16.333 Lecture # 8

16.333 Lecture # 8 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes Fall 2004 Aircraft Lateral Dynamics 16.333 7–1 • Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics ⎡ ⎤ v � � ⎢ ⎥ δa ⎢p⎥ ẋ = Ax + Bu , x = ⎢ ⎥ , u = δr ⎣r⎦ φ and ψ̇ = r sec θ0 ⎡ Yp m Yv m Yr − U 0 m g cos θ0 ⎢ ⎢ ⎢ L Lp Lr + I � N ) � � ⎢ ( �v + Izx N ) ( + I N ) ( v p � � zx zx r ⎢ Ixx Ixx A = ⎢ Ixx ⎢ ⎢ (I � L + Nv ) (I � L + Np ) (I � L + Nr ) ⎢ zx v I � � � zx p zx r Izz Izz ⎣ zz 0 1 tan θ0 where � 2 Ixx = (IxxIzz − Izx )/Izz � 2 Izz = (IxxIzz − Izx )/Ixx � 2 Izx = Izx/(IxxIzz − Izx ) and ⎡ (m) ⎢ ⎢ 0 B=⎢ ⎣ 0 0 −1 0 0 � −1 � (Ixx ) Izx � � −1 Izx (Izz ) 0 0 ⎤ ⎡ ⎤ Y Y ⎥ δ δr ⎥ ⎣ a ⎥ · Lδa Lδr ⎦ ⎦ Nδa Nδr 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Fall 2004 16.333 7–2 Lateral Stability Derivatives • A key to understanding the lateral dynamics is roll­yaw coupling. • Lp rolling moment due to roll rate: – Roll rate p causes right to move wing down, left wing to move up → Vertical velocity distribution over the wing W = py – Leads to a spanwise change in the AOA: αr (y) = py/U0 – Creates lift distribution (chordwise strips) 1 δLw (y) = ρU02Clα αr (y)cy dy 2 – Net result is higher lift on right, lower on left – Rolling moment: � b/2 � py 2 1 2 b/2 Clα cy dy ⇒ Lp < 0 L= δLw (y)·(−y)dy = − ρU0 2 U 0 −b/2 −b/2 – Key point: positive roll rate ⇒ negative roll moment. • Lr rolling moment due to yaw rate: – Positive r has left wing advancing, right wing retreating → Horizontal velocity distribution over wing U = U0 − ry – Creates lift distribution over wing (chordwise strips) 1 1 Lw (y) ∼ ρU 2Cl cdy ≈ ρ(U02 − 2U0ry)Cl cy dy 2 2 – Net result is higher lift on the left, lower on the right. � b/2 � b/2 – Rolling Moment: L = Lw (y)·(−y)dy ≈ ρU0r Cl cy y 2dy −b/2 −b/2 – For large aspect ratio rectangular wing (crude) 1 1 Lr ≈ ( to )CL > 0 6 4 – Key point: positive yaw rate ⇒ positive roll moment. Fall 2004 16.333 7–3 • Np yawing moment due to roll rate: – Rolling wing induces a change in spanwise AOA, which changes the spanwise lift and drag. – Distributed drag change creates a yawing moment. Expect higher drag on right (lower on left) → positive yaw moment – There is both a change in the lift (larger on downward wing be­ cause of the increase in α) and a rotation (leans forward on down­ ward wing because of the larger α). → negative yaw moment – In general hard to know which effect is larger. Nelson suggests that for a rectangular wing, crude estimate is that 1 CL Np ≈ ρU02Sb(− ) < 0 2 8 • Nr yawing moment due to yaw rate: – Key in determining stability properties – mostly from fin. – Positive r has fin moving to the left which increases the apparent angle of attack by rlf Δαf = (U0)f – Creates increase in lift at the tail fin by 1 ΔLf = ρ(U02)f Sf CLαf Δαf 2 – Creates a change in the yaw moment of 1 N = −lf ΔLf = − ρ(U0)f Sf CLαf rlf2 2 – So Nr = − 21 ρ(U0)f Sf CLαf lf2 < 0 – Key point: positive yaw rate ⇒ negative yaw moment. L N p <0 ? r >0 <0 Fall 2004 16.333 7–4 Numerical Results • 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 = w/U0 0.0067 ­0.0197 0.3269 ­28◦ p̂ = p/(2U0/b) ­0.0009 ­0.0712 0.1198 92◦ r̂ = r/(2U0/b) 0.0052 0.0040 0.0368 ­112◦ 1.0000 1.0000 1.0000 0◦ φ Not as enlightening as the longitudinal case. Fall 2004 16.333 7–5 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. Recall that Lp < 0 – After a disturbance, the roll rate builds up exponentially until the restoring moment balances the disturbing moment, and a steady roll is established. Disturbing rolling moment Restoring rolling moment Roll Rate p V0 -py − �' Port wing Reduction in incidence py �' V0 Starboard wing Reduction in incidence Fall 2004 16.333 7–6 Spiral Mode ­ slow, often unstable. – From level flight, consider a disturbance that creates a small roll angle φ > 0 → This results in a small side­slip v (vehicle slides downhill) – Now the tail fin hits on the oncoming air at an incidence angle β → extra tail lift → positive yawing moment – Moment creates positive yaw rate that creates positive roll mo­ ment (Lr > 0) that increases the roll angle and tends to increase the side­slip → 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. Fall 2004 16.333 7–7 Dutch Roll ­ damped oscillation in yaw, that couples into roll. • Frequency similar to longitudinal short period mode, not as well damped (fin less effective than horizontal tail). • Consider a disturbance from straight­level flight → Oscillation in yaw ψ (fin provides the aerodynamic stiffness) → Wings moving back and forth due to yaw motion result in oscil­ latory differential lift/drag (wing moving forward generates more lift) Lr > 0 → Oscillation in roll φ that lags ψ by approximately 90◦ ⇒ Forward going wing is low Oscillating roll ⇒ sideslip in direction of low wing. Fall 2004 • Do you know the origins on the name of the mode? • Damp the Dutch roll mode with a large tail fin. 16.333 7–8 −2 10 • Ailerons influence the Yaw because of the differential drag −200 −2 10 −150 −100 −50 0 50 100 150 200 10 −2 10 −1 0 10 10 1 2 −1 0 0 10 10 Freq (rad/sec) −1 10 10 Freq (rad/sec) pda | |G arg Gpda |Gβda| arg Gβda −2 −2 10 −350 −300 −250 −200 −150 −100 −50 0 10 −2 10 −1 10 0 10 1 10 −1 0 0 10 10 Freq (rad/sec) −1 10 10 Freq (rad/sec) −2 −200 −2 10 −150 −100 −50 0 50 100 150 200 10 −2 10 −1 10 0 10 1 10 Transfer function from aileron to flight variables 2 2 10 10 |Grda| arg Grda 10 −1 0 0 10 10 Freq (rad/sec) −1 10 10 Freq (rad/sec) Fall 2004 16.333 7–9 Aircraft Actuator Influence Figure 1: Aileron impulse to flight variables. Response primarily in φ. • Transfer functions dominated by lightly damped Dutch­roll mode. • Note the rudder is physically quite high, so it also influences the A/C roll. 3 −200 −2 10 −150 −100 −50 0 50 100 150 200 10 −2 10 0 10 1 2 10 10 −1 0 10 10 Freq (rad/sec) 0 10 10 Freq (rad/sec) −1 | pda 4 pda |G arg G −2 −2 10 −500 −400 −300 −200 −100 0 10 −2 10 −1 10 0 10 1 10 −1 0 10 10 Freq (rad/sec) 0 10 10 Freq (rad/sec) −1 −2 −2 10 −300 −200 −100 0 100 200 10 −2 10 −1 10 0 10 1 10 Transfer function from rudder to flight variables 2 2 10 10 | rda |Gβda| arg Gβda rda |G arg G 10 −1 0 0 10 10 Freq (rad/sec) −1 10 10 Freq (rad/sec) Fall 2004 16.333 7–10 Figure 2: Aileron impulse to flight variables. Response primarily in φ. Fall 2004 16.333 7–11 Aileron 1 deg Impulse − 2sec on then off −4 β rad 2 0 −2 −4 p rad/sec 2 0 −3 x 10 5 10 15 20 25 30 δa > 0 ==> right wing up 0 −2 −4 5 r rad/sec x 10 0 −4 x 10 5 10 15 20 25 30 Initial adverse yaw ==> RY coupling 0 −5 0 5 10 15 20 25 30 0 5 10 15 time sec 20 25 30 φ rad 0 −0.005 −0.01 Figure 3: Aileron impulse to flight variables • Aileron δa =1deg impulse for 2 sec. – Since δa > 0 then right aileron goes down, and right wing goes up → Reid’s notation, and it is not consistent with the picture on 6–4 (from Nelson). – Influence of the roll mode seen in the response of p to application and release of the aileron input. – See effect of adverse yaw in the yaw rate response caused by the differential drag due to aileron deflection. – Spiral mode harder to see. – Dutch mode response in other variables clear (1 rad/sec ∼ 6 sec period). Fall 2004 16.333 7–12 Rudder 1 deg step 0.02 β rad 0.01 0 −0.01 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 time sec 20 25 30 p rad/sec 0.1 0 −0.1 r rad/sec 0.1 0 −0.1 φ rad 1 0 −1 −2 Figure 4: Rudder step to flight variables • Rudder step input 1deg step. – Dutch roll response very clear. Other 2 modes are much less pronounced. – β shows a very lightly damped decay. – p clearly excited as well. Doesn’t show it, but often see evidence of adverse roll in p response where initial p is opposite sign to steady state value. Reason is that the forces act on the fin which is well above the cg → and the aircraft responds rapidly (initially) in roll. – φ ultimately oscillates around 2.5◦