Railway Semaphore Wire Run

What is modelled

A mechanical semaphore is held at stop by a counterweight, so a broken wire fails safe. To clear the signal the signaller pulls a steel wire that runs hundreds of metres to a crank on the post. The lever has a limited throw, and part of it is spent taking up wire length changes from temperature and the elastic stretch under tension before the arm moves at all.

Equations

• Counterweight net moment M = mc lc − ma la; wire tension at arm angle θ is T = M g cosθ / rc with crank radius r_c = 0.25 m.
• Thermal length change ΔL = α L ΔT, α = 12e-6 /K for steel.
• Elastic stretch δ = T L / (E A), E = 200 GPa.
• Lever travel for a given arm angle: x = rc θ + ΔL + δ(θ); full clear is θ = 45 degrees.
• Guide pulleys: each 20 m of run costs 2 % of the pull, so η = 0.98L/20.

Checks run in the page

• Elastic work stored, integrated along the tension ramp, against 1/2 T δ.
• Virtual work of the wire over the arm swing against the counterweight potential energy gain.
• The thermal length change integrated from α dT against α L ΔT.
• The lever travel budget against the arm angle the solver reaches.

Honest simplifications

The arm and counterweight are lumped into one net moment, the arm swing is limited to 45 degrees, the pulley loss per 20 m and the crank radius are illustrative, and the wire is assumed straight and uniformly warm. Real installations use wire compensators, adjustment screws and detailed signalling rules that are not modelled.

• Choose Long run, hot day and see the arm no longer reach full clear with the same lever throw.
• Choose Long run, frost: the wire shortens and the arm starts to rise before the lever moves much.
• Raise the counterweight: tension and lever force rise, so the wire stretches more.