《航空航天动力学》(英文版) LECTORE 14 GYRoScoPES

Y RoscoP∈5 Moo.FIED TRANSPO RT THEoREM PRECESS0小

GY RoScoPES UPTo NoW HAVE CONSIDE RED PROBLEMS RELE VANT To THE RIG ID 6oDY 0YNAMICS THAT ARE IMPORTANT To AERoSPACE VEHI CLES USEO A BoDY FRAME THAT RDTATES WITH THE VEHICLE ANOTHER IMPORT ANT CLASS oF ARo BLEMS FB0 ES SUCH A5 Gγ Ro ScopEs RoτcRuV啊 HIGH SPIN RAT∈ ESSENTIALLY MASSLESS FRAME (CARDAN) MASs CENTER FIxED BUT ROTOR CA ASSUME ANY DRIENTATION NATURAL工NTH5CASE TO USE A R6τ4TG SET oF CooRDINATES ATTACHED To THE INNER GIMBAL. G Now FRAME oF REFERENGE NOT ACTUALLY ATTACHEDTo THE BoDY( ROToR) 今 NEED To M0 IFY W USE0工 N TRANS PORT T

2-2 R∈ CALL T认 AT WE H0 A=王 AND WE SAID THAT R。R6+×H uTH8E|M后∈A6oLr∈ GULAR VELOCITY OF THE BDDY BY ASSUMPTION,山E升 LSD HAD ·工 THE GYR。CAsE,wE 5 EE THAT THER∈ARE TWo ANGULAR VELOC|T∈s F工N∈5T. 工 N TERMS6FT呢EUER ANGLE RATEs比HAVE 1 AL ANG、ecrY: :江2+6z ANEULARVELOCITY F⑩NRGM毗LAx 。中1y+ TRANS PORT THEDREM IN THIS CAsE: 6+亢x

To PRoce∈D MUST WRITE W AND NG THE BASIS VECToRs OF THE INNER GMBAL FRAME in e 2+c0sθ siNe 2xt Cos B 2y+W2 中52x +(+中cse) AND s62×+ e2,+中c05° No U ANGULAR MOMENTUM G小REM升550FGm8ALs AND ASSUME Ts MoMENT of INERTiA ABoUT sAN AXIs Df RDTDR 工七~800 T TRANSVER5E升s 中SNe NOTE: RotoR 工 "s MOING 0RT“” FRAME 工 y+ 4 cose 6ur00 -工tq5N6 8 YMMETRY,工 工 S CONSTA Th e 七 工s(+φcose)

2-4 ·50,I飞RM5 OF THE FRAME ATTACHE0T E工NN织GM6AL 工七(中5N6+c∞b) s(甲+C06-中65 cose sinE GG osb 中Ne SiNe sci+case) x=-工(5+26c6)+工56(+本C6) 工t(6- Cose siwe)+swe(+=) M2=工。(甲+c656-白日5NB) NONLINEAR 2 ORDER FULLY CoUPLED ) MRD节0suv∈fR中,6,中 GIVEN Mx,鬥y,Mx L0 K AT SOME sP∈ CIAL CASEs

-5 STEA0y PRECESSION EQUATIOAS Of MDTIOw FAR Too CoMPLEx T5 5oLvE小GEN∈风升L. 工 TERESTING SU8-PR0LEM -osin ei ASSUME )ANGLE .(NUTATION)CONSTANT. 2) ANGLE RAT∈φ( PRECESS0RTE) CONSTANT ROTOR SPIN Y CoNSTANT. eel M 2)φ 3V=C3 E D Au0y工(-2cos65)+工5申sNB(+49) 中5u[-:(种+C)-+C FOR A GWEN My,工5,工七 WE CAN PREDICT THE PRECESSI DN RATE! MAT工F6=qo

THE CASE THE ROToR SPw AXI5工5 Couple axis 工 N THE HoRz0 ATAL PLANE A Y 33 s N0 W EXPLICIT THAT工F比EAPY. MOMENT T0 A GYRD SCoPE A BoUT AN AXIS PERPENDICULAR To ITS AXIs OF SPIN THE GYRo ScDPE WILL PRECESS ABOUT AN AXIS PERPENDICULAR To BOTH 1E5 PIN AXIs AN0EAM1T升x. TORQUE ABou XIS PRECE SS ABoUT SPIN ABOUT 2, AxIS D ION OF PRECESS HON; A USES POSITIVE ENO OF SPIN AxIS T0RTT∈TouR0sP05v乇N00F MOMENT AxIs

2-1 BSERVATID心 SINCE 工 FoR A GEN EXTERN AL MOMENT,THE GREATER THE SPIN (Y),TME SLOWER THE PRECESSION() SPIN STA BILIZATION 2 BEcAUSE oF THE RELATIVELY LARGE CouPLES REQUIREO TO CHANGE THE ORIENTATION of THE SPIN AXLE. G ROScoPES CAN BE USEO TD STABILIZE TORPEDOES AND SHIPS SEE EXAMPLES