Nonholonomic rolling of a ball on the surface of a rotating cone

Borisov, Alexey V.; Ivanova, Tatiana B.; Kilin, Alexander A.; Mamaev, Ivan S.

doi:10.1007/s11071-019-05086-3

Cited by 16 publications

(6 citation statements)

References 19 publications

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“…The dynamics of the basketball on the rim is similar to that of the golf ball rolling on the edge of the hole [14,15]. The problem of a sphere rolling on different types of fixed [6,7,13,28] or rotating [5,17] surfaces contains many interesting dynamical details related nonholonomic constraints conserved quantities. The dynamics of rolling disk on a flat surface [24] has some structural similarities to that of the basketball, as well.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a basketball rolling around the rim

et al. 2021

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We analyse the dynamics of a basketball which rolls around the rim of a basketball hoop. The rolling steady motions are determined, and we investigate falling, slipping, and instability. The qualitative behaviour of the global dynamics is analysed and the possible trajectories are categorised. We investigate the effect of initial conditions which cause the basketball to fall inside or outside the basket or to remain on the rim.

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Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a basketball rolling around the rim

et al. 2021

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“…Its first studies go back to Routh, and there has recently been a renew of interest, see e.g. [19,24,9,14,13,16,8,5,6,7,2,1,11]. A rather general study of the dynamics of the system has been the object of the very recent article [11], which is the basis for the present study.…”

Section: Introductionmentioning

confidence: 94%

On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art

Fassò

Sansonetto

2022

Regul. Chaot. Dyn.

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We study some aspects of the dynamics of the nonholonomic system formed by a heavy homogeneous ball constrained to roll without sliding on a steadily rotating surface of revolution. First, in the case in which the figure axis of the surface is vertical (and hence the system is SO(3) × SO(2)-symmetric) and the surface has a (nondegenerate) maximum at its vertex, we show the existence of motions asymptotic to the vertex and rule out the possibility of blow up. This is done passing to the 5-dimensional SO(3)-reduced system. The SO(3)-symmetry persists when the figure axis of the surface is inclined with respect to the vertical-and the system can be viewed as a simple model for the Japanese kasamawashi (turning umbrella) performance art-and in that case we study the (stability of the) equilibria of the 5-dimensional reduced system.

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“…Substituting (30) into the second derivative of the effective potential energy with respect to gives 2 ( )…”

Section: Bifurcation Analysismentioning

confidence: 99%

“…It is shown in [14] that the trajectory of a ball moving on a plane under the action of constant external force and friction torque (33) is an untwisting spiral (at the final stage of motion, regardless of the initial conditions). In [30] it is shown that, depending on the initial conditions, the ball moving on a cone either moves in an untwisting trajectory (with an increase in height) or approaches the vertex of the cone (with a decrease in height). In this section, we show that, when the ball moves on a cylinder with friction (33) taken into account, the centre of mass of the ball falls (moves downward) under any initial conditions.…”

Section: Dynamics Of the Ball With The Rolling Friction Torque Taken Into Accountmentioning

confidence: 99%

Non‐holonomic rolling of a ball on the surface of a rotating cylinder

Ivanova

2020

Z Angew Math Mech

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In this paper, we investigate the motion of a homogeneous heavy ball rolling without slipping on the surface of a rotating cylinder in two settings: a setting without dissipation and a setting with rolling friction torque which is proportional to the angular velocity of the ball. In both cases we assume that there exists a non‐holonomic constraint that corresponds to the condition that there be no slipping at the point of contact. In the first case, the system of five differential equations on the level set of first integrals is reduced to quadratures. To define possible types of motion, we carry out a bifurcation analysis of the reduced system. In the case with friction we show that all trajectories shift on average downward and the ball falls.

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Nonholonomic rolling of a ball on the surface of a rotating cone

Cited by 16 publications

References 19 publications

Nonlinear dynamics of a basketball rolling around the rim

Nonlinear dynamics of a basketball rolling around the rim

On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art

Non‐holonomic rolling of a ball on the surface of a rotating cylinder

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