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the sideways flapping angle b1. As was discussed in section 3.12, the sideways
flapping can be attributed to two effects: the incidence variation due to coning and
that due to the longitudinal induced velocity distribution. As can be seen from Fig.
3.32, nearly all of the sideways flapping at low speeds is due to the longitudinal
induced velocity distribution, and calculations depend strongly on the assumptions
made. Harris19 has considered a number of simple expressions which have been used
for representing the longitudinal induced velocity distribution but none of them
shows the ‘peakiness’ evinced by the measured values. Equation 3.53 appears to give
the best agreement of those expressions examined.
Fig. 3.31 Longitudinal flapping as a function of tip speed ratio
4°
3°
2°
1°
0 0.1 0.2 0.3
b1
eqn 3.53
eqn 3.54 (K = 1.2)
4μa0/3
μ
Fig. 3.32 Lateral flapping as a function of tip speed ratio
114 Bramwell’s Helicopter Dynamics
References
1. Glauert, H., ‘A general theory of the autogyro’, Aeronautical Research Council R&M 1111,
1926.
2. Bramwell, A. R. S., ‘Some remarks on the induced velocity field of a lifting rotor and on
Glauert’s formula’, Aeronautical Research Council CP 1301, 1974.
3. Coleman, R. P., Feingold, A. M. and Stempin, C. W., ‘Evaluation of the induced velocity field
of an idealized helicopter rotor’, NACA ARR L5E10, 1947.
4. Mangler, K. W. and Squire, H. B., ‘The induced velocity field of a rotor,’ Aeronautical
Research Council R&M 2642, 1950.
5. Brotherhood, P. and Stewart, W., ‘An experimental investigation of the flow through a helicopter
in forward flight’, Aeronautical Research Council R&M 2734, 1949.
6. Heyson, H. H. and Katzoff, S., ‘Induced velocities near a lifting rotor with non-uniform disc
loading’, NACA Rep. 1319, 1958.
7. Lock, C. N. H. ‘Further developments of autogyro theory’, Aeronautical Research Council
R&M 1127, 1927.
8. Bennett, J. A. J., ‘Rotary wing aircraft’, Aircraft Engineering, March 1940.
9. Stepniewski, W. Z., ‘Basic aerodynamics and performance of the helicopter’, AGARD Lect.
Ser. 63, 1973.
10. Glauert, H. and Shone, G., ‘The disturbed motion of the blades of a gyroplane’, Aeronautical
Research Council Paper 993, 1933.
11. Bennett, J. A. J., ‘Rotary wing aircraft’, Aircraft Engineering, May 1940.
12. Horvay, G., ‘Rotor blade flapping motion’, Q. Appl. Math., July 1947.
13. Shutler, A. G. and Jones, J. P., ‘The stability of rotor blade flapping motion’, Aeronautical
Research Council R&M 3178, 1958.
14. Lowis, O. J., ‘The effect of the reverse flow on the stability of rotor blade flapping motion at
high tip speed ratios’, Aeronautical Research Council Paper 24 431, 1963.
15. Wilde, E., Bramwell, A. R. S. and Summerscales, R., ‘The flapping behaviour of a helicopter
rotor at high tip speed ratios’, Aeronautical Research Council CP 877, 1966.
16. Sissingh, G. J., ‘Lifting rotors operating at high speeds and advance ratios’, AGARD Conf.
Proc. CP–22, paper 5 (part II), 1967.
17. Stewart, W., ‘Higher harmonics of flapping on the helicopter rotor’, Aeronautical Research
Council CP 121, 1952.
18. Squire, H. B., Fail, R. A. and Eyre, R. C. W., ‘Wind tunnel tests on a 12 ft helicopter rotor’,
Aeronautical Research Council CP 2695, 1949.
19. Harris, F. D., ‘Articulated rotor blade flapping motion at low advance ratio’, J. Amer. Helicopter
Soc., January 1972.
20. Bramwell, A. R. S., ‘Stability and control of the single rotor helicopter’, Aeronautical Research
Council R&M 3104, 1959.
4
Trim and performance in axial
and forward flight
4.1 Introduction
Having obtained formulae for the rotor forces and blade flapping motion in Chapter
3, we are now in a position to solve the trim equations derived in Chapter 1. In
Chapter 1 it was possible to draw some general conclusions relating to trimmed flight
with little specific reference to the values of the flight parameters such as collective
pitch, inflow ratio, and so on. We now seek a method which enables us to calculate
these quantities for a given helicopter under given steady-flight conditions. These
values are necessary, not only because one wishes to know the control displacements
required to maintain a given flight condition, but because these, and other, parameters
are needed for performance and stability calculations.
Having solved the trim equations, one can then calculate the corresponding power
　
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