Monday, 14 March 2016

Performance estimation. part 2 of 2

F =ma part 2.

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 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.