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Rotor 93
with an average over the rotor azimuth implied, along with the integration over the radius. Lift forces
are integrated from the root cutout rroot to the tip-loss factor B. Drag forces are integrated from the root
cutout to the tip.
In coefficient form (forces divided by ρAV 2
tip) the rotor thrust and inplane forces are:
CT = σ Fz dr
CH = σ Fx sin ψ + Fr cos ψdr
CY = σ − Fx cos ψ + Fz =
Fr sin ψdr
1
2 ĉU(cℓuT − cduP )
Fx = 1
2 ĉU(cℓuP + cduT )
Fr = −β Fz + 1
2 ĉUcduR
(and the sign of CY is changed for a clockwise rotating rotor). The terms Δ Fx ∼ = Fz ˙ β and Δ Fr = − Fzβ
produce tilt of the thrust vector with the tip-path plane (CH = −CT βc and CY = −CT βs), which
are accounted for directly. The section drag coefficient cd produces the profile inplane forces. The
approximation uP ∼ = μz is consistent with the simplified method (using the function FH), hence
Fxo = 1
2 ĉU0cduT
Fro = 1
2 ĉU0cduR
CHo = σ
CYo = σ
Fxo sin ψ + Fro cos ψdr= σ
2
− Fxo cos ψ + Fro sin ψdr= − σ
2
ĉU0cd(r sin ψ + μx) dr
ĉU0cd(r cos ψ + μy) dr
where U 2 0 = u 2 T + μ2 z, and cd = cdmean/ cos Λ. Using blade-element theory to evaluate CHo and CYo
accounts for the planform (ĉ) and root cutout. Using the function FH implies a rectangular blade and no
root cutout (plus at most a 1% error approximating the exact integration). The remaining terms in the
section forces produce the inplane loads relative to the tip-path plane:
Fxi = Fx − Fz ˙ β − Fxo = 1
2 ĉUcℓ(uP − uT ˙ β)+ 1
2 ĉUcd((1 − U0/U)uT + uP ˙ β)
Fri = Fr + Fzβ − Fro = 1
2 ĉUcd(1 − U0/U)uR
CHtpp = σ Fxi sin ψ + Fri cos ψdr
CY tpp = σ − Fxi cos ψ + Fri sin ψdr
(including small profile terms from U0 = U).
Evaluating these inplane forces requires the collective and cyclic pitch angles and the flapping
motion. The thrust equation must be solved for the rotor collective pitch. The relationship between
cyclic pitch and flapping is defined by the rotor-flap dynamics. The flap motion is rigid rotation about a
central hinge, with a flap frequency ν>1 for articulated or hingeless rotors. The flapping equation of
motion is
¨β + ν 2 β +2˙αy sin ψ +2˙αx cos ψ = γ
Fzrdr+(ν
a
2 − 1)βp
including precone angle βp; the Lock number γ = ρacrefR4 /Ib. This equation is solved for the mean
(coning) and 1/rev (tip-path plane tilt) flap motion:
ν 2 β0 = γ
Fzrdr+(ν
a
2 − 1)βp
(ν 2
βc
− 1) =
βs
γ
Fzrdr
2 cos ψ 2˙αx
+
a
2 sin ψ 2˙αy