Saturday, March 10, 2012

Show some teeth

The dynamics of a bikes power delivery is not as simple as it looks. One of the first things that needs to be done after a modification that improves performance is to choose the right sprocket size, for the target speed and RPMs.

People often assume that going to a bigger sprocket automatically increases top speed, by a simple ratio (I was under this delusion too, which was removed by practical experience and some study of the physics involved)

Let's look at what it takes to go fast...

In a frictionless, vacuum, there is a simple relationship between power, acceleration and mass : 

A = P/mv
P = A/mv
v = P/Am
This is why there is no concept of "top speed" for a spacecraft. For a constant power output acceleration will asymptotically reach 0 and velocity will reach infinity (forget relativity for now, were talking Newton-ese)

Once you add gravity and friction to the picture, there is a certain constant frictional force that counters the acceleration, but there is still no limit to the maximum velocity achievable.

For a real world vehicle scenario, the most important factor comes into play next - atmosphere and air resistance.
Somewhere around 35 mph or 50 kph, this factor starts getting significant.

The formula for determining top speed from power including aerodynamic drag is involved - here is one from Mazda :

P = ((A/2 * Cd * D * v^3) + (Cr * m * g * v))

A: being the frontal area in m^2
Cd: being the drag coefficient (0.3 for sports cars, 0.5 to 0.75 for motorbikes, 1 for trucks)
D: Density of the air (1.29 kg/m^3)  
Cr: roll resistance coefficient (about 0.015)
m: Mass of the vehicle in kilograms
g: The earth's gravitational acceleration (9.81 m/sec^2)  
v: Velocity in m/sec (kph/3.6 or mph/2.2374)
P: Wheel power in W (divide by 736 to get hp)

Another common formula that works pretty well is the LRT formula :

hp = (mph / 215.39)^3.3135 x weight (lbs)

The main thing to notice is that power is proportional to the cube of the velocity. Doubling the speed needs roughly eight times the power. The drag formula seems on the button, the LRT is a bit pessimistic.

For a typical motorbike, the frontal area is about 0.5 to 0.75, including a rider. My enfield comes in at 0.75 - How do I know?
I took a head on picture of a Bullet, with fully geared rider, made a silhouette of it and counted the pixels. A few calculations based on the wheel diameter and I got a ballpark calculation.

Fully geared rider on loaded bike
Profile of above picture
Thus what we have is that a certain speed requires a certain power at the wheel. So the next step is to formulate a function for power delivery.

This is quite simple : Take a dyno chart ( I did this for the CL500 and Fireball ) and put a few numbers say every 1000 RPM into a spreadsheet, and make a chart out of it.
Then you can ask Excel to derive a second order order polynomial function F such that P = F(RPM)

The function maps the dyno data quite well, and there is only a few % difference anywhere. The CL500 chart didn't fit well, so I did a linear approximation by filling in straight line approximations between data points.

The next step is to calculate the speed at each RPM for a given sprocket and tire size. We then use the LRT and Drag formulas to determine how much power is required to actually go at that speed. From that we derive the required torque for both formulas.

Thus we now have, for a given speed and a given gearing, how much torque is needed to maintain that speed.

Now we can divide the available torque by the required torque to get a value. If this value is greater than 1.0 then the engine can increase its RPM. At the point it reaches exactly 1.0 the engine will no longer be able to rev any further.

To get a sense of what is the ideal sprocket size, we can multiply the above value by the sprocket teeth to know what is the maximum sprocket size possible to rev out at that RPM. This will obviously not be very accurate for the lower RPMs, but it gives a pretty accurate picture in the later revs.

Plugging in different values into this, we can clearly see that within a certain range, changing the sprocket does not affect top speed much, since the fag end of the power curve is quite flat. But outside that range, the top speed is always reduced.

Here is the chart for the Fireball and CL500 respectively (Google docs link, opens in new window).

The highlighted fields show the RPM range where the required sprocket size goes below the actual sprocket size. That is the rev limit.

You can easily adapt this chart for any bike if you have a dyno chart image (ask me for the Excel sheet if interested)

The main benefit of a taller sprocket is to have a slightly reduced RPM for cruising. In the fireball at 19 teeth 19", the engine will be exactly at around peak power close to 5700 RPM, hitting the "ton". Which means it's geared perfectly right.
The CL500 also hits around 128 kph at 19T, 18" wheel at 4600 revs which is the peak power RPM.

I have chosen 0.6 as the c.o.d. value, but a slight variation in it causes quite a change in the values. But at least, relatively, you can see what happens with different sprockets.

No comments:

Post a Comment