New feature in hoop dynamics: hidden jump
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ORIGINAL PAPER
New feature in hoop dynamics: hidden jump Alexander P. Ivanov
Received: 3 June 2020 / Accepted: 12 October 2020 Ó Springer Nature B.V. 2020
Abstract Dynamics of nonhomogeneous disk on a rough support is considered. Since P. Painleve´, this system becomes a popular simple model to demonstrate singularities intrinsic to multibody dynamics. It is shown that equations of motion can be ambiguous: along with continued rolling the body can jump or slide with nonzero initial acceleration. An extended model of the system is applied to stability analysis and exhibits bistability, where rolling and jump correspond to sinks, while sliding turns to be a saddle point. In previous studies, the support was treated as a horizontal plane, and such paradox was not detected. Here the cases of an inclined plane and a half-pipe are examined. In particular, such periodic motions with possible jumps are constructed, where continuous rolling is possible too. Generally speaking, ‘‘hidden’’ dynamics is an essential feature of multibody systems, not taken in account in straightforward computing. Keywords Multibody system Dry friction Painleve´ paradox Bistability
A. P. Ivanov (&) Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia e-mail: [email protected]
1 Introduction Multibody dynamics with Coulomb friction is a branch of the theory of differential equations with discontinuities. The foundations of this theory were set forth in the famous book of Filippov [1]. In particular, solutions to discontinuous systems may be nonunique. Such a situation in mechanics with static friction was discovered long before that by Jellett [2], who was the president of the Royal Irish Academy next to Hamilton. The results of Jellett went unnoticed by the scientific community as opposite to a heated discussion on so-called Painleve´ paradoxes [3] with prominent scientist including Hamel et al. [cf. 4]. The subject of discussion was kinetic friction: there are examples where solution to equations of dynamics with certain initial conditions does not exist or is nonunique. Subsequently, numerous studies of these tasks were carried out. Conventional wisdom is: in case of paradoxes, mathematical description, based on rigid body idealization, is nonsufficient, and physical content should be considered to expand the model. A century later to Painleve´, a number of papers appear devoted to Littlewood problem on a hoop with additional point mass [5]. The question is: If the hoop starts from rest with the upper position of the attached mass, will it leave the ground? Generally speaking, the hoop can roll on the ground, slide or jump. A transition from contact phase (rolling or sliding) to airborne (jump) means that reaction of the ground vanishes.
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A. P. Ivanov
Tokieda in his note [6] used the simplest model of contact: pure rolling. Then, as Littlewood noted, mathematical description is similar to dynamical equation of a heavy particle, sliding on cycloid. Simpl
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