ASC 16: Tires


Tire on Bochum’s 2017 car (photo: Anthony Dekker)

In response to my Challenger strategy post, someone asked “what about tires?”

Yes, the rolling resistance of tires is a factor with solar cars. As tires rotate, the rubber in them flexes, and some energy gets lost this way. For a top Challenger class car, with good tires, about 85% of the solar energy is lost through aerodynamic drag, and about 15% through tires. That’s why my simple car model considered only aerodynamic drag.

The force required to overcome rolling resistance is g = 9.8 times the weight of the car times a tire-specific coefficient (ranging from 0.002 for an expensive solar car tire to 0.01 for a typical car tire). For a top Challenger class car, car and driver together will weigh around 250 kg. For a four-person Cruiser, it’s more like 800 kg – about triple. Bearing in mind that Cruisers often have tires with a higher coefficient of rolling resistance, this means that rolling resistance becomes quite significant with Cruisers. That is why Cruisers will sometimes turf out passengers to cut their energy use – even when running on a flat road. Climbing the mountains, overcoming gravity will come into play, and that uses up even more energy.


In ASC race news, I have updated my information page and teams list with news about day 1 of scrutineering.


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ASC 13: Challenger strategy

This will be a lengthy post on race strategy. I will concentrate on Challenger-class cars, for which the main decision is at what speed to drive (there are also some tactical and psychological issues associated with overtaking, but I won’t get into them here).

The kind of analysis I’m doing assumes that teams have a good predictive model of how their car will perform under various conditions (collecting the data required to build such a model is yet another reason to do lots of testing). I’m using a very simplistic model of a hypothetical car – I only took half a day to build it (in R, of course); a real solar car strategy guy would take months and produce something much more detailed. I’m assuming that aerodynamic drag is the main consumer of energy. Remember that the drag force is:

F = ½ Cd A ρ v2

where Cd is the drag coefficient, A is the frontal area of the car, ρ is the density of air (about 1.2), and v is the speed of the car. The key thing here is that faster speeds burn up more energy.

I’m further assuming a solar model loosely based on the region of the American Solar Challenge, a perfectly flat road (no, that’s not realistic), a race start time of 8:00 AM, and a finish time of 5:00 PM with no rest breaks (no, that’s not realistic either). My hypothetical car has 4 square metres of solar panel with 25% efficiency, and a 5 kWh battery pack which is 80% full at the start of the day. In my first simulation, my car runs at constant speed until 5:00 PM or the battery dies, whichever comes first (solar cars are not intended to run with totally empty batteries). This chart shows what happens:

Initially, the graph is a straight line. The faster you drive, the further you go. But this only lasts up to 73.7 km/h, which takes you a distance of 663 km. Faster than that, and your battery dies before the end of the day, so that you go less far in total. In reality, of course, the driver would slow down before the battery died completely, but I’m not modelling that.

My second chart looks at varying the speed of the car. We start the day at one speed (horizontal axis), gradually speeding up or slowing down to reach a second speed at solar noon (vertical axis), and then slowly shifting back so as to end the day at the start speed:

In the chart, the distance travelled is shown by colour and number. It can be seen that, at least for my hypothetical car, the optimum is fairly forgiving: speeds between about 60 and 85 km/h are OK, as long as they average out to about 73 km/h, which means 653 km travelled. You will also notice that there is no incentive to break the speed limit – the rules have been carefully constructed to ensure that this is true.

In my third chart, I’m assuming a patch of cloud on the road ahead, between 200 km and 400 km from the start. We start the day at one speed (horizontal axis), switch to a second speed once inside the cloud (vertical axis), and then switch to a third speed when leaving the cloud (the third speed is chosen to be whatever works best with the remaining battery charge).

Two vertical stripes are noticeable on the left and right sides of the chart. At 20 km/h, the race day ends before you reach the cloud, so that the choice of “cloud speed” doesn’t actually matter. And at 100 km/h, your battery dies before you reach the cloud, so that the choice of “cloud speed” doesn’t matter either. The optimum is to run at a constant 70 km/h initially, and then to speed up to 75 km/h inside the cloud (so as to get back into sunshine that little bit sooner). If you run at an optimal 63 km/h after leaving the cloud, this lets you cover 619 km.

The final scenario changes the one above to have a moving cloud. This cloud is circular and once again 200 km in diameter, with a centre 300 km away from the start position. The edge of the cloud is initially 210 km from the road, but moving at a rapid 50 km/h towards and across it:

The optimum here is to outrun the cloud: to run at 85 km/h till the cloud is past, and then to drop back to whatever speed gives you a 73 km/h average. Not surprisingly, this gives essentially the same outcome as my first varying-speed example. A solution that is almost as good is to run at 70 km/h initially, and speed up to 80 km/h once inside the cloud. This gives you about an hour and 20 minutes inside the cloud. Slowing down to an optimal 70 km/h after leaving the cloud then lets you cover 642 km.

In real life, outrunning clouds and speeding up slightly through them can both be feasible options – if you have the right prediction software, and if know exactly what the weather is doing (which is the hard part). This is why the strategy guys in the chase car are always so busy!


Nuon’s WSC 2011 chase car (image credit)


ASC 9: Aerodynamics

For both the Challenger class and the Cruiser class, solar car racing is to a large extent about aerodynamic drag. That’s overwhelmingly what the hard-earned solar energy is being wasted on, and therefore it’s what teams need to concentrate on minimising. The drag force on a car is given by the equation:

F = ½ Cd A ρ v2

Breaking that down, v is the speed of the car, ρ is the density of air (about 1.2), A is the frontal area of the car, and Cd is the drag coefficient, a number which indicates how aerodynamic (and therefore, how energy-efficient) the shape of the car is. For Challengers, minimising Cd allows the speed v to be increased, while for Cruisers (for which the average v is essentially given), minimising Cd allows non-solar energy use to be minimised. Of course, minimising frontal area is important too (and that is the motivation behind asymmetric Challenger cars).

To give a feeling for the all-important Cd, here are some vehicles with values ranging from 0.19 to 0.57:


Drag coefficients for a selection of vehicles. Clockwise from top left: 0.57 – Hummer H2 (photo: Thomas Doerfer); 0.30 – Saab 92 (photo: “Liftarn”); 0.26 – BMW i8 (photo: “youkeys”); 0.19 – General Motors EV1 (photo: Rick Rowen)

Because Cd is so all-important, it is the one thing that solar car teams are really secretive about. Challengers generally have values under 0.1. With no need for practicality, they chase their way towards the impossible goal of Cd = 0, trying to come up with the perfect race car, which will slice through air like a hot knife through butter:


Nuon’s 2005 car, Nuna 3, with Cd = 0.07 (photo: Hans-Peter van Velthoven)

Cruisers, on the other hand, have to balance aerodynamics with practicality. Bochum’s early SolarWorld GT had Cd = 0.137:


Bochum’s 2011 car, SolarWorld GT, with Cd = 0.137 (photo: “SolarLabor”)

Eindhoven’s recent Stella Vie, with its sleek aerodynamic shape, does much better than that (but they won’t say how much better):


Eindhoven’s 2017 car, Stella Vie (photo: TU Eindhoven, Bart van Overbeeke)

I understand that Sunswift’s 2013–2015 car eVe had Cd = 0.16. Appalachian State (Sunergy) have stated that their newly-built ROSE has Cd = 0.17. PrISUm’s Penumbra has a higher value (Cd = 0.2), because of the blunt end which they chose for practicality reasons (although they did do a few clever things to reduce the impact of that blunt end). I’m not aware of the Cd values for other ASC cars.


Appalachian State’s beautiful ROSE, with Cd = 0.17 (image credit)


Mathematics in Action: Vehicle Identification Numbers

Motor vehicles have a 17-character Vehicle Identification Number or VIN on a metal plate like the one below, usually on the driver’s side dashboard, or on the driver’s side door jamb, or in front of the engine block:


A Vehicle Identification Number (VIN) plate (Photo: Michiel1972)

VINs offer an interesting example of check digit calculation. The central digit (or an X representing 10) is a check digit (calculated modulo 11) used to detect errors. Any letters in the rest of the VIN are decoded like this:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 2 3 4 5 6 7 8 1 2 3 4 5 7 9 2 3 4 5 6 7 8 9

The check digit calculation involves decoding the VIN, and multiplying the resulting numbers by the weights shown in blue, giving the products in purple:

VIN L J C P C B L C X 1 1 0 0 0 2 3 7
Decoded 3 1 3 7 3 2 3 3 10 1 1 0 0 0 2 3 7
Weights 8 7 6 5 4 3 2 10 0 9 8 7 6 5 4 3 2
Product 24 7 18 35 12 6 6 30 0 9 8 0 0 0 8 9 14

These products are added up modulo 11 (meaning the sum is divided by 11 and the remainder taken). In this case, the sum is 186 = 10 = X (mod 11), which makes the VIN valid, because it matches the original central X. What about the VIN on your vehicle?


The Sand Reckoner

In his short work The Sand Reckoner, Archimedes (c. 287 BC – c. 212 BC) identifies a number larger than what he believed was the number of grains of sand which would fit into the Universe. He was hampered by the fact that the largest number-word he knew was myriad (10,000), so that he had to invent his own notation for large numbers (I will use modern scientific notation instead).

Archimedes’ began with poppyseeds, which he estimated were at least 0.5 mm in diameter (using modern terminology), and which would contain at most 10,000 grains of sand. This makes the volume of a sand-grain at least 6.5×10−15 cubic metres (in fact, even fine sand-grains have a volume at least 10 times that).

Archimedes estimated the diameter of the sphere containing the fixed stars (yellow in the diagram below) as about 2 light-years or 2×1016 metres (we now know that even the closest star is about 4 light-years away). This makes the volume of the sphere 4×1048 cubic metres which means, as Archimedes shows, that less than 1063 grains of sand will fit.

A more modern figure for the diameter of the observable universe is 93 billion light-years, which means that less than 1095 grains of sand would fit. For atoms packed closely together (as in ordinary matter), less than 10110 atoms would fit. For neutrons packed closely together (as in a neutron star), less than 10126 neutrons would fit. But these are still puny numbers compared to, say, 277,232,917 − 1, the largest known prime!


Snakes and Ladders


Snakes and Ladders board, dated 1966, from the Auckland Museum (credit)

Snakes and Ladders is an ancient board game originating in India. It is totally random, and hence not very interesting. If players start on square #1, then after one turn, they have equal probabilities of being on squares #2, #3, #4, #5, #6, and #7. This image shows the probability distribution:

After two turns, the probability distribution is as follows (the most likely total of two dice rolls is 7, taking a player to square #8 and up a ladder to #26:

After 8 turns, players would be scattered all over the board. There is a 1% chance that any given player has won:

After 19 turns, there is a 24.7% chance that any given player has won:

This probability grows to 50.4% after 35 turns. But no matter how long you play, it remains possible (though increasingly unlikely) that nobody has won yet. Yet another reason why children tend to rapidly tire of the game.

For an alternative view of the probability analysis, see this animation:


The Game of Mu Torere

The New Zealand game of mū tōrere is illustrated above with a beautiful handmade wooden board. The game seems to have been developed by the Māori people in response to the European game of draughts (checkers). Play is quite different from draughts, however. The game starts as shown above, with Black to move first. Legal moves involve moving a piece to an adjacent empty space:

  • along the periphery (kewai), or
  • from the centre (pūtahi) to the periphery, or
  • from the periphery to the centre, provided the moved piece is adjacent to an opponent’s piece.

Game play continues forever until a draw is called (by mutual consent) or a player loses by being unable to move. Neither player can force a win, in general, so a loss is always the result of a mistake. For each player there is one “big trap” and four “small traps.” This is the “big trap” (Black wins in 5 moves):

  
The board on the left is the “big trap” for White – Black can force a win by moving as shown, which leaves only one move for White.

  
Again, Black moves as shown, which leaves only one move for White.

  
Now, when Black moves as shown, White cannot move, which means that White loses.

Here is one of the four “small traps” for White. The obvious move by Black results in White losing (but avoiding this does not require looking quite so far ahead as with the “big trap”):

Here (click to zoom) is the complete network of 86 game states for mū tōrere (40 board positions which can occur in both a “Black to move” and a “White to move” form, plus 6 other “lost” board positions). Light-coloured circles indicate White to move, and dark-coloured circles Black to move, with the start position in blue at the top right. Red and pink circles are a guaranteed win for Black, while green circles are a guaranteed win for White. Arrows indicate moves, with coloured arrows being forced moves. The diagram (produced in R) does not fully indicate the symmetry of the network. Many of the cycles are clearly visible, however: