Fixed-Chord Velocity Recursion and Maximum Speed Bounds for Articulated Chains on Archimedean and Piecewise-Smooth Paths

Abstract

A bench dragon is a long articulated chain governed by prescribed-track geometry and fixed hinge spacing. This paper develops a verifiable framework for two linked tasks: inward-winding kinematics and maximum head speed on an optimized S-shaped turn. The chain contains 223 rigid boards and 224 hinged handles. For the inward motion, the head follows a 0.55 m-pitch Archimedean spiral at a constant speed of 1 m s^-1. An arc-length ordinary differential equation is integrated by fourth-order Runge–Kutta with a 0.05 s internal step; the remaining handles are placed by fixed-chord bisection and their velocities are recovered by differentiating the constraints. The 300 s trajectory satisfies the chord constraints with a residual of approximately 1.6 × 10^-16 m. On the optimized turn, the same recursion is parameterized by path arc length. A unit-head-speed scan gives a maximum amplification factor of M_max ≈ 13.11 at s* ≈ 2.85 m, yielding an analytical head-speed bound of v_H≤ 0.153 m·s⁻¹ when all handles are constrained to 2 m  s⁻¹. A 40-particle, 120-generation particle swarm search converges to the same value. The framework separates geometry from speed control — the same fixed-chord recursion applies to both problems without modification — and the analytical speed bound provides an independent check on the particle swarm result. The approach is reusable: changing the spiral pitch, chord lengths, or path geometry requires only parameter updates, not a new derivation.