Thursday, May 28, 2009

Dover Monster tamed by math

Dover Monster Mile

The track that grabs cars as they exit the turns 2 and 4, and chews them up (crashing the cars against the outside wall)

In an interview Elliott Sadler described driving the Dover track as feeling like an amusement park roller coaster ride, however, he didn’t seem to realize he'd explained the car crushing feature of the Dover “Monster.”
It’s all about transitions, from the banking in the turns to the banking along the straight-aways. The track is 1 mile around (measured 30’ inside the outside wall, NASCAR rule), and has 24 degree banking in the turns (same as Atlanta and Charlotte—Lowe’s—) and 9 degrees on the straights; but as the track is only two-thirds size of the other two, the radius of the turns is smaller.
This photo (taken in the stands from Turn 1) shows how the track looks in person, along with section lines drawn into the picture to illustrate important points in the turn.

This cartoon sketch exaggerates the track but shows the idea, at section 1 the track is still banked at 24 degrees, by the time one gets to section 5, the banking is down to 9 degrees, as a result the runs uphill from the apex to line 1, over a crest (line 3) and then it is literally a down-hill run at the end of the turn exit (line 5).

In the traditional car chase scene over the hilly streets of San Francisco the hills are so steep that the cars literally are flying after they crest the hill. This is because the car drops at the acceleration of gravity, 32.2 ft per second per second; but if the road is dropping away more quickly relative to the forward speed of the car, then it goes air borne.
One can imagine in this sketch that if the car is moving slowly it stays on the ground, but at high speeds it does not. If one knows the curve of the hill it is easy to calculate the maximum speed that will still have the car on the ground, or conversely, the minimum speed to get it air borne (this is the essential calculation for stunt teams in doing ramp jumps).
Relative to the track at Dover, the crest of the hill for the drive path of most race cars occurs at about section line 3. Now if we look at the vertical profile of the drive path of the race car around the turn, it becomes more apparent what the effect is.
The car doesn’t need to actually become air borne to significantly change the handling of the car.If the car is still turning, the driver hasn’t straightened out yet, and the tires are pushing at their maximum side thrust to keep the car from sliding sideways, then unloading the car just a little bit, due to the road dropping away too quickly, results in the car seeming to “jump” sideways (and often into the wall).
The spring rates in a NASCAR Sprint race car are on the order of 300 lbs per inch, so if the racetrack drops away just 1 inch too much (the car continues on its arc as in the San Francisco car flight illustration, but the wheels follow the road) then the frictional load on the tires is decreased by 1200 lbs (300 lbs per inch of spring x 1 inch elongation of the spring x 4 wheels = 1200 lbs reduction in wheel load on the ground).

The teams which have 7 post rigs can predict the wheel loads for any given drive path on any track for which they have an accurate survey. It then requires a crew chief and driver to talk to the race car engineers to understand the relationship between drive path and ability of the car to turn at any given speed.

Consider an entire turn.

At point A the driver is at maximum speed from accelerating down the straight-away, and this is their lift point, the spot where they get off the throttle and onto the brakes to slow the car. As the car turns in, point B, the car is slowing and the driver is turning more to the left. By point D the car is at it slowest and the driver has turned the wheel to the maximum for this curve. From point A to D the car has been literally going down-hill due to the banking. At point D the car starts going up-hill, the driver starts accelerating and decreasing the turn of the steering wheel. As the race car exits the turn, through points E, F, and G, it is accelerating and going up hill, all the while decreasing the amount of turn in the steering wheel, the car is straightening out for the run down the next straight.

However, if by point G the car is still turning, and running as fast as the tires will allow without sliding, then as the car crests the hill at G and starts down towards section line 4, the wheels will partially unload and the car will suddenly slide sideways towards the wall: it actually appears to “jump” sideways.
Watch for TV views from the vantage point of looking down the straight-away towards a turn and see how often a car “jumps” sideways to get crunched by the Dover Monster.

Some drivers can learn intuitively what the drive path is that will allow them to defeat theDover Monster, but it’s a slow process; much easier and quicker if you have a good engineer.

All the big teams with 7 post rigs and race car engineers can calculate the path and show the driver where it is and at what speeds (engine RPM) they can negotiate the turn. The 7 post rig also allows the team to evaluate different set-up combinations, springs and shocks, for a given track, before they ever leave their race shop. This is in part why the well funded teams are the only ones that win, with the exception of a weather fluke as last week in Charlotte (Lowe’s).

Just having sophisticated equipment such as a 7 post rig is in itself not sufficient to win races, the crew chief must be technically astute enough to understand and utilize the information which engineers can provide; this is the essential skill of some one such as Chad Knaus (for the 48 car), Steve Letarte (for the 24 car) and Alan Gustafson (for the 5 car) in the Hendrick operations.


  1. This is a really fascinating post - I'm glad you put it up. The combination of real photos with your drawings is very helpful, and how you walk through those diagrams.

    It would be great if you could put this type of analysis up for the various different tracks.

    Maybe an analysis of the surface differences (concrete vs asphalt).

    Also how the track surfaces change over time due to weather, afternoon/night, rubber on the track, etc. Those would all be cool too.

  2. I teach math modeling classes and this gives a great background for my lesson next week. Thanks and send more.

  3. 36 Races,
    Thanks for the comment; interestingly enough the next blog will be about Pocono (the race after Dover). It is a curious track with three different turns each designed to mimic an existing, famous track (Trenton, now out of business; Indy; and the Milwaukee Mile). More of this in the next blog.
    prof pi

  4. Bev in Charlotte,
    Glad we can help make math and science interesting; perhaps we can infect more students with "the Knack."
    Anyone who hasn't seen the Dilbert clip and knows an engineer, must watch it.
    Those who are infected with the Knack thank all you teachers who indulged us.
    prof pi

  5. Well hello there prof! Again, another fantastic explanation of how this sport is so rich in mathematics and science. Dover is one of my favorite tracks to watch on TV (hopefully in person one day) because of the car's exit from the turns :-) Keep these comin'!

  6. M.E.,
    Thank you for the comment, glad you're enjoying the site and a somewhat technical view of race tracks and racing.

    Prof pi