Curved cone pulleys vary a belt drive’s ratio by offering different running diameters along their shafts. This calculator evaluates one selected belt station using the entered driver and driven diameters. The animation shows opposed curved profiles, an open belt and the corresponding pulley rotations.
Curved Cone Pulleys Interactive Calculator
Evaluate one selected belt station on opposed curved cone pulleys. Enter its two running diameters, input speed and fractional speed loss; see a connected open-belt drive.
Equation Used
- Parallel shafts and an open thin belt at one axial station.
- Driver contact treated as no-slip; entered speed loss assigned to driven contact.
- Equal effective belt-tension difference for geometric torque ratio.
- Surrounding cone profiles do not define a designed shifting range.
Profiles and center spacing are illustrative. No constant-length shifting design, tensioner or belt-capacity calculation.
Running diameter sets the local ratio
Hiscox figure 90 shows opposed curved cone pulleys connected by a belt that can traverse along them. At any selected station, the local diameters determine the speed ratio in the same way as ordinary belt pulleys.
This illustration holds the belt at one station. The surrounding exponential curves and automatically chosen shaft spacing provide construction context; they are not the profile of a specified machine or a solution for constant belt length. A working shifting system also needs belt guidance and tension control.
Compare a selected pair of diameters
Enter the running diameters at the intended station and the input speed. The two pulleys turn in the same direction because the belt is open rather than crossed. The motion is slowed so the faster pulley is shown at 10 rpm, while the result boxes retain the calculated speeds.
The slip input represents total fractional speed loss. To make that assumption visible, this model assigns the loss to the driven contact and treats driver contact as no-slip.
Speed, surface velocity and torque ratio
With diameters D₁ and D₂ and slip fraction s, n₂=n₁(D₁/D₂)(1−s). The entered percentage is divided by 100 to obtain s. Driver surface speed is π(D₁/1000)n₁/60 in m/s when diameter is entered in millimetres and speed in rpm.
The ideal geometric torque ratio is D₂/D₁. This follows by applying the same effective belt-tension difference at the two radii. It is not a rated torque capacity and does not account for mechanical loss or available belt tension.
Worked selected-station example
At D₁=60 mm, D₂=300 mm and n₁=1200 rpm with zero slip, output speed is 240 rpm and the speed ratio is 0.2. Driver surface speed is about 3.770 m/s. The ideal geometric torque ratio is 5.
At 5 percent speed loss, output speed becomes 228 rpm. The entered geometry still gives a torque ratio of 5 for equal effective tension difference; the speed-loss input does not establish actual transmitted force or efficiency.
A station calculation is not a complete variator design
Independent D₁ and D₂ entries do not specify an axial profile, center distance, belt length, width, permissible bending radius or tensioner. Consequently the animation does not pretend that these two numbers determine an entire constant-tension shifting mechanism.
The thin belt path is constructed tangent to the selected running circles. The curved surfaces outside that station are illustrative. Contact pressure, tracking, wear, power capacity and axial shift force are not calculated.
Curved-cone questions
Why does the belt stay at one station?
The inputs specify the diameters at that station, not a complete pair of manufactured profiles.
Why do the pulleys turn in the same direction?
An open belt connects parallel shafts without crossing.
Is belt speed reduced by the slip slider?
The displayed speed is driver surface speed. In this explanatory model the driver grips the belt and the loss is placed at the driven contact.
Does the torque ratio establish capacity?
No. It is a lever-arm ratio, without available tension or strength data.
Reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page 34, figure 90, for the opposed curved cone and traversing-belt arrangement. The selected-station calculations use belt surface-speed compatibility and torque as force times radius.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
