Lecture #AC–3 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes Copy right 2003 by Jon at h an H ow 1 Spring 2003 16.61 AC 3–2 Aircraft Lateral Dynamics ? Using a procedure similar to the longitudinal case, we can develop the equa- tions of motion for the lateral dynamics ˙x = Ax + Bu , x = ? ? ? ? ? ? ? ? v p r φ ? ? ? ? ? ? ? ? ,u= ? ? δ a δ r ? ? and ˙ ψ = r sec θ 0 A = ? ? ? ? ? ? ? ? ? ? ? ? ? Y v m Y p m Y r m ? U 0 g cosθ 0 ( L v I prime xx + I prime zx N v )( L p I prime xx + I prime zx N p )( L r I prime xx + I prime zx N r )0 (I prime zx L v + N v I prime zz )(I prime zx L p + N p I prime zz )(I prime zx L r + N r I prime zz )0 01tanθ 0 0 ? ? ? ? ? ? ? ? ? ? ? ? ? where I prime xx =(I xx I zz ? I 2 zx )/I zz I prime zz =(I xx I zz ? I 2 zx )/I xx I prime zx = I zx /(I xx I zz ? I 2 zx ) and B = ? ? ? ? ? ? ? ? (m) ?1 00 0(I prime xx ) ?1 I prime zx 0 I prime zx (I prime zz ) ?1 000 ? ? ? ? ? ? ? ? · ? ? ? ? ? Y δ a Y δ r L δ a L δ r N δ a N δ r ? ? ? ? ? 2 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 longi- tudinal 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 ? ?p -0.0009 -0.0712 0.1198 92 ? ?r 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 -welldamped. – As the plane rolls, the wing going down has an increased α (wind is e?ectively “coming up” more at the wing) – Opposite e?ect for other wing. – There is a di?erence in the lift generated by both wings → more on side going down – The di?erential lift creates a moment that tends to restore the equi- 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. 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 e?ect 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 sti?ness) → Wings moving back and forth due to yaw motion result in oscillatory di?erential 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 di?erential drag ? Impulse response for the two inputs: – Rudder input a51 β shows a very lightly damped decay. a51 p, r clearly excited as well. a51 φ oscillates around 2.5 ? ? Dutch-roll oscillations are clear. ? Spiral mode ultimately dominates φ → 0 after 250 sec. – Aileron input a51 Large impact on p a51 Causes large change to φ a51 Very small change to remaining variables. a51 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 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 16.61 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 BetaP R Phi Figure 2: Aileron impulse to flight variables. Response primarily in φ. 9