AERODYNAMIC CALCULATIONS
By Bob Hoey
Recent research on R/C bird models has produced two results that should be of interest to flying wing advocates. I found that the usual approximations for locating the aerodynamic center (ac) for a complex wing planform (like a bird wing), were producing results that were not conservative.  I went back to the basic definitions and derived a method for dividing the wing into segments and doing a piecewise integration over the semi-span of the wing. This method produces a value for the mean aerodynamic chord, and a location for the aerodynamic center (a good starting point for the cg location for a flying wing). The method has been quite helpful to me in bird model design and construction, but I would appreciate a review by some of the other aero engineers within TWITT to make sure I didn't mess up the math.
     The second result is an empirical method for determining the amount of dihedral and wing sweep necessary for a wing-alone to be stable in the lateral-directional axis. The method estimates the amount of rolling and yawing produced by the wing when in sideslip. It assumes an elliptical lift distribution across the wing span, and uses small-angle approximations for the effects of dihedral and wing sweep. The result is a "number" (which I have called "dihedral effect"), for any wing shape. This number has no physical significance, but serves as an index to compare the relative stability of the wing with that of other wing shapes.  Experience on bird models indicates that the number should be between 10 and 20 for the wing to be stable without a vertical tail.

     The MAC and aerodynamic center table was assembled rather quickly. It seems to work, but I would really like for some of the aero engineers to check it out and make sure I didn't mess up the math.

I assumed the same airfoil over the entire wing. (Cmac = constant).  I then did a piecewise integration over the wing semispan of the equation;

Cmac *q*S*MAC = (integral) Cmac *q*c^2*dy
then solved for MAC.

The location is;
CL *q*S*xbar = (integral) Cl*q*xbar*c*dy
then solved for xbar.

Any comments would be appreciated.

Bob Hoey
bobh@patprojects.org
 

Locating the Aerodynamic Center 

The aerodynamic center (ac) of a wing is a point on the wing chord which results in a constant moment when the wing angle of attack is changed. If the cg is at the ac, the forces on the tail will be minimized.
     There are several approximate methods for locating the ac of a square or tapered wing.  These methods can be misleading for the complex wing shape of a bird.  The method described here is a bit tedious, but provides the best answer for any wing.
     Using the planform for one wing, establish a reference that is perpendicular to the fuselage. This reference line should be near where the spar will be, but the fore and aft location is not important.  Divide the semi-span into 10 equal segments.  In the center of each segment measure the chord and the distance from the leading edge to the reference line.  Enter these numbers into the MAC Worksheet as shown in the example. Complete the calculations across the sheet for each segment.  Now sum the chord, chord-squared and chord-times-x columns. The "mean aerodynamic chord" (MAC) for the wing is the chord-squared sum divided by the chord sum.  The location of the leading edge of the MAC (from the reference line) is the chord-times-x sum divided by the chord sum.  (These calculations are easily done on a computer spreadsheet.)  Locate the MAC on your wing plan relative to your reference line.  Now locate the 1/4 chord of the MAC. This is the aerodynamic center of your wing, and a good starting point for the center of gravity. (Notice that merely averaging the chord values or the 1/4 chord locations will produce an ac that is .25 inches farther aft then the correct value for the Pelican.)

AERODYNAMIC CENTER CALCULATIONS
(PELICAN EXAMPLE)



 
Defining the Wing Dihedral Effect

Using the planform for one wing, establish a reference that is perpendicular to the fuselage. This reference line should be near where the spar will be, but the fore and aft location is not important.  Divide the semi-span into 10 equal segments. Construct a line that connects the 1/4 chord locations for each segment.  Using a protractor, measure the average wing sweep (angle relative to the reference line) of the 1/4 chord line for each individual segment (negative sweep is tip forward of root). Using a front view of the wing, measure the average dihedral for each of the wing segments (negative dihedral is tip lower than root).  Enter these numbers into the Dihedral Worksheet as shown in the example.  Add the dihedral and wing sweep together for each segment, and multiply the result by the dihedral factor to produce the "roll" column. (The dihedral factor is a table of non-dimensional constants for any wing that account for the spanwise distribution of lift and rolling moment). Now sum the numbers in the "roll" column.  This number is related to the dihedral effect (roll-due-to-sideslip) that will be produced by your wing shape.  This number should be between 10 and 20 for a stable airplane.  You can now use small design alterations in wing sweep or dihedral, especially near the outer third of the span, to bring the dihedral effect into the proper range.

DIHEDRAL/SWEEP CALCULATIONS
(PELICAN EXAMPLE)



 

...6/16/02
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