development of methodologies of aeroelastic analysis for the design of flexible aircraft wings

7/25/2019 DEVELOPMENT OF METHODOLOGIES OF AEROELASTIC ANALYSIS FOR THE DESIGN OF FLEXIBLE AIRCRAFT WINGS

1/203

Dissertation presented to the

In

stituto Tecnol6gico de Aeronautica: in

partial fulfillment of the requireme

nt

s for the degree of Master of Science

in

th

e

Pro

g

ram

of Ae

ronauti

cal

and

Mechanical

En

gineering

Fi

eld of Solid

Mechanics and St ructures.

Marcos sar Ruggri

DEVELOPMENT

OF METHODOLOGIES OF

EROEL STIC N LYSIS FOR THE DESIGN OF

FLEXIBLE IRCR FT

WINGS

Dissertation approved in

it

s fi

na

1 version by signatories below:

Advisor

Co-advisor

Prof

D

r

Luiz

a

rlos

San

doval Goes

Prorector of Graduate

St

udies

and

Research

amp

o M

ont

enegro

Sao Jose dos

amp

os SP - Braz

il

2015


2/203

Cataloging in Publication

Data

Documentation and

Information Division

Ru

ggeri, Marcos Cesar

Development of methodologies

of

aeroelastic analysis for the design of flexible aircraft wings /

Marcos Cesar Ruggeri.

Sao Jose dos Campos, 2015

202p.

Dissertation of Master of Science - Course of Aeronautical and Mechanical Engineering. Area of

Solid Mechanics a

nd

Structures - Instituto Tecnol6gico de Aeronautica., 2015.

Ad

visor: Prof. Dr.

Robe

rt

o Gil Annes a Silva. Co-advisor: Prof. Dr. Carlos Eduardo de Souza.

1

Asas

fl

exiveis. 2. Aeroelasticidade. 3. Corpos ftexiveis. 4.

Vibra c;ii o

aeroelastica.

5. Ca.racteristicas dinamicas. 6. Engenha.ria. a.eronautica. I. Inst ituto Tecnol6gico de

Aeronautica . II. Title.

BIBLIOGRAPHIC

REFERENCE

RUGGERI, Marcos Cesar.

Development of methodologies of aeroelastic analysis

for

the design of flexible

aircraft

wings

. 2015. 202p. Dissertation of Master of

Science - Instituto Tecnol6gico de Aeronautica, Sao

Jo

se dos Campos.

CESSION

OF RIGHTS

.AUTHORS NAME:

Marcos Cesar Ruggeri

PUBLICATION TITLE: Development of methodologies of aeroelastic analysis for t he

design of

fl

exible aircraft wings.

PUBLICATION KIND/YEAR: Disse

rt

at

ion/

2015

It is granted

to

In

stituto

Tecnol6gico de Aeronautica permission

to

reproduce copies of

this disse

rt

at

ion

and

to

only loan or

to

sell copies for academic and scienti

fic

purposes.

Th

e author reserves other publi

ca t

ion rights and no

part

of this dissertation can be

r

ep

roduced

wi

t hout

th

e authoriz

at

ion of t he author.

Marc Cesar Ruggeri

u a ~

pompo de Vasconcelos, 375

12.243-830 - Sao

Jo

se dos

Ca

mpos -

SP


3/203

iii

DEVELOPMENT OF METHODOLOGIES OFAEROELASTIC ANALYSIS FOR THE DESIGN OF

FLEXIBLE AIRCRAFT WINGS

Marcos Cesar Ruggeri

Thesis Committee Composition:

Prof. Dr. Mauricio Vicente Donadon President - ITAProf. Dr. Roberto Gil Annes da Silva Advisor - ITAProf. Dr. Carlos Eduardo de Souza Co-advisor - UFSMProf. Dr. Flavio Luiz de Silva Bussamra Internal Member - ITADr. Olympio Achilles de Faria Mello External Member - EMBRAER

ITA


4/203


5/203

v

To all my family, who gave me education

and helped me in my hard times. To

Prof. Carlos Carlassare (in memoriam),

for motivating me to never throw down

my arms and continue learning.


6/203


7/203

vii

Acknowledgments

First of all, thanks to Professor Roberto Gil Annes da Silva for trusting and giving me

the chance to study at ITA, for teaching me aeroelasticity as well as for his contributions

and ideas during my studies. Thanks to my co-advisor Carlos Eduardo de Souza also

for helping me with the development of this work and for proposing new computational

approaches from his point of view.

I must also acknowledge to the CAPES Research Promotion Agency from the Ministry

of Education of Brazil for the financial support by granting me a scholarship. In addition,

thanks to the Brazilian Air Force for giving me the opportunity to use the facilities and

services of the CTA campus, including food at the H15 building.

Among the promoters to continue with my graduate studies, I am deeply indebted to

Professors Carlos Olmedo and Miguel Bavaro from UTN-FRH, who always encouragedmy decision to pursue my masters degree in Brazil.

Lastly, to Gabrielle Leithold, a person who not only taught me to speak Portuguese

and motivated me to move to Brazil, but also whom I loved and will never forget.


8/203


9/203

ix

I can state flatly that heavier-than-air

flying machines are impossible.Kelvin, Lord William Thomson - 1895


10/203


11/203

xi

Abstract

This work deals with several computational methodologies for the aeroelastic study of

flexible aircraft wings on a preliminary design phase. An in-house vortex lattice method

code named VLM4FW has been implemented with correction of sidewash and backwash

effects to take into account the aeroelastic deformation of the wing in bending and tor-

sion. In addition, corrections on the spanwise distribution of induced drag based on the

cross-flow energy in the wake have been included. This code has been also programmed to

be coupled in a co-simulation scheme with Abaqus for aeroelastic geometrical non-linear

simulations and compute steady flight loads. Then, based on the deformed wing con-

figuration new natural frequencies and mode shapes are extracted in MSC.Nastran with

the solution sequence SOL 103. Flutter studies are next performed using the ZONA6 g-

Method in ZAERO to analyze the dynamic aeroelastic instability and evaluate the results

compared to the undeformed initial wing shape. Several case studies have been adoptedto validate the VLM4FW program with rigid and flexible wings, such as the AE-249 and

GNBA aircraft. Depending on the wing aspect ratio and flexibility, the results obtained

give a clear idea of how important is the deformed configuration for the study of dynamic

aeroelastic instabilities. The fact of considering the initial wing shape to perform a flutter

analysis can lead to large discrepancies in the estimated critical speeds, and even worse,

overestimate the real values. Flutter analyses based on geometrical nonlinear deformed

wings are assumed to be conservative for the preliminary design condition and are ex-

pected to provide better results as technological advances introduce higher aspect ratioson very flexible wings.


12/203


13/203

xiii

Resumo

Este trabalho lida com diversas metodologias computacionais para o estudo aeroelas-

tico de asas flexveis de aeronaves em uma fase de projeto conceitual. Um codigo interno

de vortex lattice method chamado VLM4FW foi implementado com correcao dos efeitos

de sidewash e backwash para ter em conta a deformacao aeroelastica da asa em flexao

e torcao. Alem disso, foram includas correcoes na distribuicao ao longo da envergadura

do arrasto induzido com base na energia do escoamento transversal. Este codigo foi tam-

bem programado para ser acoplado em um esquema de co-simulacao com Abaqus para

simulacoes aeroelasticas nao-lineares geometricas e calcular o carregamento em voo esta-

cionario. Em seguida, a partir da configuracao deformada da asa sao extradas novas

frequencias naturais e formas modais em MSC.Nastran com a solucao SOL 103. Estudos

de flutter sao realizados logo utilizando o Metodo-g ZONA6 em ZAERO para analisar a

instabilidade dinamica aeroelastica e avaliar os resultados em comparacao com a formada asa inicial nao deformada. Varios estudos de caso tem sido adotados para validar o

programa VLM4FW com as asas rgidas e flexveis, tais como as asas AE-249 e do aviao

GNBA. Dependendo da relacao de aspecto da asa e a propria flexibilidade, os resultados

obtidos dao uma clara ideia de quao importante e a configuracao de asa deformada para

o estudo de instabilidades aeroelasticas dinamicas. O fato de considerar a forma da asa

inicial para realizar uma analise de flutter pode levar a grandes erros nas velocidades crti-

cas estimadas e, pior ainda, sobreestimar os valores reais. Analises de flutter baseados

em asas deformadas com nao-linearidade geometrica sao considerados ser conservadorespara uma condicao de projeto conceitual e espera-se ainda que proporcionem melhores

resultados a medida que os avancos tecnologicos introduzam maiores alongamentos em

asas muito flexveis.


14/203


15/203

xv

List of Figures

FIGURE 1.1 Collar triangle with the interaction of aeroelastic forces. . . . . . . . 30

FIGURE 1.2 Handley-Page O/400 bomber aircraft, pioneer in the flutter phe-

nomenon.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

FIGURE 1.3 Grumman X-29 aircraft operated by NASA/USAF. . . . . . . . . . 34

FIGURE 1.4 Finite element representation for a global structural analysis. . . . . 35

FIGURE 1.5 Philosophy of computational panel methods to model lifting surfaces. 36

FIGURE 1.6 Aerodynamic and aeroelastic simulations on the EMB314 aircraft. . 37

FIGURE 3.1 Typical wing planform modeled by the Vortex Lattice Method. . . . 46

FIGURE 3.2 Geometric parameters of an aerodynamic panel in VLM. . . . . . . 47

FIGURE 3.3 Horseshoe vortex on an elementary aero panel of the right semi-wing. 48

FIGURE 3.4 Geometry of the mean camber surface for the tangential flow BCs. . 51

FIGURE 3.5 Chordwise horseshoe vortices and control points for backwash and

sidewash computation (left semi-wing). . . . . . . . . . . . . . . . . 55

FIGURE 3.6 Dependence of Theodorsens function on reduced frequency.. . . . . 67

FIGURE 3.7 Typical section in oscillatory harmonic motion. . . . . . . . . . . . . 68

FIGURE 4.1 Inter-communication between VLM4FW and Abaqus. . . . . . . . . 82

FIGURE 4.2 Software packages used for flight loads and flutter analyses. . . . . . 83

FIGURE 5.1 Right semi-wing of Garner test case. . . . . . . . . . . . . . . . . . . 86

FIGURE 5.2 Distribution of pressure coefficient on the Garner wing. . . . . . . . 87

FIGURE 5.3 Distribution of lift coefficient on the Garner wing. . . . . . . . . . . 88

FIGURE 5.4 Distribution of angles of attack on the Garner wing. . . . . . . . . . 88

FIGURE 5.5 Distribution of spanwise/chordwise lift on the Garner wing. . . . . . 89


16/203

xvi

FIGURE 5.6 Comparison of spanwise induced drag on the Garner wing. . . . . . 89

FIGURE 5.7 Right semi-wing of Saunders test case (config. = 0

). . . . . . . . 90

FIGURE 5.8 Distribution of pressure coefficient on the Saunders wing (dihedral

cases). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

FIGURE 5.9 Distribution of lift coefficient on the Saunders wing (dihedral cases). 91

FIGURE 5.10 Distribution of angles of attack on the Saunders wing (dihedral cases). 92

FIGURE 5.11 Distribution of spanwise/chordwise lift on the Saunders wing (dihe-

dral cases). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

FIGURE 5.12 Distribution of induced drag on the Saunders wing (dihedral cases). 93

FIGURE 5.13 Comparison of wing lift coefficient slope on the Saunders wing. . . . 93

FIGURE 5.14 Aerodynamic mesh of AE-249/B test case (config. 4 x 10). . . . . . 96

FIGURE 5.15 Aerodynamic mesh of AE-249/B test case (config. 6 x 10). . . . . . 96

FIGURE 5.16 Structural mesh of AE-249/B test case (config. 4 x 10). . . . . . . . 97

FIGURE 5.17 Structural mesh of AE-249/B test case (config. 6 x 20). . . . . . . . 97

FIGURE 5.18 Convergence of wing lift coefficient slope on the AE-249/B wing

(rigid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

FIGURE 5.19 Normal force distribution on the AE-249/B wing (rigid mesh 2 x 64). 98

FIGURE 5.20 Normal force distribution on the AE-249/B wing (rigid mesh 8 x 64). 99

FIGURE 5.21 Z Displacements on the AE-249/B wing (mesh 4 x 10). . . . . . . . 1 01

FIGURE 5.22 Distribution of pressure coefficient on the AE-249/B wing (mesh 4

x 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

FIGURE 5.23 Distribution of lift coefficient on the AE-249/B wing (mesh 4 x 10). 102

FIGURE 5.24 Distribution of angles of attack on the AE-249/B wing (mesh 4 x 10). 102

FIGURE 5.25 Distribution of spanwise/chordwise lift on the AE-249/B wing (mesh

4 x 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

FIGURE 5.26 Distribution of induced drag on the AE-249/B wing (mesh 4 x 10). . 103

FIGURE 5.27 Influence of ballast offset on the flutter speed of AE-249 wing (linear

theory). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

FIGURE 5.28 Distribution of pressure coefficient on the AE-249/B wing (mesh 6

x 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

FIGURE 5.29 Distribution of lift coefficient on the AE-249/B wing (mesh 6 x 20). 106


17/203

xvii

FIGURE 5.30 Distribution of angles of attack on the AE-249/B wing. . . . . . . . 106

FIGURE 5.31 Distribution of spanwise/chordwise lift on the AE-249/B wing (mesh

6 x 20).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

FIGURE 5.32 Distribution of induced drag on the AE-249/B wing (mesh 6 x 20). . 107

FIGURE 5.33 Global displacements on the AE-249/B wing (mesh 6 x 20). . . . . . 108

FIGURE 5.34 X Rotations (bending) on the AE-249/B wing (mesh 6 x 20). . . . . 108

FIGURE 5.35 Y Rotations (torsion) on the AE-249/B wing (mesh 6 x 20). . . . . 109

FIGURE 5.36 Z Rotations (in-plane bend.) on the AE-249/B wing (mesh 6 x 20). 109

FIGURE 5.37 Mises stresses on the AE-249/B wing (mesh 6 x 20). . . . . . . . . . 110

FIGURE 5.38 Mirrored deformation on the AE-249/B wing (mesh 6 x 20). . . . . 110

FIGURE 5.39 Flutter mode at initial time on the AE-249/B wing (undeformed). . 112

FIGURE 5.40 Flutter mode at semi-period time on the AE-249/B wing (unde-

formed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

FIGURE 5.41 Flutter mode at initial time on the AE-249/B wing (deformed). . . . 113

FIGURE 5.42 Flutter mode at semi-period time on the AE-249/B wing (deformed). 113

FIGURE 5.43 V-g-f plots for flutter analysis on the AE-249/B wing (undeformed). 114

FIGURE 5.44 V-g-f plots for flutter analysis on the AE-249/B wing (deformed). . 115

FIGURE 5.45 Geometric representation of the GNBA aircraft model. . . . . . . . 116

FIGURE 5.46 Aerodynamic mesh of the GNBA-12 wing. . . . . . . . . . . . . . . 118

FIGURE 5.47 Structural mesh of the GNBA-12 wing. . . . . . . . . . . . . . . . . 119

FIGURE 5.48 Distribution of pressure coefficient on the GNBA-12 wing. . . . . . . 120

FIGURE 5.49 Distribution of lift coefficient on the GNBA-12 wing. . . . . . . . . . 121

FIGURE 5.50 Distribution of angles of attack on the GNBA-12 wing. . . . . . . . 121

FIGURE 5.51 Distribution of spanwise/chordwise lift on the GNBA-12 wing. . . . 122

FIGURE 5.52 Distribution of induced drag on the GNBA-12 wing. . . . . . . . . . 122

FIGURE 5.53 Global displacements on the GNBA-12 wing. . . . . . . . . . . . . . 1 2 3

FIGURE 5.54 X Rotations (bending) on the GNBA-12 wing. . . . . . . . . . . . . 123

FIGURE 5.55 Y Rotations (torsion) on the GNBA-12 wing. . . . . . . . . . . . . . 124

FIGURE 5.56 Z Rotations (in-plane bend.) on the GNBA-12 wing.. . . . . . . . . 124

FIGURE 5.57 Flutter mode at initial time on the GNBA-12 wing (undeformed). . 126


18/203

xviii

FIGURE 5.58 Flutter mode at quarter-period time on the GNBA-12 wing (unde-

formed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

FIGURE 5.59 Flutter mode at semi-period time on the GNBA-12 wing (undeformed).127

FIGURE 5.60 Flutter mode at three-quarter-period time on the GNBA-12 wing

(undeformed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

FIGURE 5.61 Flutter mode at initial time on the GNBA-12 wing (deformed). . . . 128

FIGURE 5.62 Flutter mode at quarter-period time on the GNBA-12 wing (de-

formed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

FIGURE 5.63 Flutter mode at semi-period time on the GNBA-12 wing (deformed). 129

FIGURE 5.64 Flutter mode at three-quarter-period time on the GNBA-12 wing

(deformed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

FIGURE 5.65 V-g-f plots for flutter analysis on the GNBA-12 wing (undeformed). 130

FIGURE 5.66 V-g-f plots for flutter analysis on the GNBA-12 wing (deformed). . . 131

FIGURE A.1 Internal architecture of VLM4FW.. . . . . . . . . . . . . . . . . . . 144

FIGURE A.2 Storage of the working files for execution of Abaqus/VLM4FW. . . 146

FIGURE A.3 Architecture of the aeroelastic coupler Abaqus/VLM4FW. . . . . . 148

FIGURE A.4 Sequence of meshing in VLM4FW.. . . . . . . . . . . . . . . . . . . 150

FIGURE A.5 Regions of meshing in VLM4FW. . . . . . . . . . . . . . . . . . . . 151

FIGURE A.6 Dummy aero panels in VLM4FW to avoid gaps in the wing root. . . 152


19/203

xix

List of Tables

TABLE 4.1 Comparison of bulk data cards and keywords used in FEM models.. 83

TABLE 5.1 Model parameters of Garner wing. . . . . . . . . . . . . . . . . . . . 86TABLE 5.2 Correlation of global induced drag on the Garner wing. . . . . . . . 88

TABLE 5.3 Model parameters of Saunders wing.. . . . . . . . . . . . . . . . . . 90

TABLE 5.4 Correlation of lift coefficient slope on the Saunders wing. . . . . . . 93

TABLE 5.5 Model parameters of AE-249/B wing. . . . . . . . . . . . . . . . . . 95

TABLE 5.6 Sensitivity analysis of mesh on the AE-249/B wing (rigid). . . . . . 99

TABLE 5.7 Comparison of 1g steady flight trim condition on the AE-249/B wing.100

TABLE 5.8 Comparison of flutter results on the AE-249/B wing. . . . . . . . . 111

TABLE 5.9 Model parameters of GNBA-12 wing. . . . . . . . . . . . . . . . . . 117

TABLE 5.10 Comparison of flutter results on the GNBA-12 wing. . . . . . . . . . 132

TABLE A.1 I/O files in Abaqus/VLM4FW program . . . . . . . . . . . . . . . . 145


20/203


21/203

xxi

List of Abbreviations and Acronyms

AIAA American Institute of Aeronautics and Astronautics

AIC Aerodynamic Influence Coefficient

ANAC Agencia Nacional de Aviacao Civil

AOA Angle of Attack

BC Boundary Condition

CAPES Coordenacao de Aperfeicoamento de Pessoal de Nvel Superior

CAA Civil Aviation Authority

CFD Computational Fluid Dynamics

CG Center of Gravity

CPU Central Processing Unit

CSYS Coordinate SystemCTA Centro Tecnico Aeroespacial

CVLM Curved Vortex Lattice Method

DOF Degree of Freedom

DLM Doublet Lattice Method

EA Elastic Axis

EMBRAER Empresa Brasileira de Aeronautica

ENP Engine, Nacelle and Pylon

FAA Federal Aviation AdministrationFAB Forca Aerea Brasileira

FCS Flight Control System

FEM Finite Element Method

FID File Identifier

FSI Fluid Structure Interaction

FSW Forward-Swept Wing

GNBA Generic Narrow-Body Airliner

GVT Ground Vibration Test

I/O Input/Output

IPS Infinite Plate Spline

ISA International Standard Atmosphere


22/203

xxii

ITA Instituto Tecnologico de Aeronautica

KEAS Knots Equivalent Air Speed

KTAS Knots True Air Speed

LCO Limit Cycle OscillationLE Leading Edge

LHS Left Hand Side

LLT Lifting-Line Theory

MAC Mean Aerodynamic Chord

MTOW Maximum Take-Off Weight

NACA National Advisory Committee for Aeronautics

NASA National Aeronautics and Space Administration

NDLM Non-planar Doublet Lattice MethodNLVLM Non-linear Vortex Lattice Method

OOM Out of Memory

RHS Right Hand Side

RLE Root Leading Edge

RTE Root Trailing Edge

SHM Simple Harmonic Motion

SL Sea Level

SRF Step Response Function

TE Trailing Edge

TLE Tip Leading Edge

TTE Tip Trailing Edge

USAF United States Air Force

UTN-FRH Universidad Tecnologica Nacional - Facultad Regional Haedo

UVLM Unsteady Vortex Lattice Method

VLM Vortex Lattice Method


23/203

xxiii

List of Symbols

#

VAB Velocity vector induced by vortex filament AB#

VA Velocity vector induced by vortex filament A#

VB

Velocity vector induced by vortex filament Bm Control point m

n Global panel n

n Circulation strength of vortex filament for each panel n

xm, ym, zm xyz-coordinates of the control point m

xAn, yAn, zAn xyz-coordinate of A vertex for each panel n

xBn, yBn, zBn xyz-coordinate of B vertex for each panel n#

V(m, n) Total velocity vector at control point m induced by panel n#

Velocity influence coefficient vector

u x-component of induced velocity (backwash)

v y-component of induced velocity (sidewah)

w z-component of induced velocity (downwash)

Chordwise slope of panel (mean camber line)

Bending angle (dihedral)

In-plane bending angle (sweep)

Torsion angle

Angle of attack

Sideslip angleV Free-stream air speed

q Free-stream air dynamic pressure

Air density

j Index of chordwise panel (row)

i Index of spanwise panel (strip)

N Number of panels (modeled on right semi-wing)

Nj Number of panels chordwise (modeled on right semi-wing)

Ni Number of panels spanwise (modeled on right semi-wing)p Global panel p

BD Net circulation strength of chordwise bound vortices BD


24/203

xxiv

S Reference wing surface

c Reference wing chord

vBD Sidewash velocity at 3/4 chord on right chordwise bound vortex BD

uAB Backwash velocity at midspan 1/4 chord on spanwise bound vortex ABvAB Sidewash velocity at midspan 1/4 chord on spanwise bound vortex AB

cA Chord of elemental length of left segment A

cB Chord of elemental length of right segment Bc Average local chord of panel based on left and right segmentsls Spanwise lift induced by AB bound vortices, backwash and sidewash effects

lt Chordwise lift induced by sidewash effects

l(s)t Chordwise lift transported to midspan 1/4 chord

l Lift of each aerodynamic paneldi Induced drag of each aerodynamic panel

d0 Parasite drag of each aerodynamic panel

d Total drag of each aerodynamic panel

Sweep angle of projected bound vortex AB into X-Y plane w.r.t. Y-Z plane

ss Length of bound vortex AB projected into Y-Z plane

CL Total lift coefficient of full-span wing

CDi Total induced drag coefficient of full-span wing

Cp Pressure coefficient

Cp Difference of pressure coefficient between upper and lower wing surface

p Normal pressure

p Difference of pressure between upper and lower wing surface

n Load factor

cl Local lift coefficient spanwise

cdi Local induced drag coefficient spanwise

cdp Local parasite drag coefficient

0 Geometric angle of attack

i Induced angle of attackeff Effective angle of attack

y Width of panel strip projected into y-axis

Fa Aerodynamic force vector expressed in aerodynamic CSYS

Fbl Aerodynamic force vector expressed in local body CSYS

Fbg Aerodynamic force vector expressed in global body CSYS

Tbla Transformation matrix from aerodynamic to local body axes

Tbgbl Transformation matrix from local body to global body axes

Change in angle of attackL Change in lift due to change in angle of attack

a0 Bi-dimensional lift coefficient slope


25/203

xxv

t Time

Wagners non-dimensional time in semi-chords traveled

() Wagners function

Angular velocity of oscillationk Reduced frequency

b Semi-chord of airfoil

C(k) Theodorsens complex function

F(k) Real part of Theodorsens complex function

G(k) Imaginary part of Theodorsens complex function

ab Distance from 50% chord position down to elastic center

0 Rigid angle of attack before deformation

L Lift at aerodynamic centerM Moment at aerodynamic center

K Kinetic energy

U Potential internal energy

k Torsional stiffness

L Lagrangian of the system

Vdiv Divergence air speed

Faf Vector of flexible aerodynamic forces at structural grid

Far Vector of rigid aerodynamic forces at structural grid

G Spline matrix

AIC Matrix of aerodynamic influence coefficients at aerodynamic grid

AIC Matrix of aerodynamic influence coefficients at structural grid

x Structural deformation vector

far/a Rigid aerodynamic force derivatives w.r.t. trim variables at aerodynamic grid

far/a Rigid aerodynamic force derivatives w.r.t. trim variables at structural grid

a Trim variables vector

Kgg Stiffness matrix at structural grid

Mgg Inertia matrix at structural gridMrr Mass rigid body matrix

r Matrix of rigid body modes

e Matrix of elastic modes

ur Acceleration vector of rigid body motion

fae/a Incremental elastic aerodynamic force derivative matrix at structural grid

cr Root chord of wing

ct Tip chord of wingb Reference wingspanc Reference chord

Sw Reference wing area


26/203

xxvi

AR Aspect ratio

LE Sweep angle of leading edge line w.r.t. to y-axis

t Thickness of material

E Young modulusG Shear modulus

Tensile stress

Poisson ratio

J Torsion constant

m Structural mass

Ixx, Iyy , Izz Structural mass moment of inertia about x-x, y-y and z-z axis


27/203

xxvii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.1.1 Historical background. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3 Organization of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Recent works at ITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Theoretical background . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Steady aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Vortex Lattice Method . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2 Quasi-steady aerodynamics approach . . . . . . . . . . . . . . . . . . 64

3.3 Unsteady aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Aeroelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Static aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5.1 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5.2 Trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6 Dynamic aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6.1 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


28/203

xxviii

4.2 Proposed workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Considerations for design . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1 Cases studies overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Garner wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.1 Aerodynamic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Saunders wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Aerodynamic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 AE-249/B wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


5.4.2 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


5.5 GNBA-12 wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116


5.5.2 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1 Final remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Appendix A Architecture of Abaqus/VLM4FW . . . . . 143

A.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.2 Meshing considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Appendix B VLM4FW program code . . . . . . . . . . . . . . 153

Appendix C ABQ/VLM4FW aeroelastic coupler code 197


29/203

1 Introduction

1.1 Motivation

Aeroelasticity is considered a branch of the aerospace engineering that describes theinteraction of aerodynamic, inertia and elastic forces inherent to a flexible structure such

as an aircraft. This discipline also studies the phenomena that can result as a consequence

of the interaction of aeroelastic forces. As shown in Figure 1.1, the classical Collar aeroe-

lastic triangle (COLLAR,1978) originally from 1947 represents how several disciplines, e.g.

stability and control (flight mechanics), structural dynamics (mechanical vibrations) and

static aeroelasticity are linked as a consequence of the interrelation of two of the three

types of forces. On the center of the triangle, all three forces are required to co-exist and

interact mutually to guarantee the appearance of dynamic instability phenomena.

These interactions are important since they substantially affect the airplane loads

and can have a direct impact on several areas such as aerodynamic lift redistribution,

stability derivatives (including lift effectiveness), control effectiveness (including aileron

reversal), structural static and dynamic stability and aircraft structural dynamic response

to turbulence and buffeting, among others (WEISSHAAR, 2012).

Due to the need of airplanes to be light, they often deform appreciably under the appli-

cation of loads in service. Such deformations change the distribution of the aerodynamic

load, which in turn changes the deformations. This coupling of elastic structural, aerody-namic and inertial effects on aerospace aircraft can influence seriously the integrity of the

structure, and this makes critical the study of aeroelasticity to evaluate quantitatively the

operational limits of the flight vehicles in order to comply with design and certification

requisites. Over the aviation history these effects have had a major determination upon

the design and flight performance of aircraft, even before the first controlled powered

flight. Since some aeroelastic phenomena such as flutter or divergence can potentially

lead to structural failure, aeroelastic penaltyon heavier and more stiffened structures has

been sacrificed in order to ensure structural integrity by means of suitable variations instructural stiffness and inertia distributions.

Aeroelasticity also involves natural phenomena such as the motion of insects, fish and


30/203

30 CHAPTER 1. INTRODUCTION

FIGURE 1.1 Collar triangle with the interaction of aeroelastic forces.

Source: COLLAR, 1978.

birds, so it is not only a concern for the study of airplane lifting bodies. In addition, the

topic is also extremely relevant for the design of structures such as bridges, racing cars, he-

licopters, wind turbines, and turbomachinery, among others. For instance, propellers and

windmills are also affected by the lack of torsional stiffness, which in the first case creates

a loss of thrust due to twisting of blades, as happened with the Wright brothers when they

started to use thin propellers (GARRICK; REED III,1981). Moreover, propeller-driven air-

craft can also be subjected to another form of instability called propeller whirl flutter, due

to the presence of gyroscopic precession effects on flexibly mounted engine-propeller sys-

tems (sometimes attributed the probable cause of the Lockheed Electra airplane crashes)

(WEISSHAAR,2012).

However, other technological areas such as civil engineering also suffer from fluid-

structure interaction effects. Typical examples are some 19th century bridges which were

torsionally weak enough and collapsed from aeroelastic phenomena, as occurred with the

Tacoma Narrows Bridge in spectacular fashion in 1940. This bridge suffered a strong

torsion-bending coalescence of modes to promote the flutter, and is often misunderstood

as resonance in several undergraduate physics textbooks (BILLAH; SCALAN, 1991).

Particularly, this work deals with the study of aeroelastic computational simulations

for flexible wings, based on commercial software present in the industry as well as an

in-house aeroelastic code for nonlinear steady flight loads, which has been developed at


31/203

CHAPTER 1. INTRODUCTION 31

ITA intended to be used for very flexible aircraft. Several case studies are considered

and then results are compared for correlation and validation of the assumptions made,

together with restrictions and limitations of the theory applied.

1.1.1 Historical background

In the early 1900s, Wright brothers had already been working with the concept of

wing warpingas mechanism to induce a controlled, anti-symmetrical bi-plane wing struc-

tural twisting displacement to create rolling moments and thus get aircraft control. This

technique had also been tested by Edson Gallaudet as early as 1898, although he did not

publish or patent the original idea. The issue of warping philosophy is based on the fact

that torsionally flexible wing surfaces, distorted by the pilot, might also be distorted bythe free airstream that may produce self-excited and unintended air loads. However, by

that time most early monoplanes used wing warping roll control, including the Bleriot XI,

Bristol Prier, Fokker Eindecker aircraft among others. Airplane speeds were low enough

and structural stiffness large enough so that aerodynamic loading would not induce ex-

cessive deformations. But as engine power and airspeed increased, low torsional stiffness

created aeroelastic static divergence problems that led to catastrophic wing failures at

high speeds, even more aggravated when structural designs of wing box had a rear spar

configuration with shift of its elastic axis with respect to the aerodynamic center (Fokker

D-VIII german fighter) (GARRICK; REED III,1981).

The first fatal accidents were attributed to a possible lack of sufficient wire bracing

strength. Nevertheless, Bleriot strengthened the Bleriot XI supporting wires and in-

creased the main wing spar size and wing failures still occurred. At the time nothing

was known about such static aeroelastic phenomena (wing divergence) so that any load-

deformation interaction mechanisms were not even recognized. Soon, bi-plane designs

remained supreme after monoplanes being banned in several countries due to a series of

accidents. It was not until 15 years later, when Reissner (1926) published a first work with

a clear mathematical and physical understanding of the origin of static aeroelastic phe-

nomenon such as lift effectiveness and wing divergence. Soon after similar research papers

and reports appeared in other countries for further investigations about these aeroelastic

phenomena.

During World War I, one of the first aircraft on which a self-excited, vibratory aeroe-

lastic instability (later called flutter) problem was identified, modeled and solved was the

Handley-Page O/400 british bomber shown on Figure 1.2. Bairstow and Fage (1916), as

well as Lanchester (1916) investigated the possible cause of this disconcerting phenomenonby that time and published some reports (probably the first theoretical flutter analysis).

They later revealed that the flutter failure on the horizontal tail was caused by interaction


32/203


between fuselage twisting oscillation and the anti-symmetrical pitch rotations of the right

and left elevators (independently actuated). A workaround solution was finally found by

connecting the elevators to a common torque tube in order to eliminate potential anti-

symmetrical elevator motions and make them to move in synchrony. Similar problems oftail flutter have also been reported to occur with the Havilland DH-9 airplane years later,

with identical cure as proposed by Lanchester (1916). After these lessons learned, the

attachment of both elevators to the same torque tube became standard design practice

on aircraft industry (GARRICK; REED III,1981).

FIGURE 1.2 Handley-Page O/400 bomber aircraft, pioneer in the flutter phenomenon.

Source: Tangmere Military Aviation Museum, UK.

Binary flutter of the wing in vertical bending combined with motion of the ailerons

was also an issue on the 20s (e.g. Berkel W.B. monoplane). In the Netherlands, VonBaumhauer and Koning (1923) concluded as result of their study that mass balance of the

aileron, or even partial mass balance, could eliminate the coupling of interacting modes

to prevent wing-aileron flutter.

By the same time, several separate research groups in UK, Germany, Netherlands and

USA had started working intensively on the subject of aeroelasticity, specially in the un-

steady aerodynamics field. In Germany, Ackerman and Birnbaum (1923) published some

papers about classical vortex theory of bi-dimensional steady flows and then extended to

the modeling of harmonically oscillating airfoils, under supervision of Professor Prandtlin Gottingen. Later, Wagner (1925) continued the research work of Birnbaum and stud-

ied the growth of vorticity in the wake, as well as of the growth of lift on an airfoil in


33/203


bi-dimensional flow after a sudden change of the AOA (indicial response). Following Wag-

ners methods, Glauert (1929) considered the flat plate airfoil under harmonic oscillatory

motion. Kussner (1929) also worked in the same year publishing a paper on flutter based

on Birnbaums method. He improved convergence for reduced frequency in the order ofunity and applied this theory to the bending-torsion phenomena including aileron motion.

Approximatly at the same time, several other german researchers were also working with

flutter but using quasi-steady aerodynamics, such as Blasius (1925), Hesselbach (1927),

Blenk (1927) and Liebers (1929). On the american side, Theodorsen (1934) established

a theory for mathematically modeling bi-dimensional oscillating flat plates by separating

the non-circulatory part of the velocity potential from the circulatory part associated to

the trailing edge wake effects, published years later after intensive work in the famous

NACA Report 496 (THEODORSEN, 1935). The general model had 3 DOFs in plunge,

pitch and also aileron flap motion, and a defined function (later called Theodorsen Cir-

culationfunction) established the lags between airfoil motion and the aerodynamic forces

and moments that arise. This report is sometimes considered the fundamental basis for

the mathematical theory of dynamic instabilities and served as starting point for further

research over the next decades in the aeroelasticity field (Kussner, Wagner, Sears, Possio,

among others).

Aircraft developments still faced on the upcoming years strong couplings of aeroelastic

phenomena and were plagued by flutter problems. Gloster Grebe, Gloster Gamecock,Navy MO-1, Havilland Puss Moth, Junkers JU90 were some of the examples that are

worth to be mentioned. Nevertheless, more advanced technologies over the years allowed

to estimate with a better level of accuracy, as well as wind tunnel testing techniques

started to grow momentum in the 40s.

With the advent of flight at transonic speeds due to the jet engine development, new

challenging aeroelastic problems started to appear, many of which are still to these days.

Sweep wing introduction to allow retard the critical Mach number at transonic speeds,

together with the use of aeroelastic tailoring produced an efficient solution for the late 40sand 50s, such as was the case of the Boeing B-47. By that time, aeroelastic experiments

at transonic wind tunnels were already feasible and provided greater efficiency than earlier

methods.

On the beginning of supersonic speeds, a new type of flutter, named panel flutter also

began to play an important effect on aircraft and spacecraft, since dynamic instability

might occur involving ripples moving along skin covering leading to an abrupt local failure

of panel. Several failures of the V-2 were justified by a probable panel flutter near the

nose of the rocket, and a fighter was lost due to a failure of a hydraulic line attached toa panel that fluttered.

Starting in the early 1970s, the usage of aeroelastic tailoring had started being in-


34/203


vestigated for forward-swept wing designs with inherently low wing divergence speeds,

in order to promote a favorable bending-torsion coupling by rotating the laminate fiber

direction off-axis of the wing sweep direction (WEISSHAAR, 2012). Using forward-swept

wings with fibers aligned along the wing swept axis led to the wash-ineffect increasing theaerodynamic loading and reducing the wing divergence airspeed compared to an equiva-

lent unswept wing. In order to overcome this issue, one of the techniques first used was

adding substantial structural stiffness (which in turn penalized more weight) required to

provide enough aeroelastic stability. However, orienting composite laminate fibers slightly

off-axis permitted to intentionally couple the bending-torsion deformations in a favorable

condition known as wash-out effect, where wing bending in the upward direction now

came along with a twist in a downward direction of the LE, unloading the wing and

retarding the wing divergence phenomena without extra weight.

One of the most successful cases was the DARPA X-29 research program, whose air-

craft is shown in Figure 1.3. The Grumman X-29 was an experimental research aircraft

with a forward-swept wing and canard control surfaces configuration, leading to an inher-

ent aerodynamic instability that forced the need of a computerized fly-by-wire FCS.

FIGURE 1.3 Grumman X-29 aircraft operated by NASA/USAF.

Source: NASA, 1987.

The X-29 design made use of the anisotropic elastic coupling between bending andtwisting of the carbon fiber composite material to address this aeroelastic divergence fail-

ure, specially at high AOA. Instead of using a very stiff wing, which would carry a weight


35/203


penalty even with the relatively light-weight composite, the X-29 used a laminate which

produced intentional bending-torsion coupling needed for wash-out and so avoiding the

static divergence effect (PAMADI,2015). It was flown in a joint research projects program

between NASA and USAF from December 1984 to 1988 investigating handling qualities,performance, and systems integration on the unique forward-swept-wing research aircraft

configuration. The second prototype was used to study flight behavior at high AOA

characteristics, among other research missions.

Since the development of digital computer machines in the 60s and 70s, these events

have brought an enormous revolution for the aeroelastic computations at aerospace com-

panies and research universities. Mathematical matrix analyses of structures based on

a displacements/stiffness approach, commonly referred as Finite Element Method (AR-

GYRIS, 1966, TURNER, 1970), have started to become a standard technique in design forstructures and aeroelasticity fields nowadays. This structural analysis method quickly

spread out around the world and has since then been implemented in the aerospace in-

dustry for global and local analysis of complex aero-structures members, as seen in Figure

1.4.

FIGURE 1.4 Finite element representation for a global structural analysis.

Source: ROMMEL; DODD, 1984.

On the other hand, the computational progress in the structures area has in turn

benefited the aerodynamics field, whereas numerical methods for lifting surfaces in steady

and unsteady flows, discrete lattices and panel methods were formulated for aerodynamic

loading computations and later to be coupled in aeroelastic computational codes. The

evolution of panel codes to model potential flows has also rapidly evolved into a standard

technique for the aerodynamic discretization of 2D/3D lifting surfaces such as wings,

horizontal and vertical stabilizers, as well as main non-lifting bodies such as fuselage and

engine nacelles. A schematic representation is shown in Figure 1.5. As can be noted, one


36/203


of the main advantages of using this approach compared to other aerodynamic numerical

methods is the ability to obtain interferences automatically between different bodies.

One of the first variant of panel methods was the Vortex Lattice Method (VLM), which

is considered among the earliest numerical methods utilizing computers to actually esti-

mate aerodynamic characteristics of aircraft (MASON, 1997). Its origins were formulated

in the late 30s, and FALKNER (1943) was who named it Vortex Lattice. His idea of

discretizing the wing into a lattice (surface) with horseshoe vortices on each aerodynamic

panel can be thought as an extension of the original Lifting Line Theory (PRANDTL,1923).

However, because of the fact of being a numerical approach in which a lifting surface is

discretized into a finite number of panels, practical applications had to wait for decades

until the development of digital computers was sufficient in the early 60s. The following

decade showed a widespread adoption of the method, even leading to the organization of aworkshop at NASA exclusively for standard utilization of the method (NASA,1976) after

the publication of several reports from an original code developed at the NASA Langley

Research Center (MARGASON; LAMAR,1971).

FIGURE 1.5 Philosophy of computational panel methods to model lifting surfaces.

Source: BERTIN; CUMMINGS, 2008.

Nowadays, models used in loads and aeroelastic calculations are becoming more ad-vanced. Structural FEM models have evolved from a beam-like model to a much more

representative box-like model (WRIGHT; COOPER, 2007), giving more fidelity of the real

structure representation. On the other hand, aerodynamic models have passed over 1D,

2D strip theory to 3D panel methods as well as CFD techniques, being used as standard

tools in the aerospace industry, as seen in Figure1.6. The interest in the development of

very flexible aircraft wings as a mean of increasing fuel efficiency and flight performance

in the aerospace industry has led to the need of extending the aeroelastic computational

codes to take into account new sources of non-linearities.


37/203


FIGURE 1.6 Aerodynamic and aeroelastic simulations on the EMB314 aircraft.

Source: EMBRAER, 2006.

1.2 Objective

The objective of this work is to develop several methodologies of computational aeroe-

lastic analyses for the design of flexible aircraft wings. Several commercial finite element

and aeroelastic software packages such as Abaqus, MSC.Nastran and ZAERO are used

together with an in-house VLM code named VLM4FW, which has been programmed to

be coupled with Abaqus for aeroelastic geometrical non-linear simulations and compute

steady flight loads. The VLM4FW code takes into account the local dihedral and tor-

sion of each aerodynamic panel induced by structural deformation. In addition, it hasa correction on the aerodynamic loading for the inclusion of the sidewash and backwash

effects due to the appearance of new induced velocity components for wings out of the

X-Y plane of the aircraft. The coupling of VLM4FW with the structural nonlinear solver

Abaqus for aeroelastic studies is performed in a co-simulation scheme. Then, based on the

deformed wing configuration, flutter studies are performed in ZAERO to analyze the dy-

namic aeroelastic instability and evaluate the results compared to the undeformed initial

wing shape. Several case studies have been adopted to validate the VLM4FW program

for rigid and flexible wings, such as the AE-249 and GNBA aircraft wings.

1.3 Organization of work

This work is divided into 6 chapters. Chapter1 deals with a summarized introduction

on the subject of aeroelasticity. A motivation of study and a brief historical background

are also exposed to show how academic research has been evolving over the last century.

Chapter2gives an general overview of the main literature present as well as some recents

works published at ITA. Chapter3 starts with a theoretical background on aerodynamics

and aeroelasticity formulations implemented in the current computational codes used in


38/203


industry, as well as in the in-house VLM4FW program. Chapter4explains the methodol-

ogy proposed for the analysis of flexible wings using computational aeroelastic codes. At

the end of the chapter some extra considerations are covered regarding the simplifications

of the aeroelastic models used for design in the aerospace industry. Chapter5shows somecase studies chosen for testing, correlation and validation of results. Rigid and flexible

configurations are considered, as well as different wing planforms. The corresponding nu-

merical results are obtained after running the co-simulations in Abaqus/VLM4FW and

in MSC.Nastran/ZAERO. Lastly, Chapter 6provides final conclusions of the work with

some remarks and possible future works to continue the development of the VLM4FW

code in the area.

There are 3 appendices, which give an overview of the architecture of the VLM4FW

program code, its inter-communication with the structural FE solver Abaqus for aeroe-lastic co-simulations as well as the source codes used in this work.


39/203

2 Literature review

Several textbooks written over the past decades are still present in the aeroelasticity

literature and are excellent references for a classical theoretical background. SCANLAN

and ROSENBAUM (1951), BISPLINGHOFF, ASHLEY and HALFMAN (1955) together with

FUNG (1955)were one of the first in publishing a comprehensive mathematical treatment

of all aspects to begin the study of aeroelastic phenomena. Modern textbooks that are

worth mentioning are DOWELL et al. (2004), WRIGHT and COOPER (2007) as well as

HODGES and PIERCE (2011), which extend the classical formulation of aeroelasticity

not only for bi-dimensional airfoils, but also for aerodynamic lifting surfaces and their

interactions with elastic bodies. DOWELL et al. (2004) also presents some applications of

aeroelasticity problems in rotorcraft, turbomachinery and civil engineering.

As mentioned in Section 1.1 and illustrated in Figure 1.1, aeroelasticity deals with

both structural mechanics and aerodynamics fields. In this way, for the implementation

of aeroelastic computational codes some particular textbooks can be referenced, just to

mention a few of an enormous quantity of literature present in both areas.

On the structural side, BATHE (1996) published a comprehensive work in the theory

of finite element formulation for solving different kind of procedures. It also contains a

basic FEM program named STAP which is written in FORTRAN 77 and that, although

simplified in various areas (only valid for static linear elastic finite element analysis), serves

as an excellent base for the programming of a more general finite element program capable

of solving problems such as those of natural frequencies extraction, nonlinear statics,

dynamics, and so on. Similarly, ZIENKIEWICZ and TAYLOR (2000)also published a more

general FEM program called FEAPpv written in FORTRAN 77, intended mainly for use

in learning finite element programming methodologies and in solving small to moderate

size problems in linear and nonlinear mechanics.

On the aerodynamics side, there are also numerous textbooks and reports that have

dealt with the implementation of panel codes. Specifically for VLM, MARGASON and

LAMAR (1970) wrote a general program in FORTRAN 77 for estimating the subsonic

aerodynamic characteristics of complex wing planforms with variable dihedral and sweep.

Their idea was to correct the aerodynamic loading with the inclusion of the sidewash and


40/203

40 CHAPTER 2. LITERATURE REVIEW

backwash effects due to the appearance of new induced velocity components for wings out

of the X-Y plane of the aircraft. In parallel, KALMAN, GIESING and RODDEN (1970)

also proposed a method to correct the low values of induced drag predicted by the VLM

(sometimes even lower than the corresponding elliptical wing planform), by applying acorrecting scale factor to be introduced into the spanwise distribution of induced drag

that is based on the cross-flow energy in the wake.

Over the next decades, continuous improvements to VLM implementations have been

performed, and one of the final developments was known as the VLM4.997 published by

NASA Langley in 1997, being capable of handling up to four lifting surfaces. This NASA

code has been considered a de-facto standard program due to its general availability,

versatility and reliability (MASON, 1997). Enormous DLM improvements have also been

made, being nowadays well established in commercial software such as MSC.Nastran andZAERO for aeroelastic flutter analyses. However, most of them do not contemplate

sources of geometrical non-linearities for very flexible wings, since they are based on the

assumption of linear theory with small perturbations.

In recent years, LEE and PARK (2009) have presented an improved method to analyze

low aspect ratio wings or high angles of attack with corrections to the classical Non-linear

Vortex Lattice Method (NLVLM). The shape of the trailing vortex is curved in NLVLM

so that it can be iteratively updated to be aligned with the local direction of flow at each

panel. Thus, each trailing vortex is broken into many vortex segments of finite lengthleading to an iterative solution procedure.

A non-linear strained-based finite element approach for highly flexible aircraft has

also been formulated and implemented by SU, ZHANG and CESNIK (2009)in a simulation

framework named UM/NAST to correlate experimental results of a slender rectangular

wing. This program takes into account nonlinear flight dynamic equations coupled with

aeroelastic equations. Using this approach, the modeling and analysis of a 3D structure is

decomposed as a combination of 2D cross-sectional analysis and 1D beam analysis. The

strain-based nonlinear beam formulation previously developed for the 1D beam analysisis then coupled with finite-state unsteady aerodynamics on lifting surfaces.

On the other hand, ZHANG, YANG, LIU, XIE and others (2012-2015) have published

several works using new theoretical formulations called Curved Vortex Lattice Method

(CVLM) and Non-planar Doublet Lattice Method (NDLM), which handle aspects of ge-

ometrical non-linear aeroelasticity using closed vortex rings. As will be discussed in the

following chapters, the approach presented in this work proposes an alternative method-

ology to perform geometrical non-linear aeroelastic simulations on flexible aircraft wings,

in particular for the computation of steady flight loads and flutter.

Several authors such as YATES (1987), DESTUYNDER (1987) and ZINCHEKOV (2014)


41/203

CHAPTER 2. LITERATURE REVIEW 41

have been studying the influence of the deformed geometry from the static aeroelastic trim-

ming condition on flutter. Experimental comparison in wind tunnel have been performed

in most cases with good correlation, and some have concluded a linear approximation

seems to be conservative for the prediction of flutter phenomenon.

For more aerodynamic codes, whether using the VLM or DLM formulation, KATZ

and PLOTKIN (2001) presented several sample programs which implement 2D/3D panel

methods in steady and unsteady aerodynamics using source, doublet and vortex singular-

ities, as well as using both Neumann and Dirichlet boundary conditions. These computer

programs are also written in FORTRAN 77. Regarding to good programming practices,

MASON (1997,2006) discussed some examples on different styles, with advantages and

disadvantages for the development of aerodynamic codes written in FORTRAN.

It may be important to note although the methods described in the former classical

textbooks have been supplanted by modern computational developments, they still re-

main excellent sources of fundamental explanations, classic examples and valuable, basic

computational techniques that can be used for preliminary estimates of aeroelastic effects,

although the content is strictly theoretical and often tends to restrict coverage to static

aeroelasticity and flutter, considering in some cases very simplified cantilever wings with

a limited unsteady aerodynamics theory.

Probably one of the first academic works published at ITA regarding computational

predictions of aeroelastic phenomena was done by URBANO (1966). He implemented in

an AFIT programming language a code for estimating the critical velocity of flutter of the

IPD-6201 Universal aircraft. It was a limited tabular method as proposed bySCANLAN

and ROSENBAUM (1951) for low subsonic speed.

During the next decades, helped with the rapid evolution of the digital computers and

the industrialization of aeroelastic software, in particular MSC/NASTRAN (DOGGETT;

HARDER, 1973; RODDEN et al.,1979; RODDEN et al., 1984) various authors also published

works on the area.

2.1 Recent works at ITA

Just to cite some of the recent works, COURA (2000) and CORADIN (2010)studied the

aeroelastic phenomena on some models of cantilever and AGARD 445.6 wings. NICO-

LETTI (2006) also analyzed a 50-passenger commercial aircraft wing, but using the Strip

Theory formulation. SCHNEIKER (2010) made some parametric sensibility analyses on

the optimal weight of the wing in order to comply with the aeroelastic stabilities require-ments established by the CAA (FAA, 2004; FAA, 2014), using the Gradient Method for

optimization varying the main spars and skin thicknesses. In addition, WESTIN (2010) and


42/203

42 CHAPTER 2. LITERATURE REVIEW

PINTO (2013) correlated their aeroelastic simulations with experimental results gathered

in wind tunnel testing.

On the other hand, SOARES (2004), FREITAS (2004), SILVA (2005) and SORESINI

(2006) studied the appearance of flutter instabilities using MSC/Nastran software on

complete military and commercial aircraft configurations at different altitude and Mach

conditions, evaluating the feasibility of the operational flight envelope. None of them

used non-linear aeroelastic theories, so this kind of analysis start loosing some validity for

very flexible wings. Only SOUZA (2012) published a work on non-linearities present on

composites laminated flat plates subject to large displacements through the coupling of a

nonlinear corotational shell Finite Element formulation with a Unsteady Vortex-Lattice

Method (UVLM).

Finally,NETO (2014)studied the flight dynamics effects on flexible aircraft using an ap-

proach of general body axes. He considered the dynamic coupling of rigid-body modes with

the elastic degrees of freedom and for this implemented a Doublet Lattice Method (DLM)

on an internally developed aircraft, named as Generic Narrow-Body Airliner (GNBA).


43/203

3 Theoretical background

3.1 Introduction

For the study of flexible aircraft, the aerodynamic lift forces induce deflections ofthe aerodynamic surfaces, which in turn change the characteristics of the airflow, hence

leading to aeroelastic phenomena and affecting the dynamic loads. An understanding of

how the aerodynamic flow around three-dimensional aerodynamic bodies generates forces

and moments that are applied to aircraft during flight is very important in order to be

able to develop mathematical models that describe the aeroelastic behavior.

In the following sections of this chapter a brief overview of the aerodynamic theories

used in aeroelasticity is presented, with emphasis on those mathematical formulations

specifically implemented in the VLM4FW code and those incorporated in standard com-mercial software such as NASTRAN and ZAERO.

Generally, depending on the non-steadiness of the flow, aerodynamics study can be

split into three major categories: steady, quasi-steady, and unsteady aerodynamics. A

discussion on each particular case is considered next.

3.2 Steady aerodynamics

When aerodynamic surfaces are in steady motion (for instance an aircraft in straight

level flight), the aerodynamic variables are assumed to be constant and not changing with

time at each given position of the flow field. In this fashion, flow variables will be only

functions of spatial coordinates and not of time, as would occur in unsteady motion. For

this, several mathematical theories can be used to study a three-dimensional aerodynamic

body or even model the spanwise lift distribution of a wing under the assumption of steady

aerodynamics.

One of the simplest is the well-known Strip Theory, where the wing is considered tobe composed of a number of elemental chordwise strips. Here it is assumed that the lift

coefficient on each chordwise strip is proportional to the local angle of attack and that


44/203


45/203

CHAPTER 3. THEORETICAL BACKGROUND 45

be thought as an extension of the original Lifting Line Theory (PRANDTL, 1923), being

able to predict the normal induced velocities (downwash) over the wing using the law of

Biot-Savart and estimate the induced drag. This method is based on the solution of the

classical Laplace equation, thus built on the assumption of ideal potential, incompressible,steady, inviscid and irrotational flow (although some corrections can be done for small-

perturbations compressible flow). VLM is also subject to the same basic theoretical

restrictions that apply to classical conventional panel methods (HESS; SMITH, 1966).

Comparing the similarities of both types of methods (conventional panel methods

vs. VLM), here the singularities are also placed on a surface and the tangential non-

penetration boundary condition is guaranteed at a number of specific points, named

collocation control points. Thus, a compatible determined system of linear algebraic

equations is solved to determine singularity strengths.

Among the differences: the classical formulation of VLM does not take into account

the thickness of the lifting surface, being only suitable for slender thin bodies (neglecting

the thickness in VLM can sometimes be beneficial, as will be commented next). Further-

more, boundary conditions (BCs) are generally applied on a mean surface and not on

the actual surface. These assumptions limit the method to only compute the difference

in the pressure coefficient Cp, being unable to predict the local pressure coefficient on

the lower and upper surfaces. In addition, singularities used in the VLM theory are not

distributed over the entire surface, only horseshoe vortex filaments are disposed on eachaerodynamic panel with the circulation values as unknown of the problem.

In terms of results comparison of the VLM with other source panel methods, VLM

results often predicts the experimental data very well. This can be attributed to the

fact that VLM neglects both thickness and viscosity effects, and for most cases, the

effect of viscosity offsets the effect of thickness giving good agreement between VLM and

experiment for moderate AOAs (MARGASON et al.,1985).

Even though it is restricted to the assumptions mentioned in the preceding paragraphs,

the implementation of a VLM code can be easily corrected to take into account the effects

of sidewash and backwash, as proposed by MARGASON and LAMAR (1970) for arbitrary

complex wings planforms with variable dihedral and sweep. As will be shown later,

the VLM4FW program is based on these foundations, contemplating out of X-Y plane

corrections during the aeroelastic deformation of the wing in steady flight loads. Hence,

each aerodynamic panel is considered with a given dihedral, sweep and torsion for each

given time increment during the aeroelastic coupling.

As anticipated before, the basic approach of VLM is to discretize the continuous dis-

tribution of bound vorticity over the wing surface by a finite number of discrete horseshoe

vortices. This can be represented in Figure3.1, where the individual horseshoe vortices


46/203

46 CHAPTER 3. THEORETICAL BACKGROUND

are placed in a lattice array form, thus giving name to the vortex lattice method.

FIGURE 3.1 Typical wing planform modeled by the Vortex Lattice Method.


The bound vortex coincides with the quarter-chord line of each local panel. As re-

marked previously, in a more rigorous analysis the vortex lattice panels should be located

on the actual surface of the wing and the trailing vortices should leave the wing following

a curved path. However, for this engineering approach suitable accuracy can be obtained

yet assuming that the trailing vortices keep a straight line and extend downstream to in-

finity. These trailing vortices can be assumed to be aligned whether to the free stream orparallel to the body global vehicle axis. In this formulation the latter case will be adopted

since the formulas of the AIC computation are simpler and there is similar accuracy with

respect to the other case.

The imposition of the boundary condition requires for an inviscid flow to be tangent

to the wing surface at each control point of the panels, providing a set of simultaneous

equations in the unknown vortex circulation strengths. The control point of each panel is

centered spanwise on the three-quarter-chord line at midpoint between the trailing-vortex

legs as in DLM (KALMAN et al., 1971). This choice of the control point location at the3/4 chord has been proved to be optimum for two-dimensional flow and also results in a

high degree of accuracy for three-dimensional flow (JAMES,1969).


47/203


FIGURE 3.2 Geometric parameters of an aerodynamic panel in VLM.

It is important to outline that for the case of full span wings with general sweep,now the bound-vortex filaments of the left semi-wing will not be parallel to those of the

right semi-wing. Hence, the bound-vortex system on one side of the wing will produce

downwash on the other side of the wing, reducing the net lift and increasing the total

induced drag produced by the flow over the finite-span wing. The downwash resulting

from the bound-vortex system will be shown to be larger near the root of the wing, while

the downwash resulting from the trailing-vortex system larger near the wing tip. As a

consequence, the lift will be reduced both near the extremities of the semi-wing (root and

tip).

Considering an elementary aerodynamic panel on the right semi-wing, this will in turn

contain a horseshoe vortex as sketched in Figure3.2and3.3. Three different segments

can be divided to calculate their effects separately and the velocity induced by the vortex

filament of constant strength and equal for all segments can also be computed by the use

of the law of Biot and Savart (ROBINSON, 1956).

For the bound vortex of segment AB, which joins the left and right quarter-chord

extremities (A) and (B), the velocity induced at the control point with coordinates

(xm, ym, zm) is given as

#

VAB =n4

#

F1AB

#

F2AB

(3.1)


48/203


FIGURE 3.3 Horseshoe vortex on an elementary aero panel of the right semi-wing.


#

F1AB

= ([(ym yAn)(zm zBn) (ym yBn)(zm zAn)]

[(xm xAn)(zm zBn) (xm xBn)(zm zAn)] +

[(xm xAn)(ym yBn) (xm xBn)(ym yAn)] k /[(ym yAn)(zm zBn) (ym yBn)(zm zAn)]

2 +

[(xm xAn)(zm zBn) (xm xBn)(zm zAn)]2 +

[(xm xAn)(ym yBn) (xm xBn)(ym yAn)]2

(3.2)

#

F2AB

= +

(xBn xAn)(xm xAn) + (yBn yAn)(ym yAn) + (zBn zAn)(zm zAn)

(xm xAn)2 + (ym yAn)2 + (zm zAn)2

(xBn xAn)(xm xBn) + (yBn yAn)(ym yBn) + (zBn zAn)(zm zBn)(xm xBn)2 + (ym yBn)2 + (zm zBn)2

(3.3)

Similarly, for the left and right trailing-vortex legs which go from the corresponding


49/203


quarter-chord vertices down to infinity, segments (A) and (B)

#

VA=

n4 {CAj} + {CAk} k

#

VB=n4

{CBj} + {CBk} k

#

VA=n4

(zm zAn)+ (yAn ym)k

[(zm zAn)2 + (yAn ym)2]

1 + xm xAn(xm xAn)2 + (ym yAn)2 + (zm zAn)2 (3.5)

#

VB= n4

(zm zBn)+ (yBn ym)k

[(zm zBn)2 + (yBn ym)2]

1 + xm xBn

(xm xBn)2 + (ym yBn)2 + (zm zBn)2

(3.6)

The total velocity at some arbitrary control point (m) induced by the horseshoe vortex

of another panel (n) will be the sum of the components of those equations, which reads

#

V(m, n) =#

VAB(m, n) +#

VA(m, n) +#

VB(m, n)

Combining Equation (3.1), (3.5) and (3.6), the expression above becomes

#

V(m, n) = n1

4[{CABi}+ ({CABj} + {CAj}+ {CBj}) +

({CABk}+ {CAk}+ {CBk}) k

(3.7)

Equation (3.7) can be further compacted by putting together all the contributions

other than the strength circulation singularity n (yet unknown at this stage), in the

following fashion

#

V(m, n) = n#

(m, n) = ni(m, n)+ j(m, n)+ k(m, n)k (3.8)


50/203


The velocity influence coefficients depends exclusively on the geometry of the in-

ducing horseshoe vortex of panel (n) and its distance from the control point of panel (m).

Since it is assumed to be valid the linear superposition due to the fact of being handling

a linear system of algebraic equations, the effect from each panel is summed up togetherto finally obtain an expression for the total induced velocity at the control point (m)

#

V(m) =N

n=1

#

(m, n)n (3.9)

For full-span wings, an extra consideration must be taken into account. The left semi-

wing, even though it is not modeled geometrically, it does also contribute mathematically

and physically by inducing additional velocities on the control points of the right semi-

wing. Thus, its effect must be added in the Equation (3.9).

The derivation of the induced velocities on the left semi-wing is nearly similar to what

has been presented here, with some particular remarks. In Equation (3.1), (3.5) and (3.6),

for the left semi-wing expressions ym do not change since the control points are still at

the right semi-wing, only yAn, yBn for vertex (A) and (B) will change the sign to (-) to

represent the vortices from the contribution of the left semi-wing. Whateverxm,xAn,xBn

will remain unmodified due to X-Z symmetry assumption. In addition, all factors of terms

in formula of these expressions invert sign since those ones were derived for vortices on

right semi-wing only. It is important to highlight that here for the left semi-wing vertex

(A) means closest to wing root, and vertex (B) means closest to the wing tip.

On the other hand, it is also assumed that all singularities, belonging to each indi-

vidual left and right panel mirrored about the x-axis, are the same to guarantee the X-Z

symmetric flow in order to reduce the order of unknowns from 2N x 2N to N x N. Hence,

the aerodynamic influence coefficients for the right semi-wing, where the boundary condi-

tions of the problem are imposed, become the sum of the contribution of the right panels

over the right semi-wing and the left panels over the right semi-wing. Mathematically

(r)i (m, n) = (rr)i (m, n) +

(rl)i (m, n) (3.10a)

(r)j (m, n) =

(rr)j (m, n) +

(rl)j (m, n) (3.10b)

(r)k (m, n) =

(rr)k (m, n) +

(rl)k (m, n) (3.10c)

Finally, Equation (3.9) for full-span wings transforms to

#

V(m) =N

n=1

#

(r)(m, n)n=N

n=1

#

(rr)(m, n) +#

(rl)(m, n)

n (3.11)


51/203


Each velocity component of Equation (3.11) can be converted to matrix notation for

simpler algebraic manipulation (as will be seen later), giving place to vectors and matrices

of order N such as

{u} = [i(r)]{} (3.12a)

{v} = [j(r)]{} (3.12b)

{w} = [k(r)]{} (3.12c)

3.2.1.1 Boundary conditions

The application of the boundary conditions imposes that the flow must be tangentialat some particular points of each aerodynamic panel (named collocation control points), or

in other words, that the surface must be a streamline of the flow. Thus, since there are N

unknowns of the problem for each vortex strength, N equations of boundary conditions can

be imposed at each collocation point of the panel, leading to a determined N x N system

of equations. If the flow is tangent to the wing, the component of the induced velocity

normal to the wing at the control point must counterbalance the normal component of

the free-stream velocity.

FIGURE 3.4 Geometry of the mean camber surface for the tangential flow BCs.


As illustrated in Figure3.4,in order to guarantee the boundary condition of tangential


52/203


flow at the surface, the following relation is obtained

umsin mcos m vmcos msin m+wmcos mcos m= V sin(m m)cos m

(3.13)

Dividing each side of Equation3.13by cos m, it follows

umsin m vmcos mtan m+wmcos m= V sin(m m) (3.14)

A relationship between the chordwise slope (mean camber line) at the control point

and the torsion angle of the panel can be deducted in the following manner

m= arctandz

dx

m

= m (3.15)

Finally, after substitution of Equation (3.15), Equation (3.14) gets:

+umsin m vmcos mtan m+wmcos m = V sin(m+m) (3.16)

As a consequence, each equation will be obtained in order to satisfy the boundary

condition of the flow at each specific control point. Generalizing for all the N equations

u1sin 1 v1cos 1tan 1+w1cos 1 = V sin(1+1)

u2sin 2 v2cos 2tan 2+w2cos 2 = V sin(2+2)...

...

uNsin N vNcos Ntan N+wNcos N = V sin(N+N)

(3.17)

The Equation (3.17) can be also expressed in matrix form for convenience. After

algebraic manipulation with the help of Equation (3.12), the matrix system of equations

corresponding to the flow boundary condition can be re-arranged and becomes


53/203


sin [i(r)]{}

cos

tan [j(r)]{}

+

cos

[k(r)]{} = {V sin( + )} (3.18)Grouping together the terms with the vector of circulation as unknowns, the system

of equations is in closed-form and can be solved by inversion of the LHS matrix.

LHS sin

[i(r)] cos

tan

[j(r)] +cos

[k(r)]{} =

{V sin( + )} RHS

(3.19)

{} = [LH

development of methodologies of aeroelastic analysis for the design of flexible aircraft wings

Documents