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.

Radius-of-Curvature - sign convention

The 'sign' of the Radius-of-Curvature changes when oscillating-circle moves one side of a curve to the other side of a curve. For example, the inside to outside of a cam.

Mathematically, the sign of the 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.

Radius of Curvature Convention for Planar Cams

Note: there are papers that discuss a 'negative' radius for the Cam-Follower! This means the Cam-Follower is a 'housing' that surrounds the cam.

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

In MechDesigner, we use the Cam-Data FB to plot the Radius-of Curvature of the Cam1 and Cam2.

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 radii-of-curvatures.

Note: when the radius changes from '+ve' to '–ve', or vice versa It passes an inflection-point. At that instant, the radius-of-curvature is infinite.

The graph shows as a vertical line when the radius-of-curvature changes from +infinity to --negative, or vice versa.

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

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

Note: in MechDesigner, the cam-follower always moves with the motion as kinematically specified.

In your machine, the cam-follower's motion becomes controlled only by its contact with the cam, of course.

Thus, if the cam profile is undercut, the cam-follower cannot move with the motion as kinematically specified.

In MechDesigner, you can scrutinize the model to look for a 'dove-tail' - see model to the left.

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

Video of Undercutting - Video is very Slow!

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.

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 edge of the Cam-Follower w 'plough' into the cam - not good.

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 alignment specification.

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, so that the edge of the roller does not plough into the Cam - better.

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