Aircraft Flight Dynamics & Vortex Lattice CodesRoberto!A.Bunge! AA241X April142014 StanfordUniversity!!!!! Aircraft Flight Dynamics & Vortex Lattice Codes AA241X, April 14 2014, Stanford

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


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)


Equations of Motion

�  For Aircraft:

X =

xyzuvwθφ

ψ

pqr

!

"

#################

$

%

&&&&&&&&&&&&&&&&&

δ =

δeδtδaδr

!

"

####

$

%

&&&&

position

velocity

attitude

Angular velocity

elevator

throttle

aileron

rudder


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 )


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


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


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


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


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


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 )


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 '


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


Longitudinal Trim

�  Simple wing-tail system

L_wing

L_tail mg

h_cg h_tail

M_wing


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


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


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!


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!


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)


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


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)


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


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


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


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


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)


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


Documents

Aircraft Flight Dynamics & Vortex Lattice CodesRoberto!A.Bunge! AA241X April142014 StanfordUniversity!!!!! Aircraft Flight Dynamics & Vortex Lattice Codes AA241X, April 14 2014, Stanford