Geometric Analysis: Radius-of-Curvature

The Curvature of a curve is the degree to which it deviates from a straight line. Curvature is a property of the curve. Radius-of-Curvature is the reciprocal of Curvature.

By definition, a straight line has zero curvature [and an infinite Radius-of-Curvature] and a circle has finite curvature [and a finite Radius-of-Curvature]. I find it easier to mentally translate Radius-of-Curvature as the 'Radius-of-the-Curve'.

The Radius-of-Curvature of each point along a cam can be represented by the radius of the osculating-circle at that point. The oscillating-circle might be 'inside' or 'outside' the cam.

Two Osculating Circle at two different points. One circle is outside the cam and the other is inside the cam. 

Think of the 'Radius-of-Curvature' as the ...radius of the osculating circle.

Sign of the Radius-of-Curvature

The 'sign' of the Radius-of-Curvature changes when the oscillating-circle moves one side to the other side of a curve [or inside to outside to outside a cam], or vice-versa.

Mathematically, the side of a curve that has a 'positive' or 'negative' Radius-of-Curvature, is arbitrary.

However, with Cams [curves], the convention is:

Thus, if the Cam is a Groove Cam, then one of the cam-flanks will have a 'positive' and the other a 'negative' Radius-of-Curvature.

Note: there are papers that discuss a 'negative' radius of Cam-Follower. This means the Cam-Follower is not a roller, but a 'housing' that surrounds the cam.

Radius of Curvature Convention for Planar Cams

Radius-of-Curvature(ρ) = 1/ Curvature(κ)

In MechDesigner, we use the Cam-Data FB to plot Cam's Radius-of Curvature.

Usually, you must make sure the:

Minimum Cam Radius-of-Curvature > Cam-Follower Radius.

It is useful to consider the ratio of the Cam and Cam-Follower radii.

Rules-of-Thumb

In this graph, the olive green bar identifies the radius of the cam follower.

The graph shows that the cam-profile has both positive and negative radius-of-curvature.

Note: when the radius of a convex cam becomes concave, or vice versa, it passes a 'point-of-inflection'. The cam becomes flat  instantaneously flat. The graph is a vertical line as the radius-of-curvature changes from large positive to a large negative value, or vice versa.

If the Radius-of-Curvature of the cam is small, then you must make these design checks:

Undercutting :

If the Radius-of-Curvature of the cam is less than the Roller's radius; the 'real' [physical] Cam-Follower cannot follow the motion you have designed for it. However, the cam-follower in MechDesigner follows the motion exactly.

Contact Hertzian Stress :

As the Radius-of-Curvature gets smaller, the Contact-Stress between the cam and the cam-follower becomes larger.
(Contact Stress ≈ constant/√ρ )

Heat Treatment :

the Cam's Heat treatment is more likely to crack the steel if its Radius-of-Curvature is small.

Flat-Faced Followers :

When the cam-follower is a 'Flat-Faced Follower', the Radius-of-Curvature of the cam must be Convex (+ve) at all times.

Undercutting

If the cam's radius-of-curvature becomes less-than the radius of the cam-follower's roller, then the cam-follower cannot move with motion as specified by your design.

Note: in MechDesigner, the cam-follower always moves with the motion specified by the motion-design and the kinematic-chain.

However, in your machine, the cam-follower's motion becomes controlled only by its contact with the cam. Thus, when the cam is undercut, the cam-follower cannot move with the motion specified in the model.

Thus, YOU must make sure the cam DOES NOT UNDERCUT. You can check visually. However, the best way is to use the Cam-Data FB and a Graph FB. In the Graph, make sure the you plot the correct Radius-of-Curvature. Use the Y-axis drop-down in the open graph to select 'Radius-of-Curvature 1', or 'Radius-of-Curvature 2', for Cam 1 or Cam 2 respectively. The number of the Cam is at the contact-point in the graphic-area.

Interference

Cam's Radius-of-Curvature × -1 < Roller Radius ...the Roller will not fit into the Cam. 

Video of Undercutting - Slow!

In this case, the Radius-of-Curvature is less than the radius of the Cam Follower

However, the Cam-Follower cannot actually not follow the specified motion as the cam-follower has nothing to follow!

Too much metal has been removed from the cam.

Crown-Rollers, also called Barrel Rollers, are often used with cam mechanisms when the alignment between the cam and roller cannot be guaranteed to be within the bearings alignment specification.

If a cylindrical-roller is not aligned with the cam's rotational axis, the operating contact-stress will be significantly more than that nominal 'Hertzian' contact-stress.

The nominal Hertzian contact-stress of a crown-roller is greater than that of the cylindrical-roller.  However, a crown-roller can be misaligned to a greater extent without the contact-stress increasing.

Thus, even though the nominal contact-stress of a crown-roller is greater than that of a cylindrical-roller, it is more likely that its operating contact-stress will be less for the crown-roller when alignment cannot be guaranteed.

See Tilt and Skew Limits of Cylindrical Cam-Rollers