Introduction 1&2 2015

NANYANG TECHNOLOGICALUNIVERSITY, SINGAPORE

Tuesday, April 18, 2023

SCHOOL OF MECHANICAL AND AEROSPACE ENGINEERING

Introduction to Structural Dynamics and Aeroelasticity

Lectures 1 & 2 1/45

MA4704 - Aeroelasticity

Part I – Structural Vibrations, Static Aeroelasticity and Unsteady Aerodynamics

A/Prof Lin Rongming

Part II – Dynamic Aeroelasticity and Flutter

Prof Zhao Dan

Contact A/P Lin Rongming: Room N3.2-02-80,

Tel.: 6790 4728, E-mail: [email protected]

Consultations: Fridays 10 - 1pmCourse Website: http://edventure.ntu.edu.sg





Lectures 1 & 2 2/45

Assessments:

(1) Continual Assessment: 50% Two (2) project assignments one for Part I

and another for Part II, each contributes 25% towards overall final grade.

(2) Final Open-Book Exam: 50%

Text Book:The lecture notes are developed based on the book – Introduction to Structural

Dynamics and Aeroelasticity, Dewey H. Hodges and G. Alvin Pierce, Cambridge University Press, 2002





Lectures 1 & 2 3/45

What is Aeroelasticity ?Aeroelasticity is the study of the interactions of inertial, structural and aerodynamic forces and their effect on the structural and control performances of aircraft, space vehicles, high speed trains, etc.

Inertial Forces

StructuralForces

AerodynamicForcesStatic Aeroelasticity

DynamicAeroelasticity

Flight Dynam

icsSt

ruct

’al D

ynam

ics





Lectures 1 & 2 4/45

Why Is Aeroelasticity Important ?

The interactions of these three type of forces can lead to some major undesirable structural phenomena: Divergence (static aeroelasticity);

Flutter (dynamic aeroelasticity);

Limit Cycle Oscillations (nonlinear dynamic aeroelasticity);

Vortex shedding, buffeting and galloping (unsteady aerodynamics and aeroelasticity).





Lectures 1 & 2 5/45

Static Divergence - Wind Tunnel Test:

Wing deflection increases as airspeed increases until failure, divergence speed is lower than flutter speed here.





Lectures 1 & 2 6/45

A6 Wing Flutter - Wind Tunnel Test:

Wing vibration increases quickly at certain airspeed. Where does the wing vibration energy come from ??





Lectures 1 & 2 7/45

Limit Cycle Oscillations (LCOs):

Rectangular wing with pitch and plunge degrees of freedom. Sustained oscillation from initial disturbances.





Lectures 1 & 2 8/45

These have occurred in well controlled lab experiments, what about in practice?

PA–30 Tail Flutter observed while in flight.





Lectures 1 & 2 9/45

What is the easiest way to break a glider’s wings?

By flying it at its flutter speed !





Lectures 1 & 2 10/45

And they occur even on very expensive kit !

F-16 limit cycle oscillations observed while in flight.






How to avoid these phenomena in practice ?

These are usually carried out in practice to eliminate possible aircraft aeroelastic problems:

Structural Dynamics Design (Resonances, Damping);

Aeroelastic Design (Divergence, Flutter, Control Reversal);

Wind Tunnel Testing (Aeroelastic scaling);

Ground Vibration Testing (Modal analysis of aircraft);

Flight Flutter Testing (demonstrate that flight envelope is safe and is flutter free).






1/4 Scale F-16 Flutter Model F-22 Buffet Test Model

Wind Tunnel Testing:Wind Tunnel Testing has become an essential and integral part of modern aerospace design. Scaled or actual models are tested to evaluate their aerodynamic performances.






Ground Vibration Testing GVT: GVT and

modal analysis are carried out to establish vibration properties and to validate design calculation.

Space Shuttle GVTA340 GVT

GVT of F-35 Aircraft






Flight Flutter Testing:






A380 Flight Flutter Testing:

Final Flight Flutter Testing is needed to clear the aircraft of any flutter within the flight envelope before certification.






What are there in this course ? Introduction to Aeroelastic Modeling; Vibration Modeling of Continuous Aircraft Structures

-- Wing modeling, bending and torsional vibrations; Vibration Measurement and Modal Analysis -- To extract modal natural frequencies and modal damping from vibration test data; Modeling of Static Aeroelastic Phenomena

-- Divergence, Control Effectiveness, Control Reversal; Unsteady Aerodynamics Modeling

-- To derive lift and moment for flutter analysis; Modeling of Dynamic Aeroelastic Phenomena

-- Flutter, k-method, p-k method; Practical Aspects of Aeroelasticity

-- Aeroelastic Design, Flight Flutter Testing.






A bit of History:

The first ever flutter incident occurred on the Handley Page O/400 bomber in 1916 in UK.

A fuselage torsion mode was coupled (with very close natural frequencies) with an anti-symmetric elevator mode (the elevators were independently actuated).

The problem was solved by coupling the elevators through a torsion tube to increase the natural frequency of the anti-symmetric elevator mode.






Aileron

wing

Hinge line

Balancingmass

A bit more of History:

Control surface flutter became a frequent phenomenon during World War I.

It was solved through trial and error by placing a mass balance around the control surface hinge line.

The added mass increases the rotational mass moment of inertial and shifts the aileron resonant frequency away from flutter region.






Historical 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 – nonlinear flutter)

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






Aeroelastic Modeling:

Aircraft are very complex structures with many modes of vibration and can exhibit very complex fluid-structure interaction phenomena including divergence, flutter, LCO (nonlinear flutter), buffeting etc.

The exact modeling of these aeroelastic behaviour of an aircraft necessitates the coupled solution of:

-- The full structural vibration equations; -- The full compressible Navier Stokes equations.

As this is very difficult, we begin with something simpler – a 2DOF pitch plunge airfoil. Such a simple model has been shown to have captured most of the essential behaviour of practical complex aeroelastic systems.






The 2DOF pitch plunge airfoil:

pitch degree of freedom, h plunge degree of freedoma position of flexural axis, c chord lengthk plunge spring stiffness, kT pitch spring stiffness

x

z

OkT

Ak

h

c/2 a

I, m

Flexural axis AMass centre Ca negative A ahead of C

U tthh

C






Structural Modeling:

There are always two aspects to each aeroelastic modeling:

-- A Structural Model

-- An Aerodynamic Model

In some cases a control model is added to represent the effects of actuators and other control elements.

To develop a structural model, Lagrange’s energy method is often used. We use the method here to derive equations of motion (EOMs) for the 2DOF pitch plunge airfoil and leave the detailed discussion of the method later.






Kinetic Energy dT :

c/2 a

A

x

x

h

dx

v

hxacv 2/ smallbetoassumed

22/2

1hxacdxdT w






Kinetic Energy T :

Upon integration over the whole chord length of the airfoil, the total kinetic energy becomes,

22

2

1

2

1 IhShmT

where S and I are defined as,

maS

22

12

1amcmI AI






Potential Energy V :

The EOMs can be conveniently obtained based on Lagrange’s method, first define,

22

2

1

2

1 TkhkV

Equations of Motion:

The potential energy is simply the energy stored in the two springs,

,0h

L

h

L

dt

d

0

LL

dt

d

2T

222 k2

1hk

2

1I

2

1hShm

2

1VTL

then,






Upon substituting T and V into those 2 equations, the 2 EOMs can be established to be come,

Equations of Motion:

0 khShm

0 TkhSI

When written in matrix form, these become,

0

0

0

0

h

k

kh

IS

Sm

T






Aerodynamics Modeling:

The possible aerodynamic model to be used depends on flow regime of the problem and the simplicity to be sought.

In general, four flow regimes can be considered by aeroelasticians:

-- Incompressible -- Subsonic -- Transonic -- Supersonic

For the moment and for this course, we will deal only with incompressible modelling and that is adequate for most applications.






Oscillating airfoils leave behind them a strong vortex street. The vorticity in the wake affects the flow over the airfoil:

The instantaneous aerodynamic forces depend not only on the instantaneous position and motion of the airfoil but also on the position and strength of the wake vortices.

This means that instantaneous aerodynamic forces depend not only on the current position and motion of the airfoil but on all its motion history from the beginning of the motion.

Incompressible, Unsteady Aerodynamics:






Examples of Wake:

Wake vortex street from a pitching airfoil.






Examples of Wake:

Wake vortex street from a plunging airfoil.






Quasi-steady Aerodynamics:

Quasi-steady aerodynamic models assume that there are only four contributions to aerodynamic forces:

-- Horizontal airspeed U, at an angle of attack (t) to airfoil;

th-- Airfoil plunge speed, .

xact 2/-- Normal component of pitch speed, . -- Local velocity induced by vorticity around the airfoil.

In Quasi-steady modelling, inertial forces of the air accelerating/decelerating around the airfoil are not considered.






Lift and Moment:

The air moving around the airfoil will produce, based on Quasi-steady model, a lift L and a moment M. The lift L is defined positive upwards as a convention.

c/2 a

A

x

h M

L






x

z

O

P vx)(xzz

Effective camber line

x

vy

Lift and Moment Coefficients:Full rigorous derivation of lift will be discussed later but a simpler explanation is given here from thin airfoil theory.

The airfoil is uncambered but the pitch and plunge motions cause an effective camber as shown.

When examining a fluid particle moving along the camber line (assumed to be a streamline), its horizontal velocity is very much the free stream velocity U while its vertical velocity can be written as,

hxacvy 2/

Uvx

Hence, camber line slope becomes,






x

y

v

v

dtdx

dtdz

dx

dz

/

/ U

hxac

2/

From thin airfoil theory, lift coefficient can then be written as,

2/2 10 AAcl

where A0 and A1 are defined as,

,1

00

ddx

dzA

dndx

dzAn cos

20

cos12

c

xwhere






Substituting all these into the equation for the lift coefficient and carrying out the integration, we have,

U

ta

c

U

thttcl

42

This is the total of circulatory part of the lift acting on the airfoil. There is another portion of the lift called non-circulatory lift acting on the airfoil, which will be presented a bit later.

The moment coefficient can be accordingly derived based on thin airfoil theory,

4/4/ 21, AAcc llem

ccaccc llemAm /2/,,

About leading edge:

About flexural axis:






The coefficient A2 can be computed from the following,

After substituting all these into the lift coefficient and moment coefficient and then using these coefficients to derive the lift and moment, they become where b=c/2,

Circulatory Lift and Moment - Quasi-steady Model:

d

U

hcaA 2cos

cos2/202

(1)

a

chUbULc 4

2

Ucac

hUUcc

aMc

32

16

1

44

1

(2)






Added Inertial Mass Effect:Apart from the circulatory lift and moment whose coefficients have been obtained based on quasi-steady aerodynamics, the air around the airfoil exerts another type of force which is non-circulatory in nature.

The airfoil is forcing a mass of air around it to move (with acceleration), the air reacts and this additional force is known as the added inertial mass effect.

It can be shown later that this force is the effort required to move a cylinder of air with mass b2 where b=c/2.

This force causes both lift and moment contributions.

8/4/ 222 baUachabM nc

haUbLnc 2 (3)

(4)






These are seen by the airfoil as external forces, as a result, the complete EOM becomes,

Full Lift, Moment and Full Aeroelastic EOMs:The full aerodynamic lift and moment become,

(5)

a

chUbUhaUbL

422

Ucac

hUUcc

a 32

16

1

44

1

(6)

8/4/ 222 baUachabM

M

Lh

k

kh

IS

Sm

T 0

0

(7)






Full Aeroelastic EOMs:

Recall that the aerodynamic lift and moment are functions of pitch and plunge motions and upon substituting the lift and moment into (7), we have,

h

baa

ab

h

IS

Sm

)( 8/

122

2

hac

acUc

caca )2/)(4/()()(

4/

4/1

0

0

4/(0

10

0

0

)2

h

accU

h

k

k

T(8)






Characteristics of the Full Aeroelastic EOMs:1. The equations of motion are second order, linear, ordinary differential equations;

qDCqBΑ ][][ ][][][][ 2 Ucb

0qFE ][ ][][ 2cU

2.Notice that the equations can be written in a more general form,

where [A] and [B] are structural and aerodynamic mass matrices, [C] and [D] are structural and aerodynamic damping matrices, and [E] and [F] are structural and aerodynamic stiffness matrices. By comparing with (8), these matrices can be determined for the 2DOF airfoil ([C]=[0] since no structural damping is considered in this case).






Static Aeroelasticity: We have now established the full equations of motion for the pitch plunge airfoil based on quasi-steady aerodynamics and we are to examine these in detail;

First, we will study the static equilibrium of the system;

Static means that all the velocities and accelerations are zero;

The equations of motion in static case become,

0

0

4/0 ][ 2

2

h

accUk

cUk

T






Aerodynamic Coupling: Let us now apply a static moment M0 around the flexural axis A, then the equilibrium equations become;

02

2 0

4/0 ][ M

h

accUk

cUk

T

Upon solving these equations, we get,

accUk

M

T

4/20

][ 4/20

2

accUkk

McUh

T

0






This phenomenon is called Aerodynamic Coupling: A change in pitch angle causes a change in plunge.

This is logical since increased pitch leads to increased lift, which in turns leads to a change in plunge.

However, if we apply a static force F0 on the flexural axis in the direction of plunge, there will only be change in plunge and no change in pitch. In this case, there does not exist aerodynamic coupling:

04/0

02

2

][

Fh

accUk

cUk

T

0,0

k

Fh






Look at the static pitch displacement in the case of an applied static moment:

Static Divergence:

accUk

M

T

4/20

There is a possibility that for certain speed UD, the denominator becomes zero and as a result, become extremely large. UD is called the Divergence speed.

04/2D accUkT acc

kU T

4/D Divergence will not occur if a -c/4, i.e., the flexural axis is ahead of the aerodynamic centre, which in the case of thin airfoil, is located at quarter chord.






Practical significance of aeroelasticity;

2DOF pitch plunge airfoil model;

Structural modelling;

Aerodynamic modelling;

Full aeroelastic equations of motion;

Aerodynamic coupling;

Static divergence.

What Was the Lecture ?

In this introductory lecture, we have sought to provide a complete overview about Aeroelasticty course. Following topics were explored briefly and will be further examined in detail later in the course.

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Introduction 1&2 2015

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