Spin-It: Optimizing Moment of Inertia For Spinnable Objects

Spinning tops and yo-yos have long fascinated cultures around the world with their unexpected, graceful motions that seemingly elude gravity. Yet, due to the exceeding difficulty of creating stably spinning objects of asymmetric shape in a manual trial-and-error process, there has been little departure from rotationally symmetric designs. With modern 3D printing technologies, however, we can manufacture shapes of almost unbounded complexity at the press of a button, shifting this design complexity toward computation.

In this article, we describe an algorithm to generate designs for spinning objects by optimizing their mass distribution: as input, the user provides a solid 3D model and a desired axis of rotation. Our approach then modifies the interior mass distribution such that the principal directions of the moment of inertia align with the target rotation frame. To create voids inside the model, we represent its volume with an adaptive multiresolution voxelization and optimize the discrete voxel fill values using a continuous, nonlinear formulation. We further optimize for rotational stability by maximizing the dominant principal moment. Our method is well-suited for a variety of 3D printed models, ranging from characters to abstract shapes. We demonstrate tops and yo-yos that spin surprisingly stably despite their asymmetric appearance.

Back to Top

1. Introduction

Spinning toys have existed since antiquity as playful objects that capture the imagination. Invented independently all over the world, spinning tops are referenced in ancient Greek literature,¹² and evidence of clay tops has been found in ancient cities dating as early as 3500 B.C. Similarly, while yo-yos are believed to have been invented in China, there are many historical references, including in Mozart's The Marriage of Figaro where a yo-yo is spun to relieve stress.¹⁷ Despite the long tradition of these toys, until today creating new designs has been a trial-and-error process, calling on the intuition and meticulous patience of artists and hobbyists. Moreover, there has been little departure from rotationally symmetric designs.

Much attention has been devoted in the field of classical mechanics to explaining the motion of spinning objects; however, the focus has been primarily on analysis^{8, 9, 19, 21} rather than design. In this article, we investigate the unique geometric properties of shapes that spin, with an eye on digital modeling and free-form design. A stable spin has requirements on rotational inertia, including precise positioning of the center of mass and correct alignment of the primary axes of the body. We propose an algorithm to optimize for these inertial properties, for example, to design a spinning top that rotates smoothly and stably and can be fabricated using 3D printing.