Curvature [k] and Radius-of-Curvature [ρ]

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. [ ρ = 1 /κ ]

By definition, a straight line has zero curvature [and an infinite Radius-of-Curvature] and 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.

Radius of Curvature on Cam - Internal and External Radii

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:

Curvature of the Cam-Follower is ALWAYS Positive [+]
If the Cam is 'convex' to the Cam-Follower, then the Cam's curvature is positive.
If the Cam is 'concave' to the Cam-Follower, then the Cam's curvature is negative.

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 

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 )

RadiusofCurvature-Graph2

Usually, you must make sure the:

Minimum 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 'Ratio' =

Rcam

 

Rcam-follower

 

Radius-of-Curvature - Rule-of-Thumb

Radius of Curvature Ratio > 2


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

The graph shows is a vertical when the radius-of-curvature changes from + to –, or vice versa.

Radius-of-Curvature and Undercutting

RadiusCurvatureRatios

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.

Undercut Cam 

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.

VeryBadCam2

Interference

When the Cam is concave, its Radius-of-Curvature must be greater than the Roller's Radius.
If it is not, the Roller will not be able to 'fit' into the manufactured cam.

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

There is undercutting.
In the video, the Cam-Follower follows the motion design given to it by MotionDesigner.

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.

The cam would have a sharp corner.

Crown / Barrel Rollers and Elliptical Contact

Maximum Contact Stress of Cyclindrical and a Barrel Cam-Followers

Schematic of Maximum Contact Stress of Cylindrical and Barrel Cam-Followers

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-follower 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

Radius of Curvature Convention for Cam and Barrel Rollers

Radius of Curvature Convention for Cam and Barrel Rollers

 

Tutorial and Reference Help Files for MechDesigner and MotionDesigner 13.2 + © Machine, Mechanism, Motion and Cam Design Software by PSMotion Ltd