Table Of ContentControl of Fluid-Containing
Rotating Rigid Bodies
CommunicationsinCybernetics,SystemsScienceandEngineering
ISSN:2164-9693
BookSeriesEditor:
JeffreyYi-Lin Forrest
InternationalInstituteforGeneralSystemsStudies,GroveCity,USA
SlipperyRockUniversity,SlipperyRock,USA
Volume 2
Control of Fluid-Containing
Rotating Rigid Bodies
Anatoly A. Gurchenkov,Mikhail V. Nosov &
Vladimir I.Tsurkov
Department of Complex Systems,Computing Center of the Russian
Academy of Sciences,Moscow,Russia
CRC Press
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Version Date: 20130410
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Table of contents
Editorialboard vii
Notations ix
Abouttheauthors xi
Introduction 1
1 Controlofarotatingrigidbodywithacavitycompletelyfilled
withanidealfluid 13
1 Equations of motion of a rigid body with a cavity completely filled
withanidealincompressiblefluid 14
2 Stabilityofthesteadyrotationofasolidbodywithacavitycontaining
afluid 18
3 The dependence of the angular velocity of the perturbed motion on
themomentofexternalforces 25
4 An equivalent system of equations convenient for the consideration
ofoptimalcontrolproblems 27
5 Anexamplewithdiscontinuouscontrol 29
6 ApplicationofBellman’soptimalityprinciple 31
7 Reductionofthemainrelationtoafourth-ordersystem 34
2 Controlofarotatingrigidbodycontainingafluidwithfreesurface 37
1 Statementoftheproblem 38
2 Smalloscillationsofaviscousfluidpartiallyfillingavessel 40
3 Rotationalmotionsofarigidbodywithacavitypartiallyfilledwith
afluid 45
4 Linearizationoftheproblem 46
5 Hydrodynamicproblem 47
6 TheBubnov–Galerkinmethod 50
7 Stabilityofthefreerotationofabody–fluidsystem 52
8 Equationsofmotionofabody–fluidsysteminanequivalentform 58
3 Oscillationsofaplateinaviscousfluid: Theflatwallmodel 61
1 Anunsteadyboundarylayeronarotatingplate 62
2 Longitudinalquasi-harmonicoscillationsoftheplate 65
vi Table of contents
3 Boundarylayerstructure 67
4 Tangentialstressvector 69
5 Oscillatorysolutions 71
6 Oscillationsofaviscousincompressiblefluidaboveaporousplatein
thepresenceofmediuminjection(suction) 72
7 Motionoftheplatewithaconstantacceleration 76
4 Controlofarotatingrigidbodycontainingaviscousfluid 81
1 Smalloscillationsofaviscousfluidcompletelyfillingavessel 82
2 Equationsoftheperturbedmotionofabodywithacavitycontaining
aviscousfluid 87
3 Coefficientsofinertialcouplingsofarigidbodywithafluid:
Thecaseofacylindricalcavity 93
4 Oscillationsofaviscousincompressiblefluidinacavityofa
rotatingbody 99
5 Internalfrictionmomentinafluid-filledgyroscope 104
6 Stabilityofafluid-filledgyroscope 108
7 Equationsofmotionforarigidbodywithacavityequippedwith
fluidoscillationdampers 115
8 Anintegralrelationinthecaseofaviscousfluid 120
9 Anequivalentsystemintheoptimalcontrolsetting 123
Conclusion 127
References 129
Subjectindex 145
BookSeriespage 147
Editorial board
MichaelC.Jackson,UniversityofHull,UK
JerzyJozefczyk,WroclawUniversityofTechnology,Poland
DonchoPetkov,EasternConnecticutStateUniversity,USA
VladimirTsurkov,RussianAcademyofSciences,Russia
ShouyangWang,ChineseAcademyofSciences,P.R.China
ADVISORY EDITORIAL BOARD
C.L.PhilipChen,UniversityofMacau,P.R.China
ZengruDi,BeijingNormalUniversity,P.R.China
RaulEspejo,SynchoLtd.andWorldOrganizationofSystemsandCybernetics,UK
KeithW.Hipel,UniversityofWaterloo,Canada
BaodingLiu,TsinghuaUniversity,China
NagendraNagarur,StateUniversityofNewYorkatBinghamton,USA
JohnPourdehnad,UniversityofPennsylvania,USA
BrianHowardRudall,InstituteoftheWorldOrganisationofSystemsand
Cybernetics&BangorUniversity,UK
RudolfScheidl,JohannesKeplerUniversityofLinz,Austria
MarkusSchwaninger,InstituteofManagement,UniversityofSt.Gallen,Switzerland
Notations
x scalarvariablex;
y vectorvariabley;
J,B,L tensorsandoperators(matrices);
(x,y) scalarproductofthevectorsxandy;
x×y vectorproductofthevectorsxandy;
x˙ timederivativeofx;
x¨ secondtimederivativeofx;
f(cid:1),f(cid:1)(cid:1) spatialderivativesoff;
u∗ complexconjugatetou;
V initialvalueofthevariableV intheCauchyproblem;
0
(cid:3)x(cid:3) Euclideannormofthevectorx;
i =1,...,n i rangesfrom1ton;
u∈U ubelongstothesetU;
u∈/ U udoesnotbelongtothesetU;
U ⊂H U isasubsetofthesetH;
Rn Euclideann-space;
[a,b] closedinterval;
[a,b) half-openinterval;
(a,b) openinterval.
