THE INFLUENCE OF THE POSITIVE PITCH-FLAP … · the influence of the positive pitch-flap coupling...

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THE INFLUENCE OF THE POSITIVE PITCH-FLAP COUPLING (NEGATIVE δ δ δ 3 ) ON THE FLAPPING RESPONSE OF AN ARTICULATED TAIL ROTOR José Márcio Pereira Figueira 1 , Ronaldo Vieira Cruz 2 , Donizeti de Andrade 3 1 Flight Test Special Group, São José dos Campos, Brazil, [email protected] 2 Flight Test Special Group, São José dos Campos, Brazil, [email protected] 3 Technological Institute of Aeronautics, São José dos Campos, Brazil, [email protected] Abstract: This paper presents analyses of the effect of the positive pitch-flap coupling (negative δ 3 ) on the flapping response of an articulated tail rotor blade with geometric parameters similar to the AS355 Twin-Engine Squirrel helicopter. Key words: Helicopter, flapping, pitch-flap coupling. 1. INTRODUCTION In February 1999, a Bell M407 model helicopter was performing a level flight with 110 KIAS over a mountainous area near the city of Lagoa Santa - MG, when, suddenly, the tail rotor collided with the tail cone. The pilot was able to make a successful autorotation landing procedure. However, two passengers left the aircraft hastily, without the permission of the pilot, and were hit by the main rotor blades, which were near the ground. Both died while the pilot suffered minor injuries [1]. Bell M407 helicopter has a positive pitch-flap coupling (negative δ 3 ), i.e., a mechanical linkage in which an increase of flap angle implies in an increase of the blade pitch, as presented in the schematic drawing of the tail rotor hub shown in Figure 1. Fig. 1 – Bell M407 Tail Rotor Hub [2]. As consequence, this observation leads to the question: will this design solution yield to a flapping instability? The analysis of some documentation sent by Bell Helicopter Company concerning Bell M407 tail rotor blade flapping dynamics proves that there is no instability in forward flight. However, the point is that the tail rotor blades flapping has been underestimated by the helicopter maker at high forward speed and, therefore, the company had to modify both the tail rotor design and flight control command system in order to prove that at high forward speed and maximum tail rotor pitch angle, there would be enough clearance between the tail rotor blades and the fuselage. One of the early work involving this issue was presented by Gaffey [3] including analytical and flight tests results concerning the effects of negative δ 3 in main and tail helicopters rotors. This paper is within this context: it analyzes the influence of pitch-flap coupling on flapping response in both hover and forward flight, based on geometrical parameters of an articulated tail rotor blade similar to the one present in the AS 355 Twin-Engine Squirrel. The main contribution of this paper is the analysis of an articulated tail rotor stability in forward flight, based on the Floquet theory. 2. TAIL ROTOR At first, the tail rotor is subjected to the same phenomena encountered by the main rotor (like a small main rotor that works in the vertical plane) [2]. Indeed, its small dimensions imply in the following characteristics: a) collective pitch angle changes are controlled by the helicopter pedals; b) the effects of the pitch-flap coupling is also performed in a different way, i.e., the flapping axis and the pitch change axis are not mutually perpendicular (Figure 2); c) lack of lead-lag hinge, since the loads (forces and bending moments) are smaller and minimized by the pitch-flap coupling; and d) lack of the cyclic pitch variation (lift adjustment is only the result of the blades flapping motion, with no need of tip path plane control). In general, the pitch-flap coupling in tail rotors aims the reduction of the flap amplitude in high forward speed and in maneuvers. Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1052

Transcript of THE INFLUENCE OF THE POSITIVE PITCH-FLAP … · the influence of the positive pitch-flap coupling...

Page 1: THE INFLUENCE OF THE POSITIVE PITCH-FLAP … · the influence of the positive pitch-flap coupling (negative δδδδ3) on the flapping response of an articulated tail rotor ... tail

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THE INFLUENCE OF THE POSITIVE PITCH-FLAP COUPLING (NEGATIVE δδδδ3) ON THE FLAPPING RESPONSE OF AN ARTICULATED TAIL ROTOR

José Márcio Pereira Figueira

1, Ronaldo Vieira Cruz

2, Donizeti de Andrade

3

1Flight Test Special Group, São José dos Campos, Brazil, [email protected] 2Flight Test Special Group, São José dos Campos, Brazil, [email protected]

3Technological Institute of Aeronautics, São José dos Campos, Brazil, [email protected]

Abstract: This paper presents analyses of the effect of the

positive pitch-flap coupling (negative δ3) on the flapping response of an articulated tail rotor blade with geometric parameters similar to the AS355 Twin-Engine Squirrel helicopter. Key words: Helicopter, flapping, pitch-flap coupling.

1. INTRODUCTION In February 1999, a Bell M407 model helicopter was

performing a level flight with 110 KIAS over a mountainous area near the city of Lagoa Santa - MG, when, suddenly, the tail rotor collided with the tail cone. The pilot was able to make a successful autorotation landing procedure. However, two passengers left the aircraft hastily, without the permission of the pilot, and were hit by the main rotor blades, which were near the ground. Both died while the pilot suffered minor injuries [1].

Bell M407 helicopter has a positive pitch-flap coupling

(negative δ3), i.e., a mechanical linkage in which an increase of flap angle implies in an increase of the blade pitch, as presented in the schematic drawing of the tail rotor hub shown in Figure 1.

Fig. 1 – Bell M407 Tail Rotor Hub [2].

As consequence, this observation leads to the question: will this design solution yield to a flapping instability? The analysis of some documentation sent by Bell Helicopter Company concerning Bell M407 tail rotor blade flapping dynamics proves that there is no instability in forward flight. However, the point is that the tail rotor blades flapping has been underestimated by the helicopter maker at high forward speed and, therefore, the company had to modify both the tail rotor design and flight control command system

in order to prove that at high forward speed and maximum tail rotor pitch angle, there would be enough clearance between the tail rotor blades and the fuselage.

One of the early work involving this issue was presented by Gaffey [3] including analytical and flight tests

results concerning the effects of negative δ3 in main and tail helicopters rotors.

This paper is within this context: it analyzes the influence of pitch-flap coupling on flapping response in both hover and forward flight, based on geometrical parameters of an articulated tail rotor blade similar to the one present in the AS 355 Twin-Engine Squirrel.

The main contribution of this paper is the analysis of an articulated tail rotor stability in forward flight, based on the Floquet theory.

2. TAIL ROTOR At first, the tail rotor is subjected to the same

phenomena encountered by the main rotor (like a small main rotor that works in the vertical plane) [2].

Indeed, its small dimensions imply in the following characteristics:

a) collective pitch angle changes are controlled by the helicopter pedals; b) the effects of the pitch-flap coupling is also performed in a different way, i.e., the flapping axis and the pitch change axis are not mutually perpendicular (Figure 2); c) lack of lead-lag hinge, since the loads (forces and bending moments) are smaller and minimized by the pitch-flap coupling; and d) lack of the cyclic pitch variation (lift adjustment is only the result of the blades flapping motion, with no need of tip path plane control).

In general, the pitch-flap coupling in tail rotors aims the reduction of the flap amplitude in high forward speed and in maneuvers.

Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1052

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The influence of the positive pitch-flap coupling (negative δ3) on the flapping response of an articulated tail rotor

José Márcio Pereira Figueira, Ronaldo Vieira Cruz, Donizeti de Andrade

2

Fig. 2 – Typical Implementation Design of Tail Rotors Pitch-flap

Coupling [2].

According to Cruz [4], the conventional tail rotor may be “pusher” or “tractor”, depending on the direction of the thrust vector. If the air is blown in the opposite direction respectively to the vertical tail, it is a "pusher" tail rotor; otherwise, it is a “tractor” tail rotor.

Most of the tail rotor is of the “pusher” type because since the induced velocity grows downstream, the drag on the vertical tail is smaller in this condition and, consequently, a significant performance gain is expected. Figure 3 shows a picture of a tail rotor “pusher” type.

Despite this gain, the “pusher” tail rotor has the disadvantage of requiring special attention of the designer to the clearance between itself and the tail cone, since the blades coning is in the way of approaching the tail structure.

Fig. 3 – Schematic Diagram of a “Pusher” Tail Rotor.

3. MATHEMATICAL MODEL Due to the inherent characteristics of the tail rotor

previously presented, the dynamics for the tail rotor blade considered herein involves a one degree-of-freedom model, with an out-of-the-plane (flapping) motion.

Figure 4 presents a schematic drawing with all forces acting on an articulated tail rotor blade element during a flapping motion in hovering flight.

Fig. 4 – Schematic Representation of all Forces Involved During a

Flapping Motion of an Articulated Rotor Blade [2].

In Figure 4 dFC is the centrifugal force in the blade element, corresponding 2

dm; is the tail rotor rotation speed; dm is the mass of the blade element; dFm is the weight of the blade element, corresponding gdm; g is the acceleration due to

gravity; dFa is the blade element aerodynamic force (lift), whose expression depends on flight condition (hover or forward flight); e is the flap hinge offset; and is the blade flapping angle.

For simplicity the following assumptions are applied to

the model: - blade is considered rigid; - tail rotor downwash is constant along the blade span; - model with one degree-of-freedom (flapping); - ideal built-in-twist distribution (angle-of-attack constant along the blade span); - symmetric and constant airfoil along the blade span; - blade weight considered negligible compared to the other forces involved, and - the representative blade element is at 0,75 R, where R is the blade radius.

From Newton's second law,

β.)( BB I=M (1)

where, M(B) is the moment about B; and IB is the blade moment of inertia about B point.

Then, by using the Equation (1), the motion equation is:

cBaBB FMFMI )()(. +=β. (2)

The blade moment of inertia about B, BI , is:

3

2

0

2

0

2 Lmdxx

L

mdmxI p

LP

L

B === , (3)

where, L is the blade length; and mP is the blade mass, that has constant distribution along blade span.

The centrifugal force moment about B, M(B)FC, is:

Ω−−≅−−≅R

e

pR

eccB dr

L

mrerdFtgerFM ...).()()( 2

)( ββ , (4)

where,

R ≈ e+L.

By defining e’= e/L, then:

′+Ω−≅ eLm

FMp

cB2

31...

3

.2

2

)( β . (5)

The aerodynamic forces moment is dependent of the

flight condition, so it is necessary to decouple it in hovering and forward flight conditions.

3.1. Hovering Flight Figure 5 presents Forces and Velocities at the 0,75R

blade element in hover.

Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1053

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Fig. 5 – Blade Element in Hovering Flight [2].

The aerodynamic moment about B is:

−≅−=−≅R

ea

R

ea

R

evaB dFerdFerdFerFM )(cos)()()( β , (6)

where,

drr

erCrcdF iLa

Ω

−−−Ω≅

βφθρ α

)(

2

10

22 (7)

r

Vii

Ω=

00φ e (8)

S

FVi

ρ20 = (9)

where, CL is the lift slope coefficient of the airfoil lift curve, CL=f(); is the blade pitch angle; Vi0 is the induced velocity in hovering flight; F is the tail rotor lift force in hovering; S is the tail rotor disc area; and is the air density in the flight level.

Developing Equation (6) and considering S = c L, after some cancellations of e

2 terms and applying the model

assumptions:

+−

+

Ω

−ΩΩ≅3

41

3

81

3

4

8

1 03

)(

ee

R

VLSCFM i

LaB βθρ α . (10)

This yields to the model differential equation:

.3

4

3

81

8

2

31

33

41

83

023

223

2

Ω

=

+Ω+

+Ω+

R

VeLCS

eLm

eLCS

Lm

i

Lp

pLpp

θρ

ββρ

β

α

α (11)

The presence of a negative δ3 significantly modifies the equation of flapping, Equation (11), through the variation of the blade element pitch angle, . In mechanical terms, it is possible to obtain the following relationship:

βδθ ∆−=∆ ).( 3tg (12)

where,

Ktg =3δ is the pitch-flap coupling gain.

The negative sign in front of the gain K indicates,

physically, that flapping up decreases the blade pitch and hence the blade angle-of-attack.

This introduces the additional term "-K." in the expression of the blade element pitch angle, and consequently, the equation that calculates the aerodynamic forces moment by inserting an additional spring term in Equation (11). With this, the final equation for the tail rotor blade flap motion in hovering flight is:

Ω−

=

′+Ω+

+Ω+

+

+Ω+

R

VeLCS

eLCSKeL

m

eLSC

Lm

iLp

Lp

Lp

3

4

3

81

8

3

81..

82

31

3

3

41

83

023

322

2

32

θρ

βρ

βρ

β

α

α

α

(13)

3.2. Forward Flight

In forward flight the problem becomes more complicated, since the velocity components on the blade-element are periodic, i.e., vary with the azimuth position of the blade, :

(14) (15)

where, V0 is the helicopter velocity with respect to the air; UT is the air velocity of blade section, tangent to the tip path plane; and UP is the air velocity of blade section, perpendicular to the tip path plane.

By means of Equations (14) and (15) and Figure 5, one

can conclude that the section inflow angle, φi, which in hovering flight is constant in one blade complete revolution, in forward starts to vary with the azimuth angle of the blade.

So, the aerodynamic forces moment in forward flight about B, M(B)Fa, is:

Ω+Ω

Ω+−+−Ω+Ω−=

R

e

iLaB dr

tsenVr

tVerVtsenVrercCFM

.

cos.)())((

2

1

0

02

0)(

ββθρ α

(16)

By integrating Equation (15) and applying Newton's second law along with the other components currently obtained previously, the linear differential equation is determined with periodic coefficients:

(17)

where,

iP

T

VVerU

VrU

++−=

+Ω=

βψβ

ψ

)cos()(

sin

0

0

0])([ . =−++ ββγβ βθβ MMKM

Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1054

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The influence of the positive pitch-flap coupling (negative δ3) on the flapping response of an articulated tail rotor

José Márcio Pereira Figueira, Ronaldo Vieira Cruz, Donizeti de Andrade

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is the Lock Number, i.e., pl mcRC /3 αργ = and

K is the pitch-flap coupling gain.

Values of M, βM and M are presented by Hiatt [5]

depending on the azimuth angle of the blade during a revolution, , and the advance ratio of the aircraft, (=V0/R). Table 1 shows the values of these parameters.

Table 1: Aerodynamics coefficients according blade regions in one revolution.

Each region expressed in Table 1 represents, physically,

the following: - Region 1: the air flows from the leading edge towards the trailing edge; - Region 2: reverse flow in part of the blade span, and - Region 3: reverse flow across the blade span (only for > 1 flight regime).

For simplicity, it should be noted that this model does

not consider the blade dynamic stall which is a non-linear phenomenon.

3.3. Floquet Theory The system stability described by Equation 17 is

characterized by its eigenvalues. However, in forward flight, the value of the eigenvalues is changed according to the blade azimuth position, due to the periodic coefficients, as shown in Table 1.

Thus, the Floquet theory is used as an alternative to check the stability by extracting eigenvalues that are able to provide information about the system stability.

Assuming the solution of Equation 17 as )0()()( xttx Φ= , by the periodicity of the state it can be proved that:

CtTt )()( Φ=+Φ (18)

)()( tpetBt=Φ (19)

CT

B ln1

= (20)

where, T is the period of one complete blade revolution; p(t) is the periodic matrix; and C is the state transition matrix.

Equation (17) is numerically integrated in a blade revolution (0 to 2) using the Runge-Kutta fourth order method.

After the completion of the integration, the matrix of space state obtained corresponds to the state transition matrix, C, which contains all the information about the blade during a revolution, and, hence, their eigenvalues suggest the system stability (natural frequency and damping).

This method is only applicable to the problem of stability. It provides properties of the solution but no

solution of Equation (17) – which would be the response to the problem.

4. RESULTS AND ANALYSES Computer simulations of the mathematical model

expressed in Section 3 are made, by using geometrical parameters of the AS355 Twin-Engine Squirrel tail rotor helicopter. The following parameters are used: R = 0,94 m; zero flap hinge offset; mp=2,5 kg; c = 0,2 m; CL = 5,7 / rad (NACA 0012 airfoil); e = 2.088 rpm (218,65 rad/s). The distance between the tail cone and the tail rotor hub is 43 cm.

Based on values presented by Prouty [6] for the aircraft of the same size as AS355 Twin-Engine Squirrel---the A109 in this case---, the range for the tail rotor pitch angle actionis: minimum of - 6° and maximum of +20°.

The tail rotor lift is determined considering that all roll angle of the aircraft in hovering is to compensate the lateral drift created by the tail rotor lift. Thus, for an aircraft weight of 2,200 kgf, the roll angle of the aircraft is +3 °, resulting in F = 1150 N.

It is important to emphasize that the Twin-Engine Squirrel tail rotor is of “pusher”, “see-saw” type with a

wash-out built-in-twist torsion and δ3 = +38°, as shown in Figure 6. But the simulated model is an articulated “pusher” tail rotor, then that is the most critical configuration for the clearance of the blades in relation to the tail cone, with ideal built-in-twist and variable pitch-flap coupling.

Fig. 6 – Positive δδδδ3 of the See-saw of the AS 355 Twin-Engine Squirrel Tail Rotor.

It is also considered that the tail rotor is operating at sea

level in standard atmosphere, i.e., 0 = 1,225 kg/m3.

4.1. Hovering Flight Equation (13) is rewritten in the state-space model with

two variables: flap angle, , and flap angle angular velocity, .

β .

θβ

β

β

β

+

−−

=

afabad /

0

//

10.

..

.

, (21)

where,

3

2L

ma p= ;

+Ω=3

41

8

3 eLSCb Lα

ρ ;

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′+Ω+

+Ω= eLCSKeL

md Lp3

81..

82

31

3

322

2

αρ;

Ω−

+Ω=S

F

R

eLCSf Lp

ρθ

ρα

23

4

3

81

80

23 ; and

0 is the pitch angle of the equilibrium.

This system is simulated using the Matlab

® software

commands step and bode, considering the values of the parameters presented herein. It’s also considered a unit step input of tail rotor pitch angle, to simulate an input given by the pilot in command.

Since the most critical case of proximity to the tail cone of an helicopter tail rotor “pusher” type occurs when the pitch of the representative blade airfoil (3R/4) is at its maximum positive value, the simulations are performed considering = + 20°.

Figures 7 to 11 present the simulations results considering different values of K gain for the articulated tail blade modeled.

Fig. 7 – Unit Step Response for 3 = 38°.

Fig. 8 – Flap Angle Bode Diagram for 3 = 38°.

Fig. 9 – Unit Step Response for 3 = 0°.

Fig. 10 – Flap Angle Bode Diagram for 3 = 0°.

Fig. 11 – Unit Step Response for 3 = - 38°.

Table 2 presents an excerpt of results obtained in the

time and frequency domain for each pitch-flap coupling type proposed. Note that the tail rotor model rotation speed is 218.65 rad/s, which corresponds to the aerodynamic loads frequency on the tail blade (1/rev).

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The influence of the positive pitch-flap coupling (negative δ3) on the flapping response of an articulated tail rotor

José Márcio Pereira Figueira, Ronaldo Vieira Cruz, Donizeti de Andrade

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Fig. 12 – Flap Angle Bode Diagram for 3 = - 38°.

Table 2 –Characteristics Response for Different Pitch-flap Coupling.

δδδδ3 (°) wn

(rad/s) Magnitude

1/rev (Abs) Phase

1/rev (°)

38 233,93 0,064 0,210 -52

0 218,65 0,058 0,264 -90

- 38 202,23 0,073 0,209 -128

Contrary to what one might think from an initial

analysis, the negative δ3 pitch-flap coupling is a stable configuration in hovering flight.

Predictably, the influence of the pitch-flap coupling is in the stiffness term, which directly affects the natural frequency of motion. Observing the "spring” term of Equation (13) it’s possible to realize that negative values of K reduce the stiffness term and hence wn, which is evidenced by the simulation results.

The transient responses are very similar to each other, with low damping, , which required the same number of cycles to stabilize the flap angle in a new position.

With the maximum pitch angle, +20°, a unit step input does not compromise the integrity of the tail cone, even in a model with no physical stops. This is confirmed by the fact that the maximum overshoot reaches less than 43 cm.

It is noteworthy that even with the maximum takeoff weight, a stabilized hover flight does not require the maximum pitch angle value since civil aviation regulations require that all aircraft must have enough pedal power in a wind envelope up to 17 kt in all headings. Values of maximum pitch angle could be just usual in terms of turning on spot maneuvers, with tail to the left, considering Twin-Engine Squirrel aircraft with the maximum takeoff weight.

For flap motion without pitch-flap coupling, the dynamic system is resonant for 1/rev, operating in the natural frequency of the system, with an amplitude peak response and 90° lag. From this analysis it’s possible to affirm that the pitch-flap coupling, either positive or negative, causes the system to be no longer resonant, reducing the amplitude and altering the response phase.

As seen in Table 2, the amplitude reduction is similar to positive and negative pitch-flap coupling. Thus, if the goal is to reduce the flap amplitude response after an excitement 1/rev, the effect of two types of pitch-flap coupling is practically the same.

For the simulated model, the pitch-flap coupling type changes the blade flap angle response phase, and the maximum response corresponds to values of azimuth, in the case of a negative K, greater than 90°. This result is similar

to that obtained in a mass-spring-damper system with one degree-of-freedom and forced to oscillate at a different frequency from wn.

For the case of the main rotor, it is noteworthy that for both types of pitch-flap coupling analyzed, the system is no longer resonant, thereby causing a lateral component of force, if applied to a cyclic variation of pitch. Thus, for a pure longitudinal control, the pitch angle variation due to step command should be held "x" degrees before the longitudinal axis, where "x" corresponds to the phase of the blade flap response.

4.2. Forward Flight The solution of a linear differential equation with

periodic coefficients, as expressed in Equation (17), is quite complex, requiring the inclusion of several assumptions and simplifications.

Since the purpose of this study is to examine the stability instead of calculating the flap angle response, it’s used the Floquet method, as presented in Section 3.

Given the articulated tail rotor blade parameters values, as previously presented, it’s possible to get the results expressed in Figures 13 to15.

Fig. 13 –Floquet Analysis for 3 = 38°.

Fig. 14 – Floquet Analysis for 3=0°.

Figure 13 confirms that an articulated blade with

positive K, does not present instability in flap up to advance ratio of 1. Moreover, modern conventional helicopters, reach values of advance ratio of 0,48 at most (as for the Super Lynx 300 helicopter record), a fact that further

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confirms the dynamic stability in flapping motion of the analyzed model.

Fig. 15 – Floquet Analysis for 3 = - 38°.

It is observed that for the case of zero pitch-flap coupling, the model would lead to a slightly damped system in flapping with self-maintained oscillation.

However, Figure 15 shows that for forward flight conditions immediately above the hover throughout the flight envelope, the model would be considered unstable in flapping, in the event of a negative K (in this case, one of the roots has positive real part in the whole range of measured---modal damping---).

At first, as shown in Section 4.1, there isn’t flapping instability in hovering flight.

Besides, Gaffey [3] analytically proves that this reduction in stability may occur in the specific case of articulated tail rotor blades which can act independently in flapping. This is the case of the simulated model.

However, according to Gaffey [3] a “see-saw” tail rotor does not present this flapping instability in negative K, considering the usual values of 3. Mathematically, the damping of a configuration such as a “see-saw” tail rotor is constant with the azimuth position, while an articulated blade is a function of and .

Physically, an articulated rotor blade with negative K pitch-flap coupling is unstable thanks to the fact that there is an increase in blade pitch angle when the blade flaps up, in the presence of variable damping (function of and ). On the other hand, in a “see-saw” rotor, the flapping moments of the two blades cancel each other, by physical constraints of the set, when the blades move out of the plane. Figure 16 presents a schematic diagram of a “seesaw” and an articulated tail rotor blades flapping motion.

It is noteworthy that the tail rotor of Bell M407 helicopter is a “see-saw” type, which justifies the absence of flapping instability in forward flight. Even Gaffey [3], proved by flight tests means on a Bell 204B aircraft, that the implementation of a negative pitch-flap coupling in main and tail “see-saw” rotors is stable, with responses similar to the positive K in terms of flying qualities.

Fig. 16 – Effect of Negative δδδδ3 Pitch-flap Coupling on a See-saw Tail

Rotor and on a Articulated Tail Rotor Blade [3].

Thus, these results guarantee that the implementation of a negative K only in “see-saw” tail rotors as the Bell M407 tail rotor.

5. CONCLUSIONS In hovering flight, a negative pitch-flap coupling

implies in a stable flapping motion regardless the type of tail rotor (“see-saw” or articulated).

The transient responses for negative and positive pitch-flap coupling are quite similar, with low damping, , which requires the same number of cycles to stabilize a new position of flap angle.

Even at maximum pitch, +20°, a unit step input does not compromise the tail cone integrity for the simulated model in hovering flight.

The decreasing of the amplitude in the 1/rev frequency of excitation is similar to the positive and negative pitch-flap coupling. However, the blade flapping phase response is changed and the maximum response corresponds to values of azimuth, in the case of a negative K, greater than 90°.

For forward flight conditions immediately above the hover, throughout the whole flight envelope, an articulated tail rotor would be unstable in flapping in the event of a positive pitch-flap coupling.

However, as proved in the literature, a “see-saw” rotor does not present flapping instability for positive pitch-flap coupling in forward flight, considering the usual negative 3.

These findings justify the existence of negative 3 pitch-flap coupling, in the commercial certified helicopters, only on “see-saw” tail rotors as of Bell M407.

It is recommended, for future studies, to perform the same analysis for a “hingeless” tail rotor blade.

Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1058

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The influence of the positive pitch-flap coupling (negative δ3) on the flapping response of an articulated tail rotor

José Márcio Pereira Figueira, Ronaldo Vieira Cruz, Donizeti de Andrade

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REFERENCES

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Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 1059