Sep 14, 2026 · eess.ASJ/K move · Enter open · S save
Homayoon Beigi, Grace Conneely
Department of Mechanical Engineering, Columbia University, New York, NY, USA · Department of Electrical Engineering, Columbia University, New York, NY, USA · Recognition Technologies, Inc., South Salem, NY, USA
A finite-difference bowed-string model with implicitly resolved Stribeck friction is presented, with a regime diagnostic, the Schelleng bow-force limits on four strings, and a comparison of learned bow controllers. Implicit resolution is necessary, and quantitatively so: a lagged contact force cannot capture the string on a discrete grid, so no stick phase forms at any bow force. With friction, impedance and quality factor taken from published measurement rather than fitted, all four strings return a stick fraction of 89.1% against an ideal 90%. Schelleng's maximum bow force is recovered on every string. The minimum is not: it follows
Zvbβ−1 rather than the predicted
Z2vbβ−2, reducing both squared dependences to first powers. Six controllers at matched capacity, over four strings and twenty seeds each, place a gated recurrent network ahead of a feedforward one, by most under a mid-stroke disturbance. The feedforward network completes more strokes only from a start the model's own playability map places outside the Helmholtz region. A minimal gated variant fails because gates computed from the input alone cannot clear a latched state. Training loss selects neither the capacity nor the context length, and no learned controller improves on the lookup rule that generated its labels. That bound has a domain. Regressing the controller's score on the rule's gives a slope of 0.32, more than ten standard errors below unity, so the controller overtakes the rule where the rule fails and is bounded by it where it holds. Under a rigid finger stop the plant is provably invariant, so transfer loss between pitches belongs to the controller alone and is traced to one feature. A regime classifier without a stick test labels small-amplitude periodic slipping as Helmholtz motion, and a harmonicity measure rates a string the bow never grips above Helmholtz motion.