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12. Gessow, A. and Myers, Garry C., jnr, Aerodynamics of the helicopter, New York, Ungar, 1952.
13. Knight, M. and Hefner, R. A., ‘Analysis of ground effect on the lifting airscrew’, NACA TN
835, 1941.
14. Zbrozek, J. K., ‘Ground effect on the lifting rotor’, Aeronautical Research Council R&M
2347, 1947.
15. Prandtl, L., Appendix to Betz, A., ibid.
16. Lock, C. N. H., ‘The application of Goldstein’s theory to the practical design of airscrews’,
Aeronautical Research Council R&M 1377, 1931.
17. Durand, W. F., ibid, pp. 264, 265.
18. Theodorsen, T., ‘Theory of static propellers and helicopter rotors’, Proc. 25th Annual National
Forum American Helicopter Society 236, 1969.
19. Landgrebe, A. J., ‘Analytical and experimental investigation of helicopter rotor and hover
performance and wake geometry characteristics’, USAAMRDL TR 71–24, 1971.
20. Johnson, W., ‘Airloads and wake models for a comprehensive helicopter analysis,’ Vertica,
14(3), pp. 255–300, 1990.
21. Gray, R. B., ‘An aerodynamic analysis of a single-bladed rotor in hovering and low speed
forward flight as determined from smoke studies of the vorticity distribution in the wake’,
Princeton Univ. Aeronaut. Eng. Rep. 356, 1956.
22. Gray, R. B., ‘Vortex modeling for rotor aerodynamics – the Alexander A. Nikolsky Lecture’,
J. Amer. Helicopter Soc., 37(1), 1992.
23. Tangler, James L., Wohlfield, Robert M. and Miley, Stan J., ‘An experimental investigation of
vortex stability, tip shapes, compressibility and noise for hovering model rotors’, NASA Contractor
Rep. NASA CR – 2305, 1973.
24. Levy, M. A. and Forsdyke, A. C., ‘The steady motion and stability of a helical vortex’, Proc.
Roy. Soc. Series A, vol. 120, 1928.
25. Clark, D. R. and Leiper, A. C., ‘The free wake analysis – a method for the prediction of
helicopter hovering performance’, J. Amer. Helicopter Soc., 15(1), pp. 3–11, Jan. 1970.
26. Favier, D., Nsi Mba, M., Barbi, C. and Maresca, C., ‘A free wake analysis for hovering rotors
and advancing propellors’, Vertica, 11(3), pp. 493–511, 1987.
27. Brown, K. D. and Fiddes, S. P., ‘New developments in rotor wake methodology’, Paper No.
14, 22nd European Rotorcraft Forum, 17–19 Sept. 1996, Brighton, U.K.
28. Carradonna, F. X. and Tung, C., ‘Experimental and analytical studies of a model helicopter
rotor in hover’, Vertica, 5, pp. 149–161, 1981.
3
Rotor aerodynamics and
dynamics in forward flight
3.1 Introduction
In this chapter we examine firstly the aerodynamics of the rotor in forward flight and
then the dynamics. In order to be able to determine blade lift, drag and flapping
moment, and, ultimately, rotor performance, it is necessary, as with axial flight, to
determine the induced velocity in forward flight. Fortunately, for many important
problems a detailed description of the induced velocity distribution is not necessary,
and much useful work can be done by treating the rotor as a lifting surface with an
infinite number of blades. The dynamics of the blades and rotor are equally susceptible
to simple approaches based on straightforward ideas of induced velocity, aerofoil
characteristics and blade modelling. The effects of blade flexibility and more
comprehensive induced velocity distributions are dealt with in later chapters, as is a
discussion of the peculiarities of blade-section characteristics undergoing continually
changing conditions.
3.2 Induced velocity in forward flight
The first proposal for calculating the induced velocity for a rotor carrying a given
thrust was due to Glauert1. He regarded the rotor as an elliptically loaded circular
wing to which lifting-line wing theory could be applied, and proposed that a mean
induced velocity vi0 could be obtained from the formula
vi0 = T/2ρAV′ (3.1)
where, if V is the relative velocity far upstream of the rotor, the total velocity at the
rotor V′ = √{V2 cos2αD + (V sin αD – vi0)2}, A is the rotor area, and αD is the rotor
disc incidence.
Although a general proof of this formula has never been given, its validity was
78 Bramwell’s Helicopter Dynamics
justified on the grounds that in hovering flight it reduces to the momentum formula,
eqn 2.12, while in forward flight, when V′ ≈ V, it assumes the same form as for the
induced velocity of an elliptically loaded wing. The formula appears to be true,
however, for all loading distributions in the ‘high-speed case’2.
Glauert’s eqn 3.1 can be interpreted by imagining a circular jet of air of velocity
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