Lecture #AC-3 Aircraft Lateral dynamics Spiral, Roll, and Dutch Roll Modes Copyright 2003 by Jonathan How
Lecture #AC–3 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes Copyright 2003 by Jonathan How 1
Spring 2003 16.61AC3-2 Aircraft Lateral Dynamics Using a procedure similar to the longitudinal case, we can develop the equa tions of motion for the lateral dynamics A r+ Bu,a p, u da and v=r sec Bo os 8o +1N)(+1N)(+N) (IAL 0 tan Bo Ix=(IxxIzz-12x)/Izz IZz=(IxxIxz-I/Ixx Ix/(Is d 0 B 0
Spring 2003 16.61 AC 3–2 Aircraft Lateral Dynamics • Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics x˙ = Ax + Bu , x = v p r φ , u = δa δr and ψ˙ = r sec θ0 A = Yv m Yp m Yr m − U0 g cos θ0 (Lv I xx + I zxNv) (Lp I xx + I zxNp) ( Lr I xx + I zxNr) 0 (I zxLv + Nv I zz ) (I zxLp + Np I zz ) (I zxLr + Nr I zz ) 0 0 1 tan θ0 0 where I xx = (IxxIzz − I2 zx)/Izz I zz = (IxxIzz − I2 zx)/Ixx I zx = Izx/(IxxIzz − I2 zx) and B = (m)−1 0 0 0 (I xx)−1 I zx 0 I zx (I zz)−1 000 · Yδa Yδr Lδa Lδr Nδa Nδr 2
Spring 2003 16.61AC3-3 The code gives the numerical values for all of the stability derivatives. Can solve for the eigenvalues of the matrix A to find the modes of the system 0.0331±0.9470 0.5633 Stable, but there is one very slow pole There are 3 modes, but they are a lot more complicated than the longi tudinal case Slow mode 0.0073 → Spiral Mode Fast real 0.5633 → Roll Damping Oscillatory-0.0331±0.94702→ Dutch roll Can look at normalized eigenvectors Spiral Roll Dutch Roll B0.0067|-0.01970.3269-28 p-0.0009-0.07120.119892 0.0520.0400.0368-12 1.0001.00001.0000° Not as enlightening as the longitudinal case
Spring 2003 16.61 AC 3–3 • The code gives the numerical values for all of the stability derivatives. Can solve for the eigenvalues of the matrix A to find the modes of the system. −0.0331 ± 0.9470i −0.5633 −0.0073 – Stable, but there is one very slow pole. • There are 3 modes, but they are a lot more complicated than the longitudinal case. Slow mode -0.0073 ⇒ Spiral Mode Fast real -0.5633 ⇒ Roll Damping Oscillatory −0.0331 ± 0.9470i ⇒ Dutch Roll Can look at normalized eigenvectors: Spiral Roll Dutch Roll β 0.0067 -0.0197 0.3269 -28◦ pˆ -0.0009 -0.0712 0.1198 92◦ rˆ 0.0052 0.0040 0.0368 -112◦ φ 1.0000 1.0000 1.0000 0◦ Not as enlightening as the longitudinal case. 3
Spring 2003 16.61AC3-4 Lateral modes Roll Damping-well damped As the plane rolls, the wing going down has an increased a (wind is effectively " coming up"more at the wing) Opposite effect for other wing There is a difference in the lift generated by both wings more on side going down The differential lift creates a moment that tends to restore the equ librium After a disturbance, the roll rate builds up exponentially until the restor ing moment balances the disturbing moment, and a steady roll is estab lished Disturbing rolling moment Restoring rolling moment D吓 Port Starboard Reduction in incidence Reduction in incidence
Spring 2003 16.61 AC 3–4 Lateral Modes Roll Damping - well damped. – As the plane rolls, the wing going down has an increased α (wind is effectively “coming up” more at the wing) – Opposite effect for other wing. – There is a difference in the lift generated by both wings → more on side going down – The differential lift creates a moment that tends to restore the equilibrium – After a disturbance, the roll rate builds up exponentially until the restoring moment balances the disturbing moment, and a steady roll is established
Spring 2003 16.61AC3-5 Spiral mode - slow, often unstable From level fight. consider a disturbance that creates a small roll angle q>0 This results in a small side-slip v(vehicle slides downhill ow the tail fin hits on the oncoming air at an incidence angle B → extra tail lift→ yawing moment The positive yawing moment tends to increase the side-slip → makes things worse If unstable and left unchecked, the aircraft would fly a slowly diverging path in roll, yaw, and altitude = it would tend to spiral into the ground!! Sideslip Steadily increasing roll angle Yawing moment due to fin lift 7 Fin lift force Can get a restoring torque from the wing dihedral Want a small tail to reduce the impact of the spiral mode
Spring 2003 16.61 AC 3–5 Spiral Mode - slow, often unstable. – From level flight, consider a disturbance that creates a small roll angle φ > 0 – This results in a small side-slip v (vehicle slides downhill) – Now the tail fin hits on the oncoming air at an incidence angle β → extra tail lift → yawing moment – The positive yawing moment tends to increase the side-slip → makes things worse. – If unstable and left unchecked, the aircraft would fly a slowly diverging path in roll, yaw, and altitude ⇒ it would tend to spiral into the ground!! • Can get a restoring torque from the wing dihedral • Want a small tail to reduce the impact of the spiral mode. 5
Spring 2003 16.61AC3-6 Dutch Roll- damped oscillation in yaw, that couples into roll Frequency similar to longitudinal short period mode, not as well damped (fin less effect than the horizontal tail) Do you know the origins on the name of the mode? Consider a disturbance from straight-level fight Oscillation in yaw y(fin provides the aerodynamic stiffness) Wings moving back and forth due to yaw motion result in oscillatory differential Lift/ Drag(wing moving forward generates more lift +Oscillation in roll o that lags y/ by approximately 900 Forward going wing is low Oscillating roll- sideslip in direction of low wing Path traced by starboard wing tip in one dutch roll cycle a)Starboard wing yaws aft with wing tip high aft yaw angle as aircraft rolls through ooo(c)Starboard wing g tp low yaws forward with yaw angle as aircraft rolls wings level in negative sense Damp the Dutch roll mode with a large tail fin
Spring 2003 16.61 AC 3–6 Dutch Roll - damped oscillation in yaw, that couples into roll. • Frequency similar to longitudinal short period mode, not as well damped (fin less effect than the horizontal tail). • Do you know the origins on the name of the mode? • Consider a disturbance from straight-level flight → Oscillation in yaw ψ (fin provides the aerodynamic stiffness) → Wings moving back and forth due to yaw motion result in oscillatory differential Lift/Drag (wing moving forward generates more lift) → Oscillation in roll φ that lags ψ by approximately 90◦ ⇒ Forward going wing is low Oscillating roll → sideslip in direction of low wing. • Damp the Dutch roll mode with a large tail fin. 6
Spring 2003 16.61AC3-7 Aircraft Actuator Influence Transfer functions dominated by lightly damped Dutch-roll mode Note the rudder is physically quite high, so it also influences the A/C Ailerons influence the Yaw because of the differential drag e Impulse response for the two inputs. Rudder input ◇β shows a very lightly damped deca o p, r clearly excited as well ◇ o oscillates around2.5° Dutch-roll oscillations are clear Spiral mode ultimately dominates -0 after 250 sec Aileron input ◇ Large impact on p o Causes large change to o o Very small change to remaining variables o Influence smaller than rudder Lateral approximate models are much harder to make(see discussion in Etkin and Reid). Not worth discussing at length
Spring 2003 16.61 AC 3–7 Aircraft Actuator Influence • Transfer functions dominated by lightly damped Dutch-roll mode. – Note the rudder is physically quite high, so it also influences the A/C roll. – Ailerons influence the Yaw because of the differential drag • Impulse response for the two inputs: – Rudder input ✸ β shows a very lightly damped decay. ✸ p, r clearly excited as well. ✸ φ oscillates around 2.5◦ ⇒ Dutch-roll oscillations are clear. ⇒ Spiral mode ultimately dominates φ → 0 after 250 sec. – Aileron input ✸ Large impact on p ✸ Causes large change to φ ✸ Very small change to remaining variables. ✸ Influence smaller than Rudder. • Lateral approximate models are much harder to make (see discussion in Etkin and Reid). Not worth discussing at length. 7
pring 2003 16.61AC3-8 Rudder Impulse Beta △R oooooooo 12 14 16 time se Figure 1: RUDDER IMPULSE TO FLIGHT VARIABLES. THE RUDDER EXCITES ALL MODES. DUTCH ROLL OSCILLATIONS DOMINATE INITIALLY, THE SPIRAL MODE DOMINATES LONGER TERM
Spring 2003 16.61 AC 3–8 0 2 4 6 8 10 12 14 16 18 20 −4 −3 −2 −1 0 1 2 time sec Rudder Impulse Beta P R Phi Figure 1: Rudder impulse to flight variables. The rudder excites all modes. Dutch roll oscillations dominate initially. The spiral mode dominates longer term. 8
Spring 2003 16.61AC3-9 它 Figure 2: AILERON IMPULSE TO FLIGHT VARIABLES. RESPONSE PRIMARILY IN o
Spring 2003 16.61 AC 3–9 0 2 4 6 8 10 12 14 16 18 20 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 time sec Aileron Impulse Phi R P Beta Figure 2: Aileron impulse to flight variables. Response primarily in φ. 9