Dynamics and Control of Rotorcraft Helicopter Aerodynamics and Dynamics Abhishek Department of Aerospace Engineering Indian Institute of Technology, Kanpur February 3, 2018
Overview Flight Dynamics Model 1 Flight Dynamics Model
Section 1 Flight Dynamics Model
Subsection 1
Newton-Euler Equations - 1 F = m a CG + ω V CG + mg ˆk G M = h H dt + ω H M = [I ]{ω} + ω [I ]{ω}
Newton Euler Equations - 2 X = m( u + qw rv) + mg sin θ Y = m( v + ru pw) mg sin φ cos θ Z = m(ẇ + pv qu) mg cos φ cos θ L = I xx ṗ I zz (ṙ + pq) (I yy I zz )qr N = I zz ṙ I xz (ṗ qr) (I xx I yy )pq
Kinematic Relations Flight Dynamics Model sin θ sin φ φ = p + q + r tan θφ 0 cos θ θ = q cos φ r sin φ ϕ = q sin φ cos φ + r cos φ cos θ
Subsection 2
Small Perturbation Analysis - 1 Each velocity component is written as some of baseline (trim) value and a time dependent perturbation: u(t) = u 0 + u(t) v(t) = v 0 + v(t) w(t) = w 0 + w(t) p(t) = p 0 + p(t) r(t) = r 0 + r(t)
Small Perturbation Analysis - 2 Euler angles and forces and moment are also written in similar way φ(t) = φ 0 + φ(t) θ(t) = θ 0 + φ(t) ϕ(t) = ϕ 0 + ϕ(t) Finally we write forces and moments: X (t) = X 0 + X (t) Y (t) = Y 0 + Y (t)... N(t) = N 0 + N(t) Now we substitute these quantities in Euler equations and kinematic relations.
Small Perturbation Analysis - 3 For example: X 0 m + X 0 m = u + q 0w 0 + q 0 w + qw 0 + q w r 0 v 0 r 0 v rv 0 r v + g sin θ 0 cos θ +g cos θ 0 sin θ Small perturbation assumption is made: 2 and HOT are neglected
Small Perturbation Analysis - 4 Further the perturbation angle φ, θ are small i.e cos φ 1, sin φ φ etc. Equation becomes: X 0 m + X 0 m = u + q 0w 0 r 0 v 0 + g sin θ 0 + q 0 w r 0 v +w 0 q + g cos θ 0 θ v 0 r Setting perturbations = 0, i.e quantities = 0 corresponds to trim condition. X 0 m = q 0w 0 r 0 v 0 + g sin θ 0 This has to be subtracted from the perturbation equation above.
Small Perturbation Analysis - 5 Equation becomes: X m = u + q 0 w r 0 v + w 0 q + g cos θ 0 θ v 0 r Similarly other two force equations are: Y m Z m = v + u 0 r + r 0 u v 0 p p 0 u + g sin θ 0 sin φ 0 θ g cos θ 0 cos φ 0 φ = ẇ + v 0 p + p 0 v u 0 q q 0 u + g sin θ 0 cos φ 0 θ +g cos θ 0 sin φ 0 φ
Small Perturbation Analysis - 6 Small perturbation Moment Equilibrium Equations (with dropped): L = I x ṗ I xz ṙ + [(I z I y )r 0 I xz p 0 ]q I xz pq 0 + (I z I y )q 0 M = I y q + [2I xz p 0 + (I x I z )r 0 ]p + [(I x I z )p 0 2I xz r 0 ]r N = I z ṙ I xz ṗ + [(I y I x )p 0 I xz r 0 ]q + (I y I x )q 0 p I xz q 0 r L and M equations are coupled in ṗ and ṙ, therefore can be solved simultaneously to decouple them (as algebraic equation).
Small Perturbation Analysis - 7 Small perturbation Kinematic relations: φ = p + tan θ 0 sin φ 0 q + tan θ 0 cos φ 0 r + θ = cos φ 0 q sin φ 0 r ϕ 0 cos θ 0 φ ϕ = sin φ 0 q + cos φ 0 r + ϕ 0 tan θ 0 θ cos θ 0 cos θ 0 ϕ 0 θ cos θ 0 L and M equations are coupled in ṗ and ṙ, therefore can be solved simultaneously to decouple them (as algebraic equation).
Introduction to Stability Derivatives - 1 X, Y, Z and L, M, N are functions of a variety of states and controls, for example: X = X(u, v, w, p, q, r, φ, θ, ϕ) Rigid body motion. θ 0, θ 1s, θ 1c, θ t Pilot controls. β, ζ, θ rot Rotor states λ Inflow states Ω Propulsion system states. u FCS Flight control system states. y other states (gust etc.) t time.
Introduction to Stability Derivatives - 2 For the time being we will consider only the key parameters, rigid body motion and control inputs. Expand X in a Taylor s series about trimmed flight condition: X = X (u 0, v 0,..., ϕ 0, θ 0,..., (θ 0 ) 0, (θ 1s ) 0, (θ t ) 0 )+ X u (u u 0)+ X v (v v 0)+ Now, X -X (u 0, v 0...) = X X = X u (u u 0) + X v (v v 0) Where, X u is stability derivative, (u u 0) is perturbation u, (v v 0 ) is v
Introduction to Stability Derivatives - 3 The partial derivatives are called stability derivatives or control derivative depending on whether denominator is motion quantity or perturbation quantity. These are typically indicated as: X u = X u, Z w = Z M w, θ 1s = M θ1s... etc. Also note: X φ, X θ, X ϕ = 0 Y φ... N φ, N θ... as aerodynamic rotor loads do not change by changing φ or θ or ϕ and keeping everything else constant.
Introduction to Stability Derivatives - 4 With above definitions and again dropping for perturbation we can write our equation as: ( X =)X = X u +X v v+x w w+x p p+x q q+x r r+x θ0 θ 0 +X θ1c θ 1c +X θ1s θ 1s = X θt θ t Y= Y u u..... N = N u u + N v v +... + N θt t We can substitute these in small perturbation equation of motion.
Introduction to Stability Derivatives - 5 Using u 0 = v cos α cos β, V 0 = v sin β, w 0 = v sin α cos β u X u u (X v + ϕ cos φ 0 cos θ 0 )v (X w + ϕ sin φ 0 cos θ 0 )w X p p (X q v sin α cos β)q X r r v sin βr + g cos θ 0 θ = X θ0 θ 0 + X θ1c θ 1c + X θ1s θ 1s + X θt θ t Final set of 9 equations can be written in a matrix from like this: {ẋ} = [A]{x} + [B]{u} {x} state vector = {u, v, w, p, q, r, φ, θ, ϕ} T {u} control vector = {θ 0, θ 1c, θ 1s, θ t } T A more sophisticated model may have larger number of states involved: {x} = {u, v, w, p, q, r, φ, θ, ϕ, λ 0, λ 1c, λ 1s, β 0, β 1c, β 1s..}
Introduction to Stability Derivatives - 6 Depending on the need, the equations can be split in to logitudinal dynamics and lateral-directional dynamics separately or the entire equation can be solved. The set of equations representing longitudinal dynamics for steady level flight case: u ẇ q = θ X u X w X q g cos θ 0 u X θ0 X θ1s ( ) w θ0 Z u Z w Z q g sin θ 0 M u M w M q 0 0 0 1 0 q θ + Z θ0 M θ0 Z θ1s M θ1s 0 0 θ 1s
Introduction to Stability Derivatives - 7 A constant coefficient model is sufficiently accurate for a helicopter, if the motion is slow; i.e if the time required for rotor revolution is small compared with the period of the motion under consideration, i.e perturbation or helicopter body rates ω <1-2 rad/sec [accuracy is acceptable] 1-2< ω <10 rad/s [accuracy becomes questionable] ω >10 rad/s [inaccurate] What is typical rate generated by helicopters during maneuvers: UTTAS pull-up maneuver had highest pitch rate of 30deg/s 0.7 rad/s Choices in FD Constant coefficient stability derivatives. Time dependent stability derivatives Bigger sate vector(w/rotor states, inflow etc) but still constcoefficients.
Introduction to Stability Derivatives - 8 How to compute stability derivatives? Analytical approach Take derivatives about trim conditions. Tedious process and dependent on model. Numerical approach Numerical perturbation based calculation. Faster, easier and independent of model. Rotor model can be non-linear and comprehensive. Selecting numerical perturbation is not simple.