Cycloidal rotor aerodynamic and aeroelastic analysis

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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)−θcos ψ(Ucos γ(18)
+visin β) + sin ψ(Usin γ−vicos β)
Vy=θΩR−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
82θc1+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
82θs1+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
82θc1cos γ+ 2θs1sin γ+µ2 + sin(2γ)θc2−cos(2γ)θs2(44)
+λ2 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+µ2θ0+ 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+rµ
22
+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
22
+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)µσ
32 (54)
−(aκσ +Cdκ)
64 π2a2κ2σ2+π2C2
dκ2−32πaσθc+16(πCdκ−2πCd)µ
+64µ2+ 2π2Cdaκ2+ 8(πaκ −2πa)µσ
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
22
+πCTP
21/2!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+rµ
22
+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)µσ
32 (64)
−(aκσ +Cdκ)
64 π2a2κ2σ2+π2C2
dκ2+ 32πaσθc+16(πCdκ+ 2πCd)µ
+64µ2+ 2π2Cdaκ2+ 8(πaκ + 2πa)µσ
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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