Brown’s eccentric-cone movement combines two motions: a cone rotates off-center around a screw axis, while the screw advances through a fixed nut. A pressed roller follows the cone, so its height and rotational speed vary as the local cone radius changes.
The historical figures 262 and 263 are end and side views of the same arrangement. Both views use the same contact geometry and respond to the entered dimensions.
Eccentric Cone Change-velocity Interactive Calculator
Adjust the cone, screw and roller dimensions. Current position values follow the animation. The model follows a narrow roller rim over one screw revolution and reports axial sliding at the contact.
Equation Used
- Cone and screw rotate and translate as one rigid assembly.
- Fixed nut sets axial lead per revolution.
- Narrow roller rim stays at a fixed axial station and moves vertically.
- Circumferential rolling only; axial slip is reported.
One-turn inspection of a narrow roller rim, with circumferential rolling and axial sliding. Illustrative cone taper 0.2.
One cone, rotating and advancing
The blue cone B is rigidly attached to screw D with its geometric axis offset from the screw axis. Fixed nut E converts each screw revolution into the selected axial advance. Roller C is pressed against the surface and can move vertically while staying at a fixed axial station.
The cone’s cross-section at that station grows as the larger end approaches. Its center also circles the screw axis. Both the changing local radius and the offset cone center determine the roller’s height.
The animation shows one revolution and then pauses. Press Play to start a new inspection. The slow inspection timing is independent of the entered operating rpm; all velocity outputs use the entered rpm. Zero rpm keeps the mechanism still, and the position slider remains available.
Explore the historical geometry
The radius, eccentric offset, screw lead and roller radius all change the actual contact model. The cone has an illustrative axial length of four starting radii, with end radii 1.4 and 0.6 times the starting radius. Its taper is therefore 0.2 mm of radius per millimeter along the axis. These dimensions define the illustrated model; Brown’s description does not specify them.
The roller is treated as a narrow circular rim. Its axis stays parallel to the screw, and a schematic spring-loaded guide holds its transverse position. The end view is a section at the roller station, rather than the silhouette of the cone’s larger end.
The rim rolls circumferentially in the model, but the cone still slides axially underneath it. The calculator reports that axial sliding component explicitly. It does not assume a frictionless three-dimensional rolling contact or predict transmitted torque.
Exact cross-section contact for the narrow-rim model
Let θ be screw angle, R0 the initial cone radius, e its axis offset, r the roller radius and L the screw lead. Axial advance is a=Lθ/(2π), and local cone radius is R=R0+0.2a. The cone center in the transverse plane is (e cosθ,e sinθ).
Z=e sinθ+√[(R+r)²−e²cos²θ]
Z is roller-center height above the screw axis. The contact normal in this plane is n=(−e cosθ,H)/(R+r), where H=Z−e sinθ. Differentiating gives Z′=e cosθ+[(R+r)0.2L/(2π)+e²cosθ sinθ]/H. Vertical velocity is Z′ times 2πN/60.
The cone material velocity at the contact point and the moving roller-center velocity determine roller spin. With n=(ny,nz), its angular-speed ratio is [Z′ny−(R+e cosθ ny+e sinθ nz)]/r. Multiplying by screw rpm gives signed roller rpm. A negative value indicates the opposite rotation sense in the end view.
Axial sliding component is LN/60. Net rise is the difference between Z at 360° and at 0°. The current position outputs update as the animation or position slider moves.
40 mm starting radius and an 8 mm screw lead
The default 0.25 offset ratio places the cone axis 10 mm away from the screw axis. With a 12 mm roller radius, its initial center is 51.03 mm above the screw. After one turn the screw has advanced 8 mm, local cone radius has grown to 41.6 mm, and the roller center is 52.66 mm high.
The net rise is about 1.63 mm. At 30 screw rpm the axial sliding component is 4 mm/s. The roller also rises and falls within the turn because the cone is eccentric; its motion is not just that net rise.
Setting eccentricity to zero removes the cyclic offset effect, but a nonzero screw lead still raises the roller along the taper. Setting both eccentricity and lead to zero gives constant height and the familiar signed roller-speed ratio −R0/r.
Contact assumptions and limits
Brown’s description establishes an eccentric cone attached to a screw passing through a fixed nut, with a roller pressed against it. It does not provide detailed contact dimensions, load ratings, speed limits or a production application. Those claims are not inferred here.
The zero-width rim approximation makes the contact geometry explicit. A real finite-width roller on a tapered surface needs a separate contact and friction analysis. The model also does not calculate spring preload, material stress, wear or efficiency.
Positive lead advances the larger cone section into the roller plane. The configured one-turn travel stays on the modeled cone for every allowed input combination. Playback pauses at the end because the screw remains at its advanced axial position. Play restarts the sequence from the beginning.
Eccentric-cone questions
Are forms 1 and 2 different mechanisms?
No. Brown’s figures 262 and 263 show the same movement from the end and side.
Why does the roller not finish at its starting height?
The screw has advanced, bringing a different cone radius under the roller. It returns to the same angular orientation, not the same axial position.
Why is roller speed signed?
The sign distinguishes its rotation direction from the screw. External contact normally turns the roller in the opposite sense.
Does the roller have no slip in every direction?
No. The model imposes circumferential rolling for a narrow rim and reports axial sliding separately.
Why does the animation stop?
It is a one-turn inspection of a translating mechanism. Play explicitly restarts the experiment, and the position slider lets you inspect any point.
Historical reference
Henry T. Brown, Five Hundred and Seven Mechanical Movements, figures 262 and 263, printed page 68 and accompanying description. The figures identify cone B, roller C, screw D and fixed nut E. Figure 262, end view and Figure 263, side view.
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