from the April 1998 issue of the DC/RC Newsletter

Art Kresse
 

Way back in July '97 "Uninformed" posed this question. "Is the CG, on biplanes, put in the same place it is on monoplanes; about 25% MAC (Mean Aerodynamic Chord(1))? If so which wing does one use?

Recently on Jan 11 "just and old toy maker"responded; "....you draw a line from 25% MAC on the top wing to 25%MAC bottom wing the bisect this line using foil camber line as endpoints."

Both the question and the response are interesting and important. The answer is correct for certain special cases:

1) Both wings are the same size.

2) Both are set at the same angle of incidence.

3) The effect of the tail can be ignored.

To get rid of the first two of these restrictions, consider figure 1.

From high school physics you'all recall(?) that for an object to be at rest (or in steady flight) the sum of forces and the sum of the moments about some reference point must equal zero. You also recall that a moment is a force times the perpendicular distance to the reference. For our case the reference is REF. We must pay attention to sign (+ or -). We can arbitrarily call "up" positive and "down" negative and distances to the right of REF as positive. To simplify the algebra we can assume that the lift forces "L " are some constant k times the angle of attack "" and the wing area A. i.e. the

Lift Equation

.

If we denote "upper" and "lower" by the subscript u and l respectively, we can write two so-called equilibrium equations one for the forces and one for the moments as follows:

Force Equation:

Moment Equation:

By solving the three equations simultaneously (high school algebra) we get a very tedious expression for the location of the cg relative to REF.

,

Where is the difference between the upper and lower wing incidence. If we let


 

then

.

Furthermore, if we let

in this equation then:

which is the answer offered by "little old toy maker"

It is worth noting that a difference in incidence between the upper and lower wing can be either stabilizing or destabilizing in pitch, depending on whether the difference is positive or negative. The proof of this theorem is left as an exercise for the student.

The effect of the tail can be included with an expansion of the method shown here for the wings alone. (Another exercise.)

A final word of caution-this is an approximation which neglects some important effects like the interference between the two wings and the downwash of the wings on the tail. Hopefully however it will get you in the ballpark so the airplane can be flown safely. Any fine tuning needed can be done with the elevator trim which is why it is there in the first place-even for the big guys.


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