Typical Numerical Values

Calculations have been made to illustrate in broad fashion the ways in which parameters discussed in the foregoing analysis vary with one another and particularly with forward speed. For this purpose the following values have been used:

Rotor solidity



Blade lift curve slope



Lock number



Aircraft weight ratio


1 p(OR)2A


Parasite drag factor




The parasite drag factor is a form of expression in common use, in which f is the ‘equivalent flat plate area’ defined by:

Dp = 2 pV2f (5.88)

Dp being the parasite drag and A the rotor disc area.

Figure 5.21a shows the variation of inflow factor l with advance ratio m at two levels of thrust coefficient. As previously mentioned, l, as defined in Equation 5.39, is relative to the TPP, so is denoted by 1T in the diagram. The variation shows a minimum value at moderate m, inflow being high at low m because the induced velocity is large and high again at high m because of the increased forward tilt of the TPP required to overcome the parasite drag. The lower the thrust coefficient, the more marked the high-m effect.

Figure 5.21b shows the corresponding variation of collective thrust angle в, for CT/s = 0.2. The variations of в and l are similar in character, as might be expected from Equation 5.87.

Combination of Equations 5.39 and 5.87 leads, on elimination of l, to a direct relationship between CT and в which, using the chosen values of aircraft weight ratio and parasite drag factor in the final term, is:

в = з ^ + 2 m^ + m^3 ~2m2^ (5.89)

where B is a slowly decreasing function of m.

Подпись: -■-0.05 -*-0.15 -K- 0.3

Note that when m is zero, B = 1/3 and Vj/Vj0 = 1, so that we have Equation 3.29 as previously derived for the hover. Figure 5.22 shows variations of в with CT for different levels of m. The characteristics at low and high forward speed are significantly different. When m is zero or small the variation is nonlinear, в increasing rapidly at low thrust coefficient owing to the induced flow term (the second expression in the equation) and more slowly at higher CT as the first term becomes dominant. At high m, however, the induced velocity factor Vi/Vi0 is so small that the second term becomes negligible for all CT, so the e/CT relationship is effectively linear. The intercept on the в axis reflects the particular value of m while, more interestingly,

Figure 5.23 Variation of flapping coefficients v m

with m and s known the slope is a function only of the lift slope a. This provides an experimental method for determining a in a practical case.

A final illustration (Figure 5.23) shows the flapping coefficients a0, aj and b as functions of m. These have been calculated using Equations 5.84-5.86. The coning angle a0 varies only slightly with m, being essentially determined by the thrust coefficient. It may readily be shown in fact that a0 is approximately equal to (3CTg)/(8sa) which with our chosen numbers has the value 0.105 rad or 6.0°. The longitudinal coefficient aj is approximately linear with forward speed, showing, however, an effect of the increase of l at high speed. The lateral coefficient b1 is also approximately linear, at about one-third the value of a1. In practice b1 at low speeds depends very much on the longitudinal distribution of induced velocity (assumed uniform throughout the calculations) and tends to rise to an early peak as indicated by a modified line in the diagram.


1. Glauert, H. (1926) A general theory of the autogiro, R & M, 1111.

2. Newman, S. J., Brown, R., Perry, J. et al. (2001) Comparitive numerical and experimental investigations of the vortex ring phenomenon in rotorcraft. 57th American Helicopter Society Annual National Forum, Washington, DC, 9-11 May, Vol. 2, pp. 1411-1430.

3. Drees, J. and Hendal, W. (1951) Airflow patterns in the neighbourhood of helicopter rotors. J. Aircr. Eng., 23, 107-1112.

4. Perry, F. J. (2000) Vortex ring instability in axial and forward flight – comparisons with test, Private Technical Note 0002/2000, June.

5. Brand, A., Kisor, R., Blyth, R. et al. (2004) V-22 high rate of descent (HROD) test procedures and long record analysis. 60th American Helicopter Society Annual National Forum, Baltimore, MD, June 7-10.

6. Mangler, K. W. and Squire, H. B. (1950) The induced velocity field of a rotor, R & M, 2642.

7. Bennett, J. A.J. (1940) Rotary wing aircraft. Aircr. Eng. Aerosp. Technol., 12,139-146.

8. Stepniewski, W. Z. (1973) Basic aerodynamics and performance of the helicopter. AGARD Lecture Series, 63.

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