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lecture_8

lecture_8 - 16.333 Lecture 8 Aircraft Lateral Dynamics...

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Unformatted text preview: 16.333 Lecture # 8 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes 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 ⎥ ⎥ ⎥ ⎦ δ a , u = δ r x ˙ = Ax + Bu , x = ⎢ ⎢ ⎢ ⎣ p r φ and ψ ˙ = r sec θ ⎤ ⎡ A = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Y v Y p Y r m − U g cos θ m m ( I L v + I N v ) ( L p + I N p ) ( L r + I N r ) zx zx I zx I xx xx xx ( I L v + N v ) ( I L p + N p ) ( I L r + N r ) zx I zx zx I I zz zz zz 1 tan θ where I = ( I xx I zz − I 2 xx zx ) /I zz I = ( I xx I zz − I 2 zz zx ) /I xx I = I zx / ( I xx I zz − I 2 zx zx ) and ⎤ ⎡ ⎢ ⎢ ⎢ ⎣ ( m ) − 1 ( I xx ) − 1 I zx ⎥ ⎥ ⎥ ⎦ ⎤ ⎡ Y δ a Y δ r ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎣ ⎦ B = L δ a L δ r zz ) − 1 · zx ( I I N δ a N δ r Fall 2004 16.333 7–2 Lateral Stability Derivatives • A key to understanding the lateral dynamics is roll-yaw coupling . • L p 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: α r ( y ) = py/U – Creates lift distribution (chordwise strips) 1 δL w ( y ) = ρU 2 C l α α r ( y ) c y dy 2 – Net result is higher lift on right, lower on left – Rolling moment: b/ 2 b/ 2 L = δL w ( y ) · ( − y ) dy = − 2 1 ρU 2 − b/ 2 C l α py 2 c y dy ⇒ L p < − b/ 2 U – Key point: positive roll rate ⇒ negative roll moment. • L r rolling moment due to yaw rate: – Positive r has left wing advancing, right wing retreating → Horizontal velocity distribution over wing U = U − ry – Creates lift distribution over wing (chordwise strips) 1 1 L w ( y ) ∼ ρU 2 C l cdy ≈ ρ ( U 2 − 2 U ry ) C l c y dy 2 2 – Net result is higher lift on the left, lower on the right....
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lecture_8 - 16.333 Lecture 8 Aircraft Lateral Dynamics...

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