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Dynamics of a homogeneous ball on a horizontal plane with sliding, spinning, and rolling friction taken into account

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Abstract

We analyze the dynamics of a homogeneous ball on a horizontal plane with friction of all kinds, namely, sliding, spinning, and rolling friction, taken into account. The qualitative-analytic study of the ball dynamics is supplemented with numerical experiments.

The problem on the motion of a homogeneous ball on a horizontal plane with friction was apparently first studied in 1758 by I. Euler (Leonard Euler’s son) with sliding friction taken into account in the framework of the Coulomb model. I. Euler showed that the ball sliding ceases in finite time, after which the ball uniformly rolls along a fixed straight line and uniformly spins about the vertical. This result has long become classical and is described in many textbooks on theoretical mechanics.

In 1998, V. F. Zhuravlev considered the problem of motion of a homogeneous ball on a horizontal plane with sliding and spinning friction taken into account in the framework of the Contensou-Zhuravlev model [1, 2] and showed that the ball sliding and spinning cease simultaneously, after which the ball uniformly rolls along a fixed straight line. The Contensou-Zhuravlev theory was further developed in [3–7].

In the present paper, we consider themotion of a homogeneous ball on a horizontal plane with friction of all kinds taken into account in the framework of the model proposed in [8]. We show that, in one and the same time, both the sliding velocity and the angular velocity of the ball become zero. Our studies are based on the results obtained in [2], the properties of the friction model proposed in [8], and the method for qualitative analysis of dynamics of dissipative systems [9, 10]. The qualitative-analytic study is supplemented with numerical experiments.

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References

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Authors and Affiliations

  1. Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia

    M. V. Ishkhanyan & A. V. Karapetyan

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  1. M. V. Ishkhanyan
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  2. A. V. Karapetyan
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Correspondence to M. V. Ishkhanyan.

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Original Russian Text © M.V. Ishkhanyan, A.V. Karapetyan, 2010, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2010, No. 2, pp. 3–14.

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Ishkhanyan, M.V., Karapetyan, A.V. Dynamics of a homogeneous ball on a horizontal plane with sliding, spinning, and rolling friction taken into account. Mech. Solids 45, 155–165 (2010). https://doi.org/10.3103/S0025654410020019

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