the development of a phase locked wind turbine blade
TRANSCRIPT
Theses - Daytona Beach Dissertations and Theses
12-3-2010
The Development of a Phase Locked Wind Turbine Blade Finite The Development of a Phase Locked Wind Turbine Blade Finite
Element Model to Predict Loads and Deflections during Fatigue Element Model to Predict Loads and Deflections during Fatigue
Testing Testing
Kyle Freeman Embry-Riddle Aeronautical University - Daytona Beach
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THESIS PAPER
The Development of a Phase Locked Wind Turbine Blade Finite Element Model to Predict Loads and
Deflections during Fatigue Testing
Kyle Andrew Freeman
Master of Science in Mechanical Engineering
December 3, 2010
Embry-Riddle Aeronautical University
600 S. Clyde Morris Blvd.
Daytona Beach, FL 32114
UMI Number: EP31913
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The Development of a Phase Locked Wind Turbine Blade Finite Element Model to Predict Loads and Deflections during Fatigue
Testing
By
Kyle Freeman
This thesis was prepared under the direction of the candidate's thesis committee chairman, Darris White, Department of Mechanical Engineering, and has been approved by the members of his/her thesis committee. It was submitted to the Mechanical Engineering Department and was accepted in partial fulfillment of the requirements for the degree of Masters of Mechanical Engineering.
THESIS CQMNfiTTEE
^ ^ c Dr. Darris White Committee Chairman
Dr.^vlarc Compere Committee Member
" /
Michael Desmond Committee Member
Dr. CharlesReiriholtz Department Chaii>Mechanical Engineering
°izl*fa )r. James CunpfingRam
/Associate Vfee President for Academics
Abstract
TITLE: The Development of a Phase Locked Wind Turbine Blade Finite Element Model to Predict Loads
and Deflections during Fatigue Structural Testing
CANDIDATE: Kyle A. Freeman
DEGREE: Master of Science in Mechanical Engineering
INSTITUTION: Embry-Riddle Aeronautical University
YEAR: 2010
Full scale blade testing provides blade manufacturers with quantitative data in order to assess blade
design, manufacturing and durability. Structural testing is a requirement in order to design reliable
blades, and to develop a further understanding of the dynamics involved in a modern turbine blade.
Blade tests can be conducted in either a single axis or dual axis configurations. Historically, fatigue
testing has been performed by utilizing forced displacement systems. These systems do not allow for
the load phase angle to be controlled, and the maximum load application in the edge and flap directions
are allowed to vary. The PhLEX (Phase Locked Excitation System) under development utilizes a resonant
excitation system in order to reduce hydraulic requirements, decrease test duration and improve
distributed load matching. Control of the phase angle will allow for more accurate testing of the blade.
This thesis paper will detail the method and theory used to develop a model of a phase locked resonant
test system for structural testing of wind turbine blades.
Acknowledgements
I would like to contribute my success and express a special thanks to the following people for providing
the necessary help, encouragement, and support through my college career, including my research and
the writing of this thesis.
To my friends and family: My wife, Mother-in-Law, Father-in-Law, Dad, Mom, Sister, Brother-in-Law and friends and family everywhere
To the Professors at Embry-Riddle Aeronautical University (past and present): Darris White, Marc Compere, Charles Reinholtz, Jean-Michel Dhainaut, Bill Engblom, Sathya
Gangadharan, Glenn McNutt, Jennie Gibbs, Betty-Jane Schuk, William Barrot, Jack McKisson, Hamilton
Hagar
To the Faculty and Engineers at the NREL: Michael Desmond, Scott Hughes, Jason Cotrell, Mike Stewart, Noah Ledford, Dave Simms, Elisabeth
Frizzell, Billy Hoffman, Jeroen van Dam, Mike Jenks
IV
Table of Contents Chapter 1 Background and Introduction 1
1.1 History of Wind Power 1
1.2 Modern Wind Turbines 1
1.3 Overview of Blade Testing 5
Chapter 2 -PhLEX Model Considerations and Nomenclature 7
2.1 PhLEX Introduction 7
2.2 Finite Element Method 8
2.3 Blade Phase Angle 8
2.4 Euler-Bernoulli vs. Timoshenko 8
2.5 Nomenclature 9
Chapter 3 PhLEX Blade Properties 10
3.1 Blade Properties 10
3.2 Normalized Blade Properties 10
Chapter 4 PhLEX Blade Model 17
4.1 Model Development 17
4.2 Model Inputs and Outputs 20
4.3 Blade Loads 20
4.4 Model Convergence 21
4.5 Actuator Properties 24
4.6 Blade Path 30
4.7 Actuator Displacement 31
4.8 Natural Frequencies 32
4.9 Displacement 33
4.10 Edge and Flap Stiffness 35
4.11 Mass Sensitivity Analysis 37
Chapter 5 PhLEX and UREX Comparison 40
5.1 Introduction 40
5.2 Model Comparison 40
5.3 Natural Frequencies 41
5.4 Mode Shape Comparison 42
5.5 Deflections 43
5.6 Blade Loads 45
5.7 Flow Rate Requirements 46
5.8 Power Requirements 48
5.9 Test Duration 49
Chapter 6 Conclusion and Future Work 50
6.1 Conclusion 50
6.2 Future Work 50
v
Appendix A Source Code 52
Appendix B Works Cited 58
Table of Tables Table 4-1 - Acutator Data 25
Table 5-1 Natural Frequencies (Hz) 41
Table 5-2 - Force and Mass Comparison 44
Table 5-3-Test Duration 49
Table of Figures Figure 1-1 - Darrieus Vertical Axis Wind Turbine (14) 2
Figure 1-2 - Savonius Vertical Axis Wind Turbine (15) 2
Figure 1-3 - Horizontal Axis Wind Turbine (16) 2
Figure 1-4- Rotor Diameter Comparison (18) 3
Figure 1-5 - Annual and Cumulative Growth in the US (20) 4
Figure 1-6 - Capacity and Cost of Wind Power (22) 4
Figure 2-1-UREX Test System 7
Figure 2-2-PhLEX Test System 7
Figure 2-3 - Blade Nomenclature (35) 9
Figure 3-1 - Blade Bending Moment Directions (23) 10
Figure 3-2 - Mass per Unit Length H
Figure 3-3 - Chord Length n
Figure 3-4-Angle of Twist 12
Figure 3-5 - Flap Stiffness 13
Figure 3-6- Edge Stiffness 13
Figure 3-7-Axial Stiffness 14
Figure 3-8-Torsional Stiffness 15
Figure 3-9-Ratio of Flap and Edge Stiffness 16
Figure 4-1 -System Modeled 17
Figure 4-2 - Inputs and Outputs 20
Figure 4-3 - Natural Frequency as Nodes are Added 22
Figure 4-4 - Stiffness as Nodes are Added 23
Figure 4-5-Blade Angle as Nodes are Added 23
VI
igure 4-6- Blade Angle vs. Position 26
igure 4-7-Stiffness Added vs. Position 27
igure 4-8- Difference in Eigenvalue When Changing Stiffness 28
igure 4-9-Difference in Eigenvalue When Changing Angle 28
igure 4-10 - Difference in Eigen values (Hz) 29
igure 4-11 - Valid Regions for 55% - 95% Blade Station 29
igure 4-12 Blade Travel Path 30
igure 4-13 -Operational Actuator Length 31
igure 4-14 - Operational Actuator Angle 32 :igure 4-15 - Change in Natural Frequency 33 :igure 4-16- Normalized First Mode 34 :igure 4-17 Normalized Second Mode 34 :igure 4-18 - Flap Stiffness per Blade Station 35 :igure 4-19 - Edge Stiffness per Blade Station 36
igure 4-20 - Flap to Edge Stiffness Ratio 37 :igure 4-21 Natural Frequency change with Mass 38 :igure 4-22 - Change in Angle with Mass 39 :igure 4-23 - Change in Stiffness with Mass 39 :igure 5-1 Target Bending Loads 40 :igure 5-2-PhLEX First Mode 42 :igure 5-3 - PhLEX Second Mode 42 :igure 5-4 - UREX First Mode 42 :igure 5-5 - UREX Second Mode 42 :igure 5-6 - RTS Flap Deflections 43 :igure 5-7 - RTS Edge Deflections 43 :igure 5-8 - UREX Flap Deflections 43 :igure 5-9 - UREX Edge Deflections 43 : igure5-10- RTS Tare Loads 45
igure 5-11 RTS Edge Range Loads 45
igure 5-12 RTS Flap Range and Mean Loads 45
igure 5-13 UREX Tare Loads 45
igure 5-14 - UREX Edge Range Loads 45
igure 5-15 - UREX Flap Range Load 45
igure 5-16 - PhLEX Actuator Flow Rate 46
igure 5-17-UREX Actuator Flow Rate 47
igure 5-18 - PhLEX Power Required 48
igure 5-19 - UREX Power Required 49
VII
Chapter 1 Background and Introduction
1.1 History of Wind Power
5000 years ago wind was used to power ships have utilized the winds power to traverse rivers
and oceans (1). Wind was later used to power mills in order to grind grain or pump water (2). Windmills
were present in current Afghanistan in the seventh century which had 6 or 12 blades covered in cloth
(3)(4). In Europe, windmills first appeared in the eleventh century, and eventually became very
important tools (5). Windmills eventually fell out of favor as fossil fuel alternatives were developed, and
the availability of electricity spread (6). In 1888 Charles Brush developed a wind turbine in Cleveland,
OH that produced 12 kW (7)(4)(8). In 1891 Poul La Cour built an experimental wind turbine that drove a
dynamo to generate electricity (9). The technology developed by La Cour was improved by F.L. Smidth,
in 1941-1942 and were the first to use modern airfoils. In America, Palmer Putnam built a 53m diameter
turbine (4). In the period after World War 2, research into wind turbines dropped significantly. The oil
crisis in the 1970's spurred new interest in alternative energy. Many new prototypes were developed as
a result of the investment into wind in the 1970's (5)(8). Wind turbine power production reached
approximately 200 kW per turbine in the late 1980's. As oil costs dropped, the research into wind also
declined once again in the United States, but development continued in Europe (4). By 1994, there
were 6 TWh produced worldwide, with California producing 47% of the total, and Europe producing
34%. While Europe rapidly increased wind power installed capacity, the industry stagnated in the
United States (10).
1.2 Modern Wind Turbines
In modern configurations, wind turbines are typically installed in farm configurations. These
wind farms are then connected to the power grid where the generated power is distributed. These
utility scale wind turbines are often in the range of 1.5 MW to 5 MW machines (11). Smaller wind
turbines are typically installed in standalone applications in order to pump water or perform some other
1
type of mechanical work (12). Modern wind turbines are typically of three types, the Savonius (Figure
1-1) or Darrieus (Figure 1-2) vertical axis wind turbine (VAWT), or the horizontal (Figure 1-3) axis wind
turbine (HAWT) (13).
Figure 1-1 Darrieus Vertical Axis Wind
Turbine (14)
Figure 1 2 Savonius Vertical Axis Wind
Turbine (15)
Figure 1-3 Horizontal Axis Wind Turbin
2
Future wind turbine designs are trending towards larger and larger turbines. While land based
turbines have essentially leveled off at the 1.5 to 3 MW capacity rating, offshore wind turbines are being
developed in excess of 5 MW. Recently the Clipper Britannia 10MW wind turbine was announced. It is
designed to have a 150 m rotor diameter, while most turbines currently have a 100m rotor diameter
(17). As wind turbine output increases, the required swept area also increases (9).
10 MW WIND IN PERSPECTIVE
1980 19B5 1990 1006 2000 2005 2010 Year
Figure 1-4 - Rotor Diameter Comparison (18)
Worldwide wind installations totaled 20GW, where the United States installed over 5 GW (19).
New wind installations in 2008 have added 8,558 MW of wind capacity in the United States. Figure 1-5
shows the growth of wind installations since 1981 (20). A 2008 report suggests that 20% of the United
States energy supply could come from wind by the year 2030. It shows that during the years of 2000,
2002 and 2004 the production tax credit expired, which caused the low amount of new installations. It
is predicted that 305 GW of energy would need to be produced by wind energy to meet this goal (21).
The cost per kilowatt hour has decreased, but as larger turbines are being used, the costs are beginning
to slightly rise as shown in Figure 1-6 (22).
3
o 03 m O
E <
9 T
7 -
? 6
5 -
4 -
3
1 -
l = l Annual US Capacity (left scale)
— Cumulative US Capacity (right scale)
~ i i i
r 27
24
21
18 O
15 a a. ca
12 O m
9 J*
• I t 3
i n c o i ^ c o c ^ O T - c N i c o ^ i o c D i ^ o o o j o - ^ - c N j c o ^ i n c D r ^ c o G O O O O O O O C O ( J ) 0 ) C J ) ( J ) 0 ) C T ) C J > ( J ) 0 ) C D O O O O O O O O O
G > 0 > O ) O ) C 7 ) O ) 0 ) 0 ) 0 > C 7 > 0 ) 0 > 0 > G > 0 ) 0 > 0 > G > 0 0 0 0 0 0 0 O O O i - T - ^ T - r - ^ - T - f - i - ^ - T - T - r - t - r - C N C N J C N C N C N C N I C N C N C N
Source. iAIV&A
C
s> C
LU
O O
Figure 1-5 Annual and Cumulative Growth in the US (20)
Cost of Energy and Cumulative Domestic Capacity
Annual US Capacity
Cost of Energy
nr-T
1980 1985 •Year 2000 dollars
nnnTrfHHHIM r v 'r T
1990
r 'r 'i' 'i' 'i' T *i' T T 'r r v T 'r
1995 2000 2005
18000
16000
14000
12000
10000
8000
6000
4000
2000
*—v
o Q. TO O
Figure 1-6 Capacity and Cost of Wind Power (22)
4
1.3 Overview of Blade Testing The blades of the wind turbine are the most important part of the entire structure as they
transform the kinetic energy into mechanical power (7)(23). The purpose of performing fatigue tests on
blades is to ensure they meet their designed reliability and service life. Wind turbines are subject to
vibration, resonance, and non-deterministic loading due to their slender and flexible designs (24). There
are many blade test facilities around the world. NREL has facilities in Massachusetts and Colorado that
focus on blade structural testing which are capable of supporting 90m and 50m blades, and are planning
another facility in Texas capable of testing 100m blades (25). The NREL facilities are capable of
conducting both static and fatigue tests (26). Narec, located in the United Kingdom, has also announced
plans to build a facility capable of testing 100m blades (27).
Fatigue testing allows for a manufacturers design and material choices to be analyzed. Many
materials may be used in the construction of turbine blades, such as wood, metals, or fiberglass
composite (28). Blade tests were performed for NASA in 1977 of an 18.3 m fiberglass and foam
composite blade (29). Recent efforts at Sandia National Laboratories and the National Wind Technology
Center at NREL have produced models on a 9m scale and test results of multiple research blade designs
utilizing many different types of materials (30). As blade sizes increase, the fatigue properties of the
materials and their combinations must be tested. Comparisons of the types of composite materials
have been performed in order to predict the properties as blades are scaled up (31)(32)(33). Since wind
turbine life cycles are typically targeted to be at 20 years, these blades must be tested to see the effect
of fatigue over their lifetime (34).
Blade testing systems have consisted of force displacement systems, single axis resonant test
systems, and dual axis resonant test systems (23)(35). Forced displacement systems were very slow, and
required a large amount of energy in order to cause a blade displacement (23). Single axis resonant test
systems increased the test speed, and decreased the energy required to perform the blade test, but
5
were unable test loadings in both the edge and flap directions at the same time. Testing a single blade
first in the flap direction, and then in the edge direction could cause unforeseen blade damage in one
test that would change the results in the next direction. In current dual axis resonant test systems, the
blade is excited via multiple actuators at two different natural frequencies in both the edge and flap
directions (35). Through stochastic analysis, the phase angle between the flap and edge maximum load
occurs most frequently within a general range of values. Current dual axis test techniques allow for the
phase angle between the flap and edge loads of the blade to vary, and the test methods have no control
over phase angle.
6
Chapter 2 - PhLEX Model Considerations and Nomenclature
2.1 Phi EX Introduction The PhLEX system, which is a phase locked dual axis resonant fatigue test system, will allow
testing of a turbine blade at a predetermined fixed or constant phase angle in order to load the blade
during testing in the same manner it is loaded in the field (23). Current resonant test systems are
unable to control the phase angle, which causes loads to be randomly placed at different points of the
blade on every cycle. By allowing the phase angle to constantly change, parts of the blade designed for
low strain could be overloaded, while other parts designed for high strain could be under loaded.
Damage analysis shows that up to 50% more total damage can occur at 0 degree phase angle than at 72
degree phase angle (23). The 72 degree phase angle damage more closely matches analysis done at
NREL.
The PhLEX system modifies the current UREX test system at NREL. The UREX test system
consists of three hydraulic actuators mounted on a saddle close to the root of the blade, shown in Figure
2-1. The UREX system excites the blade at its natural frequency in both the edge and flap directions.
The PhLEX system will modify the UREX by adding an additional actuator outboard from the UREX as
shown in Figure 2-2.
The modeling was performed utilizing a MATLAB script that was developed based upon previous
methods of blade modeling. This code is able to simulate the phase locked response of a blade to a
7
dual-axis resonant test system. This paper will provide information on the modeling of wind turbine
blade phase locked dual-axis resonant test system. In order to lock the phase angle of the resonant test
system, a solution of adding a stiffener in the flap direction was proposed in order to modify the natural
frequency of the blade in flap direction, and make it approximately equal to the natural frequency in the
edge direction. Along with being able to lock the phase angle between the edge and flap directions of
the blade, it will also decrease the blade test duration by increasing the natural frequency.
2.2 Finite Element Method
The term finite element method was first used by Clough in 1960 (36). The finite element
method discretizes a continuous volume into discrete elements. Discrete elements are easily solved by
modern computer systems, even though there may be a large number of elements (37). A finite
element model using beam elements was used to develop the model (38). The model uses six degrees
of freedom per node.
2.3 Blade Phase Angle
The phase angle of the blade is the number of degrees of rotation the blade experiences
between the maximum load of the flap and the edge. A study of the phase angle yielded an average
phase angle of 72 degrees. The variability in the loads comes from the stochastic wind speed in the flap
direction, with mainly gravity and generator drive loads in the edge direction (23).
2.4 Euler-Bernoulli vs. Timoshenko
Euler-Bernoulli beam theory applies to slender prismatic beams. Euler-Bernoulli beam theory
applies to arbitrary cross section shapes (39). In 1921, Stephen Timoshenko published what is now
known as the Timoshenko beam theorem. The Timoshenko model accounts for the bending and shear
deformation of a beam (40). Previous models have shown there to be less than a 0.1 percent difference
in the frequency and deflection of the blade using a one hundred element finite element model (35).
8
2.5 Nomenclature
In order to define the nature of the wind turbine blade, the three directions of the blade are
used as defined by the International Electrotechnical Commission (IEC) (41). The spanwise direction of
the blade is defined as the direction which is parallel to the longitudinal axis of the blade. The edgewise
direction is that which is perpendicular to the spanwise blade direction, and parallel to the swept blade
profile. The edgewise direction can also be referred to as the lead-lag direction, but will be referred to
as the edge direction in this paper. The flapwise direction is perpendicular to the swept surface, and
parallel to the longitudinal axis of the blade. The flapwise direction will be referred to as the flap
throughout this paper. These directions are illustrated in Figure 2-3.
TRAILING EDGE _ TIP
- ^ " " C T E A D I N G EDGE
Figure 2-3 - Blade Nomenclature (35)
EDGEWISE
ROOT
FLAPWISE
Chapter 3 PhLEX Blade Properties
3.1 Blade Properties
Scaling the system up to large scale wind turbine blades is a large concern with this system.
Modern large scale wind turbine blades are typically constructed from fiber-reinforced glass-epoxy
compounds. The root is typically a circular shaped section which then forms an airfoil shaped section as
the length of the blade is traversed. Through many years of blade testing, a large amount of data has
been collected at the NREL facilities. As blades scale up, the mass per unit length increases, and stiffness
in both the edge and flap directions increase significantly in the root of the blade. Chord lengths also
increase significantly as blades are scaled up in size (42).
3.2 Normalized Blade Properties
Blade properties are proprietary; however an understanding of the basic characteristics of a
typical blade is important. Normalized blade properties will be presented in order to give an
understanding of the typical structure of a blade. Figure 3-1 defines the directions of the blade
properties described, as well as shows a typical cross section of a wind turbine blade.
Edge
Flap
Figure 3-1 - Blade Bending Moment Directions (23)
10
Blade Properties - Mass per Unit Length
1 0.9 1
0.8 -
0.7 -
0.6 -
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
0 -
•
I
•
•
•
.
:
: • ;
-*
..
4
--
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
Figure 3-2 - Mass per Unit Length
0.8 0.9
Blade Properties - Chord Length 1 :-
0.9 -
0.8 -
0.7 -
0.6 -
0.5 -
0.4 -
0.3
0.2
0.1 " 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Blade Station (m) 0.8 0.9
Figure 3-3 - Chord Length
11
Figure 3-2 illustrates the mass per unit length of the example blade. The plot shows that most of the
mass of the blade is located below twenty percent of the span. The blade tends to have a large amount
of mass towards the root due to mounting hardware in order to mount the blade to the hub (43).
Figure 3-3 shows the chord length of the blade. The maximum chord length occurring at twenty percent
blade station corresponds with an almost linear decrease in mass per unit length to the tip of the blade.
This would indicate that the material composition from twenty percent to the blade tip is rather
consistent, and the mass per length is only changing due to a changing geometry.
Blade Properties - Angle of Twist
0 0.1 0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
Figure 3 4 - Angle of Twist
The angle of twist in Figure 3-4 shows the relation between each element to the global coordinate axes.
These angles are used when assembling the global stiffness matrix. This particular blade starts with the
maximum twist in the blade, and ramps down to almost zero degrees of twist.
12
0.9 - 1
u.o -
0.7 -NE Z 0.6 -LU
? 0 .5-N
| 0.4 -EZ O
z U.O " 1
Blade Properties
u. I \
- Flap Stiffness
0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
Figure 3-5 - Flap Stiffness
0.8 0.9
-\
n n . 1
0.8
0.7
E z 0.6
laliz
ed E
l D
p 1
E 1/ o
~7 0.3
0.2
n i w . ,
n
Blade Properties
-
- Edge Stiffness
r^S— 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Blade Station (m) 0.8 0.9
Figure 3-6 Edge Stiffness
13
Blade stiffness defines the resistance to displacement of a given force. Due to the blade geometry, the
stiffness in the edge direction is typically higher than in the flap direction. The stiffest portion of the
blade is typically at the root, which as shown in the MPL plot contains the most mass per unit length.
The fasteners for hub attachment are embedded into the blade material at this point, making the
stiffness high in relation to the rest of the blade. The effects of the mounting area of the blade are
shown in Figure 3-5, which in the flap direction the stiffness is quite high at the root, and drops
significantly as the geometry changes (43).
In Figure 3-6 the edge stiffness has a large dip in it, which is normally not typical. This is an effect of the
design of this particular blade, and the geometry in that particular region. The geometry of the blade is
more consistent after the twenty percent blade station.
0.8 0.9 1
Figure 3-7 - Axial Stiffness
Axial stiffness of the blade resists any elongation along the length of the blade. The axial stiffness of this
blade was unknown, and estimation was provided utilizing an empirical formula that has been
Blade Properties - Axial Stiffness
0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
14
developed by the long history of blade testing at NREL. This estimation is based upon the materials used
to manufacture the blade. Axial deformation in blade should be insignificant when compared to the flap
and edge deformation that will be present in the blade test (43).
Blade Properties - Torsional Stiffness
«£ * Z in in CD
c iz
CO Cfl
c o in
\— o f -•a CD N
CT3 F v_
o z
1 :
0.9
0.8
0.7
0.6
0.5
0.4
0 3
0.2
0.1
0" 0
Figure 3-8 - Torsional Stiffness
0.2 0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
0.8 0.9
15
1.4:
1.2
1 '
g I 0.8 C/) (/) <D
| 0.6 55
0.4
0.2
0" 0
The torsional stiffness of the blade describes the ability of the blade to resist moments along the length
of the blade. Torsional stiffness of the blade drops rapidly, which allows for a coupling of the edge and
flap deformations further down the blade (43).
Figure 3-9 shows the relationship between the edge and flap stiffness along the length of the
blade. At the root, the stiffnesses are approximately equal. The ratio between the two quickly drops as
the geometry changes along the blade length. At approximately twenty percent of the blade length the
ratio between the two stiffnesses stabilizes.
Past blade tests essentially separated the flap and edge directions and the stiffness of each were
considered separately, as both directions were tested under different conditions. Since the goal of the
PhLEX test is to test the blade by causing the max load of the edge and flap to occur at approximately 72
degrees, the stiffness of each direction will get much closer together as the natural frequency of the first
and second mode converge.
16
Blade Properties - Stiffness Ratio - Flap / Edge
0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Blade Station (m)
0.8 0.9 1
Figure 3-9 - Ratio of Flap and Edge Stiffness
Chapter 4 PhLEX Blade Model
4.1 Model Development
The finite element model is developed as a lumped mass model. Each node has a given mass
and stiffness, and it is assumed that the connection between each of these nodes is massless (44). A
finite element model (FEM) utilizing Euler-Bernoulli beam theory was developed in order to find Eigen
values of the system. The resulting stiffness matrix is shown in Equation 4-1 (45). This is the local
stiffness matrix of an element comprising of the shear force and moment of each node of the element.
Fx Mx
F2
M-,
EI
V
12
61
-12
61
61
4L2
-6L
21}
-12
-6L
12
- 6 1
61
21}
-6L
4Zr
*;. 4 V i
°2
Equation 4-1 - Elemental Stiffness Matrix
Figure 4-1 diagrams the system that was modeled. A simple linear spring was placed between the
ground and the blade as a method to add stiffness in the flap direction of the blade. It is assumed that
the mass of the spring will be supported by the ground, and will not affect the blade.
L. N ~ -yA Figure 4-1 - System Modeled
17
The procedure to add the stiffness into the finite element model was performed by first determining the
support reactions of the spring element, and applying the boundary conditions to the entire system. By
applying boundary conditions of zero rotation and displacement at both the root of the blade, and the
point the spring is attached to the ground, the stiffness matrix in Equation 4-2 (46) can be modified by
only adding stiffness at a chosen node.
f5 =
' K -K
-K K _
v3
_v 5 _
Equation 4-2 Spring Stiffness Matrix
Equation 4-2 is the stiffness matrix of the spring. This matrix will be assembled along with the local
elemental matrix into the global stiffness matrix (46).
Fy '
K F2
M2
F3
M3
F4
M4
F5
EI
12
61
-12
6Z
0
0
0
0
0
6L
4L2
- 6 1
21}
0
0
0
0
0
-12
- 6 1
24
0
-12
6L
0
0
0
61
2 Is
0
8L2
-6L
21}
0
0
0
0
0
-12
- 6 1
24 + EI
0
-12
6L
-KLy
EI
0
0
61
2L}
0
8L:
- 6 1
21}
0
0
0
0
0
-12
- 6 1
12
- 6 1
0
0
0
0
0
6L
21}
-6L
M}
0
0
0
0
0
-KL}
EI
0
0
0
EI
v i
Oy
v2
e2
v3
V4
oA _v5
Equation 4-3 Assembled Global Stiffness Matrix
Equation 4-3 is the assembled global stiffness matrix. This stiffness matrix was created based upon a
three element model which would produce four nodes along the blade model. A fifth node is
18
introduced into the model to represent the ground reaction of the spring. Applying the boundary
conditions of a cantilevered connection at both the node of the mounting surface, as well as the node of
the spring that is attached to the ground will cause the first, second and ninth rows to be zero. Equation
4-4 displays the reduced global stiffness matrix (46).
~Fi M2
F3
M3
^ 4
_ M 4 _
EI
~ I?
24
0
0
SL2
- 1 2
-6L
- 1 2 - 6 1 24 +
6L
0
0
21}
0
0
EI 0
- 1 2
6L
6L
21}
0
SL2
-6L
21}
0
0
- 1 2
- 6 1
12
- 6 1
0
0
6L
2L2
- 6 1
4L2
V2
V 3
r3 v4
A.
Equation 4-4 - Reduced Global Stiffness Matrix
An Eigen analysis is performed using the global mass and stiffness matrices to determine the natural
frequencies and mode shapes (47). The Euler-Bernoulli beam model developed above was modified
using Timoshenko beam elements (48). The result of modified elements is shown in Equation 4-5.
F2
M2
^ My
F4
M4_
EI
K l + O)
24
1}
0
-12
L2
6
L
0
0
0
8 + 20>
- 6
L
2 - 0
0
0
- 1 2
~7F - 6
L 24 KL(1 +
1} + EI
0
- 1 2
~F 6
L
O)
6
Z 2-cD
0
8 + 2 0
- 6
~L
2 - 0 )
0
0
- 1 2
1} - 6
L
12
- 6
L
0
0
6
L
2 - 0 )
- 6
~L
4 + 0
v,
Equation 4-5 Timoshenko Elements
19
4.2 Model Inputs and Outputs Inputs
Blade Properties
Chord
Twist Angle
Flap Stiffness
Edge Stiffness
Mass per Unit Length
Blade Length
Blade Angle
Number of Elements
Actuator Properties
Actuator location
Actuator stiffness
Actuator angle
Saddle properties
Saddle 1 Location
Saddle 1 mass
Saddle 2 Location
Saddle 2 mass
Test properties
Target flap displacement
Target edge displacement
Flap load target
Edge load target
Outputs
Eigen Value Mode 1
Eigen Value Mode 2
Angle Mode 1
Angle Mode 2
Difference in Eigen Values
Difference in Natural Frequency
Static Blade Deflection
Range Flap Load
Mean Flap Load
Range Edge Load
Tare Load
Stiffness at actuator position
Figure 4-2 - Inputs and Outputs
Figure 4-2 - Inputs and Outputs illustrates the input and output parameters of the current PhLEX
code. The blade properties inputs are described in the PhLEX Blade Properties section. These
parameters are used in constructing the mass and stiffness matrices in the finite element model.
Actuator properties are determined by using an unconstrained nonlinear search optimization. The
saddle properties were determined by determining the mass of the UREX test system and finding the
excitation mass (23). The outboard saddle mass was determined by designing the system in CAD with
the proper materials applied and then summing the values of the component masses.
4.3 Blade Loads
The loads on the blades are generally due to wind loads, gravity, and generator loads. There are
generally two types of wind loads, a stochastic and deterministic component. The stochastic component
of wind load is due to the variability of the wind. The deterministic component is time invariant and
20
increases with height (23). When testing the blade, these loads are being applied in the test
environment. Target blade loads are typically provided by the manufacturer of a blade.
Blade Loads
Blade Station (%)
The blade loads were taken from a previous blade test in order to develop a comparison
between the two test systems after a PhLEX test is performed. The curve shape for the loads in the flap
and edge modes are very similar; however the magnitude of the flap load is approximately twice that of
the edge load.
4.4 Model Convergence
The convergence of the model was tested by increasing the number of elements until the
natural frequencies changed less than .05 percent. Both the values of the blade rotation angle and
stiffness added to the blade were recorded for each increase in the number of elements. Figure 4-3
through Figure 4-5 show the results of the test. Initially the percent change in values is large, as the
number of elements in the model increases the curve becomes linear. The linear portion of the curve
21
indicates that the model will indeed converge as more elements are added. The large spike in values
occurring at 70 elements is due to the mass of the inboard saddle being spread evenly across two nodes.
The mass is split between the nodes. How much mass is applied to each node is dependent upon the
distance the mass is to each node. Typically there is a large bias towards one of the nodes, where a
majority of the mass will applied to a single node. As the element size decreases, the size of these
spikes noticeable decrease.
Natural Frequency Change - Convergence
Nodes
Figure 4-3 - Natural Frequency as Nodes are Added
x 10 3.2:-
Stiffess Change - Convergence :
1.6" 20 40 60 80 100 120 140 160 180 200
Nodes
Figure 4-4 - Stiffness as Nodes are Added
20 40 60 80 100 120 140 160 180 200 Nodes
Figure 4-5 - Blade Angle as Nodes are Added
23
4.5 Actuator Pi operties
While the system was modeled as though a spring would be placed between the ground and
blade, in practice it will be replace with a hydraulic actuator. Throughout the rest of this paper, the
spring added in the finite element model will be referred to as an actuator. The actuator properties will
be determined by performing an unconstrained nonlinear search optimization. The fminsearch
algorithm was used from the MATLAB optimization toolbox. This algorithm performs an unconstrained
multivariable search to find the minimum of an input function (49). Fminsearch utilizes a Nelder-Mead
Simplex method in order to minimize the function. In a two variable optimization, the simplex is a
triangle, and the worst performing vertex, the largest, is rejected and replaced with another (50). The
model was put into the form of a function with the output being the difference between the first and
second mode squared. There are two inputs that are being optimized by the algorithm, which is the
stiffness being added by the actuator, and the angle that the blade must be rotated to in order to
achieve the lowest difference in natural frequency. The optimization algorithm continues to execute
until it either meets the limit of the number of executions, or until the difference of the function output
and the function inputs reach a tolerance limit. The termination tolerance of the input, ToIX, was set to
. 1 , and the termination tolerance of the function, TolFun, was also set to .1 . While these tolerance limits
may seem low, there is an order of magnitude between the two values. Table 4-1 displays the results
for the blade used. The final value of the stiffness required for a given position is typically on the order
of l e l l times that of the output value of the function.
24
Blade 1 Station
55.00%
60.00%
65.00%
70.00%
75.00%
80.00%
85.00%
90.00%
95.00%
Stiffness Added (kN/m)
5802463
196147.9
110673.5
76081.06
55191.43
42302.92
31418.89
23274.77
17048.8
Blade Angle
(Degrees)
1.271622
7.132459
8.573477
8.641049
8.210615
6.868813
5.997127
4.987597
3.812021
Natural Frequency Difference
(Hz)
0.00038
0.00134
0.00059
0.00053
0.00107
0.00090
0.00067
0.00097
0.00086 Table 4-1 - Acutator Data
The system is only valid for a certain constrained positions along the blade. If the actuator is
placed too close to the root of the blade, a solution cannot be found where the difference between the
Eigen values will fall into the required range for phase locked control. On the blade data used to analyze
the system, this position seemed to be at approximately 55% of the blade length. If the stiffness was
added at the node before it, the system would not converge to a solution. This is likely due to the
change in stiffness any closer to the root will not add enough to the entire system to change the Eigen
values significantly enough. The system also does not converge to a solution past approximately 95%
blade station. With an actuator mounted at this point, there may be additional deflection between the
root and the actuator causing other disturbances to the system.
25
Blade Angle
1 60 65 70 75 80 85 90 95
Blade Station (%)
Figure 4-6 - Blade Angle vs. Position
Figure 4-6 shows the relationship between the blade angle and the position of the actuator to
reach a converged solution. The angle of blade rotation is dependent upon the position of the spring. A
maximum is reached at approximately 67% blade station. From there the angle begins to decrease as
the spring is moved to positions further from the root of the blade.
When analyzing the results of the optimization for each position, the stiffness that is added is
also dependent upon the position of the blade. As the spring is moved along the blade, less stiffness is
required to be added at each additional position. Figure 4-7 illustrates the relationship between the
stiffness required to add to the system in order to reach a solution. The further along the blade that the
actuator is placed the less stiffness is required to be added into the blade.
26
Stiffness Added to Node Position
1
4
-t
- -+
L
65 70 75 80 85 Blade Station (%)
Figure 4-7 - Stiffness Added vs. Position
90
-t
I
J 95
The valid region that the stiffness being added to the blade and the angle at which the blade is
rotated for each position tends to be rather small. If the angle and position of the spring is held
constant and the stiffness is varied, the difference between the two Eigen values generally is very large
except for a very small area. Figure 4-8 shows the relationship between the Eigen values and the
stiffness when the blade angle and spring stiffness are held constant. As stiffness is added to the blade
in the flap direction, the difference in Eigen values rapidly decreases until the function reaches its
minimum value. When the angle is varied, the plot reflects the same results. Figure 4-9 shows the
difference in Eigen values as the angle is changed and stiffness is held constant.
27
x 10 4;-
Eigen Diff - 66.7% Station Static Angle
3.5
00-^ o CD
T3 03 1 —
CO <D 3 cd > c CD CT> LU _c CD O c CD i _
<D
3
2.5
2
1.5
1
0.5
0.5 1 1.5 2 Stiffness Added (kN/m)
2.5
x 10 Figure 4-8 - Difference in Eigenvalue When Changing Stiffness.
2.5 :-x 10
Eigen Diff - 66.7% Station Static Stiffness
CM—v O CD CO
CO CD D CO > CD O )
LU
1.5
CD O c 0
S 0.5
0"-0 5 10
Rotation Angle (Degrees)
Figure 4-9 Different => in Eigenvalue Wi c >«_n}ngi ig
15
28
CO CD
2 Oi CD Q
<
8 83 - pod
8 8295 -
8 829 -
8 8285 f
8 828
8 8275 —
8 8 2 7 ^
8 8 2 6 5 1 -
8 826 -
8 8255 " ^
Difference of Eigenvalues (Hz) at 66 67%
—0 03
9 5674 9 5676 9 5678 9 568 Stiffness Added (kN/m)
9 5682 9 5684 4
x 10 Figure 4-10 - Difference in Eigen values (Hz)
1 274
_ 1 273
I
rm
a> 1 271
* 127
1 2 6 9 ^
Difference of Eigenvalues (Hz) at 55 00° o
^ ^35 T \ ! Difference ol Eigenvalues (Hz) at 60 00° o Difference of Eigenvalues (Hz) at 65 00%
-K ~\— 0 035
5 78 5 79 5 8 5 81 5 82 5 83 Stiffness Added (kN/m) < 10
Difference of Eigenva ues (Hz) at 70 00° o
7 6074 7 6076 7 6078 7 608 7 6082 7 6084 7 6086 7 6088 7 609 Stiffness Added (kN/m) _•»
Difference ol Eigenvalues (Hz) at 85 00°o
1 9613 1 9613 1 9614 1 9614 1 9615 1 9615 1 9616 1 9616 Stiffness Added (kN/m) x 1Qs
Difference of Eigenvalues (Hz) at 75 00°o
T ' r N j
I-
[ I ~ **
8 2105' ^
£ 821 | I 2 i \ . / 9" -8 2095 8209L \ j \ _L _ t L + /
5 5182 5 5184 5 5186 5 5188 5 519 5 5192 5 5194 5 5196 Stiffness Added (kN/m)
< 10
f r~
Difference of Eigenvalues (Hz) at 90 00%
5828 r -i f --M 4 r 5 826
V v -0 ^
3 1412 3 1414 3 1416 3 14 8 3 142 3 1422 3 1424 Stiffness Added (kN/m)
!5824 / £ 5 822 [
5 82
5 818
H
2
2 226 2 22612 22622 22632 22642 22652 22662 22672 2268 Stiffness Added (kN/m) x )Q
J
1 10671 10671 10671 10671 10671 10681 10681 10681 1068 Stiffness Added (kN/m) x 1Qs
Difference of Eigenvalues (Hz) at 80 00°o
K^ 1 ^ 1 f 4 0788 4 079 4 0792 4 0794 4 0796 4 0798
Stiffness Added (kN/m) x 10»
Difference of Eigenvalues (Hz) at 95 00%
$ 3815/
/
lo
8 f
\_*_ _
F F r 1 '„
( °* 1 V 7 *
r . . . t
1 7042 70441 70461 7048 1 705 1 70521 70541 70561 7058 St ffness Added (kN/m) t
Figure 4 11 Valid Regions for 55% - 95% Blade Station
29
The ability to control the system by adding stiffness decreases as the difference in Eigen values
increase. Figure 4-10 shows the values of angle and stiffness that result in a low difference in Eigen
values. This indicates that there is a small range of values that would allow the natural frequencies to be
close enough to allow the loads of the blade to be phase locked. Figure 4-11 shows the valid regions of
stiffness and angle for blade stations of 55% through 95% at every 5%. The contour plots appear to
show an expanding area of control as the spring is moved towards the tip of the blade.
4,6 BI<i<lePdth
The blade will follow an oval path when in a phase locked configuration. The displacement of
the blade in both the edge and flap directions will follow a sinusoidal path. Combining both of these
displacements with a 72 degree phase angle will produce the plot in Figure 4-12. This is a parametric
plot of a harmonic system, also called a Lissajous figure (51).
Blade Path at 67.5% Station 0.025'- =
0.02 -
0.015
0.01
E ~ 0.005 o
8 0 Q a -0.005
LL
-0.01
-0.015 -
-0.02
-0.025 -0.03 -0.02 -0.01 0 0.01 0 02 0.03
Edge Deflection (m)
Figure 4-12 - Blade Ti a =< Path
30
4.7 Actuator Displacement
While operating the actuator length must track the displacement of the blade. The actuator
displacement was calculated by predicting the path of the blade travel. The coordinates of the blade
path were used in order to calculate the position and angle of the actuator. Figure 4-13 Operational
Actuator Length shows a range of 2.08 inches for the actuator length, and Figure 4-14 Operational
Actuator Angle shows a .54 degree range of angle.
63:-
62 5 -
Actuator Length
Actuator Length Original Position
60 5
60"-0
h > L ^ i - - [ j 0 1 0.2 0 3 0.4 0 5 0.6 0.7 0.8 0.9 1
Time (s)
Figure 4-13 - Operational Actuator Length
31
10.3; Actuator Angle
Actuator Angle
Original Angle
Time (s)
Figure 4-14 - Operational Actuator Angle
4.8 Natural Frequencies
The natural frequencies that the system wil l exhibit are modified when stiffness is added to the
blade. Since different amounts of stiffness are required at each blade station, the natural frequency will
change wi th respect to each blade position. In order to lock the phase between the maximum loading of
the edge and flap directions of the blade, the natural frequency of the flap is matched to the natural
frequency of the edge.
32
5.5 Natural Frequency
5F
4.5
f ^ o C <D 1
CD
Li
ra ZJ
to
4
3.b
3
2.5
1.5
| J. - ^ s ^ - ! i -l r 1
L__A I . . . - . . . . ! -^IU- J . . - . . I - - - . - - - I . . . . I " - ""I
X L . . . . I I J . . . . . - - J - - ^ V - - - L> _ _ I - - _|
60 65 70 75 80 85 90 95 100 Blade Station (%)
Figure 4-15 - Change in Natural Frequency
The natural frequency decreases in a linear fashion as the spring is moved along the blade.
Figure 4-15 shows the decrease along the span of the blade. The decrease in natural frequency will
cause an increase in duration of the test due to the number of cycles being decreased in a given amount
of time. Also by lowering the natural frequency there the possibility of better control of the system as it
will experience lower acceleration. The acceleration is calculated by taking the Eigen value of the first
mode and multiplying it times the deflection in that direction. The acceleration in the edge direction will
typically be lower than that in the flap.
4.9 Displacement The displacement of the blade is derived from the mode shape. The target deflections were
identified then a desired bending moment curve is developed from the deflection.
33
Normalized First Mode 1 :
0.9
0.8
§ 0 . 7 c CD
| 0.6 o
_co
% 0.5 Q
8 0-4 CO
E
Axial Disp Flap Disp Edge Disp Axial Rot Flap Rot Edge Rot
0.3
0.2
0.1
0 ' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Blade Station (m)
Figure 4-16- Normalized First Mode
Normalized Second Mode 1 :
0.9
0.8
§ 0 . 7 -c CD
I 0.6 • O
_CC
% 0.5 Q
| 0.4 -CO
| 0.3 z
0.2 -
0.1 -
Axial Disp Flap Disp Edge Disp Axial Rot Flap Rot Edge Rot
?
0'-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Blade Station (m) 0.8 0.9
f igure 4 17 Noiiroalhie r
34
4.10 Edge and Flap Stiffness
After adding an amount of stiffness to the blade, the stiffness of the entire blade was calculated
by applying a unit force where the spring is located. Taking the inverse of the stiffness matrix and
multiplying it by the unit force matrix the displacement of the blade can be found. The stiffness can
then be calculated by taking the displacement matrix and dividing the applied force by the
displacement. The resulting stiffness will then give an idea of the resistance to bending that the blade
will exhibit. Due to a decreasing amount of stiffness being added to the blade the further from the root,
the stiffness of the flap also follows the same pattern and decreases.
5000
4500
4000
3500
f 3000 I
Flap Stiffness
L 1 % 2500 -
Stiff
ne!
O
O
O
1500 -
1000 -
500 -
0 -' 5
I
5.5
I
6
| —
-
I -p I
6.5
— I
— -
i i
~4
I
7 Blade Station (m)
- n
7.5 8.5
Figure 4-18 - Flap Stiffness per Blade Station
The flap stiffness at 55% blade station is very high. It would take a large load to induce a displacement
at this station. The stiffness curve shown in Figure 4-18 shows that the stiffness at 55% station is high
when compared to that of other stations. The large value would indicate an asymptote, as adding more
stiffness at a position less than 55% results in the model being unable to reach a solution. Following the
35
asymptote, the stiffness of the blade rapidly decreases and then begins to level off. The edge stiffness in
Figure 4-19 shows a consistent decreasing stiffness.
2 5 0 -Edge Stiffness
200 -
150
CO if) CD c § 100 CO
50 -
5.5 6.5 7 Blade Station (m)
7.5 8.5
Figure 4-19 Edge Stiffness per Blade Station
36
Ratio of Flap to Edge Stiffness 2 5 " - — -r- - - T- T- -
i I I 1 | I i I I
0 - t_ r • _ L L_ _ J 60 65 70 75 80 85 90 95
Blade Station (m)
Figure 4-20 - Flap to Edge Stiffness Ratio
The stiffness of the flap and edge modes begins to converge very quickly after 55% blade
station. Figure 4-20 show that the ratio of the stiffness of the flap over the edge is 22. The ratio of the
flap and edge stiffness rapidly decreases and approaches a one to one ratio at the 66.67% blade station.
4.11 Mass Sensitivity Analysis
When the blade is tested, there will be saddles mounted on the blade in order to mount
actuators and excitation equipment to the blade. It was important to see how the model reacted to
mass being added to the blade. Since the Eigen values are dependent upon the mass of the oscillating
system these results are critical to predict the reactions of the system.
37
Si r -
Natural Frequency Change - Mass Added
4.5
N X
z 4 C CD D " CD
2 3.5 03
2.5-200 250 300 350 400 450 500
Mass (kg)
Figure 4-21 - Natural Frequency change with Mass
The position of the actuator was held constant at 66.67%. A range of mass values were input
into the optimization function and values of the stiffness and angle that would be required were found
for each given amount of mass. Figure 4-21 shows the total mass that is added to the blade in the form
of a saddle. Due to increased mass applied to the blade, the natural frequency of the first and second
modes decrease.
Figure 4-22 indicates that the angle that the blade must be rotated increases along with the
added mass. Adding mass to the saddle could allow for easier setup during an actual blade test in order
to bring the blade angle to an even angle. When mass is added to the blade, the stiffness that must be
provided by the actuator decreases as shown in Figure 4-23. Additional mass on the blade causes a
decrease in the stiffness of the blade with a unit force applied to it.
38
10: Angle Change - Mass Added
9.5 K
(f) CD
2>
S 8.5 C < I
8 r
7.5 -
7"-200
1.15c-x 10
250 300 350 Mass (kg)
400 450
Figure 4-22 - Change in Angle w i th Mass
Stiffness Change - Mass Added
500
1.1
1.05 -
CO u i CO 1 CD c
CO
0.95
0.9
0.85h-200 250 300
-350
Mass (kg)
r
400 r
450
Figure 4-23 Change in Stiffness w i th Mass
500
39
Chapter 5 PhLEX and UREX Comparison
5.1 Introduction
The PhLEX system was based the UREX test system. The PhLEX system will change how the
loads are applied to the blade. An analysis of the two system models is required in order to evaluate the
benefits of the new system when compared to the UREX.
5.2 Model ( ompai IMIII
The same blade was analyzed between the current UREX code and the developmental PhLEX
code. A 9m blade was used in both models. A target load curve used in previous 9m tests was used to
evaluate the displacements. The targeted loads are shown in Figure 5-1.
Target Bending Loads
^ - v
E z CO
• o 03 O
_J CD C -o c CD
CO •D CD N
"co E i _
o z
1 ;-
0.9 •
0.8
0.7 •
0.6 -
0.5 •
0.4
0.3
0.2 -
Flap Range Edge Range
0.1
0 0 10 20 30 40 50 60 70 80 90 100
Blade Station (%)
teure 5 1 T^rge* B^ndm^ « :»ad^
40
5.3 Natural Frequencies
The first two natural frequencies of the PhLEX are made to be equal by the addition of stiffness
at some point on the blade. In this particular example, the spring is added at the 67.5% blade station.
The first two natural frequencies for the PhLEX system have less than a .00001 Hz difference between
them. The UREX natural frequencies are where blades are currently tested. Natural frequencies up to
ten were included for reference, as these are not likely to be excited.
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode7
Mode 8
Mode 9
Mode 10
RTS
5.12719
5.12719
10.25508
18.35987
22.71142
31.47393
36.06420
37.38293
50.83082
77.82133
UREX
2.45253
5.16967
9.99421
18.76078
23.34929
32.48953
36.06153
36.82876
37.96871
47.98365 Table 5-1 - Natural Frequencies (Hz)
41
5.4 Mode Shape Comparison PhLEX Normalized First Mode
1.5
0.5 i s ' \
> " "
0 •
-0.5 • 0 20 40 60 80 100
Blade Station (%) figute5-2 - PhiFX First Mode
PhLEX Normalized Second Mode
de -=
Flap
Edge
"
..
0.25-
0.2 •
0.15 -
0.1
0.05 -
0 :-
UREX Normalized First Mode -
Flap
Edge
/ / /
,-^><^. - - J '
1 =-
0.8
0.6 -
0.4 -
0.2-
Flap
Edge
r
o--0 100
1
0.8
0.6
0.4
0.2
0 -0
20 40 60 80 Blade Station (%)
Figure 5-4 - UREX Fir>t Mode
UREX Normalized Second Mode
Flap
Edge
100
y
20 40 60 80
Blade Station (%)
Figure 5 3 - PhLEX Second Mode
The mode shapes are compared in the above figures. The mode shapes of the PhLEX are not
20 40 60 80 Blade Station (%)
Figure 5-5 - UREX Second Mode
100
easily identifiable. The edge shape in the first mode of the RTS, Figure 5-2, has the highest magnitude,
contrary to that of the UREX, Figure 5-4, in which the flap will have the highest magnitude. This is also
again mirrored in the second mode, where the flap has the greatest magnitude for the PhLEX, Figure
5-3, and the Edge has the greatest in the UREX, Figure 5-5. This comparison does not always apply, for
the PhLEX. Since the natural frequencies of the first and second mode are very close the edge and flap
will change back and forth to having the maximum magnitude for either the first or second mode.
42
5.5 Deflections PhLEX Flap Deflection
0.4- — ^ = - —=- 0 . 4 ^ UREX Flap Deflection
0.3-
I 0.2 _CD
Q 0.1 -
0 20 40 60 80 100 Blade Station (%)
Figure 5-6 - RTS Flap Deflections
PhLEX Edge Deflection 0.025:
0.02 -
0 20 40 60 80 100
Blade Station (%)
Figure 5-7 - RTS Edge Deflections
0.3 I
I 0L2-CD
CD
Q 0.1 -
0 ^ 0 20 40 60 80 100
Blade Station (%)
Figure 5-8 - UREX Flap Deflections
UREX Edge Deflection 0.025-
0 20 40 60 80 100
Blade Station (%)
Figure 5-9 - UREX Edge Deflections
The deflections of the systems are quite different in the flap direction. Due to the added
stiffness f rom the actuator in the flap direction due to the PhLEX, Figure 5-6, the deflection is much less
than that of the UREX, Figure 5-8. In the edge direction, both systems have identical displacements, as
can be seen in Figure 5-7 and Figure 5-9. The displacement in the edge direction should not be affected
by the increase in stiff in the edge direction. There is also no hardware that would l imit the deflection of
the edge. The slope of the deflection was calculated for both the PhLEX and UREX system to give an
indication of the degree of deformation in the blade when deformed.
43
UREX Force and Mass Requirements
The UREX system will be utilized in order to excite the blade at its natural frequency. Due to the
increased natural frequency in the flap direction, the actuator force and mass requirements are less in
the PhLEX. The mass and force in the PhLEX flap direction decrease significantly, as the stiffness and
natural frequency increase to be much closer to the values in the edge direction.
Flap Force (kN)
Flap Mass (kg)
Edge Force (kN)
Edge Mass (kg)
RTS
0.175
2.76
0.289
4.564
UREX
1.315
45.367
0.289
3.168
Table 5-2 Force and Mass Comparison
44
5o6 Blade Loads RiLEX Tare Loads
1S
14
12
o
6
4
0
! j ! ' !
— - - , 1 "i - *
i • P
i i ^ r i 12 25 3B 50 6B 75 BB 100
Figure 5-10 - RTS Tare Loads
PhLEX Ed^eLoads
12 25 3B 50 SB 75 eiad»afafe>n(%)
Figure 5-11 - RTS Edge Range Loads
PWJEX Rap Loads
3-ADs WITH SADGLES LOADS
F 10
! 1 1 1 ! f |
L . _ A _ j J J , J , ,
V 1 • • • ~* "• •
i L \ — | ; ; | ; -j
r - - * ^ ~ • ' 1 - - - — i -* T •
I ! _ ^ ^ J • ! j !
* — \ — ! — ^ ^ ^ c " " ' — i — ' 12 25 3B 50 6B
> &1a6on (%) 75 &B 100
Figure 5 13 - UREX Tare Loads
UREX Edge Loads
! « .
larger. Range Aotf is! Range
- - i -K n r i T r n
» i « - ^ L — «-\ i J
1
12 25- 3& 50' K 75 M 100
Figure 5-14 - UREX Edge Range Loads
UREX Flap Loads so
50
40
I * 30
1 20
10
0
1 1 1 1 1—
, _ J^_ _ 1 1 _•_ _ _ _ _•_ _ _ _ _ l _
— < 1
Targes Range
Applfed Range
Load Ratio
i V L I ' ' i i i
" * " ~ ^ r - - - - rvr^r - i - - - - - i - - - " \ i ~ - - - T - - - ^ - - - - -
i i i Sv ; \_i—--J '
Figure 5 12 RTS Flap Range and Mean Loads
12 25 3B 50 6B 75 SB 1D0< BSatfa Station (%)
Figure 5-15 - UREX Flap Range Load
45
The loads on the blade are a combination of loads due to gravity and the addition of saddles to
the blade, as well as the targeted range loads. The tare loads in Figure 5-10 and Figure 5-13 show the
UREX and PhLEX system are virtually identical, as the masses added by the saddles are equal. The edge
loads in Figure 5-11 and Figure 5-14 are both close to the target load. The load ratios of the PhLEX and
UREX systems are essentially equal. In the flap direction, the loading shows a slightly different profile.
The PhLEX load ratio in Figure 5-12 is higher up to 55% blade station when compared to the UREX load
in Figure 5-15. However, after 55%, the load is closer to the load target in RTS system than in the UREX.
5,7 Flow Rate Requirements
35:
30 -
25 -
PhLEX Acutator Flow Rates
PhLEX Flap Edge Sum
0.9 1
Figure 5-16 - PhLEX Actuator Flow Pate
46
UREX Acutator Flow Rates 2 5 : r : - : - -r : T . : z H IZ ZL
I Flap
The flow rates for both systems are displayed within a one second time frame. The flow rate
was calculated using the relationship of FR = (PA)*(S)*(f). FR is the maximum flow rate, PA is the piston
area, S is the actuator stroke, and f is the excitation frequency (23). The PhLEX system in Figure 5-16 is
operating both actuators at the same frequency of 5 cycles per second. There is a set phase difference
between the edge and flap actuators. The addition of the PhLEX actuator in the flap direction increases
the flow requirements of the PhLEX. The sum of the flow rate of all actuators reaches a maximum of 34
GPM for the PhLEX. Figure 5-17 shows the UREX actuators are running at over 2 Hz in the flap direction,
and over 5 Hz in the edge direction. Due to the lower natural frequency and lack of the fourth actuator,
the flow rates are less for the UREX system. Since all components are running slow, the UREX maximum
flow rate is 21 GPM.
47
5,8 Power Requirements
The power requirements of the system were found by taking the f low rate and multiplying it by
the hydraulic pressure (52)(53). The hydraulic pump has a pressure of 3000 psi, so the sum of the f low
rate at each cycle was mult ipl ied by the pump pressure. The PhLEX system in Figure 5-18 will take
approximately 1.6 times more total power to conduct the test than the UREX system in Figure 5-19. The
t ime reach the number of cycles wil l be decreased by a factor of 2 due to the increased natural
frequency in the flap direction.
PhLEX Power 45:-
40
35 -
30
PhLEX Flap Edge Sum
| 25
/} / u\Jf> / -V / . - -^/ J-0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (s)
Figure 5 18 - PhLEX Power Requii - J
48
30:-UREX Power
; — 7 — ^ ~~ T
Figure 5-19 - UREX Power Required
5.9 Test Duration
Assuming an uninterrupted test was to be completed, a time comparison between the two
systems is shown in Table 5-3. This time comparison is based on a one million cycle test for both the
edge and flap directions. The PhLEX system will take approximately half the time to complete an entire
test, at which point with the UREX the test would still only be half done due to the natural frequency of
the flap being lower than that of the edge.
Natural Frequency (Hz)
Hours to Complete 1 Million Cycles
Hours to Complete 2 Million Cycles
Hours to Complete 3 Million Cycles
Hours to Complete 4 Million Cycles
RTS
First Mode
5.12719
54.17737
108.3548
162.5322
216.7096
Second Mode
5.12719
54.17735
108.3548
162.5322
216.7096
UREX
First Mode
2.45253
113.2616
226.5234
339.7852
453.0469
Second Mode
5.16967
53.73224
107.4644
161.1966
214.9288
Table 5-3 - Test Duration
49
Chapter 6 Conclusion and Future Work
6.1 Conclusion
The research for this project was designed to provide for better methods to evaluate fatigue
properties of wind turbine blades. Previous blade testing methods provided the ability to place a
targeted load on the blade, however the method by which they were applied did not allow for the
control of the phase angle between the edge and flap loads. By utilizing the method of adding a spring
between the ground and blade in the finite element model, control of the phase angle was obtained by
modifying the natural frequency in the flap direction. The finite element model was developed with the
ability to work with either Timoshenko or Euler-Bernoulli beam theory. Optimization code is utilized to
find the value of the blade rotation angle and stiffness added to the blade in order to have the smallest
possible difference between the first and second natural frequencies. The optimization code utilizes the
difference of the first and second Eigen values squared as its output, and exits upon satisfaction of the
function and input tolerances. Since the edge and flap are expected to be operating at resonance, the
mode shapes from the Eigen analysis of the stiffness and mass matrices are scaled to reach the target
loading. The work of the PhLEX system is the calculated in order to compare against that of the UREX.
The work of the entire system is increased due to the additional actuator; however the test duration can
be decreased by half. The blade test can be completed with the PhLEX system in the same time as
testing the edge direction using the UREX system. Using the PhLEX system, the phase of the edge and
flap loading can be locked, and therefore the blade can be loaded as it typically is in the field.
6.2 Future Work
Future work for the PhLEX system should involve additional research into the system, and a
demonstration of the PhLEX system. A proof of concept test should be performed utilizing selected
hardware. The data collected from the test should be processed in order to validate the pre-test models
to allow correlation between the model developed and how the test system performs. A scaling study
50
to investigate the systems requirements as the blade length increases should be performed. This study
should utilize historical blade test data and results to predict how the PhLEX system will perform on
larger blades. An investigation into electro-mechanical actuators to show benefits when compared to
the current hydraulic actuator systems should also be performed in order to develop possible
alternative implementation methods.
51
Appendix A Sow ce Code clc; clear; close - 1 i ; format 1 nq; useBasicEA = 0/ plotModes = 0; numele = 160; useTimo = 1; cutlength = 1; positionl = 0; stiffnessl = 0; excMass = 100 + 58 + 22; %11. excMass = excMass * .45359237; % 'nn^err fr rr It to ^\ position2 = 5.4;
if useBasicEA == 1 stiffness2 = 121795.871031491; outboardMassChange = -42.9383567263535;
else if useTimo == 0 % P-moulli nu IT
stiffness2 = 14 6827.281315523; outboardMassChange = -23.6690480807710;
else stiffness2 = 140963.216162977;% Ti T:,sheriKO ce.rr outboardMassChange = -28.5831246716763;
end en i
rotateBladeAngle = 10;
sload(l) = (4 .306) *1000+excMass*9.8; % <IT /jn-erted to [I sloc(l) = 1.6; sload(2) = 1 . 401*1000 + outboardMassChange*9. 8; %kfl :on-r^l to II sloe (2) = position2;
[K, M, mpl, R, eL, EA] = fn_rotateBlade(rotateBladeAngle, sload, sloe, numele, cutlength, useBasicEA, useTimo);
rtsne = sloe(1)/eL+1;
R=R' ; R(:,2) = abs(R-position2); R( : , 3)=1 .'length (R) ; R = sortrows(R, 2);
if R(l,2) >= le-12 surroundingNodes = R(l:2,:); surroundingNodes = sortrows(surroundingNodes, 3); NodeA = surroundingNodes(1,3)*6-3 ; NodeB = surroundingNodes(2 , 3) * 6-3 ; K(NodeA, NodeA) = K(NodeA, NodeA) + (1-
surroundingNodes(1,2)/eL)*stiffness2; K(NodeB, NodeB) = K(NodeB, NodeB) + (1-
surroundingNodes(2,2)/eL)*stiffness2;
52
nodalNum = (NodeB - 3) / 6; else
NodeA = R(l,3)*6-3; K(NodeA, NodeA) = K(NodeA, NodeA) + stiffness2; nodalNum = R(l,3);
end
R = s o r t r o w s ( R , 3 ) ;
[V,D] = e i g ( M ( 7 : e n d , 7 : end) \K(7 : e n d , 7 :end) , ' n >ba 1 -incp ' ) ;
% Sort the eigenvalues and eigenvectors
[D,b] = sort(diag(D));
V = V ( : , b ' ) ;
V_size = size(V);
ele = 1:V size (1)/6;
xd_ele zd_ele yd__ele xr_ele yr_ele zr ele
6:V_size(1) 6:V_size(1) 6:V_size(1) 6:V_size(1) 6:V_size(1) 6:V size (1)
displacement along the length of the blade displacement in the edge direction displacement in the flap direction rotation along the length of the blade rotation about the flap direction rotation about the edge of the blade
Vs struct('x: 'zi'
'xr '
' z r '
[zeros(size(V,1),1) V(xd_ele, [zeros(size(V,1),1) V(zd_ele, [zeros(size(V,1),1) V(yd_ele, [zeros(size(V,1),1) V(xr_ele, [zeros(size(V,1),1) V(yr_ele, [zeros(size(V,1),1) V(zr ele,
max_norm_value = sqrt(abs(Vs.zd(size(Vs.zd,1),:)).A2 + abs(Vs.yd(size(Vs.yd,1),:))."2).*sign(Vs.yd(size(Vs.yd,1),:));
V norml = V ./ (ones(V size(l), 1)*max norm value);
Vs norml = struct ( -i', V_norml(xd_ele, zd', V_norml(zd_ele, _V, V_norml(yd_ele, xr', V_norml(xr_ele, }r', V_norml(yr_ele, ZJ', V norml(zr ele, )
% Get the angle at 70% station angle = atan2(Vs_norml.yd(round(size(Vs_norml.yd,1)),:), Vs_norml.zd(round(size(Vs_norml.zd,l)),:))*180/pi;
V_mag = zeros(6,size(V,2)); g = fieldnames(Vs_norml); count = zeros (7,size(Vs_norml.xd,2)) ; f r 1 = 1 .-length (g)
53
Vs_diff.(g{l}) = diff(Vs_norml.(g{l})); for r=l:size(Vs_diff. (g{l)) , 2)
V_mag(l,r) = norm(Vs. (g{l) ) (: , r) ) ; signdV = sign(Vs_diff. (g{l) ) ( : , r) ) ; for t=l:size(Vs_diff.(g{l)), 1)-1
if signdV(t) -= signdV(t+l) count(l,r) = count(l,r) + 1;
end end
end end
% Tolle^t data on ill Lhe mo J- shapes Picker = [V_mag' count (1:6, :) ' angle' D]; [a b] = max(Picker (:,1:6) ') ; Picker ( :,15) = b';
% Fini the -tiffnesr -i\ the UFEZ pu-'ition % Flap F=zeros(size(K,1)-6,1); F(rtsne*6-3) = 1; F(rtsne*6-4) = 1; displacementUREX = K(7:end, 7:end)\F; y_displaceUREX = displacementUREX(3:6:end); y_stiffUREX = l/y_displaceUREX(rtsne) % E 3ge F = z e r o s ( s i z e ( K , 1 ) - 6 , 1 ) ; F ( r t s n e * 6 - 4 ) = 1; displacementUREX = K(7 rend, 7:end)\F; x_displaceUREX = displacementUREX(2:6:end); x_stiffUREX = l/x_displaceUREX(rtsne)
% Find the jtiffnejs at the PhLEX actuator [usition % Flap F=zeros(size(K,1)-6,1); F(NodeA) = 1; displacement = K(7:end, 7:end)\F; y_displace = displacement(3:6:end); y_stiff = l/y_displace(nodalNum) % Edge F=zeros(size(K,1)-6,1); F(NodeA-1) = 1; displacement = K(7:end, 7:end)\F; x_displace = displacement(2:6:end); x stiff = l/x_displace(nodalNum)
% Find the lifference in ^igemnl ies / f re j ien'v eig_diff = (Picker(1,14)-Picker(2, 14 )) "2 freq_diff = sqrt(sqrt(eig_diff))/2/pi
% Gt.t the flao -md edge load targets segments = [o".23 1.58 3.38 5.63 7.43 9]; %Scale the loads to Lhe cut bid Je length segments perc = segments./max(segments); segments = max(R(:,1))*segments_perc;
54
rangeFlapLoadTarget = [56180 54690 39020 20660 6390 1200 0]; rangeEdgeLoadTarget = [20780 19731 14124 8103 2913 632 0]; meanFlapLoadTarget = [34330 33420 23850 12620 3910 730 0]; % Curve fit the targets rangeFlapLoadTarget = ppval(pchip(segments, rangeFlapLoadTarget), 0:eL:max(R( : , 1) ) ) ; rangeEdgeLoadTarget = ppval(pchip(segments, rangeEdgeLoadTarget), 0:eL:max(R(:,1))); meanFlapLoadTarget = ppval(pchip(segments, meanFlapLoadTarget), 0:eL:max(R(: , 1) ) ) ;
% Build acceleration vectors accelFlap = Vs.yd(:,l)*sign(Vs.yd(end,1)).*D(1)+Vs.yd(:,2)*sign(Vs.yd(end,2)).*D(2); accelEdge = Vs.zd(:,1)*sign(Vs.zd(end,1)).*D(1)+Vs.zd(:,2)*sign(Vs.zd(end,2)). *D(2); ForceFlap = accelFlap .* eL .* mpl'; ForceEdge = accelEdge .* eL .* mpl1; ForceGrav = 9.81 .* ones(size(Vs.yd(:,1),1),1) .* eL .* mpl'; for i=l:size(ForceFlap); % All three force vectors should be the same size
mTare(i) = sum(ForceGrav(I:end) .* R(1:end-(l-l),1)); MFlapRange(I) = sum(ForceFlap(I:end) .* R(1:end-(l-l),1)); MEdgeRange(l) = sum(ForceEdge(l:end) .* R(1:end-(l-l),1));
end
% Find displacement due to gravity: x=K\F forceGravV = zeros(size(K(7:end,7:end),1), 1); for i=3:6:size(forceGravV)
forceGravV(I) = ForceGrav((i+3)/6); end %forceGravM = diag(forceGravM); sag = K(7:end,7 rend)\forceGravV; sag = [0 sag(3:6:end)']';
flapDisplacement = max(abs(rangeFlapLoadTarget))/MFlapRange (1); edgeDisplacement = max(abs(rangeEdgeLoadTarget))/MEdgeRange(1); flapdisplace = flapDisplacement/2 * sum(abs(Vs.yd(:,1:2)),2); edgedisplace = edgeDisplacement/2 * sum(abs(Vs.zd(:,1:2)),2); MFlapRange = MFlapRange * flapDisplacement; MEdgeRange = MEdgeRange * edgeDisplacement; accelFlap = accelFlap * flapDisplacement; accelEdge = accelEdge * edgeDisplacement;
% Get displacement at 60% flapd = flapdisplace(nodalNum); edged = edgedisplace(nodalNum);
% Ex itation force S = 4*.0254; reqExcMass = ( (4*y_stiffUREX*flapdisplace(rtsne)*.03124) /(S*D(1) ) ) /2 excF = reqExcMass*D(1)/2*(S+flapdisplace(rtsne)*2)+reqExcMass*9.81 reqExcMassEdge = ((4*x_stiffUREX*edgedisplace(rtsne)*.03124)/(S*D(l)))/2
55
excFEdge =
reqExcMassEdge*D(l)/2*(S+edgedisplace(rtsne)*2)+reqExcMassEdge*9.81 if reqExcMass < excMass
display ( ' - • r =• s r - j n r ] i 1, r . - , 1 -, . ' ) ; else
display ( ' L\> ,„ ] \_ ,r Iss' ! ' ) ; end
% U R E A Work
FlapW = S * excF; EdgeW = S * excFEdge;
% C i l ^ u l n t e b ~ d t s i z e s r e j u i r e d f o r s ^ j J l e % G e t Lh* a s •• l e r a t i o r i ' , u - e PCI11F I D f i n J t h F_a_p = ppval(pchip(R(lrend,1), accelFlap), 0
SaddleLoad = F_a_p(sloc*100+l).*sload/9.8;
NtoLB = .224808943;
SaddleLoad = SaddleLoad * NtoLB
boltSize = 1/2;
boltArea = pi* (boltSize./2) . A 2 ;
boltStress = SaddleLoad ./ boltArea
% Plot the displacement of the blade yl = abs(Vs.yd(nodalNum,1))*flapDisplacement/2; y2 = abs(Vs.yd(nodalNum,2))*flapDisplacement/2; zl = abs(Vs.zd(nodalNum,1))*edgeDisplacement/2; z2 = abs(Vs.zd(nodalNum,2))*edgeDisplacement/2;
zm = [zl z2; yl y2; ] ; disp = [flapDisplacement; edgeDisplacement;];
dk = zm\disp; dk = [1 1];
t = 0:pi/512:500*pi; theta = 72*pi/180;
z = zl*cos(t*sqrt(D(1))+rotateBladeAngle*pi/180)+z2*cos(theta+rotateBladeAngle*pi /180+t*sqrt(D(2)));
y = y l * c o s ( t * s q r t ( D ( l ) ) + r o t a t e B l a d e A n g l e * p i / 1 8 0 ) + y 2 * c o s ( t h e t a + r o t a t e B l a d e A n g l e * p i / 1 8 0 + t * s q r t (D(2) ) ) ;
% F i n l t h e change in d ^ t a a i o r l e n g t h . a c t o l e n g t h = 6 1 . 5 4 * . 0254; % m - j r i g m a l dCt i aL r l e n g t h . % f Tn i a c t u a t o r r o t a t i n g p >int : : - l i i n ^ e ? a c t o z = - a c t _ o _ l e n g t h * s i n d ( r o t a t e B l a d e A n g l e ) ;
l r i t e r rr e h a to . 0 1 r m a x ( R ( : , 1 ) ) ) ;
56
act_o_y = -act_o_length*sind(90-rotateBladeAngle);
% Find new actuator angle and length for each position % [x y actuator_total_length theta stroke] act ( act ( act ( act ( act (
,1) = z - act_o_z; ,2) = y - act_o_y; ,3) = sqrt(act(:,1) ."2 + act (:,2) . A 2 ) ; ,4) = 90-atan2 (act (:,2),act(:,1))*180/pi; ,5) = act(:,3)-act_o_length;
[a b] = max(act(r,5)); max_force = sqrt((x_stiff * z(b))"2+(y_stiff * y(b))A2)*.224808943 % converted to kip''iii
% PhLEX Work PhW = maxfact ( r , 5) ) * maxjorce/ . 224808943;
% Total Work numCycles = le6; totalPhlex = (PhW); totalUREX = (FlapW + EdgeW); totalPhlexWork = totalPhlex + totalUREX; totalPhlexPower = totalPhlexWork * sqrt(D(l));
% UREX Only Test P = 20684271.9; % 3000 psi fxforce = 1.529161330927132e+03; exforce = 1.920389890814912e+02; stroke = .1016;
%Flap %Edge
stroke; stroke; fW + 2 * eW;
model = 15.619 mode2 = 32.793 fW = fxforce * eW = exforce * totalWork = 2 totalPower = fW * model + eW * mode2; PhlexUREX = totalPhlexPower/totalPower
% UREX ActUdtors PA = .00027; % m"2 t = 0:pi/512:l; S = S/2; FR_flap = PA*model*(S*(cos(t*model))+S)*264.172*60; FR_edge = PA*mode2*(S*(cos(t*mode2))+S)*264.172*60; P_flap = FR_flap*P/264.172/60; P_edge = FR_edge*P/264.172/60;
% PhLEX Actuator PA_p = .00075; % m"2 S_ph = max(act(:,5))12; FR_flap_ph = PA*sqrt(D(l))*(S*(cos(t*sqrt(D (1) )) )+S)*2 64.172*60; FR_edge_ph = PA*sqrt(D(l))*(S*(cos(t*sqrt(D(l))+theta))+S)*2 64.172*60; FR_ph = PA_p*sqrt(D(l))*(S_ph*(cos(t*sqrt(D(l))+pi/2))+S_ph)*264.172*60; P_flap_ph = FR_flap_ph*P/264.172/60; P_edge_ph = FR_edge_ph*P/264.172/60; P_ph = FR_ph*P/264.172/60;
57
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58
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