Tippe Top Inversion as aDissipation-Induced Instability

Advisor: Jerrold E. Marsden

Nawaf M. Bou-RabeeBloch’s Visit, 11/12/03

Figure 1: L-R non-inverted and inverted tippe-top.

Claim: Tippe top inversion is completely described by the

modified Maxwell-Bloch equations.

Dissipation-Induced Instability

A dissipation-induced instability describes a neutrally stable

equilibrium becoming spectrally (and hence nonlinearly) stable with

the addition of dissipation.

Potpourri of examples:

motion of planets in celestial mechanics

quasigeostrophic flow

tubular cantilever conveying fluid

rotating spherical pendulum (or bead on a rotating circular plate)

History

Dissipation-Induced Instabilities:

(1994, Bloch, Krishnaprasad, Marsden, & Ratiu) analysis of

phenomenon

(2001, Clerc & Marsden) normal form for phenomenon

(2002, Derks & Ratiu) effect of dissipation on families of relative

equilibria in Hamiltonian systems

History

Tippe Top:

(1905, Routh) commentary on rising of tops

(1977, Cohen) numerical simulation of phenomenon

(1994, Or) linear analysis of equations of motion

(1995, Ebenfeld) orbital analysis of relative equilibria

Modified Maxwell-Bloch Equation

Consider ODEs of the form:

q = f(q, q), q =

x

y

Linearize to obtain:

q = Aq +Bq (1)

What is the most general form of (1) invariant under SO(2) rotations?

Modified Maxwell-Bloch Equation

Gyroscopic term Spring term

z + iαz + βz + iγz + δz = 0 z = x+ iy (2)

Damping term Complex Damping term

Modified Maxwell-Bloch Equation

z + iαz + βz + iγz + δz = 0 z = x+ iy (3)

Stability criteria:

β > 0

αβγ − γ2 + β2δ > 0

α2β + β3 − αγ + βδ > 0(4)

OC

Q

q

k

y

x

z

Figure 2: Tippe top model.

Tippe Top Equations of Motion

Translational equations,

MX = FQ · ex

MY = FQ · ey

MZ = FQ · ez −Mg

(5)

Rotational equations,

L = MR2(e?)2k× k+Q× FQ (6)

Attitude equation,

k =1

IL× k (7)

Constraint,

Q · ez + z = 0 (8)

Friction Law

Force at point of contact,

FQ = Ff + Fzez

Sliding friction assumed,

Ff = −cVQ

where

VQ = VC + ω ×Q (9)

Dimensionless Equations

µx = σfx = −ν[x− e?l− σΥy + (σ − 1)(Υ · k)m]

µy = σfy = −ν[y − e?m+ σΥx − (σ − 1)(Υ · k)l]

Υ =

[

σ2µ(e?)2(Υ · k)(k×Υ) + e?σfz(ez × k) + σq× ff − σµ(e?)2(ff · (k× ez))k]

(1− µ(e?)2)σ

k = σΥ× k

(10)

(10) admits a momentum invariant,

ΥQ = Υcg · q, (ΥQ)t = 0 (11)

Linear Theory

Equilibria of (10) satisfy:

Υ× k = 0 =⇒ Υ and k are collinear

ez × k = 0 =⇒ ez and k are collinear

Therefore, equilibria satisfy:

x = y = x = y = Υx = Υy = l = m = 0, Υz = constant, n = ±1

Specify ΥQ to obtain two fixed points.

Linear Theory

Linearization yields,

VC = −νAVC − iνBΛ + iνCΦ

Λ = iνDVC + (νE + iF )Λ + (νG+ iH)Φ

Φ = −inoσΛ + iσΥozΦ,

(12)

where

VC = x+ iy, Λ = Υx + iΥy, Φ = l + im

Tippe Top Modified Maxwell-Bloch

Equations

Ignore translational effects to obtain,

Φ + iaΦ + bΦ + icΦ+ dΦ = 0 (13)

Can we reduce (13) any further?

Heteroclinic Connection

Energy of tippe top,

E = µ(x2

2+y2

2+z2

2)+σ(1−σµ(e?)2)

Υ ·Υ

2+(1− σ + σµ(e?)2)(Υ · k)2

2+µFr−1z

(14)

Energy’s orbital derivative,

Et = −ν‖vQ‖2, (15)

Invoke LaSalle’s theorem.

No Slip, No Force Problem

Dynamics of tippe top asymptotic states described by,

x = 0

y = 0

Υ =1

1− σµ(e?)2(

σµ(e?)2(Υ · k)(k×Υ) + e?fz(ez × k))

k = σΥ× k

(16)

(16) is Hamiltonian. Moreover,

vQ = 0 =⇒ n = 0 =⇒ Υz = 0 =⇒ Υ · (ez ×k) = 0 =⇒ x = y = 0

Abundance of integrals of motion.

Energy-Momentum Minimization

Find extrema of the augmented energy,

h = E + λΥQ + λk · k

where λ and λ are Lagrange multipliers.

Extrema of energy satisfy:

a0n4 + a1n

3 + a2n2 + a3n+ a4 = 0 (17)

Conclusion

Heteroclinic connection existence criteria match linear stability

criteria for inverted and non-inverted states.

Implication: modified Maxwell-Bloch equations describe orbital

stability of tippe top.

Claim: Tippe top inversion is completely described by the

modified Maxwell-Bloch equations.