A NEW APPROACH TO FORCED FLAPPING FOR THE
ORNICOPTER
D. J. van Gerwen , Th. van Holten
Delft University of Technology, Delft, 2629HS, The Netherlands Keywords: ornicopter, forced flapping, reaction torque
Abstract
The Ornicopter is a single rotor helicopter that uses forced flapping of the blades to propel the rotor and thus to prevent a reaction torque from arising on the fuselage. By changing the am-plitude of this forced flapping motion, the yaw movement of the fuselage can be controlled. As a result, the Ornicopter does not require a tail ro-tor or a similar anti-ro-torque device.
In order to limit the loads that arise on the blades due to forced flapping, and to prevent con-trollability problems, some amount of blade flap-ping freedom is required. Up to now, flapflap-ping freedom was provided by blade flexibility, but this implies that a large bending moment is in-troduced in the blade root. In the current paper an alternative solution is presented, one that gets rid of the root bending moment altogether and at the same time allows free flapping on top of the forced flapping motion.
Basic trim equations for the alternative con-figuration are derived, based on a planar model of a rigid blade rotating at constant tip speed. The hub loads in the flapping plane are analyzed and the results are compared with existing Ornicopter theory for rigid blades. Proof of concept is pvided with the help of a small electric model ro-tor.
The main advantage of the new configuration turns out to be the more beneficial blade loading, which may even allow the use of conventional he-licopter rotor blades. The positive influence on hub loads only becomes significant for relatively large Lock numbers.
1 Introduction
1.1 The Ornicopter
The Ornicopter is a single rotor helicopter that can be fully controlled without the help of a tail rotor or a similar anti-torque device. This is pos-sible because the Ornicopter rotor does not give rise to a reaction torque on the fuselage, contrary to conventional helicopter rotors.
There is no reaction torque because a propul-sive force is applied directly to the blades just like in a tip jet helicopter. However, the Ornicopter does not use tip jets to generate this propulsive force, rather it makes use of forced flapping of the rotor blades.
By forcing the rotor blades to flap up and down once per revolution (1/rev), the aerody-namic environment of the blade is manipulated in such a way that part of the lift is used to coun-teract drag. This forced flapping principle is il-lustrated in figure 1.
If the correct amplitude is chosen for the forced flapping motion, the component of the lift in the shaft plane completely counteracts the drag component, and the average torque about the ro-tor shaft is reduced to zero.
The forced flapping principle is described ex-tensively in previous work [1, 2, 3, 4], giving it a sound theoretical basis, and the concept has been proven on numerous occasions [5, 6, 7, 8, 9], on a small scale as well as in full scale.
Fig. 1 A blade element at constant pitch angle that is forced to flap once per revolution. Due to forced flapping the angle of attack changes con-tinually, and as a result the lift component pro-vides a propelling force during part of the revolu-tion.
1.2 Rotor configurations
The rotor configuration is a key component of the Ornicopter concept. It was proven in [2] that by selecting the proper number of blades and corre-sponding flapping phase for each blade, the vi-brations due to forced flapping can be all but eliminated, so that hardly any vibration is trans-ferred to the fuselage (except for those vibrations that occur in conventional helicopters as well).
For example, the demonstrator model de-picted in figure 2 uses the so-called 2x2 anti-symmetric rotor configuration. In this configu-ration, four blades are used, and opposite blades are forced to flap in the same direction. Further-more, as one pair of blades flaps up, the other pair flaps down at an exact 180◦phase difference. As a result only a 2/rev (twice-per-revolution) torque vibration remains, which is similar to the vibra-tion caused by a convenvibra-tional teetering rotor. 1.3 Ornicopter flight controls
The mechanism that generates the forced flap-ping motion (i.e. the flapflap-ping mechanism) can also be used to produce a controlled amount of torque about the rotor shaft. Hence, by changing the amplitude of the forced flapping motion of the blades, the yaw motion of the fuselage can be controlled. The forced flapping amplitude (from here on, simply ‘flapping amplitude’) thus takes over the role of the tail rotor collective pitch as the control variable for yaw.
Just like in a conventional helicopter, a swash plate is used to provide collective and cyclic pitch control. For this to work properly, conven-tional rotor dynamic behavior has to be superim-posed on the forced flapping motion of the rotor blades. This implies that the Ornicopter will re-quire some amount of flapping freedom. In previ-ous work by [10] it was shown that this superpo-sition principle holds not only for the Ornicopter in hover, but also in steady forward flight.
Flapping freedom is also necessary in order to reduce the loads imposed on the blade by forced flapping. One way to achieve this is by using ‘rigid’ blades with springs in between the blades and the flapping mechanism. In the current ra-dio controlled Ornicopter model [7], depicted in figure 2, flapping freedom is provided by the flex-ibility of the rotor blades.
Fig. 2 The radio controlled Ornicopter demon-strator model. Rotor diameter 1.7m, take-off
weight 65N.
The problem with this approach to forced flapping is that the rotor blades are loaded in a rather unfavorable way: A large fluctuating bend-ing moment is introduced at the root of the blade. This type of loading is very detrimental in terms of fatigue life, and the rotor blades will most likely need to be reinforced in order to cope with the bending stresses.
In the current paper an alternative approach to forced flapping is presented, one that gets rid of the large bending moment in the blade.
1.4 Double hinge flapping configuration The double hinge flapping configuration repre-sents an alternative to the single hinge
configu-rations used up to now. The defining characteris-tic of this configuration is that it includes an ex-tra free flapping hinge, located at some distance from the hub. This free flapping hinge prevents a flapping moment from being transferred to the blade. The blade is forced up and down by means of a forced root displacement.
In this paper the flapping configurations are represented with the help of planar blade models. Figure 3 shows four different flapping configura-tions, including a double hinge configuration.
The double hinge configuration depicted in the figure can be interpreted as a kind of dis-cretization of a flexible blade without any flexural rigidity. A flapping arm connects the hub to the blade root, and this flapping arm is loaded by a forced flapping moment. The name double hinge configuration is based on this representation be-cause it contains a flapping hinge at the hub and one at the blade root.
An alternative representation of the double hinge concept is depicted in figure 4. In this rep-resentation the root of the blade is translated up and down in axial direction. This model will be used in the remainder of the text because it al-lows investigation of an interesting limit case in which the free flapping hinge is located on the rotor shaft.
The double hinge configuration is investi-gated in order to establish the trim equations for the hover case and in order to assess whether any advantages may be offered compared with the single hinge configurations in terms of hub loads. Furthermore, it would be interesting to know whether the lack of a bending moment in the blade root would allow the use of conven-tional helicopter rotor blades. For design pur-poses the influence of flapping offset e on trim settings and loads will be taken into account. 2 Double hinge flapping equations 2.1 Model description
The analysis of the double hinge flapping rotor is based on the planar model depicted in figure 4. Only the hover condition is considered, and
the rotor is fixed in space. The model consists of a single rotor blade attached at the root to a free flapping hinge at some offset from the rotor shaft.
Fig. 4 Rigid blade model with flapping offset e and forced flapping by base displacement b. The free flapping angle is represented byβ. The blade is assumed to be prismatic with uniform mass distribution. Rotor speedΩis assumed to be con-stant.
Flapping angles are assumed to be small. The rotor blade is assumed to be rigid, homogeneous and prismatic, with a symmetric airfoil. The ro-tor is assumed to be rotating at constant speedΩ. The centrifugal load on the blade due to this ro-tor rotation is interpreted as an external stiffening moment Mc.
The baseline rotor radius is R0 and the off-set of the free flapping hinge is defined by the non-dimensional parameter e. The root displace-ment b is a once-per-revolution sinusoidal motion (1/rev forced flapping).
The aerodynamic lift L and moment Ma are found using blade element momentum theory with uniform inflow distribution. The variable r defines the position of a blade element. In this respect it is important to point out that the effects of root cutout (eR0) are taken into account in the blade element integration as well as in the area of the actuator disc.
2.2 Baseline rotor parameters and non-dimensional quantities
In the analysis that follows, a centrally hinged baseline rotor is considered for reference (e= 0). The baseline rotor is fully defined by the follow-ing characteristic parameters: rotor radius R0, tip
Fig. 3 Planar rotor blade models: (1) rigid blade, (2) flexible blade, (3) rigid blade with equivalent flexural rigidity, (4) double hinge configuration (only centrifugal rigidity)
speed Vtip, non-dimensional inflow velocity λi0,
rotor solidity σ0, Lock number γ0, disc loading DL0, average lift slope clα, average profile drag
coefficient cdp, and air densityρ.
The aircraft weight W , the blade mass m0, the chord c, rotor speed Ω, and the number of blades N can be derived from these characteris-tic parameters. This choice of baseline param-eters facilitates comparison with the basic cen-trally hinged rigid blade model from previous publications (e.g. [1]).
The actual blade parameters are functions of the offset e. For a prismatic homogeneous blade, the mass per unit length of the blade is constant and equal to m0
R0. An increase in offset, at
con-stant rotor radius, will then decrease the mass of the rotor blade. Using this fact, the rotor blade parameters can be defined as in table 1.
The model variables can be made non-dimensional with the help of the baseline pa-rameters. It is convenient to work with non-dimensional quantities because this allows a con-siderable simplification of the equations. The non-dimensional variables, denoted by lowercase characters, are defined in table 2. The method of non-dimensionalizing applied here is widely used
in conventional helicopter theory.
It is important to note that the non-dimensional force represents the ratio of the di-mensional force to the centrifugal force mul-tiplied by some constant. Similarly, the non-dimensional moment represents the ratio of di-mensional moment and the centrifugal stiffen-ing moment multiplied by some constant. This should be taken into account in the interpretation of the non-dimensional results.
To clarify this observation, the expression for the centrifugal force on a prismatic homogeneous blade may be considered. The centrifugal force on the baseline blade, due to a constant rotor speedΩ, is Fc0 = m0Ω 2R0 2 = S0Ω 2= 3 2 I0Ω2 R0 (1) Thus, referring to table 2 ,the non-dimensional force v can be expressed as
v= V R0 I0Ω2 = 3 2 V Fc0 (2) and a similar interpretation may be applied to the non-dimensional moment.
Table 1 Rotor parameters for a prismatic homogeneous blade
parameter baseline blade offset blade
length R0 Rb= R0(1 − e)
mass m0 mb= m0(1 − e)
first moment wrt hinge S0= 12m0R0 Sb= S0(1 − e)2 second moment wrt hinge I0= 13m0R20 Ib= I0(1 − e)3 Lock number γ0= clαρcR40
I0 γb=γ0(1 − e)
solidity σ0= πNcR
0 σb=σ0(1 + e)
−1
Table 2 Non-dimensional variables
variable dimensional non-dimensional
blade element coordinate r x=Rr
0 (note that dr= R0dx) root displacement b ε= Rb 0 force F f = FR0 I0Ω2 moment M m= IM 0Ω2 power P p=I P 0Ω3
induced velocity vi λi=ΩvRi0
2.3 Equation of motion
The model under consideration has a single de-gree of freedom represented by the flapping an-gleβ. The base displacement b is a forcing term. Based on figure 4, the linearized equation of mo-tion for the blade is thus found to be
Ibβ¨= Ma− Mc− Sb¨b (3) where Ma is the aerodynamic moment about the flapping hinge, and Mcis the ‘stiffening’ moment due to the centrifugal force Fc.
The non-dimensional version of this equation for a homogeneous prismatic blade is found after dividing by I0Ω2and using table 1 and table 2:
(1 − e)3β′′= ma− mc− 3
2(1 − e)
2ε′′ (4) where a prime (′) denotes differentiation with re-spect to rotor azimuthΨ:
β′′= d2β dΨ2 = d2β d(Ωt)2 = ¨ β Ω2 (5)
The external loads mc and ma are determined next.
2.4 External loads
The centrifugal force on the blade is actually a body force, but it is treated here as an external load. The centrifugal force dFc on a blade ele-ment of mass dm (where dm= m0dx) produces the following moment about the flapping hinge (using equation 1):
dMc= r sinβdFc≈ xR0βm0dxΩ2R0(e + x) (6) The non-dimensional centrifugal stiffening mo-ment is then found, after dividing by I0Ω2, by in-tegration over the blade:
mc=β Z 1−e 0 3x(e + x)dx = (2 + e)(1 − e) 2 2 β (7)
The non-dimensional aerodynamic moment mais found with the help of figure 5 which shows the velocities at the location of the blade element. The vertical and horizontal velocity components are:
and
v=Ω(eR0+ r) =ΩR0(e + x) (9) The inflow angleϕcan then be approximated, for small angles, by:
ϕ= arctanw v ≈ w v (10) so that ϕ≈ vi+ R0(˙ε+ x ˙β) ΩR0(e + x) = (λi+ε′) 1 e+ x+β ′ x e+ x (11) whereλiis the non-dimensional induced velocity. The angle of attack is
α=θ−ϕ (12)
and the lift generated by the blade element is then defined as
dL= clαα 1 2ρv
2cdr (13)
Note the assumption that w is much smaller than v so that v2+ w2≈ v2. In non-dimensional form this becomes dl=γ0 2 θ−λi+ε′ e+ x − xβ′ e+ x (e + x)2dx (14) It is convenient at this point to introduce a change of variables u = e + x, so that the non-dimensional differential lift becomes
dl=γ0 2 θ−λi+ε′ u − (u − e)β′ u u2du (15) The total lift generated by the blade is then found by integration over u (from e to 1):
l= γ0 2(1 − e) 1 + e + e2 3 θ− 1+ e 2 λi + −γ02(1 − e) 1 + e2 ε′+(2 + e)(1 − e) 6 β ′ (16) The non-dimensional aerodynamic moment about the flapping hinge due to the blade element is
dma= xdl = (u − e)dl (17)
Fig. 5 Velocities for a blade element at distance
eR0+ r from the rotor shaft
so that the total non-dimensional aerodynamic moment is found to be ma= γ0 2(1 − e) 2 3 + 2e + e2 12 θ− 2+ e 6 λi + −γ0 2(1 − e) 2 2 + e 6 ε ′+(3 + e)(1 − e) 12 β ′ (18) Now the expressions for the centrifugal stiffen-ing moment and the aerodynamic moment can be substituted into the equation of motion. The equation of motion can then be solved in order to determine the dynamic behavior of the blade. 2.5 Blade dynamics
With the help of the expressions for the external loads, the equation of motion (equation 4) be-comes: β′′= γ0 2 3 + 2e + e2 12(1 − e) θ− 2+ e 6(1 − e)λi + −γ0 2 2+ e 6(1 − e)ε ′+(3 + e) 12 β ′+ −22+ e (1 − e)β− 3 2(1 − e)ε ′′ (19) It may be worth noticing that the equation re-duces to the standard flapping equation for a cen-trally hinged rigid blade if there is no excitation εand if e= 0.
The equation of motion can be solved us-ing the method of undetermined coefficients [11]. The solution will yield a relation between the forced flapping amplitude ˆεand the free flapping amplitude ˆβ. This relation is used later on to de-termine the trim settings in hover.
In order to solve the equation of motion by the method undetermined coefficients, a sinu-soidal forced flapping motion is assumed:
ε= ˆεcos(Ψ−φ) (20) Here the phase angle φ is a design variable, whereas the amplitude ˆε is the control variable used for yaw control. For ease of computation, the latter two variables are combined to form an alternative expression in terms of E and F.
ε= E cosΨ+ F sinΨ (21) so that the amplitude of the (non-dimensional) forced flapping motion is
ˆ
ε=pE2+ F2 (22) and the phase with respect to the rotor azimuth is
φ= arctanF
E (23)
Now, given this sinusoidal excitation together with the constant aerodynamic excitation term, the flapping response is expected to be of the same form:
β=β0+C cosΨ+ S sinΨ (24) so that the free flapping amplitude is
ˆ
β=pC2+ S2 (25) Substitution of these expressions into the equa-tion of moequa-tion yields an expression for the cone angle β0 as well as a relation between the free flapping coefficients C and S and the forcing co-efficients E and F (remember that the latter two represent a control variable):
β0= γ0 12 3 + 2e + e2 2+ e θ− 2λi (26) C=C1E−C2F C3 (27) and S=C1F+C2E C3 (28) 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 offset e [%]
ratio free flapping ampl. to forcing ampl.
β
/
ε
[−]
γ0 = 5, 10, 15
Fig. 6 The ratio of free flapping amplitude ˆβto forced flapping amplitude ˆεas a function of non-dimensional flapping offset e. This relation rep-resents the dynamic behavior of a prismatic blade with 1/rev forced flapping, under the assumption
of a constant induced velocity distribution. The figure is based on constant baseline Lock number, so that the actual blade Lock number changes as a function of e. For large values of e the relation has no real physical significance, therefore e is limited to 50%. where C1= 2γ20(e3+ 4e2+ e − 6) + 1296e (29) C2= 36γ0 3+ 2e + e2 (30) and C3=γ20(3 + e)2(1 − e)2+ 1296e2 (31) Now the relation between free flapping amplitude and forced flapping amplitude can be determined, using equation 25 and equation 22:
ˆ β ˆ ε = s 4γ20(2 + e)2+ 1296 C3 (32) This relation is depicted in figure 6 for different values of the baseline Lock numberγ0. It is im-portant to keep in mind here that the actual blade Lock numberγbdecreases with increasing offset, as described by the expression in table 1.
An example of blade motion for a given offset (e= 15%), and for different values of the Lock
0 0.2 0.4 0.6 0.8 1 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
normalized radial coordinate [−]
normalized axial coordinate [−]
γ0 = 5
γ0 = 15
Fig. 7 Motion sequence for a blade during one rotor revolution in 20 equal increments. The blade offset e and the forcing amplitude ˆε are both 15% and the baseline Lock number is varied between 5 and 15. As can be expected, higher damping results in smaller tip displacement for a given forcing amplitude. Note that this rela-tively large forcing amplitude was chosen for the sake of clarity. Actual trim amplitudes are much smaller, as can be seen in figure 9.
number (γ0= 5 and 15), is given in figure 7. The figure shows a motion sequence for one revolu-tion of the rotor in equal increments.
Now that the dynamic behavior of the blade is known, it is possible to determine how large the amplitude of the forcing function should be in order to reduce the reaction torque to zero in hover conditions. That is, the trim setting for ˆε can now be determined.
2.6 Power in hover
The core of the Ornicopter concept is that the re-action torque is prevented from arising by ma-nipulation of the airflow around the rotor blades. This has a direct effect on the induced power Pi, as becomes clear when using the blade element method to calculate Pi. It appears that Pi has two components, one equal to the Piof a conventional helicopter, and one component that results from forced flapping. In mathematical form, the in-duced power can be expressed as
Pi= Piconv+ Pif lap (33)
and the total shaft power is
Psh= Pi+ Pp (34) In the trim condition the reaction torque should be zero, which implies that the (average) shaft power should also vanish. For Pshto be zero, the flapping power needs to be negative and equal to the conventional hover power:
0= Pif lap+ Piconv+ Pp (35)
The average induced power per revolution for an N-bladed rotor may be calculated as
Pi= N 2π Z 2π 0 Z Rb 0 ΩdQidΨ (36) where dQiis the differential induced torque about the hub. The non-dimensional power is found af-ter dividing by I0Ω3. The non-dimensional in-duced torque due to a blade element, for small values of the inflow angleϕ, is
dqi= (e + x)ϕdl (37) After quite some algebraic manipulation with the help of a symbolic computation package, the components of the non-dimensional induced power are found. The conventional part of the induced power is piconv = N γ0 2λi 1 − e3 3 θ− 1− e2 2 λi (38) which indeed simplifies to the conventional ex-pression for induced power if offset e is zero. Note that this part of the induced power should also equal the product of thrust (non-dimensional thrust t) and induced velocity. Ideally:
piconv= tλi (39)
This relation is used later on to determine the trim settings for hover.
The flapping component of induced power is pif lap= −Nγ0
24εˆ 2C4
C3
where C4=
γ2
0(1 − e)(3 + 7e − 16e2+ 5e4+ e5)+
+ 1944(1 − e)(1 + e + e2+ e3) (41) Now it turns out that the flapping part of the in-duced power indeed has a negative value, which concurs with the fact that energy is supplied to the system. By selecting the correct forcing am-plitude ˆε, average shaft power can thus be re-duced to zero. This forcing amplitude is referred to as the trim amplitude. Furthermore, it is conve-nient to define the flapping power per blade pf lb
to be the negative of the induced power due to flapping per blade:
pf lb = −
pi,f lap
N (42)
Note that, as an alternative to the induced power approach, the flapping power may also be cal-culated as the average power transferred to the blade by the external forces in the direction of the forcing motion:
pf lb = 1 2π Z 2π 0 Z Rb 0 (˙b + r ˙β)dLdΨ (43)
The profile drag power is also found using the blade element method. Without showing the de-tailed derivation, the resulting non-dimensional profile drag power is
pp= N γ0 8 cdp clα(1 − e 4) (44)
This expression shows that the outboard half of the blade is responsible for almost 94% of the to-tal profile drag power. Furthermore, the flapping mechanism will also generate profile drag, so that it might be more realistic simply to assume a con-stant profile drag power.
In order to quantify the trim forcing ampli-tude for hover, the trim values for the collective pitch and induced velocity need to be determined. This is done in the conventional way.
2.7 Trim condition
The induced velocity is found using simple actu-ator disc theory. The root cutout is taken into ac-count in the calculation of the disc area. In hover, thrust is equal to the aircraft weight W0, so that the induced velocity is
vi= s W0 2ρπR20(1 − e2) = vi0 1 √ 1− e2 (45) As e increases the induced velocity increases, and so will induced power. Profile drag power, on the other hand, is assumed to be independent of e because the mechanism in the root cutout will still generate profile drag. The increase in total power required for hover due to offset e is plotted in figure 8. 0 10 20 30 40 50 0 2 4 6 8 10 12 offset e [%]
increase in power required due to root cutout [%]
Fig. 8 Increase in power required for hover due to root cutout. The profile drag power is as-sumed to be independent of e. The non-lifting root cutout causes an increased induced velocity which in turn causes the power required to in-crease.
The collective pitch angleθis found with the help of the average lift per blade of equation 16:
¯l =γ0 2(1 − e) 1 + e + e2 3 θ− 1+ e 2 λi (46) So that the collective pitch in trim becomes:
θ= 3 1− e3 2 γ0 w0 N +λi0 √ 1− e2 2 ! (47)
where w0 is the non-dimensional weight. The trim forcing amplitude can now be determined by calculating the flapping power required in hover using equation 35.
From equation 40, the trim forcing amplitude then becomes: ˆ ε= s 24pf lb γ0 C3 C4 (48) An example of trim settings for a small Orni-copter can be found in figure 9.
0 10 20 30 40 50 0 1 2 3 4 5 6 7 8 offset e [%] collective θ [ ° ], ind. velocity λi [%], forcing ampl. ε [%] DL0 = 179 N/m2 γ0 = 5.68 σ0 = 0.054 V tip = 197 m/s c l,α = 6 c d,p = 0.01 θ ε λi
Fig. 9 Example of trim settings for a small Or-nicopter in hover (based on data for a Schweizer 300C two seater). The non-dimensional quanti-ties are represented as a percentage of the rotor radius R0.
Now that the trim settings are known, it is possible to evaluate the hub loads that occur due to forced flapping in the trim condition. Only the vertical force on the hub and the moments on the hub will be considered here.
2.8 Flapping loads
In order to find the flapping loads for any forcing amplitude ˆε, consider the translational equation of motion for a blade element based on figure 10. The vertical force due to a blade element is
dV = dL − (¨b + r ¨β)dm (49) and integration over the blade yields the non-dimensional vertical force on the hub:
v= l −3
2 2(1 − e)ε
′′+ (1 − e)2β′′
(50)
After substitution of the expressions for l,ε′′, and β′′, and after yet another round of algebraic ma-nipulation, the amplitude of the vertical force is found: ˆ v= ˆε· 1 12 r C5 C3 (51) where C5=γ40(1 − e)4(1 + 4e + e2)2+ + 1296γ20(1 − e)2(2 + 10e + 13e2+ 8e3+ 3e4)+ + 419904(1 − e)2(1 + e)2 (52) The moment about the hub is determined using figure 11. The free flapping hinge cannot trans-fer any moments to the hub, thus only the reac-tion forces on the flapping hinge are taken into account. The moment about the hub can be in-terpreted as a kind of flapping moment. In non-dimensional form this hub moment is
mh= ev −εfc (53) and the amplitude of the hub moment is then found to be ˆ mh= ˆε· γ0 12 r C6 C3 (54) where C6=γ20e2(1 − e)4(1 + 4e + e2)2+ + 2916(1 − e)2(1 + e)2(1 + e2)2 (55) The flapping moment about the hub, defined in the rotating hub reference frame, can be decom-posed in the body reference frame of the aircraft in order to determine the roll moment and pitch moment. This is depicted in figure 12. The non-dimensional roll moment is then defined as
mr = mhsinΨ (56) and the pitch moment is
mp= mhcosΨ (57) The hub flapping moment described by equation 53 can also be expressed as
Fig. 10 Vertical force due to a blade element.
Fig. 11 Moment about the hub (point O) due to forced flapping.
where ¯mh is an average value. The coefficients A and B can be found by evaluating equation 53. After substitution into equation 56 the roll mo-ment is found to have a constant term as well as higher harmonic terms (1/rev and 2/rev):
mr = B
2+ 1/rev + 2/rev (59) Similarly, the pitch moment is
mp= A
2+ 1/rev + 2/rev (60) Thus, a constant moment will arise in the body reference system, with magnitude equal to 12mˆh. It should be noted that it is possible to get rid of these resulting loads on the fuselage by selecting the proper number of blades and the proper flap-ping phase for each blade, as described by [2]. 2.9 Verification
The results presented here were verified using the numerical multi-body dynamics package Sim-Mechanics in the Matlab/Simulink environment.
Fig. 12 Decomposition of hub moment in the fixed reference frame and rotating reference frame.
This package employs fully nonlinear equations of motion. The simulation results were shown to be in close agreement with the theory derived in this text.
3 Comparison with single hinge flapping configuration
3.1 Single hinge Ornicopter configuration with offset
The reason for using a double hinge flapping con-figuration is that it offers several advantages with respect to the single hinge flapping configuration. The most important advantage is that it allows the use of lighter rotor blades, perhaps even conven-tional helicopter blades, because there is no large root bending moment acting on the blade. Never-theless, the flapping mechanism and hub will still be quite heavily loaded.
The question remains how the hub loads for the double hinge configuration compare to those for the single hinge configuration. It would also be interesting to know the difference in tip dis-placement in the trim condition. For these rea-sons, the double hinge configuration is compared with a single hinge Ornicopter configuration with the same rotor characteristics and offset e, as de-picted in figure 13. Instead of the forcing dis-placement, a forced flapping moment is now ap-plied at the blade root.
Fig. 13 Single hinge Ornicopter blade model model with offset and with a flapping moment ap-plied at the root, for comparison with the double hinge configuration.
3.2 Comparison of trim settings
Both blade models are based on actuator disc the-ory, therefore the power required for both config-urations is identical. The collective pitch angleθ in the trim condition depends on the induced ve-locity as seen in figure 9. As a result, the trim value for the single hinge Ornicopter configura-tion is also identical to that for the double hinge configuration.
The forced flapping amplitude required to reach the trim condition for a single hinge Or-nicopter configuration is defined in terms of flap-ping angleβs. By applying the approach outlined in this text to the single hinge model, the relation between flapping amplitude and power required in hover was found to be
ˆ βs=
s
48pf lb
γ0(3 + e)(1 − e)3 (61)
An interesting observation is that the single hinge trim flapping amplitude is independent of Lock number, becauseγ0vanishes after substitution of the expression for power required in hover. This is not the case for the double hinge configuration. It is not very meaningful to compare the sin-gle hinge flapping amplitude directly with the linear displacement amplitude ε for the double hinge configuration. A more convenient way to compare the trim flapping amplitude for the two configurations is by comparing the amplitudes of the (non-dimensional) tip displacement ˆδs. For the single hinge case this tip displacement
ampli-tude is easily determined. For small angles: ˆ
δs= Rb
R0
sin ˆβs≈ (1 − e)ˆβs (62) whereas for the double hinge configuration the maximum tip displacement depends on the phase difference between the root displacement ε and the free flapping angleβ:
ˆ δ≈ |ε+Rb R0(β−β0)| = q (E + (1 − e)C)2+ (F + (1 − e)S)2 (63) This expression may be evaluated further with the help of equations 27 and 28.
Note that, for the centrally hinged case (e= 0), the phase angle will approach 90◦ as the Lock number approaches zero, which implies that the tip displacement amplitude in this limit case would approach that for the single hinge configuration. The limit is considered because in the complete absence of damping and at zero off-set the system will be in resonance and the trim equations lose their validity.
The tip displacement amplitudes in hover for the two flapping configurations are compared in figure 14. It appears that the tip displacement for the double hinge configuration is smaller than that for the single hinge configuration. This dif-ference becomes more significant as the offset in-creases.
3.3 Comparison of flapping loads
The loads on the hub due to forced flapping can also be calculated for the single hinge configura-tion by following the approach described in this text. The amplitude of the vertical force on the hub is ˆ vs= ˆβs(1 − e) 2 12 q γ2 0(2 + e)2+ 324 (64) where the subscript s refers to the single hinge configuration. The amplitude of the non-dimensional flapping moment applied to the blade root turns out to be
ˆ mf ls = ˆβs (1 − e)2 24 p C3 (65)
0 10 20 30 40 50 0 5 10 15 offset e [%] tip displacement δ [%] DL0 = 179 N/m2 γ0 = 5.68 σ0 = 0.054 V tip = 197 m/s c l,α = 6 c d,p = 0.01 single hinge double hinge
Fig. 14 Comparison of tip displacement (as a percentage of rotor radius R0) in the trim
con-dition for the single hinge configuration and the double hinge configuration. The former is inde-pendent of Lock number, whereas the latter shifts downwards as Lock number is increased.
The moment due to flapping about the center of the hub is due to both the vertical force, at dis-tance eR0from the hub, and the flapping moment. In non-dimensional terms
mhs = vse− mf ls (66)
And the amplitude of this hub moment turns out to be ˆ mhs = ˆβs (1 − e)2 24 γ0(3 + 2e + e 2) (67)
The pitch moment and roll moment about the hub again are in the order of half the total hub mo-ment.
A comparison of the vertical force on the hub for both configurations is depicted in figure 15. The vertical force in the double hinge configura-tion is slightly lower than in the single hinge con-figuration for small offset values. This difference increases as Lock number increases, as depicted in figure 16.
The hub moments are compared in figure 17. A negligible difference is observed in the hub moments, although this difference also increases with increasing Lock number, in favor of the dou-ble hinge configuration, as depicted in figure 18.
0 10 20 30 40 50 0 2 4 6 8 10 12 14 16 18 20 offset e [%] vertical force v [%] DL 0 = 179 N/m 2 γ0 = 5.68 σ0 = 0.054 V tip = 197 m/s c l,α = 6 cd,p = 0.01 double hinge single hinge
Fig. 15 Comparison of the vertical force ampli-tudes in hover for the single hinge and double hinge configurations. 0 10 20 30 40 50 −15 −10 −5 0 5 10 15 20 25 30 offset e [%]
reduction in vertical force v [%]
γ0 = 5, 10, 15
Fig. 16 Reduction in vertical force for double hinge configuration (with respect to single hinge configuration) for different values of Lock num-ber. For rotors with higher Lock number the ad-vantage due to the double hinge mechanism be-comes more pronounced.
0 10 20 30 40 50 0 1 2 3 4 5 6 7 8 9 10 offset e [%]
flapping moment about hub m
h [%] DL 0 = 179 N/m 2 γ0 = 5.68 σ0 = 0.054 V tip = 197 m/s c l,α = 6 cd,p = 0.01 single hinge double hinge
Fig. 17 Comparison of the amplitudes of the mo-ment about the center of the hub in hover for the single hinge and double hinge configurations.
0 10 20 30 40 50 −4 −2 0 2 4 6 8 10 12 14 16 offset e [%]
reduction in hub moment m
h
[%]
γ0 = 5, 10, 15
Fig. 18 Reduction in hub moment for double hinge configuration (with respect to single hinge configuration) for different values of Lock num-ber. For rotors with higher Lock number the ad-vantage due to the double hinge mechanism be-comes more pronounced.
An important consideration in this respect is that, in reality, the internal forces in the flapping mechanism also depend on the inertia of the flap-ping arm with length eR0. This is not considered in the current analysis.
From this analysis it may be concluded the double hinge configuration offers only a very slight advantage over the single hinge configura-tion in terms of hub loads. However, for rotors with relatively high Lock number this advantage becomes more pronounced.
The tip displacement amplitude for the dou-ble hinge configuration is a few percent lower than for the single hinge case, which may prove beneficial. Variation in hub loads and trim pa-rameters at relatively low offset values is so small that no obvious optimum offset value can be es-tablished on those terms. An obvious penalty due to offset, however, is that of increased induced power. For this reason the offset should be cho-sen as small as possible taking into account the flapping mechanism geometry.
So far it appears that the main advantage of the double hinge mechanism is that it gets rid of the large fluctuating bending moment in the root of the blade. Nevertheless a fluctuating vertical force still remains.
In order to see whether it would be possible to use conventional style rotor blades for this Or-nicopter configuration, it is necessary to compare the vertical force with the vertical force fluctua-tion that arises in a convenfluctua-tional helicopter blade under cyclic pitch load.
3.4 Vertical force on a conventional heli-copter blade
In order to get some feeling for the magnitude of the vertical force on the double hinge Ornicopter blade, a comparison with the vertical force on a conventional helicopter blade is required. The vertical force that arises on a conventional free flapping helicopter blade with cyclic pitch excita-tion in hover condiexcita-tions is therefore considered.
This simple case serves the purpose of show-ing whether the vertical force on the double hshow-inge Ornicopter blade is in the same ballpark as the
vertical force on a conventional helicopter blade. Without showing the details of the derivation, the resulting amplitude of the vertical force as a func-tion of cyclic pitch amplitude ˆθc, for a centrally hinged blade, is ˆ vc= ˆθc q 4γ20+ 81 6 (68)
The cyclic pitch excitation that would be required in order to produce a vertical force equal to that found in the double hinge Ornicopter configura-tion with zero offset in hover is ˆθc= 4.1◦. Al-though this cyclic pitch value may be substantial, it is not excessive: Cyclic pitch variations of this magnitude can occur during flight in a conven-tional helicopter.
This is by no means a conclusive result, but it does show that the vertical force on the double hinge Ornicopter blade is probably in the same ballpark as the forces that can occur on a conventional helicopter blade. This suggests that it could be possible to use unmodified (or only slightly modified) conventional helicopter blades for the double hinge Ornicopter configuration. Nevertheless, a more detailed comparison with conventional helicopter blade loads is required before anything conclusive can be said on this subject.
Summarizing, it appears that the double hinge configuration can be used to get rid of the large bending moments in the blade without causing excessive vertical loads on the blade root. For relatively small Lock numbers and small offset values the difference in hub loads for the single hinge and double hinge configurations is negligible, so there is no gain in this respect, but neither is there a disadvantage in using the double hinge configuration.
Some practical issues do come to mind when thinking about implementation of the double hinge configuration. First and foremost, it would be nice to know the concept actually works.
Fig. 19 Model rotor used for proof of concept.
4 Practical considerations
4.1 Proof of concept
A proof of concept rotor was built in order to show that the double flapping hinge concept works. This 90cm diameter rotor is depicted in figure 19. The rotor was built on an exist-ing model helicopter fuselage with electric mo-tor. For simplicity the rotor was built as a single teeter model, with a free flapping hinge offset of approximately 20% (the figure shows a version with smaller hinge offset) and a fixed collective pitch angle of approximately 0◦. Commercial off the shelf wooden rotor blades were used.
A disadvantage of this single teeter configu-ration is that the inflow distribution over the rotor disk becomes skewed. This is expected to de-teriorate the effectiveness of the forced flapping. Furthermore, the relatively poor build quality of the rotor prevented the gathering of accurate mea-surements. For this reason it is difficult to use any quantitative test results for validation of the the-ory presented here. Nevertheless, the rotor was used successfully to show, in a qualitative sense, that the double flapping hinge configuration is able to reduce the reaction torque on the fuselage to zero. A reversed reaction torque could also be generated, as required for yaw control purposes.
By suspending the fuselage on a very low friction test stand, with spring action to provide a
measure for the reaction torque, the yaw motion of the fuselage could be observed. At a tip speed of 50m/s, a teeter amplitude in the order of 14◦, which is equivalent to ˆε≈ 5%, proved sufficient to reduce the reaction torque to zero. A small in-crease in the flapping amplitude caused the fuse-lage to move to a position in one direction, and a decrease caused the fuselage to move to a po-sition in the opposite direction, which proves the potential for yaw control.
The fact that a 5% forcing amplitude was re-quired even though no average lift was gener-ated is most likely due to high profile drag and also to the skewed inflow distribution that results from the teeter configuration. The existence of the skewed inflow distribution was corroborated by manually sensing the direction of air flow through the rotor disc. More detailed measure-ments need to be taken in order to investigate this phenomenon.
As described in previous work (e.g. [7]), proper selection of rotor configuration (number of blades and flapping sequence), e.g. a 2x2 anti-symmetrical configuration, will make sure that the inflow distribution can be approximated by a uniform distribution. In that case the forcing am-plitude for the trim condition should be smaller.
In any case, the concept has been proven to work. Some issues arise, however, when think-ing about practical implementation of the double hinge configuration. The most important consid-eration is that of pitch control.
4.2 Pitch control
Just like in conventional helicopters the hinge or-der is an important factor in limiting loads in the pitch control links[12]. In order to prevent large pitch moments from arising, the pitch hinge axis should be aligned with the longitudinal axis of the blade. In the hinge sequence, starting from the hub, the pitch hinge should always be last, e.g. flap-lag-pitch or lag-flap-pitch. If a conven-tional swash plate is to be used, this hinge se-quence would require a complicated mechanical linkage between the flapping mechanism and the blade.
Another possibility would be to align the pitch hinge with the forced flapping arm (i.e. the arm between the forced flapping hinge and the free flapping hinge). This would allow the use of a conventional swash plate for pitch control, but the pitch loads would increase considerably com-pared to a conventional helicopter. This can in turn be counteracted by locating the free flapping joint and the lag joint behind the pitch joint as seen from the hub: pitch-flap-lag. If the flap and lag joints coincide, effectively forming a univer-sal joint, the pitch loads are reduced to zero (in the ideal case, neglecting the aerodynamic mo-ment). A lag damper would again cause some pitching loads to be transferred to the pitch link.
A promising development that could be very useful in this respect is the use of individual blade control (IBC) techniques. IBC would allow the pitch axis and blade axis to be aligned without requiring complicated mechanical linkages to a swash plate.
All in all there are significant advantages in using a double hinge configuration in terms of blade loading, but the practical issues resulting from such a configuration will need attention. However, the practical issues raised here are not prohibitive because possible solutions are avail-able.
5 Conclusion
The Ornicopter uses forced flapping of its rotor blades to control the yaw motion of the fuselage. For control purposes the Ornicopter also needs some degree of flapping freedom.
Up to now, flapping freedom was always pro-vided by means of blade flexibility. The Orni-copter rotor blades were attached to the hub by a single hinge and a flapping moment was applied to the root of the blade. This implies that a large bending moment was introduced into the root of the blade. In order to prevent fatigue problems due to the fluctuating nature of this bending mo-ment, the blades would have to be reinforced sig-nificantly.
In order to get rid of the large bending mo-ment in the blade a new forced flapping
config-uration has been devised. This so called double hinge configuration was presented in the current paper.
The double hinge configuration was repre-sented by a planar rigid blade model with hinge offset e and translational forced root displace-ment ε, at constant rotor speed. The basic trim equations for this configuration in the hover con-dition were derived and proof of concept was pro-vided using a small electric model rotor.
The vertical force on the hub and the moment about the hub center due to forced flapping in the trim condition were analyzed, and the results were compared with the loads on an equivalent single hinge Ornicopter blade model. Examples were given, based on the characteristic rotor pa-rameters for a small two-seat helicopter with rel-atively low Lock number (γ0≈ 6).
The results were verified with the help of the numerical multi-body dynamics package Sim-Mechanics, part of the Matlab/Simulink environ-ment.
In the hover condition the trim amplitude of the root displacement ˆε, which represents the yaw control variable, was found to be approximately 5% of the rotor radius. Variation with offset e turned out to be minimal. The tip displacement for the double hinge configuration proved to be smaller than that for the equivalent single hinge configuration, although at small to moderate off-set values this difference was limited to a few per-cent at most.
The vertical force on the hub for the double hinge configuration in hover conditions turned out to be slightly smaller than for the single hinge configuration for small offset values (e< 16% for the example case).
The difference in the example case is not very large, but the difference increases considerably in favor of the double hinge configuration as the Lock number increases.
The difference in hub moment due to flapping for the single hinge and double hinge configu-ration was found to be marginal, although this difference also increases in favor of the double hinge configuration as Lock number is increased. No obvious optimum offset value could be
pinpointed based on trim settings or flapping loads, but from power considerations it is obvi-ous that the offset should be as small as possible. Using a crude model it was shown that con-ventional helicopter blades can encounter verti-cal forces of the same magnitude as found for the Ornicopter, during normal flight. Although a more detailed investigation is required in this re-spect, the result is promising and could indicate that conventional helicopter blades may be used in the double hinge configuration.
The design of the pitch control system for a double hinge Ornicopter rotor requires some at-tention in order to prevent pitch link loads from becoming too large. This issue could be cir-cumvented completely with the help of individ-ual blade control (IBC) techniques.
In conclusion, the double hinge configuration represents a viable alternative to the existing sin-gle hinge configurations. The main advantage of the double hinge configuration is the much more beneficial loading of the rotor blades.
Preliminary results show that this may even enable the use of conventional helicopter rotor blades, although more detailed research is nec-essary on this subject.
The differences between the double hinge mechanism and the single hinge mechanism in terms of hub loads become more pronounced at higher Lock numbers, in favor of the double hinge mechanism.
References
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