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

Chapter 3.0 Vehicle Design
3.2 Wing Motion and Structure Analysis
Subsequently the derivatives of θ with respect to time are given in Equations 3-15 and 3-16.
dθ / dt = -a [ sin(b t + c)] b Equation 3-15
d 2 θ / dt 2 = -a [ cos(b t + c)] b 2 Equation 3-16
For this initial 2D case it was assumed that the wing had a thickness of 1 cm and a wing section
mass (one half of a full wing) of 0.75 kg. This is based on a wing material density of 1000 kg/
m 3 . Since with wings are symmetrical the loading on each wing section will be the same. Therefore
the analysis is for only one wing section.
The chord of the wing varies along the wing length This variation in chord length with radius is
shown in Figure 3-12. Based on this figure the chord change as a function of radial station can
be calculated. A regression was performed to determine an expression for the chord length as a
function of radial location along the wing section. Equation 3-17 represents this curve fit normalized
to a wing section length of 1.
Figure 3-12: Plan-view of Entomopter Wing
c = 0.32814 + 2.61643 r – 9.1414 r 2 + 15.642 r 3 – 12.951 r 4 + 4.0584 r 5 Equation 3-17
The effect of this variable chord geometry on the mass distribution along the wing is shown in
Figure 3-13. This mass was calculated for a solid wing which is not the optimal structural
design. However this was done to demonstrate how the mass distribution is affected by the variable
chord length of the wing. The mass was calculated in ten equal increments along a 0.5m
long wing section. The figure shows the comparison in wing mass at sections along the wing
length between the variable chord wing shown in Figure 3-20 and a fixed chord wing with an
aspect ratio of 2.5. The total mass of both wing types are similar (1.0 kg for the fixed chord wing
section and 0.9 kg for the variable chord wing section). The reduction in mass on the outer portion
of the wing has a significant effect in lowering the loading on the wing. Since the acceleration
loads increase along the wing length, mass reduction on the outer portion of the wing will
have the greatest effect in reducing the overall wing loading.
53