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Master of Science Thesis
Finite element analysis of a kite
for power generation
Computational modelling of flight dynamics of a tethered
wing including non-linear fluid-structure interaction.
H. A. Bosch
Departments :
Report no
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Coaches
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Professor
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Type of report :
Date
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Precision and Microsystems Engineering (faculty of 3mE) Applied Sustainable Science Engineering and
Technology (faculty of Aerospace Engineering)
EM 12.010
Dr.-Ing. R. Schmehl & Dr. P. Tiso
Prof. Dr. Ir. D.J. Rixen
Master of Science Thesis
10 April 2012
Finite element analysis of a kite for
power generation
Computational modelling of flight dynamics of a tethered wing
including non-linear fluid-structure interaction.
Master of Science Thesis
For the degree of Master of Science in Mechanical Engineering at the
Delft University of Technology
H.A. Bosch
April 2012
Faculty of Mechanical, Maritime and Materials Engineering (3mE)
Faculty of Aerospace Engineering
Delft University of Technology
Copyright H.A. Bosch
Cover photo by Max Dereta
All rights reserved.
Delft University of Technology
Departments of
Precision and Microsystems Engineering (PME) and
Applied Sustainable Science Engineering and Technology (ASSET)
The undersigned hereby certify that they have read and recommend to the Faculty of
Mechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis
entitled
Finite element analysis of a kite for power generation
by
H.A. Bosch
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
Dated: April 2012
Supervisor(s):
Prof. Dr. D.J. Rixen
Dr.-Ing. R. Schmehl
Dr. P. Tiso
Reader(s):
Dr. A. A. Zadpoor
Preface
Working on this Master thesis was an exciting time and will bring an end to a great time of
studying Mechanical Engineering at the Delft University of Technology. From the start I was
very interested in the topic of this thesis, because it is a subject I can really relate myself
to. Working on solutions for the energy challenge that my generation faces is not an easy
task, but a very important one. I would like to think that I was able to contribute to these
developments with this thesis.
I am grateful that I had the opportunity to combine the knowledge of two research groups
from different faculties: the Research group Engineering Dynamics from the Mechanical Engineering faculty and the Kite Power research group from the Applied Sustainable Science
Engineering and Technology (ASSET) department from the Aerospace Engineering faculty.
People in both departments really showed passion for what they are doing and that inspired
me to give my best as well.
I would like to thank Prof. Dr. Wubbo J. Ockels for starting the kite powered energy concept,
giving me the opportunity to do this interesting thesis. It was also an honour to be able to
work under the supervision of Prof. Dr. Daniel J. Rixen and learn from his great expertise
in the dynamics field. My special thanks goes to my two supervisors Dr. Paolo Tiso and
Dr.-Ing Roland Schmehl. This thesis wouldn’t be here without all the hours they put into
guiding me, helping me to solve the problems and reviewing my work. I would like to thank
Dr. Amir A. Zadpoor for reading this thesis and being part of my graduation committee.
I would also like to show my gratitude to everyone in the Kite Power group. I learned a lot
from their kite experience and had many important discussions and brainstorms with almost
everyone within the group. My thanks also goes to the Precision and Microsystems Engineering (PME) department and all the feedback they gave me during the monthly presentations.
This helped me to solve difficult mechanical problems and improved my work.
My last, but most important thanks goes my parents, friends an family and all the support
they gave me throughout this and the past years of my studies.
Delft, University of Technology
April 2012
Master of Science Thesis
H.A. Bosch
H.A. Bosch
ii
H.A. Bosch
Preface
Master of Science Thesis
Summary
Airborne Wind Energy is a new field of research that focusses on sustainable electricity generation from wind with flying devices that use less material and weigh less than conventional
wind turbines. They can also access wind at higher altitudes where the power density is much
higher and the wind velocity more constant than close to the earth’s surface.
The Delft University of Technology developed a concept with a flexible kite that flies at a high
altitude and is connected to the ground with a long tether. The tether is reeled off a drum
that is connected to a generator that produces energy while the kite flies crosswind through
the air in a figure eight. The kite stops flying crosswind when it reaches its maximum altitude
and starts to hover like a parachute so that it can be reeled in with a low amount of energy.
This results in a cyclic process that will produce a net amount of energy.
A leading edge and several struts form the backbone of the kite and are made from inflatable
beams. A thin canopy between them introduces the aerodynamic forces to the structure. The
kite is connected with a set of bridle lines to a control pod that is connected to the tether.
The control pod is used to steer and control the power in the kite
Realistic computer models of kites are necessary to optimize this system, develop controllers
for a fully autonomous system, design better kites and get a better understanding of the
dynamic behaviour of kites. Existing kite models either lack the needed realism to perform
these studies by neglecting the flexible behaviour of the kite or are too computational intensive
to use when they do include the full flexible behaviour.
Therefore this thesis introduces a new realistic and reduced approach to model flying flexible
inflatable tube kites used in airborne wind energy systems. The approach should result in a
realistic model that represents the global dynamics and deformation modes of the kite, can
be steered with real steering inputs (shortening and lengthening of the steering lines) and
should be derived from physical principles. The model should be reduced to make it fast by
neglecting unimportant local effects without losing the essential dynamics and it should be
set-up so that it can be even further reduced with new model reduction techniques.
The system is split in three parts in the new approach: the kite, the aerodynamics and the
bridle lines with the tether.
Master of Science Thesis
H.A. Bosch
iv
Summary
The kite is modelled with the non-linear finite element method to stay close to its physical
properties and represent its full non-linear flexible behaviour. The inflatable beams in the
leading edge and struts are modelled with regular beam elements that represent the behaviour
to reduce the amount of degrees of freedom drastically. The canopy is modelled with shell
elements and uses a coarse mesh to reduce the computation time without losing essential
deformation modes.
The aerodynamic model of Breukels [7] is used to describe the distributed aerodynamic forces
exerted on the kite depending on its deformation. The model is based on the assumption
that the kite can be divided into a finite number of two dimensional wing sections. The
aerodynamic properties are determined for each wing section from a set of two dimension
CFD experiments and distributed over the wing section using a set of weight factors.
The finite element kite model and aerodynamic model form a fluid-structure-interaction problem together that needs to be solved iteratively. A solving algorithm is introduced that splits
the structural and aerodynamic convergence and uses a load stepping algorithm to stabilize
and speed-up convergence.
The bridles and tether are modelled dynamically as simple spring-dampers using multi-body
dynamic methods. The end of the bridles are connected to the kite at the bridle attachment
points
The most important reduction principle is the assumption that the kite reacts quasi-static
to the aerodynamic forces, because the inertia of the kite are very small compared to the
aerodynamic forces. Therefore the local inertia of the kite are neglected when solving the
finite element equations and the dynamic deformations of the kite can be approximated by
a sequence of static solutions. The quasi-static fluid-structure-interaction problem returns
forces that are exerted on the end points of the bridles in the dynamic simulation. These
forces are assumed to remain constant during the period of a time step, eliminating the
need to solve the fluid-structure-action problem in the dynamic differential equations. This
method greatly reduces the amount of computation time, because the time integration only
has to be done for the small number of degrees of freedom in the dynamic model and the
fluid-structure-interaction problem only needs to be solved once in every time-step instead of
multiple times in the time integration algorithm.
All system components were implemented in Matlab and a controller was built to fly several
figure eight tests with the model. Results show that the modelling approach leads to a fast
and realistic model. A steering input results in dynamics and a real deformation that causes
the kite to yaw comparable to real kites and other models. It can be concluded that all the
assumptions were valid and led to a model that captures the kite behaviour realistically. The
model is a factor 25-30 slower than real time, which is very fast considering the complexity of
the calculations and that it was implemented in a non compiled language. The clearly visible
distinct deformation modes and non-linear force model make it also a suitable and interesting
candidate for further new model reduction techniques. The aerodynamic model has some
shortcomings resulting in an overestimation of the lift over drag ratio and is considered to
be the main source of uncertainties in the whole model. Replacement with a better model
should further improve the model and make it possible to do an extensive validation study.
Concluding, the new proposed approach is successful: fast and realistic, flexible in its use,
able to model all type of kites, a good candidate for further reduction and can be used for
controller design and optimizing studies.
H.A. Bosch
Master of Science Thesis
Table of Contents
Preface
i
Summary
1 Introduction: kite power as solution to
1-1 The energy challenge . . . . . . . .
1-2 Thesis objective . . . . . . . . . . .
1-3 Thesis structure . . . . . . . . . . .
iii
the world
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2 Kite powered energy research
2-1 The pumping crosswind power generating kite
2-1-1 Crosswind power . . . . . . . . . . . .
2-1-2 System overview . . . . . . . . . . . .
2-2 Current research . . . . . . . . . . . . . . . .
2-2-1 Kite models . . . . . . . . . . . . . .
2-2-2 Tether models . . . . . . . . . . . . .
2-2-3 Aerodynamic models . . . . . . . . .
2-3 Conclusions and research opportunities . . . .
energy challenge
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system
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3 Thesis goal
4 Approach
4-1 Modelling options . . . . . . . . . . . . .
4-1-1 A realistic kite modelling approach
4-1-2 A reduced kite modelling approach
4-1-3 Tether and bridles . . . . . . . . .
4-1-4 Aerodynamic modelling approach .
4-2 The full model description . . . . . . . . .
4-3 Implementation . . . . . . . . . . . . . .
Master of Science Thesis
1
1
3
3
5
5
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8
13
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16
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21
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H.A. Bosch
vi
Table of Contents
5 Finite element kite model
31
5-1 The non-linear finite element method . . . . . . . . . . . . . . . . . . . . . . . .
31
5-1-1
Introduction to the finite element method . . . . . . . . . . . . . . . . .
31
5-1-2
Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5-1-3
Non-linearities in finite elements . . . . . . . . . . . . . . . . . . . . . .
33
5-1-4
Non-linearities in the kite . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5-1-5
Mathematical framework for element definitions . . . . . . . . . . . . . .
37
5-2 Kite modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5-2-1
Modelling requirements . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5-2-2
Kite selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5-2-3
Geometric dimensioning & meshing
. . . . . . . . . . . . . . . . . . . .
40
5-2-4
Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
5-2-5
Leading edge and struts . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5-2-6
Trailing edge wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5-2-7
Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5-2-8
External forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5-2-9
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5-3 Solving the non-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5-3-1
Structural solution procedure . . . . . . . . . . . . . . . . . . . . . . . .
54
5-3-2
Aero-elastic solution procedure . . . . . . . . . . . . . . . . . . . . . . .
55
5-3-3
Load control, stability and convergence procedures . . . . . . . . . . . .
58
5-3-4
Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5-4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6 Aerodynamic model
67
6-1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6-1-1
Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6-1-2
From forces to a distributed load . . . . . . . . . . . . . . . . . . . . . .
69
6-1-3
Three dimensional correction . . . . . . . . . . . . . . . . . . . . . . . .
72
6-2 Implementation in the FE model . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6-2-1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6-2-2
Aerodynamic parameters for each wing section
. . . . . . . . . . . . . .
74
6-2-3
Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6-2-4
Aerodynamic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6-3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6-4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
H.A. Bosch
Master of Science Thesis
Table of Contents
vii
7 Dynamic tether and bridles model
81
7-1 Modelling choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-1-1
7-1-2
7-1-3
Bridles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bridle attachment points . . . . . . . . . . . . . . . . . . . . . . . . . .
7-1-4 Connections . . .
7-2 Coordinates and reference
7-2-1 Coordinates . . .
7-2-2 Reference frames
7-3 Forces . . . . . . . . . .
7-3-1 Kite forces . . . .
7-3-2 Bridle springs . .
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81
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7-3-3
Tether and bridles drag . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
7-3-4
Bridle attachment points damping . . . . . . . . . . . . . . . . . . . . .
91
7-3-5
Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
7-4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
7-5 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-5-1 Wind model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-5-2 Initial values of the degrees of freedom . . . . . . . . . . . . . . . . . . .
94
94
95
7-5-3
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frames
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81
Nominal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7-6 Finite element kite model inputs . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7-6-1
Kite velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7-6-2
Wind velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7-6-3
Rotational kite velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7-6-4
Bridle attachment points . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7-7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
8 System integration
99
8-1 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
8-1-1
State equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
8-1-2
Including the FSI problem in the dynamic equations . . . . . . . . . . . .
100
8-1-3
8-1-4
8-1-5
Solver selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
105
107
8-2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2-1 Steering controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
107
8-2-2 Power controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
112
Master of Science Thesis
H.A. Bosch
viii
Table of Contents
9 Results & Discussion
113
9-1 Test flight environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9-2 Steering behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
9-2-1
Steering input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
9-2-2
The steering deformation of the kite . . . . . . . . . . . . . . . . . . . .
117
9-2-3
Angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
9-2-4
Kite speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
9-2-5 Line forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
121
9-4 Structural deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-5 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
124
9-6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124
10 Conclusions & Recommendations
125
10-1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A Geometric non-linear beam development
128
B Finite element implementation verification
134
B-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-2 Shell elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-2-1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
134
134
B-2-2
B-2-3
B-3 Beam
B-3-1
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135
139
142
142
B-3-2 Load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-3-3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
145
Load cases .
Conclusions
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C North Rhino 16
147
C-1 Detailed photos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C-2 Geometric properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
C-3 Deformation sequence figure eight flight . . . . . . . . . . . . . . . . . . . . . .
150
Bibliography
151
H.A. Bosch
Master of Science Thesis
Chapter 1
Introduction: kite power as solution to
the world energy challenge
1-1
The energy challenge
Energy is one of the most important factors in our daily life nowadays and we are using
more and more of it. According to the International Energy Outlook 2011 [1] the world
energy consumption will increase from 505 quadrillion Btu in 2008 to 770 quadrillion Btu
in 2035, an increase of 53 percent. The question that will keep humanity busy for the next
decades will be how to obtain a sustainable solution to provide enough energy for the world
demand.
Electric energy forms a large part of this consumption and the demand will increase by
2.3 percent each year from 2008 to 2035, leading to a total increase of 84 percent from 19.1
trillion kilowatthours in 2008 to 35.2 kilowatthours in 2035.
Renewable electric energy will be the fastest growing source of electricity generation
between 2008 and 2035 with an annual growth of 3.1 percent. In 2008 renewables accounted
for 19 percent of the total electricity generation, with 3.7 trillion kilowatthours.
Wind energy production has grown very fast over the past decade, from 18 gigawatts of net
installed capacity at the end of 2000 to 121 gigawatts at the end of 2008. This growth will
continue: 27 percent of the 4.6 trillion kilowatthours growth in renewable electric energy from
2008 to 2035 will be due to wind energy. In 2008 the wind energy counted for 5.7 percent
of the total renewable electricity generation with 210 billion kilowatthours and will have an
average annual growth of 7.5 percent between 2008 and 2035.
Clearly wind energy plays an important role in the future of electric energy, but it has two
problems. Firstly it can’t compete economically with fossil fuels at the moment. The operating costs of wind turbines are much lower than the operation costs for fossil fuel-fired power
plants, but the high construction costs make the total cost per kilowatthour higher. Secondly
wind energy is an intermittent technology that is only available when there is wind. New
storage techniques can be helpful to store a surplus of wind energy when there is much wind
Master of Science Thesis
H.A. Bosch
2
Introduction: kite power as solution to the world energy challenge
that can be used in periods with a shortage of wind, but only a limited amount of energy can
be stored.
Airborne Wind Energy (AWE) is new field of research and focusses on electricity generation with flying devices. These devices use a far less material and weigh much less than
conventional wind turbines, making them cheaper. Another advantage is that they can access
wind at higher altitudes. Figure 1-1 shows that the wind speed and wind power density at
high altitude is much higher than at the operating altitude of wind turbines (maximum of
200 meters) and is a potential source of much sustainable energy. Another advantage of high
altitude wind is that the wind is more constant than at low altitude. The boundary effect of
the earth slows down the wind close to the surface. These factors show a large advantage of
airborne wind energy over conventional wind energy: cheaper and access to a more constant
and denser power source.
Over the past decade the amount of research in this field has grown significantly and several
new concepts have been developed, but much more research is needed before these systems
are technically mature and can compete economically with other energy generation methods.
(a) Wind speed at high altitude.
(b) Wind power density at high altitudes.
Figure 1-1: High altitude wind power in The Netherlands.
Kite powered energy is one of these new promising concepts. A large kite as depicted in
Figure 1-2 is connected to the ground with a tether. The tether is reeled off a drum that is
connected to a generator that produces energy while flying the kite crosswind through the
air. The kite stops flying crosswind when it reaches its maximum altitude and starts to hover
like a parachute so that it can be reeled in with a low amount of energy. This results in a
cyclic process that will produce a net amount of energy as depicted in Figure 2-1). Besides
being able to easily reach high altitudes, an advantage of the system is the low construction
costs.
Many exciting research questions need to be answered in the coming years to perfect this
system and be able to make it a commercial competing product. This thesis will focus on
using kites as a solution to the world energy problem.
H.A. Bosch
Master of Science Thesis
1-2 Thesis objective
3
Figure 1-2: The kite power concept from the Delft University of Technology. Photo by Max
Dereta.
1-2
Thesis objective
Reliable computer models of kites are necessary for further development of the kite power
system. These models need to simulate the behaviour of a kite in a realistic way to further
improve the design of the kites, to develop robust controllers for system automation and to
do various system optimization studies. Kites are highly flexible structures that constantly
interact with the changing aerodynamic forces, which makes it a very difficult problem to
model. Existing kite models either lack the needed realism to perform these studies or are
too computationally intensive to use when they do include the full flexible behaviour of the
kite. Therefore the goal of this thesis is to develop a new realistic and reduced approach to
model flying flexible inflatable tube kites used in airborne wind energy systems.
1-3
Thesis structure
To develop a new modelling approach, Chapter 2 first summarizes the current state of the
research, explains the system in some more detail and states the research opportunities.
Chapter 3 formulates the goal of this thesis and explains the important requirements followed
by the description of the new modelling approach in Chapter 4. The new model exists of
three main building blocks that are explained in the subsequent three chapters: the finite
element kite model in Chapter 5, the aerodynamic model in Chapter 6 and the dynamic
tether and bridle model in Chapter 7. The three components and a controller are integrated
in one system in Chapter 8, followed by the results and discussion in Chapter 9.The final
conclusions can be found in Chapter 10.
Master of Science Thesis
H.A. Bosch
4
H.A. Bosch
Introduction: kite power as solution to the world energy challenge
Master of Science Thesis
Chapter 2
Kite powered energy research
This chapter discusses the ideas behind kite powered energy. Section 2-1 describes the research performed at the Delft University of Technology, followed by the current state of the
research on kite modelling in Section 2-2. The chapter ends with some concluding remarks
in Section 2-3 that serve as a basis for the formulation of the goal and approach of this thesis
in the next chapters.
2-1
The pumping crosswind power generating kite system
The idea for using a single crosswind flying kite powered system originates from the Laddermill
concept that was developed by Ockels [37] in the 1990’s where multiple kites are coupled
together in one system. When serious research on this subject started at the Delft University
of Technology around 2005 it was soon concluded that controlling and understanding a single
kite was already difficult enough and the idea of using crosswind power [32] became the new
focus. The first prototypes of the system were built around 2007 [29, 30, 28, 31, 27] and
the first full scale prototype of a 20kW system was built by the Kite Power research group
from the ASSET (Applied Sustainable Science Engineering and Technology) department in
2010 [12]. The last years were used to gain more insight in the system and current research
focusses on optimizing the energy output, designing better kites and fully automating the
system. This section describes the working principle behind the system in Section 2-1-1 and
the several components in Section 2-1-2.
2-1-1
Crosswind power
The principle
Figure 2-1 illustrates the basic crosswind power concept. A kite is attached to a long tether
that is connected to a power generator on the ground. The process consists of two phases:
the generation phase and the retraction phase.
Master of Science Thesis
H.A. Bosch
6
Kite powered energy research
Figure 2-1: Working principle of the kite powered system. The kite is attached to a tether that
is reeled off a drum that is connected to a generator on the ground while flying a crosswind figure
eight pattern during the power generation phase. The kite is reeled in during the retraction phase
with a minimum amount of energy.
The kites flies a figure eight pattern in the sky to obtain maximum lift forces when the system
is in the generation phase. At the same time the tether is reeled off a drum that is connected
to a generator which generates energy due to the revolving motion.
The retraction phase starts when the kite reaches a certain altitude. The kite is put in a
de-powered mode where it generates almost no force on the tether by changing the pitch
angle of the kite. The generator is used as a motor and reels in the kite to an altitude from
where the generation phase starts again. The energy needed to reel in the kite during the
retraction phase is only a fraction of the energy that is produced during the generation phase.
This results in a cyclic (pumping) process with a phase that generates energy and a phase
that costs energy, but with a net positive effect. This is illustrated in Figure 2-2 where the
mechanical power of a couple of cycles during test flight with the real system are shown.
Crosswind power
The kite flies crosswind figure eight patterns during the reel-out phase as depicted in Figure 2-1
and Figure 2-4. This crosswind principle was already described by Loyd [32] in 1980 and was
further explored by Ockels [36] and Terink [46] and nicely summarized by de Groot [11].
The kite gains more speed when flying crosswind and generates an apparent wind speed that
is higher than the normal wind speed. The lift force increases with the apparent wind speed
and therefore the power generation as well. The generated power P can be expressed in terms
of the tension force in the tether Tt and the reel-out speed of the tether Vt by P = Tt · Vt .
When the kite flies crosswind with a speed Vk it finds an equilibrium between the lift force
L, the drag force D and the tether force Tt as depicted in Figure 2-3. Loyd [32] describes
that the L/D ratio must be optimized to get a maximum power output, because the lift force
varies quadratically with the L/D ratio. The maximum power generation is obtained when
the reel out speed of the tether Vt is 1/3 of the wind speed Vw . The power production also
depends on the cycle time and the tether length [52, 55]. Several optimization studies have
shown that flying a figure eight is the most optimal trajectory [56, 18].
H.A. Bosch
Master of Science Thesis
2-1 The pumping crosswind power generating kite system
7
14
L
Location: Valkenburg
Date: 10-02-2011
Wind speed: 7 m/s
Kite size: 25 m2
12
10
Mechanical Power [kW]
8
6
D
Vt
4
Vw
2
Vk
Va
0
Tt
−2
−4
−6
0
100
200
300
400
Time [second]
500
600
700
Figure 2-2: The power production of a couple of
pumping cycles with the real system.
Figure 2-3: The force equilibrium between the lift force L, the
drag force D and the tether force
Tt in case of a horizontal crosswind flight.
Figure 2-4: Figure eight manoeuvre during a cross wind flight.
Power scale
The most important question is how much energy the system can generate to what kilowatthour price. The current system has a nominal power rating of 20kW using a 25m2 kite,
but studies have shown that it is possible to scale this system to the order of megawatts
[31]. It is not easy to predict the kilowatthour price. The structure of a kite system is much
Master of Science Thesis
H.A. Bosch
8
Kite powered energy research
cheaper than for wind turbines and uses a more constant power source, but the maintenance
costs for kite systems might be higher due to for instance the low durability of the materials.
The future will tell how much the real costs will be.
Wind window
The earth surface creates a boundary effect that slows down the wind close to its surface.
This is the reason for higher wind speeds and a more constant wind at high altitudes as was
shown in Figure 1-1a. The wind velocity is never constant and always shows variations due
to wind gusts and gaps. These effects are highly random and influence the power generation.
Robustness of flying controllers is important to safely and optimally control the kite. de Groot
[11] shows that the wind speed variation is about 10-20% in normal flight conditions. The
maximum theoretical speed that the kite can obtain is approximately the L/D ratio times
the wind speed as was shown by van den Heuvel [48], resulting in a much higher apparent
wind speed.
Figure 2-5 shows that the flying domain of a kite exists of the quarter of a sphere. The amount
of pluses and minuses shows how much power de kite generates when flying in that part of
the domain. This helps to understand how the kite can be flown to produce the maximum
amount of energy.
Figure 2-5: The wind window of a kite [18]. It can be flown in the space described by the quarter
of a sphere, but experiences different power zones (+ / -).
2-1-2
System overview
The system consists of the core components as shown in Figure 2-6a. The kite is attached
with a set of bridle lines to the control pod. The control pod contains the mechanism to
steer the kite and to power or de-power it by changing the pitch angle and has a wireless
connection with the ground station. A tether connects the control pod to the ground station.
Figure 2-6b shows the real system that was built by the Delft University of Technology.
H.A. Bosch
Master of Science Thesis
2-1 The pumping crosswind power generating kite system
9
kite
bridles
control pod
tether
ground station
(a) The full system with all
(b) An overview of the real sys-
its components.
tem (photo from Max Dereta).
Figure 2-6: System overview
Kite
Numerous types of kites exist and have been tested for power generation. Figure 2-7 shows
four different types. A division can be made between kites with and without inflatable parts.
Both have their advantages, but the ASSET system currently uses kites with inflatable parts.
The kite exists of an inflatable leading edge and inflatable struts that form the structural
backbone of the kite as can be seen in Figure 2-8. A very thin canopy forms the wing of the
kite and generates the aerodynamic forces. Four lines are attached to the four corners of the
kite to control it. The two front lines, called the power lines, handle most of the force. The
two rear lines, called the steering lines, are used the steer the kite.
The kite is an extremely flexible structure as can be seen from Figure 2-10, which makes is
hard to model it. This flexibility is necessary to give the kite its manoeuvrability. A difference
in steering line lengths deforms the kite and results in a steering movement as will be further
discussed in Chapter 9.
The length difference between the steering lines and the power lines determines the pitch
angle of the kite, as shown in Figure 2-9. If the pitch angle increases, the lift force in the kite
increases as well, called powering the kite. During the retraction phase the kite is de-powered
which means that the pitch angle is decreased. Powering the kite too much will cause the
kite to stall.
Small kites are very direct in their steering behaviour, while large kites are very slow in
Master of Science Thesis
H.A. Bosch
10
Kite powered energy research
(a) The Mutiny 25m2 inflatable
tube kite.
(c) Peter Lynn foil kite.
(b)
The
14m2 inflatable
kite.
Hydra
tube
(d) The C-shaped North
Rhino 16m2 kite.
Figure 2-7: Different kite types.
their steering behaviour. This also depends on the wind speed. The performance of the
kite is largely determined by its lift over drag ratio (L/D) as mentioned before. The L/D of
arc-shaped kites ranges roughly from 4 to 7.
Bridles
Depending on the kite type, a set of bridles is used to attach the steering and power lines to
the kite. These bridles are attached to the leading edge or struts at different points and keep
the kite in an aerodynamic optimal shape during operation. Figure 2-11 shows an example
of a bridled kite. Not all kites use bridles, the steering and power lines are directly connected
to the four corners of C-shape kites for instance, as shown in Figure 2-7d.
Control pod
The control pod, as shown in Figure 2-12, contains all the necessary electronics to steer the
kite and to (de-)power the kite. It has a wireless connection with the ground station and can
H.A. Bosch
Master of Science Thesis
2-1 The pumping crosswind power generating kite system
11
Figure 2-8: Naming conventions for an inflatable tube kite.
depowered
powered
pitch angle
wind direction
Figure 2-9: Powering and de-powering of the kite by changing the pitch angle resulting from a
difference in lengths of the steering lines and power lines.
be used in manual control or automatic control. The power for all the components is provided
by batteries. Currently they have to be recharged on the ground, but a small wind turbine
on the control pod will recharge the batteries during the flight in the future.
Tether
The tether is made from Dyneema. Depending on the length and diameter of the tether it has
an important role in the dynamics of the total system. The mass of the tether can become
much higher than the mass of the kite when flying with long and thick tethers that are needed
for large kites that generate high lift forces. The drag force on the tether increases as well
with the tether length and diameter and becomes significant. Effects like sag of the cable also
become important as can clearly be seen from Figure 2-13.
Master of Science Thesis
H.A. Bosch
12
Kite powered energy research
Figure 2-10: The extreme flexibility of a
kite.
(a) Opened control pod.
Figure 2-11: Bridle system of the Mutiny
kite.
(b) Hanging under the kite.
Figure 2-12: The control pod.
Ground station
The ground station as depicted in Figure 2-14 contains the drum, the generator and all the
needed electronics to operate them. Multiple computer systems are used to control and
monitor the kite when it is operational.
H.A. Bosch
Master of Science Thesis
2-2 Current research
13
(a) The groundstation.
Figure 2-13: Extreme sag of the tether
(photo by Max Dereta).
2-2
(b) The generator and drum.
Figure 2-14: The ground station.
Current research
Kite powered energy is a fairly new research field with many questions that still need to
be answered. This makes it a very challenging and exciting subject. Some of the current
research fields are the modelling of kites, tethers and ground stations, developing controllers,
optimizing kite designs of flying trajectories, developing aerodynamic models, implementation
strategies (air traffic, locations, legislation etc.) or studies on more durable kite materials.
This thesis focusses on the modelling part. A short overview will be given on the current
state of the research of kite modelling in Section 2-2-1, tether modelling in Section 2-2-2 and
aerodynamic modelling in Section 2-2-3
2-2-1
Kite models
Several kite models have been developed over the last years. Figure 2-15 gives an overview of
all the models based on their complexity. Some of them have been verified with some basic
experiments, but most of them haven’t been verified at all. The models will be discussed in
this section.
Master of Science Thesis
H.A. Bosch
14
Kite powered energy research
finite element
Calculation time
new model
multi body
multi plate
lumped mass
rigid body
point mass
black box
Degrees of freedom
Figure 2-15: All the currently available kite models, sorted by their complexity.
Black/grey box models are obtained with model identification techniques to fit mathematical equations to experimental data assuming no (black) or little (grey) knowledge of the
physical system. One of these models is currently being developed at the Delft University of
Technology and Erhard and Strauch [13] also developed a model without giving the details.
The advantage of such models is that their behaviour directly represents the real experiments
and that they are very fast. Nothing can be said however about the validity of the model
when situations are modelled that did not occur in the measurement data. Furthermore no
physical insight can be gained from them, nor can the effect of new kite designs be tested.
The models are very suitable to compare to other models to judge their validity.
Points mass models are the simplest physical kite models available, but limited in their use.
The kite is approximated as a point mass at the end of the tether on which lift, drag, tether and
gravity forces work. The aerodynamic forces are determined from simple plate aerodynamic
models. The flexibility of the kite and the accompanying dynamic effects are completely
neglected and controlling is introduced by directly controlling the yaw and pitch angles to
change the directions of the lift and drag forces. This makes them not very accurate and
unsuitable to study kite behaviour or test new designs. They are mostly used for preliminary
studies on trajectory optimization or system performances. These models have been developed
by several persons, for instance by Williams et al. [56].
Rigid body models model the kite in the same manner as aircrafts are modelled. Williams
et al. [56] designed a six degrees of freedom rigid model and included the moments of inertia
of the kite. Houska [21] took a different approach and neglects the local moments of inertia
of the kite by stating that they are much smaller than the combined global moment of inertia
of the kite as point mass and the tether. He uses three degrees of freedom to describe the
displacements of the kite, three degrees of freedom to describe the tether model and two
degrees of freedom to simulate the control mechanism. He superimposes bending of the arced
H.A. Bosch
Master of Science Thesis
2-2 Current research
15
shape of the kite as an additional state by introducing a second order differential equation.
These models include more of the real dynamics of a kite than the point mass models, but still
lack the flexibility of a real kite. de Groot [11] developed a rigid body model that was derived
from the multi body model of Breukels [7] that will be discussed below. He concluded that it
is very questionable whether it is possible to capture the dynamics of a kite realistically with
a rigid body model.
A particle based model was developed by Furey and Harvey [17] as depicted in Figure 2-16a. The kite structure is described by a number of lumbed mass points that are
connected with rigid rods and hinged together. The light grey lines are flexible constraints
that are used to simulate the stiffness of the canopy and the struts. The zigzag lines represent aerodynamic slices for which the aerodynamic forces are calculated. This model is able
to deform and therefore able to steer by pulling one of the steering lines, but the separate
components do not flex and therefore it still lacks the real flexibility of a kite. Furthermore
the needed properties to simulate the behaviour are quite artificial.
(a) The by Furey and Harvey [17] developed
(b) The by Williams et al. [56] devel-
particle based kite model.
oped multi plate model.
Figure 2-16: More advanced kite models.
A multi-plate model was developed by Williams et al. [53] [56] and is shown in Figure 2-16b.
The shape of the kite is approximated with a finite number of plates that are hinged together
at the leading edge. The aerodynamic forces are determined for each plate separately from
plate theory. The advantage of this model is that it includes some deformation modes of
the kite, but the disadvantage is that it still lacks the realism of a real flexible kite model,
since its elements are not able to flex. Furthermore this model has difficulties in finding an
equilibrium position and therefore brings some computational issues.
A multi-body model developed by Breukels [7] was a completely new approach and he
designed a whole framework to simulate flexible kites, as can be seen in Figure 2-17a. The
multi body framework was used to define different building blocks that are used to build a
complete kite. Rigid links are hinged together with spherical springs to form the leading
edge and struts, and linear springs are used to simulate the canopy an tether. Parameters
Master of Science Thesis
H.A. Bosch
16
Kite powered energy research
were determined from various experiments to give the model realistic properties to simulate
the correct beam stiffness and canopy stiffness. A new aerodynamic model was developed to
calculate an distributed aerodynamic loads for the surface of the kite. This model simulates
the flexibility of the kite in a very nice way and can use real steering inputs to steer te kite.
The main problem is that it still relies on many artificial elements and parameters that have
no direct physical meaning to simulate the physical behaviour of the kite.
(a) The by Breukels [7] developed
(b) The by Schwoll [43] devel-
multi body kite.
oped finite element model developed with Madymo.
Figure 2-17: More advanced kite models.
A finite element model was developed by Schwoll [43] as shown in Figure 2-17b. The
software package Madymo was used to build a very detailed model of a kite with more than
30.000 elements. This software was designed to simulate air bags and therefore has the
nice option to inflate beams. The inflatable leading edge and canopy are modelled in a very
realistic way. The great thing about the finite element approach is that the method is specially
designed to simulate structural problems and intrinsically includes all the physical material
properties in the model. There is no need for artificial parameters to simulate the physical
behaviour of the kite, everything is already there. The main disadvantage of this model is
that it is so detailed that the calculation times become very high. Furthermore, it needs
a very sophisticated aerodynamic model that produces the distributed load on the canopy.
Since this is not available, the kite doesn’t fly yet and is only used form some structural
experiments.
2-2-2
Tether models
Also several tether models were developed over the last years. The tether becomes an important part of the dynamics of the system when the tether becomes very long or thick due to
the sag, mass and drag forces. Modelling the tether is not an easy task, due to the non-linear
behaviour of the tether. Tether modelling is not the goal of this thesis and therefore only a
very brief overview is given. A more detailed summary can be found in reference [11].
A rigid tether is the simplest modelling option and can only be used for simple simulations.
H.A. Bosch
Master of Science Thesis
2-2 Current research
17
A linear spring damper represents the dynamics of a tether already quite well for a tether
shorter than 100m and when there is enough tension in the tether to keep the sag small
[5]. Stiffness and damping properties can be obtained by using Hook’s law and the material
properties.
Lumped masses connected together with springs can be used to approximate the tether
dynamics and result in many more degrees of freedom than the earlier mentioned options,
but gives a nice representation of the tether dynamics [5]. The disadvantage of this approach
is that the springs will have a high stiffness which will result in a small time step in order to
capture the high frequency behaviour and obtain a stable simulation.
Discretized inelastic rods coupled together with hinges are used by Breukels [7] and
Williams et al. [54] to model the tether. Structural damping is introduced by adding dampers
in the hinges. The problem with this approach is that very non-linear equations will arise or
constraints have to be added to the system of equations.
2-2-3
Aerodynamic models
Various methods exist to model the aerodynamics for aircrafts. Most of those methods assume
a rigid structure and are therefore not suitable for kites. Flexible kites deform under an
aerodynamic load, resulting in a different aerodynamic load. This results in a aero-elastic
problem that needs to be solved iteratively. The aerodynamic model that can be used depends
on the chosen structural model of the kite. Flexible kite models need a more sophisticated
aerodynamic model than rigid models. A short overview of available aerodynamic models
that are used by the several structural kite models.
Plate theory is the simplest way to describe aerodynamic forces and assumes that a wing can
be simulated as a rigid plate with a certain angle of attack and a lift and drag coefficient, from
which the lift and drag force and airfoil moment can be determined. The airflows around kites
are completely different from rigid wings and too turbulent to be approximated accurately
with this method [7]. More details can be found in reference [34].
The vortex lattice method is a numerical Computational Fluid Dynamics (CFD) method
that models a surface as an infinitely thin sheet of discrete vortices to compute lift and drag
forces. It gives rough approximates of these forces and is often used in the early stages of
aircraft design.
Aerodynamic derivatives are widely used in the dynamics of aircrafts. They are a fast
way to compute the aerodynamic forces based on experience and the history of the states of
the model. Details can be found in Mulder et al. [34]. This model is not applicable to flexible
models.
The Breukels model was developed by Breukels [7] and used in his multi-body kite modelling approach. He performed two dimensional CFD simulations on airfoils with different
physical properties (camber, angle of attack, thickness) and fitted functions through the results to approximate the lift, drag and aerodynamic moment curves for a two dimensional
airfoil. The aerodynamics of the kite are approximated by dividing it into a finite number
of aerodynamic sections in span-wise direction. For every section the aerodynamic forces
are determined from the fitted functions and distributed over the surface by using a set of
determined weight factors. This is the only aerodynamic model that takes the deformation of
Master of Science Thesis
H.A. Bosch
18
Kite powered energy research
the kite into account. The model has been validated to some extent, but is still a very rough
approximation of the real aerodynamic situation. This model will be discussed in Chapter 6
Computational fluid dynamics (CFD) is the most detailed analysis that can be performed
in the field of aerodynamics. The space around a structure is discretized with small volume
elements and the fluid flows and pressures are calculated using fluid dynamics theory. This
method calculates a pressure distribution of a surface over time. Computation times are low
in the two dimension case, but become very high for the three dimensional analysis. This
makes it impossible to use this method in dynamic simulations of the full kite model.
2-3
Conclusions and research opportunities
None of the discussed kite modelling approaches really satisfies, they are either too much
simplified, not representing the real deformation modes of the kite or too computational
intensive and none of them has successfully been verified
The simple models (point mass, rigid body) serve their purpose by giving a rough approximation of the kite behaviour. This is sufficient to do some preliminary optimization studies or to
design the rough outline of a controller. The models lack the needed realism to test controllers
for robustness, to see how kites behave in certain circumstances, to test new designs or to
gain more insight in the steering and dynamic behaviour.
The intermediate models (particle based, multi-plate) try to incorporate some of the flexibility
of the kites, but do that in an artificial way, neglecting the real flexibility.
The advanced models either are not able to fly and are too computational expensive to use
(finite element model) or use many artificial fitted parameters and elements so that it is
unclear how good it represents the reality (multi-body model).
It can be concluded that there is still need for better kite models that give a better representation of the highly flexible nature of the kite so that more insight can be gained in its
(steering) behaviour and better controllers and kites can be developed. New models should
have the right balance between calculation speed, amount of detail and physical validity.
Furthermore a good validation of all the currently available kite models is needed. The Kite
Power research group at the Delft University of Technology is working on this problem.
H.A. Bosch
Master of Science Thesis
Chapter 3
Thesis goal
The conclusions of Chapter 2 lead to the following formulation of the goal of this thesis. The
goal of thesis is to:
show a new realistic and reduced approach in the modelling of flying flexible
inflatable tube kites used in airborne wind energy systems.
The approach has to result in a realistic model.
• The global behaviour of the kite has to be represented realistically in the model. Local
dynamic effects like fluttering of the canopy are not important.
• A real steering input of shortening a steering line should result in a deformation of the
kite that creates a new force equilibrium and causes the roll and yaw motion. Shortening
or lengthening both steering lines should result in (de-)powering of the kite.
• The model should be derived from physical principles and not depend on artificially
introduced elements or many fitted parameters. This gives more insight in the physical
properties and behaviour of the kite, the ability to use real material properties and
naturally remains close to reality.
The approach has to result in a reduced model.
• The model should be as fast as possible to be a handy tool for controller design, performing optimizing studies and kite designers.
• The method needs to reduce the real kite to a simplified model, without losing the
essential information to capture the global dynamic behaviour.
• The model must be built so that it can be further reduced by model reduction techniques
as used in the finite element method field.
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20
Thesis goal
Two additional requirements are formulated.
• The new approach should be flexible so that it is easy to model different kite types.
• The method should not be dependent on certain computer programs, but be general
applicable.
This is an interesting and exciting research goal for several reasons.
• All the existing kite models have their shortcomings that will be solved with this approach: the model will be realistic and fast.
• This will give us more understanding of the flying and steering behaviour of a kite which
will lead to better kites designs, controllers and optimal systems.
• It will be a new tool to test new designed kites in a virtual environment before building
them.
• It poses an academically interesting challenge. An aero-elastic problem between a highly
non-linear flexible kite and an aerodynamic model coupled to a cable model of the bridles
and tether should results in a model with low computation times. The approach that
will be described in this thesis is a completely new one.
H.A. Bosch
Master of Science Thesis
Chapter 4
Approach
This chapter describes a new approach to model a kite so that it fulfils all the in Chapter 3
stated requirements. Section 4-1 discusses the several options to model the structural kite,
the aerodynamics and the tether and bridles, followed by a selection of the best options and
a full description of the modelling approach in Section 4-2. Section 4-3 describes how this
approach will be implemented.
4-1
Modelling options
This section discusses the several modelling options to model a realistic kite in Section 4-1-1,
a reduced kite in Section 4-1-1, the tether and bridles in Section 4-1-3 and the aerodynamics
in Section 4-1-4.
4-1-1
A realistic kite modelling approach
One of the main requirements states that the model has to represent the real deformations
of the kite to realistically capture the global dynamics. Only four of the described modelling
techniques in Chapter 2 include the deformations, but not all have the same level of realism.
The multi-plate model of Williams et al. [53] (option 1) and the particle model of Furey and
Harvey [17] (option 2) both introduce some degrees of freedom to include the deformation
of the kite in a very rudimentary manner. Due to the coarse discretization, the aerodynamic
forces can not be calculated very precisely. The advantage is that these models are fast,
because of their limited number of degrees of freedom, but they have two big disadvantages.
Firstly they don’t include real flexibility of the kite, because all the individual elements
are rigid and only rotate with respect to each other. This does not result in the real kite
behaviour. Secondly a large number of artificially fitted parameters is needed to mimic the
real kite properties. It is very difficult to determine those properties from measurements or
calculations, because they are not direct physical properties.
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22
Approach
The multi-body approach of Breukels [7] (option 3) captures most the global dynamic behaviour of the kite, uses a detailed aerodynamic model and is still relatively fast (20-30 slower
than real-time), but has some disadvantages. The multi-body dynamics method is primarily
used to couple rigid bodies together with constraints and not to simulate flexible structures.
The inflatable beams for instance are simulated with a series of hinged rigid elements with
rotational springs to simulate bending and torsion stiffness. This representation does not
include the axial strains. The real flexibility of the kite is thus not included. Also many
fitted parameters are needed for all the artificially introduced elements and the real physical
material properties cannot be used directly.
The finite element model of Schwoll [43] (option 4) is based on the finite element method that
is widely used to simulate structural flexible problems. The method discretizes a structure
into a large number of small finite elements and calculates the strains and stresses in all
the elements based on the real material properties. This is currently the method with the
highest accuracy for structural problems. The kite model uses a fine mesh with more than
30.000 elements to capture all the dynamic effects in the kite and even the internal pressure
of the inflatable beams is modelled in a realistic manner. However, computations can be
very slow for structures with many elements. Furthermore there is no aerodynamic model
available to produce a realistic aerodynamic pressure distribution on the canopy. The high
detail of the finite element model is useless without a correct aerodynamic model, because
the deformations will still be wrong.
Because the finite element method is specially designed to model the dynamic behaviour of
deformable bodies, it will produce the most realistic model of a kite out of the above mentioned
options and therefore fits the realistic requirement the best. All the other approaches simplify
the physical system too much. Furthermore it is a challenge to create the first flying finite
element kite.
4-1-2
A reduced kite modelling approach
The high precision model of Schwoll [43] is too detailed and computational intensive for the
new approach. This section discusses four options to reduce a high detailed finite element
model to a fast model, without losing the essential global dynamics of the kite.
Option 1: proper orthogonal decomposition with DEIM
Proper orthogonal decomposition (POD) [8, 25, 26, 38, 50] is a technique to reduce large
finite element models by finding their most important deformation modes when subjected to
their typical loads and use those modes to construct a reduction basis. The deformation of a
structure can be rebuilt from these most important deformation modes, reducing the number
of degrees of freedom from several thousands to the number of modes that are used in the
reconstruction process. The kite is probably well suited for this approach, because its shows
a finite number of well defined deformation modes when it is flying. Jellyfishing and twisting
as depicted in Figure 4-1 are the most important deformation modes that can clearly be seen
when flying a kite. The dynamic equations of a finite element model can be written in the
H.A. Bosch
Master of Science Thesis
4-1 Modelling options
23
linear and non-linear case as (this is explained in more detail in Chapter 5)
M (q)q̈ + f (q) = g(q) non-linear
M q̈ + Kq
=F
linear
(4-1)
(4-2)
where f are the internal forces, g the non-linear external forces, F the linear external forces,
M the inertia matrix, K the linear stiffness matrix and q the displacements. The displacements q can be approximated from a combination of the most important deformation modes
as
q = Ψu
(4-3)
were the matrix Ψ is an orthonormal basis and contains n vectors with the selected most
important deformations modes, called the reduction basis. u are the modal coordinates, the
reduced number of degrees of freedom that describe ’how much’ of a mode is represented
in the current deformation. The most important mode shapes can be found with the POD
method from simulation data performed with the full model subjected to its typical loads
using singular value decomposition. Combining the above formulas leads to
M (Ψu)Ψü + f (Ψu) = g(Ψu, Ψu̇) + R non-linear
M Ψü + KΨu
=F +R
linear
(4-4)
(4-5)
where the residual error R comes from the fact that not all deformation modes are used. The
projection of this error on the reduction base should be zero. This leads to the new dynamic
equations with a greatly reduced number of degrees of freedom
ΨT R = 0
T
T
T
Ψ M (Ψu)Ψü + Ψ f (Ψu) − Ψ g(Ψu) = 0 non-linear
T
T
T
Ψ M Ψü + Ψ KΨu − Ψ F
= 0 linear
(4-6)
(4-7)
(4-8)
Some remarks need to be made. Firstly the equations of the kite are non-linear. Therefore
stiffness matrix is not constant but has to be recalculated in every iteration of the NewtonRaphson method as will be explained in Chapter 5 and projected on the newly formed basis.
This is a major part of the computations and POD won’t reduce this. Krysl et al. [25] shows
that POD still can be interesting for the non-linear case when the number of modes grows
much more slowly than the number of nodes. Secondly the aerodynamic forces are non-linear
and have to be recalculated for every original degree of freedom and then projected on the
reduced basis. No speed will be gained here from the POD method. A solution could be to
use the Discrete Empirical Interpolation Method (DEIM)[9] to interpolate the external forces.
However, this method is very new and has never been used for an application like this before.
From a more practical point of view, there is no suitable finite element of the kite that could
be used for this reduction method. The model of Schwoll [43] doens’t fly because there is no
appropriate aerodynamic model and it was developed in commercial software with no option
to alter the equations of motion.
Option 2: quasi-static kite model
A flying kite can be seen as a quasi-static process. A quasi-static process happens infinitely
slow and can be seen as if it is going through a sequence of equilibrium states. The inertia of a
Master of Science Thesis
H.A. Bosch
24
Approach
Figure 4-1: Structural deformation modes of the kite [11].
kite is very small and therefore the inertia forces are very small compared to the aerodynamic
forces. This is in fact the case, because the mass of the kite is around 10kg, while the
aerodynamic force is around 4000N while flying crosswind. The behaviour of the kite is fully
determined by the aerodynamic forces. This makes the assumption valid to neglect the inertia
term in the dynamic finite element Equation 4-2 and this results in a static equation that
needs to be solved.
f (q) = g(q)
(4-9)
A distinction needs to be made between the local inertia and the global inertia. When only
the finite element model of the kite is considered without the tether and bridles, the inertia
are called the local inertia that can be neglected as was discussed above. The global inertia
of the kite are formed by a kite on a long tether and can be approximated by a point mass
on a long tether as mr 2 , which is much larger than the local moments of inertia, so those can
be omitted in the global simulation as well.
The advantage of treating the kite as a quasi-static structure is that the time-integration
algorithm doesn’t have to be used for the finite element model which saves a great amount of
computation time. The kite can be used as a force generator that exerts forces on a dynamic
model of the tether. Time integration only has to be performed on a coupled degrees of
freedom in the dynamic tether model.
The exerted forces on the bridle attachment points depend on the degrees of freedom in the
dynamic simulation. The fluid-structure-interaction problem doesn’t have to be included in
the dynamic differential equations if it is assumed that these forces remain constant during a
time-step. This means that they don’t have to be recalculated multiple times per time-step
in a dynamic solving algorithm. This will speed up the simulation with a factor 4 (depending
on the solver). This will be discussed in more detail in Chapter 8.
Option 3: black box approach
A full finite element model could be reduced by performing many simulations for various
flight conditions and trying to find a mapping between the inputs of the model (wind speed,
H.A. Bosch
Master of Science Thesis
4-1 Modelling options
25
steering input, etc.) and the outputs (forces on the bridle lines). This could be formulated
as a look-up table where the forces can be found that correspond to a certain state. This
black-box model could then be coupled to a dynamic simulation of the tether. This is not
a very flexible approach, because the whole procedure would have to be redone if the kite
model changes, but it will result in a very fast model.
Option 4: simplify the finite element mesh
The model complexity can be reduced by using a coarse mesh, because only the global dynamics are of interest and not the local dynamic effects or wrinkling of the canopy and inflatable
beams. Coarsening the mesh of the canopy is quite straightforward.
The inflatable beams can be modelled with regular beam elements that mimic the material
properties of inflatable beams, which will result in the same dynamic behaviour. Inflatable
beams have to be meshed with many small shell elements on the surface of the beam, while
a section of a normal beam can be modelled with just one element. This greatly reduces the
amount of needed elements to represent the beams and therefore speeds up the calculations.
Furthermore some simplifications of the kite can be made on parts that are not important to
model into detail, for example the connection between different parts of the kite.
4-1-3
Tether and bridles
This section discusses the options to include the tether and bridles in the simulation. The
method should be able to model the bridles of different kite types as discussed in Section 2-1-2.
Option 1: Include the tether and bridles in the finite element model
The bridles and tether can also be modelled in the finite element model as depicted left
in Figure 4-2. This can either be done by modelling them as simple spring elements or
as geometric multipoint constraints [2, 35, 24] that describe the relations between the by
the bridles constrained nodes in the finite element model. Constraints also have to be used
together with the spring elements when pulleys are used to connect multiple bridles. Some
issues arise in both approaches.
• The multipoint constraints have to be included in the dynamic equations. The stiffness
matrix will lose its symmetric properties and thereby some calculation advantages.
• The kite can not be used as a quasi-static force generator, but has to be modelled fully
dynamic.
• The resulting model has rigid body modes that introduce large rotations in the model.
The co-rotational framework has to be used in the element definition to correctly calculate the stresses and handle the rigid body modes. This method decouples the deformation modes from the rigid body modes by using different reference frames. This
would be an interesting approach, but not very practical for reasons that are discussed
Chapter 5.
Master of Science Thesis
H.A. Bosch
26
Approach
kite
kite
bridles
kite
finite element
finite element
bridles
bridles
control pod
control pod
control pod
tether
tether
tether
finite element
multi body
multi body
Figure 4-2: Three options to model the system. Model everything with finite elements (left),
model the kite with the bridles with finite elements and the tether with the pod with a dynamic
approach (middle), model the kite with finite elements and model the bridles, pod and tether
with a dynamic approach(right).
• It is difficult to use POD and DEIM in this approach.
Option 2: A finite element kite and bridle model with a dynamic tether
The tether and control pod could be modelled with a dynamic approach as discussed in
Chapter 7 and connected to a finite element model of the kite with the bridles as depicted in
the middle of Figure 4-2.
The coupling can be done in two ways. The kite with bridles can be used as a quasi-static force
generator that exerts forces on the control pod. The dynamic equations of the finite element
system can also be coupled to the dynamic equations of the tether and bridles system, forming
one large system of dynamic equations [51, 44]. This will result in asymmetric matrices and
will lose calculation advantages in the finite element equations.
Similar problems as in option 1 arise. Rotational rigid body modes are still present in the
model, although the largest rotations are now handled in the dynamic tether model which
H.A. Bosch
Master of Science Thesis
4-1 Modelling options
27
makes a co-rotational framework less important. The problem with another framework is that
the rigid body modes have to be eliminated from the equations, which is not straightforward
in non-linear analysis, because the stiffness matrix changes in every iteration.
Option 3: A finite element kite model with dynamic bridles and tether
The bridles can be modelled together with the tether in the dynamic simulation as depicted
on the right in Figure 4-2. The rigid body modes and the deformation modes are now split,
because the kite is fully constrained in its reference frame at the bridle points, removing
the necessity to use a co-rotational framework. The finite element model only has to deal
with the deformations, while the dynamic model handles the rigid body modes. A floating
reference frame is attached to the end of the bridles that rotates with the dynamic simulation
as depicted in Figure 4-3 and forms the base reference frame for the finite element model.
This method of decoupling rigid body modes and deformation modes was mentioned before
by B.Fraeijs and Veubeke [6] as the shadowing problem, because the floating frame follows
the structure as a shadow.
The coupling can again be done in two manners. Both models can be coupled together
producing one large dynamic system of equations as was explained in option 2. But the finite
element kite model can also be used as quasi-static force generator that exerts forces on the
end points of the bridles. Both models are coupled together by forces and displacements. The
dynamic model of the bridles calculates new displacements that are used as constraints in
the finite element calculations. The finite element model calculates the resulting forces on he
bridle attachment points and returns them to the dynamic bridles simulation. The forces need
to act on a mass and therefore the kite mass has to be distributed over the bridle attachment
points. The exact distribution of the mass over the bridle points is less important, because
the main dynamics are determined by the aerodynamic forces. This can be compared to a
spring-damper system with a very high damping constant and a very small mass (drag forces
and the mass of the kite), where the total dynamics of the system are mostly determined by
the damping coefficient and not by the size of the mass.
4-1-4
Aerodynamic modelling approach
It is important to use an aerodynamic model that matches the level of detail of the structural model, because the deformation of the kite fully depends on the aerodynamic forces.
Modelling the aerodynamics is very complicated and not the focus of this thesis, but an appropriate model needs to be selected to show the modelling approach. This model can always
be replaced by other models in the future.
The main criteria are that the model should be reasonably fast because the forces need to be
recalculated in every iteration step, easy to implement and give a realistic force distribution
that depends on the deformation of the kite.
Out of the available models listed in Chapter 2 only CFD and the Breukels [7] model return
a distributed load that depends on the deformation of the kite. Because CFD is much too
slow, the Breukels [7] model is the only real option. The model was verified to some extent
and produced results that were accurate enough to use in his kite modelling approach. It is
fast and easy to implement, since the description is available.
Master of Science Thesis
H.A. Bosch
28
Approach
floating kite frame
forces
displacements
Figure 4-3: A finite element kite in a floating reference frame that is attached to the end points
of the bridles in a dynamic simulation. The dynamic simulation of the bridles gives displacements
information to the finite element model that returns forces to the dynamic model.
4-2
The full model description
A combination of options from the previous sections was made that results in an approach
that fulfills all the requirements from Chapter 3 the best.
The kite will be modelled with the finite element method (option 4) and coupled to the
Breukels [7] aerodynamic model to obtain the most realistic approach, because the finite element model is based on physical properties of the system and fully incorporates the flexibility
of the kite.
This model will be reduced to a quasi-static model (option 2) with a simplified mesh (option
4) and coupled to a dynamic model of the tether and bridles (option 3) with the assumption
that the forces on the bridles remain constant within a time-step. This greatly increases the
computational speed, because the dynamic integration only has to be done for the degrees
of freedom in the dynamic model and the fluid-structure-interaction problem only has to be
solve once for each time-step, without losing the important global dynamics of the kite.
Figure 4-4 shows this approach schematically. The dynamic integration is done in the tether
and bridles model. This model calculates in every time-step the displacements of the bridle
attachment points in a floating reference frame that is the base reference frame for the finite
element model. The quasi-static fluid-structure-interaction problem between the finite element kite and the aerodynamic model is solved and returns forces to the dynamic tether and
bridles model. Those forces are exerted on the end points of the bridles.
This approach has several more advantages.
• The finite elements can be modelled with the Total Lagrangian framework.
H.A. Bosch
Master of Science Thesis
4-2 The full model description
29
• The finite element stiffness matrix remains symmetric.
• The tether and aerodynamic model can easily be changed.
• This model can even be further reduced with POD/DEIM or the blackbox approach.
• It is a flexible method that makes it easy to model a different bridle system or a kite.
The next chapters go into detail on the different parts of this approach. Chapter 5 sets up the
finite element model, followed by the aerodynamic model in Chapter 6. The dynamic tether
and bridles model will be discussed in Chapter 7. All components will be integrated in one
system in Chapter 8.
Figure 4-4: A schematic of the new modelling approach. A quasi-static finite element kite model
and an aerodynamic model form a fluid-structure-interaction problem together that is coupled to
a dynamic simulation of the tether and bridles.
Master of Science Thesis
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30
Approach
4-3
Implementation
The main criteria for the implementation are listed below.
• Speed: the result should be fast.
• Flexibility: it should be easy to alter the code and try new things.
• Independence: the model should run on every computer and not rely on certain programs.
• Interface: the model should be able to communicate easily with developed controllers,
tether models and ground station models.
Commercial finite element programs like Ansys, Abacus or Madymo can be used to build the
finite element model. These programs can interface with other programs that do the dynamic
tether and bridles calculations. The advantage of commercial software is that many element
types are available, that the calculations are optimized for speed and that the models can
easily be built in the graphical interface. The disadvantage is that the flexibility is low. There
is no option to alter the dynamic equations and try to implement new reduction algorithms.
Furthermore the model will only run on computers with the software installed. Open source
finite element programs like CalculiX or Code Aster can be used as well. These give a bit
more flexibility to alter the code, but are still not the ideal environments to play around with
new simulation techniques, because it involves profound knowledge of the program.
All the components could also be implemented in Matlab , giving more flexibility. The finite
element implementation will be slower than in commercial software, because the Matlab
language is not compiled. It has the option to compile parts of the code to (non-optimized)
C++, which will speed up the model a bit. This approach will involve much more work,
because the full finite element model has to be programmed. Fortunately parts of the code
from Tiso [47] can be used for this.
If the complete code can be written in Matlab (and compiled) it can work as a stand-alone
and can easily be coupled to different controllers that were or instance developed in Simulink
or C++.
Because flexibility is more important than speed when developing a new method, Matlab
will be used as the platform to test this new approach. If the approach turns out to be
successful, it can of course be implemented in different codes, on different platforms or be
used together with commercial finite element software.
H.A. Bosch
Master of Science Thesis
Chapter 5
Finite element kite model
This chapter describes the modelling of the kite with the finite element method. Section 5-1
briefly introduces the fundamentals of the finite element method, derives the governing equations and discusses how the non-linear behaviour of the kite is included in the model. Section 5-2 describes the modelling of the different parts of the kite: the canopy, the leading edge
and struts, the trailing edge wire, the tips, the external forces and the boundary conditions.
The solving algorithm is discussed in Section 5-3. The chapter concludes with some final
remarks in Section 5-4.
5-1
The non-linear finite element method
This section introduces the basic concept of the finite element method and the important
non-linear behaviour that the kite shows and is included in the dynamic equations.
5-1-1
Introduction to the finite element method
The finite element method is a computational technique to solve complex partial differential
equations by discretizing the domain into a finite number of finite elements. Each element
contains points at which the solution to the differential equations is calculated explicitly,
called the nodes. The unknowns are the field variables at the nodes, in the case of structural
problems as the kite the displacements. The values of the displacements can be approximated everywhere in the element via the interpolation with the shape functions. Consider
for instance the two dimensional field as depicted in Figure 5-1 with the displacement in two
directions as field variables. The field is discretized with triangular elements with three nodes.
The displacement field of an element ue (x, t) at time t can be approximated everywhere in an
element with the shape functions N e (x) = [N1 (x) N2 (x) N3 (x) N4 (x) N5 (x) N6 (x)]
and the nodal displacements of an element q e (t) = [u1 v1 u2 v2 u3 v3 ]T
ue (x, t) = N e (x)q e (t)
Master of Science Thesis
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32
Finite element kite model
u
v
u
u
v
v
Figure 5-1: A field is discretized with triangular finite elements. The displacements u and v are
the field variable. Every element has six degrees of freedom q e .
All the degrees of freedom are denoted by q. The accuracy of the obtained solution strongly
depends on the amount of used finite elements and the type of shape functions. A very
large number of different element types exist with different shapes, number of nodes, shape
functions and based on different assumptions, all suitable for different situations. The finite
element method is very powerful for calculations on complex systems. Unfortunately it can
also be a very computational expensive method for large systems with fine meshes. More on
the finite element method can be found in introductory books, like Hutton [22].
5-1-2
Equations of motion
The equations of motion inside a continuous body are described by the partial differential
equations [40]
∂
(σij + σim uj,m ) + Xj − ρüj = 0
j = 1, 2, 3
(5-2)
∂xi
and express the equilibrium of a deformed infinitesimal volume in the body. Xj are the
components of the body forces (like gravity), ρ is the mass density of the material, σ are
the stresses, xi are the three dimensional coordinates and u are the displacements in three
directions. One method to derive the discretized equations in finite elements is by writing
the potential and kinetic energies of the system in terms of the shape functions and nodal
displacements and using variational calculus to obtain the equations of motion.
The potential energy for an element with linear elastic material is expressed by
1
P(q e ) =
2
Z
εT (q e )Hε(q e )dV
(5-3)
Ve
where ε(q e ) are the stresses and H the Hooke matrix containing the linear elastic coefficients.
The strains are in general non-linear equations of the displacements.
The internal forces on the nodes of an element can be written as
f e (q e ) =
∂P
∂q e
(5-4)
The tangential stiffness matrix K t results from the second derivative of the potential energy
and depends in general on q e
∂2P
K e (q e ) =
(5-5)
∂q e ∂q e
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5-1 The non-linear finite element method
33
When small deformations are assumed ε(q e ) becomes a linear function of qe and the stiffness
matrix becomes constant.
The mass matrix M e for an element is found from the kinetic energy expression
1
T =
2
Z
ρuTe ue dV
Ve
1
= q Te
2
and can be written as
Me =
Z
Z
ρN Te N e dVq e =
Ve
ρN Te N e dV
1 T
q M eqe
2 e
(5-6)
(5-7)
Ve
This leads after assembling all the contributions from the elements to the inertia forces and
internal forces to the discretized equations of motion
M q̈ + f (q) = g(t)
(5-8)
where M is the assembled mass matrix, f the assembled internal forces and g(t) the nodal
external forces. In the linear case where the internal forces don’t depend on q, the equations
can be written as
M q̈ + Kq = g(t)
(5-9)
The direct stiffness method will be used to assemble the mass matrix, stiffness matrix and
internal force vector and is described in all introductory books on the finite element method.
The hypothesis in this thesis is that the inertia of the kite can be neglected in the local
finite element calculations due to the quasi-static assumption of the kite as explained in
Chapter 4, leaving a static problem. Furthermore the external forces, mainly coming from
the aerodynamic forces, also depend on the displacements q and the overall state of the
system X. X contains the kite velocity V k , the rotational kite velocity ω k , the velocity of
the nodes V struct , and the wind velocity V w . The external forces are explained in more detail
in Section 5-2-8. The initial configuration q 0 is prescribed, either by the solution from the
previous time step in the dynamic tether and bridles simulation or set to 0 when a single case
is solved. The boundary conditions q b contain the prescribed degrees of freedom at the bridle
attachment points, also resulting from the dynamic simulation. This altogether results in the
following static non-linear equations
(
5-1-3
f (q) = g(q, X)
q0, qb, X
prescribed
(5-10)
Non-linearities in finite elements
Most structural systems can be approximated with linear systems when the deformations
remain small. Unfortunately this isn’t the case for the kite and therefore the non-linear
equations have to be solved, which is much more computational intensive, because they have
to be solved in an iterative manner and the stiffness matrix needs to be recalculated in every
step.
Non-linearities can be recognized from a load-deflection response diagram as showed in Figure 5-2 and explained by Felippa [14]. It shows the equilibrium path of a structure that
Master of Science Thesis
H.A. Bosch
34
Finite element kite model
deforms under an applied load. Every point on the path represents a possible static equilibrium configuration of the structure. If this path is non-linear it means that the system is
non-linear. Special points can arise in the path, for instance critical points at which more
Figure 5-2: The load-deflection response diagram. Felippa [14].
equilibrium paths cross each other or when the tangent to the equilibrium path becomes
zero. Responses can get really complicated: softening, hardening, snap-through, snap-back
and buckling (bifurcation). The latter one happens often in thin structures, as is the kite.
At bifurcation points more response paths are possible, which makes it a difficult problem to
solve.
A linear system doesn’t show this behaviour. It can withstand any load and undergo any
displacement which is never realistic, but can be a good approximation around certain equilibrium configurations. However, when the configuration diverts to far from this configuration
it is necessary to look into the non-linearities.
The stiffness of a system is the tangential stiffness around a certain point of the equilibrium
path. In the linear case this is a constant, in the non-linear analysis the tangential stiffness
matrix changes along the equilibrium curve and therefore depends on the current configuration
K t (q).
In structural analysis there are four sources for non-linear behaviour. This can nicely be
seen from Figure 5-3. The figure describes the relations between the different fields in solid
continuum mechanics. In every one of these relations non-linearities can arise.
• Geometric (kinematic equations and equilibrium equations): the change in geometry is
taken into account when setting up the displacement and equilibrium equations.
• Material (constitutive equations): the material behaviour is non-linear and depends
on the current deformation, deformation history or parameters such as temperature,
pre-stress or moisture.
• Force boundary conditions: the applied forces depend on the deformation.
• Displacement boundary conditions: the applied displacements depend on the deformation of the structure.
H.A. Bosch
Master of Science Thesis
5-1 The non-linear finite element method
35
Figure 5-3: Relations between the fields in solid continuum mechanics Felippa [14].
A non-linear system has to be solved in a iterative manner, because the stiffness of the system
depends on its displacements and the other way around. During this iteration process the
stiffness matrix needs to be recalculated in every step, making solving a non-linear problem
is much more expensive than solving a linear system. More details about solving a non-linear
system are given in section 5-3.
5-1-4
Non-linearities in the kite
Three of the four mentioned sources for non-linear behaviour are present in the kite.
Geometric non-linearities are an important source of non-linearities in the kite. The
deformations of the kite are large and the stiffness changes completely when the kite changes
to a different configuration. This can clearly be seen from the canopy. The canopy is very thin
and has almost no bending stiffness in the direction of the applied out-of-plane aerodynamic
forces when it is in the undeformed state. In the linear case, this would result in very large
displacements of the canopy, because there is almost no stiffness in that direction. This is of
course not what happens. Because the canopy starts to bend, the direction of the stiffness
in the system changes and the canopy will take the load with its axial stiffness and therefore
not displace as much. This process is illustrated in Figure 5-4. These non-linearities come
from the kinematic description of the strains ε that depend non-linear on the displacements
q The difference between a linear and non-linear geometric analysis of the kite subjected to
the same load is shown in Figure 5-5.
Material non-linearities come from the inflatable leading edge and struts. The beams will
be modelled with regular beam elements as will be explained in Section 5-2-5. The material
properties of the beams become non-linear to give the beams the same behaviour as the
inflatable beams. The Young’s modulus changes depending on the deformation of the beam
and therefore the constitutive equations that describe the relations between stress and strain
are not linear anymore.
Force non-linearities come from the aerodynamic forces. The forces depend fully on the
kite displacements q and form a fluid-structure-interaction problem together with the kite.
Master of Science Thesis
H.A. Bosch
36
Finite element kite model
Faero
Faero
Fcanopy
Fcanopy
a
b
c
d
Figure 5-4: Forces an the canopy due to an aerodynamic load. a: undeformed, b: deformed, c:
linear response, d: non-linear response.
5
Linear
5
4.5
Linear
4.5
4
4
3.5
3.5
Non-linear
2.5
Non-linear
3
z (m)
z (m)
3
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
3
0
3
2
1
0
y (m)
−1
−2
2
1
0
−1
−2
−3
−3
−2
−1
0
1
2
−3
y (m)
x (m)
Figure 5-5: A geometrical linear and non-linear solution of the finite element kite model, both
subjected to the same aerodynamic loads. No scale factor was applied.
Buckling (bifurcation) is another non-linear effect seen in kites. Figure 5-6 shows an example
where the leading edge starts to buckle. These effects occur when the aerodynamic forces
get too high. Buckling of the canopy is a problem that will occur more often during the
simulations. During some flying conditions the canopy will partly be subjected to compressive
loads that will cause the canopy to buckle. This does not happen very often, because the
aerodynamic forces are most of the time directed in the outward direction. But sometimes
the tips get a negative angle of attack, inverting the directions of the aerodynamic forces,
resulting in buckling. The canopy can also start to fold in lines when the kite makes a sharp
corner. Stalling the kite will also result in buckling of the canopy. Figure 5-7 shows some
examples of this behaviour. All these buckling situations are tried to avoid, because they will
result in convergence problems.
These considerations show the necessity to do a non-linear finite element analysis of the kite.
H.A. Bosch
Master of Science Thesis
5-1 The non-linear finite element method
37
Figure 5-6: Buckling of a kite under high loading.
Figure 5-7: Some examples of buckling in the canopy in different flying conditions.
5-1-5
Mathematical framework for element definitions
Finite elements can be developed in different frameworks as described by Felippa [15]: the
Total Lagrangian, the Updated Lagrangian or the Corotational framework. Each framework
uses a different reference configuration to calculate its variables in. The finite element as
depicted in Figure 5-8a is undergoing a series of changes in load. The element starts from an
initial configuration C0 and solving the finite element equations for the different loads results
in new configurations Cn after stage n. The system drags two coordinate systems along: the
element attached system moves with the element and the node attached systems move with
the nodes. There is also a base frame that is fixed and is used to track the motion of the
elements.
In non-linear finite elements calculations there are sub-stages which the system undergoes
to reach new configuration, as shown in Figure 5-8b. To check whether one of the substages is an acceptable final solution it must be verified if it satisfies the governing finite
element equations to a certain accuracy. This needs to be done by checking the state (forces,
displacements, strains, stresses, etc.) of the new configuration with respect to a certain
reference configuration. Here a choice between the frameworks needs to be made.
1. The Total Langrangian (TL) framework: the FEM equations are formulated with respect to a fixed reference configuration that doesn’t change throughout the analysis.
Master of Science Thesis
H.A. Bosch
38
Finite element kite model
(a) A finite element going through a sequence of converged
(b) Going from one solution to a new
configurations in a stepwise manner.Felippa [15]
one, undergoing sub-stages.Felippa [15]
Figure 5-8: The solution process of a finite element.
2. The Updated Langrangian (UL) framework: the FEM equations are formulated with
respect to the last calculated accepted configuration.
3. The Corotational (CR) framework: the FEM equations make use of two systems, a fixed
reference configuration CB as in TL and a corotated reference configuration that goes
along with the element and describes the rigid body motion of CB .
All the methods have their (dis)advantages. UL is mainly used for very particular applications
such as metal forming and therefore not interesting for this project. It is more interesting to
look at the (dis)advantages of the CR framework compared to the TL framework with respect
to the modelling of a kite.
Advantages of CR compared to TL:
1. Problems with large rotations can be handled, because the stress calculation are still
correct for large rotations. In the TL model, the calculated stresses become invalid
when the rotations become too large. That the CR description can handle this, is very
advantageous for aerospace structures, thus also for a kite.
2. Decouples the material non-linearities from geometric non-linearities, which are both
present in the kite.
3. Handles anisotropic behaviour of materials without having to recalculate all the strain
directions. Ripstop material of the canopy shows this type op behaviour, although not
modelled in that way in this research.
4. Interfaces naturally with multi-body dynamics programs. So coupling with a dynamic
simulations of the tether and bridles would be easier.
5. Is well suited for the use of structural elements with rotational degrees of freedom.
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
39
Disadvantages of CR compared to TL:
1. Only small strains can be handled.
2. Can lead to unsymmetrical tangent stiffness matrices.
3. Involves difficult mathematics.
It would be nice to develop the whole model in the CR framework, because that would make it
possible to model the bridles and tether also with the finite element method. Large rotations
and displacements are no problem, because the framework decouples rigid body modes and
deformations. However, no suitable element types are available for the CR framework in
Matlab to model the canopy and developing them is not in the scope of this research. A shell
element description in the TL framework is available in Matlab from previous research done
by Tiso [47] which could be used. Furthermore there is no advantage in the CR framework
when using the proposed approach to combine a dynamic tether and bridles model with the
quasi-static finite element kite model, since the rotations of the kite will remain small in the
floating kite reference frame. Therefore it is decided to work in the TL framework. Although
for future work it might be interesting to look into developing a kite in the CR framework.
5-2
Kite modelling
This section discusses how the several parts of the kite are modelled with finite elements.
5-2-1
Modelling requirements
The main requirements for the modelling are stated as follows. The model should
• be able to capture the important global deformation modes and show the real flexibility
of the kite;
• use elements that are able to handle the geometric and material non-linearities;
• be able to handle large deformations.
• be as fast as possible without losing essential information;
• solve the fluid-structure-interaction problem between the aerodynamic forces and the
kite.
5-2-2
Kite selection
A large variety of kites exists on the market nowadays as was illustrated in Figure 2-7. The
Kite Power group from the Delft University of Technoloy has tested several of these kites
for power generation and is developing new kites specially designed for power generation.
Currently the 25m2 Mutiny kite as depicted in Figure 5-9a is used. It has a complex bridle
Master of Science Thesis
H.A. Bosch
40
Finite element kite model
system of 24 bridles with pulleys to withstand higher forces without collapsing. A different
kite will be used in this thesis: the North Rhino 16m2 as depicted in Figure 5-9b. This is a
kite with four bridles and a single curved leading edge. The kite will be used for the following
reasons.
• Since the main goal of this thesis is to propose a new modelling approach, it is more
straightforward to consider a kite that contains all the essential features and avoids
unnecessary complicated parts as a complex bridle system with pulleys and a double
curved leading edge of the Mutiny kite.
• The aerodynamic model from Breukels [7] that will be used in this research was designed
and tested for this kite. The model has some limitations that make it more difficult to
use it for kites with different shapes as will be discussed in Chapter 6.
It will be shown that the proposed modelling process can also handle more complex kites.
(a) The 25m2 Mutiny kite.
(b) The North Rhino
16m2 kite.
Figure 5-9: Two different kite types.
The North Rhino is a C-shaped kite with an inflatable leading edge and five inflatable struts.
The tips contain a thin aluminium beam and four attachment points for the bridles. The
inflatable leading edge consist of an outer shell that is made from the material Dacron an in
inside bladder (balloon) that is airtight. The trailing edge contains a thin wire to prevent
over-stretching of the trailing edge and flapping. The canopy is stitched to the top of the
leading edge and the struts and made from thin ripstop material. Appendix C shows some
pictures with details of the kite.
5-2-3
Geometric dimensioning & meshing
Geometric dimensioning
The global dimensions of the undeformed and unloaded kite are shown in Figure 5-10. Dimensioning the kite is not straightforward for several reasons. The kite has been built from
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
41
drawings, but the manufacturing is handmade and never the same as the drawings. Furthermore the shape of the kite deforms when the leading edge and the struts are filled with air,
depending on the pressure. It is also difficult to measure the kite in an unloaded state, since
it is so flexible and therefore always deformed by its own weight. The used dimensions were
based on available data from de Groot [11], but were altered on some points to produce a more
realistic kite. The leading edge and trailing edge of the kite are described by half of an ellipse,
.m
.m
.m
z
y
.m
.m
.m
x
Figure 5-10: The mesh of the North Rhino 16m2 kite.
but the span and height of the trailing edge are lower than the leading edge. This creates two
effects that are necessary for a stable, good flying kite, but this effect was not included by the
original design of Breukels [7] and de Groot [11]. Firstly it creates an initial angle of attack at
the tips and the middle section. Secondly it prevents the kite from bending too much forward
as was seen from simulations with the kite without this effect included, because the trailing
edge is shorter than the leading edge. The parameters were chosen to produce a kite with
an initial angle of attack at the tips of 6.5 deg and 12.7 deg in the middle. Kite designer use
similar values for their designs and after experimenting with different parameters, this turned
out to produce the most stable kite.
Something else that was not in the design of Breukels [7] and de Groot [11] was an initial
camber in canopy and struts. Normally struts get an initial camber for better aerodynamic
performance. This also influences the structural performance of the kite. If the struts are not
Master of Science Thesis
H.A. Bosch
42
Finite element kite model
pre-bended they will bend like a clamped beam, resulting in a negative camber. Giving the
struts an initial camber will result in better bending behaviour as can be seen in Figure 5-11a.
The canopy that is attached to the struts gets the same initial camber. More details on the
F
strut
camber
Leading edge
F Trailing edge
reality
assumption
camber
(a) Difference between bending of a strut without (top) and with (bottom) an initial camber.
camber
(b) Attachment of the canopy to the
midpoint instead of the top of the beam
results in a underestimation of the camber.
Figure 5-11: Modelling the initial camber.
dimensions of the kite can be found in Appendix C.
Meshing
The in Chapter 2 mentioned finite element kite model of Schwoll [43] contains over 30.000
elements and models the air flows in the struts and leading edge. This makes computations
very demanding and results in long cpu times. Since the goal of this thesis is to propose
a new reduced approach that only captures the global dynamics and deformations, some
simplification are made with respect to the finite element modelling.
Regular beam elements will be used to model the inflatable leading edge and struts with
similar material properties as was discussed in Chapter 4. The trailing edge wire and rods in
the tips will also be modelled with these beam elements. Shell elements will be used to model
the canopy. Details of the elements will be discussed in the next sections.
Simplifications were made in the meshing process.
• The canopy is attached to the same nodes as the struts and leading edge. This means
that the canopy is physically attached to the middle of the beams and not on top
of them. This will probably result in an underestimation of the camber as shown in
Figure 5-11b.
• The aluminium tip bar, the leading edge and the trailing edge wire are for all connected
to each other by the corner nodes. In reality this is not the case as can be seen in
Figure C-1. This poses some problems because all these elements are now coupled in
their bending behaviour. This problem is discussed in Section 5-2-7.
• The same problem occurs at the connection points of the struts, the trailing edge wire
and the canopy. They are all joined in one node, while the real situation is a bit more
complex as can be seen in Figure C-1.
• The corner nodes of the kite will be used to connect the bridles to, called the bridle
attachment points.
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
43
The main requirements for the mesh are stated as follows.
• The mesh should be as course as possible to obtain a fast model, but fine enough to
reproduce the important deformation modes of the kite.
• The aerodynamic model that will be used is based on a mesh that is divided into five
sections in the chordwise direction, so the finite element mesh will use this same division.
• The tips of the kite will be subjected to the largest deformations, are the most important
for the steering behaviour and will be subjected to buckling effects. A finer mesh will
be used for the tips to be able to catch all this behaviour and create a stable model.
• The division in span-wise direction should result in triangular elements with an aspect
ratio (height/width of the triangular elements) that is close to one to maximize the
accuracy of the elements.
A mesh generator was programmed in Matlab with some parameters that can be varied to
create the mesh as shown in Figure 5-10. This results in a model with the specifications as
given in Table 5-1.
Table 5-1: Finite element properties of the kite
Parameter
Beam elements
Shell elements
Nodes
DOF
5-2-4
Value
107
360
222
1332
Canopy
Characteristics and material properties
The canopy generates the lift and drag forces and transfers them to the beams of the kite
and acts as shear web between the struts and the leading edge and will be modelled with
triangular shell elements.
The canopy is made of very thin ripstop material and undergoes large deformations when
subjected to the aerodynamic loads. Measurements performed by EMPA materials science
and technology [33] show that the thickness of the canopy is only t = 0.08 × 10−3 m. They
also show that ripstop is an anisotropic material and therefore the stiffness depends on the
direction of the fibres. Mechanical properties based on plane stress linear elastic orhotropic
model show a stiffness of kwarp = 104.4kN/m in the warp direction and kf ill = 71.2kN/m in
the fill direction. Because modelling orthotropic material is difficult, the assumption is made
that an average of 100kN/m can be used as to estimate the canopy stiffness. This shouldn’t
have a large effect, because the kite will also fly with different materials. This results in an
Young’s modulus of
100kN/m
= 1250M P a
(5-11)
E=
0.08 × 10−3 m
Master of Science Thesis
H.A. Bosch
44
Finite element kite model
Furthermore a Possion ratio of ν = 0.3 will be used.
The ripstop material is unable to withstand compression forces and starts to wrinkle immediately when subjected to compression forces. This behaviour is not of interest in this
research and makes computations slower, but is sometimes unavoidable in simulations as was
discussed in Section 5-1-4. The canopy cannot withstand any real bending forces, but is
merely just a membrane. Pure membrane elements are not commonly used in finite element
analysis, because of their poor performance. But the bending stiffness in the element increases (decreases) with the thickness squared, while the in-plane stiffness varies linear with
the thickness. The canopy is so thin that the bending stiffness almost disappears compared
to the in-plane stiffness and the finite shell element should act in a realistic manner.
Non-linear three node triangular flat shell element
To model the canopy a three node shell element will be used that can handle the geometric
non-linear behaviour. Since the development of a geometric non-linear finite surface element is
difficult and a finite element programmed by Tiso [47] is available in Matlab , its suitability
for this thesis is investigated.
This three-node element has a membrane part for the in-plane forces and a bending part for
the out-of-plane forces.
The linear bending contribution comes from the description of Allman [3]. He describes
a triangular finite element for plate bending problems derived from the modified potential
energy principle, using a cubic displacement field. The element uses three degrees of freedom
per node, one translational and two rotational, which results in a total of 9 degrees of freedom
for the element as can be seen in Figure 5-12. The degrees of freedom of the element can be
described by
qe,bending = [w1
θx1
θy1
w2
θx2
θy2
w3
θx3
θy3 ]T
(5-12)
Figure 5-12: The triangular three node flat shell element with the membrane part (left) and the
bending part (right) as described by Tiso [47].
The linear membrane (in-plane) contribution comes from the description in Allman [4] and
Felippa [16] of the LST-3/9R element that is a member of the optimal Linear Strain Triangle (LST) family of elements, The Allmann ’88 triangle. The three corner nodes have three
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
45
degrees of freedom, two in-plane translations and one drilling rotation, as depicted in Figure 5-12. The hierarchical drilling degrees of freedom are rotational degrees of freedom that
are perpendicular to the plane of the triangle and a way to improve the performance of the element and express the higher order behaviour of the element. Cubic polynomial displacement
functions are used to describe the displacement field u and v. This element calculates the
basic and higher-order stiffness matrix and combines these as can be seen from Figure 5-13.
The degrees of freedom of the element can be described by:
q e,membrane = [u1
v1
θz1
u2
v2
θz2
u3
v3
θz3 ]T
(5-13)
The membrane and bending element can be combined in one element that is capable of
bending and in-plane deformations. This results in an element with 18 degrees of freedom:
q e = [u1
v1
w1
θx1
θy1
θz1
u2
v2
w2
θx2
θy2
θz2
u3
v3
w3
θy3 θz3 ]T
(5-14)
θx3
Figure 5-13: Hierarchical rotations due to the drilling degree of freedom Felippa [16].
The non-linear contributions come from the kinematic equations describing the axial stress ε
and curvatures χ derived from the simplified Lagrangian strain tensor and neglecting shear
effects, as described in detail by Tiso [47].




εx





εy







εxy
=
=
=
1 R
A A [u,x
1 R
A A [v,y
1 R 1
A A [ 2 (u,y
+ 12 (v,x 2 + w,x 2 )]dA
χxx =
+ 21 (u,y 2 + w,y 2 )]dA
χyy
=
+ y,x ) + 12 (w,x w,y )]dA
χxy
=
1 R
A A w,xx dA
1 R
A A
w,yy dA
(5-15)
1 R
A A w,xy dA
The element is based on isotropic linear material and the constitutive relations are described
by
σ = Am ǫ
(5-16)
with
Am
Master of Science Thesis

1 ν
Eh 
ν
1
=

1 − ν2
0 0

0

0 
1−ν
2
(5-17)
H.A. Bosch
46
Finite element kite model
The element formulation will return the elemental tangential stiffness matrix K e (q e ) and the
internal force vector f e (q e ), both depending on the element displacements q e .
Validation
The linear and non-linear version of this element have been implemented in Matlab and
compared to the SHELL63 element in Ansys in Appendix B-2. In general the results in both
cases are very similar. Only when a point load was applied, some differences were found.
This was to be expected, because a point load is a difficult load case for a plate (acting
on an infinite small area) and apparently the different elements handle this a bit different.
Figure 5-14 shows that the model is capable of handling thin canopy with an extreme low
stiffness and shows the expected deformations under an aerodynamic load. In general it can
be said that this element is suitable for its purpose and therefore will be used in the modelling
of the canopy of the kite.
Figure 5-14: The kite with extreme compliant canopy material (less stiff than the real value)
subjected to an aerodynamic load.
5-2-5
Leading edge and struts
Characteristics and material properties
The inflatable leading edge and the struts form the backbone of the kite, they provide stiffness
and transfer the lift and drag forces to the tethers. Their stiffness depends on the pressure
inside, the diameter and deflection. The outer tube is made of the material Dacron and the
inner tube is an airtight bladder from thermoplastic polyurethane (TPU). Veldman et al. [49]
discusses the modelling problems of inflatable beams that come from wrinkling behaviour,
collapses, modelling of the internal pressure field and non-linear bending. Thousands of
surface elements are needed to model inflatable beams with finite surface elements to create
a model that represents this behaviour in a realistic way. Since we are only interested in the
global behaviour the option the model the inflatable beams with regular beams with similar
material properties will be considered, thereby neglecting the wrinkling effects.
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
47
Breukels [7] performed bending and torsion experiments on inflatable beams to determine
their stiffness properties. He used 1 meter beams with different diameters and a constant
pressure and clamped them at one end as shown in Figure 5-15a. Loads were introduced at
the end of the beam while measuring the deflection at the tip. The results of this experiment
can be seen in Figure 5-15b.
(a) Set-up to measure the inflatable
(b) Results of the inflatable beam stiffness mea-
beam stiffness properties.
surements.
Figure 5-15: Experiments performed by Breukels [7] to measure the stiffness properties of inflatable beams.
The following function was fitted through the results with an R2 fit of over 0.99 percent.
h
2
3
i
Ftip = f (p, r, v) = (C1 r + C2 )p + (C3 r + C4 )
"
−
1−e
(C5 r 5 +C6 )p+(C7 r+C8 )
v
(C1 r+C2 )p2 +(C3 r 3 +C4 )
#
(5-18)
The domain where this function is valid is not completely clear, because it gives negative
stiffnesses for beams with small radii and is therefore only considered to be valid between
the radii 40mm-90mm, because all test beams were within that range. Some small radii
in the kite will therefore be increased to 40mm. The function stops being valid when the
aerodynamic forces become too high and inflatable beams collapse (see Figure 5-6). A steep
drop in beam stiffness will be seen in that case. This is outside the scope of this thesis and
will not be modelled, but this effect could be included in a later phase, because Breukels [7]
also performed studies on this topic. The non linear material behaviour can clearly be seen
from the Figure 5-15b, especially for lower internal pressures and higher deflections. These
material characteristics are converted to values that can be used for regular beam elements.
Conventional beam theory yields:
Ftip L3
3EI
Ftip L3
EI =
3v
v=
(5-19)
(5-20)
where E is the Young’s modulus, I the area moment of inertia, v the deflection and L the
length of the beam. Combining these equations yields the bending stiffness for a beam element
EI(p, v, r) =
Master of Science Thesis
Ftip (p, r, v)L3
3v
(5-21)
H.A. Bosch
48
Finite element kite model
Figure 5-16a shows this bending stiffness for a beam of 1m with a radius of 75mm to a
varying deflection. It can be seen that the bending stiffness is not constant and varies with
the deflection.
For a thin walled beam with a thickness t and round cross-sectional area the second moment
of inertia is
π(r 4 − (r − t)4 )
(5-22)
I=
4
This leads after substituting in Equations 5-21 to the Young’s modulus of a beam element
E(p, v, r) =
4Ftip (p, r, v)L3
3vr 4
(5-23)
This formula can be used to formulate the non-linear constitutive equations. The relation
between strains and stresses for a beam element depend on the current deformation. A normal
working pressure for the beams that will be used is a constant p = 0.5bar. In a first attempt to
simulate the inflatable beams with regular beam element, the non-linear material behaviour
will be approximated by assuming a fixed deflection of v = 0.03m. This will result in an
underestimation of the stiffnes for low deflections and a overestimation of the stiffness for
large deflections. This will slightly change the deflections of the whole kite, but tests with
beams with different stiffnesses show that this has a minor effect compared to the influence
of the aerodynamic forces. In a later phase the material non-linearity could be implemented.
Figure 5-16b shows how the bending stiffness of the beam varies with the radius of the beam
when the deflection is fixed at v = 0.03m.
300
700
Non−linear beam stiffness
Linear approximated beam stiffness
260
240
220
Bending stiffness EI (Nm2)
Bending stiffness EI (Nm2)
600
280
500
400
300
200
100
200
0
0.01
0.02
0.03
0.04
0.05
Deflection (m)
0.06
0.07
0.08
0
0.03
0.04
0.05
0.06
0.07
Radius (m)
0.08
0.09
(a) Beam bending stiffness varying with the
(b) Varying beam bending stiffness with
deflection for r=0.075m, t=2mm.
the radius and a deflection for v=0.03m,
t=2mm.
0.1
Figure 5-16: Stiffness of inflatable beams.
Geometric (non-)linear three dimensional two node beam element
A linear two node three dimensional Bernoulli beam without shear deflection was implemented
in Matlab and based on the description of Rixen [40].
H.A. Bosch
Master of Science Thesis
5-2 Kite modelling
49
Three dimensional geometric non linear beam elements are still a subject of research and it
is not easy to find and element that can easily be applied and coded in Matlab . Therefore
a new three dimensional beam element with two nodes was developed that is able the do
geometric non-linear analysis. The element as depicted in Figure 5-17 is based on a combined
approach of Reddy [39] and Crisfield [10] and assumes small strains, small to moderate rotations and large displacements. It is a straight element, assuming that the round leading edge
of the kite can be approximated by straight elements when the elements are small enough. The
element is basically the non-linear version of the linear beam element that was implemented
and is also based on the classic Bernoulli beam theory. It is assumed that the plane sections
in the beam remain perpendicular to the axis of the beam and rigid after a deformation of
the beam and therefore it is allowed to neglect the Poisson effect and transverse shear strains.
Furthermore a linear torsion is assumed. The non-linearity comes from the inclusion of the
in-plane forces that are proportional to the square of the rotation of the transverse normal to
the beam axis and are included in the kinematic relation between the displacements and the
strains. The axial strain ε and shear strain γ due to torsion are described by
1 dv
du
+
dx
2 dx
dφ
= ρκx
γ=ρ
dx
ε11 =
2
+
1
2
dw
dx
2
(5-24)
(5-25)
with u, v and w the displacement fields and ρ the radial distance from the axis of the beam.
The virtual work principle is used to derive the internal forces and stiffness matrix. The shape
functions for the torsion and axial displacement are taken linear and for the out of plane
displacements cubic polynomial. The full derivation of the element is given in Appendix A.
This element formulation returns the elemental tangential stiffness matrix K e (qe ) and the
internal force vector f e (qe ), both depending on the element displacements q e . The element
has a total of 12 degrees of freedom:
q Tel =
h
u1 v1 w1 ψx1 ψy1 ψz1 u2 v2 w2 ψx2 ψy2 ψz2
i
(5-26)
ψz2
ψy2
z
w2
v2
u2 ψx2
2
y
ψz1
x
ψy1
w1
v1
Z
u1 ψx1
1
Y
O
X
Figure 5-17: The geometric non-linear beam element.
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Finite element kite model
Validation
The linear and non-linear version of this element have been implemented in Matlab and were
compared to the BEAM4 element in Ansys in Appendix B-2. The Matlab beam performed
well and was slightly stiffer than the Ansys element, but besides that the results were similar.
From these tests it was concluded that the element is suitable for this project.
5-2-6
Trailing edge wire
The kite has a thin trailing edge wire stitched in the trailing edge of the canopy to provide some
extra stiffness, as showed in Figure 5-18a. This wire has two purposes. Firstly it prevents the
trailing edge from fluttering, because the wire gives the trailing edge some bending stiffness
when it is subjected to aerodynamic forces, forcing the trailing edge to stay in a nice round
shape. Secondly it prevents the struts from bending to much upward and thereby retaining
a nice aerodynamic profile and improving the aerodynamic performance.
This trailing edge wire will be added to the model by using the same beam elements that were
used for the struts and the leading edge with a radius r = 0.3mm and a Young’s modulus of
E = 100GP a (Dyneema). This simulates a thin wire with low bending stiffness, but quite
stiff in the axial direction. The trailing edge is connected to the last row of nodes that is used
to model the canopy. At some nodes the trailing edge wire is connected to both the canopy
and the end of the struts which couples the bending of the elements at those points. This
bending coupling sometimes creates unwanted side-effects, but those are avoided by keeping
the diameter of the wire small.
trailing edge wire
tip beam
Fpower
Fsteering
(a) Trailing edge wire and tip
(b) Force in
beam.
the tips due
to a steering
input.
Figure 5-18: Details of the tips, leading edge and trailing edge wire.
5-2-7
Tips
Figure C-2 and Figure 5-18a show that there is a thin, stiff beam in the tips to prevent the
tips from extreme wrinkling/folding when a steering input is applied. The beam absorbs a
part of the steering forces as depicted in Figure 5-18b.
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5-2 Kite modelling
51
This tip-beam will be modelled with the same beam elements as used to model the inflatable
beams and connected to the last row of nodes that represent the tips. The bending of the
leading edge, tip beams and trailing edge wire are now all coupled through the corner nodes.
In reality this is not the case, because there is some canopy between them that will deform
extremely. This coupled bending creates some problems. It can for instance result in an Sshaped bending of the tip. To circumvent this problem a tapered tip-beam is introduced that
has a smaller radius near the connection between the leading edge and the tip-beam to create
a more flexible connection between the tips and the leading edge that allows them to bend
more independently of each other. Since the trailing edge wire is already very thin it can’t
transfer much bending, therefore the tip-beam will only be made thinner at the connection
with the leading edge. The diameters of the tip-beam vary between 1mm at the connection
point with the leading edge to 6mm at the trailing edge side. The beams are made from
aluminium with a Young’s modulus of E = 69GP a.
The tip-beam is basically a stiff spring in its axial direction. The dynamic simulation will ’see’
this stiff spring, because the corners of the tip-beam are directly coupled to the bridles in the
dynamic tether and bridles simulation. Stiff springs result in high frequency behaviour and
can result in numerical instabilities and require very small time steps. There is no problem in
the axial direction, because the axial movement will be constraint in the dynamic tether and
bridles simulation. However, out-of-plane movements will not be constraint and the tip-beam
also acts as a spring in those directions due to its bending behaviour. Therefore the tip-beam
is modelled thinner than it is in reality, but still stiff enough to withstand the axial forces that
come from the steering bridles. This reduces the bending stiffness of the beam and therefore
removes the high-frequency behaviour.
5-2-8
External forces
The forces that act on the kite come from the aerodynamics, the structural damping, the
bridles, the gravity and lateral and rotational accelerations. These forces are described in
this section and all included in the external force vector g(q, X) as discussed in Section 5-1-2.
Aerodynamic forces
The most important and dominant applied external forces come from the aerodynamics. The
used model of Breukels [7] will be discussed in detail in Chapter 6 and returns a distributed
load F aero for the kite based on the displacements of the kite q, the velocity of the nodes
V struct in the structure, the velocity of the kite V k , the rotational velocity of the kite ω k and
the wind V w velocity.
F aero = f (q, V struct , V k , ω k , V w )
(5-27)
An example of the distributed forces is given in Figure 5-19.
Gravity forces
The gravitational forces are very small compared to the aerodynamic forces and adding them
would have almost no effect on the deformation of the kite. The kite mass is in the order of
Master of Science Thesis
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52
Finite element kite model
Figure 5-19: Distributed load from the aerodynamic model acting on the kite.
10kg and the generated lift forces in the order of 4000N when flying crosswind, resulting in a
ratio of 1/40 between them. The mass of the canopy is so low, that the gravity won’t have
any effect on its deformation. The mass of the beams is a bit higher, but they are also stiffer,
so the gravity has no effect on their deformation either.
The gravity does have influence on the global dynamics of the kite system. For instance, the
obtained velocities when flying down are significantly higher than when flying up. Therefore
the gravity of the kite will be incorporated in the dynamic tether and bridles model by
including mass at the bridle attachment points (corner points of the kite) as will be discussed
in Chapter 7.
Acceleration forces
Due to the quasi-static assumption of the kite as discussed in Chapter 4, the mass matrix M
is neglected in the finite element equations and therefore also the accompanying acceleration
forces. Other forces come from the lateral and rotational accelerations of the floating kite
reference frame in the dynamic tether and bridles simulation. Results in Chapter 9 show
that the maximum lateral speed of the kite is 30m/s and the maximum accelerations is 5
m/s2 . Resulting in a lateral acceleration force of 50N with a 10 kilo kite on a 100m line. The
maximum force due to rotational acceleration becomes
Fcentrif ugal =
10 ∗ 302
mv 2
=
= 90N
r
100
(5-28)
Both forces are small compared to the aerodynamic forces that are in the order of 4000N and
will therefore be neglected.
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5-3 Solving the non-linear equations
53
Damping forces
The finite element method solves only static problems, due to the quasi-static assumption
and therefore no damping forces are included. But damping turns out to be quite important
to create a stable model. The proposed approach will result in a sequence of static solutions
of the deformation of the kite, representing its dynamic behaviour. No applied damping
will result in oscillations in the deformations and the prescribed displacements in the bridle
attachment points sometimes result in sudden ’jumps’ of the kite from one configuration to
a completely different one with a total collapse as result.
In reality the transient movements of the kite damp out partly due to the internal structural
damping, but mostly because of the aerodynamic drag forces as will be discussed in Chapter 6.
Rayleight like damping can be introduced to add some structural damping. A damping matrix
C can be introduced that depends on the mass matrix M and the stiffness matrix K and
the coefficients µ and λ. Since de mass matrix is not available, only the stiffness matrix term
will be used.
C = µM + λK
(5-29)
The damping force is now determined by
F damp,struct = CV struct
(5-30)
The structural damping forces will be seen as part of the external forces and not separately
added to the equations of motions by writing the term C q̇ which would normally be done.
This is because q̇ and V struct are not the same. q̇ would be calculated from a finite element
time stepping scheme which is not used and V struct is determined numerically from the current
displacements q n and the displacements at the previous time step qn−1 divided by the time
step h.
V struct =
q n + q n−1
h
(5-31)
(5-32)
This numerical calculation from subsequent time steps always lacks one step behind, sometimes resulting in instabilities in the model if the coefficient λ is chosen too high. The aerodynamic damping also results in unstable models as will be discussed in Chapter 6. Therefore
some additional numerical damping is added between the bridle attachment points in the
dynamic tether and bridles model as will be discussed in Chapter 7.
5-2-9
Boundary conditions
The displacements of the four corner nodes of the kite, the bridle attachment points, are
prescribed by the dynamic tether and bridles simulation and will be denoted by qb .
5-3
Solving the non-linear equations
An appropriate solution procedure is developed in this section to solve the non-linear algebraic
finite element equations. The full solution procedure is depicted in Figure 5-24. Section 5-3-1
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Finite element kite model
discusses the basic Newton-Raphson method to solve the structural non-linear equations.
In Section 5-3-2 the solution procedure to solve the aeroelastic fluid-structure-interaction
problem between the aerodynamic forces and the structural model is discussed. A load
control algorithm is developed in Section 5-3-3 to improve the stability and convergence of
the algorithm, followed by a discussion about the convergence criteria in Section 5-3-4. This
section won’t explain how the linear finite element equations are solved, this can be found in
any introductory book on the finite elements method.
5-3-1
Structural solution procedure
The non-linear equations of the quasi-static kite that have to be solved were described by
Equation 5-10 an repeated here for convenience.
(
f (q) = g(q, X)
q0, qb, X
prescribed
(5-33)
f (q) are the internal forces and g(q, X) are the external forces that depend on the displacements q and the prescribed state of the system X consisting of the kite velocity V k , the
rotational kite velocity ω k , the velocity of the structural nodes V struct and the wind velocity
V w . The initial displacements q 0 are prescribed, either by the solution from the previous
time step in the coupled simulation with the dynamic tether and bridles model or set to
q 0 = 0 when a single case is be solved. The boundary displacements q b are prescribed by the
dynamic tether and bridles model or set by hand to solve a single case.
The Newton-Raphson iteration scheme [40, 39, 14] can be used to solve these non-linear
equations. This method is well suited for problems without difficulties in the equilibrium path
(load-deflection curve) like multiple solutions, snap-through or buckling. During normal flight
conditions these phenomena are not likely to occur very often, but they are also unavoidable
as was described in Section 5-1-4.
The Newton-Raphson method is a member of the Newton-like solving algorithms with several
variants for specific situations. The basic algorithm makes a prediction of the solution in the
prediction phase, followed by an iterative correction phase to iterate to a balance between
the internal and external forces to obtain the static solution to Equation 5-33.
First a prediction is made by saying that the new displacements q n+1 are the same as the
displacements at the end of the last time step in the dynamic tether and bridles model q n
q n = q n−1
(5-34)
Solving the static equations has nothing to do with the time tn , the time integration is handled
by the dynamic tether and bridles model. However solving the finite element equations will
result in the configuration at the next time step n + 1. For clarity the subscript n will be
omitted in the rest of this section and the newly calculated displacements are denoted by q.
In the corrective phase, the residual force r of a configuration q is formulated by
r(q) = f (q) − g(q)
(5-35)
The algorithm tries to find a balance between the internal and external forces to make the
residual zero r = 0. The prescribed boundary conditions q b will automatically be taken into
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Master of Science Thesis
5-3 Solving the non-linear equations
55
account here, since they cause internal forces as well.
Let us denote qk as an approximate value for q resulting from structural iteration k. In the
neighbourhood of this configuration, the residual can be estimated by the linear equation
r L (q k+1 ) = r L (q k + ∆q k ) ≃ r(q k ) + S(q k )∆q k = 0
with the iteration matrix
S=
∂r
∂q
=
qk
(5-36)
∂g
∂f
−
∂q
∂q
(5-37)
∂g
describes how the external forces vary with the displacements. However, the
The matrix ∂q
gradient information of the aerodynamic forces is not available and the resulting matrix would
not be symmetric. Because the solution will also converge without this matrix and to preserve
the symmetry of the iteration matrix, this matrix is omitted.
The matrix ∂f
∂q represents the tangential stiffness matrix K t which is to be recalculated for
every iteration.
The equation can now be solved iteratively for the free degrees of freedom in q by correcting
the solution q at every iteration step with ∆q k as shown in Figure 5-20.
K t (q k )∆q k = −r(q k )
q
k+1
k
= q + ∆q
(5-38)
k
(5-39)
The iteration process stops when the following criteria is satisfied which says that the residual
should be small enough compared to the internal forces.
r(q k )
kf (q k )k
< εf em
(5-40)
The final step is to obtain the boundary forces at the bridle attachment points, because those
are exerted on the endpoints of the bridles in the dynamic tether and bridles model. The
external forces at the bridle attachment nodes F b are equal to the internal forces at the
boundary nodes f b and simply become
F b = fb
(5-41)
This altogether leads to Algorithm 1 and corresponds to the blue part in Figure 5-24.
5-3-2
Aero-elastic solution procedure
The external forces g and structural displacements q both depend on each other and form a
coupled aero-elastic fluid-structure-interaction problem together.
In Algorithm 1 the solution procedure starts from a certain initial configuration q 0 and calculates the external forces for those displacements g(q 0 , X). The external forces are updated
in every iteration step with the new displacements g(q k+1 , X).
This algorithm will result in an unstable system because the aerodynamic forces are very
sensitive to changes in the displacements of the kite. In the first iteration steps, going from an
Master of Science Thesis
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Finite element kite model
Load
g
K2
K1
r
f
K0
q
∆q
q
q
qsolution Deflection
Figure 5-20: The Newton-Raphson method.
undeformed to a deformed kite, the stiffness of the canopy in the direction of the aerodynamic
forces is so low that the displacements become very large resulting in very high (unrealistic)
aerodynamic forces. This results in a very slow convergence or even un unstable system.
Therefore the iteration algorithm is split into two loops: a structural convergence loop (blue)
and a aerodynamic convergence loop (green) as depicted in Figure 5-24. The external forces
are kept fixed during the structural Newton-Raphson iterations. After obtaining the new
structural converged solution the new external forces are calculated and a new structural
convergence iteration starts. q k denotes an approximate value of the displacements q after
structural convergence step k and q̃m denotes the approximation of q after the aerodynamic
convergence step m. g m are the external forces in aerodynamic iteration step m.
The aerodynamic iterations stop when the external forces and displacements don’t change any
more. Two functions are defined that calculate the difference in configuration and external
forces after a new aerodynamic iteration as
r a,1 = q̃m+1 − q̃ m
r a,2 = g(q̃ m+1 , X) − g(q̃ m , X)
(5-42)
(5-43)
r a,1 and r a,2 should be small enough compared to respectively the displacement vector and
external forces to consider the solution to be converged.
kr a,1 k
< εaero,1
ca,1 = q̃ m+1 kr a,2 k
< εaero,2
ca,2 = g(q̃ m+1 )
(5-44)
(5-45)
This creates a very rudimentary optimizing algorithm that can end up in a situation where it
’jumps’ between two configurations, as illustrated in Figure 5-21. The initial displacements
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5-3 Solving the non-linear equations
57
Algorithm 1 Basic Newton-Raphson
Initialize structural iteration k = 0, boundary conditions q b and state vector X from the
dynamic simulation
Predictor q k = q n−1
Calculate internal and external forces
g(q k ), f (q k , X)
Calculate forces residual
r(q k ) = f (q k ) − g(q k , X)
krk
while kf
k > ε do
Corrector
k
k
∆q k = −K −1
t (q )r(q )
k+1
k
k
q
= q + ∆q
Recalculate internal and external forces
g(q k+1 , X), f (q k+1 )
Recalculate forces residual
r(q k+1 ) = f (q k+1 ) − g(q k+1 , X)
k =k+1
end while
F b = fb
q̃ 0 produce the aerodynamic force g 0 . The new calculated displacements after the structural
iteration process are q̃ 1 and result in the new forces g 1 . These forces bring the displacements
after structural convergence to q̃ 2 , which are almost the same displacements as q̃ 0 . This will
never, or very slowly, converge to the solution point (q̃ solution , g solution ).
A simple algorithm is introduced to solve this problem and speed-up the convergence. The
algorithm incrementally increases the aerodynamic forces into the new direction. It updates
the old external forces g m in the direction of the new calculated external forces (g(q̃ m+1 ) −
g(q̃ m )) with the parameter α to obtain the external forces in the next aerodynamic iteration
step g m+1
g m+1 = gm + α(g(q̃ m+1 ) − g(q̃ m ))
(5-46)
The parameter α is updated after every aerodynamic iteration. If the two residuals become
smaller, the iteration was successful and α is increased, but if the residuals increase the
iteration was not successful and α is decreased and the new obtained solution q̃ m+1 will not
be used and the iteration is redone with a smaller α parameter. This altogether leads to
Algorithm 2 and is represented by the green block in Figure 5-24.
The advantage of this split between structural and aerodynamic convergence is that the
solution method becomes more robust. There is less chance that the procedure won’t converge
Master of Science Thesis
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Finite element kite model
α= 1
g
g0
jump
α= 0.8
g1
gsolution
q
˜0
q2
˜
α= 0.1
gsolution
gsolution
q1
qsolution ˜
˜
Figure 5-21: Aerodynamic convergence problem. The algorithm can get stuck between two
static solutions (left). The α parameter was introduced to walk stepwise to the final solution.
since errors in the structural convergence won’t lead to larger errors in the aerodynamic model
that cause the solution procedure to become unstable.
5-3-3
Load control, stability and convergence procedures
A load control algorithm is implemented to speed up the convergence and improve the stability. The system won’t converge without this load control algorithm.
The problem is depicted in Figure 5-22. The displacements in the first structural iteration
step q 1 become very large when an undeformed kite is suddenly exposed to the aerodynamic
forces. The canopy has no stiffness in the out of plane direction, but the axial stiffness will
withstand the aerodynamic forces when the canopy deforms. This principle was illustrated
in Figure 5-4. The first iteration step largely overshoots the solution, resulting in a very
unrealistic large displacement of the canopy. This process will lead to a very bad convergence
or no convergence at all.
This phenomenon won’t occur as much in normal flight conditions when the kite model is
coupled to the dynamic tether and bridles models, because the solution process in every time
step starts from an initially deformed kite and results in a slightly differently deformed kite.
The initial tangential stiffness matrix has already stiffness developed in the direction of the
aerodynamic forces. The problems occurs mainly in the start-up phase of the full simulation,
or when studying single cases or when sudden large changes in flight conditions occur.
The performance can be improved by applying load control as described by Felippa [14]. The
load is increased in a stepwise manner, allowing the structure to develop stiffness in the outof-plane direction due to the bending of the canopy. This also prevents the solution process
from getting stuck in a certain configuration. When the convergence fails, the subsequent
results of the increments can be viewed by an engineer to gain more insight in the structural
behaviour to see where the problem is. This method improves the robustness of the process
and increases the speed. Even though smaller load steps have to be taken, less iterations per
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Master of Science Thesis
5-3 Solving the non-linear equations
59
Algorithm 2 Newton-Raphson with aero-elasticity
Initialize aerodynamic iteration m = 0, boundary conditions q b and state vector X, initial
configuration q̃ m = q n−1
Calculate external forces and aerodynamic convergence criteria
g m = g(q̃ m , X)
m
cm
a,1 = ca,2 = inf
α = αinit
m
while cm
a,1 > ε1 AND ca,2 > ε2 do
Initialize structural iteration k = 0
Predictor q k = q̃m
Calculate the internal forces and residual
f (q k )
r(q k ) = f (q k ) − g m
krk
while kf
k > ε3 do
Corrector
k
k
∆q k = −K −1
t (q )r(q )
k+1
k
k
q
= q + ∆q
Recalculate internal forces and residual
f (q k+1 )
r(q k+1 ) = f (q k+1 ) − g m
k =k+1
end while
q̃ m+1 = q k
Recalculate external forces and aerodynamic convergence criteria
g(q̃ m+1 , X)
kq̃m+1 −q̃m k
cm
a,1 =
kq̃m+1 k
kg(q̃ m+1 ,X)−g(q̃ m ,X)k
cm
a,2 =
kg(q̃ m+1 )k
m+1
m+1
m
if ca,1 >= cm
a,1 OR ca,2 >= ca,2 then
α =decrease α
g m+1 = gm−1 + α(g(q̃ m ) − g(q̃ m−1 ))
q̃ m+1 = q̃ m
else
α =increase α
g m+1 = gm + α(g(q̃ m+1 ) − g(q̃ m ))
end if
m=m+1
end while
F b = fb
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Finite element kite model
Load
f1
r
K0
g
qsolution
q1
Deflection
Figure 5-22: Slow convergence of the Newton-Raphson method for the kite. The undeformed
canopy of the kite has a very low stiffness in the direction of the aerodynamic forces, resulting in
an slow convergence process.
load step are needed and the configuration won’t jump between two configurations. An smart
algorithm that continuously adjusts the load step-size also speeds up the convergence.
A natural way to perform the load stepping is to slowly increase the inputs to the aerodynamic
model from the old inputs to the new inputs. Chapter 6 describes that the inputs are the
displacements q, the wind velocity V w , the kite velocity V k , the rotational kite velocity
ω k and the velocity of the nodes (structural speed) V struct all combined in the vector X.
The prescribed boundary displacements q b are also taken into account in the load control
algorithm, because they also cause a part of the load.
The inputs are step-wise increased by multiplying them by a pseudo-time factor λs at pseudotime step s that is 0 of no new load is applied and 1 if the full new load is applied. X s and
q b,s denote the values at pseudo-time step s and X 0 and q 0b are the values at the previous
time-step from the dynamic tether and bridles model and X 1 and q1b are the new inputs from
the dynamic model. The inputs at each pseudo-time-step can be written as
X s = X 0 + λs (X 1 − X 0 )
q b,s =
q 0b
+
λs (q 1b
−
q 0b )
(5-47)
(5-48)
This means that the inputs to the aerodynamic model and prescribed displacements slowly
increase from the last known situation (results from the previous time step in the dynamic
tether model) to the new situation.
A guess for the initial pseudo-time step ∆λs has to be made. If the initial step size is chosen
too low, time is lost by making too many small steps. If the initial step size is chosen too
large, time is lost in restarting the iteration process with a smaller step size. A very small
pseudo-time step size ∆λs has to be chosen when the initial configuration is undeformed.
When coupled with the dynamic simulation, an initial guess of the step size is based on the
step-size used in the precious dynamic time step.
H.A. Bosch
Master of Science Thesis
5-3 Solving the non-linear equations
61
A new step size can be chosen after each pseudo-time-step s, based on the needed number
of structural k and aerodynamic m iterations. This has to be done efficiently to obtain
the highest simulation speed. When the solution converged very fast, the step size can be
increased and vice versa. When the solution doesn’t converge within N1 iterations the process
is restarted with a smaller time step. The new step-size ∆λs+1 is determined by
µλ = (a + b ∗ l)c + d
(5-49)
∆λs+1 = µλ ∆λs
(5-50)
where µλ is a multiplier, l the total number of structural iterations and aerodynamic iterations
needed in one pseudo time step and a,b,c,d coefficients to be chosen.
The values of a,b,c,d are chosen based on the stability and stiffness of the used kite model.
Stiffer models can use larger time-steps. The characteristics as displayed in Figure 5-23
are used in this thesis and were obtained from experiments with the model. In normal
flight conditions, most time steps can be solved in just one pseudo-time step, because the
deformations will only slightly change. Therefore the influence of the multiplier µλ on the
overall speed of the simulation is low. The implementation of the algorithm improved the
robustness of the kite model very much.
5
Mulitplier µλ
4
3
2
1
0
0
5
10
15
20
Total number of iterations l
25
Figure 5-23: The characteristics of the pseudo time step multiplier µλ for the coefficients
a=9.998, b=2.220, c=-1.031, d=3.982.
This altogether leads to Algorithm 3 and is graphically shown in Figure 5-24 in the orange
part. q̄ s denotes an approximation of q after load step s.
5-3-4
Convergence criteria
There are three convergence criteria in the convergence process. One for the structural convergence, as shown in Equation 5-40 and two for the aerodynamic convergence as shown in
Equations 5-44. Stricter criteria lead in general to longer simulation times, because more
Master of Science Thesis
H.A. Bosch
62
Finite element kite model
Algorithm 3 Newton-Raphson with aero-elasticity and load control
Initialize boundary conditions q 1b , q 0b and state vector X 1 , X 0 , initial configuration q̄ s = q n−1 ,
pseudo time-step s = 1, pseudo time λs = ∆λs
while λs <= 1 AND λs−1 6= 1 do
Initialize iteration counters l = 0, m = 0
Calculate external forces and aerodynamic convergence criteria
X s = X 0 + λs (X 1 − X 0 )
q b,s = q 0b + λs (q 1b − q 0b )
gm = g(q̄ s , X s )
q̃ m = q̄ s
m
cm
a,1 = ca,2 = inf
α = αinit
m
while cm
a,1 > ε1 AND ca,2 > ε2 do
Initialize structural iteration k = 0
Predictor q k = q̃ m
Calculate the internal forces and residual
f (q k )
r(q k ) = f (q k ) − gm
krk
while kf
k > ε3 do
Corrector
k
k
∆q k = −K −1
t (q )r(q )
k+1
k
k
q
= q + ∆q
Recalculate internal forces and residual
f (q k+1 )
r(q k+1 ) = f (q k+1 ) − gm
k = k + 1, l = l + 1
end while
q̃ m+1 = q k
Recalculate external forces and aerodynamic convergence criteria
g(q̃ m+1 , X s )
kq̃m+1 −q̃ m k
cm
a,1 =
kq̃m+1 k
g(
q̃ m+1 ,X s )−g(q̃m ,X s )k
k
cm
a,2 =
kg(q̃ m+1 )k
m+1
m
m
if ca,1 >= ca,1 OR cm+1
a,2 >= ca,2 then
α =decrease α
g m+1 = gm−1 + α(g(q̃ m ) − g(q̃ m−1 ))
q̃ m+1 = q̃ m
else
α =increase α
g m+1 = gm + α(g(q̃ m+1 ) − g(q̃ m ))
end if
m = m + 1, l = l + 1
end while
q̄ s+1 = q̃ m
Update pseudo time λs according to Algorithm 4
s=s+1
end while
F b = fb
H.A. Bosch
Master of Science Thesis
5-3 Solving the non-linear equations
63
Figure 5-24: Graphical representation of the finite element solution procedure.
Master of Science Thesis
H.A. Bosch
64
Finite element kite model
Algorithm 4 Pseudo time updating algorithm
Update pseudo time
if l == N1 then
Not converged, go one step back with a smaller pseudo time-step
s=s−1
∆λs+1 = ∆λs /2
else
Increase the pseudo time-step
µλ = (a + b ∗ l)c + d
∆λs+1 = µλ ∆λs
end if
if ∆λs+1 > ∆λmax then
∆λs+1 = ∆λmax
end if
if λs + ∆λs+1 > 1 then
∆λs+1 = 1 − λs
λs+1 = 1
else
λs+1 = λs + ∆λs+1
end if
iterations are needed. The criteria can be set rather low in this research, because we are not
interested in high accuracies, but too small criteria can lead to instabilities.
Another advantage of using low criteria, is that it allows the simulation to proceed when it gets
slow because it tries so solve some small buckling/wrinkling issues in the canopy, behaviour
that is not of our interest. By setting the convergence criteria low enough, the convergence
algorithm just steps over these problem without a complete convergence.
The speed of the convergence is further increased by using lower convergence criteria for the
intermediate pseudo time steps and use strict criteria only for the last pseudo time step.
The values that were chosen to work well for this problem are:
ǫf em,end = 1 × 10−3
(5-51)
−2
(5-52)
−2
(5-53)
−5
ǫaero1,intermediate = 5 × 10
(5-54)
−2
(5-55)
ǫaero2,intermediate = 2 × 10−1
(5-56)
ǫf em,intermediate = 5 × 10
ǫaero1,end = 1 × 10
ǫaero2,end = 5 × 10
If the algorithm exceeds a certain number of maximum iterations or if the pseudo time-step
becomes too small ∆λs < 1e−5 the simulation stops.
H.A. Bosch
Master of Science Thesis
5-4 Concluding remarks
5-4
65
Concluding remarks
This chapter described the finite element modelling of a flexible kite. The non-linearities
of the kite were discussed and it was shown that it is necessary to included them in the
derivation of the non-linear quasi-static finite element equations. The canopy was coarsely
meshed with triangular shell elements and the inflatable beams with regular beam elements
with representative material properties. Also the specific modelling of difficult areas as the
tips and leading edge wire were discussed. This resulted in a reduced finite element model
with a minimum amount of degrees of freedom that is able to show the global behaviour of
the kite.
A solving algorithm was introduced to efficiently solve the aero-elastic fluid-structure-interaction
problem by separating the solving process in a structural convergence and aerodynamic converge loop. A load control algorithm was developed to make the solution process more robust
and speed up the convergence.
Problems arise when difficult behaviour as buckling/folding/wrinkling of the canopy occurs.
This behaviour is not of our interest but unavoidable in some flight conditions. The solving
algorithm was adjusted to deal with these situations as good as possible, although the model
will still fail when the deformations or buckling gets too large.
Introducing the correct damping in the model poses some problems. The damping from the
aerodynamic model results in instabilities as will be discussed in the next Chapter. Too much
structural damping also results in instabilities, because the speeds of the degrees of freedom
are not explicitly calculated in the finite element solution process due to the quasi-static
assumption but need to be determined from the previous time-steps in the dynamic tether
and bridles model.
The implementation of the elements has been verified and shows accurate results. Verification
of the implementation of the whole kite will be further discussed in Chapter 9. It will be shown
that the model performs well and shows the non-linear global deformations realistically.
Master of Science Thesis
H.A. Bosch
66
Finite element kite model
H.A. Bosch
Master of Science Thesis
Chapter 6
Aerodynamic model
The aerodynamic model that will be implemented in this thesis is the model developed by
Breukels [7] as was discussed in Chapter 4. This chapter describes the basic idea of the model
in Section 6-1, followed by the implementation in the finite element kite model in Section 6-2.
Section 6-3 verifies the results of the implementation followed by some final conclusions in
Section 6-4.
6-1
6-1-1
Model description
Aerodynamic forces
The Breukels [7] model is based on the assumption that the aerodynamics of the kite can
be approximated by seeing the kite as an assembly of a finite number of connected two
dimensional single membrane airfoils in the spanwise direction as depicted in Figure 6-1,
called the wing sections. The aerodynamic load for each wing section can be determined
separately, depending on their shape, angle of attack and velocity. The airflows around the
wing sections influence each other, because of the three dimensional nature of the kite. These
(probably significant) effects are neglected at first and afterwards reintroduced by applying
some corrections to the calculated aerodynamic forces, depending on the spanwise position of
the wing section. The shape deformation of a wing section of the kite is determined by the
following parameters, as depicted in Figure 6-2.
• The length of the chord c, measured from leading edge to trailing edge.
• The camber κ = hc is the maximum distance h from the canopy to the chord divided by
the length of the chord c.
• The thickness t =
D
L
is the diameter of the leading edge divided by the chord length c.
• The width W of a wing section.
Master of Science Thesis
H.A. Bosch
68
Aerodynamic model
D wing sections
Figure 6-1: The aerodynamic properties of the kite can be determined by the assumption that
the kite consists of a finite number of two dimensional single membrane airfoils.
The angle of attack determines the state of the kite.
• The angle of attack α is the angle between the chord and the apparent wind speed in
the XZ-plane of a wing section.
L
L
L
L
L
L
h
D
z
α
y
Vw,a
c
W
x
Figure 6-2: The camber κ = h/c, thickness t = D/c, chord length c and angle of attack α of a
two dimensional wing section.
These parameters (κ,t,α) were used as inputs to perform a large number of CFD simulations
with the program Fluent on a two dimensional airfoil for various values of the inputs. The
analysis was performed on three sets of airfoils with a 15%, 20% and 25% thickness. The
airfoils within each set ranged in camber from 0% to 12%, and the angle of attack ranged
from 0 degrees to 25 degrees. The obtained pressure distributions for each set of parameters
were compressed to a lift coefficient CL , a drag coefficient CD and an airfoil moment CM . A
fitting procedure resulted in three functions with 60 constants to map κ, t, α → CL , CD , CM .
The values of the constants can be found in Breukels [7].
CL,f it = λ1 (κ, t)α3 + λ2 (κ, t)α2 + λ3 (κ, t)α + λ4 (κ, t)
2
(6-1)
CD,f it = λ5 (κ, t)α + λ6 (κ, t)
(6-2)
Cm,f it = λ7 (κ, t)α + λ8 (κ, t)
(6-3)
(6-4)
H.A. Bosch
Master of Science Thesis
6-1 Model description
69
These functions are only valid for −20deg < α < 20deg. Highly turbulent flows arise for
higher angles of attack and the steady CFD analysis does not produce reliable data in that
domain. Empirical formulas are used to describe the aerodynamic coefficients CL and CD for
−180deg < α < −20deg and 20deg < α < 180deg, based on Spierenburg [45]. Breukels [7]
does not use an empirical formula for the airfoil moment CM for high angles of attack but
uses the fitted function for the whole domain.
CL,highα = 2 cos(α) sin2 (α)
3
CD,highα = 2 sin (α)
(6-5)
(6-6)
For numerical reasons the formulas for a low and high angle of attack are combined using a
smooth transition function σ. The transition takes place around the switching angle αsw .
σ1 =
σ2 =
1
1+
e−(α−αsw )
1
e−(α+αsw )
1+
CL = (σ2 − σ1 )CL,f it + (1 − σ2 + σ1 )CL,highα
CD = (σ2 − σ1 )CL,f it + (1 − σ2 + σ1 )CL,highα
(6-7)
(6-8)
(6-9)
(6-10)
This approach leads to the aerodynamic curves as depicted in Figure 6-3. The curves show
some peculiarities. The lift coefficient is relatively high compared to normal aircraft wings
and the drag does not always increase with an increasing camber. Furthermore the moment
coefficient curve is linear while the CFD results from Breukels [7] show non linear behaviour.
Unfortunately the original data is not available to recheck the fitting procedure and these
curves have to be used. The lift force, drag force and airfoil moment for each wing segment
can be determined by
1
2
FL = CL Sρair Vw,a
2
1
2
FD = CD Sρair Vw,a
2
1
2
M = CM Scρair Vw,a
2
(6-11)
(6-12)
(6-13)
where S = Wwingsection · c is the wetted surface determined by the width of the wing section
and the length of the airfoil, Vw,a the apparent wind speed in the XZ-plane and ρair the
density of the air.
6-1-2
From forces to a distributed load
To distribute the aerodynamic forces over the surface of the kite, each aerodynamic wing
section is divided in a finite number of chordwise subsections by n nodes as showed in Figure 6-4. The total lift and drag force will be distributed over these nodes using the constant
weight factors wi while maintaining the same airfoil moment. However, these weight factors
are not constant in changing flight conditions, but vary with the angle of attack, camber and
thickness. The factors ui are introduced to include this variation. By introducing the variable
a, the total weight factors can be expressed by using one variable a. This idea is illustrated
in Figure 6-4.
Master of Science Thesis
H.A. Bosch
70
Aerodynamic model
Aerodynamic coefficient curves
2
CL
1
0
−1
−2
−30
−20
−10
0
10
Angle of attack (s)
20
30
−20
−10
0
10
Angle of attack (s)
20
30
−20
−10
0
10
Angle of attack (s)
20
30
CD
0.4
0.2
0
−30
0.1
C
M
0.05
0
−0.05
−0.1
−30
5% camber
10% camber
15% camber
Figure 6-3: The lift coefficient CL , drag coefficient CD and airfoil moment coefficient CM .
The total forces L perpendicular to the chord and D parallel to the chord of a wing section
can now be expressed with the two requirements as
L=
D=
n
X
L(wi + ui a)
(6-14)
i=1
n
X
D(wi + ui a)
(6-15)
n
X
wi = 1
(6-16)
ui = 0
(6-17)
i=1
i=1
n
X
i=1
H.A. Bosch
Master of Science Thesis
6-1 Model description
71
w
ua
w
ua
w
ua
w
ua
ua
w
c
c
c
c
c
w ua
c
Figure 6-4: A schematic representation of the airfoil model with the distances ci and weight
factors wi + ui a
A variation of a doesn’t change the sum of the aerodynamic forces, but only changes the
distribution and therefore only the airfoil moment. The airfoil moment is used to find the
parameter a and is taken around the quarter chord point of the airfoil and can be expressed
as
n
M=
X
i=1
L(wi + ui a)(0.25c − ci )
(6-18)
The moments produced by the forces D parallel to the chord are not taken into account,
because the force and moment arm of D are small compared to L.
The parameter a can be solved from the previous equations as
a=
M − L(0.25c −
−L
n
P
n
P
wi c)
i=1
(6-19)
ui ci
i=1
The last step in the procedure is to determine the constant factors wi and ui . An infinite
number of combinations is possible. The values were obtained by Breukels [7] by comparing
the from this procedure resulting aerodynamic forces with the original CFD data. Figure 6-5
shows a comparison between the CFD data and the fitted procedure for an airfoil with 5
nodes, a thickness of 15%, a camber of 4% and an angle of attack of 10.2 degrees.
The values that are used in this thesis are given in Table 6-1.
Table 6-1: Weight coefficients for ui and wi .
i
wi
ui
Master of Science Thesis
1
0.5
1
2
0.3
2
3
0.1
-2
4
0.05
-0.5
5
0.03
0.5
6
0.02
0
H.A. Bosch
72
Aerodynamic model
Figure 6-5: A comparison of lift fractions for the approximated and direct CFD based values for
an airfoil with a thickness of 15%, a camber of 4% and an angle of attack of 10.2 degrees [7].
6-1-3
Three dimensional correction
So far, this analysis has not included any three dimensional aerodynamic effects that are
present in the kite. The aerodynamics of a kite change quite drastically due to effects as
span wise flow and tip vortices. Breukels [7] included these effects in his model by analysing
the variation of the lift coefficient along the span of the kite with the Tornado vortex-lattice
method. The results of this analysis are used to correct the in Section 6-1-1 calculated
aerodynamic coefficients for each wing section.
This three dimensional correction factor has not been included in the scope of this thesis.
The basic algorithm provides enough information to test the new kite modelling approach.
Discussions with the author of the model led to the conclusion that the uncertainty of the three
dimensional correction is so significant that it is not clear whether this correction actually
improves the model or not.
The end of the tips of the kite don’t produce any lift forces, because of the vortexes that
occur. Therefore no forces are applied to the end of the tips.
6-2
Implementation in the FE model
The aerodynamic model was specifically developed for the ADAMS kite model [7] and cannot
directly be implemented in the finite element model. This section describes how this is done.
6-2-1
Definitions
In chord wise direction, the wing sections are divided in five chord sections, because the
weight factors as defined in Table 6-1 are only available for a wing section with six nodes.
The original CFD data from [7] is not available to fit weight factors for more nodes. An
H.A. Bosch
Master of Science Thesis
6-2 Implementation in the FE model
73
interpolation could be made to extend the amount of nodes, but for now this number works
quite well with the coarse mesh that was used in the finite element model.
In span wise direction, the wing sections are defined such that they are shifted half the width
of the canopy elements relative to those finite elements as depicted in Figure 6-6, so that the
mid-line of the wing sections overlaps with a row of finite element nodes. The midpoints of
the deflected finite beam elements XA , XB in the leading edge are used as the end points of
a wing section.
XTE
wing section
c
deformed
κ
ez
ex
XA,
beam A
ΨW S
ey
XLE
undeformed
XA,
XB,node
=finite element node
XB,
z
XB,
beam B
XB,node
ΨK
y
x
Figure 6-6: The aerodynamic panels are shifted half the width of a canopy element w.r.t the
finite elements. The corners of the panel are defined by the mid points of the deformed beam
elements XA , XB of the leading edge. A straight line between those points is used to build the
reference frame ΨW S .
The mid points of the undeformed finite beam elements i in the leading edge can be calculated
K
from the average of the two beam nodes X K
i,node1 and X i,node2 in the kite coordinate frame
ΨK .
K
XK
i,node1 + X i,node2
(6-20)
XK
=
i,0
2
K of the deformed beams i can be calculated from the shape functions using
The midpoints Xi,1
the coordinate ξ = 0 that represents the midpoint of the beam, as defined in Appendix A. The
vectors u, v, w represent the nodal displacements in the local coordinate frame of the beam
element Ψl as explained in Appendix A. T K
l is the rotation matrix to rotate the displacements
Master of Science Thesis
H.A. Bosch
74
Aerodynamic model
from the local element frame to the global finite element frame.
XK
i,1


N u (ξ)uT

K
K
= X i,0 + T l  N v (ξ)v T 
N w (ξ)wT
(6-21)
The wing section coordinate frame ΨW S is defined in the global finite element kite frame ΨK
as follows.
• The y-axis points from the midpoint on finite element beam A to the midpoint on finite
element beam B.
X beamB − X beamA
eywing =
(6-22)
kX beamB − X beamA k
• A temporary vector is constructed from the node at the leading edge to the node at the
trailing edge.
X T E − X LE
(6-23)
etemp
wing =
kX T E − X LE k
• The z-axis is defined as perpendicular to the y-axis and temporary axis:
y
ezwing = etemp
wing × ewing
(6-24)
• The x-axis is defined as perpendicular to the y-axis and z-axis:
exwing = eywing × ezwing
6-2-2
(6-25)
Aerodynamic parameters for each wing section
The angle of attack α, camber κ and thickness t need to be determined for every wing section
to calculate the lift, drag an moment coefficient.
The angle of attack for each wing section is determined from the apparent velocity Va of the
wing section (opposite of the apparent wind speed), and consists of four components.
• The first part comes from the global speed of the kite, expressed by the velocity of the
finite element kite frame ΨK . This is determined from a translational component V K
k ,
K
K
a rotational component ω k and the wind speed V w .
K
K
K
K
VK
a,global = V k − V w + ω k × X LE
(6-26)
• The second part comes from the local velocity of the wing section, expressed by the
translational and rotational velocity of the frame ΨW S in the finite element frame ΨK .
Since the kite is considered to be quasi-static, these velocity terms are not available,
because only the static solution to the finite element problem is solved. However from
two subsequent time steps n and n − 1 in the dynamic simulation, the velocity of all
the degrees of freedom q can be determined in a numerical way by
VK
struct =
H.A. Bosch
q̃ s − q n−1
h
(6-27)
Master of Science Thesis
6-2 Implementation in the FE model
75
where q̃ s represent the nodal displacements after structural convergence during the
finite element solution process in time step n and qn−1 the nodal displacements at the
previous dynamic time step and h the time step size. V K
node,i will thus be updated every
time when the new aerodynamic forces are calculated during the solution procedure,
using the latest known displacements. The local translational velocity of a wing section
is now approximated by averaging the speed of the node at the leading edge V K
XLE and
trailing edge V K
.
This
term
turns
out
to
be
quite
important
for
the
aerodynamic
XT E
damping, as will be discussed in Section 6-2-4
VK
a,local =
K
VK
XT E + V XLE
2
(6-28)
The local rotational velocity of a wing section can not be taken into account. A local
rotation of a wing section results in different angles of attack within one wing section.
Since the aerodynamic model can only produce one set of aerodynamic coefficients for
a wing section, this effect can not be included.
The total apparent speed of a wing section becomes
K
K
VK
a = V a,global + V a,local
(6-29)
The angle of attack can be calculated by projecting the apparent velocity on the XZ-plane of
W S into the wing
the wing section frame ΨW S by first rotating it with the rotation matrix RK
section frame ΨW S and then calculating α from the z- and x-components. This is illustrated
in Figure 6-7.
S
WS K
VW
= RK
Va
a
(6-30)
WS
S
α = atan2(V W
a,z , V a,x )
(6-31)
With this projection, the y-velocity component is neglected. This comes from the two dimensional assumption of the airfoils. This velocity component is low compared to the x and z
components and shouldn’t not cause any problems.
The length of the chord follows from
c = kX T E − X LE k + DLE /2
(6-32)
with DLE the mean diameter of the leading edge for the wing section.
The thickness is expressed by
t=
DLE
c
(6-33)
The camber is calculated by projecting the vectors that point from from all the i nodes within
a chord to the leading edge node X LE on the z axis of the wing section frame. The maximum
of these values is the parameter h from which the camber is calculated by
κ = max
ezwing · (X i − X LE )
c
!
(6-34)
All the parameters to calculate CL , CD and CM are now defined.
Master of Science Thesis
H.A. Bosch
76
Aerodynamic model
L
FL
z
FD
x
α
D
c
VaWS
Figure 6-7: The angle of attack α is the angle between the apparent velocity of the wing section
S
VW
and the chord c.
a
6-2-3
Aerodynamic forces
The L and D force for each wing section can be determined from Equation 6-11 and are
projected on the x and z-axis of the wing section frame as depicted in Figure 6-7.
L = cos(α)FL + sin(α)FD
(6-35)
D = − sin(α)FL + cos(α)FD
(6-36)
L is used by the in Section 6-1-2 described method to reconstruct a distributed load. The
last step is to rotate the forces in the global kite frame ΨK so that they can be used in the
finite element analysis.
6-2-4
Aerodynamic damping
Both the finite element kite model and the aerodynamic model are quasi-static. The aerodynamic forces only depend on the instantaneous angle of attack α and not on the change
in angle of attack α̇ in time. When these two are coupled to the dynamic tether model, the
finite element kite model will produce a series of quasi-static solutions of the deformation in
time. This movement will normally be damping by the aerodynamic drag forces due to a
local increased or decreased angle of attack, but these are local velocities in the finite element
kite frame and not taken into account due to the quasi-static assumption. This would result
in an oscillating kite in the coupled dynamic simulation.
To simulate this aerodynamic damping either damping can be introduced in the dynamic
tether model between the bridles attachment points as explained in Chapter 7 or the local
speeds of the wing sections need to be approximated numerically to influence the angle of
attack as was done in Eq. 6-28.
H.A. Bosch
Master of Science Thesis
6-3 Verification
77
The latter method is more natural, because it represents what physically happens in the kite.
However, it was not possible to create a stable model with only this aerodynamic damping
for several reasons. The difference in nodal displacements between two time steps can be
quite large in a small time step, especially when something difficult such as buckling of the
canopy happens. This creates very high speeds in the direction perpendicular to the canopy,
resulting in large angles of attacks. The aerodynamic model did not perform very well for
these very high angles of attack and results in instabilities. Furthermore the aerodynamic
model was just not designed to include these effects. Oscillations in the kite result in vortices,
a lower lift coefficient, different drag forces and other difficult behaviour. The damping effect
was included, but only allowed to produce a small amount of damping. Additional damping
between the bridle attachment points in the dynamic tether model was required to stabilize
the model. A better aerodynamic model is needed to include these effects more realistically.
6-3
Verification
The aerodynamic model was first tested on a simple plate with three chord sections as depicted
in Figure 6-8. The used material is the same as the canopy material. Tests were performed
with different apparent velocities and directions of the panel. The model performs very well
for angles within the range 1deg < α < 20deg, which is the main flying domain of the kite.
Results look, depending on the angle of attack, as depicted in Figure 6-8.
Problems occur and force distributions become less realistic when the angle of attack approaches zero or becomes very large. An example is given in Figure 6-9a where the plate has
an angle of attack close to zero and the forces point towards different directions. In reality
this is also a complex situation, because fluttering will occur when the angle of attack is zero.
Another example is given in Figure 6-9b where the angle of attack is around 50 degrees. The
force profile starts to change significantly and is probably not very realistic any more.
The problem lies in the definition of the aerodynamic model. The distributed forces are the
result of the chosen weight factors, the CM coefficient and the parameter a. The model tries
to balance the lift forces to obtain a certain airfoil moment. The parameter a in Equation 6-19
can become very large (positive or negative) to obtain a certain airfoil moment and this results
in forces that can point in opposite directions. Also the force distribution cannot vary freely,
but is predefined to some extent by choosing the u weight factors. Furthermore according to
Breukels [7], the calculated moment coefficient is not very reliable and he also experimented
with different CM characteristics in his models, but left it unclear which one was used in
his final model. This together, makes the validity of the distribution over the canopy very
questionable.
A practical solution to solve the numerical problems that arise is limiting the a-value or
changing the weight factors. The latter one is difficult, since the original CFD data is not
available and no comparison can be made to see if the new values are realistic. Limiting the
a value means that the airfoil doesn’t always have the correct airfoil moment, but this will
not have a large effect and will therefore be implemented. Furthermore a smooth transition
function was added to vary a from zero to the calculated value between −1deg < α < 1deg.
The aerodynamic model was tested on the kite for several apparent wind velocities, with the
translational degrees of freedom of the four corners fixed. Figure 6-10 shows an example for
Master of Science Thesis
H.A. Bosch
78
Aerodynamic model
Va
Figure 6-8: An aerodynamic panel with an apparent velocity Va to test the aerodynamic model.
The elements have the same properties as the canopy in the kite.
a wind speed of 30m/s and angle of attack varying from 16 degrees at the tips to 10 degrees
in the middle. The several test cases showed that the canopy nicely gets an increased camber
and shows realistic behaviour.
(a) The test panel around α=0
(b) The test panel around α=50deg
Figure 6-9: The aerodynamic model gives unrealistic force distributions for very high and very
low angles of attack. This is an effect from the method that tries to distribute the lift forces over
the chord to obtain a certain airfoil moment.
H.A. Bosch
Master of Science Thesis
6-4 Concluding remarks
79
Figure 6-10: An example of the aerodynamic model applied to the kite, hinged at the four
corners. Both deformed and undeformed kites are displayed in all figures. The right picture shows
the kite with the canopy a factor 10 less stiff to show the effects of the aerodynamic forces.
6-4
Concluding remarks
From the tests it can be concluded that the model suits the purpose of this thesis. It represents
the general aerodynamics and returns a distributed load that can be used by the finite element
kite model. Some remarks have to be made though.
• The approach taken by Breukels [7] is a bit odd. First very detailed information is
obtained from the CFD simulations. This information is then compressed to three aerodynamic coefficients and again expanded to forces on a five nodes. A lot of information
is thrown away in this process. It would be better to use the CFD information directly
in the approach. This would make it easier to use the model for different mesh sizes of
the finite element kite, without having to redetermine all the weight factors.
• The validity of the model is questionable for multiple reasons. The coefficient curves
show some peculiarities. The apparent speed is projected on the two dimensional shape
of the airfoil and therefore ignoring one velocity component. The parameter a needs to
be limited to prevent to model from giving lift forces that work in opposite directions
w.r.t the chord, which leads to wrong airfoil moments. The obtained airfoil moment
coefficient line is questionable and Breukels [7] uses different ones, but is not clear about
that. Furthermore the three dimensional correction factor was not implemented due to
the uncertainty whether it improves the model.
• The model returns unrealistic pressure distributions around an angle of attack of α = 0
and for very large angle of attacks.
• The model becomes unstable when it is used to generate aerodynamic damping by
incorporating local velocities of the nodes in the calculation of the angle of attack.
• The validity of the kite model completely depends on the validity of the aerodynamic
model. A better aerodynamic model would therefore give more reliable results of the
total simulation. This model acts at the moment as a limitation to the total validity.
Master of Science Thesis
H.A. Bosch
80
H.A. Bosch
Aerodynamic model
Master of Science Thesis
Chapter 7
Dynamic tether and bridles model
This chapter describes the dynamic model that simulates the tether and bridles. Section 7-1
explains the modelling choices that were made, followed by the definition of the degrees
of freedom and reference fames in Section 7-2. All the forces that act on the system are
described in Section 7-3 and the full equations of motion are derived in Section 7-4 with the
initial conditions in section 7-5. The description of the inputs for the finite element model
that follow from this dynamic simulation are given in Section 7-6. The chapter concludes
with some final remarks in Section 7-7.
7-1
Modelling choices
The system to be modelled consists of a tether and four bridles: two power lines connected
to the leading edge at points A and C and two steering lines connected to the trailing edge at
point B and D as can be seen in Figure 7-1. The four bridles are connected to the control pod
that is attached to the end of the tether. The quasi-static finite element kite model exerts
forces at the four bridle attachment points. The focus of this thesis is not to develop new
cable model, therefore a model is selected that is fast and represents the global behaviour
accurate enough. This results in the model as shown in Figure 7-2. This Section discusses
the modelling choices that were made.
7-1-1
Bridles
The four bridles need to be able to shorten and lengthen to give the kite the freedom to
deform. The forces and lengths of the four tethers change quite significantly when cornering
a kite as shown by Breukels [7]. Therefore using rigid elements to model the bridles wouldn’t
be sufficient and sometimes result in pressure forces on the four tether attachment points.
This is not realistic and causes unwanted buckling effects in the canopy that are difficult
to solve for the finite element model. The next simplest option is to model them as linear
spring-dampers, which allows lengthening and shorting and gives a nice way to control the
Master of Science Thesis
H.A. Bosch
82
Dynamic tether and bridles model
z
y
A
rA rB
x
B
C D
rC r
D
z
control pod P
y
x
Figure 7-1: Set-up for the dynamic tether and bridles model.
length of the steering lines by shifting their undeformed length. The sag in the bridles can be
neglected, because the bridle lines are short, have a low mass and a small diameter resulting
in low drag forces. This allows the assumption that the forces in the bridles act in a straight
line. Concluding, a massless straight spring will be used to model the bridles.
7-1-2
Tether
The tether is much longer, thicker and heavier than the bridles, resulting in more sag and
influence on the kite dynamics. Observations from previous simulations [7, 5] show that the
sag can be neglected when flying crosswind up to 100m. Breukels [7] also shows that a kite
on a 100m line with a surface in the order of 10 square meters and a tether in the order of
2mm thickness has a ratio between the tether tension and the tether drag of only 0.02 when
flying crosswind, which means that the sag of the cable and its weight can be neglected. The
sag is also much lower while flying crosswind, than when the kite idles at its zenith position.
Since the line length in this research will stay relatively short, the kite is flying crosswind and
the main interest is the kite behaviour, the sag of the tether is neglected as well. Flexibility
of the tether is also less important to the dynamics of interest and therefore a straight rigid
link is used to connect the pod with a tether to the ground.
7-1-3
Bridle attachment points
The kite exerts forces on four masses that are attached to the end points of the bridles to
represent the mass of the kite. Most of the dynamic behaviour of the system comes from
the aerodynamic drag and lift forces and therefore the exact mass of the bridle points is
less important. It is like a spring-damper system with a very high damping constant where
most of the dynamic behaviour comes from the damper and the mass of the system is less
important. Most of the mass is located at the leading edge side of the kite, therefore the
H.A. Bosch
Master of Science Thesis
7-1 Modelling choices
83
FA
Fg
FB F
C F
D
Fdamp,kite
Fdrag, tether
Fbridles
z
FAF
BF
C
FD
y
P
x
Figure 7-2: Set-up for the dynamic tether and bridles model with the force components.
power lines get 70% and the steering lines 30% of the total mass of the kite, resulting in the
masses mA = mC = 3.3kg and mB = mD = 1.7kg.
7-1-4
Connections
In theory these should be all the necessary components to simulate the tether and bridles,
but for numerical reasons some additional elements are added. First of all, there is a stiff
aluminium bar in the tips of the kite that acts as a very stiff spring between points A-B and
C-D. This creates numerical stability problems and forces a time integration algorithm to
take very small time steps. This drastically increases the total simulation time, because the
fluid-structure-interaction problem needs to be solved for every time step. Therefore a rigid
connection is introduced between A-B and C-D in the dynamic simulation. This decreases
the flexibility of the kite a little bit, but this is no problem since the displacement in this
direction would not be more than a millimetre. An additional effect is introduced, because
the aluminium bar in the finite element simulation will bend, but is not allowed to shorten
any more and now creates axial forces in the constraint direction. These forces are also
applied to the end points of the bridles in the dynamic simulation, but shouldn’t cause any
problems since they act in the constraint direction and counteract each other. This constraint
also decreases the degrees of freedoms, which speeds up the simulation time. Secondly four
Master of Science Thesis
H.A. Bosch
84
Dynamic tether and bridles model
dampers between the bridle attachment points are added as depicted in Figure 7-10, to provide
some extra numerical damping as was discussed in Chapter 6.
7-2
Coordinates and reference frames
7-2-1
Coordinates
The system has twelve degrees of freedom and can be described by the same number of
generalized coordinates. This is preferable above more variables and the use of constraints,
since it results in a faster time integration. Furthermore the time integration solvers in
Matlab cannot deal with constraints and it is convenient to be able to use them to easily
test the effect of different algorithms. The generalized coordinates y are given by
h
y = rA
αA
βA
αB
βB
rC
αC
βC
αD
βD
αP
βP
i
(7-1)
where extrinsic Euler angles are used to describe a rotation about the space fixed z-axis with
α and a rotation about the space fixed y-axis with β and ri to describe the length of bridle A
and C as shown in Figure 7-3 for the bridle connected to point A [41]. The coordinates of the
four bridle attachment points of the kite A, B, C, D and the control pod P can be expressed
in these coordinates by
βB
βA
z
βP
rA
Ltip
xB
αB
xA
z
αA
xP
Ltether
y
R
y
E
K
z
x
ΨK
y
αP
ΨE
x
Figure 7-3: Generalized coordinates to describe
the position of bridle attachment points A and B
and the control pod P. α are the rotations about
he fixed z-axis and β the rotations about the fixed
y-axis.
H.A. Bosch
RE K
ΨE
x
Figure 7-4: The earth reference
frame ΨE , the floating kite reference
E
frame ΨK and rotation matrices RK
K
and RE to convert between them.
Master of Science Thesis
7-2 Coordinates and reference frames
85


0
 
xP = Rz (αP )Ry (βP )  0 
rP
(7-2)


0
 
xA = xP + Rz (αA )Ry (βA )  0 
rA

(7-3)

0


xB = xA + Rz (αB )Ry (βB )  0 
Ltip


0
 
xC = xP + Rz (αC )Ry (βC )  0 
rC

(7-4)
(7-5)

0


xD = xC + Rz (αD )Ry (βD )  0 
Ltip
(7-6)
(7-7)
where Rx , Ry and Rz are the rotation matrices


(7-8)


(7-9)


(7-10)
1
0
0


Rx (γ) = 0 cos(γ) − sin(γ)
0 sin(γ) cos(γ)
cos(β) 0 sin(β)


0
1
0 
Ry (β) = 
− sin(β) 0 cos(β)
cos(α) − sin(α) 0


Rz (α) =  sin(α) cos(α) 0
0
0
1
These coordinates can be combined in the vector x that contains the coordinates of all the
masses on which the forces work that are described in the next sections.
h
x = xTA xTB xTC
7-2-2
xTD xTP
iT
(7-11)
Reference frames
The simulation has two important reference frames as depicted in Figure 7-4. The first one is
the earth reference frame ΨE and the second one the floating kite reference frame ΨK . This
kite reference frame is attached to the kite and used to communicate between the finite element
kite model and the dynamic tether model. All the information that is shared between the two
simulations will be represented in ΨK : wind speed, kite speed, forces and the displacements
of the attachment points of the kite. Vectors represented in the kite reference frame get a
superscript K and vectors represented in the earth reference frame the superscript E.
Master of Science Thesis
H.A. Bosch
86
Dynamic tether and bridles model
The position and orientation of ΨK is recalculated for every time step as follows and as shown
in Figure 7-6.
E
E
• XE
0 is defined as the mid point between point xA and xC in ΨE
XE
0 =
E
xE
A + xC
2
E
E
• XE
1 is defined as the mid point between point xB and xD in ΨE
XE
1 =
• The y-axis of ΨK becomes
E
xE
B + xD
2
E
xE
A − X0 eE
y,K = E
xA − X E
0
• A vector that lies in the xy-plane is
• The z-axis of ΨK becomes
• The x-axis of ΨK becomes
E
XE
1 − X0 eE
=
temp
E
X 1 − X E
0
E
eE
temp × ey,K
eE
=
z,K
E
etemp × eE
y,K eE
x,K
eE
× eE
y,K
z,K
=
E
ey,K × eE
z,K h
iT
• The unit vectors of ΨE are given by eE
x,E = 1 0 0
h
0 0 1
iT
h
iT
, eE
y,E = 0 1 0
and eE
z,E =
The rotation matrix to rotate vectors from ΨK to ΨE is given by definition as
E
RK
E
E
E
E
E
eE
x,K · ex,E ey,K · ex,E ez,K · ex,E

E
E
E
E
E 
= eE
x,K · ey,E ey,K · ey,E ez,K · ey,E 
E
E
E
E
E
ex,K · ez,E ey,K · ez,E ez,K · eE
z,E


(7-12)
The used rotation order is yaw(ψ)-pitch(θ)-roll(φ) as is shown in Figure 7-5. This leads to the
same rotation matrix is in Equation 7-12 from which the three Euler angles can be calculated.
E
RK
= Rz (ψ)Ry (θ)Rx (φ)
= Rψ Rθ Rφ
H.A. Bosch
(7-13)
(7-14)
Master of Science Thesis
7-2 Coordinates and reference frames
87
B
X
z
ez
yaw, ψ
x
y
A
ey
D
ex
ΨK
X
roll,φ
C
pitch, θ
Figure 7-6: The calculation of
the kite reference frame ΨK from
the four bridle attachment points
A, B, C, D.
Figure 7-5: Rotations of the kite frame.
Besides using Euler angles to express the orientation of the kite and compare it to other kite
models, four Euler parameters λ0 , λ1 , λ2 , λ3 are used in the dynamic calculations to avoid
singularity problems that arise when using Euler angles as explained by Schwab [42].
λ0 = cos(µ/2)


λ1
 
λ = λ2  = sin(µ/2)ĥ
λ3
(7-15)
where µ is a rotation about a certain axis ĥ.
The rotation matrix is given by
E
= I + 2λ̃λ̃ + 2λ0 λ̃
RK
(7-16)
with λ̃ defined as the skew symmetric matrix


0
−λ3 λ2


0
−λ1 
λ̃ =  λ3
−λ2 λ1
0
(7-17)
The Euler parameters can be determined by comparing Equations 7-12 and Equation 7-16.
1q E
E
E
RK 11 + RK
22 + RK 33
2
1
E
(RE − RK
λ1 =
23 )
4λ0 K 32
1
E
λ2 =
(RE − RK
31 )
4λ0 K 13
1
E
(RE − RK
λ3 =
12 )
4λ0 K 21
λ0 =
Master of Science Thesis
(7-18)
(7-19)
(7-20)
(7-21)
H.A. Bosch
88
Dynamic tether and bridles model
7-3
Forces
This section describes all the forces that act on the tether and bridles and come from the kite,
the flexible behaviour of the tethers, the drag of the tethers, the artificial damping between
the tether attachment points of the kite and the gravity.
7-3-1
Kite forces
The finite element kite model returns four force vectors f K
kite,i at the bridle attachment points
represented in the kite reference frame ΨK that are applied to the four point masses. These
can be translated to ΨE by
E K
fE
kite,i = RK f kite,i
for i = A,B,C,D
(7-22)
The total force vector that acts on the masses of the dynamic system becomes
h
T
E
fE
kite = f kite,A
7-3-2
fE
kite,B
T
fE
kite,C
T
fE
kite,D
T
0T
iT
(7-23)
Bridle springs
Figure 7-7 shows a two dimensional perspective of the bridles that are modelled as linear
springs.
ckite
xA
xC
xA
Ltip
rA
kpower
cpower
lA,
rA
rC
xB
rB
ksteering
csteering
lB,
xP
Figure 7-7: Modelling of the bridles connecting the control pod P to the bridle attachment
points A, B, C, D in a front view (left) and side view (right) perspective.
The direction vectors of the springs can be expressed by
ei =
H.A. Bosch
xi − xP
kxi − xP k
for i = A,B,C,D
(7-24)
Master of Science Thesis
7-3 Forces
89
The speed of the change in length ri of the springs can be expressed by
ṙi =
∂ri
ẏ
∂y
for i = A,B,C,D
(7-25)
where ri can be calculated by
ri = kxi − xP k
for i=A,B,C,D
(7-26)
(7-27)
The spring forces that act on the four bridle attachment point masses can be written for all
the bridles (i = A, B, C, D) as
f bridle,i =
(
(cbridle,i ṙi + kbridle,i (ri − ri,0 ))ei
0ei
for ri >= ri,0
for ri < ri,0
(7-28)
where cbridle,i is the damping constant and kbridle,i the spring stiffness. The springs can’t exert
any forces when their length is shorter than de undeformed spring length ri,0 since cables are
not able to do this. The total force that acts on the control pod is the sum of all the bridle
forces working in the opposite direction
f bridle,P = −f bridle,A − f bridle,B − f bridle,C − f bridle,D
(7-29)
The stiffness of the bridles is given by
kbridle,i =
EA
ri
(7-30)
with E the Young’s modulus of the cable and A the cross sectional area.
The damping constant is defined by
p
cbridle,i = 2ζbridle mi kbridle
(7-31)
where ζbridle is the relative damping and mi the mass of the connected bridle attachment
point. This shows that short cables are stiffer and have less damping than long cables.
The total force that needs to be applied to the masses becomes
h
f bridle = f bridle,A T
f bridle,B T
f bridle,C T
f bridle,D T
f bridle,P T
iT
(7-32)
The bridle stiffness kbridle,i will be approximated by making it constant. Small variations
in bridle length due to steering movements are in the order of centimetres and will not
have a significant effect on the stiffness. The Young’s modulus of Dyneema cables that are
often used for flying kites is Edyneema = 100GP a and the radius of the bridles that will be
used is rbridle = 0.006m. Measured damping values of Dyneema show a relative damping
of ζbridle = 0.2, but a value of 0.8 will be used instead. This is higher, but low damping
causes oscillations that sometimes result in difficult configurations of the finite element kite
model leading to convergence problems, especially during the initialization phase. The bridle
properties with a length of 25m bridle are shown in Table 7-1. The force in a power line can
go up to 2000N, which would result in a 0.4m elongation of the line which is well within the
limits of the Dyneema cable strength that can go up to an elongation of 4%.
Master of Science Thesis
H.A. Bosch
90
Dynamic tether and bridles model
Table 7-1: Properties of the bridles
Parameter
kbridle
cbridle
7-3-3
Value
4.52×103 N/m
251.7 Ns/m
Tether and bridles drag
Drag forces are exerted on the tether and bridles while sweeping through the air. The longer
and thicker the cable, the more significant these forces become. The drag forces are important
to create a stable model and they can also be used to add some artificial damping to the model
to create a numerically stable system. Drag force on a cable can be defined as [7, 54]
fdrag,cable =
1
CD ρair Ld kV a k2
2
(7-33)
where ρair = 1.225kg/m3 is the air density, CD the drag coefficient, L the length of the
cable, d the diameter of the cable and V a the apparent wind velocity. The drag force can
be split in a tangential and perpendicular component with respect to the cable as can be
seen in Figure 7-8. The tangential component has a drag coefficient of CD = 0.0017 and the
perpendicular drag coefficient is CD = 1.065 according to Hoerner [20]. Since the tangential
part is small compared to the perpendicular part, it is neglected in this simulation.
Vk
mi
Vi
Vw
mi
FD
mi
FD
ri
Va,⊥
VP
ei
P
Figure 7-8: Speed and drag forces of a
cable.
P
Figure 7-9: Speeds and drag forces along
a cable.
The speed along the cable has a linear profile and the drag force increases quadratically with
the speed along the length of the cable as is depicted for the bridles in Figure 7-9. For this
research it is accurate enough to calculate the resultant drag force from the average apparent
speed of the bridle, which is halfway between the mass point mi and the control pod P as is
depicted in Figure 7-8.
1
(7-34)
V a = (V i + V P ) − V w
2
The part of this speed perpendicular to the cable is
V a,⊥ = V a − (V a · ei )ei
H.A. Bosch
(7-35)
Master of Science Thesis
7-3 Forces
91
with ei the unit vector in the direction from the control P to masspoint i.
The direction vector for the drag force becomes
edrag,cable = −
V a,⊥
kV a,⊥ k
(7-36)
The drag force seen by mass i is now half of the calculated drag force, this results in drag
force for a bridle applied at the endpoint of
1
f drag,cable,i = CD ρair ri d kV a,⊥ k2 edrag,cable
4
for i = A,B,C,D
(7-37)
Following the same reasoning the drag force on the tether can be calculated. The total force
that needs to be applied to the four masses becomes
h
f drag,cable = f drag,bridle,A T
7-3-4
f drag,bridle,B T
f drag,bridle,C T
f drag,bridle,D T
iT
f drag,tether T
(7-38)
Bridle attachment points damping
Four dampers are added between points A − C, B − D, A − D and B − C as depicted in
Figure 7-10 to provide some extra numerical damping, because the aerodynamic damping
turned out to be unstable as discussed in Chapter 6. The number of dampers is redundant
and two cross-dampers would have been enough, but this set-up was chosen to be able to
tweak the damping values so that they approximate the aerodynamic damping as good as
possible.
The length Lij , speed L̇ij and direction vector eij can be written for every damper acting
between masspoint i and masspoint j as
Lij = kxi − xj k
∂Lij
L̇ij =
ẏ
∂y
xj − xi
eij =
Lij
(7-39)
(7-40)
(7-41)
The forces on masses i and j become
f damp,i = cdamp,i L̇ij eij
(7-42)
f damp,j = −cdamp,i L̇ij eij
(7-43)
with the damping constant defined as
q
cdamp,ij = 2ζij (V k ) mi kkite,ij
(7-44)
where mi are the mass of a connected bridle point and kkite,ij is the equivalent spring force
between the two corner points of the kite and the relative damping
ζij (V k ) = kV k k β
Master of Science Thesis
(7-45)
H.A. Bosch
92
Dynamic tether and bridles model
where β can be used as a parameter to tweak this damping constant from zero damping to
an overdamped system. ζij is proportional to the speed of the kite V k , because the local
aerodynamic damping normally also depends on the kite speed.
The total force that is applied to the masses becomes
h
f damp = f damp,A T
f damp,B T
f damp,C T
f damp,D T
0T
iT
(7-46)
To determine the damping values, the kite is seen as if it acts as a spring between between
A − C and B − D. An experiment with the finite kite model is performed to find these spring
constants, while the kite is hinged at the four tether attachment points. First the forces at
the four tether points are measured while the kite is loaded with aerodynamic forces due
to an apparent flying speed of VkK = [−23 0 1.34]T m/s. The prescribed displacements
(boundary conditions) q b at the bridle points were taken from a simulation where the kite
was flying steady state, which means that the transient movements in the local kite reference
frame were damped out. Now extra displacements of q b,C,y = 0.1 and q b,D,y = 0.1 are applied
to points C and D in the y-direction and the resulting force at the tether points is measured
again. From this the equivalent spring constants can be determined as
∆FAC,y
19.5N
FAC,y
=
=
= 195N/m
u
0.1
0.1m
FBD,y
∆FBD,y
5.9N
=
=
=
= 59N/m
u
0.1
0.1m
kAC =
(7-47)
kBD
(7-48)
An increase of the displacement by a factor 2 also showed a force increase of approximately
a factor 2, which implies that the behaviour can be approximated with a linear spring. It is
more difficult to determine the values for the cross-dampers, so the value for β is determined
by trial and error using half of the spring stiffness of kAC and the average mass (mA + mB )/2.
βAC = 0.02, βBD = 0.002 and βAD = βBC = 0.015 were found to be good values for the
relative damping. This allows some oscillations, but makes sure the movement damps out
fast enough. The local movements of the kite are also quite well damped in reality as can be
seen from test flights. The tweaking of the damping values turned out to be quite important.
For example, if the damping of the tether points B and D at trailing edge is too high, those
points lacks too much behind in the movement of the leading edge which causes a lot of
wrinkling effects in the finite kite model and leads to convergence problems.
7-3-5
Gravity
Gravity force acting on the five mass points is
f g = −9.81 [0 0
H.A. Bosch
mA
0
0 mB
0
0 mC
0 0
mD
0
0 mP ]T
(7-49)
Master of Science Thesis
7-4 Equations of motion
93
x
z
y
cBD
B
cBC
D
cAD
A
cAC
C
Figure 7-10: Modelling of the damping between the bridle attachment points.
7-4
Equations of motion
The equations of motion are set-up using the information from the preceding sections. The
TMT method, as described by Schwab [42], uses a combination of independent generalized
coordinates, virtual power and an inertia contribution via d’Alembert forces, a method that
can easily be applied here to derive the equations of motion. This section uses Einstein
notation instead of vector notation for clarity.
According to Newton
Σfi − Mij ẍj = 0
(7-50)
Using the virtual power principle, this yields that for virtual velocities holds that
δẋi (Σfi − Mij ẍj ) = 0
(7-51)
All the coordinates of the centres of masses xi were expressed before in terms of the independent generalized coordinates yj in Equation 7-11, this can be expressed by a kinematic
transformation Ti
xi = Ti (yj )
(7-52)
and the corresponding velocities and virtual velocities become
ẋi =
∂Ti
ẏk = Ti,k ẏk
∂yk
δẋi = Ti,k δẏk
(7-53)
(7-54)
This result can be substituted in Equation 7-51 and leads to
Ti,k δẏk (Σfi − Mij ẍj ) = 0
(7-55)
Since this must hold for all virtual velocities, we can write
Ti,k (Σfi − Mij ẍj ) = 0
Master of Science Thesis
(7-56)
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94
Dynamic tether and bridles model
The accelerations can be found from differentiation of Equation 7-54
ẍj = Tj,l ÿl + Tj,pq ẏp ẏq
(7-57)
The last term is called the convective acceleration term gj
gj (ẏk , yk ) = Tj,pq (yk )ẏp ẏq
(7-58)
Substituting Equation 7-57 and Equation 7-58 in Equation 7-56 results in the equations of
motion
Ti,k Mij Tj,l ÿl = Ti,k Σfi + Ti,k Mij gj
(7-59)
Or in matrix vector notation
M̄ ÿ = f̄
(7-60)
M̄ = T T M T
(7-61)
with the reduced mass matrix
the kinematic transfer function that is used to transform between ẏ and ẋ and ÿ and ẍ and
to project forces from the body coordinates on to the generalized coordinates is
T = Ti,j
(7-62)
f̄ = T T (Σf − mg)
(7-63)
and the reduced force vector
where g is the acceleration term.
The force vector f consists of all the previous defined forces and becomes
f = f kite + f bridle + f cable,drag + f damp + f g
(7-64)
The equations of motion were symbolically derived in Matlab .
7-5
7-5-1
Initial conditions
Wind model
To analyse the behaviour of the modelling approach it is sufficient to keep the wind velocity
constant. The wind speed varies in reality quite a lot with the altitude as can be seen in
Figure 1-1, but in this modelling study the kite flies on a fixed length tether and will therefore
always fly around the same altitude. The wind speed also changes due to wind gusts or wind
gaps. These disturbances are also not taken into account, because it is better to study the
behaviour of the model first for simple conditions and add these effects in a later phase. The
flying domain of a 16m2 kite as modelled is roughly a wind speed of 5-10m/s. Therefore as
nominal parameter VwE = 5m/s will be used.
H.A. Bosch
Master of Science Thesis
7-5 Initial conditions
7-5-2
95
Initial values of the degrees of freedom
The initial values for the degrees of freedom cannot be chosen arbitrarily. First of all because
of the physical dimensions of the kite. If for example point A and C are positioned too
far apart, the finite element kite model will not be able to solve the problem or will return
unrealistic forces. Secondly because the kite needs to be positioned in a realistic position in
the wind window. If not, it will for example stall immediately. And thirdly because the bridles
suddenly change in length resulting in a different pitch angle of the kite and oscillations that
can be problematic for the finite element model. The equations to calculate the initial values
of the generalized coordinates will be described in this section.
Simulations are started from a symmetric position (mirrored in x = 0) and the length of the
power lines is prescribed. Therefore the following initial values can be assigned
αP = 0
(7-65)
αC = −αA
(7-66)
αB = −αC
(7-67)
βP = βA = βC
(7-68)
βB = βD
(7-69)
rA = rC = prescribed
(7-70)
Four unknown values remain. One of them, βP is free to choose and determines the initial
altitude of the kite. From the finite element kite model the initial distances A − C, LAC and
B − D, LBD are known, resulting in the constraint
LAC = kxA − xC k
(7-71)
From the prescribed βA , Eq. 7-71 and Eq. 7-70 the parameters αA and αC can be determined.
The remaining unknown parameters βB and αB can be found by minimizing the following
residual function R
RBD = kxB − xD k − LBD,kite
RB = kxB − xP k − rB,0
R=
2
RBD
+
2
RB
(7-72)
(7-73)
(7-74)
where rB,0 can be chosen freely and is used to give the kite an initial pitch angle.
Equation 7-74 can be solved for βB and αB by using the symbolic toolbox in Matlab .
So all the initial parameters can be determined by assuming that the kite starts from a
symmetric configuration. βA can be used to determine the initial altitude of the kite and rB,0
can be used to give the kite its initial pitch angle. Initial speeds can be set to zero.
7-5-3
Nominal parameters
All the parameters used in the dynamic tether and bridles simulation are summarized in
Table 7-2. Unless stated otherwise, these are the used values in simulations.
Master of Science Thesis
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96
Dynamic tether and bridles model
Table 7-2: Nominal parameters for the dynamic tether model
Parameter
mA , mC
mB , mC
mP
βAC
βBD
βAD , βBC
VwE s
Edyneema
rbridle
ζbridle
Ltether
Lbridles
kbridle
cbridle = cbridle,C
7-6
Value
3.5kg
1.5kg
3kg
0.02
0.002
0.015
5m/s
100GPa
0.0006m
0.8
80m
25m
4.52×103 N/m
251.7 Ns/m
Description
mass of bridle attachment point A and C
mass of bridle attachment point B and D
mass of the POD
relative damping constant between AC
relative damping constant between BD
relative damping constant between AD and BC
wind speed
Young’s modulus of Dyneema
radius bridles
relative damping coefficient bridle
length of the bridles
length of the bridles
bridle stiffness
damping coefficient
Finite element kite model inputs
The finite element kite model needs certain inputs from this dynamical tether simulation to
do its calculations, these are expressed in the kite reference frame ΨK .
7-6-1
Kite velocity
The velocity of the kite is the velocity of the kite frame ΨK . Since the origin of this frame is
defined as the midpoint between point A and B, the velocity can be calculated as
7-6-2
E
VE
A+VC
2
K E
= RE
Vk
VE
k =
(7-75)
VK
k
(7-76)
Wind velocity
The wind velocity is defined as a constant vector in the earth reference frame ΨE , in the kite
reference frame this becomes
K E
VK
(7-77)
w = RE V w
7-6-3
Rotational kite velocity
The rotational kite velocity can be calculated from the rate of the Euler parameters. To
obtain the Euler parameter rates, the time derivatives of Eq. 7-15 are taken as
∂λ0
ẏ
λ˙0 =
∂y
H.A. Bosch
∂λ1
λ˙1 =
ẏ
∂y
∂λ2
λ˙2 =
ẏ
∂y
∂λ3
λ˙3 =
ẏ
∂y
(7-78)
Master of Science Thesis
7-7 Concluding remarks
97
Since these expressions are too difficult to derive symbolically a numerical estimate will be
used by taking the their difference between time n and n − 1 divided by the time step h
λ0n − λ0n−1
λ˙0 =
h
λ1n − λ1n−1
λ˙1 =
h
λ2n − λ2n−1
λ˙2 =
h
λ3n − λ3n−1
λ˙3 =
h
(7-79)
From these Euler parameter rates, the rotational velocity of the kite frame can be determined
by [42]

 
λ˙0
λ0
λ1
λ2
λ3
"
#



0
λ0
λ3 −λ2  λ˙1 
−λ

= 2 1
(7-80)
 
K
−λ2 −λ3 λ0
λ1  λ˙2 
ωk
−λ3 λ2 −λ1 λ0
λ˙3
Euler parameters can suddenly jump between two values, which would result in very high
rates. Therefore a filter is applied to filter those outlier values.
7-6-4
Bridle attachment points
The positions of the bridle attachment points xi have to be determined in the kite reference
frame with origin X E
0
K
E
E
xK
i = RE (xi − X 0 ) for i = A,B,C,D
(7-81)
From this the boundary conditions for the finite element model can be calculated by
h
T
q b = xK
A
xK
B
T
xK
C
T
xK
D
i
T T
− q0
(7-82)
where q 0 are the original positions of the nodes at the four bridle attachment points in the
undeformed configuration.
7-7
Concluding remarks
A dynamic model to simulate the tether and bridles with twelve degrees of freedom was
developed. The bridles are modelled as linear spring-damper systems and the tether as a
rigid rod. 70% of the kite mass was put on the bridle attachment points of the leading edge
and 30% on the trailing edge points. A rigid connection between points A − B and C − D was
made to remove high frequency content. Sag in the cables and wind variations were neglected.
The equations of motion were derived with the TMT method[42].
Correct damping properties are crucial for a smooth working model. Some extra numerical damping was added between the bridle attachment points to represent the aerodynamic
damping. The aerodynamic model is not able to provide this damping due to stability issues.
The approach can easily be extended to kites with a more complicated bridle system. The
number of degrees of freedom will unavoidable increase, but this will only slightly increase
the calculation time, because solving the fluid-structure-interaction problem is the most expensive. The distribution of the mass over more bridle points will even improve the realism
of the simulation.
The initialization of the simulation is difficult, because all the component have to find an
equilibrium.
Master of Science Thesis
H.A. Bosch
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H.A. Bosch
Dynamic tether and bridles model
Master of Science Thesis
Chapter 8
System integration
This chapter describes how the developed components (finite element kite model, aerodynamic
model and dynamic tether and bridles model) are integrated in one system that solves the time
integration problem. Section 8-1 describes the time integration method, Section 8-2 describes
the developed controller and the chapter concludes with some final remarks in Section 8-3.
8-1
Time integration
This section describes the state-equations that need to be solved in Section 8-1-1 and how the
fluid-structure-interaction problem is included in these equations in Section 8-1-2, followed
by the selection of a time integration algorithm in section 8-1-3. A flow chart gives a full
overview of the system in Section 8-1-4 and the section ends with a discussion on some start
up problems in Section 8-1-5.
8-1-1
State equations
The dynamic differential equations of the dynamic tether and bridles model were derived in
Chapter 7 and resulted in Equation 7-59 which can be written in the form
ÿ = f (y, ẏ, X, t)
(8-1)
where ÿ are the accelerations that are a function of the degrees of freedom y that describe
the positions of the control pod and the bridle attachment points, ẏ the time derivatives,
X the state of the system including the kite velocity V k , the rotational kite velocity ω k ,
the velocity of the nodes V struct and the wind velocity V w as described in Chapter 5. The
undeformed spring lengths of the left and right steering lines L0,l and L0,r are added to this,
because those will be used as control variables as will be discussed in Section 8-2.
Equation 8-1 can be rewritten in the form of a standard system of two first order differential
equations by introducing
u = ẏ
(8-2)
Master of Science Thesis
H.A. Bosch
100
System integration
The resulting system of first order differential equations becomes
" #
"
ẏ
u
=
u̇
f (y, ẏ, X, t)
#
(8-3)
This system of ODE’s has to be solved to get the dynamic behaviour of the kite in time.
8-1-2
Including the FSI problem in the dynamic equations
Solving the quasi-static fluid-structure-interaction (FSI) problem results in the forces that
act on the bridle attachment points in the dynamic tether and bridles model. These forces
depend on y and X and are therefore a part of the dynamic equations that calculate ÿ as
can be seen in Figure 8-1(left) and described in Chapter 7.
Every dynamic solving algorithm does multiple function evaluations to calculate the positions
in the next time-step y n+1 . The amount of function evaluations depends strongly on the type
of solver that is used. The most computationally intensive part in this modelling approach
is solving the FSI problem. It would thus result in a very slow time integration algorithm if
the FSI problem needs to be solved for every time the accelerations ÿ are being calculated
(every function call of the solving algorithm).
Therefore the assumption is made that the forces that the FSI analysis returns can be kept
constant for the period of one time-step. The FSI analysis is removed from the dynamic
equations and only performed once for every time step and the resulting forces f F SI used
as external input to the dynamic equations by adding them to the state X as depicted in
Figure 8-1(right). This will greatly reduce the amount of time needed for the time integration.
8-1-3
Solver selection
This section discusses selection and testing of a time integration algorithm. The Matlab
code of this thesis will be written in a form so that it can be used in the integrated solver
algorithms of Matlab to be able to test with different solvers.The main criteria for the solver
are stability, speed and accuracy.
Implicit or explicit
A division in dynamic solver algorithms can be made between implicit or explicit algorithms.
Explicit solvers only need the state of the system at the current time step to calculate the
state in the next time step. The number of needed function evaluations to calculate the next
time step depends on the algorithm. The time step needs to be small enough to result in a
stable simulation.
Implicit solvers need both the state of the system at the current time step and the next time
step. This problem needs to be solved in an iterative manner. Implicit solvers have a better
stability and accuracy and are therefore often used in stiff problems. Stiff problems show
H.A. Bosch
Master of Science Thesis
8-1 Time integration
101
Figure 8-1: Schematic of the function that calculates the accelerations ÿ of the degrees of
freedom in the dynamic tether and bridles model. The left shows the original approach with the
FSI problem included in the function and the right shows the new approach where the from the
FSI analysis resulting forces acts as an input to the function.
high frequency behaviour which would require very small time steps in an explicit method to
remain stable, but can be solved with larger time steps using implicit algorithms. The time
step can be larger, but the number of function evaluations per time step will be higher. The
problem with implicit solvers is that some of them have start-up problems or need gradient
information.
The main parameter that influences the speed of the kite simulation will be the time step-size,
because the FSI problem only needs to be calculated once for every time step and the function
evaluations of the multi body equations are relatively cheap to perform. Therefore implicit
solvers are likely to perform better, because they can take a larger time steps. But this also
depends on the stiffness of the problem and it is difficult to say which solver will perform
better, as will be seen in the next sections.
Master of Science Thesis
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102
System integration
Stability
The stability and time step of the solver depend on the highest eigenfrequencies in the system.
The modelling choices in both the finite element model and dynamic model influence this.
There are three sources for high frequency content in this model.
1. The two tip beams in the kite connect both two bridle points (A-B and C-D) and are
made from aluminium. These beams act as very stiff springs between the bridle points,
resulting in high frequencies in the dynamic simulation. The dynamic equations of this
subsystem, depicted in Figure 8-2, can be written as
"
mA 0
0 mB
#"
#
"
#"
EA −1 1
ẍA
=
ẍB
1 −1
L
xA
xB
#
(8-4)
The eigenfrequencies are determined from
det(K − ω 2 M ) = 0
(8-5)
Resulting with the material properties as described in Chapter 5 in an eigenfrequency
of ω = 2 × 103 rad/s. This would for instance result in a time step h that should be h <
1.4×10−3 s for the often used explicit Runge-Kutta 4 algorithm to be stable, because the
RK4 algorithm has the stability criterion hω < 2.8. To avoid this behaviour a distance
constraint between bridle points A-B and C-D was introduced in Chapter 5 to remove
this stiff spring. This problem does not occur between the by the leading edge connected
points A-C and by the trailing edge connted points B-D. The eigenfrequency resulting
from the leading edge that acts as a spring between those points is only ω = 10.6rad/s
and even smaller for B-D.
k
A
B
xA
xB
Figure 8-2: The tip beam between bridle point A and B acts as a stiff spring.
2. The bridles in the dynamic simulation are modelled as springs and also relatively stiff.
The eigenfrequency can be calculated from the properties given in Chapter 5 by
ω=
q
k/m =
q
4.52 × 103 /1.5 = 55rad/s
(8-6)
Resulting in a maximum time step-size h = 0.056s to get the RK4 method to be stable.
3. The aerodynamics also have their effect on the frequencies in the system. When the
forces are very high and change quickly from direction, the small masses to which they
are applied accelerate and decelerate very fast. This can be another reason for high
frequent behaviour and a small time step. It is difficult to give a good estimation for
the frequencies that this will produce.
H.A. Bosch
Master of Science Thesis
8-1 Time integration
103
By removing the high frequency content generated by the kite, it can be concluded that both
implicit and explicit methods should result in reasonable time steps. The question is whether
an implicit method will still be faster now that the a reasonable time step can be obtained
using explicit solvers and what frequencies the aerodynamic forces introduce.
Error
Different numerical integration algorithms lead to different errors. This error originates from
two different sources as explained by Schwab [42]. The local truncation error is the error
after one time step, these errors add up over the total time span and the order of magnitude
depends on the algorithm. Round-off errors come from the fact that computers represent
numbers with a finite amount of errors. The global error E is the combination of both.
Figure 8-3 shows that there is always an optimal time-step and further decreasing it won’t
lead to a more accurate solution, because the round-off errors start to increase.
E
round off
error
truncation
error
Emin
hmin
log(h)
Figure 8-3: The truncation error and round-off error due to numerical integration depending on
the time step h.
Matlab uses two values to define the maximum allowable errors in its algorithms.
• The relative error (RelTol) tolerance is a measure of the error relative to the size of a
solution component. It basically controls the number of correct digits in every solution
component.
• The absolute error (AbsTol) tolerance is a threshold below which the value of a solution
component is unimportant. It determines the accuracy when the solution approaches
zero.
The Matlab solving algorithms approximate the error after every time-step and adjust the
step-size so that the error of a solution component fulfils the requirement
|e(i))| 6 max(RelT ol · |y(i))| , AbsT ol(i)))
(8-7)
where e(i) is the error and y(i) the value of solution component i. The used tolerances are
RelTol = 2×10−3 , which means that every element in the solution vector is accurate to 0.2%
and AbsTol = 1×10−6 , which means that only the first 6 digits are important.
Master of Science Thesis
H.A. Bosch
104
System integration
Time stepping
The relation between the size of the time step and the speed of the calculation is not as
obvious as it may seem.
1. For every time step, the FSI problem needs to be solved. Since this is an expensive
calculation, the time step should be as large as possible.
2. Solving the FSI problem becomes slower when the time step increases. The difference
in structural displacements between two time steps will be larger, which makes that
the FSI solving algorithm needs more iterations to solve the problem. The aerodynamic
forces and stiffness matrix of the kite need to be recalculated for every iteration, making
this a very expensive problem.
So increasing the time step has advantages and disadvantages and experiments are needed to
find out what the optimal time step is.
Several solving algorithms make use of variable time stepping to find the optimal time step
and speed-up the solution process. This approach could really speed up the process. The
disadvantage is that controllers work better with fixed time stepping algorithms.
Speed
The speed of the full simulation can be influenced by the following parameters.
• The time step size.
• The tolerances of the time integration algorithm.
• The tolerances for the finite element structural convergence process and the maximum
number of allowed iterations.
• The tolerances of the aerodynamic convergence process and the maximum number of
allowed iterations.
• Size of the finite element mesh.
• Frequency of changes in the system that come from aggressive steering inputs or wind
speed and direction changes.
• How and in what language the code is written.
Experiments with several solvers
Several Matlab solvers have been tested and compared. A 30 seconds real time simulation
was performed where the kite flies a couple figure eight trajectories with a controller, as
depicted in Figure 8-9. In the beginning of the trajectory some fast movements are needed
to follow the figure eight. The following settings were used:
H.A. Bosch
Master of Science Thesis
8-1 Time integration
105
• Computer: Intel core i5 750 @ 2.8GHz, 4GB memory,
• Matlab code with some parts compiled to C++.
• Dynamic tether and bridle model tolerances: RelTol = 2×10−3 , AbsTol = 1×10−6 .
• Structural and aerodynamic tolerances as explained in Chapter 5.
Table 8-1 shows the results. The explicit ODE45 algorithm was by far the fastest, resulting
in a speed that is 27.5 times slower than real-time. It can clearly be seen that the time-step
is not the most important parameter. The ODE45 solver used an average time-step of 0.0053
seconds, but was almost 3.5 times faster than the ODE113 solver that used an average timestep of 0.0099 seconds. The number of times the stiffness matrix K t has to be recalculated
is the most important factor as was expected. The ODE45 solver will be used in the rest of
the simulations.
Table 8-1: Tests with different numerical solvers
Solver
ODE45
ODE23
ODE113
ODE15s
ODE23t
ODE23tb
8-1-4
time (s)
825
1980
2724
1899
1335
3194
func eval
34369
68158
65202
41922
27271
99921
steps
5645
19875
30585
22023
12941
37293
failed steps
83
2844
4031
4895
2057
1523
av. h (s)
0.0053
0.0015
0.0099
0.0014
0.0023
0.0008
recalc Kt
12237
27043
31545
22854
17497
36004
Flow chart
Figure 8-4a shows the flow chart of the system with all the integrated components. Figure 8-4b
shows the same flow chart, but includes the data flows. The several components of the system
will be discussed in this section.
During the initialization phase all the parameters and initial variable values are set. A
selection of the most important ones:
• The finite element model is set up.
• Parameters for the dynamic tether and bridles model: tether length, bridle lengths,
damping values.
• The environmental parameters: wind speed.
• Time integration parameters: time step or maximum step size.
• Controller parameters: gains for power and steering control.
• Tolerances for time integration, structural and aerodynamic convergence and maximum
number of iterations.
• The initial position and speed of the kite: y, ẏ.
• The initial structural deformation q.
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System integration
(a) Flow chart of the process.
(b) Flow chart of the process including data flow.
Figure 8-4: The time integration process.
During the full system state update, all the variables that are needed in the several system
parts are calculated and stored. This full state is represented by X and contains the kite
velocity V k , the rotational kite velocity ωk , the velocity of the nodes V struct , and the wind
velocity V w as described in Chapter 5. Also all the transformation between different frames
are calculated within this step. For example transforming the bridle positions in the global
frame to bridle displacements in the kite frame.
The FSI problem receives the bridle displacements and the system state X and returns the
forces f F SI on the bridle points and adds them to the state X.
A steering controller adjusts the line lengths of the steering lines L0,r and L0,l and gives
them as an input to the dynamic equations via X. A power controller adjust the length of
the steering lines as well to power and de-power the kite.
A numerical solver does the time integration by solving the dynamic equations several times
H.A. Bosch
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107
to calculate the new positions of the pod and bridle attachment points that are represented
by y.
When the new vector y has been determined, the algorithm proceeds with the next time
step. This process repeats itself until the end time has been reached.
8-1-5
Simulation start-up
The start up of time integration is difficult. It is quite unnatural to start with an undeformed
kite somewhere in the air and suddenly expose it to wind. The following problems occur.
• The finite element model is not yet deformed. It is suddenly exposed to wind and the
deformations will change very rapidly in a short time period introducing high speeds.
If the damping is not correctly applied, deformations will get too large and the finite
element model will fail or very large oscillations will occur.
• The pitch angle of the kite has to be exactly correct. If the angle is too high, the kite
will immediately stall and fall out of the air or the leading edge will buckle, because
it bends too much backwards. If the pitch is too low, the canopy will start to flutter,
resulting in difficult buckling problems.
• The bridles that are modelled as springs are not deformed yet. Their lengths will
increase due to the introduced forces. This will happen very fast and some oscillations
will occur before it finds its equilibrium position. These oscillations will also change
the pitch angle of the kite. Another effect of these oscillations are very high rotational
speeds of the kite frame, causing instabilities in the damping effect in the aerodynamic
model.
Some experimenting with the line lengths, wind speed and temporarily switching off aerodynamic damping effects and the controllers, has led to a stable start of the system, giving the
finite element kite model and bridles some time to find an dynamic equilibrium. This start-up
sequence only has to be done once, because other simulations can start from this equilibrium
position.
8-2
Control
To be able to see whether this kite modelling approach works, it is necessary to include
controllers. The kite would immediately fall out of the sky without a steering controller,
because it is un unstable system. The pitch angle of the kite would constantly be either too
high or too low without a power controller, causing stall or a total collapse of the kite. This
section introduces a steering controller in Section 8-2-1 and a power controller in Section 8-2-2
8-2-1
Steering controller
The steering controller is based on a controller that was designed by Jehle [23] in Simulink. It
was rewritten in Matlab and modified at some points to make it work with the kite model.
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System integration
The controller is not optimized or very robust, but was just implemented to be able to fly a
trajectory and perform some experiments without having to manually make a list of steering
commands to fly a trajectory.
A figure eight on the unit sphere can be described in different ways, one option is to write it
in the parametrized form
1√
cos(t)
2
2
sin(t)2 + 1
1 √ cos(t) sin(t)
x=
2
2
sin(t)2 + 1
y=
z=
q
(a) Desired trajectory (blue, solid) and travelled trajectory (black, dashed)
1 − x2 − y 2
(8-8)
(8-9)
(8-10)
(b) Kite orientation Ψk in flight, blue = zaxis, green = y-axis, red = x-axis.
Figure 8-5: The kite model flying a figure eight.
The kite position can always be projected on this unit sphere by normalizing its position
vector. Normally during operation the tether length of the kite continuously changes. This
variable is eliminated by projecting its position on the unit sphere. Figure 8-5a shows the
desired trajectory and the actual travelled trajectory of the kite. Figure 8-5b shows the
kite-frame Ψk at certain time intervals on the trajectory.
The kite is able to freely move over the surface of the sphere and therefore has two degrees
of freedom left.
The controller uses a different frame ΨC to represent the orientation of the kite in than the
kite reference frame used by the finite element model. The z-axis is parallel to the tether.
The y-axis is perpendicular to the tether and points to bridle attachment point A. The z-axis
is perpendicular to both these axis. This is showed in Figure 8-6. The advantage of using this
frame representation is that the z-axis of the kite is always perpendicular to the unit-sphere.
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8-2 Control
109
Let us define the order of rotations to describe the orientation of the kite by first rotating
around the x-axis of the kite, then rotating around the rotated y axis and finally rotating
about the rotated z-axis. The last rotation is the yawing of the kite. The big advantage of
this rotation order is that the yaw-angle of the kite basically describes the heading of the kite
on the unit sphere! This should in theory always point in the same direction as the desired
trajectory.
y
z
rd
nd
ΨC
st
x
z
y
ΨE
x
Figure 8-6: The controller frame ΨC . Rotation order is defined as XYZ so that the rotation
around the z-axis of the kite is always the last one and indicates the heading of the kite on the
unit sphere.
The basic controller idea can be described as follows and is illustrated in Figure 8-7.
1. Find the closest point to the kite on the trajectory by solving a minimization problem,
described by the parameter s∗ in the trajectory equations 8-8
2. Determine the tangential vector t and perpendicular vector p on the unit sphere to the
trajectory at this point.
3. The new desired heading for the kite dnew is a combination of the vectors t and p.
dnew = βt + (1 − β)p
(8-11)
The parameter β depends on the maximum allowed yawing velocity and the current
forward velocity of the kite. When the kite has a low velocity, the kite is able to make a
corner with a smaller radius and vice versa. A lower value of β means a sharper corner.
4. The direction of the heading of the kite can be expressed by its yaw-angle αkite on the
unit sphere. The direction of the new desired direction dnew can be expressed by the
angle αdesired on the unit sphere.
5. The error between the yaw and desired angle is the error e = αdesired − αkite which is
the input for the controller.
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System integration
The exact mathematical expressions can be found in Reference [23]. This basic algorithm is
improved in a two ways.
• A simple form of feed forward is introduced by not looking at the point s∗ on the
trajectory, but looking at the point s∗ + n(Va ) that lies a bit further on the trajectory
with n a positive number depending on the apparent wind speed Va . The faster the
kite is flying, the further ahead it needs to look. The parameters a1 and b1 need to be
tuned.
n = a1 + b1 Va
(8-12)
• Also a simple form of gain scheduling is used. The faster the kite is flying, the higher the
steering gain Ksteering needs to be to be able to follow the trajectory. The gain varies
linear with the apparent wind speed. The parameters a2 and b2 need to be tuned.
Ksteering = a2 + b2 Va
z
y
αkite
x
e
αdesired
(8-13)
s*
t
p
dnew
Figure 8-7: The basic working principle of the steering controller.
A simple proportional controller with gain Ksteering is introduced to control the length difference ∆objective in the steering lines by comparing the desired heading αobjective and actual
heading αkite , as displayed in Figure 8-8.
∆objective = Ksteering (αobjective − αkite )
(8-14)
Another proportional controller with gain Kbridles is used to control the undeformed lengths
of the springs L0,r and L0,l that represent the steering lines, which is basically controlling the
individual line lengths.
L0,l = L0,l,init + Kbridles (∆object − (L0,l − L0,r ))
L0,r = L0,r,init − Kbridles (∆object − (L0,l − L0,r ))
H.A. Bosch
(8-15)
(8-16)
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8-2 Control
111
Figure 8-8: The proportional power controller (blue) controls the ratio of force on the steering
lines / power lines. The proportional steering controller controls the difference in lengths of the
steering lines (∆objective ).
Figure 8-9 shows the result of the working controller. It can be seen that it doesn’t follow the
trajectory perfectly. One of the properties of a proportional controller is that it will always
have an error. Furthermore the controller is not sophisticated enough to deal with the nonlinear dynamic behaviour of the kite and the delays in the system. The kite also flies with
different speeds in different parts of the trajectory that need different controller gains and
tends to slip out of the corner. The used algorithm is too rudimentary to correct for all these
effects, but was also never designed to do so. The controller does what it is supposed to do
and can be used use in some experiments.
Figure 8-9: The result from a figure eight flight. Blue is the desired trajectory and black dashed
the flown trajectory.
8-2-2
Power controller
A second controller is added to control the ratio between the forces on the steering lines and
the power lines, because the kite will buckle if the forces on the steering lines become too
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System integration
high.
rpower =
Fsteeringlines
Fpowerlines
(8-17)
Normally this ratio has a value around 0.3. This controller is not needed in the real system,
because the tether will be reeled out faster if the forces become too high. Since the tether
length is fixed in this simulation, the forces will become much higher and a controller is
needed to keep this ratio around 0.3. A simple proportional controller with a gain Kpower is
used to do this as can be seen in Figure 8-8. The length of both steering lines is controlled
simultaneously by changing their undeformed spring lengths L0 and added to the control
input from the steering controller.
L0 = L0,init − Kpower (robjective − r)
8-3
(8-18)
Concluding remarks
This chapter described an efficient way to couple the finite element kite model, aerodynamic
model, dynamic tether model and a controller together. The fluid-structure-interaction problem was removed from the dynamic equations so that it only has to be solved once per time
step, resulting in lower simulation times.
A controller was designed to steer the kite and control its power. The controller is not very
robust, but can be used to fly some test trajectories with the system.
Tests with different time integration algorithms show that the explicit Runge-Kutta algorithm
(ODE45) with a variable time step is the fastest, resulting in a simulation speed that is 27.5
times slower that real-time. This result is quite good for such a complex system programmed
in non compiled code.
Starting the simulations is a difficult phase in the time integration process, because all the
components need to find a equilibrium together.
H.A. Bosch
Master of Science Thesis
Chapter 9
Results & Discussion
This chapter shows and discusses the results of the introduced modelling approach to see
whether it leads to realistic and fast result. The individual components of the model were
already partly validated in the previous chapters.
Experiments to fully validate this model are outside the scope of this thesis, because the
primary goal of this thesis is to introduce a new modelling approach. A complete validation
would be the next step and is a difficult task. None of the currently available models has been
fully validated, because the lack of available test data. More test data has become available
over the past year, but validation methods are still in development. Comparing measurement
data and simulations is not straightforward, because for instance it is difficult to measure
the deformations of the kite or to reproduce the the exact measurements conditions in the
computer model.
Without experiments it is still possible to say something about the validity of the model by
comparing it to other models and comparing the results to known facts about kite behaviour.
This approach will be used in this chapter.
This thesis doesn’t discuss the energy production part of the kite energy system, but focusses
on the modelling of the kite. Therefore the energy generation is not a part of the results
won’t be discussed.
Section 9-1 describes the used set-up to obtain the simulation data, followed by studies on the
steering behaviour in Section 9-2, the aerodynamic behaviour in Section 9-3 and the structural
behaviour in Section 9-4. Section 9-5 discusses the results of the time integration algorithm
and the chapter concludes with some remarks in Section 9-6.
9-1
Test flight environment
A graphical user interface was designed in Matlab as shown in Figure 9-1 to perform several
test flights and study the travelled trajectory and the deformations modes of the kite.
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Results & Discussion
Figure 9-1: The Matlab user interface to analyse the results of test flight trajectories.
One representative figure eight flight was selected for analysis and is depicted in Figure 9-2
where the solid blue line shows the desired trajectory and the dashed black line the travelled
trajectory. The kite starts in the middle at point p0 and travels from p0 via p1 , p2 and p3
to p4 . It takes the kite approximately 20 seconds to fly this trajectory with a constant wind
speed of 5 m/s and a 100 m tether length.
Figure 9-2: The kite flies the selected figure eight trajectory in approximately 20 seconds. The
blue line shows the desired trajectory and the dotted line the actual trajectory.
H.A. Bosch
Master of Science Thesis
9-2 Steering behaviour
9-2
115
Steering behaviour
The main reason to model the full flexible behaviour of the kite was to obtain realistic steering
behaviour. The cornering behaviour of a kite has been subject of debate for a long time. Kite
designers want to develop high manoeuvrable kites for professional kite surfers and more stable
kites for beginners. Kites in energy generating systems also need good steering properties to
fly a figure eight. Breukels [7] was the first to answer the question how a kite actually corners.
A left corner is defined as the movement resulting from a shortening of the left steering line
and vice versa. Starting from the zenith position, this would look as a movement to the left
from the perspective of a kite surfer controlling the kite. This looks as a movement to the
right from the outer perspective taken in Figure 9-2.
When a kite is steered to the left by increasing the force on the left steering line, the angle
of attack at the left tip increases and of the right tip decreases. This results in a higher drag
force on the left tip. Kite designers thought for a long time that this was the main reason for
the cornering ability of a kite. However, this difference in drag is so small, that it only results
in a very slow yaw rate of the kite and can’t be the main reason for the cornering ability.
The lift force on the left tip also increases compared to the right tip due to the increased
angle of attack. But this is no reason for the kite to yaw, only to move sideways to the left.
The most important reason for the cornering behaviour is the asymmetric deformation of the
kite. The higher force on the left steering line causes the left tip to bend forward towards the
leading edge and the right tip to bend backward as shown in Figure 9-3. The deformation
of the kite results in an offset between the lift forces on both tips, creating a yaw moment
which makes the kite turn/yaw. This hypothesis was confirmed by Breukels [7] by comparing
the flexible kite to a rigid kite, showing that the yawing rate of a normal kite is much larger
compared to the rigid kite. These conclusions give a basis to verify the cornering properties
of the developed kite model in this section.
z
y
x
yaw
Lift left
Offset
Lift right
(a) Bottom view of the real kite.
(b) Model with magnified displacements.
Figure 9-3: Cornering of a kite is a result of the shape deformation that creates an offset in the
working lines of the lift forces at the tips. This results in a yawing motion to the left.
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9-2-1
Results & Discussion
Steering input
Figure 9-4 shows how the steering input from the controller results in a line length difference
between the steering lines. There is some delay, coming from the spring like behaviour of the
lines and the applied delay in the controller. The reason that both steering line lengths change
while making a corner is that the power controller also uses these line lengths to control the
total power in the kite. Figure 9-5 shows how the steering input relates to the yaw rate of the
kite. A line length difference of the steering lines results in a yawing motion in the correct
direction, as was expected. There is clearly a relation between yaw rate and controller input.
This proportional relationship between the yaw rate of the kite and the steering input was
found before by Erhard and Strauch [13] and Jehle [23] from different experimental datasets.
That the developed model shows the same proportional relationship as from experimental
data is a first confirmation of the validity of the model.
Length steering lines (m)
P0
P1
P2
P3
P4
0.24
Length steering line left
Length steering line right
Length difference
Controller input
25.5
0.12
25.4
0
25.3
−0.12
25.2
−0.24
0
5
10
Time (s)
15
Controller input / Line length difference (m)
Length of the steering lines
25.6
20
The steering controller input and the yaw rate
0.3
P0
P1
P2
P3
P4
1.5
0.2
1
0.1
0.5
0
0
−0.1
−0.5
Yaw rate (rad/s)
Steering controller input, steering line difference (m)
Figure 9-4: Length of the steering lines while flying a figure eight.
Steering input
−0.2
−1
Yaw rate
−0.3
−1.5
0
5
10
Time (s)
15
20
Figure 9-5: Relation between the steering input and yaw rate while flying a figure eight.
H.A. Bosch
Master of Science Thesis
9-2 Steering behaviour
9-2-2
117
The steering deformation of the kite
Figure 9-6 shows the deformation of the kite while cornering. The asymmetric deformation
can clearly be seen. While cornering to the right, the right tip bends forward and the angle
of attack increases while the left tip bends backward and the angle of attack decreases and
vice versa. This creates the resulting offset in lift forces resulting in yawing of the kite. This
looks very natural compared to Figure 9-3 and is according to the description of Breukels [7].
Figure 9-6: Deformation of the kite while cornering to the right (top) and left (bottom).
9-2-3
Angle of attack
Figure 9-8 shows that the angle of attack increases on the right tip and decreases on the
left tip while cornering to the right and vice versa. The angle of attack at the left tip even
becomes negative for a while. The higher angle of attack results in a higher lift force and
drag force at the right tip, as can be seen from Figure 9-9. The drag force is only slightly
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Results & Discussion
higher and has a minor result on the yaw rate compared to the lift force. This is exactly
what was expected and described by Breukels [7]. Figure 9-7 shows that the angle of attack
gradually varies over the kite from tip to tip. The angle of attack of the middle wing section
stays almost constant and results can be mirrored from left to right around the time of 10
seconds. Some oscillations can be seen at the tip sections right after passing point p1 . This
is the result of a fast increase in the steering input to be able to follow the trajectory. The
angle of attack of the left tip becomes negative here. Switching from a positive to a negative
angle of attack, causes the sign of the lift force to flip. This results in some oscillations. From
movies of the real system similar oscillations have been seen at the tips.
Angle of attack of every aerodynamic subsection
P0
P1
P2
P3
P4
0
5
10
Time (s)
15
20
Angle of attack (deg)
15
10
5
0
−5
Figure 9-7: The angles of attack of all the subsections of the kite while flying a figure eight.
Angle of attack at the tips while cornering
P0
P1
P2
P3
P4
5
10
Time (s)
15
20
Angle of attack (deg)
15
10
5
0
AOA left tip
AOA right tip
AOA middle
−5
0
Figure 9-8: The angles of attack of the left tip, right tip and middle section of the kite while
flying a figure eight.
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9-2 Steering behaviour
119
Lift and drag on the tips
600
P0
P1
P2
P3
P4
5
10
Time (s)
15
20
Drag right tip
Drag left tip
Lift right tip
Lift left tip
500
Force (N)
400
300
200
100
0
0
Figure 9-9: The lift and drag force on the tips of the kite while flying a figure eight.
9-2-4
Kite speeds
The kite gains a low speed in the negative y-direction when it is steered to the right in the
first seconds as can be seen in Figure 9-10. This is according to what is described by Breukels
[7] and a result from the higher lift force on the right tip than on the left tip.
The speed in the z-direction is due to the frame definition. The z-axis of the finite element
model is not defined parallel to the tether line, but slightly tilted. This results in a velocity
component in the z-direction which is part of the forward velocity. The actual velocity in the
z-direction is almost zero, because the tether length is fixed.
Figure 9-10 showspthat the forward speed of the kite varies roughly between 18m/s and 28
m/s (vf orward = vx2 + vz2 ). van den Heuvel [48] shows that the maximum flying speed of
a kite while flying crosswind is approximately the lift over drag ratio L/D times the wind
velocity. A normal L/D for kites of this type is around 5. The wind speed is 5m/s, meaning
a maximum kite speed of 25m/s. The obtained speeds are a bit higher than expected, but
in the same order. The main factor that influences the speed is the lift over drag ratio L/D
of the kite and is indeed too high as will be discussed in Section 9-3. The highest velocity is
obtained between P1 and P2 , which makes sense because there it reaches its lowest point and
converted all the potential energy to kinetic energy.
The rotational speeds in Figure 9-11 show that the pitch angle stays almost constant. It
will only change due to the action of the power controller to keep the power ratio in the kite
constant. The rolling movements of the kite are also very small. When the kite makes a sharp
corner, the kite also starts to roll a bit. The highest yawing rates are obtained right after
passing point P1 . The curvature of the figure eight there is the sharpest and the velocity of
the kite high. As a result the yawing speed needs to be high to follow the trajectory.
Besides being somewhat too high, the velocities show behaviour as expected.
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Results & Discussion
Kite speed in the kite frame ΨK
P0
5
P1
P2
P3
P4
0
Velocity (m/s)
−5
x−speed
−10
y−speed
−15
z−speed
−20
−25
−30
0
5
10
Time (s)
15
20
Figure 9-10: The speed of the kite while flying a figure eight.
Rotational kite speed in the kite frame Ψ
K
1.5
P0
P1
P2
P3
P4
roll rate
pitch rate
1
Velocity (rad/s)
yaw rate
0.5
0
−0.5
−1
−1.5
0
5
10
Time (s)
15
20
Figure 9-11: The rotational speed of the kite while flying a figure eight.
9-2-5
Line forces
As expected, the force in the right steering line is higher during the first 10 seconds of the flight
while cornering to the right as shown in Figure 9-12 and vice versa between 10-20 seconds.
The total pulling force of the kite on the tether reaches almost 6000N, which is a very high
force, but also realistic. This is in the same order of magnitude compared to experimental
data from test flights with the real system with comparable kites. The difference with the real
system is that the tether is reeled out during the powering phase, significantly decreasing the
tether force. The reel out velocity can be controlled to maintain a certain force in the tether.
Since reeling out is not implemented in the model, a one to one comparison with available
test data cannot be made. Forces are expected to be slightly too high, for the same reason
that the velocities are too high.
H.A. Bosch
Master of Science Thesis
9-3 Aerodynamics
121
Line forces in the bridles
6000
P0
P2
P3
P4
10
Time (s)
15
20
Power line left
Steering line left
Power line right
Steering line right
Tether force
5000
4000
Force (N)
P1
3000
2000
1000
0
0
5
Figure 9-12: The bridle and tether forces while flying a figure eight.
9-3
Aerodynamics
The lift over drag ratio L/D was calculated for every time step in the trajectory by calculating
the total aerodynamic forces on the kite and projecting them parallel and perpendicular to
the apparent wind vector. Figure 9-13 shows that the L/D ratio varies roughly between 7.5
and 9, while a normal value would be around 5 for this type of kite. This explains the high
speeds that were obtained. Two probable explanations can be given. Firstly the aerodynamic
model is not very accurate, it neglects tip vortices and other effects that would increase the
drag. A second reason comes from the modelling choices that were made in the finite element
model. The canopy is not attached to the top of the leading edge, but connected to the
middle of the leading edge beams. This results in a much lower camber of the airfoil. A lower
camber lowers the drag force more significantly than the lift force, resulting in a higher L/D
ratio. Furthermore the finite element model is somewhat stiffer than the real kite due to the
used coarse mesh, also resulting in less deformation of the canopy and a smaller camber.
The aerodynamics are the main source of uncertainties in the model. All the results completely
depend on the forces from the aerodynamic model. If those are not accurate, the end results
will not be accurate. Much can be gained from improving the aerodynamic model.
Figure 9-14 shows that the power controller keeps the ratio of the forces between the steering
lines and the power lines around 0.3. The controller has a low gain and was never expected
to keep the ratio at exactly 0.3, so these results show the wanted behaviour.
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Results & Discussion
Lift/Drag ratio
P0
P1
P2
P3
P4
0
5
10
Time (s)
15
20
L/D ratio
9
8
Figure 9-13: The lift/drag ratio L/Dwhile flying a figure eight.
Power ratio (force steering lines/force power lines
P0
P1
P2
P3
P4
0
5
10
Time (s)
15
20
0.34
Ratio
0.32
0.3
0.28
0.26
Figure 9-14: The power ratio (force steering lines/force power lines) while flying a figure eight.
9-4
Structural deformation
The kite shows more typical deformation modes besides the steering deformation. Figure 9-15
shows the displacements of the four bridle points in the y direction of the kite frame. It can be
seen that the bridle points A and B move to and from point C and D. This typically happens
in the real kite as well and is sometimes referred to as jellyfishing as shown in Figure 9-16.
This is another realistic structural deformation of the kite.
The maximum deformation of the nodes at both the tips is approximately 0.7 meters. This
is a total displacement of 1.4 meters with a span of the kite of 5.8 meters, resulting in a
deformation of almost 25 percent. This clearly shows the need for the non linear modelling
assumptions.
The displacements during the flight are smaller. Each bridle attachment points stays within
a region of 0.3 meters in the y-direction and the kite acts quite stable during the simulations
without much oscillations. Real kites also show this behaviour. Their nature is very flexible,
H.A. Bosch
Master of Science Thesis
9-4 Structural deformation
123
Displacements of the bridle attachment points in FE
0.8
P0
P1
P2
P3
P4
10
Time (s)
15
20
0.6
Displacement(m)
0.4
0.2
y displacement A
y displacement B
y displacement C
y displacement D
0
−0.2
−0.4
−0.6
−0.8
0
5
Figure 9-15: The displacements of the bridle attachment points in the y-direction while flying a
figure eight.
but the aerodynamic forces make it a structure that almost acts as if it were rigid. During
sudden wind gusts the flexible behaviour of the kites reveals itself. Since the simulated wind
in the model was constant, such behaviour was not expected to show up, but it would be
interesting to investigate those effects in the future. Also, the non linear material behaviour
of the inflatable beams was not implemented, which would probably result in some different
flexible modes. The course finite element mesh also introduces extra stiffness that contributes
to the total stiffness of the system.
Appendix C shows more pictures of deformations of the kite. Similar deformation patterns
can be seen in the model, confirming its validity. Movies of the real flying kite also show
similar oscillations at the tips and other dynamic behaviour. This movie material is difficult
to use as real validation confirmation, but contributes to the overall idea that the kite model
represents the global dynamic behaviour. More in depth studies of the dynamic behaviour
need to be performed to fully validate this dynamic behaviour.
Figure 9-16: Typical jellyfishing motion that appears in the kite while flying.
Master of Science Thesis
H.A. Bosch
124
Results & Discussion
9-5
Time integration
The time steps used by the dynamic Runge-Kutta explicit time integration algorithm are
shown in Figure 9-17. Small time steps are needed in the transitional phase between turning
to the right and turning to the left, where the yaw rate goes from positive to negative. Large
time steps are taken around P1 and P2 . Right after these points the highest steering input
is applied to get a peak in the yawing rate. At these points the angles of attack at the tips
show some oscillations and the lift force at one tips changes sign. This is were the time step
size also starts to oscillate, it needs a small time step when the force direction is suddenly
reversed. The average time step for this trajectory is 4.5 milliseconds and the speed of the
simulation is between 25 and 30 times slower than real time, which is fast for interpreted
Matlab code and a simulation of this size.
Multi body tether and bridle model time step
P0
P1
P2
P3
P4
0
5
10
Time (s)
15
20
Step size (s)
0.008
0.006
0.004
0.002
Figure 9-17: The used time step by the dynamic ODE45 time integration algorithm while flying
a figure eight.
9-6
Concluding remarks
The results presented in this section show a couple of things.
• The cornering behaviour of the kite model acts natural and is comparable to the model
of Breukels [7] and the results obtained from experimental test data by Erhard and
Strauch [13] and Jehle [23].
• Only a finite number of deformation modes has been observed, making this an interesting
case for further reduction algorithms.
• The speed of the algorithm is already quite fast, but compiling the code would even
further improve the speed.
• The combination of modelling choices and the aerodynamic model inaccuracies leads
to an overestimation of the lift over drag ratio L/D resulting in too high velocities and
forces.
H.A. Bosch
Master of Science Thesis
Chapter 10
Conclusions & Recommendations
10-1
Conclusions
The goal of this research was to show a new realistic and reduced approach to model flying
flexible inflatable tube kites used in airborne wind energy systems. The proposed method
exist of three main building blocks: a quasi static finite element model of the kite, a dynamic
simulation of the bridles and the tether and a quasi-static aerodynamic model. The kite
model and aerodynamic model together form a fluid-structure-interaction (FSI) problem that
needs to be solved iteratively and returns forces that are exerted on the bridle attachment
points in the dynamic simulation of the bridles and tether.
Three reduction principles are introduced without losing any information about the global
dynamic behaviour. It is assuming that the local inertia of the kite can be neglected and
that the kite responds in a quasi static manner to the aerodynamic forces. A coarse mesh
of the canopy is used together with regular beam elements to model the inflatable beams.
Furthermore it is assumed that the resulting forces from solving the FSI-problem can be held
constant during a time-step.
The (sub)goals were stated in Chapter 3. Firstly the model was examined to see whether the
results are realistic and the following conclusions can be drawn.
• The model is fully derived from physical principles by using the finite element approach
with the real material properties. The highly non-linear behaviour of the kite was nicely
captured by using geometrically non-linear finite elements.
• The global dynamics of the kite are captured very realistically. All the important deformation modes that can be seen with a real kite, are seen in the model: an asymmetric
twist while cornering and a jellyfish motion when subjected to a force disturbance.
• Results show that the kite model can steer by using the steering lines. The deformation
of the kite model is comparable to a real kite. By pulling a steering line the kite deforms
asymmetrically, creating an offset between the working lines of the lift forces on both
tips resulting in a yaw moment and cornering of the kite.
Master of Science Thesis
H.A. Bosch
126
Conclusions & Recommendations
• The deformation of the kite and the dynamics of the full model are dominated by the
forces produced by the aerodynamic model. The used aerodynamic model of Breukels [7]
was sufficiently accurate to prove this modelling approach, but has some shortcomings
that have their direct effect on the validity of this model. Firstly the model could not
be used to apply aerodynamic damping to the kite, because including the local speeds
of the wing sections resulted in unstable simulations. Extra damping in the dynamic
bridle model between the bridle attachment points was added to simulate this damping.
Secondly the model neglects some effects as tip vortices and other three dimensional
effects, resulting in an overestimation of the lift over drag ratio L/D, leading to higher
speeds and forces than expected.
• The kite has multiple flying regimes. The dynamics of a cross wind flight are completely
different from a stalling situation. The approach showed to work very well for crosswind
flights, but had difficulties simulating stall or extreme loading situations. In those
situations the canopy starts to experience more difficult dynamic behaviour such as
fluttering (buckling) or beams start to buckle. More refined calculation algorithms are
necessary to be able to solve those situations accurately. The model was primarily
designed for the crosswind flying regime.
• The realistic behaviour of the kite makes it an interesting tool for kite designers to
evaluate new designs before building them.
Secondly the model was examined on the reduction criteria.
• The kite was assumed to respond quasi-statically to the aerodynamic forces to create
a fast model, thereby neglecting all the local dynamic effects and using the kite and
aerodynamic model together (FSI-problem) as an external force generator to the dynamic tether model. This approach greatly reduced the calculation time. The dynamic
numerical integration only has to be performed on 12 degrees of freedom instead of
thousands (depending on the mesh size). This was a legitimate assumption, since the
results show a realistic model and no important dynamic effects are missed.
• The same conclusion can be drawn about the assumption that the calculated forces from
the FSI-problem can be kept constant during a time step and that the solving of the
FSI-problem does not need to be included in the dynamic differential equations. Not
having to solve the full FSI- problem multiple times per time step in the time integration
algorithm speeds up the calculations by a factor 4 (depending on the algorithm).
• The main modelling choice to reduce the finite element model was to model the inflatable
beams with regular non-linear beam elements, resulting in a massive reduction of degrees
of freedom in the model. The properties of the inflatable beams can be transferred to
normal beam elements without losing the correct dynamic behaviour.
• The reduction choices led to a model that is 25-30 slower than real time. This is very fast,
considering the amount of calculations that need to be performed simultaneously in the
a finite element model, FSI-problem and dynamic simulation all together. Matlab has
been used as programming environment, slowing things down quite drastically, because
the code is not compiled. The speed of the model can probably be increased by a factor
H.A. Bosch
Master of Science Thesis
10-2 Recommendations
127
5-10 by writing it in a compiled language. This makes it a good tool for controller
design or optimization studies.
• The simulation shows a finite amount of deformation modes. This makes the model
a good candidate for further new model reduction techniques as Proper Orthogonal
Decomposition (POD) or the Discrete Empirical Interpolation Method (DEIM).
Some additional conclusions can be drawn considering the modelling approach.
• The developed method is general applicable and flexible. It is easy to model different
kites with this method, even kites with a complex bridle system. That will only result
in some extra degrees of freedom in the dynamic model. The mass distribution of a
complex bridled kite will even be better than in the used C-shaped kite, where all the
mass of the kite is only distributed over the four bridle attachment points.
• The approach is platform independent and does not rely on commercial software, although it could also be used in combination with existing finite element software, resulting in a very powerful approach.
• This thesis shows a practical approach to combine several building blocks. Building
blocks can easily be changed, for example by introducing a new aerodynamic model.
The approach is well balanced in a way that all components use a same level of detail.
All together it can be concluded that the new proposed approach is a successful one that
fulfills the most important requirements and starts a new generation of more detailed kite
models. Some details in the modelling are open for improvements to create an even more
realistic model. It was outside the scope of this thesis, but an extensive model validation has
to be performed to be able to fully verify the model. Hopefully this will lead to better kites
that can be used to generate more energy from all the available wind high in the sky.
10-2
Recommendations
Some recommendations follow from the conclusions.
• The aerodynamic model is currently the limitation to the validity of the model. Better,
fast aerodynamic models are needed to produce more realistic results.
• The tether reel in/out should be added to compare the model to experimental data.
• The kite model can be coupled to a flexible tether model to produce more realistic
results when flying with long tethers.
• The Newton-Raphson convergence algorithm can be further optimized to converge faster
and more stable for difficult situations as buckling of the canopy.
• The finite element model can be improved to capture the kite behaviour even better:
1. The canopy needs to be attached to the top of the leading edge beams instead of
to the middle. This creates a more realistic structural model and gives a better
representation of the wing sections resulting in more realistic aerodynamic forces.
2. The non linear material behaviour of the inflated beams should be implemented.
Master of Science Thesis
H.A. Bosch
Appendix A
Geometric non-linear beam
development
In this appendix, a geometric non-linear two node straight three dimensional beam element
is developed that is used to model the inflatable leading edge and struts in the kite. The
two dimensional approaches of Reddy [39] and Crisfield [10] are combined and extended to a
three dimensional element in the Total Lagrangian framework. The non-linearity comes from
the inclusion of the in-plane forces that are proportional to the square of the rotation of the
transverse normal to the beam axis. The following (kinematic) assumptions are used:
• The element is based on the classic Euler-Bernoulli theory.
• The plane sections perpendicular to the axis of the beam remain plane, rigid and rotate
such that they remain perpendicular to the deformed axis after deformation.
• Therefore the Poisson effect and transverse stresses can be neglected.
• Large transverse displacements, small strains and small to moderate rotations are assumed.
The coordinate system of the three-dimensional beam is depicted in Figure A-1. The coordinates are described by x, y and z and the displacement field of the beam by u(x, y, z),
v(x, y, z) and w(x, y, z). The element degrees of freedom are described by
q Tel =
h
u1 v1 w1 ψx1 ψy1 ψz1 u2 v2 w2 ψx2 ψy2 ψz2
The angles and curvatures are related to the displacement fields by
dv
dw
, ψz =
dx
dx
dφ
d2 w
d2 v
κx =
, κy = − 2 , κz =
dx
dx
dx2
ψx = φ, ψy = −
H.A. Bosch
i
(A-1)
(A-2)
(A-3)
Master of Science Thesis
129
ψz2
ψy2
w2
v2
z
u2 ψx2
2
y
ψz1
x
ψy1
w1
v1
u1 ψx1
Z
1
Y
O
X
Figure A-1: The three dimensional two node geometrically non-linear beam element with elemental coordinate frame x, y and z and degrees of freedom q el .
From continuum mechanics the non-linear Green-Lagrange strains can be written as [39]
1
εij =
2
∂ui
∂uj
+
∂xj
∂xi
!
∂um ∂um
∂xi ∂xj
1
+
2
!
(A-4)
Omitting large strains, but keeping the square of the rotations of a transverse normal line in
the beam, leads to the following strain in the axial direction, also known as the von Kármán
strains. All the other strains are zero, because of the assumptions that were made. The
non-linear contribution comes from this equation.
ε11 = εx =
du
1
+
dx
2
dv
dx
2
1
+
2
dw
dx
2
(A-5)
As shown in Figure A-2 the displacement field u(x) can be expressed in the axial displacement
ū and y and z coordinate as
dv
dw
+y
(A-6)
u = ū − z
dx
dx
This leads after substitution of Eq. A-6 in Eq. A-5 to
dū
1 dv
+
dx
2 dx
= ε̄ + zκy + yκz
εx =
2
+
1
2
dw
dx
2
−z
d2 v
d2 w
+y 2
2
dx
dx
(A-7)
For the torsion a linear approach is taken as described in Reference [19]. The shear strain γ
can be written as
dφ
= ρκx
(A-8)
γ=ρ
dx
Master of Science Thesis
H.A. Bosch
130
Geometric non-linear beam development
z,w
u = −z ∂w
∂x
x, u
Figure A-2: The displacements due to bending in the zx-plane expressed by the rotation around
the y-axis dw
dx .
with ρ the radial distance from the axis of the beam.
The weak form of structural problems can now be derived by using the virtual work principle.
The virtual strain energy can be written as
δVint =
=
Z
(δεij σij + δγτ )dV
ZV Z
x A
(A-9)
(δε11 σ11 + δγτ )dAdx
(A-10)
with the strains εij , the stresses σij , the shear strain γ and shear stress τ .
Since in general holds that
∂ε
∂ε
∂ε
δu +
δv +
δw
δε =
∂u
∂v
∂w
the variation of the strains as described in Eq. A-7 and Eq. A-8 results in
(A-11)
dδū dv dδv dw dδw
+
+
+ zδκy + yδκz
dx
dx dx
dx dx
δγ = ρδκx
δε11 =
(A-12)
(A-13)
Substituting this in Eq. A-9 results in
δVint =
Z N
x
dδū dv dδv dw dδw
+
+
dx
dx dx
dx dx
+ My δκy + Mz δκz + Mx δκx dx
(A-14)
with
N=
Z
A
Mz =
Z
A
σ11 dA = EAε̄
My =
σ11 ydA = EIx κz
Z
Mx =
A
Z
σ11 zdA = EIy κy
(A-15)
τ ρdA = GJx κx
(A-16)
A
where the constitutive relations assuming elastic properties were used with the Young’s modulus E, shear modulus G and poisson ratio v.
σ = Eε
(A-17)
τ = Gγ
(A-18)
G=
H.A. Bosch
E
2(1 + v)
(A-19)
Master of Science Thesis
131
Linear shape function are used to describe the displacement field ū and torsional displacements
φ and cubic polynomial shape functions are used to describe the displacement fields w and
v. The displacement fields are described by the shape functions N i and the values at node 1
and node 2 represented by u, v, w and φ
uT = [u1 u2 ]
(A-20)
T
v = [v1 ψz1 v2 ψz2 ]
(A-21)
wT = [w1 ψy1 w2 ψy2 ]
(A-22)
φ = [ψx1 ψx2 ]
(A-23)
The displacement functions become
ū = N u (ξ)u
(A-24)
v = N v (ξ)v
(A-25)
w = N w (ξ)w
(A-26)
φ = N φ (ξ)φ
(A-27)
Let us introduce the non-dimensional variable ξ to describe the coordinate along the element
ξ=
2
x−1
L
(A-28)
The shape functions become
N Tu = N Tφ =
"
1 1−ξ
2 1+ξ
#
N Tv =
1
(1 − ξ)2 (2 + ξ)
 14L(1 − ξ)2 (1 + ξ) 
 8

 1

 4 (1 + ξ)2 (2 − ξ) 
− 18 L(1 + ξ)2 (1 − ξ)


N Tw =
1
(1 − ξ)2 (2 + ξ)
− 41 L(1 − ξ)2 (1 + ξ)
 8

 1

 4 (1 + ξ)2 (2 − ξ) 
1
2
8 L(1 + ξ) (1 − ξ)


(A-29)
Differentiation of the shape functions leads to
"
dū
−1/L
=
1/L
dx
#T
u = bu u
bφ = bu ⇒ φ′ = bφ φ
T
6(ξ 2 − 1)

dv
1 L(3ξ 2 − 2ξ − 1)

=

 v = bv v
2)


6(1
−
ξ
dx
4L
2
L(3ξ + 2ξ − 1)



6ξ

d2 v
1 
L(3ξ − 1)
=

 v = cv v
dx2
L2  −6ξ 
L(3ξ + 1)
Master of Science Thesis
(A-30)
(A-31)
(A-32)
(A-33)
H.A. Bosch
132
Geometric non-linear beam development
T
6(ξ 2 − 1)

1 −L(3ξ 2 − 2ξ − 1)
dw

=

 v = bw v

6(1 − ξ 2 )
dx
4L 
−L(3ξ 2 + 2ξ − 1)


(A-34)

6ξ

1 
d2 w
−L(3ξ − 1)
=

 w = cw w

−6ξ
dx2
L2 
−L(3ξ + 1)
κx = bφ φ
κy = −cw w
(A-35)
κz = cv v
(A-36)
This leads after substitution in Eq. A-7 and A-8 to
1
1
ε̄ = bu u + (bv v)2 + (bw w)2
2
2
γ = ρbφ φ
(A-37)
(A-38)
The internal energy from Eq. A-14 and variation of the strains in Eq. A-12 can now be written
as
δε11 = bu δu + (bv v)bv δv + (bw w)bw δw − zcw δw + ycv δv
δγ = ρbφ δφ
δVint =
=
Z
(A-39)
(A-40)
(N (bu δu + (bv v)bv δv + (bw w)bw δw) − My cw δw + Mz cv δv + Mx bφ δφ) dx
x
U Ti δu
+ V Ti δv + W Ti δw + ΦTi δφ
(A-41)
Let us write the internal force of an element as
f Tint = (U T , V T , W T , ΦT )
(A-42)
with U , V , W , Φ parts of the total internal energy from Eq. A-41 associated to the respective
degrees of freedom
U=
V =
W =
Φ=
H.A. Bosch
Z
Z
Z
Z
N bTu dx
(A-43)
(N (bv v)bTv + Mz cTv )dx
(A-44)
(N (bw w)bTw − My cTw )dx
(A-45)
Mx bTφ dx
(A-46)
Master of Science Thesis
133
The stiffness matrix now takes the form




K uu K uv K uw K uφ
K uu K uv K uw
∅
K
 K

K
K
K
K
K
∅

 vu

vv
vw
vv
vw
vφ 
K t =  vu
=

K wu K wv K ww K wφ  K wu K wv K ww
∅ 
K φu K φv K φw K φφ
∅
∅
∅
K φφ
(A-47)
and is obtained by differentiation of the internal force vector. Keep in mind that N (u, v, w),Mz (v)
and My (w) are functions of different variables.
∂U
=
∂u
Z
∂U
=
=
∂v
= K Tuv
Z
∂U
=
=
∂w
= K Tuw
Z
∂V
=
=
Z∂v
Z
K uu =
K uv
K vu
K uw
K wu
K vv
=
K ww =
=
K vw =
=
∂N
dx = EAbTu bu dx
∂u
Z
T ∂N
bu
dx = EA(bv v)bTu bv dx
∂v
Z
bTu
bTu
cTv
∂N
dx =
∂w
Z
EA(bw w)bTu bw dx
∂W
=
Z∂w
−cTw
Z
(A-50)
(A-51)
(A-52)
(A-53)
∂My
∂N
+ N bTw bw + (bw w)bTw
dx
∂w
∂w
EIcTw cw + N bTw bw + EA(bw w)2 bTw bw dx
∂V
=
Z∂w
(A-49)
∂Mz
∂N
+ N bTv bv + (bv v)bTv
dx
∂v
∂v
EIcTv cv + N bTv bv + EA(bv v)2 bTv bv dx
Z
(A-48)
(bv v)bTv
(A-54)
∂N
dx
∂w
EA(bv v)(bw w)bTv bw dx
K wv = K Tvw
Z
∂Φ
K φφ =
= GJx bTφ bφ dx
∂φ
(A-55)
(A-56)
(A-57)
The symbolic toolbox in Matlab was used to integrate the equations for the internal forces
and stiffness matrices.
Master of Science Thesis
H.A. Bosch
Appendix B
Finite element implementation
verification
B-1
Introduction
In this appendix the in Matlab developed shell element used to model the canopy and the
beam element used to model the leading edge and the struts will be verified.
Obtained solutions from the Matlab elements are compared to solutions obtained from the
finite element package Ansys. This option was preferred over comparison with some literature
cases, since this gives more flexibility in the choice of the load cases. Furthermore Ansys is
widely used finite element software and can therefore be trusted to generate reliable solutions
for these basic elements.
To verify the correctness of the linear and non-linear implementation of the stiffness matrix,
both element types will be subjected to different load cases. The maximum deflection and
|q
−q Ansys |
will be used as a measure to
mean error scaled by the maximum deflection MNatlab
dof ∗qmax
compare the results from Ansys and Matlab .
B-2
B-2-1
Shell elements
Set-up
For the tests with shell elements an aluminium plate with dimensions 1m x 0.5m x 0.001m
will be used with a Young’s modulus of 70GPa. The plate has a structured mesh with 400
elements, 231 nodes and therefore 1386 degrees of freedom.
In Matlab the plate will be modelled with the shell elements as described in Chapter 5.
In Ansys there are different shell element types, but only the SHELL63 element is capable
of using a triangular element with three nodes. This shell has both bending and membrane
H.A. Bosch
Master of Science Thesis
B-2 Shell elements
135
0.5
0.45
0.4
0.35
y (m)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
Figure B-1: Plate set-up for the shell element verification.
capabilities, six degrees of freedom for each node and geometric non-linear options. The theory
behind this element is different from the theory behind the Matlab model, so some differences
in the results are expected. The SHELL63 element is for example capable of handling large
deformations, where the Matlab element was designed for moderate deformations.
B-2-2
Load cases
Load case 1
z (m)
0
−0.05
1
−0.1
0.8
0.6
0.4
0.4
0.2
0.2
0
0
x (m)
y (m)
Figure B-2: Load case 1: setup
Load case: as in Figure B-2.
Boundary conditions: clamped on all sides.
Load: Fz = −10N for each node.
Results: Table B-1 and Figure B-3.
Table B-1: Results load case 1 shell element
Linear
Non-linear
Matlab
0.0974m
0.00522m
Ansys
0.0995m
0.0519m
Error
0.0096
0.0375
Load case 2
Load case: same as for load case 1, except the direction of the forces.
Boundary conditions: clamped on all sides.
Load: Fx = 10N for each node.
Results: Table B-2.
Master of Science Thesis
H.A. Bosch
136
Finite element implementation verification
z (m)
0
−0.05
1
0.8
0.6
0.4
0.4
0.2
0.2
0
0
x (m)
z (m)
y (m)
(a) Linear, Matlab , deformation
(b) Linear, Ansys, deformation
scale = 1, maximum deformation =
0.0974m
scale = 1, maximum deformation =
0.0995m
0
1
−0.05
0.8
0.6
0.4
0.4
0.2
0.2
0
0
x (m)
y (m)
(c) Non-linear, Matlab , deformation scale = 10, maximum deformation = 0.00522m
(d) Non-linear, Ansys, deformation
scale = 10, maximum deformation
= 0.00519m
Figure B-3: The results for load case 1 for the shell element.
Table B-2: Results load case 2 shell element
Linear
Non-linear
Matlab
3.2401×10−6 m
3.2401×10−6 m
Ansys
3.2654e-×10−6 m
3.2654e-×10−6 m
Error
0.111
0.111
Load case 3
Load case: same as for load case 1, except the direction of the forces
Boundary conditions: clamped on all sides.
Load: Fy = 10N for each node.
Results: Table B-3.
Table B-3: Results load case 3 shell element
Linear
Non-linear
Matlab
1.6401×10−6 m
1.6401×10−6 m
Ansys
1.6393×10−6 m
1.6393×10−6 m
Error
0.0665
0.0665
Load case 4
Load case: same as for load case 1, except the boundary conditions.
Boundary conditions: clamped on two smaller sides (sides in y-direction).
Load: Fz = −10N for each node.
Results: Table B-4 and Fig B-4.
H.A. Bosch
Master of Science Thesis
B-2 Shell elements
137
Table B-4: Results load case 4 shell element
Matlab
1.9678m
0.0161m
Ansys
1.9739m
0.0160m
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
z (m)
z (m)
Linear
Non-linear
Error
0.0017
0.0168
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−1.8
−2
−2
1
0.4
0.2
1
0.4
0.5
0
0
x (m)
y (m)
0.5
0
0
x (m)
y (m)
(a)
(b)
Linear,
Matlab
,
deformation
scale = 1,
maximum
deformation
= 1.9678m
Linear,
Ansys,
deformation
scale = 1,
maximum
deformation
= 1.9739m
0
−0.05
1
−0.1
z (m)
0
z (m)
0.2
−0.05
1
−0.1
0.8
0.6
0.4
0.4
0.2
0
y (m)
0.8
0.6
0.4
0.4
0.2
0.2
0.2
0
0
x (m)
0
y (m)
x (m)
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.0161m
deformation scale = 1,
maximum deformation =
0.0160m
Figure B-4: The results for load case 4 for the shell element.
Load case 5
Load case: curved shell as in Fig B-5.
Boundary conditions: clamped on two smaller sides (sides in y-direction).
Load: Fz = 500N for node 135.
Results: Table B-5 and Fig B-6.
Table B-5: Results load case 5 shell element
Linear
Non-linear
Master of Science Thesis
Matlab
0.1456m
0.0478m
Ansys
0.1472m
0.0558m
Error
0.0066
0.1303
H.A. Bosch
138
Finite element implementation verification
0.7
0.6
z (m)
0.5
0.4
0.3
0.2
0.1
1
0
0.8
0.6
0.4
0.4
0.2
0.2
0
0
x (m)
y (m)
Figure B-5: LoadCase 5: setup
Load case 6
Load case: curved shell as in Fig B-5.
Boundary conditions: clamped at the four corners.
Load: Fz = 10N for all nodes.
Results: Table B-6 and Fig B-7.
Table B-6: Results load case 6 shell element
Linear
Non-linear
Matlab
0.1344m
0.0124m
Ansys
0.1332m
0.0126m
Error
0.0126
0.0231
Load case 7
Load case: curved shell as in Fig B-5, but the material properties changed to t = 0.8× 10−3 m,
E = 0.68GP a, ρ = 625kg/m3 .
Boundary conditions: hinged on the two smaller sides (y-direction).
Load: Fz = 0.2N on all nodes.
Results: Table B-7 and Fig B-8.
Table B-7: Results load case 7 shell element
Linear
Non-linear
Matlab
0.4224m
0.01384m
Ansys
0.4234m
0.01378m
Error
0.0073
0.0243
Load case 8
Load case: curved shell with less elements and a different load as in Fig B-9.
Boundary conditions: hinged on the two smaller sides (y-direction).
H.A. Bosch
Master of Science Thesis
139
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
z (m)
z (m)
B-2 Shell elements
0.4
0.3
0.2
0.2
0.1
0.1
1
0
0.6
0
0.8
0.6
0.4
0.4
0.2
1
0
0.8
0.4
0.4
0.2
0.2
0
0
x (m)
y (m)
0.2
0
x (m)
y (m)
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 1,
maximum deformation =
0.1456m
formation scale = 1,
maximum deformation =
0.1472m
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
z (m)
z (m)
0.4
0.3
0.3
0.3
0.2
0.2
0.1
0.1
1
0
0.8
0.6
0.4
0.4
0.2
1
0
0.8
0.6
0.4
0
0.2
0
0
x (m)
y (m)
0.4
0.2
0.2
0
y (m)
x (m)
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.0478m
deformation scale = 1,
maximum deformation =
0.0558m
Figure B-6: The results for load case 5 for the shell element.
Load: Fz = 6N on node 61.
Results: Table B-8 and Fig B-10.
Table B-8: Results load case 8 shell element
Linear
Non-linear
B-2-3
Matlab
0.7203m
0.0808m
Ansys
0.7310m
0.0875m
Error
0.0102
0.1017
Conclusions
The following can be concluded from the load tests with the shell elements.
• In all the out of plane load cases the difference between the linear and non-linear case
is quite significant. This is due to the fact that the undeformed thin plate has almost
no bending stiffness. When the plate bends, more of the forces are taken by the in
plane stiffness. The geometrically non-linear element recalculates the stiffness matrix
to include this effect, the linear element doesn’t. Load case 1 shows this effect clearly.
Master of Science Thesis
H.A. Bosch
Finite element implementation verification
0.5
0.5
0.4
0.4
0.3
0.3
z (m)
z (m)
140
0.2
0.2
0.1
0.1
0
0.4
0.2
0
0
0.1
0.2
0.4
0.5
0.7
0.8
0.9
1
0
0.4
0.2
0
0
x (m)
y (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 1,
maximum deformation =
0.1344m
formation scale = 1,
maximum deformation =
0.1332m
0.5
0.5
0.4
0.4
0.3
0.2
0.9
1
x (m)
y (m)
z (m)
z (m)
0.3
0.6
0.3
0.2
0.1
0.1
1
0
0.8
0.6
0.4
0.4
0.2
0
y (m)
1
0
0.8
0.6
0.4
0.4
0.2
0.2
0.2
0
0
x (m)
y (m)
0
x (m)
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.0124m
deformation scale = 1,
maximum deformation =
0.0126m
Figure B-7: The results for load case 6 for the shell element.
• In load case 2 and 3 it can be seen that the solution is the same for the linear and nonlinear solution. This is because the deflections are so small that the tangential stiffness
matrix doesn’t need to be recalculated in the non-linear element. Both elements use
the same stiffness matrix.
• In general the deflections and deformation shapes calculated by Matlab and Ansys are
quite similar.
• In some cases the deflections are larger than allowed for the Matlab model, since it is
optimized for moderate deformations.
• Load case 5 shows quite a large difference for the non-linear solution. A probable cause
is the applied point load. Such a load is expected to give inaccurate results, since you
should never apply a point load to a plate or membrane, because this would in theory
result in an infinite displacement, because it acts on an infinite small area.
• Load case 7 and 8 show that the results are also quite similar for a thinner plate, even
though a point load is applied in case 8. This is important since the shells used in the
kite are very thin as well.
• It cannot be said whether the developed Matlab element is stiffer or less tiff than the
Matlab element. This is different for each load case.
H.A. Bosch
Master of Science Thesis
141
0.5
0.5
0.4
0.4
z (m)
z (m)
B-2 Shell elements
0.3
0.2
0.3
0.2
0.1
0.1
1.2
1.2
1
0
0.6
0.4
0.4
0.2
0.2
0.8
0
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
x (m)
y (m)
0
−0.2
x (m)
y (m)
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 1,
maximum deformation =
0.4224m
formation scale = 1,
maximum deformation =
0.4234m
0.5
0.5
0.4
0.4
z (m)
z (m)
1
0.8
0
0.3
0.2
0.3
0.2
0.1
0.1
1
0
1
0
0.8
0.6
0.4
0.8
0.6
0.4
0.4
0.2
0.2
0
0.4
0.2
0.2
0
0
x (m)
y (m)
0
x (m)
y (m)
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.01384m
deformation scale = 1,
maximum deformation =
0.01378m
Figure B-8: The results for load case 7 for the shell element.
0.5
0.45
0.4
z (m)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
1
0
0.8
0.6
0.4
0.4
0.2
0.2
0
y (m)
0
x (m)
Figure B-9: Load case 8: setup
As a final conclusion it can be said that the shell element implementation was successful and
works as supposed and is suited for the use in the canopy modelling.
Master of Science Thesis
H.A. Bosch
142
Finite element implementation verification
0.7
0.6
0.6
0.5
0.5
0.4
0.4
z (m)
z (m)
0.8
0.7
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
1
1
0.8
0.8
0.6
0.4
x (m)
0
x (m)
y (m)
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 1,
maximum deformation =
0.7203m
formation scale = 1,
maximum deformation =
0.7310m
0.5
0.5
0.4
0.4
0.3
0.3
z (m)
z (m)
0.2
0
0
y (m)
0.4
0.2
0.2
0
0.6
0.4
0.4
0.2
0.2
0.2
0.1
0.1
1
0
0.6
0.4
0.8
0.6
0.4
0.4
0.2
1
0
0.8
0
0.4
0.2
0.2
0.2
0
0
x (m)
y (m)
0
x (m)
y (m)
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.0808m
deformation scale = 1,
maximum deformation =
0.0875m
Figure B-10: The results for load case 8 for the shell element.
B-3
B-3-1
Beam elements
Set-up
For the test with beam elements a steel beam with a length of 1 meter and a circular diameter
of 0.05m was taken. The used steel properties are E = 200Gpa and ν = 0.3. The beam is
meshed with 20 elements and therefore 21 nodes with a total of 132 degrees of freedom.
In Matlab the beam will be modelled with the elements as described in section A. In Ansys
the BEAM4 element type has been chosen, since this one comes the closest to the developed
Matlab beam element, because it is based on the Bernoulli-Euler beam theory as well. An
exact agreement is not to be expected since the BEAM4 element uses some different theories.
1
3
5
7
9
11
13
15
17
19
21
Figure B-11: Load case 1: setup
H.A. Bosch
Master of Science Thesis
B-3 Beam elements
B-3-2
143
Load cases
Load case 1
Load case: as in Figure B-11.
Boundary conditions: clamped at end nodes.
Load: Fz = 10000N for node 11.
Results: Table B-9 and Figure B-12.
Table B-9: Results load case 1 beam element
Matlab
8.488×10−4 m
8.486×10−4 m
Linear
Non-linear
Ansys
8.488×10−4 m
8.488×10−4 m
0.2
0.2
0.15
0.15
0.1
z (m)
z (m)
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.1
−0.1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 50,
maximum deformation =
8.488×10−4 m
formation scale = 50,
maximum deformation =
8.488×10−4 m
0.2
0.2
0.15
0.15
0.9
1
0.9
1
0.1
z (m)
0.1
z (m)
Error
2.37×10−6
1.22×10−6
0.05
0.05
0
0
−0.05
−0.05
−0.1
−0.1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 50,
maximum deformation =
8.486×10−4 m
deformation scale = 50,
maximum deformation =
8.488×10−4 m
Figure B-12: The results for load case 1 for the beam element.
Load case 2
Load case: same as load case 1, except for the force direction.
Boundary conditions: clamped at end nodes.
Load: Fy = 10000N for node 11.
Results: Table B-10.
Table B-10: Results load case 2 beam element
Linear
Non-linear
Master of Science Thesis
Matlab
8.488×10−4 m
8.486×10−4 m
Ansys
8.481×10−4 m
8.481×10−4 m
Error
1.20×10−4
1.22×10−6
H.A. Bosch
144
Finite element implementation verification
Load case 3
Load case: same as load case 1, but applying a torsion moment instead of a force.
Boundary conditions: clamped at end nodes.
Load: Mx = 10000N for node 11.
Results:Table B-11.
Table B-11: Results load case 3 beam element
Linear
Non-linear
Matlab
0.0530deg
0.0530deg
Ansys
0.0530deg
0.0530deg
Error
6.59×10−7
6.59×10−7
Load case 4
Load case: same as load case 1, except for the force direction.
Boundary conditions: clamped at end nodes.
Load: Fy = 1M N ,Fz = 1M N for node 11.
Results: Table B-12.
Table B-12: Results load case 4 beam element
Linear
Non-linear
Matlab
0.120m
0.0556m
Ansys
0.120m
0.0561m
Error
4.73×10−6
1.70×10−3
Load case 5
Load case: same as load case 1, except for the force direction and application point.
Boundary conditions: clamped at end nodes.
Load: My = 1M N m for node 9.
Results: Table B-13 and Figure B-13.
Table B-13: Results load case 5 beam element
Linear
Non-linear
Matlab
0.1716m
0.0495m
Ansys
0.1716m
0.0514m
Error
9.89×10−7
0.0033
Load case 6
Load case: same as load case 1.
Boundary conditions: clamped at the left node.
Load: Fz = −30.000N for node 21.
Results: Table B-14 and Figure B-14.
H.A. Bosch
Master of Science Thesis
145
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
z (m)
z (m)
B-3 Beam elements
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−0.7
−0.8
−0.7
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
−0.8
1
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 4,
maximum deformation =
0.1716m
formation scale = 4,
maximum deformation =
0.1716m
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0.9
1
0.9
1
0
z (m)
0
z (m)
0
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
−0.2
−0.25
−0.25
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 4,
maximum deformation =
0.0495m
deformation scale = 4,
maximum deformation =
0.0514m
Figure B-13: The results for load case 5 for the beam element.
Table B-14: Results load case 6 beam element
Linear
Non-linear
B-3-3
Matlab
0.1630m
0.1584m
Ansys
0.1630m
0.1595m
Error
2.52×10−6
0.0018
Conclusions
The following can be seen from these load cases:
• In general the results for the Matlab and Ansys model are very similar, which leads
to the conclusion that the developed beam model is accurate enough for the purpose of
this research.
• The convergence of the Matlab element is very fast.
• The element is slightly stiffer than the Ansys element.
Master of Science Thesis
H.A. Bosch
146
Finite element implementation verification
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
z (m)
z (m)
0
−0.05
−0.1
−0.15
−0.15
−0.2
−0.2
−0.25
−0.25
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(a) Linear, Matlab ,
(b) Linear, Ansys, de-
deformation scale = 1,
maximum deformation =
0.1630m
formation scale = 1,
maximum deformation =
0.1630m
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0.9
1
0.9
1
0
z (m)
0
z (m)
−0.05
−0.1
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
−0.2
−0.25
−0.25
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x (m)
0.6
0.7
0.8
(c) Non-linear, Matlab
(d) Non-linear, Ansys,
, deformation scale = 1,
maximum deformation =
0.1584m
deformation scale = 1,
maximum deformation =
0.1595m
Figure B-14: The results for load case 6 for the beam element.
H.A. Bosch
Master of Science Thesis
Appendix C
North Rhino 16
C-1
Detailed photos
(a)
(b)
(c)
Figure C-1: Details of the North Rhino 16
Master of Science Thesis
H.A. Bosch
148
North Rhino 16
(a)
(b)
Figure C-2: The North Rhino 16
Figure C-3: Deformations while flying.
H.A. Bosch
Master of Science Thesis
C-2 Geometric properties
C-2
149
Geometric properties
Table C-1: Geometric properties of the kite
Parameter
Lstrut0
Lstrut1
Lstrut2
Ltip
rLE,mid
rLE,tip
rstrut0,LE
rstrut0,T E
rstrut1,LE
rstrut1,T E
rstrut2,LE
rstrut2,T E
Angle strut0
Angle strut1
Angle strut2
Height
Span
Value
1.93 m
1.67 m
1.08 m
0.65 m
100 mm
40 mm
50 mm
35 mm
40 mm
35 mm
35 mm
35 mm
0 deg
36.5 deg
72 deg
3.20 m
5.80 m
x
y
.m
y
.m
.m
.m
z
z
y
x
.m
.m
x
.m
.m
.m
.m
.m
Figure C-4: The dimensions of the North Rhino 16m2 kite.
Master of Science Thesis
H.A. Bosch
150
C-3
North Rhino 16
Deformation sequence figure eight flight
Figure C-5: Deformation (amplified by a factor 2) of the kite while flying a figure eight.
H.A. Bosch
Master of Science Thesis
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