Aircraft Lateral Dynamics: Spiral, Roll, and Dutch Roll Modes

Lecture #AC–3 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes Copyright 2003 by Jonathan How 1 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  ẋ = Ax + Bu , x =        v p r φ          ,u= δa δr   and ψ̇ = r sec θ0  A=             Yp Yr − U Yv g cos θ0 0 m m m L


( L
v + Izx Nv ) (
p + Izx Np ) ( L
r + Izx Nr ) 0 Ixx Ixx Ixx N


(Izx Lv + N
v ) (Izx Lp +
p ) (Izx Lr + N
r ) 0 Izz Izz Izz 0 0 1 tan θ0 where
2 Ixx = (IxxIzz − Izx )/Izz
2 Izz = (IxxIzz − Izx )/Ixx
2 Izx = Izx/(IxxIzz − Izx ) and  B=        (m)−1 0 0
−1
0 (Ixx) Izx

−1 0 Izx (Izz ) 0 0 0 2           ·  Yδ a Yδ r L δa L δr N δa N δr                   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◦ p̂ -0.0009 -0.0712 0.1198 92◦ r̂ 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 16.61 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 16.61 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 16.61 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 16.61 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 3 β shows a very lightly damped decay. 3 p, r clearly excited as well. 3 φ oscillates around 2.5◦ ⇒ Dutch-roll oscillations are clear. ⇒ Spiral mode ultimately dominates φ → 0 after 250 sec. – Aileron input 3 Large impact on p 3 Causes large change to φ 3 Very small change to remaining variables. 3 Influence smaller than Rudder. • Lateral approximate models are much harder to make (see discussion in Etkin and Reid). Not worth discussing at length. 7 Spring 2003 16.61 AC 3–8 Rudder Impulse 2 Beta P R Phi 1 0 −1 −2 −3 −4 0 2 4 6 8 10 time sec 12 14 16 18 Figure 1: Rudder impulse to flight variables. The rudder excites all modes. Dutch roll oscillations dominate initially. The spiral mode dominates longer term. 8 20 Spring 2003 16.61 AC 3–9 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 2 4 6 8 Aileron Impulse 10 time sec 12 14 16 Beta P R Phi 18 20 Figure 2: Aileron impulse to flight variables. Response primarily in φ. 9