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. |
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![]() 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 ) |
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![]() Radius of Curvature Convention for Planar Cams |
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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. 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. |
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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. |
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Interference
| Cam's Radius-of-Curvature | > Roller Radius |
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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.
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![]() 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-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. |
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![]() Radius of Curvature Convention for Cam and Barrel Rollers |
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