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MOMENTS PRODUCED BY FLAPPING
The moments about the aircraft’s center of gravity due to rotor flapping are produced by the couple at the hub as a result of the rotor stiffness, the tilt of the thrust vector perpendicular to the tip path plane, and the rotor’s inplane force. From Figure 7.12:
dMu
Atco. = (4 + HV°>^ ~ T/*
The rotor stiffness is produced by the vertical component of blade centrifugal force acting at the hinge offset; from Figure 7.10:
(P – «Ок
(r’ + ‘) C. F.„„ = ft!(3 – a„) MJg For pure longitudinal flapping:
P – a0 = – a. cos у
and the total rotor moment in the pitch direction for b blades is:
1 {2"
= ebCl2als Mk/g— cos2 J/ d\f
J0
where Mb = First static moment or
MM = – ebCt2ay Mjg
2 s
For uniform blade mass distribution, m, in slugs per foot: or
dMM з / e AhpR(CtR)2a 4 R) у
(Note: This equation could have also been derived by using the vertical shear at the hinge location associated with the incremental inertia forces acting normal to the blade elements due to the flapping motion.) For the example helicopter at sea level:
For hingeless rotors whose blades are cantilevered from the shaft, the rotor stiffness can be visualized as being produced by blade bending moments as pairs of blades are bent into an S shape. The stiffness of a hingeless rotor is usually computed from mode shape considerations at the same time the flapping natural frequencies are computed. Once the stiffness is known, an effective hinge offset can be determined:
1
3 bCt2It
4 dMJdals
For the hingeless rotor used on the Lockheed AH-56A, the effective hinge offset was approximately 13% of the radius.
For simple analyses, it is often satisfactory to assume that the total rotor vector is perpendicular to the tip path plane. It is more correct, however, to account for an effect of inflow that modifies this assumption. The geometry of this phenomenon is illustrated by a simple example in Figure 7.13. Here a helicopter is hovering with some nose-up rotor flapping, due in this case to the center of gravity being ahead of the shaft. The thrust vectors on two typical blade elements on the right and left blades are equal and tilted symmetrically with respect to the tip path plane but not to the shaft. Their contributions to the H-force, which is defined as being perpendicular to the shaft, are:
A#right = sin (ah + ф) and
AHkft= ATLsin(d, x-0)
or
A H = A Ttotal sin ax cos ф
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FIGURE 7.13 Illustration of Inflow Effect on H-Force |
Thus, when the tip path plane is tilted with respect to the shaft, the H-force is decreased by the cosine of the inflow angle in this simple example.
The effect can be derived for the entire rotor from the first equation for Сд/о derived in Chapter 3. Differentiating with respect to aXj gives:
but the second term, for ail practical purposes, is the equation for CT/<r, so that:
= CT/a + – X’
‘ 8
Thus, as the helicopter goes faster and А/ becomes more and more negative, the effect of the tilt of the thrust vector is reduced and in some extreme conditions might even change sign. This has the effect of reducing the damping that one might expect from rotor flapping.
This effect was first pointed out in reference 7.2. The equation can be written in the format of that report by assuming that the thrust coefficient is a function of the pitch at the three-quarter radius position:
Ст/°= 4 (з Є” + /
Solving for X’ and substituting in the equation for (dCH/o)/dat gives: dCH/<3 3 / ( a 0 75
————- = — Cj/о 1—————————–
dah 2 18 CT/o
This is a convenient form to use in hovering, but the one with А/ is recommended for forward flight. The reduction in the effect of the H-force due to flapping is proportional to the inflow which has to be compensated for by collective pitch.
For the example helicopter, the moment in foot-pounds produced about the center of gravity per radian of flapping is:
|
Flight |
Contribution |
Contribution |
|
|
Condition |
of Hub Couple |
of Kotor Force |
Total |
|
Hover |
200,940 |
73,500 |
274,440 |
|
115 knots |
200,940 |
123,000 |
323,940 |
|
160 knots |
200,940 |
108,000 |
308,940 |
A derivation of the rotor Y-force equation shows that the effect is the same as with the H-force; that is:
EXAMPLE HELICOPTER CALCULATIONS
Frequency ratio 457
Phase angle 459
Damping ratio 459
Acceleration coupling 460
Ratio of cyclic angle of attack to flapping 461
Azimuth constant 462
Cyclic pitch required in turn 475
Rotor stiffness 476
Moments produced by flapping 479
HOW TO’S
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The following items can be evaluated by the methods in this chapter.
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