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Old 03-04-2012, 10:28 PM   #9 (permalink)
ERTW
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I'll just post some information here, instead of creating a new thread.

Wind tunnel testing is steady state, whereas coast down tests are in transient conditions. There's a ~1% difference between acceleration/deceleration runs and constant speed runs. Wind tunnels have the problem of localised turbulence. Cd and friction are not constant, however, he assumes that they are quadratic. Drag force is quadratic when Re is above 5x10^6 (roughly 60 kph).

There are some big caveats. He coasts one kilometre with measurements taken every 100 m (double integrate a*dt using the smart phone accelerometer). He ran the cars up to 130 kph down to 50 kph. If there is a head wind over 0.5 m/s (1.8 kph), or a cross wind causes a yaw greater than 5* the data point is rejected. Everything must be up to operating temp, so that friction changes minimally with each run.

"coast down tests give reliable results if the number of tests is adequate for a statistical analysis." He did 35-45 runs. Morelli saw between 0.22 and 1.56% variation from wind tunnel Cd figures (44 and 35 runs, respectively). I can't imagine how accurate 3 runs would be...with a cross wind...on a grade

the basic governing equation is:

-m' dV/dt = A + BV^2

m = momentum
V = velocity
t = time
A = coefficient of the resistance - constant term
B = coefficient of the resistance - quadratic term

A = Wsin(g) + R

W = vehicle weight
g = the constant slope of the road
sin(g) accounts for the component of weight in the direction of motion. hilly terrain needs a d/dt
R = mean resistance: tires and driveline (found from analysis)

B = Cx'*1/2*rho*S
C'x = 2B/rho*S

rho = air density = sqrt(gRT) plus correction for relative humidity
S = frontal area

Cx' = Cx + Cr + L/ro Sum (i=1 to n) Cwi

Cx' = mean drag coefficient (found using statistical analysis)
Cx = drag coefficient (calculated)
Cr = rolling resistance coefficient, and includes wheel induced drag (measured with a special rig)
Cwi = ventilation resistance coefficient of the rotating parts (measured on a dyno)

The car needs to coast over a long enough time interval (about a minute) so that we can safely assume that most of these factors are independent of time.

the L/ro... term ranged from 0.01165-0.0119 and his cars had fairly closed wheels. Since this is the ventilation term, moon discs will assure consistent values from car to car - and let us use 0.01165 with small error.

His Cr ranged from 0.062-0.072 for the cars he tested. If we blindly use 0.08, it represents a 4-29% error in Cx (wrt to his circumstances). His baseline was Cx 0.457...so the more aerodynamic the car, the worse the assumption. However, since this method is very close to wind tunnel testing, we can do baseline testing with the published Cd and calculate Cr. It's a lot of work, and without Cr we cannot know Cx.

I hope this is enough to chew on until I post the heavy mathematics.
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