aero elasticity 01

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Introduction to Aeroelasticity

Lecture 1:

Introduction Equations of

motion

G. Dimitriadis

Aeroelasticity


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Introduction

! Aereolasticity is the study of the interaction of inertial,structural and aerodynamic forces on aircraft, buildings,surface vehicles etc

Inertial Forces

Structural Forces Aerodynamic Forces

DynamicAeroelasticity

Structural dynamics Flight Dynamics

Static Aeroelasticity


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Why is it important?! The interaction between these three

forces can cause several undesirablephenomena:! Divergence (static aeroelastic

phenomenon)! Flutter (dynamic aeroelastic phenomenon)! Limit Cycle Oscillations (nonlinear

aeroelastic phenomenon)

! Vortex shedding, buffeting, galloping(unsteady aerodynamic phenomena)


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Static Divergence

Flat plate wing in transonic tunnel before

wind is turned on

Flat plate wing in transonic tunnel withwind on - the plate is bent and touches

the tunnel wall


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Flutter

Flutter experiment: Winglet underfuselage of a F-16. Slow Mach

number increase.

The point of this experiment wasto predict the flutter Mach

number from subcritical test data

and to stop the test before flutteroccurs.


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More LCOs

Stall flutter of a wing at an angle of

attack

Torsional LCO of a rectangle


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Even more LCOs

Galloping of a bridge deck

Torsional oscillations of a bridgedeck


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Many more LCOs


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These phenomena do not

occur only in the lab

Glider Limit CycleOscillations

Tacoma NarrowsBridge Flutter

Various phenomena


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Even on very expensive kit


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How to avoid these

phenomena?

!Aeroelastic Design (Divergence, Flutter,Control Reversal)

! Wind tunnel testing (Aeroelastic scaling)! Ground Vibration Testing (Complete

modal analysis of aircraft structure)

! Flight Flutter Testing (Demonstrate thatflight envelope is flutter free)


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Wind Tunnel Testing

! scale F-16 flutter model

F-22 buffetTest model


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Ground Vibration TestingGVT of F-35 aircraft

Space Shuttle horizontal GVT

GVT of A340


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Flight Flutter Testing


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So what is in the course?

! Introduction to Aeroelastic modeling! Modeling of static aeroelastic issues and

phenomena:! Divergence, control effectivenes, control reversal,

! Modeling of dynamic aeroelastic phenomena:! Flutter

! Practical Aeroelasticity:!Aeroelastic design

!Ground Vibration Testing, Flight Flutter Testing

! Non-aircraft Aeroelasticity


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A bit of history! The first ever flutter incident occurred on the

Handley Page O/400 bomber in 1916 in theUK.

! A fuselage torsion mode coupled with anantisymmetric elevator mode (the elevatorswere independently actuated)

! The problem was solved by coupling theelevators


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More history

! Control surface flutter became afrequent phenomenon during World War

I.

! It was solved by placing a mass balancearound the control surface hinge line


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Historic examples! Aircraft that experienced aeroelastic

phenomena! Handley Page O/400 (elevators-fuselage)! Junkers JU90 (fluttered during flight flutter test)! P80, F100, F14 (transonic aileron buzz)! T46A (servo tab flutter)! F16, F18 (external stores LCO, buffeting)! F111 (external stores LCO)! F117, E-6 (vertical fin flutter)

! Read Historical Development of Aircraft Flutter, I.E.Garrick, W.H. Reed III, Journal of Aircraft, 18(11),897-912, 1981


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F117 crash


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Aeroelastic Modeling

! Aircraft are very complex structures withmany modes of vibration and can exhibit verycomplex fluid-structure interactionphenomena

! The exact modeling of the aeroelasticbehaviour of an aircraft necessitates thecoupled solution of:! The full compressible Navier Stokes equations! The full structural vibrations equations

! As this is very difficult, we begin withsomething simpler:


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Pitch Plunge AirfoilTwo-dimensional, two degree-offreedom airfoil, quite capable of

demonstrating most aeroelasticphenomena.

= pitch degree of freedom

h= plunge degree of freedomxf= position of flexural axis(pivot)

xc= position of centre of mass

Kh= plunge spring stiffnessK!= pitch spring stiffness

In fact, we will simplify evenfurther and consider a flat plate

airfoil (no thickness, no camber)


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Structural Model

! There are two aspects to eachaeroelastic models

!A structural model!An aerodynamic model

! In some cases a control model is addedto represent the effects of actuators andother control elements

! Develop the structural model


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Structural Model Details

! Use total energy conservation

xf

xc

dx

dy

x

!h


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Kinetic Energy

! The total kinetic energy is given by

where


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Potential Energy

! The potential energy is simply theenergy stored in the two springs, i.e.

! Notice that gravity can be convenientlyignored

! Total energy=kinetic energy+potentialenergy


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Equations of motion (1)

! The equations of motion can beobtained by inserting the expression for

the total energy into Lagrangesequation


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Equations of motion (2)

! This should yield a set of two equationsof the form

! or,where

Qis a vector of external forces


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Aerodynamic model

! The possible aerodynamic models dependson flow regime and simplicity

! In general, only four flow regimes areconsidered by aeroelasticians:! Incompressible! Subsonic! Transonic! Supersonic

! For the moment we will deal only withincompressible modeling


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Incompressible, Unsteady

AerodynamicsOscillating airfoils leave behind them astrong vortex street. The vorticity in the wake

affects the flow over the airfoil:

The instantaneous aerodynamic forcesdepend not only on the instantaneous

position of the airfoil but also on the position

and strength of the wake vortices.

This means that instantaneous aerodynamicforces depend not only on the current motion

of the airfoil but on all its motion history fromthe beginning of the motion.


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Wake examples (Pitch)

Pitching airfoil-Low frequency

Pitching airfoil-High frequency


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Wake examples (Plunge)

Plunging airfoil-Low amplitude

Plunging airfoil-High amplitude


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Quasi-steady aerodynamics

! The simplest possible modeling consists ofignoring the effect of the wake

! Quasi-steady models assume that there areonly four contributions to the aerodynamicforces:! Horizontal airspeed U, at angle !(t) to airfoil!Airfoil plunge speed,! Normal component of pitch speed,! Local velocity induced by the vorticity around the

airfoil, vi(x,t)


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Lift and moment

xf

c/4

!

MxfL

h

The aerodynamicforce acting on

the wing is the liftand it is placed on

the quarter chord(aerodynamic

centre). There isalso anaerodynamic

moment acting

around theflexural axis.

NB: The lift is

defined positive

downwards


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Lift coefficient

! The airfoil is uncambered but thepitching motion causes an effective

camber with slope

! From thin airfoil theory, cl=2!(A0+A1/2),where


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Lift coefficient (2)

! Substituting all this into the equation forthe lift coefficient and carrying out the

integrations we get

! This is the total circulatory lift acting onthe airfoil. There is another type of lift

acting on it, which will be presented in abit.


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Moment coefficient! The moment coefficient around the

leading edge (according to thin airfoil)

theory is given by cm=-cl/4-!(A1-A2)/4

! Therefore, the moment coefficientaround the flexural axis is given by

cmxf=cm+xfcl/c

! Substituting and integrating yields


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Added Mass

! Apart from the circulatory lift and moment, theair exerts another force on the airfoil.

! The wing is forcing a mass of air around it tomove. The air reacts and this force is knownas the added mass effect.

! It can be seen as the effort required to movea cylinder of air with mass "#b2 where b=c/2.

! This force causes both lift and momentcontributions


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Full lift and moment

These are to be substituted into the structural equations of motion:


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Full aeroelastic equations of

motion! The equations of motion are second order,

linear, ordinary differential equations.

! Notice that the equations are of the form! And that there are mass, damping and

stiffness matrices both aerodynamic and

structural


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Pitch-plunge equations of

motion

These are the full equations of motion for the pitch-plunge airfoil withquasi-steady aerodynamics. We will investigate them in more detail now.


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Static Aeroelasticity

! First, we will study the static equilibriumof the system.

!Static means that all velocities andaccelerations are zero.

! The equations of motion become


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Aerodynamic Coupling (2)! This phenomenon is calledaerodynamic coupling. Changing the

pitch angle causes a change in theplunge.

! This is logical since increased pitchmeans increased lift.! However, if we apply a force Fon the

flexural axis, the equations become


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Aerodynamic Coupling (3)

! The second equation can only be satisfied if!=0

! The first equation then gives h=F/Kh!

In this case, there is no aerodynamiccoupling. Increasing the plunge does notaffect the pitch.

! This is not the general case. The pitch-plungemodel ignores 3D aerodynamic effects

! In real aircraft bending and torsion are bothcoupled.


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Static Divergence (2)

! Static divergence in pitch occurs when themoment caused by the lift around the flexuralaxis is higher than the structural restoringforce.

! For every pitch stiffness there is an airspeedabove which static divergence will occur.

! If this airspeed is outside the flight envelopeof the aircraft, this is not a problem.

! The pitch-plunge model does not allow forstatic divergence in plunge.

! Again, this is because it ignores 3D effects.


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Windmills

Upminster Windmill (1803) Bircham Windmill (1846)Pili windmill in Kos (1800)

The position of the flexural axis is very important for windmills

aero elasticity 01

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