Cut to the chase: if you have C_d and frontal area figures; or if you have a car, a level road, a weighbridge and a stopwatch.

Measuring Drag and Predicting Performance

Introduction

Cars move down the road because they are pushed, normally by some sort of machine applying torque to the wheels. Multiplying the torque by the radius of the road wheels gives the traction force.

Meanwhile, various forces act on a car from the environment. If a car is on a slope, the weight of the car pulls it down the slope. If it is moving, various frictional forces oppose this movement. One important frictional force is that of the brakes, which are intended to slow a car down and stop it.

The other two frictional forces are air resistance and rolling resistance. Air resistance depends on body shape and on speed, and rolling resistance is proportional to weight.

Analysis of these forces is quite complex, but measuring them is remarkably easy. If we can measure the speed and weight of the car, we can plot a curve of speed against drag force.

What this page aims to do is to explain a method to obtain a table of drag against roadspeed, and use this to get performance predictions for an electric conversion of an existing car. An example is worked - for now the example uses theoretical (but hopefully useful) numbers, but in the future it will use real figures for a Mark 2 Fiesta (my chosen conversion vehicle).

Theory

The theory of what stops cars going faster is that, on a straight and level road, two sources of resistance try to slow the car down. One is called air resistance, and is caused by moving the air out of the way to get past, and the other is called rolling resistance, and is caused by the effort of deforming the tyres and road surface as the car moves along.

Air resistance is governed by something called Bernoulli's Equations: for normal road speed it is given by:-

R_a = C_d ρ A v²

where
R_a is the air resistance
C_d is the coefficent of drag, which is a constant defined by body shape
ρ (rho) is the density of air - about 1.2kg/m³
A is the frontal area - width x height
v is the speed of the car through the air

Rolling resistance is governed by the laws of friction: for normal road speed is is given by:-

R_r = F M g

where
R_r is the rolling resistance
F is the coefficient of rolling friction, which is constant for a given car
M is the mass of the car
g is the acceleration of gravity

On a level road, the total drag is given by:

R = R_a + R_r = C_d ρ A v² + F M g

For what it's worth, for a gradient of angle α (alpha) the total drag is given by

R = C_d ρ A v² + (F cos (α) + sin(α)) M g

where α is positive for uphill slopes and negative for downhill slopes.

The intention of these experiments is to determine values for F and C_d

Figures for the Mark 2 Ford Fiesta are available online.

This gives a C_d figure of 0.35, and width and height figures of 1.631m and 1.321m respectively. If the bottom of the bodywork is at the same height as the wheels, then the frontal area can be approximated by a rectangle of 1.631 x (1.321 - 0.5) = 1.339m². That's probably an over-estimate, since the side windows are sloped. So taking off the sloped bits, let's say 0.9m².

Let us also assume that the rolling resistance adds, say, 125N to the total drag, using a rolling coefficient of 0.015 and a weight of 850kg. (850kg X 9.81m/s² X 0.15 = 125N)

Power is equal to the force in newtons times the speed in metres per second. So it's the force in newtons times the speed in miles per hour divided by 0.4444.

Speed	Air Resistance	Total drag	Power
5mph	2N	127N	282W
15mph	18N	143N	950W
25mph	49N	174N	1930W
35mph	95N	220N	3427W
45mph	157N	283N	5650W
55mph	235N	360N	8807W
65mph	329N	454N	13103W
75mph	437N	562N	18748W

Experimental Method

Requirements:-
- the car, which must be a manual
- a weighbridge
- a stopwatch and an assistant to use it, 'cause you'll be driving
- a long, straight, level test track
- a windless day

First weigh the car, including you and your assistant.

Then drive along the road, reaching the maximum speed to be measured. Then put the engine in neutral and allow the car to coast to a halt. Get your assistant to measure the time taken to reach each multiple of 10mph. Obviously, if the road is not long enough for one run, do several, starting at different speeds.

Repeat the same exercise the other way to cancel out any wind or slope that you didn't notice.

Re-weigh the car.

For safety, do not turn off the engine or engage the steering lock while performing this test. Do not perform this test on a public road, and ensure that you have fulfilled all legal and training requirements to make sure you have the permissions and skills to carry out this sort of tests. You do this at your own risk - I don't see why it's my problem if you find yourself upside down in a ditch somewhere. You know how to drive, you know driving is potentially dangerous, especially when you are distracted by taking tests, so have some common sense!

Results

You should end up with a table something like this:-

Italic figures are guesses - when I have the real data I'll put it up. But I think it's something near.

For rough and ready calculations, this graph is all that's needed: it gives required motor torques and powers, and from that battery life and so range can be calculated.

Performance Predictions

This section is optional - the speed/drag/power tables above provide everything you need. This is in case you want to know how long it will take your design to do 0-60, or a standing quarter mile, or whatever.

The first thing to do is to analyse the figures. A graph can be drawn, to predict the drag at each speed, but it's worth using the lowest-speed figure to estimate the rolling friction.

For our example electric car, the maximum traction force is at 800A, which gives 108Nm through 0.3m radius wheels and a 3:1 reduction gear = a total traction force of 108 X 3 / 0.3 = 1080N per driven wheel. The total is 2160N.

This gives us a table like this:-

Examination of this chart suggests that we can expect a 0-60 time of 12.8 and a 50-70 time of 4.6. This is respectable acceleration for any car, electric or otherwise, especially the 50-70, which is very good.

Another reason why this table is so important is because it allows us to estimate the effects of adding weight to the car. The rolling friction is proportional to weight - if the maximum weight of our example car is 1475kg fully laden, that gives a rolling friction of 195x1475/913 = 319N, which means that our all-important 75mph total goes up from 553N to 677N. That, in turn, will reduce range considerably.

The other important effects of these figures is the ability to climb hills and accelerate. For a given traction figure we can calculate the maximum gradient that can be climbed, and the time taken to accelerate through each speed band. This gives important performance figures.

Assuming our fully laden vehicle weighs 1475kg our table looks like this:-

It's worth noting that this gives a 0-60 time of 22.5 seconds, which is quite slow, but probably just about bearable. On the other hand, the 50-70 time, considered the best measure of motorway performance, is 8.14 secs, which is less than the petrol equivalent in 5th gear.

It's also worth noting that these will hammer the batteries flat!

This page is part of an Open Source Electric Car Project, and is written and maintained by Simon. At this stage these pages are constantly under revision. Thoughts and comments are welcome.