Aircraft Flight Dynamics & Vortex Lattice Codes

Transcription

1 Aircraft Flight Dynamics Vortex Lattice Codes AA241X April Stanford University

2 Overview 1. Equations of motion 2. Non-dimensional EOM Aerodynamics 3. Trim Analysis Longitudinal Lateral 4. Linearized Dynamics Analysis Longitudinal Lateral 5. VLM Codes Capabilities and limitations Available codes

3 Equations of Motion Dynamical system is defined by a transition function, mapping states control inputs to future states X δ Aircraft EOM!X!X = f (X,δ)

4 Equations of Motion For Aircraft:! X = " x y z u v w θ φ ψ p q r $ % position velocity attitude Angular velocity! δ = " δ e δ t δ a δ r $ % elevator throttle aileron rudder

5 Equations of Motion (II) System of 12 Nonlinear ODEs Dynamics Eqs* Linear Acceleration = Aero + gravity + Gyro Kinematic Eqs: Relation between position and velocity m!u = X mgsin(θ)+ m(rv qw) m!v = Y + mgsin(φ)cos(θ)+ m(pw ru) m!w = Z + mgcos(φ)cos(θ)+ m(qu pv)! "!x!y!z $! = N R B (φ,θ,ψ ) % " u v w $ % Angular Acceleration = Aero + Gyro I xx!p = L + (I yy I zz )qr I yy!q = M + (I zz I xx )pr I zz!r = N + (I xx I yy )pq Relation between attitude and angular velocity!φ = p + qsin(φ)tan(θ)+ r cos(φ)tan(θ)! θ = qcos(φ) rsin(φ)!ψ = q sin(φ) cos(θ) + r cos(φ) cos(θ) *Neglecting cross-products of inertia

6 Equations of Motion (III) Sources of Nonlinearity Trigonometric projections (dependent on attitude) Gyroscopic effects Aerodynamics Dynamic pressure Reynolds Stall partial separation Uncertainties in EOM: Gravity Gyroscopic terms are straightforward, provided we can measure mass, inertias and attitude accurately Aerodynamics is harder, especially viscous effects: lifting surface drag, propeller fuselage aerodynamics

7 Lower Order EOM Sometimes lower order models suffice Very useful in mission planning: 2 DOF: only describe motion on the XY plane States: forward velocity turn rate 3 DOF: add vertical motion to previous model States: forward velocity, turn rate vertical velocity First order dynamics can be included to model the time it takes to maneuver Max values can be defined for states, to limit maneuverability (e.g. max turn rate, max climb rate) Useful for inner Dynamics Control Design: Longitudinal Lateral separation 2 nd Order Phugoid approximation

8 Non-dimensional EOMs Why? Aerodynamics scales with size of aircraft and dynamic pressure Easier to compare qualities of different designs Many textbooks and aerodynamic codes provide data in nondimensional form Non-dimensional EOMs: Non-dimensional dynamic states: Non-dimensional time: ˆt = Vt c û = u V ˆv = v V β ŵ = w V α ˆp = pb 2V ˆq = qc 2V ˆr = rb 2V Non-dimensional moments of inertia Î xx = 8I xx ρsb 3 Î yy = 8I yy ρsc 3

9 Non-dimensional Aerodynamics Aerodynamic Forces Moments Stability Control Derivatives: First derivative of non-dimensional forces and moments w.r.t. nondimensional states control, at a flight condition There are potentially 6x6 SD 6x4 CD. In practice, many are negligible Examples: C x = C y = C z = X 1 2 ρv 2 S Y 1 2 ρv 2 S Z 1 2 ρv 2 S C Zα = C Z α C l = C m = C n = L 1 2 ρv 2 Sb M 1 2 ρv 2 Sc N 1 2 ρv 2 Sb C Zδe = C Z δ e C lp = C l ˆp C lδa = C l δ a C mq = C m ˆq C nδe = C n δ r

10 Non-dimensional EOMs Example: linearized* roll equation non-dimensional form I xx!p = L Non-dimensional mapping Î xx!ˆp = Clp ˆp + C lr ˆr + C lβ β + C lδa δa + C ldr δr * Gyroscopic terms fall out, because they are 2 nd order

11 Trim Linearized Analysis Full nonlinear flight dynamics is very hard to analyze Break up into: Trim: equilibrium points of the aircraft Linearized dynamics: dynamic behavior for small perturbations Understanding both is required for Aircraft Configuration Controls design

12 Trim Analysis Flight conditions at which if we keep controls fixed, the aircraft will remain at that same state (provided no external disturbances) X trim δ trim Aircraft EOM!X = 0 For each aircraft there is a mapping between trim states and trim control inputs Analogy: car going at constant speed, requires a constant throttle position!x = f (X trim,δ trim ) = 0 X trim = g trim (δ trim ) The mapping g() is not always one-to-one, could be many-to-many!

13 Linearized Dynamics Analysis Many flight dynamic effects can be analyzed explained with Linearized Dynamics Most of the times we linearize dynamics around Trim conditions X trim + X ' δ trim +δ ' Aircraft EOM (near Trim)!X = f X X ' + f δ δ ' Useful to synthesize linear regulator controllers NOTE: regulator controllers don t help us go from one trim condition to another, they just help in staying near a Trim condition

14 An idea: Trim + Regulator Controller I. Inverse trim: to take us to the desired trim state II. Regulator: to stabilize modes and bring us back to desired trim state in the presence of disturbances X desired + - g 1 trim Linear Regulator Controller δ trim + δ ' + X δ Aircraft EOM!X X

15 Longitudinal Trim Simple wing-tail system L_wing h_cg h_tail mg M_wing L_tail

16 Longitudinal Trim (II) Moment balance: 0 = M wing h CG L wing + x tail L tail 0 =1 2ρV 2 $ c wing S wing C mwing h CG S wing C Lwing + h tail S tail C Ltail % h CG c wing C Lwing (α trim ) = h tail S tail c wing S wing C Ltail (α trim,δe trim ) C mwing Elevator trim defines trim AoA, and consequently trim CL

17 Longitudinal Trim (III) Force balance* mg = L cos(γ) L = 1 2 ρv 2 SC L V 2 mg = 1 2 ρsc L (δe trim ) θ α V γ T γ D L Trim Elevator defines trim Velocity! mg T = D + Lsin(γ) D + Lγ γ = T (δt trim ) mg 1 (L D ) (δetrim ) Elevator Thrust both define Gamma! *Assuming small Gamma

18 Longitudinal Trim (IV) How do we get an aircraft to climb? (Gamma > 0) Two ways: 1. Elevator up Elevator up increases AoA, which increases CL Increased CL, accelerates aircraft up Up acceleration, increases Gamma Increased Gamma rotates Lift backwards, slowing down the aircraft 2. Increase Thrust Increased thrust increases velocity, which increases overall Lift Increased Lift, accelerates aircraft up Up acceleration, increases Gamma Increased Gamma rotates Lift backwards, slowing down the aircraft to original speed (set by Elevator, remember!) Elevator has its limitations When L/D max is reached, we start going down When CL max is reached, we go down even faster!

19 Experimental Trim Relations Theoretical relations hold to some degree experimentally In reality: Propeller downwash on horizontal tail has a significant distorting effect Reynolds variations with speed, distort aerodynamics One can build trim tables experimentally Trim flight at different throttle and elevator positions Measure: Average airspeed Average flight path angle Gamma Phugoid damper would be very helpful One could almost fly open loop with trim tables!

20 Aircraft in constant turn: Lateral Trim φ L Vertical force balance: L cos(φ) = mg cos(φ) = m g cos(φ S 1 2 ρv 2 max ) = m C L S Centripetal force balance: Lsin(φ) = mv 2 R mv 2 R = 1 2 ρv 2 SC L sin(φ) = m 1 S 1 2 ρc L sin(φ) g 1 2 ρ(v 2 C L ) max mg mv 2 R

21 Lateral Trim (II) Minimum turn radius: R min = m S ρc Lmax sin(φ max ) It depends on: φ max, m S C L max The maximum bank angle is hard to predict Depends on Vmax, which in turn depends on propulsion system Controllability (death spiral) Structural integrity (g s < g_max) 30 ~ 40 degrees is a reasonable max roll angle Banking represents a disturbance in longitudinal dynamics Needs to be compensated with trim Elevator (increase CL) Similar to adding weight

22 Linearized Dynamics Limited to a small region (what does small mean?) In practice, nonlinear dynamics bear great resemblance We can gain a lot of insight by studying dynamics in the vicinity a flight condition We can separate into longitudinal and lateral dynamics (If aircraft and flight condition are symmetric)

23 Longitudinal Static Stability Static stability Does pitching moment increase when AoA increases? If so, then divergent pitch motion (a.k.a statically unstable) L w (α + Δα) Restoring moment L w (α) Divergent moment CG ahead CG behind CG needs to be ahead of quarter chord! As CG goes forward, static margin increases, but more elevator deflection is required for trim and trim drag increases

24 Longitudinal Dynamics Longitudinal modes 1. Short period 2. Phugoid Short period: Weather cock effect of horizontal tail Naturally highly damped Dynamics is on AoA Short period Phugoid Phugoid: Exchange of potential and kinetic energy (up- >speed down, down-> speed up) Lightly damped, but slow Causes bouncing around trim conditions Damping depends on drag: low drag, low damping! How do we stabilize/damp it? Propeller dynamics: as a first order lag Idea for Phugoid damper design: reduced 2 nd order longitudinal system

25 Lateral Dynamics Lateral-Directional modes: 1. Roll subsidence 2. Dutch roll 3. Spiral Dutch roll Roll subsidence: Naturally highly damped Rolling in honey effect Dutch roll: Oscillatory motion Usually stable, and sometimes lightly damped Exchange between yaw rate, sideslip and roll rate Spiral: Usually unstable, but slow enough to be easily stabilized Dutch Roll and Spiral stability are competing factors Dihedral and vertical tail volume dominate these Roll subsidence Spiral

26 Vortex Lattice Codes Good at predicting inviscid part of attached flow around moderate aspect ratio lifting surfaces Represents potential flow around a wing by a lattice of horseshoe vortices

27 VLM Codes (II) Viscous drag on a wing, can be added for with strip theory Calculate local Cl with VLM Calculate 2D Cd(Cl) either from a polar plot of airfoil Add drag force in the direction of the local velocity Usually not included: Fuselage can be roughly accounted by adding a + lifting surface Propeller downwash VLMs can roughly predict: Aerodynamic performance (L/D vs CL) Stall speed (CLmax) Trim relations Stability Derivatives Linear control system design Nonlinear Flight simulation (non-dimensional aerodynamics is linear, but dimensional aerodynamics are nonlinear and EOMs are nonlinear)

28 VLM Codes (III) AVL: Reliable output Viscous strip theory No GUI cumbersome to define geometry XFLR: Reliable output Viscous strip theory GUI to define geometry Good analysis and visualization tools Tornado I ve had some discrepancies when validating against AVL Written in Matlab QuadAir Good match with AVL Written in Matlab Easy to define geometries Viscous strip theory soon Originally intended for flight simulation, not aircraft design Very little native visualization and performance analysis tools