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1
1
Ma Mc ab∫ x FS x2α x (7.123)
where α is the local blade incidence, F = γ2/γ1, and the equivalent Lock’s number
for the hingeless blade, γ2 = ρ acR/E1, where E1 mS x
0
1
1
= ∫ 2 d is the generalised
mass or inertia of the blade first flapping mode.
Thus, the hub moment of eqn 7.111 can be found in terms of eqn 7.106, together
with a ‘correction’ term represented by the integral of eqn 7.123. Then to a very good
approximation22 it is found that
( m) = ( /2 1)( 1 – 1) +
2
C a a γ λ ai j1μa0
and
(Cl ) = (a/2 )( – 1)b + j Bi – j j
1
2
a 1 1 – –
2
1 2 1 3 0 4 γ λ μθ μθ μλ
(7.125)
where j1 = (a/2) 1 – G1 – F1 F D1
1
2
–
1
2
(1 – )
j FC
j FF
j FG
2 1
3 1
4 1
= 1
4
–
= 2
3
– 2
= 12
–
and D1 S x
0
1
1
= ∫ 2 d
Equations 7.124 and 7.125 are only a little more complicated than eqns 7.109 and
7.110 but, for an analysis based on one mode, are just as accurate as eqns 7.121 and
7.122 and avoid the ‘ill-conditioning’ feature.
The stability and control derivatives for a hingeless rotor have been given by
Bramwell21,22. Generally speaking, it is found that the force derivatives are only
slightly defferent from those calculated for the hinged rotor and presented in Chapter
5. The flapping motion of the blade, in spite of the elastic moment at the root, is also
very slightly different from that of the rigid hinged blade under similar conditions, so
288 Bramwell’s Helicopter Dynamics
that the principal difference between the two rotors is the moment they exert on the
hub. For the centrally hinged rotor, the pitching moment coefficient for a unit of the
rotor has already been shown to be teh and, of course, the same thrust moment applies
to the hingeless rotor.
Let us use eqn. 7.108 to find the additional elastic moment coefficient for a typical
case.
For unit flapping we have
Mc a 1 1
= ( /2γ )(λ2 – 1)
Typical values for a hingeless rotor are a = 5.7, γ1 = 7.5, λ1
2 = 1.24, so that Cm =
0.091. For teh we have already used the value 0.0214 (Chapter 4), so that the total
coefficient for the hingeless rotor is 0.1124, giving a ratio of 5.26 to 1 when compared
to the value for the centrally hinged rotor.
Thus quite a good approximation to the derivatives of a hingeless rotor helicopter
is to take the force derivatives calculated in Chapter 5 and increase the flapping
moments in some ratio determined from a calculation similar to that above.
References
1. Hildebrand, F. B., Methods of applied mathematics, Englewood Cliffs NJ, Prentice-Hall,
1956.
2. Temple, G. and Bickley, W. G., Rayleigh’s principle, New York, Dover Publications, 1956.
3. Southwell, R. V. and Gough, Barbara S., ‘On the free transverse vibrations of airscrew blades’,
Rep. Memo. aeronaut. Res. Coun. 766, 1921.
4. Bishop, R. E. D. and Johnson, D. C., The mechanics of vibration, London, Camb. Univ. Press,
1960.
5. Duncan, W. J., ‘Galerkin’s method in mechanics and differential equations’, Rep. Memo.
aeronaut. Res. Coun. 1798, 1937.
6. Duncan, W. J., ‘Principles of the Galerkin method’, Rep. Memo. aeronaut. Res. Coun. 1948,
1938.
7. Holzer, H., Die Berechnung der Drehschwingungen, Berlin, Springer Verlag, 1921.
8. Isakson, G. and Eisley, J. G., ‘Natural frequencies in coupled bending and torsion of twisted
rotating and non-rotating blades’, NASA CR–65, July, 1964.
9. Williams, R. F., Unpublished material, City Univ. London.
10. Scanlan, Robert H. and Rosenbaum, Robert, Introduction to aircraft vibration and flutter,
New York, Macmillan, 1960.
11. Houbolt, John C. and Brooks, George W., ‘Differential equations of motion for combined
flapwise bending, chordwise bending and torsion of twisted non-uniform rotor blades’, NACA
Rep. 1346, 1958.
12. Sobey, A. J., ‘Dynamical analysis of the shaft-fixed blade’, R. Aircr. Establ. tech. Rep. 73175,
1974.
13. Ormiston, Robert A. and Hodges, Dewey, H. ‘Linear flap–lag dynamics of hingeless rotor
blades in hover’, J. Am. Helicopter Soc., April 1972.
14. Done, G. T. S. and Simpson, A., ‘Dynamic instability of certain conservative and non-conservative
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