Phase II Final Report - NASA's Institute for Advanced Concepts

Chapter 3.0 Vehicle Design
3.1 Wing Sizing
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L = ρ
2
VresCl
A
Equation 3-3
w
2
Figure 3-6: Resultant Force and Velocity for the Flapping Wing
The resultant velocity will vary along the wing length, so the lift will not be constant along the
wing. The resultant velocity can be expressed by Equation 3-4, where θ is the maximum angle
the wing moves through, r i is the incremental distance along the wing, f is the flapping frequency,
and V is the free-stream (or flight) velocity. The flight velocity was chosen to be 15 m/s
for this initial analysis. This equation represents the average velocity throughout one flap of a
section of wing a distance r from the root.
V res = (θr i 4 f ) 2 + V 2 Equation 3-4
From the environmental section the density of the atmosphere near the surface is approximately
0.0145 kg/m 3 . The lift coefficient is based on the vortex formation and shedding that occurs on
each flap as well as the vented gas blowing along slots in the trailing edge of the wing. Lift coefficients
generated by insects due to flapping are on the order of 5. By adding blowing it is estimated
these lift coefficients can be increased two to three times that. For this analysis a lift
coefficient of 10 was used which represents the lower end of this estimate. The wing chord also
varies along the wing length. Therefore, the wing area of an incremental section of wing will not
be constant along the wing length and will vary based on the chord variation. From the structures
section the chord (c) as a function of wing location is given by Equation 3-5.
c = 0.32814 + 2.61643 r – 9.1414 r 2 + 15.642 r 3 – 12.951 r 4 + 4.0584 r 5
Equation 3-5
The total lift generated by one of the wing segments can be approximated by the summation
shown in Equation 3-6.
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