# Measure the Drag Coefficient of Your Car

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## Introduction: Measure the Drag Coefficient of Your Car

The purpose of this experiment is to determine your vehicle's drag coefficient Cd and coefficient of rolling resistance Crr. This is done by measuring your vehicle's speed as a function of time while coasting in neutral.

Why would you want to know Cd and Crr for your vehicle? Well, suppose you're interested in modifying your vehicle for improved fuel efficiency. You might consider modifications such as air dams, wheel skirts, removing mirrors, switching to low rolling resistance tires, etc. Cd and Crr offer a quantitative method of comparing vehicle performance before and after these types of modifications to see if you made any improvement.

For other experiments you can do on your car see my website IWillTry.org.

## Step 1: Equipment

You will need the following equipment:
• a vehicle (and someone with a driver's license)
• a clock or stopwatch
• a pen and paper (and someone other than the driver to record data)
• a flashlight (driving at night avoids traffic)
• a long stretch of flat road with little traffic or wind
• Excel or another spreadsheet application. I prefer OpenOffice Calc because I like to support open source software, but its Solver function does not handle non-linear systems (yet) so you'll have adjust input variables manually by an iterative process to fit your model to the data (it's not too hard).

## Step 2: Background Information

First, let's define some quantities:

Fd is the force on the vehicle due to air resistance (drag) in Newtons
Frr is the force on the vehicle due to rolling resistance in Newtons
F is the total force on the vehicle in Newtons
V is the vehicle's velocity in m/s
a is the vehicle's acceleration in m/s^2
A is vehicle frontal area in m^2
M is vehicle mass including occupants in kg
rho is the density of air which is 1.22 kg/m^3 at sea level
g is the gravitational acceleration constant which is 9.81 m/s^2
Cd is the vehicle's drag coefficient we want to determine
Crr is the vehicle's coefficient of rolling resistance we want to determine

Now for some formulas:

Fd = -Cd*A*0.5*rho*V^2 (formula for force due to air resistance or drag)
Frr = -Crr*M*g (formula for force due to rolling resistance)
F = Fd + Frr (total force is the sum of Fd and Frr)
F = M*a (Newton's second law)

Note that both Fd and Frr are negative indicating that these forces act opposite to the direction of the velocity. Note also that Fd is increases as the square of velocity. This is why driving at high speeds is much less efficient than driving at low speeds. Combining these formulas with a bit of algebra gives us the acceleration due to air and wind resistance as a function of velocity:

a = -(Cd*A*0.5*rho*V^2)/M - Crr*g

Note that the acceleration is negative indicating that air and wind resistance will cause the velocity to decrease.

I created a spreadsheet based on these formulas to generate a model of velocity vs time that can be compared to actual data. The model values for Cd and Crr can thus be adjusted until the model matches the data. This adjustment can be done manually, by overwriting the values of Cd and Crr with new values till the model matches the data, or it can be done using a "Solver" function.

## Step 3: Procedure

You can determine Cd and Crr from the same set of test data by measuring velocity with respect to time as your vehicle coasts in neutral. Note that Crr will not be pure rolling resistance but will include some drive-train resistance as well.

1. Drive to a flat road with little traffic or wind.

2. Have the passenger ready with stopwatch and paper to record data.

3. Have the driver accelerate up to above 70 km/h or so, and shift into neutral.

4. Record data as follows. The driver should indicate when the speed drops to exactly 70 km/h. At this time (t=0) the passenger should start the clock. The passenger should indicate every 10 seconds after that and the driver should call out the current speed to the nearest whole km. The passenger should record this value next to each time.

Aside: If you have a digital camera capable of recording several minutes of low resolution video (as most people seem to have these days), the process is much easier and more accurate. You don't need any equipment except the digital camera. Simply have your passenger record a video of your speedometer during the coast down tests, or find some way of mounting the camera so you can do the recording without an assistant. Using a free program such as Avidemux (http://fixounet.free.fr/avidemux/) you can play the video back on your computer frame by frame and view the timestamp at desired speeds.

5. Repeat the test in the opposite direction.

6. Repeat the test in both directions twice more (6 trials in all, 3 in each direction). All these values will be averaged for a more accurate analysis.

## Step 4: Results

Here are the quantities I measured for my car (a 1992 Geo Metro):

M = 1000 kg (about 850kg curb weight plus 150 kg of occupants)
A = 2.3 m^2 (a good approximation based on measurements of my car)

A plot of velocity vs time is shown below. It is based on the averages from my 6 trials. You can see that the model curve closely matches the data points. The values of Cd and Crr for the model are:

Cd = 0.370
Crr = 0.0106

Therefore, these are the drag coefficent and coefficient of rolling resistance for my car.

These values are nice to know. However, in practice, if you want to compare performance before and after making modifications to your car, you can get faster results just by measuring the time to decelerate from speed A to speed B. Pick high to medium speeds if your modifications are likely to affect drag. Pick medium to low speeds if your modifications are likely to affect rolling resistance. Don't forget to take multiple measurements in each direction and average the results.

For more experiments you can do on your car see my website IWillTry.org.

Update 2009-01-02:
I've learned a lot since originally posting this instructable 16 months ago. I've played with measuring Cd and Crr under different conditions on a number of vehicles and other experimenters have picked apart and tweaked my spreadsheet for their own uses.

My experience is that there IS a mistake in one of the underlying assumptions of the model: namely that the force of rolling resistance is constant independent of V. Vehicles are designed with negative lift (so they get pushed into the road more at higher speeds, improving handling) so the force of rolling resistance also has a component that varies with V2 like the drag force. The force of rolling resistance also includes a small component of viscous force (drivetrain) which varies with V.

The model assumes that the drag force is related only to V2 and that the force of rolling and drivetrain resistance is constant. In reality the force of rolling and drivetrain resistance is also related to V2 and V. So a better model of the force on a moving vehicle is:

F = iV2 + jV + k where i, j, and k are constants.

A curve based on that model more closely matches actual coast down data indicating it is a more accurate model. But after solving for i, j and k, there is no way to extract meaningful values of Cd and Crr since by definition, they assume i is related only to drag, and j is 0, neither of which is entirely true.

As mentioned above, If you want to compare performance of a vehicle before and after making mods, the change in coast down time itself is MUCH more meaningful than any change in Cd or Crr extracted from the coast down data.

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## 64 Discussions

I thought of the same trick years ago and am glad I'm not the only one to ponder such things. Because high speed does reveal drag better, I also considered this test: find a straight smooth road that's rarely traveled. Drive straight at about 82 mph. Have a video camera, with on-screen hh:mm:ss showing, shooting a gps. Let off the gas. Go back and watch the video. How many seconds did it take to coast down from say 80 to 40.

Top mph and mpg are influenced by drivetrain friction and rolling (tire/wheel) resistance, so leaving them in the equation isn't that bad. Another thing this factors in, is front area + Cd, or total drag, contrary to the title of the article.

Myself, I'd like to eliminate human error, even without high speed. Go to a tall hill, release the brakes with aforementioned gps / camera, and note the stopping location/time achieved. I'd do this with a super sleek rental car, and a blunt one. Look up the Cd in R/T or carfolio and interpolate to estimate your car, based on the time/distance you observed. Try to establish the baseline with semi boring cars. The interesting cars are more likely to have bogus Cd numbers.

Sorry if it was mentioned, but there are two very critical ideas I didn't see, scanning the feedback. One is turns. If the best hill you have features a subtle curve, try your best to take it smoothly and consistently. Second one is road surface. If you establish a baseline 12 years ago, and take a project car to the same hill, be on the lookout for different quality of pavement. Whether you coast, tow, or down-hill roll, a substantial difference in pavement smoothness (because of repaving) means all bets are off.

SCT has a free data logger/analyser from which you should be able to calculate this. Also, though I've not tried it yet, DashCommand.app (works with any ELM327 OBD2) has just added a logger to their app. Check the App Store to see if there are any more apps that have an OBD2 interface.

How could I "emulate" Excel solver using VBA or Javascript? If I could figure out this, I think using AppInventor I could actually create the android app which calculates the Cx and Crr I was talking about!

That's all we need to build a smartphone app which, using onboard GPS data, can calculate Cx and Crr.

Unfortuately I am not able to write it! :-(

Wow, Wow, just wow! What's going on here, the speculation on the effects of tire pressure and rolling resistance is out of control. I know this is an old topic but I can't just let this stuff lie.

Your post here and and aero civic are great by the way. Job well done and I admire your scientific approach to everything. But this tire business is irking me. I'll add as many references as I think are required but most of this I learned from print books.

TIRES!

Increased tire Pressure:
Reduces Rolling Resistance (Duh!)
Increases fuel economy (all the way to bursting point)
Decreases chances of hydro planing
http://en.wikipedia.org/wiki/Contact_patch

This is just simple physics The rest of the factors that are being discussed here have a lot more factors than just tire pressure involved

Firstly, Manufacturer tire pressure ratings are based on a multitude of performance factors, many of which have not been discussed here. Safety, cabin noise, vibration, tire compound, tire tread, tire geometry, car load, car suspension, intended car speeds, longevity, consumer market, typically expected road surfaces of main market, the list goes on. no one tire pressure is optimal regardless of what's printed on your car. These pressures just optimise a number of these factors. They are not safety limits, by any margin and 10-15% difference wont start causing crashes. Most people can barely notice a 30% change. (You just need to walk around a local parking lot to see how many people have no idea what's going on with their tires)

Tire pressure is not the only factor affecting rolling resistance and hence fuel economy. Check this out: http://en.wikipedia.org/wiki/Rolling_resistance
Anyone tempted by slick tires? Hard compounds? Weird tread patterns? Narrow tires? Different sidewall thickness? Heavier tires? Lighter tires? Different gases for inflation?

Braking and cornering
http://www.turnfast.com/tech_handling/handling_pressure
Mainly a function of 'contact patch' of tire. There's no simple answer for 'square' car tires and the sheet of paper with paint tactics mentioned here have a massive margin of error. Get a temperature gauge and have a look at the first link here if you want to start being scientific. On ashphalt, there is an optimum pressure to maximise the contact area between tire and road. Too low is as bad as too high. on rough surfaces: the lower the better. make the rubber flex into all the contours of the surface. This applys to both acceleration and braking but negatively affects cornering.
ABS systems skew all this a bit (If anyone's interested ABS systems affect all this in a variety of ways, some very good others very bad, like ice!)
Ice is a category all on it's own, whole different ball game and different ways to deal with it.

For the high pressure guys and their sceptics: this guy runs max pressure on his tires and measures the wear. http://ecomodder.com/forum/showthread.php/measured-my-tire-wear-20-000-miles-elevated-10265.html nothing out of the ordinary after 22k miles

Mythbusters tackled the fuel economy myth but didn't really deal with the safety issue other than to say it's not recommened (true scientists as always, sigh) http://ecomodder.com/forum/showthread.php/mythbusters-tests-tyre-tire-pressure-17151.html

to the author
"A simple proof: Suppose the coefficient of friction is 0.8 (typical of rubber on concrete). Let's compare a 1x1 inch square of rubber with 100 lb of weight evenly distributed on it to a 10x10 inch square of rubber with 100 lb of weight evenly distributed on it. In the first case, each square inch of rubber supports 100lb so each square inch can tolerate a shear force of 0.8*100=80lb before slipping. Multiplying this by the number of square inches (1) gives a maximum total shear force of 80lb. In the second case each square inch only supports 1lb, so each square inch can tolerate a shear force of 0.8*1=0.8lb before slipping. Multiplying this by the number of square inches (100) gives a maximum total shear force of 80lb. I say again... the size of the footprint makes no difference to the force that can be transmitted without slipping."

Yes, this is Amontom's 2nd law of STATIC friction. But car tires roll and can't be analysed in this way. Dynamically, the coefficient of friction changes according to tire load (which is not static, unless the car is. The load shifts from tire to tire with every twitch of the car) and contact patch is a factor both parallel and tangential to the wheel rotation. Here's an intro: http://en.wikipedia.org/wiki/Frictional_contact_mechanics
Conventional ideal system kinematics don't apply here. It's best just to look at reliable models for reference, F1 or motorsport enthusiasts for instance. Car and tires manufacturers never release useful data unfortunately http://technicalf1explained.blogspot.ie/2012/10/f1-tirespart-2.html

to leroy:

"So I guess the conclusions to be drawn are: (1) the larger the footprint, the less likely you are to wear out the tires (2) The size of the footprint has no effect on the rolling friction (friction retarding the car's motion in the nonskidding condition) or on the locked-up-brakes friction (skidding condition) (3) the size of the footprint also has no bearing on how long it takes to brake to a stop--for either the skidding or nonskidding case I had a look at your website."

Are you seriously suggesting we should put the narrowest tires possible on our cars, as small as can bear the car weight and this will have no effect on rolling resistance or braking distance? That race car engineers are mad for putting 12 inch wide tires on their cars. Think about that for a minute

If you video record your speedometer, then you don't have to do the timing and data recording while driving.

My biggest challenge is estimating the rolling resistance of my tires. I wish I had been able to do a test before making the mods, to calibrate things, because I know the "factory" Cd.

Oh, and if you take a frontal photograph with a reasonably long (telephoto) lens, with a measuring tape that is readable at the high point of the roof, you can insert it into a CAD program (I used DataCAD) and then trace the outline, for a pretty accurate frontal area. I included the side view mirrors, even though i believe these are excluded from the "factory" number. (One of my mods is to replace the mirrors with video cameras.)

Sincerely, Neil

I wonder if you may have better results measuring the drag due to air resistance if you started your coastdowns at a much higher speed, since at high speeds, most of the total drag is from the air.  I might start the coastdowns at something like 100 mph, then stop at around 50 mph.  This may help isolate the Cd from the Crr more effectively, since Crr is much closer to a constant force, where Cd is an exponential force.

Definitely the results will be more accurate coasting down from higher speeds, for exactly the reasons you say. But there are few places (at least near me) with level ground where one can safely do coast down testing from such high speeds.

Using a GPS device would greatly simplify the whole thing.

Hi, I think there is a mistake in your car's frontal area. All sites that I found said it is about 18 ft2 or about 1.8 m2. I think you might have added the area between the ground and the car. Here are some of these sites:
http://www.mayfco.com/geo.htm
http://www.evconvert.com/tools/evcalc/ (This calculator is not accurate)
Wikipedia Geo Metro has dimensions

So I put the correct area in your Excel file and it gave me a Cd of .47 and Crr .010. So it seems to me like there is a mistake here. I also found a PDF that shows equations for what you are doing, maybe it will help:

http://physics.technion.ac.il/~rutman/car/Roll-down%20test.pdf

3 replies

Thanks for the correction. I didn't measure the frontal area with any great accuracy, just a tape measure and some quick estimates so I would not be surprised if it is off. I've played with measuring Cd and Crr a bit more since originally posting this instructable and other experimenters have picked apart and tweaked the spreadsheet for their own uses so I'm 99.9% confident there is not a mistake in the spreadsheet.

However, I believe there IS a mistake in one of the underlying assumptions: that the force of rolling resistance is constant independent of V. The rolling resistance measured in a coast down test includes a component from viscous forces (drivetrain) which vary with V. Vehicles are also designed with negative lift (so they get pressed into the road at higher speeds for improved handling) so there is also a component of force from rolling resistance that varies with V2.

The model assumes the drag force is related only to V2 and the force of rolling and drivetrain resistance is constant. The reality is that the force of rolling and drivetrain resistance is also related to V2 and V.

So a better model of the force on a moving vehicle would be:

F = iV2 + jV + k where i, j, and k are constants.

A curve based on that model will much more closely match actual coast down data. But after solving for i, j and k, there is no way to extract meaningful Cd and Crr values since they are part of a completely different model that assumes i is affected only by Cd, and j is always 0, neither of which appear to be true.

To illustrate the point, if you do a coast down test while holding your doors open (which should affect only Cd and not Crr) you'll find that the indicated Crr changes too. That is a clear indication that the model itself is lacking. For this reason I don't advise relying on Cd or Crr values calculated from coast down data. If you really want an accurate Cd value you need to eliminate rolling resistance and viscous forces from the test (think wind tunnel). But by the same token, knowing an accurate Cd value isn't particularly useful. It will allow you to calculate the force on your car in a wind tunnel but it won't allow you to accurately calculate the force on your vehicle on the road.

If you want to compare the performance of a vehicle before/after making mods, the change in coast down time itself is MUCH more meaningful than any change in Cd or Crr extracted from the coast down data.

My analysis of your data shows a very nice linear relationship between arc tan (v/K1) and time which supports your original model where force is proportional to v2 and to a constant. Maybe you should look at my modification of your spreadsheet. Another way to measure rolling resistance would be to measure the angle of repose. That would require putting the vehicle on a ramp that could be inclined by jacking it up until the vehicle begins to move. It's easy to determine the component of the vehicle weight projected onto the ramp which would propel the vehicle forward.

It may be that the viscous force proportional to V is small, but there may still be a component of the force of rolling resistance that is proportional to V2 due to negative lift at high speeds. This should result in an overestimate of Cd and an underestimate of Crr which seems to be the case (the Cd spec for this vehicle is supposed to be around 0.36)

It would be useful to compare the rolling resistance calculated to that measured by inclined plane as you suggested, or by pulling the vehicle at constant speed (a walking pace) with a spring scale on level ground.

More on topic, do you know how to solve for Cid and Crr as a curve? I made a program to count my VSS and injector pulses and extrapolate MPG, etc, etc, and I want a button I can hit to say start timer when I drop below 60MPH and end at 30MPH and do a coast. It will record exact speed and exact time. I can simply plug back and forth with trial and error but it's proving really annoying to basically rebuild the 'Solver' on my own. Is there some integration or some formula to solve the entire curve?

3 replies

To match the entire curve you would have to write your own "solver" equivalent (ie a recursive function to keep tweaking Cd and Crr values until the error between the model and the actual data is as low as possible).

But you can get pretty good results without matching the whole curve. Rearranging the formula for drag and rolling resistance gives

M*a = -Crr*M*g -Cd*A*0.5*rho*V2

You can look at a 1 second interval near 60 MPH and another 1 second interval near 30 MPH. Using your VSS data you can calculate "V" and "a" in each case. Everything else is known except Cd and Crr. You will have two equations (one for 60MPH and one for 30MPH) and two unknowns (Cd and Crr). Use basic algebra to solve for Cd and Crr in terms of known values.

I have separated variables leading to an equation of the form:

dv/(v2 + K12) = K2dt

where K1 and K2 are constants:

K1 = sqrt(2mgCrr/(rhoACd)), and

K2 = -0.5rhoACd/m

This was integrated over v from v0 to v and over t from 0 to t. The result was:

arc tan(v/K1) = K1K2t + arc tan(v0/K1)

The assumption of values for Cd and Crr, establishes K1 and K2. Then, plotting arc tan(v/K1) versus time should produce a straight line with slope K1K2 and zero intercept arc tan(v0/K1). The least squares fit will produce estimates of slope and intercept which can be used to calculate K1 and K2. We have thus created an iterative process which should converge upon an estimate of K1 and K2. The definitions of K1 and K2 can be then used to establish the corresponding estimate of Cd and Crr. By rearranging those definitions,

Cd = -2mK2/(rhoA) , and

Crr = -K12K2/g.

Cd = 0.3902 and Crr = 0.01079 .

Then I used a guessing approach like yours to find the coefficients resulting in minimum error in the least squares fit. The results were:

Cd = 0.4261 and Crr = 0.01022 .

These differ slightly from your results:

Cd = 0.3697 and Crr = 0.01057 .

I don't know which is the best estimate of the coefficients. It's hard to argue with iterative convergence though. Maybe they are close enough to each other that it doesn't matter.

The modified Excel spreadsheet is available to anyone. I wrote it using manual iteration, since I don't know how to do recursive programming in the Excel language. (It's easy in other languages.) Also, I have a Word document that may make the above a little clearer. I'm a little restrained by the format capabilities here.

Thanks for your post. I suspect your results are more accurate than mine since I believe you've avoided the "quantization" error inherent in my simulation (my model looks at 5 second time slices, and assumes constant acceleration = F/m within each time slice). A quick modification to expand my table to use 1 second time slices yields values of Cd = 0.393 and Crr = 0.01058. My model also assumed an initial velocity equal to that of the first data point. Removing that constraint and using Excel's solver function to find the initial velocity for best fit, along with the values of Cd and Crr yields Cd = 0.4047 and Crr = 0.01039, values much closer to those you determined.

In any case, since posting this instructable, I've come to the realization that the assumed model, despite its wide acceptance is a poor one. See update I just added to the end of the instructable. Therefore I don't have much faith in the values generated by this spreadsheet except as extreme approximations.

Thank you for the nice instructional for measuring drag and rolling resistance coefficients. I haven't tried them yet, so cannot comment on their accuracy.

Overinflating vs. safety:

The main consideration is the footprint of the tire. The footprint is the amount of rubber in contact with the asphalt. It is the flattened bottom of the tire as it sits on the asphalt.

Why is the footprint so important? When force is exerted on the pavement to change the car's motion--when you hit the brakes, for example -- the force must be transmitted through the footprint to the tires and to the mass of the car. Obviously, if the footprint is partly ice, the car will go sliding when you hit the brakes or otherwise try to accelerate (in the physics sense of "change its inertia") it.

Consider the car sitting on the pavement. Its footprint is a certain size, in square meters. Consider the car driving on the pavement at 70 km/hr. The axle is moving forward at 70 km/hr.
The bottom of the tire is rolling on the pavement. Therefore its speed relative to the pavement is zero km/hr. (If it had speed relavative to the pavement, it would be skidding and would leave black rubber on the pavement. ) The tire is spinning so very, very fast to make this speed of zero on its bottom possible. So the speed of the tires' footprints is still zero km/hr.

Hence, so long as the tire air pressure hasn't changed, and the car's weight hasn't changed, then the tires' footprints are EXACTLY the same size as
they were with the car sitting at rest.

I think you can see that the larger the tires' footprints are:

(1) the easier it is to accelerate the car (e.g. braking to a stop) without something breaking

(2) the greater the car's coefficient of rolling resistance

(3) the less shear force each square cm of tire will have to bear when subjecting your car to a given acceleration (again, in the physics sense of changing your car's velocity)

The larger the area in contact with the pavement, the safer the car will be, at the expense of fuel efficiency.

Now the relevant question is, as you change the air pressure in the tires, does the tire's footprint change significantly, or does it remain about the same?

I don't know the answer. I think it would probably depend on the particular tire. However, using the principle I stated above, it would be EASY to test
whether or not the tire footprint changes with tire air pressure.

First equalize the air pressure in all four tires and measure it with a tire pressure gauge. Record the air pressure.

Get a big piece of cardboard, like a cardboard box from a refrigerator, and a can of spray paint and a Sharpie marker. (You can get cardboard for free from any retail distributor or from neighbors.) Park the car so its tires rest on the cardboard. Lie down on the cardboard with your spray paint. Spray the cardboard completely around each tire. Then move the car forward and trace the outline with your Sharpie. Mark the four outlines with #1 to signify your first air-pressure test.

When the paint dries, you're ready for your second test. Change the air pressure in your tires--either inflate it to higher pressure or let some air out. Record the air pressure. Then flip the cardboard over and repeat the test.

You can check as many different pressures as you want to see if the tire footprint changes size with air pressure--until one of your tires bursts, hahaha!