The Rigid Model -- Explaining How Camming Devices Work

As a first approximation, a camming device can be modeled as a rod wedged or ``cammed'' in a parallel crack as shown in Figure 2a below. If the rod does not slip out, a downward tug or force applied to the upper end wedges the rod harder into the crack, causing the rod to push against the walls with a normal force. Whether or not the rod slips out is determined by a third force, that of friction. Figure 2b shows the forces on the rod; the friction force on the upper left end of the rod is intentionally omitted. For camming devices that rely on friction, this upper left end corresponds to an internal axle (see Figure 1 or Figure 4) and thus experiences negligible friction.

Figure 2: The forces on a rod wedged in a crack.

If the rod shown in Figure 2 does not move, both the forces and torques must each sum to zero. These equalities and the definition of the friction force yield four equations that define the relation between the applied, normal and friction forces. R is the rod length, is the friction coefficient, and is the angle the rod makes with the horizontal.

Two important results are derived from algebraic manipulation of these equations:

Equation 1 shows the wedging effect that multiplies the applied force by a factor of 1/tan . This relationship between the applied and normal forces is invoked in the elastic model to determine cam compression. Equation 2 shows that the maximum camming angle, , at which the rod can remain in the crack is limited by the coefficient of friction, between the rod and the crack surface. For an aluminum rod in a granite crack, a measured value for the friction coefficient is .38, and the corresponding maximum camming angle is about 20 degrees; a rod tipped more than 20 degrees from horizontal would slip out of a granite crack.

To produce a camming device that works over a wide range of crack sizes, multiple rods are combined into a two dimensional surface in the shape of a cam. This shape satisfies the criterion shown in Figure 3a below: the angle between the surface and the line perpendicular to the radius must remain a constant, the camming angle, . In polar coordinates, such a shape is defined by:

Rearranging and integrating yields:

Equation 3 defines the logarithmic spiral plotted in Figure 3b. A section cut from this spiral produces the cam (Figure 3c). Typically three or four such cams are mounted together on an axle to produce a complete SLCD.