Dynamics and Simulation of Flexible Rockets 152 x 229 mm paperback | 10.6mm spine 9780128199947 Dynamics and Simulation D y n a of Flexible Rockets m i c Dynamics and Simulation s Timothy Barrows and Jeb Orr a n d of Flexible Rockets Dynamics and Simulation of Flexible Rockets provides a full state, multi-axis treatment S i of launch vehicle flight mechanics and provides the state equations in a format that can m be readily coded into a simulation environment. Various forms of the mass matrix for the u vehicle dynamics are presented. This book also discusses important forms of coupling, such la as between the nozzle motions and the flexible body. ti Timothy Barrows and Jeb Orr o n This book is designed to help practicing aerospace engineers create simulations that can o accurately verify that a space launch vehicle will successfully perform its mission. Much of f the open literature on rocket dynamics is based on analysis techniques developed during F the Apollo program of the 1960s. Since that time, large-scale computational analysis l e techniques and improved methods for generating Finite Element Models (FEMs) have x been developed. The art of the problem is to combine the FEM with dynamic models of ib separate elements such as sloshing fuel and moveable engine nozzles. The pitfalls that l e may occur when making this marriage are examined in detail. R o • Covers everything the dynamics and control engineer needs to analyze or improve the c k design of flexible launch vehicles e • Provides derivations using Lagrange’s equation and Newton/Euler approaches, allowing t s the reader to assess the importance of nonlinear terms • Details the development of linear models and introduces frequency-domain stability analysis techniques • Presents practical methods for transitioning between finite element models, incorporating actuator dynamics, and developing a preliminary flight control design Timothy M. Barrows has worked for 35 years at Draper Laboratory as a dynamicist. Early work involved analyzing the dynamic interaction between the attitude control system of the Space Shuttle and a heavy payload on its remote manipulator arm. More recent work included developing simulations for several rocket programs, most notably NASA’s Space Launch System. Dr. Barrows received a BSE in aerodynamics from Princeton and an MSE and PhD in mechanical engineering from MIT. B a Jeb S. Orr serves as Principal Staff, Flight Systems and GN&C Technical Director for Mclaurin rro w Aerospace, a small business headquartered in Huntsville, Alabama. Prior to joining s Mclaurin, Dr. Orr held technical staff positions at Draper Laboratory and SAIC. He has • O supported various research and flight development programs with an emphasis on launch rr vehicle dynamics and control. Dr. Orr received a BSE in computer engineering and an MSE and PhD in control from the University of Alabama in Huntsville. ISBN 978-0-12-819994-7 9 780128 199947 Dynamics and Simulation of Flexible Rockets_AW1.indd All Pages 02/12/2020 14:51 DYNAMICS AND SIMULATION OF FLEXIBLE ROCKETS This page intentionally left blank DYNAMICS AND SIMULATION OF FLEXIBLE ROCKETS TIMOTHYM.BARROWS JEBS.ORR Coverphoto:TheSaturnIBSA-205launchvehiclecarriesthefirstcrewedApollospacecraftinto orbitonOctober11,1968.ThisphotographwastakenfromtheAirborneLightweightOptical TrackingSystem(ALOTS)aboardaspeciallymodifiedC-135aircraft.(NASA) AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2021ElsevierInc.Allrightsreserved. MATLAB®isatrademarkofTheMathWorks,Inc.andisusedwithpermission. TheMathWorksdoesnotwarranttheaccuracyofthetextorexercisesinthisbook. Thisbook’suseordiscussionofMATLAB®softwareorrelatedproductsdoesnotconstitute endorsementorsponsorshipbyTheMathWorksofaparticularpedagogicalapproachorparticular useoftheMATLAB®software. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek permission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangements withorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency, canbefoundatourwebsite:www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedical treatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein.In usingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyof others,includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors, assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products, instructions,orideascontainedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-819994-7 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:MatthewDeans AcquisitionsEditor:CarrieBolger EditorialProjectManager:FernandaA.Oliveira ProductionProjectManager:KameshRamajogi Designer:MarkRogers TypesetbyVTeX Contents Acknowledgments vii 1. Introduction 1 2. Thesystemmassmatrix 9 2.1. Problemformulation 9 2.2. Structuraldynamics 15 2.3. Kineticenergy 25 2.4. Lagrangianaccelerations 29 2.5. Assembledequationsofmotion 32 2.6. Reducedbodymodes 39 2.7. Truncatingthesloshmotion 48 3. Sloshmodeling 53 3.1. Fluidmechanicsmodel 56 3.2. Springsloshmodelwithnonlinearterms 59 3.3. HydrodynamicmodelintheFEM 65 3.4. Summaryofhydrodynamicmodels 74 4. Pendulummodel 77 4.1. Generalpendulummodel 77 4.2. Motionequations 81 4.3. Sloshdynamicsusingthependulummodel 93 4.4. Nozzledynamicsusingthependulummodel 102 5. Forcesandtorques 109 5.1. Externalforcesandtorques 109 5.2. Fuelandnozzleoffsettorques 125 5.3. Slosh,engine,andbendingexcitation 126 5.4. Summaryofexcitationterms 138 6. Engineinteractions 143 6.1. Thetail-wags-dog(TWD)zero 143 6.2. Engine/flexinteraction 146 6.3. Definingthefiniteelementmodel 159 6.4. Bendingfrequencyshiftduetothrust 164 7. Linearization 175 7.1. Scalarequationsofmotion 176 v vi Contents 7.2. State-spacemodel 188 7.3. Distributedaerodynamics 195 8. Simulationparameters 207 8.1. Thrustdispersions 208 8.2. Finiteelementparameters 209 8.3. Transitionbetweenfiniteelementmodels 222 9. Stabilityandcontrol 233 9.1. Problemformulation 234 9.2. Designmethods 240 9.3. Actuationsystems 265 9.4. Stabilityanalysis 270 10.Implementationandanalysis 285 10.1. Numericalintegration 285 10.2. Constraints 288 10.3. MonteCarloanalysis 294 A. Listofsymbolsandacronyms 299 B. Quadruplevectorproduct 305 C. Finiteelementmodelunitconversions 307 D. Second-ordercoordinatetransformation 309 E. Angularmomentumoffree-freemodes 315 Bibliography 317 Index 319 Acknowledgments The authors are indebted to the many people that helped make this work possible.Wewouldliketothankourpresentandpastfriendsandcolleagues in the dynamics and control community at NASA’s Marshall Space Flight Center, Langley Research Center, Armstrong Flight Research Center, and theNASAEngineeringandSafetyCenter.Wemakenoattempttolisttheir names as they are too numerous. The support of systems engineers and managers during the NASA Constellation and Space Launch System programs was helpful in the ad- vancementandstandardizationofmethodsandsoftwaretoolsforanalyzing large rockets. In addition, we would like to acknowledge the many lively discussions we enjoyed among the technical staff during our tenure at the Charles Stark Draper Laboratory. Finally, we would like to recognize the contributions of Mr. Rekesh Ali, who as a graduate student researcher, contributed significantly to the typesetting of this book. vii This page intentionally left blank CHAPTER 1 Introduction Rockets, like most things, become more complicated as they grow larger. Judging from the similarity of external appearance, it might seem that go- ing from a small rocket to a large rocket would be a simple extrapolation accordingtosize.However,thisisnotthecase.Someideaofthereasonfor theaddeddifficultycanbeobtainedfromthefollowingquotefromJ.B.S. Haldane: ... consideragiantmansixtyfeethigh–abouttheheightofGiantPopeand GiantPaganintheillustratedPilgrim’sProgressofmychildhood.Thesemon- sterswerenotonlytentimesashighasChristian,buttentimesaswideand tentimesasthick,sothattheirtotalweightwasathousandtimeshis,orabout eightytoninetytons.Unfortunatelythecrosssectionsoftheirboneswereonly ahundredtimesthoseofChristian,sothateverysquareinchofgiantbonehad tosupporttentimestheweightbornebyasquareinchofhumanbone.Asthe humanthigh-bonebreaksunderabouttentimesthehumanweight,Popeand Paganwouldhavebrokentheirthighseverytimetheytookastep.Thiswas doubtlesswhytheyweresittingdowninthepictureIremember.Butitlessens one’srespectforChristianandtheGiantKiller. Inthisexample,increasingthebonecrosssectionbyafactorofahundredis notenough–itmustbeincreasedbymorethanahundred.Inotherwords, the structural weight fraction must be increased. In the design of rockets, however, the mere suggestion of increasing the structural weight fraction will produce themost pained anguish. A good portion of this extra weight will be taken out of the payload. As a typical payload weight is less than ten percent of the total rocket weight at launch, it is easy to see how the payload can disappear entirely without a stringent effort to minimize the structural weight. The result is that the design of large rockets becomes an almost desperate effort to improve structural efficiency. Fromadynamicstandpoint,asthescaleincreases,therocketgrowsflim- sier and flimsier. The natural frequencies of more and more flexible modes creepdownwardintoarangethatiswithinthecontrolbandwidth.Theop- portunities for dynamic interaction proliferate. The control engineer must verifythatalloftheseinteractionsarebenignandstable.Doingthisrequires DynamicsandSimulationofFlexibleRockets Copyright©2021ElsevierInc. 1 https://doi.org/10.1016/B978-0-12-819994-7.00006-6 Allrightsreserved.