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CYCLOIDAL ROTOR AERODYNAMIC AND

AEROELASTIC ANALYSIS

LOUIS GAGNON

Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34

Milano 20156, Italy

louis.gagnon@polimi.it

GIUSEPPE QUARANTA

Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34

Milano 20156, Italy

giuseppe.quaranta@polimi.it

MARCO MORANDINI

Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34

Milano 20156, Italy

marco.morandini@polimi.it

PIERANGELO MASARATI

Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34

Milano 20156, Italy

pierangelo.masarati@polimi.it

Abstract

Cycloidal rotors consist of an arrangement of blades that rotate and pitch about a central drum. The blades are aligned

with the drum axis. By pitching harmonically with respect to the azimuth, they create a net propulsive force which

can propel an aircraft. The aerodynamic response of a cycloidal rotor in various environments has been studied using

a simpliﬁed analytical model resolved algebraically and by a multibody model resolved numerically. An computational

ﬂuid dynamics model has also been developed and is used to understand the cycloidal rotor behavior in unsteady ﬂow.

The latter model is also used to investigate the inﬂuence of the rods used to hold the blades on the aerodynamics

of the rotor. The models are also used to calculate the eﬃciency and the stability of the cycloidal rotor in various

conﬁgurations. The three developed models are validated using a set of data obtained from experiments conducted by

three diﬀerent parties. In all the experiments, the spans and radii were in the order of a meter. The use of cycloidal

rotors as a replacement for the tail rotor of a helicopter has also been studied. Preliminary analysis suggests that they

can potentially cut down the propulsive energy demand of a helicopter by 50% at high velocities. An aeroelastic model

is used to assess the inﬂuence of ﬂexibility on the system. Diﬀerent cycloidal rotor conﬁgurations have been considered.

Keywords cycloidal rotor; tail rotor; CFD; multibody dynamics; aeroelasticity

1. Introduction

A cycloidal rotor is a device which interacts with the surrounding ﬂuid to produce forces whose direction

can be rapidly varied. A sketch illustrates the dynamics of such a rotor in Fig. 1(a); a graphical representation

is shown in Fig. 1(b). The forces produced can take any direction in a plane normal to the axis of rotation.

They have a very small latency in response time and to some extent become increasingly eﬃcient with airﬂow

velocity.

In a cycloidal rotor, a drum carries a set of wings, or blades, whose axes are aligned with the rotation

axis of the drum. Each blade can pitch about a feathering axis which is also aligned with the rotation

(a) Working principle. (b) Viewed from the side.

Figure 1. Schematic representations of cycloidal rotors.

axis of the drum. By pitching the blades with a period equal to that of the drum rotation, a net aerody-

namic force normal to the axis of rotation is generated. The direction of the force is controlled by changing

the phase of the periodic pitch. Thus, the thrust can be vectored in a plane without moving large me-

chanical parts. Cycloidal rotors have been studied for their application in vertical axis wind and water

turbines (Hwang et al.,2009;Maˆıtre et al.,2013;El-Samanoudy et al.,2010). Recent studies also investi-

gated their application in unmanned micro aerial vehicles (Jarugumilli et al.,2014;Benedict et al.,2011;

Yun et al.,2007;Iosilevskii and Levy,2006). Nowadays, their commercial use is limited to wind turbines

and marine propellers. This paper focuses on a manned aircraft application of such rotors. A hover-capable

vehicle is proposed. It makes use of a conventional helicopter rotor for lift both in hover and forward ﬂight,

and uses one or more cycloidal rotors for anti-torque, additional propulsion, and some control. In the explored

design conﬁguration, power demanding tasks like lift in hover and in forward ﬂight are delegated to a very

eﬃcient device, the main rotor.

Several researchers have experimentally explored the concept, e.g. Yun et al. (2007), IAT21 (Xisto et al.,

2014), and Bosch Aerospace as reported by McNabb (2001). They assessed the potential of cycloidal rotors

to be used in large size unmanned or small size manned vehicle propellers. Previous work by the current

authors led to the development of various models for the study of the cycloidal rotor under diﬀerent conﬁg-

urations (Gagnon et al.,2014a,c,b).

The scope of this paper is to study the implementation of a cycloidal rotor as the replacement of the tail

rotor on otherwise conventional helicopters. It has been previously shown that the conﬁguration is aerody-

namically eﬃcient. This paper further studies the implementation feasibility by studying the eﬃciency of the

cycloidal rotor airfoils under various operating conditions. The inﬂuence of the main helicopter rotor on the

rotor aerodynamics is also discussed. A computational ﬂuid dynamics model is used to evaluate the impor-

tance of the three-dimensional of the rotor ﬂow. The proposed concept is show in Fig. 2. It was chosen after

inspecting the aerodynamics of three conﬁgurations which were reported in previous works (Gagnon et al.,

2014a,b).

This research project is part of a European eﬀort to ﬁnd an optimal use of cycloidal rotors. The objective

is to study their use in passenger carrying missions. One of the main ideas is to develop a long distance

rescue vehicle which can hover and travel at higher velocities than a traditional helicopter. Four universities

and two companies constitute the Cycloidal Rotor Optimized for Propulsion (CROP) consortium; additional

information can be found on the websitea.

ahttp://crop-project.eu/

Figure 2. Proposed helicopter concept.

2. Analytical Aerodynamic Model

2.1. Deﬁnitions

The algebraic mathematical model is used to obtain the pitching schedule required to produce the wanted

thrust magnitude and direction. It can also be used to compute the resulting thrust magnitude and direction

for a given pitching schedule. The model is also used to estimate the required power. It assumes constant

slope lift coeﬃcient a=CL/α and constant drag coeﬃcient CD0, so its use is limited to small angles of attack.

It neglects blade-wake interaction. It is convenient to specialize its solution for diﬀerent ﬂight regimes. The

schematic of the model is shown in Fig. 3, which represents the side view of the cycloidal rotor, and is in the

plane of the supporting arms previously shown in Fig. 1(b). Three diﬀerent reference systems are used and

are explained in Table 1.

xb

xr

x′

r

yb

yr

Xo

Yo

R

T

Uvi

β

γ

θ

ψ

ΩR

Figure 3. Deﬁnition of reference systems.

Table 1. Description of the coordinate systems.

System name Subscript Description

Basic reference o Yopoints in the direction opposite to gravity. Xois positive towards the

right.

Rotating reference r yris directed radially outward the circular rotor path. xris tangent to

the circular path and points backwards.

Body reference b rotating reference rotated by the pitch angle θwhich is positive in the

clockwise direction.

The thrust Tproduced by the cycloidal rotor is shown in Fig. 3. Its direction is deﬁned as having an

angle βwith respect to the Yoaxis.

Thus,

To=(−sin β

cos β)T. (1)

According to momentum theory, thrust produces an inﬂow velocity vi, which moves opposite to the thrust.

Thus,

vio=(sin β

−cos β)vi(2)

The free airstream velocity Uhas an angle γwith respect to the horizontal Xoaxis. Thus,

Uo=(cos γ

sin γ)U(3)

Now that the variables U,vi,β,T, and γare deﬁned, the deﬁnition of noteworthy ﬂight regimes is given in

Table 2.

Table 2. Cycloidal rotor ﬂight regimes.

Regime Subscript Description

Hover H U= 0 (thus γis irrelevant); T > 0, vi>0, with β= 0.

Forward Flight F U > 0, with γ≈0; T > 0, vi>0, with 0 < β < π/2 because both lift

and propulsive force need to be produced.

Forward Lift L U > 0, with γ≈0; T > 0, vi>0, with β= 0 because only lift needs to

be produced.

Propulsion P U > 0, with γ≈0; T > 0, vi>0, with β≈π/2 because thrust is now

essentially aligned with forward ﬂight speed.

Reverse Propulsion RP U > 0, with γ≈0; T > 0, vi>0, with β≈ −π/2 because thrust is

now essentially aligned against forward ﬂight speed.

The other typical variables are deﬁned as follows. First, the thrust coeﬃcient is,

CT=T

ρ(ΩR)2A,(4)

where Ais the area of the hollow cylinder described by the path of the cyclorotor blades,

A= 2πRb, (5)

with bbeing the span of the rotor.

Solidity σis the ratio of the total area covered by the blades to the ring area A

σ=cbN

2πRb =cN

2πR ,(6)

with blade chord cand number of blades N.

Using the simplest momentum theory and referring to Fig. 3we have a mass ﬂow ˙mthrough the cycloidal

rotor given by

˙m=ρAeﬀq(visin(β) + Ucos(γ))2+ (Usin(γ)−vicos(β))2=ρAeﬀ qU2+v2

i+ 2Uvisin(β−γ) (7)

where Aeﬀ = 2Rb is the area of an ideal stream tube that runs across the drum, perpendicular to the drum

axis.

Since the thrust Tis equal to the rate of change of the momentum, we obtain that,

T= ˙mw = 2 ˙mvi(8)

where wis the velocity at the end of the streamtube deﬁned by the momentum theory applied to the cycloidal

rotor in a fashion inspired by Johnson (1994).

The induced velocity can be expressed as

vi=T/(2ρAeﬀ)

pU2+v2

i+ 2Uvisin(β−γ)=T /(2ρAeﬀ )

p(Ucos(β−γ))2+ (Usin(β−γ) + vi)2(9)

We deﬁne the inﬂow coeﬃcients,

µ=U

ΩR(10)

λ=vi

ΩR(11)

We also deﬁne a torque coeﬃcient starting from the torque Q,

CQ=Q

ρ(ΩR)2RA ,(12)

and a power coeﬃcient for P= ΩQ

CP=P

ρ(ΩR)3A,(13)

Still referring to Fig. 3, we deﬁne the rotation matrices that allow converting the vectors from one reference

frame to another.

To go from the rotating frame to the body frame, the rotation operator is

Rbr ="cos θ−sin θ

sin θcos θ#(14)

In general we may expect θto be smallbso as a ﬁrst approximation it is possible to use a linearized rotation

matrix of the form

Rbr ≈"1−θ

θ1#(15)

The rotation matrix to transform a vector from the basic to the rotating frame is

Rro ="sin ψ−cos ψ

cos ψsin ψ#(16)

with ψ= Ωt.

bThis assumption is needed to simplify the formulas enough to make their analytical solution feasible; however, it might not

hold in realistic operational cases, in which |θ|may grow as large as π/4.

2.2. Derivation

To start, it is necessary to compute the component of the air velocity with respect to the airfoil in the body

reference frame. The ﬂow velocity, V, as seen from the body reference frame, is thus

V=Rbr (ΩR

0)+Rro U(cos γ

sin γ)+vi(sin β

−cos β)!! (17)

As a consequence,

Vx= ΩR−cos ψ(Usin γ−vicos β) + sin ψ(Ucos γ+visin β)−θcos ψ(Ucos γ(18)

+visin β) + sin ψ(Usin γ−vicos β)

Vy=θΩR−cos ψ(Usin γ−vicos β) + sin ψ(Ucos γ+visin β)(19)

+ sin ψ(Usin γ−vicos β) + cos ψ(Ucos γ+visin β)

Assuming that ΩR≫U, ΩR≫viand |θ| ≪ 1,

Vx≈ΩR(20)

Vy≈θΩR−visin (ψ−β) + Ucos (ψ−γ) (21)

The angle of attack of the airfoil is

α= tan Vy

Vx

≈Vy

Vx

≈(θ−λsin (ψ−β) + µcos (ψ−γ)) (22)

Using the angle of attack, the lift and drag forces are computed using using a simple steady state and linear

approximation,

D=1

2ρV 2cbCD0(23)

L=1

2ρV 2cbCL/αα(24)

These two force components are in the wind reference frame (i.e. Dis parallel to the wind and Lis perpen-

dicular), so they must be transformed in force components in the body reference frame,

Fbx=−Lsin α+Dcos α≈ −Lα +D(25)

Fby=Lcos α+Dsin α≈L+Dα (26)

assuming that the angle of attack is small. Then, these two force components must be transformed in the

rotating reference frame,

Frx≈Fbx +θFby =−Lα +D+θL +θDα ≈L(θ−α) + D(27)

Fry≈ −θFbx +Fby =θLα −θD +L+Dα ≈L−D(θ−α)

So, considering that V2≈(ΩR)2,

Frx

ρ(ΩR)2cb/2=CL/α (θ−λsin (ψ−β) + µcos (ψ−γ)) (λsin (ψ−β)−µcos (ψ−γ)) + CD0(28)

=CL/α (θ(λsin(ψ−β)−µcos(ψ−γ)) −(λsin(ψ−β)−µcos(ψ−γ))2+CD0

Fry

ρ(ΩR)2cb/2=CL/α (θ−λsin (ψ−β) + µcos (ψ−γ)) −CD0(λsin(ψ−β)−µcos(ψ−γ))

The imposed pitch angle is expressed as a harmonic series truncated at the second harmonicc,

θ=θ0+

2

X

n=1

(θcn cos nψ +θsn sin nψ) (29)

The two force components can be expressed as truncated Fourier series as well,

Frx=Frx0+

N

X

n=1

(Frxcn cos nψ +Frxsn sin nψ) (30)

Fry=Fry0+

N

X

n=1 Frycn cos nψ +Frysn sin nψ(31)

Now, the constant part of the force tangential to the cylindrical path, Frx0, is computed by integration over

a period. It is required to estimate the torque and thus the power required by the engine,

Frx0

ρ(ΩR)2cb/2=CD0−1

2CL/α µ2+ 2µλ sin(β−γ) + λ2−1

2CL/α (µcos γ+λsin β)θc1(32)

−1

2CL/α (µsin γ−λcos β)θs1

and the torque coeﬃcient is

CQ

σ=NFrx0R

ρ(ΩR)2AR

A

Ncb =1

2

Frx0

ρ(ΩR)2cb/2(33)

or

CQ

σ=1

2CD0−1

4CL/α µ2+ 2µλ sin(β−γ) + λ2−1

4CL/α (µcos γ+λsin β)θc1(34)

−1

4CL/α (µsin γ−λcos β)θs1

The forces must be transformed into the basic reference frame using the deﬁnition,

T=RT

roFr(35)

which gives,

Tx= sin ψFrx+ cos ψFry(36)

Ty=−cos ψFrx+ sin ψFry(37)

The total average force is given by the constant part of Ttimes the number of blades N. Recalling the

deﬁnition of the thrust coeﬃcient, CT, and of the solidity, σ, the following relations are obtained,

CTx

σ=NTx0

ρ(ΩR)2A

A

Ncb =Tx0

ρ(ΩR)2cb (38)

=CL/α

82θc1+λ(cos β(2θ0−θc2) + sin β(2 −θs2))

+µ(cos γ(2 −θs2) + sin γ(−2θ0+θc2))+CD0

4(λsin β+µcos γ)

CTy

σ=NTy0

ρ(ΩR)2A

A

Ncb =Ty0

ρ(ΩR)2cb (39)

=CL/α

82θs1+λ(cos β(−2−θs2) + sin β(2θ0+θc2))

+µ(cos γ(2θ0+θc2) + sin γ(2 + θs2))+CD0

4(−λcos β+µsin γ)

cStrictly speaking, only the ﬁrst harmonic is needed to be able to vector the thrust. The second harmonic is considered

because it is the only higher harmonic that aﬀects the thrust, and thus could be useful for higher order actuation.

The force coeﬃcient and direction are given by

CT=qC2

Tx+C2

Ty(40)

β=−arctan CTx

CTy

(41)

The force coeﬃcient components, in wind axes, are

CTk

σ=CTx

σcos γ+CTy

σsin γ(42)

CT⊥

σ=−CTx

σsin γ+CTy

σcos γ(43)

which yield

CTk

σ=CL/α

82θc1cos γ+ 2θs1sin γ+µ2 + sin(2γ)θc2−cos(2γ)θs2(44)

+λ2 sin(β−γ) + 2 cos(β−γ)θ0−cos(β+γ)θc2−sin(β+γ)θs2

+CD0

4(λsin(β−γ) + µ)

CT⊥

σ=CL/α

8−2θc1sin γ+ 2θs1cos γ+µ2θ0+ cos(2γ)θc2+ sin(2γ)θs2(45)

+λ−2 cos(β−γ) + 2 sin(β−γ)θ0+ sin(β+γ)θc2−cos(β+γ)θs2

−CD0

4λcos(β−γ)

This concludes the deﬁnition of the analytical aerodynamic cycloidal rotor model. The following section will

present the implicit algebraic solution of these equations.

3. Algebraic Solution

For the stated purpose of using the cycloidal rotor as a replacement part for the anti-torque rotor of the

helicopter, the two most important ﬂight regimes that were considered are the Propulsion (P) and Reverse

Propulsion (RP) scenarios. These are the regimes in which the lateral cycloidal rotors work to provide torque

about the yaw axis to counteract the torque generated by the main rotor. Furthermore, the cycloidal rotors

used in this fashion can provide a net thrust. For both scenarios, the main interest is to estimate the amount

of power required to obtain the wanted torque and thrust.

In is expected that a vertical ﬂow component originating from the main rotor will impinge the cycloidal

rotors located below it. This component is not considered by the current model; owing to the characteristics

of cycloidal rotors it is assumed that such inﬂow will require a pitch angle adjustment but will not impair

the eﬃciency of the main rotor. Another ﬂight condition which has not yet been considered is the use of the

cycloidal rotor to contribute to the lift provided by the main rotor when ﬂying at low advance ratios.

The thrust required of each cycloidal rotor is estimated by the following procedure, where the original

anti-torque and forward thrust provided by the BO105 tail and main rotor are Mtand Tt, respectively.

First, the thrust required to the right and left rotors, respectively TRand TL, are

TR=Mt+TtdL

dL+dR

(46)

TL=Tt−TR(47)

The power is estimated using the previously derived Eq. (32),

P=

Frx0

ΩRN

(48)

Equation (32) is solved using the values of λand CTobtained from the procedure presented in the following

two sections. The regime is Propulsion if the requested thrust is positive and Reverse Propulsion if the

requested thrust is negative. Both regimes start from the deﬁnition of the Forward Flight mode of a cyclogyro.

They diﬀer from that regime because the perpendicular thrust component, CT⊥, is set to null. This deﬁnition

holds for cases where the main helicopter rotor takes all the anti-gravity forces.

3.1. Solving the Equations in Propulsion

In this conﬁguration, the objective is to produce CT⊥= 0 and CTk=−CTP<0, i.e. β=π/2, with µ > 0,

γ= 0. Substituting these values into Eqs. (44) and (45) yields

(−CTP

0)=σCL/α

4(µ+λP)(1

θ0)+σCL/α

4(θc1

θs1)(49)

+σCL/α

8(µ+λP)"0−1

1 0 #( θc2

θs2)

The collective pitch θ0is currently maintained at zero. Thus, a negative cosine one per revolution pitch,

θc1, is used to produce the wanted negative horizontal force. This force pushes the rotor towards the left,

opposite to the airstream direction. For now, a simple cycloidal rotor which uses only a single harmonic

pitching motion is considered. Consequently, we set θc2=θs2= 0. Then, for a null perpendicular thrust and

the conditions just stated,

θs1=−(µ+λ)θ0(50)

which indicates that θs1is null. The inﬂow coeﬃcient in this case is

λP=−µ

2+rµ

22

+πCTP

2(51)

The solution of (49) with (51) is thus,

−4CTP

σa −µ+κµ

2−θc−Cd

σa (µ−κµ

2)

(Cd

σa + 1)κ!2

−µ

22

+πCTP

2= 0 (52)

where ais CL/α and κis a term which integrates ﬂow non-uniformity and tip losses in the equations. This

term is inserted as a multiplier of the inﬂow terms when solving Eq. (49). This method is inspired by the

empirical factor of the Johnson (1994) momentum theory in hover which uses λ=κpCT/2. Parameter κin

the present work is adapted to the cycloidal rotor by an approach similar to Yun et al. (2007) which uses an

empirical multiplier on one term when solving their implicit inﬂow equation.

Now, the solution of Eq. (49) yields, when choosing the appropriate root,

θcP=1

(2 aσ) (aκ −2a)µσ + (Cdκ−2Cd)µ−(aκσ +Cdκ)p2πCTP+µ2−8CTP!(53)

if CTPis the known variable. Otherwise, if θcis the known variable we use

CTP=πa2κ2σ2

64 +πC2

dκ2

64 +aσθcP

4+(Cdκ−2Cd)µ

8+πCdaκ2+ 4(aκ −2a)µσ

32 (54)

−(aκσ +Cdκ)

64 π2a2κ2σ2+π2C2

dκ2−32πaσθc+16(πCdκ−2πCd)µ

+64µ2+ 2π2Cdaκ2+ 8(πaκ −2πa)µσ

1

2

For further interest, the torque coeﬃcient in this case is deﬁned using

CQ

σ=1

2CD0−1

4CL/α µ2+ 2µλ +λ2−1

4CL/α (µ+λ)θc1=1

2CD0+ (µ+λ)CTP

σ(55)

which yields

CQ=σ

2CD0+ µ

2+µ

22

+πCTP

21/2!CTP(56)

Looking back at Eq. (32) one can see that, for this regime, θs1and θ0have no inﬂuence on the power

consumed. This could be inspected in further detail by resolving Eqs. (44) and (45) with the intent of

increasing thrust by changing the two aforementioned angles.

The solution for the Propulsion case works for all cases where the thrust pushes the rotor against the

incoming wind and produces an inﬂow velocity viwhich has the same direction as the incoming velocity U. It

will also work for cases where the inﬂow velocity is opposed to the incoming velocity up to a condition where

the resulting velocity becomes null, such that |U|=|vi|. There, it is expected that a condition analogous to

helicopter vortex ring occurs. The Reverse Propulsion regime solution is used as soon as the required thrust

changes direction. That solution is presented in the following section.

3.2. Solving the Equations in Reverse Propulsion

In the Reverse Propulsion case, everything is kept equal to the propulsion case, with the exception that

β=−π/2. What thus happens is that the direction of the resulting mass ﬂow rate is changed, and thus, to

keep a positive value of that mass ﬂow rate, the equations (7), (8), and (9) become, when setting γ= 0 and

β=−π/2,

˙m= (vi−U)ρAeff (57)

which implies that,

T= (vi−U)ρAef f 2vi(58)

which when solved algebraically yields

vi=

−U±qU2+4T

2ρAeff

2(59)

respectively.

This latest equation has a real solution for any positive value of a thrust pointing to the right, which was

not the case when the equation was solved for the Propulsion regime.

The positive root of Eq. (59) is kept because Tand vineed to have the same sign. That logic comes from

the deﬁnitions of Fig. 3and the fact that the rate of change of the momentum is equal to the thrust. Thus,

the inﬂow parameter of the Reverse Propulsion case is,

λRP =−µ

2+rµ

22

+πCTRP

2(60)

When setting for a null thrust perpendicular to the wind we obtain,

(−CTRP

0)=σCL/α

4(µ−λRP)(1

θ0)+σCL/α

4(θc1

θs1)(61)

+σCL/α

8(µ−λRP)"0−1

1 0 #( θc2

θs2)

and thus,

θs1= (λRP −µ)θ0(62)

the equation to solve is thus,

4CTRP

σa −µ−κµ

2−θc−Cd

σa µ+κµ

2!

Cd

σa + 1!κ

2

−

µ

2!2

+πCTRP

2

= 0 (63)

where κis the same empirical term deﬁned in Section 3.2 . Finally, solving Eq. (63) yields

CTRP =πa2κ2σ2

64 +πC2

dκ2

64 +aσθcRP

4+(Cdκ+ 2Cd)µ

8+πCdaκ2+ 4(aκ + 2a)µσ

32 (64)

−(aκσ +Cdκ)

64 π2a2κ2σ2+π2C2

dκ2+ 32πaσθc+16(πCdκ+ 2πCd)µ

+64µ2+ 2π2Cdaκ2+ 8(πaκ + 2πa)µσ

1

2

or, solving for θcwhen the pitch function is wanted and the required thrust is known,

θcRP =−1

2aσ (aκ + 2a)µσ + (Cdκ+ 2Cd)µ−(aκσ +Cdκ)p2πCTRP +µ2−8CTRP (65)

where CTRP =−CTP=CT|| .

4. Calibration and Validation of the Analytical Model

The Propulsion and Reverse Propulsion ﬂight models developed are validated using hover experimental

data. The lack of experimental data in forward ﬂight in the desired scale prevented further experimental

validation. Three experimental datasets are available. They are used to calibrate and validate the analytical

propulsion models by imposing a null advance ratio. One dataset comes from Yun et al. (2007) and another

from IAT21 (Xisto et al.,2014), which is a member of the CROP consortium. The third dataset comes from

Bosch Aerospace, which ran an experimental campaign reported by McNabb (2001). For this last experiment,

the airfoils had a high drag coeﬃcient. McNabb reported it to be CD0≈0.07. The same CD0was used as

such in the current algebraic model for the Bosch dataset. The three experimental setups also diﬀered in their

three-dimensional conﬁguration. The Bosch rotor transmitted movement to the blades by rods, located at

midspan of the blades, which are covered by a cylindrical shell. The IAT21 experiments had a similar setup,

but located at both external edges of the blades. The Yun et al. setup was positioned similarly to that of

IAT21, but the rods were uncovered. Another diﬀerence between the experimental setups lies in the method

used to measure power. The power measured by Yun et al. was the supplementary power required by the

electrical drive when blades are added to the device. They obtained it by subtracting the power required by

the electric motor to rotate the rods without the blades. The power measured by McNabb is measured by

a load cell. The power measured by IAT21 is the total motor power, for which they estimated a 5% total

loss. The experimental data was thus taken as is from Yun et al. and McNabb and a 5% reduction was

applied to the power measured by IAT21. Another diﬀerence is that IAT21 used NACA0016 airfoils whereas

McNabb and Yun et al. used NACA0012. Even though the blades diﬀered, a slope of the lift coeﬃcient

a=CL/α = 6.04 is used for all three conﬁgurations. The other experimental parameters are brieﬂy described

in Table 3. Calibration was done by curve ﬁtting the power and thrust values obtained algebraically to the

measured ones. The results are shown in Figs. 4to 6.

Table 3. Description of the experimental data.

Author Radius R(m) Span b(m) Chord c(m) N. of foils N3D control

Yun et al. (2007) 0.4 0.8 0.15 6 Arms at edges

Xisto et al. (2014) 0.6 1.2 0.3 6 Cylinders at edges

McNabb (2001) 0.610 1.22 0.301 6 Cylinder at midpoint

Calibration of the model was also tried by disregarding either the power or the thrust curves. When

disregarding the power curves, the resulting kappa hover correction factor remains almost the same. However,

if the optimization is performed considering only power, the κvalue changes signiﬁcantly, yielding an excellent

match for the power at the cost of mispredicting the thrust.

0

100

200

300

400

500

600

700

200 250 300 350 400 450 500 550 600

Thrust (N)

Angular Velocity (RPM)

Model

Experim.

(a) Bosch T vs. Ω.

0

5

10

15

20

25

200 250 300 350 400 450 500 550 600

Power (kW)

Angular Velocity (RPM)

Model

Experim.

(b) Bosch P vs. Ω.

Figure 4. Comparison with the Bosch (McNabb,2001) experimental data of their 6 blade model at 25◦

magnitude pitch function. Using κ= 1.0785 and CD0= 0.07.

0

500

1000

1500

2000

200 300 400 500 600 700 800 900 1000

Thrust (N)

Angular Velocity (RPM)

Model

Experim.

(a) IAT21 T vs. Ω.

0

10

20

30

40

50

60

70

80

200 300 400 500 600 700 800 900 1000

Power (kW)

Angular Velocity (RPM)

Model

Experim.

(b) IAT21 P vs. Ω.

Figure 5. Comparison with the IAT21 experimental data of the D-DALLUS L3 model at 37.5◦magnitude pitch

function. Using κ= 1.2640 and CD0= 0.008.

Although various weightings between power and thrust were tested, it was decided to give them equal

importance. Optimizing with a strong weight on power gave an excellent match for power, but yielded a very

high κ, which is the only variable that is calibrated by the optimization. The smaller correctors κrequired

for the IAT21 and McNabb experimental data may be due to the fact that their experimental model had

full cylinders which are known to reduce the three-dimensionality of the ﬂow. To verify this hypothesis, a

previously developed three-dimensional ﬂuid dynamics model (Gagnon et al.,2014c) is used to conﬁrm the

inﬂuence of the use of endplates on the rotor. A short description of the model along with the computed

eﬀect of the endplates is presented in the following paragraph.

The OpenFOAM CFD toolkit has been relied upon to perform the ﬂuid dynamic calculations. A laminar

non-viscous solver is relied on since the main contributors to the thrust are the pressure induced forces.

0

1

2

3

4

5

6

7

8

200 250 300 350 400 450 500 550 600

Thrust (kgF)

Angular Velocity (RPM)

Model, 15deg

Model, 20deg

Model, 25deg

Model, 30deg

Experim., 15deg

Experim., 20deg

Experim., 25deg

Experim., 30deg

(a) Yun et al. T vs. Ω.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

200 250 300 350 400 450 500 550 600

Power (HP)

Angular Velocity (RPM)

Model, 15deg

Model, 20deg

Model, 25deg

Model, 30deg

Experim., 15deg

Experim., 20deg

Experim., 25deg

Experim., 30deg

(b) Yun et al. P vs. Ω.

Figure 6. Comparison with the Yun et al. (2007) experimental data of their baseline model at various pitching

function magnitudes. Using κ= 1.4804 and CD0= 0.008.

A double mesh interface which allows one to model a ﬁxed angular velocity rotor zone and six embedded

periodically oscillating zones has been developed. A moving no-slip boundary condition was also developed to

constrain the normal component of the ﬂuid velocity at the foil to match the airfoil velocity while letting the

parallel velocity uninﬂuenced. The timestep used for the simulations is variable and set to follow a Courant

number of about 10. The mesh used without endplates has 366k cells while the one with endplates has 926k

cells. The reason for such a big discrepancy is the diﬃculty of the snappyHexMesh meshing software to

mesh surfaces close to interfaces and the thus related need to have a highly reﬁned mesh in these zones. The

spacing between endplates and foils is one tenth of the chord length, which is reported by Calderon et al.

(2013) to have the same eﬀect as if it were attached to the foil.

The CFD results of the IAT21 L3 model at various angular velocities are shown with and without endplates

in Fig. 7. They conﬁrm the idea that the experimental apparatus used to transmit power will have an inﬂuence

0

500

1000

1500

2000

2500

200 300 400 500 600 700 800 900 1000

Thrust (N)

Angular Velocity (RPM)

w/o endplate

w endplates

experim.

(a) IAT21 T vs. Ω.

0

10

20

30

40

50

60

70

80

200 300 400 500 600 700 800 900 1000

Power (kW)

Angular Velocity (RPM)

w/o endplate

w endplates

experim.

(b) IAT21 P vs. Ω.

Figure 7. Eﬀect of the presence of endplates on the cycloidal rotor as conﬁrmed by the 3D CFD simulations.

on the overall performance. In the experiment, IAT21 used endplates. The instantaneous velocity ﬁelds and

streamlines of the IAT21 L3 case, where the pitch function angle magnitude is 37.5◦at the angular velocity

of 250 RPM, are shown in Figs. 8(a) to 8(d). These ﬁgures further conﬁrm the inﬂuence of the endplates

geometry. Also, Fig. 8(d) conﬁrms the fact reported by IAT21 that the ﬂow enters the cycloidal rotor over

an angle of 180◦or more and exits over an arc of roughly 90◦. Figure 8(c) agrees well with the ﬁndings of

Yun et al. (2007) that the ﬂow is deviated by the rotor.

(a) Front view w/o end-

plates.

(b) Front view with end-

plates.

(c) Side view w/o end-

plates.

(d) Side view with end-

plates.

Figure 8. Cycloidal rotor velocity streamlines from the CFD model at 250 RPM.

5. Modiﬁed Analytical and Multibody Models

Two other quickly solvable models were developed for the purpose of studying the behavior of a potential

manned vehicle which uses cycloidal rotors. One consists of a slight modiﬁcation of the previous analytical

model which can still be solved algebraically. The second model is a rigid multibody dynamics system

consisting of a main rotating drum and a set of blades having a common pitching schedule comprised of a

collective pitch and a ﬁrst harmonic pitch. These two models are validated against a set of three diﬀerent

experimental results.

5.1. Modiﬁed Analytical Model

In the previously presented analytical model, the equations were solved using only one parameter, κ, which

allowed calibrating the results to the experimental data quite well. The mathematics used made it so that

the power and thrust were each inﬂuenced diﬀerently by the adjustment factor. However, in this revised

solution, an additional factor is used and deemed necessary in order to compare the multibody and analytical

solutions using the same mathematical basis. It consists of the drag coeﬃcient of the airfoil. This theory is

used following the ﬁndings of McNabb (2001). He reported that the actual viscous drag of the airfoil used for

the experimental validation of his cycloidal rotor model was more than 10 times greater than the theoretical

one. This new approach is also conﬁrmed by the multibody model. Thus, the modiﬁed analytical model uses

an implementation of the inﬂow correction factor closer to the one of Johnson (1994). We thus have λm=κλ

and Cd,m =ζCdwhere ζis a multiplication factor for the drag coeﬃcient.

5.2. Multibody Model

The multibody model consists of a parametric implementation of a N blade cycloidal rotor. The analysis

is performed using the MBDyn open-source multibody toold. Inﬂow is calculated using momentum theory

and the blade lift and drag forces are calculated using tables for drag and lift coeﬃcients. Here, the ζ

factor is applied to coeﬃcients coming from data tables and is optimized to ﬁt the multibody model to

the experimental data. The model is run until a periodic solution is obtained; the resulting performance is

obtained by averaging over the last complete rotation of the drum.

5.3. Comparison With Experimental Data

These two models are calibrated for optimal κand Cd; the results are plotted in Figs. 9to 11. The diﬀerence

noted in the power plots is explained by the fact that at higher angles of attack the analytical model is less

dhttp://www.mbdyn.org/, last accessed September 2014.

inﬂuenced by the drag coeﬃcient due to its ﬁxed value.

Alg., 15deg

Alg., 20deg

Alg., 25deg

Alg., 30deg

Mbd., 15deg

Mbd., 20deg

Mbd., 25deg

Mbd., 30deg

Experim., 15deg

Experim., 20deg

Experim., 25deg

Experim., 30deg

0

1

2

3

4

5

6

7

8

200 250 300 350 400 450 500 550 600

Thrust (kgF)

Angular Velocity (RPM)

(b) Yun et al. T vs. Ω.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

200 250 300 350 400 450 500 550 600

Power (HP)

Angular Velocity (RPM)

(c) Yun et al. P vs. Ω.

0

1

2

3

4

5

6

7

8

200 250 300 350 400 450 500 550 600

Induced velocity (m/s)

Angular Velocity (RPM)

(d) Yun et al.Uin vs. Ω.

Figure 9. Comparison with the Yun et al. (2007) experimental data of their baseline model at various pitching

function magnitudes. Using κ= 1.4804 and CD0= 3.63 ×CD.

The experimental data available from IAT21 and Bosch are more limited in quantity and have thus been

handled with care. The diﬀerence in the induced velocity calculated for Bosch remains below one tenth.

In the absence of experimental data it is not possible to know which model gives the best approximation.

For the IAT21 simulation, the error in inﬂow velocity increases. This discrepancy may be explained by the

higher angle of attack reached by the blades of the model. The blades actually reach stall; thus, it is diﬃcult

to reproduce the same behavior between the analytical and the multibody models. The latter accounts for

static stall as tabulated in the aerodynamics look-up tables.

The angle of attack in the Bosch tailored simulations at 25◦pitch function varies between [-5◦,10◦] with

an average magnitude of 6◦. It is thus clear that the analytical constant drag coeﬃcient cannot take into

account with precision the power contributions of the viscous forces. Similarly, at 37.5◦pitch function, the

IAT21 angle of attack will oscillate between [-10◦,20◦] with an average magnitude of 8.5◦. These values are

calculated using the multibody model with a ζmultiplied drag coeﬃcient. Thus, once again, this explains

the diﬀerent drag coeﬃcient requirements for both models.

A slightly better correlation can be obtained for this revised analytical model if the drag coeﬃcient and

the κinﬂow corrector are optimized separately for the multibody model, but the best results obtained for the

analytical model come from the version previously presented in Section 2. Its use is thus limited to multibody

0

100

200

300

400

500

600

700

200 250 300 350 400 450 500 550 600

Thrust (N)

Angular Velocity (RPM)

Alg.

Mbd.

Experim.

(a) Bosch T vs. Ω.

0

5

10

15

20

25

200 250 300 350 400 450 500 550 600

Power (kW)

Angular Velocity (RPM)

Alg.

Mbd.

Experim.

(b) Bosch P vs. Ω.

Figure 10. Comparison with the Bosch (McNabb,2001) experimental data of their 6 blade model at 25◦

magnitude pitch function. Using κ= 1.0785 and CD0= 0.07.

0

500

1000

1500

2000

2500

200 300 400 500 600 700 800 900 1000

Thrust (N)

Angular Velocity (RPM)

Alg.

Mbd.

Experim.

(a) IAT21 T vs. Ω.

0

10

20

30

40

50

60

70

80

200 300 400 500 600 700 800 900 1000

Power (kW)

Angular Velocity (RPM)

Alg.

Mbd.

Experim.

(b) IAT21 P vs. Ω.

Figure 11. Comparison with the IAT21 experimental data of the D-DALLUS L3 model at 37.5◦magnitude

pitch function. Using κ= 1.2640 and CD0= 0.008.

validation purposes and the previous model is used for eﬃciency evaluation.

6. Eﬃciency Evaluation

The eﬃciency of the proposed helicopter design that was shown in Figs. 2is compared with that from a

model that resembles the Airbus Helicopters BO105 helicopter. The main advantage of this conﬁguration is

that less thrust is required from the main rotor. This is the result of using the cycloidal rotors as forward

thrust generators.

6.1. BO105 Performance Characteristics

The performance characteristics of the original helicopter are ﬁrst obtained. Diverse constant velocity ad-

vance ratios are considered. They come from a comprehensive helicopter model. The BO105 model is fully

aeroelastic and considers the dynamics of the main helicopter rotor, the tail rotor, and the airframe. Vec-

torial thrust, torque, and power ﬁgures can be computed for both rotors. Muscarello et al. (2013) gives the

detailed implementation of the model. Figure 12(a) displays the power requirement of the BO105 helicopter

undergoing a normal trim. The power contribution is broken down between the main rotor and the tail

rotor. The main rotor consumes all the power. This total power is broken down into propulsion and lifting

powers on Fig. 12(b). To do so, the BO105 model is used in a virtual wind tunnel. The lifting only power

is obtained by running the main rotor of the BO105 simulation into the virtual wind tunnel where its shaft

is held perpendicular to the ﬂow. In this conﬁguration, it is trimmed to provide the counter gravity forces

equivalent to the aircraft weight. The propulsive power is then obtained by subtracting the power required

by the main rotor shaft perpendicular to the ﬂow to the power in trim. Finally, the thrust produced by the

tail rotor, as shown in Fig. 12(c) allows the measurement of the required moment to be produced by the

cycloidal rotors.

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Total

Main Rotor

Tail Rotor

(a) Helicopter model in trim.

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Regular Trim

Lift Only

Propulsion Only

(b) Main rotor at null pitch angle.

0.6

0.8

1

1.2

1.4

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Lateral Thrust (kN)

Advance Ratio

(c) Tail rotor lateral thrust.

Figure 12. Power and thrust ﬁgures of the BO105 rotors.

6.2. Modiﬁed Aircraft

In order to evaluate the eﬃciency of the modiﬁed aircraft design, the ﬁgures obtained from the BO105 model

are used. The total power consumed by the modiﬁed aircraft consists of the sum from the left and right

cycloidal rotors and the main rotor used to generate only lift. The non-modiﬁed analytical model was used

for the simulations presented in this section. An optimization procedure using a Nelder-Mead algorithm

allows ﬁnding an optimal operating point. The points found are considered to be local optima.

A range of parameters were allowed to change during the optimization process. They are shown in Table 4.

The maximum angle of attack encountered by the blade was limited to 13.06◦. This is essential when using

Table 4. Cycloidal rotors optimization properties.

Radius R(m) Dist. dL+dR(m) Chord c(m) N. of blades NAng. vel. Ω (RPM)

Lower lim. 0.1 0.5 0.05 3 20

Upper lim. 1.275 9.5 0.5 12 2000

an analytical model which relies on a constant slope lift coeﬃcient. According to reliable airfoil data, lift

generation remains constant until 14.14◦and then drops rapidly. The maximum angle of attack is calculated

from Eq. (22) with γ= 0 and β=π/2. Combining it with Eq. (29) truncated at the ﬁrst harmonic term and

Eq. (50), Eq. (22) becomes,

α≃θo+θccos(Ψ) + θssin(Ψ) −λsin(Ψ −π/2) + µcos Ψ ≃θo(1 −(µ+λ) sin Ψ) + (θc+λ+µ) cos Ψ (66)

where the factor of θoindicates that the collective pitch should be zero. The maximum and minimum values

of the factor of cos(Ψ) correspond to Ψ = 0, π. Thus, a non null θoat these values of Ψ can only increase the

maximum angle of attack.

The maximum allowed distance between the centers of the two cycloidal rotors was 8.25 m and was

chosen as such in order to prevent the cycloidal apparatus from extending too far beyond the main rotor.

The maximum distance is derived from

dL+dR

2+b

2= 2(dL+dR)−LW≤DMR (67)

where DMR = 9.84 m is the main rotor diameter, LWis the width of the skids, and dLand dRare the

lateral distances between the center of the helicopter and the midpoint of the left and right cycloidal rotors,

respectively. Each rotor uses NACA0012 airfoils; the rotor radii were limited to avoid contact with the ground

or the main rotor. The maximum tip velocity at maximum forward velocity was limited to Mach 0.85.

The ﬁrst optimization lead to the data presented for a span of 7 m in Table 5and gives the results shown

in Figs. 13(a) to 13(c). These take into account the power saved by the main rotor. No longer having to

provide forward thrust does not require its shaft to be tilted forward. In this design, the main rotor only

pitches in order to balance the center of mass oﬀset. Figures 13(b) and 13(c) show the provided thrust and

anti-torque of cycloidal rotors and of the main or tail rotor, respectively. As required, the resulting torques

and thrusts for both the original helicopter and the proposed design are equal.

200

250

300

350

400

450

500

550

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Original Bo105

Proposed Design

(a) Total power vs original BO105.

-1

0

1

2

3

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Propulsive Thrust (kN)

Advance Ratio

Original Bo105

Cycloidal Rotor (left)

Cycloidal Rotor (right)

Cycloidal Rotor (total)

(b) Propulsive thrust.

-5

0

5

10

15

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Anti-Torque (kN m)

Advance Ratio

Original Bo105

Cycloidal Rotor (left)

Cycloidal Rotor (right)

Cycloidal Rotor (total)

(c) Anti-torque.

Figure 13. Modiﬁed aircraft design performance characteristics.

The angular velocity of the cycloidal rotors was not allowed to vary when changing advance ratio or

ﬂight regime. The pitch was restrained to single harmonic variation. The two cycloidal rotors had identical

geometry. The optimization process also led to the realization that, in order to keep a larger chord, the blade

number has to be minimized. Thus, it was set to N= 3 for the further optimization steps. The resulting

vehicle is larger than the original helicopter and it was thus chosen to constrain the maximum center to center

lateral width so as to obtain smaller rotor spans. The resulting geometries are shown in Table 5and their

corresponding power consumptions in Fig. 14. The maximum forward ﬂight velocity is limited to 72 m/s

because above that velocity it becomes nearly impossible to trim the helicopter tail rotor to provide a lateral

force large enough to oﬀset the main rotor torque. This velocity also brings the tip of the advancing blade

into the transonic regime (Mach 0.85 is reached).

Table 5. Optimization results.

Span b(m) 7.00 6.47 3.44 1.93

Chord c(cm) 35 79 14 9.8

Radius R(m) 1.25 1.27 1.21 1.24

Angular velocity Ω (RPM) 400 375 820 1310

Number of blades N3 3 3 3

6.3. Variable Angular Velocity

A quick study was conducted to understand whether a variable angular velocity could make the cycloidal

rotors more eﬃcient. This would make sense if they were powered, for example, by electric motors. It is

reasonable to assume that the angular velocity of the rotors could be controlled independently and vary with

200

250

300

350

400

450

500

550

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Original Bo105

Proposed Design

(a) 6.47 m span.

200

250

300

350

400

450

500

550

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Original Bo105

Proposed Design

(b) 3.44 m span.

200

250

300

350

400

450

500

550

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Original Bo105

Proposed Design

(c) 1.93 m span.

Figure 14. Total power consumed by the short rotor aircraft.

the advance ratio. Constraining the algorithm to use the same geometry found before and allowing each rotor

to have a diﬀerent angular velocity at each advance ratio considered yields a negligible increase in eﬃciency.

The resulting power required for this conﬁguration is shown in Fig. 15(a) and the optimal angular velocities

of the rotors are shown in Fig. 15(b). Although an irregular angular velocity pattern is shown, it was noticed

that many angular velocity conﬁgurations yield very similar results, the limiting factor being the maximum

allowed blade pitch angle. The idea of using variable angular velocity was thus not further investigated.

200

250

300

350

400

450

500

550

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Power (kW)

Advance Ratio

Original Bo105

Proposed Design

(a) Total power used by the variable design.

150

200

250

300

350

400

450

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Angular velocity (RPM)

Advance Ratio

Cycloidal Rotor (left)

Cycloidal Rotor (right)

(b) Angular velocity distribution for the variable design.

Figure 15. Variable angular velocity design test results.

7. Aeroelastic Model

This section presents some considerations on the aeroelasticity of the cycloidal rotor arrangement. In

view of the opportunity to consider low-solidity rotor designs, which are characterized by very slender, thin

blades, their low out-of-plane bending and torsional stiﬀness, and the possible presence of oﬀsets between the

pitch axis and the shear, mass and aerodynamic centers of the blade section may become a concern.

Figure 16 shows a sketch of the blade section, referred to the pitch axis; x1is the distance of the aerody-

namic center from the pitch axis; x2is the distance of the elastic axis from the pitch axis; x3is the distance

of the center of mass from the pitch axis.

The blades are loaded mainly by nearly radial, out-of-plane centrifugal and aerodynamic loads; the force

c

x1

x2

x3

Figure 16. Sketch of blade section.

per unit span is

fradial =fcentrifugal +faerodynamic ≈mΩ2r+1

2ρ(Ωr)2cCL,(68)

where mis the blade mass per unit span and r(x, t) = R+w(x, t) is the deformed radius. Both loads are

clearly proportional to Ω2and to either ror r2.

b

bent blade

Rr

w

x

Ω

Figure 17. Schematic of rotating ﬂexible blade assembly subjected to bending.

It is assumed that the blades are simply supported at both ends, as sketched in Fig. 17. Similar consid-

erations apply to diﬀerent arrangements.

The multibody aeroelastic solver has been also applied to the solidity study. The model uses ﬁnite-volume

nonlinear beam elements Ghiringhelli et al. (2000), with aerodynamics based on blade element and look-up

table static airfoil properties. Simple inﬂow models based on Momentum Theory have been speciﬁcally

developed for the analysis of cycloidal rotors Benedict et al. (2011). The results presented in Fig. 18 show

that the reduction in thrust and power related to the observed aeroelastic deformation of the blades seems

negligible at low angular velocity and low ﬂexibility. This is no longer true when the structural properties

are reduced by acting on the reference thickness.

8. Conclusions

A set of models have been developed for the study of cycloidal rotors used to propel an optimized manned

aircraft. The idea considered in the article was to replace the tail rotor of an otherwise conventional helicopter

design with cycloidal rotors located at the sides of the helicopter. In this way, the individual airfoils rotate

in a manner that makes them behave as airplane wings. The cycloidal rotors are expected to be more

eﬃcient than traditional rotors when subjected to the inﬂow from main rotor. The sensibility of the rotors

0

200

400

600

800

1000

1200

1400

1600

200 400 600 800 1000 1200 1400 1600

Thrust, N

RPM

3 blades, rigid

3 blades, flexible (nom.)

3 blades, flexible (x 0.1)

3 blades, flexible (x 0.01)

(a) Thrust.

0

5

10

15

20

25

30

35

40

45

50

200 400 600 800 1000 1200 1400 1600

Power, kW

RPM

3 blades, rigid

3 blades, flexible (nom.)

3 blades, flexible (x 0.1)

3 blades, flexible (x 0.01)

(b) Power.

Figure 18. Comparison of thrust and power vs. RPM for rigid and ﬂexible blades at 20 deg cyclic pitch.

to the geometry of the carrying rods has been conﬁrmed by the 3D CFD simulations and the magnitudes

of the correction factors of the two dimensional analytical and multibody models. The multibody model

and the analytical model correlate to satisfaction with the three sets of experimental data considered. Open

source software was used and developed as part of the research presented. Preliminary results show that

blade ﬂexibility may reduce eﬃciency. Finally, it is expected that up to 50% savings in power required for

propulsion can be obtained from the helicopter traveling at high advance ratios, with comparable eﬃciency

at lower advance ratios.

Acknowledgments

The research leading to these results has received funding from the European Community’s Seventh

Framework Programme (FP7/2007–2013) under grant agreement N. 323047.

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Calderon, D. E., Cleaver, D., Wang, Z., and Gursul, I. (2013). Wake Structure of Plunging Finite Wings. In 43rd

AIAA Fluid Dynamics Conference.

El-Samanoudy, M., Ghorab, A. A. E., and Youssef, S. Z. (2010). Eﬀect of some design parameters on the performance

of a Giromill vertical axis wind turbine. Ain Shams Engineering Journal, 1(1):85–95.

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