Exactly returning boomerangs

Abstract

Differential equations of motion of a boomerang can be integrated numerically given its aerodynamic and inertial properties and initial conditions. We use the dynamic model and experimental aerodynamic data of a typical boomerang in still air studied by Hess (Boomerangs, aerodynamics and motion. PhD thesis, University of Groningen, 1975). The trajectory size and shape are well-defined functions of five initial conditions. Beginning with a nominal guessed set, an iterative search finds release conditions that result in exact return. The distance to the point on the trajectory closest to the desired return point and its gradient with respect to the release conditions are calculated. Release conditions are then modified iteratively using Newton’s method to decrease the miss distance. Exact return conditions are presented for constant values of initial angle of attack and advance ratio. A variant of the algorithm calculates release conditions that ensure return at “turnaround” where the speed is lowest and thus catching is easiest. Although in general the set of exact return release conditions is five dimensional, it is thin in the sense that certain variables must lie in a fairly narrow range. Some initial conditions are more easily modified than others, accounting for the not-inconsiderable skill required to achieve exact return in practice. A discussion is also included of the stable asymptotic helical attractor approached by the boomerang after turnaround.