Two ideal cones on intersecting shafts can roll together along a common generator when their theoretical apexes coincide. Their speed ratio follows the sine of their half-angles. The calculator visualizes that aligned geometry and reports nominal output speed. It also retains an apex-offset comparison, but does not turn that offset into a predicted slip or wear value.
Cone-on-cone Rolling Pair Interactive Calculator
Explore two ideal rolling cones with a common apex and intersecting shafts. Change their half-angles and input speed to see the nominal speed ratio. Cone distance and apex offset are retained for the separate e/L comparison; a nonzero offset does not make this an actual-slip prediction.
Equation Used
- External cone surfaces meet along a shared generator.
- The two theoretical apexes coincide in the ideal speed model.
- The shaft angle is the sum of the cone half-angles.
- Input and output surface speeds agree without slip.
- Apex offset is reported separately; no misaligned-contact dynamics are solved.
The model shows aligned external rolling cones with fixed intersecting axes. L is the distance along a cone generator from the apex to the large end. The physical cones are truncated at 0.3L for clarity. Rotation is 100 times slower than the displayed RPM; torque, preload and traction are not calculated.
A shared apex and a common contact line
The blue cone A is the input, and the gold cone B is the output. Their axes intersect at O. The orange contact line is a common generator of the two surfaces. The physical cone tips are cut away in the illustration; dashed extensions show the theoretical apex.
A point at distance s from O along the contact line is s sin α from the A axis and s sin β from the B axis. Pure rolling requires the two surface velocities at that point to agree. Since the same s appears on both sides, the speed ratio is constant along the line.
The surface stripes rotate with each cone at the calculated ratio, using a common time scale slowed by a factor of 100. The shafts do not orbit, and the contact line stays fixed in space for these fixed shaft axes.
The input e describes a separate apex-offset magnitude. Because no offset direction, contact compliance or applied load is specified, the calculator does not draw an invented misaligned contact or claim its actual output speed. It shows e/L separately from the ideal aligned mechanism.
Connection to pitch-cone geometry
The same ideal rolling-cone construction underlies the pitch surfaces of bevel gears on intersecting shafts. NPTEL and KHK illustrate coincident pitch-cone apexes and a common contact generator. Gear teeth are not shown here: this page concerns the smooth ideal rolling surfaces.
The drawing can be used to understand how cone angles establish an ideal ratio. It does not identify a particular commercial traction drive or establish a continuously variable transmission. Fixed cone angles and fixed intersecting axes give a fixed ideal ratio.
Rolling speed ratio
At distance s along the contact generator, r_A = s sin α and r_B = s sin β. Equal rolling speed gives N_A r_A = N_B r_B, so N_B/N_A = sin α/sin β. The displayed RPM values are speed magnitudes; the shaft rotations are oriented to give matching velocity at contact.
For the external arrangement shown, shaft angle Σ = α + β. The cone distance L is measured along the generator to the large end, rather than along a shaft axis. Changing the overall size does not change the ideal angular-speed ratio.
The separate normalized offset is 100e/L percent. It describes a length comparison only. It is not a formula for slip, heat, efficiency, remaining life or an acceptable tolerance.
Worked aligned example
At α = 30°, β = 45° and 600 rpm input, the ideal speed ratio is sin(30°)/sin(45°), approximately 0.7071. Output is about 424.3 rpm and the shaft angle is 75°.
For L = 100 mm, the outer contact radii are 50 mm and about 70.71 mm. Both surfaces travel at the same tangential speed under the ideal rolling relation.
An entered offset of 0.5 mm gives e/L = 0.5%. This does not mean 0.5% slip, nor does it validate the nominal speed for that misaligned assembly. The aligned speed prediction and the geometric offset are separate results.
Keep kinematics separate from traction design
Ideal contact geometry establishes relative speed, but does not establish how much torque a real friction pair can transmit. Applied normal load, traction behavior, material properties, surface condition and deformation would be needed for that analysis.
The earlier article assigned unsupported preload ratios, surface-finish tolerances, noise levels, service lives and machine-specific applications. Those claims are not used by the calculator and have been removed from this explanation.
For a nonzero apex offset, the nominal aligned relation remains a reference calculation. This model does not prescribe how the surfaces are repositioned to restore contact, or calculate the resulting slip distribution. The visible offset bar simply compares e with L.
Rolling-cone questions
Why do the cone apexes coincide?
That gives the two external cone surfaces a shared generator and makes their speed ratio consistent along the ideal contact line.
Does cone length change the speed ratio?
No. With fixed half-angles, scaling the geometry scales both contact radii equally. L still affects the normalized offset e/L.
Is the reported offset percentage a slip percentage?
No. It is only a ratio of lengths. Actual misaligned contact and slip are not solved.
Why are the cone tips absent?
The physical surfaces are truncated for a clearer shaft view. Dashed lines extend to the shared theoretical apex.
Are these bevel gears?
No teeth are shown. These are the ideal rolling cone surfaces related to bevel-gear pitch geometry.
Does the animation run at full RPM?
No. Both cone speeds are slowed by the same factor of 100, preserving their ratio.
Geometry references
K. Gopinath and M. M. Mayuram, NPTEL / IIT Madras, Machine Design II, Lecture 13, Figure 13.2 illustrates pitch cones with intersecting axes, a common apex and generator contact.
KHK, Bevel Gears, Figure 8.1 states the equal-contact-speed relationship and the sum of the pitch-cone angles. These references establish the ideal geometry; the smooth-cone illustration does not model gear teeth or gear strength.
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