Geometric Analysis: Radius-of-Curvature

Curvature [ κ ] of a curve is the degree to which it deviates from a straight line. Curvature is a property of the curve. Radius-of-Curvature [ ρ ] is the reciprocal of Curvature. [ ρ = 1 /κ ]

By definition, a straight line has zero curvature [and an infinite Radius-of-Curvature]. By definition also, a circle has a 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 around the cam. 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 the 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 convention of the Radius-of-Curvature, is arbitrary, but should remain consistent with each analysis.

However, the convention with Cams is:

Thus, with a Groove Cam, one flank is 'positive' and the other flank is 'negative' Radius-of-Curvature.

Note: it is possible to have a 'negative' radius for the Cam-Follower! The Cam-Follower is a 'housing' that surrounds the cam.

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

Usually, you must make sure the:

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

It is useful to consider the ratio of the Cam Radius against the Cam-Follower Radius.

Radius-of-Curvature - Rule-of-Thumb

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

In this graph, the green identifies the radius of the cam-roller.

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. This is when the cam is flat and the radius-of-curvature is infinite [∞].

The graph shows a vertical line when the radius-of-curvature changes from +∞ to –∞, 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, then for a given Load, the Contact-Stress between the cam and the cam-follower becomes larger.
( Contact Stress ≈ constant / √ρ )

Heat Treatment :

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 +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 | > Roller Radius

Video of Undercutting - Video is very Slow!

The Radius-of-Curvature is less than the radius of the cam-roller.

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.

Whether to use a Cylindrical-Cam-Follower or a Barrel Cam-Follower is a function of the type of machine and how well the follower and cam's rotational-axes can be aligned.

Cylindrical Roller - miss-alignment

When the rotational axis of a cylindrical-roller is not parallel to the rotational-axis of a cam, the cam-follower will tilt relative the cam's surface, and roll along its edge. - see the bottom and left example in the image to the left.

The contact is distorted. The maximum value of Contact-Stress is significantly greater than the nominal value that is calculated for 'line-contact'.

Barrel Rollers.

If the rotational-axis of the cam-roller is not parallel to that of the cam, the cam-follower will tilt relative to the cam's surface.

In this case, the contact moves across the cam surface, but when the tilt-angle is within limits, it does not roll along its edge.

The contact is not distorted much. The maximum value of Contact-Stress is not more than the nominal value that is calculated for 'elliptical-contact'.

The nominal maximum contact-stress of a barrel-roller is greater than that of a cylindrical-roller.  However, when a cam-follower tilts the actual maximum contact-stress of a cylindrical-roller is much greater than that of a crown-roller.

The permissible tilt angle of a cylindrical-roller is very small, ( <0.1° ), and thus a barrel-roller is a good design option in many cases.

See Tilt and Skew Limits of Cylindrical Cam-Rollers