Cam Geometry 101

Climbing cams, as invented by Ray Jardine in the '70s, are one of the most useful items on the rack. Various designs and brands have their own particular flavor, but they all share one thing in common: the exponential spiral profile where the metal meets the rock. This short article will explain how this shape works in the context of a camming device, and will explain the mathematical equation that defines the profile.

The general image of a typical cam in a crack is shown below. The rounded cams touch the rock, and the stem bar between them is where you clip in. The axle bolt is the circle where they all meet. Notice that the axle bolt is higher than where the cams touch the rock. This is what provides the (approximately) 4:1 mechanical advantage that keeps the cam secure, even in a smooth, parallel crack. Also, notice that even though the cam on the left is rotated to a different angle than the one on the right, the same "camming angle" appears on both sides. This is due to the special profile shape of the cams. No matter what angle the cam is rotated to, it will have the same angular relationship to the crack.

To obtain this shape, a little math is required. If you are up to it, you can derive the proper equation yourself by solving the calculus differential equation for a curve with a constant angle between the radial and tangent vectors (HUH?). Or not. The formula is rather simple, even if it does require some Higher Algebra to understand.

The diagram shows a typical cam profile against an X-Y cartesian coordinate system. Since the profile is a curved shape, it's much easier mathematically to use what are known as as polar coordinates. Basically, instead of saying that you go right X inches and then up Y inches, each point on the curve is defined by saying you go out R inches, then rotate the R line the appropriate angle. For each angle, there is a specific radius, R; and vice versa. Notice that the formula involves the use of the exponential function, and that the angles must be in radians. You can modify the formula to work in degrees by using the conversion factor of degrees to radians, as shown.

To generate an actual cam profile, you can use a programmable calculator. By entering a series of angles (say, 10, 20, 30, etc. degrees), your program can save the angle, calculate the radius, retrieve the angle and then perform a polar to rectangular conversion. You can then plot the X-Y points on graph paper. You can also use the points you get to draw the curve in weak CAD programs (like AutoCAD 2000) using a spline function; you should draw the points well beyond the curve you want, on both ends, so that the "end-effects" of the spline wind up outside what you want to keep. Or, if you have a better CAD system, like MicroStation, you can automatically generate the proper curve profile.