Flat Belt Drive

What is modelled

A belt transmits torque only because the tension on its tight side is larger than on its slack side. Friction on the wrapped arc lets the tension grow exponentially along the pulley, which sets the largest pull before the belt slides. The faster the belt, the more the centrifugal tension takes away from the useful tension.

Equations

• Capstan law (T1 − m v2) / (T2 − m v2) = eμθ with T1 + T2 = 2T0.
• Wrap on the small pulley θ = π − 2 arcsin((R2−R1)/C); belt length 2C cos α + R1(π−2α) + R2(π+2α).
• Power P = (T1 − T2) v; at the tension limit the power peaks at v = √(Tmax / 3m).
• Friction law with slip F = Fmax tanh(s / 0.04) and driven pulley dynamics J ω′ = F r − τload − bω.

Checks run in the page

• The belt element equation integrated around the arc.
• Tangent points, straight length and belt length from explicit coordinates.
• The best belt speed by golden-section search.
• The slip after a stepped run against the force balance.

Honest simplifications

The belt is inextensible except for a smooth creep law, the tension sum is held at 2T₀, the friction coefficient is constant, the belt is thin, and the inertia and damping of the driven shaft are rough numbers. A real leather belt also creeps, stretches and changes its friction with age.

• Press Overloaded machine and watch the driven pulley stall.
• Press Slack belt to see how little pull a loose belt carries.
• Press Very fast belt and see the centrifugal tension steal the useful tension.