Thursday, 10 September 2015

Performance estimation. Part 1 of 2



F=ma is probably the single most important equation in everyday physics and mechanics. It describes all of our physical actions and those of our motorcycles as well. So why are we discussing it here? The answer will become apparent as we delve deeper into understanding what makes our motorcycles perform as they do.

A few years ago I wrote some software for simulating straight line performance. The motivation was to help with gear ratio selection to optimize performance on a race trace.  It has proved useful in this context and several race teams now use it.  Since then it has found a new and unexpected use. We are all familiar with the performance road tests that most if not all car and bike magazines feature with each new model, but have you ever wondered where the riders go to ride at insane top speeds and burn rubber and clutches to get their numbers?  Some acceleration tests might be done on a drag strip but a ¼ mile is not long enough to wind out to top speed, at least 2 or 3Km is needed for that.  Even most race tracks do not have straightaways long enough. There are a few special high speed test tracks in the world where you can get to and hold top speed for as long as the tyres, machine and rider can take it.  The problem with these tracks is that there are very few of them, they are expensive to hire and are in constant use.  What is the solution?  It has been a fairly common practice for magazines and their test riders to use a favourite section of public road which is both reasonably flat and smooth, as well as being generally out of sight and with little traffic. Such venues are getting harder to find with ever increasing population and traffic. 

Any modern machine with even the minimum of pretensions to being a sport bike will do 150mph and several are pushing 200mph.  With the possible exception of the German autobahns those speeds are well in excess of the maximum speed limits in any country. Obviously, that means that performance tests carried out on public roads are without doubt illegal, not to mention increasingly dangerous as bikes get faster.  Many magazines have been breaking the law and getting away it mostly up to now.  The editor of a US magazine, asked me whether it was possible to make reasonable estimates of top speed and acceleration with software.  It just so happens that a few years ago I developed some software for that very purpose.  Many tests and comparisons since then have shown that given the right information it produces figures which agree pretty well with physical testing. 

Engineers have always tried to calculate the performance possibilities of whatever they might be designing, be it bridges, ships, planes, cars or motorcycles.  With the advent of affordable powerful computers such calculations have reached a high level of sophistication over the past few decades.  Builders of aircraft and other vehicles such as motorcycles have a very good idea of the performance envelope of future models long before they leave the proverbial drawing board.  So how does it work?

To answer that we return to the one single simple equation which answers all – yes it is F=ma. To understand more let us look at just what it means.  In words it can be expressed as Force equals Mass multiplied by Acceleration.  For our purposes it is acceleration that is of most interest to us.  If we know the acceleration history of the motorcycle we can calculate its velocity at any given point in time, which is exactly what we are after.  So it makes more sense if we rearrange the formula to give us the acceleration.  It then becomes a=F/m, that is; Acceleration equals Force divided by Mass.  This means that the acceleration at any point in time is dependent of the nett force pushing the motorcycle and the total mass of the motorcycle, including rider, fuel and luggage.  Simple?  Well it would be if we knew the driving force over time and total equivalent mass. Let us look at each of these terms, and some others, to determine whether complications exist in the determination of these factors.

Velocity or speed.

Velocity put a number on how fast we are traveling.  In the metric system it is usually specified as m/s (metres per second) or km/h (kilometres per hour) and in the imperial system as ft/s (feet per second) or mph (miles per hour).

How velocity varies over time is the essence of what our performance estimations are all about.  This what we want to know.  All the calculations that we do, or get a computer to do for us, are ultimately directed to this single parameter.


Acceleration tells us how quickly the velocity changes with time, either up or down.  Although when the velocity decreases (braking) the acceleration is often referred to as a deceleration, but is more likely to be considered as a negative acceleration in calculations.  Acceleration is expressed in units of velocity divided by time or in the metric system as (m/s)/s or m/s² and (ft/s)/s or ft/s² in the imperial system.  Another unit of acceleration which is common to both measurement systems is g.  This is a unit which compares acceleration to that which gravity gives to a free falling object which is 32.2 ft/s² or 9.807 m/s².

 An example of an acceleration curve.  The short dips are due to the removal of driving force due to gear changes.  Note how the acceleration decreases as the speed increases as we change up through the box.

An important thing to understand about acceleration is that it is not about velocity, it is about the “change” in velocity over a given time period.  Consider as an example that we have an acceleration such that our speed increases by 20Km/h over 1 second.  Then if we are traveling at 30 Km/h at the start of that second then at the end of 1 second we will be traveling at 30 + 20 or 50Km/h.  However, if we were doing 200Km/h and experienced the same acceleration for 1 second then we would end up at 200 + 20 or 220Km/h.  In each case our speed changed by 20Km/h, the starting velocity doesn’t affect that.

This leads us into how we can calculate velocity over time if we have knowledge of how the acceleration varies with time.  This done by means of a mathematical technique known as integration.  Analytically this is a continuous process but digital computers do not work like that, they work in discrete steps.  We all know that acceleration of a vehicle is not constant over the whole speed range from zero to maximum.  As we change to a higher gear acceleration reduces and even within a single gear acceleration will vary depending on where we are on the power curve.  However, if we divide our period of acceleration into much smaller time increments then within each of these increments the acceleration will be close to being constant.  The shorter the increment the closer the acceleration will be to constant.  This prompts the question of just what time increment is appropriate for our purposes.  Modern sport bikes accelerate very quickly, taking only a few seconds to go through the full range of the 5 or 6 gears.  So obviously 1 second intervals will be way too long for our calculations but might be fine for a heavily loaded steam train.  Whilst developing my software I evaluated time increments ranging from a tenth of a second to a thousandth  of a second.  I choose a period of one hundredth because there was no significant change in the results by using a thousandth and it reduced the number of individually calculations by a factor of ten.

 This is the velocity curve which derives from the acceleration curve above.  Although gear changes stop acceleration for short periods the effect on velocity is put small steps in the velocity curve.

If we know our velocity at the start of any time interval and acceleration history then we can calculate the new velocity at the end of that time interval.  We simply multiply the average acceleration during the interval by the period of that interval and add that to our initial velocity.  If we repeat this over the total period of acceleration then we will build up a picture of how velocity varies with time.

Hence the importance of the expression “a=F/m”, we need to calculate this as a precursor to calculating velocity.

Mass (also called inertia)

This is one of only two parameters needed to calculate acceleration.  “Mass is just the weight of the bike – right”?  Wrong!  It is a little more complex I am afraid.  Without going into the frequent misunderstandings about the difference between mass and weight, let us consider the complexities.  In performance calculations we are concerned with two types of mass or inertia.  The most obvious is what can be called the translational inertia and the less obvious one is rotational inertia.  The word translation simply means a linear or non-rotating motion. Rotational “inertia” is referred to as “Moment of Inertia”, which I’ll abbreviate to MoI in the following text. 

Just as it takes a force to accelerate an object in a linear (translating) fashion it takes a torque or moment to accelerate a rotating object.  In order words if we want to increase the spin rate of a wheel we have to apply a torque to it, in order to accelerate its MoI.  Imagine a motorcycle wheel resting on the ground, now if we apply a horizontal force at the axle we will start to accelerate the wheel along the ground but as well as accelerating its mass along the ground we also have to accelerate its MoI in rotation.  Thus, the force that we apply at the axle is not all available to accelerate the mass of the wheel in translation, some of that force is split off to produce a torque to increase the rotational speed of the wheel.  Looking at “a=F/m”  we see that reducing the force will reduce the acceleration.  However, we can also reduce the acceleration by using a higher value for mass in the formula.

There are several ways of dealing with the two types of inertia in calculations but I find the simplest is to use the concept of an “equivalent mass”.  The equivalent mass is a value for the combination of linear and rotational (or angular) masses which gives the same results as if we calculated the proportional split of the driving force needed for the rotational and translational accelerations.  We might well ask how much difference ignoring wheel rotational  inertia makes to our performance calculations, is it really significant?  The answer depends on the parameters of each individual model but a rough general guide is that without wheel MoI our acceleration estimates would be high by over 10%.

Measuring the non-rotating masses of the motorcycle is easy, we just put the machine and rider, ready to go, on some scales, but what about the rotational inertia of the wheels?  There are no simple scales for that.  Well, there are several methods that could be employed to measure MoI, which vary in difficulty and potential accuracy.  The wheel MoI is important for reasons other than acceleration, steering in general and lean-in rate in particular are highly dependent also.  I won’t now go into how the MoI will be measured, that will be the subject of a future article.

As I explained above, it is useful to convert MoI into an equivalent translational mass, fortunately that is quite simple, we just divide the MoI by the wheel radius squared.  Once we have done that for both wheels we just add those values to the overall mass as determined on the scales.  Now we can modify our formula “a=F/m” to “a=F/me” , where me is the equivalent mass.  That is fine when both wheels are on the ground, but when accelerating hard the front wheel may be airborne and so we should not add in its rotational inertia until it touches down again.

If only things remained that simple.  The crankshaft, clutch and other parts in the drive train rotate and have to be accelerated as well.  We are faced with problems getting accurate data.  Firstly, that sort of data will be hidden away in the engineering files of the manufacturer and is not offered in the sales documentation offered to the public or press.  Unlike the relatively simple task of measuring the MoI of the wheels, it just is not reasonable to expect to have to dismantle engines and gearboxes to do the same with the innards.  Modern sport bikes generally have small diameter crankshafts and so compared to a wheel and tyre their MoI might be small.  However, engine components rotate faster than the wheels by a factor equal to the overall gear ratio.  That means that they have to be accelerated faster also and that requires more torque, reducing that available for the linear acceleration. Accounting for the gear ratio we can calculate the effect of the engine components by adding an effective or equivalent MoI  to the actual wheel MoI.  Unfortunately, there is a square law relationship between the true crankshaft MoI and its effective MoI when transferred to the wheel.  To see what this means consider this example using the gear ratios from an Suzuki SV650 (I just happened to have those on my desk).  In 6th gear the overall gear ratio is 5.2:1, when we square that we get 27.  That means that we need to multiply the crankshaft MoI by 27 to calculate its effect at the wheel.  So even though the crank’s MoI may be relatively low it starts to get significant when multiplied by 27 or so.  It gets worse, the overall ratio of the same bike in 1st gear is around 3 times more at 15:1, with its square equal to 225.  The clutch spins slower than the crankshaft and hence its effect is less, for the same motorcycle the multipliers to use on the clutch MoI would be 6 when in 6th gear and 51 when in 1st, still quite significant multipliers. You can see why some people lighten cranks and flywheels to increase acceleration.

The SV650 is in the middle of the range of crankshaft size and the range of gear ratios.  A large cruiser V-twin will have a much heavier crankshaft/flywheels but will be slower revving with lower overall gear ratios which will tend to cancel out the effect of the heavier crank.  On the other hand high revving multi-cylinder engines will have lighter cranks.  So the MoI of the rotating engine components cannot be ignored when making performance calculations.


Force is the second of our two necessary parameters.  The engine produces a torque which is transferred to the rear wheel after being multiplied by the overall gear ratio along the way.  The torque at the wheel is resisted by generating a horizontal force at the contact patch of the tyre on the road.  That force is what drives the bike forward, but it is not all available for acceleration, firstly we have to deduct a bunch of parasitic losses.  It is just like taxes, you earn $50,000 but you are only left with $30,000 to $40,000 to put to good use.  So the “F” in “a=F/m” has to be the nett force after deducting tax.

At high speed the most onerous tax is aerodynamic, that is air resistance, this is a tax on speed.  Tyre rolling resistance is a base tax throughout the speed range but makes up most of the tax burden at low speeds.  Ground slope and wind can be very important factors in determining the net force available for acceleration but these can be ignored for our purposes if we are on a horizontal surface with no head or tail wind.  Tyre slip is another factor that is very important, particularly in the lower gears.  This detracts a little from the accelerating force in the sense that air resistance does, and can also be considered as putting a limit on the maximum possible driving force.  Load transfer under acceleration increases that limit.

Rolling resistance is relatively easy to handle because tyre manufacturers do have values for a rolling resistance coefficient.  Multiplying this coefficient by the loaded weight of the motorcycle will give us  the force that must be deducted from the driving force.  Typical coefficients for motorcycle tyres are around 0.015 to 0.02 at low speeds, depending on inflation pressure.  That means that a 250Kg loaded motorcycle needs around 5Kgf of push to move the wheels around, if that sounds a bit low it is because when we move a bike about we also have to push against chain and brake drag etc.

Information on aerodynamic drag for a specific motorcycle is generally harder to obtain.  Occasionally manufacturers will publish some data when they feel it will be useful for sales, but in general this is rare.  We are left with no alternative but to apply some educated guesses as to what values to use in our calculations.  Motorcycles tend to be grouped into distinct performance/weight/size categories and within such groups the variation in drag numbers is relatively small.  Magazines are a good source of performance data acquired from years of countless road tests.  So in the absence of a wind tunnel the next best thing is to search for performance data from tests of similar size and shape motorcycles.  Then we can reverse engineer those similar models in the software to obtain aerodynamic characteristics that should be reasonably close to those of the bike under consideration.

Torque/power curves (and how to measure them)

The starting base for determining the driving force is the torque produced by the engine throughout the RPM range, this can be measured by a dynamometer of which there are many types.  The most common is known as an inertia dynamometer.  The rear wheel sits on a heavy steel drum and the bike is ridden up through the gears as if riding on the road.  A certain gear is selected for doing the actual measurement, typically 4th or 5th in a 6 speed box, slower bikes will probably be done in their top gear.  The “rider” will then accelerate the drum throughout the required RPM range.  A PC together with some instrumentation will measure the acceleration of the drum in small increments over the test duration.

This type of measurement has the advantage that the acceleration of the drum is due to the torque on the rear wheel and so transmission losses are inherently eliminated from our calculations, removing at least one uncertainty.  However, it has the disadvantage that it does not directly measure torque, instead it measures drum acceleration.  The dyno manufacturer knows the MoI of the drum and so he can say “To accelerate our drum at this rate the torque had to be xxx”.  We have seen above that acceleration depends on wheel and engine components MoI which varies  with which gear we are in.  So in addition to the drum MoI we really need to add in a value to account for the motorcycle’s rotational inertias as well.  Of course your local dyno shop will not have that information for all the different bikes that he gets to test.  To make some attempt to account for this, some dyno suppliers include what we might call “fiddle factors” into their calculations which effectively add to the drum MoI.  This inflates the power and torque figures which the customer gets and so he goes away happy.  This approach is not entirely without merit because if the fiddle factor is accurate then we get power and torque curves without the effect of engine inertias, which we know varies with which gear we are in.  The problem is that there is no fiddle factor that is applicable to all bikes in all gears on all dynos.  There is no single drum size which is optimal for the full range of motorcycles likely to be tested.  In other words a simple drum inertia dyno, although being a very useful tool is full of compromise and any results obtained need to viewed in the light of such knowledge.  However, there is a method using an inertia dyno by which we can get better estimates engine MoI.  I’ll elaborate on that in another article.

To summarize

In addition to the initial and current use of the software for general performance calculation and gear ratio optimaization, the increasing difficulty of economically and legally doing actual performance tests has directed us to consider “virtual road tests”.  Although the basis of such calculations is as simple as “a=F/m” we have seen that there are various complexities and uncertainties in the specification of both F and m.  That does not mean that we are prevented from getting reasonable results.  Experience has shown that the software that I developed can produce results for top speed and acceleration which match closely with practice.  The next article in this series will look at the software in detail and view various.  In the meantime you can see more details on my web site at

© Tony Foale August 2015

Thursday, 9 July 2015


Over the past few years I have had a number of people asking me to give a motorcycle dynamics seminar within Europe. For various reasons this never happened. However, providing that there is sufficient interest I intend to give one during August or September in Barcelona. It will be a two day event and details of subject matter etc. can be found on my web site Email me on to express an interest and/or request more info. Also indicate any preferred dates and whether weekend or week days are better. Barcelona is a beautiful city with much to see and do so come and enjoy. BTW the seminar will be given in English. Please share this post.

Friday, 6 February 2015

I have recently been contemplating the optimization of the performance of drum brakes for my classic racing Aermacchi. I have also been in discussions with a couple of friends regarding improvements to a drum brake for a road bike.
To put this work on a more technical but easy to use basis I have developed a small computer programme to evaluate the effects on braking torque of changes to the overall drum and backplate geometry and changes to the coefficient of friction.
As this software might be of use to others I have made it freely available and it can be downloaded from the freeware section of my web site at
It is a single standalone .exe file so just download it to a directory of choice and run it directly or make an appropriate shortcut to it.  No installation procedure is necessary.