|DOE Modeling Time Trial
|Pinewood Grand Prix
|2 November 2002
Primary: Michael Lastufka
Assistant: Arin Lastufka
This article contains the abstract and a brief version of the full report. The full report is available on request to those wishing to duplicate the experiment.
This time trial pitted five factors, wheel/body clearance (Cw), Frontal Cross-section (A), Nose Length (N), Wheelbase (B), and Center of Mass measured from the front (CMf), directly against one another and in paired combination using five settings each. Leveraging Design Of Experiment (DOE), simple mathematical models of these pinewood car performance factors emerged for two time trial cars in 30 configurations. In particular, the models detail how each factor or pair of factors contributes to speed and stability. Specially designed cars enabled factor isolation for adequate reduction of race time variation. Each factor passed a previous Screening DOE Time Trial.
Two final sets of runs with a new configuration checked the veracity of the models. They indicated along with statistical measures that the blue car model is a better predictor of actual times than the yellow model. Most of the blue car data fit noiseless theoretical time predictions slightly better than the actual average time model and both accurately predicted the 30th configuration race time.
The DOE method provided relatively simple models of both the noiseless theoretical races and the actual races. Comparison shows instabilities working through wheel play, and using the nose and wheelbase as levers. Air resistance proved a stronger factor than previously thought and wheel play insignificant. The factors of nose length, wheelbase and center of mass demonstrated interdependence and regions of instability. Some configurations should be avoided as unstable; notably ones with short wheelbases and front-weighted configurations with no short nose and long wheelbase.
This experiment follows the DOE Screening Time Trial of March 16, 2002. Of the ten pinewood car speed factors found to be significant, five were selected for this modeling study. 29 configurations of three runs each gathered data on five settings for each factor; a five factor, three-level experiment design. Wheels prepared using the Maximum Velocity Pro Hub wheel tool allowed the November 2, 2002 time trial to produce two valid mathematical models after two previous failed attempts.
In the DOE Screening Time Trial, it was noted that a few of the trial car configurations were not as stable as most. The unstable configurations arose from combinations of nose, base and center of mass settings. Thus, the three factors couple to some degree in unstable car configurations. This experiment showed the interdependence of such effects.
The investigators' special cars and assessories are adjustable to the required configurations while leaving the unadjusted speed factors undisturbed.
Two sets of times recorded the preformance of two adjustible five ounce cars in the DOE Modeling Time Trials. 30 speed factor configurations ran enough times to insure statistically valid results, one of which was a check for the mathematical model developed from the data.
One practical strength of the DOE methodology allows mixed factor configurations. This greatly reduces the number of trials needed. Randy Lisano, a certified Six Sigma Blackbelt (professional experiment designer), generated a set of 29 configurations for this modeling DOE time trial. The time trial car was reconfigured before each set of three runs.
A few of the configurations are pictured below. The descriptions only include visible factors, not wheel-body clearance.
Front weighted, short wheelbase
Centered weight, "normal" car
"Fast" check configuration: rear weight, short nose, long wheelbase, low profile
From the two sets of 30 configurations and three repititions each, 180 data points produced a statistical model for both the average time and race time variation of both time trial cars. The actual times and variations for each configuration were subtracted from the predictions of these models to determine how far off they were. If the models are good, the actual measurements and variation are highly correlated with the model predictions. The yellow car data was twice as noisy as the blue, but both still provided a significant average time model. Only the blue car had a significant time variation model.
In the end, the blue car average time model proved to be much more accurate predicting the 30th configuration time, but its race time variation model failed. One reason the yellow car average time model failed was that it predicted that the rear weighting of configuration 30 would slow it down; it didn't.
To study non-deterministic effects, like drift and median collisions, a deterministic prediction model, the RaceIt computer program derived from theory, produced a DOE style model for comparison. The predictive model agreed on stable configurations better than the DOE model produced from the data itself. The remaining configuration data were then analyzed for patterns and reasonable explanations for their deviations.
Finally, removing the unstable configurations from the data produced a DOE average time model much more like the deterministic RaceIt one.
Comparing deterministic theory to actual pinewood run data helped identify variation in run times due to non-deterministic behavior. This behavior has only a few causes through four interfaces, air flow and car, tread and track, lane median and inner wheel surface, and wheel / axle / body. Pressure from air flow, roughness of the track and lane median, poor alignment of interfaces, roughness in materials and off center or out-of-round wheels power the gyrations of non deterministic behavior.
The wheel bore pits itself against the axle and a small body surface; the only car parts in relative motion. Through this interface, replicated four times, forces act on the body and the wheels competing for control of the car's path. The resulting chaos, collision, shaking and rubbing rob energy from the car and increase the time needed to arrive at its destination. We found it necessary to tame this interface just to get data good enough to produce any significant results.
Vibrations from pressure drag act through the lever arms offered by the nose and wheelbase around CMf, subject to 'play' between the wheels and the body. Wheelbase provides the greatest leverage and becomes a prime source of time variation in an unstable car.
The blue car closely follows deterministic physical behaviors as long as it is stable. Unstable configurations become sensitive to irregularities in the track, the air and the wheel bore interfaces amplifying them through various levers (nose and wheelbase), joints (wheel/body clearance) and around fulcrums (center of mass). All of these effects showed up in the blue average time model and more so in the yellow car model.
One ever challenging goal of a pinewood racing team is to eliminate anything that causes instability in the car design. This study showed at least two types of configurations to avoid:
Surprisingly, the blue car analysis suggests that a stable car's interactions among its body levers around the center of mass decrease to zero and a linear time model emerges. Here is the extremely acurate, linearized, RaceIt DOE Model for the Blue Car in this trial:
predicted time = 2.8423766 + 0.02655 A - 0.007875 N - 0.0041875 B - 0.0082381 CMf
where A is the frontal cross section of the car in square inches,
N is the car's nose length in inches,
B is the wheelbase in inches and
CMf is the distance from the front of the car to the center of mass.
Note, a negative coefficient in front of a factor means it decreases the time when the factor is increased. Only increasing A increases the time, all other factors reduce the time when increased.
Compare this with the complicated closed form of the deterministic model in Tracking Down Solutions, http://www.lastufka.net/lab/cars/why/summry.htm
This formula will not likely apply exactly to any other pinewood car. Tracks and other factors not considered in this trial (like lubrication) will be different. However, the signs and size of its coefficients should be comparable to other stable pinewood cars.
The analysis of this experiment suggests that instabilities may be further reduced in the following ways though some may not be statistically significant.
|Lastufka, Michael, DOE Screening Time Trial Pinewood Grand Prix, 16 March 2002, Lastufka Labs available on request.