The iPACENOTES steering wheel
24.04.2016. / Knowledge
Although we've promised to explain complete iPACENOTES language in this blog, we have to step back and explain how to actually make a tool for measuring corners. iPACENOTES will allow sharing pacenotes and thus it is critically important that all drivers make pacenotes using the same principles and the same cornering scale. So, here it is, the iPACENOTES steering wheel:
In order to make one, find nearest automotive accessories dealer and buy steering wheel cover which fits on your car's steering wheel. Then, cut 14 pieces of duct tape of the approximate size 1cm x 3cm. Use white or any onther lighty colored duct tape.
Now comes the complicated part. We want to measure corner's curvature radius with the steering wheel. But, the problem is that different cars require different steering angle to go through the same corner. In other words, if we want to create the same pacenotes with Porsche 911 and Subaru STi we have to use different markers on our steering wheel.
We need a formula which relates corner's curvature radius and steering angle, and this formula must be somehow related to the particular car model. However, it turns out that there is no such formula (at least not useful for our purposes), but there is one which is very close to what we want:
corner's curvature = track / 2 + wheelbase / sin(steer angle / steering ratio)
We've borrowed the formula from The Steering Bible (see http://www.carbibles.com/steering_bible.html). Here, the wheelbase is the horizontal distance between the center of the front wheels and the center of the rear wheels, while the track is the distance between the centerline of two wheels on the front axle.
Steering ratio is the ratio of the number of degrees turned at the steering wheel vs. the number of degrees the front wheels are deflected. For example, if the steering wheel is turned 20° and the front wheels only turn 1°, that gives a steering ratio of 20:1.
One simple way to determine steering angle of your car is to set the front wheels straight ahead, then rotate the steering wheel by exactly 360 degrees (one full circle). Now measure deflection angle of both of your front wheels. These angles will be different (the outer wheel will deflect less than the inner one due to the Ackerman angle). Sum the degree change of both front wheels and divide by two to get the average angle. Finally, divide the number 360 by the average wheel angle change and you'll get the steering ratio.
To sum up, in order to make iPACENOTES steering wheel for your car you need to measure the width of your car, the wheelbase and the steering ratio. Having these measured (or read from your car's technical specification), you need to calculate steer angles which will be the locations of the 7 stickers.
The 7 different iPACENOTES corner's curvature radiuses are 216m, 110m, 65m, 41m, 29m, 20m and 14m (we'll explain later why). The steer angle labels 1-7 correspond to these 7 curvature radiuses. In other words, 'Left 1' in iPACENOTES vocabulary means left corner with curvature radius of 216 meters, 'Left 2' is left corner with curvature radius of 110m and so on.
The formula for calculating steer angle corresponding to iPACENOTES label M for your car is:
angle M = arcsin(wheelbase / (corner radius - track / 2)) * steering ratio
Although the formula may look complicated, it really isn't. Let's calculate explicitly 7 angles for Subaru STi Wagon, model 2011. Wheelbase for STi is 2.624m, track is 1.529m and steering ratio is 15. Now we have:
angle 1 = arcsin(2.624 / (216 - 1.529 / 2)) * 15 = 10°
angle 2 = arcsin(2.624 / (110 - 1.529 / 2)) * 15 = 21°
angle 3 = arcsin(2.624 / (65 - 1.529 / 2)) * 15 = 35°
angle 4 = arcsin(2.624 / (41 - 1.529 / 2)) * 15 = 55°
angle 5 = arcsin(2.624 / (29 - 1.529 / 2)) * 15 = 80°
angle 6 = arcsin(2.624 / (20 - 1.529 / 2)) * 15 = 116°
angle 7 = arcsin(2.624 / (14 - 1.529 / 2)) * 15 = 171°
The duct tape stickers have to be placed very precisely on the steering wheel cover. For Subaru STi the angles between neutral (blue sticker) and the stickers at positions 1, 2, 3, ..., 7 are 10°, 21°, 35°, 55°, 80°, 116°, 171°. Since it is easier to measure distances then the angles on steering wheel, we'll use plain clothier’s tape measure and simple formula.
Let's say the diameter of your steering wheel is 37 cm (14.5"), which is pretty standard. The formula for converting angle to distance is (diameter × angle) / 360 * 3.14. So, in our case we have:
Sticker 1: 37 × 10 × 3.14 / 360 = 3.2 cm
Sticker 2: 37 × 21 × 3.14 / 360 = 6.8 cm
Sticker 3: 37 × 35 × 3.14 / 360 = 11.3 cm
Sticker 4: 37 × 55 × 3.14 / 360 = 17.7 cm
Sticker 5: 37 × 80 × 3.14 / 360 = 25.8 cm
Sticker 6: 37 × 116 × 3.14 / 360 = 37.4 cm
Sticker 7: 37 × 171 × 3.14 / 360 = 55.2 cm
Now use clothier’s tape to measure offsets from the blue sticker to other stickers by placing tape on the outside edge of steering wheel cover.
The obvious question here is why the distance between stickers is not the same? Why we didn't simply put equidistant stickers and avoid all the above math? The answer is, unfortunately, not exact nor deterministic.
Ultimately, corner level (1 through 7) loosely maps to the maximal speed through corner. The maximal speed through the corner is determined by the centrifugal force at the corner. The speed is maximal when the grip (frictional force) is equal to the centrifugal force. The centrifugal force is equal to:
centrifugal force = (mass of the car × speed^2) / radius of curvature
Let's say that the maximal speed through the corner 7 (which has curvature radius of 14m) the maximal speed is 45 km/h. From the centrifugal force formula and the curvature radius for the corner 6 we get that the maximal speed through the corner 6 is 54 km/h. Going further down, maximal speeds through corners 5, 4, 3, 2 and 1 are 65 km/h, 77 km/h, 97 km/h, 126 km/h and 176 km/h respectively. Does the change between consecutive maximal speeds correspond to changes in some gearbox ratios? No. We didn't want to put gearbox and gear ratios into the game because these numbers are very different for various cars. It would just be too messy.
At the end, we've actually determined corner levels 1-7 by measuring steer angle on a real road. We drove one specific road having lots of different corners (which, by the way, is used for one special stage of a certain rally) and we simply measured all different corners. Afterwards we have verified this scale on many different roads and made small corrections. The whole "hard" mathematics above serves just for the purpose of making equivalent iPACENOTES wheel for different cars.
The iPACENOTES steering wheel is the fundamental tool which we'll use in iPACENOTES system. To understand and to be prepared for the subsequent parts of iPACENOTES language, we invite you to make your iPACENOTES steering wheel and to start measure corners on public roads. This simple excercise will unveil lots of "secrets" of rally pacenotes system and, equaly important, it will open many new questions which we'll going to address in the following blogs.
In the previous blog we've mentioned Hayden Paddon, a young Kiwi WRC driver who is using something similar to iPACENOTES steering wheel system for making pacenotes. Today (April 24th, 2016) Hayden won his first ever WRC rally, Rally Argentina. Big congratulations to Hayden!
Check also this short video https://youtu.be/9q7EYVJL9XU where Hayden explains how he is making pacenotes.