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Dynamics of Coupled Flap-Lag Motion by the Helicopter Rotor Blades
Copyright © IFAC Automatic Control in Aerospace, Saint-Petersburg, Russia, 2004
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DYNAMICS OF COUPLED FLAP-LAG MOTION BY THE HELICOPTER ROTOR BLADES S.A. Mikhailov, E.I. Nikolaev, N.A. Shilova Kazan Tupolev State Technical University 10 K. Marx Sir., Kazan 420111 Russia nata_ [email protected]
Abstract: Small oscillations of isolated stiff blade in the rotation and flap planes about the steady-state motion of the rotor blade in horizontal flight and boundaries of instability of these oscillations, the so-called chord flutter, are considered. The method of determination of the main rotor chord flutter critical rate in forward flight is presented. Blade torsion is taken into account by means of a flapping compensator kinematic operation. Copyright © 2004 IFAC Keywords: periodic motion, dynamics, stability analysis, differential equations, numerical methods.
I. INTRODUCTION
Resultant
For newly designed stiff main rotor helicopters with blades on the elastic element the problem of the stability of coupled blade flap and leadllag motion is urgent. Blade flapping or out-of-plane motion and leadllag or in-plane motion are coupled because of Coriolis and aerodynamic forces. These coupled oscillations can become unstable and can lead to undesired consequences at some regimes of a helicopter flight.
at
the
blade
U= ~Ui + U;.
The
q> = arctg(Up / UT)
= arctg(Up / UT) ,
relative
element
inflow
angle
is is
and angle of attack of the blade section is a = e - cp. The blade pitch angle taking into account flap compensator is equal to:
where eo, eh e2 values are determined from helicopter trim calculation.
2. EQUATIONS OF COUPLED FLAP-LAG BLADE MOTION
Aerodynamic lift L = pU 2 ccJ2 is perpendicular and resistance force D = pU 2 ccd /2 is parallel to resultant speed U. Lift and resistance force components in the rotation-plane-fIxed coordinate system are defIned as:
Using the rotation-plane-fIxed coordinate system, consider airspeeds of the stream flowing around the blade section and aerodynamic forces acting on the blade section (Johnson, 1983b). Relative airspeed is decomposed into 3 components. Tangential speed UT is parallel to rotation plane and is in the same direction as resistance force, vertical speed Up is normal to rotation plane and directed downward, radial speed UR is directed towards blade tip:
F;=Lcoscp-Dsincp=pc(cPT -cdU p)U/2; Fx' = Lsincp+Dcosq> = pc(cPp +CdUT)U
/2.
While deriving equations of coupled flap-Iag blade motion the integrals of these linear forces taken along blade radius will be used.
UT =r +lJ.sin(rot)-r~/ro-r:,uR Up = A. + r~ / ro + ~IlCOs(rot) ; u R = IlCOS(rot),
speed
Since small speeds are characteristic of low rotor loading, the following assumption about angle smallness can be made. Suppose that values e, cp, Cd/Cl are small compared to unity. Hence, it
(1)
where relative non-dimensional speeds up = Up / roR, UR = U R / roR, UT = UT / roR are used.
565
follows that
et
hinge axis. The following linear forces are acting on the blade element (Johnson, 1983a):
and Up/UT are also small that is
U ':::J UT and cp ':::J Up/UT = Up/UT' Supposing the angle smallness, lift-curve slope is usually considered constant and stall is neglected. Then lift coefficient is Cl =c;a=C;(S-Up/UT ) which gives:
F.'
I. Inertia force
(r - el;) about the lag binge; 2. Centrifugal force mro 2r, milial directed, acting at a distance e~x = ~ TJ~~ from the lag hinge;
= pC;cU; (S-Up /UT )/2
F.' = pc;c(UpUTS-U; +cd / c; Ui)/2.
3. Aerodynamic force Fx' acting at a distance (r - ei;) from the lag hinge; 4. Because of a spring in flap hinge the moment Ki;C, is acting on the blade element; 5. The Coriolis force 2roiz'm=2ro~~TJpTJ;m directed as inertial force, acting at a distance (r - el;) from the lag hinge.
(3)
By using formulas (1) and (3), the following expressions are obtained:
Since z = TJIlP then its derivatives are defmed as: dz r-ep . dz R Z = - = - - R /3; z' = - = - - ~ . Integmting dt R-ep dr R-ep linear torques along the blade mdius gives the torque equilibrium condition about lag hinge axis:
Consider the equilibrium of moments about the equivalent flap hinge and equivalent leadllag binge axes. The following linear forces genemting the flap moments are acting on the blade element: I . Inertia force
mz = mTJp ~
(j~, (r-e,)mm- )~ +Oe,~, mm-)"",+K" + C,~
acting at a distance
+O~. ~~~ Rmm-}"'PJi~ l(r-e,)F;m-
(r - ell) about the flap hinge; 2. Centrifugal force mro 2r acting at a distance Z = TJIlP from the flap binge; 3. Aerodynamic force F.' acting at a distance (r- ell) from the flap binge; 4. Because of a spring in flap binge the moment KIl(P - Pconstr) is acting on the blade element; 5. Coriolis inertia force 2roxm = 2ro ~ TJ~ m acting at a
the lag motion equation, where q is derivative of the damper torque with respect to angular mte of blade rotation about the lag binge. To simplify form of expressions the dash is cut out while writing further non-dimensional values. Having made some transformations of equation (6), the equation of moments about flap hinge axis can be written as:
Turn to non-dimensional values in equilibrium equations: r=r/R; ~=ep/R; ~=edR;
;:; = ~= r -~ . 'I~
R
I_~
Integmting
linear moments along the blade mdius gives the following moment equilibrium condition about flap hinge axis: R
By transforming equation (7), the equation of moments about lag binge axis is written as follows:
R
fTJp (r-ep)m~dr + fTJprm~ro2dr+ Kp~-
..,
..,
R
R
..,
..,
-Kp~"""'tr - f2roTJ~ ~mTJp~dr = J(r-ep)F; dr, (6) where
mode
shapes
(7)
Hub of the binge rotor has the mechanical damper in the lag bing€;. Therefore, a term C~ ~ is added into
distance z = TJIl~ from the flap hinge, directed opposite to accelemtion (inward, along the mdius) (Johnson, 1983a).
;:; = TJp = r -~ . 'Ip R I-~'
mx = mTJ ~ ~ acting at a distance
where
TJp = (r-ep )/(I-~);
TJ~ =(r-e~)/(I-~) are blade modes as rigid body oscillations about flap and lag hinges respectively. Consider the equilibrium of moments about the lag
566
differential equations with periodic coefficients. There are no universal and rigorous practical methods of stability analysis of such equations. It can be shown that the expressions of the following type
.
p. = b; + ~)b: cos(krot)+b: sin(koot); 1=1
~. = z~ + ~::Cz: cos(koot) + z: sin(koot)
3. LINEARIZATION OF THE EQUATIONS
(12)
1=1
It is assumed that lag hinge and flap binge axes coincide with the main rotor shaft, that is ell = e~ = O. In this case fundamental modes are 1111 = 11~ = r. By applying the above assumptions and expressions (1), (4), (5) to equations (8) and (9), the following blade oscillation equations are obtained:
form the partial solution of the system (10), (11) and defme undisturbed blade motion (Mill, et al., 1966). The linearization is based on the suggestion that variables 13 and ~ during the dynamic process change so that their displacements from steady-state values p. and ~. remain sufficiently small during the whole process. Introduce the perturbed motion 13 = f3" + .1.13, ~ =~. +.1.~ into the equations of motion (10), (11). Here .1.13 and .1.~ are perturbation angles, that is angles of displacement from steady-state blade motion. Assume that condition of sufficient smallness is satisfied for perturbation angles of the process under review. Retaining the perturbation terms and discarding the terms of high order smallness, that are products and powers of perturbation angles .1.13, .1.~ and their derivatives, give the linear perturbation equations about the steadystate motion of the blade.
2 P2 +(v; + YIl cos(rot)+ YIl Sin(2rot») 13 + ro 6 8 2 1 Il. ) y---cos ~ YIl 2 + ( -+-sm(rot) (rot)P~8 6 ro 4 -(2+ YIl COS(rot»)13 ~ _ (ty1l2 6 ro 4
COS2(rot)~2-
_I ~~2 _ YIl cos(rot) ~ ~ _ (ty £+ YA + 2 8 ro
ro
6
8 ro
6
YAIl + 4 sinerot)
+ [91l sin(2rot)/4+ ecos(rot)/3 - Acos(rot)/4]y1l~ + ell . e A) y---cos(rot)~-~ {tyll ~ + ( -sm(rot)+--3 4 6 ro 3 ro -{ty1l2 sin 2(rot)/4 - {tyllsin(rot)/3 - (ty/8 + (10)
P
- 2+ ro
[1~· 8 00
YIlcos(oot»)~· 2 cos(200t) + 1 r. 2 --YIl .., +v + 6 ro 8 ~
[( - 2
2 +[V2 +5L YIl cos(rot)+5L YIl sin(2rot)+ ~ ca 3 ca 4 y y
+
p.
_ YA + (tyllsin(oot) + (ty]i+[ cos(oot) 6 3 4 00 6 00
+(~_~_ ell Sin(rot»)Y ~ + Yllecos(rot) 13 ~ + ro
6
2 YIlcos(oot) + YIl Sin(2rot)] ,..,+ A [ y~. ---+ 6 8 800
+(-2- YIlCOS(rot»)p. _ {ty ~. _ (tyIlCOS(oot) (_ 6 4 00 3
{tyJJA. ( rot ») ..,+--+ r Y ~2 We cos( rot) ~ ..,+ r +--oos 4 8 ro2 ro 6 386
ro
lloos(200t)+ Il p. _ 800s(oot) ~. 8300 -ell( cos(2rot) + 1) (/4 - Acos(oot)/4+
+[A -9/3 -ellsin(rot)/2] YIl cos(rot)p+ 2
2 C YIl + YIl cos 2(rot)p2 -~-COS2(rot)~2 + 4 4 2
c;
.
+ellsin(2rot)/4+ecos(rot)/3]YIl~
.2
ro2
_5L YIl cos(rot)~ ~ + {ty ~~ + {ty1l2 oos2 (rot)p~cay 3 ro 8 ro2 4 2 2 . 2 (rot)Cd Y Cd YIl . YA Cd YIl ------sm(rot)+----sm 8 3 4 C; 4
(13)
8 ro
6
cay 4 00
+ (tyA] ~ + [8ooS(rot) + 8Ilcos(2rot)+8 Il ._ p 6ro 6 ro 8
p.
C;
-{tyA/6 - (tyJJA.sin(oot)/4 = O.
= 0;
~+[{ty p. + {tyllcos(oot p. _5L1 ~. _
YIl 13 Cd Y ~ +(2+-COS(rot) ) 13-----3 ro cay 8 ro2
C;
1
IlCOS(rot) ..,r. +-+ Il Sin(oot)] Y-+ p 6 8 6 00
(11)
Cd
Cay
As it 1S shown, these equations are non-linear
567
oos(oot);-. Cd Ilcos(200t)+ll r • 2 .., .., +v + 300 cay 4 I;
+
Cd
COS(oot)
Ca
3
y
+
Cd
J.lsin(2oot)
ca
4
y
+
8Acos(oot)] 4
· l'ler e )...1 ~ O'IS zo' zepz,p "" z"",z,n' C ommon mu1bp discarded. 3. The value of n' - the number of harmonic components taken into account in steady-state motion is assigned. By using expressions (12) and with • •• • b' regard for values bo' bel' b,p ... , b"", m ,
r
yJ.l~+
y~' (2 + YJ.l COS(oot») 13' +--+ Sy ~. + --+ 400 3 800
[
+ SyJ.lcos(oot) ~. + yf... _ Sy _ SyJ.lSin(oot)J ~ + 6 3 8 6 00
Z~, z;p z;p ...,
trim
+ [( 2+ YJ.lCOS(oot»)~· -+YJ.l 2 (COS(2oot) + 1)13' + 3004
+ YJ.lf... cos(oot)/2 - SyJ.l2 sin(2oot)/8 (14)
Here and thereinafter, perturbations about steadystate blade motion are equation variables. Symbol f1 is omitted to simplify expression form. 4. THE STABILITY ANALYSIS For determination of chord flutter critical angular rate it is necessary to analyze equations (13), (14) for stability of trivial solution, that is to show that ~ -+ 0, ~ -+ 0 at t -+ +00. The solution of the system of linear differential equations with periodic coefficients can be written down in the form (Mill, et al., 1966):
expressions
for
W, ~', 13',
Here A2, AI, Aa are matrices of order m X m, which elements depend on main rotor angular rate 0>, helicopter' s flight parameters and blade constructive parameters, and K is a column of coefficients, KT = [bo bel ·b.I .. h en bm Zo Zel Z.I" 'Z"" Zm] ~ 0 .
13 = e1..·1 (bo + ~)b,* cos(koot) + bSk Sin(koot»);
7. Roots of equation det(Azf...; +AIf...R +Ao)=O are determined. For the linearized system under study (13), (14) characteristic number AR defmes areas of stability and instability (Mikhailov, et al., 2003), that is the chord flatter boundaries. 8. The quantity r=maxRe(f... Rk ), (k=I,2xm) is a
k ~1
~ = e)..·1 (zo + :t (z,* cos(koot) + Zsk Sin(koot»)
calculation,
~', ~', ~', deflning steady-state blade motion are written down. The latter expressions are substituted into the system of algebraic equations. Standard trigonometric transformations are applied to the system coefficients. 4. Coefficients at the similar harmonic components and at the terms free from harmonic are grouped together for each of the system equations and then are equated to zero. 5. Equations for free terms and junior n harmonics are selected from obtained ones and their coefficients are grouped according to powers of characteristic numberf...R. 6. Obtained equations are written in the matrix form
+ SyJ.lcos(oot)~· +SyJ.l2 (COS(2oot)+I)~. + 6 00 8
SyJ.lC~S(oot) Jf3 = O.
z:, z: , determined from helicopter
(15)
k =1
Here n is constant number equal to the number of harmonic components retained in the linearized equation solution. It is obvious that trivial solution will be stable only at Re(f...R) < O. And the case is considered when coefficient vector is not zero.
k
criterion function of the algorithm: r < 0 means stability; r> 0 means instability. 5. CRITICAL RATES OF MAIN ROTOR IN THE ANSAT FORWARD FLIGHT
General algorithm for stability analysis of blade flap-Iag motion in forward flight, that is deflnition of chord flutter conditions, is presented.
The above method for defIning stability of disturbed blade flap-Iag motion in forward flight was assumed as a basis for calculation of chord flatter critical angular rate for well-known Russian helicopter Ansat. To carry it out and make numerical calculations Maple 6 and MATLAB software systems were used. The results of calculations made for different values of flight speed and lag hinge spring rate Kr, are presented in Figs. 1-4.
1. At given blade construction and flight parameters the linearization of system of non-linear differential equations (10), (11) is carried out. As a result, the linear differential equations (13), (14) are obtained presenting the system of perturbation equations of the blade in the forward flight. 2. The value of n - the number of harmonic components retained in the linearized equation solution is assigned. Expressions (15) and their derivatives are substituted into the obtained linear system and linear algebraic system is formed concerning m = 4n + 2 expansion coefficients of blade oscillation angles bo, bepb,p "" ben ' bm '
Nondimensional advance ratio J.l is a measure of forward flight speed and its values from 0 to 0.4 cover the entire speed range for the Ansat helicopter. Nominal operating rotor angular rate is denoted by 000 , For the Ansat helicopter 000 , = 38.26 rad/sec.
568
12.00
]
(a)
8.00
J3
i
4.00
~
.~
0.00
~r Oh
= 0
1 c..
-,
-4.00 -8.00 1.20
]
0.80
Oh
t
0.40
= 0
0.00
~ ic..
-0.40
, I,I,
(b)
ij
.~
-0.80 -1.20 0.25
]
t
Oh ij
0.05
.~
-0.05
1
-0.15
= 0
c..
""
(c) 0.15
"
\I
-0.25 0.15 (d)
]
t
Oh
0.05
ij
= 0
.~
1
-0.05
c..
-0.15 0.1
]
t
Oh ij
0.05 0
= -0.05 0
c..
1\
-0.1
v
I
""
(e)
f'
Y\IV'" Ir{''v,,' '.f'i.. ',/",)(\ __ .-- --- ---i\ ~,/\ v',' I
, ,, I ,, , , ,: \
I, I,
.~
1
A,'~
,
I'
"\I
'\
\.,
'
v~,
--
V
I
-0.15
I
2
0
t,
Fig. 1. Perturbations about the steady-state motion of the rotor blade; (c) Jl = 0.56; (d) Jl = 0.54; (e) Jl = 0.50.
569
3
4
sec 00/000
= 0.6; (a) Jl = 0.60; (b) Jl = 0.58;
Results of numerical modeling of the perturbed blade motion in forward flight are shown in Fig. I. The plot (c) shows the blade perturbations at the boundary of stability. The method gives that for the ratio ro/roo = 0.6 such a point for flight speed is J.1 = 0.56. The plots (a) and (b) demonstrate instable flight regimes when flight speed exceeds the boundary value; the plots (d) and (e) present stable flight regimes under the boundary speed value. Fig. 4 shows critical values of main rotor rate evaluated for different flight speed. Stability region is located under the curve and instability region is located above the curve. The results are presented for full and simplified mathematical models. The second one is obtained after 9 1 and 92 are taken equal to zero in the expression for blade pitch angle 9.
~ ~I
0.0 ~~
'-'\1\
-0.2
I
-
if
~\. /~
.. -0.4
\ \
-0.6
---------
/
_\ I
/
11
0 0.1 -
-e- 0.2 -
j
\
:::.
I
-0.8
0.3 _ 0.35
-
0·1
I
1.2
0.4
0.0
Fig. 2. Effect of an advance ratio on the flap-Iag motion stability; K~ = 4.10.5 Nmlrad.
6. CONCLUSIONS
0.4
The method for determination of the main rotor chord flutter critical rate in helicopter's forward flight was presented. A detailed chord flutter parametric investigation for Russian helicopter Ansat has been developed using the method. Because of lack of space only few numerical results are presented in the paper. The following conclusions are made based upon presented investigation results:
.. -0.4
I . No chord flatter effect exists at operating rates of the Ansat helicopter main rotor over the entire flight speed range. These results are in agreement with experimental'data. 2. Helicopter flight speed has a significant influence on flap-Iag stability in forward flight. Stability margin decreases with increasing advance ratio J.1. 3. Increasing lag binge spring rate Kc, has a strong stabilizing effect on flap-Iag blade motion. 4. It is important to include harmonic components of blade pitch angle in mathematical model; otherwise the model overpredicts the blade stability.
'-+- 2.5
-0.8
1.2
0.4
0.0
Fig. 3. Effect of the lag hinge spring rate on the flap-Iag motion stability; J.1 = 0.4.
t r~
0.7
~'v
0.6
If
0.5
1
- ~ fi' "\ 1\ "'-"'
;
0.4 -
0.3
--
"-.......
\
I'"
\
"
8 contains harmonics 8 1 = 0; 82 = 0 I
I
I
I
I
REFERENCES
I I
0.0
V ~-
",
1.2
Fig. 4. Stability limits; K~ = 4.105 Nmlrad.
570
Johnson, W. (1983a). Helicopter Theory. Book L pp. 364-367. Mir, Moscow. Johnson, W. (1983b). Helicopter Theory. Book II, pp. 510-515. Mir, Moscow. Mikhailov, S.A., E.1. Nikolaev, N.A. Shilova and RM. Aleev (2003). Mathematical model and numerical method of flutter design of helicopter main rotor blades. In: Proceedings of the International Conference on Dynamical System Mode/ing and Stability Investigation, pp. 266-267. Kyiv, Republic of Ukraine. Mill, M.L., A.V. Nekrasov, A.S. Braverman, L.N. Grodko and M.A. Laykand (1966). Helicopters. Calculation and Design. Book 1, Aerodynamics, pp. 358-395. Mashinostroyenie, Moscow.
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
DYNAMICS OF COUPLED FLAP-LAG MOTION BY THE HELICOPTER ROTOR BLADES S.A. Mikhailov, E.I. Nikolaev, N.A. Shilova Kazan Tupolev State Technical University 10 K. Marx Sir., Kazan 420111 Russia nata_ [email protected]
Abstract: Small oscillations of isolated stiff blade in the rotation and flap planes about the steady-state motion of the rotor blade in horizontal flight and boundaries of instability of these oscillations, the so-called chord flutter, are considered. The method of determination of the main rotor chord flutter critical rate in forward flight is presented. Blade torsion is taken into account by means of a flapping compensator kinematic operation. Copyright © 2004 IFAC Keywords: periodic motion, dynamics, stability analysis, differential equations, numerical methods.
I. INTRODUCTION
Resultant
For newly designed stiff main rotor helicopters with blades on the elastic element the problem of the stability of coupled blade flap and leadllag motion is urgent. Blade flapping or out-of-plane motion and leadllag or in-plane motion are coupled because of Coriolis and aerodynamic forces. These coupled oscillations can become unstable and can lead to undesired consequences at some regimes of a helicopter flight.
at
the
blade
U= ~Ui + U;.
The
q> = arctg(Up / UT)
= arctg(Up / UT) ,
relative
element
inflow
angle
is is
and angle of attack of the blade section is a = e - cp. The blade pitch angle taking into account flap compensator is equal to:
where eo, eh e2 values are determined from helicopter trim calculation.
2. EQUATIONS OF COUPLED FLAP-LAG BLADE MOTION
Aerodynamic lift L = pU 2 ccJ2 is perpendicular and resistance force D = pU 2 ccd /2 is parallel to resultant speed U. Lift and resistance force components in the rotation-plane-fIxed coordinate system are defIned as:
Using the rotation-plane-fIxed coordinate system, consider airspeeds of the stream flowing around the blade section and aerodynamic forces acting on the blade section (Johnson, 1983b). Relative airspeed is decomposed into 3 components. Tangential speed UT is parallel to rotation plane and is in the same direction as resistance force, vertical speed Up is normal to rotation plane and directed downward, radial speed UR is directed towards blade tip:
F;=Lcoscp-Dsincp=pc(cPT -cdU p)U/2; Fx' = Lsincp+Dcosq> = pc(cPp +CdUT)U
/2.
While deriving equations of coupled flap-Iag blade motion the integrals of these linear forces taken along blade radius will be used.
UT =r +lJ.sin(rot)-r~/ro-r:,uR Up = A. + r~ / ro + ~IlCOs(rot) ; u R = IlCOS(rot),
speed
Since small speeds are characteristic of low rotor loading, the following assumption about angle smallness can be made. Suppose that values e, cp, Cd/Cl are small compared to unity. Hence, it
(1)
where relative non-dimensional speeds up = Up / roR, UR = U R / roR, UT = UT / roR are used.
565
follows that
et
hinge axis. The following linear forces are acting on the blade element (Johnson, 1983a):
and Up/UT are also small that is
U ':::J UT and cp ':::J Up/UT = Up/UT' Supposing the angle smallness, lift-curve slope is usually considered constant and stall is neglected. Then lift coefficient is Cl =c;a=C;(S-Up/UT ) which gives:
F.'
I. Inertia force
(r - el;) about the lag binge; 2. Centrifugal force mro 2r, milial directed, acting at a distance e~x = ~ TJ~~ from the lag hinge;
= pC;cU; (S-Up /UT )/2
F.' = pc;c(UpUTS-U; +cd / c; Ui)/2.
3. Aerodynamic force Fx' acting at a distance (r - ei;) from the lag hinge; 4. Because of a spring in flap hinge the moment Ki;C, is acting on the blade element; 5. The Coriolis force 2roiz'm=2ro~~TJpTJ;m directed as inertial force, acting at a distance (r - el;) from the lag hinge.
(3)
By using formulas (1) and (3), the following expressions are obtained:
Since z = TJIlP then its derivatives are defmed as: dz r-ep . dz R Z = - = - - R /3; z' = - = - - ~ . Integmting dt R-ep dr R-ep linear torques along the blade mdius gives the torque equilibrium condition about lag hinge axis:
Consider the equilibrium of moments about the equivalent flap hinge and equivalent leadllag binge axes. The following linear forces genemting the flap moments are acting on the blade element: I . Inertia force
mz = mTJp ~
(j~, (r-e,)mm- )~ +Oe,~, mm-)"",+K" + C,~
acting at a distance
+O~. ~~~ Rmm-}"'PJi~ l(r-e,)F;m-
(r - ell) about the flap hinge; 2. Centrifugal force mro 2r acting at a distance Z = TJIlP from the flap binge; 3. Aerodynamic force F.' acting at a distance (r- ell) from the flap binge; 4. Because of a spring in flap binge the moment KIl(P - Pconstr) is acting on the blade element; 5. Coriolis inertia force 2roxm = 2ro ~ TJ~ m acting at a
the lag motion equation, where q is derivative of the damper torque with respect to angular mte of blade rotation about the lag binge. To simplify form of expressions the dash is cut out while writing further non-dimensional values. Having made some transformations of equation (6), the equation of moments about flap hinge axis can be written as:
Turn to non-dimensional values in equilibrium equations: r=r/R; ~=ep/R; ~=edR;
;:; = ~= r -~ . 'I~
R
I_~
Integmting
linear moments along the blade mdius gives the following moment equilibrium condition about flap hinge axis: R
By transforming equation (7), the equation of moments about lag binge axis is written as follows:
R
fTJp (r-ep)m~dr + fTJprm~ro2dr+ Kp~-
..,
..,
R
R
..,
..,
-Kp~"""'tr - f2roTJ~ ~mTJp~dr = J(r-ep)F; dr, (6) where
mode
shapes
(7)
Hub of the binge rotor has the mechanical damper in the lag bing€;. Therefore, a term C~ ~ is added into
distance z = TJIl~ from the flap hinge, directed opposite to accelemtion (inward, along the mdius) (Johnson, 1983a).
;:; = TJp = r -~ . 'Ip R I-~'
mx = mTJ ~ ~ acting at a distance
where
TJp = (r-ep )/(I-~);
TJ~ =(r-e~)/(I-~) are blade modes as rigid body oscillations about flap and lag hinges respectively. Consider the equilibrium of moments about the lag
566
differential equations with periodic coefficients. There are no universal and rigorous practical methods of stability analysis of such equations. It can be shown that the expressions of the following type
.
p. = b; + ~)b: cos(krot)+b: sin(koot); 1=1
~. = z~ + ~::Cz: cos(koot) + z: sin(koot)
3. LINEARIZATION OF THE EQUATIONS
(12)
1=1
It is assumed that lag hinge and flap binge axes coincide with the main rotor shaft, that is ell = e~ = O. In this case fundamental modes are 1111 = 11~ = r. By applying the above assumptions and expressions (1), (4), (5) to equations (8) and (9), the following blade oscillation equations are obtained:
form the partial solution of the system (10), (11) and defme undisturbed blade motion (Mill, et al., 1966). The linearization is based on the suggestion that variables 13 and ~ during the dynamic process change so that their displacements from steady-state values p. and ~. remain sufficiently small during the whole process. Introduce the perturbed motion 13 = f3" + .1.13, ~ =~. +.1.~ into the equations of motion (10), (11). Here .1.13 and .1.~ are perturbation angles, that is angles of displacement from steady-state blade motion. Assume that condition of sufficient smallness is satisfied for perturbation angles of the process under review. Retaining the perturbation terms and discarding the terms of high order smallness, that are products and powers of perturbation angles .1.13, .1.~ and their derivatives, give the linear perturbation equations about the steadystate motion of the blade.
2 P2 +(v; + YIl cos(rot)+ YIl Sin(2rot») 13 + ro 6 8 2 1 Il. ) y---cos ~ YIl 2 + ( -+-sm(rot) (rot)P~8 6 ro 4 -(2+ YIl COS(rot»)13 ~ _ (ty1l2 6 ro 4
COS2(rot)~2-
_I ~~2 _ YIl cos(rot) ~ ~ _ (ty £+ YA + 2 8 ro
ro
6
8 ro
6
YAIl + 4 sinerot)
+ [91l sin(2rot)/4+ ecos(rot)/3 - Acos(rot)/4]y1l~ + ell . e A) y---cos(rot)~-~ {tyll ~ + ( -sm(rot)+--3 4 6 ro 3 ro -{ty1l2 sin 2(rot)/4 - {tyllsin(rot)/3 - (ty/8 + (10)
P
- 2+ ro
[1~· 8 00
YIlcos(oot»)~· 2 cos(200t) + 1 r. 2 --YIl .., +v + 6 ro 8 ~
[( - 2
2 +[V2 +5L YIl cos(rot)+5L YIl sin(2rot)+ ~ ca 3 ca 4 y y
+
p.
_ YA + (tyllsin(oot) + (ty]i+[ cos(oot) 6 3 4 00 6 00
+(~_~_ ell Sin(rot»)Y ~ + Yllecos(rot) 13 ~ + ro
6
2 YIlcos(oot) + YIl Sin(2rot)] ,..,+ A [ y~. ---+ 6 8 800
+(-2- YIlCOS(rot»)p. _ {ty ~. _ (tyIlCOS(oot) (_ 6 4 00 3
{tyJJA. ( rot ») ..,+--+ r Y ~2 We cos( rot) ~ ..,+ r +--oos 4 8 ro2 ro 6 386
ro
lloos(200t)+ Il p. _ 800s(oot) ~. 8300 -ell( cos(2rot) + 1) (/4 - Acos(oot)/4+
+[A -9/3 -ellsin(rot)/2] YIl cos(rot)p+ 2
2 C YIl + YIl cos 2(rot)p2 -~-COS2(rot)~2 + 4 4 2
c;
.
+ellsin(2rot)/4+ecos(rot)/3]YIl~
.2
ro2
_5L YIl cos(rot)~ ~ + {ty ~~ + {ty1l2 oos2 (rot)p~cay 3 ro 8 ro2 4 2 2 . 2 (rot)Cd Y Cd YIl . YA Cd YIl ------sm(rot)+----sm 8 3 4 C; 4
(13)
8 ro
6
cay 4 00
+ (tyA] ~ + [8ooS(rot) + 8Ilcos(2rot)+8 Il ._ p 6ro 6 ro 8
p.
C;
-{tyA/6 - (tyJJA.sin(oot)/4 = O.
= 0;
~+[{ty p. + {tyllcos(oot p. _5L1 ~. _
YIl 13 Cd Y ~ +(2+-COS(rot) ) 13-----3 ro cay 8 ro2
C;
1
IlCOS(rot) ..,r. +-+ Il Sin(oot)] Y-+ p 6 8 6 00
(11)
Cd
Cay
As it 1S shown, these equations are non-linear
567
oos(oot);-. Cd Ilcos(200t)+ll r • 2 .., .., +v + 300 cay 4 I;
+
Cd
COS(oot)
Ca
3
y
+
Cd
J.lsin(2oot)
ca
4
y
+
8Acos(oot)] 4
· l'ler e )...1 ~ O'IS zo' zepz,p "" z"",z,n' C ommon mu1bp discarded. 3. The value of n' - the number of harmonic components taken into account in steady-state motion is assigned. By using expressions (12) and with • •• • b' regard for values bo' bel' b,p ... , b"", m ,
r
yJ.l~+
y~' (2 + YJ.l COS(oot») 13' +--+ Sy ~. + --+ 400 3 800
[
+ SyJ.lcos(oot) ~. + yf... _ Sy _ SyJ.lSin(oot)J ~ + 6 3 8 6 00
Z~, z;p z;p ...,
trim
+ [( 2+ YJ.lCOS(oot»)~· -+YJ.l 2 (COS(2oot) + 1)13' + 3004
+ YJ.lf... cos(oot)/2 - SyJ.l2 sin(2oot)/8 (14)
Here and thereinafter, perturbations about steadystate blade motion are equation variables. Symbol f1 is omitted to simplify expression form. 4. THE STABILITY ANALYSIS For determination of chord flutter critical angular rate it is necessary to analyze equations (13), (14) for stability of trivial solution, that is to show that ~ -+ 0, ~ -+ 0 at t -+ +00. The solution of the system of linear differential equations with periodic coefficients can be written down in the form (Mill, et al., 1966):
expressions
for
W, ~', 13',
Here A2, AI, Aa are matrices of order m X m, which elements depend on main rotor angular rate 0>, helicopter' s flight parameters and blade constructive parameters, and K is a column of coefficients, KT = [bo bel ·b.I .. h en bm Zo Zel Z.I" 'Z"" Zm] ~ 0 .
13 = e1..·1 (bo + ~)b,* cos(koot) + bSk Sin(koot»);
7. Roots of equation det(Azf...; +AIf...R +Ao)=O are determined. For the linearized system under study (13), (14) characteristic number AR defmes areas of stability and instability (Mikhailov, et al., 2003), that is the chord flatter boundaries. 8. The quantity r=maxRe(f... Rk ), (k=I,2xm) is a
k ~1
~ = e)..·1 (zo + :t (z,* cos(koot) + Zsk Sin(koot»)
calculation,
~', ~', ~', deflning steady-state blade motion are written down. The latter expressions are substituted into the system of algebraic equations. Standard trigonometric transformations are applied to the system coefficients. 4. Coefficients at the similar harmonic components and at the terms free from harmonic are grouped together for each of the system equations and then are equated to zero. 5. Equations for free terms and junior n harmonics are selected from obtained ones and their coefficients are grouped according to powers of characteristic numberf...R. 6. Obtained equations are written in the matrix form
+ SyJ.lcos(oot)~· +SyJ.l2 (COS(2oot)+I)~. + 6 00 8
SyJ.lC~S(oot) Jf3 = O.
z:, z: , determined from helicopter
(15)
k =1
Here n is constant number equal to the number of harmonic components retained in the linearized equation solution. It is obvious that trivial solution will be stable only at Re(f...R) < O. And the case is considered when coefficient vector is not zero.
k
criterion function of the algorithm: r < 0 means stability; r> 0 means instability. 5. CRITICAL RATES OF MAIN ROTOR IN THE ANSAT FORWARD FLIGHT
General algorithm for stability analysis of blade flap-Iag motion in forward flight, that is deflnition of chord flutter conditions, is presented.
The above method for defIning stability of disturbed blade flap-Iag motion in forward flight was assumed as a basis for calculation of chord flatter critical angular rate for well-known Russian helicopter Ansat. To carry it out and make numerical calculations Maple 6 and MATLAB software systems were used. The results of calculations made for different values of flight speed and lag hinge spring rate Kr, are presented in Figs. 1-4.
1. At given blade construction and flight parameters the linearization of system of non-linear differential equations (10), (11) is carried out. As a result, the linear differential equations (13), (14) are obtained presenting the system of perturbation equations of the blade in the forward flight. 2. The value of n - the number of harmonic components retained in the linearized equation solution is assigned. Expressions (15) and their derivatives are substituted into the obtained linear system and linear algebraic system is formed concerning m = 4n + 2 expansion coefficients of blade oscillation angles bo, bepb,p "" ben ' bm '
Nondimensional advance ratio J.l is a measure of forward flight speed and its values from 0 to 0.4 cover the entire speed range for the Ansat helicopter. Nominal operating rotor angular rate is denoted by 000 , For the Ansat helicopter 000 , = 38.26 rad/sec.
568
12.00
]
(a)
8.00
J3
i
4.00
~
.~
0.00
~r Oh
= 0
1 c..
-,
-4.00 -8.00 1.20
]
0.80
Oh
t
0.40
= 0
0.00
~ ic..
-0.40
, I,I,
(b)
ij
.~
-0.80 -1.20 0.25
]
t
Oh ij
0.05
.~
-0.05
1
-0.15
= 0
c..
""
(c) 0.15
"
\I
-0.25 0.15 (d)
]
t
Oh
0.05
ij
= 0
.~
1
-0.05
c..
-0.15 0.1
]
t
Oh ij
0.05 0
= -0.05 0
c..
1\
-0.1
v
I
""
(e)
f'
Y\IV'" Ir{''v,,' '.f'i.. ',/",)(\ __ .-- --- ---i\ ~,/\ v',' I
, ,, I ,, , , ,: \
I, I,
.~
1
A,'~
,
I'
"\I
'\
\.,
'
v~,
--
V
I
-0.15
I
2
0
t,
Fig. 1. Perturbations about the steady-state motion of the rotor blade; (c) Jl = 0.56; (d) Jl = 0.54; (e) Jl = 0.50.
569
3
4
sec 00/000
= 0.6; (a) Jl = 0.60; (b) Jl = 0.58;
Results of numerical modeling of the perturbed blade motion in forward flight are shown in Fig. I. The plot (c) shows the blade perturbations at the boundary of stability. The method gives that for the ratio ro/roo = 0.6 such a point for flight speed is J.1 = 0.56. The plots (a) and (b) demonstrate instable flight regimes when flight speed exceeds the boundary value; the plots (d) and (e) present stable flight regimes under the boundary speed value. Fig. 4 shows critical values of main rotor rate evaluated for different flight speed. Stability region is located under the curve and instability region is located above the curve. The results are presented for full and simplified mathematical models. The second one is obtained after 9 1 and 92 are taken equal to zero in the expression for blade pitch angle 9.
~ ~I
0.0 ~~
'-'\1\
-0.2
I
-
if
~\. /~
.. -0.4
\ \
-0.6
---------
/
_\ I
/
11
0 0.1 -
-e- 0.2 -
j
\
:::.
I
-0.8
0.3 _ 0.35
-
0·1
I
1.2
0.4
0.0
Fig. 2. Effect of an advance ratio on the flap-Iag motion stability; K~ = 4.10.5 Nmlrad.
6. CONCLUSIONS
0.4
The method for determination of the main rotor chord flutter critical rate in helicopter's forward flight was presented. A detailed chord flutter parametric investigation for Russian helicopter Ansat has been developed using the method. Because of lack of space only few numerical results are presented in the paper. The following conclusions are made based upon presented investigation results:
.. -0.4
I . No chord flatter effect exists at operating rates of the Ansat helicopter main rotor over the entire flight speed range. These results are in agreement with experimental'data. 2. Helicopter flight speed has a significant influence on flap-Iag stability in forward flight. Stability margin decreases with increasing advance ratio J.1. 3. Increasing lag binge spring rate Kc, has a strong stabilizing effect on flap-Iag blade motion. 4. It is important to include harmonic components of blade pitch angle in mathematical model; otherwise the model overpredicts the blade stability.
'-+- 2.5
-0.8
1.2
0.4
0.0
Fig. 3. Effect of the lag hinge spring rate on the flap-Iag motion stability; J.1 = 0.4.
t r~
0.7
~'v
0.6
If
0.5
1
- ~ fi' "\ 1\ "'-"'
;
0.4 -
0.3
--
"-.......
\
I'"
\
"
8 contains harmonics 8 1 = 0; 82 = 0 I
I
I
I
I
REFERENCES
I I
0.0
V ~-
",
1.2
Fig. 4. Stability limits; K~ = 4.105 Nmlrad.
570
Johnson, W. (1983a). Helicopter Theory. Book L pp. 364-367. Mir, Moscow. Johnson, W. (1983b). Helicopter Theory. Book II, pp. 510-515. Mir, Moscow. Mikhailov, S.A., E.1. Nikolaev, N.A. Shilova and RM. Aleev (2003). Mathematical model and numerical method of flutter design of helicopter main rotor blades. In: Proceedings of the International Conference on Dynamical System Mode/ing and Stability Investigation, pp. 266-267. Kyiv, Republic of Ukraine. Mill, M.L., A.V. Nekrasov, A.S. Braverman, L.N. Grodko and M.A. Laykand (1966). Helicopters. Calculation and Design. Book 1, Aerodynamics, pp. 358-395. Mashinostroyenie, Moscow.