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Rope, wheel and load
The rope is driven by friction alone. The tension can rise by a factor e^(mu theta) around the wheel, no more; beyond that the rope slides.
Model checks
The science
What is modelled
A detachable-gondola terminal drives the haul rope by friction around a lined bullwheel. A hydraulic tensioner holds the slack side at a set tension and the motor torque adds the difference to the tight side, which carries the loaded uphill cabins. The page combines the Euler-Eytelwein friction relation, the force balance of the line and the motor and brake mechanics. Rope speed, tensioner and brake ranges follow the 3D page's sliders.
Equations
- Capstan
T1 / T2 <= e^(mu theta), so the largest drive force isT2 (e^(mu theta) - 1). - Drive force
F = (m_up - m_down) g sin(alpha) + c_r g cos(alpha) m_total. - Shaft power
P = F v / eta; wheel speedn = v / (pi D). - Motor torque
F (D/2) / (i eta). - Braking
a = (F_brake + F) / m_total,s = v2 / 2a; the brake force is limited to the friction capacity.
Checks run in the page
- Capstan law by integrating the tension around the wrap.
- The slip threshold by bisection.
- Emergency stop by time stepping.
- Gear and wheel kinematics and a marker moved round the rope loop.
Honest simplifications
A quasi-static model: constant rope tension on the slack side, no rope elasticity or creep, no tensioner dynamics, a single friction coefficient (no wet or worn lining model), a lumped rolling resistance of 0.02 and efficiency of 0.9, a 800 m line with equal ropes, rope and cabin masses as typical values, no wind and no cabin swing. Real ropeways have certified safety factors and redundant brakes.
What to try
- Press Slippery lining (wet) and watch the safety factor fall below 1 as the rope slides.
- Press Low tension: the slack-side tension sets how much force the wheel can pass.
- Raise the brake force beyond the slip limit and see the stopping distance stop improving.