16.333 Lecture #8 Aircraft Lateral Dynamics Spiral, Roll, and dutch roll Modes
16.333 Lecture # 8 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes
Fa2004 16.3337-1 Aircraft Lateral dynamics Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics Ax+ Bu T d yb= rsec go m g cos o 4=(+M)(+层N)(+层N)0 (ILLr+ tan e ere xxZ xxz I=Iax/(xI-盈) and 0 B 0
� � � � Fall 2004 16.333 7–1 Aircraft Lateral Dynamics • Using a procedure similar to the longitudinal case, we can develop the equations of motion for the lateral dynamics ⎤⎡ v ⎥ ⎥ ⎥ ⎦ δa , u = δr x˙ = Ax + Bu , x = ⎢ ⎢ ⎢ ⎣ p r φ and ψ˙ = r sec θ0 ⎡ ⎤ A = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Yv Yp Yr m m m − U0 g cos θ0 ( I L � v + I� Nv) ( Lp + I� Np) (Lr + I� Nr) 0 zx zx I� zx I� xx xx xx (I� Lv + Nv ) (I� Lp + Np ) (I� Lr + Nr ) 0 zx I� zx zx I� I� zz zz zz 0 1 tan θ0 0 where I� = (IxxIzz − I2 xx zx)/Izz I� = (IxxIzz − I2 zz zx)/Ixx I� = Izx/(IxxIzz − I2 zx zx) and ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ (m) −1 0 0 0 (I� xx) −1 I� zx ⎥ ⎥ ⎥ ⎦ ⎡ ⎤ Yδa Yδr ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ B = ⎣ L ⎦ δa Lδr zz) −1 · zx 0 I (I Nδa Nδr 0 0 0
Fa2004 16.3337-2 Lateral stability Derivatives a key to understanding the lateral dynamics is roll-yaw 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 ar( y)=py/Uo Creates lift distribution(chordwise strips) SLu(y=spuoCl ar(y)cydy Net result is higher lift on right, lower on left Rolling moment b/2 L SLu(3) (y)dy=-DpUb Cla tra-cydy= Lp<0 /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=Uo Creates lift distribution over wing(chordwise strips) Lu(y) spU Cicdy x 5Pl00-2Uory)Cicydy Net result is higher lift on the left, lower on the right Rolling Moment: L=/ Lu(y). (-y)dy a pOor/ Cic, For large aspect ratio rectangular wing(crude Lr≈( to Cl>0 Key point: positive yaw rate= positive roll moment
� � � � 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) = ρU0 2 Clααr(y)cydy 2 – Net result is higher lift on right, lower on left – Rolling moment: b/2 b/2 L = δLw(y)·(−y)dy = −2 1 ρU0 2 −b/2 Clα py2 cydy ⇒ Lp < 0 −b/2 U0 – 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) ∼ ρU2 Clcdy ≈ ρ(U0 2 − 2U0ry)Clcydy 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 Clcyy2 dy −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
Fa2004 16.3337-3 p 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 a)and a rotation(leans forward on down- ward wing because of the larger a).- negative yaw moment In general hard to know which effect is larger. Nelson suggests at for a rectangular wing, crude estimate is that Nn≈pUSb(-)<0 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 b △ f Creates increase in lift at the tail fin by △L )fS/CLn,△a Creates a change in the yaw moment of N=-△r=-p(U1SCr7 P(UOS Cl <0 Key point: positive yaw rate= negative yaw moment. LN P r>0<0
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 because of the increase in α) and a rotation (leans forward on downward 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 Np ≈ ρU0 2 Sb(−CL) < 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(U0 2 )fSfCLαf Δαf 2 – Creates a change in the yaw moment of 1 N = −lfΔLf = −2 ρ(U0)fSfCLαf rl2 f 1 – So Nr = 2ρ(U0)fSfCLαf lf − 2 < 0 – Key point: positive yaw rate ⇒ negative yaw moment. L N p < 0 ? r > 0 < 0
Fa2004 16.3337-4 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 ow mo 0.0073 Spiral Mode Fast real 0.5633 Roll Damping Oscillatory-0.0331±0.9470→ Dutch roll Can look at normalized eigenvectors Spiral Roll Dutch Rol B=/00 00067001970.3269-28° p=p/(20/6b)-000907120.1992° 个=7/(20/b)000520004000368-112 1000100010000° Not as enlightening as the longitudinal case
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◦ pˆ = p/(2U0/b) 0.0009 0.0712 0.1198 92◦ rˆ = r/(2U0/b) 0.0052 0.0040 0.0368 112◦ φ 1.0000 1.0000 1.0000 0◦ Not as enlightening as the longitudinal case