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90 Rotor
The forces acting on the hub are the thrust T , drag H, and side force Y (positive in z, x, yaxis
directions, respectively). The aerodynamic analysis is conducted for a clockwise rotating rotor,
with appropriate sign changes for lateral velocity, flapping, and force. The analysis is conducted in
dimensionless form, based on the actual radius and rotational speed of the flight state. The inplane hub
forces are produced by tilt of the thrust vector with the tip-path plane, plus forces in the tip-path plane,
and profile terms (produced by the blade drag coefficient). The orientation of the tip-path axes relative
to the shaft axes is then C PS = Xrβs Y−βc. Then
⎛
⎝ CH
CY
CT
⎞
⎠ = C SP
⎛
⎝
0
0
CT /C SP
33
⎞ ⎛
⎠ + rCY tpp
0
⎝ CHtpp
⎞ ⎛
⎠ +
⎝ CHo
rCYo
0
The inplane forces relative to the tip-path plane can be neglected, or calculated by blade-element theory.
Note that with tip-path plane command and CHtpp and CY tpp neglected, it is not necessary to solve for
the rotor collective and cyclic pitch angles. In general the inplane forces relative to the tip-path plane
are not zero, and may be significant, as for a rotor with large flap stiffness. Figures 11-3a and b show
respectively the tip-path plane tilt and thrust vector tilt with cyclic pitch control (no-feathering plane tilt)
as functions of flap stiffness (frequency), for several rotor thrust values. The difference between tip-path
plane tilt (fig. 11-3a) and thrust vector tilt (fig. 11-3b) is caused by tilt of the thrust vector relative to the
tip-path plane.
The profile inplane forces can be obtained from simplified equations, or calculated by blade-element
theory. The simplified method uses:
CHo
CYo
= σ
8 cdmeanFH
μx/μ
−μy/μ
where the mean drag coefficient cdmean is from the profile power calculation. The function FH accounts
for the increase of the blade-section velocity with rotor edgewise and axial speed: CHo =
1
2 σcdU(r sin ψ + μ)dr = 1
2 σcd(u 2 T + u2 R + u2 P )1/2 (r sin ψ + μ)dr; so (from Harris)
with V 2 = μ 2 + μ 2 z.
FH =4 1
2π
2π 1
0
∼= 1+V 2
0
⎞
⎠
2 2 2 1/2
(r + μ sin ψ) +(μcos ψ) + μz (r sin ψ + μ) dr dψ
3μ + 1
4 μ3 V 2 − 1
(1 + V 2 ) 2
+ μμ 2 z + 3
4 μ3
ln
11-4.3 Blade-Element Theory
√
1+V 2 +1
Blade-element theory is the basis for the solution for the collective and cyclic pitch angles (or flap
angles) and evaluation of the rotor inplane hub forces. The section aerodynamics are described by lift
varying linearly with angle-of-attack, cℓ = cℓαα (no stall), and a constant mean drag coefficient cdmean
(from the profile power calculation). The analysis is conducted in dimensionless form (based on density
ρ, rotor rotational speed Ω, and blade radius R of the flight state). So in the following σ, ν, and γ are
for the actual R and Ω, and a =5.7 is the lift-curve slope used in the Lock number γ. The blade-section
aerodynamic environment is described by the three components of velocity, from which the yaw and
V