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Roberto A . Bunge
AA241X
Apr i l 14 2014 S tanford Univers i ty
Aircraft Flight Dynamics & Vortex Lattice Codes
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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 14 2014, Stanford University Roberto A. Bunge
Equations of Motion
� For Aircraft:
X =
xyzuvwθφ
ψ
pqr
!
"
#################
$
%
&&&&&&&&&&&&&&&&&
δ =
δeδtδaδr
!
"
####
$
%
&&&&
position
velocity
attitude
Angular velocity
elevator
throttle
aileron
rudder
AA241X, April 14 2014, Stanford University Roberto A. Bunge
Equations of Motion (II)
� Dynamics Eqs* Linear Acceleration = Aero + gravity + Gyro Angular Acceleration = Aero + Gyro
*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 14 2014, Stanford University Roberto A. Bunge
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
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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 ¡ 2nd Order Phugoid approximation
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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 non-
dimensional form
� Non-dimensional EOMs:
¡ Non-dimensional dynamic states:
¡ Non-dimensional time:
¡ Non-dimensional moments of inertia
u = uV
v = vV≈ β
w = wV≈α
p = pb2V
q = qc2V
r = rb2V
I xx =8IxxρSb3
t = Vtc
I yy =8IyyρSc3
AA241X, April 14 2014, Stanford University Roberto A. Bunge
Non-dimensional Aerodynamics
� Aerodynamic Forces & Moments
� Stability & Control Derivatives: ÷ First derivative of non-dimensional forces and moments w.r.t. non-
dimensional states & control, at a flight condition ÷ There are potentially 6x6 SD & 6x4 CD. In practice, many are negligible ÷ Examples:
Cx =X
12ρV 2S
Cy =Y
12ρV 2S
Cz =Z
12ρV 2S
Cl =L
12ρV 2Sb
Cm =M
12ρV 2Sc
Cn =N
12ρV 2Sb
CZα=∂CZ
∂α
Clp=∂Cl
∂p
Cmq=∂Cm
∂q
CZδe=∂CZ
∂δe
Clδa=∂Cl
∂δa
Cnδe=∂Cn
∂δrAA241X, April 14 2014, Stanford University Roberto A. Bunge
Non-dimensional EOMs
� Example: linearized* roll equation non-dimensional form
I xx !p =Clpp+Clr
r +Clββ +Clδa
δa+Cldrδr
Ixx !p = L
* Gyroscopic terms fall out, because they are 2nd order
AA241X, April 14 2014, Stanford University Roberto A. Bunge
Non-dimensional mapping
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
AA241X, April 14 2014, 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)
� 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!
Aircraft EOM
Xtrim
δtrim
!X = 0
!X = f (Xtrim,δtrim ) = 0 Xtrim = gtrim (δtrim )
AA241X, April 14 2014, 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
� 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
Aircraft EOM (near Trim)
Xtrim + X'
δtrim +δ'
!X = ∂f∂X
X ' +∂f∂δδ '
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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
Xdesired
δtrimg−1trimAircraft
EOM
X
δ
!X∫
Linear Regulator Controller
δ '+
+
+
-
X
AA241X, April 14 2014, Stanford University Roberto A. Bunge
Longitudinal Trim
� Simple wing-tail system
L_wing
L_tail mg
h_cg h_tail
M_wing
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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 14 2014, Stanford University Roberto A. Bunge
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 14 2014, 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 14 2014, 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 14 2014, Stanford University Roberto A. Bunge
Lateral Trim
� Aircraft in constant turn: � Vertical force balance:
� Centripetal force balance:
L
mg
φ
mV 2
R
Lcos(φ) =mg
→ cos(φ) = mS
g12ρV 2CL
⇒ cos(φmax ) =mS
g12ρ(V 2CL )max
Lsin(φ) = mV2
R
⇒ R = mV 2
12ρV 2SCL sin(φ)
=mS
112ρCL sin(φ)
AA241X, April 14 2014, Stanford University Roberto A. Bunge
Lateral Trim (II)
� Minimum turn radius:
� It depends on:
� 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
Rmin =mS
112ρCLmax
sin(φmax )
φmax,mS&CLmax
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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)
AA241X, April 14 2014, 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)
� 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 14 2014, Stanford University Roberto A. Bunge
Longitudinal Dynamics
� Longitudinal modes 1. Short period 2. Phugoid
� Short period: ¡ Weather cock effect of horizontal tail ¡ Naturally highly damped ¡ Dynamics is on AoA
� 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
2nd order longitudinal system
Short period
Phugoid
AA241X, April 14 2014, Stanford University Roberto A. Bunge
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
Dutch roll
Spiral Roll subsidence
AA241X, April 14 2014, 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 14 2014, 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 14 2014, 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 14 2014, Stanford University Roberto A. Bunge
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