Opposed stepped pulleys change the speed ratio as a belt moves between paired steps. Calculate the ideal open-belt length for four pairs and see why equal diameter sums do not make every belt path exactly the same length.
Stepped-Pulley Belt-Length Calculator
Enter shaft center distance, a paired diameter sum and four driver pitch diameters. Each driven diameter is S-D. The worksheet reports four exact zero-thickness open-belt path lengths and their spread; it does not select or tension a real belt.
Equation Used
- Parallel shafts and coplanar circular pitch lines form an open belt drive.
- One fixed center distance applies to all four axial steps.
- Each driven diameter equals the entered constant sum minus its paired driver diameter.
- The belt is inextensible and has zero thickness; groove datum, tension, slip, power and alignment limits are excluded.
The four pairs share one shaft center distance. Exact tangent-and-arc calculations show the remaining belt-length difference between steps with equal diameter sums.
Opposed steps share one movable belt
As the belt is shifted axially, a larger driver step pairs with a smaller driven step and vice versa. Constant paired diameter sum holds the dominant circumference term nearly constant.
The diagram identifies the four pulley pairs; the numerical lengths come from ideal tangent-and-arc geometry. The belt lines in the drawing are schematic.
Check the constant-sum approximation
Use the exact path spread to see why equal diameter sums do not guarantee perfectly identical belt length. Final selection must use the belt manufacturer’s datum and installation method.
Exact open-belt pitch path
Four-step example
At 400 mm center distance with a 175 mm paired diameter sum, the 125/50, 100/75, 75/100 and 50/125 mm pairs have equal middle circumference terms but different exact path corrections. The calculator reports the difference between the longest and shortest paths.
Exact geometry is not belt selection
Equal paired diameter sums keep the main circumference term constant, but exact circular geometry also includes a diameter-difference correction. Real belts also require catalog pitch datum, standard lengths, wrap, tension and power checks.
Stepped-pulley questions
Does constant diameter sum make all exact paths equal?
No.
Are these outside diameters?
No; they are declared pitch diameters.
Does the smallest spread select a belt?
No.
Belt-drive references
- IIT Kharagpur / NPTEL, Belt Drives: open-belt nomenclature and belt-length geometry.
- Gates Heavy Duty V-Belt Drive Design Manual: center distance, pitch diameters and catalog belt-length selection procedure.
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