[6] G.T.Walker[7] 1896 P I II I II M.Pascal[10] G.T.Walker A.P.Markeev[11] M.Pascal A.D.Blackowiak [12] H.K.Moffatt T.Tokieda[15] A.P.Markeev M.Pascal

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1 viscous Nature Moffatt & Shimomura [1][2] 2005 [3] [4] Ueda GBC [5] : 2: Wobble stone 1

2 [6] G.T.Walker[7] 1896 P I II I II M.Pascal[10] G.T.Walker A.P.Markeev[11] M.Pascal A.D.Blackowiak [12] H.K.Moffatt T.Tokieda[15] A.P.Markeev M.Pascal M.Pascal A.P.Markeev H.Bondi[13] G.T.Walker n c concordance discordance G.T.Walker M.Pascal A.P.Markeev H.Bondi n c A.Garcia M.Hubbard [14] H.Bondi 2

3 T.R.Kane [8] R.E.Lindberg R.W.Longman[9] A.Garcia M.Hubbard Coulomb A.Garcia & M.Hubbard Coulomb 1 n 0 n 0 n c1 n c2 2 3 a b a a b (viscous M.Pascal n c1 n c2 3

4 2 2.1 m g 3 3: 4: X, Y Z Z X, Y, Z a cm, b cm,c = 1 cm x, y z δ P x p = (x, y, z) ω P u n n 4 Ru Rf F x p F F Coulomb viscous Coulomb viscous f = µv p µ v p R f v p d dt v p u L 4

5 v g = v p + x p ω d dt L = x p F (1) m d dt v g = F mgu (2) F = Rµv p + Ru (3) ( ) d R = mg + m dt (x p ω) u (4) 2.2 ω v p G.T.Walker H.Bondi ω x p x y n u ω x y x, y a b a < 1, b < 1 n 1 µ ( p, q, s z 1 p ( x ) 2 2 c + q xy ( y ) ) 2 c c 2 + s 2 p, q, s a, b, δ ) n 2, np v p x, y x, y, n M.Pascal Ẍ + ν 2 1X + a 1 Ẋ + a 2 Ẏ = 0 Ÿ + ν 2 2X + a 3 Ẋ + a 4 Ẏ = 0 I 3 ṅ k 1 XẌ + k 3XŸ k 2Y Ẍ + k 4Y Ÿ s 1 XẊ s 2XẎ s 3Y Ẋ s 4Y Ẏ = 0 X, Y x, y I 3 z a i a 1 = m 1 nk 1, a 2 = m 2 nk 2, a 3 = m 3 + nk 3, a 4 = m 4 + nk 4 m i 1 µ k i, s i a, b, δ n, x, y

6 5: n(t) 6: x(t) n(t) 7: y(t) n(t) x(t), y(t) A(t), B(t) M.Pascal I 3 ṅ = k 1ν A2 + k 4ν B2, Ȧ = a 1 2 A Ḃ = a 4 2 B (5) 2.4 n c1 e a1t, e a4t a 1, a 4 n a 1 = 0, a 4 = 0 n n c1+ n c1 n c1 n c1+ a 1 > 0, a 4 > 0 k 1 > 0 n c1 n c1+ n c1+ = m 1 k 1 n c1 = m 4 k 4 = f 1 cos 2 θ + f sin 2 θ J 2 µ sin θ cos θ = f cos2 θ + f 1 sin 2 θ J 2 µ sin θ cos θ k 1 < 0 θ δ f J2 J 1 J 1,2 2.5 n c2 n c1 n n 0 > 0 n 0 n 1 < 0 n 1 6

7 5 n 1 = n 0 + 2m 1 n 1 n 1 < 0 n 0 > 2m 1 k 1 n 0 < 2m 1 k 1 n c1+ = m 1 k 1 n c1+ < n 0 < 2m 1 k 1 2m 1 n c2+ k 1 n c2 = 2m 4 k 4 n c2 < n 0 < n c1 n c1+ < n 0 < n c2+ n c2± k r n r n K 0 (r, ρ) K 0 (r n 1, ρ) < 1 h 1 < K 0 (r n, ρ) r 1 + ( 1) i i= ( 1)i+1 ρ 2 h 1 2m 1 k 1 n 0, h 4 2m 4 k 4 n 0, ρ h 4 h 1 ρ, h 1 1 r n = 2[ h 1 (ρ + 1) ], r 1 n = 2[ h 1 (ρ + 1) ] + 1 [x] x 8 r n δ θ µ 2 h 1 (ρ+1) 2 h 1 (ρ + 1) = J 2µ n 0 sin θ cos θ L(f), L(f) f f f 2 7

8 8: µ, δ, n 0 δ 1 δ L(f) a J 2 f J 1 b f f = a = 10 b 1 10 L(f) b b = 1 10 b 3 viscous 8

9 10: µ = 100 9: a=10 b 1 viscous Coulomb Coulomb v p v p = 0 Λ Coulomb f = µ v p v p (Λ) v p (Λ) vp1 2 + v2 p2 + Λ2 [5] Λ Coulomb 2 a > b J 2 < J 1 2 z z x, y θ δ = 0 3 v p 9

10 [1] H.K.Moffatt and Y.Shimomura. Spinning eggs - a paradox resolved Nature V.416, (2002) [2] H.K.Moffatt, Y.Shimomura and M.Branicki Dynamics of ans axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic approximation Proc.R.Soc. Lond. A 460, (2004) [3] Y.Shimomura,M.Branicki and H.K.Moffatt. Dynamics of an axisymmetric body spinning on a horizontal surface.ii. Self-induced jumping Proc.R.Soc. A 461, (2005) [4] T.Mitsui,K.Aihara,C.Terayama,H.Kobayashi and Y.Shimomura. Can a spinning egg really jump? Proc.R.Soc. A 462, (2006) [5] T.Ueda, K. Sasaki and S. Watanabe Motion of the Tippe Top : Gyroscopic Balance Condition and Stability SIAM Journal on Applied Dynamical Systems 4, (2005) [6] [7] G.T.Walker. On a Dynammical Top,Quarterly journal of pure and applied mathmatics Vol.28, (1896) [8] T.R.Kane and D.A.Levinson Ralistic Mathematical Modeling of the Rattleback, I.J.Non- Linear Mechanics, V.17,No.3, (1982). [9] R.E.Lindberg and R.W.Longman On the Dynamic Behavior of the Wobblestone, Acta Mechanica, V.49,81-94(1983). [10] M.Pascal Asymptotic Solution of the Equations of Motion for a Celtic Stone, Journal of Applied Mathematics and Mechanics.V.47,No2, (1983) [11] A.P.Markeev. On the dynamics of a solid on an absolutely rough plane, Journal of Applied Mathematics and Mechanics.V.47,No4, (1984) [12] A.D.Blackowiak,R.H.Rand and H.Kaplan The dynamics of the celt with second order aberaging and computer algebra, Proc. ASME DETC 97/VIB-4103(1997) [13] H.Bondi. The rigid body dynamics of unidirectional spin, Proc. R. Soc. Lond. A 405, (1986) 10

11 [14] A.Garcia and M.Hubbard Spin reversal of the rattleback:theory and experiment Proc.R.Soc. Lond. A148, (1998) [15] H.K.Moffatt and T.Tokieda. Celt reversals:a prototype of chiral dynamics, Proc. R. Soc. Edinburgh. A 138, (2008) 11

64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k

63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5