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