Dynamics and Control of RotorcraftHelicopter Aerodynamics and Dynamics
Abhishek
Department of Aerospace EngineeringIndian Institute of Technology, Kanpur
February 3, 2018
Rotor LoadsTrim and Rotor Response
Overview
1 Rotor LoadsRotor Loads Calculation
2 Trim and Rotor ResponseTrim
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Section 1
Rotor Loads
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Subsection 1
Rotor Loads Calculation
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 1
H-force: in-plane alongfreestream rotor drag,positive towards tail
Y-force: in-plane, positivetowards right
T: rotor thrust,perpendicular to rotor disk
Mx : roll moment
My : pitch moment
Q: rotor torque
NOTE: these are in fixed-frame !
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 2
T =
Nb∑i=1
∫ R
0dFzi Q =
Nb∑i=1
∫ R
0ydFxi
H =
Nb∑i=1
∫ R
0(dFxi sinψi + dFyi cosψi )
Y =
Nb∑i=1
∫ R
0(−dFxi cosψi + dFyi sinψi )
Mx =
Nb∑i=1
∫ R
0dFzi y cosψi
My =
Nb∑i=1
∫ R
0dFzi y sinψi
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 3
The periodic nature of rotor: ψi = ψ + (i − 1) 2πNb
. The sectionalforces in rotating frame are given by:
dFx = dLφ+ dD
dFy = −dLβ + dFR
dFz = dL
To obtain steady forces in fixed frame:
T =1
2π
∫ 2π
0T (ψ)dψ
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 4
In non-dimensional form it is given as:
CT
σClα
=1
2
∫ 1
0(u2
T θ − uPuT )dy
The non-dimensional velocities are given by:
uT = r + µ sinψ
uP = r∗β + λ+ βµ cosψ
uR = µ cosψ
Similarly non-dimensional H-force, Y-force, torque Q and roll andpitch moments can be obtained.
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 5
Consider a rigid rotor (no flapping), with three blades (Nb = 3)
Fz =Fz
ρcClalpha(ΩR)2R=
1
2
∫ 1
0(u2
T − uPuT )dy
Assuming the cyclic pitch variation and assuming uniform inflow:
Fz = F0 + F1c cosψ + F1s sinψ + +F2c cos 2ψ + F2s sin 2ψ
+F3c cos 3ψ + F3s sin 3ψ
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 6
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 7
The total force due to all three blades is:
Rotor acts like a filter, allowing only Nb/rev harmonics to fixedframe. Rest all harmonics are blocked. Nb/rev originates fromNb ± 1/rev
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 8
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 9
For rotor loads, inertial forces also need to be included. Let mo beblade mass/length.
Blade root shear forces (rotatingframe)
sx =
∫ R
e[dFx −mo(y − e)ζ]dy
sy =
∫ R
e[dFy + moyΩ2]dy
sz =
∫ R
e[dFz −mo(y − e)β]dy
Hub forces in rotating frame are:fx = sx , fy = sy , and fz = sz
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Rotor Forces and Moments - 10
Blade root bending moments (rotating frame)
nf =
∫ R
e[dFz −mo(y − e)β −moyΩ2](y − e)dy
nl =
∫ R
e[dFx −mo(y − e)ζ](y − e)dy
nt =
∫ R
edMxdy
Hub moments (rotating frame) are given by: mx = nf + esz ,my = nt , and mz = −nl − esx
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Numerical Integration of Loads
Aerodynamic forces and moments are calculated for eachblade element and integrated over the length of the blade toobtain root loads (shears and bending moment)
Root loads are integrated over the whole rotation for eachblade to get mean loads for trim analysis
Rotor LoadsTrim and Rotor Response
Rotor Loads Calculation
Numerical Integration of Loads
Aerodynamic forces and moments are calculated for eachblade element and integrated over the length of the blade toobtain root loads (shears and bending moment)
Root loads are integrated over the whole rotation for eachblade to get mean loads for trim analysis
Rotor LoadsTrim and Rotor Response
Trim
Section 2
Trim and Rotor Response
Rotor LoadsTrim and Rotor Response
Trim
Subsection 1
Trim
Rotor LoadsTrim and Rotor Response
Trim
Helicopter Trim - 1
Trim means equilibrium: all forces and moments are balanced.Need to draw FBD of the helicopter in free flight!
Rotor LoadsTrim and Rotor Response
Trim
Helicopter Trim - 2
xcg , ycg ≡Cg location wrt hub.
h ≡ height of hub.
lt ≡ tail location.
T = Rotor thrust, D = airframe drag, H= rotor drag, Y =rotor side force
YF = Tail rotor thrust
αs = longitudinal shaft tilt
φs = lateral shaft tilt.
Helicopter forces are in body frame and need to be transformedfrom xbybzb to xEyE zE . This involves rotation about xb followedby φs and then about yi by αs .
Rotor LoadsTrim and Rotor Response
Trim
Helicopter Trim - 3
The transformation matrix for the two rotations are given by:
[T1] =
1 0 00 cosφs sinφs0 − sinφs cosφs
[T2] =
cosαs 0 − sinαs
0 1 0sinαs 0 cosαs
Net transformation from body axis to earth axis is:
[T1] = [T1][T2] =
cosαs 0 − sinαs
sinαs sinφs cosφs cosαs sinφscosφs sinαs − sinφs cosαs cosφs
Rotor LoadsTrim and Rotor Response
Trim
Helicopter Trim - 4
So,FxeFyeFze
= [T ]
HYT
=
cosαsH − sinαsTsinαs sinφsH + cosφsY + cosαs sinφsTcosφs sinαsH − sinφsY + cosαs cosφsT
Rotor LoadsTrim and Rotor Response
Trim
Helicopter Trim - 5
So, vertical force equilibrium equation is:W − T cosαs cosφs + Y sinφs − H sinαs cosφs + Yf sinφs +D sin θfp = 0Similarly:
D cos θfp + H cosαs − T sinαs = 0
(Y + Yf ) cosφs + T cosαs sinφs + H sinαs sinφs = 0
My + Myf −W (xcg cosαs − h sinαs)− D(xcg sinαs + h cosαs) = 0
Mx + Mxf + Yf h + W (h sinφs − ycg cosφs) = 0
Q − Yf lt = 0
In addition we have a equation of inflow (if using uniform inflow):
λ = µ tanαs +kCT
2√µ2 + λ2
Rotor LoadsTrim and Rotor Response
Trim
Trim Variables
Inputs are: (θ75, θ1c , θ1s , αs , φs , θtr , λ0)Outputs are: (T ,Y ,H,Mx ,My ,Q, λ)Question is how do we solve non-linear trim equations?Popular approach is Jacobian based i.e using Newton Raphson.
Rotor LoadsTrim and Rotor Response
Trim
Trim Procedure
Define residual for each equation:R1 = res(1) = T −WR2 = res(2) = D + H − Tαs
R3 = res(3) = Y + Yf − TφsR4 = res(4) = My + Myf + w(hαs − xcg )− hDR5 = res(5) = Mx + Mxf + w(hφs − ycg ) + Yf hR6 = res(6) = Q − Yf ltR7 = res(7) = λ− µ tanαs + kCT
2√µ2+λ2
R1 through R7 should be zero.We can do Taylor’s series expansion for same:R1 + ∂R1
∂θ1∆θ1 + ∂R1
∂θ2∆θ2 + ....+ ∂R1
∂θ7∆θ7+HOT = 0
R2 + ∂R2∂θ2
∆θ2 + ....+ ∂R1∂θ7
∆θ7+HOT = 0
Rotor LoadsTrim and Rotor Response
Trim
Newton Raphson Iteration
The iterative procedure that NR-algorithm is given as :
θn+1 = θn − J−1R
where J - Jacobian of the system which is calculated by
Jij =∂R i
∂θj
≈Ri(θj+∆θj ) − Ri(θj )
∆θj
where θj is the j th element of input control θR i is the i th element of residual R∆θj is perturbation in θj
and n - is the number of iteration
Rotor LoadsTrim and Rotor Response
Trim
Coupled Trim Algorithm
Rotor LoadsTrim and Rotor Response
Trim
Numerical Integration Methods
Trapezoidal rule:∫ ba f (x)dx ≈ (b − a) (f (a)+f (b))
2Gaussian quadrature:∫ b
af (x)dx ≈ b − a
2
n∑i=1
wi f
(b − a
2xi +
a + b
2
)
i wi xi1 0.3607615730481386 0.6612093864662645
2 0.3607615730481386 -0.6612093864662645
3 0.4679139345726910 -0.2386191860831969
4 0.4679139345726910 0.2386191860831969
5 0.1713244923791704 -0.9324695142031521
6 0.1713244923791704 0.9324695142031521
Rotor LoadsTrim and Rotor Response
Trim
Thank You