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Aeroelasticity & Experimental Aerodynamics (AERO0032-1)

Lecture 7 Galloping

T. Andrianne

2015-2016

Motion of a linear structure in a subsonic, steady flow Described by :

Dimensional analysis (from L6)

1

AD

Ur

mr

Re

ηS

Source FIV1977

VIV (Lecture 6) Galloping

→ AD = F Ur, Re,mr,ηS( ) → UC

r,AD( ) = F Re,mr,ηS( )

•  Reduced velocity

•  Dimensionless amplitude

•  Mass ratio

•  Reynolds number

•  Damping factor

Today’s lecture

2

Galloping : definition and examples Transverse galloping

Mechanism of galloping Non-linear aspects Flow analysis

Torsional galloping Conjoint galloping and VIV

Today’s lecture

3




U∞ !y

U∞ − !y

Vrel

On the fluid side U∞ Force

Vrel Force !y

Fy

Non-symmetric geometry (e.g. iced conductor)

à Velocity and Force in the same direction à Energy transfer from the flow to the structure

Circular Cylinder

à Velocity and Force in OPPOSITE directions

Galloping vs. VIV

4

U∞ !y

U∞ − !y

Vrel

On the fluid side U∞

Force

Vrel

Force

!y Fy

Galloping

5

GALLOPING = velocity-depend, damping controlled instability à transverse or torsional motions

Davison (1930) : “dancing vibrations” Parkinson (1971): “galloping horse”

•  Transverse galloping

•  Torsional galloping (Torsional Flutter)

•  Quartering wind

•  Stall Flutter

U∞ !y

U∞

Longjiang Bridge

α

α

U∞

!y

Galloping examples

6

Galloping examples

7

Today’s lecture

8




DRAG

LIFT

Body free to oscillate around its static equilibrium

Mechanism of galloping

9

!yVrel

α Fx

-‐Fy

y x

U∞

U∞

y x

At a particular instant, the body velocity is à  Effective velocity Vrel at


10

!yVrel

α Fx

-‐Fy

y x

U∞

!y

α = tan−1 !yU∞

#

$%

&

'(

Quasi-steady assumption Meaning : The motion of the structure is slow compared to the motion of the fluid à Flow has time to adapted to the motion of the structure

à Lift and Drag in the course of oscillation are the same at each α as the value measured statically during wind tunnel experiments

Criterion: Large reduced velocity

Ur =U∞

fD

Quasi-steady assumption

11

Quasi-steady assumption à The motion of the structure is slow compared to the motion of the fluid


Ur =U∞

fD

Ur = 10 Ur = 3.3


12

Quasi-steady assumption à The motion of the structure is slow compared to the motion of the fluid


Ur =U∞

fD

What is the meaning of “Large” ?

Fung (1955) : “any disturbance experienced by the oscillating body at a certain point of its motion must be swept downstream sufficiently far, by the time the body comes back to that same point (one period later), of the disturbance to no longer affect the flow around the body”

U∞

D U∞

fn

Fung proposed : U∞/fn > 10 D à Ur =U∞

fnD>10


13

Blevins (1977) : “the frequency of the shed vortices (= fVS from the Strouhal relation) must be at least twice as large as the oscillating frequency fn” à  fVS > 2 fn From Lecture 6 about VIV : fVS = St U∞ / D à St U∞ / D > 2 fn Ur = U∞/(fn D) > 2/St Assuming a Strouhal number St = 0.2 à Ur > 10 = idem Fung’s criterion

(despite different physical reasoning) Nakamura & Mizota (1975) : (for a square prism)

if not respected, interaction between VIV and galloping

Ur =U∞

fnD> 2 U∞

fnD( )cr

= critical reduced airspeed for VIV

D

L

At a particular instant, the body velocity is à  Effective velocity Vrel at

Interested in the transverse force : Fy

Stability problem à interested in the variation of Fy with α


14

!yVrel

α Fx

-Fy

y x

U∞

!y

α = tan−1 !yU∞

#

$%

&

'(

Fy = −Lcosα −Dsinα

α

α

dFydα

< 0 Meaning : dα > 0 à dFy < 0, i.e. increasing motion leads to decreasing force à STABLE system

> 0 Meaning : dα > 0 à dFy > 0, i.e. increasing motion leads to increasing force à UNSTABLE system

Using From wind tunnel testing : measurement of Lift and Drag vs α


15

dFydα

=ddα

−Lcosα −Dsinα( ) = − dLdα

cosα + Lsinα − dDdα

sinα −Dcosα

= 12ρU∞

2S CL −dCD

dα#

$%

&

'(sinα + −

dCL

dα−CD

#

$%

&

'(cosα

)

*+

,

-.

Fy = −Lcosα −Dsinα

Around 0° CL = 0 dCL/dα ~ -5 CD ~ 2.2 dCD/dα ~ 0

dFydα

≈ −12ρU∞

2S dCL

dα+CD

$

%&

'

()cosα

≈ − 12ρU∞

2S dCL

dα+CD

$

%&

'

()

~ 1 for small α, i.e. !y

à Instability criterion (Den Hartog’s criterion) : dFydα

> 0→ dCL

dα+CD < 0

L = 12ρU∞

2SCL

D =12ρU∞

2SCD

Source : NAKAMURA1975

Back to the single degree of freedom system :


16

U∞

y

-Fy m!!y+ c!y+ ky = Fy = −Lcosα −Dsinα ≈ −L −Dα

≈ − L0 +dLdα

α#

$%

&

'(− D0 +

dDdα

α#

$%

&

'(α

≈ −L0 −dLdα

+D0#

$%

&

'(α

≈ −L − dLdα

+D#

$%

&

'(!yU∞

m!!y+ c+ dLdα

+D!

"#

$

%&1U∞

(

)*

+

,- !y+ ky = −L

= Total damping

= Aerodynamic damping

= Structural damping

= 0 around the equilibrium position

Taylor expansion

α <<

Neglecting α2 terms

m

k c

In terms of aerodynamic force coefficients :


17

m!!y+ c+ dLdα

+D!

"#

$

%&1U∞

(

)*

+

,- !y+ ky = 0

L = 12ρU∞

2SCL

D =12ρU∞

2SCD

m!!y+ c+ dCL

dα+CD

!

"#

$

%&12ρU∞S

(

)*

+

,- !y+ ky = 0

Galloping occurs when total damping vanishes at the critical galloping velocity : à Den Hartog criterion is a sufficient condition (but not necessary, see later)

Galloping = velocity-depend, damping controlled instability

U∞C

c+ dCL

dα+CD

!

"#

$

%&12ρU∞

CS = 0→U∞C = −

2cdCL

dα+CD

!

"#

$

%&ρS

In terms of non-dimensional quantities :


18

m!!y+ c+ dCL

dα+CD

!

"#

$

%&12ρU∞S

(

)*

+

,- !y+ ky = 0

Assuming lightly damped system : δ= 2 πζ (δ = log decrement)

→UrC =

U∞C

fnh= −

4dCL

dα+CD

$

%&

'

()

mρh2

δ

ωn =km

ζ =c

2mωn

!!y+ 2ωn ζ +dCL

dα+CD

!

"#

$

%&ρU∞h4mωn

(

)*

+

,- !y+ω 2

ny = 0

→U∞C = −

4mζωn

ρh dCL

dα+CD

$

%&

'

()

Mass ratio

mρh2

δ = mass-damping parameter

→ UCr,AD( ) = F Re,mr,ηS( ) (similarly to VIV phenomenon)

Today’s lecture

19




Until here : critical airspeed based on a linear approach

à  How to obtain the galloping amplitude ?

à  Need for an expression of the full aerodynamic forces (not only α<< )

Non-linear aspects

20

→ UCr,AD( ) = F Re,mr,ηS( )`

Fy = −Lcosα −Dsinα→CFy = −CL cosα −CD sinα

CL and CD both α dependent

Source : NAKAMURA1975 Source : Parkinson1964

CFy ≈dCFy

dαα +O(α 2 )

→CFy =CFy α,α2,α 3,α 4,α 5,...( )

Equation of motion : Using appropriate non-dimensional variables :

Non-linear aspects

21

m!!y+ c!y+ ky = Fy =12ρU∞

2hCFy

→Y ''+ 2ζY '+Y = Fy = nU2CFy

Source : Parkinson1964

CFy = AY 'U!

"#

$

%&−B

Y 'U!

"#

$

%&3

+C Y 'U!

"#

$

%&5

−D Y 'U!

"#

$

%&7

o : Measured, statically in wind tunnel ----- : Polynomial fit

y =Yh, τ=ωt, U∞ =Uωh, n= ρh2

2m, ω 2 = k

ml, 2ζω= c

ml

Where A,B,C,D are fitting parameters A=2,69 B=168 C=6270 D=59900

Non linear equation of motion:

Non-linear aspects

22

Y ''+ 2ζY '+Y = Fy = nU2 A Y '

U!

"#

$

%&−B

Y 'U!

"#

$

%&3

+C Y 'U!

"#

$

%&5

−D Y 'U!

"#

$

%&7(

)**

+

,--


Y ''+Y = nA U −2ζnA

"

#$

%

&'Y '−

BAU"

#$

%

&'Y '3+

CAU 3

"

#$

%

&'Y '5−

DAU 5

"

#$

%

&'Y '7

(

)*

+

,-

U0 =2ζnA

“A” must be positive

= Den Hartog’s criterion

CFy = AY 'U!

"#

$

%&−B

Y 'U!

"#

$

%&3

+C Y 'U!

"#

$

%&5

−D Y 'U!

"#

$

%&7

A =dCFy

d Y '/U( ) Y '/U=0=dCFy

dα α=0

A > 0→dCFy

dα> 0→ dCL

dα+CD < 0


Non-linear aspects

23

Y ''+ 2ζY '+Y = Fy = nU2 A Y '

U!

"#

$

%&−B

Y 'U!

"#

$

%&3

+C Y 'U!

"#

$

%&5

−D Y 'U!

"#

$

%&7(

)**

+

,--



"

#$

%

&'Y '−

BAU"

#$

%

&'Y '3+

CAU 3

"

#$

%

&'Y '5−

DAU 5

"

#$

%

&'Y '7

(

)*

+

,-

U < Uc = U0 à  Stable system U0 < U < U1 à One stable and one unstable limit cycles


Non-linear aspects

24

Y ''+ 2ζY '+Y = Fy = nU2 A Y '

U!

"#

$

%&−B

Y 'U!

"#

$

%&3

+C Y 'U!

"#

$

%&5

−D Y 'U!

"#

$

%&7(

)**

+

,--



"

#$

%

&'Y '−

BAU"

#$

%

&'Y '3+

CAU 3

"

#$

%

&'Y '5−

DAU 5

"

#$

%

&'Y '7

(

)*

+

,-

U1 < U < U2 à  2 stable branches and 2

unstable branches à  Hysteresis loop

U > U2 à One stable and one unstable limit cycles

Non-linear aspects

25


Wind speed for VIV resonance nA2β

U =nA2β

12πSt

Comparison with experiments : four different damping values (β) à Universal galloping curve (Parkinson & Smith (1964))

Discrepancies for low damping (β=0.00107) x symbols

Reason : QS fails

NL QS model : à Critical airspeed ✔ à Amplitude ✔ à Hysteresis ✔

Non-linear aspects

26


Universal galloping curve (Parkinson & Smith (1964)) Remember : the multiplicative factor can be

written :

nA2β

δ = 2πβ

mr =mρh2

n = ρh2

2m

→

"

#

$$$

%

$$$

β =δ2π

n = 1mr

"

#

$$

%

$$

nA2β

=πA2mrδ

=πASc

Sc = Scruton number à  Galloping is driven by :

•  Aerodynamic characteristics : A •  Scruton number

Today’s lecture




27

Flow analysis

28

Effect of afterbody on galloping = part of the body downstream of the points of separation

dCFy

dα> 0

Den Hartog

Source : FSI2011

h

d

Turbulence changes everything !

Flow analysis

29

dCFy

dα> 0

Den Hartog

Source : FSI2011

h

d

Effect of afterbody on galloping = part of the body downstream of the points of separation

Source : FSI2011

Flow analysis

30

In the absence of after-body à No galloping Long after body à No galloping

h

d

104 < Re <105

Flow analysis

31

Effect of turbulence

Increasing Tu Interference with TE cornerì Earlier re-attachment (d/h)max î

Flow analysis

32

Source : PARKINSON1971

Static Small Oscill.

Large Oscill.

High suction peak Low suction peak


6° 13° 0°

Inflection Points in the CFy curve : flow re-attachment

Flow analysis

33

Den Hartog criterion is a sufficient condition (not necessary)

Inflection Points in the Cfy curve à  Responsible for the hysteresis behaviour

Today’s lecture




34

Torsional galloping or Torsional flutter, around an hinge point (rotation center)

Torsional galloping

35

Bridge deck section in the Ulg’s wind tunnel

Much more difficult than transverse galloping ! •  Fluid forces depend on both the angle and angular velocity •  Phasing between fluid forces and motion depends on the flow

velocity •  The relative flow velocity, Vrel(x) varies from point to point

à  No equivalent static configuration

Torsional galloping

36

Vrel

α U∞

Vrel(x)

α U∞

!y

x

!θ!y(x) = x !θ

Transverse galloping Torsional galloping

Define a reference radius rr à  Unique transverse velocity, representative of motion of the whole

body

à  Reference angle of attack α and relative velocity Vrel

Torsional galloping

37

U∞

θ

M

Rotation center

rr

rr !θ

γ r

Reference angle of attack α Relative velocity Vrel

Torsional galloping

38

θ −α = tan−1 rr !θ sinγ rU∞ − rr !θ cosγ r

#

$%

&

'( Vrel

2 = rr !θ sinγ r( )2+ U∞ − rr !θ cosγ r( )

2

α ≈θ −rr !θ sinγ rU∞

≡θ −R !θU∞

R = rr sinγ r Vrel ≈U∞

for α <<

Source:FSI2011

Reference angle of attack α Relative velocity Vrel

The choice of rr is not obvious and γr is a function of θ à R=R(θ) à  α is a non-linear function of θ Origin of rr

¾ chord point in airfoil flutter analysis à Ok for attached flows … but not really for bluff body flows

Example of choice : - Rectangular section around the geometric center : R = d/2 à  α corresponds to the instantaneous angle at the leading edge

Torsional galloping

39

α ≡θ −R !θU∞

R = rr sinγ r

Vrel ≈U∞

U∞

θ

Torsional galloping

40

The equation of the torsional motion : J = mass moment of inertia c = torsional damping coefficient k = torsional stiffness coefficient

Similarly to the transverse galloping, Taylor expansion of CM

J !!θ + c !θ + kθ =M =12ρU∞

2h2CM

CM =CM α=0+dCM

dα α=0

α +...

!!θ + 2ζω + 12ρU∞Rh

2

JdCM

dα"

#$

%

&' !θ + ω 2 −

12ρU 2

∞Rh2

JdCM

dα"

#$

%

&'θ = 0

c = 2ζωJ

ω 2 = k / J

α ≡θ −R !θU∞

CM ≈dCM

dαθ −

R !θU∞

$

%&

'

()

Torsional galloping

41

The equation of the torsional motion :

!!θ + 2ζω + 12ρU∞Rh

2

JdCM

dα"

#$

%

&' !θ + ω 2 −

12ρU 2

∞Rh2

JdCM

dα"

#$

%

&'θ = 0

Torsional galloping Static divergence

ω 2 −12ρU 2

∞Rh2

JdCM

dα= 0

→U∞ =ω 2

12ρRh2

JdCM

dα

Condition : dCM

dα> 0

2ζω + 12ρU∞Rh

2

JdCM

dα= 0

→U∞ =−2ζω

12ρRh2

JdCM

dα

Condition : dCM

dα< 0

Torsional galloping

42

Slope of the torsional moment coefficient More elongated bodies à Higher and negative values of the slope

à More sensitive to torsional flutter

Source:FSI2011

Torsional galloping

43

Because of the flow complexity, QS is not adapted to model Torsional Flutter à  Need for more advanced Fluid-Structure analysis

Example : Tr-PIV on the upper surface of a rectangular cylinder (h/d=4)

Torsional galloping

44

Example : DVM simulations of a rectangular cylinder (h/d=4) free to oscillate Stable response (i.e. below critical airspeed)

Torsional galloping

45

Example : DVM simulations of a rectangular cylinder (h/d=4) free to oscillate Unstable response (i.e. above critical airspeed) : TORSIONAL GALLOPING

Torsional galloping

46

Example : Torsional galloping around the vertical axis of a bridge deck

Torsional galloping

47

Example : Torsional galloping around the vertical axis of a bridge deck

Today’s lecture




48

Threshold for galloping : typically UG > 10 (quasi-steady assumption) Threshold for VIV : typically Uv ~ 1

But for low damping (ζ<<) and/or low-density structures (n>>) UG can decrease up to unity or even below à  Interaction between VIV and galloping

If Uv=UG à resulting oscillation amplitude higher than a single phenomenon prediction

Conjoint galloping and VIV

49

UG =2ζnA

UV =1

2πSt

Conjoint galloping and VIV

50

Famous example of conjoint galloping (torsional) and VIV (heaving) Tacoma Narrow bridge failure

Source: Larsen2000

Transversal (vertical) galloping = velocity dependent, damping controlled instability

à Can be modeled using Quasi-Steady theory (condition : Ur > 10) à Motion becomes an equivalent AOA :

LINEAR APPROACH

à Den Hartog criterion (pay attention to turbulence effects !)

à Estimate of the critical galloping velocity : NON LINEAR APPROACH à To complete the linear analysis

à Higher order polynomials for CFy

à Amplitude of the galloping oscillations

Summary

51

!y α = tan−1 !yU∞

#

$%

&

'(

U0 =2ζnA

Transversal (vertical) galloping NON LINEAR APPROACH à  Different types of galloping behaviour

à  Importance of after-body length (d/h) à  Den Hartog is sufficient for galloping

to occur but not necessary

Summary

52

Torsional galloping à  Much more difficult than transversal galloping

–  Fluid forces depend on both the angle and angular velocity –  The relative flow velocity, Vrel(x) varies from point to point

à  No equivalent static configuration

à  Tentative to apply the QS theory = not efficient (flow separation) à Elongated bodies are more prone to torsional flutter Conjoint transverse galloping and VIV à  Potential combination of the two phenomena à Difficult to model / understand

Summary

53

References

54

•  FIV1977 ‘Flow-induced vibration’, R.D. Blevins •  FSI2011 ‘Fluid-structure Interactions : cross-flow induced instabilities’, M.P. Paidoussis,

S. J. Price, E. de Langre. Cambridge University Press •  NAKAMURA1975 Nakamura, Y. and T. Mizota (1975). "Torsional flutter of rectangular

prisms." Journal of the engineering mechanics division, ASCE 101(EM2): 125-142. •  WASHIZU1978 Washizu, K. and A. Ohya (1978). "Aeroelastic instability of rectangular

cylinders in a heaving mode." Journal of Sound and Vibration 59(2): 195-210. •  PARKINSON1964 Parkinson, G. V. and J. D. Smith (1964). "The square prism as an

aeroelastic non-linear oscillator." The Quaterly Journal of Mechanics and Applied Mathematics 17: 225-239.

•  PARKINSON1971 Parkinson, G. V. 1971 Wind-induced instability of structures. Philosophical Transactions of the Royal Society (London) 269, 395–409.

•  LARSEN2000 A. Larsen, Aerodynamics of the Tacoma Narrows Bridge : 60 years later, Structural Engineering International. 10/2000; 10(4):243-248

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