FLAPPING EQUATIONS IN FORWARD FLIGHT

The complete equations for rotor flapping in forward flight can be derived by equating the increments of hinge moment due to aerodynamic, centrifugal, inertial, and weight forces to zero. The resultant equations will be similar to those derived for the trim values of cyclic pitch in Chapter 3 and are based on similar assumptions:

• Aerodynamic forces are considered to act from the hinge to the tip.

• The reverse flow region is ignored.

• The airfoil lift characteristics are linear and free of stall and compressi­bility effects.

• The blade motion consists of only coning and first harmonic flapping.

• Small-angle assumptions are valid.

The use of these assumptions has been found to be justified for conventional helicopters flying within conventional flight envelopes. For special applications, these assumptions can be relaxed at the expense of simplicity.

Since the blade motion consists only of coning and first harmonic flapping, the centrifugal forces may be assumed to he in the plane of rotation of the blade element, as shown in Figure 7.10. This assumption also eliminates all inertial moments due to flapping from the analysis. The increment of moment about the flapping hinge due to centrifugal force is:

ДЛГСР. = – AC. F. h

where

AC. F. = mAr(r’ + e)Cl2

and

h = r’a0 +

(f’P-гЧ)

For most rotors, the hinge offset ratio, e/R, will be small enough that it can reasonably be eliminated from those terms inside the square brackets.

The final contribution to the hinge moment that must be considered is that due to the blade weight, Mv. But:

R-e

mgr’dr’

I

The summation of the constant portions of the various contributions to the hinge moment can be used to give the equation for coning at the flapping hinge as it did in Chapter 3 for the rotor without hinge offset:

+ Мл + M-iw — О

For most applications, the second term in each denominator is’ small with respect to the first term. Taking it as negligible, the equation becomes:

The first term may be thought of as the basic flapping that is independent of hinge offset. The second term represents cross-coupling due to hinge offset and is significant when the determination of cross-coupling is a primary objective of the analysis. The similar equation for the lateral flapping, bxr, is:

– 0ojli + 20xp – Bx f 1 + – p2 ] + 2p ( pa, – Cr/a — 3 2 2py

The primary use of these equations in the analysis of stability and control is to yield flapping derivatives as a function of changes in flight conditions and control inputs. These flapping derivatives can then be converted into pitch and roll moment derivatives using the relationships between moments and flapping. The evaluation of the derivatives will be discussed in detail in Chapter 9.