Brown item 204 shows waisted rollers transferring rotation between oblique shafts by rolling contact. Explore this arrangement and calculate ideal driven speed from two entered effective contact radii, assuming equal tangential speed without slip.
Ideal Rolling-Contact Speed Calculator
Enter driver speed, two declared effective contact radii and a schematic shaft angle. The worksheet applies equal tangential speed at an ideal no-slip contact only; it does not generate the three-dimensional roller surfaces or rate a transmission.
Equation Used
- The two declared effective contact radii identify one ideal common rolling contact.
- Tangential-speed magnitudes are equal and rotation directions are opposite at that contact.
- The entered angle is a schematic projection parameter, not a solved spatial surface definition.
- Roller profile generation, contact-line geometry, spin, creep, traction, slip, preload, friction, Hertz stress, bearings, efficiency, heat, power, strength, wear, manufacture and safety are excluded.
The ratio follows from equal tangential speed at the two entered contact radii. The illustration shows waisted roller forms; their three-dimensional contact surfaces are not calculated.
Waisted rollers transmit rotation between oblique shafts
Henry T. Brown item 204 describes rotary motion passing from one shaft to another arranged obliquely to it by rolling contact. Its engraving shows two waisted roller bodies.
The schematic animates opposite rotation using an ideal equality of tangential speeds at the entered contact radii.
A historical transmission principle
The arrangement illustrates rotary transmission between oblique shafts through rolling contact. Applying the principle requires roller surfaces and contact conditions that suit the actual shaft geometry and loads.
Declared ideal contact-speed equality
The negative sign records opposite rotation. The equation assumes independently entered effective contact radii; it does not generate the spatial roller surfaces that would provide that contact.
Declared radius example
For n1=480 rpm, r1=45 mm and r2=60 mm, ideal driven speed is -360.0 rpm, ratio magnitude is 0.750:1, declared contact speed is 2.262 m/s, and driven period is 0.167 s. The 20° angle changes the illustration only.
Ideal speed ratio and a working friction drive
Equal tangential speed gives a radius ratio for ideal rolling contact. Shaft angle alone does not specify the required roller surfaces or their effective radii. Traction, slip, contact stress and losses require a more complete model before a working drive can be rated.
Brown oblique-roller questions
Does shaft angle determine the ratio here?
No. It changes the schematic only because a full surface definition is absent.
Are r1 and r2 dimensions from Brown?
No. They are declared effective contact radii.
Does this calculator design a working friction drive?
No.
Oblique-roller references
- Henry T. Brown, Five Hundred and Seven Mechanical Movements, item 204: public-domain engraving and caption for oblique-shaft rotary transmission by rolling contact.
- OpenStax University Physics, section 10.3: standard tangential-speed relation v=rω used in the ideal speed calculation.
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