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Rotor 101
11-5.1.3 Twin Rotors
For twin rotors, the induced power is determined by the induced velocity of the rotor system, not
the individual rotors. The induced power is still obtained using Pi = κPideal = κfDTvideal for each rotor,
but the ideal induced velocity is calculated for an equivalent thrust CTe based on the thrust and geometry
of both rotors. The profile power calculation is not changed for twin rotors.
In hover, the twin-rotor induced velocity is vi = κtwin T/2ρAp, from the total thrust T and the
projected disk area Ap =(2−m)A. The overlap fraction m is calculated from the rotor hub separation
ℓ. A correction factor for the twin-rotor ideal power is also included. For a coaxial rotor, typically
κtwin ∼ = 0.90. So the ideal inflow is calculated for CTe =(CT1 + CT 2)/(2 − m).
In forward flight, the induced velocity of a coaxial rotor is vi = κtwinT/(2ρAV ), from the total
thrust T and a span of 2R. The correction factor for ideal induced power (biplane effect) is κtwin ∼ = 0.88
to 0.81 for rotor separations of 0.06D to 0.12D. The ideal inflow is thus calculated for CTe = CT 1 + CT 2.
The induced velocity of side-by-side rotors is vi = κtwinT/(2ρAeV ), from the total thrust T and a span of
2R+ℓ, hence Ae = A(1+ℓ/2R) 2 . The ideal inflow is thus calculated for CTe =(CT 1 +CT 2)/(1+ℓ/2R) 2 .
The induced velocity of tandem rotors is vF = κtwin(TF /(2ρAV )+xRTR/(2ρAV )) for the front rotor and
vR = κtwin(TR/(2ρAV )+xF TF /(2ρAV )) for the rear rotor. For large separation, xR ∼ = 0 and xF ∼ = 2;
for the coaxial limit xR = xF =1is appropriate. Here xR = m and xF =2− m is used.
To summarize, the model for twin-rotor ideal induced velocity uses CTe = x1CT 1 + x2CT 2 and the
correction factor κtwin. In hover, xh =1/(2 − m); in forward flight of coaxial and tandem rotors, xf =1
for this rotor and xf = m or xf =2− m for the other rotor; in forward flight of side-by-side rotors,
xf =1/(1 + ℓ/2R) 2 (x =1/2 if there is no overlap, ℓ/2R >1). The transition between hover and forward
flight is accomplished using
x = xf μ 2 + xhCλ 2 h
μ 2 + Cλ 2 h
with typically C =1to 4. This transition is applied to x for both rotors, and to κtwin.
With a coaxial rotor in hover, the lower rotor acts in the contracted wake of the upper rotor.
Momentum theory gives the ideal induced power for coaxial rotors with large vertical separation (ref. 13):
Pu = Tuvu, v 2 u = Tu/2ρA for the upper rotor; and Pℓ =(¯αs/ √ τ)Tℓvℓ, v 2 ℓ = Tℓ/2ρA for the lower rotor.
Here τ = Tℓ/Tu; ¯α is the average of the disk loading weighted by the induced velocity, hence a measure
of nonuniform loading on the lower rotor (¯α =1.05 to 1.10 typically); and the momentum theory solution
is
¯αs
√ τ = 1
2τ 3/2
1 + 4(1 + τ) 2 ¯ατ − 1
The optimum solution for equal power of the upper and lower rotors is ¯αsτ =1, giving τ = Tℓ/Tu ∼ = 2/3.
Hence for the coaxial rotor in hover the ideal induced velocity is calculated from CTe = CTu for the
upper rotor and from CTe =(¯αs/ √ τ) 2 CTℓ for the lower rotor, with κtwin =1. Thus xh =1/(2−m) =1/2
and the input hover κtwin is not used, unless the coaxial rotor is modeled as a tandem rotor with zero
longitudinal separation.
11-5.2 Table Performance Method
The induced power is calculated from the ideal power: Pi = κPideal = κfDTvideal. The induced
power factor κ is obtained from an input table (linearly interpolated) that can be a function of edgewise
advance ratio μ or axial velocity ratio μz, and of blade loading CT /σ.