; fracture; tribology; experimental mechanics; biomechanics and machinedesign.Themedianlevelofpresentationisthefirstyeargraduatestudent. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors.Potentialauthorswhowishtosubmitabookproposalshouldcontact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, e-mail: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink. More information about this series at http://www.springer.com/series/6557 Nikolay Banichuk Alexander Barsuk (cid:129) (cid:129) Juha Jeronen Tero Tuovinen (cid:129) (cid:129) ä Pekka Neittaanm ki Stability of Axially Moving Materials 123 NikolayBanichuk Alexander Barsuk Russian Academy of Sciences State University of Moldova Institute for Problems in Mechanics Chisinau, Moldova Moscow,Russia Tero Tuovinen JuhaJeronen Faculty of Information Technology Faculty of Information Technology University of Jyväskylä University of Jyväskylä Jyväskylä, Finland Jyväskylä, Finland Pekka Neittaanmäki Faculty of Information Technology University of Jyväskylä Jyväskylä, Finland ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid MechanicsandIts Applications ISBN978-3-030-23802-5 ISBN978-3-030-23803-2 (eBook) https://doi.org/10.1007/978-3-030-23803-2 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface In this book, we discuss a variety of problems involving analyses of stability in mechanics,focusingespeciallyonthestabilityofaxiallymovingmaterials.Thisisa specialtopicencountered, forexample,inprocessindustry applications,suchasin papermaking. In theoretical terms, the field of axially moving materials is located halfway between classical solid mechanics and fluid mechanics. The object of interest is a solid, but it flows through the domain of interest such as a particular sectionofapapermachine,motivatingtheuseofanEulerianviewpoint.Invibration problemsinmechanics,alossofstabilityisoften(butnotalways)accompaniedbya bifurcationinthecomplexeigenvaluecurvesdescribingthebehaviorofthesystem under the action of a quasistatically increasing external load. Thus in many of the chaptersof this book, we will consider bifurcationsin some form. Inouropinion,atthepresenttime,analyticalandsemianalyticalapproachesare undervalued. They can serve as a basis for fundamental theoretical understanding, butalso importantly,asabasis forfast numericalsolversforrealtimeapplications, such as online prediction and control of industrial processes. In this book, a focus on analytical approaches is a recurring theme, sometimes with a change in per- spective, or with an unconventional application of a known general result such as the implicit function theorem. Some of the results presented in this book are new, andsomehaveappearedonlyinspecializedjournalsorinconferenceproceedings. Some appear now for the first time in English. The book is organized into nine chapters. Chapters 1 through 4 discuss bifur- cations in mechanics, introducing the basic ideas, approaches and methods used throughoutthebook.Manytopicsarediscussedviatheuseofparticularexamples. Chapter 1 plays an introductory role, discussing prototype problems and bifurca- tions of different kinds. This chapter concisely summarizes the typical simplest bifurcation problems that arise in a setting of classical solid mechanics. We con- centrate on finding the eigenvalues (critical stability parameters) and eigenmodes, characterizing the shape of stability loss, of the appropriate spectral problem. Chapter 2 is devoted to the bifurcation analysis of algebraic polynomial equa- tions. The parametric representation of the solutions of considered equations and their bifurcations are considered. Bifurcation analyses of the cubic equation and v vi Preface of the fourth-order polynomial equation are presented in detail. This has applica- tionsinthestabilityanalysisofone-dimensionaldifferentialequationmodelswhose characteristic equations reduce into polynomials. Chapter3dealswithnon-conservativedynamicsystemswithafinitenumberof degrees offreedom. The system is investigated under small perturbations, charac- terized by a small parameter. The characteristic polynomial and a series expansion withrespecttothesmallparameterareusedfortheeigenvaluesandeigenvectorsto evaluate the critical stability parameter and to study ideal and destabilizing per- turbations. Sufficient conditions for stability are obtained and described. Some examplesofidealperturbationsandthestructureofthecorrespondingperturbation matrices are considered. The stability of the systems subjected to deficient pertur- bations is also investigated, and the determination of the deficiency index is pre- sented. Ziegler’s double pendulum is considered as a classical example of a non-conservative system. We give a detailed exposition, deriving the governing equations starting from the principle of virtual work, analyze the stability of the system, and consider some special cases. We present numerical simulations using new visualization techniques, in which the bifurcation behavior can be seen. Chapter 4 deals with the general methods of bifurcation analysis, applied to continuous systems, and some methods of optimization of the critical stability parameter. We will briefly introduce the different types of stability loss and then lookatconditionsunderwhichmergingofeigenvaluesmayoccur.Wewilllookat a problem where applying symmetry arguments allows us to eliminate multiple (merged) eigenvalues, thus reducing the problem to determining a classical simple eigenvalue. We then discuss a general technique to look for bifurcations in prob- lemsformulatedasimplicitfunctionals.Thisisusefulforawideclassofproblems, includingmanyproblemsinaxiallymovingmaterials.Attheendofthechapter,we will consider a variational approach to the stability analysis of an axially moving panel (a plate undergoing cylindrical deformation). Chapters 5 through 8 form the main content of the book, concentrating specif- ically on axially moving materials. We start by introducing the theory of axially moving materials in a systematic manner, the aim being to give a complete over- viewofthefundamentals.Bypresentingthetheoryasaself-containedunit,itisour hope that this chapter may especially help the student or new researcher just entering the research field of axially moving materials. Specialists, on the other hand, can benefit from the discussion on the effects of the axial motion on the boundary conditions, a topic that has received relatively little attention. This nat- urallyleadstoamixedformulation,whichbothreducesthecontinuityrequirements on the solution and clearly shows how the boundary conditions arise, contrasting theclassicaltreatmentofthetransversedeformationsofaxiallymovingelasticand viscoelastic materials using fourth- and fifth-order partial differential equations. Chapter 5 starts by considering the general balance laws of linear and angular momentum.Wederivesomegeneralequationsforbeamsandspecializethemtothe small-displacement regime. We discuss linear constitutive models for elastic and viscoelasticmaterials,andhighlighttheconnectionbetweenbeamsandpanels.We thenintroduceaxialmotioninasystematicmanner,viaacoordinatetransformation, Preface vii and discuss how this affects the boundary conditions. We consider the dynamic linear stability analysis of axially moving elastic and viscoelastic materials. We numericallylookatbifurcationsinthestabilityexponents.Asaresult,wefindthat the small-viscosity case behaves radically different from the purely elastic case. Because no real material in papermaking is purely elastic, this has important practical implications for the correct qualitative understanding of real physical systems. Chapter 6 treats bifurcations of axially moving elastic plates made of isotropic andorthotropicmaterials.Astaticstabilityanalysisisperformedtofindthecritical axialdrivevelocityandthecorrespondingshapeinwhichthesystemlosesstability. As a special topic of particular interest for process industry applications, we then lookattheaxiallymovingisotropicplate,butwithanaxialtensiondistributionthat variesalongthewidth.Itisseenthatasfarasthecriticalvelocityisconcerned,the classical simplification assuming homogeneous tension is acceptable, but the eigenmode is highly sensitive to even minor variations in the axial tension distri- bution in the width direction. Chapter 7 deals with the theoretical analysis of bifurcations of axially moving strings and beams. Critical velocities of bifurcations of the traveling material are determined for torsional, longitudinal, and transverse vibration types. The stability analysis of the axially moving string with and without damping is performed. We also comment on exact eigensolutions for axially moving beams and panels. The stability analyses of an axially moving web with elastic supports, and when sub- jected to a uniform gravitational field, are also considered. Chapter 8 deals with bifurcations in fluid–structure interaction in the context of axially moving materials. This chapter includes a brief introduction to fluid mechanics, after which we look at analytical solutions for two-dimensional potentialflowinfluid–structureinteractionwithanaxiallymovingpanel.AGreen’s functionapproachisusedtoanalyticallyderivethefluidreaction pressure interms of the panel displacement function. This simplifies the numerical problem to solvingaone-dimensionalintegrodifferentialmodel.Anadded-massapproximation of the derived solution is discussed. Numerical results based on the original exact solution are presented, for both elastic and viscoelastic axially moving panels subjected to a potential flow. The book concludes with Chap. 9, considering optimization problems in ther- moelasticity.Wefindtheoptimalthicknessdistributionforabeamtoresistthermal bucklingandtheoptimalmaterialdistributionforabeamofuniformthickness,with the same goal. Finally, we derive a guaranteed double-sided estimate that governs energy dissipation in heat conduction in a locally orthotropic solid body, which holds regardless of how the local material orientation is distributed in the solid body. Connecting to the main theme of this book, this has applications in the analysisofheatconductioninpapermaterials,whichisimportantwhenconsidering the drying process in papermaking. This book is addressed to researchers, specialists, and students in the fields of theoretical and applied mechanics, and of applied and computational mathe- matics. Considering topics related to manufacturing and processing, the book can viii Preface also be applied in industrial mathematics. We hope that contents should also be of interest to applied mathematicians and mechanicians not currently in these fields, who may nonetheless be stimulated by the material presented. It is our hope that the various solution techniques touched upon across the chapterswillbenefitthereader,whetherintheiroriginalcontextorinanunexpected newapplication.Wealsohope,viadetailedexpositionandexamples,tohavemade the theory of axially moving materials more accessible to a new generation of researchers. Although the field of axially moving materials was established over a century ago, new exciting applications await, for example, in printable electronics andmicrofluidics.Thismakesthefieldworthyofstudynotonlyfromafundamental academicviewpoint butalso for those primarily interestedin applications. Moscow, Russia Nikolay Banichuk Chisinau, Moldova Alexander Barsuk Jyväskylä, Finland Juha Jeronen Jyväskylä, Finland Tero Tuovinen Jyväskylä, Finland Pekka Neittaanmäki Acknowledgements The research presented in this book was supported by the Academy of Finland (grant no. 140221, 301391, 297616); RFBR (grant 14-08-00016-a); Project RSF No. 17 19-01247; RAS program 12, Program of Support of Leading Scientific Schools (grant 2954.2014.1); Jenny and Antti Wihuri Foundation. The authors would like to thank Matthew Wuetrich for proofreading. For discussions and valuable input, the authors would like to thank SvetlanaIvanova,EvgenyMakeev,ReijoKouhia,TyttiSaksa,andTuomoOjala. Contents 1 Prototype Problems: Bifurcations of Different Kinds . . . . . . . . . . . . 1 1.1 Rigid Column with Elastic Clamping. . . . . . . . . . . . . . . . . . . . . . 1 1.2 Elastic Column and Its Optimization . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Elastic Rod Under Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Divergence and Optimization of Wings . . . . . . . . . . . . . . . . . . . . 18 1.5 Stability of Tensioned Cantilever Beam. . . . . . . . . . . . . . . . . . . . 22 1.6 Accelerating Motion of Rod (Rocket, Missile) Under Follower Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Bifurcation Analysis for Polynomial Equations. . . . . . . . . . . . . . . . . 33 2.1 Bifurcation and Parametric Representations . . . . . . . . . . . . . . . . . 34 2.2 Analysis of a Cubic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Analysis of a Quartic (Fourth-Order) Polynomial Equation . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3 Nonconservative Systems with a Finite Number of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1 Critical Parameters and Destabilizing Perturbations. . . . . . . . . . . . 69 3.2 Characteristic Polynomial and Series Expansions . . . . . . . . . . . . . 71 3.3 Ideal Perturbations and Sufficient Conditions for Stability (n¼2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4 Matrices and Examples of Ideal Perturbations . . . . . . . . . . . . . . . 75 3.5 Stability of Systems Subjected to Deficient Perturbations and Determination of the Deficiency Index . . . . . . . . . . . . . . . . . 78 3.6 On the Stability and Trajectories of the Double Pendulum with Linear Springs and Dampers . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.1 Problem Setup and Derivation of the Model . . . . . . . . . . . 80 3.6.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6.3 Nondimensional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 ix
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