Thrust

Подпись: 2pU2 • c dr • CL 1 p • pR2 •{OR)2 Подпись: c pR Подпись: {5.45)

Following the derivation of the hover analysis in Chapter 3 we write an elementary thrust coefficient of a single blade at station r as:

and for N blades, introducing the solidity factor s and non-dimensionalising,

Подпись: {5.46)dCT = s • uTCL dx

Подпись: CL Подпись: aa Подпись: a в Подпись: uP uT Подпись: {5.47)

On expressing CL in the linear form (i. e. the blade is unstalled):

from which (5.46) becomes:

Подпись: {5.48)dCT = sa(euT~uPuT) dx

Подпись: CT Подпись: ■»2p (виТ—uPuT) dC 0T Подпись: dx Подпись: {5.49)

For the hover we were able to write uT = x, uP = l: in forward flight, however, uT and uP and in general в also, are functions of azimuth angle C The elementary thrust must therefore be averaged around the azimuth and integrated along the blade. It is convenient to perform the azimuth averaging first and we therefore write the thrust coefficient of the rotor as:

To expand the terms within the inside brackets, we recall from Chapter 4 that the flapping angle b may be expressed in the form:

Подпись:Подпись: {5.51)b = a0—a1 cos C—b1 sin C

from which also we have:

—- = a1 sin C—b1 cos C

dc

For the feathering angle в a similar Fourier expansion (Equation 4.9) can be used: however, there is always one plane, the plane of the swashplate or no-feathering plane (NFP), relative to which there is no cyclic change in в; for our analytical solution therefore this will be used as the reference plane. Thus we have в = в0, constantin azimuth, and following the same procedure as
for hover we shall assume an untwisted blade, giving в0 constant also along the span. Averaging round the azimuth will make use of the following results:

*2p

sinC dC = 0

J0

*2p

cosC dC = 0

J0

*2p

sinCcosC dC = 0 (5.52)

J0

*2p

sin2C dC = p

J0

*2p

cos2C dC = p

Подпись: (5.53)

J0

= в0 ■ x2 m

Подпись: (5.54)

while:

Подпись: —lx Подпись: dx Подпись: (5.55)

all other terms cancelling out after substituting for b and (d^/dC) and integrating. Hence finally,

This is the simplest expression for the lift coefficient of a rotor in forward level flight. The assumptions on which it is based are those assumed for hover in Chapter 3, namely uniform induced velocity across the disc, constant solidity s along the span and zero blade twist. As before, it may be assumed that for a linearly twisted blade, Equation 5.55 can be used if the value of в is taken to be that at three-quarters radius. Also in Equation 5.55 the values of в and 1 are taken relative to the non-feathering plane as reference. Bramwell (p. 157) derives a significantly more complex expression for thrust when referred to disc axes (the TPP) but since the transformation involves the assumption that actual thrust, to the accuracy required, is not altered as between the two reference planes, the change is a purely formal one and Equation 5.55 stands as a working formula.

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