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