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Quadratic Torque Curve

The analysis presented in the previous section was all based on the assumption of a constant torque output from the engine.  This can be used to provide some interesting insights and a straightforward means of estimating motor car performance.

In this section, a slightly more realistic expression for engine torque will be considered.  Choosing such a form is not easy – there are at least four conditions that need to be imposed on that torque curve:

1.                         1.  The torque curve will have a maximum value;

2.                  2.  The maximum will occur at some specified engine speed;

3.                  3.  The engine power will have some maximum value;

4.                  4.  The maximum power will occur at some defined engine speed.

Choosing an analytic form for the torque that can be fitted to all four constraints, and which can be substituted back into the equation of motion and integrated, is not particularly simple.  To make life easy, a quadratic expression is chosen:

where w, is the angular velocity (equal to 2pR_S), and a, b and c are constants that need to be determined.  The power P is then given simply by

It can be seen that these expressions only contain three independent constants, and so they can only be fitted to three of the conditions listed above.  It runs out to be slightly easier to fit to conditions 1, 2 and 4.  The end result is the following expression for the torque:

where:  w_T = angular velocity at which the peal torque occurs;

              w_P = angular velocity at which peal power occurs;

               t₀ = maximum value for the torque.

An expression of this type is said to be “homogeneous” in angular velocity.  As a consequence, the formula can be used with w, w_P and w_T expressed in any convenient units (e.g. revs per second, revs per minute, etc), provided that all three use the same units.

The equation of motion to be solved therefore takes the following form:

where:

The equation of motion can be rearranged to the following form:

where:

Acceleration Time and Example Application

After a degree of extremely tedious algebra, this becomes:

and there we have it, an analytic expression for the acceleration times of a car, between velocities v_i and v_f.  For my SEAT Toledo, the parameters characterising the torque and power are as follows:

            Maximum torque (t₀) of 208 lb-ft at 2000 RPM (w_T);

            Maximum power of 135 BHP at around 4200 RPM (w_P).

The torque and power curves derived using the quadratic approximation are shown below:

While these curves look plausible, it can be seen that the torque at low RPM is too high – below about 1400 RPM the turbocharger is not active, and the true torque at these revs will be much lower.  Similarly, the tail off of power over around 4800 RPM is much more dramatic than is indicated in this graph.  Nevertheless, for the rev range 1500 RPM to 4500 RPM, the representation of the torque curve is reasonably accurate.

The following table indicates the results obtained for acceleration from 30 mph upwards in third gear, with all parameters as given elsewhere in this article:

It can be seen that the time to travel from 30 mph to 50 mph is around 3.8 sec, and the time to travel from 30 to 70 mph is given as being about 8.6 seconds.  Both of these figures seem to be in accord with my general experiences of driving the car, and would seem to confirm that the analytic expression given above can usefully be used for estimating car performance.

Of course, the best way to calculate acceleration times, given an arbitrary torque curve, is to perform a numerical integration, using perhaps the Runge Kutta method as the integration technique – however, I’ve said quite enough and will not go into that here!

THE END