In addition to defining the relationship of forces in SLCDs, it is important to understand the geometry of the devices. The geometry of Black Diamond’s size # 3 Camalot was studied.
First, the curve of the device needs to be defined. To define this, the lobe was traced on a piece of paper. Then a line was drawn from the axle of rotation to a point on the curve. The angle of this line was defined as phi = 0. Then, the radius of the lobe from the axle of rotation was measured as a function of phi.
This data was plotted onto the graph below in order to show the shape of the curve measured.
This data was also fitted to a linear curve and an exponential curve to test its appropriateness.
Both the linear fit, with an R-squared value of .999, and the exponential fit, with an R-squared value of .9987, seem to be eligible fits for the actual Black Diamond lobe shape. To have a better idea of how these two curves look, the linear and exponential fits were extrapolated over a range of phi values and plotted as shown below.
Observing the curves, one finds at small angles phi, the linear fit and exponential fit appear very similar, explaining why both the linear fit and exponential fit have very high R-squared values. This makes it impossible to determine weather Black Diamond actually uses a curve defined by an exponential or linear relationship.
Because Black Diamond claims to use an exponential relationship, let us further examine this measured curve, defined by radius = 1.3663e ^ (0.2673 * phi).
The equation for a logarithmic spiral is given by:
r = ae ^ (b * theta), where r is the distance from the origin, phi is the angle from the x-axis, a is a sizing constant, and b is a growth constant.
Geometry not detailed here shows that the angle between the radial line extending from the spiral’s origin and the tangential line is equal to gamma = arctan (1 / b). The camming angle is equal to 90 – gamma; therefore for the measured value of b = .2673, the camming angle is determined to be 14.9653 degrees.
If you’re interested, you should continue learning more about the force analysis and the cam angle analysis of SLCDs at Matt’s research page .