Aeroelasticity & Experimental Aerodynamics. Lecture 7 Galloping. T. Andrianne

Transcription

1 Aeroelasticity & Experimental Aerodynamics (AERO0032-1) Lecture 7 Galloping T. Andrianne

2 Dimensional analysis (from L6) Motion of a linear structure in a subsonic, steady flow Described by : A = F U D ( r, Re, m r,η S ) ( U C, A r D ) = F ( Re, m r,η S ) Reduced velocity Dimensionless amplitude Mass ratio 1 Reynolds number Damping factor U r A D m r Re η S Source FIV1977 VIV (Lecture 6) Galloping

3 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 2

4 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 3

5 Galloping vs. VIV Circular Cylinder Force!y On the fluid side!y V rel V rel!y Force F y à Velocity and Force in OPPOSITE directions Non-symmetric geometry (e.g. iced conductor) On the fluid side Force!y!y V rel V rel!y F y Force à Velocity and Force in the same direction à Energy transfer from the flow to the structure 4

6 Galloping GALLOPING = velocity-depend, damping controlled instability à transverse or torsional motions Davison (1930) : dancing vibrations Parkinson (1971): galloping horse Transverse galloping!y Torsional galloping (Torsional Flutter) α Quartering wind Stall Flutter!y α Longjiang Bridge 5

7 Galloping examples 6

8 Galloping examples 7

9 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 8

10 Mechanism of galloping Body free to oscillate around its static equilibrium y x - Fy LIFT y V rel α x!y DRAG Fx 9

11 Mechanism of galloping At a particular instant, the body velocity is à Effective velocity V rel at #!y α = tan 1 % $ & ( '!y y x - Fy V rel α!y Fx 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 U r = fd 10

12 Quasi-steady assumption Quasi-steady assumption à The motion of the structure is slow compared to the motion of the fluid Criterion: Large reduced velocity U r = fd Ur = 10 Ur =

13 Quasi-steady assumption Quasi-steady assumption à The motion of the structure is slow compared to the motion of the fluid Criterion: Large reduced velocity U r = 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 D fn Fung proposed : /f n > 10 D à U r = fn D >10 12

14 Quasi-steady assumption Blevins (1977) : the frequency of the shed vortices (= f VS from the Strouhal relation) must be at least twice as large as the oscillating frequency f n à f VS > 2 f n From Lecture 6 about VIV : f VS = St / D à St / D > 2 f n U r = /(f n D) > 2/St Assuming a Strouhal number St = 0.2 à U r > 10 = idem Fung s criterion (despite different physical reasoning) ( )cr Nakamura & Mizota (1975) : U r = fn (for a square prism) D > 2 fn D = critical reduced airspeed for VIV if not respected, interaction between VIV and galloping 13

15 Mechanism of galloping At a particular instant, the body velocity is à Effective velocity V rel at #!y α = tan 1 % $ Interested in the transverse force : F y F y = L cosα Dsinα & ( '!y y V rel α x -Fy L α!y D α Fx Stability problem à interested in the variation of F y with α df y dα < 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 14

16 Mechanism of galloping Using F y = L cosα Dsinα df y dα = d ( L cosα Dsinα) = dl dd cosα + Lsinα sinα Dcosα dα dα dα = 1 2 ρu )# 2 S C L dc & # D % (sinα + dc L $ dα ' dα C &, + % D (cosα. * $ ' - From wind tunnel testing : measurement of Lift and Drag vs α Around 0 L = 1 2 ρu 2 SC L D = 1 2 ρu 2 SC D ~ 1 for small α, i.e.!y C L = 0 dc L /dα ~ -5 C D ~ 2.2 dc D /dα ~ 0 df y dα 1 2 ρu $ 2 S dc L dα + ' & % CD )cosα ( 1 2 ρu $ 2 S dc L dα + ' & % CD ) ( à Instability criterion (Den Hartog s criterion) : Source : NAKAMURA1975 df y dα > 0 dc L dα + C D < 0 15

17 Mechanism of galloping Back to the single degree of freedom system : -Fy m!!y + c!y + ky = F y = L cosα Dsinα L Dα α << k y m c # % L 0 + dl $ dα α & # ( % D 0 + dd ' $ dα α & (α ' # L 0 dl dα + D & % 0 (α $ ' # L % dl $ dα + D & (!y ' Taylor expansion Neglecting α 2 terms (! m!!y + c + # dl " dα + D $ * & 1 ) % = Structural damping = Total damping + -!y + ky = L, = 0 around the equilibrium position = Aerodynamic damping 16

18 Mechanism of galloping In terms of aerodynamic force coefficients : (! m!!y + c + # dl " dα + D $ * & 1 ) % + -!y + ky = 0, L = 1 2 ρu 2 SC L (! m!!y + c + dc L dα + C $ # D & 1 " % 2 ρu + * S-!y + ky = 0 ), D = 1 2 ρu 2 SC D Galloping occurs when total damping vanishes at the critical galloping velocity :! c + dc L dα + C $ # D & 1 " % 2 ρu C S = 0 U C = à Den Hartog criterion is a sufficient condition (but not necessary, see later)! # " 2c dc L dα + C D $ &ρs % C Galloping = velocity-depend, damping controlled instability 17

19 Mechanism of galloping In terms of non-dimensional quantities : (! m!!y + c + dc L dα + C $ # D & 1 " % 2 ρu + * S-!y + ky = 0 ), ω n = k c ζ = m 2mω n (!!!y + 2ω n ζ + dc L dα + C $ # D & ρh + * -!y +ω 2 n ) " % 4mω n, y = 0 Assuming lightly damped system : δ= 2 πζ U C 4mζω = n $ ρh dc L dα + C ' & D ) % ( (δ = log decrement) U C r = U C f n h = 4 $ dc & L % dα + C D ' ) ( m ρh 2 δ m δ = mass-damping parameter 2 ρh Mass ratio ( U C, A r D ) = F ( Re, m r,η S ) (similarly to VIV phenomenon) 18

20 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 19

21 Non-linear aspects Until here : critical airspeed based on a linear approach à How to obtain the galloping amplitude? ( ) = F Re, m r,η S U C r, A D ` ( ) à Need for an expression of the full aerodynamic forces (not only α<< ) F y = L cosα Dsinα C Fy = C L cosα C D sinα C Fy = C Fy ( α,α 2,α 3,α 4,α 5,...) C L and C D both α dependent C Fy dc Fy dα α +O(α 2 ) Source : NAKAMURA1975 Source : Parkinson

22 Non-linear aspects Equation of motion : m!!y + c!y + ky = F y = 1 2 ρu 2 hc Fy Using appropriate non-dimensional variables : y = Yh, τ =ωt, =Uωh, n= ρh2 2m, ω 2 = k ml, 2ζω= c ml Source : Parkinson1964 Y ''+ 2ζY '+Y = F y = nu 2 C Fy! C Fy = A Y ' $! # & B Y ' $ # & " U % " U % 3! + C Y ' $ # & " U % 5! D Y ' $ # & " U % 7 Where A,B,C,D are fitting parameters A=2,69 B=168 C=6270 D=59900 o : Measured, statically in wind tunnel : Polynomial fit 21

23 Non-linear aspects Non linear equation of motion: (! Y ''+ 2ζY '+Y = F y = nu 2 A Y ' $! # & B Y ' $ * # & )* " U % " U % 3! + C Y ' $ # & " U % 5! D Y ' $ # & " U % (" Y ''+Y = na U 2ζ % " $ 'Y ' B % " $ 'Y ' 3 C % " + $ 'Y ' 5 D % + * $ 'Y ' 7 )# na & # AU & # AU 3 & # AU 5 - &, A must be positive 7 + -,- U 0 = 2ζ na! C Fy = A Y ' $! # & B Y ' $ # & " U % " U % 3! + C Y ' $ # & " U % 5! D Y ' $ # & " U % 7 A = dc Fy = dc Fy d ( Y '/U) Y '/U=0 dα α=0 A > 0 dc Fy dα > 0 dc L dα + C D < 0 Source : Parkinson1964 = Den Hartog s criterion 22

24 Non-linear aspects Non linear equation of motion: (! Y ''+ 2ζY '+Y = F y = nu 2 A Y ' $! # & B Y ' $ * # & )* " U % " U % 3! + C Y ' $ # & " U % 5! D Y ' $ # & " U % (" Y ''+Y = na U 2ζ % " $ 'Y ' B % " $ 'Y ' 3 C % " + $ 'Y ' 5 D % + * $ 'Y ' 7 )# na & # AU & # AU 3 & # AU 5 - &, 7 + -,- U < Uc = U 0 à Stable system U 0 < U < U 1 à One stable and one unstable limit cycles Source : Parkinson

25 Non-linear aspects Non linear equation of motion: (! Y ''+ 2ζY '+Y = F y = nu 2 A Y ' $! # & B Y ' $ * # & )* " U % " U % 3! + C Y ' $ # & " U % 5! D Y ' $ # & " U % (" Y ''+Y = na U 2ζ % " $ 'Y ' B % " $ 'Y ' 3 C % " + $ 'Y ' 5 D % + * $ 'Y ' 7 )# na & # AU & # AU 3 & # AU 5 - &, 7 + -,- U 1 < U < U 2 à 2 stable branches and 2 unstable branches à Hysteresis loop U > U 2 à One stable and one unstable limit cycles Source : Parkinson

26 Non-linear aspects Comparison with experiments : four different damping values (β) à Universal galloping curve (Parkinson & Smith (1964)) Source : Parkinson1964 Wind speed for VIV resonance na 2β U = na 1 2β 2πSt Discrepancies for low damping (β= ) x symbols Reason : QS fails NL QS model : à Critical airspeed à Amplitude à Hysteresis 25

27 Non-linear aspects Universal galloping curve (Parkinson & Smith (1964)) " $ $ δ = 2πβ m r = m 2π Remember :# ρh 2 # the multiplicative factor na can be $ $ 2β $ % $ n = ρh2 2m " $ % $ β = δ n = 1 m r written : na 2β = π A 2m r δ = π A Sc Source : Parkinson1964 Sc = Scruton number à Galloping is driven by : Aerodynamic characteristics : A Scruton number 26

28 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 27

29 Flow analysis Effect of afterbody on galloping = part of the body downstream of the points of separation Source : FSI2011 d h Turbulence changes everything! Den Hartog dc Fy dα > 0 28

30 Flow analysis Effect of afterbody on galloping = part of the body downstream of the points of separation Source : FSI2011 d h Den Hartog dc Fy dα > 0 29

31 Flow analysis In the absence of after-body à No galloping Long after body à No galloping 10 4 < Re <10 5 d h Source : FSI

32 Flow analysis Effect of turbulence Increasing Tu Interference with TE cornerì Earlier re-attachment (d/h) max î 31

33 Flow analysis Inflection Points in the C Fy curve : flow re-attachment Source : Parkinson1964 Static Small Oscill. Large Oscill. Source : PARKINSON Low suction peak High suction peak 32

34 Flow analysis Inflection Points in the Cfy curve à Responsible for the hysteresis behaviour Den Hartog criterion is a sufficient condition (not necessary) 33

35 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 34

36 Torsional galloping Torsional galloping or Torsional flutter, around an hinge point (rotation center) Bridge deck section in the Ulg s wind tunnel 35

37 Torsional galloping Transverse galloping Torsional galloping V rel α U!y V rel (x) α θ!!y(x) = x! θ x 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, V rel (x) varies from point to point à No equivalent static configuration 36

38 Torsional galloping M r r γ r r r! θ θ Rotation center Define a reference radius r r à Unique transverse velocity, representative of motion of the whole body à Reference angle of attack α and relative velocity V rel 37

39 Torsional galloping Source:FSI2011 Reference angle of attack α Relative velocity V rel # θ α = tan 1 r! θ sinγ r & % r! ( V 2 rel = ( r θ! sinγ r ) 2 + r θ! cosγ r $ θ cosγ r ' for α << α θ r! θ sinγ r θ R θ! R = r sinγ r V rel ( ) 2 38

40 Torsional galloping = r sinγ r Reference angle of attack α Relative velocity V rel α θ R θ! R V rel The choice of r r is not obvious and γ r is a function of θ à R=R(θ) à α is a non-linear function of θ Origin of r r 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 θ 39

41 Torsional galloping The equation of the torsional motion : J!! θ + c θ! + kθ = M = 1 2 ρu 2 h 2 C M J = mass moment of inertia c = torsional damping coefficient k = torsional stiffness coefficient Similarly to the transverse galloping, Taylor expansion of C M C M = C M α=0 + dc M dα α=0 α +... C M dc $ M dα θ R θ! ' & ) % ( α θ R! θ c = 2ζωJ ω 2 = k / J "!! θ + 2ζω ρ $ Rh2 # J dc M dα % '! " θ + ω 2 1 & 2 ρ U 2 $ Rh2 # J dc M dα % 'θ = 0 & 40

42 Torsional galloping The equation of the torsional motion : "!! θ + 2ζω ρ $ Rh2 # J dc M dα % '! " θ + ω 2 1 & 2 ρ U 2 $ Rh2 # J dc M dα % 'θ = 0 & Torsional galloping Static divergence 2ζω ρ Rh2 J dc M dα = 0 ω ρ U 2 Rh 2 J dc M dα = 0 = 2ζω 1 2 ρ Rh2 J dc M dα = ω ρ Rh2 J dc M dα Condition : dc M dα < 0 Condition : dc M dα > 0 41

43 Torsional galloping Slope of the torsional moment coefficient Source:FSI2011 More elongated bodies à Higher and negative values of the slope à More sensitive to torsional flutter 42

44 Torsional galloping 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) 43

45 Torsional galloping Example : DVM simulations of a rectangular cylinder (h/d=4) free to oscillate Stable response (i.e. below critical airspeed) 44

46 Torsional galloping Example : DVM simulations of a rectangular cylinder (h/d=4) free to oscillate Unstable response (i.e. above critical airspeed) : TORSIONAL GALLOPING 45

47 Torsional galloping Example : Torsional galloping around the vertical axis of a bridge deck 46

48 Torsional galloping Example : Torsional galloping around the vertical axis of a bridge deck 47

49 Today s lecture Galloping : definition and examples Transverse galloping Mechanism of galloping Non-linear aspects Flow analysis Torsional galloping Conjoint galloping and VIV 48

50 Conjoint galloping and VIV Threshold for galloping : U G = 2ζ na typically U G > 10 (quasi-steady assumption) Threshold for VIV : typically U v ~ 1 U V = 1 2πSt But for low damping (ζ<<) and/or low-density structures (n>>) U G can decrease up to unity or even below à Interaction between VIV and galloping If U v =U G à resulting oscillation amplitude higher than a single phenomenon prediction 49

51 Conjoint galloping and VIV Famous example of conjoint galloping (torsional) and VIV (heaving) Tacoma Narrow bridge failure Source: Larsen

52 Summary Transversal (vertical) galloping = velocity dependent, damping controlled instability à Can be modeled using Quasi-Steady theory (condition : U r > 10)!y à Motion becomes an equivalent AOA : #!y α = tan 1 % $ & ( ' LINEAR APPROACH à Den Hartog criterion (pay attention to turbulence effects!) à Estimate of the critical galloping velocity : U 0 = 2ζ na NON LINEAR APPROACH à To complete the linear analysis à Higher order polynomials for C Fy à Amplitude of the galloping oscillations 51

53 Summary 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 52

54 Summary Torsional galloping à Much more difficult than transversal galloping Fluid forces depend on both the angle and angular velocity The relative flow velocity, V rel (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 53

55 References 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): WASHIZU1978 Washizu, K. and A. Ohya (1978). "Aeroelastic instability of rectangular cylinders in a heaving mode." Journal of Sound and Vibration 59(2): 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: PARKINSON1971 Parkinson, G. V Wind-induced instability of structures. Philosophical Transactions of the Royal Society (London) 269, LARSEN2000 A. Larsen, Aerodynamics of the Tacoma Narrows Bridge : 60 years later, Structural Engineering International. 10/2000; 10(4):