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NATIONAL ADVISORY COMMITTEF: F O R AERONAUTICS TECHNICAL MEMORANDUM 1312 CALCULATION O F THE BENDING STRESSES I N HELICOFTER ROTOR BLADES* By P. de Guillenchmidt INTRODUCTION The problem of determining the s t a t i c and dynamic s t r e s s e s on the blades of rotary-wing a i r c r a f t i n forward f l i g h t has occupied the atten- t i o n of many engineers of every country f o r a long time. U p t o within t h e last f e w years, however, no satisfactory solution has been found f o r blades having a distribution of mass and r i g i d i t y varying along the span.
I n France, M r . Dorand had f o r many years (1932 with the Breguet-Dorand gyroplane) used a graphical method which gave s a t i s f a c t o r y results; how- ever, it was too long and called f o r precise diagrams on account of the graphic double derivations involved. More recently, American engineers have developed several methods which, while affording correct solutions i n the general case, a l s o require a volume of calculation, which increases a t an appalling r a t e when the number of points examined on the blade a r e or higher harmonics f o r the external forces acting on the t o be increased it has been necessary t o intro- blade are t o be introduced. Accordingly, duce more approximate methods, and it i s these methods which a r e usually employed i n design.
The purpose of the present report i s t o describe a comparatively rapid method of calculation which gives a correct t h e o r e t i c a l solution of the problem i n the most general case. This method i s the r e s u l t of collaboration between the Bureau of Calculation of the Helicopter Division of the National Societies of Airplane Construction f o r South Eastern and f o r Central France, s e t up within the Committee of Rotating Wing Units of the French Association of Aeronautical Engineers and Technicians, (A.F.I.T.A.), on the instigation of Col. G a r r y , Chief of the Section of Rotating Wing Units of the Technical Service Division of the A i r Ministry.
The method i s based on the analysis of the properties of a vibrating beam, and i t s uniqueness l i e s i n the simple solution of the d i f f e r e n t i a l equation which governs the motions of the bent blade. It i s applicable, whatever the limiting conditions may be (blades hinged, blades clamped, blades f i t t e d with dampers, etc. . . .). It requires, s t r i c t l y speaking, t h e preliminary calculation of the natural frequencies and mode shapes of S.N.C.A.C. Report, ""Calcul en Flexion de Pales de Giravions."
Document He3-0.03, December 23, 1948.
2 NACA rm 1312 the blade i n rotation. This calculation can be reduced, however, as w i l l be shown later, i n a certain number of cases, t o the calculation of the n a t u r a l frequency of the first model, which reduces the calculation required t o some extent.
For the explanation of the method, l e t us take the case of a hinged blade i n flapping motion and impose the usual r e s t r i c t i v e asumptions, which are: (a) The twisting deflections of the blade a r e negligible.
(b) The blade i s r i g i d i n i t s plane, t h a t i s t o say, the drag deflections are negligible with respect t o the deflections normal t o the plane of the blade.
( c ) The deflections and the flapping angles a r e small, hence t h e i r higher powers can be disregarded and we may assume
cos p - 1 s i n p - t a n p - p
(d) The bending deflections of the blade do not influence the aero- dynamic forces acting on the blade. Included, however, i s a term f o r the aerodynamic damping due t o the f a c t t h a t the vibra- t i o n of the blade produces, f o r each element, a change i n the
r e l a t i v e velocity, and consequently e? the angle of attack. *
SYMBOLS R radius of rotor distance of blade element from axis of rotation r X abscissa, along axis OX o f t h e r i g i d blade, of an element of the e l a s t i c a l l y deflected blade, with the flapping axis as origin ( f i g . 1) 5. abscissa along axis OX ordinate of a point of the e l a s t i c a l l y deflected blade along Y an axis OY normal t o axis OX lThe "mode 0" i s t h a t which corresponds, f o r a hinged blade, t o a vibration without bending, t h a t is, t o flapping f3. The m o d e 1 f o r such a blade i s then t h a t which corresponds t o a vibration of one node.
%his damping term has, obviously, no significance when there i s separation of flow, because beyond angle of separation the normal l i f t coefficient is p r a c t i c a l l y independent of the angle of attack.
NACA m 1312 3 a distance of the flapping hinge from the axis of rotation of the rotor flapping angle of the r i g i d blade B m' mass of the blade per u n i t length a t a point under consideration 2 blade chord E Young's modulus of the blade material I a normal section of the blade moment of i n e r t i a of (0 angular velocity of the blade t t i m e $ = a t azimuth angle of the blade a t t i m e t phase difference of the azimuth angle due t o damping 3 1 n V forward speed of helicopter V resultant velocity of the air on a section of t h e r i g i d blade normal component of velocity v VN tangential component of velocity v VT e angle of attack of a blade section angle of the resultant velocity v with the normal plane cp angle of velocity v with the resultant velocity on the 4) f cos cp
deflected blade ( )
a angle of the normal plane with the forward velocity r a t i o of advance P
(" : ; "1
air density P dCZ/di lift-curve slope of the p r o f i l e acceleration of gravity 6 5 NACA TM 1312 s t a t i c moment w i t h respect t o the a x i s of rotation of the p a r t the abseissa x of the blade located beyond deflection of the blade. (See cp coefficient of damping due t o text. ) natural deflection function of order i ?li natural frequency of the order i of the nonrotating blade v1,o natural frequency of the order i of the blade caused by v i , w r o t a t i o n with angular velocity o a u x i l i a r y functions, dependent on time only h i n order of a harmonic i n Fourier s e r i e s subscripts of terms i n cosine and sine of a Fourier s e r i e s na, nb THEl EQUATION O F DEFLECTION O F THE BLADE L e t us consider the forces which act a t S on a blade element of span dx i n the plane YOX. (See fig. 1.) These forces are: (2) A corrective term of t h e damping of the l i f t force due t o t h e f l e x u r a l e l a s t i c deformation of the blade. This term i s of the form (3) The w e i g h t dp = m'g dx, the components of which along OX and OY are, respectively m'g s i n ! 3 dx (negligible) h i s elementary l i f t already includes, according t o the definition of VN, a term due t o the aerodynamic damping of the flapping r i g i d blade.
NACA TM 1312 5 (4) The centrifugal force of rotation ml$r dx, the components of which along OX and OY are, respectively
m f o 2 ( a + x cos N mt(u2r d~
and
m t a ? ( a + x s i n p ) a ~ z r n t d r p d~
( 5 ) The force of i n e r t i a of flapping: - m t i j d~
(6) The force of i n e r t i a of deflection: -m'y d~
S t r i c t l y speaking, the following forces should a l s o be included: (7) The centrifugal force of flapping: m t x b 2 d~ (8) The Coriolis force due t o the simultaneous action of blade deflection and flapping motion: - 2 & d dx however, we s h a l l disregard them i n r e l a t i o n t o the centrifugal force of rotation.
For the blade element d t t o be i n equilibrium, it i s necessary t o add t o these forces the unknown actions of the adjacent elements on the element under consideration.
Calculation of Ki
vT - cur + V cos a s i n $
v = - - cos cp cos cp
i cos cp
- -
cu1: + V cos a s i n $
b NACA TM 1312 It becomes dC
K = E 1 3 (m + v cos a s i n $)
2 d i Since the calculation can be made only when the coefficient of the damping term due t o the deflection i s independent of t i m e , the periodic p a r t i n V s i n $ w i l l be disregarded. This reduces the problem t o the corresponding mean speed of the air with respect t o the blade element.
Hence The d i f f e r e n t i a l equation of the blade then reads The dots s i g n i e the derivatives with respect t o t i m e .
This equation of the p a r t i a l derivatives must be completed by four limiting conditions which define the integration constmts, namely, y = O f o r x = O d2Y E I - = O f o r x = O a n d f o r x = X - a dx2 = O f o r x = R - a dx NACA TM 1312 METHOD OF SOLUTION The foregoing deflection equation i s composed of a first member with terms dependent on the deflection, and of a second member which is inde- pendent of deflection. This second member comprises the aerodynamic forces, the weight and the forces of i n e r t i a of rotation, and flapping acting on the r i g i d blade.
These forces a r e e a s i l y computed f o r each point of the blade and f o r each one of i t s azimuth positions after the equation of flapping p of the r i g i d blade has been solved. W e s h a l l waive t h e i r calculation and identify the second member of the preceding equation by t h e function F t d ( x , t ) . W e further put
-ICR m'(a + 6)dE = s
s o t h a t the preceding equation reads This i s an equation of p a r t i a l derivatives with second member, representing the forced vibrations of the blade with damping.
To resolve it, w e introduce the natural functions of the deflections of the blade, that is, the vibrations obtained by solving the foregoing equation of t h e p a r t i a l derivatives above without second member.
Consider first, f o r simplicity, a s t a t e of forced vibrations with- out damping a r i s i n g from t h e deflections of the blade. ( I n other words, t h e term K$ i s disregarded.)
When we consider the moment, Mrig, of the forces exerted on t h e assumedly r i g i d blade, t h i s moment is, a t a point of the blade, a func- t i o n of t i m e only and can therefore be developed i n s e r i e s of periodic functions of $ = ClTt 8 NACA TM 1312 This moment is none other than the moment of the distributed outside forces appearing i n the second member of the preceding equation.
F'd Let us resolve these forces at each i n s t a n t i n series o f distribu- t i o n s such t h a t they each impart t o the blade a deflection taking the form of a natural mode of deflection of the corresponding order.
It can be shown t h a t an a r b i t r a r y deflection of the blade may always be resolved i n s e r i e s of natural functions by reason of the relations of orthogonality existing between the natural functions of continuous beams, whatever t h e i r limiting conditions. (Physically, it means that the dif- ferent natural vibrations a c t independent of each other without mutual interactions. ) The proposed resolution has the form
where gi i s a function of time only and v i i s the natural function
of the deflection of the blade of the order i.
The deflection of the order i of the blade a f f e c t s then, a t each
instant, the form of the function v i , t h a t is, it will be given by
where hi i s a function of the time only. The t o t a l deflection is, by v i r t u e of the relations of orthogonality invoked above, obtained by superposition of the various natural deflections of the blade vibrating a t the corresponding natural frequencies, w i t h amplitudes and phase dif- ferences defined by the function hi, which, i t s e l f , is obtained by putting the expression i n the equation of the p a r t i a l yi = hivi derivatives.
Hence, for a frequency of the order i NACA TM 1312 Now it i s known t h a t an equation of t h e form representing the natural vibrations of a f r e e l y vibrating blade and involved i n rotation with an angular velocity w permits an i n f i n i t e number of solutions of the form q s i n v t , satisfying qi i s the natural func- and the limiting conditions. Every solution t i o n of order i corresponding t o the natural frequency vi,w.
Therefore, when vi i s a natural function, it simultaneously
s a t i s f i e s the equation (4) and the equation i s the corresponding natural frequency of the blade actuated where Vi,co with a speed of rotation w.
Hence, a f t e r simplification Since the functions hi and g i are periodic with respect t o Jr, they can be developed i n harmonic s e r i e s 10 NACA TM 1312 The d i f f e r e n t i a l equation
Li 5 ( . I %) - him2 d2 ( s 2) + '* him' q ] = gim'qi
i =O dx dx dx i = O therefore resolves i t s e l f by identification of the coefficients, and limiting i.t t o the t h i r d harmonic, it reduces t o the system
hi, 2b(v2i,w - h2) = gi, 2b
Each function g i i s well defined.
To determine it, simply multiply the two members of equation (3) by q i and integrate Over the blade.
Owing t o the conditions of orthogonality NACA TM 1312 it leaves hence The bending moment exerted on the e l a s t i c a l l y deformable blade i s computed next.
"he e l a s t i c deformation due t o a single harmonic of the outside forces, such as i s , as shown previously, O n replacing the terms by t h e i r values obtained from equation (?), h i the corresponding bending moment reads 12 NACA TM 1312 O n making the exact calculation of the natural functions qi, it i s found t h a t they vary very l i t t l e with LU and t h a t a natural function of the order i, f o r w = 0, can be compared with the sane natural function f o r the normal speed LU.
This simplification is not exact, but the resulting e r r o r i s small (less than 3 percent f o r the first natural function i n the case of the blade c i t e d i n t h e example hereinafter), being of the same order as those committed i n the d i s t r i b u t i o n of the masses and the flexural s t i f f n e s s of the blade.
I n t h i s case, the natural function q i s a t i s f i e s both and Hence d27 i
E1 - by i t s value i n ( 7 ) , the coefficient of the On replacing
dxz harmonic n of the e l a s t i c moment which, i n fact, bends the f l e x i b l e blade, i s given by the expression NACA TM 1312 1 3
The value g i is computed by (6) f o r the values of Jr f o r which F'd
is given. The development of gi i n Fourier series defines the coefficient
The natural functions vi and the integrals m y q i dx ds can be
J
computed by a c l a s s i c a l method such as the i t e r a t i o n method (Stodola) , o r the Galerkin method.
There i s no occasion t o be preoccupied with normalizing the natural functions. For the amplitude of qi, any convenient scale is suitable; the e f f e c t of the scale disappears later i n the product The calculation f o r c u = 0 is made while remembering the previous statement that the deflection i s p r a c t i c a l l y unmodified by rotation.
If the i t e r a t i v e method is used f o r computing the natural functions of the deflection, the natural frequencies of the blade not rotating and of the blade rotating a t angular velocity 0) can be computed by applying Rayleigh's method t o the obtained natural deflection. This method affords rigorous solutions, converges rapidly, and avoids the solution of n equa- t i o n s with n unknowns t o which the Gtzlerkin method leads.
- Note: The bending moments could a l s o be computed by direct
application of equation (7). This method i s predicated on the d 2 q i / b 2 which prohibits the use of the approxi- exact knowledge of
mation vi,,, =
because a slight e r r o r i n a function can cause q i , O a substantial e r r o r i n i t s second derivative. The calculations of the natural deflections are quite complicated. O n the other hand, the function y being defined by dots, it i s not possible t o derive it d i r e c t l y t o obtain d2y/aX2.
14 NACA TM 1312
SIMPLIFICATION OF TRE CALCULATIONS Numerous calculations made on blades of various helicopters with d i f f e r e n t plan forms and distributions of masses and d i f f e r e n t amounts
of r i g i d i t y have shown that, i n certain cases - blades of moderately
conical shape, l i t t l e twist, and l i g h t l y loaded a t the t i p - the natural
functions, other than the first, exert l i t t l e influence on the maximum bending moments exerted on the blade, and consequently on the maximum t o which the blade i s subjected. This is alternating fatigue stresses F'd = f(F) because the distributions of the outside forces (curves f o r J I ) f o r such blades represent t h e behavior of the first the various natural distribution (curve m ' q = f ( F ) ) i n a s a t i s f a c t o r y manner and also because the natural functions of higher orders present a l l the ''loops'' and "nodes" i n continuously increasing number, matching poorly the greater deflection which the maximum moments produce.
It i s only i n cases of small deflections t h a t the "parasite remainder" of the higher frequencies, a r i s i n g from the f a c t t h a t the distributions of the outside forces never have a curve exactly i d e n t i c a l with the first natural distribution, can play a significant part.
When the blades have pronounced camber and twist and a r e l i g h t l y loaded a t the t i p s , the forces of i n e r t i a can become more important than the aerodynamic forces a t the blade t i p i n the e n t i r e sector of the swept disk corresponding t o the advancing blade. The distributions of the out-
side forces F t d = f(r) can assume, therefore, the curves approached by
the second natural distribution (curve m ' q 2 ) f o r an e n t i r e s e r i e s of J I , and, i n t h a t case, the second and sometimes the t h i r d natural functions must be taken i n t o consideration i n the calculation of the maximum moments exerted a t the blade.
Examination of the curves permits one t o determine, F'd = f(r) with a l i t t l e practice, when resolution of the outside forces can be l i m i t e d t o the first natural d i s t r i b u t i o n and the deflection of the blade t o t h a t corresponding t o the first natural function.
I n the latter case, the calculations a r e considerably simplified.
The d i s t r i b u t i o n of the outside forces i s reduced t o NACA TM 1312 "he bending moment on the e l a s t i c blade becomes O n comparing t h i s bending moment with the moment of the outside force d i s t r i b u t i o n F'd exerted on the assumedly i n f i n i t e l y r i g i d blade, we get seen t h a t the bending moment on the e l a s t i c blade i s obtained It i s by multiplying the coefficients of the harmonics of the same order o f moment on the r i g i d blade by a f a c t o r
l , o
An =
v2 - n*w2
1,UJ which i s constant Over the blade for a given harmonic.
It i s no longer necessary t o calculate the natural functions
%
and the natural frequencies V l,o and Vi,cu. "hey can be readily and
closely approximated by the Rayleigh method applied t o a curve repre- senting approximately the deflection of the blade while s t i l l satisfying t h e limiting conditions rigorously.
A s regards the various harmonics t o be kept f o r the moments on the r i g i d blade, it seems t h a t no advantages a r e gained by going beyond the third, which is already r e l a t i v e l y small.
The bending moment, a t a point on the abscissa x, on the r i g i d blade is then given by the expression
Mrig = MQ + Ma cos $ + Mb s i n Jr + M2a cos 24f +
M2b s i n 2 4 r + M3a cos 3$ + M3b s i n 3 4 f
16 NACA TM 1312 The effective moment bending the actual e l a s t i c blade a t the same point i s obtained by the simple r e l a t i o n 2 2
l , o l , o
( M ~ cos + + M b s i n +) t
Melast. = - Mo +
V 2 1 , V21,u3 - u j !
CALCULATION INCLUDING A E R O D Y N A M I C D A M P I N G DUE TO
DEFLECTION - GENERAL METHOD
disregarded i n equation (2) must be I n t h i s case, the term Ki included.
It has been shown t h a t
P dc Z
r The term - 2 - i s homogeneous t o a distributed mass.
2 d i Putting Zr dCz
q X ) = E - -
2 m t d i NACA TM 1312 gives
i s a dimensionless coefficient. Since, as a r u l e 2 - r where
%) ' 2 di i s not proportional t o the distributed mass m ' , the coefficient @ (XI i s a function of x.
Having already permitted one approximation f o r K by including only the corresponding mean speed on a blade element, another simplifica- along the t i o n i s effected by bringing only the mean value of 9x1 span of the blade i n t o the first member of the equation of motion.
Accordingly, equation (2) reads With the symbols already employed, we get The foregoing system of equations ( 5 ) i s therefore replaced by another of t h e form
hi,na@*i,Lu - I12&) + hi,nbn& = &i,na
hi, nb(v i, u) - I12uj?)- hi,nanG@ = gi,nb
l a
NACA TM 1312 whence Putting the re fore = s i n @i,n NACA TM 1312 C ons e quent ly The calculation i s then carried out i n the same way as without damping; the deflection due t o a single harmonic n of t h e outside forces being always i= CQ s i n nq) i=l Replacing h i by i t s value gives Hence, as before, the term of the harmonic n of t h e moment which, i n fact, bends t h e f l e x i b l e blade
Ei , na cos 4Ji+ -
i=l gi, nb
20 NACA rn 1312
o r a l s o with I f , as previously, the d i s t r i b u t i o n of the outside forces i s limited t o the f i r s t d i s t r i b u t i o n and the deflection of the blade t o t h a t corre- sponding t o the first natural function, the following r u l e results: The e f f e c t i v e bending moments acting on an e l a s t i c blade, with damping, are obtained from the bending moments acting on the assumedly r i g i d blade, each harmonic of which i s modified as follows: (1) Its amplitude i s multiplied by a coefficient equal t o (2) There i s a forward phase difference of qn, s o t h a t NACA m 1312 21 PRACTICAL PROCElSURE OF CALCULATION I N THE C A S E O F THE GENERAL METHOD (The Double Resolution of the Outside Forces Being Limited t o the Third Natural Distribution and t o t h e Third Harmonic i n $) The first step i s t o determine the natural functions of the
blade ql, q2, and q3 = f(r), as well as the corresponding natural
frequencies v ~ , ~ , V 2 , 0 , and v ~ , ~ of the nonrotating blade and t h e v l , , , , v2,,,, and v3 of the blade rotating a t angular velocity LD.
9LD Next, it i s necessary t o evaluate the quantities and then p l o t the curves
-
against r.
Then determine the outside forces on the r i g i d blade F t a = f ( Y ) Jr (eight positions spaced 43' apart must be explored). f o r different Next evaluate for each position Thence one obtains f o r each position.
22 NACA TM 1312 g17 g27 g3 i n Fourier series (by the Runge method, f o r example) and stop with the t h i r d harmonic of \Ir: whence g1,o %,a gl,b * g1,3b . . .
Q2,o g2, a '2, b g2, 3b g3, 0 g3,a . '3,b '3,3b Calculate the bending moments on the blade f o r d i f f e r e n t s t a t i o n s x = K.R by the formula i = 3
~x = 7 Mi,x
i=l with where gi, nb t a n qi,n =
g i , na
NACA m 1312 PRACTICAL ExAMpIz;E Hereinafter follows an application of t h i s method t o the calculation of the bending moments exerted on the blade of the helicopter N.C. 2001 i n forward f l i g h t , with p = 0.43.
The mechanical characteristics of the blade, which has a radius of R = 6.85m, a r e given i n figure 3. Figures 4 and 5 give the natural functions q and the blade distributions m'q. Figure 6 gives the natural frequencies of t h e nonrotating blade and of the blade rotating a t angular velocity u) f o r the different modes of vibration. Figure 7 gives the d i s t r i b u t i o n of the aerodynamic forces, and figure 8 the dis- t r i b u t i o n of the t o t a l outside forces on the r i g i d blade f o r Fld several azimuth positions of the l a t t e r .
According t o figure 8, the distributions of the outside forces f o r t h e positions cp = 45O, go0, l 3 7 O , and 180° have c l e a r l y the shape of t h e second natural distribution m'q2, which explains, as w i l l be seen i n figures 9 t o 12, the importance of the bending moments computed with t h e second natural function included.
Figure 13 shows the bending moment d i s t r i b u t i o n f o r several radii p l o t t e d again $, and figure 1 4 the enveloping curve of the maximum bending moments exerted on the blade. Figure 14 a l s o shows the maximum bending moment curve acting on the blade of the S.E. 3000 helicopter
( R = 6m, p = 0 . 4 1 ) , whose mechanical characteristics a r e shown a l s o i n
figure 3. It i s seen t h a t f o r t h i s blade, which presents an average camber and is l i g h t l y loaded at the t i p , the e r r o r made i n the maximum bending moment by limiting the distributions of the outside forces t o t h e first d i s t r i b u t i o n does not exceed 6 percent, which j u s t i f i e s the simplification of the calculation indicated previously.
For the blade of the NC.2001, the maximum bending moment i s severely subjected t o the influence of the second distribution, but, by way of compensation, it i s p r a c t i c a l l y c l e a r of t h a t of the t h i r d distribution (except at the t i p ) .
However, it should be noted that the l a t t e r assumes a significant p a r t i n the evaluation of t h e negative maximum bending moment, expressed 24 NACA TM 1312 Mfin (see figs. 10 and l3), and consequently in the appraisal of by the maximuin alternating fatigue, defined by is the resistant modulus of the particular section.
where W - I
- v
Translated by J. Vanier National Advisory Committee for Aeronautics NACA m 1312 Y ~ / S' a t x c o s p = a t x = r t I Figure 1.
I VT Figure 2.
NACA rm 1312 I .Blade N.C.2001 m'
--
m' m'( 0.5) E1 E1(0.5) Figure 3.- Plan forms and distribution of mass and rigidity of the S.E . 3 000 and N.C.2001 helicopter rotor blades.
1.0 .8 .6 .4 .2
-
r -.2
- .4
-.6 \ \ -.8 - 1 . 0 0 .I .2 .3 .4 .5 -6 -7 -8 -9 1.0 Figure 4.- N.C.2001 blade - natural elastic deflection curves of the blade.
NACA TM 1312
Figure 5.- N.C.2001 blade - natural distributiork m'v2q =
for Y = 1.
Y ,i 0 1 2 3 Figure 6.- Natural frequencies of the N.C.2001 blade.
28 NACA TM 1312 4 00 3 00
-
r -100 L I I I I I I 0 . 1 . 2 . 3 .4 .5 .6 -7 -8 .9 I . ' ( )
Figure 7.- N.C.2001 blade - distribution of aerodynamic forces F' on
A the rigid blade.
-2 0 -4 0 -60 -8 0 -100 -120 -140 *1 -2 -3 .4 .5 .6 .7 .8 -9 1 . 0 Figure 8.- N.C.2001 blade - total forces F'd distributed over the rigid blade.
NACA TM 1312 M ( d g ) included only ____ F i r s t and second natural modes included ---_-__ F i r s t , second, and t h i r d natural modes included 14 0
-
0 r 0 .1 . 2 . 3 .4 .5 .6 .7 .
Figure 9.- N.C.2001 blade - blade bending moments f o r ~r = 0.
M(mW -
r -20 -4 0 -60 0 .1 . 2 . 3 .4 .5 .6 .7 .8 -9 1 . 0 Figure 10.- N.C.2001 blade - blade bending moments for ~r = 90'.
NACA TM 1312 M 1 0 -10 -2 0 0 .1 .2 . 3 . 4 .5 .6 .7 .8 .9 1.0 Figure 11.- N.C.2001 blade - blade bending moments for J I = 180'.
Figure 12.- N.C.2001 blade - blade bending moments f o r Jr = 270'.
NACA TM 1312 NACA TM 1312 F i r s t natural mode only
-- -___ N.C.2001 blade
__ _ _ - - - - S .E. 3000 blade F i r s t and second modes included
- - - - - - N.C .2001 blade
S .E. 3000 blade F i r s t , second, and t h i r d modes included N. C .2001 blade - _ _ _ _ _ _ _ _ _ _ S .E. 3000 blade 1 4 0 .3 .4 .5 .6 .7 .8 .g 1.0 0 .1 .2 Figure 14.- Maximum bending moments.
NACA - Langley Field, Va.