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

- 172 Downloads

## Abstract

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.

## 1 Introduction

The motion of bodies on surfaces is a classical problem of mechanics [1, 2] and was investigated in various statements (see, for example, [3, 4, 5, 6]). In the most simple case, a point particle instead of a rigid body could be considered. That kind of problems appears when we study the dynamics of mechanical systems with rotating parts performing different operations such as the mixing, grinding, and drying, of diverse substances. Based on results of computer simulations [7, 8], it is possible to investigate the dynamics of systems with a large number of particles. However, the output of such a simulation usually does not represent any analytical results. That is why it is reasonable to consider simple systems, such as a particle moving on a surface under the action of a friction force. Even in these simple cases, some complex dynamic effects can be discovered. If there is a friction force and the surface doesn’t move, the system will come to rest. However, if we assume that the surface rotates with a constant angular velocity, steady and periodic motions can appear in the system. This fact also makes it possible to use such problems to identify the friction coefficient. The problem of the motion of a point particle on a rotating surface was studied in [9]. The motion of a particle on a rotating table was investigated analytically and numerically in [10].

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.

## 2 Definition of the problem and equations of motion

Let *P* be a heavy particle of mass *m* which moves on a two-dimensional sphere of radius \(\ell \) under the action of a viscous friction force, with the coefficient of friction being *c*. The sphere rotates with a constant angular velocity \(\varvec{\omega }\) about a fixed axis. It is assumed that the axis passes through the center of the sphere *O*.

*P*can be described by the following system [15]:

## 3 Equilibria positions and their stability

### Theorem 1

Thus, with increasing \(\lambda \), the stable equilibrium becomes unstable, but the image point stays in an \(\varepsilon \)-neighborhood of the equilibrium. With decreasing \(\lambda \), the equilibrium becomes stable again and the image point returns to the equilibrium, as shown in Fig. 2. The behavior of the system is reversible with respect to \(\lambda \).

The proof of this theorem has been described in detail [13, 14, 16, 17].

### Theorem 2

(Subcritical Hopf bifurcation) If \( a=L_1(\lambda _0)>0\) and \( d={\mathrm{d}R}/{\mathrm{d}\lambda }_{\lambda =\lambda _0}<0\), then (13) has a non-trivial asymptotically unstable stationary solution \(r=(d (\lambda -\lambda _0)/a)^{1/2}\) for \(\lambda <\lambda _0\). For \(\lambda >\lambda _0\) the trivial solution is unstable and no nearby stable stationary solution exists.

Thus, with increasing \(\lambda \) the stable equilibrium becomes unstable, and the point leaves the neighborhood of it. When \(\lambda \) is decreased again below \(\lambda _0\), the point will usually not return to the equilibrium. The behavior of the system is therefore irreversible with respect to \(\lambda \). A typical situation is shown in Fig. 3 for a locally subcritical, but globally supercritical Hopf bifurcation: If \(\lambda \) increases beyond the critical value \(\lambda _c\), the system quickly moves to the stable large amplitude oscillation and remains there, even if \(\lambda \) decreases below \(\lambda _0\) again. Only if it reaches the limit point cycle (“LP”), it jumps back to the stationary state, exhibiting a hysteretic behavior.

## 4 Numerical investigation of the bifurcating solutions

During the path-following along the periodic solutions, we encounter a singularity of the differential equations due to the use of spherical coordinates: Close to the Hopf bifurcation point the solutions are periodic in \(\theta \) and \(\varphi \); but after the trajectories pass through the south pole of the sphere, the azimuthal angle \(\varphi \) increases by \(2\pi \) during one period. In order to overcome this difficulty, a different coordinate system (Cartesian or spherical coordinates along the rotation axis) has been used close to the crossing of the south pole, when the equations become singular.

### 4.1 Limiting behavior for small friction coefficients

When the parameter \(\chi \) is set to zero, all gravitational and damping forces vanish and the mass can move freely on the sphere, tracing out arbitrary great circles. For small values of \(\chi \), or equivalently, for large rotation speeds \(\omega \), the numerical calculations indicate that the periodic solution approaches the equator of the rotating sphere and rotates with the same speed as the sphere. That behavior is also expected by mechanical reasoning: If we neglect the gravitational force, the friction and centrifugal force will cause the mass to move along the equator; a small gravitational force causes a periodic excitation.

*z*-axis. Since now the gravity acts in the direction \(\varvec{e}_g=(\sin \alpha , 0, -\cos \alpha )^\mathrm{T}\), its potential becomes

*p*: At leading order, we obtain from (15d)

## 5 Conclusions

The Hopf bifurcation of a moving mass on a rotating sphere has been investigated. By transforming the system to Jordan Normal Form, calculating the Center Manifold and simplifying the system using Normal Form theory we obtained a simple expression for the first Lyapunov coefficient. Since this coefficient is negative, the periodic solutions bifurcate supercritically from the steady state. These bifurcating solutions are also computed numerically, and their limiting behavior for vanishing friction force is studied.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW).

## References

- 1.Pars, L.: A Treatise on Analytical Dynamics. Heinemann, London (1965)zbMATHGoogle Scholar
- 2.Routh, E.J.: Dynamics of a System of Rigid Bodies. Dover Publications, New York (1955)zbMATHGoogle Scholar
- 3.Sinopoli, A.: Unilaterality and dry friction: a geometric formulation for two-dimensional rigid body dynamics. Nonlinear Dyn.
**12**(4), 343–366 (1997)MathSciNetCrossRefGoogle Scholar - 4.Karapetyan, A.V.: The movement of a disc on a rotating horizontal plane with dry friction. J. Appl. Math. Mech.
**80**(5), 376–380 (2016)MathSciNetCrossRefGoogle Scholar - 5.Ehrlich, R., Tuszynski, J.: Ball on a rotating turntable: comparison of theory and experiment. Am. J. Phys.
**63**, 351 (1995)CrossRefGoogle Scholar - 6.Udwadia, F.E., Di Massa, G.: Sphere rolling on a moving surface: application of the fundamental equation of constrained motion. Simul. Model. Pract. Theory
**19**(4), 1118–1138 (2011)CrossRefGoogle Scholar - 7.Fleissner, F., Lehnart, A., Eberhard, P.: Dynamic simulation of sloshing fluid and granular cargo in transport vehicles. Veh. Syst. Dyn.
**48**(1), 3–15 (2010)CrossRefGoogle Scholar - 8.Alkhaldi, H., Ergenzinger, C., Fleissner, F., Eberhard, P.: Comparison between two different mesh descriptions used for simulation of sieving processes. Granul. Matter
**10**(3), 223–229 (2008)CrossRefGoogle Scholar - 9.Brouwer, L.E.J.: Collected Works, vol. II. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
- 10.Akshat, A., Sahil, G., Toby, J.: Particle sliding on a turntable in the presence of friction. Am. J. Phys.
**83**, 126–132 (2015)CrossRefGoogle Scholar - 11.Burov, A.A.: On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle. Acta Mech.
**212**(3), 349–354 (2010)CrossRefGoogle Scholar - 12.Burov, A.A., Yakushev, I.A.: Bifurcations of relative equilibria of a heavy bead on a rotating hoop with dry friction. J. Appl. Math. Mech.
**78**(5), 460–467 (2014)MathSciNetCrossRefGoogle Scholar - 13.Bautin, N.N.: Behaviour of Dynamical Systems Near the Boundaries of the Stability Region. Gostekhizdat, Moscow (1949)Google Scholar
- 14.Lyapunov, A.: General Problem of the Stability of Motions. CRC Press, Moscow (1992)CrossRefGoogle Scholar
- 15.Shalimova, E.S.: Steady and periodic modes in the problem of motion of a heavy material point on a rotating sphere (the viscous friction case). Moscow Univ. Mech. Bull.
**69**, 89–96 (2014)CrossRefGoogle Scholar - 16.Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995)CrossRefGoogle Scholar
- 17.Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
- 18.Oberle, H.J., Grimm, W., Berger, E.: BNDSCO, Rechenprogramm zur Lösung beschränkter optimaler Steuerungsprobleme. Techn. Univ, München (1985)Google Scholar
- 19.Seydel, R.: A continuation algorithm with step control. In: Numerical Methods for Bifurcation Problems. ISNM 70. Birkhäuser (1984)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.