Brown item 259 shows a V-grooved pulley notched to improve contact with a round band. Explore the ideal geometry of a circular band between two straight groove flanks, including clearance above an independently entered root relief.
Declared Round-Band and V-Groove Geometry
Enter a declared round-band diameter, included groove angle, root-relief depth and pulley pitch diameter. The worksheet reports ideal tangent-circle geometry only; it does not predict adhesion, slip, torque, wear or safe manufacture.
Equation Used
- The band cross-section is an undeformed circle tangent to both straight symmetric groove flanks.
- The groove angle is the included angle and the root relief lies below the hypothetical sharp-flank intersection.
- The pulley pitch-diameter input is currently unused by the calculations and drawing. It does not determine speed, torque or slip.
- Material, deformation, preload, wrap angle, friction, speed, centrifugal effects, heat, wear, tolerances, strength, adhesion and safety are excluded.
The results assume a circular band tangent to both straight V flanks. The illustration is schematic: its drawn flank angle does not track the entered groove angle, so use the numerical results for the entered geometry.
A notched V-groove carries a round band
Brown item 258 shows a smooth V-grooved pulley for a round band. Item 259 modifies that form by notching the groove to increase the adhesion of the band. Brown supplies no dimensions, force law or performance rating.
The viewer shows a rotating pulley and a schematic section. The calculations describe ideal circle-and-flank tangency, not a groove specification for a particular belt.
A generic historical belt-pulley form
Brown contrasts the notched form with the smooth V-grooved pulley in item 258. Choosing a groove for a working drive requires the belt material, loads and manufacturer guidance as well as its cross-sectional geometry.
Ideal tangent-circle geometry
This describes an undeformed circle tangent to two straight flanks and a separately declared relief below their hypothetical intersection.
Declared geometry example
For a 4 mm round band in a 40° included V with 0.50 mm declared root relief, the centre lies 5.85 mm above the sharp apex, the ideal band-to-apex gap is 3.85 mm, span at the band centre is 4.26 mm, and total clearance to the relieved root is 4.35 mm.
Geometry, grip and belt forces
Belt tension, flank contact force and transmitted torque are different quantities. The tangent-circle calculation establishes clearances and dimensions, but does not solve forces or deformation. Grip and slip also depend on the actual band, groove, contact conditions and drive layout.
Notched V-groove questions
Does this calculator predict adhesion?
No.
Is the 0.50 mm relief a recommendation?
No. It is a declared display default.
Does pitch diameter determine output speed here?
No. A complete drive needs both pulley pitch diameters and a validated belt model.
Notched V-groove references
- Henry T. Brown, Five Hundred and Seven Mechanical Movements, items 258-259: primary public-domain source identifying the smooth V-groove for a round band and the notched modification for increased adhesion.
- Gates, Round Endless Belts: manufacturer confirmation that round endless belts use application-specific sheaves rather than one universal groove rule.
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