APPENDIX A
APPENDIX A HELICOPTER ROTOR BLADE FLAPPING RESPONSE TO CYCLIC FEATHERING INPUT IN HOVERTNG FLIGHT The governing equation o f motion for the flapping response of a rotating blade due t o cyclic feathering input is given in reference 5 as: where Theref ore,
f
PI ,static
f
6 1 = - 3 = tan
APPENDIX B
APPENDIX B DERIVATION OF HINGELESS-ROTOR HUB MOMENT EQUATIONS FOR HOVER FLIGHT USING EQUIVALENT HINGED BLADE ANALYSIS
The equivalent uniform blade hinge offset 5, and hinge spring restraint KvF
for a given hingeless-rotor cantilever .blade configuration can be established by using equations (3) and (4). The summation of the once-per-rotor-revolution moments about the virtual flapping hinge is The expression for blade centrifugal force flapping moment is The expression f o r blade inertia flapping moment is The independent variable can be changed from t to t , b as follows: and equation (B3) becomes MI,F = -'v31 035) The expression f o r blade flapping moment due to gyroscopic forces induced by aircraft angular velocity is
MG,F = 2517(IV + e+) (B6)
APPENDIX B
APPENDIX B where
-
T = p cos IC/ - q sin IC/
The expression for blade flapping moment due to aircraft angular acceleration is (B7) where The expression f o r blade flapping moment due to the flapping hinge spring is MS,F = -KVFPl (B8) I f the aerodynamic drag moment is neglected, the expression for the thrust moment may be written with the aid of figure 4 as
M ~ , ~ = BR 2 PU2c(r - ev)cz d r
e V Let Substituting equations (B10) into equation (B9) gives k pacCZ2R4lO 1 u2xa dx
M ~ , ~ = - i . 3
APPENDIX B
APPENDIX B where
k3 = (B4 - tB3<{
u = x
J
: The inflow angle may be found from the following equations U C$ = tan -1 P,s 'T,s
= -v - rP + q r cos + + p r sin @
UT,s = Qr The section angle of attack can be found from the following relation: C Y = @ + @
= eo + 0tx - A1 cos @ - B1 sin @ + C$
The first-harmonic thrust moment about the virtual flapping hinge is where pacR4
'yv = -
IV Substituting these moments into equation (Bl) and dividing by IvQ2 gives the gen- e r a l first-harmonic flapping equation of motion as follows:
APPENDIX B
APPENDIX B where CV KIF = 1 + ev - IV and
p i = -a1 cos Q - b l sin Q
-
p1 = a1 sin Q - b l cos Q
- - p1 = a1 cos Q + b l sin Q Equating sin Q t e r m s gives: Equating the cos Q t e r m s gives: where The longitudinal flapping motion increments due t o cyclic feathering, aircraft angular velocity, and aircraft angular acceleration are as follows:
APPENDIX B
I I1111 I 1 I 1 1 1 1 1 1 1 1 I, 111 APPENDIX B The lateral flapping motion increments due to cyclic feathering, aircraft angular velocity, and aircraft angular acceleration a r e as follows: Ab1 A a l -=- P ;I
APPENDIX B
APPENDIX B The structural bending moment transmitted across the virtual flapping hinge is MVF = KVFPl The total hub moment at the rotor center is The first harmonic hub moments due to shear offset a r e developed in the same manner as the flapping moments about the virtual hinge. This development is,as follows: where q
cos + - ~1 sin + - j ? + - cos + +
(B30) 52 52 and R
W = r m g d r
ev Substituting for 7 in equation (B32) gives
evSG,F = e . ( . . + e, $(252p cos rC/ - 252q sin q)
Therefore, the general equation for the total offset shear moment per blade is q
cos + - B1 sin + - + - c o s + + - 2S2qk2uvev sin q
APPENDIX B
APPENDIX B where Separating sin q and cos rc/ terms, dividing through by IvS22, and combining at the virtual flapping hinge with equations (B27), (B21), (B22), and (B23) for moments give the following equations for longitudinal rotor-hub-moment increments: r I where
APPENDIX B
APPENDIX B and The following equalities exist between the lateral and longitudinal hub moments: B.
P q
APPENDIX C
APPENDIX C DERIVATION OF ONCE-PER-ROTOR-REVOLUTION CHORDWISE STRUCTURAL BENDING-MOMENT EQUATION USING E Q W A L E N T HINGED BLADE ANALYSIS The equivalent uniform blade hinge offset in the chordwise (or lagging) degree
tV
of freedom is the same as in the flapping degree of freedom and the spring restraint ' about the virtual lagging hinge KVL is given by equation (9). The summation o f moments about the virtual lagging hinge is The expression for blade lagging moment due to Coriolis forces is MC,L = -21v!22a0& The expression for blade lagging moment due to centrifugal force is The expression for blade lagging moment due to inertia force is The expression for blade lagging moment due to the virtual lagging hinge spring is If the aerodynamic drag moment contribution is neglected, the expression for the aerodynamic lagging moment in hover flight is (fig. 4) MT,L -Je BR -pU2crc2 1 sin @ d r V or for small angles
APPENDIX C
, APPENDIX C The first'harmonic t e r m s in the product CY@ are retained from the multiplication of equations (B13) and (B16) as follows: where A s = - - V (For hover) (C9) s2R Given
T = -q cos + - p sin @
(C10) Substituting equations (C7), (C8), and (C10) into equation (C6) along with equations (B18), (B21), and (B10) and combining t e r m s gives The general lagging equation o f motion becomes Assuming f irst-harmonic sinusoidal motion gives the following general expression f o r lagging motion: where 1-1 I f the flapping hinge spring restraint is neglected, which is generally small com- pared to centrifugal stiffening at operating rotor speed, is neglected,
I
APPENDIX C
APPENDIX C Substituting equation (C15) into equation (C13) gives The structural moment transmitted across the virtual lagging hinge is Theref ore Equation (C18) represents the superposition of the force inputs in the lagging
degree of freedom due to PI, 01, and T which a r e seen as functions of + by the
rotating blade as follows:
P1 = P1 cos * - Qc/p,l +
- I I (
Therefore, equation (C18) can be written as a function of azimuth angle as follows:
APPENDIX C
APPENDIX C and During a hovering maneuver the following t e r m s in equation (C20) are functions of time t: q in equation (C20) is also a function of time t The aircraft normal load factor during the maneuver.
REFERENCES 1. Gustafson, F. B.: Powered-Lift Research at Langley Field. J. Roy. Aeron. Soc., vol. 67, no. 630, June 1963, pp. 371-377.
J. Am. Helicopter SOC., 2. Cresap, W. L: Rigid Rotor Development and Flight Tests.
vol. 7, no. 2, Apr. 1962, pp. 32-41.
3. Statler, W. H.; Heppe, R. R.; and Cruz, E. S.: Results of the XH-51A Rigid Rotor Nat. Forum, Am.
Research Helicopter Program. Proc. Nineteenth Ann.
Helicopter SOC.,May 1963, pp. 119-133.
4 . Huston, Robert J.: An Exploratory Investigation of Factors Affecting the Handling Qualities of a Rudimentary Hingeless Rotor Helicopter. NASA TN D-3418, 1966.
5. Young, Maurice I.: A Simplified Theory o f Hingeless Rotors With Application t o Tandem Helicopters. Proc. Eighteenth Ann. Natl. Forum, Am. Helicopter SOC., Inc., May 1962, pp. 38-45.
6. Yntema, Robert T.: Simplified Procedures and Charts for the Rapid Estimation of Bending Frequencies of Rotating Beams. NACA TN 3459, 1955. (Supersedes NACA RM L54GO2.)
7 . Brooks, George W.: On the Determination of the Chordwise Bending Frequencies of Rotor Blades. J. Am. Helicopter SOC.,vol. 3, no. 3, July 1958, pp. 40-42.
8. Gessow, Alfred; and Crim, Almer D.: A Method for Studying the Transient Blade- Flapping Behavior of Lifting Rotors at Extreme Operating Conditions. NACA TN 3366, 1955.
9. McCloud, John L., III; and Biggers, James C.: Full-scale Wind-Tunnel Tests of a Nonarticulated Helicopter Rotor. NASA TN D-2392, 1964.
10. Ward, John F.; and Huston, Robert J . : A Summary of Hingeless-Rotor Research a t NASA - Langley. Proc. Twentieth Ann. Natl. Forum, Am. Helicopter Soc., Inc., May 1964, pp. 76-83.
11. Lockheed-California Co.: Investigation of Elastic Coupling Phenomena of High Speed Rigid Rotor Systems. TRECOM Tech. Rept. 63-75 (Lockheed Rept. No. 17013), U.S. Army Transportation Res. Command (Fort Eustis, Va.), June 1964.
12. Huston, Robert J.; and Ward, John F.: Handling Qualities and Structural Character- istics of the Hingeless-Rotor Helicopter. Conference on V/STOL and STOL Air- craft, NASA SP-116, 1966, pp. 1-16.
NASA-Langley, 1966 L-4939 “The aeronautical and space activities of the United States shall be
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