Page 25Algebra Comes to the Rescue

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The power curves
take a lot of time to create.
The chart on the previous page had 12
sets of power level readings.
At three or more readings per power level, at one minute intervals, that can be 40 minutes.

If we use some algebra, this time can be reduced dramatically.

When we ask the computer to fit a curve to the data points it comes up with
an equation of the type ………
y= ax2 + bx + c
A “hilltop” chart. As we saw on pages 21 and 23.

Now, back in my days at school, we were taught differentiation.

So differentiating the equation above. .....dy/dx = 2ax + b

And this represents the gradient of the power curve for any value of
x, the engine speed.

But at the peak of the curve, the gradient is zero, so
2ax + b = 0 for the peak power point.
Or x = peak power engine speed
= – b / 2a

Wouldn’t it be good if we can find values for
a and b.

We can – they specify
the downward slope of the torque chart. (a is negative)

So, if we plot torque vs engine speed
and ask the computer to do a straight line fit,
it will give us the values for a and b, See the chart opposite.

From the experimental measurement viewpoint, getting data for a straight line
is far easier than for a curve. Only two points are needed. So, for experimental rigour,
how about we take readings for three load points on the engine.
Three points in a straight line prove that there is no curve.

Three measurements at each load gives nine readings.
At one minute per reading, the job is over
in ten minutes!

On the torque chart above, “Mr are we sure about this?”, has run seven load settings.

The slope of the line is downward, so
a is negative. a = – 0.051

b is the intercept on the y axis when x=0. ie b= 6.078

Please note that this chart is Newton meters vs Radians per second.
(Multiply the radians by 10 to get an approx RPM value)
This is the SI system (Le Système International) – no 360 degrees, no
minutes, no ft-lbs.
Instead - Radians, seconds and Newtons. Let’s use “this century” science.

From above,
we said the peak power speed is at – b / 2a = – 6.078 / (2 x – 0.051)
The minus signs cancel and
6.078 / 0.102 = 59.58 Rads/sec

And with 9.549 radians to 360 degrees, peak speed is at
569 RPM.