Geometric-Phase Transport of Torsion-Induced Lateral Motion in a Bottom-Fixed Slinky

Torsion-induced motion in helical springs is governed by coupled bending, contact, friction,and large-rotation effects, making the post-release response of a bottom-fixed Slinky difficultto infer from imposed twist alone. This study formulates a stacked planar Kirchhoff/Cosseratreconstruction framework for the Dancing Slinky phenomenon. The model keeps the naturalplanar curvature fixed, separates material-frame twist from spatial centerline rotation, andretains the n = 1 translational branch as a zero-stiffness lateral mode transported bya geometric phase. Five experiments were processed from synchronized top-center andmarker tracking data. Relative to a fixed-direction n = 1 baseline, the phase-transportedreconstruction increased the mean top-center trajectory R2 from 0.357 to 0.868, reducedthe mean FFT NRMSE from 10.33% to 4.95%, and produced phase-assimilated FFT R2values between 0.912 and 0.979. These results indicate that the loop-like lateral motion isprimarily a phase-transported translational response rather than a twist-dependent change innatural curvature. The reported validation is a diagnostic observation-model test because thegeometric phase is reconstructed from the post-release marker-angle history; independentrelease-state, contact, friction, and impulse measurements remain necessary for a fullypredictive forward simulation.