# Toys_ Toys_ Toys_

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```							     Toys! Toys! Toys!
The Dynamics of Some Curious Toys
Bard Ermentrout

CHS
November 2002

Toys! Toys! Toys! – p.1/20
Introduction

Many mechanical, magnetic, toys and
carnival rides
Toys often illustrate fundamental physical
principles.
Easy to produce the models – diff eqs
Can produce very complex dynamics.
Accessible to undergrads with just a little
math

Toys! Toys! Toys! – p.2/20

A ball bouncing on a hard surface suggests a simple model
Let −vn be the downward velocity of the nth bounce
Assume that it hits the surface and reverses course with damping
The new velocity is rvn . . .
. . . and when it returns, it is −rvn so we get

vn+1 = rvn

Example: v0 = 5, r = .8,
vn = 5(0.8)n

Toys! Toys! Toys! – p.3/20
Sugar-water oscillator

ρs                filled: ρs gh = ρw gH
h eq = r H
h    H
L
ρw
ρ    empty: ρsg(h −L) =ρw g(H − L)
r= s
ρw             h eq = r H − (1−r)L
How does it work?
Archimedes principle
When tube ﬁlled with sugar – goes to ﬁrst equilibrium
When tube ﬁlled with water – goes to second equilibrium
Both equilibria unstable – Rayleigh instability
Thus, it oscillates

Toys! Toys! Toys! – p.4/20
Equations

dh
= µ(heq − h)
dt
Flowing down:
µ is resistance of sugar water
heq is ﬁlled
Flowing up:
µ is resistance of water
heq is empty
Switch when rate, dh/dt becomes small in magnitude
Simulation

Toys! Toys! Toys! – p.5/20
Extensions

More than one cup?
Assumption that H is ﬁxed
Actually changes inversely with h
Leads to Coupled SWO – oscillate out of phase

Toys! Toys! Toys! – p.6/20
The Drinking Bird

A favorite since
1946!
Another oscillator

Toys! Toys! Toys! – p.7/20

1. What causes the ﬂuid to rise up the toy bird’s
neck?
2. What causes the bird to dip and then erect
himself
3. What is the purpose of the fuzzy head of the
bird? the water?
4. How do temperature & humidity affect
operation?
5. How can he fail?

Toys! Toys! Toys! – p.8/20
Explaining the bird

Evaporative cooling lowers pressure
Pressure seal broken, ﬂuid returns to base

Toys! Toys! Toys! – p.9/20
Euler-Lagrange Equations

x = a sin θ                           .
a   y = a(1 − cosθ ) K.E. m . 2 . 2 = m 2 θ2
y                                     2 (x + y )  2 a
θ       (x,y)          P.E mgy = mg a(1 − cosθ )
L = K.E. − P.E.
x

d ∂L     ∂L
=
dt ∂vj   ∂xj
˙
For the pendulum, v = θ, x = θ, so
d2
ma2 2 = −mga sin θ
dθ
Toys! Toys! Toys! – p.10/20
A little model

P.E.: (m2 gl2 − m1 gl1 ) cos θ
2 ˙
K.E.: 1 (m2 l1 + m2 l2 )θ 2
2   1
2
2
m2
l2                     Evaporative cooling:
˙
m2 = r|θ|; Upper mass
˙
l1    θ max   rises proportional to
swinging speed;
m1
θ                 M = m1 + m 2 .
θ min
Dip: When θ hits a critical
value, m2 reset and
velocity set to zero
Simulation

Toys! Toys! Toys! – p.11/20
Chaos & beyond

Many toys and carnival rides are in fact
chaotic
Some are intrinsically chaotic, but
Most depend on being periodically driven
Others are “quasi-periodically” driven

Toys! Toys! Toys! – p.12/20

What if the ball is bouncing on an oscillating
platform?
Simulation – Try ω = 2, 3; look at sensitive
dependence!
Can study a simpliﬁed model on a pocket
calculator!

Toys! Toys! Toys! – p.13/20
Reduction to a map

New velocity is old velocity damped and augmented by the
platform: a cos f tn
tn is the time it hits the platform which is prior time plus time in the
air, T = 2v/g
Thus we get an iteration:

tn+1     = tn + 2vn /g
vn+1     = rvn + a cos(f tn+1 ) = rvn + a cos[f (tn + 2vn /g)]

Simulation

Toys! Toys! Toys! – p.14/20
Fractal attractors

Successive snapshots are “self-similar” another
hallmark of chaos

Toys! Toys! Toys! – p.15/20
The Tilt-a-whirl

It is the random tipping and spinning
of the Tilt-A-Whirl’s seven cars that
make this ride so much fun. The Tilt-
A-Whirl is a platform type ride that is
based on a circular track of bridge type
construction.   The seven Tilt-A-Whirl
cars are ﬁxed to a pivot pin on each
platform. The revolving platforms roll
over hills and valleys causing centrifu-
gal and gravitational forces upon the
cars causing them to tip and spin ran-
domly.

Toys! Toys! Toys! – p.16/20
A model tilt-a-whirl

P.E.: mgz
m
K.E.:     (x2
2 ˙     + y2 + z 2 )
˙    ˙
φ
r2
Angles α, β are small so
θ=ωt
r1
C
2
2d φ                          2      dφ
mr2 2       =      −mr1 r2 ω sin φ − ρ
z                dt                                   dt
β
+      mgr2 (α sin φ − β cos φ)
α
x,y
α =        α0 − α1 cos 3θ
β   =      3α1 sin 3θ

Toys! Toys! Toys! – p.17/20
Simulation & Explanation

Simulation
This, like the ball problem, is a periodically forced, damped oscillator
At each turn of the big wheel, we plot the angular velocity and angle: Poincare Map

’
φ(2πΚ)

1

0.5

0

-0.5

-1

-1.5

1      2      3            4   5   6
φ(2πΚ)

Toys! Toys! Toys! – p.18/20
Other toys

Dolphins! – Like the bouncing ball &
Tilt-a-Whirl
Magnetron – Autonomous transient chaos
Zipper – Quasiperiodically forced damped
pendulum

Toys! Toys! Toys! – p.19/20
Conclusions

Lots of simple toys have very interesting
dynamics.
Try: pendula driven to sit upside down,
multiple pendula, lava lamps, and so on ...
Most of all, have fun!

Toys! Toys! Toys! – p.20/20

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