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3-2 D'ALEMBERTS PRINCIPLE AuuSυ s TO CoNVI∈ RT DYNAMICS TO STATIcs TREAT THIS AS升“FRcE”工NTHE STATIC FORCE BALANCE MASS TIMEs A REVERSED ACCELERATION k FORM F60 m又kx D'ALEMBERT FORCE 弓00 FORCE BALANCE STATiCS F-k×-c义-Mx=D ∑F=0 SIMPLIFIES FORMVLATIDN OF EOM
3-2 EXAMPLE:ROTAT(N& OISK (MGR)WITH AMAss NvER6BY拜 SPRIN 6 R A55UME CIkTANT ANGULAR RATE w K R。↓ MASs M HELD BY 2 sPAWGS ATMM| JAL POsIT10人R。FR ENTER-升000 AL MOTI0 DENOTED BY R DEFINE SEcONO FRAME ATTACHE 0 TO MGR WITH X-AxIs (x")IN DRECTION of THE SPRINGs A·R+R 0N0 TE ThE P6τtoN OF THE MASS. R, N NEVTRAL DIS PLACEMENT F SPRINGs WILL ASSUME THAT MAss SLIDES T A RADt AL SLoT FRICTION WITH MASS M0T1D小F“R", THE TWD SPRIN6S WILL EXERT A RESTORING FoRcE oF -2KR WHICH WILL ACT IN THE X DIRECTION 7 IN THE SECONO FRAME (CALL TISM") R
LET US FIA0EA6s6LT升 CcELER升T0Ab F THE MASS AND WRITE THAT IN THE SECowD FRAME AS WELL 汝F+25x(“+5x(x6) →C5E1U5EmEM"FRA吣 E Now U.5E MATRIX NOTATIoN 2 十山AWf R 0 D R-W(R+R) 2RW KNOW THAT M=mA文 2Kk.n(-“)R+(2k-u D 2 2wR MAss MOTION- COMP A RED TD MOTION of MASS WIMMOUT THE MGR
3-4A sLτo心oFR+2R。R。u2 u?0 OF THE FoRM RAL),RW A Cost A St Ao. A, RELATED To THE INITIAL CONDITIONS RlO) ANo &(o) SOLUTION IS SINVSo IDAL OSCILLATION OsCILLATION FREQvENCy AT訓A四T创0sTvE√ SINCE 2K 2k女小AT以 AL FREQ0F0C70 NO SPIN BuTN0 E THAT THERE I升sNR山 FOR WHICH 02 r w> Wc SOLUTION CHAA∈ESTe 810
EXAMPLE: MoRE ON FRAME SELECTION MGR ROTAT IAG AT RAT MASS LESS STRING WRAPPED RRA0MGR,山1 TH A MASS吖 ATrACHE O MGR RADIUS a MGss工 WITIALLY LOCATE0Hr“A: 工小1ME七, POINT A HAS ROTATED 只0T%· AND TE MAS5Hh5 SWUNG OUT T0THE WMAT Is THE ACcELERAToN of OIFFERENTAL ERUATU FoR > VERY SIMILAR To SPACECRAFT DEVICE USE0 TO SLOW SPIN · FRAME SELECT0,-N0TETH升uEc升 ASSUME T奸 E STRING IS的川∈E川P川T8, NEED To TRACK PoiNT 8 AND WAAT AN EAsy 认 AY TD SPEc1 Fy LECAToN oF Mm CHODSE SECOND FRAME So THAT Y-AxIs PASSES THROVGH B FRoM 0 POSITION oF RT 小 a FRAME ASSUME ONLY FoRCE ACTING ON THE Mss Is THE TENSIOI THE Rofe f
3-6 soFR叫,6u以HTx5 THE ANGVLAR KATE BETWEEN THESE TWo FRAMES LET BE THE ANGLE FRoM 1x To 2y ANGLE FR0AAT。A √lt AuGLE FRo从A 6+ MN.了· LEUGTH OF STRING UNWOUND→keEm FROM →合=比+上 ABOUT Z-AxIs R+xF+2bx产+心 CM0 E TO WRITE THIS工 V FRAME2→咋EM升TKX FORM + 0-Phk a+2 + T 0 a+2
50,FR WE MUST SoL√E 广(M+子)-(九+去) +2 a VL 以七 七+C AT七 七+C NTEGRATE BTH SIDES L50 士a七 WOULD Y0U FWD T? SOLUTLoN SIMPLI FIEd BY APPRoPRIATE SELECTION oF FRAME2
EXAMPLE RoTATIN& PLATFoRM CARRYING ATUBE WITH A MASS IN IT THAT IS HELD BY丹sRuG, THE PL肝 FORM TS ROTATWNG AT A GIEN RAT巩AN0 THE TBE y CAN OS THE MGR WITH THE 27 RT升 CH FRAME⊥T0 ￥- kIs ALIG小E0b1H THE TUBE SELECT丹s6coN0 FRAME TH,T工5升TT件cHE0T 牦 TUBE. GET FRoM FRAME4T02wT丹 ROTATIon OF 9 AlouT 4,(2x) ASSUME THr4 MHE NEUTRAL角5千 R THE SPRING 千=0→AA5 LOCAT10N以RTo工s τ0 TAL ANGULAR VELDCITY0 F FRAME2凵 RT INERT阳L PLAN To COM PVTE ACCELERATIONS ANO REPRESENT THEM IN THE 2-FRAME ( ROTATION6Y6 C S 况se Se C9M 0 √c0B