Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin. If spun in, say, the clockwise direction, the rattleback will exhibit stable rotary motion. If spun in the counter-clockwise direction, the rattleback’s rotary motion will transition to a rattling motion, and then reverse its spin resulting in clockwise rotation. This counter-intuitive dynamic behavior has long been a favored subject of study in graduate-level dynamics classes.
Previous literature on rattleback dynamics offer insight into a myriad of advanced topics, including three-dimensional motion, sliding and rolling friction models, stability regions, nondimensionalization, etc. However, it is the current authors’ view that focusing on these advanced topics clouds the students’ understanding of the fundamental kinetics of the body. The goal of this paper is to demonstrate that accurately simulating rattleback behavior need not be complicated; undergraduate engineering students can accurately model the behavior using concepts from introductory dynamics and numerical methods.
The current paper develops an accurate dynamic model of a rattleback from first principles. All necessary steps are discussed in detail, including computing the mass moment of inertia, choice of reference frame, conservation of momenta equations, and application of kinematic constraints. Basic numerical techniques like Gaussian quadrature, Newton-Raphson root-finding, and Runge-Kutta time-stepping are employed to solve the necessary integrals, nonlinear algebraic equations, and ordinary differential equations.
Since not all undergraduate engineering students are familiar with 3D dynamics, a simpler 2D rocking semi-ellipse example is first introduced to develop the transformation matrix between an inertial reference frame and a body-fixed reference frame. This provides the framework to transition seamlessly into 3D dynamics using roll, pitch, and yaw angles, concepts that are widely understood by engineering students. In fact, when written in vector notation, the governing equations for the rocking ellipse and the spin-biased rattleback are shown to be the same, enforcing the concept that 3D dynamics need not be intimidating.
The purpose of this paper is to guide a typical undergraduate engineering student through a complex dynamic simulation, and to demonstrate that he or she already has the tools necessary to simulate complex dynamic behavior. Conservation of momenta will account for the dynamics, intimidating integrals and differentials can be tackled numerically, and classic time-stepping approaches make light work of nonlinear differential equations.