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

- 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).
- The spreadsheet I created to analyze the results. Download here: Drag_Coefficient.xls

## 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.

7. Download the spreadsheet I created Drag_Coefficient.xls and enter all your data following the instructions included. The spreadsheet averages data from all 6 trials to create a single data set representing velocity (V actual) as a function of time. It then generates it's own model for velocity (V model) based on entered constants and initial guesses for Cd and Crr. Excel's "Solver" function can be used to adjust Cd and Crr in order to minimize the error between the model and actual data. If you are using OpenOffice Calc (which I highly recommend and which you can download for free from http://www.openoffice.org), unfortunately, the solver function currently only handles linear systems, so you will have to adjust the input values manually to minimize the error between the model and the data. Once the error is minimized and the model data matches the actual data as best it can, then Cd and Crr are correct.

## 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 V^{2} 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 V^{2} and that the force of rolling and drivetrain resistance is constant. In reality the force of rolling and drivetrain resistance is also related to V^{2} and V. So a better model of the force on a moving vehicle is:

F = iV^{2} + 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.

## 65 Discussions

10 years ago on Introduction

Hi really good effort.. I have some doubts in that!! 1.How u calculate the frontal area of the car( Near to windshield area or Front bumper area) 2. how u calculate Cd & Crr .please give explanation.

Reply 10 years ago on Introduction

1. The frontal area is the cross-sectional swept area of the car. Imagine you drove your car through a snow bank. The frontal area is the area of the hole that it would leave behind. There are lots of ideas for how to measure it accurately. Try doing a search on "measure frontal area" in the forums at gassavers.org or ecomodder.com. For my car I just made a rough estimate based on simple measurements. 2. Technically I did not "calculate" Cd and Crr. I determined them by trial and error. I guessed some initial values. Using Excel I plotted a curve of what the velocity of the car would do based on those values. Then I adjusted the values (using Excel's solver function) until the predicted velocity curve matched the actual data. All the formulas are given, and you can download and check the Excel spreadsheet to see how I used them.

Reply 3 months ago

I'm confused like where did u get the model values on that graph and also how to determined by trail n error

actually my graduation project is based on coefficient of drag

i'm going to build up a car model in software and then in reality and then i have to compare the software reults with actual

it would be really kind of u if u can help out

4 years ago on Introduction

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.

4 years ago on Introduction

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.

5 years ago on Introduction

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!

5 years ago on Introduction

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! :-(

5 years ago on Introduction

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 corneringhttp://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

7 years ago on Introduction

This is a great post mate. Thanks for sharing this to car enthusiasts.

7 years ago on Introduction

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

8 years ago on Introduction

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.

Reply 8 years ago on Introduction

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.

9 years ago on Step 4

Using a GPS device would greatly simplify the whole thing.

10 years ago on Introduction

Hi, I think there is a mistake in your car's frontal area. All sites that I found said it is about 18 ft

^{2 or about 1.8 m}2. 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

Reply 10 years ago on Introduction

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 V

^{2}.The model assumes the drag force is related only to V

^{2}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 V^{2}and V.So a better model of the force on a moving vehicle would be:

F = iV

^{2}+ 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.

Reply 10 years ago on Introduction

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 v

^{2}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.Reply 10 years ago on Introduction

I would like to see your spreadsheet. I'll private message you.

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 V

^{2}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.

10 years ago on Introduction

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?

Reply 10 years ago on Introduction

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*V

^{2}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.

Reply 10 years ago on Introduction

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

dv/(v

^{2}+ K1^{2}) = K2dtwhere 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 = -K1

^{2}K2/g.I modified your Excel spreadsheet accordingly. Using your data, the process converged upon

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.