Tippe Top Inversion as a Dissipation-Induced Instability · Tippe Top Inversion as a...
Transcript of Tippe Top Inversion as a Dissipation-Induced Instability · Tippe Top Inversion as a...
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.