roberto!a.bunge aa241x april132015...
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Roberto A . Bunge
AA241X
Apr i l 13 2015 S tanford Univers i ty
Aircraft Flight Dynamics
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
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Equations of Motion
� Dynamical system is defined by a transition function, mapping states & control inputs to future states
Aircraft
EOM
X
δ
!X
!X = f (X,δ)
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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States and Control Inputs
X =
xyzuvwθφ
ψ
pqr
!
"
#################
$
%
&&&&&&&&&&&&&&&&&
position
velocity
attitude
angular velocity
δ =
δeδtδaδr
!
"
####
$
%
&&&&
elevator
throttle
aileron
rudder
AA241X, April 13 2015, Stanford University Roberto A. Bunge
There are alternative ways of defining states and control inputs
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Full Nonlinear EOM
� Dynamics Eqs.* Linear Acceleration = Aero + gravity + Gyro Angular Acceleration = Aero + Gyro
*Assuming calm atmosphere and symmetric aircraft (Neglecting cross-products of inertia
� Kinematic Eqs.: Relation between position and velocity Relation between attitude and angular velocity
Ixx !p = l + (Iyy − Izz )qrIyy !q =m+ (Izz − Ixx )prIzz !r = n+ (Ixx − Iyy )pq
!x!y!z
!
"
###
$
%
&&&= NRB
(φ,θ ,ψ )
uvw
!
"
###
$
%
&&&
• System of 12 Nonlinear ODEs
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)
!φ = p+ qsin(φ)tan(θ )+ rcos(φ)tan(θ )!θ = qcos(φ)− rsin(φ)
!ψ = q sin(φ)cos(θ )
+ r cos(φ)cos(θ )
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
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Flight Dynamics and Navigation Decoupling
Roberto A. Bunge AA241X, April 13 2015, Stanford University
Xdyn =
uvwθφ
pqr
!
"
##########
$
%
&&&&&&&&&&
δdyn =
δeδtδaδr
!
"
####
$
%
&&&&
Flight Dynamics
Xnav =
xyzψ
!
"
#####
$
%
&&&&&
δnav =
uvwθφ
pqr
!
"
##########
$
%
&&&&&&&&&&
Navigation
Flight Dyn. δdyn Xdyn = δnav
Nav. Xnav
� Flight Dynamics is the “inner dynamics” � Navigation is “outer dynamics”: usually what we care about
The full nonlinear EOMs have a cascade structure
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Longitudinal & Lateral Decoupling
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� 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)
For a symmetric aircraft near a symmetric flight condition, the Flight Dynamics can be further decoupled in two independent parts
Xlon =
uwθ
q
!
"
####
$
%
&&&&
δlon =δeδt
!
"#$
%&
Longitudinal Dynamics
Xlat =
vφ
pr
!
"
####
$
%
&&&&
δlat*=δaδr
!
"#
$
%&
Lateral-Directional Dynamics
* As we shall see, throttle also has effects on the lateral dynamics, but these can be eliminated with appropriate aileron and rudder
Lon.
Lat.
Xlon
Xlat
δlon
δlat
Flight Dynamics
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Alternative State Descriptions
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� 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
u =V cos(α)cos(β)v =V sin(β)w =V sin(α)cos(β)
CL = ao(α −αLo)
γ =θ −α
Transformations:
CL ≈ −m
12ρS
accelzV 2
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Simplified Models
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� 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
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
Many more models! …. …. …. ….
x, y,ε
V N
E
ε
!ε!x =Vo sin(ε)!y =Vo cos(ε)
x, y, z,ε !ε,γ !x =Vo sin(ε)cos(γ )!y =Vo cos(ε)cos(γ )
!z = −Vo sin(γ )
x,y,z: North, East, Down position coordinates : course over ground ε
x, y, z,γ,ε V,CL,φ!ε = 1
2ρ mS−1VCL sin(φ)
!γ = 12ρ m
S−1VCL cos(φ)−
gcos(γ )V
x, y, z,γ,ε,V T,CL,φ !V =Tm−gmsin(γ )− 1
2ρ mS−1V 2CD (CL )
“Everything should be made as simple as possible, but not more” (~A. Einstein)
D
V γ
ε = !ε dt∫ !ε ≤ VoRmin
γ ≤ γmax
V ≤VmaxCL <CLmax
φ ≤ φmax
T ≤ Tmax
+ dynamics + variables + details
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Aerodynamics
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� 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 = 12ρV 2SCL
dyn. pressure size Aircraft and flow geometry, and Reyonolds
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Aerodynamics II
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� The dimensionless forces and moments are a function of:
i. Aircraft geometry (fixed): AR, taper, dihedral, etc.
ii. Control surface deflections
iii. Relative flow angles: iv. Reynolds number:
� 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
α ≈wV, β ≈
vV, λ =
VΩR
, p = pb2V, q = qc
2V, r = rb
2V
δe,δa,δr
CL,CD,CY ,CT ,Cl,C m,Cn
Re = ρcVµ
if the variation of speeds is small, it can be assumed constant and factored out
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Stability and Control Derivatives
Roberto A. Bunge AA241X, April 13 2015, Stanford University
Stability Derivtaives Control Deriva1ves Nonlinear/Trim
CL
CD
CY
Cl
Cm
Cn
CT
CLq CLδe
CDq
Cmq
!α β p rq δe δa δr α λ
CT (λ)
CL (α,λ)
CD (α,λ)
Cm (α,λ)
CnβCnp
Cnr
Cmδe
CnδaCnδr
Cmα
ClβClp
CYβ
CDα
CLα
CDδe
CYr
Clr
CYδr
ClδaClδr
Cn (λ)
Cl (λ)
are small angular deflections w.r.t. a zero position, usually the trim deflection
δ
is an angle of attack perturbation around !α
α
~zero
Minor importance
Estimate via calculations
Estimate via calculations or flight testing
Estimate or trim out via flight testing
Hard to estimate
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Stability and Control Derivatives II
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� Examples of force and moment expressions:
� Example dimensionless pitching equation*:
Cl =Clpp+Clδa
δa +Clββ +Clδr
δr
CL =CL (α0,λ)+CLα!α +CLq
q+CLδe(δe −δe0 )
Cm =Cm (α0,λ)+Cmα!α +Cmq
q+Cmδe(δe −δe0 )
Lift:
Rolling moment:
Pitching moment:
I xx !q =Cm =Cm (α0,λ)+Cmα!α +Cmq
q+Cmδe(δe −δe0 )
* Neglecting gyroscopic terms
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Aerodynamics IV
Roberto A. Bunge AA241X, April 13 2015, Stanford University
Bihrle, et. al. (1978) LSPAD, Selig, et. al. (1997)
Zilliac (1983)
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Trim Analysis
� Flight conditions at which if we keep controls fixed, the aircraft will remain at that same state (provided no external disturbances)
� 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
� 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
Aircraft EOM
Xtrim
δtrim
!X = 0
!X = f (Xtrim,δtrim ) = 0 Xtrim = gtrim (δtrim )
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
Xdesired δtrimTrim Relations/Tables
Aircraft EOM
δ
Linear Regulator Controller
δ '+
+
+
-
X
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Longitudinal Trim
� Simple wing-tail system
L_wing
L_tail mg
h_cg h_tail
M_wing
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Longitudinal Trim (II)
� Moment balance: 0 =Mwing − hCGLwing + xtailLtail
→ 0 =1 2ρV 2 cwingSwingCmwing− hCGSwingCLwing
+ htailStailCLtail#$
%&
⇒hCGcwing
CLwing(αtrim ) =
htailStailcwingSwing
CLtail(αtrim,δetrim )−Cmwing
Elevator trim defines trim AoA, and consequently trim CL
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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Longitudinal Trim (III)
� Force balance*
L
D mg
T γ
γ
V θ αmg = Lcos(γ ) ≈ L = 1
2ρV 2SCL
⇒V 2 =mg
12ρSCL (δetrim )
T = D+ Lsin(γ ) ≈ D+ Lγ
⇒ γ =T(δttrim )mg
−1
(LD)(δetrim )
Trim Elevator defines trim Velocity!
Elevator & Thrust both define Gamma! *Assuming small Gamma
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
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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
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Turning Maneuver
� Centripetal force balance:
� Minimum turn radius:
� Depends on:
L
mg
φ
mV 2
R = mV !εLsin(φ) = mV
2
R
⇒ R = mV 2
12ρV 2SCL sin(φ)
=mS
112ρCL sin(φ)
AA241X, April 13 2015, Stanford University Roberto A. Bunge
Rmin =mS
112ρCLmax
sin(φmax )
φmax,mS&CLmax
0"
10"
20"
30"
40"
50"
60"
0" 10" 20" 30" 40" 50" 60" 70" 80"
turn%ra
dius%(m
)%
roll%angle%(deg)%
Minimum%Turn%Radius%(wing%loading%=%35%g/dm^2)%
CL"="0.6"
CL"="0.8"
CL"="1.0"
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Turning Maneuver II
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� 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:
¡ Assume that before the turn we have trimmed the aircraft in level flight at the desired alfa (CL):
I xx !q =Cm (α,λ0 )+Cmqq+Cmδe
(δe −δe0 )
⇒Cmδeδe0 =Cm (α,λ0 )
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Turning Maneuver III
Roberto A. Bunge AA241X, April 13 2015, Stanford University
� The pitch rate is the projection of the turn rate onto the pitch axis:
� To maintain the same alfa (CL), extra elevator
is required to counter the pitch rate
� To take advantage of elevator throw, horizontal
tail incidence has set appropriately, otherwise turning ability might be limited
δe = −Cmq
csin(φ)2R
q = !ε sin(φ) = VRsin(φ)
⇒ q = qc2V
=csin(φ)2R
L φ
q = !ε sin(φ)!ε = V
R
0"
2"
4"
6"
8"
10"
12"
0" 5" 10" 15" 20" 25" 30" 35" 40" 45"Exra%elevtor%defl
ec.o
n%(deg)%
Turning%radius%
Extra%elevator%deflec.on%required%in%a%turn%to%maintain%CL%=%0.8%
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Linearized Dynamics Analysis
� Many flight dynamic effects can be analyzed & explained with Linearized Dynamics
� Most of the times we linearize dynamics around Trim conditions
� 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”
Aircraft EOM (near Trim)
Xtrim + X'
δtrim +δ'
!X = ∂f∂X
X ' +∂f∂δδ '
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
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Longitudinal Static Stability
� Static stability ¡ Does pitching moment increase when AoA increases? ¡ If so, then divergent pitch motion (a.k.a statically unstable)
� 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
Lw (α)
Lw (α +Δα)
CG ahead CG behind
Restoring moment
Divergent moment
AA241X, April 13 2015, Stanford University Roberto A. Bunge
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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
� 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
2nd order longitudinal system
Short period
Phugoid
AA241X, April 13 2015, Stanford University Roberto A. Bunge
ω ph ≈ 2 gVo
ζ ph ≈12CD0
CL0
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Lateral Dynamics
� Lateral-Directional modes: 1. Roll subsidence 2. Dutch roll 3. Spiral
� 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
� Note: see “Flight Vehicle Aerodynamics”, Ch. 9 for more details
Dutch roll
Spiral Roll subsidence
AA241X, April 13 2015, Stanford University Roberto A. Bunge
σ roll ≈12ρVoSb
2
IxxClp
σ spiral ≈12ρVoSb
2
Izz(Cnr
−Cnβ
Clr
Clβ
)
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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
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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)
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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
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Recommended Readings
Roberto A. Bunge AA241X, April 13 2015, Stanford University
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)