|Grand Prix Racing -||The Science of Fast Pinewood Cars|
Besides, "What lubricant is best?", the most common question seems to be this one about where to put the weight. When there are factors like these that you have a lot of control over, you certainly want to do what is best with them.
But once again, the answer is not a simple one in all cases. See the "How to" manual for a summary. The more important considerations are listed below.
- Track Shape - Placement doesn't matter on an all ramp track, others do
- Flat Length - Height doesn't matter if the flat is 4 times the ramp length
- Short Flat - Forward weight might win if the track is "S" shaped
- Rear Axle Position - As close to the rear as possible
- Wheel Base - As long as possible gives best stability
But before we get into details, let's agree on two places where the weight generally should NOT BE PLACED. The first is in front of the front axle, and the second is behind the rear axle! If you can weight your car with something really dense like gold or osmium you might be able to make it work so that the bumper doesn't hit the lane median when the other end lifts off the track spontaneously as the starting peg drops! But those of us who can't afford it would have a dragging bumper.
So weight placement means putting the center of mass somewhere between the axles, above or below them. Be aware, placement too high and forward can cause front bumper rubbing on the ramp and though high and behind the rear axle works on the ramp, a wheelie ensues on the flat with the rear bumper on the median.
In order to prove which placement is best on which tracks, let's follow this plan. First, find the best weight placement on the ramp track. This is the easiest case and should get you thinking in the right direction for the next track shape.
Adding a flat to the ramp, we get a "typical" pinewood race track. The simple closed race model applies and we use one of its deriving models to obtain results from derivatives with respect to total race time. Then to verify the results, we stage a virtual race on a smooth transition track using a computer simulation to measure possible advantage in inches at the finish line.
Add a sudden drop to the ramp from the starting line and you get an "S" shaped track. This configuration is the most difficult to analyze because of the many geometric factors involved. A piece-wise linear model is not sufficient to produce correct results, so we have to resort to computer simulation. But what parameters should be used? Stay tuned, it's complicated!
A race that begins on a ramp and ends on the same ramp is easy to analyze. The ramp equations of the race model derived in this manual are sufficient to model it. But the trajectory length equation, L1, must change. The change that is required is to make the trajectory length exactly equal to the ramp length! This is because the center of mass begins in front of the starting peg a distance of Nose + Wheel_Base - Center_of_Mass_x and finishes the race that same distance in front of the finish line.
The height of the center of mass is constant on the ramp and serves only to raise or lower the trajectory; it's angle above the track does not change and is equal to that of the ramp track.
Though, the center of mass position does not matter on a ramp track, other factors might help determine its place by default. For example, making a low profile for less air resistance will place the center of mass low. Consequently, if placed higher, air resistance is likely to be increased.
Unlike the ramp track, the trajectory of the car's center of mass on this track shape is not simple. It is shorter or equal to the total length of the track surface. It changes in angle, and is not strictly speaking a line at the transition. Other factors can be ignored to facilitate finding an answer.
One of these factors is air resistance. Although air resistance is usually greater on a car that has a higher center of mass, it need not be the case. Two cars can be built that have the same aerodynamic drag but very different center of mass heights. Yes, air resistance could possibly change the findings here to some extent, but as you will see, the change is not significant. We find the same to be true for all sources of friction!
Assuming no friction, the appropriate model equations can be solved for the best race time with respect to the placement of the center of mass. What we find is that
Of course, this is for no friction on the "zero" transition track. On a smooth transition track with friction, there are a few things that happen.
On the smooth transition, the front wheels are on a flatter slope than the rear wheels and so experience more friction than the rear wheels under equal axle force and friction coefficients. The difference factor is the cosine of the difference in slope. On a slope, the weight shifts toward the front axles as the weight is raised. As a result, raising the center of mass shifts weight to the front axle where the effect of friction is greater than it would be if the same weight shifted rearward. The result is more total friction - as the weight raises, the car meets with more resistance than it would otherwise. (This is one way in which a front-weighted car is generally disadvantaged.)
Also, as the weight is raised, the trajectory becomes shorter because the center of mass is closer to the loci of the centers of curvature of the track. On a semicircular track the center of mass is closer by delta CMh (dCMh) (CMh is the height of the center of mass) and its trajectory is shorter by
(R-CMh0)O - (R-CMh)O = dCMhO
for CMh0 the original value of CMh, R is the radius of the circle and O is the angle of arc subtended by the semicircular track in radians.
For dCMh = 0.75 in and O = 0.365, the trajectory shortens by 0.27 inches which is consistent with other calculations. At the usual speeds, this means shortening the time by nearly 0.002 seconds. I believe this equation approximates any concave trajectory delta given O, the ramp angle fairly well.
Both together indicate good reason to expect that to some extent, on some ramps, the higher center of mass might be better! That's what models are for! It would be difficult to isolate this behavior measured in thousandths of a second in real experiments. The higher center of mass leads to more friction on the transition. In general, friction does not change the behavior noted in the frictionless model, but should move the "break even" point down from the theoretical flat length of 4 times the ramp length.
With these considerations in mind, let's race 6 cars each with a center of mass in a different location, but otherwise being identical. Three cars will have high positions, three low. Two cars each will have a front position, two a middle position and two a rear position. The table reflects the center of mass coordinates used for each car.
|High||h:2.4 x:5.8||h:2.4 x:2.9||h:2.4 x:0.0|
|Low||h:-0.59 x:5.8||h:-0.59 x:2.9||h:-0.59 x:0.0|
Race Specification File: cm.xml =================== Race Standings =============================== Car Name Time Split Distance Speed Accel Lane Place ---------------- End Of Ramp ---------------------------- Front, Low 1.084 0.000 132.24 128.61 1 1 Front, High 1.084 0.000 132.24 128.61 2 2 Middle, Low 1.084 0.000 132.24 128.61 3 3 Middle, High 1.084 0.000 132.24 128.61 4 4 Rear, Low 1.084 0.000 132.24 128.61 5 5 Rear, High 1.084 0.000 132.24 128.61 6 6 ---------------- Transition 10ft 8in ---------------------------- Rear, Low 1.460 0.000 162.64 37.91 5 1 Rear, High 1.461 0.258 162.53 36.04 6 2 Middle, Low 1.462 0.284 161.06 32.48 3 3 Middle, High 1.463 0.557 160.87 30.55 4 4 Front, Low 1.463 0.589 159.42 27.07 1 5 Front, High 1.466 0.933 158.92 24.72 2 6 ---------------- Finish Line ---------------------------- Rear, High 2.949 0.000 144.33 -9.72 6 1 Rear, Low 2.952 0.524 144.65 -9.72 5 2 Middle, High 2.974 3.577 141.77 -9.85 4 3 Middle, Low 2.977 4.034 142.13 -9.85 3 4 Front, Low 3.002 7.435 139.87 -9.72 1 5 Front, High 3.002 7.462 139.21 -9.72 2 6
This is an interesting race! Do you see how all the cars were even at the end of the ramp? Then the low weighted cars took a few hundredths of an inch lead, only to be out paced at the end - except for the front weighted cars. It was the high front weighted car that would have the most drag on the transition, and it looks like that made the difference in the front-weight pairing. In reality, it would have been worse because the high-front weighted car would have raised its rear wheels off the track on the ramp and transition causing the front bumper to drag on the lane median.
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|Grand Prix Racing -||The Science of Fast Pinewood Cars|
|Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.|