I can't believe that it was back in September that I posted the first part, I wrote this next part soon afterwards but forgot to post it. So here it is at long last.
In the first part we had a quick look at
some of the theory and difficulties associated with estimating the straight
line performance of motorcycles and other vehicles. This time we will look at the software, see how it works and work
though some examples.
As we saw last
time we can arrange the performance equation, F=ma to a =
F/m, we also noted some difficulties in specifying both the nett F
and equivalent m, which depends on inertia right back to the
crankshaft. Firstly let’s see how the
software arrives at values for F and a .
The nett F
is influenced mainly by engine torque and overall gear ratio, with aerodynamic
drag subtracting from this at higher speeds. Tyre rolling resistance also
reduces the value of F available but is reasonably constant
throughout the speed range and assumes relatively more importance at lower
speeds. There is also another effect
which reduces the average force applied through the tyre for acceleration which
is often forgotten, and that is the time that the torque is removed during gear
changes. Gear change times typically
range from 0.1 seconds for a very fast change with a quick shifter fitted to
around 1 second for a more leisurely foot movement. After the throttle closing and reopening during changing gears,
most engines require a certain time period for the fuel mixture to settle down
again before delivering full power, although modern EFI engines are much better
in that regard than older models. This
is not usually easy to quantify. The
first thing that we need to do in the software to start an estimation is to
enter the power or torque data, this will vary with RPM and we enter the data
into a table at appropriate values of RPM as shown in fig.1.
Fig.1 Data entry screen for
power/torque data. Either torque or
power numbers can be used and the software will calculate the other
appropriately. These curves are from my
own Suzuki SV650.
There is a “What
if?” option to allow the power curve to be increased by a given percentage
which gives a quick way of seeing the effect on performance of more or less
power. Once the power data has been
prepared I would normally move to enter data about the elevation of the road
surface, but as this is a virtual test on a perfectly flat and horizontal road
we can skip that step and proceed to the main data entry window shown in fig.2.
Fig.2 Main data entry
window. The graph is just a repeat of
the power/torque curves for reference.
The data entry boxes are for the multitude of parameters needed to get
the best performance estimate.
Working down the
left side. Initially, there is the
aerodynamic data. Looking at these in order:
CdA – this is a standard way of specifying the data from which to
calculate drag. The Cd is a coefficient
which puts a number on the efficiency of the shape regardless of size and A is
the frontal area which adds the size information. This value affects the top speed and high speed acceleration.
CP height – Centre of Pressure height. This is the height of the point
through which we consider the drag forces to act. At high speeds, the drag force creates a moment which lifts
weight off the front and transfers it to the rear tyre, thus creating more
potential traction. Aerodynamic load
transfer is also highly dependent on the shape of any fairing fitted which may
cause down force or lift, but without wind tunnel data we have little choice
but to ignore that factor. However, it
is at low speed in low gears when traction is needed most and so the effect of
this parameter is quite small and does not have to be particularly
accurate. This is especially true for
heavier and/or slower motorcycles.
Air density – Drag is directly proportional to air density. Fortunately, this is a value that is easily
calculated from temperature, barometric pressure and relative humidity. In our performance calculations I will use a
standard air density and so all virtual tests will use the same figure.
The next data
grouping is m (mass) or weight related, again let us consider
each in order;
CG height – The height of the centre of gravity. This parameter controls the load transfer, front to rear, due to
the acceleration. Up to a point this
transfer aids maximum traction and so if we have enough torque available, as in
the lower gears, it will aid acceleration.
The limit is if the load transfer is sufficient to unload the front
wheel completely leading to a tip over wheelie unless the applied torque is
reduced. The software automatically
calculates when a terminal wheelie is imminent and limits the torque to the
maximum usable.
Load on tyre,
front and rear – This is the static weight on each
tyre of the loaded motorcycle. This
gives us the mass of the bike without the rotary inertia inertia effects of the
wheels and rotating engine parts.
Wheel MoI,
front and rear – Moments of inertia of each
wheel. MCN now have the apparatus to
measure this.
Crank MoI – This is the moment of inertia of the crank and other engine
parts. This is difficult to measure without
an engine strip, but for the future I have asked the MCN testers to get the
dyno runs in at least two gears, from those I can make a calculation of the
whole drive train inertia.
Wheelbase
– This is obvious, but it is necessary for the performance calculations because
in conjunction with the CG and CP heights it controls the load transfer and
hence potential traction.
Next there is a
grouping of mainly tyre related parameters:
Rolling force – This is the force necessary to overcome friction and rolling
resistance of the tyres. It could be
measured by measuring the force to pull the bike along a flat tarmac surface,
but in most cases it will be estimated from published data on tyre
characteristics, as I mentioned last month.
Tyre maximum
µ – (µ is the symbol for coefficient of
friction) – This value is dependent on both tyre properties and the road
surface. For most purposes on dry
tarmac the value will be around 1.0.
Quality sports tyres may be around 1.1 and cheap no-name tyres maybe as
low as 0.8 or worse. This figure will
only have a significant effect on our estimates for those motorcycles with
enough torque to spin the tyre. This is
rarely a problem in the higher gears and so top speed estimates will not be
affected but getting off the line on a sports bike will be highly dependant on
maximum µ, so low speed
acceleration calculations depend on how accurately we specify this number.
Peak rear
tyre slip – It is necessary for any tyre to slip to
a certain extent in order to produce a driving force. This not the same as the tyre spinning wildly such as when doing
a burnout, in that case the tyre is slipping much beyond the amount of slip
which produces the maximum drive. A
typical value of slip which produces the maximum traction is around 15%. That means that the tyre will be rotating
15% faster than necessary to match the road speed. The relationship between slip and driving force is extremely
complex and the software uses the so called “Magic formula” tyre model by
Pacejka which is commonly used in mathematical models of vehicle
behaviour. In common with the
coefficient of friction the value of peak slip will not affect top speed
estimates by much. However, at low
speeds the slip acts to effectively lower the gear ratio and this must be accounted
for in the calculations. Just as well
that these days we have computers to do the donkey work for us.
Tyre radius,
front and rear – This parameter for the front tyre
is used to calculate the angular or rotational acceleration of the wheel which
with the MoI of the wheel enables the programme to calculate the effective mass
of that wheel. The same applies to the
rear but in addition that tyre radius is used to calculate the relationship
between engine RPM, gear ratio and road speed.
The final data input
grouping on the left side specifies the testing requirements and conditions as
follows:
Maximum
distance – Within reasonable limits the maximum
speed of a vehicle depends on the length of the testing surface. The longer the test strip the higher the speed
that we can obtain. This parameter
specifies the cutoff length for our virtual test. There is an option to select ¼ mile length which is of course a
popular distance for acceleration testing.
Start
velocity – This was originally programmed into the
software for performance simulation on the race track so that the entry speed
onto a straightaway could be specified.
It could be used in our simulations to get roll-on acceleration values.
Final
velocity and Include braking – Both of these
were also designed to be used for race track simulation and will not be used in
our performance estimations.
Head wind – The function of this is obvious but will not be used here. Our estimates will assume no head wind.
Moving over to
the top centre of the window we have a grouping of gear ratio data. This is information that is readily
available from owner’s manuals or other easily accessible sources and can be
entered with 100% accuracy. There is no
need to go into detail about this but the last two items need comment.
Gear change
time – This is the period that driving force from
the engine is removed during changes.
Typically it will range between 0.1 to 1.0 seconds. The actual time can have a very large effect
on acceleration as we will see in following examples, and is the motivator
behind Honda’s and Yamaha’s development of the so called seamless change gear
boxes used in the motoGP world championships.
RPM for
change – This is the RPM which we select to be when
we up-shift. This determines which part
of the power curve we use. A low value
puts us lower down the power curve and the maximum potential acceleration will
not be reached, of course this reflects the reality of how we use our bikes
under normal riding conditions. Modern
bikes are too fast and accelerate too quickly to require riding techniques
which demand maximum use of the power available. Having said that it is also true that many people want to know
the maximum potential of their machines or of those that might represent a
future purchase. There is a lot of
misinformation around about the best RPM to change gear at to achieve maximum
performance, let’s consider a couple of options that get suggested.
Change gear at the RPM of peak torque – Of course this is based on a =
F/m, F is proportional to engine torque and so we get maximum
acceleration at the point of peak torque, right? Wrong, well it’s both right and wrong. It applies only if we are considering the acceleration in a given
gear. It is the torque at the wheel not
at the engine which determines acceleration and we can increase torque at the
wheel by simply using a lower gear and revving the engine more.
Change gear at the RPM of peak power – If we had an infinitely variable gear
box the maximum acceleration at any given speed will occur if the gearing is
set to give the engine RPM for peak power not peak torque. In practice we do not usually have
infinitely variable gear boxes and so we have a spread of RPM in each gear. For maximum acceleration we need to place
that RPM spread over the power curve such that the area under the power curve
is maximized. This will occur when the
peak power RPM is somewhere near the middle of our RPM spread depending on the
rate of increase or decrease of the power curve either side of the peak. So changing gear at the RPM for peak power
is also not the optimum, to enclose the maximum area we need to change gear
somewhere passed the peak. However,
there are further complications in the determination of the “exact” best
RPM to change at. One is because not
all gears in the box have the same proportional gap between adjacent gears, so
the RPM spread in each gear is likely to be different. Another is because in any given gear, the
acceleration is less at the high end of the RPM spread than at the lower
end. This means that we spend more time
at the high end of the spread and so it would optimize performance if that is
where the peak of the power curve was concentrated. In fact if we plotted the power curve against time instead of
against RPM then maximizing the area under that curve would truly give us the
optimum conditions. However, before we
can map the RPM based power curve onto a time based one we have to know the
performance of the bike first but that is what we are trying to calculate. A classic chicken and egg situation. If we wanted to go to such extremes we would
have to adopt an iterative approach.
That is; firstly make a guess at the optimum change RPM and calculate
the performance, then map the power curve to the time base and re-evaluate the
change RPM and repeat the calculation.
Of course this would have to be repeated for each gear. In practice this would be going beyond
sensible limits to refine the values and placing the power peak between 2/3 to
¾ up the RPM spread will be very close
to giving the best performance, but before revving beyond peak power it would
be sensible to check the redline RPM for engine safety.
Fig.3 The RPM spread of 5th
gear superimposed on the power vs. RPM graphs for gear change RPMs of 7250,
9200 and 9800.
Let us see just
what difference the gear change RPM value makes. Fig. 3 shows the RPM spread of 5th gear superimposed
on the power curve for 3 different change RPMs. viz; 7250 which represents
changing gear on reaching peak torque RPM, 9200 for changing gear when peak
power is reached and 9800 which straddles the peak power RPM. Fig. 4. shows the
effect of these different change points on the velocity over time. Fig. 5 shows the effect on the distance covered
in the same time.
It is clear the
large difference between changing at peak torque RPM and near peak power RPM.
Fig. 5 shows
that when changing at 7250 it takes nearly a second longer to accelerate up to
500 metres. Fig. 4 shows that an extra 2 seconds is needed to reach
100mph. There is a much smaller
difference between changing on reaching peak power RPM and revving passed the
peak. For engine longevity it would be better to change at 9200 as the
performance penalty is minimal. The
power curves for this engine are fairly flat around the peak power area, on a
highly tuned engine the curve will be much steeper and the difference between
the 9200 and 9800 cases would be emphasized in comparison.
Top speed will
not be affected by the up-change RPM and so the velocity curves in Fig. 4 will
converge eventually.
Don’t forget to check out my web site at www.tonyfoale.com and buy the software .
Fig. 4 Plot of velocity vs. time showing the effect of
the up-shift RPM. The green curve is for a change RPM of 7250, the blue is for
9200 and the red 9800 RPM. The 9200 and
9800 cases reach 100mph (160 km/h) in around 9.2/9.3 seconds but shifting at
7250 would take us 11.3 seconds or 2 seconds longer.
Fig. 5 Graph of distance covered when up-shifting at 3 different RPM
values. It would take us another second
to cover 500 metres from a standing start with the early shift. However, engine life and fuel economy would
both be served better with the more gentle riding technique.
