Download - Coupled Nonlinear Aeroelasticity and Flight Dynamics · PDF file27.08.2014 · Aeroelasticity and Structural Dynamics Research Laboratory Coupled Nonlinear Aeroelasticity and Flight

Aeroelasticity and Structural Dynamics Research Laboratory

Coupled Nonlinear Aeroelasticity and Flight Dynamics of Highly Flexible

AircraftWeihua Su

Assistant ProfessorDepartment of Aerospace Engineering and Mechanics

The University of AlabamaTuscaloosa, AL

Applied Modeling & Simulation Seminar Series NASA Ames Research Center

Moffett Field, CAAug. 27, 2014


Overview

• Introduction– Background– Motivation

• Theoretical formulation– Geometrically nonlinear beam– Unsteady aerodynamics– Flight dynamic modeling

• Numerical studies• Concluding remarks• Ongoing and future developments

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Aerodynamic Efficiency andWing Aspect-Ratio

3

F-22aAR = 2.36

ETA (Germany)AR = 51.33

B787-8AR = 11.08

0

1

or ln WLR ED W

Large wing aspect-ratio to achieve high aerodynamic efficiency

High aerodynamic efficiency


What about Structural Design?

• U.S. Air Force Sensorcraft studies– High-altitude, long-endurance– Unmanned vehicles– Sensor platform– Very high fuel fractions (up to 60%)

• Very light structures– Not necessarily carry fuel, but…

4

AeroVironment’s Helios>24 hrs

0

1

or ln WLR ED W

Highly flexible aircraft

Low structural weight fraction

High-aspect-ratio wings

+


What’s Challenging?

• Large wing deformation– Linear solution might not be sufficient– Nonlinear solution needed

• Coupling between wing oscillation and rigid-body motion– Coupled transient response– Body freedom flutter

• Other effects– Low Reynolds flights– Local transonic effects

5

Need an integral solution for nonlinear aeroelasticity + flight dynamics


Objectives

• Create a low-order aeroelastic and flight dynamic framework– Effectively represent dynamic behavior of highly flexible vehicle– Efficient solution– Facilitate active aeroelastic tailoring and control studies

• Explore structural, aerodynamic, and control techniques to enhance flight efficiency and performance– Reduce drag– Reduce power consumption– Suppress instability– Reject air disturbance

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Coupled Nonlinear Aeroelasticity and Flight Dynamics

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CoupledAeroelasticity &Flight Dynamics

• Reduced-order• Wing bending / twist • Airfoil camber d.o.f.

Geometrically Nonlinear BeamGeometrically

Nonlinear Beam

Structural DynamicsStructural Dynamics

• Composites • Active materials

Cross-Sectional Analysis


Solid Mechanics

Solid Mechanics

• Potential flow theory• 3-D corrections• Stall models• Gust/turbulence models

Finite-State Inflow TheoryFinite-State

Inflow Theory

AerodynamicsAerodynamics

Rigid-Body DynamicsRigid-Body Dynamics

Flight Dynamics

Flight Dynamics

A simplified aeroelastic/flight dynamics simulation system


Reduced-Order Structural Modeling

• From 3D elastic problem to 2D beam cross-sectional analysis and 1D beam model

• Dimensional reduction using the Variational-Asymptotic Method:– Active thin-walled solution (mid-line discretization)– VABS (finite-element discretization)– User defined stiffness constants

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X YZX YZ

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

x x

x x

y y

z z

F K K K KM K K K KM K K K K

K K K KM

Cross-Section Stiffness and Inertial Properties


Basic Coordinate Systems

• Global frame (G)• Body frame (B) – origin not

necessary to be C.G. of vehicle• Body frame motion variables

• Local beam frame (w)• Auxiliary local frame (b)

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B

B

B B

B B

B B

B B

pb

p vb

p vb


Strained-Based Geometrically Nonlinear Beam Formulation

• Geometrically nonlinear beam formulation[1]

• Four local strain degrees-of-freedom (ε): extension, twist, flatwise bending, and chordwise bending

• Constant-strain elements• Capture large complex deformations with fewer

elements – computationally efficient• Isotropic and anisotropic constitutive relations

10

[1] Su, W., and Cesnik, C.E.S., “Strain-Based Geometrically Nonlinear Beam Formulation for Modeling Very Flexible Aircraft,” International Journal of Solids and Structures, Vol. 48, No. 16-17, 2011, pp. 2349-2360. (doi: 10.1016/j.ijsolstr.2011.04.012)

Sample element deformations with constant strain

Strains () and body velocities ()are independent variables


Formulation Based on Principle of Virtual Work

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0

( ) ( ) ( , , ) ( , , ) 0( ) ( ) ( , , ) ( , , ) 0 0

0

FF FB FF FB FFF

BF BB BF BB B

TT TF p F disthFF

TTB pbhb

M M C C RKM M C C Rb

R JJ JKNg B F

R JJ

T TpM dist pt ptTT Tpbb b

J JB M F M

JJ J

0ii

W

Equations of Motion

Generalized Mass Generalized Damping Generalized Stiffness

Generalized Force

Internal VW External VW

+

• Inertia force, internal strain, and strain rate• Gravity loads, distributed loads, and point

loads


Recovery of Nodal Displacement

• Solution of displacement-strain equation:

• Marching kinematics in complete aircraft

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0( ) ( )0 0

( ) ( ) ( )

( ) A s s s G s

h s A s h ss

h s e h e h

Prescribed root

s0, h0

s, h(t)

B Frame

B.C.


Unsteady Aerodynamics

• 2-D Theodorsen-like unsteady aerodynamics (Peters et al., 94, 95)

• Glauert expansion of inflow velocityas function of inflow states, λn

• Finite state differential equation is transformed to independent variables and

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2 2 201

2 2 2 20 4

12 22

1 28

mc

mc

zl b z y d by b d bc yy y y

m b b yz dy y b c y

Inflow velocity

01

12 n n

n

b

1 2 3 4 1 2 3E E z E E F F F


Finite-State Inflow Theory: Modifications

• Aerodynamic coefficient modifications based on XFoil (Re effects) or CFD calculations

• Compressibility accounted for by Prandtl-Glauert correction

• Spanwise aerodynamic corrections(3-D effects)

• Simplified stall model

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Additional aerodynamic development in progress


• Fixed region in space• Amplitude distribution

– Peak at center and zero at boundary– Possibly different distribution in East

and North directions– Smooth transition

• Time variation: 1-cosine withdifferent temporal durations

Discrete Non-uniform Gust Model15

2 21( , , ) 1 cos 2 cos sin2 c E N

g

tA r t A A At

00 0

( ) sin 1 , ( ) sin 1 , 02 2

E Nn n

E Nr rA r A r r rr r


Dryden Gust Model

• Gust PSD function

• ωm: Frequency component (rad/s)• U0: Free stream velocity (m/s)• Lw: Scale of turbulence (m), determined by altitude (m)• Superposition of all frequency components with random phase

16

22

0

22

00

1 3

( )

1

w mw w

w m

w m

LLU

LUU

22

0

22

00

2 1 12

( )

1 4

w mw w

w m

w m

LLU

LUU

1

( ) ( ) cos( )w m m mm

w t t

MIL-F-8785C MIL-HDBK-1797


PSD and Time History of Gust Velocity

• Frequency band [0.1~6] Hz• adjusted to obtain enough

wing deformation• Uniform spanwise distribution

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w

Power concentrated at the lowfrequency range


Flight Dynamics Modeling18

The trajectory and orientation of a fixed body reference frame, B, at point O, which in general is not the aircraft’s center of mass


Full Air Vehicle Model for Flight Simulations

• Elastic equations of motion

• Finite-state 2-D unsteady aerodynamics

• Body reference frame propagation

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( ) ( ) ( )M C , , K R , , , , , , ,ub

1 2 3F F F

Strains (4 by m structural d.o.f.)

Body velocities (6 flight dynamic d.o.f.)

Inflow states (N by m aerodynamic d.o.f.)

Control inputs

12

0GBBP C

Inertial velocities (6 d.o.f.)

Frame orientation(4 quaternions)


Numerical Studies

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Flutter of Constrained Vehicle

• Similar to constrained wind-tunnel model (no body DOFs)• Fixed root angle of attack (8 deg)• Free stream velocity 1% higher than flutter speed

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Coupled out-of-plane bending/torsion/in-plane bending mode


Blended-Wing-Body (BWB) Model

• Properties inspired from HiLDA (High Lift over Drag Active Wing) wind-tunnel model

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Elevon: 25% chord


Comparison of Flutter Modes with Rigid-Body Constraints

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All cases trimmed for 6,096 m (20,000 ft) altitude, same fuel condition

Fully constrained rigid-body DOFs

Additional plunge DOF

With pitch and plunge DOFs (“same” for free flight – 6 DOFs)

Flutter Speed Frequency

Fully constrained

dof’s172.52 m/s 7.30 Hz

+ plunging 164.17 m/s 7.07 Hz

+ pitching and plunging 123.17 m/s 3.32 Hz

Free flight 123.20 m/s 3.32 Hz

Traditional wind-tunnel setup maybe non-conservative – need rigid-body DOFs in the aeroelasticanalyses, simulations, and tests


Highly Flexible Flying Wing Model

• Representative of Helios prototype[2]

– Five engines and three pods– Payloads applied at center pod– Empty gross mass: 726 kg

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[2] Patil, M.J., and Hodges, D.H., “Flight Dynamics of Highly Flexible Flying Wings,” Journal of Aircraft, Vol. 43, No. 6, 2006, pp. 1790-1798.


Trim Results and Flight Stability

• Speed: 12.2 m/s at sea level; Payload: 0 – 227 kg (at center pod)• Linearization about each trimmed condition with increase of payloads• Root locus for phugoid mode (left: flexible, right: rigid) • Unstable phugoid mode for payload > 152 kg

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Payload

[2]

Flexible Rigid

Payload

Zero payload: span-loaded

Full payload: center-loaded

Nonlinear aeroelastic/flight dynamic characteristics dependent on trim conditions


Non-symmetric Gust Input and Response –Fully-Loaded Configuration

• Payload: 227 kg; gust region radius: 40 m;maximum gust center amplitude: 10 m/s

• Non-symmetric discrete gust distribution:– gusts mainly applied on right wing

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2 s gust duration 4 s gust duration 8 s gust duration

Gust duration impacts after-gust flight path


Instantaneous Vehicle Positions and Orientations

• Positions and orientations at 0, 5, 12, 18, 24, and 30 s, respectively

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Flight Direction

8-s gust4-s gust

2-s gust

Illustration of unstable Phugoid mode


Animation of Vehicle Motion with Gust Perturbations

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2-s gust4-s gust8-s gust


Concluding Remarks

• Framework for modeling and analyzing highly flexible aircraft– Coupled nonlinear aeroelastic/flight dynamic simulation– Strain-based geometrically-nonlinear beam– Incompressible unsteady aerodynamics (with compressibility corrections

and stall models)– Rigid-body flight dynamics

• Highly flexible aircraft have radically different behavior than conventional aircraft– Coupling between aircraft deformation and rigid-body motions changes

flutter boundaries– Flutter boundary in free flight condition may be different from constrained

flight– Finite amplitude gust can excite instabilities

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Concluding Remarks (Cont’d)

• What did we learn from the physics of highly flexible aircraft?– Operating (trim) condition should be the basis in weight, structural,

and stability analyses• Deformed geometry other than the undeformed shape

– Traditional linear solution to highly flexible aircraft aeroelasticity might not be sufficient

• Nonlinear solution is required– Coupling between aeroelasticity and flight dynamics needs to be

considered• Aeroelastic models should incorporate the rigid-body motion, and vice

versa• Individual solutions might not be appropriate

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Active Aeroelastic Tailoring and Control

• Traditional approach for aerodynamic/flight control

• Drag due to control surfaces• Conformal wing shape changes

– Integral strain actuation of bending/twist

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NASA Langley MFC Actuator


Wing Camber Change

• NASA VCCTEF

• Jointly proposed by UA/GA Tech/OSU/MSU– Full variable camber wing

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Recent Development

• Wing cross-sectional warping– Plate-like modeling capability with beam model– Augmented EoM with camber degrees (finite-section modes)

• Impact on aeroelasticity, flight dynamics,and control -> on-going

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Shell S-Beam, no Warping S-Beam, warping

( ) ( , , )nn n n q

b

RM q C q K q R

b R

Camber shape control for higher flight efficiency


Linear Strain Modes

• Approximate solutions using strain modes

• Modes from elastic EOM

• Only take the elastic components of the modes

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( , ) ( ) ( )s t s t

0 00 0 0

FF FB FF

BF BB

M M KM M b

FC

B

( ) Fs


Modal Equations35

1 2 3

FF FB FF FB FF F

BF BB BF BB B

M M C C K R

M M C C R

F F F

1 1 2 2 3

( , , , , )

( , , , , )FF FB FF FB FF F

BF BB BF BB B

F B F B

M M C C K R

M M C C R

F F F F F


Highly Flexible Wing

• Beam properties:

• Nonlinear flutter speed: 23 m/s

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Ref. 3* Current (linear) Current (nonlinear)

Velocity (m/s) 32.2 32.2 23.3Frequency (Hz) 3.60 3.60 1.61

[3] Patil, M.J., Hodges, D.H. and Cesnik, C.E.S., “Nonlinear Aeroelasticity and Flight Dynamics of High-Altitude Long-Endurance Aircraft,” Journal of Aircraft, Vol. 38, No. 1, 2001, pp. 88-94.

Length (m) 16Chord (m) 1Mass per length (kg/m) 0.75x-sectional c.g. position 50% chord x-sectional shear center 50% chord Rotational inertia (kg·m) 0.1Flat bending rigidity (N·m2) 2.00 × 104

Edge bending rigidity (N·m2) 4.00 × 104

Torsional rigidity (N·m2) 1.00 × 104


Modal-Based Static Solution

• Convergence of static solutions with different number of modes

• For more discussion:

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Modes about undeformed shape Modes about deformed shape

2% error

0.5% error 0.5% error

Fewer modes required if modes are obtained about deformed shape

[4] Su, W., and Cesnik, C.E.S., “Strain-Based Analysis for Geometrically Nonlinear Beams: a Modal Approach,” Journal of Aircraft, Vol. 51, No. 3, 2014, pp. 890–903. (doi: 10.2514/1.C032477)


Multi-disciplinary Simulation of Flight Vehicles38

Closed-loopAeroelasticity &Flight Dynamics

• Reduced-order• Wing bending / twist • Airfoil camber d.o.f.

Geometrically Nonlinear BeamGeometrically

Nonlinear Beam

Structural DynamicsStructural Dynamics

• Piezoelectric actuator

Bending/Torsion Actuation

Bending/Torsion Actuation

Electro-Mechanics

Electro-Mechanics

• Composites• Active materials



Solid Mechanics

Solid Mechanics

• Optimal wing shape• Stability control• Rejection of air

disturbance• Trajectory control

Control AlgorithmControl

Algorithm

ControlControl

• Potential flow theory• 3-D corrections• Stall models• Gust/turbulence models

Finite-State Inflow TheoryFinite-State

Inflow Theory

AerodynamicsAerodynamics

Rigid-Body DynamicsRigid-Body Dynamics

Flight Dynamics

Flight Dynamics

Wing Morphing Mechanism

Wing Morphing Mechanism

Wing MorphingActuation

Wing MorphingActuation

An active aero-servo-elastic simulation system


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