Lovington High School

1
What Causes Spin to Reverse Itself? A Study and
Explanation of the Rattleback (Celt).

• Team 038
• Lovington High School

Team Members

Nicholas Tobkin, Dustin
Graham, Elizabeth Myers,
Jeremiah
Giese Sponsoring Teacher
Mrs. Pamela Gray
2
Final Presentation What Causes Spin to Reverse
Itself?
3
Problem Definition

• What is the rattleback?
• Its unique properties
• Wondered how and why it displayed such a
  strange behavior
• We want to be able to mathematically analyze
  and simulate the object

An image of the path the rattleback takes when
spun (used paint to mark the paper).
4
Computational Plan

• Obtained several papers discussing physics and
  the mathematics behind the rattleback.
• Dr. John Russell has aided in the understanding
  of complex formulas and development of programs.
• Application of Newtons Law results in having to
  solve six coupled nonlinear ordinary differential
  equations.
• We can work with a regular ellipsoid, using the
  same a, b, c, center of gravity location and Is.
• We can vary initial position through initial
  selection of a, b and g.

5
Constraints

• Analysis assumes sufficient friction so no slip
  occurs at the point of contact, ellipsoid can
  only roll.
• Since there is no sliding at this point the
  velocity of the point of contact relative to the
  surface is zero.

6
Program (input parameters)

• Specify the rotating object
• - Dealing with portion of ellipsoid
• - Ellipsoid has ellipse as cross-section
• - a, b, and c specify size and shape
• The analysis allows us to deal with any portion
  of an ellipsoid where the cut is parallel to the
  x, y plane. The variable h and how we calculate
  the Is takes this into account.

7
Program (input parameters cont.)
- Volume V 1/2 ((4/3)pabc) - Density r
g/cm3 - Mass in grams, M rV - Location of
center of gravity x y 0 and z z
(3/8)c
8
Program (input parameters cont.)

• Mass resists linear accelerations
• Mass moments of inertia about each axis resist
  rotational acceleration about that
  axis
• a) Ixx (1/5)M(b2 c2)
• b) Iyy (1/5)M(a2 c2)
• c) Izz (1/5)M(a2 b2)

9
Program (input parameters cont.)

• Need to describe where it is when it starts to
  spin - g Rotation about initial z axis - a
  Rotation about new x axis (after g
  rotation) - b Rotation about new y axis
  (after a g rotation) - g
  is angle between the final z axis line
  perpendicular to surface (x or y axis)

10
Program (input parameters cont.)
- Must specify how the spinning is started -
These are the angular spin rates a) w1 (about x
axis) b) w2 (about y axis) c) w3 (about z
axis) - Also factor in torque due to air
(proportional to spin rates) a) Tx
-sw1 b) Ty -sw2 c) Tz -sw3
By introducing a term assumed proportional to the
spin rate through the constant s we can account
for a dissipative affect (slowing down).
11
Program (input parameters cont.)
- Time will also be factored into the program to
limit how long the analysis will run. We will
run it approximately 20
seconds. - Recap of input variables a, b, c, z
Ellipsoid shape, initial center of
gravity M Mass of
ellipsoid Ixx, Iyy, Izz Mass moments of
inertia a, b, g Initial
orientation w1, w2, w3 Initial spin
rates s Air
resistance coefficient tmax
Length of simulation
12
Angular positions definitions of a, b, g
g - First rotation about z (z stays the same,
z1) X g and becomes x1 Y g and
becomes y1 a - Second rotation about
x1 (x stays the same, x11) y1 a and
becomes y11 z1 a and becomes z11
b - Third rotation about y11 (y11 stays the
same, y) x11 b and becomes x z11 b
and becomes z
13
Description of Kinematic Analysis

• Kinematic - Relates various coordinates
  (x1,x2,x3) positions (a, b, g) velocities
  (v1,v2,v3) spin rates (w1,w2,w3) to each other
  through geometry and/or constraints -
  Here the contact point (x1,x2,x3) as measured
  from the ellipsoid center is related to (a, b,
  g) the position of the ellipsoid, indirectly
  through the direction cosines (m1, m2, m3) -
  Contact point has 0 translational velocity as a
  result of assuming there is sufficient
  friction to keep it from sliding Vcg r x
  w

14
Description of Kinetic Analysis

• Kinetics - Relates, various translational
  rotational accelerations to forces moments
  caused by these forces From Newton

d d t
(M v)CG f - Mg h
- Where f is the vector sum of the force normal
to the surface two components of friction tangent
to the surface Mg weight
d d t
(h) r x f (where h is the angular momentum)
15
Kinematic Equations

• m1 cos a sin b
• m2 sin a
• m3 cos a cos b
• e (a m1)2 (b m2)2 (c m3)21/2
• (Direction cosines -- cosines of angle between a
  perpendicular to the plane and the three axes.

16
Kinematic Equations (cont.)

• x1 a2m1 / e
• x2 b2m2 / e
• x3 c2m3 / e
• m1 w3m2 - w2m3
• m2 w1m3 - w3m1
• m3 w2m1 - w1m2

(Location of contact point relative to where
ellipse is defined -- x, y, z.)
(Rate of change of direction cosines with time.)
17
Kinematic Equations (cont.)

• e (a2m1m1 b2m2m2 c2m3m3) / e
• x1 a2(em1 - em1) / e2
• x2 b2(em2 - em2) / e2
• x3 c2(em3 - em3) / e2

(Time rate of change of location (velocity of
contact point) in rotating system.)
18
Kinematic Equations (cont.)

• v1 w2(h-x3) w3x2
• v2 -w3x1 - w1(h-x3)
• v3 -w1x2 w2x1
• d1 w2(v3-x3) - w3(v2-x2)
• d2 w3(v1-x1) - w1(v3-x3)
• d3 w1(v2-x2) - w2(v1-x1)

(Relative velocity minus defining values. Three
velocities in rotating coordinates. h - distance
from ellipsoid to center of gravity (cg).)
19
Kinetic Equations
(First time forces are introduced.)

• F1 -sw1 Mg(x3-h) m1- x3m3
• F2 -sw2 Mg(h-x3) m1 x1m3
• F3 -sw3 Mg(x2m1 - x1m2
• R1 Dw1 (B - C) w2 w3
• R2 (C - A) w1 - Dw2 w3
• R3 D(w22 - w12) (A - B) w1w2

20
Kinetic Equations (cont.)
Mass Moments of Inertia about the (x, y, z) Axis
through the Center of Gravity.

• A (IXX)CG B
  (IYY)CG
• C (IZZ)CG D (IXY)CG

21
Kinetic Equations (cont.)
Mass Moments of Inertia Translated to the Point
of Contact.

• I11 A Mx22 (h-x3)2
• I22 B Mx12 (h-x3)2
• I33 C M(x12 x22)
• I12 I21 D - Mx1x2
• I23 I32 M(h-x3)x2
• I31 I13 M(h-x3)x1

These are the terms which couple the three
rotations together so energy can be transferred
from one to another causing rattleback NOTE
All the mixed numbers I12, I23, I13--these are
the culprits.
22
Kinetic Equations (cont.)

• S1 M(h-x3) d2 x3d3
• S2 M(x3-h) d1 - x1d3
• S3 Mx1d2 - x2d1
• Q1 F1 R1 S1
• Q2 F2 R2 S2
• Q3 F3 R3 S3

If each of these equations are sequentially
substituted into the next, the Qs are only
functions of a, b, g, w1, w2, and w3 (the six
unknowns).
23
Kinetic Equations (cont.)

• E1 Q1 I12 I13
• Q2 I22 I23
• Q3 I32 I33
• E2 I11 Q1 I13
• I21 Q2 I23
• I31 Q3 I33

Determinants for w1, w2, w3--the angular
accelerations
24
Equations (cont.)

• E3 I11 I12 Q1
• I21 I22 Q2
• I31 I32 Q3
• G I11 I12 I13
• I21 I22 I23
• I31 I32 I33

25
Six Nonlinear Ordinary Differential Equations to
be Solved
Integrate a w3 sinb w1 cosb b (-w3 cosb
w1 sinb) tan a w2 g (w3 cosb - w1 sinb) sec
a w1E1 / G w2E2 / G w3E3 / G
Note a d a b d b
g d g d t
d t d t
26
Output
Plot a, b, g (Ellipsoid
orientation) w1, w2, w3 (Ellipsoid
spin rates) As functions of time. We can also
plot d where d cos-1 m3
27
Output (cont.)
All equations refer to Kanes paper. Kane, T.R.
Realistic Mathematical Modeling of the
Rattleback. International Journal of Nonlinear
Mechanics. 1982. Vol. 17, No. 3, pp.175
28
Progress

• Made substantial progress
• Researched rattleback and other related
  documentation
• Located papers from Kane, Schultz, and Mitiguy
• With the aide of Dr. John Russell, we have found
  and developed equations representing the
  objects unique properties
• Extensive knowledge in physics and calculus

29
Results

• We created a program in MATLAB to run a
  simulation
• It displayed rattleback behavior.
• Conditions can be changed in the program,
  different experiments performed.
• Graphs the behavior of rotation, velocities
  around 3 axes, and d angle.
• Discovered rattleback performs multiple reversals.

30
Results

• Roll angle, a, 20 second time period.
• Rapidly rocking along the short axis

• Angular velocity of roll (rad/sec)
• Shows velocity of rocking.

31
Results

• Pitch angle, b, same time interval
• Rocks along long axis, increases, then returns to
  0

• Angular velocity of pitch (rad/sec)
• Decays to a low value as the rattleback starts
  rotating around the z-axis.

32
Results

• Yaw angle, g, shows spin reversal.
• Begins spin in the negative direction, but
  reverses.

• Angular velocity of yaw.
• Velocity increases up to reversal point, then
  decays.

33
Results

• Delta (d), angle between the vertical axes of the
  ellipsoid and the surface.
• Rocks back and forth corresponding to other
  angles, velocities

34
Results

• These graphs show the yaw angle and velocity over
  a 100 sec. period.

• The rattleback rotates, and reverses at around 6
  seconds.

35
Results

• These graphs show the yaw angle and velocity over
  a 100 sec. period, rotating counter-clockwise.

• The rattleback never dips below zero, since it
  does not reverse.

36
Results

• This run shows the effect of adding air
  resistance, sigma (s).
• The roll, pitch, and yaw dissipate to zero.

37
Results

• In this run, the rattleback was started by
  tapping the end.

• The change in beta still produces rotation and a
  reversal in the yaw angle.

38
Future Application

• Satellites - Development testing -
  Movement - Guidance system programming

39
Knowledge Gained

• Better acquainted with C programming language
• Basics of MATLABÒ
• Advanced physics and calculus
• Advantages of teamwork
• Personal determination and satisfaction in
  completing difficult projects