Team 038

Lovington High School

Team Members: Nicholas Tobkin, Dustin Graham, Elizabeth Myers, & Jeremiah Giese

Sponsoring Teacher: Mrs. Pamela Gray

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

Final PresentationWhat Causes Spin to Reverse Itself?

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).

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 Newton’s 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 I’s.

• We can vary initial position through initial selection of , and .

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.

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 I’s takes this into account.

Program (input parameters cont.):

- Volume V = 1/2 ((4/3)abc)

- Density = g/cm3

- Mass in grams, M = V

- Location of center of gravity

x = y = 0 and z = z = (3/8)c

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)

Program (input parameters cont.):

• Need to describe where it is when it starts to spin- Rotation about initial z axis- Rotation about new x axis (after rotation)- Rotation about new y axis (after &

rotation) - is angle between the final z axis & line perpendicular to surface (x or y

axis)

Program (input parameters cont.):- Must specify how the spinning is started

- These are the angular spin rates:

a) 1 (about x axis)

b) 2 (about y axis)

c) 3 (about z axis)

- Also factor in torque due to air (proportional to spin rates)

a) Tx = -1

b) Ty = -2

c) Tz = -3

By introducing a term assumed proportional to the spin rate

through the constant we can account for a dissipative affect (slowing down).

Program (input parameters cont.):

- Time will also be factored into the program to limithow 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 gravityM Mass of ellipsoidIxx, Iyy, Izz Mass moments of inertia

,, Initial orientation1, 2, 3 Initial spin rates

Air resistance coefficienttmax Length of simulation

Angular positions definitions of

- First rotation about z (z stays the same, z1)X and becomes x1

Y and becomes y1

- Second rotation about x1 (x stays the same, x11)y1 and becomes y11

z1 and becomes z11

- Third rotation about y11 (y11 stays the same, y)x11 and becomes x

z11 and becomes z

Description of Kinematic Analysis

• Kinematic - Relates various coordinates (x1,x2,x3) positions (, , ) velocities (v1,v2,v3) spin rates (1,2,3) to each other through geometry and/or constraints

- Here the contact point (x1,x2,x3) as measured from the ellipsoid center is related to (, , ) the position of the ellipsoid, indirectly

through the direction cosines (1, 2, 3) - Contact point has 0 translational velocity as a result of assuming there is sufficient

friction to keep it from slidingVcg = r x

Description of Kinetic Analysis

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

From Newton: d

d t

(h) = r x f (where h is the angular momentum)

(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

Kinematic Equations

1 = cossin

2 = sin

3 = cos cos

= [(a 1)2 + (b 2)2 + (c 3)2]1/2

(Direction cosines -- cosines of angle between a perpendicular to the plane and the three axes.

Kinematic Equations (cont.)

x1 = a21 /

x2 = b2/

x3 = c2/

1 = 3

=

=

(Location of contact point relative to where ellipse is defined -- x, y, z.)

(Rate of change of direction cosines with time.)

Kinematic Equations (cont.)

= (a211 + b222 + c233) /

x1 = [a2(1 - 1)] / 2

x2 = [b2( - )] / 2

x3 = [c2( - )] / 2

(Time rate of change of location (velocity of contact point) in rotating system.)

Kinematic Equations (cont.)

v1 = 2(h-x3) + 3x2

v2 = -x1 - h-x3)

v3 = -x2 + 2x1

1 = 2(v3-x3) - (v2-x2)

2 = (v1-x1) - (v3-x3)

3 = (v2-x2) - (v1-x1)

(Relative velocity minus defining values. Three velocities in rotating coordinates. h - distance from ellipsoid to center of gravity (cg).)

F1 = -1 + Mg[(x3-h) 1- x33]

F2 = - + Mg[(h-x3) 1 + x13]

F3 = - + Mg(x21 - x1]

R1 = [D1 + (B - C) 2]

R2 = [(C - A) 1 - D

R3 = D(1

2) + (A - B) 1

(First time forces are introduced.)

Kinetic Equations

Kinetic Equations (cont.)

A = (IXX)CG B = (IYY)CG

C = (IZZ)CG D = (IXY)CG

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

Kinetic Equations (cont.)

I11 = A + M[x22 + (h-x3)2]

I22 = B + M[x12 + (h-x3)2]

I33 = C + M(x12 +x2

2)

I12 = I21 = D - Mx1x2

I23 = I32 = M(h-x3)x2

I31 = I13 = M(h-x3)x1

Mass Moments of Inertia Translated to the Point of Contact.

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.

Kinetic Equations (cont.)

S1 = M[(h-x3) 2 + x3

S2 = M[(x3-h) 1 - x1

S3 = M[x1- x2

Q1 = F1 + R1 + S1

Q2 = F2 + R2 + S2

Q3 = F3 + R3 + S3

If each of these equations are sequentially substituted into the next, the Q’s are only functions of , , , 1, 2, and 3 (the six unknowns).

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 the angular accelerations

Equations (cont.)

E3 = I11 I12 Q1

I21 I22 Q2

I31 I32 Q3

G = I11 I12 I13

I21 I22 I23

I31 I32 I33

Six Nonlinear Ordinary Differential Equations to be Solved

Note:

= d = d = d

d t d t d t

Integrate: = 3 sin + cos = (-3 cos+ sin) tan +

= (3 cos- sin) sec

G

E2 / G

3=E3 / G

Output

Plot:

, , (Ellipsoid orientation)

1, 2, 3 (Ellipsoid spin rates)

As functions of time.

We can also plot where

= cos-1 3

Output (cont.)

All equations refer to Kane’s paper.

Kane, T.R. Realistic Mathematical Modeling of the Rattleback. International Journal of Nonlinear Mechanics. 1982. Vol. 17, No. 3, pp.175

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 object’s unique properties

• Extensive knowledge in physics and calculus

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 angle.

• Discovered rattleback performs multiple reversals.

Results:

•Roll angle, , 20 second time period.

•Rapidly rocking along the short axis

•Angular velocity of roll (rad/sec)

•Shows velocity of rocking.

Results:•Pitch angle, 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.

Results:•Yaw angle, , shows spin reversal.

•Begins spin in the negative direction, but reverses.

•Angular velocity of yaw.

•Velocity increases up to reversal point, then decays.

Results:

•Delta (), angle between the vertical axes of the ellipsoid and the surface.

•Rocks back and forth corresponding to other angles, velocities

Results:

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

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

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.

Results:

•This run shows the effect of adding air resistance, sigma ().

•The roll, pitch, and yaw dissipate to zero.

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.

Future Application:

• Satellites:- Development testing

- Movement- Guidance system

programming

Knowledge Gained:

• Better acquainted with C programming language

• Basics of MATLAB

• Advanced physics and calculus

• Advantages of teamwork

• Personal determination and satisfaction in completing difficult projects