Qualitative investigation of the dynamics of a top on a plane with friction

Abstract. The existing results about inversion of a tippe top (TT) establish stability of asymptotic solutions and prove inversion by using the LaSalle theorem. Dynamical behaviour of inverting solutions has only been explored numerically and with the use of certain perturbation techniques. The aim of this paper is to provide analytical arguments showing oscillatory behaviour of TT through the use of the main equation for the TT. The main equation describes time evolution of the inclination angle θ(t) within an effective potential V (cos θ, D(t), λ) that is deforming during the inversion. We prove here that V (cos θ, D(t), λ) has only one minimum which (if Jellett's integral is above a thresholdI3 < 1 holds) moves during the inversion from a neighbourhood of θ = 0 to a neighbourhood of θ = π. This allows us to conclude that θ(t) is an oscillatory function. Estimates for a maximal value of the oscillation period of θ(t) are given.

Section: The Tippe Top Modelmentioning

confidence: 99%

“…This is because even the most simplified model for the rolling and gliding TT is a non-integrable dynamical system with at least 6 degrees of freedom. The focus in many works has been on the asymptotics of the TT [1,5,9,10,13] or on numerical simulations for a TT [3,11,18].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of an Inverting Tippe Top

Rauch-Wojciechowski¹

2014

“…which is the form for the potential for steady motions of an axisymmetric sphere, derived in [10,11].…”

Section: Special Solutions Of Tt Equationsmentioning

confidence: 99%

Tippe Top Equations and Equations for the Related Mechanical Systems

Rutstam¹

2012

Abstract. The equations of motion for the rolling and gliding Tippe Top (TT) are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis3 during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT. The Tippe Top (TT) is known for its counterintuitive behaviour; when it is spun, its rotation axis turns upside down and its center of mass rises. Its behaviour is presently understood through analysis of stability of its straight and inverted spinning solution. The energy of TT is a monotonously decreasing function of time and it appears that it is a suitable Lyapunov function for showing instability of the straight spinning solution and stability of the inverted spinning solution [1,7,10,11,19]. The energy is also a suitable LaSalle function to conclude that for sufficiently large angular momentum L directed close to the verticalẑ-direction the inverted spinning solution is asymptotically attracting [1,19].Since the 1950s [9] it is also known how TT has to be built for the inversion to take place, more precisely that if 0 < α < 1 denotes the eccentricity of the center of mass then the quotient of moments of inertia γ =has to satisfy the inequality 1−α < γ < 1+α. It is also understood that the gliding friction is necessary for converting rotational energy to potential energy.When it comes to describing the actual dynamics of TT, there is very little known. In several papers [5,7,16,22] numerical results of how the Euler angles (θ(t), ϕ(t), ψ(t)) change during inversion are presented, but an understanding of qualitative features of solutions (as well as details of physical forces and torques acting during the inversion) remains a rather unexplored field.There is also an alternative approach [18,21] through the main equation of the TT that shows the source of oscillatory behaviour of θ(t) and of the remaining variables ϕ(t), ψ(t). The main difficulty in making further progress lies in the complexity of the TTs equations of motion.The TT is described, in the simplest setting, by six nonlinear dynamical equations for six variables (θ(t),θ(t),φ(t), ω 3 (t), ν x (t), ν y (t)). Analysis is slightly simplified by the fact that these equations admit one integral of motion, the Jellett integral, and that the energy function is monotonously decreasing in time. To make further progress in reading off the dynamical conte...

“…This equation is more complicated but also involves ω3 and (ωẑ) like equation (7). From this linear system we determine ω3 and substitute into theḊ expression.…”

Section: Integrals Of Motionmentioning

confidence: 99%

“…The energy of the TT has, under assumption that the friction force F f acts against the direction of the sliding velocity v A , negative time derivativeĖ = v A F f (v A ) < 0 and is decreasing monotonously. These two features of the TT equations allow, under some additional assumptions about the reaction force, for complete description of all asymptotic motions of the TT and for analysis of their stability [7,5,12,1]. The asymptotic solutions of the TT constitute an invariant manifold satisfying the conditions v A = 0 andv A = 0 and it consists of vertically spinning motions and of tumbling solutions having constant inclination θ of the symmetry axis, so that the sphere is rolling along a circle around fixed center of mass.…”

Section: Introductionmentioning

confidence: 99%

Phase Space of Rolling Solutions of the Tippe Top

Glad¹

2007

Abstract. Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables (θ, ϕ, ψ) these integrals give separation equations that have the same structure as the equations of the Lagrange top. It makes it possible to describe the whole space of solutions by representing them in the space of parameters (D, λ, E) being constant values of the integrals of motion.