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 Presentation What 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/cm 3 - 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) I xx = (1/5)M(b 2 + c 2 ) b) I yy = (1/5)M(a 2 + c 2 ) c) I zz = (1/5)M(a 2 + b 2 )

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) T x = -  1 b) T y = -  2 c) T z = -  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 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 I xx, I yy, I zz Mass moments of inertia , ,  Initial orientation  1,  2,  3 Initial spin rates  Air resistance coefficient t max Length of simulation

Angular positions definitions of   - First rotation about z (z stays the same, z 1 ) X   and becomes x 1 Y   and becomes y 1  - Second rotation about x 1 (x stays the same, x 11 ) y 1   and becomes y 11 z 1   and becomes z 11  - Third rotation about y 11 (y 11 stays the same, y) x 11   and becomes x z 11   and becomes z

Description of Kinematic Analysis Kinematic - Relates various coordinates (x 1,x 2,x 3 ) positions ( , ,  ) velocities (v 1,v 2,v 3 ) spin rates (  1,  2,  3 ) to each other through geometry and/or constraints - Here the contact point (x 1,x 2,x 3 ) 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 sliding V cg = 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 = cos  sin   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.) x 1 = a 2  1 /  x 2 = b 2   /  x 3 = c 2   /   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.)  = (a 2  1  1 + b 2  2  2 + c 2  3  3 ) /  x 1 = [a 2 (  1 -  1 )] /  2 x 2 = [b 2 (   -   )] /  2 x 3 = [c 2 (   -   )] /  2 (Time rate of change of location (velocity of contact point) in rotating system.)

Kinematic Equations (cont.) v 1 =  2 (h-x 3 ) +  3 x 2 v 2 = -   x 1 -    h-x 3 ) v 3 = -   x 2 +  2 x 1  1 =  2 (v 3 -x 3 ) -   (v 2 -x 2 )  2 =   (v 1 -x 1 ) -   (v 3 -x 3 )  3 =   (v 2 -x 2 ) -   (v 1 -x 1 ) (Relative velocity minus defining values. Three velocities in rotating coordinates. h - distance from ellipsoid to center of gravity (cg).)

F 1 = -  1 + Mg[(x 3 -h)  1 - x 3  3 ] F 2 = -   + Mg[(h-x 3 )  1 + x 1  3 ] F 3 = -   + Mg(x 2  1 - x 1   ] R 1 = [D  1 + (B - C)  2 ]   R 2 = [(C - A)  1 - D     R 3 = D(     1 2 ) + (A - B)  1   (First time forces are introduced.) Kinetic Equations

Kinetic Equations (cont.) A = (I XX ) CG B = (I YY ) CG C = (I ZZ ) CG D = (I XY ) CG Mass Moments of Inertia about the (x, y, z) Axis through the Center of Gravity.

Kinetic Equations (cont.) I 11 = A + M[x (h-x 3 ) 2 ] I 22 = B + M[x (h-x 3 ) 2 ] I 33 = C + M(x 1 2 +x 2 2 ) I 12 = I 21 = D - Mx 1 x 2 I 23 = I 32 = M(h-x 3 )x 2 I 31 = I 13 = M(h-x 3 )x 1 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 I 12, I 23, I these are the culprits.

Kinetic Equations (cont.) S 1 = M[(h-x 3 )  2 + x 3    S 2 = M[(x 3 -h)  1 - x 1    S 3 = M[x 1    - x 2    Q 1 = F 1 + R 1 + S 1 Q 2 = F 2 + R 2 + S 2 Q 3 = F 3 + R 3 + S 3 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.) E 1 = Q 1 I 12 I 13 Q 2 I 22 I 23 Q 3 I 32 I 33 E 2 = I 11 Q 1 I 13 I 21 Q 2 I 23 I 31 Q 3 I 33 Determinants for         the angular accelerations

Equations (cont.) E 3 = I 11 I 12 Q 1 I 21 I 22 Q 2 I 31 I 32 Q 3 G = I 11 I 12 I 13 I 21 I 22 I 23 I 31 I 32 I 33

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    E 2 / G  3 =E 3 / 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 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