University:
Massachusetts Institute of Technology
Course:
2.141 | Modeling and Simulation of Dynamic Systems
Academic year:

2020
Views:

283

Pages:

3
Author:

Sporkz

Capstan—a mechanical amplifier Photograph removed due to copyright restrictions. ω τ θ r Fnormal F Δθ v control Fcontrol v out F + ΔF Fout A schematic diagram of a basic capstan and a force diagram for a small segment of the rope are shown in the figures. Fnormal = F sin ∆θ/2 + (F+∆F) sin ∆θ/2 In the limit of small angles Fnormal = F dθ/2 + (F+dF) dθ/2 Assuming continuous slip and Coulomb friction between rope and drum, dF = µFnormal = µ 2F+dF dθ 2 dF = F dθ or d ln F = µ dθ Integrating from 0 to θ Fout = eµθ Fcontrol Note that this relation is only valid if ωr ≥ vcontrol From continuity: vcontrol = vout Torque required of capstan drive: τ = (Fout - Fcontrol) r = (eµθ -1) r Fcontrol Power dissipated: Pdissipated = τ ω + Fcontrol vcontrol - Fout vout Pdissipated = (eµθ -1) r Fcontrol ω - (eµθ -1) Fcontrol vcontrol Note that Pdissipated ≥ 0 with Pdissipated = 0 if ωr = vcontrol A bond graph follows: Fcontrol v out 0 1 Coulomb R friction 0 Drum rim speed TF :r Fout This is a three-port resistor. R Typical boundary conditions result in the following causality. Fcontrol Se v out 1 0 Coulomb R friction 0 TF :r Sf Constitutive equations are ⎡ Fout ⎤ ⎢ τ ⎥ ⎢ ⎥ v ⎣ control⎦ ⎡ =⎢ ⎣ 0 0 eµθ 0 0 (eµθ-1)r 1 0 0 v ⎤⎡⎢ out ⎤⎥ ⎥⎢ ω ⎥ ⎦⎣Fcontrol⎦ Fout Sf