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 sin(φ) cos(φ) Izzr! = n + (Ixx − Iyy )pq 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 Deriva ves 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. Bunge Longitudinal Trim (III)
L Force balance* γ 1 2 mg = L cos(γ) ≈ L = 2 ρV SC α L θ mg V ⇒ V 2 = γ 1 ρSC (δe ) 2 L trim T D Trim Elevator defines trim Velocity! mg
T = D + Lsin(γ) ≈ D + Lγ T 1 ⇒ γ = (δttrim ) − mg (L ) D (δetrim )
Elevator & Thrust both define Gamma!
*Assuming small Gamma AA241X, April 13 2015, Stanford University Roberto A. Bunge 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!
AA241X, April 13 2015, Stanford University Roberto A. Bunge 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!
AA241X, April 13 2015, Stanford University Roberto A. Bunge Turning Maneuver
φ Centripetal force balance: L mV 2 Lsin(φ) = R mV 2 = mVε! mV 2 m 1 R ⇒ R = = 1 2 1 2 ρV SCL sin(φ) S 2 ρCL sin(φ) mg
Minimum turn radius: Minimum%Turn%Radius% (wing%loading%=%35%g/dm^2)% m 1 60" Rmin = S 1 C sin( ) 50" 2 ρ Lmax φmax 40"
30" CL"="0.6" m CL"="0.8" turn%radius%(m)% φ , & C 20" CL"="1.0" Depends on: max Lmax S 10"
0" 0" 10" 20" 30" 40" 50" 60" 70" 80" roll%angle%(deg)%
AA241X, April 13 2015, Stanford University Roberto A. Bunge 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:
Iˆ q!ˆ = C (α, λ )+C qˆ +C (δ −δ ) xx m 0 mq mδe e e0
¡ Assume that before the turn we have trimmed the aircraft in level flight at the desired alfa (CL): ⇒ C δ = C (α, λ ) mδe e0 m 0
AA241X, April 13 2015, Stanford University Roberto A. Bunge Turning Maneuver III
φ The pitch rate is the projection of the turn rate L onto the pitch axis: V q = ε!sin(φ) = sin(φ) q = ε!sin(φ) R qc csin(φ) V ⇒ qˆ = = ε! = 2V 2R R
To maintain the same alfa (CL), extra elevator Extra%elevator%deflec.on%required%in% is required to counter the pitch rate a%turn%to%maintain%CL%=%0.8% 12" 10" csin(φ) 8" C 6" δe = − mq 2R 4" 2" 0" 0" 5" 10" 15" 20" 25" 30" 35" 40" 45" To take advantage of elevator throw, horizontal Exra%elevtor%deflec.on%(deg)% tail incidence has set appropriately, otherwise Turning%radius% turning ability might be limited
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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 + X ∂f ' ∂f ' trim X! = X + δ Aircraft EOM ∂X ∂δ (near Trim)
' δtrim +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”
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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
AA241X, April 13 2015, Stanford University Roberto A. Bunge Longitudinal Static Stability
Static stability ¡ Does pitching moment increase when AoA increases? ¡ If so, then divergent pitch motion (a.k.a statically unstable) L (α + Δα) w Restoring Divergent moment moment Lw (α)
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
AA241X, April 13 2015, Stanford University Roberto A. Bunge Longitudinal Dynamics
Longitudinal modes Short period 1. Short period 2. Phugoid Phugoid Short period: ¡ Weather cock effect of horizontal tail ¡ Usually highly damped, if you have a tail ¡ Dynamics is on AoA
Phugoid: ¡ Exchange of potential and kinetic energy (up- >speed down, down-> speed up) ¡ Lightly damped, but slow ¡ Causes “bouncing” around pitch trim conditions g ω ph ≈ 2 ¡ Damping depends on drag: low drag, low Vo damping! 1 C ¡ How can we stabilize/damp it? ζ ≈ D0 ph C 2 L0 Propeller dynamics: as a first order lag Idea for Phugoid damper design: reduced 2nd order longitudinal system
AA241X, April 13 2015, Stanford University Roberto A. Bunge Lateral Dynamics
Lateral-Directional modes: Dutch roll 1. Roll subsidence 2. Dutch roll 3. Spiral
Roll subsidence: Roll subsidence 1 2 ¡ Naturally highly damped 2 ρVoSb Spiral σ roll ≈ Cl ¡ “Rolling in honey” effect p Ixx
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 1 2 ρV Sb Cl σ ≈ 2 o (C −C r ) spiral I nr nβ C zz 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 AA241X, April 13 2015, Stanford University Roberto A. Bunge 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
AA241X, April 13 2015, Stanford University Roberto A. Bunge 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)
AA241X, April 13 2015, Stanford University Roberto A. Bunge 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
AA241X, April 13 2015, Stanford University Roberto A. Bunge 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)
AA241X, April 13 2015, Stanford University Roberto A. Bunge