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7/29/2019 Aero Elasticity 01
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Introduction to Aeroelasticity
Lecture 1:
Introduction Equations of
motion
G. Dimitriadis
Aeroelasticity
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
More LCOs
Stall flutter of a wing at an angle of
attack
Torsional LCO of a rectangle
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Introduction to Aeroelasticity
Even more LCOs
Galloping of a bridge deck
Torsional oscillations of a bridgedeck
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Introduction to Aeroelasticity
Many more LCOs
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Introduction to Aeroelasticity
These phenomena do not
occur only in the lab
Glider Limit CycleOscillations
Tacoma NarrowsBridge Flutter
Various phenomena
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Introduction to Aeroelasticity
Even on very expensive kit
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
Wind Tunnel Testing
! scale F-16 flutter model
F-22 buffetTest model
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Introduction to Aeroelasticity
Ground Vibration TestingGVT of F-35 aircraft
Space Shuttle horizontal GVT
GVT of A340
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Introduction to Aeroelasticity
Flight Flutter Testing
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
Structural Model Details
! Use total energy conservation
xf
xc
dx
dy
x
!h
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Introduction to Aeroelasticity
Kinetic Energy
! The total kinetic energy is given by
where
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
Equations of motion (1)
! The equations of motion can beobtained by inserting the expression for
the total energy into Lagrangesequation
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
Full lift and moment
These are to be substituted into the structural equations of motion:
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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
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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Introduction to Aeroelasticity
Windmills
Upminster Windmill (1803) Bircham Windmill (1846)Pili windmill in Kos (1800)
The position of the flexural axis is very important for windmills