APPENDIX A
APPENDIX A ANGULAR-VELOCITY RESPONSE OF A HINGELESS ROTOR HELICOPTER By Robert J. Huston and William J. Snyder For a simultaneous two-axis displacement-and-hold type of control input, the equa tions of motion about the lateral and longitudinal axes, respectively, for an aircraft with t direct control coupling and gyroscopic cross-coupling are:
( MY 6y )i
Mq H , + - q - - p = 6 x p ) + 6 y ’ y I Y I Y The solutions to equations (Al) a r e given by the following two equations for the initial conditions of p = q = 0 when MP Mq 4H2 2 o
( G - T J - E
H2 -k MpMq
+ tiyR] + cosh(k$?)l..p(- i B g + Fxk) [$ sinh(i@]exp(- B ) (A2b)
where
APPENDIX A
APPENDIX A
when e-?Y- 4H2
0, t h solutions to equations (Al) are given by IXIY P =
H2 +MpMq H2 + MpMq
- tan-l @]exp(- -B
� $ + lxp) + 6 y k ] [-& sink@lexp(- \ B) (A3b)
where At t = 00, equations (A2) and (A3) simplify to P, = 6, ( A 4 4 The longitudinal stick displacement required to t r i m the final pitching velocity to zero during a steady-state rolling maneuver, determined by setting equation (A4b) equal t o zero, is
APPENDIX A
APPENDIX A and in a similar manner, the lateral stick displacement required to trim the final rolling velocity to zero during a steady-state pitching maneuver is For the case of no direct control coupling (My6, = M,6y = 0 ) and for a single axis control input, the ratio of final angular velocities obtained from equation (A4) is for 6y = 0 and for 6, = 0
APPENDIX B
APPENDIX B COMPUTED CHARACTERISTICS OF THE HINGELESS ROTOR OH- 13 HELICOPTER For the hingeless rotor OH-13 helicopter, the cyclic pitch input due to longitudinal and lateral stick motion, with control retardation, can be written as The ingredients for the equations of motion of appendix A (eqs. Al) may be written as
APPENDIX B
APPENDIX B The value of the hub moment increments in equations (B2) to (B8) is calculated by using the procedures of references 3 and 4 for y = 3.85 and wlNR/S2 = 0.185 and is pre 17 and 18. The direct hub moment increments a r e increased to account sented in figures for thrust vector tilt due to cyclic pitch and thrust vector lag due to pitching and rolling velocities. For these calculations, the thrust vector tilt and lag were considered to be 100 percent effective; that is, there was no reduction in tip-path-plane tilt or following rate due to rotor flapping stiffness. The complete moment increments, including the effect of thrust vector lag and tilt, a r e known from the following equations: Th A1 A(%)Al = A r H 9 $ ) IV' IV' A1 when - = - - Ab'
27 (1.0 - 0.29
P Y '
MH, = A( IVJq +%pi?) A
when
g A a ' = - z ( l . 0 - 0.29
Y '
APPENDIX B
APPENDIX B A($), = A( M ~ , I,,), B MH, A = A( IVJP The computed hub moment increments with respect to cyclic pitch and pitching and rolling velocities are presented in the following table: _ _ Contribution due to direct hub moment Ingredients for equations (B2) to (B8) (figs. 17 and 18) Arv:F)B1 = 0.155 = 0.155 AC:AF)A1 = 0.210 A ( S ) A l = 0.250 A r S ) = -0.210 A ( S ) B l = -0.250 B1 A r H 7 t ) = 0.155 IVn A1 A("Itr."), = -1.60 A($)p = -1.73 A C F f ) q = -1.60
A - = -1.73
A : : $ ) , = 0.80
A - = -0.80
A ( : : $ ) p = -0.80
APPENDIX C
I APPENDIX C EFFECT OF BLADE MASS CONSTANT (LOCK NUMBER) ON THE CONTROL RESPONSE CHARACTERISTICS OF A HINGELESS ROTOR HELICOPTER It became apparent during the analysis for this paper that the design choice for a value of the blade Lock number is a factor which may significantly affect the handling qualities of a hingeless rotor helicopter. The blade Lock number is the ratio of blade air
pacR forces to blade m a s s forces - . An increase in blade weight, other factors being
( Iv4)
held constant, results.in a reduction in blade Lock number.
The effect of combinations of blade Lock number and blade stiffness on the nondi mensional rotor hub control moment per unit of cyclic pitch and damping moment p e r unit of aircraft angular velocity is presented in figures 17 and 18. These two figures a r e reproduced from reference 3 and are the source of the ingredients for equations (B2) to (B8). Current cantilever blade designs have a nonrotating flapwise frequency ratio olNR/S2 on the order of 0.2 and there is little variation between designs. The range of Lock numbers at standard air density, however, varies from values below 4 to more than 8.
The data of figure 17 can be used to show that there is little effect of a change in control moment blade Lock number on the total available for the normal values of flapwise natural frequency. However, a reduction of blade Lock number from a value of 8 to 4, for example, results in a substantial increase in the coupling moment. This increased coupling would require a corresponding increase in the required control retardation angle previously discussed. It should be noted that a change in blade Lock number thereby results in a different level of the ratio of damping to critical damping, where the damping referred t o is the damping of the first flapwise mode of the blade.
The effect of a reduction in blade Lock number on the aircraft angular velocity The direct damping moment is substantially increased damping is shown in figure 18.
with a reduction in blade Lock number from 8 to 4, for example. An adverse effect, how ever, is the increase in the damping moment t e r m that produces a gyroscopic control coupling from essentially zero t o approximately half of the direct damping moment.
APPENDIX C
APPENDIX C
=P
26-
Y
k rl a - "1NR "lNR
- -
R n Figure 17.- Effect of blade Lock number and blade stiffness on rotor hub control moments.
Three blade rotor.
-_ "lNR
-
n Figure 18.- Effect of blade Lock number and blade stiffness on rotor h u b damping moments. Three blade rotor.
APPENDIX C
APPENDIX C The rudimentary nature of the hingeless rotor helicopter used in this flight inves tigation resulted in a particularly unfortunate choice of blade Lock numbers (y = 3.85).
The dynamics associated with a Lock number this small can be expected t o present a much more serious gyroscopic coupling problem to a pilot than a rotor with a blade Lock of 8 .0, for example.
number An even more important point for consideration in future designs is the effect of density altitude on control characteristics and control response.
The blade Lock number is directly proportional to the mass density of the air. Therefore, large variations in operating density altitude, which may be expected of turbine powered helicopters under operational use, can result both in severe direct cross-coupling and in severe gyroscopic coupling at altitudes even if these effects a r e not present at standard air density. It appears that in order to eliminate completely all coupling effects accompanying changes in density altitude, special provision would have to be made to vary control retardation with density altitude and to provide some type of feedback device which is sensitive t o aircraft angular velocity. A design giving zero coupling at some intermediate altitude may minimize the effects at both extremes of altitude sufficiently for many applications and minimize the requirements for special devices in other cases.
The equations of motion presented in appendix A may be used to study the effects of density altitude in order to minimize the handling qualities problems presented by the cross-coupled response of a hingeless rotor helicopter.
_- REFERENCES Rigid Rotor Development and Flight Tests. Paper No. 62-17, Inst.
1. Cresap, W. L.: Aerospace Sci., Jan. 1962.
2. Huston, Robert J.; and Tapscott, Robert J.: Results of Some Wind Tunnel and Flight Studies With Helicopters at NASA. Vertical Take-Off and Landing (VTOL) Air craft. Ann. N.Y. Acad. Sci., vol. 106, art. 1, Mar. 1963, pp. 57-69.
3. Ward, John F.; and Huston, Robert J.: A Summary of Hingeless-Rotor Research at NASA - Langley. Proc. Twentieth Ann. Nat. Forum, Am. Helicopter SOC.,Inc., May 1964, pp. 76-83.
4. Young, Maurice I.: A Simplified Theory of Hingeless Rotors With Application to Tandem Helicopters. Proc. Eighteenth Ann. Nat. Forum, Am. Helicopter SOC.,Inc., May 1962, pp. 38-45.
5 . Garren, John F., Jr.: Effects of Gyroscopic Cross Coupling Between Pitch and Roll on the Handling Qualities of VTOL Aircraft. NASA TN D-812, 1961.
Helicopter Flying and Ground Handling Qualities; General Requirements for.
6. Anon.: Mil. Specification MIL-H-8501A, Sept. 7, 1961.
NASA-Langley, 1966 L-4828 39 I “The aeronautical and space activities of the Uizited States shall be conducted so as i o contribute . , . to the expansion of himan hiowl edge of pheiionzena in the atmosphere and space. The Administration shall provide for the widest practicable and appropriate dissemination of information conceriiiiig its actiriities and the resrtlts thereof .” -NATIONAL A N D SPACE ACT OF 1958 AERONAUTICS
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