A friction roller driven by a rotating conical drum changes speed when it moves to a different drum diameter. This calculator compares two contact settings, A and B, at the same drum speed. It reports relative speeds rather than absolute RPM, and checks whether each contact diameter lies between the drum end diameters.
Conical Drum Variable Rotary Roller Interactive Calculator
Compare one friction roller at two positions on a conical drum. A and B are separate settings of the same drive. Set the drum end diameters and the contact diameter for each setting to compare their relative speeds. Contact diameters outside the drum are explicitly flagged.
Equation Used
- No slip for valid contact positions.
- Same fixed roller diameter for A and B.
- Same drum speed for both settings.
- Ratios remain arithmetic only if a requested diameter is outside the drum.
- The animation uses an illustrative common drum speed, not a specified operating RPM.
The illustration uses a straight cone approximation and an illustrative 200 mm axial span. Roller diameter equals the small-end diameter in the animation only, allowing its revolution count to represent the normalized ratio. A requested contact diameter must lie between the entered end diameters.
One roller at two contact positions
The drum turns about its horizontal shaft. A roller touches its tapered surface and turns about a shaft aligned to keep its rim tangent to that surface. Moving the roller lengthwise changes the drum diameter at contact.
The two drawings are comparisons of separate settings, not two simultaneous rollers sharing one output shaft. Both drums turn at the same illustrative speed. When contact is valid, each roller turns at the corresponding ideal speed.
The straight drum profile is a geometric simplification of the variable-friction principle illustrated by Brown, figure 265. The animation uses a roller diameter equal to the small-end drum diameter as a reference. This choice makes the displayed revolutions match the normalized speed ratio; it is not a calculated hardware dimension.
A variable-friction principle
Brown describes variable roller rotation obtained by traversing a friction roller along a rotating conical drum. This page illustrates that principle and compares contact diameters. Traction and load capacity require the contact materials, preload and drive construction of an actual transmission.
Real machines require guides, preload, bearing support and a defined range of travel. The depicted supports and shafts establish the motion arrangement but are not a detailed assembly design.
Relative speed from contact diameter
For a fixed roller diameter D_r and drum speed N_d, equal tangential speed gives N_r = N_d D_contact/D_r. Comparing this with operation at the small-end diameter D_s eliminates both N_d and D_r: N_r/N_small = D_contact/D_s.
Thus setting A gives D_A/D_s, setting B gives D_B/D_s, and B relative to A gives D_B/D_A. The maximum geometric span between drum ends is D_l/D_s.
A ratio can be computed arithmetically even if the requested diameter is outside the drum. Such a setting is not physically available on the entered drum. It is flagged in the diagram, and its gray roller stops to avoid implying valid contact.
Worked comparison
For a 60 mm small end and 180 mm large end, the ideal full speed span is 180/60 = 3. A contact diameter of 80 mm gives 80/60 = 1.333 times the small-end reference speed.
At 120 mm contact diameter, the relative speed is 120/60 = 2. Setting B is 120/80 = 1.5 times as fast as setting A. Both contacts lie within the 60–180 mm drum range.
If the small end is changed to 100 mm while A remains 80 mm, A is outside the drum. Its 0.8 arithmetic ratio remains visible, but the warning makes clear that the specified contact is unavailable.
What the ratios leave out
The calculation assumes rolling contact without slip. It does not determine normal force, transferable torque, contact stress, temperature, efficiency or wear. A diameter ratio alone cannot establish those quantities.
The end diameters determine whether the selected contact positions exist. The drawing keeps invalid selections visible as an extrapolation with a gray roller and a no-contact label. The indicated position remains at the diameter you entered.
Axial length is not an input, so the displayed 200 mm span is illustrative. Changes in the input diameters set the shown taper. The wheel plane follows the surface normal so its ideal rim contact remains tangent at each valid setting.
Conical-drum questions
Are there two rollers on the machine?
The drawings compare two settings of one drive. They are not a simultaneous two-roller assembly.
Do the results give output RPM?
No. They are relative speed ratios. Actual RPM also requires drum speed and roller diameter.
Why is a roller gray?
Its requested contact diameter lies outside the entered small-to-large drum range. The diagram shows where the extrapolation would be but does not imply physical contact.
Why does the roller speed up toward the large end?
At constant drum RPM, a larger contact diameter has greater surface speed. A fixed-size roller must turn faster to match that speed without slip.
Is the roller diameter a design recommendation?
No. The illustration uses the small-end diameter as its roller reference solely to visualize the normalized ratios.
Mechanism reference
Henry T. Brown, Five Hundred and Seven Mechanical Movements, printed pages 66–67, figure 265 describes a friction roller traversing a rotating conical drum to obtain variable roller rotation. The scanned drawing uses a curved profile; this calculator explicitly uses a straight-profile approximation consistent with its diameter inputs.
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