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

Chapter 3.0 Vehicle Design
3.3 Wing Aerodynamics
edge vortices, which contribute significantly to the forces in the case of flapping wing flight.
Hence, blade element theory also cannot be used.
3.3.3.2.6 Vortex Theory
This approach is probably the most applicable for use with the flapping wing, but here again no
physics based modeling for vorticity and wakes exist, and empirical data provided by Robin
Gray, Landgrebe in accordance with the Biot Savart Law and the Kutta Jowkousky Theorem is
the basis for implementation of this model. The data on free wake modeling is all experimental
and is specific to the point solutions by different researchers. Hence, here again experimental
data is required for the case of Entomopter. Also, this model only considers the trailing edge vortices
(mainly strong tip vortices), but leading edge vortex is still not included in this model.
3.3.3.2.7 Aeroelasticity Theory
Finally, the theory of aeroelasticity was considered to model the flapping wing aerodynamics.
Most of the researchers in the past have used this theory to predict the flapping wing performance.
Mainly Theodore Theoderson [254] has given closed form time-dependant expressions
for two dimensional flat plate lift and moment. I. E. Garrick [99] has given a closed form expression
based on the Theoderson work for average thrust computation, and Azuma [255] has given
closed form time dependant expression for lift, moment, thrust and power based on this theory.
The theory mainly addresses the mechanism of flutter instability and is used to find the structural
deformations and loads in conventional fixed wing flight. Even though the aeroelastic
deformations are generally smaller in nature under normal aerodynamic loads, this theory provides
reasonable estimates for larger deformations as suggested by above referenced researchers.
This theory takes into account four degrees of freedom, which are as follows:
1. Flapping or bending
2. Pitching or torsion
3. Aileron deflection
4. Tab deflection
For the Entomopter we need only consider two degrees of freedom: flapping and pitching. This
approach takes into account the trailing edge vortices and forces produced by flapping and pitching
in time domain, but still falls short of taking into account the strong leading edge vortex
effect. Also, this approach can only calculate forces in two dimensions and also does not depend
on the airfoil shape, instead assuming a thin flat plate. So, this model needs to be modified to
account for three dimensional finite wing effects. Many researchers in the past have tried to
arrest the finite wing effects based on this model, and more than twenty treatments exist for the
same. Some have given solutions for an elliptic wing, while others have assumed a uniform rectangular
wing. The treatment given by Eric Reissner [222] is geared toward a variable planform
shape and can help simulate the wing of Entomopter. This approach was selected to create an
analytical model for Entomopter.
3.3.3.3 Basic Methodology
The aerodynamic model proposed by Theodore Theoderson and I. E. Garrick for flutter analysis
(aeroelasticity) was taken as the baseline model [254]. But this model is only applicable for two
dimensional analysis, hence the three dimensional planform effects of the Entomopter will be
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