03 lect 14 aeroelastic

8/4/2019 03 Lect 14 Aeroelastic

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Aeroelastic effects

Wind loading and structural response

Lecture 14 Dr. J.D. Holmes


2/17

Aeroelastic effects

Very flexible dynamically wind-sensitive structures

Motion of the structure generates aerodynamic forces

Positive aerodynamic damping : reduces vibrations - steel lattice towers

if forces act in direction to increase the motion : aerodynamic instability


3/17

Aeroelastic effects

Example : Tacoma Narrows Bridge WA - 1940

Example : Galloping of iced-up transmission lines


4/17

Aeroelastic effects

Aerodynamic damping (along wind) :

Relative velocity of air with respect to body = xU

Consider a body moving with velocity in a flow of speed Ux


5/17

Aeroelastic effects

Aerodynamic damping (along wind) :

xUbCUb2

1C

)U

x2(1Ub

2

1C)xUb(

2

1CD

aD

2

aD

2

aD

2

aD

Drag force (per unit length) =

U/xfor small

aerodynamic damping term

xUbCxcaD

total damping term :

along-wind aerodynamic damping is positive

transfer to left hand side of equation of motion : D(t)kxxcxm


6/17

Aeroelastic effects

Galloping :galloping is a form of aerodynamic instability caused by negative

aerodynamic damping in the cross wind direction

Motion of body in z direction will generate an apparent reduction in angle of attack,

From vector diagram : U/z


7/17

Aeroelastic effects

Galloping :

Aerodynamic force per unit length in z direction (body axes) :

Fz = D sin + L cos = )cosCsinb(CU2

1LD

2

a

(Lecture 8)

)cosd

dCsinCsin

d

dCcosb(CU

2

1

d

dF LL

DD

2

az

For = 0 : )d

dCb(CU21

ddF L

D

2

az

)d

dCb(CU

2

1F LD

2

az


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Aeroelastic effects

Galloping :

Substituting, U/z

)U

z)(

d

dCb(CU

2

1F LD

2

az

z)d

dCb(CU

2

1 LDa

For , Fz is positive - acts in same direction as0)d

dC(C LD z

negative aerodynamic damping when transposed to left-hand side


9/17

Aeroelastic effects

Galloping :

den Hartogs Criterion0)d

dC(C LD

critical wind speed for galloping,Ucrit , occurs when total damping is zero

0z)d

dCb(CU

2

1zc LDcrita

)d

dCb(C-

2cU

LDa

crit

)d

dCb(C-

mn8U

LDa

1crit

Since c = 2(mk)=4mn1 (Figure 5.5 in book)

m = mass per unit length n1 = first mode natural frequency


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Galloping :

Cross sections prone to galloping :

Square section (zero angle of attack)

D-shaped cross section

iced-up transmission line or guy cable

Aeroelastic effects


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Aeroelastic effects

Flutter :

Consider a two dimensional body rotating with angular velocity

Vertical velocity at leading edge : d/2

Apparent change in angle of attack : Ud/2

Can generate a cross-wind force and a moment

Aerodynamic instabilities involving rotation are called flutter


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Aeroelastic effects

Flutter :

General equations of motion for body free to rotate and translate :

per unit massHHzHm

(t)Fzz2z 321

z2

zzz

AAzAI

M(t)2 321

2

per unit mass moment of inertia

z

Flutter derivatives


13/17

Aeroelastic effects

Flutter :

Types of instabilities :

Name Conditions Type of motion Type of section

Galloping H1>0 translational Square sectionStall flutter A2>0 rotational Rectangle, H-

section

Classical flutter H2>0, A1>0 coupled Flat plate, airfoil


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Flutter :

1

2

-0.1

-0.2

A2*

0.1

0

1

2

6

4

2

0

8

1

2

H2*

-2A

0.4

0.3

0.2

unstable

stable

stable

Flutter derivatives for two bridge deck sections :

A1*3

2

1

00 2 4 6 8 10 12

1

2

-6

-4

-2

00

H1*

2 4 6 8 10 12

12

U/nd

U/nd

Aeroelastic effects


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Aeroelastic effects

Flutter :

Determination of critical flutter speed for long-span bridges:

Empirical formula (e.g. Selberg)

Experimental determination (wind-tunnel model)

Theoretical analysis using flutter derivatives obtained experimentally


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Aeroelastic effects

Lock - in :

Motion-induced forces during vibration caused by vortex shedding

Frequency locks-in to frequency of vibration

Strength of forces and correlation length increased


17/17

End of Lecture 14John Holmes

225-405-3789 [email protected]

03 lect 14 aeroelastic

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