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Linköping Studies in Science and Technology.Dissertations, No. 1500
Analysis of Dynamics of theTippe Top
Nils Rutstam
Department of MathematicsLinköping University, SE–581 83 Linköping, Sweden
Linköping 2013
Linköping Studies in Science and Technology.Dissertations, No. 1500
Analysis of Dynamics of theTippe Top
Nils Rutstam
Division of Applied MathematicsDepartment of Mathematics
Linköping UniversitySE–581 83 Linköping
Sweden
ISBN 978-91-7519-692-3 ISSN 0345-7524
Copyright © 2013 Nils Rutstam
Printed by LiU-Tryck, Linköping, Sweden 2013
iii
They have proven quite effectivelythat bumblebees indeed can fly
against the field’s authority.-Einstürzende Neubauten
Abstract
The Tippe Top is a toy that has the form of a truncated sphere with a small peg.When spun on its spherical part on a flat supporting surface it will start to turnupside down to spin on its peg. This counterintuitive phenomenon, called in-version, has been studied for some time, but obtaining a complete description ofthe dynamics of inversion has proven to be a difficult problem. This is becauseeven the most simplified model for the rolling and gliding Tippe Top is a non-integrable, nonlinear dynamical system with at least 6 degrees of freedom. Theexisting results are based on numerical simulations of the equations of motion oran asymptotic analysis showing that the inverted position is the only asymptot-ically attractive and stable position for the Tippe Top under certain conditions.The question of describing dynamics of inverting solutions remained rather in-tact.
In this thesis we develop methods for analysing equations of motion of theTippe Top and present conditions for oscillatory behaviour of inverting solu-tions.
Our approach is based on an integrated form of Tippe Top equations thatleads to the Main Equation for the Tippe Top (METT) describing the time evolu-tion of the inclination angle θ(t) for the symmetry axis of the Tippe Top.
In particular we show that we can take values for physical parameters suchthat the potential function V(cos θ, D, λ) in METT becomes a rational functionof cos θ, which is easier to analyse. We estimate quantities characterizing aninverting Tippe Top, such as the period of oscillation for θ(t) as it moves froma neighborhood of θ = 0 to a neighborhood of θ = π during inversion. Resultsof numerical simulations for realistic values of physical parameters confirm theconclusions of the mathematical analysis performed in this thesis.
v
Populärvetenskaplig sammanfattning
En Tippe Top är en liten leksak, byggd som en trunkerad träkula med en kortpinne som handtag. När man snurrar den på den runda delen, tillräckligt snabbtpå ett plant underlag, vänder den sig upp och ner så att den börjar snurra på pin-nen. Detta överraskande fenomen har studerats ett bra tag, men många aspekterhar visat sig svåra att analysera. Svårigheterna ligger i att studera spinn-rörelse,rullande rörelse samt glidfriktion samtidigt och visa hur de samverkar i det dy-namiska system av 6 frihetsgrader som Tippe Top:en definerar.
I denna avhandling presenteras ekvationerna för Tippe Top i flera former; vistuderar dem dels i vektor-form, dels i koordinat-form och dels i formen av enseparationsekvation, bestämd utav energin för systemet. Dessa former belyserolika egenskaper av vändningsrörelsen som en Tippe Top utför och hjälper ossatt förklara både varför den vänder sig och hur man ska beskriva vändningenasymptotiskt. Det bevisades redan på 1950-talet att glidfriktionen mellan TippeTop och underlaget är den huvudsakliga mekanismen som driver vändningsrö-relsen. Man kan vidare visa att om Tippe Top:en är konstruerad på ett korrektsätt och den startar med tillräckligt hög vertikal rotationshastighet så kommervändning alltid ske. Även om detta svarar på frågorna varför och när vändningav Tippe Top sker har vi ingen ordentlig beskrivning på dynamiken i vändnings-rörelsen, det vill säga hur den vänder på sig.
Vi utvecklar matematiska metoder att studera differentialekvationer som be-skriver Tippe Tops rörelse och visar att den centrala frågan om rörelse av vinkelnθ(t) mellan Tippe Tops symmetriaxel och vertikalaxeln kan reduceras till en stu-die av en enklare ekvation för θ(t).
Denna huvudekvation för Tippe Top liknar en separabel energiekvation somuppstår i specialfallet av en rullande Tippe Top.
För en rullande och glidande Tippe Top är dock rörelsekonstanter i dennaekvation tidsberoende, vilket försvårar analys. Vi kan ändå visa hur potential-funktionen i ekvationen deformeras för en Tippe Top som vänder på sig. Speci-ellt så kan vi hitta parametervärden så potentialen blir en rationell funktion somär mycket enklare att analysera. Vi uppskattar relevanta storheter som karak-täriserar en vändande Tippe Top, bland annat svängningstiden för θ(t) då denoscillerar under vändningen.
vii
Acknowledgments
First and foremost, I’d like to thank my supervisor Prof. Stefan Rauch. Histenacity, enthusiasm and unyielding support have made these years a delightfulride.
Great many thanks goes to my assistant supervisor Hans Lundmark, for hisexceptional range of knowledge, his ability to explain when I’m stuck and forhelping with numerical simulations.
Further I thank all of my past and present colleagues at MAI, in particularall of my fellow PhD-students, for making MAI a great place to work.
I would like to take this opportunity to thank the various teachers and pro-fessors who have taught and inspired me during the years in Lund, Waterlooand Linköping. These include Tomas Claesson, Christer Bennewitz, Olivia Con-stantin, Per-Anders Ivert, Brian Forrest, William Gilbert, Andreas Rosén, Bengt-Ove Turesson and many others.
My wonderful family just could not be more perfect and supportive even ifthey tried, thanks to all of them.
Finally, I thank my friends for making everything else in my life awesome.This includes Mats, Daniel, Nina, Anders, Emma, Marcus, Pernilla, Evelina,Cristoffer, Leo, Erika, Anna, Boel, Adam and others. Thank you for keeping mesane, but mostly for keeping me insane.
ix
Contents
I Introduction 11 Overview of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 The Tippe Top Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 The frictional force . . . . . . . . . . . . . . . . . . . . . . . . 104.2 The dynamical system for the TT model . . . . . . . . . . . 114.3 A note on TT models . . . . . . . . . . . . . . . . . . . . . . . 12
5 Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 The Main Equation for the Tippe Top . . . . . . . . . . . . . . . . . 147 Special case of rational METT . . . . . . . . . . . . . . . . . . . . . . 158 Numerical solving of the TT equations . . . . . . . . . . . . . . . . 169 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
II Papers 25
1 Tippe Top equations and equations for the related mechanical systems 271 Notation and a model for the TT . . . . . . . . . . . . . . . . . . . . 30
1.1 The vector and the Euler angle forms of equations of TT . . 341.2 Equations in Euler angles with respect to the rotating ref-
erence system K . . . . . . . . . . . . . . . . . . . . . . . . . 371.3 The main equation for the TT . . . . . . . . . . . . . . . . . . 38
xi
xii Contents
2 Special solutions of TT equations . . . . . . . . . . . . . . . . . . . . 433 Reductions of TT equations to equations for related rigid bodies . 47
3.1 The gliding HST . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Transformation from TT to gliding HST . . . . . . . . . . . . 493.3 The gliding eccentric cylinder . . . . . . . . . . . . . . . . . 51
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2 High frequency behaviour of a rolling ball and simplification of theMain Equation for the Tippe Top 571 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Simplified form of the separation equation for the rolling axisym-
metric sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 The rational form of the separation equation . . . . . . . . . . . . . 63
3.1 High frequency behaviour of a rolling sphere . . . . . . . . 643.2 Estimates for general nutational solutions . . . . . . . . . . 68
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Dynamics of an inverting Tippe Top 751 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 The Tippe Top model . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 The Main Equation for the Tippe Top . . . . . . . . . . . . . . . . . 81
3.1 The rational form of the METT . . . . . . . . . . . . . . . . . 844 Convexity of the rational potential V(z, D, λ) . . . . . . . . . . . . . 84
4.1 Estimates for position of minimum of V(z, D, λ) . . . . . . . 865 Oscillation of θ(t) within the deforming rational potential V(cos θ, D(t), λ) 89
5.1 Estimates for the period of oscillation . . . . . . . . . . . . . 926 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Part I
Introduction
1
Introduction
1 Overview of Papers
This thesis consist of an introduction and three papers.
Paper 1
Tippe Top equations and equations for the related mechanical systems
We model the Tippe Top (TT) as an axially symmetric sphere rolling and glidingon a flat surface according to the Newton equations for motion of a rigid body.The equations of motion are nonintegrable and are difficult to analyse. Theonly existing arguments about TT inversion are based on analysis of stabilityof asymptotic solutions and on a LaSalle type theorem. These arguments donot explain the dynamics of inversion. To approach this problem we reviewand analyse here the equations of motion for the rolling and gliding TT in threeequivalent forms; a vector form, an Euler angle form and an integrated form,each one providing different bits of information about motion of TT. They leadto the Main Equation for the tippe top, which describes well the oscillatorycharacter of motion of the symmetry axis 3 during the inversion. We show alsothat the equations of motion of TT in a suitable limit give rise to equations ofmotion for two other simpler mechanical systems: the gliding heavy symmetrictop and the gliding eccentric cylinder. Analysis of equations for these systemsis useful for understanding the dynamics of the inverting Tippe Top.
3
4 Introduction
Paper 2
High frequency behaviour of a rolling ball and simplification of the MainEquation for the Tippe Top
The Chaplygin separation equation for a rolling axisymmetric ball has an alge-braic expression for the effective potential V(z = cos θ, D, λ) which is difficultto analyse. We simplify this expression for the potential and find a 2-parameterfamily for when the potential becomes a rational function of z = cos θ. Then thisseparation equation becomes similar to the separation equation for the heavysymmetric top. For nutational solutions of a rolling sphere, we study a high fre-quency ω3-dependence of the width of the nutational band, the depth of motionabove V(zmin, D, λ) and the ω3-dependence of nutational frequency 2π
T . Theseresults have bearing for understanding the inverting motion of the Tippe Top,modeled by a rolling and gliding axisymmetric sphere.
Paper 3
Dynamics of an inverting Tippe Top
In this paper we tackle the difficult question of describing dynamics of invertingsolution of the Tippe Top (TT) when the physical parameters of the TT satisfy1 − α2
< γ < 1 and the initial conditions are such that Jellett’s integral satisfies
λ > λthres =
√mgR3 I3α(1+α)2√
1+α−γ. Our approach is based on studying an equivalent
integrated form of TT equations that leads to the Main Equation for the TippeTop (METT), an equation describing time evolution of the inclination angle θ(t)of inverting TT. We study how the effective potential V(cos θ, D, λ) in METT de-forms as TT is inverting and show that its minimum moves from a neighborhoodof θ = 0 to a neighborhood of θ = π. We formulate conditions for oscillatorybehaviour of θ(t) when it moves toward θ = π . Estimates for the maximal valueof the oscillation period of θ(t) are given.
2 Background
A Tippe Top is a small toy, built as a truncated ball with a peg as a handle.When it is spun fast enough on a flat surface with the handle pointing upward,the top will start to turn upside down in a nutating manner until it ends upspinning on its handle. This counterintuitive and fascinating phenomenon iscalled inversion.
In this thesis we study a model of the Tippe Top with the aim of understand-ing how the dynamics of inversion follows from the properties of the modelingequations.
It turns out that even the most simple comprehensive model of a Tippe Topwhich exhibits the inversion phenomenon constitute a non-integrable, non-linear
2 Background 5
dynamical system of (at least) six degrees of freedom. This makes dynamics ofthe Tippe Top a challenging problem to analyse.
Due to the fascinating nature of the problem, analysis of Tippe Top inversionhas a rich history. In the 1950s several papers [3, 10, 13, 21, 22] established aworking physical model for the Tippe Top, essentially reducing it to a rollingand gliding axisymmetric sphere. There was also some debate over what is themain cause of the inversion phenomenon, if it is the shift of the center of massalong symmetry axis in the spherical part, or the gliding friction between theTippe Top and the flat supporting surface. Del Campo [8] has shown definitelythat the gliding friction is the only mechanism, within the model, giving rise toinversion.
This result was later affirmed by Cohen [7] in a paper where he presentedthe first numerical simulations of Tippe Top inversion.
Since the 1990s the focus has shifted to analysis of the mathematical natureof the problem; analysis of the integrable subcase of a rolling Tippe Top [11, 15]and analysis of asymptotic behaviour of the Tippe Top [2, 6, 9]. These worksprovided among other things criteria for how the Tippe Top has to be builtand how fast it has to be spun in order to enable inversion. The equationsfor purely rolling Tippe Top are known to be integrable since Chaplygin andRouth [4, 25]. When however friction between Tippe Top and the supportingplane is introduced the dynamical equations acquire two additional variablesand become a non-integrable dynamical system of 6 degrees of freedom. Arigorous analysis of these equations and of the dynamics of inversion remainsan unexplored field.
Most of the work in that direction focused on numerical simulations for var-ious initial conditions and for various assumptions about the friction [20, 26].
This thesis reviews the existing results regarding the asymptotics and dy-namics of the model of the Tippe Top and provides analysis of the dynamics ofinversion. We study in particular the separation equation for the rolling TippeTop as a starting tool for understanding inverting solutions of the rolling andgliding Tippe Top. We show that for a certain range of parameters this equationcan be simplified so that precise estimates of behaviour of nutational solutionscan be found.
In this introduction we review the basics of rigid body motion, illustrate itfor the well known example of a Heavy Symmetric Top and summarize the mainpoints in our study of dynamics of the Tippe Top. Finally, to illustrate the inher-ent complexity of Tippe Top inversion, we numerically integrate the equations ofmotion for the rolling and gliding TT for a set of physical parameters and initialconditions corresponding to an inverting TT. The resulting graphs of evolutionof relevant functions and variables are discussed at length.
6 Introduction
3 Preliminaries
The basic notation we use in this thesis to describe the motion for rigid bodiesis similar to conventions used in Landau [16] and Goldstein et al. [12]. We shalluse the Euler angles (θ, ϕ, ψ) to describe the orientation of the body.
We consider a rigid body of mass m with instantaneous contact with thesupporting plane at the point A. In this thesis we will exclusively consideraxisymmetric bodies. Let CM be the center of mass of the body and O thegeometric center (not necessarily the same).
We choose a fixed inertial reference frame K0 = (X, Y, Z) with X and Y
parallel to the supporting plane and with vertical Z. We place the origin of thissystem in the supporting plane. Let K = (x, y, z) be a frame defined throughrotation around Z by an angle ϕ, where ϕ is the angle between the plane spannedby X and Z and the plane spanned by the points CM, O and A (see figure 1).
C M
O
3
1
2
z
x
yZ
X
Y A
s
K0K
K
Figure 1: Diagram of the three reference systems K0, K0 and K for an axisymmetricbody. Note that 1 and 3 are parallel to the plane spanned by x and z.
The third reference frame is K = (1, 2, 3), with origin at CM, defined througha rotation around y by an angle θ, where θ is the angle between Z and thesymmetry axis. Note that this frame is not fully fixed in the body. We let 3be parallel to the symmetry axis. The angle ψ is the rotation angle of the bodyabout the symmetry axis 3.
The Euler angles for this body relative to the vertical z will be denoted(θ, ϕ, ψ) and the angular velocity of the frame K w.r.t. K0 is ωre f = θ2 + ϕz =
−ϕ sin θ1 + θ2 + ϕ cos θ3. The total angular velocity of the body is found byadding the rotation around the symmetry axis 3: ω = ωre f + ψ3 = −ϕ sin θ1 +
θ2 + (ψ + ϕ cos θ)3, and we shall refer to the third component of this vectoras ω3 = ψ + ϕ cos θ. In this thesis we will always let the "dot-notation" meanderivative w.r.t. time.
Let s denote the position vector of CM w.r.t. K0. Let ρ be the vector fromCM to an arbitrary point P in the body. The velocity of the point P w.r.t. K0 can
3 Preliminaries 7
then be written asv = s + ω × ρ. (1)
Note that ρ = ω × ρ.If we think of a rigid body as composed of fixed masses mi (with the total
mass m = ∑i mi) at positions ρi w.r.t. CM then the angular momentum of thebody about the CM is defined as
L = ∑i
miρi × vi, (2)
where vi denotes the velocity w.r.t. K0 of the ith mass. From (1) we see that thisexpression can be written as
L = ∑i
miρi × (s + ω × ρi) = ∑i
mi(ρi × s) + ∑i
miρi × (ω × ρi).
But by the definition of the center of mass, ∑i miρi = 0, so only the second sumremains. We can then define a linear operator acting on a vector u in the body:
Iu = ∑i
miρi × (u × ρi).
We call I the inertia operator (or inertia tensor). We then write the angular mo-mentum as L = Iω. The inertia operator is symmetric and positive, whichmeans we can find an ON-basis where I is diagonal. The components of I inthis basis are called the principal moments of inertia and the corresponding vec-tors the principal axes. Since the body is axisymmetric, the principal axes are notunique; to the principal moments of inertia I1 = I2 there corresponds a wholeplane of principal vectors. Therefore, the vectors 1, 2 in the moving frame K willbe principal vectors, along with 3.
The motion of each mass mi is described by the Newton equation mvi = Fi,where Fi is a force acting on the mass mi. Then by summing these equations forthe motion of the center of mass s(t) we have obtained the Newton equation
ms = F,
where F = ∑i Fi is the net external force acting on the body. Since ω = I−1L, the
time-evolution of the angular velocity ω(t) can be found by solving the equationobtained when differentiating (2):
L =∑i
(miρi × vi + miρi × vi)
=∑i
mi(ω × ρi)× (s + ω × ρi) + ∑i
ρi × (mivi)
=(s × ω)× ∑i
miρi + ∑i
ρi × (mivi) = ∑i
ρi × Fi,
where Fi is the force acting on the i:th particle. The right hand side in thisequation is the sum of all external torques acting on the body w.r.t. CM (if the
8 Introduction
angular momentum is written w.r.t. another point in the body, the sum shouldbe the sum of external torques acting on the body w.r.t. that point).
In the following example we will look at a simple application of derivingthe equations of motion for a specific rigid body, the Heavy Symmetric Top(HST). We also find the separation equation for the motion of this body. This isinstructive since the technique of deriving the separation equation for the TippeTop involves the same steps. The equations for the HST are also of interestsince in paper 1 we derive equations for a non-integrable generalization of thisrigid body (the gliding HST). In paper 2 the method used in [1, 12] to analysenutational motion and to estimate relevant quantities for the HST is applied instudy of certain nutational solutions of the model for the rolling Tippe Top.
Example 3.1
The Heavy Symmetric Top (also called the Lagrange Top) is an axially symmetricrigid body with one point fixed, moving under the action of vertical gravitationalforce. We let the fixed point A coincide with the origin of the inertial system K0.As we see in figure 2 the center of mass CM is at distance l from the origin andon the 3-axis.
θ
C M
3
1
z
x
y
s=l 3
A
Figure 2: Diagram of the heavy symmetric top.
We note that the forces acting on the HST are the reaction force F applied atthe point A and the force of gravity acting at CM (we let g = 9.82 m · s−2 be thegravitational acceleration, as usual). Here we have s = l3 = −a, so the equationsof motion in vector notation are
ms = F − mgz, LA = s × (−mgz). (3)
The inertia tensor w.r.t. the principal axes is I = I111t + I222t + I333t, but I1 = I2since the HST is axisymmetric (here the superscript t means the transpose of avector). Using Steiner’s theorem [1], we also have the "shifted" inertia tensorw.r.t. the fixed point A: I
∗ = I∗1(11t + 22t
)+ I333t, where I∗1 = I1 + ml2. The
3 Preliminaries 9
angular momentum w.r.t. the point A is written as LA = I∗ω. To make the
system of equations (3) closed we need to add the kinematic equation for themotion of the symmetry axis:
˙3 = ω × 3 =((I∗)−1LA
)× 3 =
1I∗1
LA × 3.
The dynamics of the HST is found by solving the closed system of equations
LA = −mgl3 × z, ˙3 =1I∗1
LA × 3.
The solution to the system gives the value of the total reaction force F = ms +
mgz = ml ¨3 + mgz needed to keep the tip of the HST fixed at the point A.
It is well known that the HST admits three integrals of motion, the energyE = 1
2 ω · LA + mgs · z and the angular momentum projected on the 3 and z-axisL3 = LA · 3, Lz = LA · z. These integrals of motion make it possible to find aseparation equation for system (3). Since the angular velocity ω = −ϕ sin θ1 +θ2 + (ψ + ϕ cos θ)3, the integrals are:
E =12
LA · ω + mgs · z =I∗12(θ2 + ϕ2 sin2 θ) +
I3
2(ψ + ϕ cos θ)2 + mgl cos θ, (4)
Lz = LA · z = (I∗1 sin2 θ + I3 cos2 θ)ϕ + I3ψ cos θ, (5)
L3 = LA · 3 = I3(ψ + ϕ cos θ). (6)
By solving equations (5), (6) for ϕ, ψ and by eliminating ϕ, ψ from the energyintegral (4) we obtain a separable differential equation for θ:
E =I∗12
θ2 +L2
3
2I3+
(Lz − L3 cos θ)2
2I∗1 sin2 θ+ mgl cos θ.
By renaming constants, this equation becomes
2E
I∗1= θ2 +
I∗1 a2
I3+
(b − a cos θ)2
sin2 θ+ β cos θ, (7)
where
β =2mgl
I∗1, a =
L3
I∗1, b =
Lz
I∗1.
The solution to (7) gives the function θ(t), which makes it possible to find ϕ(t)and ψ(t) expressed by quadratures in terms of elliptic functions. The trajectoryof the solutions is then represented by a curve on the unit sphere S2 that isdrawn by the symmetry axis 3.
10 Introduction
4 The Tippe Top Model
The model for the Tippe Top (TT) that we shall use is a rigid, eccentric ball asan approximation of the truncated ball with handle. The truncation shifts thecenter of mass of the sphere from its geometric center along its symmetry axis.
Thus we consider a sphere with radius R and axially symmetric distributedmass m so that the center of mass CM is shifted from the geometric center Oalong the symmetry axis 3 by αR, where 0 < α < 1. The TT is in instantaneouscontact with the supporting plane at A, and we consider the TT’s movements asit rolls and glides in the plane. We therefore choose the same reference frames asin the previous section; the vector s is the position of CM with respect to K0 anda = R(α3 − z) = R(sin θ1 + (α − cos θ)3) is the vector from CM to A (figure 3).
2
θ
OC M
3
z
AvA
a
αRs
K0
Figure 3: Diagram of the TT. Note that a = Rα3 − Rz.
Newton’s equations for the TT describe the motion of the CM and rotationaround the CM:
ms = F − mgz, L = a × F, ˙3 = ω × 3 =1I1
L × 3, (8)
where F is the total force acting on the TT at the point A. The third equation isthe kinematic equation describing the motion of the symmetry axis 3. Here theangular momentum is taken w.r.t. CM, L = Iω, where I = I1
(11t + 22t
)+ I333t
and ω = −ϕ sin θ1 + θ2 + ω33 is the angular velocity.In our model we assume that the point A is always in contact with the sup-
porting plane, which is expressed as the contact constraint z · (s(t) + a(t))t≡ 0.
This is an identity with respect to time t, so all its time-derivatives have to vanishas well. In particular, z · (s + ω × a) = 0.
Thus the gliding velocity vA(t) = s + ω × a, which is the velocity of thepoint that is in instantaneous contact with the plane at time t, will have zero z-component. In the frame K, the gliding velocity has components vA = νx cos θ1+νy2 + νx sin θ3, where νx, νy are the components in the 2 × z and 2 direction.
Equations (8) also admit Jellett’s integral of motion λ = −L · a [25].
4 The Tippe Top Model 11
4.1 The frictional force
We assume that the force acting at the point A has the form F = gn z − µgnvA. Itconsists of a normal reaction force and of a friction force of viscous type, wheregn and µ are non-negative coefficients. Other frictional forces due to spinningand rolling are ignored. The contact criterion gives that the vertical componentgn z is dynamically determined, but the planar component of F has to be specifiedindependently to make equations (8) fully determined.
The original equations of motion (8) can then be written as
mr = −µgnvA, L = a × (gn z − µgnvA),˙3 =
1I1
L × 3, (9)
where r = s − sz z. The way of writing the frictional part of the reaction force,−µgnvA = −µ(L, 3, s, s, t)gn(t)vA, indicates that it acts against the gliding veloc-ity vA and that the friction coefficient can in principle depend on all dynamicalvariables and on time t.
The assumption that the friction force is of viscous type is common in mostworks about the motion of the TT [6, 9, 18, 19, 24]. Or [20] discusses the possibleinclusion of a Coulomb-type friction force, −µCgnvA/|vA|, along with the vis-cous friction. Bou-Rabee et al. [2] argues that a Coulomb-type friction force onlycauses algebraic destabilization of the initially spinning TT, whereas viscous-type friction gives exponential destabilization.
The model for the external force, which we use in equations (9), immediatelyimplies that the energy E = 1
2 ms2 + 12 ω · L + mgs · z is decreasing monotonically
for the rolling and gliding TT. A short calculation using the equations (9) givesE = F · vA = −µgn|vA|2 < 0. But since the CM of TT is lifted by 2Rα dur-ing inversion, the potential energy is increased. This means that kinetic energydecreases during inversion. An argument due to Del Campo [8], which we re-peat in paper 1, shows that the planar part of the reaction force is providingthe essential mechanism which allows the transfer of energy to occur within theaccepted model for the rolling and gliding TT. Thus the gliding friction is theforce which gives rise to the inversion phenomenon.
4.2 The dynamical system for the TT model
Since we have specified the friction force for the model of the rolling and glidingTT, equations (9) can be written in Euler angle form and solved for the highest
12 Introduction
derivative of each of the variables (θ, ϕ, ω3, νx, νy). We get the system
θ =sin θ
I1
(I1 ϕ2 cos θ − I3ω3 ϕ − Rαgn
)+
Rµgnνx
I1(1 − α cos θ), (10)
ϕ =I3θω3 − 2I1θ ϕ cos θ − µgnνyR(α − cos θ)
I1 sin θ, (11)
ω3 =− µgnνyR sin θ
I3, (12)
νx =R sin θ
I1
(ϕω3 (I3(1 − α cos θ)− I1) + gnRα(1 − α cos θ)− I1α(θ2 + ϕ2 sin2 θ)
)
− µgnνx
mI1
(I1 + mR2(1 − α cos θ)2
)+ ϕνy, (13)
νy =− µgnνy
mI1 I3
(I1 I3 + mR2 I3(α − cos θ)2 + mR2 I1 sin2 θ
)
+ω3θR
I1(I3(α − cos θ) + I1 cos θ)− ϕνx, (14)
which, if we add the equation ddt (θ) = θ, becomes a dynamical system (θ, θ, ϕ, ω3, νx, νy) =
(h1(θ, θ, ϕ, ω3, νx, νy), . . . , h6(θ, . . . , νy)). The value of the normal force gn can bedetermined from the contact constraint (a + s) · z = 0 and we have
gn =mgI1 + mRα(cos θ(I1 ϕ2 sin2 θ + I1θ2)− I3 ϕω3 sin2 θ)
I1 + mR2α2 sin2 θ − mR2α sin θ(1 − α cos θ)µνx. (15)
The system has effectively 6 unknowns and is complicated to solve. In section 8of this introduction we study numerical integration of this system to discern thebehaviour of the variables more clearly.
4.3 A note on TT models
The rigid body modeling the TT in this thesis is an axisymmetric sphere, whichis a rigid body that has been studied extensively in the past [4, 5, 25]. This isalso the model for the TT used almost exclusively in the articles related to thesubject. A valid objection against using this model is that it does not take intoaccount the peg the TT ends up spinning on after inversion. There are articleswhere an adjusted model for the TT is considered in order to study the ”jump”a TT does from rolling on its spherical part to spinning on its peg.
Hugenholtz briefly describes (in the appendix to [13]) the process where thespherical part and the peg is in contact with the plane at the same time at pointsA and A′ respectively. The motion of the TT rising up on its peg is characterizedby decreasing normal force on A and increasing normal force on A′. It is shownnumerically under simplified circumstances that if ψ is initially large and ϕ small(or ϕ initially large) there exist solutions such that the normal forces changesfrom a TT resting entirely on its sphere to a TT resting entirely on its peg.
5 Asymptotic results 13
Karapetyan and Zobova [27] studied a model for a TT where the top is con-structed from two spherical segments joined by a rod, the smaller of the seg-ments representing the end of the peg. This model requires two sets of equationsof motion, similar to (8), each for when the supporting plane is in contact withthe corresponding spherical segment. It also requires equations for the bound-ary case when both segments touch the plane simultaneously, which occurs forsome inclination angle during inversion. The dynamics for a TT based on thismodel has not been explored; the existing works study the stability of steadymotion, i.e. when the TT spins in the upright or inverted position or when itrolls on one of its segments, with the center of mass fixed.
These models are mainly an extension of the model based on a rolling andgliding axisymmetric sphere, essentially using two such spheres to model the TT.Thus it seems more interesting to fully understand the dynamics of one rollingand gliding axisymmetric sphere before the model is adjusted to better describea real TT.
5 Asymptotic results
There is a special subclass of solutions to equations (9) for a rolling and glidingTT with F = gn z − µgnvA, satisfying the condition vA = 0. This gives F = gn zand equations of motion
mr = 0, L = Rαgn3 × z, ˙3 =1I1
L × 3, (16)
where r = s − sz z and the third coordinate is determined by msz = gn − mg.The solutions to this system are important since they describe asymptotic
solutions to the equations of motion (9) for the rolling and gliding TT.In [9] and [24] it is shown that for these solutions the center of mass (CM)
remains stationary (s = 0). These are the situations where the TT is eitherspinning in the upright position, spinning in the inverted position or rollingaround the 3-axis in such a way that the CM is fixed (tumbling solutions).
In paper 1, we derive these solutions from the equations of motion in Eulerangle form and show that when 1 − α < γ = I1/I3 < 1 + α, the stationarytumbling solutions are admissible for all values of cos θ ∈ (−1, 1).
In [9], the relative stability (in the sense of Lyapunov) of a given spinningand tumbling solution is derived as a relation between the value of the Jellettintegral λ and the physical characteristics of the TT. This analysis thus specifieshow fast a given TT should be spun to make solutions to (16) stable.
In particular, if |λ| is above the threshold value λthres =
√mgR3 I3α(1+α)2√
1+α−γ, only
the inverted spinning position cos θ = −1 is stable [9, 24] (the sign of λ is deter-mined by the direction of the initial spin, so without loss of generality we mayassume λ > 0).
The solutions to (16) characterized here are a subset of the precessional so-lutions to the system (8) for the rolling TT. But as the analysis in [24] confirms,
14 Introduction
these solutions are the asymptotic solutions to (9). A theorem of LaSalle type[24] shows that each solution to these equations, satisfying the contact criterionz · (a + s) = 0 and such that gn(t) ≥ 0 for t ≥ 0, approaches exactly one solutionto (16) as t → ∞. Thus the solution set to (16) can be seen as an asymptotic setfor the inverting solutions of the TT equations.
This application of LaSalle’s theorem [17] says that trajectories of the sys-tem (9) approach the asymptotic set, which consists of spinning and tumblingsolutions. If a TT satisfy 1− α < γ < 1+ α, the whole interval of θ ∈ (0, π) is ad-missible as tumbling solutions. If moreover λ is above the threshold value λthres,only the inverted position is stable. Thus we have conditions for when an in-verted spinning solution is the only attractive asymptotic state in the asymptoticLaSalle set. Therefore a TT satisfying these conditions has to invert.
6 The Main Equation for the Tippe Top
If we consider a different model where the supporting surface is such that theTT is not gliding, then the force F at the contact point A is dynamically de-termined and the equations of motion (8) for the TT become equations for arolling axisymmetric sphere. The problem of determining the motion of thiswell known [5] integrable system can be reduced to solving a separation equa-tion in θ. In case of rolling and gliding TT a similar equation has been introducedin [23] where certain parameters are allowed to depend explicitly on time. Thisequation, which we call the Main Equation for TT, provides a way to study thedynamics of inversion for a rolling and gliding TT.
A purely rolling condition for TT implies that the gliding velocity vanishes,vA = s + ω × a = 0. Thus we have s = −ω × a, which allows to eliminate Ffrom (8), so that we get the equations of motion:
d
dt(Iω) = ma ×
(− d
dt(ω × a) + gz
), ˙3 = ω × 3. (17)
This system admits three integrals of motion; the Jellett integral
λ = −L · a = RI1 ϕ sin2 θ − RI3ω3(α − cos θ),
the energy
E =12
mR2[(α − cos θ)2(θ2 + ϕ2 sin2 θ) + sin2 θ(θ2 + ω2
3 + 2ω3 ϕ(α − cos θ))]
+12
[I1(ϕ2 sin2 θ + θ2) + I3ω2
3
]+ mgR(1 − α cos θ), (18)
and the Routh integral
D = ω3
[I1 I3 + mR2 I3(α − cos θ)2 + mR2 I1 sin2 θ
]1/2:= I3ω3
√d(cos θ), (19)
where d(cos θ) = γ + σ(α − cos θ)2 + σγ(1 − cos2 θ), σ = mR2
I3and γ = I1
I3.
7 Special case of rational METT 15
The existence of three integrals of motion makes it possible to reduce the sys-tem (17) to a separable differential equation for θ. We eliminate ω3 = D
I3
√d(cos θ)
and ϕ =λ√
d(cos θ)+RD(α−cos θ)
RI1 sin2 θ√
d(cos θ)from the energy (18), so we get essentially the
Chaplygin separation equation [5]:
E = g(cos θ)θ2 + V(cos θ, D, λ), (20)
where g(cos θ) = 12 I3(σ((α − cos θ)2 + 1 − cos2 θ) + γ
)and
V(z, D, λ) = mgR(1 − αz) +(λ√
d(z) + RD(α − z))2
2I3R2γ2(1 − z2)+
(R2D2 − σλ2)
2R2 I1(21)
is the effective potential for the TT depending algebraically on z = cos θ.The solution θ(t) of (20) stays between the two turning angles θ0 and θ1
determined from the equation E = V(cos θ, D, λ). The solution curve (θ(t), ϕ(t))will trace out oscillating trajectories between these angles on the unit sphere S2.
For the rolling and gliding TT, λ is still an integral of motion, but D and themodified rolling energy E (the part of the energy not involving vA) are no longerintegrals of motion. Due to equations (10)–(14), they satisfy
d
dtD(θ, ω3) =
γm
α√
d(z · 3)(z × a)× vA,
d
dtE(θ, θ, ϕ, ω3) = m(ω × a) · vA.
If we consider the (modified) energy E and D as given functions of time, we canmake the same elimination of ϕ and ω3 from the energy expression as in therolling case and get the Main Equation for the Tippe Top (METT):
E(t) = g(cos θ)θ2 + V(cos θ, D(t), λ). (22)
It has the same form as equation (20), but it depends explicitly on time throughthe functions D(t) and E(t). This is a first order time-dependent ODE whichcan be analysed given some quantitative knowledge about the functions D(t)and E(t).
Equation (22) describes, for generic values of λ and D, oscillatory motion ofthe angle θ(t) much in the same way as the separation equation for the rolling TT(20). Since D(t) is time-dependent, the potential is deformed during inversion.In this thesis we study the METT for inverting solutions in detail and we deducebehaviour of θ(t) for this type of solutions.
7 Special case of rational METT
The effective potential V(z, D, λ) is an algebraic function in z (due to the pres-ence of
√d(z) in (21)), which is complicated to analyse. But in paper 2 we find
16 Introduction
that if 1− α2< γ < 1, which is in the interval (1− α, 1+ α) needed for inversion,
we can choose a positive value for the physical parameter σ = 1−γγ+α2−1
such that
V(z, D, λ) becomes rational:
V(z, D, λ) =mgR(1 − αz) +(λ(α − (1 − γ)z) + RD
√γ + α2 − 1(α − z))2
2I3R2γ2(γ + α2 − 1)(1 − z2)
+R2D2(γ + α2 − 1)− (1 − γ)λ2
2R2 I1(γ + α2 − 1).
The advantage of a rational potential becomes obvious if we rearrange theseparation equation as
z2 =(1 − z2)
g(z)(E − V(z, D, λ)) := f (z).
Real motion of the rolling TT occurs when f (z) ≥ 0, z ∈ [−1, 1], so analysis ofnutational and precessional solutions for a rolling axisymmetric sphere is thenreduced to analysis of roots to the rational function f (z) in this interval. Thenumerator is a third degree polynomial with negative z3-coefficient.
When f (z) has two (unequal) roots z0, z1 in (−1, 1), it corresponds to nuta-tional motion of the symmetry axis between two bounding latitudes on the unitsphere S2. Simple analysis shows that the third root is < −1.
In paper 2 we look at high frequency behaviour of nutational motion fordifferent initial conditions. Thus we assume that the angular velocity alongthe symmetry axis ω3 is large and consider such quantities as the width of thenutational band, the frequency of nutation, the average velocity of precessionand the difference between the energy and the minimum of the potential. Howthese quantities change as functions of ω3 can be estimated in a precise butstraightforward manner, both in the case of a TT with no initial precessionalvelocity ϕ(0) = 0 (similar to the case of a falling HST [1, 12]) and also when thisvelocity is small.
The rational form of the potential V(z, D, λ) is further used in paper 3, wherewe look especially at how to show that an inverting solution θ(t) to METT isoscillating. We find that we can estimate a maximal period of nutation for a TTwith rational potential. When the time of inversion is an order of magnitudelarger than this maximal period, the angle θ(t) is oscillating during inversion.
8 Numerical solving of the TT equations
The aim of solving the dynamical system describing the TT equations numeri-cally is to compare properties of the calculated solutions with the features de-duced from the qualitative analysis of TT equations studied in this thesis.
For relating with the published results by Cohen [7] and Ueda et al. [26],we have chosen values of physical parameters close to the values used by theseauthors. They are summarized in table 1. Other published numerical results by
8 Numerical solving of the TT equations 17
Or [20] or Bou-Rabee et al. [2] use parameters based on a dimensionless versionof the TT equations.
Source m [kg] R [m] α I3 [kg · m2] I1 [kg · m2]
Our data 0.02 0.02 0.3 25 mR2 131
350 mR2
Cohen [7] 0.015 0.025 0.2 25 mR2 2
5 mR2
Ueda, Sasaki andWatanabe [26]
0.015 0.015 0.1 25 mR2 2
5 mR2
Table 1: Physical parameter values used for TT
We present here numerical solutions for the special values of physical pa-rameters of TT that give the rational potential V(z, D, λ) studied in this thesis(σ = (1 − γ)/(γ + α2 − 1) gives I1 = 131
350 mR2). We take initial values θ(0) = 0.1rad, ω3(0) = 155.0 rad/s (corresponding to a value of λ twice the thresholdvalue), ϕ(0) = 0, ϕ(0) = θ(0) = 0 and νx(0) = νy(0) = 0. We further let µ = 0.3.But the same calculations performed for close generic values of parameters, giv-ing an algebraic (non-rational) potential, provide similar graphs. In paper 1,figure 4 we have plotted the surface V(cos θ, D, λ) as function of (θ, D) for a TTwith similar values of physical parameters, giving a non-rational potential, withthe difference that I1 = 47
125 mR2 there.The graph for the inclination angle θ(t) in figure 4.a displays the main feature
of an inverting TT. It has a general form of logistic type curve superposed withsmall amplitude oscillations well visible in the diagram.
In the graphs 4.b and c, we have plotted ψ(t) and ϕ(t) that, together withθ(t), give ω3 = ψ + ϕ cos θ. These graphs show, in vicinity of time 3 seconds,a non-obvious phenomenon of sudden change of amplitude of ψ, ϕ indicatingrapid exhange of angular velocity between these two variables while ω3 changesvery slowly and smoothly (figure 5.a).
An educated guess is that this phenomenon (also visible in correspondinggraphs published in [7, 26]) markes initiation of inversion. The exact mechanismof interaction between forces and torques acting on TT remains unexplained yet.This behaviour of ψ(t) and ϕ(t) draws attention to the neighborhood of time atapproximately 3 seconds in the θ(t) graph, where we now notice that initiallysmall oscillations acquire larger amplitude so that the nutational band for θ(t)almost extends to the north pole of the unit sphere S2.
The graph in figure 5.a shows that ω3(t) is an (surprisingly) almost monotonouslydecreasing function from the initial value of 155 rad/s to about −85 rad/s. Itis unclear if the derivative ω3(t) changes sign at all. The decreasing behaviourof ω3 reflects inversion of direction of axis 3 and the frictional loss of rotationalenergy. The curve ω3(t) is in the beginning and at the end almost horizontal.During the rapid inversion of TT between 4 and 6.5 seconds the symmetry axis 3turns upside down, so the projection ω · 3 of predominantly vertical angular ve-locity ω on 3 changes sign. The moment of initiation of inversion of TT at about 3seconds noticed at graphs for ψ(t) and ϕ(t) is also visible in the graphs 5.b for the
18 Introduction
0 1 2 3 4 5 6 7 8Time [s]
0
π2
π
θ [
rad]
a)
0 1 2 3 4 5 6 7 8Time [s]
−1000
−750
−500
−250
0
250
500
750
1000
ψ [
rad/s
]
b)
0 1 2 3 4 5 6 7 8Time [s]
−1000
−750
−500
−250
0
250
500
750
1000
ϕ [
rad/s
]
c)
Figure 4: Plots obtained by numerically intergrating equations (10)-(14) and equa-tion (15) using the Python 2.7 open source library SciPy [14]. Parameter values arechosen from the first row of table 1 and µ = 0.3. Initial values are θ(0) = 0.1 rad,ω3(0) = 155.0 rad/s, ϕ(0) = 0, ϕ(0) = θ(0) = 0 and νx(0) = νy(0) = 0. Plota shows the evolution of the inclination angle and plots b and c show the evolutionof the angular velocities ψ(t) and ϕ(t).
8 Numerical solving of the TT equations 19
0 1 2 3 4 5 6 7 8Time [s]
−100
−50
0
50
100
150
200
ω3 [
rad/s
]
a)
0 1 2 3 4 5 6 7 8Time [s]
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Velo
cit
y [
m/s
]
b)
0 1 2 3 4 5 6 7 8Time [s]
0.015
0.020
0.025
0.030
0.035
0.040
0.045
Energ
y [
J]
c)
Figure 5: Plots obtained as in figure 4. Plot a shows the evolution of the angularvelocity ω3(t), plot b shows the gliding velocities νx(t) (blue) and νy(t) (green)
and plot c show the evolution of the energy functions E(t) (green) and E(t) (blue).
20 Introduction
−0.0008 −0.0006 −0.0004 −0.0002 0.0000 0.0002 0.0004 0.0006 0.0008D [Kg m2 /s]
0.015
0.020
0.025
0.030
0.035
0.040
0.045En
ergy
[J] (D0 ,E0 )
(D1 ,E1 )
a)
0 1 2 3 4 5 6 7 8Time [s]
−15
−10
−5
0
5
10
15
θ [r
ad/s
]
b)
Figure 6: Plot a shows the curve (D(t), E(t)), calculated by integrating equations(10)-(14) and equation (15) as for figure 4. Plot b shows the evolution of the angularvelocity θ(t).
gliding velocities νx(t), νy(t). Both velocities are oscillatory but initially νx (blue)is larger than νy (green). In a neighborhood of time 3 seconds νy becomes largerthan νx and is rapidly, oscillatory increasing by 1–2 orders of magnitude whenthe inversion takes place. During the inversion the amplitude of νx also radicallyincreases an order of magnitude but νx becomes predominantly negative. Theloss of total energy during inversion is illustrated in graph 5.c where the greencurve represents monotonous decrease of the total energy E = −µgn|vA|2 andthe blue curve represent the modified rotational energy E(t), which is duringthe inversion consistently larger than E(t). This does not contradict the conser-vation of energy when we look closer at E = E − 1
2 mv2A + mvA · (ω × a). For an
inverting TT the angular velocity ω remains close to be parallel with the z axisand the product ω × a points behind the plane of the picture in figure 3. Thedirection of rotation of TT (with ω almost parallel to z) causes νy = vA · y to bepositive and to point also behind the plane of the figure.
9 Conclusions 21
Thus the term
12
mvA · (2(ω × a)− vA) ≈12
m(νx x + νyy) ·(2(ω × a)y − (νx x + νyy)
)
≈ −12
mν2x +
12
mνy(2|ω × a| − νy) (23)
may be positive if νy > νx and 2|ω× a| − νy = 2|ω||a| sin θ − νy > 0 is sufficientlylarge. In figure 5.b we cannot estimate the size of each term, but as we see infigure 3, |a| is growing during inversion, |ω| is decreasing and sin θ is growinguntil the axis 3 passes θ = π
2 . This may altogether keep the term (2|ω × a| − νy)positive and sufficiently large to make (23) positive. This is consistent with thegraph 5.c where the difference E − E becomes largest in the middle of inversionwhen θ ≈ π
2 .The figure 6.a illustrates the curve (D(t), E(t)) when it goes from the bound-
ary value at the start of TT inversion (D0 ≈ 9.381 · 10−4, E0 ≈ 4.912 · 10−2) tothe final boundary value (D1 ≈ −6.667 · 10−4, E1 ≈ 1.734 · 10−2) after inverting.The shape of the curve reflects the oscillatory behaviour of E(t) while D(t) isdecreasing almost monotoneously since D(t) = I3ω3(t)
√d(cos θ(t)).
The last graph in figure 6.b shows how the angular velocity θ(t) is changingbetween positive and negative values, thus confirming the oscillatory characterof θ(t) during inversion.
9 Conclusions
In this thesis we have approached the question of qualitative description of dy-namics of the TT. The general question of deducing dynamics from equationsof motion for a system with 6 degrees of freedom is an impossible task unlessequations are integrable.
Since the TT equations are not integrable it is only possible to approach thisquestion for specific types of initial conditions where the main features of thedynamical behaviour are weakly dependent on perturbation of initial conditions.
This is the case of inverting behaviour, which is persistent and easily re-producible for a toy TT. So the point is to explain the features of dynamicalequations and of initial conditions that are responsible for the observable effect.
In this thesis we have shown that this is a consequence of the METT equationthat is fulfilled by every solution of TT equations with suitable D(t), E(t) thatcan be found for any given solution.
In case of inverting TT motion, D(t) and E(t) are not exactly known, but wecan qualitative describe how they are changing from the initial values (D0, E0)to the final values (D1, E1).
As usual this work opens more questions than it answers. First, all estimateshave been made in the case of rational potential V(z, D, λ), not an algebraicone. The restriction on parameters, which make
√d(z) polynomial in papers 2
and 3 allow us to get estimates of relevant quantities describing dynamics of an
22 Introduction
inverting TT. An extension to algebraic V(z, D, λ) seems to be feasible, howevertechnically more complicated.
The connection between the choice of friction law, the value of the frictioncoefficient µ and the time of inversion Tinv has not been discussed here.
The numerical simulations agree well with the analytically deduced featuresof inverting solutions and these numerical solutions preserve its character for anopen set of values about the values taken here. This is in agreement with thepersistent nature of inversion observed in toy experiments.
These numerical solutions indicate also that there may exist a more specificmechanism of how friction works for TT. The graphs show a clear initiation timefor an inversion when ψ, ϕ start to oscillate strongly and νx, νy change signs.The mathematical nature of this mechanism remains unclear, but the mechanismseems to be generic and somehow ”has to sit” in the equations.
In this thesis we have clarified the logic of the LaSalle type statements. Wehave explained what they say about inversion of TT and what they does notsay about inversion. We have cleared up the relation of the TT equations tothe METT and to equations of two other rigid bodies. We have found that theparameter limit Rα = −l, R → ∞ of TT equations gives equations for the glidingheavy symmetric top and that for initial conditions with D = 0, λ = 0 werecover equations for a gliding eccentric cylinder from the TT equations. Forbetter understanding of how to work with TT equations we have analysed therational case.
Finally we have shown that inverting solutions θ(t) with D(t), E(t) satisfyingthe boundary conditions (D(0) ≈ D0, E(0) ≈ E0) and (D(Tinv) ≈ D1, E(Tinv) ≈ E1)have to oscillate provided that the time of inversion is sufficiently long.
References
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, second ed.,Springer Verlag, 1989.
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[3] C. M. Braams, On the influence of friction on the motion of a top, Physica 18(1952), 503–514.
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[5] , On a ball’s rolling on a horizontal plane, Math. collection of theMoscow Math. Soc. 24 (1903), 139–168, English transl.: Regul. Chaotic Dyn.7 (2002), no. 2, 131-148.
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References 23
[7] C. M. Cohen, The Tippe Top Revisited, Am. J. Phys. 45 (1977), 12–17.
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[11] S. T. Glad, D. Pettersson, and S. Rauch-Wojciechowski, Phase Space of RollingSolutions of the Tippe Top, SIGMA 3, Contribution to the Vadim KuznetsovMemorial Issue (2007), 3247–3259.
[12] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, third ed., Addison-Wesley, 2002.
[13] N. M. Hugenholtz, On tops rising by friction, Physica (1952), no. 18, 515–527.
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[15] A. V. Karapetyan, Qualiative investigation of the dynamics of a top on a planewith friction, J. Appl. Math. Mech. 55 (1991), no. 4, 563–565.
[16] L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press Ltd., 1960.
[17] J. P. LaSalle, Some Extensions of Liapunov’s Second Method, IRE TransactionsOn Circuit Theory 7 (1960), 520–527.
[18] H. K. Moffatt and Y. Shimomura, Spinning eggs – a paradox resolved, Nature416 (2002), 385–386.
[19] H. K. Moffatt, Y. Shimomura, and M. Branicki, Dynamics of an axisymmetricbody spinning on a horizontal surface. i. stability and the gyroscopic approxima-tion., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), 3643–3672.
[20] A. C. Or, The Dynamics of a Tippe Top, SIAM J. Appl. Math. 54 (1994), no. 3,597–609.
[21] D. G. Parkyn, The inverting top, Math. Gaz. 40 (1956), 260–265.
[22] W. Pliskin, The Tippe Top (Topsy-Turvy Top), Am. J. Phys. 22 (1954), no. 28,28–32.
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24 Introduction
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