nonlinear approximate aeroelastic analysis of flapping wings in hover and forward flight

197 Pages·2011·5.31 MB·English

NONLINEAR APPROXIMATE
AEROELASTIC ANALYSIS OF FLAPPING
WINGS IN HOVER AND FORWARD
FLIGHT
by
Abhijit Gogulapati
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Aerospace Engineering)
in the University of Michigan
2011
Doctoral Committee:
Professor Peretz P. Friedmann, Chair
Professor Carlos E. Cesnik
Associate Professor Luis P. Bernal
Associate Professor Bogdan Epureanu
Abhijit Gogulapati 2011
©
All Rights Reserved
To Amma, Nanna, Madhu, and Vidya
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ACKNOWLEDGEMENTS
I will be ever grateful to my advisor Prof. Peretz Friedmann for being a friend and
a mentor through the various stages of my doctoral studies. His emphases on clarity
of thought, quality of research, and thoughtful presentation, have provided valuable
lessons that I am unlikely to forget.
I acknowledge support from the Air Force Office of Scientific Research (AFOSR),
which sponsored this research through the Multi-University Research Initiative
(MURI) grant with Dr. Douglas R. Smith as program director. I am thankful also to
Prof. CarlosCesnik, Prof. LuisBernal, andProf. BogdanEpureanu, forservingonmy
thesis committee.
I have been fortunate to collaborate with several excellent researchers during
the course of this work. I thank them for their vital contributions to this thesis: Dr.
Hikaru Aono for generating the CFD based results for the rigid wings; Dr. Patrick
Trizila, Dr. Li Liu, and Ashwani Padthe, for the rigid airfoil cases; Eugene Kheng
and Paul Davidson for prompt help in conducting experiments on the CAPRAN film;
Dr. Satish Chimakurthi for providing results for the isotropic wings; and Dr. Pin Wu
and Robert Love from the University of Florida at Gainsville for providing samples
and results for the anisotropic wings, material properties, and discussions. Special
thanks also go to Prof. Carlos Cesnik, Prof. Anthony Waas, Dr. Hikaru Aono, and
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Chang-Kwon Kang for the illuminating discussions and timely inputs that nudged
me in the right direction, and Prof. Wei Shyy for his initiative in bringing together
several research groups that were working on the flapping wing problem.
It was an absolute pleasure to share the working environment with Dr. Li Liu,
Nicholas Lamorte, Ashwani Padthe, Dr. Bryan Glaz, and Eric Muir. I could not have
asked for better officemates and thank them for putting up with my ’thinking-out-
loud’ sessions. Denise Phelps, Suzanne Smith, Cindy Enoch, and Michelle Shepherd,
also deserve considerable praise for their ever friendly attitude and making sure the
administrativesideofthingswentwithoutincident. ThanksalsotoDaveMcCleanfor
excellent computer-related advice.
I owe a great debt of gratitude to friends - Amit Salvi, Paul Davidson, Christian
Heinrich, Nalin Chaturvedi, Shiladitya Basu, and several more - who have made my
stay in Ann Arbor so enjoyable, enriching, and memorable. Life was always upbeat
and never dull with you around. Finally, to my parents, sister Madhulika, and fiance
Vidya, who have been my source of strength throughout this journey: none of this
would have been possible without the unconditional support and love you have given
me.
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TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
CHAPTERS
I. Introduction, Literature Review, and Objectives . . . . . . . . . . 1
1.1 Existing Micro Air Vehicles . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Aeroelastic Analysis of Flapping Wings . . . . . . . . . . . . 4
1.2.1 Structural Dynamic Considerations . . . . . . . . . . . 5
1.2.2 Unsteady Aerodynamic Mechanisms . . . . . . . . . . . 7
1.3 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Modeling Wing Flexibility . . . . . . . . . . . . . . . . . 9
1.3.2 Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . 12
1.3.3 Aeroelastic Studies and Effect of Wing Flexibility . . . 17
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Novel Contributions of the Dissertations . . . . . . . . . . . . . . 24
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II. Structural Dynamic Modeling. . . . . . . . . . . . . . . . . . . . . . . 26
2.1 Wing Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Representation of Rigid Body Rotations . . . . . . . . . 26
2.1.2 Implementation in MARC . . . . . . . . . . . . . . . . . 30
2.1.3 Kinematic and Rigid Body Constraints . . . . . . . . . 31
2.2 Modeling Composite Materials . . . . . . . . . . . . . . . . . . . . 33
III. The Approximate Aerodynamic Model . . . . . . . . . . . . . . . . . 36
3.1 Summary of the Original Model . . . . . . . . . . . . . . . . . . . 36
3.1.1 Overview of the Aerodynamic Formulation . . . . . . . 37
3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3 Modifications Introduced . . . . . . . . . . . . . . . . . . 40
3.2 Description of the Modified Approach . . . . . . . . . . . . . . . . 41
3.2.1 Extension to Forward Flight . . . . . . . . . . . . . . . . 41
3.2.2 Airfoil Degrees of Freedom and Spanwise Flexibility . 45
3.2.3 Quasi-steady Vorticity and Circulation . . . . . . . . . 48
3.2.4 Effect of Fluid Viscosity . . . . . . . . . . . . . . . . . . . 55
3.2.5 Determination of Shed Vorticity . . . . . . . . . . . . . 56
3.2.6 The Vortex Wake Model . . . . . . . . . . . . . . . . . . 60
3.3 Calculation of Aerodynamic Loads . . . . . . . . . . . . . . . . . . 62
3.3.1 The Vortex Impulse Method . . . . . . . . . . . . . . . . 62
3.3.2 Unsteady Bernoulli Equation . . . . . . . . . . . . . . . 63
3.4 Limitations of the modified aerodynamic model . . . . . . . . . . 66
IV. Aeroelastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 The Updated Lagrangian Approach . . . . . . . . . . . . . . . . . 68
4.2 Fluid-structure Coupling in MARC . . . . . . . . . . . . . . . . . 74
V. Structural Dynamic Model: Validation and Comparisons . . . . 76
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5.1 Implementation of Wing Kinematics . . . . . . . . . . . . . . . . 76
5.2 Centrifugal Stiffening Effect . . . . . . . . . . . . . . . . . . . . . 81
5.3 Comparison of Mode Shapes and Frequencies for
Anisotropic Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1 Tensile Tests on the CAPRAN Membrane . . . . . . . 83
5.3.2 Comparison of Mode Shapes and Frequencies . . . . . 85
5.4 Flapping Tests in Vacuum: Comparison of Tip Displacements 92
VI. Verification of the Aerodynamic Model . . . . . . . . . . . . . . . . 95
6.1 Airfoil Cases: Comparison with CFD for separated flow . . . . 98
6.2 Rigid Zimmerman Wings in Hover . . . . . . . . . . . . . . . . . 105
6.3 Rigid Zimmerman Wing in Forward Flight . . . . . . . . . . . . 113
VII. Aeroelastic Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Preliminary Calculations using the Aeroelastic Interface . . . . 119
7.2 Flexible Flapping Wings in Hover . . . . . . . . . . . . . . . . . . 120
7.3 Flexible Flapping Wings in Forward Flight . . . . . . . . . . . . 136
VIII. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . 140
8.1 Principal conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.2 Recommendations for future work . . . . . . . . . . . . . . . . . . 144
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
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LIST OF FIGURES
Figure
1.1 Insect, birds, and MAVs [3] . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Insect and bio-inspired wings . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Insect flapping stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Mid-plane of a 4-noded shell element and local coordinate system. . . 35
3.1 Schematic of the aerodynamic formulation. . . . . . . . . . . . . . . . . 38
3.2 Spanwise sections and wing-fixed axes. . . . . . . . . . . . . . . . . . . 38
3.3 Radial chords used in the original approach. . . . . . . . . . . . . . . . 40
3.4 Coordinate system that is fixed to the stroke plane. Free stream
velocity vector and cylindrical surface are also shown. . . . . . . . . . 42
3.5 Normal cylinder described by the wing section. . . . . . . . . . . . . . 42
3.6 Flapping wing vehicle in forward flight. Shaded region indicates shed
wake surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7 Coordinate systems used in the computation of airfoil camber. . . . . 44
3.8 Degrees of freedom of the airfoil and coordinate systems used. . . . . 46
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3.9 Topviewofstrokeplaneshowingthetimedependentradiusofnormal
cylindrical surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.10 Component of u∞ normal to the instantaneous position of the wing. . 49
3.11 Airfoil and corresponding circle showing shear layers.. . . . . . . . . . 50
3.12 Distances and angles that are used in the computation of velocity
potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.13 Circle, corresponding airfoil, and roots of the inverse transform . . . 60
3.14 Contours of integration for computing velocity potential from bound
and shed vorticity from an airfoil. For clarity, shed vortices are indi-
cated by solid circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Formulation of the aeroelastic equations (left) and implementation of
the aeroelastic model in MARC (right). . . . . . . . . . . . . . . . . . . . 69
5.1 Rectangular plates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Implementation of rigid body rotations in MARC. . . . . . . . . . . . . 79
5.3 Comparison of tip displacements for an accelerating plate. Results 1
and 2 (solid lines) from Ref. [70]; result 3, circles, MARC current study . 81
5.4 Anisotropic wing configurations, from Ref [69]. . . . . . . . . . . . . . . 82
5.5 Painted (left) and unpainted wings (right). . . . . . . . . . . . . . . . . 83
5.6 Tensile test specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8 Specimens of the film. Dense speckle (heavy dots) - left and light
speckle (light dots) - right. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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