Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction

Burov, A.A.; Yakushev, I. A.

doi:10.1016/j.jappmathmech.2015.03.004

Cited by 11 publications

(10 citation statements)

References 12 publications

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“…Taking into account (11) and (12), the second and the fourth conditions are also satisfied. The last condition R > 0 can be rewritten as follows:…”

Section: Equilibria Positions and Their Stabilitymentioning

confidence: 99%

“…The problem of motion of a heavy bead on a circular hoop rotating about its vertical diameter has been studied in [11]. The similar problem for a circular hoop rotating about some other vertical axis has also been investigated [12]. In the present paper, a three-dimensional analogue of this problem is studied under the assumption that there is viscous friction between the point and the sphere.…”

Section: Introductionmentioning

confidence: 99%

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On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

2019

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We investigate the Hopf bifurcation of a mass on a rotating sphere under the influence of gravity and viscous friction. After determining the equilibria, we study their stability and calculate the first Lyapunov coefficient to determine the post-critical behavior. It is found that the bifurcating periodic branches are initially stable. For several inclination angles of the sphere's rotation axis, the periodic solutions are calculated numerically, which shows that for large inclination angles turning points occur, at which the periodic solutions become unstable. We also investigate the limiting case of small friction coefficients, when the mass moves close to the equator of the rotating sphere.

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“…Taking into account (11) and (12), the second and the fourth conditions are also satisfied. The last condition R > 0 can be rewritten as follows:…”

Section: Equilibria Positions and Their Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

2019

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“…Указанные соответствия и свойства исследовались, вообще говоря, для изолированных решений. В механике систем с сухим трением наблюдают как множества неизолированных относительных равновесий [2,22,23], так и множества неизолированных установившихся движений [9,10,17]. Для таких систем указанные выше соответствия, насколько известно автору, не изучались.…”

Section: е с шалимоваunclassified

On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis)

Shalimova¹

2016

Nelin. Dinam.

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The motion of a heavy point on the surface of a rotating sphere is considered. It is assumed that the rotation axis does not coincide with the vertical diameter of the sphere and the angular velocity of the sphere is constant. The Lagrange equations for this system are derived. Sets of relative equilibria are found and their dependence on the parameters of the system is studied in extreme cases when the magnitude of the angular velocity or the distance between the rotation axis and the center of the sphere is large. The results are represented in graphic form. The same graphic series are also numerically plotted in the general case.

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Section: On the Occasion Of The 65th Anniversary Of Acad V V Kozlomentioning

confidence: 99%

“…In [3] the problem of motion of a heavy bead on a circular hoop rotating about its vertical diameter was studied. A similar problem for a circular hoop rotating about some other vertical axis was investigated in [4].In these papers the dependence of the nonisolated equilibrium sets of a bead on system parameters was considered. The fact of existence of these sets for systems with dry friction is well known (see [5,6]).…”

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confidence: 99%

On the motion of a heavy material point on a rotating sphere (dry friction case)

Burov

Shalimova

2015

Regul. Chaot. Dyn.

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The problem of motion of a heavy particle on a sphere uniformly rotating about a fixed axis is considered in the case of dry friction. It is assumed that the angle of inclination of the rotation axis is constant. The existence of equilibria in an absolute coordinate system and their linear stability are discussed. The equilibria in a relative coordinate system rotating with the sphere are also studied. These equilibria are generally nonisolated. The dependence of the equilibrium sets of this kind on the system parameters is also considered.MSC2010 numbers: 70F40, 70K42, 70K50 a particle on a sphere, absolute and relative equilibria, bifurcations of equilibria On the occasion of the 65th anniversary of Acad. V. V. Kozlov, a prominent scientist and teacherBased on the results of computer simulation (see [1,2]), it is possible to investigate the dynamics of systems with a large number of moving parts. However, the output of such a simulation usually does not represent any qualitative results. That is why it is reasonable to consider some simple problems such as the motion of a material particle on some surface under the action of friction force. In [3] the problem of motion of a heavy bead on a circular hoop rotating about its vertical diameter was studied. A similar problem for a circular hoop rotating about some other vertical axis was investigated in [4].In these papers the dependence of the nonisolated equilibrium sets of a bead on system parameters was considered. The fact of existence of these sets for systems with dry friction is well known (see [5,6]). The investigation of the existence and stability of nonisolated equilibria in gyro systems with friction [7, 8] laid a foundation for the development of the stability theory for systems with dry friction [9]. Methods of stability analysis of equilibria of this kind based on the general theory of systems with discontinuous right-hand sides were later developed in [10][11][12]. An original approach to studying the dependence of nonisolated equilibrium sets on the parameters of a system for two-dimensional and three-dimensional problems was suggested by A. P. Ivanov [13,14]. The same kind of bifurcations, as well as sufficient conditions for stability of equilibrium sets, were considered in [15]. Remark 1. The motion of rigid bodies on moving surfaces is generally investigated in the presence of nonholonomic constraints and under the assumption of no slippage (see, for example, [16]). The same problems for nonholonomic systems under the assumption of dry friction are still very poorly understood.

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Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction

Cited by 11 publications

References 12 publications

On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis)

On the motion of a heavy material point on a rotating sphere (dry friction case)

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