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Rattlebacks for the rest of us Simon Jonesa) Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803 Hugh E. M. Hunt Engineering Dynamics and Vibration, University of Cambridge, Cambridge CB21PZ, United Kingdom (Received 21 November 2018; accepted 18 June 2019) Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin (i.e., a spin-bias). If spun in one direction, the rattleback will exhibit seemingly stable rotary motion. If spun in the other direction, the rattleback will being to wobble and subsequently reverse its spin direction. This behavior is often counter-intuitive for physics and engineering students when they first encounter a rattleback, because it appears to oppose the laws of conservation of momenta, thus this simple toy can be a motivator for further study. This paper develops an accurate dynamic model of a rattleback, in a manner accessible to undergraduate physics and engineering students, using concepts from introductory dynamics, calculus, and numerical methods classes. Starting with a simpler, 2D planar rocking semi-ellipse example, we discuss all necessary steps in detail, including computing the mass moment of inertia tensor, choice of reference frame, conservation of momenta equations, application of kinematic constraints, and accounting for slip via a Coulomb friction model. Basic numerical techniques like numerical derivatives and time-stepping algorithms are employed to predict the temporal response of the system. We also present a simple and intuitive explanation for the mechanism causing the spin-bias of the rattleback. It requires no equations and only a basic understanding of particle dynamics, and thus can be used to explain the intriguing rattleback behavior to students at any level of expertise. VC 2019 American Association of Physics Teachers. https://doi.org/10.1119/1.5115498 I. INTRODUCTION rattleback being particularly confounding. As such, a model of a planar, rocking semi-ellipse is first introduced in Sec. II Rattlebacks, also known as celts or wobblestones, are semi- to acquaint the reader with the necessary concepts used in a ellipsoidal tops that have a preferred direction of spin (i.e., a more complex dynamics analysis. Undergraduates can intui- spin-bias). If spun in, say, the counter-clockwise direction, the tively understand the motion of this body and the equations rattleback will exhibit seemingly stable rotary motion. If spun are less intimidating since they are only two-dimensional. in the clockwise direction, the rattleback’s rotary motion will The dynamics are governed by conservation of linear and transition to a rattling motion and, subsequently, it will reverse angular momenta coupled with basic kinematic constraints, its spin direction resulting in counter-clockwise rotation, as concepts taught in most post-secondary introductory physics depicted in Fig. 1. This counter-intuitive dynamic behavior has classes. Rather than limit the investigation to a roll without long been a favored subject of study in graduate-level dynam- slip condition, which is used in the bulk of the previous liter- ics classes. ature on rattlebacks, the derivation accounts for a conditional Previous literature on rattleback dynamics offer insight stick/slip constraint using a Coulomb friction model. into a myriad of advanced topics, including chaotic motion, The rattleback is introduced in Sec. III, where it is shown energy decay, stability regions, and nondimensionalization the dynamic analysis is virtually identical to that of the rock- 1–11 to name but a few. However, it is the current authors’ ing semi-ellipse. The only significant difference involves the view that focusing on these advanced topics may obscure the use of a mass moment of inertia tensor rather than the scalar understanding of the fundamental kinetics for students new value used in the planar example. Finally, a physical explana- to three-dimensional rigid body dynamics. The goal of this tion for the spin-biased behavior of the rattleback is presented paper is to demonstrate that accurately simulating rattleback in Sec. IV. The explanation is intuitive and can be used to behavior need not be complicated; undergraduate physics accurately explain the spin-bias behavior using a simple dem- and engineering students have all the necessary tools to onstration without the need for complex equations. accurately model the behavior using concepts from introduc- tory dynamics, calculus, and numerical methods courses. II. A ROCKING SEMI-ELLIPSE Furthermore, it appears that no previous literature has pro- vided a simple means of explaining the spin-biased behavior Leaping directly into the seemingly complex dynamics of of the rattleback; the explanations are generally closely linked the rattleback can be intimidating. Many of us do not have to minutia of the complex governing equations. While these reliable intuition when it comes to three-dimensional dynam- complex explanations are accurate, they are too involved to ics, which makes it difficult to assess when simulation results be useful when explaining rattleback behavior to the inter- are correct. As such, a simplified planar analogy to the rattle- ested layman. As such, this article will introduce an intuitive back, the rocking semi-ellipse shown schematically in Fig. 2, explanation for the spin-biased behavior that can be under- will be analyzed first to assist in deriving the governing stood by laymen with only a basic understanding of physics. equations of motion for the rattleback. It is intuitive that if Three-dimensional rigid body dynamics can be a confus- one releases the semi-ellipse from some perturbed state, and ing, counter-intuitive area of study, with the behavior of the one assumes roll without slip and no energy loss terms, the 699 Am. J. Phys. 87 (9), September 2019 http://aapt.org/ajp VC 2019 American Association of Physics Teachers 699 Fig. 1. Schematic showing the spin-bias of a rattleback. semi-ellipse will rock back and forth indefinitely with con- elliptical perimeter of the body such that it is always in stant amplitude. This example will act as a validation case contact with the ground. This style of problem is often and provide confidence in the approach. It should be noted, taught at the end of the introductory dynamics class for though, that the roll without slip condition is too rigid a con- undergraduate engineers as “relative-motion analysis using straint in the current investigation; the effect of sliding on rotating axes.”12 the response of the system is also of interest. Thus, slippage Define the system of Cartesian axes X, Y, Z associated will be allowed at the contact point via a Coulomb friction with the inertial reference frame R and Cartesian axes x, y, z model. While this planar behavior does not seem as exciting associated with the body-fixed reference frame B. In the as that of a spin-biased rattleback, it will be shown that the body-fixed frame, the parametric definition of the semi- vector form of the governing equations is virtually identical elliptical surface is to that of the rattleback, thus this simplified model acts as a useful and intuitive introduction to rattleback dynamics. x2 y2 þ ¼ 1 À a x a À b y 0; (1) The current authors have found that students, undergradu- a2 b2 ate and graduate alike, often struggle to fully comprehend the derivations presented in previous literature on rattle- where a is the major axis radius and b is the minor axis backs. There can be confusion regarding which reference radius. frame certain vectors are defined in, how to approach deriva- tives in rotating reference frames, and how to parametrically A. Notation and important kinematic concepts define the geometry of the system. As such, the derivation below is presented in a pedantic manner in order to clarify The section below is included for clarity of notation and to these topics. review important concepts regarding the use of rotating ref- AsshowninFig.2, an inertial reference frame R is erence frames. These concepts can be reviewed in under- defined with origin O. Note, the exact location of O is not graduate level dynamics texts12,13 if further information is important and can be left arbitrary. A body-fixed reference required. frame B is defined with origin C centered on the upper line (1) Define an arbitrary vector function BuðtÞ as follows: of the semi-ellipse, where b^1 is colinear with the top of the ^ ^ semi-ellipse. In this planar example, the k axis of R and b3 X3 B axis of B are parallel. The rotation of the body-fixed uðtÞ¼ ujb^j ¼ u1b^1 þ u2b^2 þ u3b^3; (2) frame relative to the inertial frame is measured via h, j¼1 defined positive in the counter-clockwise direction, as shown in Fig. 3. The center of mass for the semi-ellipse is whose scalar components ujðtÞ¼uðtÞb^j, j ¼ 1, 2, 3 are found at point G. The contact point between the semi- scalar functions of time. The pre-superscript B signifies ellipse and the ground is P, which is the location of the the vector is defined in the rotating, body-fixed reference resultant force vector f. frame. For clarity, bold lowercase letters are used for For clarity, points C and G are fixed in the rigid body, general vectors while a hat over a lowercase letter signi- while point P moves on the body. Point P can be thought of fies a unit vector. as a massless and frictionless slider moving along the Similarly, an equivalent definition could be written for the vector in the inertial reference frame R R uðtÞ¼uxi^þ uyj^þ uzk^: (3) Fig. 3. Body-fixed reference frame B rotated by angle h relative to inertial Fig. 2. Schematic of the semi-ellipse showing the body-fixed reference frame. reference frame R.

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