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Old 07-12-2008, 12:07 PM   #1 (permalink)
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help with simplified cDA/cRR formula

I saw a coastdown calculator and wanted some help understanding it before I try and integrate it into the mpguino.

Basically you (or your computer) times a coastdown at high speed (say 55mph to 50mph) and another at low speed (say 7mph to 2mph) and put the speeds and times and weight in and it tells you your cDA and rolling resistance. But I'd like to understand what it is doing a little better. I appreciate that they have approximated things like pressure and temperature for the sake of simplification, but can anyone determine what the "6" on the cDA line represents or the "28.2" on the rr line actually is?
The crux of it is:

a1=(va1-vb1)/t1;
a2=(va2-vb2)/t2;
v1=(va1/2)+(vb1/2);
v2=(va2/2)+(vb2/2);
cDa=((6*m)*(a1-a2))/(v1*v1-v2*v2);
rr=(28.2*(a2*v1*v1-a1*v2*v2))/(1000*(v1*v1-v2*v2));

abbreviations:
a1: high speed deceleration
a2: low speed deceleration
v1: high speed average velocity KPH
v2: low speed average velocity KPH
cDA: coefficient of drag * area
rr: Rolling resistance
va1: high speed start KPH
vb1: high speed end KPH
va2: low speed start KPH
vb2: low speed end KPH
m: mass in Kilograms

Other conversion factors
1 mile = 1.609344 KM
1 pound = 0.45359237 kilograms

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Last edited by dcb; 07-12-2008 at 01:10 PM..
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Old 07-13-2008, 07:21 PM   #2 (permalink)
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shameless bump (maybe this belongs in instrumentation?)
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Old 07-14-2008, 03:16 PM   #3 (permalink)
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Quote:
Originally Posted by dcb View Post
I saw a coastdown calculator and wanted some help understanding it before I try and integrate it into the mpguino.

Basically you (or your computer) times a coastdown at high speed (say 55mph to 50mph) and another at low speed (say 7mph to 2mph) and put the speeds and times and weight in and it tells you your cDA and rolling resistance. But I'd like to understand what it is doing a little better. I appreciate that they have approximated things like pressure and temperature for the sake of simplification, but can anyone determine what the "6" on the cDA line represents or the "28.2" on the rr line actually is?
The crux of it is:

a1=(va1-vb1)/t1;
a2=(va2-vb2)/t2;
v1=(va1/2)+(vb1/2);
v2=(va2/2)+(vb2/2);
cDa=((6*m)*(a1-a2))/(v1*v1-v2*v2);
rr=(28.2*(a2*v1*v1-a1*v2*v2))/(1000*(v1*v1-v2*v2));

abbreviations:
a1: high speed deceleration
a2: low speed deceleration
v1: high speed average velocity KPH
v2: low speed average velocity KPH
cDA: coefficient of drag * area
rr: Rolling resistance
va1: high speed start KPH
vb1: high speed end KPH
va2: low speed start KPH
vb2: low speed end KPH
m: mass in Kilograms

Other conversion factors
1 mile = 1.609344 KM
1 pound = 0.45359237 kilograms
Well, the numbers that you have quoted are for 'metric' units. Read on for explanations...

In standard metric units (mks - metres/kilograms/seconds) there should be 2/1.2 in stead of "6" and 1/9.8 instead of "(28.2.../(1000...".

So, re-written:
cDa=((2*m)*(a1-a2))/((1.2)*(v1*v1-v2*v2))
rr=(a2*v1*v1-a1*v2*v2)/(9.8*(v1*v1-v2*v2))

From the (hand-derived) math:
- the "2/1.2" is the reciprocal of the standard 0.5*'rho' ['rho' being the density of air @ 1.2 kg/m^3]
- the 1/9.8 is the reciprocal of acceleration due to gravity [9.8m/s^2]

As you have stated the speed units as km/h, there are factors of 3.6 (1 km/h = 1000 m / 3600 s = 1m/3.6s). So, used appropriately, one will achieve the original equations with the "6" and "28.2".

And, as you have already suspected, the results will be an indication of real values but not absolute. I don't feel like doing the calculus to get the exact results - too time consuming and the external influences when taking the readings will not justify its precision!

Also note that there is another component that is often overlooked: a 'viscous' drag - a velocity component in the equation:
F = 'rho'/2*cDA*v^2 + cV*v + m*g*cF

BTW = this is bringing back memories of my master's thesis...
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Old 07-15-2008, 12:21 AM   #4 (permalink)
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Thanx CO, it sounds like it might be good enough for our purposes then. I hope to take a lot of the timing/speed variables out of the equation by having the computer do all that, but I also didn't want to make the user jump through too many hoops about current temp/humidity/pressure/locust density/etc
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Old 09-04-2008, 02:54 PM   #5 (permalink)
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Viscous Drag

Quote:
Originally Posted by co_driver View Post
Also note that there is another component that is often overlooked: a 'viscous' drag - a velocity component in the equation:
F = 'rho'/2*cDA*v^2 + cV*v + m*g*cF
Is there a simple way to get the viscous drag component? I have seen where people get the rolling resistance by warming up their car and stopping in a level area (parking lot), and then either push or pull the car with a scale to see what the resulting rolling resistance would be. Would a person be able to get the rolling resistance this way, and then calculate the viscous drag from the equation above? It should work. Is there a more simple way? Does viscous drag account for much, or is it able to hide in the cDA component ealily?
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Old 09-04-2008, 04:23 PM   #6 (permalink)
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Quote:
Originally Posted by wyatt View Post
Is there a simple way to get the viscous drag component? I have seen where people get the rolling resistance by warming up their car and stopping in a level area (parking lot), and then either push or pull the car with a scale to see what the resulting rolling resistance would be. Would a person be able to get the rolling resistance this way, and then calculate the viscous drag from the equation above? It should work. Is there a more simple way? Does viscous drag account for much, or is it able to hide in the cDA component ealily?
As one is bound to find (from all of the discussions in this forum), there are no simple solutions. But there are rules of thumb.

In a spreadsheet that I have developed (its a beast!), I do include the viscous component - and guestimated it to be ~1 for a 2-wheel drive vehicle (where F(viscous) = cV*v (v in [m/s]). It could be ignored, but will have an influence on the standard pair of coast-down tests (high speed and low speed). Hint: (greater than) 3-speed coast down tests will show this minor factor (did this in uni many years ago w/ driveshaft torque measurement and data logger).

This is notably small when compared to the other two inputs:
eg. '90 Firefly:
@ 25 m/s:
F(friction) = 81.3N (cF = 0.010, m = 830kg)
F(viscous) = 25.0N
F(aero) = 236.3N (cD = 0.36, A = 1.75m^2)

[Disclaimer: YMMV, above numbers may or may not simulate real life, ...]

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