The motions of the celt on a horizontal plane with viscous friction

Transcription

1 The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and Technology Sae Universiy), munisyna@gmail.com ABSTRACT The problem of he moion of a cel on a fixed horizonal plane wih viscous fricion is considered. On he plane of he parameers of he problem, regions of sabiliy of uniform roaions abou he verical are consruced. The dynamics of ransien processes from unsable moions o sable ones is sudied. 1 Inroducion The Cel is a convex solid body, one of is principal cenral axis of ineria being perpendicular o he surface of he body, and he direcions of he principal curvaures of he surface a he poin of inersecion wih his axis are no parallel o he oher wo principal axes. I is well known ha he sabiliy of he roaions of his body around he verical axis depends from he direcion of roaion. In mos of he papers devoed o his propery, he non-holonomic formulaion of he problem is considered;i is assumed ha he velociy of he poin of conac beween he body and he plane is zero see, for example, [1], []). In he paper [3] he moion of he cel on a plane wih fricion is considered, and he consisency of his formulaion of he problem wih full-scale experimens is confirmed. In he presen paper, he invesigaion of [1] is coninuing, in which i is assumed ha he viscous fricion force acs on he sone from he side of he plane. This model of fricion allows us o carry ou no only numerical, bu also analyical sudies in he problem. In addiion, when he coefficien of viscous fricion srives for infiniy, he force of viscous fricion is realized he non-holonomic consrain []. Saemen of he problem We will inroduce he following variables: v is he velociy of he mass cenre of he cel, ω is is angular velociy and γ is he uni vecor of he rising verical. The slipping velociy is given by he relaion u=v+[ω,r], where r is he radius vecor of he poin of conac of he body wih he plane, defined by he equaion γ = grad fr)/ grad fr). A graviaional force P= mgγ, he normal componen of he reacion of he suppor plane N=Nγ and he fricion force F= mku k is he coefficien of viscous fricion) ac on he cel. The equaions of is moion in he moving coordinaes have he form m v+[ω,mv]=n mg)γ+ F 1) J ω+[ω,jω]=[r,nγ+ F] ) γ+[ω,γ]= 3) v+[ω,r],γ)= ) where J=diagA 1,A,A 3 ) is he cenral ineria ensor or he cel. Eq. 1)) is he heorem of he change of he momenum of he cel, Eq. ) is he heorem of he change of he angular momenum, Eq. 3) is he condiion for he vecor γ o be consan in an absolue coordinaes, and Eq. ) is he condiion for he cel o be in conac wih

2 1 ω - κ Fig. 1: A 1 = 3 kg m, A = kg m, A 3 = kg m, a 1 = m, a = m, a 3 = 3 m, m=1 kg, δ =.7; J = 7 kg m, ω =.97 s 1 he supporing plane. The sysem 1) ) is closed wih respec o he variables v, ω, γ and N. From his sysem, he normal reacion of he supporing plane is deermined hen he sysem 1) 3), aking ino accoun Eq. ) is considered The resuling sysem of equaions has soluions of he form N = mg+[r, ω]+[ṙ,ω],γ))+[ω,r],[ω,γ])) ) v 1 = v = v 3 =, γ 1 = γ =, γ 3 = 1, ω 1 = ω =, = ω ω R). 6) They correspond o uniform roaions of he el around he principal axis of ineria which is normal o is surface and coincides wih he verical. The equaion of he body surface a γ 3 = 1 can be represened in he form fr)=x 3 + a 3 x 1 cosδ+x sinδ) a 1 x 1 sin δ x cos δ) a + O 3 x 1,x ), where a 1, a are he main radii of curvaure of he body surface a he poin of conac, a 3 is he heigh of he cener of mass, δ is he angle beween he vecors of principal curvaures and principal axes for cel he relaions are saisfied A 1 A, a 1 a, δ modπ/)). 3 Sabiliy condiions Linearized equaions of perurbed moion of he sysem in he neighborhood of soluions Eq. 6) and he corresponding characerisic equaion are in [1].In he case of non-holonomic saemen of he problem k + ) he sabiliy condiions have he form [1],[] A 1 < A < A 3, a 1 > a > a 3, <δ < π a1 J =A 1 +A A 3 ) + a ) ma 3 a 3 a 3 3 7) a1 + a ) + a ) 1 a > 8) a 3 a 3 a 3 a 3 ω <, ω > ω = mg Ja 3 a 1 a 3 )a a 3 ) 9) Eq. 7) means ha he roaion occurs around he axis of he greaes momen of ineria, and he corresponding equilibrium ω = ) is sable. Eq. 8) imposes consrains on geomeric and dynamic parameers of he body. Eq. 9) means ha only roaions in he negaive direcion and wih a enough large angular velociy are sable. In he case of an arbirary coefficien of viscous fricion, he linearized equaions of he perurbed moion of he sysem in he neighborhood of he soluions Eq. 6) are raher cumbersome [1] and he analyical analysis of

3 ω -1 κ Fig. : A 1 = 3 kg m, A = kg m, A 3 = kg m, a 1 = m, a = m, a 3 = m, m=1 kg, δ =.7; J = 3 kg m, ω = 3.13 s 1-1 κ Fig. 3: A 1 = 3 kg m, A = kg m, A 3 = kg m, a 1 = m, a = m, a 3 = m, m=3kg, δ =.7, J = 1 kg m sabiliy condiions is difficul. A Fig. 1, Fig., Fig. 3 he sabiliy regions for some parameers of he problem are given. In he case shown a Fig. 1, for sufficienly large fricion coefficiens, here are wo regions of sabiliy. One corresponds o a non-holonomic saemen of he problem below he doed line), he second is locaed in a neighborhood of zero. As k decreases, hese regions merge ino one, conaining almos all negaive values of ω and a small range of posiive values ω. A very small k he sabiliy region have some symmery wih respec o he horizonal. In he case shown a Fig. which differs from he previous only by he heigh of he cener of mass), he sabiliy region is divided ino wo pars: he region corresponding o he non-holonomic case and he region in he neighborhood of equilibrium. In he case shown a Fig. 3 body mass increases) only he region of sabiliy in he neighborhood of equilibrium remains. Numerical experimens Numerical experimens were carried ou for a cel wih parameers A 1 = kg m, A =. 1 3 kg m, A 3 = kg m a 1 =.661 m, a =.73 m, a 3 =.98 m, m=.1 kg, δ =.1, J =.7 kg m. These parameers correspond o he model of a sone having he shape of an ellipsoid, and all obained resuls of numerical experimens coincide wih full-scale experimens. The region of sabiliy of permanen roaions for his model is presened a Fig.. Numerical experimens 3

4 -1 κ Fig..73. Fig. : γ 3 )= and γ 3 )=.999 were carried ou on a plane wih a coefficien of fricion k= s 1. The iniial condiions had he form γ )=, ω 1 )=ω )=, u)=. 1) A Fig. on he lef are he resuls of experimens wih differen iniial angular velociies of roaions wih a very small iniial deviaion from he verical γ 3 )=.99999), on he righ for greaer deviaion γ 3 )=.999). The area of sabiliy of roaions wih a seleced coefficien of viscous fricion is highlighed in gray. As we see, sable roaions wih a posiive angular velociy of roaion have a very small region of aracion. We noe ha in he non-holonomic saemen of he problem for he chosen parameers here are no sable roaions J < ). In he case of he plane wih fricion and a small posiive iniial angular velociy of roaion, here is a change in he direcion of roaion of he sone, wih a subsequen exi o he sable uniform roaions. The resuls of numerical experimens wih sufficienly large iniial angular velociies are presened a Fig. 6. Here such propery of a sone, as he ransiion of roaional moions o vibraional moions and vice versa is observed. The final movemen wih a negaive iniial speed can be roaion in boh he negaive and posiive direcions. Transien processes For sudy he ransien processes shown a Fig. he equaions 1) 3) up o second-order erms in he variables v 1,v,ω 1,ω,γ 1,γ are considered. ẋ=ak,δ, ) x+bx,k,δ, )+O 3 x) 11)

5 1 γ 3 - Fig Fig. 7: δ =, γ 3 )=.999 The changing of variables v 1 = ν cosψ, v = ν sinψ, ω 1 = ρ cosϕ, ω = ρ 1 sinϕ 1, γ 1 = ρ 1 ξ 1 cosϕ 1, γ = ρ ξ sinϕ is performed, where ξ 1, = mga 1, a 3 )/A,1. Averaging over variables ψ, ϕ 1 and ϕ is carried ou. Then he equaions Eq. 11) have a view = k m a 1 cos δ a sin δ A 3 ξ1 v= kv, ρ 1 = k ma 3 ρ 1, ρ = k ma 3 ρ, A A 1 ρ1 + a cos δ a ) 1 sin δ ξ ρ + mg A 3 sinδa 1 a ) ρ 1 ξ 1 ) 1) ρ ξ For he case of he coincidence of he direcions of he principal axes of he body wih he direcions of he principal curvaures δ = ) he numerical soluions of he averaged sysem Eq. 1) Fig. 7, doed curves) and leanerized sysem Fig. 7, solid curves) have a good coincidence. Solving he averaged sysem Eq. 1) for example, under he iniial condiions Eq. 1), we have v, ρ 1 = ρ 1 )exp mka 3 /A )), ρ and equaliy = )e α 1ρ 1 ρ 1) ), α 1 = a 1 A a 3 A 3ξ 1 13) is fair. Under iniial condiions differing only in he direcion of roaion, he moions are compleely analogous and differ only in sign, and he corresponding curves ) Fig. 7 are symmerical wih respec o he horizonal.

6 ν - ρ 1 ρ Fig. 8 This propery is no preserved for he Cel δ ), and he equaliy Eq. 13) akes he form = )e α 1ρ1 ρ 1) ) ga 1 a )sin δ + ka 1 cos δ + a sin δ) ) 1 e α 1ρ1 ρ 1) )a 1 cos δ+a sin δ)/a 1 1) In his case, he soluions of he averaged sysem, differing only in he iniial direcion of roaion, are no symmeric wih respec o he horizonal. However, he displacemens arising in his case are sufficienly small, and, depending on he iniial condiions, hey can be direced boh o he lower and upper half-planes. In his case, he soluions of he sysem Eq. 11) deviae significanly from he soluions of he averaged sysem Fig. 8), bu in realiy he displacemen is direced o he lower half-plane. The changing in he direcion of roaion of he sone is explained by he deviaions of he exac soluion from he averaged soluion, and he final value of he angular velociy of roaion is always in a small neighborhood of zero. 6 Conclusions Thus, he resuls of modeling he ineracion of he Cel wih he supporing plane by he force of viscous fricion are consisen wih he known properies of is dynamics, and i makes sense o invesigae he considered problem furher. Acknowledgemens This research was suppored financially by he Russian Foundaion for Basic Research , ) and Program No.9 Advanced Topics of Roboic Sysems of he Presidium of he Russian Academy of Sciences. 6

7 References [1] A. V. Karapeyan, Sabiliy of seady moions, in Russian). Ediorial URSS, [] A. P. Markeev, The dynamics of he body osculaing wih a solid surface, in Russian). Insiue of Compuer Sciences, [3] K. D. Zhuravlev V.P., Global moion of he cel, Mechanics of Solids, vol. 3, no. 3, pp. 3 37, 8. [] A. V. Karapeyan, On realizing nonholonomic consrains by viscous fricion forces and celic sones sabiliy, Journal of Applied Mahemaics and Mechanics, vol., no. 1, pp. 3 36,