Aircraft Flight Dynamics

Transcription

1 Aircraft Flight Dynamics AA241X April Stanford University

2 1. Equations of motion Full Nonlinear EOM Decoupling of EOM Simplified Models 2. Aerodynamics Dimensionless coefficients Stability Control Derivatives 3. Trim Analysis Level, climb and glide Turning maneuver Overview 4. Linearized Dynamics Analysis Longitudinal Lateral

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 States and Control Inputs! 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 There are alternative ways of defining states and control inputs

5 Full Nonlinear EOM 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 *Assuming calm atmosphere and symmetric aircraft (Neglecting crossproducts of inertia 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)

6 Nonlinearity and Model Uncertainty Sources of nonlinearity: Trigonometric projections (dependent on attitude) Gyroscopic effects Aerodynamics Dynamic pressure Reynolds dependencies Stall partial separation Model uncertainties: 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 Flight Dynamics and Navigation Decoupling X dyn = u v w θ φ p q r! " $ % δ dyn = δ e δ t δ a δ r! " $ % Flight Dynamics X nav = x y z ψ! " $ % δ nav = u v w θ φ p q r! " $ % Navigation Flight Dyn. δ dyn X dyn = δ nav Nav. X nav Flight Dynamics is the inner dynamics Navigation is outer dynamics : usually what we care about The full nonlinear EOMs have a cascade structure

8 Longitudinal Lateral Decoupling For a symmetric aircraft near a symmetric flight condition, the Flight Dynamics can be further decoupled in two independent parts Flight Dynamics δ lon Lon. X lon δ lat Lat. X lat Longitudinal Dynamics! u $ w X lon = θ " q %! δ lon = δ $ e " % δ t Lateral-Directional Dynamics! v $ φ X lat = p " r %! δ lat * = δ $ a " % δ r * As we shall see, throttle also has effects on the lateral dynamics, but these can be eliminated with appropriate aileron and rudder Although usually used in perturbational (linear) models, many times this decoupling can also be used for nonlinear analysis (e.g. symmetric flight with large vertical motion)

9 Alternative State Descriptions Translational dynamics: 1. {u, v, w}: most useful in 6 DOF flight simulation 2. {V, alfa, beta}: easiest to describe aerodynamics Longitudinal dynamics: 1. {V, alfa, theta, q}: conventional description 2. {V, CL, gamma, q}: best for nonlinear trajectory optimization 3. {V, CL, theta, q}: all states are measurable, more natural for controls Transformations: u = V cos(α)cos(β) v = V sin(β) w = V sin(α)cos(β) C L = a o (α α Lo ) C L γ =θ α m 1 2 ρs accel z V 2

10 Simplified Models Many times we can neglect or assume aspects of the system and look at the overall behavior Its important to know what you want to investigate N ε V V γ x,y,z: North, East, Down position coordinates ε : course over ground E D Model States Controls EOM Constraints 2 DOF navigation+ 1 DOF point mass 3 DOF navigation + 1 DOF point mass 3 DOF navigation + 2 DOF point mass 3 DOF navigation + 3 DOF point mass x, y,ε!ε x, y, z,ε!ε,γ x, y, z,γ,ε V,C L,φ!x = V o sin(ε)!y = V o cos(ε)!x = V o sin(ε)cos(γ)!y = V o cos(ε)cos(γ)!ε = 1 2 ρ m 1 S VC L sin(φ)!z = V o sin(γ)!γ = 1 2 ρ m 1 S VC L cos(φ) gcos(γ) V x, y, z,γ,ε,v T,C L,φ!V = T m g m sin(γ) 1 2 ρ m S 1 V 2 C D (C L ) Many more models!.... ε =!ε dt!ε V o R min γ γ max V V max C L < C Lmax φ φ max T T max + dynamics + variables + details Everything should be made as simple as possible, but not more (~A. Einstein)

11 Aerodynamics In the full nonlinear EOM aerodynamic forces and moments are: Given how experimental data is presented, and to separate different aerodynamic effects, its easier to use: Dimensional analysis allows to factor different contributions: Dynamic pressure Aircraft size Aircraft geometry Relative flow angles Reynolds number X,Y, Z,l, m, n L, D,Y,T,l, m, n L = 1 2 ρv 2 SC L dyn. pressure size Aircraft and flow geometry, and Reyonolds

12 Aerodynamics II The dimensionless forces and moments C L,C D,C Y,C T,C l,c m,c n are a function of: i. Aircraft geometry (fixed): AR, taper, dihedral, etc. ii. Control surface deflections δ e,δ a,δ r iii. Relative flow angles: iv. Reynolds number: α w V, β v V, λ = V pb, ˆp = ΩR 2V, Re = ρcv µ qc ˆq = 2V, if the variation of speeds is small, it can be assumed constant and factored out rb ˆr = 2V Alfa and lambda: dependence is nonlinear and should be preserved if possible The rest can be represented with linear terms (Stability and Control Derivatives) At low AoA some stability derivatives depend on alfa, and at high angles of attack all are affected by alf

13 Stability and Control Derivatives CL Stability Derivtaives Control Deriva1ves Nonlinear/Trim α! β ˆp ˆq ˆr δ e δ a δ r α λ C Lα C Lq C Lδe C L (α, λ) δ are small angular deflections w.r.t. a zero position, usually the trim deflection CD C Dα C Dq C Dδe C D (α, λ) α! is an angle of attack perturbation around α CY C Yβ C Yr C Yδr ~zero Cl C lβ C lp C lr C lδa C lδr C l (λ) Minor importance Estimate via calculations Cm C mα C mq C mδe C m (α, λ) Estimate via calculations or flight testing Cn C nβ C np C nr C nδa C nδr C n (λ) Estimate or trim out via flight testing Hard to estimate CT C T (λ)

14 Stability and Control Derivatives II Examples of force and moment expressions: Lift: Pitching moment: Rolling moment: C L = C L (α 0, λ)+ C Lα α! + C Lq ˆq + C Lδe (δ e δ e0 ) C m = C m (α 0, λ)+ C mα α! + C m q ˆq + C mδe (δ e δ e0 ) C l = C lp ˆp + C lδa δ a + C lβ β + C lδr δ r Example dimensionless pitching equation*: Î xx!ˆq = Cm = C m (α 0, λ)+ C mα α! + C m q ˆq + C mδe (δ e δ e0 ) * Neglecting gyroscopic terms

15 Aerodynamics IV Zilliac (1983) Bihrle, et. al. (1978) LSPAD, Selig, et. al. (1997)

16 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! If internal dynamics are stable, then flight condition converges on trim condition

17 An idea: Trim + Regulator Controller I. Inverse trim: set control inputs that will take us to the desired state II. Regulator: to stabilize modes and bring us back to desired trim state in the presence of disturbances X desired Trim Relations/Tables δ trim + δ ' + δ Aircraft EOM X Linear Regulator Controller + -

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

19 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

20 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

21 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!

22 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!

23 Turning Maneuver Centripetal force balance: φ L Lsin(φ) = mv 2 R mv 2 R = 1 2 ρv 2 SC L sin(φ) = m S ρc L sin(φ) mg mv 2 R = mv!ε Minimum turn radius: R min = m S ρc Lmax sin(φ max ) 60" 50" Minimum%Turn%Radius% (wing%loading%=%35%g/dm^2)% Depends on: φ max, m S C L max turn%radius%(m)% 40" 30" 20" CL"="0.6" CL"="0.8" CL"="1.0" 10" 0" 0" 10" 20" 30" 40" 50" 60" 70" 80" roll%angle%(deg)%

24 Turning Maneuver II What are the constraints on maximum turn? 1. Elevator deflection to achieve high CL in a turn 2. Do we care about loosing altitude? 3. Maximum speed and thrust 4. Controls: maneuver can be short lived, so high bandwidth is require for tracking tracking 1. Roll tracking, etc. 2. Sensor bandwidth 5. Maximum G-loading 6. Maximum CL and stall 7. Aerolasticity of controls at high loading Elevator to achieve CL: The pitching moment balance equation in dimensionless form: Î xx!ˆq = Cm (α, λ 0 )+ C mq ˆq + C mδe (δ e δ e0 ) Assume that before the turn we have trimmed the aircraft in level flight at the desired alfa (CL): C mδe δ e0 = C m (α, λ 0 )

25 Turning Maneuver III The pitch rate is the projection of the turn rate onto the pitch axis: q =!ε sin(φ) = V R sin(φ) φ L q =!ε sin(φ) ˆq = qc 2V = csin(φ) 2R!ε = V R To maintain the same alfa (CL), extra elevator is required to counter the pitch rate δ e = C mq To take advantage of elevator throw, horizontal tail incidence has set appropriately, otherwise turning ability might be limited csin(φ) 2R Exra%elevtor%deflec.on%(deg)% 12" 10" 8" 6" 4" 2" Extra%elevator%deflec.on%required%in% a%turn%to%maintain%cl%=%0.8% 0" 0" 5" 10" 15" 20" 25" 30" 35" 40" 45" Turning%radius%

26 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 Provide stability in the face of uncertainty in different dynamic parameters They help in rejecting disturbances They can also help in going from one trim state to the another, provided they are not too far away

27 Linearized Dynamics Limited to a small region (what does small mean?) Especially due to trigonometric projections and nonlinear alfa dependences 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) Linearized models also provide some information about trim relations

28 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

29 Longitudinal Dynamics Longitudinal modes 1. Short period 2. Phugoid Short period: Weather cock effect of horizontal tail Usually highly damped, if you have a tail 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 pitch trim conditions Damping depends on drag: low drag, low damping! How can we stabilize/damp it? Propeller dynamics: as a first order lag Idea for Phugoid damper design: reduced 2 nd order longitudinal system ω ph 2 g V o ζ ph 1 2 C D0 C L0

30 Lateral Dynamics Lateral-Directional modes: 1. Roll subsidence 2. Dutch roll 3. Spiral Dutch roll Roll subsidence: Naturally highly damped Rolling in honey effect σ roll 1 2 ρv o Sb 2 I xx C lp Roll subsidence Spiral 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 σ spiral 1 2 ρv o Sb 2 C (C nr C lr nβ ) I zz C lβ Dutch Roll and Spiral stability are competing factors Dihedral and vertical tail volume dominate these Note: see Flight Vehicle Aerodynamics, Ch. 9 for more details

31 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

32 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)

33 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

34 Recommended Readings 1. Fundamentals of Flight, Shevell Big picture of Aerodynamics, Flight Dynamics and Aircraft Design 2. Dynamics of Flight, Etkin Very good development of trim and linearized flight dynamics and aerodynamics. Some ideas for control 3. Flight Vehicle Aerodynamics, Drela Great mix between real world and mathematical aerodynamics and flight dynamics. No controls. Ch. 9 very clear and useful development of linearized models 4. Automatic Control of Aircraft and Missiles, Blakelock In depth description of flight EOMs and many ideas for linear regulators 5. Low-speed Aerodynamics, Plotkin Katz Great book on panel methods (only if you want to write your own panel code)