investigations into the effects of 3-dimensional geometric

Investigations into the Effects of 3-Dimensional GeometricParameters on the Structural Design of an Adaptive Morphing

Wingtip.

André Ferreira Tribolet de Abreu

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Dr. Afzal SulemanCo-supervisor: Dr. Srinivas Vasista

Examination CommitteeChairperson: Prof. Dr. Fernando José Parracho LauSupervisor: Prof. Dr. Afzal SulemanMembers of the Committee: Dr. José Lobo do Vale

October 2014

”Don’t cry because it’s over, smile because it happened”

- Dr. Seuss

iii

Acknowledgments

I would like to thank Prof. Afzal Suleman for the opportunity he gave me to develop my thesis at the DLR in

Braunschweig which enabled me be to have a great work and life experience.

I would like to acknowledge the DLR institution for providing the facilities and conditions to complete my

thesis. A special thanks to Dr. Srinivas Vasista for guiding me throughout the six months I spent at the DLR whose

support and advice enabled me to complete my work. Also to Dipl.-Ing. Bram van de Kamp who received me

when I arrived at the DLR and got me started.

This project was the culmination of the last five years of completing my degree which wouldn’t be possible

without the friends made in this time. So a very special thanks to my classmates who accompanied me throughout

the several challenges that arose and without whom this time wouldn’t have been as much fun and fruitful as it

was. A special thanks to Pedro Isidro who also went to Braunschweig to complete his thesis and accompanied

me throughout the challenges of moving to a new country (namely finding a house!), meeting new people and

experiencing the German culture.

To my family, I am deeply grateful for the continued support in all my endeavors, who are greatly responsible

for where I am now and for always being a solid foundation to whom I could turn to whenever necessary.

Finally I would like to thank Beatriz Bento, who never gets tired of all my faults and makes hard work easier

every day.

v

Abstract

As part of the EU FP7 project NOVEMOR, a droop-nose adaptive morphing wingtip (AMWT) is being designed

at the DLR (German Aerospace Center), with the potential to reduce drag and substitute classical wing control

surfaces. The AMWT is activated via a compliant mechanism and the tools used for its design, in their current

status, are not suitable for highly 3D geometries (which is the case of wingtips with tapering and sweep).

The present thesis proposes a methodology to investigate how the 3D geometric parameters affect the output

of the compliant mechanism and the shape morphing of the wingtip. Furthermore the methodology is applied to a

modeled wingtip similar to the one being used in project NOVEMOR in order to aid and provide data for its design

process. The 3D parameters considered in this thesis are sweep angle, tapering in the chord direction and tapering

in the thickness direction.

The analysis is divided into two parts: a first analysis focused on the effects of the 3D geometrical parameters

on the morphing shape of the wingtip comparing it to a defined ideal scenario; a second analysis focused on the

effects on the morphing mechanism.

Keywords: compliant mechanism, droop-nose, morphing wingtip, topology optimization

vii

Resumo

Como parte do projeto NOVEMOR da EU FP7, um bordo de ataque da extremidade da asa adaptativo (AMWT)

esta a ser desenvolvido pelo DLR (Centro Aeroespacial Alemao), com o potencial para reduzir a resistencia da asa

ao escoamento e de substituir superfıcies de controlo tradicionais. O AMWT e ativado atraves de um mecanismo

flexıvel (compliant mechanism) e as ferramentas utilizadas para a sua concepcao nao levam em conta os parametros

geometricos 3D presentes na extremidade de uma asa, como o afilamento e o angulo de varrimento.

O presente trabalho propoe uma metodologia para investigar como os parametros geometricos 3D afetam o

funcionamento do mecanismo flexıvel e a geometria do bordo de ataque adaptativo. Alem disso, a metodologia

e aplicada a uma extremidade de asa modelado semelhante ao que esta a ser usado no projeto NOVEMOR, a fim

de ajudar e fornecer dados para o processo de desenvolvimento do AMWT. Os parametros 3D considerados nesta

tese sao angulo de varrimento, afilamento da corda e afilamento da espessura do perfil.

A analise e dividida em duas partes: uma primeira analise que se concentra nos efeitos dos parametros

geometricos 3D sobre a forma da asa quando atuada comparando-a com um cenario ideal definido; uma segunda

analise qie incide sobre os efeitos no mecanismo flexıvel.

Palavras-chave: mecanismo flexıvel, bordo de ataque adaptativo, extremidade da asa adaptativa, optimizacao

topologica

ix

Contents

Acknowledgments v

Abstract vii

Resumo ix

List of Figures xiv

List of Tables xv

Acronyms xvii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 AMWT Design Concepts 3

2.1 AMWT Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Wing Shape Morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Definition, Advantages and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.2 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Compliant Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Definition, Advantages and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Design Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Methodologies Chosen 11

3.1 Analysis Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 CAD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 FE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Result Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Analysis of the Effects of 3D Geometrical Parameters on Skin Morphing Shape - SMAN 23

4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Sweep Angle Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xi

4.1.2 X-taper Ratio Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.3 Y-taper Ratio Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Displacement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.2 Maximum Stress and Strain and Reaction Forces . . . . . . . . . . . . . . . . . . . . . . 38

5 Analysis of the Effects of 3D Geometrical Parameters on the Compliant Mechanism - CMAN 41

5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 0◦ sweep angle, 0.5 X-taper ratio and 1.0 Y-taper ratio . . . . . . . . . . . . . . . . . . . 42



5.1.4 35◦ sweep angle, 1.0 X-taper ratio and 1.0 Y-taper ratio . . . . . . . . . . . . . . . . . . 49

5.1.5 35◦ sweep angle, 0.5 X-taper ratio and 0.5 Y-taper ratio . . . . . . . . . . . . . . . . . . 51

5.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.1 Profile Node Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.2 Compliant Mechanism Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Conclusions and Future Work 57

6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Analysis Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 64

xii

List of Figures

1.1 3D geometry of the Embraer wingtip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1 Classification of shape morphing wing concepts according to [6] . . . . . . . . . . . . . . . . . . 4

2.2 ‘Batwing’ UAV developed by NextGen Aeronautics (source: NextGen Aeronautics) . . . . . . . . 5

2.3 UMAAV with three different span length configurations [16] . . . . . . . . . . . . . . . . . . . . 6

2.4 Possible Gull configurations in a Gull morphing wing [47] . . . . . . . . . . . . . . . . . . . . . 7

2.5 (a) Camber morphing with shape memory alloys actuation [48]. (b) Camber morphing with force

introduction points for transmitting the actuator force to the wing skin (Patent DE2907912-A1,

Dornier company, 1979). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Examples of (a) rigid body and (b) compliant crimping mechanisms [15] . . . . . . . . . . . . . . 8

2.7 Large deflection beam (left) and its pseudo-rigid body model (right) [15] . . . . . . . . . . . . . . 9

3.1 (a) Reference leading edge profile with reference axis for scaling in thickness and chord direction.

(b) Wingtip stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Wingtip with 0.5 (a) chord wise taper ratio and (b) thickness wise taper ratio and (c) with a 20◦

sweep angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 CAD model of a wingtip with 20◦sweep angle and 0.4 taper ratio in both chord and thickness

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Zoom in of the parameterized compliant mechanism where it connects to the wingtip stringer . . . 15

3.6 Result of meshing of a uniform wingtip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.8 Wingtip and Mechanism boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.9 Ideal versus actual morphing shape of P4 of a wingtip with 0.5 X and Y taper ratio . . . . . . . . 20

4.1 Node identification and axis orientation relative to a wingtip profile . . . . . . . . . . . . . . . . . 24

4.2 Percentile 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Maximum z direction displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Maximum Stress and Strain for varying sweep angle wingtips . . . . . . . . . . . . . . . . . . . . 25

4.5 Reaction Forces for varying sweep angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.6 Percentile 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.7 Error distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

xiii

4.8 0.5 x taper ratio wingtip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


4.10 Maximum Stress and Strain for varying taper ratio relative to the x axis . . . . . . . . . . . . . . 29

4.11 Reaction Forces for varying taper ratio relative to the x axis . . . . . . . . . . . . . . . . . . . . . 29

4.12 Total force for varying taper ratio relative to the x axis . . . . . . . . . . . . . . . . . . . . . . . 30

4.13 Percentile 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.14 Error distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.15 0.5 y taper ratio wingtip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


4.17 Maximum Stress and Strain for varying taper ratio relative to the y axis . . . . . . . . . . . . . . 33

4.18 Reaction Forces for varying taper ratio relative to the y axis . . . . . . . . . . . . . . . . . . . . . 34

4.19 Total force for varying taper ratio relative to the y axis . . . . . . . . . . . . . . . . . . . . . . . 34

4.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.21 Percentile 90 for varying both types of taper ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.22 Percentile 90 for varying sweep angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.23 von Mises stress distribution for different wingtips . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.24 Wingtip surface area for varying 3D geometrical parameters . . . . . . . . . . . . . . . . . . . . 40

5.1 Displacement error distribution (1500 0 50 100) . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Distribution of the z component of the mechanism node displacement for LC2 (1500 0 50 100) . 44



5.5 Displacement error distribution (1500 0 50 50) . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.6 Distribution of the z component of the mechanism node displacement for LC2 (1500 0 50 50) . . 48

5.7 Displacement error distribution (1500 35 100 100) . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.8 Distribution of the z component of the mechanism node displacement for LC2 (1500 35 100 100) 50



xiv

List of Tables

3.1 Magnitude of applied forces for each specific type of Profile . . . . . . . . . . . . . . . . . . . . 19

4.1 Displacement error Percentile 90 for applied displacements of 1mm and −10mm in the x and y

directions respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Displacement error Percentile 90 for applied displacements scaled according to taper ratio values . 36

4.3 Droop angles of wingtips resulting from scaled and not scaled applied displacements . . . . . . . 37

5.1 Percentile 90 values of the profile node displacement errors (1500 0 50 100) . . . . . . . . . . . 43

5.2 Average mechanism node displacement (1500 0 50 100) . . . . . . . . . . . . . . . . . . . . . . 43

5.3 x and y components of the displacement of the nodes located at the control points for LC2

(1500 0 50 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45




(1500 0 100 50) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.7 Percentile 90 values of the profile node displacement errors (1500 0 50 50) . . . . . . . . . . . . 47

5.8 Average mechanism node displacement (1500 0 50 50) . . . . . . . . . . . . . . . . . . . . . . . 48


(1500 0 50 50) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.10 Percentile 90 values of the profile node displacement errors (1500 35 100 100) . . . . . . . . . . 49

5.11 Average mechanism node displacement (1500 35 100 100) . . . . . . . . . . . . . . . . . . . . . 50


(1500 35 100 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51




(1500 35 50 50) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.16 Variation [%] in the Percentile 90 values of the profile node displacement error when the compliant

mechanisms are introduced (from LC0 to LC1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.17 Variation [%] between the ideal and the actual target displacements . . . . . . . . . . . . . . . . . 54

xv

List of Acronyms

2D Two Dimensions

3D Three Dimensions

AMWT Adaptive Morphing Wingtip

CAD Computer Aided Design

CMAN Compliant Mechanism Analysis

DLR German Aerospace Center

EU FP7 European Union’s Seventh Framework Programme for Research

LC0 Load Case with applied displacements scaled according to tapering without compliant mechanism

LC1 Load Case with applied displacements scaled according to tapering with compliant mechanism

LC2 Load Case with applied force located on the compliant mechanism where the actuator would be attached

P2 Profile 2 located at 3/4 of the span

P4 Profile 4 located at 1/4 of the span

SB Compliant section of the parameterized compliant mechanism

SG Support section of the parameterized compliant mechanism

SIMP Solid Isotropic Material with Penalization for intermediate densities

SMAN Shape Morphing Analysis

SR Section of the parameterized compliant mechanism where it attaches to the wingtip stringer

List of Symbols

αi – ith Design variable

αi – ith Lower bound of the design variable

αi – ith Upper bound of the design variable

λX,Y – Taper ratio relative to X,Y axis

cT – Profile chord at tip

cR – Profile chord at root

e – Node displacement error

xvii

f – Objective function

gj – jth Inequality constraint

hk – kth Equality constraint

hT – Profile thickness at tip

hR – Profile thickness at root

m – Number of inequality constraints

n – Number of design variables

r – Number of equality constraints

uAi – Actual node displacement along axis i

uIi – Ideal node displacement along axis i

x, y, z – Global coordinate axes

X,Y, Z – Tapering coordinate axes

xviii

Chapter 1

Introduction

1.1 Background and Motivation

As part of the EU FP7 project NOVEMOR, a droop-nose adaptive morphing wingtip (AMWT) is being designed.

The AMWT is composed by a fiberglass composite material skin with an optimized thickness distribution and a

double-L stringer whose position is also optimized. The actual morphing mechanism introduces the actuation force

via the stringer, where it connects with the wingtip.

The 3D geometry of the wingtip is given by Embraer and as can be seen from Fig.1.1, it can be characterized

by a sweep angle and tapering in chord and thickness directions. The presence of these parameters result in a

complex morphing design and manufacture processes.

The adoption of morphing technology on the wingtip has the potential to reduce drag and substitute classical

wing control surfaces. The lift distribution may be controlled to best suit each flight condition and aeroelastic

problems (such as aileron efficiency loss) can be overcome [49].

Figure 1.1: 3D geometry of the Embraer wingtip

The use of a compliant mechanism to achieve morphing is one of the key features of the AMWT, bringing the

potential associated advantages such as weight savings, reduced part and assembly costs, and the elimination of

backlash [49].

The tools available and used for the design of the morphing mechanism are valid for a 2D geometry (profile)

of a specified wing section and thus it becomes necessary to analyze the effects of the 3D geometrical parameters

1

on the morphing of the wingtip in order to extrapolate the 2D design results suiting them for the actual 3D wingtip.

This thesis proposes a method to analyze the above mentioned effects and implements it on a modeled wingtip

with similar dimensions and geometrical parameters to the Embraer wingtip.

The analysis is divided in two parts: a first analysis focused on the effects of the 3D geometrical parameters

on the morphing shape of the wingtip comparing it to a defined ideal scenario; a second analysis focused on the

effects on the morphing mechanism. In the first analysis the parameters are dealt with individually by creating

sets of wingtips with the presence of the same parameter but with different values. In the end, wingtips with

combinations of the parameters are also discussed. In the second analysis only one value of each parameter is

considered and wingtips containing combinations of the parameters with those values are analyzed.

1.2 Thesis Layout

Following is the layout of the thesis with a brief description of the contents of each chapter:

• Chapter 2: Description of the process being used for the design of the AMWT and a literature review on

the associated concepts (morphing technologies and compliant mechanisms).

• Chapter 3: Explanation of the analysis process developed and the reasons for the methodology chosen.

• Chapter 4: Analysis of the effects on the morphing shape of several wingtips with different configurations

of 3D geometrical parameters with discussion of the results obtained.

• Chapter 5: Analysis of the effects on the compliant mechanism for wingtips with 3D geometrical parameters

similar to the Embraer wingtip.

• Chapter 6: Final conclusions and future work.

2

Chapter 2

AMWT Design Concepts

In this chapter a description of the methodology being used for the design of the AMWT is given (still under

development) and a literature review on the main concepts used in the methodology is made in order to provide

contextualization and facilitate the understanding and visualizations of the proposed analysis method described

and implemented in the following chapters.

2.1 AMWT Design Process

The design process of the AMWT is currently being developed at the DLR and the following description is based

on the paper that is being written on the topic [49] regarding the construction of a demonstrator model for the

droop-nose morphing device. The leading edge of the AMWT must be able to assume two different shapes: the

clean target shape without actuation input and the droop target shape when actuated (approximate droop angle of

2◦).

The structural design of the AMWT goes through three stages: design of the wingtip skin; design of the

compliant mechanism and finally the design of the support for the compliant mechanism.

For the skin design stage, the DLR design tool [20] is used and data regarding the design domain geometry,

location of the connection points, target displacements and output forces to be delivered by the mechanism, and the

stiffness of the skin are obtained. This data is then used as input for the second stage where the compliant mech-

anism is designed via topology optimization using the solid isotropic material with penalisation (SIMP) material

model [9]. Data obtained from this stage is then transfered to the design of the support, namely design domain

geometry, material and thickness (as the design is monolithic), reaction forces from the compliant mechanism and

the actuator and their locations. With this data the support is designed, again using topology optimization.

2.2 Wing Shape Morphing

2.2.1 Definition, Advantages and Challenges

In the aeronautical industry, the term morphing is used when referring to ‘a set of technologies that increase a

vehicle’s performance by manipulating certain characteristics to better match the vehicle state to the environment

3

and task at hand’ [50]. The exact type or extent of the geometrical changes necessary to qualify a structure

for the title of ‘shape morphing’ has no consensus between researchers in the area since the above definition

can also include established technologies such as flaps, slats, ailerons or retractable landing gear while the term

morphing contains a connotation of ‘radical shape changes or shape changes only possible with near-term or

futuristic technologies’ [6].

Given the review made in [6], wing morphing concepts can be classified into three major groups according

to which wing parameter is affected as illustrated in Fig.2.1: planform alteration (changes in sweep, span or

chord), out-of-plane transformation (twist, dihedral/gull and spanwise bending) and airfoil alterations (camber and

thickness).

In this thesis, the shape morphing being analyzed lies in the ‘airfoil alteration’ category where the curvature of

the leading edge has a clean (for cruise) and a droop (for take-off and landing) configuration.

Figure 2.1: Classification of shape morphing wing concepts according to [6]

The great advantage of the use of shape morphing wings is the potential to radically expand an aircraft’s

flight envelope. A morphing aircraft will be more competitive compared to conventional aircrafts as more mission

tasks are added to their requirements [6]. In order to be able to fly at a range of flight conditions, wings are

designed in order to satisfy several different (many times opposing) requirements thus often leading to sub-optimal

performance at each flight condition [6]. An ideal example of a morphing wing is a wing capable of continuously

adjusting its airfoil shape increasing its lift/drag ratio for each different flight condition [45].

Several challenges occur with the use of morphing technologies, most of them rely on the existence of a flexible

skin with conflicting requirements: it has to be sufficiently soft to allow shape changes but at the same time stiff

enough to maintain the desired shape under aerodynamic loads. Also, the strictness of these requirements change

for different flight conditions [6].

Furthermore, according to [33], morphing concepts can bring additional weight, complexity and power con-

sumption (required by the actuation systems) to the aircraft. Finally, from [29], there is ‘a strong need to understand

the scalability of morphing wing concepts to achieve sufficient structural stiffness, robust aero-elastic designs, and

an adequate flight control law to handle the changing aerodynamic and inertia characteristics of morphing vehi-

cles’.

4

2.2.2 State-of-the-Art

In this section a general review of the main types of morphing is made in order to give an idea of what is being done

in the aeronautical industry and to situate the wing shape morphing that is dealt with in this thesis. This review is

mainly based on the findings in [6] (where an extensive review on morphing technologies is done) and in [44].

Sweep

The introduction of sweep angle delays the rise in drag at transonic speeds caused by compressibility effects [4]

enabling supersonic flight with subsonic designed wings. This is important because wings designed for supersonic

flights are highly inefficient at low speeds. Naturally the idea of a variable sweep wing arises to combine efficient

high-speed requirements with efficient low-speed requirements [6] which is not possible for a fixed sweep angle

wing due to the contradictory nature of the requirements. Furthermore with a variable sweep the structural loads

can be redistributed along the span reducing the bending moment requirements at root sections [26].

The main concept used for designing a variable sweep wing is a rigid wing that rotates around a pivot [6, 26]

this enables changes in wing area, span and aspect ratio which was done by NextGen Aeronautics [3, 12] who

developed a UAV (called BatWing) with sweep change capability during flight. The NextGen Aeronautics wing

concept was extended by [13] where optimizing actuator orientation for rigid and flexible wings was the main goal.

To attain the target morphing shape some studies use shape memory polymers [51] and shape memory alloys

[43].

Figure 2.2: ‘Batwing’ UAV developed by NextGen Aeronautics (source: NextGen Aeronautics)

Span

Aircrafts that possess wings with large span (high aircraft aspect ratio) have lower maneuverability and cruise

speeds but better range and fuel efficiency compared to aircrafts with a high aspect ratios [27]. Thus a variable

span wing has the potential give an aircraft the advantages of both large and short spans. Increasing the span

decreases the spanwise lift distribution (for the same lift) leading to a decrease in wing drag but on the other hand

the wing-root bending moment will increase considerably so it is necessary to take into account aerodynamic and

aeroelastic properties when designing a variable span wing [6].

Most concepts use telescopic wings to enable span change and several studies on developing this type of

wing morphing are being made (list of studies can be found in [6]). Aerovisions Inc. developed the ‘Unmanned

Morphing Aerial Attack Vehicle’ (UMAAV) where the wing consisted of several sliding segments.

5

Another approach is to use a scissor-like mechanisms to vary the wing span. Studies on this approach can be

found in [18, 17, 10].

Figure 2.3: UMAAV with three different span length configurations [16]

Chord

Chord morphing technologies are mainly used in rotary-wing aircrafts due to the structural complexity of a fixed

wing where fuel tanks, spars and other components are present [6]. Chord variations change the wing area and the

aerodynamic load distribution.

In [32] an inter-penetrating rib mechanism to change the chord length by means of miniature DC motors and

lead screws was used although the added weight and complexity of the design were big disadvantages for its

application [6].

In rotor blades several investigations on the concept of ‘static extended trailing edge’ where a flat plate is

extended through a slit trailing edge can be found in [24, 23, 19].

Twist

‘If the angles of attack of spanwise sections of a wing are not equal, the wing is said to have twist’ [34]. This

parameter changes the lift distribution along the span so control over the twist can enhance flight performance in

different flight conditions. Also, varying twist can be used as a roll control mechanism.

In this type of morphing, the main fields of study rely on using the wing aeroelastic flexibility for a net benefit

(traditionally this wing property is treated as an obstacle to overcome) by using control surfaces to promote favor-

able twist. The energy of the airstream is used to twist the wing in favor of the desired control outcome instead of

opposing the traditional generated control forces [30]. This concept has been implemented and tested in the Active

Aeroelastic Wing research program (funded by NASA and the US air force) where an F/A-18 fighter was modified

and used [6] in the early 2000’s.

The Active Aeroelastic Aircraft Structures research project in Europe also was created to develop and design

concepts to use aeroelastic flexibility as a positive behavior for aircraft performance [6].

However, for the practical implementation of active aeroelastic concepts the development of methods, algo-

rithms, software, analytical and experimental investigations should be carried out [22].

Investigations on the use of piezoelectric materials and shape memory alloys as actuators to induce wing

twisting is also being done.

6

Dihedral/Gull

Control over dihedral/gull can change the aerodynamic span, replace traditional control surfaces, change the vor-

ticity distribution and improve stall characteristics of an aircraft.

In [47] a mechanism is proposed where it is possible to extend the span of the wing and once extended it can

obtain a gull configuration (Fig.2.4). This mechanism is highly inspired on bird’s capabilities of changing the

shape configuration of their wings.

Figure 2.4: Possible Gull configurations in a Gull morphing wing [47]

Dihedral/Gull morphing can be achieved by two methods: either using folding wings (example showed in

Fig.2.4) or using variable cant and toe angle winglets [44].

Camber

Camber morphing is the ability to change the curvature of an airfoil. It can be done over the whole airfoil or on

specific parts such as the leading or trailing edge.

Traditional methods of camber control rely on elevators, rudders, ailerons and flaps which are used in most

modern aircrafts. A lot of research is currently being focused on the design of seamless and gapless morphing

concepts (contrary to the traditional camber control methods used) for airflow laminarization in order to obtain

significant drag reductions [20] which comes closer to the connotation, mentioned before, that is attributed to

morphing (‘radical shape changes or shape changes only possible with near-term or futuristic technologies’ [6]).

Actuation of seamless and gapless camber morphing concepts can be done using piezoelectrics, shape memory

alloys and conventional actuators such as servo and ultrasonic motors and pneumatic and hydraulic devices.

(a) (b)

Figure 2.5: (a) Camber morphing with shape memory alloys actuation [48]. (b) Camber morphing with forceintroduction points for transmitting the actuator force to the wing skin (Patent DE2907912-A1, Dornier company,1979).

7

A relatively recent concept of camber morphing actuation uses compliant mechanisms. This concept was

chosen for the design of the AMWT (topic of the present thesis). Discussion on the design methodology and

workings of this type of morphing was done in the beginning of the current chapter.

2.3 Compliant Mechanism

2.3.1 Definition, Advantages and Challenges

A compliant mechanism, as defined in [21], is a single-piece flexible structure that delivers the desired motion by

undergoing elastic deformation (as opposed to using movable joints) that can be designed to obtain any desired

input/output force/displacement characteristics. Traditional mechanisms are designed to be strong and stiff and are

usually assembled from discrete components [21]. A compliant mechanism is designed to be flexible enough to

transmit the desired motion and at the same time to be stiff enough to withstand the external loads [25].

Compliant mechanisms bring several advantages in the aeronautic industry, namely when dealing with appli-

cations such as shape change in aircraft wings [25]. The fact that they are constituted by a single-piece structure

eliminates backlash error (backlash: the maximum distance or angle through which any part of a mechanical sys-

tem may be moved in one direction without applying appreciable force or motion to the next part in mechanical

sequence [5]) leading also to a reduction in production and maintenance costs associated with mechanisms con-

taining multiple parts [25]. Furthermore, compliant mechanisms present smooth deformation fields reducing stress

concentrations [25].

Another big advantage of compliant mechanisms (although out of the scope of this thesis) is their ability to be

miniaturized suiting them for use in microelectromechanical systems [28].

Several challenges also arise with the use of compliant mechanisms, namely the added complexity in analyz-

ing and designing the mechanism. It is necessary to combine knowledge of mechanism analysis methods with

knowledge of deflection of flexible members (due to large deflections, in most cases linearized beam equations

are no longer valid). Although the theory of analysis and design of compliant mechanism is continuously being

developed, it is still typically more difficult than the analysis and design of rigid body mechanisms [11].

(a) (b)

Figure 2.6: Examples of (a) rigid body and (b) compliant crimping mechanisms [15]

Fig.2.6 illustrates the different types of mechanism mentioned previously emphasizing the visual simplicity of

the single-piece compliant mechanism as opposed to the more complex multi-part rigid body mechanism.

8

2.3.2 Design Methods

Several methods exist for the design of compliant mechanisms and most of them can be divided into methods based

on the pseudo-rigid body model or based on optimization [1].

The pseudo-rigid body model is used to model the deflection of flexible members by using rigid body compo-

nents (attached with pin joints) that have equivalent force-deflection characteristics connecting rigid-body mecha-

nism theory with compliant mechanism theory. For each flexible segment of a compliant mechanism a pseudo-rigid

body model is built and springs are added to the model in order to simulate the force-deflection relationships be-

tween those flexible segments [15]. This approach allows the design of compliant mechanisms without concern

for the energy storage in the flexible members which is useful for systems with concentrated compliance [1] where

large deflections are necessary only in localized areas of the system [2].

Figure 2.7: Large deflection beam (left) and its pseudo-rigid body model (right) [15]

Optimization based design methods view flexible mechanisms as flexible continua and are used to design

mechanisms with distributed compliance where large portions of the structure deform when it’s loaded [1].

A standard optimization problem seeks to minimize (or maximize depending on what is best for the current

problem) an objective function f(x) by varying one or more design variables xi between specified bounds and

subject to a number of equality or inequality constraints (hk(x) and gj(x) respectively).

minx

f(α) α = [αi, αi, . . . , n]T ∈ Rn

subject to gj(α) ≤ 0, j = 1, 2, . . . ,m

hk(α) = 0, k = 1, 2, . . . , r

αi ≤ αi ≤ αi, i = 1, 2, . . . , n

(2.1)

Topology optimization is the most general level of structural optimization of a continuum mechanism [15] (fol-

lowed by shape and size optimization) where given a design domain, the algorithms created consider all possible

ways of distributing the material in the domain in order to obtain the desired output [1].

The most popular numerical FE-based topology optimization method is the Solid Isotropic Material with Pe-

nalization for intermediate densities (SIMP) method [37] (used in the design of the compliant mechanism for the

AMWT).

When optimizing the topology of a structure the goal is to determine the optimal placement of a given isotropic

material (in the design domain) [9]. If the design variable is a parameterized material density (ρ), for example,

9

where ρ = 1 corresponds to a region with material and ρ = 0 to a region with no material, then the optimal

solution will contain mostly elements with intermediate densities. In the SIMP method a penalization is given to

intermediate densities thus removing them from the optimal solution.

The penalization approach can be justified by introducing manufacturing costs (in order to obtain intermediate

thicknesses, the structure must suffer some machining process) in order to obtain suitable penalization values

[36, 35]. Finding ranges of microstructures to generate adequate values of penalizations is another way of justifying

the SIMP approach which is demonstrated in [8]. Finally the use of penalizations as a computational tool in discrete

value optimization is a standard method in nonlinear optimization [36] as can be seen in [39, 38].

The SIMP method is computationally efficient since only one free variable is used per element, it is robust since

it can be used for any combination of design variables, the penalizations can be adjusted freely and it is conceptually

simpler than other optimization methods since the algorithms used do not involve complex derivations [36].

In terms of disadvantages, the SIMP method depends greatly on the degree of penalizations used and it does

not necessarily converge to the optimal solution [46]. The fact that it depends on the mesh used [7] can be viewed

as a disadvantage but can be mostly avoided by constraining the length of the internal boundaries [14] or by using

mesh-independent filtering methods [40, 41]. On the other hand, for a simple topology, mesh-dependence can be

beneficial by enabling the proof that a topology converges to a known exact analytical solution [36] which was

demonstrated in [31, 42].

Other topology optimization methods exist such as the OMP and NOM methods but as shown in [36] the SIMP

method provides substantial advantages over them.

10

Chapter 3

Methodologies Chosen

Several obstacles arise when analyzing the effects of the 3D geometrical parameters on the shape morphing of the

wingtip and on the compliant mechanism. This chapter enumerates those obstacles and discusses the solutions

found and methods used in order to obtain a reliable and feasible analysis. It is important to note that there are

several variables to consider in an analysis of this type and sometimes the path chosen is not necessarily the only

solution possible and so what is presented in this thesis is a proposed method to analyze the above mentioned

effects based on the choices that were considered as most appropriate.

As stated before, two analyses were made. The first, regarding the effects on the shape morphing of the wingtip,

will be denoted as SMAN (shape morphing analysis) and the second, regarding the effects on the compliant mech-

anism, will be denoted as CMAN (compliant mechanism analysis).

3.1 Analysis Concept

As a first step, a parameterized CAD model of the geometry of the wingtip is constructed where it is possible to

change parameters such as sweep angle and taper ratio. The next step is to apply a loading case to various wingtips

with different 3D geometrical parameters, simulating the actuation of the morphing mechanism (this is done using

finite element analysis software). Finally the obtained morphing shape is compared to an ideal case to determine

the differences in shape.

The first problems that arose were how to define an ideal morphing shape for a certain profile of the considered

3D wingtip and how to obtain results comparable between different wingtips. To solve these problems, it is

assumed that each wingtip will contain two morphing mechanisms, one located at 1/4 of the span (Profile 4 or P4)

and the other located at 3/4 of the span (profile 2 or P2). The 3D geometrical parameters that were considered are

sweep angle, tapering in the chord direction and tapering in the thickness direction.

Now it is necessary to define the ideal morphing shape of the considered profiles. This was done by building

uniform wingtips (no sweep angle nor tapering) with a cross-section identical to the profile where the displacement

is being applied in the non-uniform wingtip. So, for example, when considering a wingtip with taper ratio of 0.8,

it is necessary to consider a uniform wingtip scaled to 0.85 of the root profile (to match P2) and another one scaled

to 0.95 of the root profile (to match P4). The same displacements are then applied to those uniform wingtips and

11

the resulting morphing shape is what will be considered as ideal.

For CMAN it was necessary to also build a parameterized CAD model of the compliant mechanism to in-

corporate with the different wingtips. In SMAN, to simplify the analysis and be able to analyze a bigger variety

of different wingtips, only the desired output of the mechanism is taken into account, substituting it by desired

displacements located where the mechanism would be attached to the stringer. With this simplification it is pos-

sible to eliminate the step of designing the mechanisms for each different wingtip. The same displacements are

applied on both sections of the wingtip (P2 and P4) assuming that for each different profile considered, a morphing

mechanism could be designed to obtain those displacements.

In CMAN, results regarding the shape morphing of the wingtip are also discussed in order to validate the above

mentioned simplification made in SMAN.

3.2 CAD Models

Wingtip

In order to test several different wingtips, a parameterized geometrical model was built, using CATIA software,

based on the inboard profile of the wingtip given by Embraer (Fig.3.1a). Since the aim of the NOVEMOR project

is to design a droop-nose adaptive morphing wingtip, only the leading edge of the wingtip is modeled and so in

future references, when using the term wingtip, it is meant the leading edge of the wingtip.

(a) (b)

Figure 3.1: (a) Reference leading edge profile with reference axis for scaling in thickness and chord direction. (b)Wingtip stations

The orange line in Fig.3.1a defines the chord direction and was obtained via two points: the first (on the right)

is the midpoint between the top and bottom edge of the profile; the second (on the left) is located at 50% of the

length of the profile curve. The origin of the axises (in red in Fig.3.1a) is the midpoint between the two above

mentioned points. Scaling the profile in the chord and thickness direction is then defined by the X and Y axes

shown in Fig.3.1a respectively.

From the reference profile, four more are created defining the 3D geometry of the wingtip (Fig.3.1b). All 5

profiles are parallel to each other and are equally distanced. The distance between the first and the last profile

defines the span of the wingtip and is set to a value of 1500mm to match the span of the Embraer wingtip. The

second and fourth profiles are located at 1/4 and 3/4 of the span respectively (where the displacements will be

applied) and their shape, before and after morphing, are the basis of the analysis that will be done.

Translating and scaling the profiles allows to change the geometrical parameters of the wingtip namely sweep

angle and tapering.

12

λX =cTcR

(3.1) λY =hThR

(3.2)

The definition of the taper ratio parameter is given by Eq.3.1 (in the case of tapering in the chord direction)

and Eq.3.2 (in the case of tapering in the thickness direction) or simply put into words, it is the ratio between the

considered dimension at the tip and at the root of the wingtip. The sweep angle is defined as the angle between the

leading edge of the uniform wingtip and the actual leading edge maintaining a constant span.

(a) (b) (c)

Figure 3.2: Wingtip with 0.5 (a) chord wise taper ratio and (b) thickness wise taper ratio and (c) with a 20◦ sweepangle

Fig.3.3 gives an example of a possible wingtip with the surface uniting all 5 profiles included.

Also in Fig.3.3 are listed the six parameters that control the geometry of the wingtip. The first two change

the scaling of the root profile allowing to model different uniform wingtips for the purpose of obtaining the ideal

morphing shape. The third parameter defines the span and as said before is set at a fixed value of 1500mm. The

fourth parameter changes the sweep angle and finally, the last two parameters, control the tapering in the chord and

thickness direction (relative to the dimensions of the inboard profile which are defined by the first two parameters).

Figure 3.3: CAD model of a wingtip with 20◦sweep angle and 0.4 taper ratio in both chord and thickness direction

The CAD model created allows to rapidly obtain different wingtip geometries for future morphing analysis.

Each of the 5 profiles of a wingtip is defined by 101 points whose coordinates are saved in a file for posterior

handling by different software.

13

It is important to note that the global coordinate system (used for position and for force and displacement

direction) is not identical to what is shown in Fig.3.1a, which was obtained by the method chosen to implement

tapering. Although all profiles still belong in the xy plane, the x axis of Fig.3.1a is rotated 5.958◦ anti-clockwise

relative to the x axis of the global coordinate system.

To simplify the language used in the following sections, the terms X-taper and Y-taper are used to refer tapering

in the chord direction and in the thickness direction respectively while the lowercase symbols, x and y, are used to

refer to the global coordinate system.

Compliant Mechanism

The first task was to determine how to model the compliant mechanisms for the different wingtips that are consid-

ered. It is not feasible to go through the entire topology optimization process (which requires data regarding the

geometry of the wingtip) to obtain the adequate mechanism for each different profile and it is necessary to have

flexibility in terms of the load cases that will be applied (topology optimization results in a compliant mechanism

optimized for a specified displacement case) thus it was decided to use a compliant mechanism already modeled

for a profile of the Embraer wingtip [49] and adapt it to fit the different wingtips that will be considered.

(a) Optimized Mechanism

(b) Adapted parameterized mechanism

Figure 3.4

Fig.3.4a illustrates the optimized compliant mechanism for a profile of the Embraer wingtip. The model was

then adapted in order to fit P2 and P4 of the wingtips that will be analyzed. The resulting mechanism is shown

in Fig.3.4b. The green section (SG) is a stiff truss structure, thereby holding the mechanism in its place. This

replacement was done in order to simplify the geometry of the mechanism for future meshing, since there is

no need to support the actuator and it will not influence the output of the compliant mechanism. It was not

removed completely because its absence would change the resulting off-plane displacements of the mechanism

when actuated. This section can be scaled in the X and Y directions in order to easily fit wingtips with X and/or

Y tapering.

14

The blue section (SB) is the actual compliant mechanism that transfers the actuation force to the stringer in

the predefined manner. This structure remains the same with the added possibility of scaling. In this case, it is

only possible to scale along both directions at the same time in order to maintain the dimension aspect ratios of the

structure, allowing the same load transfer path for all mechanisms.

The red section (SR) is where the mechanism connects to the stringer. This sections’ position defines the posi-

tion of the whole mechanism. Point A (Fig.3.5) defines the overall position of the mechanism, the coordinates of

this point are parameters that can be changed and are set to coincide with the location of the applied displacements

on the stringer as defined in the previous chapter. Point B (Fig.3.5) defines the orientation of the red section in

order to align it with the stringer.

Finally, the yellow section is the connection between the final shape of the red and blue sections.

For short, the parameterized mechanism model created allows for scaling of the green section in both X and

Y directions, scaling of the blue section in all directions at the same time, positioning of Point A and positioning

of Point B in order to align the red section with the stringer of the wingtip it will be inserted in.

Figure 3.5: Zoom in of the parameterized compliant mechanism where it connects to the wingtip stringer

Consider the case of a wingtip with a 0.5 X-taper ratio as an example of the practical application of the

parameterized mechanism model. For P2, SG will be scaled with a value of 0.625 in the X direction (0.825

for P4), SB is scaled with the same value in all directions and SR is positioned with the appropriate coordinates

taken from the wingtip CAD model.

3.3 FE Models

Wingtip

As stated before, simulating the shape morphing of a wingtip is done by applying displacements to the stringer at

profile 2 (P2) and profile 4 (P4). To do so it is necessary to make use of finite element software.

To simplify the meshing step and at the same time enable uniform meshing for different wingtips, a Matlab

script was created (created by Dr. Srinivas Vasista for the Embraer wingtip and adapted for the current parameter-

ized wingtip by the author of this work) that determines the nodes and respective coordinates defining the wingtip

skin geometry based on the coordinate data produced from the CAD model. The script also identifies and defines

the elements of the mesh identifying what nodes constitute each element (four nodes define one element). Further-

more, the script adds the stringer to the model by defining its height and position, the stringer is connected to the

15

skin perpendicularly to the tangent of the profile contour at the point where it is specified to be located.

In this step, several parameters can be modified namely stringer position, height and foot length at each of the

5 profiles, number of nodes defining the stringer, number of nodes along the profile contour and number of nodes

along the span of the wingtip. For the purpose of obtaining comparable results, for all the various wingtips to be

analyzed the above mentioned parameters are set to the same values:

• Stringer position at each profile: 40% of the profile contour length counting from the bottom section;

• Stringer height at each profile: 20mm;

• Stringer foot length: 15mm (uniform along span);

• Number of nodes along stringer height: 8;

• Number of nodes along stringer foot: 13;

• Number of nodes along profile contour before stringer foot: 50;

• Number of nodes along profile contour after stringer foot: 70;

• Number of nodes along span: 101.

The stringer height was chosen so it would fit inside the wingtip for all the values of taper ratio to be used.

All other parameter values were predetermined by the project supervisor and allow for a fine enough mesh for the

purpose of this thesis.

The number of nodes along the stringer foot parameter allows for a larger number of nodes per unit length in

that region since it is where the stringer is attached to the skin.

Figure 3.6: Result of meshing of a uniform wingtip

Fig.3.6 illustrates more clearly what is meant by the above mentioned parameters. In purple is represented the

foot region of the stringer, in blue the stringer, in green and yellow the regions before and after the stringer foot

respectively.

The data regarding node coordinates and node and element identification is saved in a file to be read by the FE

software (ANSYS 15.0).

Using ANSYS Parametric Design Language, a macro was created that reads the node and element data, builds

the FE model of the wingtip, applies the displacements and saves the results for future analysis. The following

paragraphs describe the tasks done by the macro justifying the choices made.

16

The element type chosen was SHELL181 which is a four-node element with six degrees of freedom at each

node (translation and rotation for x, y and z axes) being suitable for the thin shell structure that is the considered

wingtip.

Structure thickness and material properties are also determined. An isotropic material was chosen since the

purpose of this work is to analyze geometrical parameter effects, where the added complexity of an orthotropic

material would deviate from that purpose. The elastic moduli was set to 42GPa and the Poisson’s ratio to 0.26.

These values were chosen to simulate the actual fiberglass composite material that is being considered for the

construction of the wingtip. Finally a uniform thickness of 2mm was chosen for the whole structure since it is

close to the values obtained in the skin optimization that is being done.

As boundary conditions, the nodes on the edges of the wingtip that mark the end of the leading edge are set to

have zero degrees-of-freedom in terms of translation and rotation, this is illustrated in Fig.3.7a.

To simulate the actuation of the morphing mechanism for SMAN, as said before, displacements are applied to

the stringer at 1/4 and 3/4 of the span. It was decided to apply these displacements on the first bottom node of the

stringer. In Fig.3.7a, the applied displacements are marked as the blue triangles seen on the stringer and Fig.3.7b

shows the node layout of a profile and locates the exact node where the displacement is applied. The figure also

indicates the axes orientation that is used.

(a) Boundary conditions (b) Reference axes system and location of applied displacement

Figure 3.7

Finally a nonlinear (including large-deflection effects) static solution is obtained for a specified load case. The

resulting node displacement, stress, strain and reaction force values are saved into files for posterior analysis.

Compliant Mechanism

For CMAN, the mechanism CAD models for P2 and P4 of a wingtip were imported into the already existing

wingtip FE model described above. The meshing step cannot be done in the same automatic manner as before

due to the complex geometry of the mechanism. The automatic mesh tool available in ANSYS was used with an

element size of 1mm. The mesh was then refined in certain areas in order to guarantee the existence of rows with

at least 2 elements in every section of the mechanism. Prior to meshing, three hard points were created to guarantee

the creation of nodes located at the two connecting points with the stringer and a third located where the actuator

would apply the force.

The element type chosen was SHELL181 and the thickness of the mechanism was set to 5mm. An isotropic

material was used with an elastic moduli of 70GPa and a 0.35 Poisson’s ratio in order to simulate the material

properties of Aluminum 7075, which is the material being considered for the construction of the mechanism.

17

As boundary conditions, all nodes in the trailing edge of the mechanism were fixed in all six degrees of freedom

i.e. no translation nor rotation. Furthermore, the node where the actuator would apply the force only has two free

degrees of freedom: translation along the x axis and rotation around the z axis simulating the presence of a

linear actuator connected to the mechanism. Fig.3.8 illustrates the applied boundary conditions and the red arrow

represents the location and direction of the force that the actuator would apply (mid-point of the edge).

Figure 3.8: Wingtip and Mechanism boundary conditions

Load Cases

In order to obtain comparable results in SMAN, the applied displacements are equal for P2 and P4 and for all the

different wingtips. The selected values for displacement were 10mm in the negative y axis direction and 1mm in

the positive x axis direction. These values are in the order of magnitude of what is needed to obtain a droop angle

similar to what is required for the NOVEMOR project.

During the discussion of the results in SMAN it was found that another load case should also be considered,

LC0, where the magnitude of the displacements applied were scaled, from the above mentioned values, according

to the tapering of the wing. So if the wingtip has an X-taper ratio of 0.5 then P2 and P4 are scaled by 0.625 and

0.875 in the X direction, respectively, relative to the root profile thus the applied displacements in that direction

are scaled by the same values. Notice that the scaling of the displacements should be relative to the tapering axes

X and Y and not relative to the x and y axes but since the difference was found to be negligible and in order to

simplify the process, the applied displacements were scaled in the x/y direction with the same value as the profile

X/Y scaling when tapering is present.

Relative to CMAN, two load cases were considered. The first, LC1, is to apply displacements with the same

magnitude and location as in SMAN in order to compare results in terms of ideal and actual morphing shape of

the wingtip skin with and without the mechanism. The second, LC2, is to apply a force in the x direction where

the actuator would be connected, in order to better simulate the actual behavior of the compliant mechanism when

inserted in a wingtip with 3D geometrical parameters.

Since the mechanisms to be used were not optimized for each different profile, it was not known if any desired

value of displacements for LC1 was possible. For the same reason, for LC2, it was not possible to vary the force

18

magnitude to try and obtain the same desired stringer displacement between different wingtips since they were not

designed for that purpose being simply scaled from the original mechanism.

It was found that, for LC1, it was not possible to obtain converged solutions for all considered wingtips when

applying displacements of −10mm in the y direction and 1mm in the x direction. Since the main goal of this

loading case is to compare with the results obtained in SMAN, it was decided to apply displacements scaled

from the above mentioned values according to the scaling of the considered profile. This case was found to be

satisfactory resulting in converged solutions for all considered wingtips.

For LC2 it was found that for certain values of the applied force, the solution would also not converge. In order

to obtain results that are comparable between the different wingtips under LC2, it is necessary to define a constant

parameter. The ratio between applied force and mechanism area (the area of SGwas not taken into account since it

has only support purposes and will not influence the load path significantly) was chosen as the most suitable control

parameter since the mechanism area is the simplest measurable property of the different compliant mechanisms

used for the different wingtips. The value of the force-to-area ratio was chosen as the largest at which the FE

solutions of all the wingtips converge.

Profile X-scaling Profile Y-scaling SR+SB [mm2] Applied force [N ]1.0 1.0 1675.86 201

0.625 1.0 654.63 790.875 1.0 1283.08 1541.0 0.625 654.63 791.0 0.875 1283.08 154

0.625 0.625 654.63 790.875 0.875 1283.08 154

Force-to-area ratio [N/mm2] 0.12

Table 3.1: Magnitude of applied forces for each specific type of Profile

3.4 Result Analysis Tools

With all the data that is possible to obtain, it becomes necessary to create tools that can filter the relevant data

for the purposes of this work. Four main MATLAB scripts were written (’ansys read.m’, ’displ wing analysis.m’,

’SSF analysis.m’ and ’displ error analysis.m’) that extract and analyze the relevant data and their tasks will be

described in the following paragraphs.

The first script, ’ansys read.m’, reads the files created by the FE model. It extracts the resulting displacement

values of the nodes of P2 and P4 and calculates their deformed coordinates. Also the maximum stress and strain

values of the wingtip and the reaction forces where the determined displacements are applied are extracted from

the available data for posterior comparison. This script is applied to all wingtips, including all different uniform

wingtips that are necessary, in order to have data regarding ideal and actual morphing shapes. All in all, the script

serves as a filter to the data obtained from the FE model.

Now it is necessary to be able to compare the actual morphing shape of the wingtip containing 3D geometrical

parameters with what is considered to be the ideal shape. For this, script ’displ wing analysis.m’ was created. As

was stated before, to analyze one wingtip containing some 3D geometrical parameters, it is necessary to consider

two more which contain the ideal morphing characteristics. For the sake of simplicity, the wingtip that is being

19

analyzed will be called wing1 and the wingtips with ideal morphing of P2 and P4 will be called wing2 and wing4

respectively.

e =uIi − uAiuIi

, i = x, y (3.3)

The ’displ wing analysis.m’ script reads the filtered data from wing1, wing2 and wing4 and plots them together.

Fig.3.9 illustrates the concept being used to analyze the effects of 3D geometrical parameters. Of course it is not

possible to quantify the effects visually so the script also calculates the node displacement errors, defined by

Eq.3.3 for each node. At this point it is important to note that the x and y displacement components will be treated

differently than the z component. Since the applied displacements are only in the x and y directions, the ideal z

displacement (from a uniform wingtip which approximates to 2D behavior) will be several orders of magnitude

smaller compared to what happens once a 3D geometrical parameter is introduced, hence relative error values,

such as defined by Eq.3.3, will have little relevance in terms of evaluating the effects on morphing shape and so it

was decided that analyzing the absolute values of the z displacement component instead will be done.

This data is once again saved in files for posterior comparison between wingtips with different 3D geometrical

parameters in order to try and find a trend in morphing behavior.

Figure 3.9: Ideal versus actual morphing shape of P4 of a wingtip with 0.5 X and Y taper ratio

Until now, the scripts described, are targeted towards the analysis of data for an individual wingtip. The

purpose of the ’displ error analysis.m’ script is to compare displacement results between sets of wingtips with one

changing parameter in order to isolate and describe its effects.

The big problem present is how to quantify how much different the actual morphing shape is relative to the

ideal shape since the available data are the errors for each node that compose a profile. A first approach would be

20

to calculate the average displacement error of a profile and use it as a quantifying parameter. As several wingtips

were analyzed, it was seen that the distribution of errors for a profile can be greatly dispersed meaning that, for

the same profile, errors could have values between 0% and 10% for some nodes while others could have values

over 100%. This leads to the unreliability of an average value. Thus it was decided that the percentile of the errors

would be a suitable statistical measure to quantify the global profile morphing error.

It is not the purpose of this thesis to determine at which point does the inclusion of 3D parameters invalidate

the actual morphing shape regarding its aerodynamic functions, but rather to analyze how the morphing changes

due to the presence of 3D geometrical parameters, so the choice of using percentiles becomes appealing since it

is a flexible measure regarding performance standards. If, for example, it is found that a morphing shape is only

valid if 80% of the node errors are below a certain value than the percentile 80 of the errors is used and once the

established criteria is passed than the 2D morphing mechanism design is no longer valid for the wingtip.

In the same line of reasoning it is also useful to visualize the distribution of error values of a given pro-

file and analyze how it changes when changing a certain geometrical parameter. This is also performed by the

’displ error analysis.m’ script.

The ’SSF analysis.m’ script was created to analyze the behavior of maximum stress, maximum strain and

reaction forces when a 3D geometrical parameter changes. The script plots the data for a good visualization of

what is happening for posterior analysis.

21

Chapter 4

Analysis of the Effects of 3D Geometrical

Parameters on Skin Morphing Shape -

SMAN

In order to isolate the effects of the variation of each parameter individually, 3 sets of wingtips were created: a set

of wingtips with different sweep angles ranging from 0◦ to 40◦ in steps of 5◦; a set of wingtips with X-taper ratios

ranging from 0.4 to 1.0 in steps of 0.1; and a set of wings with Y-taper ratios ranging from 0.4 to 1.0 in steps of

0.1.

The results for each set of wingtips are presented in the subsequent sections based on 3 aspects:

• Displacement errors;

• Maximum Strain and Stress;

• Reaction forces due to the applied displacements;

A node displacement relative error is defined as the difference between the ideal and actual displacement

divided by the ideal displacement of the node (Eq.3.3).

When referring to reaction forces, it is intended to refer to the resulting forces located at the nodes where

the displacements are applied. It is important to not that since the applied displacements are only in the x and y

direction, there will be no z component of the reaction force.

Each node of a given skin profile is identified by a number, counting from the bottom section up to the top

section. Fig.4.1 illustrates key node identification, namely the first and last node (nodes number 1 and 133), the

skin node coinciding with the beginning of the spar (node number 57) and finally the node dividing the top from

the bottom section (node number 73). As boundary conditions, nodes 1 and 133 are fixed unable to be displaced

and thus are not accounted for in the statistical parameters that will be discussed.

23

Figure 4.1: Node identification and axis orientation relative to a wingtip profile

4.1 Results

4.1.1 Sweep Angle Variation

This set consists of 9 wingtips with different sweep angles ranging from 0◦ to 40◦ in steps of 5◦.

Figure 4.2: Percentile 90

Displacement Errors

As can be seen from Fig.4.2, the percentile 90 of the node relative errors increases almost linearly with increasing

sweep angle and for the biggest sweep angle considered (40◦) the percentile 90 remains below 10%. Furthermore,

this linear evolution is similar for both x and y displacements.

The maximum z direction displacement also increases in an almost linear relation with increasing sweep angle

(Fig.4.3) reaching a displacement of 1.7mm for a 40◦ sweep angle.

It is also noticeable that both profiles behave similarly which is to be expected since they both have identical

shapes.

Due to the straightforward behavior of the displacements with regard to increasing sweep angle it is not neces-

sary to scrutinize further data regarding displacement errors.

24

Figure 4.3: Maximum z direction displacement

Maximum Stress and Strain

It is necessary to determine the strains and stresses acting in the structure during morphing in order to verify

whether the skin material will allow such deformations.

Figure 4.4: Maximum Stress and Strain for varying sweep angle wingtips

Fig.4.4 displays the evolution of maximum stress and strain for the different wingtips. It can be seen that

increasing the sweep angle, both stress and strain also increase.

Reaction Forces

From Fig.4.5a and 4.5b it is possible to infer that increasing sweep angle also increases the absolute values of the

reaction forces. It is noticeable that the x component of the reaction force is around 2 orders of magnitude larger

than the y components, even though the applied x direction displacement is one order of magnitude smaller.

25

(a) x component (b) y component

Figure 4.5: Reaction Forces for varying sweep angles

4.1.2 X-taper Ratio Variation

This section presents the results obtained from a set of wings with X-taper ratios ranging from 0.4 to 1.0 (the latter

returns to the case of a uniform wing) in steps of 0.1.


Displacement Errors

The evolution of the percentile 90 of node displacement errors for varying X-taper ratio (Fig.4.6) shows that this

type of geometrical feature has a greater influence on the morphing shape than sweep angle. Also the inboard

profile (Profile 4) is much more affected going up to almost 80% for the 0.4 tapered wing while the outboard

profile (Profile 2) is less affected by varying taper ratio by staying in the 0% to 10% interval of percentile 90

without showing a tendency to continuously increase.

Since Fig.4.6 shows high values for the percentile 90, for this set of wingtips it is helpful to look at the error

distributions (Fig.4.7).

Each graph in Fig.4.7 represents the percentage of nodes, of a certain profile of a certain wingtip, that display

26

(a) x error component - Profile 2 (b) y error component - Profile 2

(c) x error component - Profile 4 (d) y error component - Profile 4

Figure 4.7: Error distributions

displacement errors contained in the interval specified by the horizontal axis of the graph.

Fig.4.7 reinforces the statement that P4 is much more affected by varying X-taper ratio than P2. For P2, errors

never exceed the 20% to 30% error interval while for P4, errors become more distributed throughout the intervals

with decreasing value of taper ratio. Also from Fig.4.7, it can be seen that the x direction displacement is slightly

more affected than the y direction displacement since, for the same wingtip, the percentage of nodes included in

the 0% to 10% is larger for y displacement than for x displacement (valid for both profiles).

It is also important to visualize where, in the profile, are the largest errors occurring. Fig.4.8 depicts the errors

and normalized displacements for a wingtip with 0.5 X-taper ratio. This wingtip was chosen because it is repre-

sentative of all the wingtips considered in this section, regarding the node location of errors. The displacements

in Fig.4.8b are relative to Profile 4 of the wingtip and are normalized by the maximum ideal displacement for the

profile (in this case with a value of 10.86mm).

It can be seen from Fig.4.8a, that after node 80 (shortly after beginning of the top section of the profile) the

displacement errors increase almost linearly along the skin for P4, also between nodes 60 and 80 there is a peek

(more accentuated for the x error in P4) that corresponds to the nose area of the profile where higher curvature

is present. In Fig.4.8b the ideal and actual normalized displacements start to misalign significantly after node 60

(right after the stringer).

27

(a) Node relative error (b) Profile 4 normalized displacements

Figure 4.8: 0.5 x taper ratio wingtip

As for the z component of the displacement, P2 and P4 both display a tendency to increase the z direction

displacement as the value for the taper ratio decreases (Fig.4.9). Once again, P4 is more affected by varying taper

ratio, having the displacement increase more steeply reaching 0.16mm for a 0.4 taper ratio compared to 0.13mm

displacement in P2 for the same wingtip.



The maximum stress and strain evolve in a similar manner for varying taper ratios (Fig.4.10). Both parameters

increase at an almost constant rate for wingtips between 1.0 to 0.6 taper ratio and then the slope increases for each

step of taper ratios from 0.6 to 0.4 tapered wingtips.

28

Figure 4.10: Maximum Stress and Strain for varying taper ratio relative to the x axis

Reaction Forces

It can be seen from Fig.4.11a that P2 requires a larger force in the x direction than P4 while from Fig.4.11b the

contrary happens, P4 needs a larger force in the y direction than P2. The force difference between each profile

widens as the value of taper ratio decreases.


Figure 4.11: Reaction Forces for varying taper ratio relative to the x axis

Since the components of the reaction force evolve differently for each profile, it is necessary to also look at the

behavior of the total reaction force. Fig.4.12 shows that P2 requires a larger total force than P4 for the same values

of applied displacements. Overall, the total reaction force has a tendency to increase for decreasing values of taper

ratio.

29

Figure 4.12: Total force for varying taper ratio relative to the x axis

4.1.3 Y-taper Ratio Variation

This section presents the results obtained from a set of wings with Y-taper ratios ranging from 0.4 to 1.0 (the latter

returns to the case of a uniform wing) in steps of 0.1.


Displacement Errors

Out of the 3 parameters evaluated, the y taper ratio seems to affect the morphing shape the most. From Fig.4.13,

it can be seen that the percentile 90 of P2 increases almost linearly with decreasing value of taper ratio, while P4

seems to peek between 0.6 and 0.7 taper ratio (with a percentile 90 between 40% and 50%). The Y-taper ratio has

a significantly larger effect on the displacement errors for P2 than for P4, with the P2 percentile 90 displacement

errors reaching 160% for a 0.4 y-taper ratio. Also from Fig.4.13, for the same profile, the percentile 90 for both x

and y displacement errors evolve similarly with varying taper ratio.

Once again, it is necessary to visualize the error distributions to have a better idea of the influence of varying

Y-taper ratio.

Fig.4.14 shows how this type of taper ratio heavily affects the morphing shape of the wingtip. With only a 0.9

30

(a) x error component - Profile 2 (b) y error component - Profile 2

(c) x error component - Profile 4 (d) y error component - Profile 4

Figure 4.14: Error distributions

taper ratio, the percentage of nodes with errors between 0% and 10% goes down to around 75% for both profiles

and both components of the errors. Furthermore, it is emphasized the larger influence of taper ratio over P2 where,

from 0.6 taper ratio to lower values, errors larger than 100% start to appear.

The component of the error that is more influenced by taper ratio is different for each profile. In P2, the x

component presents smaller errors than the y component while in P4 the opposite occurs.

To visualize where in the profile are the largest errors occurring, the wingtip with 0.5 taper ratio was chosen

since it is representative of the error location for this set of wingtips. The displacements in Fig.4.15a and 4.15b are

normalized by the maximum ideal displacement for the profile (in this case with a value of 10.32mm for P2 and

10.68mm for P4).

Fig.4.15c displays a similar error evolution as seen for the X-taper ratio (Fig.4.8a) with the difference that the

behavior of P2 switches places with P4 and that the actual values of the errors are higher.

It is after node 60 (after the stringer) that the ideal and actual displacement curves begin to misalign significantly

and comparing Fig.4.15a to Fig.4.15b it can be noticed that the actual displacement in one profile goes in the

opposite direction than in the other profile relative to the ideal displacement for both x and y components. In other

words, when the actual displacement in P2 is lower than the ideal displacement, then in P4 the actual displacement

will be larger than the ideal.

31

Also, for P2, after node 100, the actual displacement has a different sign than the ideal which means that those

nodes are being displaced in the opposite direction that was intended (this starts to happen for taper ratios lower or

equal to 0.6).

(a) Profile 2 normalized displacements (b) Profile 4 normalized displacements

(c) Node relative error

Figure 4.15: 0.5 y taper ratio wingtip

As for the z component of the displacement, P2 and P4 both display a tendency to increase the z displacement

component as the value for the taper ratio decreases (Fig.4.16). P2 and P4 display very similar values of z direction

displacement being the largest difference between them smaller than 0.05mm for a 0.4 taper ratio (around 10% of

the value of the displacement).

32



The maximum stress and strain grow less with varying Y-taper ratio than varying X-taper ratio. Fig.4.17 shows

how maximum stress and strain evolve in a similar manner from taper ratios of 1.0 until 0.6 but for lower taper

ratios maximum strain grows at a higher rate than maximum stress. Overall, decreasing the value of taper ratio

increases maximum stress and strain.

Figure 4.17: Maximum Stress and Strain for varying taper ratio relative to the y axis

Reaction Forces

It can be seen from Fig.4.18a that decreasing the Y-taper ratio actually decreases the x component of the reaction

force. In P2 the reaction force almost drops to 0N . On the other hand, Fig.4.18b shows that the absolute value

for the y component increases with decreasing value of taper ratio. It is noticeable that the y component of the

reaction force for P4 has opposite sign of what would be expected. This suggests that the displacement applied

on P2 is enough to displace the spar in P4 more than intended so a force in the opposite direction is necessary to

compensate the over displacement.

33


Figure 4.18: Reaction Forces for varying taper ratio relative to the y axis

Again it is necessary to look at the total force required to find an overall trend of how much force is necessary

to morph the wingtip. From Fig.4.19, it can be seen that the variation of total force required for both profiles is

kept between 600N and 740N which is a small interval compared to the previous 2 sets of wingtips where the

total force would range from 700N to 1500N in the case of varying sweep angle (to 1100N in the case of X-taper

ratio). Furthermore, for P2 the total force actually decreases with taper ratios from 1.0 to 0.6, from there the force

increases with continuing decreasing taper ratio. The force required in P4 displays a slight increase with taper

ratios from 1.0 to 0.8 but from there shows a tendency to decrease with decreasing values of taper ratio.

Figure 4.19: Total force for varying taper ratio relative to the y axis

4.2 Discussion of Results

The results displayed in the previous sections will now be discussed comparing how the 3 different varying param-

eters affect the morphing shape of the wingtip.

34

4.2.1 Displacement Errors

In terms of the x and y components of the displacement errors, it is possible to infer that sweep angle affects the

morphing shape at a much lower level than both types of tapering considered. Of course one cannot say that a

step of 5◦ in sweep angle is a directly comparable geometrical parameter change to a 0.1 step in taper ratio but

with a global view of the scope in which the parameters varied the comparison is possible. The percentile 90 for

varying sweep angle has an almost linear behavior staying below 10% for both profiles and components of the

displacement (Fig.4.2).

Now comparing the effects between X and Y-taper ratio, one can see how one affects more significantly a

different profile than the other. From the percentile 90 data obtained (Fig.4.6 and 4.13) it is clear that X-tapering

has a greater affect on the inboard profile (P4) while Y-tapering has a greater affect on the outboard profile (P2).

It is important to keep in mind that X-tapering decreases the ratio between the horizontal and vertical dimen-

sions of the profile along the span of the wingtip while Y-tapering increases this profile aspect ratio. So the fact

that remains true for both types of tapering is that for the same wing, the profile which presents a bigger aspect

ratio is going to display larger errors (larger values for the percentile 90).

(a) Profile aspect ratio definition (b) Relation between tapering and profile aspect ratio

Figure 4.20

Also, comparing both types of tapering, it can be concluded that in general, Y-tapering will result in larger

errors than X-tapering. This can be due to the fact that a 0.1 X-taper ratio step does not result in an equal profile

aspect ratio step then for the case of a 0.1 Y-taper ratio step. Fig.4.20b shows how aspect ratio behaves when

tapering is introduced, it can be seen that the green curve has a higher slope than the blue curve (using a linear

curve to best fit the data in a least-squares sense, the green curve has a slope of −2.97 and the blue curve has a

slope of 2.05) meaning that a variation of 0.1 Y-taper results in a bigger aspect ratio change than a 0.1 X-taper

step. Still, this difference between slopes only results in around 5% difference between the aspect ratio percentual

increase (or decrease) of a 0.1 X and Y-taper step. Thus it does not seem to account for the larger errors shown in

Y-tapering, reinforcing the conclusion stated at the beginning of this paragraph.

To remove the profile aspect ratio variable from the effects of tapering, Fig.4.21 shows the evolution of the

percentile 90 of the displacement errors for the case when both types of taper ratios are varied at the same time in

equal steps.

35

With the profile aspect ratio variable controlled, it can be seen that the inboard profile displays higher errors

than the outboard profile. A possible explanation is the fact that for both profiles, the applied displacements have

equal values even though the area bounded by the inboard profile is larger than the area bounded by the outboard

profile. In other words, for the same displacement, the larger profile of a wingtip will display larger errors when

tapering is present.

Figure 4.21: Percentile 90 for varying both types of taper ratios

To investigate the correlation between profile size and the magnitude of applied displacements, three tapered

wings were analyzed by applying LC0. So if the wingtip has a X-taper ratio of 0.5 then P2 and P4 are scaled by

0.625 and 0.875 in the X direction, respectively, relative to the root profile and thus the applied displacements in

that direction are scaled by the same values. The first wingtip has a X-taper ratio of 0.5, the second wingtip has a

Y-taper ratio of 0.5 and the third wingtip has a X and Y-taper ratio of 0.5.

Table 4.1 displays the percentile 90 of the displacement errors for the above mentioned wingtips for the case

of constant applied displacements (same results as shown in Fig.4.6, 4.13 and 4.21) while Table 4.2 displays

the percentile 90 for the same wingtips but for the case where the applied displacements are scaled according to

tapering.

Percentile 90 [%]X/Y taper 0.5/1.0 1.0/0.5 0.5/0.5Component x y x y x yP2 4.42 3.24 128.46 133.06 45.81 42.51P4 58.1 57.46 34.68 32.26 82.47 73.28

Table 4.1: Displacement error Percentile 90 for applied displacements of 1mm and −10mm in the x and ydirections respectively

Percentile 90 [%]X/Y taper 0.5/1.0 1.0/0.5 0.5/0.5Component x y x y x yP2 4.51 3.14 53.23 52.74 2.58 2.40P4 58.42 57.79 14.78 14.09 7.48 6.62

Table 4.2: Displacement error Percentile 90 for applied displacements scaled according to taper ratio values

36

Comparing both tables, it is noticeable how scaling the applied displacement in the X axis has so little affect

on the resulting displacement errors having a maximum of 3% change between the percentile 90 values while in

the case of scaling the applied displacements along the Y axis, the percentile 90 values drop to less than half.

This indicates that the applied displacements in the Y axis are the main source of the morphing shape errors

(which would be expected since the Y component of the displacement is one order of magnitude larger than the X

component) and its value will be a decisive criteria when deciding if the 2D optimization will remain valid for the

3D wingtip.

Furthermore, Tables 4.1 and 4.2 show that scaling both components of the displacements reduces the displace-

ment error percentile 90 to less than 1/10 of its original value. This reduction can be explained by the decrease in

the value of the Y component of the applied displacement, but as seen before, it would only account for reducing

the percentile 90 to around half of its original value. Another factor that can explain the rest of this reduction is the

resulting droop angle of the wingtip. Having a large difference between the target droop in P2 and P4 may cause

higher displacement errors.

Droop Angle [◦]X/Y taper 0.5/1.0 1.0/0.5 0.5/0.5Displacement not scaled scaled not scaled scaled not scaled scaledP2 1.622 1.621 1.025 0.6367 1.631 1.010P4 1.153 1.152 1.011 0.882 1.157 1.008

Table 4.3: Droop angles of wingtips resulting from scaled and not scaled applied displacements

Cross checking Table 4.3 with Tables 4.1 and 4.2, it seems that the difference between P2 and P4’s target

droop angle is also correlated to displacement errors. In the case of a X-tapered wing, the droop angle difference

between profiles doesn’t significantly change when the X component of the applied displacement is scaled (also

the Y component of the applied displacement remains constant) thus the percentile 90 of the displacement errors

also does not change significantly. In the case of Y-tapering, the droop angle difference increases, when scaling the

applied displacements, but the Y component of those displacements decrease with the scaling and so the percentile

90 of the errors decreases suggesting that the magnitude of the Y displacement has a higher influence on the

morphing shape than the droop angle difference. Finally in the case of X and Y-tapering both the Y component of

the applied displacements and the droop angle difference decrease when the applied displacements are scaled thus

resulting in a large displacement error percentile 90 drop.

To view the effect of having all three 3D geometrical parameters present in a wingtip, the sweep angle was

varied for two different wingtips, one with 0.8 X and Y-taper ratio and another with 0.5 X and Y-taper ratio.

Fig.4.22a and 4.22b show that for tapered wings, the introduction of sweep angle can actually help decrease the

percentile 90 of the displacement errors, specially for the outboard profile.

37

(a) 0.8 taper ratio relative to both x and y axis (b) 0.5 taper ratio relative to both x and y axis

Figure 4.22: Percentile 90 for varying sweep angles

Regarding the location of the errors in a profile, the data consistently shows that, for both types of tapering,

errors start to grow significantly along the top section of the profile and in the bottom section (before the stringer)

errors are usually low, under 10%. Meaning that the top section’s morphing shape is the area in the profile most

affected by tapering.

As for the z component of the displacement, there is a tendency for its value to increase with the increasing

geometrical change resulting from varying parameters. Between the two types of tapering, Y-tapering results in

displacements usually two times larger than X-tapering. It is noticeable that, in this case it is the sweep angle that

increases the z displacement the most, displaying values of one order of magnitude higher than for the two cases of

tapering. Still, the highest z displacement component has a value of 1.7mm which is very small when compared to

the span of the wingtip (1500mm). In the end, the importance of how much displacement occurs in the z direction

will depend on how much strain and stress will result in the morphing compliant mechanism.

4.2.2 Maximum Stress and Strain and Reaction Forces

Following is a brief relative comparison between the varying of the different parameters. The maximum stress

and strain and the reaction forces are not of importance when discussing the effects of tapering and sweep angle

on morphing shape. However, they are of importance as criteria to choose adequate skin material and compliant

mechanisms.

The results obtained for maximum stress and strain give a similar discussion to what was done for the z com-

ponent of the displacements. An increase in geometrical change resulting from varying parameters will increase

the values of maximum stress and strain. The X-tapering shows a larger increase in stress and strain than for the

other two parameters which both behave in a similar manner.

The stress and strain distributions for a given wingtip are similar, hence Fig.4.23 displays only the stress

distribution as representative of both distributions. For all wingtips, stress and strain peaks appear where the

morphing displacements are applied which is expected. However, it is noticeable how X-tapering also leads to

stress and strain peaks on the most outboard profile in the upper region where it is fixed while for Y-tapering the

peaks appear in the same region but for the most inboard profile. Furthermore, Y-tapering leads to a high stress and

38

(a) 0.5 X-taper, 1.0 Y-taper and 0◦ sweep angle (b) 1.0 X-taper, 0.5 Y-taper and 0◦ sweep angle

(c) 1.0 X-taper, 1.0 Y-taper and 35◦ sweep angle (d) 0.5 X-taper, 0.5 Y-taper and 35◦ sweep angle

Figure 4.23: von Mises stress distribution for different wingtips

strain region in the nose of the outboard section of the wingtip, this can be explained by the small curvature radius

present due to Y-tapering. The introduction of a sweep angle leads to a peak region in the bottom fixed section of

the wing tip.

Combining all 3 parameters seems to alleviate the above mentioned stress and strain peak regions (excluding

the peaks located where the morphing displacements are applied) although the maximum values are higher.

Regarding the reaction forces, it was found that the parameter that has the least affect is Y-tapering where for

some values of taper ratio the forces are even lower than for the case of a uniform wing. In this type of tapering,

the total reaction forces vary in a relatively small interval between 600N and 740N . For the other two parameters,

as for maximum stress and strain, increasing the geometrical change in the wingtip will increase the values of the

total reaction forces.

It could be expected that if the surface area decreases than the reaction forces would also decrease since there

is less material to displace. Fig.4.24 illustrates how a 0.4 taper ratio decreases the surface area by 2.2%, in the case

of Y-tapering, and by 27.0%, in the case of X-tapering. Also, a 40◦ sweep angle increases the surface area only

by 3.2%. So, there seems to be no direct correlation between surface area and reaction force, since in the case of

X-tapering, the area decreases and the reaction forces increase while for the case of sweep angle the area increases

an the reaction forces also increase. Therefore it seems that the change in reaction forces has to do solely with the

geometric configuration of the wingtip.

For the case of X-tapering, it is clear that the distance between the applied displacement and the fixed boundary

edge of the wingtip decreases, resulting in a lower bending moment for the same force. Thus to achieve the same

39

(a) (b)

Figure 4.24: Wingtip surface area for varying 3D geometrical parameters

displacement it would be expected to be necessary to increase the force. For the same reason, it would also be

expected that P2 would require a higher force than P4, and as can be seen from Fig.4.12 this is the case.

For the case of Y-tapering, the distance between the applied displacement and the fixed boundary edge of

the wingtip remains constant, which can explain the fact that the reaction force values change in a much smaller

interval than for the case of X-tapering. Also, as said in the previous section, it seems that the force necessary

to displace P2, when Y-tapering is present, is enough to displace P4 to the point where P4 actually needs a y

component of the force in the opposite direction to compensate. Another noticeable occurrence in this type of

tapering is the fact that the x component of the reaction forces almost goes to zero for P2 (Fig.4.18a). Since the

profile becomes thinner when Y-tapering is introduced, the bending of the profile (due to the y displacement) will

result in a larger x direction displacement leading to lower values of the resulting reaction force.

For the case of increasing sweep angle, the x component of the force is the main contributor to the total reaction

force increase. The fact that introducing sweep angle results in the x direction no longer being perpendicular to

the fixed edges of the wingtip, can explain this increase in the value of reaction force.

40

Chapter 5

Analysis of the Effects of 3D Geometrical

Parameters on the Compliant Mechanism

- CMAN

In the previous chapter, the analysis made excludes the actual presence of the morphing mechanisms in the wingtip.

Tapering and sweep angle will not only affect the resulting morphing shape but will also affect how the compliant

mechanism transfers the actuation force to the stringer where it is attached.

In this chapter, wingtips containing compliant mechanisms will be subjected to two different morphing load

cases (LC1 and LC2) in order to analyze the influence of the mechanism on the morphing shape (comparing to

results obtained in the previous chapter) and in order to analyze the effects of the 3D geometrical parameters on

the mechanism.

For the purpose of this analysis, it is not necessary to consider the same range of different wingtips as in the

previous chapter since the effects of sweep and tapering on the morphing shape have already been discussed and

now the inclusion of the mechanism serves to validate the simplification made in SMAN and to investigate the

change in displacement behavior of the compliant mechanism in a wingtip similar to the one given by Embraer.

Thus the following five wingtips will be analyzed in this chapter:

• 0◦ sweep angle, 0.5 X-taper ratio and 1.0 Y-taper ratio (1500 0 50 100);





The code in brackets after every wingtip listed above is used in future plots and tables in order to identify which

wingtip is the data referring to. The first number is the span length of the wingtip in mm, the second number is the

sweep angle, the third and fourth numbers are the X and Y taper ratio respectively in percentage units.

41

It is important to state that for each of the above mentioned wingtips, it is necessary to have two additional

models of uniform wingtips with no sweep angle and with profiles identical to P2 and P4 to serve as the ideal

morphing cases (same logic as used in the previous chapter). In total, 12 models of different wingtips including

compliant mechanisms are created.

The values chosen for sweep angle and taper ratio are similar to the 3D geometrical parameters present in the

Embraer wingtip. So the last wingtip listed above (with all 3 parameters) is the closest to the actual wingtip of the

NOVEMOR project.

Each wingtip was subjected to both load cases (LC1 and LC2) and data regarding node displacement of the

wingtip skin and of the mechanism is analyzed.

Two types of results are shown: the first type regards the shape morphing of the profile, similarly to what is

done in the previous chapter; the second regards the displacement of the mechanism. The first type contains results

that are comparable to what was obtained in the previous chapter and the use of LC0 is made when referring to

the load case with scaled applied displacements to a wingtip without the presence of the morphing mechanisms.

In the second type of results, when referring to displacements it is meant to refer to the off-plane displacements

(along the z axis) since the analysis of the skin morphing will already reflect the in-plane errors of the mechanism

and it is the off-plane displacements that are of interest when quantifying the effects of the 3D parameters on the

2D optimized mechanism.

Since each mechanism was meshed individually as described in Chapter 3, there is no logical node numbering

associated with its position in the mechanism (as found for the wingtip skin and stringer). Furthermore, due to

the slim nature of sections of the mechanism a fine mesh was necessary to guarantee reliable solutions leading to

between 4500 and 7000 nodes on each mechanism. With this in mind, it is necessary to display the mechanism

displacement data in a suitable manner. This is done by providing the average value of the off-plane displacements

and by displaying displacement contour plots of the mechanisms.

For the first analyzed wingtip, a more detailed explanation of the meaning of the plots and tables displayed is

given. Since for all wingtips the plots and tables are the same (only the values change), for the other four analyzed

wingtips the explanation will be omitted and only the data provided by the plots and tables is analyzed.

For brevity, M2 and M4 refer to the morphing mechanisms at P2 and P4 respectively.

5.1 Results

5.1.1 0◦ sweep angle, 0.5 X-taper ratio and 1.0 Y-taper ratio

In this wingtip, for LC0 and LC1 the applied displacements along the x axis are 0.625mm and 0.875mm for P2

and P4 respectively and −10mm for P2 and P4 along the y axis. For LC2 the applied force is 79N at M2 and

154N at M4.

42

(a) (b)

Figure 5.1: Displacement error distribution (1500 0 50 100)

Fig.5.1 gives a comparison between the displacement error distributions of the profile morphing shape of LC0

and LC1. As can be seen, the presence of the morphing mechanisms has very little effect on the error distribution.

The most noticeable difference is for the x component on P2 where approximately 10% of the nodes shifted their

displacement error from the first to the second relative error interval.

Component Profile Percentile 90 [%]LC0 LC1

x P2 4.51 9.64P4 58.42 59.00

y P2 3.15 5.40P4 57.79 58.30

Table 5.1: Percentile 90 values of the profile node displacement errors (1500 0 50 100)

Table 5.1 confirms what was stated before, the percentile 90 of the x displacement errors on P2 is the most no-

ticeable difference, more than doubling with the presence of the morphing mechanism. Globally, the introduction

of the mechanisms result in an increase of the percentile 90 of the profile displacement errors for both profiles and

for both components. Finally it is important to note that the percentile 90 values of P2 suffered a larger increase

than those of P4 (both in relative and absolute terms).

Now relative to the displacement results of the mechanism, Table 5.2 displays the average off-plane displace-

ment values. For both LC1 and LC2, the presence of X-tapering increases the average off-plane displacement of

M2 and M4. Furthermore all averages are negative meaning that the presence of this 3D parameter displaces both

mechanisms on average in the outboard direction.

MechanismAverage z displ. [mm]

LC1 LC2Ideal Actual Ideal Actual

M2 −0.0105 −0.0343 −0.0007 −0.0018M4 −0.0166 −0.6790 −0.0052 −0.0079

Table 5.2: Average mechanism node displacement (1500 0 50 100)

Fig.5.2 illustrates the distribution and magnitude of the off-plane displacements in each mechanism. Only LC2

43

is shown since it represents the most similar load case to reality in terms of the off-plane behavior of the morphing

mechanism. In LC1 it is assumed that the required displacement at the stringer can always be reached regardless

of the mechanism so the output of the mechanism at that point is always the same while in LC2, it is possible to

check the difference in behavior of the mechanism due to the presence of 3D geometric parameters given the exact

same input conditions which is the intended goal. The scale used in Fig.5.2 (and all the following similar plots

for the different wingtips) are different for each mechanism but the same for the ideal and actual case of the same

mechanism. The highest/lowest value of the scale corresponds to the highest/lowest displacement present between

the ideal and actual cases.

(a) LC2 - M2 - ideal (b) LC2 - M2 - actual

(c) LC2 - M4 - ideal (d) LC2 - M4 - actual

Figure 5.2: Distribution of the z component of the mechanism node displacement for LC2 (1500 0 50 100)

Although the average displacements are negative (outboard), Fig.5.2 shows the different behavior between

different sections of the same mechanism. For M2 and M4, when X-tapering is introduced, the bottom part of SB

increases its off-plane displacement towards the outboard while the SR reverses the direction of its displacement

towards the inboard of the wingtip.

The mechanisms are designed to give a certain output, namely a specified displacement at control points where

it is attached to the stringer. Table 5.3 displays the resulting x and y displacements at those points (same location as

in LC1 where the displacements were applied) in order to check how the 3D parameter affects the output given the

same force input. It is noticeable how the magnitude of the resulting displacements at the control points increase in

44

M2 but decrease in M4 leading to the conclusion that design corrections to the mechanism will depend on where

it is located along the span.

MechanismResulting Displacement [mm]

x yIdeal Actual Ideal Actual

M2 0.037 0.046 −0.695 −0.760M4 0.165 0.110 −2.078 −0.749

Table 5.3: x and y components of the displacement of the nodes located at the control points for LC2(1500 0 50 100)


In this wingtip, for LC0 and LC1 the applied displacements along the y axis are −6.25mm and −8.75mm for P2

and P4 respectively and 1mm for P2 and P4 along the x axis. For LC2 the applied force is 79N at M2 and 154N

at M4.

(a) (b)


From Fig.5.3 one can see how the presence of the mechanism actually improves the morphing behavior by

lowering both components of the displacement errors of the profiles. This is specially accentuated in P4 where the

percentage of nodes with errors below 10% goes from around 75% to 100%.


x P2 53.23 51.01P4 14.78 8.78

y P2 52.74 51.13P4 14.09 7.24


Table 5.4 confirms the improvement of the skin displacement errors. All percentile 90 values decrease with the

introduction of the mechanisms, with P4 having the most noticeable change: the percentile 90 goes down by 41%

for the x component of the displacement errors while it goes down 49% for the y component.

45

Turning to results relative to the mechanisms, Table 5.5 shows how the average off-plane displacements in-

crease for both load cases and both mechanisms by at least one order of magnitude when Y-tapering is introduced.

Their values are negative indicating an overall displacement of the mechanism towards the outboard of the wingtip.



M2 −0.0017 −0.0952 −0.0020 −0.0176M4 −0.0071 −0.1371 −0.0012 −0.0287


Fig.5.4 shows the difference in displacement distribution in the mechanisms when Y-tapering is introduce.

Contrary to what was seen in an X-tapered wingtip, the SR section of the mechanism increases its displacement

towards the outboard (not the inboard) along with the bottom part of the SB section. Furthermore the maximum

displacements occur in the bottom part of SB.




The resulting x and y displacements located at the control points are displayed in Table 5.6. For both mecha-

nism not only do the displacements decrease with the introduction of Y-tapering but also they decrease at almost

identical rates: the y component decreases by 40% and 42% in M2 and M4 respectively and the x component

46

decreases by 29% and 28% in M2 and M4 respectively. It seems that for Y-tapering, the effects on the mechanism

output does not depend on its location along the span.



M2 0.128 0.091 −1.392 −0.835M4 0.213 0.153 −2.389 −1.386




and P4 respectively and along the x axis are 0.625mm and 0.875mm for P2 and P4 respectively. For LC2 the

applied force is 79N at M2 and 154N at M4.

(a) (b)



x P2 2.58 1.59P4 7.48 2.34

y P2 2.40 1.50P4 6.62 2.21


From Fig.5.5 one can see that the presence of the mechanism has almost no effect on the distribution of

the profile node displacement errors. The only difference is in the x component of the displacement errors in

P4 where less than 1% of the nodes contain errors in the 10% to 20% interval when the mechanism is introduced.

Regardless, the percentile 90 values displayed in Table 5.7 clearly show that, for both profiles and both components,

the presence of the mechanisms decreases the profiles node displacement error.

47

Regarding the tapering effects on the mechanism, Table 5.8 displays the average mechanism off-plane dis-

placements for both load cases. Again there is a consistent increase in the average off-plane displacement towards

the outboard. Furthermore, M4 is clearly the most affected mechanism when both types of tapering are introduced

presenting the largest increase in average off-plane displacement.



M2 −0.0372 −0.0760 −0.0049 −0.0102M4 −0.0686 −0.1249 −0.0162 −0.0207


Relative to the displacement distribution, it can be seen from Fig.5.6 that the bottom part of SB suffers a big

change in the magnitude of its off-plane displacement. Also in SR a change is present but with a distinction

between its upper and bottom halves (higher off-plane displacements on the bottom half than on the top).




Table 5.9 display the in-plane resulting displacements located at the control point for the X and Y-tapered

wingtip. In this case there is an increase in displacements in M2 and decrease in M4. Leading to the same

conclusion as for the X-tapered wingtip where the effects on the mechanism output depend on the its location

48

along the span.



M2 0.047 0.059 −0.474 −0.683M4 0.136 0.122 −1.289 −1.051



In this wingtip, for LC0 and LC1 the applied displacements along the x axis are 1mm and along the y axis are

−10mm for P2 and P4. For LC2 the applied force is 201N at M2 and M4.

(a) (b)



x P2 7.11 11.52P4 6.84 9.81

y P2 7.37 11.60P4 7.89 10.24


For this wingtip, the presence of the mechanism increases the percentile 90 of the node displacement error for

both profiles and both components (Table 5.10). There is an increase of 60% in P2 for both components and in P4

the percentile 90 of the x component increases by 40% while for the y component there is a 30% increase. The

distribution of errors shown in Fig.5.7 shows how there is an increase in the number of nodes with errors in the

10% to 20% interval when the mechanism is introduced.

Looking at the effects of sweep angle on the mechanisms, it is noticeable how the average off-plane displace-

ment actually changes direction (Table 5.11) going from outboard to inboard contrary to the wingtips analyzed so

far.

49



M2 −0.0858 0.1452 −0.0239 0.0143M4 −0.0866 0.1361 −0.0241 0.0148


From the displacement distribution (Fig.5.8), it can be seen that the inboard displacement occurs in SR and the

front part (towards the leading edge) of SB. The highest inboard displacements occur in the top half of the section

where the mechanism connects to the stringer (SR). It is also noticeable how both mechanisms have practically

equivalent displacement distributions which is expected since both mechanisms are identical and are located in

identical profiles.




Relative to the in-plane displacements located at the control points, Table 5.12 shows how the introduction

of sweep angle leads to a decrease in the magnitude of the target displacement for the same applied force. Also,

consistent with the previous results, both mechanisms have the same rate of decrease of those displacements.

50



M2 0.202 0.128 −1.714 −1.301M4 0.202 0.131 −1.716 −1.298




and P4 respectively and along the x axis are 0.625mm and 0.875mm for P2 and P4 respectively. For LC2 the

applied force is 79N at M2 and 154N at M4.

(a) (b)



x P2 4.21 4.54P4 14.72 14.24

y P2 4.14 4.84P4 13.81 13.38


In the wingtip containing all three 3D parameters, the inclusion of the compliant mechanism has little effect on

the profile node displacement error distribution (Fig.5.9). Table 5.13 shows that for P4 the percentile 90 decreases

by 3% for both components while for P2 it increases by 8% and 17% for the x and y components respectively.

Overall the inclusion of the mechanisms has less effect on this wingtip than on all the others considered previously.

Relative to the results of the mechanisms, once again the average off-plane displacements invert their direction

(Table 5.14) now for the case where all 3D parameters are present suggesting that the direction of these displace-

ments is mainly governed by the sweep angle since, from the previous cases, only the wingtip with this parameter

showed similar behavior.

51



M2 −0.0372 0.0746 −0.0049 0.0011M4 −0.0686 0.0888 −0.0162 0.0062


From Fig.5.10 it can be seen that the displacement distribution is similar to what was seen in the wingtip with

only sweep angle where the most affected regions of the mechanism are the top half of SR and the front part of

SB. The difference is that now the magnitude of the displacements is different between M2 and M4 (due to the

tapering).




Table 5.15 displays the resulting in-plane displacements located at the control points and a general decrease

of their values occurs with the presence of all three 3D parameters. In this case the percentage of decrease in the

in-plane displacements seem to depend on the position of the mechanism along the span which can be related to the

presence of X-tapering since it is the only parameter that resulted in this type of behavior (in the previous wingtips

with a single 3D parameter).

52



M2 0.047 0.036 −0.474 −0.432M4 0.136 0.076 −1.289 −0.694


5.2 Discussion of Results

In the previous sections the results obtained were presented and analyzed individually for each different wingtip. In

this section a comparison between those individual results is made in order to derive conclusions on the effects of

3D geometrical parameters on the compliant mechanisms and consequently on the shape morphing of the wingtip.

5.2.1 Profile Node Results

Table 5.16 shows the variation of the percentile 90 of the profile node displacement errors when the compliant

mechanism is introduced in the model. Considering the wingtips with the presence of only one of the three 3D

geometrical parameters (1500 0 50 100, 1500 0 100 50 and 1500 35 100 100), the first noticeable difference is

that for the Y-tapered wingtip the percentile 90 values decrease when the mechanisms are introduced while for

the X-tapered and swept wingtips the percentile 90 values increase, meaning that in the analysis made in the

previous chapter, the simplification made of not including the mechanisms overestimates the errors for Y-tapering

but underestimates the errors for X-tapering and sweep angle.

With this, the important question rises of if the simplification made in the previous chapter invalidates the

results obtained. Looking at Table 5.16, for the first three wingtips, two groups of variations can be made: small

variations (considered as less than 5%) and large variations (considered as larger than 30%). Considering the

presence of the large variations, it would be sensible to assume that the absence of the mechanisms is not a feasible

simplification but with a more detailed view (considering also the absolute values of the percentiles in Tables 5.1,

5.4 and 5.10) one can see how the large variations occur when the value of the percentile 90 is small to begin

with (lower than 15%) and as the percentile 90 values are larger, the variation of going from LC0 to LC1 becomes

smaller.

In conclusion, the simplification made in the previous chapter is valid with the limitation that for profiles with

small displacement errors it is much less precise than for profiles with large displacement errors. In other words

the results obtained without the mechanisms give a valid description of the order of magnitude of the errors which

is in line with the purpose of investigating the effects of 3D geometrical parameters on the morphing shape of the

wingtip.

Considering the X and Y-tapered wingtip (1500 0 50 50) and comparing Table 5.7 with Table 5.16, it can be

seen that for low percentile 90 values, their variation when introducing the mechanisms is relatively high (larger

than 37%) confirming what was stated above. Also, it is noticeable how there was a decrease in the percentile

90, meaning that the presence of both types of tapering leads to an overestimation of the errors obtained when the

mechanism is not present.

53

P2 P4Wingtip x y x y

1500 0 50 100 113.75 71.43 0.99 0.881500 0 100 50 -4.17 -3.05 -40.60 -48.621500 35 100 100 62.03 57.40 43.42 29.791500 0 50 50 -38.37 -37.50 -68.72 -66.621500 35 50 50 7.84 16.91 -3.26 -3.11

Table 5.16: Variation [%] in the Percentile 90 values of the profile node displacement error when the compliantmechanisms are introduced (from LC0 to LC1)

Relative to the wingtip containing all three geometrical parameters it is interesting to note how the percentile

90 values are low (Table 5.13) and yet their variation when going from LC0 to LC1 is also relatively low (Table

5.16), leading to the conclusion that the presence of the mechanisms has little effect on the morphing shape when

X-taper, Y-taper and sweep angle are present, at least for this specific case (0.5 X and Y taper, 35◦ sweep angle).

Also noticeable is how in P2 the percentile 90 values increase while in P4 they decrease. In all other wingtips the

percentile 90 variation has the same direction for both profiles.

5.2.2 Compliant Mechanism Results

From the results obtained it is clear that the introduction of 3D geometrical parameters will alter the compliance

behavior of the mechanism by increasing the magnitude of the off-plane displacements. It is noticeable how X-

tapering and sweep angle deviate the SR of the mechanism towards the inboard while Y-tapering deviates it towards

the outboard.

When X and Y-tapering are combined, the SR deviates towards the outboard having a similar off-plane be-

havior to when only Y-tapering is present. When sweep angle is also added, the mechanism displays an off-plane

displacement distribution similar to when only sweep is present.

Furthermore, with tapering, the bottom part of SB deviates more than SR creating a ”belly” region in the

mechanism while with sweep angle the off-plane displacements continuously increase going from the bottom part

of SB to SR. Again when all three geometrical parameters are present the distribution becomes similar to when

only sweep is present.

This leads to the conclusion that sweep angle is the leading parameter to determine the mechanism’s general

off-plane shape. However the presence of tapering decreases the magnitude of the displacements compared to

when only sweep is present.

M2 M4Wingtip x y x y

1500 0 50 100 25.5 9.3 -33.7 -64.01500 0 100 50 -28.8 -40.0 -28.1 -42.01500 35 100 100 -36.4 -24.1 -35.1 -24.41500 0 50 50 24.2 44.1 -10.0 -18.41500 35 50 50 -23.6 -8.9 -43.8 -46.2

Table 5.17: Variation [%] between the ideal and the actual target displacements

Relative to the effects on the mechanism output, Table 5.17 displays the change in the in-plane displacements

of the nodes located where the mechanism is designed to displace with a specified value for the different considered

54

wingtips.

As stated before, sweep angle and Y-tapering result in almost identical variations of the target displacements

between M2 and M4 suggesting a non dependence on the location of the mechanism along the span (which does

not happen with X-tapering).

When all parameters are put together in the same wingtip, the variations are different between M2 and M4

suggesting that the dependence on mechanism location along the span, shown with X-tapering, is the dominant

factor. Thus design corrections made will be different for M2 and M4. Finally the combination of all three

parameters result in a decrease of the target displacements for both mechanisms.

55

Chapter 6

Conclusions and Future Work

6.1 Achievements

The main purpose of this thesis was to investigate how 3D geometric parameters (tapering and sweep) would

affect the morphing design of the AMWT being done, as part of the EU FP7 project NOVEMOR, since the tools

which consider highly 3D geometries are currently in development at the DLR. To accomplish this a methodology

was developed in order to be able to compare wingtips with different geometrical parameters and define an ideal

morphing behavior. Furthermore, the methodology was applied in order to obtain data relevant to the AMWT

design process being developed.

The first analysis made (SMAN) allowed to determine tendencies in the shape morphing behavior of a wingtip

(similar to the one being developed) when 3D geometrical parameters are introduced. This data is useful in the

wingtip skin design stage where it is necessary to obtain a skin thickness distribution (and skin material properties

distribution) that permits morphing to obtain the target shape.

The second analysis (CMAN) determined the effects on the mechanism’s compliance behavior (without having

to go through all of the mechanism’s design phases) which is of importance in order to correct or adapt the topology

optimization of the mechanism stage. Furthermore this analysis served to validate the simplification made in

SMAN where the compliant mechanisms are replaced by applying only their desired displacement output to the

stringer where they would be connected.

The analyses made (SMAN and CMAN) aided the development of the tools to design the AMWT providing

a better knowledge on how to approach the problem considering the 3D geometric nature of the wingtip. Further-

more, it suggests there are some additional constraints which the aerodynamic group need to include in designing

the target morphing shapes namely the overall skin surface area needs to remain unchanged from clean to droop

even if morphing is non-uniform along the span (i.e. twist is present) and the target droop shapes perhaps should

be defined as profiles perpendicular to hinge axis instead of parallel to the root profile in order to decrease the

off-plane displacements.

The following section summarizes the conclusions obtained from both analyses.

57

6.2 Analysis Conclusions

SMAN

In terms of displacement errors it was found that, out of the 3 considered parameters, Y-tapering affects the target

morphing shape the most leading to larger displacement errors.

Also, it was found that for the same wingtip, the profile with the largest aspect ratio (between P2 and P4) will

contain larger displacement errors. When the profile aspect ratio is maintained constant for a wingtip, the profile

with the largest errors will be the inboard profile (P4). These two trends are valid for both cases of scaled and not

scaled applied displacements.

In terms of the input given to achieve morphing, it was found that the magnitude of the Y component of

the applied displacements is strongly correlated to the resulting displacement errors where increasing the applied

Y displacements increases the errors. The difference between droop angle of P2 and P4 was also found to be

correlated with the resulting displacement errors in the same way although to a lesser extent meaning that if both

factors grow in different directions, the overall displacement error change will be according to the change in

magnitude of the Y component of the displacement.

For a given profile, it was found that the errors start to increase significantly after the stringer meaning that its

position will influence the overall morphing shape error and that the top section of the profile is the most influenced

by tapering.

With tapering present, it was found that the introduction of a sweep angle can actually decrease the resulting

displacement errors, which is a positive sign since in general wingtips will contain combinations of all 3 considered

parameters.

Finally, the introduction of any of the 3D parameters will increase the undesired z component of the displace-

ment.

In terms of stress and strain, it was found that introducing a 3D geometrical parameter will invariably in-

crease the maximum strain and stress. However combining both types of tapering results in a more uniform stress

distribution, where the only peak regions are located where the displacements are applied.

In terms of reaction forces, it was found they will solely depend on the geometrical configuration imposed by

the changing 3D parameter. For the case of X-tapering and sweep angle, the change in geometrical configuration

will result in an increase in the reaction force, while for Y-tapering, the geometrical configuration change can

actually result in lower values for the reaction forces.

CMAN

The simplification, made in SMAN, of substituting the compliant mechanism by applying displacements where

it would be attached, was found to be a valid solution in order to enable the analysis of the morphing shape of a

larger scope of different wingtips with comparable load cases. With the reserve that when the profile displacement

errors are small, the results obtained are less precise.

Relative to the effects on the mechanism, sweep angle was found to be the most determinant 3D geometrical pa-

rameter on the resulting off-plane displacement behavior. Although tapering will influence the magnitude of those

displacements and curiously, with all three parameters present in a wingtip, the magnitude of the displacements

58

are smaller than in the wingtip with only sweep.

Finally, relative to the output variation of the mechanisms, it was found that the introduction of 3D param-

eters will generally decrease the target displacements with the exception of M2 in wingtips 1500 0 50 50 and

1500 0 50 100 (where there is an increase in the target displacement), but since the AMWT contains all three

parameters it would be expected that a traditionally designed compliant mechanism will require a larger actuation

force in order to obtain the same target displacement. Also, depending on the position of the mechanism along the

span (M2 or M4) the change in the desired output will be different.

6.3 Recommendations for Future Work

In order to simplify and speed up the methodology developed, the different tools built should be integrated in order

to create a single automatized analysis tool. Furthermore, more parameters can be added to the wingtip CAD

model to enable different geometries of the base profile so it can be applied to a bigger variety of wingtips.

The position of the mechanisms along the span and their orientation (in the analysis made in this thesis, the

mechanism were positioned always in the xy plane at 1/4 and 3/4 of the span) are variables that can be taken into

account and changed in order to optimize the compliant mechanism’s morphing capacity.

The stringer’s position can also be an added variable since as seen in SMAN, profile displacement errors start

to increase significantly aft the stringer.

Relative to CMAN, an analysis on the changes in stress distribution should also be made (this was not done

due to time constraints).

Finally, as future work (which is being done at the DLR), it is necessary to adapt the design process of the

AMWT taking into account what was learned from the analysis made in this thesis.

59

Bibliography

[1] Albanesi, A. E., Fachinotti, V. D., and Cardona, A. Design of compliant mechanisms that exactly fit a desired

shape. Asociacion Argentina de Mecanica Computacional, 28:3192–3205, 2009.

[2] Albanesi, A. E., Fachinotti, V. D., and Pucheta, M. A. A review on design methods for compliant mechanisms.

Asociacion Argentina de Mecanica Computacional, 29:59–72, 2010.

[3] Andersen, G. R., Cowan, D. L., and Piatak, D. J. Aeroelastic modeling, analysis and testing of a morphing

wing structure. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confer-

ence, 2007.

[4] Anderson, J. D. Modern Compressible Flows. McGraw-Hill International Editions, 1999.

[5] Bagad, V. S. Mechatronics. Technical Publications Pune, 4th edition revised edition, 2008.

[6] Barbarino, S., Bilgen, O., Ajaj, R. M., Friswel, M. I., and Inmanl, D. J. A review of morphing aircraft.

Journal of Intelligent Material Systems and Structures, 22:823–877, 2011.

[7] Bendsoe, M. P. Optimal shape design as a material distribution problem. Structural Optimization, 1:193–202,

1989.

[8] Bendsoe, M. P. and Sigmund, O. Material interpolation schemes in topology optimization. Archive of Applied

Mechanics, 69:635–654, 1999.

[9] Bendsoe, M. P. and Sigmund, O. Topology optimization: Theory, methods and applications. Berlin Heidel-

berg: Springer-Verlag, 2003.

[10] Bharti, S., Frecker, M. I., Lesieutr, G., and Browne, J. Tendon actuated cellular mechanisms for morphing

aircraft wing. Proceedings of SPIE Modeling, Signal Processing, and Control for Smart Structures, 6523,

2007. San Diego CA.

[11] BYU. Brigham young university - compliant mechanisms research: Intro to compliant mechanisms.

URL http://compliantmechanisms.byu.edu/content/intro-compliant-mechanisms. (Accessed

2014).

[12] Flanagan, J. S., Strutzenberg, R. C., Myers, R. B., and Rodrian, J. E. Development and flight testing of a

morphing aircraft, the nextgen mfx-1. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,

and Materials Conference, 2007.

61

http://compliantmechanisms.byu.edu/content/intro-compliant-mechanisms

[13] Grady, B. Multi-objective optimization of a three cell morphing wing substructure. Master’s thesis, University

of Dayton OH, 2010.

[14] Haber, R. B. and Bendsoe, M. P. A new approach to variable-topology shape design using a constraint on the

perimeter. Structural Optimization, 11:1–12, 1989.

[15] Howel, L. L. Compliant Mechanisms. John Wiley & Sons Inc., 2001.

[16] Inc. AeroVisions. Unmanned morphing aerial attack vehicle. URL http://www.canosoarus.com/

05UMAAV/UMAAV01.htm. (Accessed 2014).

[17] Johnson, T., Frecker, M., Abdalla, M., Gurdal, Z., and Lindner, D. Nonlinear analysis and optimization of

diamond cell morphing wings. Journal of Intelligent Material Systems and Structures, 20:815–824, 2009.

[18] Joo, J. J., Sanders, B., Johnson, T., and Frecker, M. I. Optimal actuator location within a morphing wing

scissor mechanism configuration. Proceedings of SPIE Smart Structures and Materials 2006: Modeling,

Signal Processing, and Control, 6166, 2006.

[19] Khoshlahjeh, M., Bae, E. S., and Gandhi, F. Helicopter performance improvement with variable chord

morphing rotors. Proceedings of 36th European Rotorcraft Forum, 2010. Paris, France.

[20] Kintscher, M., Wiedemann, M., Monner, H. P., Heintze, O., and Kuehn, T. Design of a smart leading edge

device for low speed wind tunnel tests in the european project sade. Institute of Composite Structures and

Adaptive Systems, German Aerospace Center, DLR, Braunschweig, Germany, 2011.

[21] Kota, S., Hetrick, J., Li, Z., and Saggere, I. Tailoring unconventional actuators using compliant transmissions:

Design methods and applications. IEEE/ASME Transactions on Mechatronics, 4:396–408, 1999.

[22] Kuzmina, S., Amiryants, G., Schweiger, J., Cooper, J., Amprikidis, M., and Sensberg, O. Review and outlook

on active and passive aeroelastic design concepts for future aircraft. Proceedings of International Congress

of the Aeronautical Sciences (ICAS 2002), 2002.

[23] Leon, O. and Gandhi, F. Rotor power reduction using multiple spanwise-segmented optimally-actuated

trailing- edge flaps. Proceedings of 35th European Rotorcraft Forum, 1309, 2009. Hamburg, Germany.

[24] Leon, O., Hayden, E., and Gandhi, F. Rotorcraft operating envelope expansion using extendable chord

sections. Proceedings of American Helicopter Society 65th Annual Forum, 2009. Grapevine TX.

[25] Lu, K.-J. and Kota, S. Design of compliant mechanisms for morphing structural shapes. Journal of Intelligent

Material Systems and Structures, 14:379–291, 2003.

[26] Mattioni, F., Gatto, A., Weaver, P. M., Friswell, M. I., and Potter, K. D. The application of residual stress tai-

loring of snap-through composites for variable sweep wings. 47th AIAA/ASME/ASCE/AHS/ASC Structures,

Structural Dynamics, and Materials Conference, 2006.

[27] McCormick, B. W. Aerodynamics, Aeronautics and Flight Mechanics. Wiley, New York, 2nd edition, 1995.

62

http://www.canosoarus.com/05UMAAV/UMAAV01.htm

http://www.canosoarus.com/05UMAAV/UMAAV01.htm

[28] MIT. icampus: Comet, the compliant mechanism tool. URL http://icampus.mit.edu/projects/

project/?pname=CoMeT. (Accessed 2014).

[29] Moorhause, D., Sanders, B., von Spakovsky, M., and Butt, L. Benefits and design challenges of adaptive

structures for morphing aircraft. Aeronautical Jornal, Royal Aeronautical Socieaty, London, 110:157–162,

2006.

[30] Pendleton, E. W., Bessette, D., Field, P. B., Miller, G. D., and Griffin, K. E. Active aeroelastic wing flight

research program: Technical program and model analytical development. Journal of Aircraft, 37, 2000.

[31] Prager, W. and Rozvany, G. I. N. Optimal layout of grillages. Journal of Structural Mechanics, 5:1–18, 1977.

[32] Reed, J. L., Hemmelgarn, C. D., Pelley, B., and Havens, E. Adaptive wing structures. Proceedings of

SPIE Smart Structures and Materials 2005: Industrial and Commercial Applications of Smart Structures

Technologies, 5762, 2005. San Diego CA.

[33] Reich, G. W. and Sanders, B. Introduction to morphing aircraft research. Jornal of Aircraft, 44:1059, 2007.

[34] Roskam, J. and Lan, C. T. Airplane Aerodynamics and Performance. DAR Corporation, 1997.

[35] Rossow, M. P. and Taylor, J. E. A finite element method for the optimal design of variable thickness sheets.

AIAA journal, 11:1566–1569, 1973.

[36] Rozvany, G. I. N. Aims, scope, methods, history and unified terminology of computer-aided topology opti-

mization in structural mechanic. Struct Multidisc Optim, 21:90–108, 2001.

[37] Rozvany, G. I. N. A critical review of established methods of structural topology optimization. Struct

Multidisc Optim, 37:217–237, 2009.

[38] Shin, D. K., Guerdal, Z., and Griffin, O. H. A survey of methods for discrete optimum structural design.

Computer Assisted Mechanics and Engineering Sciences, 1:27–38, 1994.

[39] Shin, D. K., Guerdal, Z., and Griffin, O. H. A penalty approach for nonlinear optimization with discrete

design variables. Engineering Optimization, 16:29–42, 1999.

[40] Sigmund, O. Design of material structures using topology optimization. Ph.d thesis, Dept. Solid Mechanics,

Technical University of Denmark, 1994.

[41] Sigmund, O. and Peterson, J. Numerical instabilities in topology optimization: a survey on procedures

dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 16:68–75, 1998.

[42] Sigmund, O., Zhou, M., and Rozvany, G. I. N. Layout optimization of large fe systems by optimality criteria

methods: aplications to beam systems. Concurrent Engineering Tools and Technologies for Mechanical

System Design (held in Iowa), 1992.

[43] Sofla, A. Y. N., Meguid, S. A., Tan, K. T., and Yeo, W. K. Shape morphing of aircraft wing: Status and

challenges. Materials and Design, 31:1284–1292, 2007.

63

http://icampus.mit.edu/projects/project/?pname=CoMeT

http://icampus.mit.edu/projects/project/?pname=CoMeT

[44] Sousa, F. M. C. Topology optimization of a wing structure: Studies on morphing compliant edges. Master of

science in aerospace engineering, Instituto Superior Tecnico, September 2013.

[45] Spillman, J. The use of variable camber to reduce drag, weight and costs of transport aircrafts. Aeronautical

Jornal, 96:1–9, 1992.

[46] Stolpe, M. and Svanberg, K. On the trajectories of penalization methods in topology optimization. Struct

Multidisc Optim, 21:128–139, 2001.

[47] Subbarao, K., Supekar, A. H., and Lawrence, K. Investigation of morphable wing structures for unmanned

aerial vehicle performance augmentation. American Institute of Aeronautics and Astronautics, 2009. Seattle,

Washington.

[48] Vale, J. Development of computational and experimental models for the synthesis of span and camber mor-

phing aircraft technologies. Phd degree in aerospace engineering, Instituto Superior Tecnico, 2012.

[49] Vasista, S., Riemenschneider, J., and Monner, H. P. Design and testing of a compliant mechanism-based

demostrator for a droop-nose morphing device. SciTech, 2015. appearing in press.

[50] Weisshaar, T. A. Morphing technologies - new shapes for aircraft design. RTO-MP-AVT-141, Neuilly-sur-

Seine, France, 2006.

[51] Yu, Y., Li, X., Zhang, W., and Leng, J. Investigation on adaptive wing structure based on shape memory

polymer composite hinge. Proceedings of SPIE International Conference on Smart Materials and Nanotech-

nology in Engineering, 6423, 2007. Harbin, China.

64

investigations into the effects of 3-dimensional geometric

Documents