Derivation of the differential equation of motion

Transcription

1 Divion of h iffnil quion of oion Fis h noions fin h will us fo h ivion of h iffnil quion of oion. Rollo is hough o -insionl isk. xnl ius of h ll isnc cn of ll (O) - IDU s cn of gviy (M) θ ngl of inclinion of IDU olling ngl of h ll β olling ngl of h IDU γ slop of h goun O oigin: fix on goun poin wh h ll ouchs h goun cn of h ll cn of ss of h IDU i, j fnc uniy vcos, η uniy vcos fo inclin goun uniy vco in h icion ss of h ll ss of h IDU I ini of h ll ou ini of h IDU ou I s of i n j. θ β j i O γ Figu 1: Noions β Th son fo inoucing β lis in h fc h i is his ngl, which is conoll y h oos, n no θ, which is n ipon p of h sys u no icly conoll. In ou clculions will poc s follows: using f oy ig, h ngul onu lnc will win ou poin n solv h oin xpssion fo. Th uni vcos, η n fis xpss in θ η Rfnc lin ch o h ll cosγ sinγ i, j sinγ η cosγ i, j sinθ sin( θ γ ) cosθ cos( θ γ ), η i, j (1) Th inics inic, o which s h vco fs. In h following, if nohing ls is nion, vcos lwys f o h i, j s

2 g g η U S Figu : Th f oy ig of Rollo Th f oy ig is shown in figu. Th h focs cing on h ll n IDU (hough of s on sucu) h wigh of h ll, g, h wigh of h IDU, g n cion foc of h goun, S, h is no known. Expssing h ngul onu lnc ou poin will sn l, hn S is no n o clcul. Th ngul onu lnc ss, h h su of ll oqus h c on h sys is qul o h of chng of ngul onu of h sys, ll of his liv o poin in spc. If no oqu is ppli, h ngul onu (h spin of h sys) will no chng. If, on h oh hn, h is oqu, h spin will ih cs o incs, pning on h icion of h oqu. Mhiclly, h ngul onu lnc is win lik his: M H & () o, showing h finiions i F (3) i / i i/ i i i wh H is h ngul onu, M sns fo onu, h o inics h i iviv n is h fnc poin w will consi. Sinc h foc F in figu cs on, i os no c oqu ou his poin. Th is why i won pp in h following quions. Fis h su of ll ons clcul, M g j g j η g j η g j / ( ) / ( ) ( ) ( ) ( ) (4)

3 wh fo xpl / is h vco poining o n hving s is oigin. Susiuing h spciv xpssions fo η n (1) yils h following sul: M [ ( ) g sinγ g sin( θ γ )] k (5) onsiing h igh-hn si of h ngul onu lnc quion 1 & & & H H( ll) H( IDU ) [ ( ) && ] [ ( ) && I k I k] / / θ (6) wih n ing h cclion in poin n, spcivly. Thy cn clcul s follows using (1): ( O O / ) ( / / ) cos γ ( η) && && sin γ (7) ( O) ( O ) && / / / / ( ) cos γ && θ cos( θ γ ) θ& (8) && sinγ && θ sin( θ γ ) θ& cos( θ γ) 0 Susiuing (7) n (8)ck ino (6), xpssing s lik / in s of, η n n hn finlly using (1), yils: & && H ( I ) k { sin( θ γ ) sin γ} k { && sin && sin( ) & γ θ θ γ θ cos( θ γ )} { cos( θ γ ) cos γ} { && cos && cos( ) & γ θ θ γ θ sin( θ γ )} k I && θk (9) Now h suls cn plugg fo & H n M ck ino h ngul onu k o g h iffnil lnc quion () n o-uliply oh sis wih quion of oion of h oo. f qui lnghy pocss of siplifying n cncling s, his is wh pops ou:

4 [ I ( cosθ) ] θ&& [ I ( cos θ] && θ& [ sin θ] g ( )g sin γ 0 (10) Full ol of h ll oo If o sophisic syss wn o conciv o s h ll s ovn, full scl ol of h oo is n h inclus oos, ucion gs, i.. vyhing. Sinc h oos hv n conoll ov h volg ppli vi pulswih-oulion, i woul nic if h syss cion coul xpss o ny V oo (), h is, quions n which yil, &, θ n so on in s of h only inpu V oo (). If his cn chiv, h Rollo s ovn cn pic fo ny inpu. F oy igs of h ll cov n h IDU shown in figu 3. No h now hs wo sp ois h inc wih ch oh whil in h pvious nlysis h oo hs n consi s whol. F1 F 1 g F F g S Figu 3: F Boy Digs of Rollo s cov (lf) n h IDU (igh) Th f oy ig (FBD) fo h ll consiss of 4 focs: h wigh of h ll g, foc F 1 wn ig ck whl n IDU, h cion foc of h goun S n h iving foc F h h oos supply. Th whls of h oos cully un on ck whls wih ius. Th oo on his si ovs gs gins h fix ck whl cusing h wigh of h ll (IDU) o shif, which ks Rollo ov fow (lik hs whl). Th ck whl on h oh si is f o ov. oo on his si uns h ck whl ch o sll g whl on h i uning Rollo. Th IDU s FBD is shown on h igh of Figu 3. No h h only 3 focs h: h IDU s wigh g, foc F 1 wn ig ck whl n IDU n gin F. Now F cn clcul using ngul onu lnc ou poin (u his i of h IDU only). F 1 osn coniu o h su of ons.

5 M H & (11) IDU ( gj) F k && θi k ( ). (1) Using (1), o-uliplying wih k n solving fo F F θ&& (I ) && cosθ g (13) nowing h h whls hv ius w n h ucion io of h oo s g is, h oqu liv y h oos is givn y τ oo (14) w [ θ&& (I ) & cosθ g ] This oqu cn now lso xpss in s of h volg ppli o h oo n i s ngul vlociy {1}. Sinc τ oo I (15) wh is oo consn n I is h u cun n lso V ω β & (16) oo w wh is scon oo consn n V is h u volg, h oo volg is givn y τ oo Voo RI V R w β &. (17) H, R is h lcic sisnc of h oo. Solving fo τ oo yils τ oo Voo w R β & (18) Now (14) cn s n (18) qul o on noh n solv fo β && (hin insi θ && of (14)) o oin

6 β && V oo R β& w && w [ I ( cosθ) ] I g (19) (10) ogh wih (19) now sci h sys in wo scon o nonlin iffnil quions. No wy nyon cn solv h nlyiclly, u i cn gin us s spc o nuiclly clcul soluions. Th fou s vils, v &, β n v β β & inouc. S quions shown low: v& V θ& oo sin θ I β& w wr I I g ( ( g )g sin γ cosθ) (0) k& v& β V oo β& w wr && [ I ( cosθ) ] I g (1) & &β v v β Th k & ll o h igh in (0) hs gin n o ccoun fo viscous ficion. If his is clos o liy is noh qusion. Bu sill n ovll viscous ficion ss sonl n is sy o ipln.