Phase Space of Rolling Solutions of the Tippe Top

Glad, S.T.

doi:10.3842/sigma.2007.041

Cited by 7 publications

(13 citation statements)

References 10 publications

(28 reference statements)

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“…The inversion process of the Tippe top is the result of the action of the system of interrelated inertial torques generated by the rotating mass. The intuitive statement of the researchers that that frictional force acting on the Tippe top resulting in its inversion is not correct [5][6][7][8][9][10][11][12][13][14][15][16]. The frictional force acting on the spinning top activates the inertial torques of the Tippe top.…”

Section: Resultsmentioning

confidence: 99%

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The Mathematical Model for the Tippe Top Inversion

Usubamatov

Bergander

Kapayeva

2021

Advances in Mathematical Physics

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The dynamics of rotating objects is an area of classical mechanics that has many unsolved problems. Among these problems are the gyroscopic effects manifested by the spinning objects of different forms. One of them is the Tippe top designed as the truncated sphere which is fitted with a short, cylindrical rod for rotation. The unexplainable gyroscopic effect of the Tippe top is manifested by its inversion towards the support surface. Researchers tried to describe this gyroscopic effect for two centuries, but all modelings were on the level of assumptions. It is natural because the Tippe top has a more complex design than the simple spinning disc, which gyroscopic effects did not have an analytical solution until recent time. The latest research, in the area of gyroscopic effects, reveals the action of the system of several interrelated inertial torques on any spinning object. The gyroscopic inertial torques are generated by their rotating mass. These inertial torques and the variable ratio of the angular velocities of the spinning object around axes of rotations constitute the fundamental principles of gyroscope theory. These physical principles of dynamics of rotating objects enable to description and compute of any gyroscopic effects and also the Tippe top inversion.

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Section: Resultsmentioning

confidence: 99%

“…This gyroscopic effect of the spinning Tippe top manifests other designs of the rotating objects as hardboiled eggs, an American football, round smooth ovals, etc. [7][8][9]. All similar spinning objects with the property of inversion attracted physicists and mathematicians [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

The Mathematical Model for the Tippe Top Inversion

Usubamatov

Bergander

Kapayeva

2021

Advances in Mathematical Physics

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“…It is useful however to consider D(θ(t), ω 3 (t)) = I 3 ω 3 (t) d(cos θ(t)) being now a time dependent function. Its derivative we calculate using the equations of motion (3)- (7) for the rolling and gliding TT:…”

Section: The Main Equation For the Tippe Topmentioning

confidence: 99%

“…3 we see this. The plots a and b present a graph of θ(t) and a curve (θ(t),θ(t)) obtained by solving equations of motion (3)- (7) for the rolling and gliding TT, with g n given by equation (8) and with values of parameters taken from Example 1. So the effective potential V given by (13) is rational.…”

Section: Estimates For Position Of Minimum Of V (Z D λ)mentioning

confidence: 99%

Dynamics of an Inverting Tippe Top

Rauch-Wojciechowski¹

2014

SIGMA

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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.

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“…In Karapetyan and Kuleshov [12] the same potential is found as a special case of a general method for calculating the potential of a conservative system with n degrees of freedom and k < n integrals of motion. The potential for the rolling TT in the above form was derived and analysed in [8].…”

Section: The Main Equation For the Ttmentioning

confidence: 99%

Tippe Top Equations and Equations for the Related Mechanical Systems

Rutstam¹

2012

SIGMA

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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...

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Phase Space of Rolling Solutions of the Tippe Top

Cited by 7 publications

References 10 publications

The Mathematical Model for the Tippe Top Inversion

The Mathematical Model for the Tippe Top Inversion

Dynamics of an Inverting Tippe Top

Tippe Top Equations and Equations for the Related Mechanical Systems

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