F=MA
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.
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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.
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.
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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.
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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.
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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.
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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
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 www.tonyfoale.com
