Simplified procedures for estimating flapwise bending moments on helicopter rotor blades. Part 1 - Procedures and charts

NASA CR- 1440 SIMPLIFIED PROCEDURES FOR ESTIMATING FLAPWISE BENDING MOMENTS ON HELICOPTER ROTOR BLADES Part I - Procedures and Charts By Anton J. Landgrebe Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.

Prepared under Contract No. NAS 1-7880 by UNITED AIRCRAFT RESEARCH LABORATORIES East Hartford, Conn.

for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Cleoringhouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 C_N_ Page No.

SUMMARY ............................... I TRANSFER FUNCTION CONCEPTS APPLIED TO THE BENDING MOMENT PROBL_4 . . i0 Flapwise Bending Moments .................... lO Sensitivity Studies to Select Final Blade Design Parameters . . 21 Comparison with Experimental and Computer Analysis Results 33 Effect of Advance Ratio and Design Changes on Stress . . . 34 lil CONTENTS (Continued) Page No.

Hingeless Rotor Applications ....... ............

Sample Stress Calculation ..................

Effect of Advance Ratio and Design Changes on Stress ....

Nonuniform Hingeless Blade Application ...........

Steady Hub Moment Applicatlon. ...............

Approximate Analys is Results ...................

Articulated Blade Results ..................

Hingeless Blade Results ...................

Teetering Rotor Applications ...................

CONCLUS IONS ..............................

5O APPENDIXES: A: An Approximate Method for Determining Flapwise Moment Transfer Coefficients ....................

B: UAC Normal Mode Transient Analysis ..............

C: Determination of Parameter Values for Sensitivity Study . . . 70 REFERENCES ..............................

TABLES : I: Table of Blade Designs and Blade Numbers ..........

8O II: Organization of Figures for Transfer Coefficient Charts • • III: Articulated Blade Designs, Controls, and Lift Coefficients for the Applications Study .................

IV: Hingeless Blade Designs, Controls, and Lift Coefficients for the Applications Study .................

FIGURES : 52-177 Transfer Coefficient Charts (see Table II for organization) 144 iv

SIMPLIFIED PROCEDURES FORESTIMATING FIAPWISE BENDING

MOMENTS ONHELICOPI_R ROTOR BLADES

PART I - PROCEDURES ANDCHARTS

By Anton J. Landgrebe

United Aircraft ResearchLaboratories

SUMMARY

Simplified procedures are developed for predicting the flapwise bending

momentsacting on low stiffness articulated and hingeless helicopter rotor blades..

It is shownthat such momentsare basically linear functions of several indepen-

dent rotor parameters and can be computedby transfer function, superposition

techniques. Transfer coefficients relating independent rotor parameters to

harmonics of moment are derived using a computeranalysis which numerically

integrates the blade flapwise equation of motion. These coefficients are

presented in the form of design charts for a wide range of blade design parameters

and operating conditions of interest. Detailed procedures for using the charts

in conjunction with the performance charts of NASA CR-114are described and

illustrated with sample calculations. The accuracy of the methodis demonstrated

by several comparisonsofbending moment(or stresses) computedfrom the charts

with full scale rotor experimental results andwith results from a more complete

numerical analysis in which chordwise and torsional degrees of freedom are

included. A complementaryanalysis for predicting closed-form, approximate

expressions for the transfer coefficients is also developed.

The transfer coefficient charts are shownto provide quantitatively accurate

flapwise bending moments for low stiffness articulated and hingeless helicopter

blades at advanceratios from 0.25 to 0.5 and qualitatively accurate results

(useful for preliminary design) for advanceratios above 0.5. In addition, the

approximate analysis is found to provide qualitatively accurate transfer

coefficient expressions which facilitate analysis of results.

The results are presented in two parts. Part I contains the transfer

coefficient charts and describes their developmentand application. Part II _

presents tabulations of transfer coefficients which, if desired, allow the

coefficients to be determined to a greater degree of accuracy than is possible

with the charts of Part I.

SeeNASA CB-1441, 1969.

INTRODUCTION

The design of helicopter rotor blades has always represented a relatively

more difficult structural design problem than that of fixed wings. Several

factors have contributed to this increased difficulty. First, rotor blades,

becauseof their rotation, operate continuously under nonsteady flow conditions

in forward flight; hence fatigue loads assume proportionately greater impor- tance. Second, at some flight conditions, the blade bending moments are greatly influenced by the helical wake of the rotor. This is particularly true in the O to lO0 kt speed regime where predicted blade bending moments are less reliable when such wake effects are neglected. Third, rotor blades have comparatively small structural stiffness values (for their span), being supported in flight principally by centrifugal forces. The net bending moments on blades therefore tend to be relatively small differences of large aerodynamic, inertial, and centrifugal moments, and accurate representation of such component moments is mandatory. Lastly, the relative flexibility of blades results in increased coupling between the airloads and the blades' elastic deflections.

It is not surprising, therefore, that until recently the design of rotor blades has generally proceeded on a semi-empirical basis with heavy reliance placed on flight-test substantiation of designs. However, with the development of modern computer technology, significant advances have been and are being made to place the design of rotor blades on a firmer analytical foundation. For example, digital computer programs for predicting the fully coupled response of an elastic blade at any steady flight condition have been developed which include the effects of stall, compressibility, and reverse flow, as well as blade geo- metry, mass, and stiffness variations. One such program has been developed at the United Aircraft Research Laboratories and a version of this program is described in Ref. 1. Correlations of the results of this program with NASA and Sikorsky Aircraft flighttest data indicate that critical blade bending moments can be computed with reasonable accuracy for those flight conditions which principally determine blade fatigue life; i.e., steady level flight at speeds above approximately 100 kts. At low flight speeds, the prediction of accurate moments is presently hampered by the relatively rudimentary state of the art of predicting rotor wake effects. Work in this area is, however, proceeding on many fronts (e.g., Refs. 2 through 6).

Unfortunately, lengthy computer programs are not without certain disadvan- tages. They are not, for example, ideal tools for preliminary design where rapid methods of reasonable accuracy are desired. In addition, closed-form solutions are not obtained and, as a result, the mechanisms by which particular design parameters influence the final results may often be obscured. Consequently, it would be highly desirable if simplified procedures for predicting bending moments could be developed which would: (1) be rapid in application; (2) retain much of the accuracy associated with refined digital methods; and (3) permit identifica- tion of mechanisms by which various design parameters influence the final results.

Reported herein are the results of an analytical investigation, conducted

at the United Aircraft ResearchLaboratories, to develop simplified procedures

for estimating flapwise bending moments on helicopter rotor blades. These pro-

cedures are based on transfer function, superposition techniques. The prin-

cipal objectives of this investigation are to: io Provide quantitatively accurate bending moment transfer functions for use (in conjunction with the performance charts of Ref. 7) in the preliminary design of articulated and hingeless rotors, .

Develop complementary approximate transfer function expressions to facilitate qualitative analysis of results, and 1 Demonstrate the utility of the l_rocedures developed through sample design applications.

More specifically, the intent of the first objective is to provide a set of design charts for predicting moments for a wide variety of constant chord rotor blade configurations operating in steady flight at substantially unstalled conditions, advance ratios from 0.25 to 1.4, and conventional tip Mach numbers.

Included in Part I of this report are descriptions of (i) transfer function techniques as applied to the rotor blade bending moment problem, (2) parameters considered in the analysis, (B) the transfer coefficient sensitivities to various blade design parameters, (4) transfer coefficient design charts, (5) sample design applications using the transfer coefficient charts, and (6) an approximate analysis which can be used to analyze results. Item (_) includes charts of harmonic transfer coefficients at various blade radial stations for nine uniform blade designs for articulated and hingeless rotors over an advance ratio range from 0.25 to 1.4. These charts permit the determination of flapwise moments for a wide range of uniform blade designs and, as will be described, for blade designs having a limited degree of nonuniformity. Item (5) includes com- parisons of flapwise moments (or stresses) predicted by the transfer function approach with results from experiment and from a more accurate digital computer analysis which considers the fully coupled flapwise, chordwise, and torsional degrees of freedom of the blade. Tabulations of transfer coefficients are presented in Part II.

The guidance and assistance provided to this investigation by Mr. Peter J. Arcidiacono, Chief, Aerodynamics (UARL), is gratefully acknowledged.

This includes the initial formulation of the bending moment transfer function concept and the derivation of the approximate analysis presented in Appendix A.

Also acknowledged is the assistance provided by Mr. Marvin C. Cheney, Supervisor, Rotary Wing Technology (UARL), in establishing the techniques to be used for the hingeless blade analysis. The NASA technical monitor for this investigation was Mr. John Ward.

SYMBOLS Section llft curve slope, I/radian Longitudinal flapping coefficient for the nth harmonic; coefficient of -cos n @ term in Fourier series expansion of blade flap angle ( B ) with respect to the rotor shaft (Eq. (AI2)), radians except where noted in deg Blade coning angle, radlans except where noted in deg 0 o LongitudSnal flapping coefficient for the ist harmonic; coefficient GI of -cos @ ter_, in Fourier series expansion of blade flap angle with respect to the rotor control axis, deg A Dynamic amplification factor (Eq. (A27)) AR Blade aspect ratio (radius/chord) Cosine component of first harmonic cyclic pitch; coefficient of AI s -cos _ term in Fourier series expansion of the blade pitch angle with respect to the rotor shaft, radians except where noted in deg b Number of blades Lateral flapping coefficient for the nth harmonic; coefficient bn.

of -sin n _ term in Fourier series expansion of blade flap angle ( _ ) with respect to the rotor shaft (Eq. (AI2)), radians except where noted in deg Lateral flapping coefficient for the Ist harmonic; coefficient bl of -sin _ term in Fourier series expansion of blade flap angle withrespect to the rotor control axis, deg B Tip loss factor; % of blade radius outboard of which the lift is assumed zero Sine component of first harmonic cyclic pitch; coefficient of BI s -sin _ term in Fourier series expansion of the blade pitch angle with respect to the rotor shaft, radlans except where noted in deg Local value of blade chord, ft; cosine component of nth harmonic Co Blade chord at reference station (0.55 R), ft E Ratio of local chord to chord at reference station, C/C o U2C Cd Section drag coefficient, d/+ f, Transfer coefficient relating the amplitude of the nth harmonic C( ),n bending moment to independent parameter ( ) (See Eq. (17)), where ( ) can represent e?5 , e I , k c or Xs , Als,Bls or _B , 1/deg except nondimensional for k c and k s (does not apply to Eq. (2)) Rotor drag coefficient, O/p_RZ (2R) 2 C D Rotor drag coefficient/solidity for minimum ½ peak-to-peak stress Rotor lift coefficient, L /p ?TR 2 (2 R) 2 C L Blade lag damper coefficient, ft-lb/rad/sec CL o CT Rotor thrust coefficient, T/p'trR2(2R) 2 Blade modal constant (Eq. (A7)) Ci d Section drag force per unit span, lb/ft D Rotor drag, lb Flap hinge offset ratio; distance from center of rotation to flap hinge divided by rotor radius E Young's modulus of elasticity, lb/in. 2 EI Ratio of local flapwise stiffness to reference value, EI/EI o Nondimensional force coefficients fin Eq. (A20) R 2 FP Frequency parameter, EIo/mo(_LR) 2 Frequency parameter corresponding to an advance ratio equal to FPp,_< 0.5 or less than 0.5 Frequency parameter corresponding to an advance ratio above 0.5 FP/_ > 0.5 Acceleration due to gravity, ft2/sec g Acceleration due to gravity divided by _2R g Subscripts denoting flapwise bending modes i,i I Flapwise section area moment of inertia, in. 4 I Blade flapping moment of inertia, slug-ft 2 IF Flapwise section area moment of inertia at reference station, in.

I0 Damping constant for ith mode (Eq. (Al_)) ID i Integrals for ith mode (Eq. (A13)) Ik,i k ExPonent of Constants appearing in Eq. (C2) and (C_) KI o , Km o Blade spanwlse loading, lb/in.

!

L Rotor lift, lb Mass per unit span, slug/ft m Mass per unit span at reference station, slug/ft m o m Nondimensional mass per unit span, m/m o Flapwise bending moment, positive when upper surface is in M compression, ft-lb except in.-lb where noted m M Nondimensional flap_ise bending moment, M/(ET/R) Transfer function relating nondlmensional bending moment to M() independent parameter ( ) where ( ) can represent 875, e I , X c or Xs , Al, , Bls or _B , 1/deg except nondimensional for X c and Xs Transfer coefficient relating the nth cosine or sine harmonic of M"( ),rl,c or $ nondimenslonal bending moment to independent parameter ( ) where ( ) can represent e75, e I , k c or k s , Ai= , Bis , or _B , 1/deg except nondlmenslonal for k c and M Nondimensional effective mass constant (Eq. (ll)), CII/(MP)O MP Mass parameter, pRCo/2m o Advancing tip Mach number : Mach number at the tip of blade at MI,90 _ = 9o deg n Harmonic number appearing in Fourier expansion PM Rotor pitching moment, positive nose up, ft-lb Amplitude of chordwise bending mode qv Amplitude of ith flapwise bending mode qw i Transfer coefficient relating the nth cosine or sine harmonic of qwi( ),n,c or s the ith modal amplitude to independent parameter ( ) where ( ) can represent 875 , el, kc or X s , Als , 81s , or _B, (;q.

Transfer coefficient relating the nth cosine or sine harmonic of G'Wj( l,n_c or s the ith modal amplitude to independent parameter ( ) where ( ) can represent e75 , e! , k c or k s , Ais , 81s , or _9, (Eq. (A29)) Ratio of local section radius to rotor radius R Rotor radius, ft or in.

RM Rotor rolling moment, positive advancing side down, ft-lb S Sine component of nth harmonic Nondimensional aerodynamic force per unit span (Eq. (A3)) SA SDP Spar depth parameter ; spar cavity depth at blade tip divided by value at blade root (Eq. (16)) T Rotor thrust, lb Spar wall thickness, ft tw TWP Tip weight parameter: tip weight divided by blade weight (Eq. (i_)) U Total section velocity, ft/sec Up Ratio of velocity component at blade section parallel to shaft to _R Ratio of velocity component at blade section normal to blade u T span and up to _R ) ?

V Forward speed, knots w Blade weight per unit span, lb/in.

Blade weight, lb WBLADE Blade tip weight, lb WTIP WTP Wall thickness parameter; spar wall thickness at blade tip divided by value at blade root (Eq. (15)) Distance from blade neutral axis to outermost fiber of spar measured normal to chord, ft Spar cavity depth (see Fig. 3), ft Ys Z Total flapwise displacement of blade section divided by R Rotor control angle of attack; angle between axis of no feathering cl c and a plane perpendicular to flight path, positive nose up, deg Rotor shaft angle of attack; angle between shaft axis and a plane perpendicular to flight path, positive nose up, deg Blade flap angle with respect to plane normal to shaft, radians except where noted in deg Preconing angle for hingeless blade, radians except where noted _B in deg Blade Lock number, pOCoR4/I F 7' Chordwise bending mode shape Yv Flapwise bending mode shape Yw Damping ratio of ith mode (Eq. (A19)) ® Total pitch angle at any blade station (Eq. (A4)), radians Oi Amplitude of linear blade twist, positive when tip angle is larger, radians except where noted in deg Blade pitch angle at the 0.75 R station, radlans except where e75 noted in deg Rotor inflow ratio; ratio of velocity parallel to control axis _C (axis of no feathering) to _R , positive up A S Rotor inflow ratio; ratio of velocity parallel to shaft axis to R , positive up Rotor advance ratio; ratio of forward velocity component in plane of rotor to _R Air density, slug/ft 3 P Spar material density, lb/in. 3 Ps _r Rotor solidity, bc/_R Flapwise stress at maximum blade thickness, positive when upper _r F surface is in compression, lb/in. 2 0he-half peak-to-peak value of _F , lb/in'2 _r'F_ pTp Minimum value of _F'pTp (Eq. (19)), lb/in. 2 O'F M IN _.pT P Angle by which the response of the nth harmonic for the ith mode _i,n lags the excitation force, radians Blade azimuth angle measured from downstream blade position in direction of advancing blade, radians except where noted in deg u Ratio of chordwise modal frequency to Cdv Ratio of flapwise modal frequency to OJ w Rotor rotational frequency, rad/sec Miscellaneous ( )/, ( )11 Indicates first and second derivatives, respectively, of ( with respect to x (xx) (), Indicates first and second derivatives, respectively, of ( with respect to IP, 2P, etc. Abbreviations for once per revolution, twice per revolution, etc.

TRANSFER FUNCTION CONCEPTS APPLIED TO

THEBLADE SENDING MOMENT PROBLEM

Flapwise Bending Moments

The simplified procedures developed for calculation of flapwise bending

moments are based on transfer function, superposition concepts. The use of

such conceptsbasically implies that the dependent variable of interest (in

this case blade bending moment)is linearly related to several of the indepen-

dent variables of which it is a function. Under such conditions, if the

relating functions (i.e., partial derivatives or transfer functions) can be

determined, solutions for any combination of the independent variables can

rapidly be obtained by simple scaling and superposition procedures. Bailey

(Ref. 8) employedsuch procedures in developing his "t " coefficients, which

are simply transfer coefficients operating on the rotor flight parameters to

yield such items of interest as rotor thrust and blade flapping angle.

The application of transfer function concepts to the flapwise bending

problem can best be illustrated by considering the following simple analysis

of bending moments for an articulated blade. The equations have been simpli-

fied to permit emphasison the basic linearity of the flapwise bending response

of the blade with certain parameters. A more complete analysis of the bending

moment problem (referred to as the approximate analysis) is presented in

AppendixA. For the present purposes, consider the blade response when the

following assumptionsare made: II Variations in llft curve slope due to stall, compressibility, and reverse flow effects are negligible.

2. Only rigid body flapping motion influences the airloads.

3. Blade twist is linear.

_. The induced velocity over the rotor disc is constant.

5. The blade is rigid torsionally.

o The blade elastic bending deflection can be represented by a single flapwise bending deflection mode.

7. Gravity is neglected.

Under the above assumptions, it can be shown that the differential equation of motion governing the elastic deflection of the blade is dZ qw + E..k_qw = PCoR '_ a s {[e.ts + (_._ 0.75)81](?+,=sin_,)z

z

2m ° ]_li_yzd _. _ E Yw

(1)

+ (r" +p'sin')IXc-('-')dB.d_ -F'Sc°s_ where qw, Yw ' and _w represent, respectively, the amplitude, shape, and natural frequency of the assumed bending mode. Since it has been assumed that the bending deflections of the blade do not influence the airloads, the rigid body flapping motion of the blade, _ , in Eq. (1) is (for a given blade) a linear function of the independent parameters: 875 (pitch at the three-quarter span), 81 (blade linear twist rate), and X c (inflow ratio) (Ref. 8).

Equation (1) can then be expressed in the following form.

(2)

+ _'_qw = ceTseTs + c8,8, + C4Xc d_ z where C87s, C8, , CXc are harmonic functions of the blade azimuth angle, _ .

By assuming a negative Feurier expansion for qw , Eq. (1) can be solved to yield the following general expression for the bending mode amplitude.

GW = (3) + o...

In Eq. (3), the coefficients qw87_ , etc. can be thought of as coefficients operating on the basic rotor parameters ( 875 , etc.) to yield blade bending harmonic amplitude coefficients. It is shown in Appendix A that the expression for bending moment at a blade radial station, _ , and azimuth position, _ , due to a single flapwise mode is : M(_, _,) : E.T.(7) _l(?)qw($) (4) R in which R is the blade radius, and EI (_) and ywll(T) ar_ the blade structural stiffness and the blade curvature (second derivative of the blade mode shape) at radial station, r , respectively. Substituting qw in harmonic coefficient form from Eq. (3) in the bending moment expression of Eq. (4) and nondimensionalizing yields M[_ ,_) - - _[(_eTs,n,c875 _(T ,_)= E.[/R = MeTs'°e't5 + Me''°81 + MXc'°kc -n

(5)

+ T_e,,n,cS, + MXc,n,c Xc)C°S n_ + (Me,,,n,s875 + _te,,n,sS, + _Xc,n,sXc) sin m k in which the M's are defined as Me.,5, n' = ytw"(_ }_ c or s e75,n,c or s (6) Me,, n,c or s w _ r Jqo,,n,c or s

r xo,n,c s : or s

and will be referred to as transfer coefficients of flapwise bending moment.

If the summ_tions over n are carried out, the expression for bending moment for each point on the blade becomes : (7) where now the functions M s_,s , etc. are termed transfer functions and operate on 875, etc. to yield the azimuthal variation of the blade bending moment at each spanwise station (_) of the blade. It is thus shown that the flapwise bending moment distribution of an articulated blade can be approximated as a linear function of three parameters: collective pitch, 875 , blade linear twist, 8_ , and inflow ratio, k c . These parameters will hereafter be referred to as independent bending moment parameters, or more briefly, "independent parameters."

It is shownin Appendix A that the flapwise moments of a hingeless blade

can also be analyzed by techniques generally similar to those utilized in

determining articulated blade moments. The transfer function equation for the

hingeless blade bending momentis given by Eq. (8).

ET(T) M (-_, _) R [i_eT.(7'_')87s+ Me, (T'_)8_ + _),,(T,_,)Xs

(8)

+ )A,.+ +

As indicated, the flapwise moment for the hingeless blade is not only linearly related to 875, e i , and k S , but also to blade cyclic pitch ( Als and Bis ) and built-in coning angle ( _B ) which will also be termed independent param- eters for hingeless blades. Note that in the articulated rotor equations cyclic pitch angles were eliminated as independent parameters by assuming small offset and expressing inflow ratio in the control axis system ( X c ). Hinge- less rotor controls will be related to the shaft axis system, which is standard practice since the tip path plane is generally perpendicular to the shaft.

Conversion from one system to the other is important, however, when using the performance charts of Ref. 7, and this procedure will be discussed in the hingeless rotor application section of this report.

For a given blade design, flapwise bending moment directly determines blade flapwise stress, _F , according to the relation: _(_,_,) = y(F) M(_,_,) (9) T('F) Emphasis is placed on bending moment transfer functions rather than stress transfer functions in this study as the former are more g_eral in applicabil- ity, being independent of such detail design quantities as spar thickness.

It has been shown above that the flap_ise deflections of rotor blades are, under the previously defined assumptions, linear functions of certain indepen- dent rotor parameters. Of the assumptions made, several are not critical from the linearity standpoint, having been made principally for the purpose of simplicity in demonstrating the approach. For example, any twist distribution could have been assumed, in which case the linear twist rate, e I , would be replaced by some other measure of blade twist. Compressibility effects could also have been included to the extent that the lift-curve slope, a , could be allowed to vary wlth radius and azimuth angle. Inclusion of the aerodynamic forces due to the elastic blade response also does not affect the llnearlty.

On the other hand, several assumptions cannot be eliminated without introducing nonlinear effects. For example, blade stall effects, in general, cannot be included as these cause the llft-curve slope to be a function of 875 , e_ , and X c . However, since rotor stall occurs only over a limited portion of the disc, it is possible that reasonable estimates of blade bending moments can be obtained by transfer function techniques for those flight conditions where only moderate amounts of stall are present. Variable inflow effects cannot be included, but these effects generally decrease with increasing forward velocity and may be neglected for the purposes of this investigation. In general, inclusion of torsional deflections also introduces nonlinearities as they are functions of blade bending; however, below about 200 kts torsional deflections are small (particularly if significant stall is not present) and can be neglected. Operation at speeds above 200 kts implies high advance ratio operation where large torsional deflections can occur with conventional blades due to the destabilizing torsional moments produced in the reverse flow region.

However, the application of transfer function techniques to the higher flight speed conditions is feasible if it is assumed that satisfactory rotor operation at such conditions will probably require that excessive torsional deflections be eliminated by approximate blade design changes.

To summarize, in the important i00 to 200 kt flight speed range of modern high performance helicopters, the flalm_Ise bending deflections of rotor blades are substantially linear functions of several independent rotor parameters ; as a result, the application of transfer function, superposition techniques to the problem is possible. At higher speeds, blade torsional coupling effects may give rise to nonlinearities. However, transfer functions for high speed conditions may be of interest in preliminary design to provide a reference point for further more detailed analysis.

Chordwise Bending Moments Final blade designs are, of course, dependent on chordwise bending stress_ as well as flapwise stresses. The former are, however, nonlinear functions of 875 , ej , and k c as indicated in Eq. (10) (which applies to articulated blades). • too,Q, R 3J'o i_ .yvZdF Similar nonlinearities exist for hingeless blades. Hence, the transfer function approach cannot be conveniently applied to simplify chordwise bending calcula- tions. However, experience with articulated rotor blades has shown that because of the large chordwise stiffness of such blades, chordwise stresses are only about 15 percent of flapwise stresses and, unless critically phased, generally represent a significantly smaller percentage of the total blade stress.

(Note that blade chordwise response is generally lightly damped and small chordwise stresses are contingent on avoiding resonant conditions). Errors introduced by the neglect of the chordwise stresses are therefore probably acceptable for the purposes of preliminary design. Such an assumption was made in the design study of Ref. 9. A more conservative alternate treatment of chordwise stress effects is that of Ref. lO, where the maximum allowable design fatigue stress was arbitrarily reduced by the maximum anticipated chordwise stress. Should more accurate estimates of chordwise stresses be desired, a closed-form solution to Eq. (lO) could be obtained. Such a solution would be relatively accurate since the aeroelastic coupling effects associated with chordwise deflections are small. However, for hingeless rotors where chordwise stresses can become critical, still more refined digital methods would probably be necessary.

In view of the above, the methods presented herein consider only rotor blade flapwise bending deflections. Hingeless rotor blades are included as it is believed that the transfer function approach, while perhaps not covering the entire blade moment problem, will still provide valuable insight into the basic causes of the important flapwise stresses encountered on such blades.

METHODS FOR CALCULATING TRANSFER FUNCTIONS It has been shown that transfer function techniques, properly applied, may be expected to yield reasonably accurate solutions to the rotor blade flapwise bending problem. Two methods for calculating these functions will now be considered.

Approximate Transfer Function Analysis One method for calculating transfer functions involves the use of an appropriate simplified analysis of the type discussed in a previous section.

Such au analysis, while yielding only approximate, qualitative results, has the distinct advantage of providing a closed-form solution to the problem. Thus it is useful in analyzing trends established by more refined analyses and permits identification of mechanisms by which blade characteristics influence the resultant bending moment.

The basic assumptions on which the approximate analysis developed in this study is based are : ii The lift curve slope is constant (compressibility effects, stall, reverse flow neglected).

. The induced velocity over the rotor disc is constant.

, Blade motions for the articulated blade are limited to the rigid body flapping mode (assumed known) and two flapwise bending modes.

Blade motions for the hingeless blade are limited to two flapwise bending modes. The natural frequency of the first mode is assumed close to 1P so that the dynamic response of this mode can be approximated by that of a rigid flapping mode of an articulated rotor.

, Flapping harmonics above the second and third are negligible for the articulated and hingeless rotors, respectively.

.

Time varying aerodynamic damping and spring terms as well as intermodal coupling effects are neglected, except where noted.

7. The blade twist is linear.

8. The chord is constant.

9. The flap hinge offset is zero.

With these assumptions, it is possible to reduce the basic flapwise equation of motion of the rotor blade to two linear, differential equations with constant coefficients. The airloads become known periodic functions and the solution to the equations can be obtained from simple vibration theory (Ref. ll) by assuming a periodic solution for the response of each bending mode. By such an approach, the harmonic coefficients for the i th bending mode are given by = Ai,n,JFf,( i,n,c, or s (ll) qwi,( ),n,cors _,'E i _wi 2 where the subscript ( ) is used to denote the independent parameters e75 , el, etc.

Equation (ll) is a classical equation representing the response of any linear second order dynamic system to periodic excitation. M is an effec- tive mass constant, A is the dynamic amplification factor, _w is the natural frequency ratio, and F is the amplitude of the nondimensional excitation force. The analysis given in Appendix A is devoted to defining the relation between these general dynamic parameters and specific parameters defining the rotor blade system.

Computer Analysis To obtain transfer functions suitable for quantitative calculations, the assumptions regarding compressibility, reverse flow, number of modes retained, and aeroelastic coupling must be eliminated and, hence, digital computation programs must be employed. It should be emphasized that the elimination of these assumptions does not destroy the basic linearity of the problem but merely multiplies the mathematical operations involved manyfold. A refined version of the UAC Normal Mode Transient AnaJLysis, described in Ref. l, was available for this purpose. A brief description ofthis program is given in Appendix B. A convenient feature of the program is the capability of either including the combination of flapwise, chordwise, and torsion modes (fully coupled) or include- ing only flapwise modes. Several comparisons of theoretical and experimental results which serve to demonstrate the generally accurate results obtainable with the program are presented in Figs. 1 and 2. For brevity, hereafter this analysis will be referredto as the computer analysis.

Transfer functions of flapwise bending moment are obtained from this program by (1) suppressing chordwise and torsional mode effects, (2) independently vary- ing the input parameters e75 , e I , k c or X s (also Als , Bls , and _B for hingeless rotors), and (3) suitably normalizing the resulting computed bending moments. For example, the collective pitch transfer function is simply (12) _ R ) M(T,_) The harmonic transfer coefficients presented herein were determined by the computer analysis. The ability to determine transfer coefficients by the computer analysis as well as by the previously described approximate analysis results in a relatively ideal situation where the advantage of computer-type solutions (accuracy) is combined with that of closed-form solutions (insight).

Q PARAMETERS AFFECTING TRANSFER FUNCTIONS The several parameters which influence articulated rotor blade transfer functions can be determined from inspection of Eq. (1). By definition, the transfer functions are those functions multiplying the independent parameters, e75 , e i , and k c in the final solution to Eq. (1) and, consequently_ are affected by the remaining parameters appearing in Eq. (1). These are listed below : List of Parameters Affecting Articulated Blade Transfer Functions 1. Advance ratio, 2. Mass parameter, MP = pCoR/2m o m B • Blade natural frequencies, w w i 4. Blade mode shapes (and derivatives), Ywi ( Yw/,)'w// ) 5. Blade mass distribution, m = m/m o 6. Blade chord distribution, _ = C/C o 7. Flap hinge offset, 8. Tip loss factor, B 9. Llft-curve slope (which depends upon tip Mach number, Reynolds number, airfoil section, and stall), S lO. Blade flapping, ll. Froude number, _2R/@ The above list of parameters, excluding items 7 and lO, is applicable to hingeless rotor transfer functions with the addition of transfer functions relating cyclic pitch angles, Als and Bls , and built-ln coning, _B" It may also be shown (Ref. 12) that ww and Yw are uniquely determined if the dimensionless quantity Elo/mo_R 4 (frequency parameter) is specified along with the blade mass distribution, m , stiffness distribution, £-_ , root restraint condition and offset. In addition, blade flapping, B , is principally determined by e75 , X c , _ , tip loss factor, offset, and mass parameter.

The mass parameter, MP, may be identified In Eq. (1) as a multiplier of the aerodynamic terms, and is related to the conventional Lock number, y , of a uniform blade by the equation: MP : pCoR = --K- Y (13) 2m o 6a The basic parameters influencing the transfer functions are therefore given in the list below: List of Parameters Affecting Articulated and Hingeless Blade Transfer Functions i. Advance ratio, 2. Mass parameter, MP = pCoR/?_m o • = R B Frequency parameter, FP EIo/mo(nR) 2 4. Mass distribution, _ = m/m o 5. Stiffness distribution, El = El/El o 6. Chord distribution, _ = C/C o 7. Flap hinge offset, e (articulated only) 8. Tip-loss factor, B 9. Tip Math number, MI,90 i0. Airfoil section ii. Reynolds number 12. Blade loading (i.e _, stall effects) iB. Froude number, _2 R/g As indicated above, the blade mass, stiffness, and chord distributions have been normalized with respect to values at a particular radial station. The reference radial station is arbitrary and was selected as the 55% station for all blades considered in this investigation.

SELECTION OFPARAMETERS FORCHARTS

For maximum utility, the transfer coefficient charts should, if possible,

be sufficiently extensive to permit assessmentof the effects of major design

parameters on bending moment. The thirteen parameters affecting the transfer

functions, enumeratedabove, are obviously too manyto permit the generation

of all-incluslve charts. The numberof parameters must be reduced to a

manageablesize if a start is to be madeon the problem. This section discus-

ses the procedure followed in selecting the parameters to be included in the

transfer coefficient charts.

Parametersof SecondaryImportance

Several of the parameters which affect the transfer functions are of

secondaryimportance and can be eliminated from further consideration at this

time. Theseare : l.

Froude number. Gravitational effects are generally small and this factor was neglected.

, Reynolds number. For the range of applicability of the transfer functions (i.e., substantially unstalled flight conditions), this parameter will have little effect. Reynolds number would, of course, be useful as a guide in defining the range of applicability of the resulting functions for a given size rotor.

Airfoil section. The effect of choice of airfoil section on the o transfer functions is n_gliglble provided that the llft curve slope - Mach number relationship is unchanged. This would substantially be the case when the airfoil is operating below critical Mach number.

As such conditions are the only ones to which transfer function techniques are applicable, the choice of airfoil, like Reynolds number, principally determines the operating conditions over which the transfer functions apply. The use of cambered sections presents no problem provided the angle for zero llft is either constant and/or varies linearly along the blade. In such situations, the transfer functions are unchanged and the effective collective pitch and linear twist used to scale the functions are simply redefined.

2O Constant Parameters The following parameters are sufficiently constant for most rotors that they can be eliminated as variables.

io Tip loss. For constant inflow analyses a tip loss factor, B s of 0.97 is generally assumed and was used in this study.

.

Flap hinge offset ratio. For articulated rotors, the flap hinge offset ratio, e , is small -- typically 0 to 0.05. (For such small values, this parameter could equally well be considered to be of secondary importance.) A value of zero was selected on the basis of compatibility with Ref. 7.

.

Tip Mach number. For rotors operating in the 0.25 to 0.5 advance ratio range, this parameter is determined principally by the choice of rotor tip speed, _R , a quantity which, in turn, is governed by conflicting retreating blade stall and advancing blade compress- ibility requirements. A typical tip speed value of 672 fps was used for advance ratios equal to or less than 0.5. At higher advance ratios, rotor tip speed (and hence tip Mach number) is usually determined by compressibility limitations on the advancing blades.

In this investigation, the rotor tip speed for advance ratios greater than 0.5 was based on the requirement that the advancing tip Mach number be equal to 0.9. It should be noted that only one value of rotor tip speed was considered at each advance ratio. However, it is believed that the transfer functions so determined could also be applied to obtain good qualitative results for rotors operating at other tip speed values.

Chord distribution. A nominally constant chord distribution compatible with that of Ref. 7 was assumed in this investigation since nearly all existing rotor blades are of constant chord design.

Sensitivity Studies to Select Final Blade Design Parameters With the elimination of the parameters mentioned above as variables, the remaining parameters affecting the transfer functions are: i. Advance ratio, 2. Mass parameter, MP = pCoR/2m o R 2 B- Frequency parameter, FP = E r0/mo(nR) _

_. Massdistribution, m = mlm o

m 5. Stiffness distribution, E! = El/El o This list still represents a formidable number of potential parameter combina- tions (especially when one considers the large number of combinations of mass and stiffness distribution which could be considered). It is not feasible to generate transfer functions for all possible combinations -- nor is it particularly desirable from an efficiency standpoint, as certain combinations of parameters would probably represent impractical blade designs. As an example, an investigation of uniform blades of conventional material construc- tion having reasonable aspect ratio and large frequency parameter values would be of little value since large frequency parameters imply large airfoil thick- ness-to-chord ratios. On a uniform blade, such large thickness ratios over the entire blade radius would be impractical from an aerodynamic standpoint.

A further example would be the probable impracticality of a blade having a large inverse taper in mass distribution (mass increasing toward the tip) combined with a large conventional taper in stiffness distribution (stiffness decreasing toward the tip). It is evident, then, that some care must be exercised in selecting parameter combinations investigated if the most infor- mation is to be obtained from the number of calculations which are considered feasible at this time.

It was decided to limit the scope of the present investigation to the development of transfer coefficient charts which would include the effects of advance ratio and two of the four blade design parameters listed above. In addition, only blades having low structural stiffness compared to centrifugal stiffness would be considered. Such blades are used on nearly every current helicopter.

As an aid in selecting two of the four remaining blade design parameters (mass and frequency parameters, and mass and stiffness distributions) for inclusion in the transfer coefficient charts, a sensitivity analysis was conducted in an attempt to establish the relative importance of each at an advance ratio of 0.5. For this purpose, transfer coefficients were determined for a reference blade having uniform chord, mass and stiffness as well as a mass and frequency parameter combination typical of existing rotors. The sensitivity of these reference transfer coefficients to changes in mass param- eter, frequency parameter, and mass and stiffness distributions were then determined for both articulated and hingeless blades. To insure that the mass and stiffness distributions selected were reasonably compatible and bore some resemblance to practical blade distributions, it was decided to relate the distributions to specific changes in the blade structure. In accordance with this philosophy, a conventional D-spar blade was set up as a reference and simultaneous changes in mass and stiffness distributions such as would result from linearly tapering either the wall thickness or the depth of the blade spar

with radius were considered. Also, the change in massdistribution resulting

from the addition of a concentrated weight at the blade tip (tip weight) was

considered. These three specific changesin blade structure may be defined by

three newblade design parameters - tip weight parameter, TWP,wall thickness

parameter, WTP,and spar depth parameter, SDP. Theseare defined by the

following equations : TWP = TIP WEIGHT = WTIP (14) BLADE WEIGHT WBLADE WTP SPAR WALL THICKNESS AT TIP _ twTIP (15) SPAR WALL THICKNESS AT ROOT tWROOT SOP = SPAR CAVITY DEPTH AT TIP ° YST_P (16) SPAR CAVITY DEPTH AT ROOT YSROO T Figure 3 indicates schematically the type of blade designs considered.

To conduct the sensitivity study, it was necessary to define values of the blade design parameters for the reference blade and ranges over which the various parameters could be expected to vary. The parameter reference values and ranges used in the study for both the hingeless and articulated blades were defined from considerations given in Appendix C and are listed below.

Parameter Reference Value Range I. Mass Parameter, MP O.3 O.1 to 0.5 2. Frequency Parameter, FP 0.0025 O.OO1 to 0.O15 3z Tip weight Parameter, TWP O.O 0.0 to 0.5 4. Wall thickness Parameter, WTP 1.O 0.O to 1.O 5. Spar depth Parameter, SDP 1.O O.0 to 1.O To facilitate identification of the numerous combinations of blade design parameters studied, blade numbers were assigned to each. A list of blade numbers and corresponding parameter combinations is given in Table I. The mass and inertia distributions for the various blades are presented in Figs. 4 and 5 and discussed in more detail in Appendix C. Note that the distributions have been normalized by the values at the 55% radial station. The natural frequen- cies of the principal flapwlse bending modes for each blade considered are given in Figs. 6 and 7.

Results of the Articulated Blade Sensitivity Study. - To illustrate the type of results obtained in the sensitivity study for articulated blades, the variation of the collective pitch transfer coefficients at the 55% radial station (critical station) with mass parameter is shown in Fig. 8. As mention- ed previously, the transfer coefficients are simply the harmonic coefficients defining the various harmonics of nondimensional flapwise moment produced by a unit change in one of the independent parameters (in this instance, e75).

Results similar to those of Fig. 8 were generated for all blade design param- eters (FP, MP, etc.) and all of the independent parameters ( e75 , e I , X c ).

To facilitate comparing the sensitivity of the transfer coefficients to changes in all of the blade design parameters, the amplitudes of the transfer coefficients for each harmonic were determined for the 55_ radial station.

For example, the amplitude of the collective pitch transfer coefficient for a given harmonic, n , is defined as: (17) C8_6' , M'876 n,c) 2 n : , +( 8,,,n, s) The variation of transfer coefficient amplitude with each blade design parameter is presented in Figs. 9 through ll, for the three independent parameters.

Rather than completely nondlmenslonallzing the transfer coefficients, for convenience, the units per degree have been retained with the obvious exception of the inflow ratio transfer coefficient. Arrows have been used to indicate the design parameter values for the reference blade. For purposes of compar- ison, the same transfer coefficient amplitude scales have been retained within each figure. The sensitivity of the transfer coefficients to a given blade design parameter is indicated by the variation of coefficient amplitudes fron_ reference blade values.

Considering the entire range of parameter values presented, it is shown that the blade design parameters can essentially be divided into three levels of decreasing importance : 1. Mass parameter Spar wall thickness parameter 2. Frequency parameter Tip weight parameter 3. Spar depth parameter That is, the transfer coefficients are (i) most sensitive to variations in mass parameter and spar wall thickness variations, (2) less sensitive to frequency parameter and tip weight parameter variations, and (3) relatively insensitive to variations of spar depth parameter.

Analysis of the results indicates that the above order of importance is determined primarily by the fact that the blade's mass characteristics play a greater role in determining the blade dynamic response than its structural stiffness characteristics. This is due to the presence of the large centrifugal forces which tend to dominate the total stiffness characteristics of the blade.

As a result, it is not surprising that the spar depth parameter is the least important for, as shown in Figs. 4 and 5, it principally influences blade structural stiffness. The remaining parameters all influence the effective mass (M) of the blade in Eq. (ll) and alter the dynamic response of the system.

Transfer coefficient sensitivity results for other radial stations were examined and, in general, do not indicate any significant departures from the characteristic trends shown in Figs. 9 through ll for the 55% radial station.

From the sensitivity results based on the nondimensional bending moment transfer coefficients, frequency parameter and mass parameter were selected as the two parameters for inclusion in the transfer coefficient charts. Although the sensitivity to spar wall thickness parameter was found to be greater than that to frequency parameter over the entire wall thickness parameter range presented, the transfer coefficients are relatively insensitive over the impor- tant range from 0.5 to 1.O. Since blade designs having wall thickness taper ratios approaching zero are considered extreme, and since frequency parameter is a primary variable even for conventional blades, the frequency parameter was selected. The sensitivity of the coefficients to tip weight variations is of the same order as that due to frequency parameter variations; however, frequency parameter was selected because blades having tip weights are considered to represent a special design category. Although the selection of mass and frequency parameter implies that the transfer coefficients generated are, strictly speaking, restricted to uniform blade applications, the relative insensitivity of the transfer coefficients to wide variations of spar depth and moderate variations in wall thickness permits their application to certain nonuniform blade designs as well.

Results of the Hingeless Blade Sensitivity Study. - As in the sensitivity sZudy for articulated rotors, blade parameters analyzed for hingeless rotors included mass and frequency parameters, and nonuniform blade mass and stiffness distributions as described by tip weight parameter, wall thickness parameter, and spar depth parameter. The blade design parameters, based on available information for existing hingeless blades, were found to lie within the parameter ranges considered for the articulated designs. Thus, for consistency, the same reference values and parameter ranges used in the articulated blade analysis were chosen for the hingeless blade analysis. Transfer coefficients relating nondimensional bending moment to collective pitch ( e75 ), twist (el) , inflow ratio ( _c ), preconing ( _B ), and cyclic pitch ( Als and Bls ) were computed for the reference blade at an advance ratio of 0.5. The sensitivity of these results to variations in blade design parameters are presented in Figs. 12 through 17 for the critical station ( T = 0). The results indicate that the steady and first harmonic transfer coefficients are generally an order of magnitude greater than the second and third harmonic coefficients. Since, during normal operation much of the steady and first harmonic would be elimi- nated by the use of preconing and cyclic pitch, the higher harmonics comprise a significant portion of the net moment on the blade. For this reason, the ordinate scale of these harmonics have been expanded by a factor of lO.

Unlike the transfer coefficients for articulated blades, it is difficult to generalize the sensitivity results for the hingeless blades. It is shown in Figs. 12 through 17 that the transfer coefficients are sensitive to all non- dimensional blade design parameters including spar depth parameter, a parameter found to be relatively unimportant in the articulated blade analysis. However, as for articulated blades, the coefficients are relatively insensitive to spar wall thickness if extreme variations are excluded. Although the sensitivity of the hingeless blade transfer coefficients to each of the blade design parameters was found to be significant (except possibly for wall thickness parameter), mass parameter and frequency parameter were selected as the two parameters for further analysis. These were chosen largely on the basis that the results would be compatible with the articulated blade results.

Advance Ratio Selection Since the intent of this investigation was to develop a set of flapwise moment charts compatible with the performance charts of Ref. 7, the following advance ratios were selected for analysis: = 0.25, 0.4, 0.5, 0.7, 1.0, 1.4 From the previously defined variation of rotor tip speed with advance ratio, the flight velocities corresponding to each advance ratio can be computed.

The relationship between these three quantities is shown in Fig. 18 where the specific combinations covered in the transfer coefficient charts are noted.

Frequency Parameter Variation for High Advance Ratios It should be noted that rotor tip speed is constant for advance ratios up to 0.5 and, as a result, the frequency parameter for a given blade design is also constant. In contrast to this, at higher advance ratios, the frequency parameter for a given design varies with _ because of the reduction in tip speed necessary to avoid compressibility losses. To account for this, the frequency parameters found to be representative in the low advance ratio range were simply ratioed using the following relation: (18) = F P_.<o.5(0.447) ( I -t- p. ) z F P/,. >o.5 = FP_ _<o.5 t 'O'R_ <-0.5 12 _R_>o.s The constant (0.447) is based on sea-level standard conditions, an advancing tip Mach number of 0.9, and a tip speed for advance ratios below 0.5 of 672 ft/sec (see Eq. (C6)). Figure 19 shows the variation of frequency param- eter with _ for the three values of FP_<o. 5 used in the transfer coefficient design charts.

Summary of Parameters Selected for Charts The parameters selected for the transfer coefficient charts are summa- ri_ed below.

Independent bending collective pitch, @75 moment parameters blade twist, 81 inflow ratio, k c (for articulated blades), k s (for hingeless blades) cyclic pitch, Als) cyclic pitch, Blslhingeless blades only preconing, _B Blade design param- mass parameter, MP = 0.i, 0.3, 0.5 eters (Blades 1-9) frequency parameter, FP = O.001, 0.0025, 0.01 (for _% 0.5); FP/(l+ _ )2 = 0.oo0447, 0.00112, 0.00447 (for @ > 0.5) Advance ratio - _ = 0.25, 0.4, 0.5, 0.7, 1.0, 1.4 Harmonics n = O, i, 2, 3 for _ _< 0.5 n = O, i, 2, 3, 4, 5 for _ > 0.5 Blade stations - complete radial distribution (for articulated blades); _ = 0 and _ = 0.55 (for hingeless blades) DESIGN CHARTS CF _RNDING MOMENT TRANSFER COefFICIENTS Transfer coefficient charts for uniform, low stiffness articulated and hingeless blades were determined for all combinations of the parameters listed in the above summary. The complete set of charts are presented at the end of this report in Figs. 52 through 177. (Tabulations of all transfer coefficients, including five harmonics for all a_vance ratios and six stations for both articulated and hingeless blades, are presented in Part II of this report.)

The organization of the figures forming the charts is outlined in Table II.

For discussions purposes, sample charts are presented in Figs. 20 and 21.

The radial distribution of the collective pitch transfer coefficients for articulated blade design 5 (reference blade) are presented for three harmonics at advance ratios 0.25, 0.4, and 0.5 in Fig. 20. A sample chart for hingeless blades is presented in Fig. 21, where the variation of the collective pitch transfer coefficients at _ = O with advance ratio (_-- 0.25 to 0.5) is presented for blade designs having a frequency parameter of 0,0025 and mass parameter values of O.i, 0.3, and 0.5. The charts were automatically processed by a computerized plotter. To simplify the reading of the charts, grid lines have been spaced at one-half inch intervals.

The transfer coefficients for articulated blades have been plotted versus radial station to permit the determination of bending moment at any point along the blade. The transfer coefficients for hingeless blades have been plotted versus advance ratio for two specific radial stations (_ = 0 or 0.55) because the extreme sensitivity of the coefficients to radial location precluded presentation on the same plot (see Fig. 22).

Scales for the articulated blade charts are constant for all harmonics for a given blade design. However, the scales were varied, as required, for each design for increased accuracy of reading. Differing scales for different blade designs were desirable because although transfer coefficient values for a given design may appear small relative to those for another, the resulting bending moments are not necessarily also small. For example, the transfer coefficients generally decrease in magnitude with increasing frequency param- eter; however, if the frequency parameter was increased by increasing the modulus of elasticity through a material change, the bending moments may actually be increased because the transfer coefficients are multiplied by EI/R to obtain bending moment. Thus, the scales were generally determined by the magnitude of the maximum harmonic transfer coefficient for each blade design.

For hingeless blades, the zero and first harmonic transfer coefficients can be an order of magnitude greater than the contribution of the higher harmonics. However, due to the previously explained trim considerations, the higher harmonics are equally important. Thus, two sets of scales were used for

the hingeless blade transfer coefficients; i.e., within each figure, one scale

was used for the zero and first harmonic transfer coefficients and a second

for the higher harmonic coefficients.

Finally, due to the sensitivity of the transfer coefficients to advance

ratio, the charts have been divided into two advanceratio groupings : _ = 0.25

to 0.5 and _ = 0.5 to 1.4. This permits the lower advanceratio charts to

be read with a greater degree of accuracy. Also, it will be shownthat this

divides the transfer coefficient charts into (1) those which are quantitatively

accurate (i.e., those for @ = 0.25 to 0.5), and (2) those which are less

accurate due to the sensitivity of the flapwise moments to control settings

and torsional coupling at high speeds (i.e., those for @ _ 0.2). Three

harmonics are included in the charts for the lower advance ratios and five

harmonics for the higher advance ratios. The fourth and fifth harmonic transfer coefficients are negligible for the lower advance ratios.

APPLICATIONS OF TRANSFER COEFFICIENTS In this section of the report several examples illustrating various applications of the transfer coefficient charts and techniques are presented.

Included in the articulated rotor applications are (1) a sample calculation illustrating the detailed procedures for using the charts to compute stress, (2) a correlation with experimental and computer analysis results for the H-34 rotor, (3) an investigation of the effect of advance ratio and blade design changes on vibratory stress, and (_) the development of sample stress contours for use in conjunction with the performance charts of Ref. 7.

Included in the hingeless rotor applications are (1) an investigation of the effect of advance ratio and blade design changes on vibratory stress, (2) an application of the transfer coefficient charts to a nonuniform hingeless blade, and (3) a brief discussion of procedures for including steady hub moments.

Several comparisons are then presented of results obtained from the transfer coefficient charts and from the approximate analysis developed in Appendix A.

The section is then concluded with a discussion of possible techniques for applying the transfer coefficient charts to estimate teetering rotor bending moment s.

Articulated Rotor Applications Sample Stress Calculation. - To demonstrate detailed procedures involved in the use of the transfer coefficient charts in conjunction with the perfor- mance charts of Ref. 7, the following example is presented. In the example, it is required that the flapwise stress distribution at the 65% radius of an H-34 rotor blade operating at the following flight condition be determined.

150 kts

Forward sl_ed, V

8500 lb

Lift, L

-650 ib (propulsive force)

Drag, D

650 ft/sec

Tip speed, _R

o.39 Advanceratio, O.O02195 slugs/ft 3

Density, p

0.79

Tip M_chnumber,M l, 90

_ae rotor characteristics are : Nu_er of blades, b 28 ft Radlus, R 1.37 ft Chord, c 0.0622 Solidity, Blade t_rist, e I -8 dog Flap hinge offset/radius, 0.05 M_ss and stiffness distributions see Fig. 23 The blade bending stresses are determined using the following procedure: i. From the weight and flapwise moment of inertia distributions, deterge representative values of blade mass per unit length (m o ) and structural stiffness ( ETo ). From Fig. 23, w = o.44 ib/in. ( m o = 0.164 slugs/ft) I o = 1.47 in. 4 ( EI o = i07(1.47) = 1.47 x 107 ib-in. 2) Note that for articulated blades, the high mass and stiffness region at the root end of the blade _ be neglected for the purposes of determining moments in the more critical outboard regions of the blade.

2. Calculate the us and frequency parameters.

POoR = 0.002195 (I.37)(28) = 0.26 MP = 2rn o 2 (0.164) (I.47 x I0 "/'}/144 El: 9 = 0.00188 ,¢ FP = m oRZ(,t"),R )z 0.164(28) z (650) 2 3. Check the applicability of the transfer coefficient charts to the current problem of interest.

a. Articulated or hingeless root restraint ?

YES (articulated) YES b. Nominally constant chord?

c. Tip speed near value in Fig. 18 YES (65Ors 672 fps) for same _ ?

d. Small flap hinge offset? YES e. Advance ratio, mass parameter and frequency parameter within limits of charts? YES f. Mass and stiffness distributions reasonably represented by uniform blades ?

YES (see Fig. 23) (As an aid in answering this question the results from the sensitivity studies may be used (Figs. 9 through ii)).

If the answers to any of these questions are negative, the application of the charts may still be of interest to establish qualitative trends for preliminary design.

4. Nondimensionalize rotor lift and drag.

CL L 8500 = = = 0.0596 o" pbc R(,O, R) 2 (0.002195)(4) (I.37)(28)(650) z C o = D - 650 = = - 0.00457 o- pbc R (,Q,R) z 0-002195(4)(1'37)(28)(650)z 5- Use the performance charts in Ref. 7 to obtain 875 and a C .

Figures 7, 13, and 14 of Ref. 7 are used to interpolate for _ = 0.39 and M_,90 = 0.79 for 8t = -8 deg. The correction for solidity ( _ = 0.0622 vs = O.i of Ref. 7) is included. The resulting control angles are: a c = -10.9 deg 875 = 8.5 deg Check operating point to insure that significant stall is not present.

r e Calculate thrust coefficient CT and inflow ratio (X c ).

# r(CL/_)C05=C + (Co/or) sin=c] = 0.00358 CT= J L Xc = vsin°q cT = - 0.078 D,R Z_ (Caution: k c in the control axis system is required for the articulated transfer coefficient charts; however, k s in the shaft axis system is required for the hingeless blade charts.)

7. Using the transfer coefficient charts, interpolate to determine the transfer coefficients at the required radial station for the given mass param- eter, frequency parameter, and advance ratio.

The articulated rotor charts for the low advance ratio range (Figs. 52 through 78) were used to interpolate to MP = 0.26, FP = 0.00188, _ = 0.39, and _ = 0.65 to obtain the following transfer coefficients.

Transfer Coefficients N) C ors MS?s,n)cors ,n,cors Mkc, n, cor s 0 0.0008 0.0092 -0.53 l)c -0.0118 0.0027 -0.85 l,s 0.0105 =0.0104 0.94 2,c 0.0063 0.0019 O.61 2,s -0.0029 0.0058 -O.23 3,c -0.004 0.0025 -0.03 3,s -0.0065 -0.002 -0.47 8. Compute the transfer function for each independent parameter ( 875, 81 , k¢ ) by performing a harmonic summation of the transfer coefficients.

M( )= M( ),0- _-(M( ),n,c cosn_+M(),n,sSinn_) I1=1 _ Note that ( ) is used to symbolize one of the independent parameters. The resulting transfer functions are presented in Fig. 24.

9. Multiply the transfer functions M-87s ' Me,-- , and --Mxc by the values of the respective independent parameters ( 875, 81 , and X¢ ), and by the multi, plying factor for stress (Ey/R ) to obtain the contribution to stress of each independent parameter. Then, calculate the total stress distribution by summing the contributions of each of the independent parameters.

Ey - 29400 PSI R °'F _ E y -

- + e, + %c)

Note that if bending moments were desired, the multiplying factor would be ET/R . The resulting contributions of each of the independent parameters to the stress distribution, for the current example are presented in Fig. 25.

The total stress distribution is presented in Fig. 26b.

If only the total, and not the components of stress du_ to the indepen- dent parameters, were of interest, it would have been more efficient to retain the component contributions in harmonic form and perform the harmonic summation once in Step 9 rather than for each parameter as in Step 8.

Comparison with Experimental and Com_uter Analysis Results. - The transfer coefficient charts were used to predict the flapwise stress distributions for the H-34 rotor operatingat llO, 150, and 175 kts. These results were compared with experimental results (NASA/Ames wind tunnel test) reported in Ref. 13, and with the results of the computer analysis described in Appendix B. Chordwise and torsional flexibilitywere included in the computer analysis and the controls ( 875, kc ) were adjusted until the correct lift and drag were computed (when used in this manner, the computer analysis will be referred to as "fully coupled").

The results are compared in Fig. 26 for the three flight speeds and two radial stations. As shown, the results obtained using the transfer function approach are only slightly less accurate than those from the computer analysis (fully coupled) _ud both agree well with experiment.

The characteristic stress distribution is generally the same for the three flight speeds and two radial stations presented. The azimuth locations for the peak stresses are approximately 120 and 250 degrees, although in some cases the positive peak value is located at 0 degrees. To obtain some insight for the characteristic stress distribution, the transfer functions and the components of stress representing the independent contributions of collective pitch, twist, and inflow ratio presented in Figs. 24 and 25 were examined. The distributions presented are for the 150 kt cc_dition and the 0.65 radial station. It is shown in Fig. 25 that the contributions due to 875 and kC are comparable in magnitude and opposite in sign and thus tend to cancel. As a result, bl_ie twist contributes significantly to the total stress amplitude.

Effect of Advance Ratio and Design Chan6es on Stress. - The transfer coefficient charts were used to compute the variation of blade stress with advance ratio for several blade designs operated in a possible compound helicopter mode of operation. Table III lists the blade designs considered and the flight conditions at which stresses were calculated. The basic design variations which were made involved increasing twist, 81 , from O to -8 deg and increasing frequency parameter from 0.0025 to O.O1 through the use of advanced fiber reinforced materials such as boron-epoxy. The Ey/R values used to convert the transfer coefficients to stress are representative of blades having spar thicknesses equal to 12% of the blade chord and aspect ratios of 19. Flexural moduli (E) of lO 7 and 2.8 x lO 7 for aluminum and boron- epoxy, respectively, were used.

The compound helicopter mode of operation considered was defined by the following requirements.

i.

The aircraft propulsive force is provided by some means other than the rotor so that the rotor may be operated at an angle of attack of zero degrees.

e The rotor generates a lift coefficient/solidity (CL/_) of 0.051 at advance ratios up to 0.5. At higher advance ratios, the CL/_ of the rotor is equal to 85% of the lower stall value as determined from the charts of Ref. 7 (values of CL/_ are given in Table III).

e Rotor tip speed is equal to 672 fps at advance ratios up to 0.5 and is reduced at higher advance ratios to produce an advancing tip Mach number of 0.9.

_e Rotor collective pitch values used to scale the transfer coefficients were determined from the charts of Ref. 7 and are listed in Table III.

Sample results of the calculations for the reference aluminum blade design with zero twist are _resented in Fig. 27 in the form of azimuthal time histories of stress for the 55% radial station. These results indicate large increases in overall vibratory stress level and higher harmonic stress content as the advance ratio of the rotor is increased. Also presented in Fig. 27 are stress time histories computed using the computer analysis (fully coupled) including two blade edgewise and one torsional degrees of freedom and trimming rotor control settings ( 8_5 and k¢ ) to produce the CL/_and _c values corresponding to the rotor mode of operation described previously. Blade edgewise and torsional stiffness characteristics were selected so as to provide the following frequency ratios.

Natural Frequency Ratios Advance Ratio Aluminum Blade o.25-o.5 o.7 1.o 1.4 0.242 0.242 0.243 0.243 1st edgewise mode 3.353 3.587 3.949 4.446 2nd edgewise mode 1st torsional mode 7.142 8.095 9.483 i1.356 Boron-Epoxy Blade 0.243 0.244 0.244 0.244 1st edgewise mode 5.274 5.855 6.717 7.904 2nd edgewise mode 1st torsional mode 7.142 8.095 9.483 11.356 Flapwise frequencies can be obtained from Fig. 6a.

The results of Fig. 27 indicate that while the qualitative accuracy of the transfer function approach is good at all advance ratios, its quantitative accuracy decreases with increasing advance ratio. It was determined that this loss in accuracy was due principally to the increased elastic torsional deflections of the blade which occur at high _ because of adverse reverse flow effects. These deflections influence airloads as well as the control parameters required to trim the rotor at the required C c /_ . To demonstrate this point, additional calculations were made retaining only flapwise modes and using trim control values from Ref. 7 in the computer analysis. These results are also presented in Fig. 27 and for the higher advance ratios compare much more favorably with the transfer function results. Table III indicates that when controls from Ref. 7 are used in the computer analysis the CL/_ differs somewhat from the prescribed values.

Results similar to those of Fig. 27 are presented in Figs. 28 through 30 for the other blade designs considere_and substantiate the previously reached conclusions regarding the adverse influence of advance ratio on stress magni- tude and the quantitative accuracy of the transfer function approach.

The results of Figs. 27 through 30 are summarized in Figs. 31 and 32 where the vibratory stress amplitudes predicted by the various approaches for different desgins are compared. Figure 31 shows the effect of introducing blade twist for a given blade material while Fig. 32 presents a more direct comparison of stresses for the untwisted aluminum and boron-epoxy blades for the complete advance ratio range. It is evident from the results presented that increasing blade twist significantly increases blade stress at a given advance ratio. Considering that the vibratory fatigue stress limit for aluminum can be of the order 5,000 psi, a uniform blade with -8 degrees of twist would be limited to flight at advance ratios less than 0.4. Much higher advance ratios are achievable with untwisted blades, as indicated in Fig. 32. In fact, neglecting torsional effects (which account for the higher stresses shown by the fully coupled computer analysis), the vibratory stress for the untwisted blade reaches a maximum at an advance ratio of about i.O and then declines.

This decline is due primarily to the reduction in rotor lift and hence rotor collective pitch at the high advance ratios. A similar peaking in the stress for the -8 deg twist blades would not occur because blade twist is, of course, independent of advance ratio. Further analysis of the effects of blade twist are presented in a later section.

The results obtained also show that blade stresses for the boron-epoxy blade are approximately two times those for the aluminum blade. The advantage of the boron composite material results from its allowable vibratory stress which is more than double that of aluminum. If the blade dynamic response (and hence blade strain, assuming the same thickness) for the two blades were identical, the stresses for the boron blade should be higher by a factor equal to the ratio of the elastic moduli for the two materials (2.8); it is evident then that the dynamics of the blade response have, in fact, been changed by increasing the frequency parameter from 0.0025 to O.O1. Comparison of the transfer coefficients for the two frequency parameter values indicates that the 2P response of the blade is reduced significantly as the frequency param- eter is increased to O.O1. This reduction is due to the increase in the blade's first flapwise frequency ratio from 2.61P to 2.97P (see Fig. 6a), which implies both that the blade is stiffer and that the amplification of the 2P response is reduced. (The reduction in response may be estimated with reasonable accuracy using Eq. (ll) assuming that F- and M are constant).

Because of the relatively large damping of the first flapwise mode (about 20% of critical) there is little increase in the amplification of the 3P response so that the overall dynamic response of the blade is reduced significantly.

Stress Contour Plot. - As a further example of the utility of the transfer function approach, the transfer coefficients for one articulated blade design were used in conjunction with one of the performance charts of Ref. 7 to determine stress contour lines as a function of rotor performance.

A typical flight condition and blade design, defined by the following, were selected.

Advance ratio, # 0.4 Tip Mach number, Mi,90 0.8 Mass parameter, MP 0.3 Frequency parameter, FP 0.0025 Multiplying factor for stress, Ey/R 31400 psi Blade twist, 8 i -8 deg The one-half _eak-to-peak flapwise stress (vibratory stress) contours were computed for the critical 55% station and are presented in Fig. 33. The performance chart was taken directly from Fig. 14b of Ref. 7, and the stress contours have been superposed on the basic chart in increments of 250 psi.

Examination of the results indicates that for a given drag coefficient (Co/_) vibratory stress increases with increasing lift coefficient ( CL/_ ). Also, for a given lift coefficient, there exists a drag coefficient for which the vibratory stress is a minimum. This occurs where a stress contour line is tangent to a constant lift coefficient line. The locus of such points forms the minimum stress line indicated on Fig. 33. For those operating conditions where the rotor is providing an appreciable propulsive force,the vibratory stress is much less sensitive to changes in lift or propulsive force than for conditions where the rotor is contributing substantial drag.

The nature of the vibratory stress variation with C D /_ for one value of CL /_ is more graphically shown by the top graph in Fig. 34. For the Cd_ chosen, the optimum CD/_ (from stress considerations) is about -0.006. The bottom graph in Fig. 34 shows how the character of the stress time history changes as CO /_ varies. Analysis indicates that the minimum vibratory stress will occur at the CD/_ value for which the governing positive stress peak in the stress time history shifts sharply from one general azimuthal location to another. In this particular example, as C0/_ is decreased through the optimum value of -0.OO6, the positive stress peak shifts from a of 240 deg to a _ of 0 deg. Some of the factors causing this shift may be deduced from analysis of Fig. 35 where the contributions of the independent parameters to stress are presented.

Returning to Fig. 34, it is shown that the rate of change of peak-to-peak stress with drag coefficient for a given lift coefficient is reasonably linear on either side of the minimum stress point. This suggests the possibility of reducing the stress-performance information provided by contour plots to a more concise form through a linearization of the contour results. This is accomplished by describing the rate of change of ½ peak-to-peak stress with drag coefficient by two partial derivatives -- one for drag coefficients greater than the coefficient for minimum stress and one for coefficients less than the coefficient for minimum stress. The partial derivatives are presented in Fig. 36 as functions of CL/_ along with the minimum ½ peak-to-peak stress and the drag coefficient at which the minimum stress occurs. The vibratory stress may thus be expressed by the following linear equation relating stress to the drag coefficient for a given lift coefficient.

$7 (19) All values on the right hand side of the equation, (except the drag coefficient which, of course, must be specified) are presented in Fig. 36.

From the linear nature of the curves with C L/_ in Fig. 36, it is evident that the linearization process could be carried still further if desired by describing the curves in terms of their slopes and Intercepts. If this were done, it appears that stress contour plots of the type shown in Fig. 33 could be described with good accuracy by eight constants representing the slopes and intercepts of the four lines in Fig. 36.

The ability to express vibratory stress contour plots in terms of rotor performance characteristics has only been investigated for the contours presented in Fig. 33. Further investigation of this approach for other blade designs and advance ratios was beyond the scope of this study.

Hingeless Rotor Applications Sample Stress Calculation. - In this section, sample calculations are presented to illustrate the use of the transfer coefficient charts in conjunc- tionwith the performance charts of Ref. 7 to compute the effect of advance ratio and blade design changes on hingeless blade stresses. To apply the hingeless rotor transfer coefficient charts, it is necessary to determine control settings ( e75, 81, etc.) for a given flight condition. The performance charts, Ref. 7, were originally developed for articulated rotors and were used to provide control se_tings for such rotors in the preceding section. How- ever, the charts can be used with equal facility for hingeless rotors if trimmed, zero hub moment, flight is assumed. The equivalence, for performance considerations, of flapping coefficient, Of,for the articulated blade in Ref. 7, and cyclic pitch coefficient, BIS , for the hingeless blade allows rapid determination of the control parameters 875, ks , and B1s. This equivalence of a i for a pure flapping rotor and Bts for a feathering hingeless rotor is independent of the hingeless blade stiffness or its first bending mode phase angle, assuming zero hub moment (therefore zero first harmonic response) and neglecting higher harmonic responses. The tip path plane must be the reference plane when equating the two systems in order to produce identical aerodynamic forces relative to the free stream. Considering the above assumptions, the tip path plane of the hingeless rotor, therefore, is independent of its stiffness. The difference between the two systems exists $8 only in the alignment of the shaft under the rotor. With the pure (IP) flapping rotor, neglecting coning and lateral motions, there is only flapping with respect to the shaft ( O l ). To convert this system to a pure feathering rotor, without altering the tip path plane, the shaft is adjusted through the angle of to eliminate flapping with respect to the shaft. To maintain similar blade aerodynamics, theblades must be feathered, with respect to the shaft, an equal amount 01 (or, more properly, 81s). An additional parameter which affects hingeless blade moments is cyclic pitch coefficient, Als , and the articulated rotor equivalence of this parameter, assuming zero hub moment, is the flapping coefficient, -b I , which can be obtained with sufficient accuracy from Ref. 7 or Ref. 8, or alternately can be calculated using the following equation.

(2o) Eq. (20) was derived using an approximate analysis similar to that described in Appendix A. It should be noted that the value of -b I obtained from Ref. 7 should be multiplied by the quantity y/8 since the data presented in that report are for a Lock number, ¥ , of 8.0.

Since hingeless rotor steady hub moments are extremely sensitive to control parameter changes, it is difficult, using the performance charts of Ref. 7, to determine cyclic pitch controls with the precision required for trimmed zero moment flight. As a result, residual 1P moments (steady hub moments) may be present in the total flapwise moment time history obtained by scaling and superposing transfer coefficients. The size of the residual l_ moments can be significant relative to the higher harmonics; however, it will be shown that, for low stiffness hingeless rotors, cyclic pitch controls, Ais and BJs , do not significantly affect higher harmonic moments. Additional evidence of the relative independence, at constant lift, of the higher harmonic moments (or stresses) to the 1P moments produced by cyclic pitch is shown in Fig. 37. Here the stress distributions from the computer analysis are presented for two cyclic pitch settings for a sample rotor at an advance ratio of 0.5. Ai s cyclic pitch has been changed by 1 deg to effect a large change in 1P stress (equivalent to lO,OO0 ft-lb change in hub moment). The total stress distributions (as computed), and distributions which result when the first harmonic component of the total stress is omitted, are shown.

Although large differences are seen in the total stresses, nearly identical distributions result when the first harmonic is removed. It is possible, therefore, to arbitrarily remove the residual IP stress to obtain a good approximation to the final trimmed stress distribution. The accuracy of this approximation will decrease as blade pitch is increased to the point where IP flapwise moments are required to compensate for IP chordwise moments to achieve trim.

The general procedures described above were used to compute the effect of advance ratio on blade stress for several blade designs operated in the com- pound helicopter mode described previously for the articulated rotor (see page 34). Listed in Table IV are the specific designs considered and the advance ratios at which stresses were computed. As with the articulated rotor, th_ basic design variations involve increased twist and increased frequency param- eter (through the use of boron-epoxy). To illustrate the steps involved in obtaining the control settings from Ref. 7 the following example is presented.

Flight Conditions and Blade Characteristics = 0.25

M,,9o = 0. 5

Cu/_ = 0.051 as=O =0.i FP = 0.0025 MP= 0.3 ( y= 10) 8 i = -8 deg BS " 3 deg The control parameters, @75, 8t s , and k s are obtained from the performance charts, Ref. 7, as follows: For hingeless rotors, OrS : QC + 8111 01 : Bl: 4O Therefore, for a s= O, Bls = -aC = Gl • Figures 4 and 5 of Ref. 7 present the performance results for advancing tip Mach numbers of 0.7 and 0.8, respec- tively. @? 5 and Bis are obtained from each figure and linear interpolation is used to obtain the values at 0.75 Mach number. For this example, where 01 = - a c , points on Fig. 4(a) and 4(b) of Ref. 7 are determined where O I = - o¢ and CL/_ , = O.051. (If some other prescribed aS were used, such as 2 deg, then corresponding points would be found at o I = 2 -a c ). A rapid method for locating the correct G I is to crossplot points from Fig. 4(b) at C L /_ = 0.051 for two or more values of ac on Fig. 4(a) at corresponding values of G I .

For example, points on the C L /_ = 0.051 curve at a c = -2 deg and - 4 deg (Fig. 4(b)) can be placed on Fig. 4(a) at 01 = 2 deg and 4 deg and connected with a straight line (the corresponding points for the a s = -2 deg condition would be placed at O a = 4 deg and 6 deg). The intersection of this line with the C L /_ = 0.051 curve represents the O I (or Bls ) solution which satisfies the a s and Cd_ constraints. From this point, rotor drag coefficient, C0/_ , can be obtained and the equivalent point on Fig. 4(b) gives the collective pitch, 875. For the example under consideration, and after averaging the results obtained from Figs. 4 and 5, O I = 8% = 2.b_l deg 875 = 4.0 deg The inflow ratio, k s , is obtained using the conventional momentum equation assuming ks 2 << _z and C L = C T for low angles of attack: CT k s - p.tana s - _ = O.OlOZ (21) The longitudinal cyclic pitch, Als , is obtained from Eq. (20) as -0.9 deg.

The calculated control parameters are then used to obtain the transfer functions for { = 0 (from Figs. 107, llO, ll3, ll6, ll9, and 122) and then combined to form the total stress time history shown in Fig. 38. The upper figure compares the resulting total stress, which is seen to contain 1P components (and therefore isnot exactly trimmed), with the computer analysis results assuming flapwise modes only. The major discrepancy is seen to occur in the steady stress which results from the lower lift coefficient obtained with the more accurate analysis using control para/eters from Ref. 7 (see Table III). In addition, the absence of gravity effects in the transfer function charts produces slightly higher steady stresses at the same lift level (approximately l_O0 psi for this condition). The lower figure represents the trimmed results where the same stress time histories have been replotted excluding the 1P components. Shown also on this figure are the stress results using the fully coupled computer analysis where typical hingeless blade

chordwise and torsional characteristics were assumedandwhere the rotor has

been retrimmed (through collective pitch changes) to the correct CL/_ •

Effect of Advance Ratio and Des!_n Chan6es on Stress. - Calculations similar to those described above were performed to compute the stresses for the several designs and advance ratios listed in Table IV. Sample stress time histories at different _'s are presented in Figs. 38 through 40 for the twisted and untwisted reference aluminum blade. At the higher _'s useful results using the fully coupled computer analysis could not be obtained due to the extreme torsional deflections which were encountered. Also, the accuracy of the transfer function approach is generally reduced at the higher _'s due partly to the basic limitations of the method and also by the inability to obtain accurate control parameters from the performance charts of Ref. 7.

This is reflected in the lower lift levels which are normally produced by the computer analysis when the control parameters of Ref. 7 are used. It was found, for example, at an advance ratio of 1.4 that a 0.i deg error in 875 produced nearly a O.O1 change in CL/_ . This is over 25_ of the desired C L /_ and serves to demonstrate the extreme sensitivity that exists at the high advance ratios as well as the necessity for exercising considerable care in using the performance charts of Ref. 7 to obtain control parameters.

Emphasis should therefore be shifted from the quantitative to qualitative features of the results for the high advance ratio conditions.

The variations in the vibratory stresses (excluding 1P component) with advance ratio are summmrized in Figs. 41 and 42 where the effect of the twist and frequency parameter (material) design changes are shown. The results are generally consistent with the previously presented articulated rotor results in that higher stresses are produced by both increased twist and the use of boron-epoxy. It should be noted that the -8 deg of twist increases the stress at a given _ by about 30 to 50 percent for the hingeless rotor, whereas with the articulated blade the increase is about 300% (see Fig. 31). The reason for this will be discussed in a later section.

Nonuniform Hin_eless Blane Application. - In practice, it would not be possible to construct a hingeless rotor having blades uniform from the tip to the tenter of rotation due to the necessary offset produced by the finite shaft diameter and the variations in blade stiffness which result from blade retention and feathering bearing mechanisms. The principal effect that these nonuniformities have is to increase the effective flapwise stiffness of the blade. This higher stiffness is reflected primarily in the first mode natural bending frequency, _Wl , which has the major influence on the blade response and therefore the blade aerodynamics.

The frequency parameter calculated using the uniform outboard section of the blade would generally result in a first mode frequency below the frequency determined using the actual nonuniform stiffness distribution, including the

inboard effects. Whenusing the frequency parameter corresponding to an

improper frequency, the aerodynamicsand consequently the blade moments

determined from the transfer coefficient charts would be in error. The error

can be readily avoided, however, by entering the charts with an effective

frequency parameter corresponding to the knownfirst modefrequency, _wl.

This, in effect, assumes that the moment distribution on the nonuniform blade

is approximately the sameas that on a uniformblade having the samefirst

modenatural frequency. The procedure is best demonstratedby a sample

calculat ion.

Flight Condition and Blade Characteristics

p = o.oo2378 slugs/ft 3 F = o.25 CLI_ = 0.051 Col= = -0.0033 R = 17.5 ft C = 1.125 ft e_ = -4 deg 2R = 650 fps mo = 0.13 slugs/ft -- = 1.118 wW I UP = 0.18 The control parameters are obtained from Fig. 21 of Ref. 7 and Eqs. (20) and (21).

81s : 01 = 3.1 deg 875 = 6.0 deg Ais = -0.83 deg k s = -o.oe97 The frequency parameter corresponding to the first mode bending frequency, _wl , obtained from Fig. 7a, is O.O1. The transfer coefficients are then obtained from the charts in the normal fashion, and the total nondimensional moments, M , are found by using the control parameters and summing the harmonics. The actual blade moments are obtained by solving for an effective uniform blade El/R compatible with the frequency parameter and using the result to dimensionalize the coefficient M .

E_._ = ( FP)m (,Q,R)z R R = (0.O1)(O.13)(650)2(17.5) - 9600 ft-lb El M =-_-M = 9600 _ ft-lb m = i15,000 M in.-lb This uniform blade moment, as stated previously, is assumed then to be equal to the nonuniform blade moment.

The results of this method are compared in Fig. 43 to the results obtained from the computer analysis using the actual nonuniform blade charac- teristics. The moment time histories are presented with and without the residual 1P component and good agreement in the vibratory moments is indicated.

Again, the major discrepancy occurs in the steady component which is due primarily to the absence of gravity in the transfer function charts.

If the first mode bending frequency of the blade under consideration is not known, Ref. 12 provides a rapid and relatively accurate method for determining flal_wise bending frequencies for nonuniform hingeless blades.

Stead_ Hub Moment Application. - When a flight condition requires that a steady hub moment be provided to balance some moderate fUselage moment, the steady moment may be converted to a IP moment in the blade coordinate system at _= 0 using Eqs. (22) and (23) which can then be added directly to the higher harmonic moment distribution to obtain a good approximation of the total moment.

M,, c, ?= 0 = -6" (PM) (22) M,, s, i"=0 : _ (RM) (23) In addition, an estimate of the cyclic pitch required to produce this moment can be obtained by using the first harmonic transfer coefficients at _ = O and solving the following equations for Z_AI$ and ABf s .

"2(PM)R (2k) b E T _ MAIse, C_AIs + MB,s,I,C/k BIs Z(RM )R (25) -_ _A,s,I,SAA,S+ Me,s,l.sAe,s bEI Approximate Analysis Results In this section, transfer coefficients and total stress distributions are computed using the approximate analysis (Appendix A) for two blade designs operating at one advance ratio. The results are compared with those obtained using the transfer coefficient charts. The purpose is to demonstrate both the general accuracy of the approximate analysis and its usefulness in providing insight into some of the factors contributing to observed blade stress characteristics. The following rotor flight condition parameters were considered.

Articulated Hingeless MP 0.3 O.3 FP 0.0025 0.01 0.5 0.5 c,./o- 0.o51 o .o51 Co/o" -0.001 -0.0044 critical r 0.55 0 e7 5 2.2 deg 12 deg el -8 deg -4 deg k c, k s -0.OO5 -O.15 In performing the calculations, blade flapping derivatives obtained from the computer analysis neglecting edgewise and torsional effects were utilized.

Such derivatives should closely approximate those which can be derived from the tables of flapping coefficients presented in Ref. 7. The average lift curve slope appearing in the approximate analysis was taken as 6.87 -- a value equal to the unstalled lift curve slope at the 75_ radial station at a blade azimuth angle of 0 deg.

Articulated Blade Results. - The transfer coefficients computed by the approximate analysis for each radial station are compared with those from the charts in Fig. 44, where the approximate analysis is seen to yield gener- ally excellent qualitative results. In Fig. 45 the total stress for the 55% radial station computed by scaling and superposing the transfer coefficients of Fig. 44 are compared. The total stress results are broken down into components in Fig. 46 to illustrate the contributions due to inflow, collective pitch, and twist. (Note that, for increased clarity, a different scale for the inflow component has been used). Again, the approximate analysis results are in good qualitative agreement with the more accurate transfer coefficient chart results. In addition, it is evident that, for this particular flight condition, the stress component due to twist is the dominant one. This component can be analyzed further to define its harmonic content and the relative contributions of the first and second bending modes. The approximate analysis results indicate that the stress due to twist for this condition can be expressed as

-[

(_F)#_ --2600 + 260C0S_ - 3220sin_ + 1700 COS2_ +1560sin2_ +lOOcosS_ l +[160 + 80COS_ + 170sin_ - 290COS2_ + 20sin2_ - 200sin3_ i=1 + 60cos3_ - 30sin3_]i = z where the i subscript denotes the contribution of each bending mode. The dominant stress harmonics are the steady, 1P and 2P and the first bending mode is the principal contributor. Factors contributing to these harmonics can, in turn, be defined by examining the generalized force expressions given in Appendix A. Such examination indicates that the principal stress harmonics induced by blade twist are caused directly by the harmonics present in the following airload term (see Eq. (A10)).

(T-o.'/5)e, ul .z = el(_-.0.75) IF_'+2FPsin+ + -'2 (I -COS By contrast, the airloads due to the flapping motions induced by twist are negligible. From the form of the above expression, it is evident that air- loads exciting the first mode are proportional to the square of the local

velocity, and thus the largest and smallest response of the blade should be

expected on mhe advancing and retreating portions of the rotor disc, respec- tively. The question may be raised as to why the first bending mode is so responsive to blade twist airloads. The reasons for this are: (1) the natural frequency of the first mode (2.61P) is closer to the principal alrload excitation frequencies and (2) the radial distribution of airloads due to twist is such that considerable energy can be transmitted to the first mode.

To illustrate the latter point the radial distribution of blade airloads at of 90 deg produced by i deg of positive twist is compared in Fig. 47 with the mode shape of the first bending mode. The striking similarity of the two curves is apparent and implies that the twist airloads are largely orthogonal to (and hence do not excite) all modes but the first.

It should also be noted that the twist airload expression presented above contains vibratory terms involving first and second powers of advance ratio In the low advance ratio range (_ s 0.5) the @sin@ term is the largest term. This accounts for the generally linear increase with _ of the stress increment due to twist shown in Fig. 31.

Hingeless Blade Results. - The transfer coefficients at the root of a hingeless blade as predicted using She approximate analysis and the transfer coefficient charts based on the computer analysis are presented in Fig. 48 for the critical radial station. The corresponding stress distributions (excluding 1P components) for the flight condition of interest are presented in Fig. 49.

The stress results are broken down into components in Fig. 50 to indicate the relative contributions due to inflow, collective pitch, and twist. For greater clarity, the ordinate scale for the twist component has been expanded.

Examination of the results of Figs. 48 through 50 indicates that the approximate analysis results are in good qualitative agreement with the more accurate results obtained using the transfer function charts. The total stress comparison (Fig. 49) tends to be somewhat better than the component comparisons (Fig. 50) because of compensating errors in the e_5 and k S com- ponents. Figure 50 also indicates that the twist component is by far the smallest, having an amplitude of only about 5_ of the e75 and k s component amplitudes and about 20% of the total stress amplitude shown in Fig. 49. This is in direct contrast to the articulated blade situation where twist effects are much larger. The difference is due directly to the elimination of the large 1P component of stress due to twist when the hingeless rotor is trimmed so that no steady moments are transmitted to the fuselage.

The large collective pitch stress component was briefly examined to determine the relative contribution of the various harmonics and bending modes.

The stress component as given by the approximate analysis is: = [elTO0 - 15ZOOCosZ_ + 600osin?._ - 1890c0s3_- 750sin3_ ]i, i + [3170 + 5650c0sZ_ - 1400sin2_ - 7050cos_+4OOOsin3_ - 950cos4_ - 490$in4_1 _= • z where, as before, the i subscript denotes the contribution of each bending mode. As might be expected from their natural frequencies (1.12 for the first mode, 3.37 for the second), the first mode has large 2P components while the second mode has large 3P components. Both modes, of course, exhibit steady components due to the steady airloads acting on the blade. The considerably larger steady response of the first mode is due to its particular mode shape which as shown in Fig. 51 is quite similar to the radial distribution of steady airloads. The large steady stress computed for this hingeless blade could of course be reduced by the introduction of preconing.

Teetering Rotor Applications It is believed that a good approximation to the bending moments experienced by uniform teetering rotors can be made by combining selected transfer coefficients from the articulated and hingeless rotor charts. The elastic deflection of a teetering rotor contains both articulated mode compo- nents (resulting from odd harmonic excitation) and hingeless or cantilever mode components (resulting from steady and even harmonic excitation).

Assuming that interharmonic coupling effects present in the articulated and hingeless transfer coefficient charts are small and/or approximately the same for both types of rotors, then the total response of a teetering rotor can be obtained by simple superposition. This assumption may be reasonable for low stiffness articulated and hingeless blades since their natural mode shapes and frequencies do not differ greatly. The formal procedure for obtaining teetering rotor bending moments from the charts is summarized below.

i. Calculate the mass and frequency parameters from the blade weight and structural characteristics (see steps 1 and 2 of the Articulated Rotor Applications section).

2. Determine the control parameters e75, k c , O I, and b i from Ref. 7 or other source for the flight condition under consideration (see steps 5 and 6 of the Articulated Rotor Applications section).

3. Obtain the odd harmonic transfer coefficients for the independent

parameters, 875, kc , and 81 from the articulated rotor charts. Note that the only blade stations consistent with the hingeless rotor charts are 0 and 0.55R; however, data for other stations are available from the tables of Part II.

4. Compute the transfer function for each articulated mode independent parameter. Note that the summation is performed only for the odd harmonics M( ) = - _ (M() n,c cOsn_+ M( ),n,s sinn _) n ODD where ( ) is used to symbolize one of the independent parameters.

5. Multiply the transfer functions M87_ , -MXc , and MS, by the values of the respective independent parameters to form the total odd harmonic Fourier series of the nondimensional moment.

6. Dimensionalize by EI/R to obtain bending moment, or Ey/R to obtain stress.

7. If the frequency for the first cantilever mode is known, use Fig. 6 to determine the effective frequency parameter. If not, use the same frequency parameter calculated in step 1.

8. Determine the hingeless rotor controls, Als , Bas , and k s from the following identities.

AIs = - b I BI s = QI ks = XC + FGI ( kc calculated in step 2) 9- Obtain the steady and even harmonic transfer coefficients for the independent parameters, eTS , X S , _ , Als , Bis , and _B from the hinge- less rotor charts. Data are available for the root station and the 0.55 station from the charts and for additional stations fr_n Part II.

lO. Compute the transfer function for each hingeless mode independent parameter.

M'( ): M( ),0- _. (M(),n,cCOSn_+'M(),n,sSinn_) nEVEN ii Multiply the transfer fuz_ctions M--_ , -- -- -- u7 Mks , .M_ , MA, , MB,' and M_e b; the values of the respective independent parameters to fogm the %otal even harmonic (including steady) Fourier series of the nondimensional moment.

12. If the frequency parameter was obtained from Flg. 6, solve for the

effective El/R from the frequency parameter as described in the Nonuniform

Hingeless Blade Application section. Using thls value, or the actual EI/R

if the blade was uniform or the first hlngeless modebending frequency was

unknown,multiply by M(), to obtain the bending moment. Alternately, multi-

ply M( }by Ev/R to obtain stress.

lB. Add the results from steps 6 and 12 to form the total teetering rotor bending moment Fourier series for each blade station desired.

It should be noted again that this technique is only an approximation to teetering rotor moments because of the previously mentioned Interharmonic coupling assumption. Evaluation of the accuracy of the method through compar- ison with experimental results or results of other, more complete, analytical methods was beyond the scope of thls study.

C ONC LUSI ONS i. For unstalled conditions, the flapwise bending moments for an articulated rotor blade can be expressed as a linear function of the indepen- dent parameters : collective pitch, blade twist and inflow ratio. Additional independent parameters for a hingeless blade are cyclic pitch and preconing.

Thls linear representation lends itself to transfer function, superposition techniques which provide a simplified procedure for analyzing the bending moment problem.

2. The transfer function approach In combination with the performance charts of Ref. 7 yields accurate flapwlse bending moments for low stiffness articulated and hingeless rotors operating at advance ratios between 0.25 and 0.5. In thls advance ratio range, the influence of blade chordwise and torsional response on flapwise moments Is small.

3. Above an advance ratio of 0.5, the increased sensitivity of the results to small errors in the independent parameters determined from Ref. 7 reduces the quantitative accuracy of the transfer function approach. In addition, the approach cannot be expected to yield reliable results for those blade designs which may experience large torsional deflections at hlgh advance ratios.

4. In computing trimmed moments for hlngeless rotors, residual first harmonic components resulting from small errors in the independent control parameters do not significantly affect the steady and higher harmonic moments acting on the rotor blades and can be neglected.

5- The transfer coefficient design charts developed specifically for uniform blades can be used wi%_in reasonable limits to obtain preliminary results for blades having nonuniform spanwise characteristics.

6. An approximate analysis, based on a simplified aerodynamic theory, provides closed-form transfer coefficient expressions which are qualitatively accurate and useful in gaining insight into the mechanisms by which helicopter blade flapwise moments are produced.

_IX A

AN APPROXIMATE METHOD FC_ OBTAINING FIA/_SE MOMENT TRANSFER COEFFICIENTS By Peter J. Arcidiacono United Aircraft Research Laboratories Articulated Rotor It is shown in Ref. 14 (pgs. 95-98) that the nondimensional response of a rotating blade normal to its plane of rotation can be expressed as a series summation of natural radial shape functions (mode shapes), Yw , suitably scaled by time-dependent generalized coordinates, qw , as in Eq. (A1) cO (T,_) = _ ywi(F)qw i (@) (A1) i,l In such a situation, the differential equation of motion for the blade can be expanded (see Ref. lh, pg. 95) into a series of dynamically uncoupled equations - one for each mode - of the following form:

(A2)

where SA is the nondimensional aerodynamic force per unit span normal to the blade span. Following the development of Ref. 15, pg. 189, SA can be expressed as :

(A3)

°,°I

: Zmo +

It will now be assumed that stall, compressibility and reverse flow effects may be neglected so that the llft curve slope, o , may be considered constant.

The total pitch at any blade station can be expressed in terms of the collec- tive pitch of the 75% radius station, 875 , the linear twist rate of the blade, 81 , and the cyclic pitch angles introduced by the control system, Als and Bi s , as

(A4)

= COS _# sin _ + e I (? - 0.75) 0 eT, 5 - A15 - Bis Also, the nondimensional velocity components ( up and uT ) defining the local section angle of attack of a flexible blade can be obtained directly from those derived in Ref. 15, pg. 188, for a rigid blade by substituting the x CO x I flexible blade velocity, _ = Z Yw , and slope, (_)i cO qw i = _ Yw. qw i ' for their xi=l I i:l t rigid blade counterparts: -F_ and _ . "The resulting expressions for Up and U T are : cO i - cO )'wi qwi - M-,Z 7wiqwi cos VJ (A5) Up : ;ks i_E_l: " : t U T : T + /.I. sin (A6) Substituting Eqs. (A4) through (A6) and defining (A7) cj i • d_ _0 1 Z = Zwi Eq. (A2) becomes (qwi + (_wi qWi)Cli 2toO 0 _Ywi{[e75-Als COS qJ -Bls sin Xx _ 2 : (pOoR) _oB + e,(T- 0,75)](F z + 2M-T sin @ + F z sin z @) (A8) + (W + /_ sin _ ;ks- i_:, 7"wi' _l'wi' - /_" i'=, 7"wi' qwi'

)} C cos dF - )'wi

It is convenient at this point in the derivation to recognize that an articulated blade having zero flap hinge offset has a rigid body mode commonly referred to as the flapping mode. Thus, Eq. (A1) can be written as i,, Ywi qwi i;I Ywiqwi and (A9)

x _ , 2

i:l Y'Wi qwi i:l Y'Wi qwi

z, :B+T.y w

-' , ': i qwi i:l i qwi where the subscript i in Eq. (Ag) and hereafter in this analysis is now taken to denote the flexible blade modes only. Note also that in the final form of Eq. (A9) the flexible blade response has been assumed to be represented adequately by only two modes. By substituting Eq. (A9) into (.4.8) and assuming a constant chord blade, ( _ = 1), the basic equation of motion governing the response of the blade's ith flexible mode is

mo °/o

(_o)

+ ks(F+ /_ sin _) -_" + _. Ywi I qw i il= I - /_ sin _ F + i= - _/_ cos _ B + T_ywi, qwi i t= I 2 sin 2_ _ + _. ¥wi Iqw i i'=l P¢oR )O B _ d7 - 2m o L )'wi

To obtain a simple closed-form solution to Eq. (AIO) the following

assumptions are made :

(a) The flapping motion of the blade is known in advance from previous

analyses such as that of Ref. 7, (b) The flexible modes are uncoupled aerodynamically so that terms such _-I l I as YWi YWi_Wi, can be approxlmated by E: 2w!

qw i , (c) The time-varying aerodynamic spring and damping terms due to blade flexibility (such as _ _Wi' qwi' COS _ ) can be neglected, (d) The gravity term, g , can be neglected as second order.

With these assumptions, Eq. (AlO) is _wi \ 2mo,/ pC.oR) B = 2too' afo r*i_L[eT_-A_s cos _, -e_s sin _, (All) By representing the flap angle as a Fourier series defining the constants, (A_3) _0 B _ k Ik, i = yw i r d?

os 2 _ d_ ZDi = Ywi and noting the relationship between shaft axis and control axis quantities for a zero offset rotor (from Ref. 15, log. 168): = + Ol al s BI s b I = his- Ais

(Azs)

a 2 = 02s b 2 = b2s Xc = Xs - F BIs Eq. (All) can be expressed as x + _ Z Cl i _w i wi wi % \--Z_-0) I0i = \ 2m 0 /,_ _Xc (El, i P. T0, i sin + 8, [T3, i - 0.T5 I2, i + 2F(I2, i - 0.75 T,,i) sin9 + T (:r,,i - 0.75 T0,i)(I-C0S 29) + OO -/_Ze,i COS 9- T TO,i sin 29

t( "t < 1

+ a I - I2, i- _'p. Io, sin 9 +#Ii, i cos 29 + _- Io, i sin 39 + b_ [ z,i +_- Io, cos9 +FIt, i sin 29- _-/_ Io_COS 39] • (,-co, Now, in the absence of stall, the various flapping coefficients can be shown to be 14near functions of 875 , el , and k c (see Ref. 8) and can be written in the following form: (A_7) etc Substituting Eq. (A17) into Eq. (A16) and defining the blade mass parameter and modal damping ratio as pCoR (AI8) (MP) : 2m 0 (A19) Eq. (A16) can be written as CI i (MF_Q qw i + 2_imwiqw i + Wwi- qw i xx x 2 ) = (A20) 11=0 + (_L,O-tS,n_s075+ Fii,01,n,s + Fi, Xc,n,s _c)sin nq/] 5"t where %he non&imensional force coefficients _re iefine& as follOWS : = I-Z,i + _ ZO,i + 4 b2 875 Z,,i + b, I_2, _+ -_ Io_ - °o875 FI''_

07_

F"" 875,t, C = - 02075 0_875 _,eT_,_c - - y 3:o;_ + bzB7_ z _ __I_-b_gv 6 4 _o,_ Fi,875,3,C = 02875 Z -to, _

_,87s,4,c, = - bzS7 _ "_"

£,,i z;:4-

- = ?--_l,t- b2_75 2 - Ote75 Fi,875 J,s " 2 _i,8.rs,Z, c = _ 028757- $Z,i + b_875_l_'i - 00875"-_" T`°'i z _,Ov._,_,s = bzevs_2/-'-='-'t,i + o_e7 5 "_ _'O_i _,875,4, s : 02075 4 _,,e,.o.c- i3 - .75i_,,+ _ (I,,,-.,_i_ + b2e,_io, ,I,c = - azel _ Ig + bwei -(Iz, i + /_2 :_ _- Io, V Oo# I/z Ii, _ _ /z 2 Fi,el,2,c - 2 (ll,i- .75 Io,i) + b2el 2 Iz, i + a,e, _l,, i Fii,8,,3,c = 028, _ P-Ii, i- b,81 4 F--2 TO, i

_ _

Fi,8 a ,4,C =-bzsj "_" 1o,i (A2.2) m Fi,e I,S,c = 0

,,.e,.,.s : 2_ < _,-.,5 I,.,>-,2e, __,,,-°,e,(.2_-_Io,/

Fi,si_,s = - a28i 2T2,i + b,81 F It,i - 0o8 i -_- Io, i

-

Fi,el,5,s = b2ol _- F Ii,i + Olol _- Io,i Fi, ol,4,s = 020, __2 !O,i Fi,Xc,O,C : Ii, i + bzXc 4 V_ Io,i

- _-

Fi, kc'''c= 02kc_" I', i+ blkcIi2,i+ /_2 i o,l, O0>,c F I', i _i,Xc,Z, c = b2x c 2 Iz.,i + olx c F Ii,i

}

_,Xc,_,c = a2Xc__ Ii,i- b,×c_- Zo,i

(_3)

_i,kc,O, s = 0

E

_i,kc,_,s = - O2X c 2 I2_ + blkc F II,i - aox c 2 I°,i Fi,Xc,%s = bzXc -}-p. I,,i + o,X c 4 _-' Io, i

E

F_Xc,4,s = a2;k c 4 Io, i Equation (A20) can be solve& in a straightforwazd manner by assuming a solution of the form: qwi n =o

_ _. [i%,,,.o., e,, . %,,.o., %,o.o.o

_' * _°/''°(°*- _,,o/1

+ /qwi,875,n,s, 875 + qwi,Si,n,s qwi,Xc,n,c 6O Substituting Eq. (A24) into (A20) and employing simple vibration theory (e.g., see Ref. ii, pg. 73) yields the following expressions for the various qw coefficient in Eq. (_4): = Fi,( ),n,c Ailn

(_5)

= Fi,( ),n,s Ai,n (A26) qwi,( ),n,s _w2C,iJ(MR 0 The dynamic amplification factor, Ai, n , and the phase angle by which the response lags the excitation are given by Eqs. (A27) and (A28), respectively, I

Ai, n = (A27)

_i,n = ton-' (_._.8) °Jwi Z_.22 '

-(' 1

w i It is now desirable to express the blade response in terms of a Fourier series referenced to _ = 0 and employing a sign convention compatible with well established helicopter notation, i.e., qw i

q,',i ,e,,,o ez5 + qwi ,e, oe, + q,'i,xc,o xc

(_9) 4 _ -n_=, [(qw, ,O,,,n,c 075 + qwi ,Sl,n,c O, + qwi,Xc ,n,cX c ) COS n,] --_n=, [(qwi,_s,n,s 075"t'q'i,',,n, s O, ÷qwi,X¢,n,s Xcl$inn,]

The qw coefficients in Eq. (A29) are related to the qW coefficients in

Eqs. (A25) and (A26) by the following equations.

(A30)

qwi,(),o ; qwi,(),o

(A3Z)

qwi,(),n,c = -qwi,(),n,c COS_)i,n + qwi,(),n,sSin_)i,n (A32) : COS_i, n _lwi,( ),n,, -qwi,( ),n,c sinq_i ,n - qwi,( ),n,, where ( ) indicates 075 , O I , or k c as the case maybe.

on the Finally, the nondimensional flapwise moment at any radial station (A30) blade can be expressed in terms of the response coefficients of Eqs.

through (AB2) as follows:

-. =_-"R = ,_= _._" q.,-=_:,Z Y.," {_w,._.o o._ +_._._ e, +q._._.o Xc

(A33a)

-_,[G,.o.:.c e.. +_w,._,o,c o, ÷ q.,,_°,o,c _c)_

+(_w, ,_,o,. e.. +..,,,_,o,. e, + q.,,,,:,._,c )._oo_.]}

An alternate expression for M in terms of nondimensional bending moment transfer coefficients can be defined as: M

= MeT,,o 87s + Me,,oe, + Mxc,o Xc

(A33b) - _ [(_o..,,,,c o.o + _e,,.,c o, + .,,o,,,,°Xc) _o_ r,_, 13=1

+( _o..,.,. o.. + _e,,._ e, + _c,o,.Xc ),Jr,' ,,_,]

Equating like trigonometric terms in Eqs. (A33a) and (A33b) yields the following final expressions for the desired flalXrlse bending moment transfer coefficients.

Mc).o = ._ _'*i"q*i ,( ),o

I=1 4_ //_ : "Zwi qwi,( ),n,c (A34) M( ),n,C i=i

_( ),.,s : _ Y*i"q*i( ) .,s

i:l where, again, ( ) indicates 875 , 81 , or k c as the case may be.

Hingeless Rotor The derivation of the flapwise moment transfer coefficients for the hingeless rotor follows generally that described previously for the articulated rotor. One basic assumption which differs for the hingeless rotor is the assumption that the first mode response for low stiffness hingeless rotors can be approximated bythe flapping mode response for an articulated rotor. With this assumption, the equations of motion are solved to obtain the response of the second hingeless mode. Details of the procedure follow.

The nondimensional displacement of a hingeless blade having a fixed built-in coning angle _S is _) z (A35) Z" : K/gB + Ywi qwi _ _',Se + _Yw iqw i i:l I:l The first bending mode ( i = i) response is now assumed to be essentially a rigid flapping mode whose motion is referenced to the undeformed position of the blade. Thus:

_'.,q., = _(#-_B)

Substituting in Eq. (A35) gives (A36) : K/3 + 7'wzq.2 where _ is assumed to be given by Eq. (AI2). The basic equation of motion for the second hingeless mode is given by Eq. (AI0) modified to include the bending moment produced by the combined effects of built-ln coning and centrifugal force. The modizication to Eq. (A2) is developed in a straight- forward manner from the analysis of Ref. 14 by noting that the flexible mode of a hlngeless blade is no icager orthogonal to the function _ . The resulting 2nd mode equation is (qXw2 + _w22qw2)_O B m_'w22dK = _o B 7w2(SA-g)d_"-_B ._o'mKYw2 d_" (A37) With the definition of Z given in Eq. (A36), the Eq. (A5) takes the following form: X X I (A38)

up : Xs - _/_ - 7"w2qw 2 - _(B + ×wzqw2)cos q_

Substituting Eqs. (A3), (A4), (A6), (A7), and (A38), assuming constant chord, and neglecting gravity as before, Eq. (AB7) can be written as lr X

+ e(_ -o.7_)] (_÷ _sin _)2 + (F÷ _ _io _,)[x_-_ ->,.2q,_

(A39)

-_(P + )'wJqw_)CO.V, ]}d_-

spring and damping terms are neglected, Eq.

(A39) If the time varying qw 2 becomes: (PCoRIx B

(_l'w 2 + _w22qw2)C,2 +

' P-rno aqw2/o "fTw_zdT ('pCoR I f {[(875 AImCOS _ Blssin _) : 2mo 0 Tw a - -- (A40) + 8,(F - 0.75)](F + F sin _k)" +(T +/_sin_k)(,ks-_ -/_-/_ cos V_)} d'f This equation is identical to Eq. (All) developed for the articulated rotor with i set equal to 2. The solution to Eq. (A40) is thus the same as that to Eq. (All) so that the results given by Eqs. (A21) through (A32) are all applicable provided that i is set equal to 2 and that Yw and its derivatives correspond to those for the second hingeless mode.

The flapwise moment equation applicable to the hingeless rotor as represented in this analysis is _ MR M E, = Yw," (/_-/gB) + Ywz"qwz (A41) : Yw'"'[O0 -n=lZ (Ons COS nV/ + bns sin n_)-/_e ] + ?'w_' [qwz. 0 -n___ ' (_Wz,n,c COS n_k + _w2,., s sin n_ )] Using Eq. (Al5), (A41) is: = MR _ yw_' [O0-/_B -(O,-B,s)COS_ - (b, + A,s) sin _, El (A42)

" [

-n=zZ(an COS n_+ bn sin n_)] + Xw_' i_wz, o - =, (qwz,n,c cos n_ + qw2,n,s sin nV_)] Finally, substituting Eqs. (AI7) and (A29) ( i = 2) yields the following expression for M :

M. :[

M = -_ -- 7w OoeT_ eta + aoe, e, + Coxc Xc -_S - (0=8Te87_ + aae, Sa + alXcXC -B, s)cOs - (b,eT_eTs + b,e,e, + b,xcX c + A,s ) sin _, -n_z= <(anezse'_ + One,8' + °nxc)'c) c°s- n_, (A43) + (bneT_eT_ + bne, e, + bnxcXc)sin n_,)]

+ e.. + e, +

- _- <(qw2,Sr_,n,c eTS + qwz,8,,n,c e, + q"*2 ,;kc,n ,c Xc ) COS n_/ n:l + (qwz,8"ts,n,s 075 + qw2,Sl,n,s el + _wz,Xc,n,s X c) sin n_>] Expressing M in the general format of Eq. (A3Bb) as M = _eTe, o875 + Rie,,o8_ + DXc,O ),c + Ri/3s,O Be + ]_A,, ,o A,s + Rie,s,OB,s

(A_4)

+ Msi,n,c 81 + MXc,n,cXC + MA,s,n,c A,s + Ms,s,n,cB's)COS n_ e,.

+ MS,,n,s 8, + MX¢,n ,s Xc + MA,s,n,s A,s + _s,,,n,sB,s)Sin n_ + (gOTs,n,s 07_ and equating llke trigonometric terms in Eqs. (A4B) and (A44), yields the final expressions for the nondimensional bending moment transfer coefficients for the hlngeless rotor. Note, in order to e_press the results in conventional hlngeless rotor terminology (ks instead of k C ), the following identities were used: )'c : Xs-F Ba s -z;s =' , : MBis: aB_ _ "_-/= _ F C)Xc The resulting coefficients are: -- HQ It M( ho : 'Tw I o( ) + ')"w2 T_wz,( ),o II II -- M( ),t,c= Yw I Ol( ) + 7'w2 qw2,( _,l,c II II M( 1,2,c = )'Wl 02( ) + 7w 2 qw2, ( _,2,c . H O + H- M( ),3,c = Twm 3(_ 7"w2qw2,( ],3,c II II _ M( ),4,c = 7"w I 04( ) + 7'w2 qw2,( 1,4,c

(_5)

II- M( _,_,s = Yw_/b_( ) + 7'wz qwz,( )j,s II II -- M( )_,s = 7"w I bz_ ) + )'wz qwz,( 1,z,s .,p ii-

M( ),3,s: 9"_ b3( _ Yw2 qwz,( ),_,s

II-- II + _'wzqw2,( ),4,s MB.,o : -.Tv_; = _ II MAIs _l,S. w I M%s,l,c ='Yw_ _- F_xs'_'c MBIs,n_6,,C = -/_n _t "),s,n,c MBis,n, s : -#.L_ n ]_Xs,n,s In Eq. (A45) the subscript ( ) indicates 875 , 81 , or X S as the case may be. Also, to reiterate, the "a" and "b" flapping coefficients are assumed -- w S to be known from Refs. 7 or 8 and the qw2 are given by Eqs (A30) through (AS2).

API_NDIX B UAC NORMAL MODE TRANSIENT ANALYSIS The transfer coefficient charts presented in Figs. 52 through 177 were computed using the UAC Normal Mode Transient Analysis (referred to as computer analysis in the text and figures). This analysis was also used to provide bending moment and stress results which were used as a standard against which the accuracy of results obtained using the charts could be evaluated. A brief discussion of the analysis is presented below. A detailed description of the equations of motion used is given in Ref. 1.

Method of Solution The computer analysis constitutes a numerical method for solving the differential equations governing the fully coupled flapwise, edgewise, and torsional response of flexible rotor blades. A normal mode approach combined with a finite difference, step-by-step integration of the equations is employed. The former is widely used throughout the aircraft industry and assumes that the deflections of the rotor blade can be represented by summa- tions of the blade's natural vibratory modes. Although an elastic body has an infinite number of such modes, the normal mode approach owes its success to the fact that only a relatively small number of modes is generally required to adequately approximate the elastic deformations of interest. The use of normal modes also results in certain simplifications of the equations of motion (such as elimination of certain dynamic coupling terms) thereby facilitating the solution of the equations through numerical integration techniques on a digital computer.

Technical Features of the Computer Analysis The analysis has the following capabilities and advantages: 1. The program includes the effects of up to 5 flapwise, 2 chordwise, and 2 torsional elastic modes as well as the flapping and lagging rigid body modes for an articulated rotor.

2. The chordwise and torsion modes can be suppressed to permit applica- tions for which only the influence of the uncoupled flapwise modes are of interest.

3. The computer program can be applied equally well to blades having many types of root-end boundary conditions by simply providing as input the appropriate mode shapes and natural frequencies. Any degree of hinge restraint from fUlly articulated (with flapping hinge offset) to rigid in either the flapwise or chordwise direction can be studied and both lag hinge damping and control system flexibility can be included.

4. The elastic deflections of the rotor blade and the aerodynamic forces generated by it are fully coupled. Aerodynamic forces in the blade equations can be considered to be composed of (1) components independent of blade motion (forcing functions), (2) components dependent on blade deflections (spring forces), and (B) components dependent on the rate of change of blade deflec- tions (damping forces). Analyses which do not couple blade deflections and blade loads, therefore, consider only the first of these three components.

At high forward speeds the contributions of the coupled aerodynamic loads become increasingly important. It has been shown, for example, that the flapping instability of an articulated rotor cannot be predicted unless these effects are considered. It is also apparent that the equivalent aerodynamic spring constants and damping coefficients in the second and third components are dependent upon blade azimuth angle and rotor flight condition. For example, twisting of the rotor blades in reversed flow can produce effects on blade loads which are opposite to the effects in conventional flow. Also, aero- dynamic damping, which is closely related to airfoil lift-curve slope, will change markedly when either stall or compressibility effects are encountered.

A primary advantage of the numerical method of solution employed in the pro- gram is that it takes full advantage of the capabilities of a high-speed com- puter to include all of these coupling effects in a straightforward, physically meanln_l maune_ o 5. The computer program can be used to compute time histories of blade motion even when such motion is unstable. Such conditions of instability or near-instability can exist at high advance ratio flight conditions where either retreating blade torsional divergence or blade flapping instability imposes a limit on maximum speed.

6. Dynamic coupling forces between the flapwise, chordwise, and torsional degrees of freedom have been included.

7. Airfoil lift, drag, and pitching moment data at angles of attack from O to 360 deg and at M_ch numbers from O to 0.9 _ are employed (see Ref. 7 for lift and drag data used).

8. Very high frequency blade response in the primary modes can be predicted. This can be particularly important at high forward speeds and low rotor speeds when the entire blade enters the reversed flow regime.

APPENDIX C DETERMINATION OF PARAMETER VALUES FC_ SENSITIVITY STUDY The factors considered In arriving at parameter values for the reference blade and ranges over which the parameters might be expected to vary are discussed below.

Mass Parameter The values of mass parameter, MP = ,oCoR/2m 0 (Cl) were determined by selecting a referemce value typical of existing blade designs, and considering practical variations based on probable variations of aspect ratio and altitude that might occur. Several existing articulated and hingeless blade designs were examined to establish the reference mass param- eter. Although these designs were not all inclusive due to the limited amount of detailed blade data available, they are believed to be representative of modern blade designs. Blade aspect ratio ranged from 15 to 21, and mass per unit length at the 55% radial station from 0.35 to 0.78 lb/in. The mass distribution of these blades over the outer 70 to 80% of the radius was found to be fairly uniform, with local variations generally not exceeding plus or minus 10% of the average (except in the vicinity of the tip and root). For example, the weight distribution for a Sikorsky H-34 blade is presented in Fig. 23. The mass parameter values of these existing blade designs (for sea level standard conditions) range from O.18 to O.B2 with most lying between 0.28 and 0.32. A reference mass parameter value of 0.3 was therefore selected.

The reference mass parameter selected differs from that of the blade used in Ref. 7 which, on the basis of the information quoted in that reference, is estimated to have a mass parameter of 0.23. This value is not representative of typical blade values unless rotor operation at approximately 7000 ft (as opposed to sea level) is assumed. The use of a reference mass parameter of O.B in the present investigation implies operation of typical blades at sea level and is believed Justified by (1) the wider interest in such operation, and (2) the fact that results of the type presented In Ref. 7 have been shown in Ref. 8 to be insensitive to wide variations in Lock number (and hence mass parameter, see Eq. (13)).

7O

A relation betweenmassparameter and aspect ratio (AR) maybe obtained

by assuming constant air density, p , and geometrically similar blades made of the same material. The mass per unit length is then proportional to the square of the chord, and the mass parameter may be expressed as : MP= P CQR PCoR = ('-"_-P _(A R) (C2) 2 rl"t 0 = KmoCoz \ Km o / Thus, for similar structural design and material, the mass parameter is pro- pe_tional to aspect ratio. Typical aspect ratios of current designs are between 15 and 21. Aspect ratios below l0 are not likely unless one is considering stowed rotor cc_figurations, while aspect ratios greater than about 25 are not likely because of static stresses. Assuming 19 to be a typical aspect ratio for the reference mass parameter of 0.3, and a probable aspect ratio range to be lO to 25, the following mass parameter variation results : Aspect ratio - i0 19 25 Mass Parameter - 0.16 0.3 0.4 Since MP is also directly proportional to air density, mass parameter decreases with increasing altitude for a given blade. For example : Altitude (ft) - 0 10,000 20,000 Density ratio - 1 0.74 0.53 Mass parameter - 0.3 0.22 0.16 From these considerations, the mass parameter values for the sensitivity study were selected as 0.1, 0.3, and 0.5.

Frequency Parameter The values of frequency parameter, FP : EZ..o/mo(_R) z Rz (C3) selected for the sensitivity study were determined by choosing a reference value typical of existing blade designs, and considering practical variations based on probable variations of aspect ratio, material, blade thickness ratio, and rotor tip speed.

The sameset of existing blade designs used for the determination of mass

parameterwere considered to establish a reference frequency parameter value

The blades considered were of aluminumspar construction and normally operate

at tip speeds of 620 to 700 ft/sec. Their flapwise area momentsof inertia

range from 0.7 to 7.2 in. _ at the 55%radial station. Variations in inertia

from the reference value were generally within 25%of the average values

except near the root. The frequency parameter values of these existing blade

designs range from 0.O019to 0.0028. A reference frequency parameter of

0.0025 was selected.

An alternate expression for frequency parameter involving aspect ratio

can be derived if one again assumes geometrically similar blades of the same

material. For such blades, the moment of inertia and mass per unit length at the reference station is proportional to the fourth and second power of the chord, respectively. The frequency parameter may then be expressed as: (_R) z R 2 KrtloCo2 LkKrno ) (,O,R) (AR) z <C4) Considering the aspect ratio range from IO to 25, the following frequency parameter variation results : Aspect ratio - i0 19 25 Frequency parameter - 0.009 0.0025 0.00145 If new materials are considered, while maintaining a constant tip speed and radius as well as similar structural geometry, the frequency parameter is proportional to the ratio of the modulus of elasticity to the density of the material. That is, FP GEIo E mo Ps The following is a comparison of the modulus-to-density ratios and the corresponding frequency parameters for (1) a composite boron-epoxy material utilizing cross plies to reduce the modulus, (2) an isotropic metal (such as aluminum), and (B) a high modulus uniaxlal boron-epoxy composite. (Note that the ratio of E/p s for all isotropic metals is approximately constant.)

Material Uniaxial boron-

90 deg Cross ply Isotropic metal

composite (aluminum)

epoxy composite O.B x i07 i x 10 7 2.8 x 107

£ , psi

0.075 o.1 0.075 PS' ib/in'3 - 0.4 x 108 1 x lO 8 4 x 108 £/PS ' in. -

o.oo25 O.Ol

Frequency parameter- 0.001 Consideration of a new structural geometric configuration implies a change in the inertia-mass ratio ( IO/m _ brought about by a change in spar depth for a given material, chord, tip speed, and radius. Since the reference values I O andmoare based on the 55_ radial station, large increases in spar depth are not likely because of drag considerations. Also, large reductions of spar depth could result in inefficient aerodynamic sections. If it is assumed that I 0 is proportional to thickness ratio squared and the mass per unit length is assumed to be held constant, then the following frequency parameter variation results from a thickness ratio variation of 8 to 18_ chord: Airfoil thickness ratio - 0.08 O.12 0.18 Frequency parameter - 0.0011 0.0025 0.0056 Frequency parameter is inversely proportional to the square of the tip speed. For the advance ratios up to 0.5 a tip speed of 672 ft/sec was selected. Major departures from this value would not be meaningful because corresponding variations in Mach number effects are not accounted for in the transfer functions. A tip speed variation of plus or minus lOO ft/sec is therefore considered as covering the range of interest.

Considering the low advance ratio range ( _ = 0.25 to 0.5) the following frequency parameter variation results if blade material, thickness ratio, and radius are held constant : Tip speed fps - 572 672 772 Frequency parameter - 0.0035 0.0025 O.O019 Tip speed values higher than 772 ft/sec or lower than 572 ft/sec are normally not of interest because of forward speed limitations imposed by the advancing tip Mach number and retreating blade stall, respectively. Of course, for the high advance ratio range ( _ > 0.5), lower tip speed values become mandatory to avoid compressibility losses. Lower tip speeds were considered for the high advance ratio range as shown in Fig. 18 and their effect is discussed below.

From the consideration of existing blade designs and independent changes of aspect ratio, material, thickness ratio, and tip speed, the nominal frequency parameter values selected for the low advance ratio range of this investigation are O.OO1, 0.0025, and O.O1. The additional frequency parameters of 0.005 and 0.015 were included in the sensitivity study. Although combined variations of aspect ratio, material, etc. could be made to produce frequency parameters greater than O.O1, the structural stiffness of the resulting blades would start to become large relative to the centrifugal stiffness. Consider- tlon of such blades falls outside the scope of this investigation. It should be noted that in the low advance ratio range ( _ _< 0.5), where the rotor tip speed is constant, the frequency parameter for a given blade design is also constant. In contrast to this, at higher advance ratios the frequency param- eter for a given design varies with _ because of the tip speed variation.

To allow for this, the frequency parameters found to be representative in the low advance ratio ratio range were simply ratioed according to the following relation to define the values for the higher FL 's.

F P/_>.5 = FP#. <.5 ,Q,R#. > .s ,GR_<,5 11 +u.)

= FP/_ <.s --=-- - (C6)

1 '

OMI,90 672 ( I +,u.)

(1117)(0.9) : FPF<. 5 (0.4468)(I +,u,) z The actual values of frequency parameter used at the different advance ratios considered are presented in Fig. 19.

Tip Weight Parameter One of the variations from a uniform blade design that was considered involves the addition of a tip weight over the outer three percent of the blade (as shown in Fig. 4). To nondimenslonalize the tip weight, a new blade design parameter was introduced called tip weight parameter (TWP). The tip weight parameter is defined as the ratio of tip weight to blade weight ; that is TWP = WTIP (C7) WBLADE The tip weight affects the blade mass distribution but not the stiffness distribution of the blade. The following five tip weight parameter values were selected for investigation: TWP = 0.O, 0.05, O.15, 0.3, 0.5 The last two values are probably unreasonable from a practical viewpoint (unless tip mounted engines are considered) and were included to provide an appreciable range in the sensitivity analysis. The mass distribution corresponding to the tip weight parameters selected are presented in Fig. 4.

Wall Thickness Parameter The second departure from a uniform blade design consists of a simulta- neous change in mass and stiffness distribution resulting from linearly taper- ing the spar wall thickness, as illustrated in Fig. 3. The ratio of wall thickness at the blade tip to wall thickness at the blade root was defined as the wall thickness parameter (WTP), tWT_p WTP = (C8) tWROOT Only wall thickness parameter values representing conventional thickness variations (thickness decreasing toward the tip) were considered. Specifically, the following values were selected : WTP = 0.0, 0.5, 1.0 The value of 0.0 probably does not represent a practical design, and was selected principally because it is an absolute limit on WTP. The mass and stiffness (inertia) distributions corresponding to the above WTP values and referenced to the 55% radial station are presented in Figs. 4 and 5. In computing these distributions, thin wall spars (wall thickness small compared to spar depth) were assumed, and spar depth was assumed constant_ The nora- structural mass distribution was assumed constant and equal to three-tenths of the structural mass at the reference station. This is equivalent to 23% of the total mass per unit length at that station and represents an average of the existing blades considered.

Spar Depth Parameter

The final nonuniform blade configuration considered results from linearly

tapering the spar depth as illustrated in Fig. 3. The ratio of the depth of the spar cavity (see Fig. 3) at the blade tip to the corresponding value at the blade root is defined as spar depth parameter (SDP), YSTIP SOP = (C9) YS.ooT Spar depth parameter values greater than 1.O (inverse taper) were not considered. The following spar depth parameter values were selected.

SDP-O, 0.33, 0.7, i.O

The spar depth parameter value of 0.33 corresponds to a 6% thickness ratio tip section and an 18% thickness ratio root section. The spar depth param- eter value of 0.7 corresponds to a two to one variation in inertia from root to tip. The mass and inertia distributions corresponding to the above $DP values and referenced to the 55% radial station are presented in Figs. 4 and 5.

As before, the distributions were computed assuming a thin wall spar, a constant wall thickness, and a constant non-structural mass/total mass ratio of 23% at each radial station. As shown in Figs. 4 and 5, tapering the spar cavity depth independently from the _wall thickness produces a large change in the inertia (i.e., stiffness) distribution but only a small change in mass distribution.

REFERENCES

lo Arcidiacono, P.J.: Prediction of Rotor Instability at High Forward Speeds. Vol. I, Differential Equations of Motion for a Flexible Helicopter Rotor Blade in Steady Flight Including Chordwise Mass Unbalance Effects. USAAVLABS Technical Report 68-18A_ U. S. Army, February 1969.

2, Piziali, R. A.; and DuWaldt, F. A.: A Method for Computing Rotary Wing Airload Distributions in Forward Flight. TRECOMReport No. TCREC TR 62-44, November 1962.

.

Miller, R.H.: On the Computation of Airloads Acting on Rotor Blades in Forward Flight. Journal of the American Helicopter Society, April 1962.

_o Tararine, S.: Experimental and Theoretical Study of Local Induced Velocities Over a Rotor Disc. Proceedings of CAL/TRECOM Symposium on Dynamic Load Problems Associated with Helicopters and V/STOL Aircraft. Vol. I, June 1963.

5.

Willmer, M. A. P.: The Loading of Helicopter Rotor Blades in Forward Flight. R.A.E. Report Naval 2-N-76935. No. 8, April 1959.

.

Landgrebe, A. J.: An Analytical Method for Predicting Rotor Wake Geometry. AIAA Paper No. 69-196, AIAA/AHSVTOL Research, Design and Operations Meeting, February 1969.

.

Tanner, W. H.: Charts (Tables) for Estimating Rotary Wing Performance in Hover and High Forward Speeds. NASA Contractor Report CR-114, (CR-115) 1965.

o Bailey, F. J., Jr.: A Simplified Theoretical Method of Determining the Characteristics of a Lifting Rotor in Forward Flight. NACA Technical Report TR 716, 1941.

.

Rasumoff, A.; and Amer, K. B.: Helicopter Configuration and Propulsion Systems Study. Part IV: Analytical Procedure for Predicting Rotor Blade Weight. WADC Technical Report 57-583. November 1957.

lO.

Fradenburgh, E. A.: High Performance Single Rotor Helicopter Study.

TREC Technical Report 61-44. April 1961.

ll.

Scanlon, R. H.; and Rosenbaum, R.: Introduction to the Study ofAircraft Vibration and Flutter. MacMillan Co., New York, 1951.

REFERENCES (Continued)

12.

Yntema, R. T.: Simplified Procedures and Charts for the Rapid Estimation

of Bending Frequencies of Rotating Beams. NACA TN 3459, 1955.

13. Rabbott, J. P., Jr.; Lizak, A. A.; and Paglino, V.M.: A Presentation of

Measured and Calculated Full-Scale Rotor Blade Aerodynamic and

Structural Loads. U. S. Army Aviation Materiel Laboratories Report

TR 66-31, July 1966.

14.

Bisplinghoff, R, L.; Ashley, H.; and Halfman, R. L.: Aeroelasticity.

Addison-Wesley Publishing Co., Inc., Cambridge h2, Massachusetts, 1955.

15.

Gessow, A.; and Myers, G. C., Jr.: Aerodynamics of the Helicopter.

MacMillan Co., New York, 1952.

TABLE I TABLE OF BLADE DESI(_S AND BLADE Blade Design Parameters Wall Spar MasB Tip Weight Thickness Frequency Depth Paremeter Blade Parameter Parameter Parameter Parameter MP FP TWP WTP Number SDP 0.1 0.O01 Oi 1.0 1.0 2 .o 0.0025

i

O.O1 0.001 0.3 4(Referenc_ _ g N 0.0025

I

0.O1

_ _ _

0.O01 0.5 7 .,-4 .,.'1 0.OO25 i

I

0.O1 I 9 ,' ¢= 0.3 o.005

I

o.oz5 0.05 12 0.O025 0.15 0.3 0.5 15 _ s I O

0.5 16 _ I_

O !

I

18 §

0.7 o.33 19 o 20 V (o._47)(l +_ )2.

* Listed FP values are for _ <_0.5.

For _A >0.5, FP = FPA_<O 5 FPp.>0.5 F P_._O.S

0.001 o.ooo_7 (l +_)2

0.0025 o.oon2 (1 + _)2 0.01 o.oo4_7 (1 + _)2

TABLE II

ORGANIZATION OF FIGURES FOR TRANSFER COEFFICIENT CHARTS (a) Articulated Blades Advance Ratio Independent Blade Design Figure Parameter Numbers Numbers Range i

52 - 60

875 ito9 Low

(_= 0.25 to 0.5)

8, 61 - 69

7O - 78

X¢ ito9 976 79 - 87

_gh

i

(_--0.5 to 1.4) 8, 88 - 96

i

,i

_¢ 97 - 105

8O Table II (Concluded) (b) Hingeless Blades Radial Advance Ratio Independent Blade Design Figure Station Numbers Parameter Numbers Range 106- 108 @75 i,&,7;2,5,8;3,6,9 8! 109- lll Low 112- 114 _S 115 - 117 (N= 0.25 to 0.5) AI$ ,118 - 120 81 s 121- 123 B.

_=O 124 - 126 87@ 1,4,7;2,5,8;3,6,9

8, 127 - 129

130 - 132

High _s

133 - 135

(_ = O.5 to 1.4)

AIS 136 - 138 8! s 139 - 141

Be

142 - 144 1,4,7;2,5,8;3,6,9 145 - 147 8, Low ),S 148 - 150 151 - 153 (_= 0.25 to 0.5) 154 - 156 IS 157 - 159 = 0.55 160 - 162 875 1,4,7;2,5,8;3,6,9

8, 163 - 165

166 - 168

High

169 - 171

(_ = 0.5 to 1.4)

A IS

172 - 174

BI s 175 - 177

/3o

,-I ,_ e¢') o'_o"/o'_ O00 • e • 0000 l I 000 ¢3 000000 000 0_ ..._ o'_ OJ _ 0 ..._ ._. ,-4 ,-_ ['-00 %0'.O'.0

_._6._ o_o o

.J 000000 000 000000 000

a!

g.

__ _r_O 000 _¢_ 040 0

._._,_-_0_ _o __op._

.... _._.

l¢% 0 0 0 0 0 0

dododo ddd c;ddodd ddd

_,oo _o o o_. _. _oo |

• _._. . .

,,< v 0 0 0 _

°99999 °99 °99999

,,, _

_ o

°b- i_ (_ ..,,_" I._ l"- 0 ,._

:t dddd_ ddd o o o o,_,_

_, _®

O0 I!

I/ U

"6"

v .,4

d

VI VI r'_ O_

d_d

_dd_d

II II II II II II II _>- ¢r b

1I

II j _._. : i : ..... ::: ...

O0 I s I O0 000 i I I 000

_, , , oo _,_- o_

• _._. : : : ...

,-J

_,_ o_, __ o_

_S_ _ oo _

.J t)

m

_ __ _ oooo oo_

00000 O0 000000 000 i,f'% _ _ C'_O',O _'- 0 _ _'_0 C_O_O I_'- 0 _0_ #% o • . ._o.

0 0 0 0 0

oo oo<7, ooo ooo

s

|

i o 0 000 ,.-4 e,'_ o_ 000

_ _,_o,_ _o_

_ _=_oo

.... _._. . . .

iiiii Z • . _._._.o_ _._.

_,_ .

O0 000 o r"_ ,"_ 0 O0 ddS_ 0 o _, II ii II i,_

f_

v II

!i °

Ci Vl u_ !=, al

_g_o_ f,l-,i _ 0 r-I

ti II II II II II II I II II II

t i ,,,_1_ '>

_3

COMPUTER ANALYSIS 149 KTS O MEASURED DATA m

v= o.s77

' '= °.'- //F '_

,8 0 90 180 270 360 BLADE AZIMUTH ANGLE, _, DEG

Figure 1.- Typical comparisons of bending moments predicted by the computer

analysis (fully coupled) with measured data.

COMPUTER ANALYSIS O MEASURED DATA H-34 ROTOR V= 110 KTS O _n O J I z D z C o _1 z o o F- o -1200 90 180 270 DYNAMICALLY SCALED 4 FT DIAMETER MODEL /_= 1.0 CT/O= 0.0347 120q o o

o/'7 o

.i )0 no o I- .J -1200 z o no o I- () -2400 0 90 180 270 360 BLADE AZIMUTH POSITION, _, DEG Figure 2.- Typical comparisons of torsional stresses and moments predicted by the computer analysis (fully coupled) with measured data.

SP,R,,L. _,,C*NESS.'w_ LSP,RC_'T*O_TM.*.

..,.._:,;.:.:..**:.:.:.:..

\

SPAR NON-STRUCTURAL WEIGHT TOP VIEW ALL BLADES (CONSTANT CHORD AND SPAR WIDTH)

F]!_iii_!]_iiiii_iii_ii_i_ii_ii_ii_i_iiIiiiiiii_iiiii_iiiiiiiii_iiii

EDGE VIEWS UNIFORM BLADE (BLADES I-II) UNIFORM BLADE WITH TIP WEIGHT (BLADES 12-15) TIP WEIGHT BLADE WiTH TAPERED WALL THICKNESS (BLADES 16, 17) BLADE WITH TAPERED SPAR DEPTH (BLADES 18-20) TIP ROOT Figure B.- Three vle_ illustrations of blade designs.

2,O I !

= SPAR I WALL THICKNESS DEPTH PARAMETER, PARAMETER

\

BLADE N_._ O. WT_.P SD._P 5 (RE F.) 1,0 1,0 16 o5 "1o0

\

17 0 1.0 18 1.0 .7 I t'5

\

19 1.0 .33 20 1.0 0

\

IE

a, % A url II "-------- -- _ _ BLADE 5 (REFERENCE 1.0 z

/ / / _'"_=-_ =-'E--:-- .___'--:-_-

o I I- I-- ILl u z uJ n,.

w u.

u.I .5 % o I-

I I I I

.J .2 .4 .6 .8 1.0 u o ..J TIP WEIGHT o BLADE 15 PARAMETER w o i TW. P BLADE NO_ I- 15 0 5 (REF.)

.05 BLADE 14 .15 14 .3 15 .5 BLADE 13 _ T G.

BLADES ( R ;;;;;N;2E ,_ I

.L L_..--_n

.2 ,4 ,6 .8 1.0 BLADE RADIAL STATION, "/"

Figure 4.- Mass distributions for blades with tip weight, tapered wall thickness

or tapered spar depth.

5.0 W i I l U Z WALL THICKNESS SPAR DEPTH mmJ

\

a¢ PARAMETER e PARAMETER.

mc,m BLADE NO. W T.__P.P SDP L W 5 (REF.) 1,,0 1.0

\

16 .5 1.0 4.O W 17 0 1.0 .,I 18 1.0 .7 19 1.0 .33

\

20 1.0 0 i=.

k f BLADE 20 i_ 3.0 A O II Z b v W Z O O < 2.0 ul I.- t_ G¢ IL .J BLADE 5 (REFERENCE) _" "_ -.,. _'_ U.

.u 1.0 U O -I U.

O O i I-

I i I

.2 .4 .6 .8 1.0 BLADE RADIAL STATION, T

Figure 5.- Flapwise area moment of inertia distributions for blades with tapered

thickness or spar depth.

i:ii: :,rl i!L !i!: _i_ !!!

!T!I: H:i ;Uq :;;: _:::: {ii! :i_! ........

!:!k ?;_ ;id iU iiU i;iii!ii

ii_:.i

:qHti., iitl .... gil!

$;Ii t'it ii',! i_il ;ill :::; i','._ _!i! iU,; :i:i

........ ili! iii::i

.... :!u 1i11 :::: i :::

:dii*ii

,:.x

• !_

:::! !h_ !T:i :::: i:i; !!ii Hi! ;:I;

i!i

illl; :::_ !!ii _!: :.:: !!!: !il] I_; tli_ ::Z:

::_!ui:.i

;if:it' ::;: I:;; ,t;i; ',. : o .... t t;r 7;'; i':: I--

i

.... i: iH! !ii: i!iiii ': ..... !; U _'l' :" # .... t' ' Z ix::!ii IM t:_t :: IM

}i_il

I& .;:; i_ i. :;_: :::: !_-_:Hi -s 7:!1 L2L ,_i I-"

ii_!l

.:;;!

Z 7:1 _:_ ,..

.,..| .... i .... i :!;i !F_ :::: I','.11 .... | :ri; ,_n !!i!

.... !

iitt_il _:,:

['.T_l

H4 ::,'.T':.

.... i ':!! :_U i;!!

.... | .H "'_ ..... 44; ..;; ,.;;!

. .._| !!tt ....

:!;: :::; if::l t;t 'tlt '_;" .... !

.... 4, ;HI '.It tt! ....

:::;iiI] "t:! t_t _tf ttt ....

*;il ill _'' 7!

:1 I 11 14 HII £: ,,+

-- ii!

iiit ,;I it_ HI i!il

0 .015 .020 .025 FREQUENCY PARAMETER, FP (a) Articulated blades.

Figure 6.- Variation of natural frequency ratio with frequency parameter.

1.4 :ii :I1 .,.

1.2 !II NI _4 :I| :;I 1.1, _4 1,0 .025 ,020

!ilii!

iii !ii _!

g__...

_t e.- U Z ;:1 ::: !:: IU i:: :'.: ::: ili !i: ?!

UJ E

.!

U.

:,' _i iii

..d :;_ 2;I ;I 'it :_! :+ H_ E !:! ::: i[_ l-- 4[ ::I i;t *l U1 i':: !i Z

::' i!i ii

Ht !U i'. :: I! ii ,..i:

_i !i! i!

_!I :I: ii

;!! H: !

_H it: :S ;i!

it: it: !_i ii" :2: I_t 11: U?

l!i lil !ii

i!i _:! !i!

,,, i_: H!

_: ii_ _*.

...... H

_,i !:,i !_+

i{i ;i'"

ta; _i i:i iil ::*

1_: !i

7!

I,; !t+ ,,

lil !il ill

tit

.025 ,020 (b) Hingeless blades.

Figure 6.- Concluded.

9O

;A tl! :I:)1 !it1 +_+ ...... '*' i!ii !;ii iJ:_t..........

,,+ ql *l+ )ti iil;; _:_: :_:: :,.: ..:: i::i

il

.... !i_ _:ii ::!i i!_i ::i _, !i!! " :t!

12: ma

!Ii i_

d 'f)l., 1 .i!: :::) !::: V- !:t 11 FI) 12!:

;;i L2

u x m,m !:! ,, , i_; o 1!;1 .

U.I :)_ !ii ..... s ......

::_ :![ mm l!! ::t .J _III_ :' " 'T _ 4-;- _.r, ..-t-, lit :ii l-- :;i ii) z !11 ,,_ 11) :_i! i?i .1 .....

,_,_ '._ _ _ 2: .11 ::.! ' ;i'. ::l:'t: :!! 11: 'i ::t tt I ()! _)) ') !); _:) ,,r

:,!_'[!f='i':':i!r:ll!.i:[!!!'l

_i,:tll:l,.i: i 0 ' ,1 .3 .a $ TIP WEIGHT PARAMETER, TWP 0 .2 .4 .6 .8 1.0 WALL THICKNESS PARAMETER, WTP, AND SPAR DEPTH PARAMETER, SDP (a) Articulated blades.

wall Figure 7.- Variation of natural frequency r_tio with tip weight parameter, thickness parameter and spar depth parameter.

I0 • 1 .2 .3 .4 .5 TIP WEIGHT PARAMETER, TWP .2 .4 .6 .8 1.0 WALL THICKNESS PARAMETER, WTP, AND SPAR DEPTH PARAMETER, SDP (b) Hingeless blades.

Figure 7.- Concluded.

!

m +_ J

\

or-I O

\

/

O_

\

/

\

-_ e3 *r-I _-4 II D.

o _4 o 0 o o

! ! l"

,-4 _ fl uJ _ " °,.--I D.

3[ u_ _3 .J e_ o[3 q) 0 c_ :E LU k- _J % ,% U I:E F- O_ m 4-_ m ry C_ U_ ©

\

¢4

\

I

!

© b_ °,.-_ o _t o _0 • U '_L_w '1N:II:)I:I:1303 _IY:ISNV_I1 H311d :I^11397-I0:) HARMONIC, n ADVANCE RATIO, _ = 0.5 INDICATES REFERENCE BLADE 0OR2 BLADE RADIAL STATIONa'F= .55 .... 1OR3 n--2ANDn=3 n--0AND n--1 .06 .06 f/ J .04 •04 1 S s s" t / .O2 .02 t I_ I I I 0 . 1 :2 .3 .4 .5 0 MP MASSPARAMETER LA .06 -- .06-- _ , ' ' 1 .04 .04 "_ _ -- 1 .02

!

W _'_ , I • • 005 .01 •015 015 0 0 .00S .01 F RE _0_

,."7 FREQUENCY PARAMETER, FP

I I I .06 _)u .06 . , , ' / .04 z .04 ...

•02 .02 I I I l___ • 1 .2 .3 .4 .5 ,5 0 ;" 0 .1 .2 .3 .4,.p., Wl TIP WEIGHT PARAMETER, TWP .061 I J _ _- ....

• 06 , ' ' '

I

"04 F,_... _ a i i _ _ [ I i 1 I .4 .6 .8 I.0 0 .2 .4 .6 .8 1.0 I- .2 _1 _SPARWALL THICKNESS PARAMETER, WTP , .06 , , , r :- -I e. .06 I _E .,( .04 .O4 .O2 , -- -- ........ !

I i i I L I i I i .2 .4 .6 .8 1.0 0 .4 .6 .8 i.0 .2 SPAR DEPTH PARAMETER, SDP to blade design Sensitivity of collective pitch transfer coefficients Figure 9•- parameters for an articulated blade• HARMONIC+ n ADVANCE RATIO, /.L =0.5 i INDICATES REFERENCE BLADE -- OOR2 BLADE RADIAL STATION,T= 0.55 ----- 10R3 n=0ANDn=I n=2 AND n= 3 • 03 -- .03 f i I 1 [ ....... T _ t I / .02 .02 -1 / f/ .01 .01 I 1 I I .1 .2 .3 .4 .5 .1 .2 .3 .4 .5 o MASS PARAMETE R,MP I&l ¢3 .03, IM 0.

.02 t/I .02

I- .o I ;--+

t .01 -= .01 __ I I u .005 .010 .015 0 .005 .010 .015 FREQUENCY PARAMETER, FP z .O3 lu 1 .03 I I F..... T I 1 D u M.

U.

- .02 .02 UJ o u _.. i ,< l J___ I I I "r -- - -l- ......

,,v 0 .1 .2 .3 .4 .5 0 .1 .2 .3 .4 .5 o- TIP WEIGHT PARAMETER TWP I- .03 u,.t, ----T-----T .... T--- --F---_ .03 ..... _t-- " +_ F " -I -_' "' .02 - .02 a

-

< -- i

m .0 1 - -_ 0.1 -_ " I O '_ J _ I 1 1 I --1_ J 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 I- .03 ' SPAR WALL THICKNESS PARAMETER, WTP o. _ v r r .03 1 _ r I ,< .02 - - .02 - .01 - - .01 --'- , I I I I . _ I L I 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP Figure lO.- Sensitivity of blade twist transfer coefficients to blade design parameters for an articulated blade.

HARMONIC, n INDICATES ADVANCE RATIO, /J, ----0.S -- 0OR2 REFERENCE BLADE BLADE RADIAL STATIONw'r=.55 ---- 1OR3 n=2ANDn=3 n=0 AND n=l 4 _ I '[ I 1 4 I /I /

/ 3 ' ' I ' l

3 / / / / 2 / - 2 / / / 1 // - 1 •1 .2 .3 .4 .5 0 .1 .2 .3 .4 .5 0 MP "u MASS PARAMETER u 3 _ I i 3 I I l % Z 2 "" ......

U.

M.

l tU U I I I I 0 .OOS .010 .015 0 .005 .010 015 14.

FREQUENCY PARAMETER, FP Z 2_ i J J _ '--- | :.. .....

P

P

I I I I I I I L l O .4 .5 0 .1 .2 .3 .4 .S 0 .I .2 .3 TIP WEIGHT PARAMETER TWP IL I I I I '._ I I L 3 _.

a I- ,.J a,, I I ; £ I 0 .2 .4 .6 .8 1.0 .2 .d .6 .8 1.0 SPAR WALL THICKNESS PARAMETER, WTP I • 8 1.0 0 .2 .4 .6 .8 1.0 0 .2 SPAR DEPTH PARAMETER, SDP Figure ii.- Sensitivity of inflow ratio transfer coefficients to blade design parameters for an articulated blade.

HARMONIC, n 0OR2 ADVANCE RATIO,/./. = 0.S l INDICATES REFERENCE BLADE ----- 1OR3 BLADE RADIAL STATION, T'= 0 n= 0 AND n= I n= 2 ANDn= 3 1.2 .12 .08 .04 I I [ I • 3 .4 .5 0 .1 .2 .3 .4 .5 MAS PARAMETER, MP I .12 1 I .08 \\ .04 I I .005 .010 .01 0 • 005 .010 015 FREQUENCY PARAMETER, FP 1 i .12 I I I •08 t I f f .04 I I I I i [ ] I .I .2 .3 .4 .5 0 .1 .2 •3 .4 .5 TIP WEIGHT PARAMETER, TWP I I r l .12 i I I I .08 .04 I l __ i i_ I 1 •6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR WALL THICKNESS PARAMETER, WTP ! I I I .12 , I T i .08 - .4 .04 I I I I • 2 .4 .6 0 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP 12.- Sensitivity of collective pitch transfer coefficients to blade design parameters for a hingeless blade.

igure HARMONIC, n ADVANCE RATIO, _ = 0.5 INDICATES _ 0OR 2 REFERENCE BLADE ----- 1 OR 3 BLADE RADIAL STATION,'F= 0 n= 2 AND n= 3 n= 0 AND n= 1 .2O .20 i I l .15 .15 / / / .10 .10 / / / /

.os _ .os

O /// _ ....

o- 0 .1 .2 .3 .4 .S 0 .1 .2 .3 .4 .5 MASS PARAMETER MP

_"',_' ' L " l' ' /

.10 \ _ .10 - -1

"""" - | .05 |

0 .0S .010 .015 0 .0S ,010 .015 FREQUENCY PARAMETER, FP .15 -----'1- i I 1 ! .15 ' ' ' ' i

.1o . ol I

" ,-i--:-t

0 .1 .2 .3 .4 .S 0 .1 .2 .3 .4 • TiP WEIGHT PARAMETER, TWP .10 - -- ......... ...... _ .10 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR WALL THICKNESS PARAMETER, WTP

l ' '

I , , , , / _-+--,--_--,-_

0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .6 1.0 SPAR DEPTH PARAMETER, SDP Figure 13.- Sensitivity of blade %vls% transfer coefficients %0 blade design parameters for a hlngeless blade.

HARMONIC, n 0OR2 ADVANCE RATIO, _ = 0.5 INDICATES ---- 1OR3 REFERENCE BLADE BLADE RADIAL STATION, T= 0 n=2ANDn=3 n=0ANDn=I 60 _ i I W

' L I !

__J

ffl L , t ( I J t I I t Z .! .2 .3 .4 .5 0 0 .1 .2 .3 .4 .S 0 MASS PARAMETER, MP Z i.i ! I \ Z

6°l '

4o _ Z c w U 1 I I Z .OlO .005 .010 .015 .015 0 0 .005 FREI UENCY PARAMETER, FP i I I 6 1 60 I I I I n U n, W L Z n_ b- I [ J 1 I I i I 0 .3 .4 .S 0 .1 .2 .3 .4 0 .1 .2 I.- TIP WEIGHT PARAMETER TWP I 6 ' i i i r 60 w I I ..J t 4O u.

Z .__._.

2O a L t l l 1 I I L .6 ,8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 G. SPAR WALL THICKNESS PARAMETER, WTP I i 60 1 I 6 u I t 4 4O f J f f I I f 2O ---- jl I . J I I j i I ] 0 .2 .6 .8 1,0 0 .2 .4 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP Figure 14.- Sensitivity of inflow ratio transfer coefficients to blade design parameters for a hingeless blade.

HARMONIC. n _0OR2 ADVANCE RATIO, _ = 0.5 INDICATES ------1OR3 REFERENCE BLADE BLADE RADIAL STATION,T= 0 n= 2 AND n= 3 n=0ANDn=I .06 .6 I 1 # I

I

.O4 .4 o su .O2 o .2- ¢¢ lu i G.

L k I [ _J .I .2 .3 .4 .S I*- 0 .1 .2 .3 .4 .S 0 MASS PARAMETER MP , , .06 .6 l 1 .4 .04 < u

\

I.- .2 .02 z -1 I I

g

i ,, l .01 .01 015 .005 .015 FRE( UENCY PARAMETER, FP o .6 I I !

I i .06 u L .04 .4 z K

t

- .02 .2 z u i I -r_'T

E i

I I i .4 .5 0 .1 .2 .3 .4 .5 .I .2 .3 u 0 TIP WEIGHT PARAMETER TWP J u .6 I 1 ] I , 0.6 | I I >.

u .04 .

.4 .< u.

o w .02 .2 I- I I - _ "I_ _- I i i .J IL 1.0 • 6 .8 1.0 0 .2 .4 .6 .8 SPAR WALL THICKNESS PARAMETER, WTP .< .6 I i .06 i I" T I I .4 .04 ----_ "- "_ "_- .02 .2 I 1 I _ 1 I .6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP

Figure 15.- Sensitivity of Als cyclic pitch transfer coefficients to blade design

parameters for a hingeless blade.

HARMONIC, n ADVANCE RATIO, _ = 0.5 -- 0OR2 INDICATES BLADE RADIAL STATION, F'=0 REFERENCE BLADE 1OR3 n=0ANDn=I n=2ANDn=3 .6 I _ I I .06 .4 - ,04 ,.....

t I / .2 - ___--- .02 f / I I I I t&l I .1 .2 .3 .4 .5 0 .I .2 .3 .4 .5 n, MASS PARAMETER MP Ua .6 n 1 I I .06 F- .04

z .4

"c

.2 .02 n", u i I I I

o

• 005 .010 015 0 .005 .010 .015 z .i FREQUENCY PARAMETER, FP

.6

I I I , .06 l I I I ii L W 0 .4 .04 U In U. .2 ,02 Z n, I I I I I- I I I .I .1 .2 .3 .4 .5 0 .1 .2 .3 .4 .5 Z 0 U TWP TIP WEIGHT PARAMETER I- .6 = .06 J I = I I I I

E

u .04 -J .4 u >.

u .02 m .2 L o I 1 I I J w I I I a 0 .2 .4 .6 .8 1.0 0 .2 ,4 .6 .8 1.0 SP_R WALL THICKNESS PARAMETER, WTP

-_ .6

1 i i i .06 i i i i .J G.

.04

.2 o2 -

I I I I ""_] I I L 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP

Figure 16.- Sensitivity of Bls cyclic pitch transfer coefficients to blade design

parameters for a hingeless blade.

HARMONIC, n ADVANCE RATIO, /.1 --0.S / INDICATES -- 0OR 2 r 0 REFERENCE BLADE -- ---- 1 OR 3 BLADE RADIAL STATION,-= n= 0 AND n= 1 n=2 AND n= 3 .6 , ' 1 ' .06 ' ' 1 ' .4 - .04 .2 .02 O m L____-- __J___ I----- ----I I I 0 .1 .2 .3 .4 .5 0 .I .2 .3 .4 .5 n, MASS PARAMETER MP " .6 ' I .06 [ I T | !

I-

; .4 - .o4

E " .2 - .02 m 0 .005 .010 .01S 0 .00S .010 .015 FREC,UENCY PARAMETER, FP i:" .6 i i i r .06 _ _ I 1 U.

UJ U .4 .04 iv u.I u.

•2 ,02

t

Q_ l._ ___ .L_ __L _-- I "- I I I , J z 0 .1 .2 .3 .4 .5 0 .1 .2 .3 .4 .S • _ TIP WEIGHT PARAMETER TWP 0 .6 i i I I ,06 I I T U I III n, .4 .04 G.

U.

m .2 _" .02 a I...

"; __ .__l __ _____ ----I----r---I- --r-- " 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 < SPAR WALL THICKNESS PARAMETER, WTP • 6 i i , i .06 i I . . . . . . _ . . . . . T .4 .04 .2 ----_ .02 v 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 SPAR DEPTH PARAMETER, SDP Figure 17.- Sensitivity of preconing transfer coefficients to blade design parameters for a hingeless blade.

bo +_ or-I C_ ,,-I +_ or-I © or-I I © llJ ,-cl o I o °r.4 _3S/1.-I '_+ 033dS ell1 _1010_1 .026

/

.024 0 VALUES USED IN C PRRENT NVESTIGATION .022 . .020 ,011 U. .01_

=-

IM .014 == < 6, )- .012 U 2[ mLm IM .010 U.

.008

J

.006

J

I

.004 .002

(_-

.2 • 3 .4 .5 .6 .7 .8 .9 i.0 1.1 1.2 1.3 1.4 ADVANCE RATIO, Figure 19.- Variation of frequency parameter with advance ratio.

.015

BLADE 5 M075 , O MP = 0.3, FP = 0.0025

I

.010

.t 0.25 _] 0.4

. OO5

<) 0.5 w (I.

Z -.005

m

-.0t0

u c

-.0t5

Z

-.0=O

U L !

n w M075, 1, s U

= 0

Z < -°_5 _r U 0.

-.O]O tu > I- U j -.015 .-I O U -.020

-.025

-.a3o -° ol]5 O .Z ._ o6 .B 1.0 O .Z ._ .6 .U I.Q BLADE RADIAL STATION, T (a) Zero and first harmonics.

Figure 20.- Sample design chart: collective pitch transfer coefficients for articulated blade 5 at advance ratios 0.25, 0.4, and 0.5.

.025 , .020 ,

_s, 2, = MeTs, 2, *

A o.2s ! .015 I

.020 13 0.4

O.S

.015 .OzO

o tU a

._5

"' .010 ,, UJ a.

.OQ5

D m ev

o 0

u C

_ -.1115 -.010

II z

-.Os5

_ -.010

.11 I.O 0 .;_ .I; .e .I_ ! .0

0 ._ .I .B

U.

.. .020 -_5

i W o _0 3, • MO7S , 3, c u S ' ,,J

w .01$

Ik t#l z < e_

_" .010

-.OD5

-r U °: t.,- 0,.

-.O]O

,. ._S

> t-

\ Z

u ILl

\ /

..J ,.

..J u

-.020

_.0o5

-.o10 -.a25

-.o15 -.03Q

0 .Z .ll .l .II 1.0 0 .Z .tt .ts .u I.o

BLADE RADIAL STATION, (b) Second and third harmonics.

Figure 20.- Concluded.

.7

7=0 M075, O BLADE MP

.6

I A 2 0.1 f _] 5 0.3

_, 8 o.s

.S

O F P : 0.0025 O

.4

f O.

m Z w ns O

.2

c

.l

Z .i D

_25 .¼0 .q5

.30 .3S .50

u_

U.

U. .6 .6

O M875 , 1, c M075, I, s= U

/

n_

I

/ 1

i,,

,. .5 .5

.3 .3

j I,---'-

_ .2 . .2

f

o ._

U

.----E

.I -I i

0 0

.! .... .!

_5 .SO .3S qO qS SO 2S SO 3S qO qS 50

ADVANCE RATIO,/l

(a) Zero and first harmonics

Figure 21.- Sample design charts: collective pitch transfer coefficients for

hingeless blades 2, 5 and 8, advance ratios 0.25 to 0.5 and _ = O.

•06 i . Oq _

MP BLADE J_075 , 2, © M075,2, •

.05 "E] s o.3- / .03

_1, e 0.5 /

/

F P = 0.0025 ,_

.0_ , / .02

a _ .0l

' t

m

.q5 .SO

L

.07 .07

M()75' 3, c M075, 3, • U

W I

J ' .06

I& .06

z < n,.

.OS

I-

.05

-i- u I-

/ / .o_

.o_ /

lu

/

I,- U

.03 ;//

.I

.O3

.J u

o_ ///Jr/

.O2

.ol /_/

.Or

Y /

J

I

.25 .S0 ._5 .q0 .qS .S5 ._ .q5 .SO .50 ._0

ADVANCE RATIO,/L

(b) Second and third harmonics

Figure 21.- Concluded

1.2

I I I I

M075, MP= 0.3 FP = 0.0025 (FOR/z <_ 0.5) UNIFORM BLADE I-I 0.25

+ 0.4

A 0.5

X 0.7

0 1.o

, 1.4

I i I i

0 .4 .6 .8 1.0 .2

I I t .4

M075, 1, c M075, 1, s

I I I l _ I

.2

-.2

, I I I i _

0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 BLADE RADIAL STATION, F

(a) Zero and first harmonics

Figure 22.- Variation of Collective pitch transfer coefficients with radial

position for the reference hingeless blade and advance ratios 0.25 to 1.4.

•6 .8 1.0 .2 .4 .6 .8 i.0 I I I I I

' I

s M075,3, M075 , 3, c

I I I I I

I I I

.8 1.0 0 .2 .4 .6 .8 1.0 .2 .4 .6 BLADE RADIAL STATION, T

(b) Second and third harmonics

Figure 22.- Continued

.4 .4- !

| I ' ! " I I I ' • n o.2s M075 • 4, • 4" 0.4 0.5 X 0.7 .2-0 1.0 _ 1.4 -- .2 Q a.

o o

w -.2

I I I I I I I I

• 2 .4 .6 .8 1.0 .2 .4 .6 .8 i.0 I I I - I I I I I M075 , 5, c Me75 , 5, s .21--- -.21--

I I I

I I

.8 1.0 .2 .4 .6 .8 !.0 BLADE RADIAL. STATION, F

(o)

Fourth and fifth harmonics Figure 22.- Concluded

o.2_

0.S0_

L I

0.751

L

A 1.0 _ a um I-

4! :i t

O L WEIGHT DISTRIBUTION, LB/IN Z < I'- LU U x uJ v ,.,I U h- I- roll ..I .J !h_1.1_ 1'0.87 I FLAPWISE AREA MOMENT, IN .1(

_u

Ik' _' {" T.I .FT. '7.07 1.9 i ;i "i i' ( --- ---'_ )'_l"_N_"_'_"_ "'' '1 ' ' ; I 1.471 L | 1.439. I \\_J_'

6 )2'o 4b 6'o 8'o 160 l_o 1,kol_,o 1_0 26o 2_0 2_0 2_0 _0 3603_0_0

_.._ t. A P HINGE ENTER OF ROTATION BLADE RADIUS, INCHES Figure 23.- Weight and moment of inertia distributions for an H-34 blade.

.03 1 I I I I .02 .01 _) -.01 -.02 I I I I I , .03 i I i I i .02 o .01 lJl i/m

, I I i I I

a. -.01 ,< -I u. 2 I I l I I _c I-- i -I 0 60 120 180 240 300 360 BLADE AZIMUTH POSITION, ¢, DEG Figure 24.- Collective pitch, twist and inflow ratio transfer functions for the H-34 rotor at 150 kts and _ = 0.65.

6 ,XlO 3 I I I I _ I COLLECTIVE PITCH CONTRIBUTION j" 875= 8.5 ° (FROM REF.

-2 I -4 -- 2[ x103 I I I I I g" I BLADE TWIST CONTRIBUTION ._ B.

_1 L I

-, I I I I I

6 XlO 3 I I I I I INFLOW RATIO CONTRIBUTION _. = -.078 (FROM REF.7) 4-- 2-- 0"_._ --2 u

-4 I I I L I

60 120 180 240 300 360 BLADE AZIMUTH POSITION, ¢, DEG Figure 25.- Contributions of collective pitch, twist and inflow ratio to the total stress of the H-34 rotor at 150 kts and _ = 0.65.

H-34 ROTOR 110 KTS aS=-5 e L= 8300 LB D=-750 LB EXPERIMENT (REF.13) -- -- TRANSFER FUNCTION APPROACH (CONTROLS FROM REF. 7) COMPUTER ANALYSIS (FULLY COUPLED) 4 XIO 3- 7_ fl.375 2" f a.

-2 i, F- W D.

*C L 2 ,,,_0 3 I F= 0.650 , ×___. _,

!

-2 -4 0 30 60 90 120 150 180 210 240 270 300 330 360 BLADE AZIMUTH POSITION, _, DEG.

(a) ii0 kts Figure 26.- Comparison of predicted and measured stress distributions for the H-34 rotor at _ = 0.3T5 and 0.65.

H-34 ROTOR 150 KTS aS=.5 ° L= 8500 LB D =-650 I..B EXPERIMENT (REF.13) TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) COMPUTER ANALYSIS (FULLY COUPLED ) I(I0"*

I

-- 0.375

-- f

,f f

", _

j S

--2 \.-_.

m --4r L n, I-- 4 X10 "_ ILl

I

7= 0.650 11.

-J ek

/ \\ /

-2 -4 -6 270 300 330 30 60 90 120 150 180 210 240

BLADE AZIMUTH w, os_'r_oN, ¢, OEO

(b) 15o kts

Figure 26.- Continued H-34 ROTOR 175 KTS as= -5o L= 7100 LB D= -250 LB EXPERIMENT (REF.13) TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) COMPUTER ANALYSIS (FUL'LY COUPLED ) 4 _,v m -4 n LL -6 I/I gJ n,, I-- MJ i/) 7= 0,650 f n .< / .,I / ii 0 L

t

/i

\

\

-4

.J

-6

I

30 60 90 120 150 180 210 240 270 300 330 360 BLADE AZIMUTH POSITION, _s, DEG

(c) 175 kts

Figure 26.- Concluded

_)1 -- 0 "r= 0.55 -- TRANSFER FUNCTION APPROACH (CONTROLSFROM REF.7 ) COMPUTER ANALYSIS (FULLY COUPLED) - _ COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM _EF.7 ) 4FXi03 , , 1 i ,.._\ , 'lIj 1°3 '/1= o.2s ....

/ #=0.5 ,/_,

_",< ii \ ,i

\ ,,/ ",,,.,,"

o \ /

',\ _ .// 2 "'" I \_ l I e.

6-X103 --_-- T--T r r ,, :6IX103 --'r 1 ; r T--

/

\ F= 1.0

I.

,,, , , , ,,

!7

i-- } { j

0 _ _

-I

Il li /l !

, . Y _.1//

-2 , LJ'k /,

_i,,_-e/i '

I I,._ _ _ / -4 I%% #l I %% _ . o S

I

_6 _ _6 _ J- 0 60 120 180 240 300 360 0 60 120 180 240 300 360 BLADE AZIMUTH POSITION,_, DEG

Figure 27.- Comparison of azimuthal stress distributions predicted by the

transfer function approach and the computer analysis at advance ratios 0.25, 0.5, 0.7 and 1.0 -- reference articulated blade (aluminum), 91--0 , _ = 0.55.

_1-'8 DEG T=0.55 ..... TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7 ) ..... COMPUTER ANALYSIS (FULLY COUPLF.L_ I _COMPUTER ANALYSIS (FLAPWlSE MODES, CONTROLS FROM REF.7 ) 8 -Xl03 _ /_= 0.5 i I /z = 0.25 .2 2 -i', -4

:::t be',, t

0 60 120 180 240 300 360 -6 BLADE AZIMUTH POSITION,_, DEG I -8 I I

-1°o 60 d,o 1so 2,_o 300

BLADE AZIMUTH POSITION,_, DEG Figure 28.- Comparison of azimuthal stress distributions predicted by the transfer function approach and the computer analysis at advance ratios 0.25 and 0.5 -- reference articulated blade (aluminum), gl = -8 deg, _ = 0.55.

_1 = 0 T= 0.55 TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) ...... COMPUTER ANALYSIS (FULLY COUPLED_/ - _ COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM REF.7) t I 8 Xl03 i w 8IX103j_ , t i i i , #/ # 4/| /_ = 0.25 4 -4 I/I I_. -4 u. -8 I i i i 1 I l 1 I *Jl IIpXlO 3 ....

m 12-XiO 3 I-- h lld ,- !,_ /s.= 1.o . _ = 0.7 ,,,'-_ I/1 8- .i I1.

-4 1 I I 81 i I i I -8 0 60 120 180 240 300 360 0 60 120 180 240 300 360 BLADE AZIMUTH POSITION,_, DEG Figure 29.- Comparison of azimuthal stress distributions predicted by the transfer function approach and the computer analysis at advance ratios 0.25, 0.5, 0.7, and 1.O -- boron-epoxy articulated blade, @l = O, r = 0.55.

r 0.55 _1 ---8.0° -= TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7 ) ..... COMPUTER ANALYSIS (FULLY COUPLED) - --COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS =ROM REF.7 ) 16[--X103 I i T r T ........

I

#= 0.5 12 t I I t I 8 -X 103---r ....... _ ...... T_...... _.....

--1

y. = 0.25 m

4 4

LI.

i f -4

' 1

\.t I I -8 -8 I

!

L .... J__ .... _ j j L

I

0 60 120 180 240 300 !

-12 -4 BLADE AZIMUTH POSITION,_', DEG

!

i -16 I i I I I -20 60 120 180 240 300 36O BLADE AZIMUTH POSITION, _', DEG Figure 30.- Comparison of azimuthal stress distributions predicted by the transfer function approach and the computer analysis at advance ratios 0.25 and 0.5 -- boron-epoxy articulated blade, gl = -8 deg, _ = 0.55.

12 Xl0 3 I I I REFERENCE ARTICULATED BLADE (ALUMINUM) TRANSFER FUNCTION APPROACH(CONTROLS FROM REF.7) COMPUTER ANALYSIS (FULLY COUPLED) _ COMPUTER ANALYSIS (FLAPWISE MODES) CONTROLS FROM REF.7) el _1=0 _n I-- O.

I I I !

i1 t_ 16 Xl0 3 I I MJ e_ BORON ARTICULATED BLADE I- lIJ v') D.

iL a.

I-- el (_1 =-8° _1=0

I I I I

.2S .30 .3S .40 .4S .50 ADVANCE RATIO,

Figure 3L.- Comparison of 1/2 pee_-to-pee_ stress of articulated blades

predicted by the transfer functlon approach and the computer analysis for

advance ratios 0.25 to 0.5 -- reference blade (aluminum) and boron-epoxy

blade, gl = 0 and -8 deg, _ = 0.55.

TRANSFER FUNCTION APPROACH, (CONTROLS FROM REF.7) COMPUTER ANALYSIS(FULLY COUPLED) COMPUTER ANALYSIS (FLAPWlSE MODES, CONTROLS FROM REF.7) 01=0 7= 0.55 10 Xl03 I I

\

m BORON BLADE 8 N o.

MP= 0.3 rt FP= 0.01 I.- -_'= 87900 PSI LI.

6- IM I-- s/t LlU sd_ 4- Q.

.J U.

O,.

I-- REFERENCE ALUMINUM BLADE e= MP= 0.3 F P : 0.0025 ER_'= 31400 PSI

I I I I I I

0 0.2 0.4 0.6 08 1.0 1.2 !.4 ADVANCE RATIO, /1

Figure 32.- Comparison of i/2 peak-to-peak stress of articulated blades

predicted by the transfer function approach and the computer analysis for

advance ratios 0.25 to 1.4 -- reference blade (aluminum) and boron-epoxy

blade, @l = O, _ = 0.55.

PSi CONTOURS OF 1/2 PTP FLAPWISE STRESS, (TF 1/2 PTP' FROM TRANSFER FUNCTION APPROACH PERFORMANCE CURVES FROM REF.7, FIG. 14b MINIMUM STRESS LINE (O'FMIN1/2 pTp) .020 ,016 .012 -.020 Figure BB. Contours of 1/2 peak-to-peak stress.

X10 3 6.0

I , /

/ c L = . -= .

W I 5.5 W I o. 5.0 < ¢N J

i

4.5

I I I I

4.0 03 -.02 -.01 0 .01 .O2 DRAG COEFFICIENT/SOLIDITY, CD/o 8 X103 I I r 0.55 C L/_r = 0.05 -= MINIMUM PEAK-TO-PEAK STRESS CD/O -.016 -.008 -.006 .008

I I I I I

60 120 180 240 300 360 BLADE AZIMUTH POSITION,_, DEG Figure 3_.- Variation of stress with drag coefficient for a constant lift coefficient 0.05.

I iX10 4 J I I I I eL ALL CD/O" I BLADE TWIST, 81 , CONTRIBUTION _I = -8 o o , v¢ ikl O= I-- if1

-I_ I I

uJ t/,) a,.

.< .,J

%

u.

2 X10 4 I I I INFLOW RATIO,_c,CONT RIBUTION • 008 .042 0 -.046 -.125 016 -.191 -I -2 I i I 0 60 120 '180 240 300 360 BLADE AZIMUTH POSITION, _', OEG

Figure 35.- Variation of compone_ts of flapwise stress with azimuth position

and drag coefficient for a constant lift coefficient 0.05.

5 Xl03

J

u_

I

f

O'FMIN 1/2 P_TP "-_ 13..

J

..I

Z

.oo4

4 i

LL L _ UJ W

Du

I'- V') (CD/°') MIN 1/2 PTPo'_: a.

I" o -,008 3 0.

F" z -.012 m A. IU W -

z_

U. W

o_

I,U II_ I-- U.

._uJ acO i.u U FOR CD/° ((CD/O_ I_l_ V2 pWP o F Oo n,, n_ tur', >

• . .o3

• 04 .05 .06 .07 .08 ROTOR LIFT COEFFICIENT/SOLIDITY, CL/O"

Figure 36.- Constants in linear equation representing i/2 peak-to-peak stress

as function of rotor lift and drag.

T=O p.= 0.5 BORON-EPOXY HINGELESS BLADE _I = 0 I I

I

30XI0 3_ AI s '1.05 DEGo HUB MOMENT=t6800 FT-LB A I• 2.05 DEG, HUB MOMENT= 6800 FT-LB TOTAL STRESS 2O

/ \

RST HARMONIC OMITTED

/

/

O. \

/

\

\

ill n, l-

\

u4 i/I

\

n 4_

\

.J 14=

\

\

I I I I I 0 60 120 180 240 300 BLADE AZIMUTH ANGLE, _, DEG Figure 37.- Effect of a one degree change in Als cyclic pitch on hingeless blade stress.

12Xi0 3 I I I I I TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7 ) COMPUTER ANALYSIS k (FULLY COUPLED)

/

COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM REF. 7 )

/

\

\

B U- -4 I I I I I u¢ 11) U,l l- v'J IM 8X10 3 I I I I I a.

FIRST HARMONIC OMITTED ,.I II.

-

-4 I I I I i 0 60 120 180 240 300 360 BLADE AZIMUTH POSITION, _, DEG Figure 38.- Comparison of azimuthal stress distributions predicted by the transfer function approach and the computer analysis at advance ratio 0.25 -- reference hingeless blade, e I = -8 deg, _ = O.

TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) .... COJ'APUTERANALYSIS (FULLY COUPLED) COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM REF.7) 30 XI03 I I I I I 0 60 120 180 240 300 BLADE AZIMUTH POSITION,_, DEG

Figure 39.- Comparison of azimuthal stress distributions predicted by the

transfer function approach and the computer analysis at advance ratio 0.5

-- reference hlngeless blade, 01 = -8 deg, _ = O.

TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) COMPUTER ANALYSIS (FLAPWlSE MODES ONLY ) X10 3 _ , , , , --- I0 FIRST HARMONIC OMITTED

/\ i,= o.z

BLADE AZIMUTH ANGLE,_, DEG Figure 40.- Comparison of azimuthal stress distributions predicted by the transfer function approach and the computer analysis at advance ratios 0.7, 1.O and 1.4 -- reference hingeless blade, @l = 0, _ = O.

20XI03 I I I I 81= -8 DEG FIRST HARMONIC OMITTED TRANSFER FUNCTION APPROACH ( CONTROLS FROM REF. 7 ) COMPUTER ANALYSIS ( FULLY COUPLED ) 16 m - _ COMPUTER ANALYStS { FLAPWlSE_ MODESo CONTROLS FROMREF.7) BORON BLADE FP= 0.01 L ALUMINUM BLADE F P = 0.0025 Ill P UI L

I I I I

.25 .30 .35 .40 .45 .50 L L P 12 X 10 3 • I I I I J I ALUMINUM BLADE FIRST HARMONIC OMITTED F- .25 .30 .35 .40 .45 .50 ADVANCE RATIO,_ Figure kl.- Comparison of i/2 peak-to-peak stress of hlngeless blades predicted by the transfer function approach and the computer analysis for advance ratios

0.25 to 0.5 -- reference blade (aluminum) and boron-epoxy blade, @l = 0 and

-8 deg, _ = 0.

TRANSFER FUNCTION APPROACH (CONTROLS FROM REF. 7) COMPUTER ANALYSIS (FULLY COUPLED) --- -- COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM REF. 7) 28X10 3 I I I I I I 81=0 FIRST HARMONIC OMITTED a.

Q.

I- n tJ_ IAI I-.

\

IM BORON BLADE s FP= O.

/ \ \

/ --I L \ , a.

\

\ I-- a.

ALUMINUM BLADE X ¢'N 8 FP=0.0025 / 1 I I I 0 .2 .4 .6 .8 1.0 1.2 1.4 ADVANCE RATIO, # Figure 42,- Comparison of i/2 peak-to-peak stress of hingeless blades predicted by the transfer function approach and the computer analysis for advance ratios 0.2.5 to 1.4 -- reference blade (aluminum) and boron-epoxy blade, @l = O, _ = O.

E1 = 115000 IN.-LB NONUNIFORM HINGELESS BLADE MP= C.18 FP = 0.01 81= -4 DEG R F=: 0 F = o.2s CL/O' = 0.05 CD/O' = -0,0033 _B = 2.8 DEG -2 X10 3 I I I

, \

\

\

\

\

\

\

\

TRANSFER FUNCTION APPROACH (CONTROLS FROM REF.7) COMPUTER ANALYSIS (FULLY COUPLED) k COMPUTER ANALYSIS (FLAPWISE MODES, CONTROLS FROM REF.7} _,

\

i , I I i FIRST HARMONIC OMITTED

/,¢'-<x \ -

60 120 ! 80 240 300 360 BLADE AZIMUTH POSITION, _, DEG Figure 43.- Comparison of azimuthal moment distributions for a typical nonuniform hlngeless blade predicted by the transfer function approach and the computer analysis at advanoe ratio 0.25 -- @l = -_ deg, r = O.

TRANSFER COEFFICIENTS FROM' CHARTS APPROXIMATE ANALYSIS - -., c OR, g,_ M075 ' n, c OR• M01" ¢, n, c ORs o o -. o.01_ .

I,¢ 0 -.02 .!

.o4- .o2 I .o2- ff'-'_._ .o,

,.o o Y'- '

-.02 -- -.01 ._ o_ .o, r .. _ ,f . ../_, o o,f-----, .

_:,e o .o,L ..,_

0.2 • 1 3,c 0 .... -.-" "" _" "" ..02 _ -. -I -o, -.o, _ • ' ' , -,I I L_',_._J , 0 .2 .4 .6 .8 i.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .a 1.0 BLADE RADIAL STATION, F Fi_Ire 4_.- Comparison of transfer coefficients predicted by the approximate and computer analysis for a articulated blade (MP = 0.3, FP • 0.0025) at advance ratio 0.05.

REFERENCE ARTICULATED BLADE MP= 0.3 FP= 0,0025 _1 = -8 DEG Iz= 0.5 ";= 0.55 CL /O= 0.05 ac= 0 8X103 I i CHARTS "" APPROXIMATE ANALYSIS l TRANSFER FUNCTIONS FROM: o.

\

I-

/

w

/

o.

\

.,I

/

M.

\

/

\

\

/

\

\

I I L I

0 60 120 180 240 300 360 BLADE AZIMUTH POSITION, _./, DEG Figure 45.- Comparison of an azimuthal stress distribution predicted using transfer functions from the approximate and computer analyses for the reference articulated blade at advance ratio 0.5.

TRANSFER FUNCTIONS FROM; CHARTS APPROXIMATE ANALYSIS F= 0.55 4 Xl0 3 • T I I | COLLECTIVE PITCH CONTRIBUTION I

I o,s -- 2.5 OEG

i__ _ - -2 "_ .... "" L-. "_, I I 4 Xl0 3 , * I BLADE TWIST CONTRIBUTION I 1

/

_i=-80EG 2 I

= -2 I- \\ / /

_ iiI/ , i

o I _ J I I I I I xl0 INFLOW RATIO CONTRIBUTION I I I I _c = -o.oo5_ _1_ I I I I I 0 60 120 1 B0 240 300 360 BLADE AZIMUTH POSITION, _, DEG Figure _6.- Comparison of the collective pitch, twist and inflow ratio stress contributions predicted using transfer functions from the approximate and computer analyses for the reference articulated blade at advance ratio 0.5.

I MODE SHAPE BLADE LOADING FOR(_I= 1 DEG

\

\

I I .2 .4 .6 BLADE RADIAL STATION, r Figure 47'.- Comparison of first mode shape and advancing blade loading distri- bution due to twist for the reference articulated blade at advance ratio 0.5.

TRANSFER COEFFICIENTS FROM= APPROXIMATE ANALYSIS [ J COMPUTER ANALYSIS (CHARTS) • 08 -- J

.06-

o g m £ .2 .04 -

o

uu

(.(.__

- 0 ""I,M _,,.r ,,,3: • . _ .06

r-o

I z_: _uJ u_ .06, • N _ .04 - .041-- w D v .021-- Ol- (-) (-) ,_,J'-"l (-) (-) (_) (r'_ _

[_//JI oL

-.J-'l --J-I

_.o,,_, 0

z

_- _ 20 I

o c 10

j 4! 1 21-

a_ 0 1,c 1,• 2,c 2, s 3,c 3, s HARMONIC COMPONENT, n, cOR • (a) Collective pitch, twist and inflow ratio transfer coefficients.

Figure 48.- Comparison of transfer coefficients predicted by the approximate

and computer analyses for a hingeless blade (MP = 0.3, FP = O.Ol) at advance

ratio 0.5 and _ = O.

TRANSFER COEFFICIENTS FROM= APPROXIMATE ANALYSIS COMPUTER ANALYSIS (CHARTS) .03 .15

°i .10

m - .01 - _- .05 - .02 - o (o)(-___)

(olr'l (o) _ (o)n (o)_

,_ 0 ill N n,.

O .O3

"T .3

(-) lU n_ O .2 .02 - O u ill (-) (-) (-) .01 - o i::_ = .I m

z t

8.

Ig//_"._ I- Z .O6 z (°) ill if/J(-) ¥/Ak,..= m W I.

.O4

_ _ .lo-

0 u U .O2 _ =_ .os.

z l'/,/i (l_g__ r_Jl r_ I.- f/ll (o)__ (o) (o) 0 (o) (o) r-1 (o) 0 ,,' 1,c 1,s 2, c 2, • 3, c 3,$ HARMONIC COMPONENT, ne c OR • (b) Cyclic pitch and preconlng transfer coefficients.

Figure _8.- Concluded BORON-EPOXY HINGELESS BLADE MP=0.3 FP=0,01 E__R =87900 PSI _1= -4 DEG /_= 0.5 _= 0 CL/G= 0.05 =S = 0 /_S = 0 FIRST HARMONIC OMITTED I I I I I TRANSFER FUNCTIONS FROM:

10 OLlO'

CHARTS APPROXIMATE ANALYSIS _. 7.5 _ 5.0 --I ), IIL 2.5

_

I I f I I 60 120 180 240 300 360 BLADE AZIMUTH POSITION, _, DEG Figure 49.- Comparison of an azimuthal stress distribution predicted using transfer functions from the approximate and computer analyses for a boron- epoxy hingeless blade, @l = -4 deg, at advance ratio 0.5.

TRANSFER FUNCTIONS FROM : CHARTS APPROXIMATE ANALYSIS 35 Xl04

I w I I I

COLLECTIVE PITCH CONTRIBUTION 2,_

/

I I I I I

60 120 180 240 300 360 A_ I I I I I L

I

20 Xl0 3 BLADE TWIST CONTRIBUTION _1 = -4 DEG I W4 wJ -10 .J

- I I I I I

w.

0" 60 120 180 240 300 360 INFLOW RATIO CONTRIBUTION L l I I I I -10 104 XS=-0"1495 -25

I I I I I

0 60 120 lg0 240 300 360 BLADE AZIMUTH POSITION,_4 DEG

Figure 50.- Comparison of the collective pitch, twist, and inflow ratio stress

contributions predicted using transfer functions from the approrlxmate

and computer analyses for the boron-epoxy hlngeless blades, @l = "h deg, at advance ratio 0.5.

HINGELESS BLADE MP= 0.3 FP=001 #=0.5 lO 1.0

I l I I

MODE SHAPE STEADY LOADING FOR__TS = 1 DEG 8 i m n_ ..I a o _s Z o z N < Z w ,6 6 J W ul ..J m Q.

J 0 I- I- Z 4 w I1_ ,4 Z u.

U Ill 0..

.r w LM a ,2 2 _

I I 1 I

.2 .4 .6 .8 1.0 BLADE RADIAL STATION, 7 Figure 51. Comparison of the first mode shape and steady loading distribution due to collective pitch for the boron-epoxy hingeless blade at advance ratio O.5.

@

.OOB

8LADE 1

I

MP=0.1, FP=0.001 _075 o 0

I

A 0.25

Pl 0.4

0.5

I

M075, 1, •

I

-.I]]E -.00_

.6 .8 l.O

o d;2 ._ .5 .e 1.0 o .2 .u

BLADE RADIAL STATION, _" (a) Zero and first harmonics.

Collective pitch transfer coefficients for articulated blade 1

Figure 52.-

at advance ratios 0.25, 0._ and 0.5.

.008

.OO8

M#75, 2, s

I

.00E A 0.25

.005

Ill 0.4 _t' 0.5 ,,_k.

.00q, I,U a n,.

.00E

I,M O..

-.00z

c

ff

-. 00u, Z

-.006

U D .2 .U

.6 .El 1.0 O .E .U .a

.8 t.Q

,, -QQB

-QQ8

UJ M(_75, 3, c M075 , 3, s U iv

I

I .l_E

.00fi

U.

Z _- .O0q, Z U I- n

., .002

F- U

J.Ap .11-

a Ol

.J U

-°002

_e,-J

-° 0ot, -. oo_,

-.oQfi

-.1_6

Q .2 .11 .6

• 0 I.Q 0 .2 ._ .1_ .B !.0

BLADE RADIAL STATION,, _" (b) Second and third harmonics.

Figure 52.- Concluded.

.(3O8

i BLADE 2 D M075 , 0 MP = 0.1, FP = 0.0025 & 0.25 0.4 0.5 W a 0.

\ /

-.01_

I" Z W

-.nOB

13 ,2

.u .5 .8 t.Q

u_

A_

.QQB

u. . (313B

I I U,I _75 ' 1, c Me75, 1, • U n,

I

.nn5

uJ .CX3E

U.

Z .< E

.QD_

._

U I- a.

uJ .0132

I- U

"k

SY

u,I .J O U

_-- -.oQa

-o002

-.BI_

-.005

-.0QE

.2 .U .E .B I.Q 13 .Z .E .8 .B I.Q

BLADE RADIAL STATION, T (a) Zero and first harmonics.

Figure 53.- Collective pitch transfer coefficients for articulated blade 2

at advance ratios 0.25, O.& and 0.5.

.nnB

I | I MO-/5 , 2, c M_75, 2, • & 0.25

I

.006

.nn6 E9 0.4

_1' O.S

nn_

i,i ev

.OQ2

.002

w O.

,,?

ev

o -. 0[]2

g

,.IY

-.00_

I_E Z t,,

-.OQB

U .g ._ .E

.g I.Q [1 .2 .tl .B .8 t.Q

i, .Q[]B

i i m

_e75, 3, c Me7s, 3, •

U n., t_

. I:]Dfi I

.[1)s

U.

.< I,-

.B_IL

X U i.-

._32

wl i I- U ,,.J .J U

-.OD2 -.13132

--..... j

-. C£1_

-. OI]E -.OOE

I) .2 .11 .6 .e I.Q 0 .2 .IJ, .6 .B 1.0

BLADE RADIAL STATION. F (b) Second and third harmonics.

Figure 53.- Concluded.

BLADE 3 i_P = 0.1, FP = 0.01

.O03

• 0.25

e, o.5

.oro

a

•001

W 0,.

I- n,.

0 -. OQz

w

-.0(_

X t_

-.00_

O .2 .U .B •8 t.Q

u_

_L

u. .0oq, , .00_ I

tkl _75, 1, c U

M_75,I, •

I oos

w ._3

tl.

Z

/

.001,

_" .Q02

Z U I- ra a.

°nn3

uJ °(_| U

• .nn2

.J U

.0oi

-.QOi

-.Q02

-.ODI

0 .2 .II .6 .B l.O Q ,E .tl .6 .g I.Q BLADE RADIAL STATION, F (a) Zero and first harmonics.

Figure 54.- Collective pitch transfer coefficients for articulated blade B

at advance ratios 0.25, O.4 and 0.5.

.00_

.00_

m ' !

M075 , 2, ': Me75 , 2, • 0.25

I

.003

I

.003

rq 0.4

o.5

.OQZ

.002

W n*

.OQI

UJ a.

t,- Bt

-.om -.oo, "--_ J

W C

ff-.oo_

-.OQ2

I=E Z ILl

-.003

-.DO3

U

O .;_ .tl .E .B l.O Q .2 .It .fi

.O l.O

u.

.001

u.

.00; | u.I O M075 , 3, c U M075, 3, • ,v

uJ I

0 • f_ . I .

.003

u.

Z ,<

\ ,//

I-"

-.OOl

.1112

-r U I- ra O.

-.OQ2

UJ .OOZ

>.

I-,.

U

\ !

uJ

.J -.003

.J O U

-.QO_

-.001

-.nn5

-.11112

-.111:16

-.003

.2 .Ii .6 .O l.O O .2 .U .6 .8

1.Q

BLADE RADIAL STATION° _" (b) Second and third harmonics.

Figure 54.- Concluded.

.Or5

i BLADE 4 ]_075 , O MP = 0.3, FP = 0.001

I

.Ol0

& 0.25

rq 0.4

_1' 0.5

.005

) a ILl

-.005

-.010

t c

-.015

Z

-. QL:K]

D

O .2 ._ .B .0 L.Q

u_

._5

I .03Q !

L 0 _#75 ' 1, • _075,1, c U

I .r_s

U.

Z < //.= /,, U

m \

0.

.j -.0]0 > U UJ

.OZO

-J -.015

-I O U

.11115

-.020

-.025

-.0_0 -.005

.B I.Q

0 .Z .U .8 .O I.Q Q .Z .li .ll

BLADE RADIAL STATION. _" (a) Zero and first harmonics.

55.- Collective pitch transfer coefficients for articulated blade 4

Figure

at advance ratios 0.25, 0._ and 0.5.

.O25 .OEO

I ' M_5 , 2, c M_)75. 2. •

& 0.25 .I

I

.02O

.Q15

"_ 0.4 _1' 0.5

.015 .010

o Q

.OlO

.QOS

w o.

i .OOS

f

tv

o 0

U C

-.010

r_

,,z, -.QIQ -.015 O

O .2 ._ .a .0 I.O

.2 .LI .8 .0 l.O

.0_0 . OQS

-- M075 , 3. • U

-=, .ms o: _ J

z "'--e......__.

_- .o_o -.ms "_ \ /A

' Irk e'7 5 , 3,¢ ' ,,. '

-_ o ' - _ -.ms

-®s

-.01O -.0_5

-.U15 -.(3313

O .2 .U, .6 .0 1.0 O .Z ." .6 ,D I.Q BLADE RADIAL STATION, F (b) Second and third harmonics.

Figure 55.- Concluded.

•OrS • _LAOE 5

]_075 00 MP = 0.3, FP = 0.0025 .

0.5

P-- _005 _

75 ' 1, •

-.Og5 '

- -._S

.U .IS .B t.Q .6 .B I.Q Q .Z G .Z .U BLADE RADIAL STATION, T (a) Zero and first harmonics.

Figure 56. - Collective pitch transfer coefficients for articulated blade 5

at advance ratios 0.25, 0.4 and 0.5.

.0_S .02Q , 5 '2, c _07$,2,•

, o._ I .o, I

.ego _} 0.4

q, 0.5

.0'/5 .OlO

-.o]0

-.025

-.Olg -.o3a

.z • U B .11 l.O Q .Z .U .6 .B l.O

BLADE RADIAL STATION, F (b) Second and third harmonics.

Figure 56. - Concluded.

BLADE 6 ]_075,0 MP = 0.3, FP = 0.01

I

.Q!5

• 0,25 rlt o.d i

.010

0.5 w o

w .Q05

w @.

O!

-.005

o

-.010

l- z

-.0tS

m

.U .6 .8 L.Q

u_

u.

.0_0

I I w O 1_075 , i, c _ie7S, 1, • U E

I

.015

" .015

Ik Z < n_

.OZO

_" .010

'5" U I" m G.

.OOS

,- .005

>.

I- U w •J I1' ,j .

o u

-.005

-. Qlti

-.OIQ

-.UlS

-.015

.g !.0 O .Z .li .IS .IS t.O D ,Z .U ,IS BLADE RADIAL STATION, T (a) Zero and first harmonics.

Figure 57. - Collective pitch transfer coefficients for articulated blade 6

at advance ratios 0.25, 0.4 and 0.5.

.1_0

- I °_ ...... / t_5 , 2, c M#75,2, •

.Q15 a o.25 I .ms I

l'q 0.4 _1' O.S

.QIO .OlQ

Z

-_ -.015 • -.015

.z .I .e .e 1.0 0 ._ .u .a .e 1.o

u o

u.

- .0_0

I .1_0 I

ILl U M675, 3, c M#75 ' 3,, s

w .015

.015

eL .41 _" .01O

.010

-I- U I,'-

_u .005 .005

I-- U oJ

-.I]1O -.01O

-.QI5 -.015

II .Z .U .B .n I.Q O .Z .LI .6 .1_ l.O BLADE RADIAL STATION, T (b) Second and third harmonics.

Figure 57.- Concluded.

BLADE 7 MP = 0,5, FP = 0,001

.0s

• 0.25 rq 0.4 0.5 o w o

e_ .0]

w eL -.01

° /

.0t , • I

UIll U #75 ' I, c M075, I, •

_/ -t

-.01 .oz ._

-.OS -.0_ -. 06 -. 03 O .Z .U .IS .8 l.O 0 .Z .U .U .8 l.O BLADE RADIAL STATION, _" (a) Zero and first harmonics.

Figure 58.- Collective pitch transfer coefficients for articulated blade 7 at advance ratios 0.25, 0.4 and 0.5.

-. O2 -. 05 -Z .tl .IS .O 1.0 0 ._ is .iS .D l.[l BLADE RADIAL STATION, _" (b) Second and third harmonics.

Figure 58.- Concluded.

15'/ I BLADE 8 _e75, o MP = 0.5, FP = 0.0025

I

.Q_

& 0.25 n4

.0_

0.5

.010

-.Oso

t

-.020

,_

-.0_10 .... 8 t.O

U .lws

o '

1_75

i -.o_o k

_ X

_ --_° _ ----

_._n \ ,

-. o,10 _v,r -.020

-.OSQ -.[]_Q .ll .IS .8 l.Q Q .Z ,¼ .8 ,IS 1.0 0 .Z BLADE RADIAL STATION, _" (a) Zero an_ first harmonics.

Figure 59.- Collective pitch transfer coefficients for articulated blade 8

at advance ratios 0.25, 0.4 and 0.5.

. OV, O i . OU, O

p _07s, 2. _ t_e7s, 2. •

.0_0 a, o.2sl _ J ._0 . J

0.5

.02O .02O

Q 0¢

,. .OIO

a.

J-

. o__ o"__--_.___"

Z

w_ -. O3O -o _o

U__ 0 .2 .IL .6 .B I.Q Q .2 .U .8 .8 I.Q

ll..

.OqO , . • .030 , U M075 , 3, c M075 , 3. s n,,

,., I [

. .030 .020

Z ,< n_

i-- .OZO .O10

-r U I- ra el w

° _ "--_/

-.OlO -.020

-.Q3Q -.1_o

O .Z .IL .§ .tl l.O 0 .Z .U, .IS ._ l.O BLADE RADIAL STATION, (b) Second and third harmonics.

Figure 59.- Concluded.

BLADE 9 !

_e7S, O MP = 0.5, FP = 0.01

I

.015

• 0.25 _] 0.4 .010 .'Ik I_1; w -- o ku Q

.OO5

w w eL I-- tJ

-.OtO

z

-.015

w

L.I

0 .2 .U .E .8

u

L W Ye75 , , U

y i.Z o'

.020

w -.005

L Z

/ \

_e

i- -.nlo

z u I- D

/7, .o,o

a,. -.015

M,I > D I- u Z-'I .., M,I .j -. I_1:] ,.J o u

y,./l o y-

-.0_5

\ / "

-.m0 _ ,'/ -.oos

-.1_5 -.OlO

0 .Z .¼ .E .e 1.0 0 .Z .U .5 .1_ 1.0 BLADE RADIAL STATION.

Zero and first harmonics.

(a)

Figure 60.- Collective pitch transfer coefficients for articulated blade 9

at advance ratios 0.25, 0.4 and 0.5.

.020 - i o020

i

M_5,2. c Me75, 2, $

I

& 0.25 I .015

.015 '1_ 0.4

0.5

¢ .005 .005

_,,,... -.DOS -.005 I_ __ :, /

E-.o_o -.too _1 /

7.

U.I -.015 -.o15

O .Z .IA .fi -8 1.0 O .2 .U .a .8 l.O

u_

.015 , .OZ0 , W M075 , 3, c Me75,3, s U

.o10 .o15 I

m _6 Z o_

. OO5 . Olo

-r U D- _6 m,l

U 0 _ .oos

-.005 /

.J J U

_o,o __ ¢ _.=_ ----

\ / _.o,o

-.t_l] -.QI5

0 .Z .li .§ .11 l.O 0 ._ .U .15 ._ 1.O BLADE RADIAL STATION, F (b) Second and third harmonics.

Figure 60.- Concluded.

BLADE 1

_.o

MP=0.1, FP=0.001

I

.o_

• 0.25 rq 0.4

.om

_1' 0.5

.o_

UJ a n_ UJ G.

Z

k

O

j'/

\

u

c OI I" Z

Q .2 .u

LU

_B .8 t.O

U.

MO 1 , , • U.

UJ O

I

u o_3

E lu L Z

-.002 /

< .OOg

I,- I,-

_.., \\,\jr_

b" ua a .d m 0 m -...

-.11111