The string shooter as an axially moving geometrically exact Kirchhoff rod: A unified framework for dynamics and spectral stability

Abstract

This study presents a mathematically rigorous derivation and spectral analysis of the string shooter system, a closed-loop continuum driven at high axial speeds. While often regarded merely as a scientific toy, the system exhibits rich nonlinear dynamics located at the intersection of drag-dominated string behavior and stiffness-dominated rod mechanics. To capture this duality, we establish a unified continuum framework modeling the loop as an axially moving, geometrically nonlinear Kirchhoff rod. The derivation proceeds from first principles using an Arbitrary Lagrangian–Eulerian (ALE) kinematic description, strictly separating kinematic constraints from balance laws to resolve the interplay between gyroscopic transport terms, aerodynamic drag, and flexural rigidity. A distinct novelty of this work lies in the rigorous treatment of the system’s topology: we formulate a dimensionless strong-form finite difference scheme that enforces C1-geometric continuity for the loop closure while simultaneously permitting physical discontinuities in shear and tension forces at the drive singularity. This approach allows for a direct solution of the generalized eigenvalue problem, revealing the spectrum of the underlying steady-state solution. We validate the nonlinear boundary value problem for the steady-state and its modal analysis against simple (analytical) benchmarks and present examples that characterize the transition from stable stationary equilibria to flutter instabilities (self-excited vibrations) driven by non-conservative drag forces.