A cone-and-disk friction drive changes its speed ratio by moving a friction wheel to a different contact radius on a cone. In this calculator the cone is the input. Under ideal rolling contact, increasing the cone contact radius increases the disk output speed. The illustration shows a straight cone and a disk rim kept tangent to its surface.
Cone-and-disk Variable Speed Interactive Calculator
Move a friction disk along a straight cone and compare their ideal speeds. The cone is the input and the disk is the output. The disk rim remains tangent to the cone, and both animations use the same slowed time scale.
Equation Used
- Cone input and disk output follow the labels consistently.
- No slip at the ideal contact point.
- The contact radius varies linearly with axial position on the straight cone.
- Disk shaft is parallel to the cone generator; its rim is tangent at the orange point.
- Ideal torque ratio assumes no power loss and is not a torque-capacity rating.
- Animation is slowed 100 times; both angular speeds preserve the calculated ratio.
The straight cone and tangent wheel are an ideal contact model. The 200 mm illustrated axial span covers the position slider. Bearings, wheel thickness and support fittings are schematic. No preload, contact stress, available traction, wear or loaded slip is solved.
Equal surface speed at one contact
The cone turns about its horizontal shaft. The disk turns about an inclined shaft parallel to the straight cone generator. Its rim touches the cone at the orange point. Sliding the disk changes the axial contact position and therefore the cone radius at that point.
The contact velocities must agree if the surfaces roll without slip. The larger the cone contact radius, the farther its surface travels per input revolution. A disk of fixed radius must therefore rotate faster at the larger-radius position.
The three-dimensional view shows both shafts and the common contact. The cone and disk rotation markers use the same slowed time scale. The disk does not float above the cone, and its axis is not incorrectly shown perpendicular to the cone axis.
This is an ideal straight-cone illustration. Historical variable-friction drives also used curved profiles and more elaborate contact arrangements. Practical drives need support, preload and traction design beyond the geometry displayed here.
Variable friction transmission
Traversing a friction wheel over a changing-radius surface provides a continuous geometric speed ratio. Hiscox illustrates a related variable rotary drive with a traversing pulley on a concave conical drum. The straight profile used here is a simpler model that matches the calculator inputs.
This page does not assign the generic arrangement to particular textile machines, conveyors or vehicle models without machine-specific evidence. The illustration demonstrates a transmission principle, not a rated commercial product.
Correct ratio for a cone input
Let r_s be the small-end radius, x the axial position and α the cone half-angle. The contact radius is r_c = r_s + x tan α. Let the disk radius be R_d = D_d/2.
No slip requires equal tangential speed: 2Ï€N_in r_c = 2Ï€N_out R_d. Thus N_out/N_in = r_c/R_d. With the cone as input, moving toward the larger end increases output speed. The earlier code used the reciprocal relation; that has been corrected.
For ideal lossless power transfer, T_out/T_in = N_in/N_out = R_d/r_c. This is a torque ratio, not an estimate of transferable torque. Contact load, materials, traction and efficiency would be required for that estimate.
Worked speed-ratio example
Use a 20 mm small-end radius, 10° cone half-angle and 120 mm axial contact position. The cone contact radius is 20 + 120 tan(10°), approximately 41.16 mm. A 160 mm diameter disk has 80 mm radius.
At 1200 rpm cone input, disk output is 1200 ×41.16/80, approximately 617.39 rpm. The speed ratio is 0.5145, and the ideal output/input torque ratio is about 1.9437.
At the small end, x =0, output is 1200 ×20/80 =300 rpm. Moving to x =200 mm raises the contact radius to about 55.27 mm and output to about 828.98 rpm. These results are geometric predictions without slip or losses.
Geometry is only part of a friction drive
The model resolves the contact position, shaft alignment and ideal speed relation. It does not resolve elastic deformation, creepage, finite contact-patch effects, normal load, bearing load, temperature or wear.
The wheel plane contains the cone surface normal and the circumferential direction at contact. Its axis is parallel to the cone generator. This keeps the ideal wheel rim tangent as the wheel traverses the straight profile.
Extreme slider positions remain visible and mathematically connected, including the ends of the illustrated cone. A physical machine would require edge clearance and a defined travel stop. The schematic does not certify those end positions as usable hardware settings.
Cone-and-disk drive questions
Which part is the input?
The cone. The disk is the output throughout the calculator, equations and animation.
Does the disk speed up at the small end?
No, not with the cone as input and disk radius fixed. The disk speeds up as the cone contact radius increases.
Why is the disk shaft inclined?
In this ideal straight-cone model the shaft is parallel to a cone generator. That places the wheel rim tangent to the cone surface.
Does the torque ratio give the maximum output torque?
No. It is a lossless power ratio. Available traction and actual torque capacity are not calculated.
Are the displayed shaft speeds real-time?
The numeric RPM values are the calculated speeds. The animation is slowed 100 times to make the motion readable while preserving their ratio.
Historical mechanism reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), page37, figure103 shows a traversing friction pulley on a concave conical drum. It establishes the related variable-radius friction-drive principle. This calculator explicitly uses a straight cone, not an exact reconstruction of that curved historical profile.
The speed equation follows directly by equating the two tangential velocities at the contact. The torque ratio follows from ideal power conservation.
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