chord lengthsD = dragE = tensile modulusec = unit vector along chord of blade elementeD = unit vector drag on blade elementeL = unit vector along lift on blade elementer, eφ, ez = polar coordinate system triadex, ey, ez = global coordinate system triadeξ, eη, eζ = local mechanical coordinate system triadeξ, e
�η , e
�ζ = local blade-element coordinate system triad
F = Prandtl’s momentum loss factorG = shear modulusIt = cross-sectional torsion constant
Iη, Iζ = cross-sectional moments of inertia along eη and eζJ = advance ratio, v0∕n2RTL = liftl = length of blade axisM = internal momentMξ = torsional moment along blade axisMζ = bending moment along major principal
axis of cross sectionMη = bending moment along minor principal
axis of cross sectionn = propeller revolutions per secondNB = number of propeller bladesNξ = axial force along blade axisP = braking powerQ = braking forceqa = aerodynamic loadingqc = inertial loadingr, φ, z = coordinates along basis vectors er, eφ, and ezRH = hub radiusRT = tip radiusra = vector from blade axis to center of pressures = arc lengthT = thrustU = Prandtl’s stress functionu = arc length integration parameterv = total airstream velocityv0 = advance velocityv� = velocity in plane of blade-element cross sectionWPL = lower limit of working conditionsWPU = upper limit of working conditionsx, y, z = coordinates along basis vectors ex, ey, and ezz0 = blade axis tip coordinate along ezα = angle of attackβ = blade pitch angleβASD = allowable stress design blade pitch angleδc∕4 = distance from blade axis to quarter-chord pointε = drag-to-lift ratio, CD∕CLη = propeller efficiencyϑ = angle from chord to minor principal axis
of cross sectionξ, η, ζ = coordinates along coordinate system eξ, eη, eζρ = fluid densityρb = density of blade material
ρη = curvature of η vs J curveρη, ρζ = blade axis curvature in direction of eη and eζσ = normal stressσa = maximum allowed stress in cross sectionσeq = von Mises equivalent stressτ = shear stressϕ = blade axis parameterφ0 = blade axis tip coordinate along eφϕ� = inflow angleχ = nondimensional radius, r∕RTω = propeller angular velocity� = nondimensional propertiy0, 0 0 = differentiation with respect to s, d�·�∕ds
I. Introduction
A DESIGN procedure taking into account the flexibility ofpropeller blades is proposed. Traditional aerodynamic design
procedures treat propeller blades as rigid, neglecting the flexibilityand aeroelastic characteristics of the blades.However,when the bladeflexibility is taken into account, it can be exploited in order toimprove propeller performance over a range of operating conditions.Indeed, by choosing an appropriate blade axis (BA), one can eitherimprove or reduce the performance of the flexible blades over theselected range of working conditions. It was already noted by Munk[1] in his patent application in 1949 that already relatively smallelastic changes in blade pitch angle noticeably improved propellerperformance.In the last decades, several aeroelastic models have been suggested
by Rosen and Friedmann [2], Hodges and Dowell [3], and Ahlström[4], among others, mainly in order to analyze the aeroelasticity of thehelicopter rotors and wind-turbine blades.There have been attempts to develop adaptive flexible propellers.
Sandak and Rosen [5] proposed to insert a flexible element betweenthe hub and the root of the blade. Exposed to aerodynamic loads, theflexible insertion undergoes torsional deformations that yield afavorable change in pitch angle at the root of the blade. They reportedthat swept blades could benefit in terms of performance from thisapproach.Another approach to passively adaptive propellers was suggested
byHeinzen et al. [6]. Their idea is somewhat different. The blades arestill treated as rigid, but they are allowed to pivot freely around theradial axis at the hub of the propeller. The aerodynamic properties ofthe blades are tailored in such a way that the equilibrium of theaerodynamic loads is attained at the optimum inflow angle.Alternatively, in the case of flexible blades, they have a fixed pitch,
but they are designed in such amanner that their elastic axis and pitchdeform into a more favorable configuration when the aerodynamicand/or inertial loads change.A complete design procedure consisting of aerodynamic
optimization, allowable stress design, and BA optimization yieldingan optimized propeller blade taking advantage of the BA flexibilitywith the intention of improving its aerodynamic performance has notyet been suggested.It is known that variable-pitch propellers effectively solve the
problemof propeller efficiency at different advance ratios. However aflexible fixed-pitch propeller has a number of advantages over itsvariable-pitch counterpart: it reduces the complexity and weight ofthe propulsion system; furthermore, variable-pitch propellers are notallowed in the light sports aircraft category.In what follows, a complete mathematical model of flexible
propeller blades is presented: first, a geometrical description of theblades is given, followed by a detailed explanation of how theclassical blade-element momentum model proposed by Adkins andLiebeck [7] was extended to take into account blades with a bladeaxis of arbitrary geometry; in the end, the structural part of themathematical model is given. Then, a propeller design algorithm isdescribed in Sec. III. It is shown there how the mathematical modelwas cast into an actual design tool. The description of the designalgorithm is followed by the description of the experimental setupand comparison and discussion of the numerical and theoretical
results in Secs. IVand V, respectively. All the important findings andobservations are concentrated in the Conclusions (Sec. VI).
II. Mathematical Model of the Flexible Propeller Blades
To determine the aerodynamic and structural properties ofthe flexible blades, a mathematical model was composed. Aero-dynamic properties were described by extending the blade-elementmomentum (BEM) model of propeller action proposed by Adkinsand Liebeck [7]. The extension of the BEMhad to be able to take intoaccount blades of arbitrary geometry. The structural behavior of theblades was treated by the Euler–Bernoulli beam theory, accountingfor bending deformations; and the Saint–Venant theory of torsion,accounting for torsional deformations. The structural integrity of theblades was assessed according to the von Misses stress theory.
A. Geometry of the Blade
BAwas defined by two coordinate functions, φ�r� and z�r�, in thepolar coordinate system, as can be seen from Fig. 1. Coordinatefunctions could be, in principle, any arbitrary functions. For the sakeof clarity and ease of parameterization, it was decided to define themas cubic splines:
φ�r� � φ0
2
�3
�r − RHRT − RH
�2
−�r − RHRT − RH
�3�;
z�r� � z02
�3
�r − RHRT − RH
�2
−�r − RHRT − RH
�3�
(1)
satisfying the boundary conditions given by
φ�RH� � φ 0�RH� � φ 0 0�RT� � 0; φ�RT� � φ0;
z�RH� � z 0�RH� � z 0 0�RT� � 0; z�RT� � z0 (2)
All boundary conditions, with the exception of φ�RT� and z�RT�,were kept constant; φ�RT� and z�RT� were treated as parameters φ0
and z0, respectively, in the BA optimization procedure, which isdescribed in more detail in Sec. III.The arc length s and the entire length of the BA lwere determined
by following:
s�r� �ZRT
RH
���������������������������������1� r2φ 02 � z 02
qdr; l � s�RT� (3)
The arc length was then taken as a new independent variable. The BAwas then calculated in the global coordinate system (GCS) defined bybasis vectors ex, ey, and ez by the following expression:
Fig. 1 Definition of blade axis coordinate functions.
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y � y�s�; z � z�s�
x�s� � Rh �Zs
0
���������������������������1 − y 02 − x 02
qds; 0 ≤ s ≤ l (4)
The position of the BA and orientation of the GCS are depicted inFig. 2. Along the BA, two more local coordinate systems (LCSs) hadto be defined in order to allow for the vectorization of the BEMequations and to allow for solving the equations of beam mechanics(LCSM). LCS was used to position the blade sections with respect tothe BA and determine their angle of attack with respect to theincoming airstream. The aerodynamic load induced by the individualblade section is also naturally expressed in theLCS,which is depictedin Fig. 2. Themathematical definition of the LCSbasis vectors eξ, e
�η ,
and e�ζ is taken from Drazumeric and Kosel [8]:
eξ �����������������������������1 − y 02 − z 02
qex � y 0ey � z 0ez;
e�η � −y 0ex �z 02 � y 02
��������������������������1 − y 02 − z 02
py 02 � z 02 ey
−y 0z 0 �1 −
��������������������������1 − y 02 − z 02
p�
y 02 � z 02 ez;
e�ζ � −z 0ex −y 0z 0 �1 −
��������������������������1 − y 02 − z 02
p�
y 02 � z 02 ey
� y02 � z 02
��������������������������1 − y 02 − z 02
py 02 � z 02 ez (5)
LCSM was used to disassemble the loading acting on individualblade elements into components along the principal axis of the crosssections. LCSM is defined by basis vectors eξ, eη, and eζ . The vectorcomponent eξ is the same as in the LCS, pointing along theBA,whileeη and eζ follow the relations
LCS, LCSM. and the quantities used in Eqs. (2) and (3) are explainedby Fig. 2. In addition, β and ϑ represent the pitch angle from e�η to thechord of the airfoil c and from the chord to the minor principal axis ofthe airfoil, respectively.It is assumed that the BA runs through the center of gravity of the
blade-element cross section, which then represents the blade’s elasticaxis as well. In general, the elastic axis does not coincide with the
cross-sectional center of gravity; however, since the blade-elementcross section is closed and singly connected, the assumption that theBA and elastic axis coincide is reasonable.
B. Extended Blade-Element Momentum Model
Aerodynamic properties of the blades were determined byextending the blade-element momentum (EBEM) model proposedby Adkins and Liebeck [7]. The BEM model was selected for itsreported reliability of results in comparison to experimental data,numerical efficiency, and versatility, as has been show by Gur andRosen [9], Hepperle [10], and Whitmore and Merrill [11], amongothers. Extending the BEM requires one to rewrite the BEMequations in vector form, therewith allowing for BA geometries ofarbitrary shape and orientation in space and correcting the equationsdescribing the interference factors az and aφ.The blade-element model is based on the properties of the cross
section, namely, on the airfoil used. To calculate the contribution ofindividual blade elements to the total thrust and torque, it is necessaryto calculate the distributions of aerodynamic forces,dL∕ds anddD∕ds, along the BA. To do so, it is necessary to determine thevelocity in the plane of the cross section v� and the angle of attack ofindividual blade elements α. The problem is illustrated in Fig. 3.The total velocity v and corresponding ϕ at a given s are
determined by
v � −ωr�1 − aφ�eφ − v0�1� az�ez (7)
tan ϕ � v0�1� az�ωr�1 − aφ�
(8)
Velocity v� is a projection of v onto the plane of the cross section. It isassumed that the cross-section plane is always perpendicular to theBA, and therewith to eξ as well. Therefore, the projection is carriedout by using a double vector product:
v� � eξ × �v × eξ� (9)
The inflow angle ϕ� is calculated in a similar fashion by defining aunit vector along the v and projecting it onto the adequate vectors ofLCS: e�η and e�ζ , respectively:
tan ϕ� ��cos �ϕ�eφ � sin �ϕ�ez� · e�ζ�cos �ϕ�eφ � sin �ϕ�ez� · e�η
(10)
The angle of attack is finally calculated as α � β − ϕ�. Lift and dragdistribution along the BA are given by
Fig. 2 Definition of local coordinate system on the blade axis.
Fig. 3 Loads on a blade element.
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dL
ds� 1
2Nbρv
�2cCL (11)
dD
ds� ε
dL
ds(12)
where CL and ε are read out of the airfoil CL vs α and CD vs αcharacteristics at Re � 2 · 105 measured by Lyon et al. [12]. Weopted to use the Clark Y as the airfoil of the cross section.Lift acts in a direction eL perpendicular to v�, whereas drag acts
along eD, which in turn points in the direction of v�. The directions of
the lift and drag forces are expressed, respectively, as
eL � − sin �ϕ��e�η � cos �ϕ��e�ζ (13)
eD � cos �ϕ��e�η � sin �ϕ��e�ζ (14)
The vectors eL and eD are used in determiningCZ andCφ, which canbe cast as
CZ � CL�eL − εeD� · ez (15)
Cφ � CL�−eL � εeD� · eφ (16)
As a result, dT∕ds and dQ∕ds are expressed easily, regardless of theshape of the BA:
dT
ds� 1
2ρv�2NBcCZ (17)
dQ
ds� 1
2ρv�2NBcCφ (18)
The momentum model is based on conservation of the linear andangular momenta. The amount of momentum imparted on the fluidpassing through the propeller disk is accounted for by the interferencefactors. These in turn are calculated by relating themomentummodelto the blade-elementmodel. At this point, it is important to realize thatforces and moments on blade elements are all derived with respect tos. On the other hand, forces and moments calculated with themomentummodel are all derived with respect to r. Namely, the bladeelement of the width ds at position s generates forces and momentsthat are related to the forces and moments produced in an annulus ofwidth dr�s� at radius r�s� by themomentummodel. Furthermore, theinterference factors have to be corrected in such away that they retaintheir physical meaning in the momentum equations, which is thevelocity increase in the axial and angular directions.Momentum equations are independent of the BA shape. The
equations used are the same as proposed by Adkins and Liebeck [7]:
dT
dr� 2πrρv0�1� az�2v0azF (19)
dQ
dr� 2πrρv0�1� az�2ωraφF (20)
Combining Eqs. (15–20) and realizing that holds dr∕ds � er · eξ,the corrected equations for az and aφ are derived as
az �NBcCzer · eξ
8πrF�eL · eφ�2 − NBcCzer · eξ(21)
aφ �NBcCφer · eξ
−8πrF�eL · eφ��eL · ez� � NBcCφer · eξ(22)
where
F � 2
πarc cos�e−f�; f � NB�1 − χ�
���������������������������1� χ2 tan2 ϕ
p2χ tan ϕ
(23)
To obtain the total T and P, their respective distributions [Eqs. (17)and (18)] have to be integrated along the blade axis:
T �Zl
0
dT
dsds (24)
P � ω
Zl
0
rdQ
dsds (25)
Finally, CT , CP, and η are calculated as
CT �T
ρn2D4(26)
CP �P
ρn3D5(27)
η � CTJCP
(28)
C. Structural Model of the Blade
The propeller blades considered in this work can be thought of asslender, pretwisted cantilever beams of variable cross sections builtout of homogenous and isotropic elastic materials. Furthermore, it isassumed that blades in the un-deformed state are straight enough thatno special treatment for nonstraight beams is necessary. Due to theallowable stress design performed on the blade, it is expected that allthe stresses are within the elastic region of the material at all times.The blade’s cross sections are topologically the same along the
entire length of the BA. The cross sections are, however, scaleduniformly in size. Therefore, a unit cross section was introduced.Airfoil chord was used as a normalization scale. Nondimensionalcoordinates were introduced in the LCSM as �η � η∕c and �ζ � ζ∕c,allowing for calculation of important cross-sectional properties,namely, domain area, principal moments of inertia, and torsionalmoments of inertia, in nondimensional forms. Nondimensionalproperties and their dimensional counterparts are calculated as
�A �Z
�Ad �A; A � c2A;
�Iη �Z
�A
�ζ2 d �A; Iη � c4 �Iη;
�Iζ �Z
�A
�η2 d �A; Iζ � c4 �Iζ;
�It � −4Z
�A
�U d �A; It � c4 �It (29)
where �U is the Prandtl’s stress function, which was also calculatedin nondimensional form by solving Poisson’s equation with anappropriate boundary condition [13]:
Δ �U � 1; �Uj∂ �A � 0 (30)
Aerodynamic and inertial loading along with internal stresses arepresented in Fig. 4. Aerodynamic loading qa is obtained by addingtogether lift and drag distributions along the BA, given by Eqs. (11)and (12), respectively. The resulting qa is then expressed as
qa �1
Nb
dL
ds�eL − εeD� (31)
whereqa acts in the center of pressure. To obtain the accurate internalstress due to aerodynamic loads, it is necessary to calculate the
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distance from the c.p. to the BA. First, a unit vector along the airfoilchord is defined as ec � cos βe�η � sin βe�ζ . Therefore, ec is used todefine the radi vector ra pointing from the BA to the c.p., where ra isdefined as
ra � c�δc∕4 �
CM;c4
CL�cos α� ε sin α�
�ec (32)
where δc∕4 represents the distance from the c.g. to the quarter-chordpoint along the ec measured from the leading edge of the airfoil.Inertial loads comprise the centrifugal force acting on the blade. Eachblade element experiences centrifugal force that is proportional to itsmass, radial position, and the angular velocity. Consequently, theinertial loads are expressed as
qc � ρbc2 �Aω2rer (33)
It is important to notice that the centrifugal force acting on everyblade element points in the radial direction, namely, er, regardless ofthe BA shape and its position in space. Both qa and qc are expressedin the GCS. Hence, also the total internal momentM is expressed inthe GCS by adding together qa and qc, and integrating along the BA:
M �Zl
s��rq�u� − rq�s�� × �qa�u� � qc�u�� � ra�u�
× qa�u�� du (34)
The first integral term in Eq. (34) represents the contribution of bothaerodynamic and inertial loadings, qa and qc, to the internal momentM due to the shape of the BA. The second term of Eq. (34), however,represents the contribution of qa to theM due to the misalignment ofthe airfoil’s c.p. and BA. To evaluate stresses and deformations, themomentM is decomposed into three components along the BA andthe two principal axes of the cross section:
Mξ �M · eξ; Mη �M · eη; Mζ �M · eζ (35)
Deformations of the blade are composed out of two mutuallydependent processes: bending and torsion. Bending deformations aretreated according to the Euler–Bernoulli theory, taking into accountlarge deflections. In this respect, large deflections are accounted foraccurately in a geometrical sense. Torsional deformations areaccounted for by applying the Saint–Venant theory of torsion,neglecting warping effects. All the cross sections along the blade aresolid. Consequently, warping of the cross sections along the bladedue to torsion is expected to be small. Therefore, warping is neglectedin the calculation of the deformed shape of the blade. On the otherhand, blades are rigidly attached to the hub. Consequently, there isnormal stress acting on the blade’s root cross section preventing itfromwarping. However, since the blades and the hub are made out ofthe same material and warping is already considered negligible,normal stress due to warping can also be neglected in the stresscalculations [14].
Torsional and bending deformations were related to appropriatetwisting and bending moments by studying changes in pitch angle βand curvatures of the BA in the direction of the principal axis ofthe cross section, ρη and ρζ, respectively. To calculate the (un)deformed geometry of the BA, the following constitutive equationswere used [8]:
β 0 − β 0DP �Mξ −MξDP
GIt;
1
ρη−
1
ρηDP� −
Mη −MηDP
EIη;
1
ρζ−
1
ρζDP� Mζ −MζDP
EIζ(36)
Herein, the curvatures ρη and ρζ at the corresponding section of theBAwere expressed with the basis vectors of the LCSM as [14]
1
ρη� e 0ξ · eζ;
1
ρζ� e 0ξ · eη (37)
By inserting Eqs. (5), (6), and (37) into the constitutive model[Eq. (36)], the nonlinear mathematical model of the blade bendingand torsional deformations is obtained as a system of differentialequations:
β 0 � β 0DP −Mξ −MξDP
C4G �It;
y 0 0 ��
1
ρηDP�Mη −MηDP
c4E �Iη
�eζ · ey
��
1
ρζDP−Mζ −MζDP
c4E �Iζ
�eη · ey;
z 0 0 ��
1
ρηDP�Mη −MηDP
c4E �Iη
�eζ · ez
��
1
ρζDP−Mζ −MζDP
c4E �Iζ
�eη · ez (38)
Governing equations (38) are used to calculate both the un-deformedblade geometry in the unloaded state as well as the deformedgeometry of the blades at the offdesign point working conditions. Toobtain the un-deformed shape of the BA, theMξ,Mη, andMζ are setto zero. Quantities marked with subscript DP in Eqs. (36) and (38)refer to values of the corresponding quantities in the design point.The state of stress is described by normal stress σ and shear stress τ
as follows:
σ � Nξ
c2 �A�Mη
c3 �Iη�ζ −
Mζ
c3 �Iζ�η (39)
τ � 2Mξ
c3 �Itj∇ �Uj (40)
There are three contributions to σ: axial force along the BA; Nξ; andthe two bending moments Mη and Mζ, pointing along the principalaxis of the cross section. On the other hand, τ consists only of atorsional moment along the BA. Knowledge about the state of stressis important in performing the allowable stress design during thepropeller design algorithm.It is important to point out that the presented structural model is
nonlinear as far as the treatment of the geometry of the BA isconcerned. However, the constitutive relations between stresses andstrains are still linear. The nonlinear geometrical model of the BAwasselected for two reasons:1) The structural model was to be used in an optimization
procedure, so the optimized geometry of the BA was not known inadvance. In principle, the optimized BA could have a much higher
Fig. 4 Aerodynamic and inertial loadings with corresponding axial
force, bending, and torsional moment in the cross section.
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curvature. It would also be possible to increase the number of freeoptimization parameters. As a result, the BA could assume anarbitrary shape.2) From the mathematical perspective, linearization of the
geometrical model of the BA would not simplify the problemsignificantly. Onewould still have to solve a boundary value problemin the structural part of the algorithm iteratively and iterate thestructural results with the aerodynamic part of the optimizationalgorithm. Therefore, significant gains in the numerical effectivenessdue to the linearization of the structural model are not expected.
III. Propeller Design Algorithm
The proposed mathematical model was used to create a designprocedure for flexible propeller blades. The aim of the designprocedurewas to find a complete propeller geometry that would yieldthe highest efficiency in converting braking power into useful thrustover a preselected range of working conditions.A schematic diagram of the proposed algorithm is depicted in
Fig. 5. In the beginning, the design point and two work points,WPUand WPL, have to be selected. These points are used later in thealgorithm in evaluating the objective function. After the workingconditions are selected, an initial BA geometry is selected and a pureaerodynamic optimization of the c�s� and β�s� is performed. Thegoalof the optimization is to minimize induced losses of the blade.However, such an aerodynamic optimization yields blade geometriesthat cannot support the induced aerodynamic and inertial loads sincethe chord length at the root of the blade tends toward zero, which isclearly a structurally unfeasible solution. Therefore, it is important tocarry out an ASD that resolves this issue.The ASD was based on the von Mises stress theory, taking into
account σ and τ as given by Eqs. (39) and (40). The von Mises stresstheory is built around the idea of calculating the equivalent stress σeq,which has to satisfy the following two relations:
σeq �������������������σ2 � 3τ2
p(41)
max�η;�ζ∈∂ �A
σeq�s; �η; �ζ� � σa ⇒ c � c�s� (42)
To perform the allowable stress design correctly, it is important toidentify the operational conditions at which the blade is exposed to
the greatest stress. Stress along the BA depends on aerodynamic andinertial loads, and it critically depends on their interaction with oneanother. Depending on the shape of the BA, the aerodynamic andinertial loads can either add together or subtract fromone another. Forthe sake of clarity, suppose that the inertial load is constant (as in thecase of the constant-speed propellers) and the greatest stress incurredon individual blade elements can occur at either the greatest orsmallest aerodynamic loads. What is even more interesting, for agiven BA, is that different sections of the blade can experience thegreatest stresses at different working conditions. Interaction betweenaerodynamic and inertial loadings can be deduced from Fig. 4.Furthermore, as a result of theASD, a new chord distribution along
the BA is obtained. Since the change in the chord distribution resultsin the changed loading conditions along the BA as well, the ASD hasto be included in an iterative procedure together with aerodynamicanalysis/optimization.After the ASD and the following aerodynamic optimization have
converged, the obtained blade geometry undergoes the proposedstatic aeroelastic analysis. During the analysis, the blade geometry isdeformed in accordance with the aerodynamic and inertial loads intheWPL andWPU. Since the aerodynamic loads heavily depend onthe shape of the blade, the static aeroelastic analysis is anotheriterative procedure, consecutively calculating aerodynamic andinertial loads on one hand and deformations on the other hand untiladequate convergence is reached. However, in order to reachconvergence, underrelaxation of the iteration scheme had to beemployed. The calculated datawere then used to evaluate the initiallyselected BAwith respect to the selected objective function.As an optimization objective function, the curvature of the η-vs-J
curve was selected. The idea behind selecting such an objectivefunction is presented in Fig. 6. It was expected that, by minimizingthe curvature of the η-vs-J curve around the design point (DP), theefficiency characteristic with respect to the advance ratio wouldincrease. The curvature of the η-vs-J curve was evaluated using
First and second derivatives, η 0 � dη∕dJ and η 0 0 � d2η∕dJ2,respectively, were determined using second-order central finitedifferences, taking into account values of J and η in the design pointand in the lower and upper limits of theworking conditions (WPL and
Fig. 5 Design procedure philosophy.
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WPU, respectively). From a plethora of optimization routines, aresponse surface analysis was selected to perform a BA optimizationsince a better understanding of the parametric space was desired.However, any other BA optimization method could have beenemployed at this stage.After the optimum BAwas selected, the un-deformed shape of the
BA has to be calculated. The search for the un-deformed geometry ofthe blade was mandatory should one want to manufacture a physicalmodel of the blades for testing or carry out more detailed fluid–structure interactions by means of coupled computational fluid andstructural dynamics.
IV. Verification Through Experimental Investigation
Selected blade geometries were manufactured via stereo-lithography. Therefore, the proposed design procedure wasperformed using the material properties of the resin to be used,which were obtained from the datasheet provided by the resin
manufacturer. The resin used wasWatershed®XC 11122.¶ Based onthe results of the design procedure, three different blade geometrieswere selected andmanufactured: a straight blade (SB), a blade havingminimum curvature [backward–swept blade (BB)], and a bladehavingmaximumcurvature of the η-vs-J curve [forward–swept blade(FB)]. The SBwas selectedmainly as a reference geometry. A designprocedure was set according to the blade geometrical requirementsand theWPL, DP, andWPU requirements presented in Tables 1 and 2,respectively.Manufactured blades are shown in Fig. 7. The blades were
manufactured already attached to one-third of the hub. The completepropeller was then assembled via keys and locks in a puzzlelikefashion. The keys and locks are also visible in the Fig. 7. Bymanufacturing blades this way, it was ensured that the blades wereattached to the hub properly and accurately without any stress ormisalignment in the joints, and they were perfectly staticallybalanced.The blades were tested in Clarkson University’s aeronautical wind
tunnel. Thewhole experimental setup with BBsmounted is shown inFig. 8. The wind tunnel is of the Eiffel type, and it is capable ofproducing wind speeds of up to 70 m∕s in the center of the testsection. The wind tunnel has a test section of 1.22 m wide, 0.9 m tall,
Fig. 6 Curvature of the η-vs-J curve as an optimization objectivefunction.
and 1.67 m long. Forces and moments were measured with a straingauge reaction force and torque balance, which were capable ofmeasuring all three components of the reaction force and torquevector.One of the blade tips was colored in white and monitored with a
high-speed camera. The camera was set to film at a rate of 2110frames per second and with a frame exposition time of 45 μs. As canbe observed from Fig. 8, the camera was positioned normal to thevertical plane that contains the propeller axis of rotation. Examples ofthe captured and processed images at different advance ratios arepresented in Fig. 9. The tip of the blade is clearly seen from theimages. Furthermore, the out-of-plane deflection of the tip can clearlybe distinguished. That was also the reason for positioning the camerain such a manner.
V. Results and Discussion
A. Blade Axis Optimization
BAwas described as a two-parametric curve. The two parameters(φ0 and z0 of the tip of the BA) were describing the tip position in ananalogous manner to the coordinates in the polar coordinate system.A sweep of the parameters was performed, and a response surface ofthe objective function was obtained (Fig. 10).The response surface presented in Fig. 10 was obtained with the
blade geometrical requirements; and DP, WPL, and WPU settingspresented in Tables 1 and 2, respectively. For the sake of clarity, theoperating points were chosen as if the propeller was operated in aconstant-speed mode. As a result, only the aerodynamic loadschange, whereas the inertial loads stay constant. Consequently, it was
easier to manage the problem and properly understand the theoreticaland experimental results.It is important to point out that a more realistic operating scenario
can also be modeled without any changes to the design algorithm.One would only need to change the WPL and WPU parameters toreflect the more realistic operating conditions. Furthermore, onecould also change the whole objective function in order for thecalculated design to better comply with the requirements. Thepresented objective function was mainly selected because it reflectsblade efficiency characteristics over a range of operating conditionsand because it requires a relatively low number of blade characteristiccalculations for its evaluation, Blade characteristic calculations onlyhave to be performed in the three selected operating points:WPL, DP,and WPU, respectively.From the combined response surface and contour plot, one can
clearly see that a minimum in the objective function exists.Furthermore, it is clearly seen that the maximum of the objectivefunction is also very pronounced.By examining the response surfaces of the η atWPL, DP, andWPU,
presented in Fig. 11, it is clearly seen that the biggest differencebetween different geometries of the BAs occurs at WPU, namely, athigher advance ratios. It is also interesting to note that all of the BAdesigns achieve almost the same η at the DP.BA designs marked with BB, SB, and FB in Fig. 10 were selected
for further analysis and comparison. It is interesting to observe howthe differences in the values of the objective function translate intodifferences in the CT , CP, and η-vs-J characteristics. Thesecharacteristics are presented in Figs. 12–14. The FB geometryobviously undergoes an unfavorable deformation as the J isincreased, which leads to an earlier and quicker transition of thepropeller from the propeller into the brake mode with respect to thereference SB. Comparing SB andBB geometries, it is also noticeablethat, even though there is not much difference in the value of theobjective function for the twoBAgeometries, one can observe quite adistinctive difference in the CT , CP, and η characteristics of the twogeometries. BBs deform favorably as the J is varied.
B. Influence of Blade Rigidity
The influence of the blade rigidity on the blade’s performancecharacteristics is presented. The blade rigidity is primarilydetermined by the material properties the blade is made out of.Important material properties that influence the blade rigidity are thetensilemodulusE and the value of the allowable stress σa. The tensilemodulus influences the blade rigidity directly since it represents aratio between the strain and the stress. The allowable stress influencesthe blade rigidity indirectly through the allowable stress designprocedure. At given loads, the allowable stress determines theminimum size of the cross sections along the BA.To evaluate the influence of the blade rigidity on its performance
characteristics, the allowable stress σa was varied, while the tensilemodulus E was kept constant. It is expected that the higher the σavalue, the lower the blade rigidity. For each value of σa, the optimum
Fig. 9 BBs at a) rest, b) J � 1.1, and c) J � 1.8.
Fig. 10 Response surface of the optimization objective function.
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geometry of the BA was found. These blades were then comparedagainst one another and their perfectly rigid counterparts. Theinfluence of the σa value on the optimization objective function ispresented in Fig. 15. The selected BA geometries are summarized inTable 3. It is evident that the change in the σa value affects mainly theout-of-plane deflection z0 of the BA in its un-deformed state. As theσa decreases from 60 to 20 MPa, the z0 drops by almost 40%, from−39 to −24 mm. On the other hand, the sweep angle φ0 of the BA in
its un-deformed state is not affected considerably by the change of theσa value. As the σa is changed, theφ0 changes only by 1% or 0.2 deg.Furthermore, it is interesting to observe that, as the blade rigidity is
decreased, the value of the optimization objective function ρη isimproved. As the σa increases from 20 to 60MPa, the ρη is decreasedby almost 30%, from 0.28 to 0.20. Therefore, it is expected thatthe blade efficiency characteristics should improve considerably,especially in comparison to the ideally rigid blade.
Fig. 11 Response surface of η at a) WPL, b) DP, and c) WPU .
Fig. 12 CT of the selected BAs at different advance ratios. Fig. 13 CP of the selected BAs at different advance ratios.
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Performance characteristics of BB20, BB40, BB60, and the ideallyrigid blade are presented in Figs. 16–18. The performancecharacteristics of the ideally rigid blades are very much alike.Therefore, for the sake of clarity, only the performance characteristicof the ideally rigid BB20 is shown in the figures. It can be observedthat, as the rigidity of the blades is decreased, the effect of staticaeroelastic optimization becomes more pronounced. The moreflexible blades exhibit significantly better performance at the advanceratios above the DP advance ratio. The coefficientsCT ,CP, and η areincreased by 67, 47, and 10%, respectively, at J ∼ 1.8 as the rigidityof the blades is decreased. It can be deduced from the efficiencycharacteristics presented in Fig. 18 that the operational range of theflexible propeller is also significantly extended. The rigid BB20
transitions into the break mode at J ∼ 1.9, whereas the elastic BB60
transitions into the breakmode at J ∼ 2.1, which represents an almost10% increase in the upper limit of operational advance ratios.Furthermore, a slight increase of maximum efficiency of the elasticblades can be observed as the blade rigidity is decreased.It is also interesting to note that the proposed optimization
procedure and the blade rigidity do not affect the blade characteristicsmuch at advance ratios below the DP advance ratio. Below the DPadvance ratio, the performance characteristics for all the presentedcases are very much alike.
C. Experimental Results
Due to safety precautions, the experiment was conducted at lowerturning rate than the blades were designed for. The reason primarilylies in the uncertainty of the material properties and the materialbehavior under dynamic stress. Potential structural failure of theblades and damage of the wind tunnel and the rest of the measuringequipment were prevented by lowering the turning rate of the bladesfrom 3000 to 2000 rpm.The blades were tested in the range of advance velocities from 15
up to 30 m∕s, which at a given turning rate of 2000 rpm correspondsto a range of advance ratios from 1 up to 2. Propeller thrust, brakingtorque, and the blade’s tip out-of-plane deflection were measured.The measured results were then compared with theoretical results.
Fig. 14 η of the selected BAs at different advance ratios.
Table 3 Optimum blade axisparameters for different values of σα
Fig. 16 Influence of the σα on the CT for the selected BA geometries.
Fig. 17 Influence of the σα on the CP for the selected BA geometries.
Fig. 18 Influence of the σα on the η for the selected BA geometries.
Fig. 15 Response surface of the optimization objective function fordifferent values of σα.
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The measurement uncertainty was determined according to therecommendations given byGuide to the Expression of Uncertainty inMeasurement (GUM) [15]. The measurement uncertainty for themeasured quantities is determined at the lower and upper limits and inthe middle of the measurement range of velocities. Measurementpoints selected for the evaluation of the measurement uncertainty aregiven in Table 4. They are named consistently with the conventionused at the static aeroelastic optimization procedure: WPL, DP, andWPU, respectively. Measurement uncertainty was determined for v0,J, CT , CP, η, and z0. The relative values of the measurementuncertainty for the selected quantities are presented in Table 5.An overall reasonably good agreement between experimental and
theoretical results was observed. As the J is increased, the differencebetween the measured and calculated results increases. Generallyspeaking, theoretical results, except in the case of the tip deflection ofthe FB blades, overpredicted the measured values. This is consistentwith the validation results presented by Sodja et al. [16]. Theexperimental results reflect the predicted behavior of the selected BAgeometries. Although, it is important to point out that, according toFig. 10, the FB blades should exhibit theworst η-vs-J characteristics.However, during the experiment, the worst η-vs-J characteristic wasexhibited by the SB geometry. As the advance velocity and turningrate are changed, the aerodynamic and inertial loads change in anonlinear fashion and in their absolute value, yielding differentblade deformations, and therefore different blade characteristics.Consequently, the advance ratio J cannot be thought of as a measureof similarity anymore. The proposed analysis algorithm also predictsthis different behavior, as can be seen from comparing η-vs-Jcharacteristics, depicted in Figs. 19–21.Figure 22 depicts the blade tip out-of-plane deflections. The
deflections were determined by analyzing images taken by the
high-speed camera. Deflections predicted for SBs and BBs slightlyexceeded those measured; however, in the case of the FBs, thepredicted tip deflections were underpredicted in comparison tomeasured values.In the case of FBs, aerodynamics and inertia provide counteracting
loadings. The inertial loads tend to deform the blade forward, in thedirection of rotation, and toward the propeller plane. The aero-dynamic loads, on the other hand, deflect the FBs backward, oppositethe direction of rotation, and away from the propeller plane.Examining theCT characteristics of the FBs in Fig. 19 reveals that theFB propeller transitions from propeller mode into brake modequicker than predicted by the numerical analysis. Therefore, the
Table 4 Measurement points selected forevaluation of measurement uncertainty
Fig. 19 Experimental results of CT for the selected BA geometries.
Fig. 20 Experimental results of CP for the selected BA geometries.
Fig. 21 Experimental results of η for the selected BA geometries.
Fig. 22 Experimental results of blade tip out-of-plane deflection forselected BA geometries.
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aerodynamic forces are much smaller than those predicted, and theactual deflection of the tip is driven mainly by inertial loads at higheradvance ratios.In the case of SBs and BBs, the inertial and aerodynamic loads act
together, deflecting the blades in the same direction. Therefore, theoverpredicted loads also reflect in the overpredicted tip deflections.
VI. Conclusions
An extension of the classical BEM model, the EBEM, wasdevelopedwith the purpose of analyzing and optimizing high-aspect-ratio propeller blades having arbitrary blade axis geometries. Theextension is based on vectorizing the BEM governing equations andthe blade geometry, and correcting the governing equations regardingthe interference factors. As a result, the EBEM is easily connectedwith structural analysis, forming a static aeroelastic analysis tool.A mathematical model based on the EBEM and beam theory for
the analysis and design of the flexible propeller blades was proposed.Themodel treated aerodynamic and inertial loads as static. Dynamicsand aeroelastic stability of the blades were not treated. To analyzethese phenomena, one should implement a separate time-dependantanalysis. The proposed design procedure performs an aerodynamicoptimization of the blade in conjunction with the ASD in thedeformed state of the blade. Optionally, a static aeroelastic analysiscan be included to estimate the blade performance in the offdesignpoints as well. In the end, the un-deformed blade geometry iscalculated.The proposed design and analysis procedure were combined into
BA geometry optimization. A two-parametric BA optimization basedon response surface analysiswasperformed.Therewith, an effect of theBA on the propeller performance was assessed and experimentallyconfirmed. The effect of flexibility was most noticeable at the advanceratios above the design point advance ratio. The study shows thepotential to significantly expand the efficiency envelope of thepropeller performance as compared to the fixed-pitch rigid propellers.
Acknowledgments
The authorswould like to thank the EUResearch ExecutiveAgency(REA) for supporting theA2-Net Teamproject under FP7MarieCurieGrant 269190. The A2-Net Team project established a connectionamong the authors of the paper and created an environment in whichthe authors have done much of the work presented here. Furthermore,the authors would like to thank Daniel Valentine at ClarksonUniversity for all the fruitful discussions on the models of propelleraction and for providing the authors with an important insight into thechallenges hidden behind propeller modeling.
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