helicopter blade twist optimization in forward flight · helicopter blade twist optimization in...
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Helicopter Blade Twist Optimization in Forward Flight
Marco Lonoce
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisor(s): Prof. Filipe Szolnoky Ramos Pinto Cunha
Examination Committee
Chairperson: Prof. Fernando José Parracho LauSupervisor: Prof. Filipe Szolnoky Ramos Pinto Cunha
Member of the Committee: Prof. João Manuel Gonçalves de Sousa Oliveira
November 2016
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To my father...
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Acknowledgments
I want to start by thanking Professor Filipe Cunha for the theme of thesis and for his help during the
work. Every meeting was really important to understand the correct direction to the end of this thesis.
My university in Italy, Politecnico di Torino, that gives me the opportunity of a Double Degree Project
with the Instituto Superior Tecnico in Lisbon, Portugal. It was fundamental the heavy work between the
coordinator of the two universities to create this new project.
The important help that the government of Italy and Piedmont region gave me as scholarship and student
residence every year.
My family and specially my father that couldn’t see the end of my studies. They believed me in every
choice that I took.
Finally, all of people that were not mentioned but gave a contribution to this thesis.
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Resumo
O incremento da eficiencia do helicoptero e um aspecto fundamental a ter em consideracao no
desenvolvimento inicial do helicoptero. Este ponto pode ser conseguido de varias maneiras. Em relacao
a potencia consumida ha certos aspectos ligados a aerodinamica do rotor que precisam de ser tomados
em conta, em especial a potencia induzida aquela que e necessaria fornecer ao rotor para este gerar
a propulsao. E possıvel minimizar esta potencia uniformizando a velocidade induzida ao longo da pa
para todas as posicoes azimutais. Nesta tese e explorada a ideia de modificar a torcao na pa para cada
condicao de voo de maneira a minimizar a potencia induzida. Sao considerados as seguintes hipoteses:
uma seccao com torcao linear, uma seccao com torcao quadratica, duas seccoes com torcoes lineares,
e tres seccoes com torcoes lineares. Por outro lado esta tese tem tambem como objectivo determinar
quais os conceitos apresentam uma boa oportunidade para a aplicacao de uma controlo activo da
torcao da pa. As simulacoes foram realizadas tendo como base o Sikorsky UH-60A Black Hawk, para o
qual todos os parametros do rotor estao disponıveis.
Palavras-chave: Controlo activo da torcao, reducao da potencia induzida, estruturas adap-
tativas, optimizacao do rotor principal, actuadores piezoeletricos, materiais compositos em fibras.
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Abstract
Improving the efficiency of the helicopter is one of the main objective in helicopter design. Several
ways are already taken in account to achieve this purpose. In relation to the power consumption there are
some aspects connected with the aerodynamic of the main rotor, specially the induced power, the power
used to generate the thrust needed to fly. It’s possible to minimize this power trying to uniform the inflow
along the blade for all the azimutal positions. In this thesis the idea is to modify the blade twist in each
flight conditions to obtain the minimum induced power. The twist distribution concepts considered are
one segment linear twist, quadratic twist, two linear twist segments with different divisions of the blade in
inner and outer parts and a three linear twist segments with different airfoils. This thesis has the purpose
to understand which concepts represent good opportunity for active twist control implementations. With
the results of the simulations a simple active twist control concept is developed. An objective is to
understand which piezoelectric actuators work better for this purpose, where they have to be placed and
how they have to be actuated. All the simulations are done on the Sikorsky UH-60A Black Hawk where
all the main rotor parameters are available.
Keywords: Active Twist Control, Induced power reduction, morphing, main rotor optimization,
piezoelectric actuators, macro fibers composite materials.
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Nomenclature xvii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Active Blade Twist Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 5
2.1 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Xfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Inflow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Flapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Fmincon and Global Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Implementation 17
3.1 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Effect of Blade twist on Main Rotor Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Results 31
4.1 Optimized Blade Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 One section with linear twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 Quadratic Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
xi
4.1.3 Two sections with linear twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.4 Two sections with linear twist and different airfoils . . . . . . . . . . . . . . . . . . . 36
4.1.5 Three linear segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Optimum Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Concept 41
5.1 Smart Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Piezoelectric Actuation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Conclusions 51
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography 52
A Appendix: Matlab Code 59
A.1 Main Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.3 Optimization function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.4 Flapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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List of Tables
3.1 UH-60A Tail Rotor Characteristics [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 UH-60A Data [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Reduction of power between Fixed linear twist and simulations . . . . . . . . . . . . . . . 39
5.1 Actuators Properties [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Actuators and Concepts Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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List of Figures
2.1 Flow Model Momentum Theory Hovering . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Blade Element Theory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Forces around NACA 23015 - Xfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 CP distribution NACA 23015 - Xfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Local Momentum Analysis BEM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Velocity distribution Hovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Inflow and Thrust Hovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Velocity distribution Forward Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Flapping Hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Sikosrky SC 1095 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Sikosrky SC 1094 R8 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Error Analysis lift curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Error Analysis drag curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Interpolation lift curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Interpolation drag curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.7 Blade Elements for Hovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.8 Blade Elements for Forward Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.9 Number of Azimuthal Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.10 Flight Test Data Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.11 Power consumption with different twists - Hovering . . . . . . . . . . . . . . . . . . . . . . 27
3.12 Power consumption with different twists - Low Speed Forward Flight . . . . . . . . . . . . 28
3.13 Power consumption with different twists - Medium Speed Forward Flight . . . . . . . . . . 28
3.14 Power consumption with different twists - High Speed Forward Flight . . . . . . . . . . . . 29
3.15 Comparison among different twist behaviours . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Linear Twist - Only one section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Parameter a - Quadratic twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Root Tip difference - Quadratic twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Two sections with linear twist 40 - 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 Two sections with linear twist 50 - 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
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4.6 Two sections with linear twist 60 - 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Two sections with linear twist 70 - 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.8 Two sections, two airfoils and linear twist 50 - 50 . . . . . . . . . . . . . . . . . . . . . . . 36
4.9 Blade planform UH-60A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.10 Three sections with linear twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.11 Sikorsky UH-60A Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.12 Comparison Flight Test Data and Three linear segments . . . . . . . . . . . . . . . . . . . 39
4.13 Twist Behaviours Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Active Twist Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Piezoelectric Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Continuous Electrode vs Interdigitaded Electrode . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Continuous Electrode vs Interdigitaded Electrode Electric Field . . . . . . . . . . . . . . . 45
5.5 Free Strain Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6 Micro Fiber Composite Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.7 Electric Field Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.8 Blade Model with black box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.9 Airfoil with Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.10 Torsion Angle of the Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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Nomenclature
Roman Symbols
A Disk Area
Aed Equivalent Wetted Area
Ainf Infinity Volume Control Area
B Coefficient for effective Blade Radius
c Chord
Cl Coefficient of Lift
Clα Slope of coefficient of lift - alpha curve
Cd Coefficient of Drag
Cdp Coefficient of Pressure Drag
Cm Coefficient of Moment
CT Coefficient of Thrust
CP Coefficient of Power
D Drag
e Eccentricity of the flapping hinge
f Tip or Hub loss parameter
F Correction Factor for tip or hub loss
fdrag Drag Parameter
kx Cosinus component for linear inflow model
ky Sinus component for linear inflow model
Ib Mass moment of Inertia
L Lift
m Mass Flow
P Power
r Adimensional radius
S Control Surface Area
T Thrust
UP Component of velocity perpendicular to the rotor
UT Component of the velocity parallel to the rotor
V Forward Speed
xvii
vi Induced Velocity
Vtip Blade Tip Velocity
y Coordinate along the blade
w Wake velocity
Greek Symbols
α Angle of Attack
αij Thermal Coefficient
β Flapping Angle
β0 Coning Angle
β1c Cosinus first harmonic flapping Angle
β1s Sinus first harmonic flapping Angle
γ Shear Strain
∆ES Spacing between Electrodes
χ Skew Angle
λ Inflow velocity
σ Solidity
µ Advanced Ratio
φ Inflow Angle
ρ Density
ψ Azimuth
θ Blade Twist
Ω Rotor Shaft Speed
Abbreviations and Acronyms
AFC - Active Fiber Composite
BEMT = Blade Element Momentum Theory
BET = Blade Element Theory
CFD = Computational Fluid Dynamics
MFC = Macro Fiber Composite
PZT = Zirconate Titanate Piezoceramic
UAV - Unmanned Aerial Vehicles
xviii
Chapter 1
Introduction
1.1 Motivation
A helicopter is a low speed, short range and low altitude aircraft. One of the main objectives in
helicopter design is improving the efficiency that is related with endurance, range, maximum forward
speed and also ceiling.
The helicopter power consumption is essentially divided in four parts: main rotor induced power, main
rotor profile power, fuselage parasitic power and tail rotor power [1]. All of these aspects are strongly
connected with the aerodynamic of the helicopter that, unfortunately, depends of the flight condition.
Recent studies [2, 3, 4] are focused on main rotor performance enhancement, because of its strong
influence on over-all helicopter performace (for example: hover and forward flight performance, noise
emission and vibration transmission at the hub). Unfortunately, the flight envelope of a helicopter is too
broad and for this reason, it’s impossible to meet all the constraints and requirements to have always
an optimum condition. The old design philosophy privileges just some flight conditions, in general the
hovering [5].
At this moment, the possibility to adaptively modify the design parameters to achieve an optimal con-
figuration for each flight condition is understudied. To reduce vibration, noise or minimize the induced
power for every forward speeds it’s possible to modify the shape, or the other parameters, of the blade.
These systems are called smart helicopter blades [6] that integrate an active mechanisms able to modify
the blade characteristics in function of the flight condition:
• Flaps: in helicopter blades, flaps are not a primary control surface but are used only to local
modified the lift to reduce vibration and noise [6, 3, 7, 8].
• Morphing blades: this idea shows how important is the biomimetic from the engineering point of
view. Birds modify their wings to adapt in each flight condition, in the same way a morphing blade
is able to change the shape to obtain improvements in aerodynamic loads, vibration and noise.
The possibilities are a modification of the leading or trailing edge, the airfoil camber or an active
twist control [6, 9, 10, 11, 12].
1
• Active Flow Control: the main object is to improve the lift on a profile. Every points after the sepa-
ration point don’t produce lift, so the idea is to bring the separation point as closer as possible to
the trailing edge to reduce the area without lift generation. The solutions found are a re-energising
or suction of the boundary layer on the top part of the airfoil [6, 13].
All of these systems need a correct integration with the blade and bring several challenges or prob-
lems that have to be fixed [14]. The helicopter blades are designed to work under large centrifugal loads
and, for this reason, a higher part of the section is filled with structural material. The space where it’s
possible to install an actuator is very limited. Another aspect is the weight that, if increased, amplifies
the centrifugal loads to the hub. Also the active system is influenced by the centrifugal loads and has
to be designed thinking that it has to be able to work under these loads. Of course, after all the design
aspects, if the complexity increases the possibility of failure also is higher. But on the other hand, these
systems could reduce the power consumption, vibration (really important for comfort on board for pilot
and passengers and fatigue loads on the helicopter) and noise.
1.2 Active Blade Twist Control
When a blades rotates, each point travels at different speed that is a function of the radial position
and, in forward flight also, azimuth position. This means that the contribution of lift and drag of each
point is also different, going higher at the tip. The consequence is that the lift distribution along the
blade is not constant. With a energy balance it’s easily understandable that the condition with uniform
lift distribution has the minimum induced power consumption [1, 15]. The blade twist is now introduced,
to achieve this purpose. Unfortunately, each flight condition has a different blade twist distribution that
minimizes the induced power. Increasing blade twist in hover and decreasing it in high speed forward
fight is well recognized in helicopter rotor design [1, 15]. According to Leishman [1], the inflow equation
in hovering condition has a special solution that gives uniform inflow:
θ(r) =θtipr
(1.1)
where θ represents the twist.
Equation 1.1 shows a hyperbolic behaviour of the twist. Unfortunately, this solution is physically
unrealisable for r → 0. In pratical sense it’s possible to approximate this solution with a linear twist
distribution because of the hub and root cut-out. According to recent studies, for example a rotor blade
optimization at Eurocopter [16], the twist distribution is related to the mission profile of the helicopter
considered. They optimized the rotor twist in each flight condition between hover and high speed forward
flight. After that, in function of the mission profile it’s chosen an intermediate condition that is a weighted
average between time in hover and forward flight. Other recent studies [17, 18] try to build a main rotor
that it is optimized for each flight condition. To achieve this purpose it’s necessary to use an active twist
control able to generate a torque moment to re-distribute the twist along the blade.
2
In the beginning the active twist control was utilized to reduce rotor vibrations [19], in fact in 1990s Chen
and Chopra did some tests in hover and forward flight in a wind tunnel [20, 21]. They used a smart rotor
model with individual blade control and embedded piezoceramic actuators. They wanted to achieve
a target value around 2 but they got a maximum twist amplitude of 0.5. This result was enough
to demonstrate that the concept was feasible and that there were advantages in term of vibrations
and noise [20, 21]. Other studies were conducted for example, the active rotor of NASA / Army / MIT
demonstrated that on fixed frame the active system was able to reduce vibratory loads [9]. Thakkar and
Ganguli [11] studied how to reduce vibration, delay flow separation and alleviate dynamic stall. The most
recent studies done by Sikorsky Aircraft and Cheng and Celi [4, 22] demonstrated less power needed
of 1-2% in wind tunnel tests. Another study done by Chopra [23] coupled computational fluid dynamics
and computational structural dynamics analysis on a Sikorsky UH-60A Black Hawk helicopter rotor with
an active twist control. The lift to drag ratio increased by 7.3% and the corresponding power decreased
by 3.3%, the system required a target of 4 twist control to achieve this result. Due to the constraints of
weight, space and mechanical actuations a possible solution for an active twist control is a piezoelectric
actuation system. In general, piezoelectric materials have the property to convert mechanical energy in
electrical energy and vice-versa [6]. A piezoelectric material is polarized and when an electric field is
applied in the direction of polarization it extends in that direction and contracts in the other two (in general
symmetrically), if is applied in the opposite direction it extends in the two other directions and contracts
in the main one [24]. But, if is applied an electric field perpendicular to the direction of polarization the
effect is a shear deformation mode. Research has been conducted on manufacturing a twisting actuator
that utilises actuators assembled in circle that can directly provide rotation and torque [25, 26].
1.3 Objectives
This thesis is focused on the application of variable blade twist to obtain reductions in power con-
sumption. Different twist distributions are considered and how they have to change to minimize the
induced power in each flight condition is studied. The purpose is to understand what happen in the
different conditions and if some of them present advantages related to the others.
1.4 Thesis Outline
This thesis is divided in 6 chapters:
• Chapter 2 : It’s the theoretical background behind the thesis. The software Xfoil used to obtain the
airfoils information, the models for the aerodynamic characteristics, the inflow and the algorithms
for the optimization.
• Chapter 3: Shows how the theoretical background is used to implement the model. The code is
tested doing a comparison with the flight data of the Sikorsky UH-60A Black Hawk and it’s used,
also, to show the effect of the blade twist on the total power consumption of the helicopter.
3
• Chapter 4: Presents the results of the simulations and a comparison among the solutions.
• Chapter 5: Preliminary design of an active twist control.
4
Chapter 2
Background
To study the aerodynamics of the rotor CFD analysis based on free wake methods are commonly
used [27, 28]. They calculate the vortical wake structure to predict the rotor performance. These meth-
ods have a high computational cost, for this reason it’s preferable to use some simple methods to analyze
the rotor with lower accuracy but with a much lower computational cost. Three theories are applicable
with this characteristics [1]: Momentum theory, blade element theory and blade element momentum
theory, that is the combination of the previous two. For each condition, hovering, climb, descent, forward
or lateral flight the aerodynamic is different and also each model has to be used in a different way. The
momentum theory gives analytical solutions, the blade element theory is more realistic but in some con-
ditions the approximations cannot simulate the reality with the necessary accuracy. The main aspect
that affects the quality of the results is the main rotor wake that is not stationary ever in stationary flight,
in fact it interacts with the tail, fuselage and control surfaces.
2.1 Momentum Theory
The momentum theory was established by Rankine in 1865 [29] to analyse marine propellers. The
generalization was done by Glauert in 1935 [1]. The theory idealizes the rotor as an infinitesimally thin
actuator disk over which a pressure difference exists. It’s the same to consider an infinite number of
blades with zero thickness. The actuator disk supports the thrust generated by the movement of the
rotor. The power needed is related to the torque of the rotor shaft and also an induced power related
to the gain in kinetic energy of the rotor slipstream. The theory is independent of the characteristics of
the rotor, it just takes in account the rotor diameter. All the calculations are done considering a control
volume around rotor and its wake with the applications of conservation laws[1].
This theory is based on some approximations:
• The airflow is incompressible
• Between the two surfaces of the actuator disk there is a pressure jump.
• There is no swirl velocity in the airflow
5
Figure 2.1: Flow model for momentum theory analysis of a rotor in hovering condition[1]
• The airflow before and after the actuator disk is uniform.
From the conservation of mass [1]:
m =
∫∫∞ρ−→V · d
−→S =
∫∫1
ρ−→V · d
−→S (2.1)
where m is the mass flow, ρ is the density,−→S is the control surface area and
−→V represents the velocity
for each control area.
m = constant→ m = ρA∞(w + Vc) = ρA(vi + Vc) (2.2)
With the conservation of momentum [1]:
T =
∫∫∞ρ(−→V · d
−→S )−→V −
∫∫0
ρ(−→V · d
−→S )−→V = m(Vc + w)− mVc = mw (2.3)
where A is the control surface area, w represent the velocity in far wake called wake velocity, vi is the
inflow velocity calculated on the rotor/disk plane and Vc is the Climb Velocity.
Finally an equation govering the conservation of energy in the flows can be written as [1]:
6
P =
∫∫∞
1
2(ρ−→V · d
−→S )|−→V |2 −
∫∫0
1
2(ρ−→V · d
−→S )|−→V |2 =
1
2m(Vc + w)2 − 1
2mVc
2 =1
2mw(2Vc + w) (2.4)
This is a scalar equation, it simply states that the work done on the fluid by the rotor is manifested as a
gain in kinetic energy of the fluid in the rotor slipstream per unit time. This is the power that the system
needs to create the pressure jump near the actuator disk.
With a combination between the result of the conservation of momentum and energy:
v1 = v2 =w
2(2.5)
In hovering condition (Vc = 0) the thrust, T , is calculated as:
T = mw (2.6)
and power is:
P = Tvi =1
2mw2 (2.7)
In the wake below the rotor the flow velocity increases, due to continuity considerations the area of
the slipstream has to decrease. From the conservation of mass in hover condition the ratio between the
cross-sectional area of the wake to the area of the rotor is:
A∞A
=1
2(2.8)
The area of the wake is the half of the rotor disk area. In term of radius is r∞ = 0.707R.
According to Leishman [1] the induced velocity at the rotor disk in hovering condition vh can be obtained:
vh =
√T
2Aρ(2.9)
With this equation, the ideal power consumed is:
P = Tvh = T
√T
2Aρ= 2ρAvh
3 (2.10)
The power required to hover increases with the cube of the induced velocity at the disk. To make a rotor
with a given thrust but with a minimum induced power it’s necessary to increase the mass flow through
7
the disk and this consequently requires a large rotor disk area [1].
Thinking in term of adimensional parameters, it’s possible to define the coefficient of thrust and the
coefficient of power.
CT =T
ρ(Vtip)2A(2.11)
CP =P
ρ(Vtip)3A(2.12)
2.2 Blade Element Theory
The Blade Element Theory (BET) is the base of analysis of helicopter rotor aerodynamics because
it provides estimation of blade aerodynamic loading in function of azimuthal and radial position [1].
In general, BET divides the blades in several sections and analyses each one as 2D airfoil able to
generate aerodynamic forces. Some corrections as tip and hub loss are taken in account with empirical
corrections. The BET can be used as basis to help design of rotor blades because, unlike the simple
momentum theory, all the characteristics of the blade are part of the theory [1].
Figure 2.2: Blade Element Theory Model[1]
In the blade element theory the angle of attack is defined as:
α = θ − φ φ = arctg(UPUT
) (2.13)
where α is the angle of attack, θ is the twist angle, φ is the inflow angle, UP is the component of the
velocity perpendicular to the rotor and UT is the component of the velocity paralell to the rotor.
8
For each section it’s possible to compute the lift L and drag D:
dL =1
2ρU2cCldy dD =
1
2ρU2cCddy (2.14)
where c is the chord and dy represents the width of the infinitesimal blade element.
Next step is the calculation of the perpendicular, Fz and parallel Fx forces to the rotor plane.
dFz = dLcosφ− dDsinφ (2.15)
dFx = dLsinφ+ dDcosφ (2.16)
To calculate the total forces it is necessary to compute the contribution of each blade. The infinitesi-
mal coefficients of thrust and power are computed as:
dCT =dT
ρA(ΩR)2=
NbdFzρA(ΩR)2
=1
2σ(Clcosφ− Cdsinφ)r2dr (2.17)
dCP =dP
ρA(ΩR)3=NbdFxΩy
ρA(ΩR)2=
1
2σ(Clsinφ+ Cdcosφ)r3dr (2.18)
where Ω - Rotational Velocity of the Rotor, Nb - Number of Blades and σ - Solidity of the rotor and σ and
r are calculated using:
σ =Nbc
πRr =
y
R(2.19)
The aerodynamic coefficients, in particular the lift Cl and the drag Cd used in the theories are calcu-
lated with the software Xfoil.
2.3 Xfoil
The software Xfoil was developed by Mark Drela from MIT [30]. It’s able to analyse isolated subsonic
airfoils taking in account Reynolds and Mach numbers. The main outputs that the software gives are the
distribution of forces around the airfoil, how the pressure coefficient changes in both top and bottom of
the airfoil and how the lift and drag coefficients change in function of the angle of attack α.
The software is based on 2D panel method, ”analyzes using integral equation instead of differential
equation” the velocity field near the airfoil. The only input that the software requires is a database of
coordinates that represents the geometry of the profile.
Other outputs are available after the analysis:
• The coefficient of moment Cm
• The transition point between laminar and turbolent flow for bottom and top
9
• The coefficient of pressure drag Cdp
Figure 2.3: Xfoil: Forces distribution around NACA 23015 airfoil
Figure 2.4: Xfoil: CP distribution around NACA 23015 airfoil
10
2.4 Blade Element Momentum Theory
The blade element momentum theory (BEMT) is a combination between blade element and momen-
tum approach first proposed for helicopter use by Gustafson and Gessow (1946). This theory allows the
estimation of the inflow velocity along the blade for hovering condition[1].
Figure 2.5: Annulus of rotor for local momentum analysis for hovering condition[1]
As shown in Figure 2.5, the theory consists first in the application of the conservation laws to an
annulus of the rotor disk. Each annulus gives an increment in thrust dT and it’s possible to calculate
this just with a simple momentum theory with two assumptions: 2D and no mutual effects between
successive annuli. To solve the problem of 2D restriction the tip-loss effect by Prandtl is enough. For
each annulus the area considered:
dA = 2πydy (2.20)
The incremental thrust of the annulus is the product between mass flow rate and twice the induced
velocity [1]:
dm = ρdA(Vc + vi) = 2πρ(Vc + vi)ydy (2.21)
dT = 2vidm = 2ρ(Vc + vi)vidA = 4πρ(Vc + vi)viydy (2.22)
This is the Froude - Finsterwalder equation. It’s more convenient to work with adimensional quantities:
dCT =dT
ρ(πR2)(ΩR)2= 4
Vc + viΩR
viΩR
y
Rdy
R(2.23)
Writing everything in function of total inflow λ = Vc+viΩR , climb inflow λc = Vc
ΩR and induced inflow
λi = viΩR with the adimensional radius r = y
R :
dCT = 4λλirdr = 4λ(λ− λc)rdr (2.24)
With equation 2.24 is possible to compute the coefficient of induced power:
11
dCPi = λdCT = 4(λ)2(λ− λc)rdr (2.25)
The calculation of the total thrust and induced power is done by the integration along the blade:
CT =
∫ r=1
r=0
dCT = 4
∫ 1
0
λ(λ− λc)2rdr (2.26)
and
CPi =
∫ r=1
r=0
λdCT = 4
∫ 1
0
λ2(λ− λc)2rdr (2.27)
2.5 Inflow Models
As the dynamic representation of the rotor system reaches a certain level of sophistication in term
of degrees of freedom of the blade motion, it becomes apparent that a comparable level of detail must
be used for the aerodynamic part [31, 1]. The helicopter aerodynamic is strictly related to the induced
velocities at and near the main rotor. In the past, due to the limited computational capability, the induced
inflow was considered uniform. Now, there are several non-uniform representations for each flight con-
dition. Some models can represent better the dynamic effects, others just the aerodynamic loads over
the rotor. For high accuracy simulations, CFD models able to analyse the wake are used [27, 28].
In hover condition the inflow can be determinated directly using BEM theory [1]. The principle is the
equivalence between the circulation theory of lift and the momentum theory of lift. The inflow model for
hovering is [1]:
λ(r, λc) =
√(σClα
16− λc
2)2 +
σClα8
θr − (σClα
16− λc
2) (2.28)
Equation 2.28 is function of the rotor characteristics as σ the solidity or Clα the lift-curve-slope and the
blade twist θ how the inflow changes along the blade.
From the other point of view, in forward flight, the induced velocity field is no longer axisymmetric. The
effects of the individual tip vortices produce an highly non-uniform inflow over the rotor disk, specially
in the range 0.0 ≤ µ ≤ 0.1 (where µ = VΩR . Introducing now the azimuthal angle ψ defined as the
angle of the blade relative to the main axis of the vehicle After that range, in high speed forward flight,
the time-averaged longitudinal inflow becomes more linear and can be approximately represented by a
linear variation in function of azimuthal angle ψ and adimensional radius r [31, 1].
λi = λ0(1 + kxx
R+ ky
y
R) = λ0(1 + kxr cosφ+ kyr sinφ) (2.29)
12
The coefficient λ0 is the average induced inflow at the centre of the rotor given by the momentum theory:
λ0 =CT
2√µ2 + (λ0)2
(2.30)
To solve this last equation it’s necessary to impose an iterative method. The two coefficients kx and ky
are related to the wake skew angle χ that is the angle between the wake and the rotor disk.
χ = arctan(µx
µz + λ0) (2.31)
The two advanced ratios considered here are just the parallel and perpendicular component to the rotor
plane of the advanced ratio µ. If atpp is define as the angle between the rotor plane and the air speed,
the advanced ratios are: µx = µ cos atpp andµz = µ sin atpp. There are several estimations to calculate
the coefficients of the equation 2.29. One of the more recent that also gives good correlations with flight
test data is the Pitt-Peters model (1981) [32]:
kx =15π
23tan(
χ
2) (2.32)
and
ky = 0 (2.33)
In hovering condition the velocity distribution perpendicular to the blade leading edge is axisymmetric
and it’s showed in Figure 2.6.
Figure 2.6: Velocity distribution in hovering condition. All the azimuthal positions show the same distri-bution
The inflow velocities and the generated thrust depend only by the radial position and for this reason
13
are axisymmetric as showed in Figure 2.7.
Figure 2.7: Inflow velocities and thrust in hovering condition [33]
Here, the thrust is defined to be negative because in the opposite direction of the inflow velocity. Near
the hub, where the inflow velocities and also the tangential velocities are smaller, low thrust is produced.
The higher part of the thrust is distributed between 50% and 90% of the blade length. Near the tip the
thrust decreases really fast due to the tip losses.
In forward flight there is no axisymmetry in the velocity field. There are three aerodynamic aspects
to be considered: dynamic stall, reverse flow and transonic effect. In the retreating blade near the tip the
blade increases a lot the angle of attack to compensate the reduction of lift produced by the decreasing
of the local airspeed. When the forward flight increases above a certain level the blade encounters the
dynamic stall. In the reverse flow region, the rotational velocity is slower than the aircraft airspeed and
the air flows from the trailing to the leading edge of the airfoil. In the other side, the blade near ψ = 90·
has a velocity that is the sum of the rotational and forward speed V = ΩR + Vforward. The velocity
distribution is showed in Figure 2.8.
Figure 2.8: Velocity distribution in Forward Flight with Reverse Flow Region
14
2.6 Flapping
Helicopter rotors are articulated with flapping and lead-lag hinges at the root of each blade. Modern
rotor hubs are hinge-less but allow motion about a virtual hinge location. The blade, essentially, presents
three mechanical hinges that allow three different movements: flapping, lead or lag and feather. The
blade flapping is the up and down movement of the rotor blade important to reduce the dissymetry of lift
due to the different local speed between advancing blade side and retreating blade side. The lead-lag
hinge allows in-plane motion of the blade in response to the Coriolis accelerations and forces produced
by the flapping. Finally, the feather is the ability to change the blade pitch. These hinges allow each
blade to independently flap and lead or lag with respect to the hub plane [1]. In hovering condition the
blades reach a steady equilibrium position under the action of aerodynamic and centrifugal forces. The
equilibrium angle, with respect to the hub plane, is called coning angle β0 and it’s calculated with the
equilibrium of aerodynamic moment, Mβ , and centrifugal moment MCF , in relation to the flapping hinge
[1]:
Mβ +MCF = 0 (2.34)
and the coning angle:
β0 =3∫ ReRLydy
MΩ2R2(1 + e)(2.35)
where L is the lift, y is the radial position, M is the mass of the blade (equally distribuited), R is the total
radius of the blade and e is the eccentricity of the flapping hinge.
This equation is valid for any form of aerodynamic load over the blade [1].
In forward flight the blades flap up and down in a periodic manner with respect to azimuth due to
cyclically varying air-loads [1]. Considering Figure 2.9, the flapping angles are functions of aerodynamic,
centrifugal and inertial forces.
∫ R
0
d(MCF ) +
∫ R
0
d(I) +
∫ R
0
d(Mβ) = 0 (2.36)
The equation of flapping motion becomes:
Ibβ + IbΩ2β =
∫ R
0
Lydy (2.37)
where Ib is the moment of inertia of the blade about the flap hinge.
Considering that the azimuthal position ψ = Ωt and the rotational speed is constant:
β(t) = Ω?
β(ψ) β(t) = Ω2??
β(ψ) (2.38)
15
Figure 2.9: Flapping hinge with aerodynamic, centrifugal and inertial loads [1]
So the final equation for flapping is [1]:
??
β(ψ) + β(ψ) =1
IbΩ2
∫ R
0
Lydy (2.39)
In forward flight the aerodynamic forces act at multiples of the rotor frequency. For this reason the blade
flapping motion can be represented by an infinite Fourier series [1]:
β(ψ) = β0 +
∞∑n=1
(βnc cosnψ + βns sinnψ) (2.40)
For simple analysis it’s possible to assume that the solution for blade flapping motion is given by the
first harmonics only [1]:
β(ψ) = β0 + β1ccosψ + β1ssinψ (2.41)
2.7 Fmincon and Global Search
Fmincon is a Matlab function able to find the minimum of constrained non-linear multi-variable func-
tion [34]. It’s part of the optimization toolbox of Matlab. The reason to use an optimization function is
the reduction of calculation time. Another solution is a parametric analysis but just with a few variables
the computational cost becomes really high. The function finds a local minimum near the initial condi-
tions given in input. In all the engineering applications an optimized condition corresponds to a global
minimum. To solve this problem Fmincon works together with a Global Search that generates a family
of initial conditions to calculate all the local minima in the range considered and after that the algorithm
selects the global minimum [35].
16
Chapter 3
Implementation
3.1 Numerical Model
To analyse the aerodynamic characteristics of the helicopter a model based on Blade Element Mo-
mentum theory and Blade Element theory is used. The characteristics of the airfoils are taken from the
software XFoil [30], a 2D simulation code based on panel method developed by Mark Drela in MIT. In
this thesis all the simulations are done on a Sikorsky UH-60A Black Hawk. For this reason the airfoils
analysed are the two used on the UH-60A main rotor: SC 1095 and SC 1094 R8 [36]. In Figure 3.1 and
Figure 3.2 are presented the characteristics of UH-60A airfoils with angle of attack α = 3.
Figure 3.1: Sikosrky SC 1095 Airfoil with α = 3
The two main parameters, for this kind of analysis, are the coefficients of lift and drag:
Cl = f(α) (3.1)
17
Figure 3.2: Sikosrky SC 1094 R8 Airfoil with α = 3
and
Cd = g(α) (3.2)
The results don’t represent continuous functions, they are given for several angles of attack α with
∆α = 0.25 as step. To implement an optimization code the input data have to be continuous so an
interpolation is required. The database is interpolated using polynomial equations and the order is
chosen to achieve the best fitting possible. The total error for each polynomial order is studied and the
results are showed in Figure 3.3 and Figure 3.4.
Figure 3.3 and Figure 3.4 represent how the total fitting error changes with the order of the polynomial
equation. The two points chosen represent a good compromise between computational cost and error.
The results of the interpolation are presented in Figure 3.5 and Figure 3.6:
In general, simple models use a linear equation to represent lift curve and a quadratic one for the
drag curve. High order polynomial equation can also represent stall condition and drag bucket to really
take in account all the airfoil informations.
To describe hovering and forward flight conditions two models are taken in account. In hover, using the
equivalence between the circulation and momentum theories of lift, Blade Element Momentum Theory
(BEMT) allows the estimation of the inflow distribution along the blade. Considering no climb velocity
the simplified model is [1, 31]:
λ =σClα16F
(
√1 +
32F
σClαθr − 1) (3.3)
Using this model, it’s possible to estimate the inflow in function of the twist distribution. In forward
flight, the helicopter must provide a lifting force and a propulsive force in opposition of weight and air-
frame drag. The rotor moves through the air and all the blade sections encounter a periodic variation
in local velocity. There are some consequences as blade flapping, unsteady effects, non-linear aerody-
18
Figure 3.3: Error Analysis CLvsα
Figure 3.4: Error Analysis CDvsα
namics, stall, reverse flow and an higher interference between rotor wake and the main rotor itself [1].
The induced velocity field is no longer axisymmetric and the effects of the individual tip vortices tend
to produce a highly non-uniform inflow over the rotor disk specially during the transition from hover into
forward flight, within the range 0.0 ≤ µ ≤ 0.1.
19
Figure 3.5: Interpolation CLvsα with 10 order polynomial equation
Figure 3.6: Interpolation CDvsα with 12 order polynomial equation
In higher speed forward flight (advanced ratio higher than 0.15) the time averaged longitudinal inflow
becomes more linear and can be approximately by [31, 32, 1]:
λi = λ0(1 + kxr cosψ + kyr sinψ) (3.4)
20
The estimated values of first harmonic inflow considered in this paper is that one from Pitt and Peters
(1981) that has a good representation of the inflow gradient as functions of the wake skew angle and the
advanced ratio when compared to the experimental data [1, 32].
The BEM theory assumes that the blades can be divided into small elements that operate aerody-
namically as 2D airfoils and the aerodynamic forces can be calculated considering just the local flow
conditions [1]. Due to the simplicity of the theory the assumption that the airflow field around the airfoil is
always in equilibrium is necessary. To understand what is the minimum number of blade elements that
the analysis requires, it’s necessary to analyse how the total power changes in function of the number
of blade elements. Hovering and high speed forward flight conditions are considered. So, to compute
the thrust and power of the main rotor the component of the velocity parallel to the rotor and normal to
the blade leading edge UT and the component of the velocity perpendicular to the rotor UP have to be
considered:
UT (y, ψ) = Ωy + µΩR sinψ (3.5)
UP (y, ψ) = λiΩR+ yβ(ψ) + µΩRβ(ψ) cosψ (3.6)
Where y is the radial coordinate, Ω the rotor shaft speed and R the maximum radius of the blade. The
angle of attack α can be expressed in function of the twist angle θ and the inflow angle φ :
α = θ − φ φ = arctan(UPUT
) (3.7)
The incremental lift dL and drag dD are:
dL =1
2ρU2cCldy dD =
1
2ρU2cCddy (3.8)
where c is the chord, Cl = f(α) = f2(y, ψ) and Cd = g(α) = g2(y, ψ). So the equations of thrust T and
power P can be written as:
T =
∫∫NbdFz =
∫∫Nb(dL cosφ− dD sinφ) (3.9)
P =
∫∫NbdFxΩy =
∫∫Nb(dL sinφ+ dD cosφ)Ωy (3.10)
whereNb is the number of blades, dFx the force parallel to the rotor disk and dFz the force perpendicular.
Replacing equations 3.8, 3.5, 3.6 inside thrust and power equations 3.9 and 3.10 also the thrust and
power are expressed in function of azimuth angle ψ and radial position y. So, in a general form:
dFz = Fz(y, ψ)dydψ dFx = Fx(y, ψ)dydψ (3.11)
So the integrals become:
21
T =
Rmax∫Rmin
2π∫0
NbFz(y, ψ)dydψ (3.12)
P =
Rmax∫Rmin
2π∫0
NbΩyFx(y, ψ)dydψ (3.13)
To implement these equations in a computational code the integrations have to be replaced with
summations. So the blade is divided in a finite number of sections with a width of ∆y.
∆y =1− Rmin
Rmax
N(3.14)
Where Rmin is the minimum radius of the blade, Rmax the maximum radius and N the number of the
elements considered. To understand the minimum number of blade elements two analysis in hovering
and forward flight conditions are considered.
Figure 3.7: Power consumption of UH-60A in hovering
Also in forward flight the aerodynamic field for each azimuthal position is different. From the math-
ematical point of view it’s enough to perform an azimuthal integration between 0 to 2π to obtain exactly
the power consumption and the induced power. With a computational code it’s not possible to consider
infinite number of blade elements, so in the next error analysis it’s presented what is the minimum num-
ber of azimuthal position to have a good compromise between quality of the results and computational
time.
Figure 3.9 shows that at least the code needs the aerodynamic characteristics of 100 blades to
22
Figure 3.8: Power consumption of UH-60A in forward flight
Figure 3.9: Power consumption of UH-60A convergence increasing the azimuthal positions
represent the rotor power consumption.
There are also some corrections to take in account like hub loss [37, 1], tip loss [37, 1] and reverse flow
[1]. The only correction that is not considered is the compressibility effects in high speed forward flight
that increase the drag. At higher rotor advance ratios, there is a considerable amount of reverse flow on
the retreating side of the rotor disk. The region with reverse flow is characterized by UT ≤ 0. To take
in account the effects of the reverse flow it’s enough to change the sign of the drag in this region. The
23
coefficient of profile power can be computed as:
CPo =σCdo2π
2π∫0
1∫0
(r + µ sinψ)3drdψ − σCdo2π
2π∫π
−µ sinψ∫0
(r + µ sinψ)3drdψ (3.15)
For the hub and tip loss two corrections are considered, because BEM theory permits a finite lift to
be produced at the blade root and tip that it’s unrealistic. There is a factor B [37, 1] used to represent
the effective blade radius that can produce lift. The parameter is around 0.95 for the Hub relation while
for the tip loss the Prandtl tip-loss function is considered. The latter considers a solution to the problem
of the loss of lift near the tips taking in account the induced effects related with a finite number of blades.
So, in the equation there is the number of the blades Nb. The equation is considered for each azimuthal
position and then averaged. The two relations for hub and tip are pretty similar [1]:
f =Nb2
1− rrφ
(3.16)
f =B
2
r − rminrφ
(3.17)
and the correction factor F [1] is calculated for each situation as:
F =2
πarccos(e−f ) (3.18)
Another model taken in account is one for the fuselage drag that is important to understand how the
coefficient of thrust has to change to compensate weight and drag of the helicopter. The model used is
[1]:
D =1
2ρV 2fdragAeq (3.19)
where the drag parameter fdrag and the equivalent wetted area Aeq are related with the helicopter type.
The model has also a flapping consideration [1]. The hinge offset is neglected to simplify the calculation
and the second order differential equation is [1]:
??
β + β =1
IbΩ2
R∫0
Lydy (3.20)
where for definition the flapping β is function of the azimuthal position ψ:
β = β(ψ)
The code is implemented in Matlab and it’s able to compute, using some optimization tools, the
twist distribution that minimizes the power for each flight condition. The function from the optimization
tool of Matlab is ’Fmincon’ [34] combined with a global search [35] for the minimum. This function can
24
calculate a local minimum given initial conditions while the global search creates a system of different
initial conditions to obtain all the minimum solutions and after it will take the global one. To obtain the
flapping solution an ordinary differential equation solver ’ode45’ is used. This solver is based on Runge-
Kutta methods, a family of implicit and explicit iterative methods used in temporal discretization for the
approximate solutions of ordinary differential equation (ODE) [38].
3.2 Verification and Validation
To validate the method the flight data of the UH-60A helicopter is used [36, 39, 40, 41]. To make a
comparison between simulations and flight data all the typologies of power have to be taken in account.
The code calculates the main rotor induced power, main rotor profile power and fuselage parasitic power.
So an estimation of the tail rotor power is necessary. In table 3.1 some characteristics of the tail rotor
are presented:
Table 3.1: UH-60A Tail Rotor Characteristics [19]
Tail Rotor ParametersTail Rotor Radius 1.6764 m
Nominal tail rotor speed 124.62 rad/sTail Rotor Blade Chord 0.2469 mTail Rotor Blade Twist Linear −18
Blade Airfoil SC 1095Number of Blades 4
The tail rotor has to be able to generate a force to counterbalance the torque in the main rotor. Four
conditions are considered: hovering, low, medium and high speed forward flight. For each of these
conditions the coefficient of thrust of the tail and its power consumption is calculated. In all of these
simulations the power consumption of the helicopter increased of 5%. The distance between the hub
centre of the tail rotor and the rotor shaft in the UH-60A is 9.926 m. In all the calculations the weight of
the helicopter is W = 8322.4 kg [19].
Table 3.2: UH-60A Data [19]
Main Rotor ParametersMain Rotor Radius 8.1788 m
Nominal Main Rotor Speed 27.0 rad/sBlade Chord Lenght 0.5273 m
Blade Twist NonlinearBlade Airfoil SC 1095 / SC 1094R8
Number of Blades 4Blade Mass per unit length 13.92 kg/m
The blade twist of the UH-60A is non linear and presents some transition areas for the presence of
different airfoils. To simplify the calculation that behaviour is approximated by a linear blade twist of −16
25
and only the SC 1095 airfoil. Figure 3.10 shows the comparison between flight test and the results from
the method. Due to the limitation related to the linear model applied in low advanced ratio range, the
calculations are done for advanced ratio higher than 0.075.
Figure 3.10: Comparison Power consumption of UH-60A between Flight Test Data and Blade Elementtheory
The predictions, using this simple model, are in good agreements with the flight test data and for that
reason it’s verified the application of this method in the analysis of the helicopter performance.
3.3 Effect of Blade twist on Main Rotor Power
High twisted blades improve hover, vertical climb and low speed performance, for example, for mil-
itary helicopters nap-of-the-earth performance capability [42, 43]. From the aerodynamic point of view
in hovering condition, the result is a more uniform downwash velocity in the far wake that corresponds
to a reduction of induced power required [42]. In 1987 Keys et al. [42] conducted a test to quantify
the effect of twist on performance and aicraft vibrations. They considered a four bladed rotor with Mach
scaled composite blades and they tested it in a wind tunnel with two linear twist distributions: −11.5
and −17.3. Increasing the blade twist, in hovering condition, showed a reduction of 2.4% on power
required that corresponds in a 5% increase in useful load [42]. The experiment also showed that the
new redistributed downwash velocity in the inboard part of the rotor increased the aerodynamic load on
the fuselage of 6%. So, the benefit of the twist was reduced of 15% [42].
26
According to the theory [1], in hovering situation an hyperbolic variation of twist θ0y has the minimum
induced power. This solution is not physically possible because it’s not feasible to build a blade with this
shape (the angle near the root would be too big). Nevertheless a linear twist distribution can improve
the performance is a similar way as the hyperbolic twist variation.
The characteristics of UH-60A Sikorsky table 3.2 are used to show the effect of blade twist using
BEM theory for hovering condition and BET for forward flight. There are some differences from the real
helicopter: in this analysis only the SC 1095 airfoil is considered and the twist distribution is linear or at
least without twist. In fact the comparison is among 5 blade twist behaviours: no twist, −4 linear twist,
−8 linear twist, −12 linear twist ,−16 linear twist.
Figure 3.11: Total power for different linear twist distributions in Hovering
Figure 3.11 shows the effect of different twist distributions in hovering condition. This example used
a linear twist variation and the same profile SC 1095 on the blade. The coefficient of power decreases if
the linear twist slope is increased. This is not true for the simulation with -16 because when the blade
twist is increased, the inner part of the blade has an angle of attack higher than the angle of attack in
stall condition. If this section is between the hub and the root cut-out the effect is not presented but if the
blade twist increases above a certain value that section will produce less lift and more drag. Considered
a helicopter and a twist distribution along the blade there is only an optimum condition that minimizes the
power consumption. In the case of the UH-60A Sikorsky with SC 1095 airfoil and linear twist behaviour
the optimized solution is -13.5. So a solution with -16 presents higher twist slope than the optimal one.
This solutions came from an analysis with BEM theory with tip and hub losses and airfoil characteristics
from the simulation of Xfoil. Also, the reduction of the benefit of the download [42] is not considered.
In forward flight the effect of blade twist is different. In 1948 there was a study [44] that indicated that
higher blade twist reduced forward flight power based on flight test data. This conclusion looks wrong
27
but in that period the early helicopters were limited to 130km/h, in low speed forward flight highly twist
blades are able to reduce the coefficient of power as shown in Figure 3.12.
Figure 3.12: Total power for different linear twist distributions in low speed forward flight
Modern helicopters can easily reach a speed of 300km/h. The required blade twist distribution to
minimize the power consumption slowly decreases if the forward flight increases. The results are showed
in Figure 3.13 and Figure 3.14.
Figure 3.13: Total power for different linear twist distributions in medium speed forward flight
In Figure 3.13 it’s evident that after µ = 0.18 the linear twist distribution with −8 of slope becomes
the solution that requires less power. Finally, in Figure 3.14, in high speed forward flight a linear twist
behaviour of −4 presents better results.
Keys et all in 1987 [42] studied the four bladed rotor with Mach scaled composite blades also in
forward flight with the two linear twist distributions of −11.5 and −17.3. The solution with high twisted
blade presented a measured power increment of 5% at 330km/h. They also calculated that the helicopter
with −17.3 linear twist distribution presented the same power consumption of the −11.5 at 330km/h
around 322.5km/h. So the performance penalty due to twist was approximately 7.5km/h.
28
Figure 3.14: Total power for different linear twist distributions in high speed forward flight
In 2015 Han et all [19] studied the effect of variable rotor speed and variable blade twist to reduce
rotor power and improve helicopter performance. They developed an empirical aerodynamic model and
a CFD model. The empirical one includes a main rotor model, a fuselage model, a tail rotor model and
a propulsive trim method. The model developed was complex, in fact took in account also an elastic
deformation of the rotor blades. The inflow model was the Pitt-Peters [32]. They considered all of these
effect in the equations of motion based on the generalized force formulation and used the Newmark
integration method to integrate in time domain. With this model they analysed the UH-60A Sikorsky
according to the characteristics of the table 3.2. The result is presented in Figure 3.15.
Figure 3.15: Comparison among different twist behaviours [19]
29
Figure 3.15 confirmed the results found in the simulations with BEM theory and BET, specially the
solutions of the empirical model. In hovering the CFD model doesn’t show high differences among the
different twists but in high speed forward flight the negative effect of high twisted blade is shown.
30
Chapter 4
Results
4.1 Optimized Blade Twist
An optimal blade twist behaviour, in function of the forward flight, is determined by what kind of twist
is imposed. Some examples use a linear twist, others a non-linear blade twist, and for cases of a blade
with more than one airfoil each blade section could have a different blade twist. The more complex
is the function that describes that the blade twist more advantages are possible to achieve in term of
power reduction. According to recent studies [45], three sections blade is a solution to take in account
the different aerodynamic environments along the blade for each flight condition. For example in the
British Experimental Rotor Program (BERP) [45] the blade is divided in three sections with three different
airfoils. The central section has the main lifting airfoil, the RAE 9645, with a maximum lift coefficient of
about 1.55. However, this high lift coefficient is obtained at the expense of higher pithing moments that
is counterbalanced by the airfoil in the inner part, the RAE 9648, where the high maximum lift is not
so important. In the tip the airfoil is the RAE 9643 that is a low thickness-to-chord ratio to increase the
divergence Mach number. This is really important to increase the maximum speed of the helicopter. So
a solution with three different segments is commonly used and for each section there is a particular airfoil
that needs a different twist behaviour for the optimization. About the simulations, for each flight condition
between hovering condition and high speed forward flight with µ = 0.35 the twist slope that minimizes the
power consumption is found. The Pitt-Peters inflow model [32] starts to be in agreement with the flight
data around µ = 0.1, so there is a lack of simulations between hovering and this value. Another main
aspect is the accuracy of the model developed to simulate the helicopter, it is based on BEM and blade
element theory and cannot achieve the accuracy of models based on free vortex methods. ElQatary et
all [46], did a comparison between CFD and BEMT models. The differences in power consumption were
in a range between 2.2% and 7%, so the error has an order of some kWatt. Now, considering a flight
condition, for example µ = 0.2, and two solutions really close one to the other, with just a small difference
in the twist distribution, the difference in power consumption has an order of Watt or at least hundreds of
Watt. So, the power consumption will be really close and this will not affect the comparison with the flight
data and the validity of the code but the accuracy it’s not enough to determine which solution minimizes
31
the power consumption. So, the results present some fluctuations and it’s necessary to interpolate the
simulations to obtain the twist behaviours.
4.1.1 One section with linear twist
The first analysis considers a rectangular blade with only the SC 1095 Airfoil. The blade twist distri-
bution is linear from the root until the tip. In Figure 4.1 is showed how the twist slope should change to
minimize the power consumption.
Figure 4.1: Optimum linear twist slope for each advanced ratio µ for UH-60A
The hovering condition requires −13.5 of twist slope along the blade. The required slope decrease
really fast around µ = 0.1 and it’s around −8. Around µ = 0.2 the twist slope reaches −6 and it
increases really slowly with the advanced ratio.
4.1.2 Quadratic Twist
A quadratic twist behaviour along the blade with just one segment is also considered. The twist
follows a parabolic equation. There are three possible applications: the first considers the minimum
of the parabola exactly in the tip of the blade and the others a minimum inside the blade or without a
minimum. To simplify the calculation the minimum is considered at the end of the blade.
θ = ar2 + br + c (4.1)
where θ is the twist, r the adimensional radius and a,b,c the three coefficients that characterize the shape
of the parabola.
Imposing the first derivative egual to zero in the tip:
dθ(r = 1)
dr= 2a+ b = 0 b = −2a (4.2)
32
Equation 4.2 shows the dependence between the constants a and b.
Figure 4.2: a parameter of the parabolic equation - Optimum quadratic twist in function of advanced ratioµ for UH-60A
Considering a minimum exactly in the tip reduces the number of variables. So, it’s possible to study
a non-linear twist distribution without increasing the computational time to obtain the results.
Figure 4.3: Difference tip and root twist - Optimum quadratic twist in function of advanced ratio µ forUH-60A
In Figure 4.2 the change of this ’a’ parameter with forward speed is shown. And also in Figure 4.3 is
plotted the difference in degree between the twist in the root and the one in the tip. In hovering condition,
the concept with one section and linear twist requires −13.5 of twist slope that has to be reduced to
−6. So, the active twist control has to reduce the twist slope of 7.5. Figure 4.3 shows the difference of
the twist between root and tip for the quadratic twist distribution. In hovering the value is around 7 and
33
in high speed forward flight is less than 3. So, the active twist control concept for the quadratic twist has
to reduce the difference from root and tip only of 4. Figure 4.2 shows how the a parameter changes
with forward speed. This coefficient characterizes the shape of the parabola, in hovering is around of
20, so the twist has to change really fast increasing a little bit the forward speed.
4.1.3 Two sections with linear twist
In the following condition two linearly twisted segments are considered. In a situation like this one
there is an extra parameter that is how to divide the blade in two segments. In order to understand the
effects of the different division four conditions are presented: 40%, 50%, 60% and 70% of the blade
length for the inner part.
Figure 4.4: Optimum two linear twist slopes in function of advanced ratio µ for UH-60A. Blade division in40% - 60%
Figure 4.4 considers a division in 40% for the inner part and 60% for the outer part. Increasing the
forward flight in this condition weakly affects the inner part that looks almost constant and the outer has
the theoretical behaviour where the twisted ratio is reduced in high speed forward flight condition. For the
hovering condition was not possible to find the minimum using the optimization code due to the strongly
non-linear equation that describes the inflow along the blade (when the optimization code, during the
iteration, gets complex numbers fails). For this reason and due to the fact that all the simulations in
forward flight present almost constant twist, the hovering condition is calculated fixing the inner part of
the blade to the value of the forward flight and optimizing only the outer part. This solution couldn’t
represent the minimum power consumption in hovering but gives advantages in term of active twist
control implementation.
All of these simulations present higher oscillations that the other behaviours taken in account due to
the increase of one variable in the optimization code (root and tip twist angle for the one segment linear
twist and ’c’ and ’a’ parameters in the non-linear twist ratio).
34
Figure 4.5: Optimum two linear twist slopes in function of advanced ratio µ for UH-60A. Blade division in50% - 50%
Figure 4.5 divides the blade exactly in half. Also here it’s possible to assume only one linear twist for
all the flight conditions in the first segment.
Figure 4.6: Optimum two linear twist slopes in function of advanced ratio µ for UH-60A. Blade division in60% - 40%
Figure 4.6 is the opposite of Figure 4.4 and the approximation looks feasible yet.
Figure 4.7 considers 70% of the blade for the inner part and just 30% for the outer part. The inner part
has a different behaviour and it’s not possible to consider only one linear twist for all the flight conditions.
35
Figure 4.7: Optimum two linear twist slopes in function of advanced ratio µ for UH-60A. Blade division in70% - 30%
4.1.4 Two sections with linear twist and different airfoils
All of these simulations consider only the SC 1095 airfoil, in Figure 4.8 the blade it’s exactly divided
in two equal parts but two airfoils, the SC 1095 and the SC 1094R8 are used:
Figure 4.8: Optimum two linear twist slope in function of advanced ratio µ for UH-60A. Two airfoils SC1095 and SC 1094R8 with blade division 50% - 50%
Also here, the hovering condition is calculated fixing the inner part to the forward flight value and just
optimizes the outer part twist slope.
The results are quite interesting for the engineering point of view. Thinking about an implementation on
a real rotor could be possible to fix the inner part of the blade and introduce an active twist control only
in the outer part. This kind of solution could reduce the power consumption of the helicopter without
36
introducing too much weight and other complex elements inside the blade. The concepts with linear or
quadratic distributions require an active twist system that has to modify the twist behaviour from the root
to the tip of the blade. According to some studies [18, 17], a possible solution is to use piezoelectric
actuators that have to be spread along the blade. Each one required a certain amount of power to
generate the electric field to use the actuators. But, these concepts with two segments present an inner
part of the blade that doesn’t require a new redistribution of the twist and for this reason it’s necessary to
place the actuators only in the outer part. The system requires less actuators that means less electrical
power and less weight.
4.1.5 Three linear segments
The last simulation is directly related to the blade shape of the UH-60A: the idea is to consider three
different sections with linear twist and two different airfoils SC 1095 and SC 1094 R8. The real helicopter
has a non-linear twist behaviour divided in three sections but in this analysis three linear segments are
considered. In Figure 4.9 [41] the blade shape of the Sikorsky UH-60A Black Hawk is presented.
Figure 4.9: Blade planform and shape of UH-60A
To simplify the model all the transitions areas are not considered and also the final taper is neglected.
The tip shape is related to compressibility effects that in this model are not considered. So, the inner
part represents the 50% of the blade, then the central part the 35% and the outer part the last 15%.
The solution shows a behaviour similar to the two linear segments, the inner part presents a constant
twist for all the flight conditions. The hovering condition is calculated fixing the inner and outer part with
the same values of the forward flight for the reasons presented before. Also in this simulation the inner
and outer parts present twist distribution almost constant. The central segment shows a twist behaviour
that decreases the slope increasing the forward speed.
This simulation presents, as the other, some oscillations in the results, the complexity of the model in
this condition is related with four parameters that have to be optimized and these are the tip, the root and
the two nodes that connected the three segments. Increasing the forward flight all the results have less
wiggling because the inflow model used describes in a better way the reality with high speed forward
flight.
37
Figure 4.10: Optimum 3 linear twist slopes in function advanced ratio µ for UH-60A
Thinking about an engineering application of this solution is necessary to implement an active twist
control just for the central segment of the blade and it’s possible to fix inner and outer parts.
4.2 Optimum Solution
The Sikosky UH-60A Black Hawk has a blade divided in three segments with two different airfoils.
The real twist is presented in the following picture:
Figure 4.11: Twist behaviour Sikorsky UH-60A Black Hawk [? ]
The twist in the real helicopter is non-linear but can be approximated by a linear twist distribution with
-16 of slope. Now, it’s possible to compare the power consumption between the flight test data and the
38
three linear segments simulation.
Figure 4.12: Comparison Flight Test Data and Three linear segments
The coefficients of power in function of the advanced ratio µ. The picture with the three linear seg-
ments simulation shows that an optimized twist for each flight condition could really reduce the power
consumption of an helicopter are presented in Figure 4.12.
This is an example to show that there are some advantages, now could be interested calculated how
large are the reductions of power for each solution considered. In the following table some results are
presented:
Table 4.1: Reduction of power between Fixed linear twist and simulations
Simulation concept Reduction of PowerLinear twist -1.34%
Quadratic twist -8.65%2 Segments with 40% - 60% blade division -4.86%2 Segments with 50% - 50% blade division -4.46%2 Segments with 60% - 40% blade division -3.44%2 Segments with 70% - 30% blade division -2.55%
2 Segments, 2 airfoils with 50% - 50% blade division -2.21%3 Segments, 2 airfoils with 50% - 35% - 15% blade division -6.09%
Table 4.1, presents a comparison among all the simulation concepts with a fixed linear twist distribu-
tion of -13.5. The reference twist uses the optimal condition in hovering of the UH-60A with only the SC
1095 airfoil. Using a linear twist with active twist control can reduce the power consumption of 1.34%.
The other solutions give better results, in fact the concepts with two segments can reach a reduction of
4.46%. All of these concepts fix the inner part twist and use an active twist control only for the outer
part. The two segments with 40% - 60% blade division can get higher performance in comparison to
the others because the controlled part is higher (the 60%). At the end there are two concepts: quadratic
39
twist behaviour and three segments with linear twist that are the two best conditions with 8.65% and
6.09% of power consumption reduction.
In Figure 4.13 linear twist, quadratic twist and three linear twist segments are presented.
Figure 4.13: Power consumption in function of advanced ratio µ for different twist behaviours.
The simulations show, for each flight condition, the power consumption of the UH-60A in function of
the advanced ratio µ. The blade twist changes according to the behaviours calculated in the previous
section, to minimize the power consumption.
The figure represents a graphic point of view of the power reduction achievable with the different con-
cepts. Already is known that the linear twist distribution gives the lowest benefit while the other conditions
the highest. Looking only to the hovering the three linear twist segments gives the lower power consump-
tion but after µ = 0.25 the quadratic twist distribution has better values. In term of power consumption
reduction the non-linear twist represents the best condition but from an engineering point of view there
are some other aspects:
• The non-linear concept requires an active twist control that involves all the blade from the root to
the tip while the three linear twist segments can just use a system for the second segment for only
35% of the blade.
• In the non-linear concept the twist morphing requires higher variations of the parameters that could
produce problems related to fatigue and deformations.
• The weight, the power consumption and the forces required of the actuator in the three linear twist
segments are lower because the system has to control only the 35% of the blade.
After these considerations, looks easier to implement an active twist control for the three linear twist
sections than for the non-linear twist distribution.
40
Chapter 5
Concept
5.1 Smart Blades
Active systems can adapt the blade aerodynamic depending on the flight conditions. These are
integrated inside a blade to enhance the performances of modern helicopters. Most concepts improve
the lift to balance the retreating side, the stall behaviour, vibrations, noise or power consumption. There
are three main ideas: flaps, that modify the flow with tabs and slats, morphing systems that change the
blade shape and finally the flow control around the blade.
On helicopter blades, flaps are not designed as a primary control surface but they modify the pro-
duced lift to reduce vibrations [3, 7, 8]. For example an active trailing edge flap can control dynamically
[3, 47], with a deflection of few degrees, the load on the rotor disk. Due to the cyclic variation of loads,
specially in forward flight, the vibration effects occur at 1, 2, 3 and 4 times the rotational frequency of
the blade (for a 4 blades rotor) [48]. For this reason, the actuator has to be actuated with the same fre-
quencies of the loads to cancel them. The maximum reduction obtained was around the 80% [49, 47].
Another application for the trailing edge flap is the noise reduction produced by the blade vortex interac-
tion (BVI) [8, 47, 50]. This is possible with an individual control of each blade.
Another kind of trailing edge system is the Gurney flap that deploys at 90 degrees a small tab that cre-
ates low pressure to bring the separation point closer to the trailing edge [51] but with a drag penalty
[51, 52].
The second concept is the active flow control, instead of modify the geometry it acts directly on the
boundary layer by a re-energization or a suction to bring the separation point closer to the trailing edge
[13]. The actuators are placed in cavities with a jet system. The wind tunnel tests have shown that these
jets can improve the aerodynamic performance if used at a specific frequency [13].
The third and last concept is based on morphing blades. The idea is to modify the aerodynamic
characteristics by a continuous change of the blade shape in the same way the birds and other flying
animals do. The first concept is a variable droop leading edge, it wants to modify the curvature of the
airfoil for high angle of attack to alleviate the dynamic stall [6]. This system can reduce of 50% the drag
produced by the dynamic stall in the retreating blade region. The system reduces the angle of attack
41
to avoid the dynamic stall but in this way the helicopter maximum speed is reduced due to a decrease
in lift. Another way to control the produced lift is a system that can change the camber of the airfoil.
The lift is increased for the same chord length [47]. The last morphing concept is the active twist, it can
be used to modify statically the blade twist to optimize the rotor at each flight condition or a cyclic twist
control to actively damp vibrations [53]. Early experiments tried to change the twist in the root of the
blade [9, 11, 53], after consider a distributed actuation system along the blade [9, 53]. Almost all the
studies demonstrate a vibration reduction, Thakkar shows up to 69% of reduction [11] and wind tunnel
test a 95% of reduction [9].
Figure 5.1: Active Twist Control [17]
According to these studies [17], an on-blade active control is feasible and efficient to fix typical rotor
problems. Despite this, the helicopter industry has some uncertainties about it due to the high devel-
opment costs, complexity and the risk for a manned flight vehicle. Beside that these developments
represent an important milestone for the Unmanned Aerial Vehicles (UAVs) where the risk is reduced
[17].
5.2 Constraints
All the smart systems embedded on a blade present some constraints due to the blade motion and
related to a long term use of them. Helicopter rotors are designed to work under very high centrifugal
loads. For this reason the weight has to be the minimum possible [6]. Another aspect is the space
available for a smart system, the structural material fills the bigger part of the section and the only space
available is near the trailing edge. Also, all the weight introduced behind the aerodynamic centre has to
be balanced by an extra mass in the leading edge [2]. Very distributed and light systems as the active
twist control can satisfy easily this kind of constraint but for example the droop leading edge needs a
heavy mechanism to deform the leading edge of the profile [54].
In the design of active systems it’s not possible to ignore that the actuator has to work under a centrifugal
acceleration that strongly depends from the radial position. The systems have to be placed near the tip to
maximise the effect on the blade aerodynamic but unfortunately the centrifugal forces there are higher.
42
The only solution is to use a very light system. In addition to the centrifugal loads there are a lot of
vibrations that have to be taken in account to avoid premature failure of the component.
Helicopter blades are designed for 10 000 flight hours [55], an active system need to have a design
life superior to the blade without losing performance through the operational life. In typical applications
both pneumatic and hydraulic system are used and also recently electrical mechanisms have reached
equivalent level of safety [56]. Beside the design life of a component there is also the event of failure.
This could be really dangerous for all the systems that modify the geometry of the airfoil and it’s important
to guarantee that the helicopter remains stable and controllable. This is not the case of the active twist
control because the only effects in event of failure are reductions in performance and control of vibrations
[6].
The last aspect is connected to the power needed to operate the actuator. It’s possible to provide a
certain amount of power with a negative effect related to the increase of the rotor hub complexity [57].
Large helicopter blades include a de-icing system for high altitude and a typical power it’s around 1kW
for each blade. This could give an idea about the maximum power consumption for the active systems.
5.3 Piezoelectric Actuation Systems
At the end of the 19th century, Pierre and Jacques Curie discovered that some materials have the
property to produce electrical energy if deformed [58]. This is the direct piezoelectric effect. It’s also
possible the opposite, a piezoelectric material subjected to an electric field deforms according to the
magnitude of the electric field and this is called converse piezoelectric effect.
It’s possible to characterize a piezoelectric material by strain constants dij where i represents the elec-
tric field direction and the j the deformation direction. Piezoelectric materials present below the curie
temperature TC a dielectric moment along one direction. In Figure 5.2 is the axis number 3:
Figure 5.2: Piezoelectric Deformation [6]
If the electric field is applied in the direction of the polarization the material extends along that direc-
tion and contracts in the other two directions. Of course, changing the direction of the electric field the
behaviour is the opposite. In almost all the piezoelectric materials the two other directions perpendicular
to the moment of polarization have the same strain constants.
43
Taking in account the analytical formulation with only external electric field applied, the strain vector can
be expressed as:
ε1
ε2
ε3
γ23
γ31
γ12
=
0 0 d31
0 0 d32
0 0 d33
0 d24 0
d15 0 0
0 0 0
·
Ev1
Ev2
Ev3
(5.1)
where the Evi are the components of the electric field, εi and γij the normal stress and shear compo-
nents.
According to previous studies [47, 59, 60], the zirconate titanate piezoceramic (PZT) constitutes
the best materials for actuation applications because they have better displacement capabilities but
they are not convenient for integration. For this reason the Active Fibre Composite (AFC) and Macro
Fiber Composites (MFC) are preferred for the active twist control. They consist of piezoceramic fibers
embedded inside polymer substrate. AFC has circular fibers while MFC rectangular fibers to improve
the contact with the electrodes. These electrodes are glued on the top and the bottom of the composite
material and there are two ways to build the actuator in function of which dij constant the concept needs.
Figure 5.3: Continuous Electrode vs Interdigitaded Electrode [61]
The continuous electrode has a direction of polarization perpendicular to the composite fibers and
works in d31 mode while the new shape of the interdigitated electrode generate an electric field in the
direction of the fibers. The electric fields are shown in Figure 5.4.
The piezoelectric properties are complex to calculate, there are some homogenization techniques
[59, 61]] to derive the macroscopic behaviour of the component.
The different shape between the fibers in AFC and MFC greatly influences the composite performances.
The MFC actuator with rectangular fibers has more area in direct contact with the interdigitated elec-
trodes, which minimizes electric field attenuation by the low dielectric epoxy matrix [18]. In Figure 5.5 is
44
Figure 5.4: Continuous Electrode vs Interdigitaded Electrode Electric field [61]
comparized the free-strain between MFC and AFC.
Figure 5.5: Free Strain Comparison MFC and AFC [18]
The Macro Fibers Composite actuator has performance 2.5 times higher than the Active Fibers Com-
posite.
A new technology is developed by Physik Instrumente that uses a d33 patch actuator as shown in Fig-
ure 5.6.
A. Paternoster [6] did, in his thesis, a comparison between the MFC and the d33 patch actuator from
Physik Instrumente. The difference is the electric field generated in the two conditions: the MFC electric
field is not homogeneous and it depends by the shape of the interdigitated electrodes while the patch
actuator electric field goes straight through the material as shown in Figure 5.7.
The actuators that Paternoster considered, are a MFC from Smart-Material and a PICMA piezoelec-
tric stack actuator from Physik instrumente. All the properties are listed in the following table Table 5.1:
The d33 coefficient is a parameter related to the chemical capability of the piezoelectric material,
the equivalent one takes also in account the superior design of the patch actuator. The MFC with the
45
Figure 5.6: Micro Fiber Composite Actuator [6]
Figure 5.7: Electric Field Comparison between MFC and patch actuator
interdigitated electrode cannot efficiently apply an electric field in the same way of the patch actuator.
The comparison gives some observations:
• The Physik Instrument d33 patch actuator requires 10 times lower voltage to achieve the same
strain of the MFC, the explanation is related to the small spacing between the electrodes of the
patch actuator.
• The equivalent d33 coefficient of the patch actuator is a little bit higher than the MFC actuator.
• The free strain of the MFC actuator is higher than the patch actuator due to the higher maximum
voltage that it’s possible to apply to the actuator.
In conclusion the two actuators present similar performance, the patch actuator requires 10 times
less voltage to obtain same strain and could be a good solution if there are integration problems to
transmit large voltages through the rotor hub. The MFC requires higher voltage but also has an higher
free strain at the maximum operational voltage.
5.4 Application
In chapter 4, the results show that implement an active twist control can reduce the induced power.
The solution with three linear twist segments is the most convenient because requires an active twist
control only for the 35% of the blade and reduces the power consumption of 6.09% in comparison to a
fixed twist blade. According to bibliography [18, 17], the actuators have to be placed along the blade with
46
Table 5.1: Actuators Properties [6]
MFC 4010-P1 PI d33 actuatorActive length [mm] 40 15.5Active width [mm] 10 4.6
Active thickness [mm] 0.3 0.3Length [mm] 54 36Width [mm] 22 8
Maximum Voltage [V] 1500 130Electrode Layer Spacing [µ m] 500 55
d33 coefficient [pC/N] 460 400equivalent d33 coefficient [pC/N] 440 450
Capacitance [nF] 1 100Free strain per volt [µ strain/V] 0.87 8.2
Max free strain [µstrain] 1300 1100
an orientation +45 or −45 to maximize the effect. Figure 5.8 represents a UH-60A rotor blade and the
black box is the 35% where the active twist control has to be placed.
Figure 5.8: UH-60A rotor blade with black box to identify the active part
The simulation for the three linear twist segments for the UH-60A Black Hawk shows in the second
section a twist slope range from −5 and −1 from µ = 0 to µ = 0.35. So, there are three main concepts:
• Build the blade optimized for hovering condition and an active twist control that reduces the twist
slope increasing the forward flight. The actuators have to be placed oriented from the leading edge
to the trailing edge for the top of the blade and the opposite for the bottom (always with 45). This
is important because the higher coefficient d33 it’s only for the elongation. With an inversion of
the electric field the deformation is related to the d31 coefficient. In hovering condition the active
twist control doesn’t require power and in case of failure the rotor remains optimized for hovering
condition.
• Build a blade optimize for high speed forward condition and then increases the twist slope for all
the other flight situations. The orientations of the actuators is the opposite of the previous case.
• Build a blade optimize for an intermediate condition around −3. The advantage is to use the
actuators in the two directions but only for half of the range needed in the two other cases. The
actuators in both top and bottom are placed in the same directions but top and bottom are going
47
to work always with different mode: one in d31 and the other d33. In this way the system works
exactly in the same condition with negative or positive polarization.
The active twist control has to be implemented on a section with length = 2.86 m and the concepts
consider distribution of actuators in the top and bottom of the blade as shown in Figure 5.9.
Figure 5.9: Airfoil with actuators
It’s possible to do a simplify calculation with the three concepts. In the analysis is considered the
best condition where every actuator can reach the maximum strain. This is important to understand the
theoretic minimum number of actuators for each concept.
Table 5.2: Actuators and Concepts Comparison
MFC 4010-P1 PI d33 actuatorMaximum Rotation for each Actuator [] 0.1 0.0325
Concept 1: Required Number of Actuators 40 123Concept 2: Required Number of Actuators 40 123Concept 3: Required Number of Actuators 20(one side) - 28(two sides) 62(one side) - 84(two sides)
Maximum Number of Actuators 158 714
The maximum rotation for each actuator represents the changing in twist due to an actuator placed
on the top or on the bottom of the blade. It considers free strain elongation of the actuators and the
distance from the rotational axis of the airfoil. The first two concepts need 4 of changing twist in total
and require a minimum number of actuators to achieve this. The last concept requires only 2. The
effect in d31 mode is around an half the d33 mode. The minimum number is done considering the same
number of actuators for bottom and top and in total are 28 for the MFC and 84 for the patch actuator.
This calculation shows that the patch actuator can achieve the same result with an high number of ac-
tuators but less voltage and less dimensions. The MFC will occupy the 25% of the first two concepts
and the 18% for the third while the patch actuator requires the 17% and the 11% of the available space.
About the power consumption using the patch actuator the electrical power consumption is reduced of
70%.
Until now, the concept is not complete but as future work the idea is to do a FEM analysis to under-
stand the twist changing that the system can induce. There are some studies, for example by Koalovs,
Barkanov and Gluhihs [18], that consider all the material properties of each part of the blade (spar, lead,
foam and skin) with embedded the MFC actuator. They did a 3D finite element model in ANSYS and
analysed the piezoelectric effect with the thermal analogy between piezoelectric effect and thermoelastic
effect. So, the applied electric field is modelled as a thermal load. The analogy is:
48
αij =dij
∆ES(5.2)
where α is the thermal strain, d the piezoelectric coefficient and ∆ES the spacing between the elec-
trode. The steady-state thermal analysis determines the torsion angle of the rotor blade as shown in
Figure 5.10.
Figure 5.10: Torsion Angle of the Thermal Analysis [18]
49
50
Chapter 6
Conclusions
In this thesis variable blade twist is used to reduce rotor power and improve helicopter performance.
A simplified model based on BEM and blade element theory to calculate the aerodynamics loads of the
helicopter is used [1]. The database of the airfoil is obtained by a simulation using Xfoil for 2D profiles
[30]. The inflow models are analytical, one for hovering condition [31] and the Pitt - Peters model [32]
for forward flight that gives higher correlation with flight data for advanced ratio higher than 0.1. All the
effects due of tip and hub loss, reverse flow and flapping behavior are also considered [37, 1]. Several
solutions are taken into account and all of those can reduce the power consumption and optimize the
helicopter in each flight condition: one section with linear or quadratic twist behaviour, two sections with
linear twist distributions but different blade divisions with one or two airfoils and finally the three sections
linearly twisted with two airfoils. The best solution comes from a compromise between complexity, weight
and power consumption of the active twist control system and benefit produced by the actuation.
In term of power consumption reduction a comparison among all the concepts is done. The reference
condition is a fixed linear twist distribution of −13.5 of slope. The highest power consumption reduction
comes from the simulation with one section and quadratic twist distribution and the result is 8.65%. The
concept with linear twist along one section gives the lowest power consumption reduction around 1.34%.
Another concept with good result is the three linear sections that can provide a power consumption
reduction around 6.09%.
In current studies, ATC system is used primarly to reduce the vibrations and it’s a piezoeletric com-
posite material able to create torques if connected with electricity.
All the concepts with two or three sections show something interesting, the twist slope of the inner part
of the blade remains almost constant changing the advanced ratio and in the three sections concept also
the outer part remains almost with constant slope. So they don’t require an active twist control system
that has to be spread along all the blade. Taking in account the main interesting concepts, the three
sections and the quadratic there are some differences: the non-linear concept requires an active twist
control that involves all the blade from the root to the tip while for the three linear sections it’s enough to
placed the ATC in the central section for just 35% of the blade length. So, there are advantages about
electrical power consumption and reduction of the weight of the actuators.
51
The ATC concept is based on piezoelectric actuation. The idea is to place the actuators in both top
and bottom of the blade to maximize the effect. The two actuators that could be used are the Micro
Fiber Composites (MFC) and the patch actuator from Physik Instrumente. The difference between the
two actuators is in the electric field generated, the first one has a not homogeneous electric field that
depends by the shape of the interdigitaded electrodes while the patch actuator electric field goes straight
through the material. Due to this property, the patch actuator requires 10 times less voltage to achieve
the same strain of the MFC and it is also smaller.
Finally, an application of these systems on the three linear twist segments is done. The actuators
have to be placed only for the 35% of the blade with an orientation of +45 or −45 to produce the
torque needed. In hovering condition the required twist slope in the central segment is around −5 while
in high speed forward flight is −1. So the range for the actuators is around 4 of twist slope. There are
three concepts: optimize the blade in hovering condition and then actuation for all the other situation,
optimize for high speed forward flight or a middle condition that requires only half of the range but in the
two directions.
A simple analysis to understand the theoretical minimum number of actuators is done. The hypotesis is
that each actuator can reach the maximum stain (the free strain) also if it is placed on the blade. The
first two concepts with MFC require the 25% of the blade area, the last one in a middle condition only
the 18% while the patch actuator requires the 17% and 11%. In term of electrical power consumption
the patch actuator can achieve a reduction of the 70% in comparison to the MFC actuator.
6.1 Future Work
The first step, in future works, is analysis of the MFC and patch actuators embedded in a blade and
the torque and morphing that are able to produce. This is necessary to understand what are the number
of actuators that are needed. The calculations done in this thesis took in account only the Sikorsky
UH-60A Black Hawk and the results strongly depend on initial parameters that characterize helicopter
and main rotor. So, a possible application has to be done only on this helicopter or on a scaled model.
There are some aspects that are not taken in account for example how to provide electrical power to the
blade and how the inboard computer has to control the actuators.
52
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58
Appendix A
Appendix: Matlab Code
A.1 Main Code
%========================================================================
% Blade Element Momentum Theory (BEMT) and Blade Element Theory (BET)
% Marco Lonoce - November 2016
%
%========================================================================
%% Acquisition of the informations about the Airfoil
clear all %#ok<CLSCR>
close all
clc
global p CT real solidity matrix dr d2r r matrix phi Ut azimut lambda PP solidity Nb CT req CL real ...
slope exp r p clalpha p cdalpha p clalpha2 p cdalpha2 theta mu R min R max
% Open Xfoil file - Read Only
fid = fopen('sc1095.txt', 'r');
for l=1:12
tline = fgetl(fid); %#ok<NASGU> % Skipping the first 12 rows
end
a = fscanf(fid,'%f', [7 Inf]);
fclose(fid);
clear tline l fid
%% Characterstics of the Airfoil
alpha = a(1,:); %Angle of Attack
CL = a(2,:); %Coefficient of Lift
59
CD = a(3,:); %Coefficient of Drag
clear a
%% Calculation of Cl alpha - Linear Regression
x = alpha(35:111)';
y = CL(35:111)';
slope exp = (x\y)*(180/pi);
clear x y
%% Interpolation
p clalpha = polyfit(alpha,CL,5);
p cdalpha = polyfit(alpha,CD,7);
%% Other airfoil
% Open Xfoil file - Read Only
fid = fopen('sc1094r8.txt', 'r');
for l=1:12
tline = fgetl(fid); %#ok<NASGU> %
end
a = fscanf(fid,'%f', [7 Inf]);
fclose(fid);
clear tline l fid
%% Characterstics of the Airfoil
alpha = a(1,:); %Angle of Attack
CL = a(2,:); %Coefficient of Lift
CD = a(3,:); %Coefficient of Drag
clear a
%% Interpolation
p clalpha2 = polyfit(alpha,CL,5);
p cdalpha2 = polyfit(alpha,CD,7);
%% Input Data
n = 200; %Number of radial stations to calculate
Nb = 4; %Number of blades
c tip = 0.5273; %Blade Chord Length - tip - meters
60
RPM = 257.831; %Rev / Min
W = 8322.4; %Mass of the helicopter in kg
R max = 8.1778; %Radius in meters
R min = 1.39; %Root cut-off in meters
h = 0; %Altitude in meters
Vc = 0; %Climb velocity - m/s
%% Preliminary Calculations
rho = 1.225*((288-0.0065*h)/288)ˆ4.2561; %Density kg/mˆ3
rev = RPM*(2*pi)/60; %Radians per second
dr = (1-R min/R max)/n; %Radial increment - adimensional
r = ((R min/R max):dr:1); %preallocating the r-range
Ad = pi*((R max)ˆ2); %Disk Area - mˆ2
d2r = pi/180; %deg to radians
CT req = (W*9.807)/(rho*Ad*(rev*R max)ˆ2); %Thrust coefficient requested in hovering
taper = 1; %Vector with all the taper ratios
lambda climb = Vc/(rev*R max); %Adimensional inflow velocity for climb
%% Characteristic of the blade for Flapping
mass meter = 13.92; %Mass for meter kg/m
Ib = sum(mass meter*R max*((r*R max-R min).ˆ2)*dr); %Inertial Moment
%% Parameters for the convergence and the parametric analysis
toll = 0.0025; %tollerance in deg for the calculation of the hovering condition CT
iteration = 5000; %number of iteration for the convergence
toll result = 0.075;
step = 0.5; %Step for parametric analysis of the twist (changing in the root)
t s = 5; %Fixed starting value for the tip
theta start = t s; %Starting value for alpha in section root - in deg
theta final = theta start+15;%Deg - Final value in the root section
row index = length(taper)*(length(theta start:step:theta final));%Solutions Matrix
format long
%% Forward Flight - Input Data
Vf = linspace(0.1*rev*R max,0.25*rev*R max,20); %Flight Range
Vf = [0 Vf]; %Including hovering condition in the analysis
graph mat = zeros(length(Vf),10);
61
hbar = waitbar(0,'Calculation phase'); %Wait Bar
%Initialization of the flapping coefficents
beta0 = 0*d2r;
A = 0*d2r;
B = 0*d2r;
for i 2 = 1:length(Vf)
if Vf(i 2) == 0
mu = 0;
end
%Do the calculation only if there is forward flight
if Vf(i 2)~=0
mu = Vf(i 2)/(rev*R max); %Advanced Ratio
f drag = 0.020; %Coefficient of drag for parasitic drag
A eq = 4.65; %Wetted Area
drag ext = 0.5*rho*(Vf(i 2)ˆ2)*f drag*A eq; %Parasitic Power Estimation
a tpp = -atan(drag ext/(W*9.807)); %Inclination of the TPP - function of the forward speed
azimut = linspace(0,360,100); %Azimut in degree
azimut = azimut*d2r; %Azimut in radians
% Flapping Behavior
%Control to avoid NaN if the previous iteration fails
if isnan(beta0)
beta0 =0;
end
if isnan(A)
A = 0;
end
if isnan(B)
B=0;
end
beta = beta0 - A*cos(azimut) - B*sin(azimut);
dbeta = A*sin(azimut) - B*cos(azimut);
% Coefficient of Parasitic Power
CP parassite = (drag ext*Vf(i 2))/(rho*A eq*(rev*R max)ˆ3);
end
62
% Forward Flight - Inflow Model Calculations
if Vf(i 2)~=0
%Average Inflow Model - Momentum Theory
syms lam in
fun = @(lam in) lam in - CT req/((2*cos(a tpp))*(sqrt((mu*cos(a tpp))ˆ2+(lam in)ˆ2)));
lam2 = solve(fun,lam in);
lam2 = double(lam2);
lam = lam2(2);
%Linear Model - Pitt & Peters (1981)
skew angle = atan(mu*cos(a tpp)/(lam - mu*sin(a tpp)));
Kx = (15*pi/23)*tan(skew angle/2);
lambda PP = lam*(1+Kx*(r')*cos(azimut));
%Velocities
r matrix = (r')*ones(1,length(azimut));
azimut matrix = ones(length(r),1)*azimut;
ones mat = ones(length(r),length(azimut));
Ut = rev*r matrix*R max + Vf(i 2)*sin(azimut matrix);
Up = (lambda climb*ones mat+lambda PP)*(rev*R max) ...
+ (r')*dbeta*R max + mu*rev*R max*ones(length(r),1)*(beta.*cos(azimut));
phi = atan(Up./Ut);
mean phi = mean(mean(phi));
dim phi = size(phi);
%Adimensional Velocity
dim = size(r matrix);
dim1 = dim(1);
velocity adim = r matrix+mu*ones(dim1,1)*sin(azimut);
end
% Optimization phase
if mu == 0
solution matrix = zeros(row index,7);
riemp hor = 1;
%Cycle for different taper
for i = 1:length(taper)
63
tap = taper(i); %Taper considered in this iteration
%Chord and solidity are defined with taper
c = c tip + r*(c tip*tap - c tip);
solidity = (Nb*c)/(pi*(R max)); %Local Solidity in function of the chord
%Parametric Analysis for Hovering
for k = theta start:step:theta final
error cond = -1; %Flag for calculation of hovering condition
CT real = 1; %initial value --> too big
contatore = 0; %initialization for convergence
slope fix = t s - k;
k2 = k;
while (error cond > toll result | | error cond < 0) && contatore < iteration
theta1 = k2 +0*r*0.5;
theta2 = k2 +slope fix*r*0.35;
theta3 = k2 + slope fix - 1.5*r*0.15;
dim r = size(r);
theta hov = zeros(dim r(1),dim r(2));
for i boom=1:dim r(1)
for j=1:dim r(2)
if r(i boom,j) < 0.5
theta hov(i boom,j) = theta1(i boom,j);
elseif r(i boom,j)>=0.5 && r(i boom,j)<=0.85
theta hov(i boom,j) = theta2(i boom,j);
elseif r(i boom,j)>0.85
theta hov(i boom,j) = theta3(i boom,j);
end
end
end
F=1;
%Iteration for tip and hub losses
for lambda index=1:10
lambda = ((solidity.*slope exp)./(16.*F)).*(sqrt(1+ ...
((32*F)./(solidity.*slope exp)).*theta hov.*r*d2r)-1);
lambda(end) = lambda(end-1);
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alpha real = theta hov*d2r - lambda./r; %alpha in radians
alpha real = alpha real/d2r; %alpha in deg
CL real = polyval(p clalpha,alpha real);
CD real = polyval(p cdalpha,alpha real);
for i boom2=1:dim r(1)
for j=1:dim r(2)
if r(i boom2,j)>=0.5 && r(i boom2,j)<=0.85
CL real(i boom2,j) = polyval(p clalpha2,alpha real(i boom2,j));
CD real(i boom2,j) = polyval(p cdalpha2,alpha real(i boom2,j));
end
end
end
f = (Nb/2)*((1-r)./lambda);
F = (2/pi)*acos(exp(-f));
f2 = (0.95/2)*((r-R min/R max)./lambda);
F2 = (2/pi)*acos(exp(-f2));
end
CT real = sum(0.5*solidity.*F.*F2.*CL real.*(r.ˆ2)*dr);
error cond = (CT real - CT req)/CT req;
%Repeat the Iteration if the CT is not guarantee
%Original Code converges faster
if error cond < 0
k2 = k2 + toll;
elseif error cond >= toll result
k2 = k2 - toll;
end
contatore = contatore + 1;
end
if contatore == iteration
fprintf('Error in the convergence, %.2f %.4f %.4f \n', tap,k2,slope fix);
else
%Induced coefficient of power
CP ind = sum(lambda.*0.5.*solidity.*CL real.*(r.ˆ2)*dr);
%Profile drag
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CP profile = sum(0.5*solidity.*CD real.*(r.ˆ3)*dr);
%Drag
CP tot = CP ind + CP profile;
if riemp hor == 1
CP ref = CP tot;
end
CP err = (CP tot - CP ref)*100/CP ref;
%Solution Matrix
solution matrix(riemp hor,:) = [tap k2 slope fix ...
CP ind CP profile CP tot CP err];
end
riemp hor = riemp hor + 1;
end
end
else
tap = taper; %Taper considered in this iteration - Rename
c = c tip + r*(c tip*tap - c tip);
c matrix = (c')*ones(1,length(azimut));
solidity = (Nb*c)/(pi*(R max));
solidity matrix = (solidity')*ones(1,length(azimut));
error cond = -1;
cont = 0;
while abs(error cond) > toll result
while abs(error cond) > toll result | | exitflag == -2 | | fval < 0 | | cont > 5
objectivefunction = @funtominimize;
x0 = [10 8 6 4 ];
LB = [0 0 0 0];
UB = [20 20 20 20];
constraintfunction = @fmc constraint;
gs = GlobalSearch;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon','x0',x0,'objective',objectivefunction, ...
'nonlcon',constraintfunction,'lb',LB,'ub',UB,...
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'options',opts);
[x,fval,exitflag] = run(gs,problem);
error cond = (CT real - CT req)/CT req;
cont = cont+1;
end
f = (Nb/2)*((1-r matrix)./(r matrix.*abs(phi)));
F = (2/pi)*acos(exp(-f));
f2 = (0.95/2)*((r matrix-R min/R max)./(r matrix.*abs(phi)));
F2 = (2/pi)*acos(exp(-f2));
MofLift = sum(0.5*rho*(velocity adim.ˆ2)*((rev*R max)ˆ2) ...
.*CL real.*F.*F2.*(R max*r matrix - R min)*dr,1)/(Ib*revˆ2);
p = polyfit(azimut,MofLift,4);
[t, y] = ode45(@odefunflap,azimut,[0 0]);
nuovo = y(:,1)';
beta0 = mean(nuovo);
A = -2*mean(cos(azimut).*nuovo);
B = -2*mean(sin(azimut).*nuovo);
beta = beta0 - A*cos(azimut) - B*sin(azimut);
dbeta = A*sin(azimut) - B*cos(azimut);
Up = (lambda climb*ones mat+lambda PP)*(rev*R max) ...
+ (r')*dbeta*R max + mu*rev*R max*ones(length(r),1)*(beta.*cos(azimut));
phi = atan(Up./Ut);
alpha real = theta*d2r - phi; %alpha in radians
alpha real = alpha real/d2r; %alpha in deg
dim ut = size(Ut);
for ixt=1:dim ut(1)
for jxt=1:dim ut(2)
if abs(Ut(ixt,jxt)) < 10
alpha real(ixt,jxt) = -0.72;
end
end
end
f = (Nb/2)*((1-r matrix)./(r matrix.*abs(phi)));
F = (2/pi)*acos(exp(-f));
f2 = (0.95/2)*((r matrix-R min/R max)./(r matrix.*abs(phi)));
F2 = (2/pi)*acos(exp(-f2));
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CL real bis = polyval(p clalpha,alpha real);
CD real = polyval(p cdalpha,alpha real);
CT profile = 0.5*solidity matrix.*CD real.*velocity adim.ˆ2*dr;
CT lift = 0.5.*solidity matrix.*F.*F2.*CL real.*(velocity adim.ˆ2)*dr;
CFz = CT lift.*cos(phi)-CT profile.*sin(phi);
CL real2 = sum(CFz,2)/length(azimut);
CT real = sum(CL real2);
error cond = (CT real - CT req)/CT req;
%CT lift represents a coefficient of power
CT lift = 0.5.*solidity matrix.*F.*F2.*CL real.*(velocity adim.ˆ3)*dr;
CT lift2 = CT lift.*sin(phi);
CP reverse = -0.5*solidity matrix.*CD real.*velocity adim.ˆ3*dr;
for k=1:dim(1)
for l=1:dim(2)
if velocity adim(k,l)>0
CP reverse(k,l) = 0;
end
end
end
CP profile2 = sum(0.5*solidity matrix.*CD real.*velocity adim.ˆ3*dr ...
+ CP reverse +CT lift2,2)/length(azimut);
CP profile = sum(CP profile2);
CP ind2 = sum(lambda PP.*0.5.*solidity matrix.*CL real ...
.*(velocity adim.ˆ2)*dr,2)/length(azimut);
CP ind = sum(CP ind2);
fval = CP ind + CP profile;
end
end
if mu == 0
minimo cp = min(solution matrix(:,6));
i 3 = find(minimo cp == solution matrix(:,6));
power = minimo cp*rho*Ad*((rev*R max)ˆ3);
power tail = power*1.05;
tip angle = solution matrix(i 3,3) + solution matrix(i 3,2);
tip2 = tip angle - 1.5;
graph mat(i 2,:) = [mu Vf(i 2) solution matrix(i 3,2) solution matrix(i 3,2) ...
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tip angle tip2 minimo cp minimo cp power power tail];
else
cpowertot = fval + CP parassite;
power = cpowertot*rho*Ad*((rev*R max)ˆ3);
power tail = power*1.05;
graph mat(i 2,:) = [mu Vf(i 2) x(1) x(2) x(3) x(4) fval cpowertot power power tail];
end
waitbar(i 2 / length(Vf));
end
close(hbar);
%% Create a file with the results
fileID = fopen('result 1.txt','w');
fprintf(fileID,'sc1095 cutroot 1.39 - taper: %.3f \r\n \r\n', taper);
fprintf(fileID,'%.4f %.4f %.6f %.6f %.6f %.6f %.12f %.12f %.12f %.12f\r\n',graph mat');
fclose(fileID);
A.2 Constraints
function [c, ceq] = fmcconstraint(x)
global CT real CP ind dr d2r r matrix phi solidity matrix mu azimut p cdalpha ...
p cdalpha2 Nb CT req p clalpha p clalpha2 lambda PP Ut R min R max
theta1 = x(1) -(x(1)-x(2))*r matrix*0.5;
theta2 = x(2) -(x(2)-x(3))*r matrix*0.35;
theta3 = x(3) -(x(3)-x(4))*r matrix*0.15;
dim r = size(r matrix);
theta = zeros(dim r(1),dim r(2));
for i=1:dim r(1)
for j=1:dim r(2)
if r matrix(i,j) < 0.5
theta(i,j) = theta1(i,j);
elseif r matrix(i,j)>=0.5 && r matrix(i,j)<=0.85
theta(i,j) = theta2(i,j);
elseif r matrix(i,j)>0.85
theta(i,j) = theta3(i,j);
end
end
end
alpha real = theta*d2r - phi; %alpha in radians
alpha real = alpha real/d2r; %alpha in deg
dim ut = size(Ut);
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for i=1:dim ut(1)
for j=1:dim ut(2)
if abs(Ut(i,j)) < 10
alpha real(i,j) = -0.72;
end
end
end
CL real = polyval(p clalpha,alpha real);
CD real = polyval(p cdalpha,alpha real);
for i=1:dim r(1)
for j=1:dim r(2)
if r matrix(i,j)>=0.5 && r matrix(i,j)<=0.85
CL real(i,j) = polyval(p clalpha2,alpha real(i,j));
CD real(i,j) = polyval(p cdalpha2,alpha real(i,j));
end
end
end
f = (Nb/2)*((1-r matrix)./(r matrix.*abs(phi)));
F = (2/pi)*acos(exp(-f));
f2 = (0.95/2)*((r matrix-R min/R max)./(r matrix.*abs(phi)));
F2 = (2/pi)*acos(exp(-f2));
%Profile drag
dim = size(r matrix);
dim1 = dim(1);
velocity adim = r matrix+mu*ones(dim1,1)*sin(azimut);
CT profile = 0.5*solidity matrix.*CD real.*velocity adim.ˆ2*dr;
%lift
CT lift = 0.5.*solidity matrix.*F.*F2.*CL real.*(velocity adim.ˆ2)*dr;
CFz = CT lift.*cos(phi)-CT profile.*sin(phi);
CL real2 = sum(CFz,2)/length(azimut);
CT real = sum(CL real2);
CP ind2 = sum(lambda PP.*0.5.*solidity matrix.*CL real.*(velocity adim.ˆ2)*dr,2)/length(azimut);
CP ind = sum(CP ind2);
c = [-CP ind; x(2)-x(1); x(3)-x(2); x(4)-x(3)];
ceq = [-CT real + CT req];
end
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A.3 Optimization function
function app fmc = funtominimize(x)
global solidity matrix dr d2r Nb r matrix phi mu azimut lambda PP CL real ...
theta p clalpha p cdalpha p cdalpha2 p clalpha2 Ut R min R max
theta1 = x(1) -(x(1)-x(2))*r matrix*0.5;
theta2 = x(2) -(x(2)-x(3))*r matrix*0.35;
theta3 = x(3) -(x(3)-x(4))*r matrix*0.15;
dim r = size(r matrix);
theta = zeros(dim r(1),dim r(2));
for i=1:dim r(1)
for j=1:dim r(2)
if r matrix(i,j) < 0.5
theta(i,j) = theta1(i,j);
elseif r matrix(i,j)>=0.5 && r matrix(i,j)<=0.85
theta(i,j) = theta2(i,j);
elseif r matrix(i,j)>0.85
theta(i,j) = theta3(i,j);
end
end
end
alpha real = theta*d2r - phi; %alpha in radians
alpha real = alpha real/d2r; %alpha in deg
dim ut = size(Ut);
for i=1:dim ut(1)
for j=1:dim ut(2)
if abs(Ut(i,j)) < 10
alpha real(i,j) = -0.72;
end
end
end
CL real = polyval(p clalpha,alpha real);
CD real = polyval(p cdalpha,alpha real);
for i=1:dim r(1)
for j=1:dim r(2)
if r matrix(i,j)>=0.5 && r matrix(i,j)<=0.85
CL real(i,j) = polyval(p clalpha2,alpha real(i,j));
CD real(i,j) = polyval(p cdalpha2,alpha real(i,j));
end
end
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end
%Profile drag
dim = size(r matrix);
dim1 = dim(1);
velocity adim = r matrix+mu*ones(dim1,1)*sin(azimut);
CP reverse = -0.5*solidity matrix.*CD real.*velocity adim.ˆ3*dr;
f = (Nb/2)*((1-r matrix)./(r matrix.*abs(phi)));
F = (2/pi)*acos(exp(-f));
f2 = (0.95/2)*((r matrix-R min/R max)./(r matrix.*abs(phi)));
F2 = (2/pi)*acos(exp(-f2));
CT lift = 0.5.*solidity matrix.*F.*F2.*CL real.*(velocity adim.ˆ3)*dr;
CT lift2 = CT lift.*sin(phi);
for k=1:dim(1)
for l=1:dim(2)
if velocity adim(k,l)>0
CP reverse(k,l) = 0;
end
end
end
CP profile2 = sum(0.5*solidity matrix.*CD real.*velocity adim.ˆ3*dr ...
+ CP reverse +CT lift2,2)/length(azimut);
CP profile = sum(CP profile2);
%Drag
CP ind2 = sum(lambda PP.*0.5.*solidity matrix.*CL real.*(velocity adim.ˆ2)*dr,2)/length(azimut);
CP ind = sum(CP ind2);
app fmc = CP ind + CP profile;
end
A.4 Flapping
function dbdpsi = odefunflap(t,y)
global p
dbdpsi = [y(2); -y(1) + p(1)*tˆ4 + p(2)*tˆ3 + p(3)*tˆ2 + p(4)*t + p(5)];
end
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