Turning radius: as a function of the rear PA angle

*^{The effect of changing the rear axis angle to the turning radius of the board}*

I got deeper into this topic when a couple of regulars in a longboarding forum decided to reverse their rear trucks to see if there are specific advantages or disadvantages to this. My explanation on the relation between wheelbase, angles and turning radius (image right), and therefore about what one can and can’t expect, didn’t really register; they thought they would reverse their trucks and something uncanny would happen. I also tried here, in very simple terms and with animated images, to show that reversing is merely de-wedging. With this as a starting point, this post is to show, in very practical terms and without abstract models, that reversing the rear truck is simply an option available for when we need it, not some fanciful eccentricity. I aim to derive a very simple rule-of-thumb for those keen to experiment.

The numerator is the wheelbase (here it's "100). The denominator has the sine of the tilt of the deck (here it's 10°) times the sum of the tangent of the front p.a.a. (here's it's 60°) plus the tangent of the rear p.a.a. (x). — ^{The numerator is the wheelbase (here it is 100). The denominator has the sine of the tilt of the deck (here it is 10°) times the sum of the tangent of the front PAA (here’s it is 60°) plus the tangent of the rear PAA (x).}

The most useful rule-of-thumb, in my opinion, would be to use the turning radius as a function of the rear pivot axis angle (PAA) to calculate the change of the turning radius (in percentage) per 1 degree of change of the PAA. I start with a wheelbase of 100 and a front PAA of 60° (round numbers within the scope of real-life applications) and change the rear PAA between -30° and 30° (i.e. numbers which, I assume, are within our reach using actually existing trucks, brackets, etc).

First I graphed the function between the two parameters (image right). Notice that when calculating the ratio between two turning radii, the wheelbase and the tilt are removed by the operation. This means that the change of radius is dependent quite neatly only on front and rear PAA’s.

Rule of thumb: when your deck has a front pivot axis angle of around 60° and rear pivot axis angle in the region from 30° to -30°, for every 1° of change you get roughly 1% difference in your turning radius.

The next graph shows the incremental increase of the turning radius of a deck [of any length, tilted to any degree (other than 0°)] with a front truck PAA of 60°, in steps of 1° reduction of rear truck pivot axis angle. So, y-axis is turning radius change in % (100% would imply no change) and x-axis is steps of 1° dewedging, from +30° to -30°. These changes add up, so it doesn’t mean a 10° change gives 110% the radius. It’ll be more (because each 1% difference refers to the previous length).

And so we arrive to our rule of thumb: when your deck has a front PAA of around 60° and rear PAA in the region from 30° to -30°, for every 1° of change you get roughly 1% difference in your turning radius.

Disclaimer: I would think that the derivative of the function would remove wheelbase and tilt, like calculating the ratios between different radii does, but it doesn’t. I have to come back to this some day.

There was an interesting debate in a forum about the possible meaning of the word “gear” in regards to skateboarding. I argued that “gearing” should imply only the length of the turning radius and others that the concept should also include equipment quality etc. Those keen can check it here.