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 Measuring the forces on your car

 Reducing forces by 20% reduces your fuel costs 20%

If there were no forces tugging at your car on the freeway, you could maintain freeway speed with a bike pedals. But there are at least three strong forces at work: aerodynamic drag, rolling resistance, and drivetrain friction. The sum of these three can be hundreds of pounds and countering them requires considerable power.

By changing the car and/or our driving habits, each of these forces can be reduced. Without knowing the relative strengths of the forces, however, it's difficult to prioritize your attack on the problem or to measure the results of your changes.

This note describes a quick way to separate and quantify the forces involved.


   wiki's: lse, pivoting, 
   list of cars and their Cd x A values. sq ft   
   spreadsheet for measurements.
   drag equation
   drag coef
   "The Physics of Automobile Energy Consumption", D.Koon, St.Lawrence Univ, July 2005.
       force calculations for connection to whpm. 3000 lb car example.
   Least Squared Error links, glossary
   conventions. x vs *
   program to solve LSE for this problem.
   glossary about EVs, forces etc

The a,b,c's of forces

It's difficult and costly to measure the individual forces affecting a moving car but it's easy to measure the deceleration which, in turn, can give us the total force. The components of this total force are our real interest, however. For reasons described later, I'll be categorizing these components on how they relate to speed, written 'V'. Component forces which are proportional to the square of the speed will be separated in this process from those which are proportional to just 'V'. The 3rd category will be components which don't change with 'V' at all; they're often called 'constants'. I'm aiming to represent the total force, 'F', as

The 'F' is the answer, the total force we're looking to understand. We know the V's because we supply them in our measurements. a,b, and c may 'just be numbers' but they're numbers we don't know. A technique called Least Squared Error, or 'LSE', will give them to us in exchange for our measurements.

Aerodynamic Drag
The following shows the expression for calculating this force, assuming you have numeric values for all the listed parts. The key thing to note here is that it's proportional to the squared of the speed; that's big.

LSE doesn't want to know about the inner definition of 'a', we just have to remind ourselves later that we know a lot about 'a'. For LSE, this term merely looks like

Cd and A are valuable, individual concepts which each suggest different ways of addressing aerodynamic drag. However, when working with the numbers themselves, I believe they should be treated as a single item, 'CdA', not Cd and A individually. For more, see the glossary entry for Cd.

rolling resistance

Rolling resistance, called 'Fr' here, is with you at all speeds; in fact, it's almost the same at all speeds. Some articles suggest that it doesn't change at all with speed, that Fr depends only on 'Wt' (the car's weight) and RRC.

     Fr = Wt * RRC     
Others papers say Fr decreases a little at higher speeds, as in
     Fr = Wt * RRC - something * V

drivetrain friction

Drivetrain friction, which we'll call 'Fd', is the friction from your transmission, the bearings in your driveshaft assembly, the differential 'rear end', and the wheel bearings. It's unchanged with speed thus it's one of our constants, our 'c' components.

In gas-powered cars, Fd is roughly 10% of the total energy loss. That's significant because the energy inefficiencies in a typical car are very large. 10% of a large number is worth consideration.

To review, only one force, aero drag, affected the 'a' term while two forces, Fr and Fd, supplied constant values for our 'c'. So we have the desired polynomial needed by LSE but we understand that the simple looking a,b,c values really contain information about our forces. We see:

And, depending on which papers you believe, maybe Fr also has something to say about 'b'. The uncertainty about the 'b' term is interesting enough that have LSE solve both of the following equations:

When the LSE gives us our a,b,c values (or maybe just a & c), we will still have to separate the 'a' into 'Cd' and 'A'. We'll do that by measuring 'A', finding the air density and solving for Cd.

Likewise with 'c' which is the sum of Fr and Fd. Because Fr is a function of weight, we'll get a 'c' for the car in its heavier state. We'll then know how much bigger Fr got and can then solve for Fd.

Measuring the total force at different speeds

You drive an assistant, a special stopwatch, a notebook back and forth on a nearly level road on a day with little or no wind. The road must be able to legally handle high speeds and conditions must allow you to coast at low speeds (25) without endangering anyone.

You'll be measuring the deceleration, the rate at which it slows down. Indirectly you'll be measuring the sum of the forces which are trying to slow the car. The easiest way to get a usable deceleration value is to measure how long it takes the car to slow 5 mph. You'll start your measurements at a range of speeds, 60, 50, 40, 30, etc. To make a measurement, say at 60 mph, get the car moving at about 63 mph, put the car into neutral and let it coast. When the needle hits 60 mph, your assistant should start the stopwatch. When the needle hits 5 miles per hour slower (55 in this example), the assistant should stop the watch and record the speeds, direction, and time value. It's often useful to note where on the road this measurement took place.

You'll immediately be impressed with how far your car can coast at slow speeds. And you'll be equally surprised to see how quickly it decelerates at the very high speeds.

I suggest that you repeat the same speeds moving in the opposite direction as it is really difficult to visually determine if a road is truly level. The effect of a slight slope will especially apparent in your low speed measurements.

It's also informative to repeat the whole set of measurements, say, 3 times. Ideally the recorded numbers would all be the same - but they won't be. And you won't really know which are more correct than others.

The gathered time measurements can quickly give the deceleration rates:

    deceleration, mph/sec = 5 mph / measured seconds

    Say your measurement is 4.6 seconds:

    deceleration = 5 mph / 4.6seconds 

                 = 1.08 mph/sec

    This means that, at the measured speed, the forces would cause
    your car to slow down 1.08 mph every second.  (At a lower 
    speeds, the car doesn't slow down so quickly).
To convert this deceleration into the total force slowing the car, you need to fold in the weight of the car, 'Wt', in lbs:
                                         21.8 mph 
    F,lbs = Wt x deceleration, mph/sec / --------

                 5 mph                   21.8 mph 
          = Wt x ----------------      / --------
                 measurement secs          sec    

                 5 mph     
          = Wt x -------- / measurement
                 21.8 mph 

          = Wt x 0.23 / measurement    (if 5mph used for decel measurement)

    Say your measurement is 4.6 seconds and your car weight 3800 lbs.

    F,lbs = 3800 x -----------

          = 3800 x 0.05

          = 190 lbs
You can find the weight of your car on the web, or in your owners manual, or by having the car weighed. Your measurement notes should include this value. You can guestimate the weight of your assistant if that's important.

To finally separate Fr from Fd, you'll have to repeat the entire set of measurements at 2 different car weights. This second measurement set can be done on the same day as the first measurements. Ideally the two measurement sets should correspond to the lightest and heaviest you can make your car (without hurting anything).

You can also make experiments to measure other effects. For instance, the Fr is less if the tires are fully inflated. So, make a measurement set with tires at 'typical' inflation and another measurement set at full inflation. Obviously, you'll need to note which is which. If you plan this in advance, you might come up with the following measurement plan:

  1. empty the car of excess 'stuff'. weigh the car if possible. Leave the tire pressure at its typical value. Take a set of measurements.
  2. record the current tire pressures then fill up the tires to their correct values. Take and mark a set of measurements.
  3. Now add friends and 'stuff' of known weight. Take a set of measurements.

A sample measurement set from my Mustang; it will be used whenever an example seems useful.

Measurement Set #1
normal weight
mph secs where
85-80 5.75 880SB,Oakld
75-70 6.65 880SB,Oakld
65-60 8.9 880SB,Oakld
40-35 14.50 Evelyn, SB
40-35 12 Evelyn, SB
30-25 10.65 Evelyn, b4 mary
30-25 14.04 Evelyn, retn
Measurement Set #2
1000 lbs heavier
mph secs where
85-80 5.75 880SB,Oakld
75-70 6.65 880SB,Oakld
65-60 8.9 880SB,Oakld
40-35 14.50 Evelyn, SB
40-35 12 Evelyn, SB
30-25 10.65 Evelyn, b4 mary
30-25 14.04 Evelyn, retn

Using LSE to get a,b,c

plug into spreadsheet or use the C program
retrieving a,b,c; troubleshooting.
derivation: lse link
2x2 subset link

Converting 'a' to 'Cd' and 'A'

divide out the 
measure A. for the mustang, it's 74"x54"/ 144 = 27.75 sq.ft.
Cd = 'a' * 391 / A

Splitting 'c' to get 'Fr' and 'Fd'

The 'c' from your 'heavy (measurement) run' will be renamed
here to be 'c2'. The other 'c1'. The '2' '1' suffixes will
also be used on car weights, Fr etc.

        c1 = Fd + Fr1
        c1 = Fd + Wt1 * RRC

we know that Fr2 is Fr1 x Wt2/Wt1 so 'c2' is also

        c2 = Fd + Fr2
        c2 = Fd + Wt2 * RRC

flipping and combining expressions
        c1 - Wt1 * RRC = Fd = c2 - Fr1 * Wt2/Wt1
        c1 - Wt1 * RRC 
        c2 - Wt2 * RRC

      subtracting the 2 eqns
        c1 = Fd + Wt1 * RRC
        c2 = Fd + Wt2 * RRC 
     c2-c1 = Fd-Fd + (Wt2-Wt1) * RRC
     c2-c1 =         (Wt2-Wt1) * RRC

and, finally: