Tuesday, 25 August 2015

Some basic terms related to Aircraft - Part 8 - Mean Aerodynamic Chord

12. Mean Aerodynamic Chord (MAC)

  • Understanding this term is important to correctly set the CG of your scratch built model, if you have deviations from the original design.
  • Moreover this will help us to cross check the CG location suggested by the designer.
  • All plans indicate that the Center of Gravity (CG) should be located at a certain point of the Mean Aerodynamic Chord. This figure is usually given in percentage, but is sometimes a measurement.
  • The CG can be measured from any point along the span from the leading edge of the wing, if the wing is a rectangular (constant chord with no sweep) plan form.
  • If the wing is tapered then you must locate the MAC before you can locate the CG.

Mean Aerodynamic Chord (MAC) - which is the average for the whole wing.

  • The MAC calculation requires rather complicated mathematics, so a simpler method called 'Geometric Mean Chord'(GMC) or 'Standard Mean Chord'(SMC) can be used as given in the figures below. [ MAC is only slightly bigger than GMC except for sharply tapered wings. Taper ratio = tip chord/root chord.]

How to locate the Mean Aerodynamic Chord on a Tapered or Delta Wing
  • Measure the root and tip chord.  Then draw the following lines on the plans.
  • At the root of the wing, draw a line parallel to the center-line of the fuselage extending forward from the leading edge and rearward from the trailing edge.  Both lines should be the length of the tip chord.
  • Do the same thing at the tip but drawing the lines the length of the root chord.
  • Connect the ends of the lines so that they create an "X" over the wing panel.
  • Where the two lines intersect is the span wise location of the Mean Aerodynamic Chord.

To calculate MAC of a tapered wing, the following simplified equation may be used:
MAC = root chord * 2/3 * ((1+T+T2)/(1+T))    Where T is the wing's taper ratio.
The MAC distance from the center line may be calculated as follows:
Distance = half span * (1+2*T)/(3+3*T)

For a delta wing the CG should be located 10% ahead of the geometrically calculated AC point as shown above.
To Locate the Mean Aerodynamic Chord on an Elliptical Wing
The span wise location of the MAC is 1/2 wingspan x 0.4244.
The MAC is 0.9055 x the Root Chord.

The MAC of an elliptical wing is 85% of the root chord and is located at 42.4% of the half wingspan from the root chord.   Elliptical wing's area = pi * wingspan * root chord/4.
The AC location for biplanes with positive stagger (top wing ahead of the bottom wing), is found according to the drawing below.

If the plan indicates that the CG should be located at some percentage of the MAC, then measure the MAC and put the CG the given percentage back from the leading edge along the MAC.

For example, if the MAC is 10" and the plan indicates the CG should be 25% back from the leading edge, then the CG is 2-1/2" back from the leading edge at the MAC.

Note:  The lines cross at the span wise location of the MAC.  It is not the fore/aft CG location (unless the CG happens to be located at 50% MAC).
  • For conventional designs (with main wing and horizontal stab) the CG location range is usually between 28% and 33% from the leading edge of the main wing's MAC, which means between about 5% and 15% ahead of the aircraft's  Neutral Point NP. This is called the Static Margin, which is expressed as a percentage of MAC.
  • When the static margin is zero (CG coincident with NP) the aircraft is considered "neutrally stable“. However, for conventional designs the static margin should be between 5% and 15% of the MAC ahead of the NP.
  • The CG location as described above is pretty close to the wing's Aerodynamic Center (AC) because the lift due to the horizontal stab has only a slightly effect on the conventional R/C models.
  • However, those figures may vary with other designs, as the NP location depends on the size of the main wing vs. the stab size and the distance between the main wing's AC and the stab's AC.
  • The simplest way of locating the aircraft's NP is by using the areas of the two horizontal lifting surfaces (main wing and stab) and locate the NP proportionately along the distance between the main wing's AC point and the stab's AC point.
For example, the NP distance to the main wing's AC point would be: D = L • (stab area) / (main wing area + stab area) as shown on the picture below:
  • There are other factors, however, that make the simple formula above inaccurate.
  • In case the two wings have different aspect ratios (different dCL/d-alpha) the NP will be closer to the one that has higher aspect ratio.
  • Also, since the stab operates in disturbed air, the NP will be more forward than the simple formula predicts.
The figure below shows a somewhat more complex formula to locate the NP - but would give a more accurate result using the so called Tail Volume Ratio, Vbar.
This formula gives the NP position as a percentage (%) of the wing's MAC aft of the wing's AC point.

For those who are not so keen on formulas and calculations there is the online - Aircraft Center of Gravity Calculator (http://adamone.rchomepage.com/cg_calc.htm), which automatically calculates the CG location as well as other useful parameters based on the formula above.
Note:- If you are building your RC-Plane based on a well proven model and plan, then all these calculations are not required, just follow the designers instruction, but having an idea about these terms and calculations will give you more confidence when you fly the aircraft and observe something wrong about the flight characteristics of your model. You can analyse what went wrong during construction of your model and can take the corrective action as required.

Scratch built Futana 3D Plane - Balsa Wood Frame and Polythene Film Skin 

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