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Aircraft Flight Dynamics Aircraft Flight Dynamics Roberto A. Bunge AA241X April 13 2015 Stanford University AA241X, April 13 2015, Stanford University Roberto A. Bunge Overview 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 4. Linearized Dynamics Analysis ¡ Longitudinal ¡ Lateral AA241X, April 13 2015, Stanford University Roberto A. Bunge Equations of Motion Dynamical system is defined by a transition function, mapping states & control inputs to future states X Aircraft X! EOM δ X! = f (X,δ) AA241X, April 13 2015, Stanford University Roberto A. Bunge States and Control Inputs ! x $ # & # y & position # z & # & # u & !δe $ elevator # v & velocity # & # & #δt & throttle # w & δ = X = # θ & #δa & aileron # φ & attitude # & # & "δr % rudder # ψ & # p & # & # q & angular velocity # & r "# %& There are alternative ways of defining states and control inputs AA241X, April 13 2015, Stanford University Roberto A. Bunge Full Nonlinear EOM • System of 12 Nonlinear ODEs Dynamics Eqs.* Kinematic Eqs.: Linear Acceleration = Aero + gravity + Gyro Relation between position and velocity mu! = X − mgsin(θ)+ m(rv − qw) ! x! $ ! u $ # & # & y! = N RB v mv! = Y + mgsin(φ)cos(θ)+ m(pw − ru) # & (φ,θ,ψ ) # & # z! & # w & mw! = Z + mgcos(φ)cos(θ)+ m(qu − pv) " % " % Angular Acceleration = Aero + Gyro Relation between attitude and angular velocity φ! = p + qsin(φ)tan(θ)+ r cos(φ)tan(θ) Ixx p! = l + (Iyy − Izz )qr θ! = qcos(φ)− rsin(φ) Iyyq! = m + (Izz − Ixx )pr I r! n (I I )pq ! sin(φ) cos(φ) zz = + xx − yy ψ = q + r cos(θ) cos(θ) *Assuming calm atmosphere and symmetric aircraft (Neglecting cross- products of inertia AA241X, April 13 2015, Stanford University Roberto A. Bunge 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 AA241X, April 13 2015, Stanford University Roberto A. Bunge Flight Dynamics and Navigation Decoupling The full nonlinear EOMs have a cascade structure X X δdyn dyn = δnav nav Flight Dyn. Nav. Flight Dynamics Navigation ! u $ # & ! u $ v # & # & ! $ v # w & δe ! $ # & # & x # w & # & δ # & θ # t & # y & # & X = # & δdyn = X θ dyn φ # & nav = δ = # & # & δa # z & nav φ # & # & # & # p & # & # & "δr % "# ψ %& p q # & # & # q & # r & # & " % " r % Flight Dynamics is the “inner dynamics” Navigation is “outer dynamics”: usually what we care about AA241X, April 13 2015, Stanford University Roberto A. Bunge 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 δlon Lon. Xlon δlat Lat. Xlat Longitudinal Dynamics Lateral-Directional Dynamics !u $ !v $ # & # & φ !δa $ w !δe $ # & # & Xlat = δlat * = # & Xlon = δlon = # & # & # & p "δr % θ "δt % # & # & r "q % " % * 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) AA241X, April 13 2015, Stanford University Roberto A. Bunge Alternative State Descriptions Translational dynamics: Transformations: 1. {u, v, w}: most useful in 6 DOF flight simulation u = V cos(α)cos(β) 2. {V, alfa, beta}: easiest to describe v = V sin(β) aerodynamics w = V sin(α)cos(β) Longitudinal dynamics: 1. {V, alfa, theta, q}: conventional C a ( ) description L = o α −αLo 2. {V, CL, gamma, q}: best for nonlinear trajectory optimization γ =θ −α 3. {V, CL, theta, q}: all states are m accelz measurable, more natural for controls CL ≈ − 1 2 2 ρS V AA241X, April 13 2015, Stanford University Roberto A. Bunge Simplified Models ε Many times we can neglect or assume aspects of the system N V V and look at the overall behavior γ Its important to know what you want to investigate x,y,z: North, East, Down position coordinates ε : course over ground E D Model States Controls EOM Constraints x! V sin( ) 2 DOF navigation+ = o ε V x, y,ε ε = ∫ ε!dt ε! ≤ o 1 DOF point mass ε! y! = Vo cos(ε) Rmin 3 DOF navigation + x! = V sin(ε)cos(γ) x, y, z,ε o 1 DOF point mass ε!,γ z! = −Vo sin(γ) γ ≤ γ max y! = Vo cos(ε)cos(γ) + dynamics −1 V ≤ V + variables 3 DOF navigation + ε! = 1 ρ m VC sin(φ) max 2 S L + details 2 DOF point mass x, y, z,γ,ε C < C V,C ,φ L Lmax L 1 m −1 gcos(γ) γ! = 2 ρ S VCL cos(φ)− V φ ≤ φmax 3 DOF navigation + T g 1 ! 1 m − 2 x, y, z,γ,ε,V T,CL,φ V = − sin(γ)− 2 ρ V C (C ) T ≤ Tmax 3 DOF point mass m m S D L Many more models! …. …. …. …. “Everything should be made as simple as possible, but not more” (~A. Einstein) AA241X, April 13 2015, Stanford University Roberto A. Bunge Aerodynamics In the full nonlinear EOM aerodynamic forces and moments are: X,Y, Z,l, m, n Given how experimental data is presented, and to separate different aerodynamic effects, its easier to use: L, D,Y,T,l, m, n Dimensional analysis allows to factor different contributions: ¡ Dynamic pressure ¡ Aircraft size 1 2 ¡ Aircraft geometry L = 2 ρV SCL ¡ Relative flow angles dyn. pressure size Aircraft and flow geometry, and Reyonolds ¡ Reynolds number AA241X, April 13 2015, Stanford University Roberto A. Bunge 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 δe,δa,δr w v V pb qc rb iii. Relative flow angles: α ≈ , β ≈ , λ = , pˆ = , qˆ = , rˆ = V V ΩR 2V 2V 2V ρcV if the variation of speeds is small, it can be iv. Reynolds number: Re = assumed constant and factored out µ 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 AA241X, April 13 2015, Stanford University Roberto A. Bunge Stability and Control Derivatives Stability Derivtaives Control Derivaves Nonlinear/Trim α! β pˆ qˆ rˆ δe δa δr α λ δ are small angular deflections w.r.t. a zero position, usually the trim CL C CL CL Lα q δe CL (α, λ) deflection CD C α ! is an angle of attack CD Dq CD CD (α, λ) α δe perturbation around α CY C C C Yβ Yr Y δr ~zero Minor importance Cl C C C C Cl C (λ) lβ lp lr lδa δr l Estimate via calculations Cm C C Cm Estimate via calculations mα mq δe Cm (α, λ) or flight testing Estimate or trim out via flight testing Cn Cn Cn Cn Cn Cn Cn (λ) β p r δa δr Hard to estimate CT CT (λ) AA241X, April 13 2015, Stanford University Roberto A. Bunge Stability and Control Derivatives II Examples of force and moment expressions: Lift: C = C (α , λ)+C α!+C qˆ +C (δ −δ ) L L 0 Lα Lq Lδe e e0 Pitching moment: C = C (α , λ)+C α!+C qˆ +C (δ −δ ) m m 0 mα m q mδe e e0 Rolling moment: C = C pˆ +C δ +C β +C δ l lp lδa a lβ lδr r Example dimensionless pitching equation*: Iˆ q!ˆ = C = C (α , λ)+C α!+C qˆ +C (δ −δ ) xx m m 0 mα m q mδe e e0 * Neglecting gyroscopic terms AA241X, April 13 2015, Stanford University Roberto A. Bunge Aerodynamics IV Zilliac (1983) Bihrle, et. al. (1978) LSPAD, Selig, et. al. (1997) AA241X, April 13 2015, Stanford University Roberto A. Bunge Trim Analysis Flight conditions at which if we keep controls fixed, the aircraft will remain at that same state (provided no external disturbances) Xtrim ! Aircraft X = 0 EOM δtrim 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 g ( ) X = f (Xtrim,δtrim ) = 0 trim = trim δtrim 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 AA241X, April 13 2015, Stanford University Roberto A. Bunge 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 δtrim + Trim δ Aircraft X Relations/Tables EOM + δ ' Linear Regulator Controller + - AA241X, April 13 2015, Stanford University Roberto A. Bunge Longitudinal Trim Simple wing-tail system L_wing h_cg h_tail M_wing L_tail mg AA241X, April 13 2015, Stanford University Roberto A. Bunge Longitudinal Trim (II) Moment balance: 0 = M wing − hCG Lwing + xtail Ltail → 0 =1 2ρV 2 #c S C − h S C + h S C % $ wing wing mwing CG wing Lwing tail tail Ltail & h h S CG C ( ) tail tail C ( , e ) C ⇒ Lwing αtrim = Ltail αtrim δ trim − mwing cwing cwingSwing Elevator trim defines trim AoA, and consequently trim CL AA241X, April 13 2015, Stanford University Roberto A.
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