Stability and Control, Apuntes de Ingeniería Aeroespacial

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Stability & Control

Aerospace Science & Technology

F. Mellibovsky

Escola d’Enginyeria de Telecomunicaci´o i Aeronautica de Castelldefels Universitat Politecnica de Catalunya

March 2, 2011

Summary

Basic Concepts

Longitudinal Flight Longitudinal Equations Longitudinal Equilibrium Longitudinal Balance Longitudinal Stability Static Stability Dynamic Stability

Lateral Flight Lateral Equations Lateral Equilibria Lateral Stability Static Stability Dynamic Stability

Definitions

I (^) Flight Equilibria: All forces and moments are balanced I (^) Flight Parameters: Speed, ascent/descent rate, turn radius... I (^) Aircraft Control / Piloting: Ability to prescribe flight parameter values and modify equilibria I (^) Performances: Equilibrium optimisation I (^) Flight Dynamics: Response to unbalanced force/moment I (^) Time-evolution of flight parameters I (^) Stability I (^) Stability conditions: Unstable, Neutral, Stable I (^) Static Stability: Initial response to perturbation Restoring Force I (^) Dynamic Stability: Dynamical behaviour Restoring Force Damping Force I (^) Handling & Flying Qualities: Dynamical response optimisation I (^) Manoeuvrability: Response celerity to command input I (^) Stability Augmentation: Automatic Control

Body Axes

Angular velocities: I (^) x: p (roll) I (^) y: q (pitch) I (^) z: r (yaw) Aerodynamic forces: I (^) xa: −D (drag) I (^) y: FY (lateral force) I (^) za: −L (lift) Aerodynamic Moments & Commands: I (^) x: Ml , δl I (^) y: Mm, δm I (^) z: Mn, δn

symmetryplane

"wings"plane

Secondary Control Surfaces & Control Systems

Secondary Control Surfaces I (^) High-Lift Devices: I (^) flaps, slats, slots I (^) Trim tabs: I (^) elevator trim I (^) rudder and aileron trims (large aircraft) I (^) Spoiler / Air Brakes I (^) Others: flaperons, stabilators... Flight Control Systems: I (^) Mechanical I (^) Hydro-mechanical I (^) fly-by-wire

Other Surfaces

Empennage: I (^) Tailplane: I (^) Horizontal Stabiliser I (^) Elevator I (^) Elevator trim tab I (^) Fin: I (^) Vertical Stabiliser I (^) Rudder I (^) Combined: V-tail (with ruddervators) Canards I (^) Lifting-canard I (^) Control-canard Winglets

Longitudinal Equations

Hypothesis: I (^) Aircraft reflexion-symmetry plane is vertical −→ φ = 0 I (^) No roll or yaw rotation velocities −→ p = r = 0 I (^) Aerodynamic Forces within the symmetry plane −→ β = 0 I (^) Propulsion Forces parallel to the symmetry plane I (^) Aerodynamic Moment orthogonal to the symmetry plane I (^) Propulsion Moments orthogonal to the symmetry plane Consequences: I (^) Longitudinal Equations decoupled I (^) Flight occurs in the vertical plane

Longitudinal Equations

Equations (aerodynamic axes projection): I (^) Propulsion equation (P): m V˙ = − 12 ρSV 2 CD + T cos α − mg sinγ I (^) Lift equation (L): −mV γ˙ = − 12 ρSV 2 CL−T sin α + mg cos γ I (^) Pitching Moment equation (Q): Iy q˙ = 12 ρSlV 2 Cm+MT I (^) Pitch Velocity equation (QV): α˙ + ˙γ = q I (^) Altitude equation (H): h˙ = V sin γ Simplifying Hypothesis: I (^) T~ and V~ parallel −→ P and L decouple regarding T I (^) MT = 0 −→ P and Q decouple regarding T I (^) Small γ (cos γ ' 1, sin γ ' γ) −→ P and L decouple regarding γ

Longitudinal Balance

Centre of Lift (xL): I (^) Place where Lift is Applied I (^) No Pitching Moment Cm(xL) = 0 I (^) Changes with α

Aerodynamic Centre (xAC ): I (^) Where Lift change is applied I (^) Constant Pitching Moment (∂Cm/∂α)(xAC ) = 0 I (^) Fairly independent of α

Pitching Moment Coefficient (with respect to xG ):

Cm = Cmα (α − α 0 ) = Cm 0 + Cmα α

Pitching Moment Gradient Model:

Cmα =

xAC − xG l

CLα

but the aircraft is yet to be balanced...

Longitudinal Balance

I (^) Elevators (stick or yoke command): δm I (^) Modify lift (small effect): CL = CLα (α − α 0 ) + CLδm δm I (^) Modify pitching moment: Cm = Cm 0 + Cmα α + Cmδm δm I (^) Q: Cm = Cm 0 + Cmα α + Cmδm δm = 0 I (^) δm ↔ α: δm = −(Cm 0 + Cmα α)/Cmδm I (^) Depends on: centering (xG − xAC ) I (^) Independent of: h, V , γ, m I (^) To sustain α: elevator trim (secondary control surface). I (^) Pitching moment due to engine axis neglected.

Longitudinal Balance

I (^) L: mg cos γ = 12 ρSV 2 CL(α) −→ V 2 (α − α 0 ) = ct I (^) α ↔ V I (^) Depends on: h, m I (^) Independent of: centering (xG − xAC ) I (^) P: T = 12 ρSV 2 CD (α) + mg γ −→ ∆T (δx ) = mg ∆γ I (^) δx ↔ γ I (^) Depends on: h, m, V I (^) Independent of: centering (xG − xAC ) I (^) P/L: T = mg (1/E + γ) I (^) V ↔ γ I (^) Depends on: h, m, V , Regimes I (^) Independent of: centering (xG − xAC )

Longitudinal Balance

I (^) Power setting: δx ↔ γ I (^) Elevator deflection: δm ↔ α ↔ V ↔ γ Flight Regimes: T − D = mg γ I (^) R1: ∆V · ∆γ < 0 I (^) R2: ∆V · ∆γ > 0

Drag (Required Thrust) vs. Speed:

D = DP + Di = AD V 2 + BD /V 2

AD =

ρSCD 0 BD =

kW 2 1 2 ρS Power vs. Speed:

P = PP + Pi = AD V 3 + BD /V

Available Thrust:

T (δx ) = kT ρV λT^ δx

Dynamic Stability

I (^) Five longitudinal equations I (^) Three longitudinal modes: I (^) Pitching Oscillation: oscillatory, involving I (^) α: angle of attack I (^) q: pitch angular velocity I (^) Phugoide: oscillatory, involving I (^) V : velocity I (^) γ: slope I (^) Return to straight & level flight: aperiodic, involving I (^) h: altitude

Pitch Oscillation

Pitching oscillation (α ↔ q) with respect to a point ahead of G

I (^) Decoupling Hypothesis: V , h ' ct I (^) Spring / Restoring effect: Cmα (pitching moment due to α) I (^) Damping effect: Cmq (horizontal stabiliser), CLα (wing) I (^) Properties I (^) Short period: ωn = V

√ 1 2 ρSl/Iy

√ −Cmα = (1/ρx )

√ g (xG − xAC )

√ CLα /CL I (^) Stable while periodic, then aperiodic, then unstable I (^) Static margin ms = (xAC − xG )/l: G forward → ω ↑, constant damping I (^) Attention to Pilot-Induced Oscillations! Do not try to pilot it!