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