Wind-Induced Dynamic Instabilitiesof Flexible Bridges Nikolaos Nikitas Department of Civil Engineering University of Bristol A dissertation submitted to the University of Bristol in accordance with the requirements for the degree of Doctor of Philosophy in the Faculty of Engineering. April 2011 Abstract The wind-induced vibrations of flexible bridges and their components have long been a major concern. Although a great level of sophistication has been reached in wind- resistant design, there is still a significant threat from the wind. Most intriguingly, often when one problem is solved a new one seems to arise in its place. This study examines a selection of such peculiar aerodynamic issues involved in the routine ‘life’ of a bridge. All of them share in common the need for further explanations to address previous modelling omissions and weaknesses and offer new understanding of the underlying phenomena. Starting with flutter, an inverse scheme was employed to identify the flutter deriva- tive description of aeroelastic loading using actual response measurements of a full-scale suspension bridge. As expected for ambient data, the identification produced noisy es- timates of the parameter values, yet clear trends could be distinguished. Encouragingly the trends were in reasonably good agreement with results from wind tunnel tests on similar cross-sections. Evidence of aeroelastic coupling between vertical and torsional vibrations was identified from the recorded bridge data and the flutter wind speed for the single-degree-of-freedom torsional instability case was estimated. It is believed this is the first time this has been achieved based solely on full-scale data. The study shows the viability of the method to identify the flutter derivatives from full-scale data, which has rarely been attempted previously and never with such clarity of the results. This is potentially useful for identifying the actual aeroelastic behaviour and safety of bridges in service, particularly as uncertainties of wind tunnel tests, such as Reynolds number dependence are overcome. Next, quasi-steady galloping theory was revisited aiming to address previous incon- sistencies and clarify the correct generic equations and their implications. The case of a section free to vibrate in two orthogonal directions was considered, subject to flow at an arbitrary angle of attack to the principal structural axes. Putting forward the correct galloping criterion for this non-classical case, it was possible to quantify differences from previous incorrect analysis of the galloping condition and structural damping demand. The effects of the structural parameters on galloping thresholds were addressed, again overcoming former shortcomings. Finally there was an attempt to elucidate the influence of critical Reynolds number on the apparent galloping instability of dry circular cables. This instability per se cannot fully fit the earlier classical galloping description. Through a series of wind tunnel tests flow conditions responsible for excessive motion were identified. These conditions were more perplexing than the single-sided laminar separation bubble and 2 the associated steady lift that is generally believed to dominate the critical Reynolds number range. For cylinders normal to the flow, discontinuities in the aerodynamic loading appear to act like a quenching intermittency, effectively inhibiting response. For other cable inclinations the unusual flow structures that emerge seem to be related to the observed dynamic instability of the cable. According to this thesis, the transitional behaviour of the boundary layer is entirely responsible for the large amplitude vibration events of bridge cables in dry conditions attributed to so-called dry galloping. Acknowledgements I owe a great debt of gratitude to my supervisor John Macdonald. His support, his trust in me, his advice and his critical views have been more than inspiring for this work. I am sure that our endless, always unscheduled, discussions can be heard on every page of this thesis. He introduced me to the world of bridge monitoring and he made me piece of the legacy of the Clifton Suspension Bridge (CSB). Without him this thesis would have never been this thesis. I would also like to thank Jasna Jakobsen for all background information and support provided when performing the identification analysis for the CSB. It was a pleasure to work with her and learn from her. Further all the ‘team’ that made possible the wind tunnel tests at the National Research Council (NRC) of Canada deserve a special thank. Terje Andersen, Mike Savage, Brian McAuliffe, Guy Larose, along with the technical staff at NRC have been more than helpful partners. I gratefully acknowledge the support of The Clifton Suspension Bridge Trust during the CSB site tests and the financial support from EPSRC during my PhD course (under John Macdonald’s Advanced Research Fellowship). Last but not least, I want to thank my family in Drama for their understanding and love throughout the period of my research at Bristol. Especially my father, a real modern Ulysses in my mind, has a unique contribution to my work for he has never stopped advising me on how to defeat my strongest enemy; myself. From being to becoming Author’s Declaration I declare that the work in this dissertation was carried out in accordance with the regulations of the University of Bristol. The work is original except where indicated by special reference in the text and no part of the dissertation has been submitted for any other degree. Any views expressed in the dissertation are those of the author and in no way represent those of the University of Bristol. The dissertation has not been presented to any other University for examination either in the United Kingdom or overseas. Signed: Dated: Contents 1 Introduction 1 2 The Aeroelasticity Framework 5 2.1 Wind-induced structural loading . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Static loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Wind buffeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Vortex shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.4 Galloping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.5 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.6 Wake-induced loading . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.7 Rain-wind Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Circular galloping: myth or true? . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 Reynolds number effects . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Inclination effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.3 Instability mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 State of the art in bridge wind design . . . . . . . . . . . . . . . . . . . 35 2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Identification of flutter derivatives from full-scale data 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 The case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Wind characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Response and modal parameters . . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Response Characteristics . . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Flutter derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.1 Flutter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.2 Identification Method . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.3 Application to the Clifton Suspension Bridge . . . . . . . . . . . 57 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 i ii CONTENTS 4 Quasi-steady galloping analysis revisited 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Quasi-steady derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.1 Relevance to uniform continuous systems . . . . . . . . . . . . . 74 4.3 Application: quantifying differences . . . . . . . . . . . . . . . . . . . . . 75 4.4 The detuning effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Experiments on galloping vibration of a circular cylinder 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Wind tunnel tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.2 Setup details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3.1 Overview and large responses . . . . . . . . . . . . . . . . . . . . 99 5.3.2 Pressure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.1 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . 106 5.4.2 Mechanism implications . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6 Conclusion and outlook 115 Publications 121 References 123 List of Tables 5.1 Orientation angles for studied cases. . . . . . . . . . . . . . . . . . . . . 93 5.2 Position details for the model. For rings and lowest cable end ‘distance from floor’ refers to stagnation points, while for cobra probes ‘distance from model’ refers to along-wind distance. . . . . . . . . . . . . . . . . . 98 iii List of Figures 2.1 Mapping of Strouhal number against Reynolds number in the subcritical range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Free vibration tests for a circular cylinder. . . . . . . . . . . . . . . . . . 10 2.3 Circular cylinder vibration phenomena past the lock-in range U=5m/s. Top: vertical response record with f c =9Hz. Bottom: surface pressure at transverse tap, f v =13Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Storebælt Bridge vortex-induced vertical motion at max (left) and min (right) of the amplitude cycle. Encircled is a parked van with its view distorted due to motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Typical galloping response curve for a rectangular prism. Inset the clas- sical galloping mechanism illustrated. . . . . . . . . . . . . . . . . . . . . 15 2.6 Response of a rectangular prism with side ratio 2/1 against reduced velocity for varying critical structural damping ratio. The prism has Sr=0.081 that sets the relevant U r for vortex resonance at ≈12.34. . . . 16 2.7 (a) Displacements and aeroelastic forces on a thin airfoil; (b) Displace- ments and aeroelastic forces for a bridge section. . . . . . . . . . . . . . 17 2.8 (a) Quantitative difference of response characteristics for a full bridge under different flow conditions. (b) A ∗ 2 from wind tunnel tests for a tor- sionally unstable bridge section under laminar and turbulent flow condi- tions. Changes appear minimal to sustain any substantial modification in the flutter behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.9 (a) Rivulet formation on the circular cable section; (b) Inclination ge- ometry of the inclined and yawed to the flow cable. . . . . . . . . . . . . 22 2.10 Upper water rivulet mean angular position during rain-wind vibrations and lift force, displacement time series for a different large response configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.11 Resume of wind tunnel test results on the proposed cables in the Higashi- Kobe Bridge. Improvement in aerodynamic performance is apparent for the solution with protuberances under all conditions. . . . . . . . . . . . 25 v vi LIST OF FIGURES 2.12 Mean drag coefficient versus Reynolds number. On top, transitions (Tr) from laminar (L) to turbulent (T) flow are presented in relation to sep- aration points (S) and boundary layers (BL). . . . . . . . . . . . . . . . 28 2.13 (a) The axial flow, evidenced by light flags positioned inside the wake, act towards inhibiting communication between shear layers and promot- ing a secondary circulatory flow. The function described, simulates the galloping of a circular cylinder equipped with a long splitter plate. (b) Enhanced vortices are produced when axial vortices from the inclined cable, mix and interact with ordinary K´ arm´ an vortices. . . . . . . . . . 31 2.14 Dry cable instability design criteria together with real-bridge unstable records. Dotted lines are due to the uncertainty in defining structural damping values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1 Clifton Suspension Bridge (CSB) elevation showing monitoring instru- ment locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Sketch of the CSB cross-section . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Vertical and torsional modes of CSB . . . . . . . . . . . . . . . . . . . . 47 3.4 (a) Histogram of wind speeds during the 2003-04 recording period. (b) Polar plots of 1h mean wind velocities from both anemometers. . . . . . 49 3.5 (a) 1h average wind speed over the monitoring period. (b) 1h RMS vertical accelerations at the reference location over the monitoring period. 51 3.6 (a) RMS vertical accelerations σ v , in relation to wind speed for all 1h records. (b) Same as (a) for 1h records dominated by wind loading, with RMS vertical accelerations now divided by the vertical turbulence intensity. The power-law approximating the obtained trend is also plotted. 52 3.7 PSDs for different loading conditions for (a) vertical (b) torsional and (c) lateral accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.8 (a) PSDs of filtered data for first vertical and torsional modes for the maximum wind speed record. (b) The evolution of the coupling action is evident in the vertical PSD for wind speeds above 11m/s. . . . . . . . 53 3.9 Decolouring process. In the 1DOF system, filter application will produce corrected spectra, see dashed line, simulating white noise loading. For the 2DOF system, filtering with F L will erroneously modify the true aeroelastic coupling, see dashed line versus greyed area. . . . . . . . . . 57 3.10 Example covariance functions (scaled with variance) for the combined two degrees-of-freedom plotted against time lag. . . . . . . . . . . . . . . 58 3.11 Flutter derivatives of Clifton Suspension Bridge from full-scale data, compared with wind tunnel extracted flutter derivatives for various cross- sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 LIST OF FIGURES vii 3.12 Identified H ∗ 1 flutter derivative from each 15-minute record. . . . . . . . 61 3.13 Flutter derivative A ∗ 2 with additional points from A ∗ 4 . . . . . . . . . . . . 63 3.14 H ∗ 1 for different H-sections and a possible Reynolds number based ex- planation for the H ∗ 1 inversion. Crossovers can initiate when Reynolds number changes alter the multiplier of U r in Eq.(3.6). . . . . . . . . . . 64 4.1 Geometry of a bluff section indicating lift and drag forces (L, D), relative angle of attack (α) and principal structural axes (x, y). (a) The general case with α 0 = 0 and the 2DOF motion potential. (b) The special case for 1DOF across-wind oscillations. . . . . . . . . . . . . . . . . . . . . . 70 4.2 Sections used in the galloping analysis. . . . . . . . . . . . . . . . . . . . 76 4.3 Non-dimensional aerodynamic damping coefficients (S DH , min(S xx , S yy ), S 2D ) as a function of angle of attack. Negative values indicate unstable behaviour (in the absence of structural damping). . . . . . . . . . . . . . 78 4.3 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Comparison between the erroneous S sc and the correct value for the 1D rotated y-axis case, S yy , for (a) the square in Fig.4.2(d) and (b) the triangle in Fig.4.2(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Evolution of the aerodynamic damping solution for different values of detuning, κ, for (a) the section in Fig.4.2(j) and α = 123 ◦ and (b) the section in Fig.4.2(k) and α = 30 ◦ . The lower branch is the important one. In (a) for perfect tuning the solution is unstable (negative aerody- namic damping) and for detuning above about 7% it approaches the 1D solution, which in this case is stable. In (b), on the other hand, for per- fect tuning the solution is stable and for detuning above 1% it becomes unstable while moving towards the 1D solution. . . . . . . . . . . . . . . 83 4.6 Modal trajectories corresponding to Fig.4.5(a) for (a) κ = 1, (b) κ = 1.005, (c) κ = 1.05 and (d) κ = 1.1. The applicable S detuned value is also indicated. Unstable modes are plotted with solid lines while stable ones are dotted. Note for comparison that S xx = 0.45, S yy = 0.06. For all plots the structural damping value was c = 0. . . . . . . . . . . . . . . . 85 5.1 Transformation from real cable to wind tunnel model. . . . . . . . . . . 92 5.2 View of the NRC wind tunnel facility and its test section with the model in place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Turbulence intensity and mean velocity across the wind tunnel section. . 95 5.4 Elevation of cable model showing instrumentation arrangement. . . . . . 97 5.5 Typical frequency response curves for three pressure taps. . . . . . . . . 98 viii LIST OF FIGURES 5.6 Three examples of motion traces during records of instability; Setup 2A and 2C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.7 Proportion of total variance from 20 POMs (from all pressure tap data) against Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.8 First Proper Orthogonal modeshapes for a set of dynamic and static tests103 5.9 Spectra of the lift coefficient on setup 2A, averaged over all four pressure rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 Correlation functions of lift coefficients between pressure rings. . . . . . 105 5.11 Mean pressure coefficient distribution around cylinder for a large re- sponse case. Model setup 2A. . . . . . . . . . . . . . . . . . . . . . . . . 107 5.12 Drag evolution of the half-section with Reynolds number. Model setup ϕ=60 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.13 Drag evolution of the half-section with Reynolds number. Model setup ϕ=90 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.14 Ring 1 C L and C D transitional avalanche-like behaviour. Setup ϕ=90 ◦ . 110 5.15 Pressure distributions during a transitional state succession. Model setup ϕ=90 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Nomenclature The precise interpretation of notation and abbreviations must be obtained from the local context in which it is used and in which it will be explained. As a further guide, the following is a non-exhaustive list of commonly used terms Notation A ∗ i , H ∗ i , P ∗ i Flutter derivatives (i=1-6) a 1 , γ, b 1 Wake oscillator fitted parameters B Representative sectional dimension (chord length) b Half chord length C(k) Theodorsen circulation function C D,L,M Static force coefficients for D, L, M C D 1/2 Static drag coefficient from half the cylinder section C ij Covariance estimate between i, j series C p Static pressure coefficient c Structural damping D Static drag force D b Buffeting drag force D se Self-excited drag force d Across-wind dimension F L (f) Filter function for lift force F x,y Force along direction x or y f Frequency f c Structural frequency f v Vortex shedding frequency G ij Modal integral of modes i, j H j (f) Frequency Response Function of mode j h(s, t), p(s, t), α(s, t) Vertical, lateral and torsional displacement h i (s), p i (s), α i (s) Vertical, lateral and torsional i th mode shape I i Turbulence intensity along component i ix x NOMENCLATURE I i Generalised inertia of mode i ı Imaginary unit |J j (f)| 2 Joint acceptance function of mode j K Reduced cyclic frequency based on the full chord length k Reduced cyclic frequency based on the half chord length L Static lift force L se Self-excited lift force L b Buffeting lift force x,y,z L i Turbulence length scale parameters l Characteristic length ℓ Maximum lags for covariance estimates M Static overturning moment M b Buffeting overturning moment M se Self-excited overturning moment m Mass N Number of samples n Number of lags in correlation function Q i Generalised aerodynamic loading of mode i q i , ˙ q i , ¨ q i , Generalised displacement, velocity and acceleration of mode i R ij Correlation function Re Reynolds number r g Radius of gyration ˜ S Power Spectral Density S c Scruton number S 2D Generalised two tuned degree-of-freedom galloping criterion S DH Den Hartog galloping criterion S sc Non-generalised galloping criterion S ij Dimensionless aerodynamic damping along i due to motion along j Sr Strouhal number s Space variable U Wind velocity U n Normal wind component U r Reduced wind velocity U rel Relative wind velocity u, v, w Longitudinal, transverse and vertical directions x, y, z Directional motion variables α Angle of attack β Orientation angle β ∗ Wind-cable related angle NOMENCLATURE xi δ ij Kronecker delta ε Roughness characteristic diameter δ ij Kronecker delta Θ(f) Sears function θ Torsional motion ϑ Cable inclination angle θ r Rivulet angular position ζ Critical damping ratio κ Detuning ratio ν Kinematic viscosity ρ Density of air σ i Standar deviation of i τ Non-dimensional time variable φ Mode shape function ϕ Cable-wind angle χ 2 (f) Aerodynamic admittance function ω Cyclic frequency Abbreviations 1DOF One Degree of Freedom 2D Two-dimensional 2DOF Two Degree of Freedom 3D Three-dimensional 3DOF Three Degree of Freedom CEV Complex Eigenvalues Analysis CHBM Covariance Block Hankel Matrix CSB Clifton Suspension Bridge DOF Degree of Freedom ERA Eigenvalue Realization Algorithm FHWA Federal Highway Administration FRF Frequency Response Function HDPE High Density Poly-Ethylene IRF Impulse Response Function IWCM Iterative Windowed Curve fitting Method MIV Motion Induced Vortices NRC National Research Council xii NOMENCLATURE PSD Power Spectral Density POM Proper Orthogonal Mode RMS Root Mean Square SDOF Single Degree of Freedom SVD Singular Value Decomposition WIV Wake Induced Vibrations Chapter 1 Introduction Structural disasters have always served engineering as a trigger for moving forward. The image of the Tacoma Narrows Bridge (TNB) collapsing back in 1940, under the action of only moderate winds, has been one of the most striking failures ever recorded and markedly influenced design practice. Since then bridge spans have expanded greatly with no catastrophic event similar to the Tacoma Narrows failure recurring. This should be attributed to the great wealth of knowledge that was developed in addressing this notorious incident. Yet there are still remaining doubts about the exact mechanism that instigated the dynamic instability. Most wind-structure interaction phenomena, especially where bluff bodies are concerned, possess an intriguing duality. Explaining them in a certain way does not always rule out alternate interpretations. Cooperative phenomena seem to form and it is exactly when such hybrid cases are encountered that our abilities of prediction break down. In any case the design framework currently in use for aerodynamic effects on bridges has generally proved successful, combining theory and wind tunnel experimentation. The latter acts as an effective ‘safety net’ that can warn us of potential aerodynamic problems. But apart from that, many in- novative solutions have been conceived in wind tunnels. The form of many modern bridges are actually owed to them. Through the wind tunnel testing for the Severn Bridge the streamlined bridge section was invented, which allowed more efficient aero- dynamically stable bridges. Nevertheless in a wind tunnel not all parameters of the full-scale conditions can be faithfully reproduced, while size-effects could yield any links between prototype and model inaccurate. Ultimately definitive information about the wind resistance of a bridge can only be taken from the actual bridge. In the current work a combination of full-scale monitoring, theory and wind tunnel tests is utilised in investigating dynamic instabilities that threaten a bridge’s structural integrity. The research begins by considering the old question of ‘flutter’ (flutter being the widely accepted dynamic instability blamed for the TNB collapse) on an existing long 1 2 Chapter 1. Introduction suspension bridge - the historic Clifton Suspension Bridge (CSB). As a matter of fact the CSB belongs to the era of the first European steps in building suspension bridges. Many examples of the time (e.g. Menai Straits Bridge, Brighton Chain Pier Bridge, and Roche-Benard Bridge) had severe wind-induced vibrations leading to destruction or serious damage. Still the issue addressed is not only the specific CSB application but a more generic approach to answering challenges such as ‘Can we predict instabilities for bridges that were not designed against them?’ and ‘Can we predict the true safety margins to compare with our design idealisations?’. Field monitoring of dynamic re- sponses combined with identification methods on the ambient data were performed in order to estimate the vulnerability of the bridge to wind action. Exploring the potential to identify actual full-scale behaviour rather than relying on wind tunnel tests is the the main target of this section of the thesis. Subsequently the interest shifts to another bridge-related issue; the dynamic in- stability of bridge cables and the explanation of their excessive motion first witnessed and reported on Japanese bridges in the early 1980’s. To justify this research further, the TNB collapse is reconsidered. A recent explanation of the collapse says that just before switching into an uncontrollable twisting motion, a pair of the middle cross-tie cables on the bridge had snapped. Such an explanation could well mean that even local parameters can have a major effect on the overall response, so before addressing the overall design of the bridge the response characteristics of the discrete bridge parts should be fully recognised first. It has long been thought that a circular cable, due to perfect symmetry, cannot gallop (as in classical Den Hartog galloping of iced transmission lines). This view has recently been challenged and this study aims to uncover the details characterising the instability generation mechanism known as dry galloping. The specific objective adopted is to analyse the Reynolds number effects, which were previously overlooked. For this purpose a series of large-scale wind tunnel tests was carried out with an aeroe- lastic cable model inclined at different angles and equipped with pressure measuring taps. There have been very few similar large-scale tests in the past and none of them had equipment to measure the aerodynamic forces on a moving model. The layout of the thesis is as follows: • Chapter 2 introduces the aeroelasticity framework necessary to follow all concepts discussed in this work. The literature is critically reviewed while information on both loading characteristics and modelling practices are provided. The descrip- tion of conventional well-known self-excitation phenomena such as galloping and flutter is followed by a review of the more sparse research on dry galloping vibra- 3 tions. A series of controversial points are highlighted and additional justification is given for the main study that follows. • Chapter 3 presents the flutter derivative identification scheme performed on the CSB full-scale ambient vibration data . The chapter commences by introduc- ing background information on the monitoring procedure and the wind condi- tions encountered on the CSB. Subsequently the stochastic subspace identifica- tion method used is explained. Flutter derivative results are compared with other similar sections and eventually an estimate for the flutter critical speed is made. • Chapter 4 is devoted to addressing shortcomings of the current literature regard- ing galloping, quantifying the differences in the results that can be observed with different conditions or assumptions. The correct two-degree-of-freedom quasi- steady galloping analysis is put forward for the first time, identifying the true effect of the structural parameters on the dynamic instability. The analysis is useful in understanding the characteristics of galloping and in interpreting the observed response traces in the following chapter. • Chapter 5 addresses the dry-galloping vibrations of circular cables that differ from the instability considered in the previous chapter. It starts with describing the experimental setup used and the large responses encountered. Next, a discussion of the results regarding the influence of Reynolds number on the dry galloping mechanism is made. The inclination angle appears to have key role in the oc- curence of unstable motion. Discontinuities in the aerodynamic forces originating from intermittent jumps between different flow states along with unusual flow structures are unique features found due to the transitional behaviour. • Chapter 6 concludes the thesis summarising the main findings and contributions of the present research. It also includes suggestions for further work to advance the present investigation. Chapter 2 The Aeroelasticity Framework Long-span bridges are exposed to severe wind action and the main target of this chapter, before getting into the detailed analysis of this work, is to present the main attributes of wind-structure interaction that are essential for the aerodynamic bridge design and highlight yet unresolved issues that are later confronted. Background information at- tempts to present both the loading characteristics as well as modelling practices that have been employed in constructing wind-resistant structures. The interest centres on self-excitation phenomena, which inherently introduce a feedback effect since the actual aerodynamic forcing becomes a function of the flow-induced motion. Such phenomena, where cause and effect are interlinked, naturally pose additional hurdles in their treat- ment. Formulations and theoretical knowledge used in the later chapters are mostly provided here. A key feature is to also present the influence of Reynolds number on flows past cir- cular cylinders, which later is proved to be an essential piece in the puzzle of excessive bridge cable vibrations. For cable vibrations specifically, a number of inconsisten- cies underlying the proposed explanation mechanisms are illustrated for the first time. Throughout the chapter, real bridge examples are given in relation to the instability issues discussed to further justify the needs for and benefits of the current research. 2.1 Wind-induced structural loading In general, the wind approaching a structure can be decomposed into a steady and a fluctuating vector component. In flexible bridges the gust loading falls well in the range of their many low natural frequencies and on top of that even the uniform flow is capable of imposing unsteady pressures, which can dynamically interact with structural motion causing either a detrimental or a beneficial effect. The two mechanisms interact 5 6 Chapter 2. The Aeroelasticity Framework with the net effect being in some cases a nonlinear one, which cannot be fully perceived in the realm of simple linear superposition. There is also the static effect from the flow, which is capable too of leading to failures, as in the case of torsional divergence. What follows should be treated as a synopsis of the basic aerodynamic and aeroelastic theory as included in the classical textbooks of Blevins [1], Simiu and Scanlan [2], Zdravkovich [3] and Dowel et al. [4]. 2.1.1 Static loads The mean wind around a structure engenders static lift (L) and drag (D) forces together with an overturning moment (M). The L, D, M expressions per unit length are given by L = ρBU 2 2 C L , D = ρBU 2 2 C D , (2.1) M = ρB 2 U 2 2 C M , where ρ is the air density, B is a representative sectional dimension, U is the mean wind velocity and C L,D,M are the static lift, drag and moment coefficients respectively. C L,D,M are in probably all shapes functions of the angle of attack α. The scenario of torsional divergence earlier referenced is brought about when C M has the tendency to increase with increasing α. Thus for a structure with finite torsional stiffness, above a critical value, wind will cause an ever-increasing α ultimately leading to destruction. For dynamic loading a quasi-steady formulation considers that Eqs.(2.1) are still valid with U replaced by the relative wind velocity U rel , which accounts for the influence of either fluctuating wind components or structural motion. Complications with this idealisation do exist, since evidently the introduction of rotational velocity waives the uniqueness of such a U rel . 2.1.2 Wind buffeting The unsteady loading imposed by wind turbulence is termed wind buffeting. Wind turbulence, although chaotic and complex in its nature, is briefly characterised by a number of parameters. These are: • The turbulence intensity I i = σ i /U where σ i is the Root Mean Square (RMS) of the fluctuating wind component along i = u, v, w the longitudinal, transverse and vertical directions relative to the wind. 2.1. Wind-induced structural loading 7 • The relevant auto-Power Spectral Densities (PSDs) ˜ S i (f) and cross-PSDs ˜ S i 1 i 2 (f), i 1 = i 2 , where f is the frequency variable and it holds σ 2 i = _ ∞ 0 ˜ S i (f)df. Various PSDs have been proposed in the literature for design purposes [5–7], wherein most cases an efficient description of the fluctuations of strong winds is accomplished through the von K´ arm´ an model of isotropic turbulence [8]. • The length scale parameters x,y,z L i , nine in total, which designate a directional (x, y, z) average size of turbulent eddies in each wind component i. The classical treatise for buffeting loading, utilising the statistical concepts from ran- dom vibration theory was first applied by Liepmann [9] to aircraft wings and later by Davenport to long-span bridges [10, 11]. The formulation, expressing the force com- ponents in quasi-steady terms, attempts to first establish the sectional loading PSDs from the sectional (or point) directional wind PSDs. Further the generalised loading of each mode is recovered by integration along the length, utilising the modal shape and wind correlation information. In the frequency domain the PSD expression for vertical response in each mode j is given by [11] j ˜ S z (f) = (ρBU) 2 4 _ _ C D + dC L dα _ 2 ˜ S w (f) + 4C 2 L ˜ S u (f) _ χ 2 (f)|H j (f)| 2 |J j (f)| 2 , (2.2) where H j (f) is the Frequency Response Function (FRF) of mode j, |J j (f)| 2 is the joint acceptance of mode j and χ 2 (f) is the admittance function. The joint acceptance function is the means of translating the point-like load to a span-wise load, while the admittance function should be deemed to be a correction factor to compensate for the frequency dependence of the instantaneous aerodynamic load [12, 13] and for its sectional variation. The first to theoretically evaluate such a function was Sears [14]. For a vertical gust w that is a sinusoidal time function of the form w = w 0 e ı2πft (note ı here is the imaginary unit), Sears derived the corresponding oscillatory lift on an airfoil as L(t) = ρBU 2 2πw 0 Θ(f)e ı2πft , (2.3) where Θ(f) is called the Sears function. Restating Eq.(2.3) in the frequency domain it becomes ˜ S L = (ρBUπ) 2 χ 2 (f) ˜ S w , (2.4) yielding quite easily that |Θ(f)| 2 = χ 2 (f). Many other empirical options exist for envisaging χ 2 (f), and there remains an ambiguity over which is the most appropriate to use. In studies with long bridges with very low fundamental natural frequencies the contribution from aerodynamic admittance is usually conservatively neglected [15], though in some cases it is essential for the reliable response estimation [16]. As presented 8 Chapter 2. The Aeroelasticity Framework in Hay’s [17] analysis for the Wye and Erskine Bridges, during narrow band vertical response for the vertical displacement RMS values σ z , the buffeting action should set σ z ∝ IU 2.83 . 1 (2.5) The recovered relations from full-scale monitoring of the actual bridge responses were actually containing a somewhat lower exponent than 2.83. In any case Eq.(2.5) shows that response is proportional to turbulence intensity and is only asymptotically di- verging. A final factor that is worth mentioning is the effect of signature turbulence, which consists of wind fluctuations imposed not by the approach natural flow but by the actual submerged structure or elements ahead of it. Any shape, unless very well streamlined, will produce signature turbulence; however its effect in standard buffeting analysis is generally ignored. 2.1.3 Vortex shedding Vortices trailing behind bluff bodies are a very common picture in nature. Of course the picture is not alone but is accompanied by forces with well defined dynamic char- acteristics. Strouhal [18] was the first to observe and suggest that the frequency of the wake oscillations follow Sr = f v d U , (2.6) where Sr is the shape-dependent Strouhal constant, f v the vortex shedding frequency and d the across-wind dimension. An across-wind force with frequency f v is exerted on the submerged bluff body and as logically expected if the body is allowed to move across- wind with a structural frequency f c , then the two frequencies’ coalescence will result in resonance. Actually the classical conception of the wake structure (two counter-rotating vortices shed alternately from each side during one cycle) implies that an along-wind force also exists with twice the shedding frequency and is able too of causing resonance. Unfortunately there are slight inconsistencies in this idealisation. The example of the circular cylinder will allow to effectively present them. For a circular cylinder Sr is strongly a function of Reynolds number (Re) Re = Ud ν , (2.7) where ν is the kinematic viscosity. For subcritical values (i.e Re<≈ 10 5 ) customarily Sr is considered to have a value of 0.2. Still this is not strictly true, the actual behaviour is illustrated in Fig. 2.1 and is governed by discontinuity intervals as well as a strange 1 Hay [17] does not explicitly state which component the turbulence intensity I refers to. 2.1. Wind-induced structural loading 9 inversion of the monotonicity. The critical range behaviour is of far greater interest (and variation) in large-scale engineering applications and will be discussed in detail later. 0.00 0.17 0.19 0.21 0.23 S r 0.02 0.04 √ Re 1/ 2E4 2E3 1000 500 300 Re retarded transition parallel shedding 5000 1300 240 230 180 360 0.06 Figure 2.1. Mapping of Strouhal number against Reynolds number in the subcritical range. Adapted from Fey et al. [19]. A great number of experiments have revealed unique features of the aeroelastic char- acter of the vortex shedding loading and response. The tests by Feng [20] were among the first to demonstrate a series of intriguing nonlinear features, such as the capture of f v from the structural vibration frequency in an extended range near resonance during the phenomenon called ‘lock-in’. Seminal reviews on vortex phenomena refer to this study [21–24]. Feng performed his wind tunnel tests with freely oscillating (restrictively across-wind) cylinders for varying structural damping values in the subcritical range. He measured the vibration amplitude ¯ Y (scaled with cylinder’s diameter d), f v , f c , the phase difference η of the two throughout the lock-in region and for limited runs the across-wind force coefficient dynamic amplitude ¯ C Y . Part of his results are presented in Fig. 2.2, where all recorded variables are plotted as functions of the reduced velocity U r =U/f c0 d, with f c0 being the still air structural frequency. As shown in the figure the lock-in behaviour establishes itself when approximately reaching the Strouhal res- onance indicated earlier. From then on, f v remains equal to f c until the point where it jumps back to f v ≈ 1.4f c following naturally Eq.(2.6). For other shapes the lock-in re- gion is double-sided extending also to frequencies <f c . It is noteworthy that hysteresis exists in the self-limiting amplitude response of Fig(2.2), with magnitude depending on whether the wind velocity is increasing or decreasing. The upper branch attained for increasing wind cannot be reached from rest displaying the interesting feature of depen- dence on the ‘loading’ history. For the few runs that ¯ C Y is recovered it is obvious that 10 Chapter 2. The Aeroelasticity Framework its value is strongly amplified. Although not shown here, similar amplification holds also for the along-wind dynamic forcing. It should be pointed out that the extreme re- sponse and forcing do not match the initiation of the lock-in zone but they are situated close to its middle. Finally the phase difference values along with their discontinuous jump right in the heart of the synchronised regime, have become a great matter of controversy and added trouble in the modelling task. In any case phase values indicate both in phase and in quadrature forcing components. The detailed reasoning on phase jumps was provided by Williamson and Roshko [22], who identified different modes of vortex shedding and abrupt transitions between them in the amplitude-structural frequency-wind velocity parameter space. Later work of Williamson with Jauvtis [25] and Morse [26] completed, for the time being, the shedding mode characterisation. Sr =0.198 Y - U 5 6 7 8 4 Y - U r =U/f c0 d 0.5 1.0 1.5 0.4 0.2 0.6 C Y 1 2 0 0˚ 50˚ 100˚ η f v f c η Y - Y - from rest - d C Y - f v,c /f c0 Figure 2.2. Free vibration tests for a circular cylinder. Adapted from Feng [20]. 2.1. Wind-induced structural loading 11 Outside the 1:1 (f c :f v ) synchronisation range there is also a multitude of interesting phenomena. Williamson and Roshko [22] identified a region of 1:3 subharmonic reso- nance and reasoned that wake stability considerations prevent 1:2 similar occurrences. As a matter of fact they point that subharmonic resonance should be possible for any 1:n ratio, with n being an odd number. Experiments from Durgin et al. [27] partly con- firm this view by finding large response of free vibrating cylinders in such 1:3 regimes but not in 1:2 or elsewhere. Still it is worth noting that such phenomena are very rare and according to classical synchronisation theory should readily vanish when n>3 [28]. A final unresolved feature picked for this synopsis consists in the force-displacement relation outside lock-in. As presented in Fig. 2.3 the displacement, top signal, is virtu- ally a pure sinusoid (at f c =9Hz) of very stable magnitude. On the other hand the force causing this displacement, illustrated in terms of the representative transverse pres- sure tap trace in the lower signal, shows a strong modulation and a different frequency (i.e. f v =13Hz). Thus a question surfaces on how these two signals can combine in a cause-effect relation. According to Minorsky [29] this seems to be a classical example of asynchronous excitation. A system possessing a stable focus point (i.e. initial equi- librium point) followed by two adjacent limit-cycles (such can be a system expressed by a polynomial of at least fifth order) could bifurcate and rest to the outer limit-cycle when a suitably sized periodic action of random frequency is applied. Figure 2.3. Circular cylinder vibration phenomena past the lock-in range after Ferguson and Parkinson [30]. U=5m/s. Top: vertical response record with f c =9Hz. Bottom: surface pressure at transverse tap, f v =13Hz. Modelling of vortex wake phenomena is far from complete. Leaving aside the purely computational treatises of discrete vortex potential flow models and the numerically 12 Chapter 2. The Aeroelasticity Framework solved Navier Stokes equations, what remains to be the most efficient attacking tool for our descriptive low-dimensional studies are the so-called wake-oscillator models. Their conception belongs to Bishop and Hashan [31, 32] who were the first to suggest that a cylinder’s wake behaves like a mechanical oscillator. Taking into account the non-linear behaviour proven above, and particularly the lock-in and limit-cycle (the stationary cylinder vortex shedding is what is idealised as limit-cycle) attributes, a ‘wise’ compatible modelling choice from the world of mechanics would be a Van der Poll oscillator [33, 34]. Such a model was first implemented by Hartlen and Currie [35] and acquires the form d 2 Y dτ 2 + 2ζ dY dτ +Y = ρd 2 ω 2 0 8π 2 Sr 2 m C Y , d 2 C Y dτ 2 −a 1 ω 0 dC Y dτ + γ ω 2 0 ( dC Y dτ ) 3 +ω 2 0 C Y = b 1 dY dτ , (2.8) where Y is the across-wind motion scaled with d, C Y is the instantaneous across- wind force coefficient, τ=2πf c t is the non-dimensional time variable, ζ is the critical structural damping ratio in still air, m is the mass per unit length, ω 0 is the ratio of shedding to structural frequency f v /f c and a 1 , γ, b 1 are empirical constants to be fitted from experiments. This archetypal form of the model assumes nonlinear phenomena originating from the fluidic Van der Poll oscillator and being driven, in the case of motion, by a linear coupling motion-dependent term. Hartlen and Currie employed the first approximation solution of Kryloff and Bogoliuboff [33] and recovered parts of the behaviour in Feng’s [20] experiments. Amplitude or phase hysteresis was not apparent in analytical results but this is only due to the solution method employed. Later variants of the model altered the fluid stiffness term, the fluid damping term, the structural damping term or the form of the coupling-forcing term in Eqs.(2.8) improving each time the match to experiments,(for a review see [36–39]). The same model was also used on a less phenomenological basis, being derived from first principles, having though C Y substituted by a ‘hidden-flow’ variable [1]. The sophistication of the model although extensive has been little concerned with a great branch of phenomena, such as chaotic and quasi-periodic oscillations. The preceding discussion tacitly assumed smooth flow conditions. The introduction of turbulence would make vortices lose coherence along the body length and resonant peaks in the C Y spectrum degenerate, broaden, and evidently waive their efficacy in setting up motion. Somewhat similar effects can be brought upon by surface protu- berances, surface roughness, long splitter plates or more complex additions such as wavy separation lines, or spirally arranged bumps [40, 41]. Earlier it was seen that the ‘enhancement’ of vortex shedding during large response leads to amplification of 2.1. Wind-induced structural loading 13 the dynamic loading. Similarly when the vortices shed by a body start losing their strength, the mean pressure drag C D is expected to decrease. The most prominent example of this rule is the circular cylinder and its behaviour along the Re range. For bridges, although the British ‘revolution’ of streamlining sections was thought to adequately handle vortex phenomena [42], this was not actually the case. Vortex- induced vibrations have been quite systematic in bridges. The Wye Bridge had such occurrences, but only of small amplitude, while the very similar Erskine Bridge did not [17]. The discrepancy was reasoned in view of the higher turbulence intensities measured at site for the latter, which as referenced earlier could disorganise the vortex formation and propagation processes 2 . The Kessock Bridge [44] had similar observa- tions with winds of only quite low turbulence intensities exciting moderate amplitude bending oscillations. The extremely wind-prone Deer Isle Bridge [45, 46] sustained vor- tex oscillations in probably all of its configurations (being retrofitted or repaired in many instances). For the Storebælt Bridge Larsen et al. [47] presented the excessive form the phenomenon acquired, with Fig. 2.4 being indicative of amplitudes observed. The Shanghai Lupu Bridge [48] is a unique reference, being an arch bridge with vortex issues while on operation. Figure 2.4. Storebælt Bridge vortex-induced vertical motion at max (left) and min (right) of the amplitude cycle, after Larsen et al. [47]. Encircled is a parked van with its view distorted due to motion. The presentation in this section was mainly based on the circular cylinder paradigm and one should expect that flexible bridges possessing cables, that can most of the times fall into the circular cylinder category (disregarding only for the time being any incli- nation), should have issues regarding them. Fortunately as presented by Virlogeux [49] 2 This is not to be confused with the case of turbulence strengthening vortices as Matsumoto et al. [43] observed on some bridge section types. Their analysis also includes non-classical Motion- Induced-Vortices (MIV) making up for the seeming inconsistency. 14 Chapter 2. The Aeroelasticity Framework it can be easily shown, by considering typical values, that large amplitude vortex shed- ding should not be a concern for bridge cables, since lower mode excitation is restricted to quite low wind speeds of consequently limited energy content. Nevertheless, higher- modes would receive large vortex forcing but due to their higher damping would not produce excess motion. But the stresses exerted in the cables apart from being func- tions of amplitude, they are also depending on curvature, which evidently increases in higher modes. Thus even for low-amplitudes but higher mode persistent response, large stress loading cycles will result, raising fatigue concerns. The record of a specific cable in the Saint-Nazaire Bridge that initially sustained fatigue damage, was replaced and later was found to be driven in large higher mode vortex oscillations, should be deemed to be an indicative example. 2.1.4 Galloping Classical galloping refers to across-wind motion arising due to the so-called ‘incidence effect’, which translates to a wind forcing contribution originating from variations in the effective flow incidence angle. It is considered equivalent to the condition dC L dα +C D < 0 . (2.9) Rotational asymmetry is evidently a basic requirement in the operating mechanism. A simplistic view of galloping is provided in the inset sketch of Fig. 2.5, where a body in a flow of velocity U is moving downward with velocity ˙ y, thus altering according to definition, the effective flow incidence. The shear layer on the lower side moves closer to the body, therefore getting more curved, while the upper side shear layer moves away from the body and becomes less curved. As a result a net downward pressure force is acted across the side faces (in contrast to an upward force that a streamlined airfoil would experience) further assisting motion. It will be shown in Chapter 4, Eqs.(4.7&4.8), that in this classical scenario the threshold reduced wind velocity U 0 for setting off galloping, calculated by means of linear theory, is proportional to structural damping. Thereof a diverging response should result. In reality a limiting mechanism operates to set response into a steady state. Parkinson et al. [50, 51] were among the first to implement nonlinear concepts to evaluate steady galloping amplitudes that well match experimental observations. Their analysis recovers that galloping increases roughly proportional to reduced velocity U r , as shown in Fig. 2.5, with also hysteresis effects and jump behaviour emerging in the range noted by a dashed line and bounded between arrows. Later Novak [52–54] proposed the notion of ‘universal response curves’ after rescaling axes in Fig. 2.5 with 2.1. Wind-induced structural loading 15 Figure 2.5. Typical galloping response curve for a rectangular prism. Inset the classical galloping mechanism illustrated. S c =ζm/ρd 2 . This dimensionless factor, named after Scruton, was first proposed by Scruton [55] as the influencing parameter against most aeroelastic instabilities. Novak extended on the distinction between ‘hard’ and ‘soft’ galloping oscillators, soft being the ones able to gallop from rest while hard the ones in need of an initial hard ‘push’ to get into motion, presenting also a number of special cases where stable sections can turn into weak hard ones. Galloping forces in general terms should be considered as the product of flow- afterbody interaction. Parameters altering any of the two constituents inevitably will affect the instability characteristics. Introduction of turbulence for instance can turn an unstable section to stable or the opposite, with any changes being strongly shape- dependent. Similarly insertion of a splitter plate in the near wake can be quite dramatic concerning the separation process [56, 57] and induce galloping in cases where it would not be expected, as in the examples of a rectangular section with along to across- wind dimension ratio higher than 3 and the perfectly symmetric circular cylinder. The splitter plate influence on galloping is the opposite of the influence earlier quoted on vortex shedding, thus a question is raised on the link between galloping and vortex shedding. In most practical applications in wind, the regions where the two phenomena become dominant are well separated, with vortex shedding being confined in relatively low reduced velocities and galloping appearing later only for much higher U r values. A typical example of aeroelastic response would be expected in the form of Fig. 2.6, with vortex shedding giving a response ‘hump’ near U r =5 and after a quiescent transition period galloping taking over. 16 Chapter 2. The Aeroelasticity Framework 0 5 10 15 20 0.1 0.2 U r Damping ratio ζ: 0.37 % 0.76 % 1.40 % 2.12 % 4.40 % U 1 Y - Y - 2 Figure 2.6. Response of a rectangular prism with side ratio 2/1 against reduced velocity for varying critical structural damping ratio. The prism has Sr=0.081 that sets the relevant U r for vortex resonance at ≈12.34. Adapted from [53]. Actually the specific example of Fig. 2.6 although explained in this expected way even in well respected textbooks, see [1], it contains some paradoxical features. Con- trary to the aforesaid descriptions what seems like galloping here starts abruptly at U r ≈11 for all different values of structural damping, while in the ζ=4.40% case it attains a rather unexpected decay. Taking into account that the rectangular prism in hand has Sr=0.081, setting the relevant U r for vortex resonance at ≈12.34, this paradigm is probably a good indication of the rather complicated form that hybrid vortex-galloping oscillations may obtain. For ζ=0.37% the galloping threshold is eval- uated at U 0 =5.2 [53] but oddly is inhibited up to where vortex shedding is regularly located. On the other hand for ζ=4.40% it follows U 0 =61, which explains the non- increasing response character. The interaction range includes many more possibilities to be pursued in later chapters. In the context of the present work the term galloping will be used to characterise any motion-triggered translational aeroelastic instabilities regardless of orientation and extending to also cover the combined participation of orthogonal motion. Galloping tends to be considered of less interest in cable-supported bridges, still there were a number of instances in design or construction where it did show up. Ac- cording to Virlogeux [58], the hexagonal bunches of strands to be installed on the Normandy Bridge were abandoned due to galloping concerns that arose while in wind tunnel testing. Instead a circular strand distribution was promoted, which was also cov- ered by a high density polyethylene (HDPE) sheath with helical fillets. Wardlaw [59] 2.1. Wind-induced structural loading 17 reports the wind tunnel investigation of the single tower of the Aratsu-Ohashi cable- stayed Bridge where an apparent galloping susceptibility was documented throughout the construction stages. A more controversial appearance should be the one docu- mented in Deer Isle Bridge with deck motion amplitudes reaching up to 6100mm [46]. Although also other loading mechanisms could be held responsible for this incident the characteristics of the bridge as presented by Cai et al. [60] readily support the galloping scenario. Galloping events and the countermeasures for their treatment, for a series of arch and truss bridges e.g. Burton Bridge, Bras d’Or Bridge and Commodore Bary Bridge are presented by Wardlaw [59, 61]. Finally the case with excessive vibrations recorded on the circular iced hangers of Storebælt Bridge [62], reminiscent of iced trans- mission line galloping, is another phenomenon worthy of engineering attention along with other less evident similar instances. 2.1.5 Flutter Flutter is the aeroelastic instability that naturally follows galloping. It is nominally of divergent character, rapidly building up, and has provided engineering history with some of the most spectacular failure pictures. In different forms it is thought to have been met first in slender bridges, dating as long as two centuries ago, and much later on the torsionally weak wings (and tails) of World War I fighter aircraft, while nowadays it is still one of the most serious concerns of aerodynamic design. Although many classifications have been used by different authors, here the broadest categorisation into coupled (or classical) flutter and Single-Degree-of-Freedom (SDOF) flutter will be introduced. Figure 2.7. (a) Displacements and aeroelastic forces on a thin airfoil; (b) Displacements and aeroelastic forces for a bridge section Coupled flutter was initially used to name the combined torsional-bending insta- bility of airfoils. The phenomenon necessitates for a torsional and a bending mode to oscillate at the same frequency but with a decisive phase difference between them, that allows their cooperation to extract energy from the wind. Theodorsen [63] laid 18 Chapter 2. The Aeroelasticity Framework the basis for ensuing flutter analysis by estimating the self-excited lift force (L se ) and pitching moment (M se ) of a thin airfoil section immersed in an incompressible fluid flow, while performing small amplitude harmonic vibrations of cyclic frequency ω. The model problem is illustrated in Fig. 2.7(a), where typically the drag force and displace- ment are unimportant and additionally it is assumed that the shear centre and chord centre coincide. Theodorsen’s derivation established L se , M se as linear functions of translation h, rotation α and of their first and second time derivatives, ˙ h, ¨ h, ˙ α, ¨ α, where each overdot denotes one differentiation with respect to time. His well-known solution is given as follows [64, 65] L se = πρb 2 [U ˙ α + ¨ h] −2πρbUC(k)[Uα + ˙ h + b 2 ˙ α] , M se = −πρb 2 [U b 2 ˙ α + b 2 8 ¨ α] +πρb 2 UC(k)[Uα + ˙ h + b 2 ˙ α] , (2.10) where k=bω/U is the reduced cyclic frequency, based on the half chord length b, and C(k) the complex Theodorsen circulation function that is assigned to time delays, and is analytically expressible in terms of Bessel functions. C(k) becomes 1 for static conditions, i.e. k →0, reverting to the quasi-steady formulation. It was early realised [66–68] that direct application of Eqs.(2.10) to the analysis of bridges, owing to their generally bluff sections, will yield inaccurate results. Hence the rationale of expressing aeroelastic forces as linear k-dependent motion functions was preserved but C(k) was replaced by a more reliable empirically determined substitute. Many options [69–71] toward defining a reliable such substitute were formulated but the one originating from Scanlan and co-workers [72, 73] became dominant. The Scanlan approach in its latest amended form [74], expresses the self-excited force and moment components for the bridge section in Fig. 2.7(b) as L se = 1 2 ρUB _ KH ∗ 1 ˙ h U +KH ∗ 2 B ˙ α U +K 2 H ∗ 3 α +K 2 H ∗ 4 h B +KH ∗ 5 ˙ p U +K 2 H ∗ 6 p B _ , D se = 1 2 ρUB _ KP ∗ 1 ˙ p U +KP ∗ 2 B ˙ α U +K 2 P ∗ 3 α +K 2 P ∗ 4 p B +KP ∗ 5 ˙ h U +K 2 P ∗ 6 h B _ , M se = 1 2 ρUB 2 _ KA ∗ 1 ˙ h U +KA ∗ 2 B ˙ α U +K 2 A ∗ 3 α +K 2 A ∗ 4 h B +KA ∗ 5 ˙ p U +K 2 A ∗ 6 p B _ , (2.11) where K=Bω/U is the reduced cyclic frequency, based on the full deck length B this time, and H ∗ 1−6 , A ∗ 1−6 , P ∗ 1−6 are the so-called flutter derivatives, which are functions of K and are derived by means of sectional free or forced wind tunnel tests. No inertial contributions are considered, since for heavy bridge decks in air they should 2.1. Wind-induced structural loading 19 be minimal. P ∗ derivatives linked with D se were a later addition to the formulation, which actually proved critical for explaining the aeroelastic behaviour of the Akashi Kaikyo full bridge model [75]. The linearisation concept utilised in deriving Eqs.(2.11) should be applicable for small structural deflections, corresponding to only incipient instability action, and in the absence of concerted vortex shedding with its subsequent strong nonlinear characteristics. The main attribute enabled by coupled flutter is having motions (one of them being necessarily torsional) which when autonomously considered would behave stably, i.e. have positive effective damping, but in common operation would exchange energy between them through coupling terms H ∗ 2,3,5,6 , A ∗ 1,4,5,6 , P ∗ 2,3,5,6 with a total positive net energy effect for their system. Well streamlined bridges such as Severn, Lillebælt, Burrard Inlet, Humber and Bosporus should exhibit catastrophic coupled flutter at very high wind speeds, when interestingly part of their direct aeroelastic forces (velocity products with H ∗ 1 , A ∗ 2 , P ∗ 1 ) contribute positive aerodynamic damping that adds to the structural damping [76]. Note that these outcomes were derived only in wind tunnel tests, and design wind speeds nowadays by far exceed the recorded operational envelopes of modern real long span bridges, making the coupled flutter phenomenon a quite improbable event. Still, complying with the strict flutter design guidelines has imposed shape modifications on major modern bridge prototypes, referring in short to the slot and stabiliser additions in the Akashi Kaikyo Bridge, the slot addition in Zhejiang Xihoumen and Tsing Ma bridges [77], and the fairing addition to the Ting Kau Bridge [78]. SDOF flutter should more accurately for the onomatology adopted in this work be termed as torsional flutter, since the galloping convention introduced earlier should cover any exclusively translational self-excited response. The mechanism operation re- lies, as in incipient galloping, on an aerodynamic negative damping effect which reduces structural damping and ultimately turns their sum’s sign to negative. Apparently from Eqs.(2.11) A ∗ 2 becoming positive serves this goal. This function which is a typical characteristic of many bluff sections previously used in bridges (e.g. H-sections [73]), constitutes the most striking difference between airfoil and bridge behaviour. Thus historically SDOF flutter was realised as a separated-flow phenomenon. Model stud- ies have shown that there could be cases where a strongly positive A ∗ 2 coexists with a tendency toward torsional-bending coupling. Then the intrinsic proclivity for torsional flutter may drive the participation of vertical motion too, establishing optical (but not functional) resemblance with coupled flutter. Further on bridge flutter, its interaction with turbulence remains mystifying. Name- ly a retardation of divergence of the full bridge response was early witnessed [12, 76] as qualitatively illustrated in Fig. 2.8(a). The phenomenon was initially accredited to the 20 Chapter 2. The Aeroelasticity Framework R M S d i s p l a c e m e n t Wind velocity U Laminar flow Turbulent flow (a) 0.2 -0.2 0 2 4 Turbulent flow Laminar flow fU/B A 2 * (b) Figure 2.8. (a) Quantitative difference of response characteristics for a full bridge under different flow conditions. Adapted from [59, 76]; (b) A ∗ 2 from wind tunnel tests for a torsionally unstable bridge section under laminar and turbulent flow conditions. Changes appear minimal to sustain any substantial modification in the flutter behaviour. Data after [79]. effect of turbulence on flutter derivatives themselves, but a series of later studies [74,79] proved that any alterations were minimal. A specific example for a torsionally weak bridge section exposed to both laminar and turbulent flow conditions, is presented in Fig. 2.8(b). Shown changes in A ∗ 2 are negligible, with even a minutely earlier negative crossing for the turbulent scenario that should contradict experience. Other forms of reasoning the flutter-turbulence interaction consist of multi-modal behaviour that ‘spreads’ the fluttering mode’s energy into other modes with various damping and frequency characteristics, essentially setting up a complex tuned damping system [12, 76]. Finally explanations were acquired by means of the distracting action of turbulence on the span-wise coherence of aeroelastic loading [80], although Scanlan also pointed out that there is a possibility for such inhomogeneity to act detrimentally if local extreme excitation regions, e.g. of very positive A ∗ 2 , are introduced. 2.1.6 Wake-induced loading Wake-induced loading refers to wind forces exerted on a body when this is situated in the unsteady wake of another upstream bluff body. Meeting a ‘structured’ disturbed stream of diffused and convected vorticity, when not far downstream of the source, seems quite different than facing directly a fully developed turbulent flow, which also has unsteady components but any localised imprints in it have been smeared out. Notwithstanding the intuitive discrepancy, the forcing mechanism has been historically perceived as only a special case of the previously presented instabilities and have been explained on the same grounds. The term wake galloping is customarily employed to 2.1. Wind-induced structural loading 21 characterise any excessive response events where wake loading seems to occur, still Assi et al. [81] recently, revisiting an old problem set forward by Zdravkovich [82], exempli- fied the case where a tandem pair of circular cylinders, not far apart (distance/diameter, l/d=4), performs vibrations inherently different from common galloping. Galloping should be successfully attained by quasi-steady theory, which strongly relies on the mean wind velocity distribution. In the Wake-Induced-Vibrations (WIV) of Assi et al. the unsteadiness brought upon by the vortices of the upstream obstacle are the critical driving parameter and not the mean wind velocity distribution, while on the other hand the phenomenon is non-resonant and induces a wake stiffness (or frequency) attribute that will not match vortex shedding as presented earlier. Such phenomena, and any interaction phenomena in general, will not be pursued further, but it is of relevance for this work to shortly present them. A very interesting feature in modelling wake interference effects is that ordinary modelling ‘tools’ can capture their non-conservative nature once endowed with a ‘mem- ory’ effect, which assumes the form of an artificially induced time-lag between motion and aeroelastic force [83, 84]. This is striking when considering the discontinuous, or even bistable [85] character observed. Bridge cables in parallel arrangements and on the lee side of bridge towers, for certain wind directions, could evidently fall in the realm of wake-induced response. Preventing such events consists of spacing cables far apart. A distance of more than five or six diameters is thought to be an adequate remedy ac- cording to Matsumoto et al. [86] and Tokoro et al. [87]. Records of large responses that should be possibly accredited to wake effects can be found in many modern bridges. In the Akashi Kaikyo Bridge, during the construction stage, hangers spaced at l/d=9 vibrated violently [88]. In another case of inclined stays this time, on the Second Severn Crossing with even greater inter-cable distances (approximately 4m), relatively large cable vibrations were witnessed in February 1999. Wind was blowing almost in parallel to the cable fans, exciting all cables into first mode large transverse motions. The phenomenon looks similar to tube arrays response, as in ‘breathing’ [49] for instance, though the large spacing discourage engineers of linking it to a wake-related source. On the Øresundsbron Bridge where inclined twin cables of d=250mm spaced 670mm apart were used as stays, many extreme cable vibrations occurred [89]. The small spacing, determined through a series of scaled wind tunnel tests though, is thought to be of some connection to the events. Yet it is quite difficult to conclude, when even crude details (e.g. of the vortex shedding process) due to the individual moving cable’s inclination remain in ambiguity. 22 Chapter 2. The Aeroelasticity Framework 2.1.7 Rain-wind Instabilities The cooperative action of rain and wind can produce forcing, which exceeds the previous categorisation. The vibrations of yawed cable-stays in the Meiko-Nishi Bridge during erection, reported by Hikami and Shiraishi [90], were the first to be attributed the designation rain-wind instabilities. The name derives from the initial observation that as soon as either rain or wind ceases, forcing dies down. Actually the contribution of rain was even earlier postulated to be strongly influential in aeroelastic loading owing to the monitoring work of Hardy and Bourdon [91] on transmission lines. Nowadays most of the vibration recordings in many cable-stayed bridges concern the cables and are classified as such phenomena, so evidently they deserve dedicated space in this study. Figure 2.9. (a) Rivulet formation on the circular cable section; (b) Inclination geometry of the inclined and yawed to the flow cable. What seems to accompany the instability is the formation of water rivulets along the cable length, as in Fig. 2.9. The formation is a process governed by many param- eters such as the rainfall intensity, the cable inclination, the wind yaw, the treatment and material of the cable surface and naturally the wind velocity and motion frequency. There still remains some abstruseness on the subtle characteristics of these vibrations, while in some cases contradicting results have been brought forward. Hikami and Shi- raishi [90] performing wind tunnel tests on cable geometries with α=45 ◦ , β=±45 ◦ , according to Fig. 2.9(b), observed that it is the upper rivulet motion that contributes the excitation force with a lower rivulet only adding positive damping. Similarly Fla- mand [92] for α=25 ◦ , β=20 ◦ –50 ◦ attained large vibrations when an upper rivulet (with no lower one though) formed and oscillated in the circumferential direction. Further attaching false fixed rivulets at the place where the real ones were previously seen, no 2.1. Wind-induced structural loading 23 instability was found, thus yielding that it is the synchronised motion of the rivulet that causes vibrations and not simply its appearance. Oppositely Bosdogiani and Oli- vari [93] adopted the view that the motion of the liquid rivulets is not indispensable, and by attaching rigid bars, of realistic size and shape, where rivulets would normally form proved that classical galloping could emerge. On the same grounds Matsumoto et al. [94] using a horizontal (α=0 ◦ ) circular cylinder, generically yawed (β=0 ◦ –45 ◦ ) to the flow, performed an extensive parametric study varying the angular position of an artificial glued upper rivulet, in order to assess the effectiveness of the rivulet place- ment. It is interesting to note that they obtained an unstable response even for the typical non-yawed, non-inclined cross-flow scenario. Their results do not agree with the rain-wind tests of Cosentino et al. [95] on a geometrically similar set-up. Still the impli- cation of rain-wind vibrations on non-inclined cables was also made by Verwiebe [96], who quotes the examples of hangers in two arch bridges that sustained such events just before being commissioned. Verwiebe for cable orientations with α=30 ◦ , β=0 ◦ –90 ◦ presents three different vibration mechanisms underlying the rain-wind phenomenon. In two of them the participation of both upper and lower rivulets is mandatory, with resulting trajectories being planar across or along wind, depending on the symmetry of the rivulets’ movement. For his remaining third mechanism only the lower rivulet drives response resulting in an elliptic motion close to across-wind that can ultimately cross over to purely across-wind when an upper rivulet forms. Notwithstanding the discrepancies there are features unanimously accepted. Inci- dents show up in a velocity restricted range visually similar to the vortex shedding response of Fig. 2.2. Yet unlike vortex shedding the bounds of this range are inde- pendent of the motion frequency [90, 96] becoming functions of simply the wind speed. Close to this observation lies that Reynolds numbers are of the order of 10 4 –10 5 , which is nominally subcritical or very early critical. Amplitudes can reach up to several meters i.e. >10d. There is some consistent repeatability of vibrations for a range of effective β ∗ (see Fig. 2.9 for the angle definition) between approximately 20 ◦ and 35 ◦ , but separately α and β are also relevant due to the gravity force influence on the water rivulet motion. Flamand and co-workers [95, 97] recorded the water thickness and lift force characteristics during large rain-wind induced excitation, as presented in Fig. 2.10. Both plots are noisy and have intermittent ‘firing’ intervals, raising questions as to how synchronisation can be feasible in such erratic waveforms. The self-exciting character of rain-wind vibrations renders them similar to galloping and alike formulations have been used for their modelling. Yamaguchi [98] employing quasi-steady theory, showed that a sliding upper rivulet can oscillate along the cir- cumference with a frequency originating from aerodynamic stiffness. As this frequency measure varies with increasing wind speed, it approaches the cable frequency with their 24 Chapter 2. The Aeroelasticity Framework 12 7 2 -3 1 2 3 4 Lift force Displacement time (s) c m , N / m 10 12 14 16 18 20 22 24 26 28 30 17 19 21 23 time (s) p o s i t i o n ( ° ) r i v u l e t U=11.5m/s, α=25°, β=50° α=25°, β=30° Figure 2.10. Upper water rivulet mean angular position during rain-wind vibrations and lift force, displacement time series for a different large response configuration. Adapted from Flamand et al. [97]. coalescence setting off instability. This model along with its later successors does not produce the random-like features shown in Fig. 2.10, however it has been deployed with relative success in modern cable-stays (e.g. [99]) for treating a problem that is not yet resolved in field. A long list of large-scale real events can be presented. The examples of Puente Real Bridge, Veteran’s Memorial Bridge, Fred Hartman Bridge, Erasmus Bridge, Dubrovnik Bridge, Farø and Higashi-Kobe Bridge are only indicative of how widespread the phenomenon is. The actual case of Higashi-Kobe Bridge is of particular interest. Fig. 2.11 from Kitazawa et al. [100] presents a summary of the preliminary cable wind tunnel tests that were performed in order to mitigate the rain-wind response of the proposed bridge design. Assuming that regular spaced protuberances along the cable circumference will inhibit formation and motion of rivulets, it was found that in- deed they are very effective in restraining large motions, keeping response low relative to the plain smooth cylinder option. This was the verdict for all the tested wind speed range and for both rainy and dry conditions. Thus such an aerodynamic measure was applied to the newly erected bridge for first time ever. Still there is another feature worthy of note in Fig. 2.11. For the cable encased in a smooth circular duct, large vibrations were attained not only under rain but also in dry conditions. The frequencies (at around 1Hz) are far off to reason direct K´ arm´ an vortex shedding resonance, and no water on the cable surface exists to form aerodynamically unstable rivulets. Thus the origin of this new type of response sets forward a new puzzle to later consider. Closing up this section it should be referred that large vibrations on the protuberance-equipped cables of Higashi-Kobe Bridge did occur for an extreme velocity outside the tested range of the preliminary tests (i.e. around 40m/s) [101]. It is the empiricism in the current state of knowledge that clearly necessitates for additional testing and understanding of underlying mechanisms in order to get into a state of successfully predicting problems. 2.2. Circular galloping: myth or true? 25 Figure 2.11. Resume of wind tunnel test results on the proposed cables in the Higashi- Kobe Bridge. Improvement in aerodynamic performance is apparent for the solution with protuberances under all conditions. Adapted from Kitazawa et al. [100]. 2.2 Circular galloping: myth or true? The unheralded diverging response illustrated for the plain cable in Fig. 2.11, seems to resemble the classical galloping of Fig. 2.5. However a nominally perfect circular body cannot fit galloping per se. Due to perfect symmetry any incidence effect is expected to cancel out. As Parkinson argues in his cogent galloping review [23] for such shapes “their afterbodies do not interfere with the separated shear layers and the subsequent vortex formation, so that only vortex-induced vibration from rest will occur for elastically-mounted cylinders, and galloping is not an issue”. This statement puts forward a fundamental challenge: Is galloping possible for a circular cylinder? and if not what is this new phenomenon captured by Fig. 2.11? Answering these questions is far from obvious and entails first providing a short background on aerodynamics specific to the circular cylinder. Actually Parkinson’s quote is stripped from influences brought upon by Reynolds number transitions and three-dimensionality of the flow, so it is natural to pursue discrepancies over these parameters. 2.2.1 Reynolds number effects For circular cylinders aerodynamic characteristics were early found to be decisively al- tered by Reynolds number. Further to the subcritical Sr changes illustrated in Fig. 2.1, the later critical and post-critical regions embody many interesting features (not only 26 Chapter 2. The Aeroelasticity Framework in terms of Sr variation) that could well be held responsible for complex dynamic behaviour of structures. Bearman [102] was the first to systematically map the evolution of all main aero- dynamic features in the Re range 10 5 to 7.5×10 5 . He discovered that a discontinuous jump in time-averaged base pressure, and concomitantly in time-averaged pressure drag 3 force, takes place at Re ≈3.4×10 5 suggesting that this is brought forward by the establishment of a laminar separation bubble only on one side over the complete length of the cylinder. Interestingly the bubble formed consistently on the same side, despite the very smooth cylinder finish and the absence of apparent asymmetries in flow conditions. A large steady lift force (i.e. C L ≈1.3) also resulted due to this one-sided bubble. At the same time the frequency characteristics for lift, acquired through wake velocity measurements, changed in an intermittent manner. For Re=3.55×10 5 , when the bubble was thought to be unstably bursting, Sr was transiting between two well defined values at 0.23 and 0.32. Subsequently for a small Re increase the bubble sta- bilised and acquired a single Sr ≈0.32. At larger Re a bubble similar to the first formed on the other side of the cylinder bringing an end to the asymmetry-induced lift and the so-called critical Reynolds region. Sr after this was about 0.48 with intensity more than an order of magnitude lower than in any previous state. Bearman finally noted that ‘contamination’ of the surface, e.g. by a dust particle, would trip the flow, alter the flow uniformity over a considerable length, and transform the periodic vortex shedding regardless of Sr to a wide-band process. An argument along the same lines, concerning the high sensitivity while in near-critical Re, was earlier suggested by Humphreys [103], who found that the addition of few light silk threads on the stagnation line can promote regular 3D flow-patterning. A substantial contribution to this insightful schema would come more than ten years later. Kamiya et al. [104] performing similar experiments to Bearman, measured the pressure distribution at a near-middle section and acquired plots translating meticu- lously the laminar separation bubble notion to pressure profiles. In addition to before, they obtained one-bubble states in alternate sides, waiving any uncertainty left that the lift appearance may be an artefact caused only by geometric imperfections. They also captured a hysteresis effect on all their recorded transitions (i.e. zero to one bubble, one to two bubbles, and vice versa). This designates that the history of the flow is crucial in the critical Re region. The actual transitions illustrated a transient period, where steady lift coefficients would not jump directly to a large (or zero) lift value but first gradually increase (or decrease) and this way reduce the amplitude of the ensuing discontinuity. In terms of the formation of bubbles this could probably be seen as a 3 Reference to drag throughout this study concerns the pressure part of drag, which is dominant for moderate to high Re numbers. 2.2. Circular galloping: myth or true? 27 gradual growing with increasing growth rate. This conceptual view was also shared by Almosnino and McAlister [105] who described the phenomenon as a supercritical bifurcation. Conversely, Schewe [106] in his seminal work, where he first proposed the similar- ity of flow transitions to bifurcations he recorded sudden abrupt flow-state jumps and consequently termed the bifurcations subcritical. In any case the bifurcations implied by both Schewe and Almosnino and McAlister require higher order polynomials (at least fifth, see [107]) for their accurate description. Schewe also suggested a number of innovating ideas. Among them, he quotes that the observed transition phenomenon is a hydrodynamic instability that should be treated in the framework of phase (or critical) transitions. One particular contribution to the initial Bearman description includes the recovery of the so-called critical fluctuations. Following Schewe, before any discontinuous drag or lift jump takes place, the periodic shedding moves away from its well defined Sr value, and acquires low frequency components that designate the so-called critical slowing inherent in the proximity of any critical point. Another interesting feature Schewe finds is that in the one-bubble regime, lift and drag PSDs show the same pronounced frequency (Sr=0.33), which does not match the classical shedding mode, where they should have a 1:2 relation. He estimates that this origi- nates from vortices being strong only from the one side where a bubble has not yet formed. Further he completes the characterisation of flow for Re up to 7.1×10 6 , in what should be deemed as the supercritical and transcritical Reynolds range. In the course of increasing Re, a series of new overlapping transitions are postulated, where bubbles progressively disappear towards reaching the fully turbulent state. In this final turbulent state ordinary shedding seems to have revived at Sr ≈0.27, very close to the subcritical value, a finding which has been known for many decades due to the work of Roshko [108]. Roshko also quoted that the obtained mean pressure distributions for these high Reynolds numbers are insensitive to the addition of a splitter plate in the wake. A concise schematic description of all the above was devised by Zdravkovich [109], and is illustrated in Fig. 2.12. As shown, the time-averaged drag coefficient C D varies as the position of the laminar-turbulent transition travels from the wake toward the stagnation point. The one and two bubble states earlier defined, are mapped to the designated TrBL1 and TrBL2-3 regimes. Most importantly Zdravkovich notes that the whole classification should be valid only for a disturbance-free flow. The effects of freestream turbulence or surface roughness could drastically alter the image. Although most of the time both these influences are modelled as a simple shift to the left of the C D curve in Fig. 2.12, the actual modifications are more subtle. The changing operations concern exclusively the states where turbulence intrudes the shear and boundary layers, 28 Chapter 2. The Aeroelasticity Framework Figure 2.12. Mean drag coefficient versus Reynolds number. On top, transitions (Tr) from laminar (L) to turbulent (T) flow are presented in relation to separation points (S) and boundary layers (BL). Adapted from Zravkovich [109]. termed TrSL and TrBL respectively. Large enough roughness values can for instance completely preclude separation bubbles and thus sweep away the greatest part of TrBL, whereas freestream turbulence would also very efficiently relocate and contract TrSL2. All this description is exclusive to a static cylinder. Motion similarly perturbs the flow by bringing vortex formation nearer to the cylinder surface, altering many details of the resultant aerodynamic force. According to Humphries [110], who performed water- tunnel tests on a large-scale flexible cylinder, motion can preserve vortex shedding unaltered (i.e. without Sr transitions) throughout the critical Reynolds number range. 2.2.2 Inclination effects Imagine a circular cylinder generically inclined to the flow. In this case, there is not only the normal to the body wind component but also an axial wind contribution running along the span-wise direction. This heuristic approach, where the flow is decomposed into two independent orthogonal parts (i.e. independence principal), although appeal- ing on its simplicity is not always accurate. The actual flow is far more complicated and 3D features emerging from the complex fluid-structure interaction can render such a two-component flow partition invalid. Bursnall and Loftin [111] performed one of the very few test studies on yawed cylin- ders inside the critical Reynolds number range. Their results suggest that the wind component normal to the cylinder U n , is not enough to adequately characterise the mean drag evolution. For their tested cases with cylinders inclined at 90 ◦ , 75 ◦ , 60 ◦ , 45 ◦ 2.2. Circular galloping: myth or true? 29 and 30 ◦ to the freestream, they produced maps of U n versus the normal to the cylinder- axis mean drag force, and obtained five different curves. They found that the lower the inclination the lower the U n at which the critical drag-drop occurs. Namely for the vertical cylinder the drag begins declining at 3.7×10 5 , whereas rotating 45 ◦ or 60 ◦ (transiting to more shallow configurations) this value becomes 2.3×10 5 and 1.06×10 5 respectively. The situation is also preserved if the total wind is used instead of just the normal wind component. Additionally there are vast discrepancies in the final supercritical mean drag coefficients from different inclination set-ups. However, these differences would much reduce when the coefficients are estimated based on the actual wind speed. This sets a fundamental question on which would be the most appropriate wind measure to use in our descriptions. Further, acquired pressure distributions re- vealed that increasingly deviating from the vertical case, bubbles became less apparent and less stable. Ramberg [112] when treating inclined cylinders, focused on vortex shedding and much lower Reynolds numbers. He found that for Strouhal number calculations, the use of the normal wind component is a convincing approximation for a wide range of inclinations. He also presented the increased geometric sensitivity that is inherent in the inclined cylinder flow, by producing variously slanted, multistable wakes for only minimal boundary alterations. These results were also corroborated by Shiraishi et al. [113], who set off to test the validity of considering the yawed circular section as analogous to an elliptical one. They disproved this view and with flow visualisations illustrated that indeed streaklines bend and cross the static inclined cylinder at almost right angles, which should readily support the use of simply the normal wind com- ponent. Following Ramberg, they found that the addition of end-plates would alter the shedding strength and uniformity. Resonant peaks in lift PSDs do not have the sharpness seen in the non-inclined scenario. Yet, Ruscheweyh [114] testing a range of inclined cantilevered (where probably also tip vortices contribute in loading) cylinders, showed that the vortex shedding response would reduce due to inclination only for higher Scruton numbers. Depending on the end-plate spacing, lift PSDs would also attain a low frequency content, probably similar to what was previously referred as critical transitional fluctuations. Actually Ramberg captured different wake modes co- existing during his tests. Hence, critical-like fluctuations could result by transitions, not on the boundary layers this time, but between the referenced wake modes. On an- other aspect, exploiting the along-length inhomogeneity, Hayashi and Kawamura [115] uncovered a pressure gradient on the lee side of their cylinder, regardless of boundary conditions. Its sign promotes flow directed from downstream toward upstream, which noticeably is opposite to the axial wind component. Detailed characteristics of this combined axial flow are largely undefined. 30 Chapter 2. The Aeroelasticity Framework 2.2.3 Instability mechanisms So is the circular cylinder response exemplified in Fig. 2.11 galloping? In terms of defi- nitions it was actually named ‘dry galloping’, mainly due to its seeming divergence with increasing wind speed, but whether it truly holds any resemblance to classical galloping is a view seriously disputed. A series of recent studies [116–118] assign the phenomenon to ordinary vortex shedding origin, and refute the galloping characterisation. In any case, the study of such phenomena has lately received a lot of research interest. Large vibrations of bridge cable-stays that were initially suspected to be due to rain, are now clarified to have occurred under dry conditions. Although it is clear that to mitigate the previously presented rain-wind vibrations one should inhibit the rivulet formation and motion, the required countermeasures for dry wind vibrations are a disturbing mystery. To answer it, it is essential to uncover the mechanisms stoking the instability. Matsumoto and his co-workers with a series of seminal papers [94, 119, 120], were the first to attempt an explanation of such unexpected wind behaviour. Initially it was sug- gested that three different types of response can occur. A galloping type, which could be either diverging or velocity restricted, a vortex shedding type with long period, and their mixed type. It was postulated that rain-wind phenomena would also fall into this broad framework. Rivulets when formed would only amplify these identical instability types. Amplification is usually envisaged in a quasi-steady manner, simply imposing an added geometric asymmetry. Galloping type response The galloping type response emerges due to the axial flow that runs in the lee side of the inclined circular body. The early suggestion [94, 119] was that this axial flow is a non-vortex flow, close in value to the approaching wind axial component. Acting as an air-curtain it would simulate a long-rigid splitter plate that as indicated earlier it will induce classical galloping on a plain circular section. The intensity of the axial flow, is the critical parameter that decides the vibration occurrence. Evidently, according to this rule, non-yawed cables are not susceptible to dry galloping. As a matter of fact, Matsumoto added an external artificial uniform flow in the wake of a horizontal cylinder, and observed large response similar to the yawed equivalent. This was consid- ered a convincing proof of the suggested theory. The schematic representation of this archetypal idea is given in Fig. 2.13(a). A splitter plate in principle has a dual oper- ation. It reduces the mean drag force and inhibits vortex formation in the near wake. A more accurate wording of the latter effect is that vortex formation is postponed to far downstream, where it becomes ineffective in feeding back substantial forcing [121]. It was experimentally observed by Matsumoto [101], that when dry galloping occurs, 2.2. Circular galloping: myth or true? 31 Figure 2.13. (a) The axial flow, evi- denced by light flags positioned inside the wake, act towards inhibiting communica- tion between shear layers and promoting a secondary circulatory flow. The func- tion described, simulates the galloping of a circular cylinder equipped with a long splitter plate. (b) Enhanced vortices are produced when axial vortices from the in- clined cable, mix and interact with ordi- nary K´ arm´ an vortices. Adapted from Mat- sumoto et al. [94, 119, 123]. classical vortex shedding has become weak and intermittent. As a matter of fact it was also stated that K´ arm´ an vortices are capable of suppressing large low frequency mo- tion, but no supporting explanation was given on this. In the latest theory amendment, Matsumoto et al. [122] consider vortex mitigation to be an indication of the axial flow intensity. When axial flow is strong then it faithfully resembles a long rigid splitter plate that will completely inhibit the vortex forcing and readily cause galloping. On the other hand the vortex appearance would mean that the axial flow is closer to a less efficient perforated splitter plate, which could even become unable to set off the instability. Subsequently unstable and diverging galloping are distinguished in terms of the efficacy of the splitter plate analogue. Vortex type response This type of response is more ambiguous and probably controversial. Its original con- ception was based on the observation that for both wind tunnel tests and field record- ings, large events seemed to cluster at discrete reduced velocity ranges, at around 20, 40, 80, 120 etc. These figures look like following a certain pattern of multiples of the reduced velocity that corresponds to ordinary shedding (i.e. 5). Still, no ev- ident reason existed for this connection. The work of Bearman and Tombazis [124] around the same period, provided a plausible explanation. They introduced a mild three-dimensionality in the wake of an ellipse with a blunt trailing edge and acquired 32 Chapter 2. The Aeroelasticity Framework wake velocity PSDs with span-wise distinct frequency peaks. The spatial transitions between alternate shedding frequencies were accommodated by so-called vortex dis- locations (or splittings). These dislocations were then associated a characteristic low frequency of switching-states. Matsumoto et al. [120] suspected that a similar source for low frequency loading could exist behind cables. Evidently the axial flow has to become a vortex flow in this scenario. Acquiring wake velocity PSDs, Matsumoto et al. found that even for a non-yawed cylinder there is a slight shedding variation along the length. This among other justification, encouraged the belief that even normal to the flow cylinders, latently have the ability of producing large response. In this explanation attempt however, the frequency forcing characteristics in all cases could not be proved to follow as submultiples of ordinary vortex shedding. Bearman and Tombazis with their imposed three-dimensionality, could accurately control their dislocations’ posi- tions but in Matsumoto’s et al. cables, dislocations, if any, are randomly distributed. Thus although promising, this mechanism was abandoned for not fitting the specific details. Yet the idea of a vortex type response was not invalidated. Matsumoto et al. [123] came back soon afterwards claiming that the mechanism could be founded on subharmonic resonance. As earlier noted Durgin et al. [27] pre- viously observed strong 1:3 resonant vortex shedding response for a vertical cylinder. Shirakashi et al. [125] argued that this was only an end effect, but Matsumoto et al. married the two views and showed their applicability on inclined cable test results. Flow visualisations presented an axial vortex originating from the upstream end and propagating towards the cable’s middle. Once shed it interacted with ordinary K´ arm´ an vortices for enhancing every third produced vortex. Taking into account that for an inclined cable Sr ≈0.15, the quoted timeliness results U r =20, giving a well match to the earlier said reduced velocity ranges. An illustration of the above description is given in Fig. 2.13(b). Subsequently this mechanism successfully entered design guidelines for bridge cables [49]. Two marked features found, are that turbulence can enhance this instability and that axial flow is again the regulating parameter. When the latter is increased diverging galloping will emerge. This makes disputable the actual distinction between unstable galloping and the vortex-response type, which seem to overlap (at least for inclined sections). As a matter of fact in his later published work Matsumoto et al. [122] seem to view this mechanism as redundant. Thus all comes down to the simple rule that complete mitigation of vortex shedding designates diverging galloping while intermittent and partial mitigation unstable galloping. Even responses found inside the critical Re region are considered to be due to the K´ arm´ an vortex weakening that takes place. Intriguingly, bluntly turning this rationale to prevention measures, it could mean that vortex shedding suppression devices are galloping inducers. This norm is obviously served by ordinary splitter plates. 2.2. Circular galloping: myth or true? 33 Yeo and Jones [116, 117] and Zuo and Jones [118] are still ardent advocates of the vortex type response. According to them the phenomenon is exclusive to inclined cylinders and will be controlled by reduced wind velocity. Actually suggesting control from reduced wind velocity, is another way of stating that any motions occurring, are primarily forced vibrations. Yeo and Jones employed a hybrid numerical turbulence modelling scheme to simulate the flow past a skewed horizontal circular cylinder at Re=1.4×10 5 (calculated on freestream velocity), and recover the aerodynamic force functioning. Increasing skew angle up to a critical value (with the flow thereafter fun- damentally changing), K´ arm´ an shedding is gradually suppressed and the axial velocity component rises. The weakened shedding interacts with the axial flow and develops so- called swirling structures, advancing along the length in organised patterns. Shedding among others, would also have to serve as a signal carrier. The resultant travelling forces, when considered sectionally, have PSDs with low frequency content, and look both frequency and amplitude modulated. With added skewness, the lowest acquired peak in the across-wind PSDs, draws away from the ordinary K´ arm´ an value. It is pos- tulated that such forcing component starts up a long period vortex resonance, which could further get amplified due to motion. Eventually it is suggested that an efficient plan for counteracting the whole process is to fully eliminate K´ arm´ an vortex shed- ding. This will deprive axial flow from the potential to nucleate swirling structures. Awkwardly this is exactly the opposite from what Matsumoto et al. [122] advise. Critical Reynolds number and galloping In the previous section, critical Reynolds number was assigned a secondary role in cable instabilities influencing them only due to contributing to vortex shedding mitigation. Yet historically there have been cable-like examples which ideally fit reasoning exclu- sively on Reynolds number. The high Re, along-current vibrations of an oil jetty at Immingham [126], could convincingly be captured if the instantaneous drag force act- ing on it, steeply decreases with Re, exactly as shown for the mean C D in Fig. 2.12. Steam generator tubes in a number of nuclear power plants had identical issues [127]. Martin et al. [128] modelled in a quasi-steady way the flow-structure interaction and obtained what looks like the benchmark mass-on-moving-belt problem [129]. Imagine a circular cylinder oscillating while sweeping a smooth critical drag drop region. During the part-cycle that it accelerates against the flow, Re becomes larger due to the relative velocity increasing. Therefore, the drag force acting on the cylinder has progressively smaller values, establishing a drag differential that points in the direction of the cylin- der motion. Likewise, when the cylinder velocity reverses, the new difference in drag is again in the direction of motion. This process will continuously pump energy into the 34 Chapter 2. The Aeroelasticity Framework system until a stable limit-cycle is reached. Being in need of negative dC D /dRe, such oscillations are evidently feasible only in the critical Reynolds number range. The unusual feature with the similar vibrations of stranded power conductors over the River Severn, was that motion could also be close to the vertical plane [130]. Richards [131] showed that Re related, shape-induced lift in skew winds and its subse- quent changes (i.e. derivatives) would have a crucial function in the aeroelastic forcing. Covering the strands with tape to form a smooth finish the instability disappeared. Prophetically probably, Richards also warned about the actual non-zero lift measured on the modified smooth cables. Macdonald et al. [132] applied a newly proposed gen- eralised galloping theory [133–135], which quasi-steadily accounts for Reynolds number effects, and analytically proved the source of the stranded Severn conductor problem to lie on excessively negative dC D /dRe and C L dC L /dRe terms. Note that charging the instability on Re derivatives could not work for purely across-wind vibrations. Re would then remain unchanged, thus any Reynolds effect in a first approximation negates. On a plain circular section steep gradients in both C D and C L exist at Re nominally around 1-3×10 5 , where laminar separation bubbles start forming. Accord- ingly excessive force derivatives with respect to Re can potentially cause the same type of galloping instability found in stranded cables. However, there is much hesitation in accepting Re effects as a major piece in the dry galloping puzzle. One of the main arguments against this is that Reynolds numbers in most recorded events fall short for being characterised critical. Typically they are of the order of 10 4 , which with some (but not absolute) certainty classifies them as subcritical. A second more insightful reason has to do with a fundamental objection. When a system approaches a critical transition point, as this can be envisaged the case here, it will get increasingly slower in recovering from small perturbations [136]. At any instant the dynamic state carries the additional influence of adjacent (or even far apart) previous states, establishing a memory effect. Quasi-steady theory, which idealises the instantaneous force to be only a function of the instantaneous relative velocity, cannot naturally cope with this requirement. Put into simpler words, Carassale et al. [137] used the hysteresis evidence of Kamiya et al. [104] and Schewe [106] to question the inclusion of Re derivatives in quasi-steady formulations. Such derivatives could be non-unique or even indeterminate. However, they also utilised purely geometric quasi- steady theory inside the critical Re region, where in principle this could be equally inapplicable. Promoting a simple Reynolds number explanation for dry galloping, is thought by many equivalent to proving the quasi-steady theory validity throughout the critical range. Concluding this part, a paradoxical feature should be noted. The strands-induced instability of the Severn conductor was cured by surface smoothening. In modern cable-stayed bridges the instability-control tactics are to effectively roughen 2.3. State of the art in bridge wind design 35 the already smooth cables. More intriguingly the roughening measures could even seem to resemble the old conductor’s shape, making up a clear inconsistency as commented by Tanaka [138]. This extensive review of proposed mechanisms is a sine qua non for the scope of the current thesis. The many contradicting elements presented, not adequately stressed in many points, are the best indication that phenomena for which theory is only now getting shaped are addressed. The diversity shown in explanations, dominates also in field-observations. Records spread over a wide range of wind conditions, geometric configurations, and structural characteristics perplexing any analysis. At the Iroise Bridge, a long monitoring campaign indicated large, unforeseen cable vibrations in what categorically was said to be the critical Re range [139]. On the other hand Zuo and Jones, reporting on the monitoring of the Fred Hartman and Veterans Memorial Bridges [140], identify all sorts of large cable vibrations but none in the nominal critical Re regime. Interestingly they record that a particular non-ordinary-K´ arm´ an event type occurs for similar wind conditions, with similar modal features, for both dry and rainy conditions. A connection is made with U r , which seems to consistently fall near 40, and subsequently the scenario that rain-wind and dry-wind vibrations are the same forced phenomenon is advanced. Finally, Matsumoto et al. [101, 122, 141] summarise a number of large-scale cable incidents in a selection of Japanese bridges. Most of their events have Re>10 5 , concentrate in a narrow defined cable-wind angle domain (i.e. near ≈60 ◦ ), refer to single low mode motion and occur at U r >100. Ordinary such high reduced wind velocities would designate indubitable prevalence of quasi-steady theory, but in this instance there also seem to be complications from complex unsteady effects. 2.3 State of the art in bridge wind design After a long section focusing on aerodynamic issues only recently being explored, it is time to illustrate how today’s bridge engineering design deals with them as well as with the earlier presented older and more rigorously studied wind problems. The biggest concern in bridge aerodynamic performance is flutter. Its interaction with gust loading is inevitable in any real-case consideration. There are yet more interaction phenomena, and fortunately many of them have reached an elaborate state of treatment. To this end only the basic framework for combined flutter-buffeting evaluations, essential for the following chapter, will be given. In §2.1.5, flutter was discussed in view of sectional aeroelastic forces, but no connec- tion was made to the estimation of stability limits when a full-bridge, with subsequent multi-modal behaviour is the application in hand. Following the succinct derivation of 36 Chapter 2. The Aeroelasticity Framework Jain et al. [142], for the most general case of a geometrically complex bridge with mixed modes, analysis proceeds as follows. The deflection components, shown in Fig. 2.7(b), are written by use of the dimensionless mode shapes h i (s), α i (s), p i (s) as h(s, t) = i h i (s)Bq i (t) , α(s, t) = i α i (s)Bq i (t) , p(s, t) = i p i (s)Bq i (t) , (2.12) where s is the distance along the deck span and q i is the ith mode generalised displace- ment. The equation of motion becomes I i _ ¨ q i + 2ζ i ω i ˙ q i +ω 2 i q i ¸ = Q i , (2.13) where ω i is the modal cyclic frequency, I i is the generalised inertia and Q i is the generalised aerodynamic force. The latter two are given by I i = l _ 0 _ m(s)h 2 i B 2 +I(s)α 2 i +m(s)p 2 i B 2 _ ds , Q i = l _ 0 _ Lh i (s)B +Mα i (s) +Dp i (s)B _ ds , (2.14) m(s) and I(s) being the mass and mass moment of inertia (about the section’s centre of gravity) respectively and l is the deck span length. Let the lift, moment and drag per unit span be linearly decomposed into the sum of self-excited and buffeting components L = L se +L b , M = M se +M b , D = D se +D b . (2.15) In general there is also self-induced buffeting action, which is herein discarded. For the self-excited parts, Eqs.(2.11) are employed additionally assuming that all flutter derivatives are constant along l. For the modal integrals it is written l _ 0 h i (s)a j (s) ds l = G h i α i , (2.16) together with the rest five obvious h i (s), α i (s), p i (s) permutations. When Eqs.(2.13) are Fourier-transformed into the reduced frequency (K) domain they can be expressed in matrix form as E˚q = ˚ Q b , (2.17) 2.3. State of the art in bridge wind design 37 where ˚ denotes the Fourier transformation. The components of E and ˚ Q b are given by E ij = −K 2 δ ij +ıKA ij (K) +B ij (K) , A ij (K) = 2ζ i K i δ ij − ρB 4 lK 2I i _ H ∗ 1 G h i h j +H ∗ 2 G h i α j +H ∗ 5 G h i p j +A ∗ 1 G α i h j +A ∗ 2 G α i α j +A ∗ 5 G α i p j +P ∗ 1 G p i p j +P ∗ 2 G p i α j +P ∗ 5 G p i h j ¸ , B ij (K) = K 2 i δ ij − ρB 4 lK 2 2I i _ H ∗ 4 G h i h j +H ∗ 3 G h i α j +H ∗ 6 G h i p j +A ∗ 4 G α i h j +A ∗ 3 G α i α j +A ∗ 6 G α i p j +P ∗ 4 G p i p j +P ∗ 3 G p i α j +P ∗ 6 G p i h j ¸ , ˚ Q b i = ρB 3 U 2 I i l _ 0 _ ˚ L b h i (s) + ˚ M b α i (s) + ˚ D b p i (s) _ ds , (2.18) with δ ij =1 for i = j, δ ij =0 for i = j. The multi-modal flutter critical condition is determined by solving the homogeneous equivalent of Eq.(2.17). The evaluation consists of varying K and deriving different sets of ω. The values of K and ω for which both the real and imaginary parts of the determinant of matrix E become simultaneously zero, are the effective negative- damping thresholds. The minimum wind speed calculated by these pairs, define the flutter velocity. Thereon expressing Eq.(2.17) in PSDs form and utilising standard concepts of random vibrations, the characteristics of the combined buffeting response can be estimated. As seen by considering only the homogeneous solution, the flutter condition does not include explicitly the buffeting influence. A complimentary analysis accounting also for buffeting was presented by Scanlan [13]. He considers the classical case of two interacting modes and calculates the average rate of change in their total energy. This turning to positive may be taken as a sign of instability. Additionally a slight variation regarding the generalised self-excited forces was proposed. According to it, the flutter derivative products of A ij (K) and B ij (K) in Eqs.(2.18) are no more functions of a single K. Instead they are evaluated at the aerodynamically modified K j corresponding to the relevant participating mode j (see also [12]). This latter analysis variant will be used in an inverse way for estimating the values of flutter derivatives for the CSB in the next chapter. The described flutter framework found application in the latest and most important bridge designs, in-short referring to the examples of the Storebælt Bridge [143], the Akashi Kaikyo Bridge [15] and the Messina Strait planned bridge [144]. A great deal of specialised analysis also exists for the rest individual elements of a bridge, and particularly for the versatile cable-stays. Treatments are mostly based on quasi-steady aerodynamic theory with any unsteady concerns being confined to ordi- 38 Chapter 2. The Aeroelasticity Framework nary K´ arm´ an vortex shedding. As illustrated in §2.1.7 and §2.2.3, for typical circular cables, many alarming phenomena occur beyond the reach of classical galloping and classical vortex shedding. Subsequently they would be unaccounted in any typical de- sign. Not having as yet hard evidence on the forcing attributes of such instabilities, the only true solution for engineering practice is to recur to experiments. This obviously serves both current and future design, since an empirical basis is constructed which could then lead to the development of analytical tools. The core knowledge in prac- tically dealing with dry cable galloping is expressed by Fig. 2.14, where curves in the S c –U r parameter space bound the regions of safety. The initial dynamic tests of Saito et al. [145], on an inclined bridge-cable replica, designate that the necessary S c (mainly seen as a structural damping requirement) for restraining vibrations is approximated by 35 √ S c = U r . Such a connection implies that cables will always become unstable for sufficiently high U r . The unrealistically high damping values imposed for most cases by the Saito et al. relation, led to a new test campaign sponsored by the Federal Highway Administration (FHWA) [146]. FHWA results, indicate that the threat should partly be released since for Sc>3 no large events were recorded. The latter is reminiscent of the actual design guideline against rain-wind phenomena. There Sc>10 [138], similarly on purely empirical grounds. Figure 2.14. Dry cable instability design criteria together with real-bridge unstable records. Dotted lines are due to the uncertainty in defining structural damping values. Adapted from Matsumoto et al. [122, 141] and Kumarasena et al. [146] Thus the current state of understanding comprises two far apart instability criteria without clear indication of which is the most appropriate. Large-scale records from real bridges [122, 141], due to uncertainty over the exact structural damping values, deceivingly seem to agree with both. Recently Matsumoto supported that the two cri- 2.4. Concluding Remarks 39 teria should be equally valid [122]; Saito’s et al. criterion designates unstable galloping (cf.§2.2.3), while the FHWA criterion diverging galloping. FHWA data near S c =4.5 in Fig. 2.14 do not seem to agree with this assertion. It should be mentioned that Fig. 2.14 crudely groups points corresponding to different configurations, and most importantly that some of these (especially unstable ones) may be non-repeatable. In §2.1.3 it was noted in passing that alternative numerical methods can be used for dealing with the wind-structure interaction modelling. The two alternatives quoted (i.e. discrete vortex models and solution of the Navier Stokes equations) have many variants depending on their computational implementations. In any case, for bridge structures where the combination of wind conditions and size results in very high Reynolds num- bers, there are noticeable difficulties with the operation of such schemes. Discrete vortex methods suffer from inconsistencies due to the poor knowledge of how to ad- just the vortex arrangement to accurately describe the turbulent flow. On the other hand Navier Stokes solvers need to resolve a wide range of turbulence scales, which numerically becomes extremely tedious. Often simplifications are put forward such as time or spatial averaging of the equations, which necessitate explicit turbulence mod- els. Tuning turbulence models for large separation problems remains ambiguous and introduces non-robustness into calculations. A detailed review of functioning charac- teristics and specific attributes of the methods is outside the scope of the current thesis. Yet it is informative to briefly refer to a number of bridge applications that illustrate the use and value of these analytical tools. For the flutter analysis earlier presented the flutter derivatives, which are yet to be determined, are possible to calculate with any of the two methods. Larsen et al. [143, 147] and Starossek et al. [148] numerically established sensible agreement with experimental flutter derivatives for a large number of both streamlined and bluff cross sections. For the circular cylinder flow the near critical Reynolds regime still remains rather challenging to simulate. Recently Yeo and Jones [116, 117] have treated the high Re inclined cylinder flow and their results are of specific interest to the current study. A general conclusive point to be drawn out is that although the trust in such computational methods has seriously increased over the last two decades, they are still in need of validation. The simplifications and as- sumptions made in order to reduce computational costs create uncertainties that can only be waived through comparisons with experimental results. 2.4 Concluding Remarks Apart from laying the ground for the analysis to follow, this chapter additionally tried to establish questions. All of them should naturally fall into the broadest themes of: ‘How 40 Chapter 2. The Aeroelasticity Framework efficient and reliable is the current state of knowledge?’ and ‘Are there new elements to complement this knowledge?’. This thesis will attempt to touch upon both. As presented, the best available defence that bridge engineering has against wind is multi-modal flutter analysis. It aims primarily at keeping sound the deck, which is probably the heart of a bridge. Although the method has reached a great level of sophistication, concerns will always exist as to whether results are representative and realistic. This is not surprising for a method that has developed inside wind tunnels, based on scaled models. Jones et al. [149] and Katsuchi et al. [150], quote the significance that full-scale ambient recorded data should have in a verification scheme. Actually modern bridges cope quite well with flutter. Any size-effect or inconsistency of modelling, if is there, does not seem to show up due to the structure operating really far from the conditions where the phenomenon was estimated to unfold. Unfortunately the best available real-scale example that could, when well instrumented, give us a great lesson on flutter and expose potential flaws, lies 80m below water in the bottom of Puget Sound (i.e. Tacoma Narrows). In any case, all monitoring attempts should be seen as a chance of putting more reality into bridge modelling. This is exactly the objective sought in the following chapter. Further, it is also essential to understand the limitations and shortcomings of other analytical tools currently in hand. Simple details (e.g. the exact geometric arrange- ment), when correctly accounted for, can bring straightforward explanations on what might have looked as an unforeseen aerodynamic event. And this is by no means an easy task. In the context of this research, the quasi-steady galloping framework is re- visited in order to produce an original contribution that recognises the true limits of the galloping analysis and points out all past omissions and defects. Still there are also truly new wind-structure interaction phenomena that cannot be cast into the existing knowledge. For a latest F/A-18 fighter jet, unexpected unsteady events rose from the newly shaped wings [151]. Similarly in long-span bridges new events of alarming am- plitude came through their smooth cables, which were paradoxically shaped like that to prevent old well-known dynamic instabilities. A number of proposed mechanisms were devised to explain these large cable vibrations, hoping to become precursors of a better analytical framework. Intriguingly, very few common grounds can be found between different approaches. As illustrated, they seem to disagree even on basic prin- ciples, often leading to exactly opposite results. The wind tunnel tests presented and analysed in a later chapter, attempt to offer a different interpretation of this unique phenomenon. The critical Reynolds number range by means of the associated com- plex transitional behaviour is carefully examined to reveal all the attributes compatible with the instability. Stepping towards a holistic approach of the aerodynamic bridge 2.4. Concluding Remarks 41 design it is essential that the unique parts that compose it are adequately clarified and understood. And this is exactly the notion that this thesis attempts to serve. Chapter 3 Identification of flutter derivatives from full-scale data The estimated response of large flexible bridges to severe wind loads is prone to mod- elling uncertainties that can only ultimately be assessed by full-scale testing. To this end, ambient vibration data from full-scale monitoring of the historic Clifton Suspen- sion Bridge (CSB) have been analysed in order to capture elements of true wind-bridge interaction. The multi-modal flutter framework earlier presented, is herein employed in an inverse investigation. Flutter derivative identification, which has rarely previously been attempted on full-scale data, was performed to seize any trends towards aerody- namic instability. The chapter does not intend to be a meticulous dynamic description of CSB. Instead, a number of useful notes for today’s aeroelasticity will be drawn out, while there is a clear potential for the old outdated CSB to become the test bed for future advances. 3.1 Introduction For large-scale structures the most rational way to proceed with predictions on the reliability and operational safety, includes identification methods from response only measurements. Especially for existing bridges, the risk of flutter can substantially be verified in this way. A bluff bridge cross-section, unlike a flat plate or an airfoil, has no analytical expression for the fluid forces exerted on it while in motion. Identifying the critical wind speed for instability inevitably has to adopt some experimental, semi- empirical or numerical foundation. Most commonly wind tunnel tests of scale models are used for reproducing the flutter phenomenon leaving the question of the effects of scaling. It is well established that minor details such as deck railings or roadway grills and vents can strongly alter the aerodynamic performance (see Scanlan and Tomko [73], 43 44 Chapter 3. Identification of flutter derivatives from full-scale data Jones et al. [152] and Matsumoto et al. [153]). Hence, analysis of the response of the real bridge can clarify the validity of wind tunnel tests and even reveal aspects, which either due to modelling assumptions or to loading irregularities, were previously concealed. Aeroelastic parameters have rarely been obtained from full-scale bridge data. Okau- chi et al. [154] were the first to attempt something relevant. Building a bridge section model (at a large scale, roughly 1/10) and setting it on-site against real wind conditions, they compared results with smaller wind tunnel equivalents. They suggested that, al- though for turbulent flow the relative differences in the turbulence details can impose inconsistencies, in general wind tunnel models produce representative results. Jakobsen and Larose [155] addressed the problem on the H¨ oga Kusten Bridge and presented a comparative analysis with wind tunnel results using a subspace identification technique for extraction of flutter derivatives. Costa and Borri [156] essentially applied the same identification routine, both on numerically simulated responses and on measured data from the Iroise Bridge, quoting good performance of the method in each case. For all of these bridge studies, the identification routine itself was found to be reliable, when tested using simulated data produced with added variously coloured noise. Compar- isons between full-scale and wind tunnel results were not unreasonable, but since the full-scale bridges were far from flutter, the trends in flutter derivatives were not clear. Another approach to the problem of identifying the aerodynamic effects on full-scale bridge vibration characteristics was used by Macdonald [157] on the Second Severn Crossing. Variations of effective damping ratios and natural frequencies with wind speed were found, and some indications of aeroelastic modal coupling were identified on the partially constructed bridge. In other full-scale studies, Littler [158] and Brownjohn [159] on the Humber Bridge, Bietry et al. [16] on the Saint-Nazaire Bridge, Jensen et al. [160] on the Great Belt Bridge, Ge and Tanaka [161] on the H¨ oga Kusten Bridge during construction and Nagayama et al. [162] on the Hakucho Bridge all found some trends of effective aerodynamic damping with wind speed, but coupling between modes and flutter derivatives were not pursued. The limited number of full-scale studies from which aeroelastic parameters have been found, makes any new cases useful for extending knowledge on the viability of system identification from site data and for interpreting actual bridge behaviour. 3.2 The case study As part of this work, analysis is performed on full-scale vibration measurements from the historic CSB, shown in Fig. 3.1. The CSB spans the Avon Gorge in Bristol, UK and was designed by I.K. Brunel in 1830, although it was not completed until 1864 3.2. The case study 45 (Barlow [163]). It was one of the longest suspension bridges of that time, with a main span of 214m. Wrought iron chains provide the suspension system, being the common practice for such early long-span bridges. In the light of modern understanding of bridge aerodynamics, the bridge cross-section (Fig. 3.2) and its light weight make it potentially susceptible to wind-induced vibrations. Indeed, on a few occasions in its lifespan large amplitude vibrations in strong winds have been reported. FIGURES Leigh Woods Clifton Maintenance craddle Accelerometer reference cross-section Accelerometer cross-section Anemometer 214m 80.2m 107m 46m Figure 3.1. Bridge elevation showing instrument locations. Based on figure after Barlow [163], with permission from Thomas Telford Publishing. On Christmas Day 1990 there was evidence of vertical motion at the bridge ends of the order of 250mm, which translates to even larger amplitudes within the bridge span. Both vertical and torsional deck motions were evident on a video recording of the bridge towards the end of a storm. A similar large vibration event was reported on 3 December 2006. Although no wind recordings exist from the bridge site itself on these occasions, data from the nearest weather stations imply that the maximum 1h mean wind speeds could have been around 20m/s. For recordings on site with wind speeds up to 16m/s, coupling action between the first vertical and torsional modes seemed to occur and the maximum vertical displacement at the ends of the bridge was 35mm (maximum measured elsewhere 92mm). The coupling action between modes and the rapid growth of vibration amplitudes for a modest increase in wind speed, indicate strong aeroelastic effects and make the bridge behaviour rather interesting. Such characteristics are reminiscent of features observed, in a more severe form, on the Tacoma Narrows (Farquharson et al. [164]) and Deer Isle (Kumarasena et al. [165,166]) bridges. It is worth noting that ten suspension bridges from the same era as the CSB failed due to wind between 1818 and 1889, including the Menai Straits Bridge, the Wheeling Bridge and a span of the Brighton Chain Pier (Farquharson et al. [164]). In contrast the CSB has survived virtually unscathed for over 140 years. According to empirical 46 Chapter 3. Identification of flutter derivatives from full-scale data Figure 3.2. Sketch of the bridge cross-section. estimates, similar in most bridge design rules, it is potentially susceptible to flutter with an estimated critical wind speed of not more than 20m/s. Therefore adverse aeroelastic effects are expected to become significant in moderately strong winds. For the current study, the wind conditions and bridge response recordings over two winter periods, from November 2003 to March 2004 [167] and from December 2007 to February 2008, are used. The site monitoring was conducted by Macdonald initially under a contract from the Clifton Suspension Bridge Trust. Later the monitoring was continued for research purposes with the assistance of the author. The data include several occasions with moderately strong winds, and reasonable ranges of wind speeds and directions, thus enabling a meaningful assessment of the wind effects on the bridge dynamics. Two ultrasonic anemometers were mounted 61m either side of midspan, more than 5m above road level. Nine accelerometers were positioned at a series of cross-sections along the bridge during an earlier analysis of the CSB dynamic response, enabling mode shapes to be identified (Macdonald [168]). The first three vertical and torsional modes are shown in Fig. 3.3. For the records considered here, two sets of three accelerome- ters were positioned at midspan and at a cross-section slightly off centre (26.8m from midspan) as illustrated in Fig. 3.1. This location was chosen as the reference cross- section since all significant vibration modes could be measured there. All motion related measurements below, refer to this location. Signals from all instruments were passed through anti-aliasing filters with a cut-off frequency of 4Hz and were recorded at a sampling rate of 12.5Hz. The primary aim here is to uncover the underlying mech- anism of large amplitude response that the bridge was found to produce for certain wind conditions, and explore the flutter potential. To do this the variation of modal characteristics with wind velocity has to be determined. Modal parameter estimates from a frequency based curve fitting technique (devised by Macdonald [157]) are used here, together with a subspace stochastic identification formulation especially modified to extract flutter derivatives (Jakobsen [169]). Due 3.3. Wind characteristics 47 f v1 =0.293Hz, f v2 =0.424Hz, f v3 =0.657Hz, ζ v1 =3.31% ζ v2 =1.99% ζ v3 =2.12% f t1 =0.356Hz, f t2 =0.498Hz, f t3 =0.759Hz, ζ t1 =2.60% ζ t2 =3.44% ζ t3 =2.16% Vertical Torsional Figure 3.3. Experimentally obtained vertical and torsional modes for CSB, adapted from Macdonald [168]. to lack of wind tunnel data from a scale model, flutter derivatives of other typical bridge cross-sections, as presented by Scanlan and Tomko [73], are used for comparative assessments. Cross-sections employed for this purpose are chosen to represent both the low structural depth and the high parapets (perforated on the CSB), of the section in hand. In the following section, first a short background of the acquired wind measure- ments is given. Typical wind speeds and wind turbulence conditions are described and the local terrain effects are discussed. Subsequently attention is moved to the bridge response details. All necessary modal background is provided, and the wind parameter on them is distinguished. The last part containing the flutter derivative identification scheme, starts by shortly commenting on the employed Covariance Block Hankel Ma- trix (CBHM) formulation. Eventually a flutter velocity estimate is sought to compare with theoretical predictions. 3.3 Wind characteristics The topography around the CSB has a considerable effect on the local wind attributes. As shown in the polar plots from both anemometers in Fig. 3.4, the wind speeds follow certain trends with orientations. (True North is at 31 ◦ relative to the axes shown). These trends differ markedly from the general wind pattern in the region away from the local influences. The strongest winds in the absence of topographic effects are typically from the south-west direction (at about 250 ◦ on Fig. 3.4 axes). The on-bridge measured stronger winds, are aligned along the gorge and can be probably explained by funnelling of the flow in these orientations and sheltering due to the high ground near the bridge ends. Evidently strong winds from the south-west are greatly attenuated at the bridge site, and virtually no wind from the north-east quadrant is experienced. It was also 48 Chapter 3. Identification of flutter derivatives from full-scale data observed that the correlation of the wind directions and wind speeds measured by the two anemometers, was strongly influenced by the wind orientation. In particular, for wind directions close to along the gorge even a small change in wind direction, results a large but consistent variation in the ratio between the two individual anemometer wind speed values. A typical assumption is made that that wind loading is approximately stationary for records up to one hour [2]. The maximum wind speed, averaged over one hour (unless stated all wind information hereafter refers to 1h means), did not exceed 16m/s at the bridge site, although higher speeds were measured at the nearest weather stations, for winds from the south-west. A histogram of 1h average wind speeds at the bridge is given in Fig. 3.4. Maximum 1s gusts were of the order of 26m/s for both anemometers. In addition, the wind turbulence and angle of attack parameters were considered. For wind turbulence there was a strong dependence on wind direction and a weaker one on wind speed. High levels of turbulence (up to 60% longitudinal turbulence intensity, I u ) were experienced, particularly for wind not along the gorge and for lower wind speeds. In winds over 8m/s, which only occurred along the gorge, approximately normal to the bridge, the mean longitudinal turbulence intensity was 21% and the mean vertical turbulence intensity I w , 10%. The vertical and across-wind (I v , in the deck- wind plane) turbulence intensities followed very similar patterns to the longitudinal turbulence. For longitudinal turbulence up to 40%, the across-wind turbulence was approximately equal to it and the vertical turbulence intensity approximately half the value. These are typical relationships between the three components of turbulence. For higher turbulence intensities measured, generally in lower wind speeds, the vertical and across-wind turbulence intensities were relatively larger. For the vertical angle of attack there was also strong dependence on the wind direction, and there were noticeable differences in the measurements from the two anemometers. The presence of the bridge itself is likely to have affected these measurements, as well as the topography of the gorge, since the anemometers were relatively close to the deck. High vertical angles of attack occurred, up to approximately ±10 ◦ . It should be reminded that these values are averaged over one hour periods. There was no significant difference in vertical angles of attack for different wind speeds. A wind aspect significant for the subsequent analysis, refers to the frequency content of wind buffeting. Although the traffic loading seems to be reasonably well captured by a white noise approximation (predominantly from step loading as vehicles drive onto or off the suspended span), the same does not hold for wind. By comparing spectral estimates deduced for various combinations of wind and traffic, it was found that above 3.3. Wind characteristics 49 Figure 3.4. (a) Histogram of wind speeds during the 2003-04 recording period. (b) Polar plots of 1h mean wind velocities from both anemometers. 50 Chapter 3. Identification of flutter derivatives from full-scale data approximately 0.25Hz the wind loading spectra tailed off as f −ǫ with ǫ around -8/3, producing a general loading spectrum of the form ˜ S load (f) = ˜ S w f −8/3 + ˜ S t , (3.1) where ˜ S w is a constant for a given record, being a function of the wind parameters, and ˜ S t is the magnitude of white noise traffic loading specific for each record. The frequency power exponent of -8/3 when compared with the -5/3 value corresponding to isotropic K´ arm´ an turbulence (referring to both I u and I w ) customarily employed in design, it implies that the product of the aerodynamic admittance and joint acceptance functions in Eq.(2.2) should be inversely proportional to f. 3.4 Response and modal parameters 3.4.1 Response Characteristics Fig. 3.5 shows the 1h average wind speeds over the whole of the first monitoring pe- riod, and the corresponding Root Mean Square (RMS) vertical accelerations at the reference cross-section. The RMS amplitudes normally show a clear daily cycle with the varying traffic load, with a maximum vertical response of approximately 0.02m/s 2 . By comparison it can be seen that only in wind speeds exceeding approximately 8m/s does the response noticeably exceed the maximum traffic-induced response. The maxi- mum wind-induced acceleration measured was approximately four times the maximum traffic-induced acceleration. The torsional and lateral acceleration responses at the reference cross-section followed very similar patterns to the vertical response over the monitoring period, although the magnitudes of the responses were lower. The maxi- mum instantaneous value of each component was found to be approximately six times the 1h RMS value. Dynamic displacements were calculated from the measured accelerations by double integration and it was noticed that the response is dominated by low frequency modes. The dominance of the low frequency modes is caused by the relatively higher wind loading at low frequencies and the effect of the integration, which exaggerates low fre- quency components. Whereas the maximum RMS acceleration due to wind loading was approximately four times the maximum due to traffic loading, in terms of displacement the maximum RMS response to wind was approximately 10 times that for traffic. The influence of wind loading on the measured vertical accelerations is shown in Fig. 3.6. Similar figures were obtained for the lateral and torsional accelerations. The 3.4. Response and modal parameters 51 10 17 24 1 8 15 22 29 5 12 19 26 2 9 16 23 1 8 15 22 0 2 4 6 8 10 12 14 16 November December January February March Date (2003-04) 1 h r a v e r a g e w i n d s p e e d ( m / s ) (a) 10 17 24 1 8 15 22 29 5 12 19 26 2 9 16 23 1 8 15 22 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 November December January February March Date (2003-04) R M S v e r t i c a l a c c e l e r a t i o n ( m / s 2 ) (b) Figure 3.5. (a) 1h average wind speed over the monitoring period. (b) 1h RMS vertical accelerations at the reference location over the monitoring period. scatter of results, particularly at low wind speeds, is largely due to the varying traf- fic contribution. The varying wind turbulence intensity also proved, as expected, to have an effect. Excluding records dominated by traffic, and normalising by the verti- cal turbulence intensity, gives a much clearer relationship with wind speed as shown in Fig. 3.6(b). Vertical turbulence intensity is considered more appropriate for such scaling, when bridge sections with relatively low mean lift coefficient C L (cf. Eq.(2.2)) are considered. Normalised RMS responses become a power law functions of the wind speed as shown in Eq.(2.5). The power exponent is very close to the theoretical 2.83 buffeting value, which could imply that response is solely due to buffeting. Yet as shown in Fig. 2.8 for turbulent conditions, a full bridge experiences the combined buffeting- flutter action with a power law too. Also it is notable that no sharp peaks, that could signify vortex-induced response, exist in Fig. 3.6(b). 3.4.2 Modal Analysis Modal parameters were previously calculated from curve fitting acceleration Power Spectral Densities (PSDs), using the Iterative Windowed Curve fitting Method (IWCM) of Macdonald [157]. IWCM was specifically developed for the analysis of ambient vibration data with in general non-white loading spectrum. The method iteratively curve fits in the frequency domain a series of idealised single-degree-of-freedom (SDOF) transfer functions, taking into account the shape of the loading spectrum, the effect of windowing on the spectra (both from the measured data and from the idealised transfer functions) and the contributions of multiple modes (in a linear sense). Measurements were only taken on the suspended bridge deck, but all modes in- evitably involve vibrations of other parts of the structure, particularly the chains. 52 Chapter 3. Identification of flutter derivatives from full-scale data 0 2 4 6 8 10 12 14 16 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Wind speed (m/s) R M S v e r t i c a l a c c e l e r a t i o n , σ v ( m / s 2 ) (a) 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 Wind speed (m/s) N o r m a l i s e d R M S v e r t i c a l a c c e l e r a t i o n , σ v / I w ( m / s 2 ) 12:30am - 6:30am Wind speed >8m/s No vehicles No vehicles or pedestrians ∝ (Wind speed) 2.8 (b) Figure 3.6. (a) RMS vertical accelerations σ v , in relation to wind speed for all 1h records. (b) Same as (a) for 1h records dominated by wind loading, with RMS vertical accelerations now divided by the vertical turbulence intensity. The power-law approximating the obtained trend is also plotted. Analysis was performed for frequencies up to 3Hz with twelve vertical, eleven tor- sional and four lateral modes being identified in this range, based on measurements in low wind speeds (Macdonald [168]). Typical PSDs for three different loading sce- narios for vertical, torsional and lateral accelerations are given in Fig. 3.7 to present the effect of wind loading on the bridge response. An important detail is the proxim- ity of the first vertical and torsional modes, with natural frequencies of 0.293Hz and 0.356Hz respectively (ratio 0.82). These are the first antisymmetric modes of each type as depicted in Fig. (3.3). It appears that in the stronger winds they start to couple in a possibly incipient flutter motion as evidenced by the small hump at 0.35Hz in the vertical spectrum seen in Fig. 3.7(a) and more clearly in Fig. 3.8. For further analysis it was desirable to reduce the actual multi-mode response to a simpler two-degree-of-freedom equivalent, concerning only the relevant first vertical and torsional modes. Low-pass filtering could not fully remove the contributions of the second mode of each type, since they were very closely located. However, using the measured responses at both accelerometer cross-sections (see Fig. 3.1) together with the known mode shapes, the following operation was performed. With the second section in the midspan, the first pair of modes being anti-symmetric is showing as zero. The second pair of modes on the other hand, containing symmetric modes that maximise there, could be readily identified. Transferring these modal displacement values to the reference cross section and subtracting them from the total signal, results in almost pure first mode motions. Fig. 3.8(a) hence shows responses of the first pair of modes (z vertical and θ tor- sional) for the highest recorded 1h wind speed (15.3m/s). The peak in the vertical PSD 3.4. Response and modal parameters 53 0 0.5 1 1.5 2 2.5 3 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 Frequency (Hz) V e r t i c a l a c c e l e r a t i o n P S D ( ( m / s 2 ) 2 / H z ) Maximum wind, with traffic Minimal wind, Rush hour traffic Moderate wind, only (a) 0 0.5 1 1.5 2 2.5 3 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 Frequency (Hz) T o r s i o n a l a c c e l e r a t i o n P S D ( ( r a d / s 2 ) 2 / H z ) Maximum wind, with traffic Minimal wind, Rush hour traffic Moderate wind, only (b) Figure 3.7. PSDs for different loading condi- tions for (a) vertical (b) torsional and (c) lat- eral accelerations. 0 0.5 1 1.5 2 2.5 3 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 Frequency (Hz) L a t e r a l a c c e l e r a t i o n P S D ( ( m / s 2 ) 2 / H z ) Maximum wind, with traffic Minimal wind, Rush hour traffic Moderate wind, only (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 x 10 −3 f (Hz) P S D ¨ θ ( r a d / s 2 ) 2 / H z 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.03 0.06 0.09 f (Hz) P S D ¨z ( m / s 2 ) 2 / H z (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 10 −4 10 −3 10 −2 10 −1 Frequency (Hz) V e r t i c a l a c c e l e r a t i o n P S D ( ( m / s 2 ) 2 / H z ) 15.3m/s 13.8m/s 11.9m/s 9.8m/s 7.9m/s 6.1m/s 1 st Torsional mode (b) Figure 3.8. (a) PSDs of filtered data for first vertical and torsional modes for the maximum wind speed record. (b) The evolution of the coupling action is evident in the vertical PSD for wind speeds above 11m/s. 54 Chapter 3. Identification of flutter derivatives from full-scale data at the torsional frequency is strong evidence of coupling action. The coupling was not evident for low winds and became stronger only for higher winds (Fig. 3.8(b)), setting it most probably to be an aeroelastic effect. The coherence between the vertical and torsional accelerations showed similar evidence, with the value around 0.35Hz rising to approximately 0.6 in high winds compared with typical values below 0.4 in low winds and at other frequencies. It is also worth noting that the second pair of modes showed a similar tendency for coupling action in strong winds, due to their also similar shapes (see Fig. 3.3) and close natural frequencies (ratio 0.424/0.498≈0.85). The next section discusses in more detail the identification of the CSB flutter derivatives, so as to be able to quantify the observed coupling signs and the tendency towards flutter. 3.5 Flutter derivatives 3.5.1 Flutter Analysis According to the semi-empirical Selberg [170] equation for bridge sections resembling flat plates, an estimate for the flutter speed is given by U f f θ0 B = C ¸ r g m ρB 3 [1 −( f z0 f θ0 ) 2 ] , (3.2) where U f is the flutter speed, B the deck width, r g the radius of gyration, m the total mass per unit length, C a constant depending on the section’s behavioural resemblance to a flat plate, ρ the air density and f z0 and f θ0 the still air vertical and torsional natural frequencies. For the first pair of natural frequencies, described above, substi- tuting ρ=1.2kg/m 3 , B=9.46m, C=2, m=5370kg/m, r g =4m, the flutter onset speed is estimated at approximately 18m/s, or 14m/s using the more conservative variant of Eq.(3.2) in the British design rules [171]. These very low values are due to the low torsional natural frequency, the close neighbourhood of vertical and torsional modes and the low mass per unit length. However, the bridge cross-section is not a flat plate and the uncertainty in the value of C (containing an experimental correction factor) means Eq.(3.2) is not very reliable. Still, it gives a rough first approximation for the flutter speed and this is within the range of wind speeds encountered at the site. For evaluating the flutter behaviour, the classical multi-modal formulation presented in §2.3 is employed for expressing the aeroelastic forces. Further it is assumed that the considered modes have no considerable mixed components. Substituting z and θ to the generalised displacements h 1 , α 2 in Eqs.(2.18) (whereas α 1 =p 1 =h 2 =p 2 =0), the 3.5. Flutter derivatives 55 motion-dependent lift and moment, L se and M se become L se = 1 2 ρU 2 B _ KH ∗ 1 G zz ˙ z U +KH ∗ 2 G zθ B ˙ θ U +K 2 H ∗ 3 G zθ θ +K 2 H ∗ 4 G zz z B _ M se = 1 2 ρU 2 B 2 _ KA ∗ 1 G θz ˙ z U +KA ∗ 2 G θθ B ˙ θ U +K 2 A ∗ 3 G θθ θ +K 2 A ∗ 4 G θz z B _ , (3.3) where following the convention of Scanlan [12,13], the reduced frequencies K(=2πfB/U) and all flutter derivatives H ∗ i , A ∗ i are calculated based on the frequency of the mode they refer to. Any amplitude or structural damping dependence of flutter derivatives was not accounted for. In this basic 2D formulation, the motion-induced drag force, D se , and the effects of the along-wind motion are ignored, meaning that P ∗ i deriva- tives are disregarded. Actually the standard linear decomposition of D se was recently questioned by Starossek et al. [148]. According to them for a bridge section symmetric along the centreline, both positive and negative rotations will result the same exposed area to the wind. Thus it is more natural that drag forcing with double the motion frequency will result. Their assertion was corroborated by numerical simulations. In any case along-wind displacements were quite low for all records in CSB. 3.5.2 Identification Method A state space formulation of the dynamic problem was assembled using the Covariance Block Hankel Matrix Method (CBHM method), which is founded on the widely used Eigenvalue Realization Algorithm (ERA) described by Juang and Pappa [172]. The formulation in CBHM is identical to ERA with the exception that instead of the Markov parameters containing Impulse Response Functions (IRFs), covariance estimates of output measured random data are employed. Jakobsen [169] first applied CHBM in the estimation of flutter derivatives from wind tunnel tests and the method has since found extensive use in aerodynamic applications and testing. Peeters and De Roeck [173], Qin and Gu [174] and Siringoringo and Fujino [175] all describe the matrix derivation in detail. Brownjohn et al. [176] found a relative advantage of the method over other operational modal analysis approaches. The method is based on the Singular Value Decomposition (SVD) and appropriate factorisation of a Hankel matrix built up by covariance estimates of the output time series (i.e. displacements or accelerations in this application). If y stands for the displacement matrix with z and θ in rows, then the unbiased sample cross-covariance matrix to be used in the Hankel block construction is given by C yy (i +n, i) = C yy (n) = 1 N −n N−n y(i +n)y T (i), n = 0, 1, .., ℓ . (3.4) 56 Chapter 3. Identification of flutter derivatives from full-scale data where n is the number of sampling intervals for the discrete time delay n∆t, N is the number of samples in the time series, ℓ is the maximum number of lags considered and i is a counting index. The biased estimate, which only differs in the use of the denomi- nator N instead of N −n, can be used instead with negligible differences for long time records. The method exclusively handles white noise loading but here it was attempted to also account for the actual shape of the loading spectra given in Eq.(3.1). Imagine the SDOF system of Fig. 3.9 with frequency response function H i (f). When loaded by coloured noise, indicated by the non-flat PSD ˜ S load , then the produced response PSD will not preserve the shape of H i (f) as the identification method necessitates. A remedy would be to apply a filter function on the response PSD with magnitude equal to the inverse square root of the loading PSD and zero phase lag. Thus the corrected PSD, dashed line in Fig. 3.9, would look as if produced by a fictitious white noise load. For the two-degree-of-freedom system similarly, ordinary filtering on the response data in the frequency domain could account for the coloured lift and moment spectra. A new attribute that has to be taken care of though, is preserving after filtering the exact cou- pling between modes, that was artificially altered. The lift filter function F L in Fig. 3.9 for instance, will introduce increased coupling. This is not realistic because buffeting action for each mode is assumed uncorrelated to the coupling action between modes due to self-excited forcing. The correction for this deficiency, was to modify the final extracted coupled derivatives with the ratio of the filter values in order to try reverting to the initial coupling. Thus, the obtained H ∗ 2 and H ∗ 3 in the illustrated example, are adjusted by dividing them with the ratio of F L (f θ )/F L (f z ). Additional information on the performance of identifications with coloured noise and specific attributes on the relative sensitivity of each derivative were described by Jakobsen [169]. H ∗ 2 , H ∗ 4 and A ∗ 4 were found to be affected the most from such a correction procedure. Interestingly H ∗ 4 is not a coupling derivative. In practice, for the specific problem in hand the effect of the actual wind spectra on the flutter derivatives was found to be insignificant. The decomposition of the Hankel matrix recovers simultaneously all parameters of the discrete time realisation. Knowing the modal stiffness and damping matrices for the in wind and still air cases (pure structural stiffness and damping contributions) allow one to separate the flutter derivatives. The whole method (having here the dimensionality for the problem already decided as trivially two degrees-of-freedom) relies on the choice of two parameters; the length of the individual time record N and the number of time delays ℓ for which the covariance matrix is evaluated and stored in the block Hankel matrix. The choice of both is investigated through a sensitivity analysis together with inspection of the time evolution of the auto and cross-covariance functions. 3.5. Flutter derivatives 57 ˜ S load (f) f f f |H i (f)| 2 ˜ S resp (f) f z f θ F L (f) f ˜ S z (f) f z f θ f F L (f θ ) F L (f z ) 1DOF 2DOF Figure 3.9. Decolouring process. In the 1DOF system, filter application will produce corrected spectra, see dashed line, simulating white noise loading. For the 2DOF system, filtering with F L will erroneously modify the true aeroelastic coupling, see dashed line versus greyed area. Adapted from Jakobsen [169]. 3.5.3 Application to the Clifton Suspension Bridge The proximity of the fundamental vertical and torsional modes seems to encourage some coupling action, which could potentially be the initial sign of classical flutter. The PSDs in Fig. 3.8 indicate some non-negligible values of the H ∗ 2 or H ∗ 3 flutter derivatives, since coupling occurs in the vertical PSD at the torsional motion frequency. The flutter derivative identification was performed in one case with recorded acceleration data and in another with displacements evaluated by double integration of the accelerations in hand. Both cases produced identical results. For the N and ℓ identification parameters, time records from 10 minutes to 1 hour and lag ranges between 10 and 40 seconds were used, preserving analogies with sim- ilar previous treatments. Example covariance functions, for moderately strong wind, are plotted against time lag in Fig. 3.10. As previously demonstrated by Jakobsen et al. [177], the suitable number of maximum time lags is strongly influenced by the response character at different wind speeds. Higher wind speeds usually allow only a shorter meaningful portion of the covariance function for accurately reproducing the two-degree-of-freedom interaction, e.g. due to high aerodynamic damping of the pure vertical response in the case of streamlined box-girders. For the example in hand an optimum set of values, producing representative results, was found to be the combi- nation of 15 minute records (N=11×2 10 ) with time delays up to 20 seconds (ℓ=250). To justify the choice, such a lag value is slightly higher than the beating period of the 58 Chapter 3. Identification of flutter derivatives from full-scale data two frequencies in hand, while as seen in Fig. 3.10, C zθ peaks at approximately 10s. Sensitivity of the identification for the ℓ range quoted, was only weak 0 10 20 30 40 −1 −0.5 0 0.5 1 C z z 0 10 20 30 40 −1 −0.5 0 0.5 1 C z θ 0 10 20 30 40 −1 −0.5 0 0.5 1 C θ z lag (s) 0 10 20 30 40 −1 −0.5 0 0.5 1 C θ θ lag (s) Figure 3.10. Example covariance functions (scaled with variance) for the combined two degrees-of-freedom plotted against time lag. Results for the CSB flutter derivatives are given in Fig. 3.11. No wind tunnel tests have been undertaken on the CSB deck section, but where possible the site data are compared with available wind tunnel results of other deck cross-sections. Sign conven- tions for aerodynamic forces are as in Fig. 2.7, i.e. lift force and vertical displacement pointing downwards and overturning moment with rotation positive for the windward side of the bridge girder moving upwards. A sensitivity analysis on the measured wind characteristics, such as the turbulence and the angle of attack, proved not to be able to reproduce a clear picture of their effect. The identified trends of flutter derivatives remained unaltered, but data were insufficient to quantify a systematic impact of the investigated parameters. Some of the derivatives in Fig. 3.11 appear to have an offset for still air wind conditions. This has also been encountered in previous treatises both in wind tunnel [169], and on site [155, 156]. Here it can mostly be attributed to effects such as the distortion from traffic, influencing both the loading and the mass distribution, as well as uncertainties in the modal masses and inaccuracies in the still air structural matrices. The non inclusion of P ∗ i derivatives may also be of some influence. For off-diagonal still air values there is no direct control on this offset, which can be purely a noise artefact even in the absence of gyroscopic terms [178]. For diagonal values there was 3.5. Flutter derivatives 59 Figure 3.11. Flutter derivatives of Clifton Suspension Bridge from full-scale data, compared with wind tunnel extracted flutter derivatives for various cross-sections (after Scanlan and Tomko [73] and adapted to Eqs.(3.3)). A ∗ 1 and A ∗ 3 for section G5 are negligible. H ∗ 4 and A ∗ 4 were not measured in the wind tunnel tests. Identified values correspond to binned and averaged identified values. 60 Chapter 3. Identification of flutter derivatives from full-scale data an attempt to minimise the offset since for damping especially, the absolute values are of great importance. Although the identified flutter derivatives are noisy, unsurprisingly for full-scale am- bient data, some trends are apparent. Consistent with the observed bridge behaviour, the results indicate that, within the range of wind speeds recorded (maximum 15.3m/s), the bridge is not susceptible to torsional flutter (so called ‘damping-driven flutter’ as presented by Matsumoto et al. [179]), which was the reason for the famous Tacoma Narrows Bridge collapse. Neither galloping 1 seems an issue. This is due to having close to negative A ∗ 2 and H ∗ 1 (direct damping derivatives), which will probably need a further increase in reduced wind speed to initiate such alarming behaviour, if indeed it does occur. However, H ∗ 1 apparently shows a steep positive gradient near the highest wind speed recorded, suggesting it could become positive for higher wind speeds, pos- sibly leading to instability. This trend persists regardless of the selected identification parameters (N and ℓ), indicating it is not due to numerical errors, although the last few points in the figure are from averages over few records, so their accuracy may be limited. Fig. 3.12 shows the H ∗ 1 flutter derivative estimates from each 15-minute record, before the averaging used for Fig. 3.11. Although only the last few points show the apparent positive gradient, these points depart significantly from the trend at lower reduced velocities and the differences are greater than the general scatter of points, implying this is most probably a real effect. If this is confirmed, the effect of possible positive H ∗ 1 (i.e. negative aerodynamic damping of vertical motion) could provide a feasible explanation for the occasional observations of large vibrations of the bridge in strong winds. Actually such reversing H ∗ 1 behaviour, tending to self-excited oscillations, was first met for the Tacoma Narrows Bridge section by Scanlan [73]. It was postulated then, that the phenomenon is vortex shedding related. Yet oddly in the same study, an H- section, with depth to width ratio equal to the Tacoma Narrows Bridge did not produce any pronounced H ∗ 1 turn. Results from later wind tunnel tests by Neuhaus et al. [180], plotted in Fig. 3.12, show an almost ever decreasing H ∗ 1 . Additionally the expected reduced wind velocity for vortex oscillations on Tacoma should be U r = U/fB≈0.55 −1 according to [143]. This sets vortex shedding quite far apart (Fig. 3.12) to be capable of causing lock-in and imposing a strong effect on H ∗ 1 . Analogously for the CSB, if the captured phenomenon is assumed the same, the high turbulence intensities on-site make quite improbable the explanation on a vortex-shedding basis. Thus, this monotonicity inversion or in some cases simply multi-valuedness, seems an interesting unresolved 1 Purely translational instabilities, according to the definition introduced in Chapter 2, are termed galloping. However, the instability described here may elsewhere be found with the name SDOF translational flutter. 3.5. Flutter derivatives 61 0 1 2 3 4 5 6 7 −15 −10 −5 0 5 U rz (= 2π/K = U f z B ) H ∗ 1 identified Tacoma [73] Tacoma [180] Tacoma nominal S −1 r Figure 3.12. Identified H ∗ 1 flutter derivative from each 15-minute record. H ∗ 1 for the Tacoma Narrows Bridge from Scanlan and Tomko [73] and from Neuhaus et al. [180] are given for comparison. issue. It is worth noting that the numerical derivation of H ∗ 1 for the Tacoma Narrows section, practised by Larsen [143], did not reproduce the experimental convexity. Dimensionally assessing the possibility of unstable motion in the pure vertical re- sponse, a value of H ∗ 1 ≈6 needs to be reached. This estimate is based on the low amplitude structural damping ratio of ζ=3.3% for the vertical mode [168] and on the as- sumption that no beneficial amplitude-dependent increase in structural damping takes place. Structural damping is believed to be so high, compared with modern suspen- sion bridges, because of the many joints in the structure, particularly the wrought iron suspension chains. The possibility that bias errors could have lead to erroneous overestimation should be ruled out, since IWCM handles efficiently most artefacts. The H ∗ 2 and H ∗ 3 derivatives, which control the coupling from torsional to vertical motion, have small values. However at the higher wind speeds there is a noticeable growing negative trend in H ∗ 3 , in line with the curves for other bridge profiles, which potentially explains the previously illustrated coupled spectra in Fig. 3.8. The relative influence of H ∗ 3 for the most coupled response record, translates to a vertical force approximately 1/10 of the peak restoring elastic force for the mode. The evolution of H ∗ 4 (aeroelastic direct vertical stiffness) reflects a reduction of vertical natural frequency with increasing wind speed, although this could alternatively be due to an amplitude dependence rather than the wind. Similarly A ∗ 3 (aeroelastic direct torsional stiffness) illustrates a reduction in the torsional natural frequency with 62 Chapter 3. Identification of flutter derivatives from full-scale data increasing wind speed. The highest identified values of A ∗ 3 and H ∗ 4 translate to an actual frequency drop of less than 7% in each case (cf. variation of up to 4% from traffic [168]). Such values are higher than the 1%-3% found in [73], which could imply that the recorded frequency shifts in CSB are actually due to a combinatory cause. In any case variations are quite low. This reinforces previous observations that unlike airfoil flutter, for bridges with bluff sections aeroelasticity influences more the damping than the frequencies of the modes (see Scanlan and Tomko [73] and Billah and Scanlan [181]), although in this study coupling could also become apparent. In any case, from the sections in hand, a qualitative similarity was found with the bluff section of Tacoma Narrows. For a proper estimation of the critical flutter wind speed, through Complex Eigenvalue analysis (CEV) described in §2.3, data inclusive of higher wind speeds are needed to extend the plots of Fig. 3.11. However, considering that the Tacoma Narrows bridge failed under pure torsional flutter, due to A ∗ 2 (negative torsional damping), focus is also put on this flutter derivative. Actually an estimate of its value at higher wind speeds, can be obtained by utilising the relationships between flutter derivatives initially proposed by Matsumoto [179] and here written as in Scanlan et al. [182] H ∗ 1 = KH ∗ 3 , H ∗ 4 = −KH ∗ 2 , A ∗ 1 = KA ∗ 3 , A ∗ 4 = −KA ∗ 2 . (3.5) These suggested relationships are based on the assertion that twisting θ and the apparent angle of attack associated with the bridge girder vertical velocity generate similar motion dependent forces. Scanlan et al. [182] present a simple elegant proof based on the classical Theodorsen flutter treatise [63]. The relationships should be mostly applicable for streamlined sections but were found to yield also accurate match for many bluff cross sections too [179]. In [182] the streamlined section of the Tsurumi Bridge was found to comply well with Eqs.(3.5), while the bluff section from Golden Gate showed a much worse fit. Eqs.(3.5) are here employed to investigate the possible extension of estimates of A ∗ 2 to higher reduced wind speeds through the measured values of A ∗ 4 . This can be achieved since the scaling of the reduced wind speed for A ∗ 4 uses the lower vertical frequency giving higher reduced wind speeds than A ∗ 2 , which is expressed in relation to the torsional frequency. Consequently it is possible to better review the possibility of single-degree-of-freedom torsional flutter on the CSB. Fig. 3.13 shows the A ∗ 2 initial data with the additional points. Fitting the polynomial A ∗ 2 =0.12U rθ (0.3U rθ – 1) provides an estimate for pure torsional flutter at U rθ ≈6.3 i.e. at U≈21m/s. This estimate is based on the low amplitude structural damping estimate for the torsional mode where ζ=2.6%. No allowance for a potential beneficial increase of structural damping with amplitude was considered. This, combined with the uncertainty in the estimation of small aerodynamic forces represented by A ∗ 4 and with the uncertainty in 3.6. Concluding Remarks 63 the relationships in Eqs.(3.5), expand the error margins of the estimation. Thus the actual flutter wind speed could be higher, although results of this magnitude are still significantly below today’s standards. For comparison, the modern Ting Kau Bridge has an expected flutter velocity of more than 60m/s [78], while the latest Sutong and Stonecutters bridges raise this value to an impressive 88m/s and 140m/s respectively [77]. 0 1 2 3 4 5 6 7 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 U rθ (= 2π/K = U f θ B ) A ∗ 2 A ∗ 2 A ∗ 4 /(−K) (Matsumoto) 0.12U rθ (0.3U rθ − 1) Tacoma Figure 3.13. Flutter derivative A ∗ 2 with additional points from A ∗ 4 as suggested by Matsumoto et al. [179]. The fitted polynomial, indicated by the broken line, was used for estimation of a pure torsional flutter wind speed. 3.6 Concluding Remarks A similar flutter derivative identification scheme has been attempted also for the Ting Kau Bridge, which benefits from a state of the art health monitoring system. Its flutter derivatives H i ,A i for i=1,2,3,4 were previously measured during scaled sectional wind tunnel tests, thus providing the analysis a reference to compare with. Additionally, data records are inclusive of the period that the bridge was hit by Typhoon York. Evidently it should have been expected that the Ting Kau Bridge case study is far more interesting and didactic than CSB. As a matter of fact it is not. The maximum wind speed acquired at deck level on the Ting Kau Bridge was only 24.1m/s, too low in comparison with the modelled flutter capacity of around 60m/s. Any flutter signs are too weak and probably get masked by other influences, which contaminate the signal parts assigned to flutter derivatives. Further all structural modes are three-dimensional with their mixed character seriously complicating any analysis. 64 Chapter 3. Identification of flutter derivatives from full-scale data On the other hand the CSB is the simplification anyone would try to have for the full-scale flutter problem. Modal complexities and unknowns are reduced, and on top of that the operational wind velocities span an extensive regime closing to what could be an instability threshold. It was proved that the historic bridge showed some similar trends with the Tacoma Narrows Bridge, in terms of both the direct damping flutter derivatives H ∗ 1 and A ∗ 2 . Although it is now well accepted by the engineering society that single-degree-of-freedom torsional flutter led the Tacoma Narrows Bridge to destruction, there have also been some different views. K´ arm´ an’s initial diagnosis was that vortex shedding was the reason for the catastrophe, while Plaut [183] employing nonlinear logic argued that initial small damage to a cross-tie introduced an asymmetry without which the bridge would never go into unstable torsional motion, whatever many wind tunnel tests unanimously support. Such ideas attempt to refute the whole bridge flutter analysis in use today. CSB being a unique large scale example seemingly close to flutter, can be used in exploring how the phenomenon truly unfolds (progress or die down by leaking energy to other modes) and answer many remaining questions. U r H ∗ 1 dC L dα +C D Re 2 < 0 dC L dα +C D Re 1 = c 1 > 0 dC L dα +C D Re 0 = c 0 > c 1 Figure 3.14. H ∗ 1 for different H-sections [73] and a possible Reynolds number based expla- nation for the H ∗ 1 inversion. Crossovers can initiate when Reynolds number changes alter the multiplier of U r in Eq.(3.6). With the opportunity of the CSB identification another aerodynamic feature was highlighted. Positive A ∗ 2 is the signature sign of H-sections. Yet, although often ne- glected, H ∗ 1 for many H-profiles seems to change its gradient sign, also tending to a positive unstable value, see the left of Fig. 3.14. Such behaviour was explained by Scanlan as lock-in from vortex shedding. This would be quite unlikely for the inversion in CSB that has not been seen to experience vortex shedding in any case. An alternate explanation never suggested before could be founded on Reynolds number effects as 3.6. Concluding Remarks 65 follows. H ∗ 1 could be quasi-steadily estimated by [184] H ∗ 1 (U r ) = − _ dC L dα +C D _¸ ¸ ¸ ¸ α 0 U r 2π , (3.6) where C D is the mean drag coefficient, α the angle of attack and α 0 the equilibrium angle of attack to which all flutter derivatives refer to. This linear estimate was found to be very close to the true H ∗ 1 value for a number of bridge sections [169,184]. Suppose that the critical Reynolds number Re 1 is reached. Then C D will reduce and Eq.(3.6) will result a less steep approximation. This is shown in Fig. 3.14. Potentially at Re 1 a crossover could occur that replicates the H-section behaviour. This could even lead to a sign change for H ∗ 1 , when the Reynolds number change induces the classical galloping condition, cf. Eq.(2.9). A C D drop with concomitant Strouhal number increase was previously recorded in wind tunnel tests of the Storebælt bridge section by Schewe [185]. His attempt was to reason the 20% higher Strouhal number that was experienced for the bridge on-site. In general, this work attempted to contribute to the literature on the analysis of aeroelastic effects from ambient vibration data on full-scale bridges, being one of very few similar studies. On the specific issue of CSB’s flutter potential, the identified A ∗ 2 was extrapolated at higher reduced velocities by use of the A ∗ 4 results, and an apparent trend was recovered. On this basis, a wind speed estimation for pure torsional flutter has been made. Although uncertain, it is believed this is the first time this has been achieved based solely on the actual full-scale dynamic performance of a bridge. Concluding this chapter a future perspective can be given. It was early seen that small shape modifications could enforce dramatic changes over flutter derivatives, at least on scale models. Scanlan and Tomko [73] presented results from tests on two truss- stiffened bluff sections for which A ∗ 2 on the addition of railroad tracks, or a short middle traffic barrier became positive in where they were negative. This was considered a wind tunnel artefact. For the CSB application recorded data after restoration work has taken place on the bridge, indicate that there have been changes in flutter derivatives. Note that none of the works induced any explicit shape alteration to the deck. The process of understanding if such changes are truly of aerodynamic source is still ongoing. In any case this could hold the key for increasing the safety limit of a true bridge monument. Chapter 4 Quasi-steady galloping analysis revisited The flutter derivative based definition of aeroelastic loads, shown in the previous chap- ter, entails a series of laborious dynamic tests. Fortunately such descriptions become redundant in the case of galloping vibrations, where simpler static tests of the bluff body in hand together with quasi-steady theory can accurately capture the ensuing dynamic behaviour. Specifically the condition U r ≫ Sr −1 is considered sufficient for enabling this simplification. Yet classical galloping analysis deals mostly with cases of across-wind vibrations, leaving aside the more general situation where the wind and motion may not be normal. This can arise in many circumstances, such as the motion of a power transmission cable about its equilibrium configuration, which is swayed from the vertical plane due to the mean wind, or for a tall slender structure in a skewed wind. Furthermore the generalisation to such situations, when this had been made, has not always been performed correctly. This chapter aims at elucidating such shortcomings, thus naturally it is separated from the introductory Chapter 2. Initially the correct equations for the quasi-steady aerodynamic damping coefficients for the rotated system or wind are re-derived, and differences from other variants are highlighted. Motion in two orthogonal structural planes is considered, potentially giving coupled translational galloping, for which pre- vious analysis has often been limited or has even arrived at erroneous conclusions. For the two-degree-of-freedom case, the behaviour is dependent on the structural as well as aerodynamic parameters, in particular the relative natural frequencies in the two planes. Differences in the aerodynamic damping and zones of galloping instability are quantified, between solutions from the correct perfectly tuned, well detuned and clas- sical Den Hartog equations (and also an incorrect generalisation of it), for a variety of typical cross-sectional shapes. The presentation intends to introduce concepts essential for the subsequent parts of this thesis. 67 68 Chapter 4. Quasi-steady galloping analysis revisited 4.1 Introduction Quasi-steady theory allows aerodynamic problems to be simplified vastly by replacing the actual unsteady condition in hand to an equivalent static one, where only the relative flow velocity is considered for capturing the relevant aerodynamic forcing. Its most famous application is probably the galloping criterion put forward by Den Hartog [186, 187] setting the condition for dynamically unstable behaviour of a single degree- of-freedom (DOF) oscillator as: F ′ = sin α _ −L +D ′ _ + cos α _ L ′ +D _ < 0 (4.1) which is often (e.g. see Holmes [188, p117], H´emon and Santi [189, p856]) expressed in terms of static force coefficients as: S sc = sin α _ −C L +C ′ D _ + cos α _ C ′ L +C D _ < 0 (4.2) where α is the angle of attack and L, D, C L , C D are the static lift and drag forces and static lift and drag coefficients respectively, assumed for nominally 2D flow to be functions only of α, and the prime indicates the derivative with respect to α. The criterion is only a statement to avoid an undamped oscillator becoming negatively damped due to aerodynamic action. Thus the whole problem reduces to determin- ing the aerodynamic damping contribution and setting it equal and opposite to the available structural damping. Eq.(4.2) presented as such is strictly not valid since its trigonometric terms only apply for α = 0, corresponding to 1D across-wind oscillations, which is tacitly ignored. In the general case (i.e. α = 0), the principal structural axes may not be aligned along the flow direction and normal to it, for example for a vertical section in skewed wind (or for a horizontal section in inclined wind), or considering the static sway of a catenary due to the mean wind force. Then Eq.(4.2) fails to accurately account for the effect of aerodynamic damping, as in the across-wind galloping scenario, and even if employed in its correct Den Hartog stated form then it fails to describe the true condition. The correct treatise, although partly presented in the literature already (Richardson and Martuccelli [190], Blevins [1]) is generally not followed in practice and the resulting errors in calculations rising from the mishandling have not previously been quantified. To this end a number of benchmark cross-sections are considered to illustrate the differences emerging in defining instability bounds. In addition, the full extension of a generalised translational galloping criterion of sections with principal axes arbitrarily inclined to the flow, has to cover motion in both axes, including their coupling, which is especially important when they have close natural frequencies. Such an analysis has previously been performed by a number of 4.1. Introduction 69 authors with specific scopes and sometimes with erroneous conclusions. In particular, Jones [191] addressed coupled motion in two planes in some detail, but only for the special case of α = 0 with identical natural frequencies in the vertical and horizon- tal directions, and she concludes that no vertical galloping can occur when horizontal motion is restrained. Although suggesting that this may be a reason for experimental behaviour observed, it results only from mistreating boundary conditions and neglecting the effect of the envisaged restraining force in the analysis. Liang et al. [192] and Li et al. [193] used the formulation in terms of body co-ordinates, following Davenport [194], and covered the seemingly more general case of α = 0 2D perfectly tuned coupled motion. The fact that the frequencies are restrained to be tuned renders α arbitrary, thus the results should only be equivalent to Jones’ [191] case and not a generalisa- tion of it. By using force coefficients defined in body co-ordinates rather than wind co-ordinates, the connections of their work to the Den Hartog criterion are unclear. In the analysis a special behavioural subcase is missed and the inaccurate quote is put forward that 2D coupled galloping oscillations may occur only when the fundamental natural frequencies of a structure in the two orthogonal principal axial directions equal each other. Macdonald and Larose [134, 135], focusing on the dry inclined galloping of circular cables, accurately provided the full 2D aerodynamic damping contribution, including also terms due to Reynolds number and 3D geometric effects. Also, both resonant and non-resonant conditions between vibrations in the two planes were taken into account. A similar treatise, though waiving (and questioning) the use of Reynolds number dependent terms, was presented by Carassale et al. [137], who utilised Kro- necker products and matrix calculus to derive a full aerodynamic damping matrix. In both these research works the interest focused on circular cylinders and the objective was beyond deriving a simple Den Hartog analogue for motion in two orthogonal planes and testing it against the 1D requirement as is pursued here. Especially in Carassale et al. [137], due to concentrating on circular sections, derivatives with respect to α were not included in the analysis, rendering the formulation inapplicable to other clas- sical galloping cases. It is worth referring also to Luongo and Piccardo [195], who use bifurcation analysis to capture the limit cycle behaviour of detuned configurations, again limited to Jones’ [191] schematic case. These references so far, alongside broader 3DOF treatises 1 , with different perspectives and not focusing on subtle translational interaction details (e.g. [197, 198]), roughly cover the full range of available literature on modelling two-degree-of-freedom (2DOF) galloping vibrations. Other previous analysis of explicit 2DOF coupled quasi-steady instabilities has pri- marily been concerned with the combination of translational and rotational motions e.g. [1, 199–201]. Still, there is an inherent incompatibility of the quasi-steady formula- 1 Flutter galloping according to the terminology of Chabart and Lilien [196]. 70 Chapter 4. Quasi-steady galloping analysis revisited tion with rotations. There would always be some arbitrariness on selecting the point to which a unique effective wind vector refers to. On the other hand, in many cases there are similar structural conditions for translational motions in the two orthogonal planes normal to the cylinder axis, rotational motion may not occur simultaneously, and there is no complication in extending the classical 1DOF galloping treatment. These render natural the analysis of 2DOF galloping. The work should be very relevant to the later studied bridge cable motions. In what follows the correctly modified version of the Den Hartog criterion for an arbitrary angle of attack for 1D motion and a solution for the more generic motion in two orthogonal planes are re-derived. For the analysis the full aerodynamic damping matrix is formed and the scenario of coupled galloping oscillations is considered, which is a function of the structural parameters as well as the aerodynamic ones and can lead to elliptical trajectories. 4.2 Quasi-steady derivations The novel contribution of this chapter is not to propose new galloping criteria but, em- ploying the current state-of-the-art [134, 135, 190], to quantify the difference between the generalised 2DOF galloping scenario shown in Fig.4.1(a) and the normally con- sidered special case in Fig.4.1(b) for pure across-wind motion. For completeness, the quasi-steady aerodynamic damping derivations, yielding galloping criteria, are briefly repeated hereafter with an added intention of highlighting the errors in Eq.(4.2). For succinctness the variation of the force coefficients with Reynolds number is neglected, but its incorporation is straightforward. Figure 4.1. Geometry of a bluff section indicating lift and drag forces (L, D), relative angle of attack (α) and principal structural axes (x, y). (a) The general case with α 0 = 0 and the 2DOF motion potential. (b) The special case for 1DOF across-wind oscillations. The classical Den Hartog derivation starts typically by writing the mean aerody- namic force, per unit length, along the y-axis in Fig.4.1, as a function of the mean drag 4.2. Quasi-steady derivations 71 and lift forces: F y = Lcos α +Dsin α . (4.3) where L = 1 2 ρU 2 rel BC L , D = 1 2 ρU 2 rel BC D , ρ is the fluid density, B is a reference di- mension of the section and U rel is the relative velocity. For motion limited to the y direction, expanding F y around ˙ y=0 the standard damped equation of motion becomes m¨ y +c ˙ y +mω 2 y y = F y = F y | ˙ y=0 + ˙ y · dF y d ˙ y ¸ ¸ ¸ ¸ ˙ y=0 , (4.4) where m is the cylinder mass per unit length, ω y is the circular natural frequency (in the absence of wind), c is the structural damping coefficient, and dots represent differentiation with respect to time. Noting that exclusively when the free-stream wind velocity U and the motion velocity ˙ y are orthogonal and thus α = arctan(−˙ y/U) (Fig.4.1(b)), dF y d ˙ y ¸ ¸ ¸ ¸ ˙ y=0 = − 1 U F ′ y ¸ ¸ α=0 . (4.5) From Eq.(4.3), the derivative of F y with respect to α, which can be used for Taylor expanding around any initial inclination α 0 , is F ′ y = L ′ cos α −Lsin α +D ′ sin α +Dcos α . (4.6) In Eq.(4.4), the aeroelastic force (the last term on the right hand side), is equivalent to a linear viscous damping force. The condition for dynamic instability is simply that the total effective damping coefficient is negative. Hence from Eqs.(4.5&4.6), noting that for α = 0, U ′ rel = 0, it is easily shown that the galloping criterion is − dF y d ˙ y ¸ ¸ ¸ ¸ ˙ y=0 = ρUB 2 _ C ′ L +C D _ < −c , (4.7) where the threshold U is proportional to c as noted in §2.1.4. For zero structural damping, this reduces to the classical Den Hartog criterion presented earlier in Eq.2.9 S DH = _ C ′ L +C D _ < 0 . (4.8) which agrees with Eq.(4.2) for α = 0. If the motion is not normal to the wind direction, Eq.(4.7) does not hold and there are two problems with Eqs.(4.1&4.2). Firstly α = arctan(−˙ y/U) so Eq.(4.5) is not valid, which affects both Eqs.(4.1&4.2). Secondly in finding L ′ and D ′ , U ′ rel = 0, so extra terms are introduced. For the general case and for extending the analysis to cover two orthogonal translational DOFs (see Fig.4.1(a)), which potentially can lead 72 Chapter 4. Quasi-steady galloping analysis revisited to coupled response, the derivation follows. Eq.(4.3) still holds and also F x = −Lsin α +Dcos α. (4.9) Expanding F y and F x around zero motion, to find equivalent viscous damping terms, similar to before, F y = F y | ˙ x=˙ y=0 + ˙ x · dF y d ˙ x ¸ ¸ ¸ ¸ ˙ x=˙ y=0 + ˙ y · dF y d ˙ y ¸ ¸ ¸ ¸ ˙ x=˙ y=0 , F x = F x | ˙ x=˙ y=0 + ˙ x · dF x d ˙ x ¸ ¸ ¸ ¸ ˙ x=˙ y=0 + ˙ y · dF x d ˙ y ¸ ¸ ¸ ¸ ˙ x=˙ y=0 . (4.10) For evaluating the derivatives the chain rule is employed d() d ˙ x = ∂() ∂U rel · dU rel d ˙ x + ∂() ∂α · dα d ˙ x , (4.11) and similarly for ˙ y. Keeping in mind the relations U rel = _ (U y − ˙ y) 2 + (U x − ˙ x) 2 , tan α = U y − ˙ y U x − ˙ x , tan α 0 = U y U x , (4.12) finally the unit length full 2×2 aerodynamic damping matrix of a bluff section is ob- tained: C aero = _ _ c xxa c xya c yxa c yya _ _ = _ ¸ ¸ ¸ ¸ _ − dF x d ˙ x − dF x d ˙ y − dF y d ˙ x − dF y d ˙ y _ ¸ ¸ ¸ ¸ _ ˙ x=˙ y=0 = ρBU 2 _ _ S xx S xy S yx S yy _ _ , (4.13) with S xx = C D (1 + cos 2 α 0 ) −(C L +C ′ D ) sin α 0 cos α 0 +C ′ L sin 2 α 0 , (4.14a) S xy = −C L (1 + sin 2 α 0 ) + (C D −C ′ L ) sin α 0 cos α 0 +C ′ D cos 2 α 0 , (4.14b) S yx = C L (1 + cos 2 α 0 ) + (C D −C ′ L ) sin α 0 cos α 0 −C ′ D sin 2 α 0 , (4.14c) S yy = C D (1 + sin 2 α 0 ) + (C L +C ′ D ) sin α 0 cos α 0 +C ′ L cos 2 α 0 . (4.14d) The derivation is also valid for wind skewed to the cylinder axis, by employing the wind component normal to the cylinder and the force coefficients with respect to that component, as long as the independence principle is a viable approximation. 4.2. Quasi-steady derivations 73 The 1DOF instability thresholds for galloping in the x or y planes are simply when the diagonal terms of C aero become negative (or more generally equal to minus the structural damping coefficient). Evidently the non-dimensional aerodynamic damping coefficients, S xx and S yy in Eqs.(4.14a&d), differ from S sc in Eq.(4.2), confirming that it is incorrect. For α 0 = 0, S yy in Eq.(4.14d) reduces to S DH in Eq.(4.8) (as does S xx in Eq.(4.14a) for α 0 = ±90 ◦ ), as expected. For the instability condition of the coupled response, an eigenvalue analysis has to be performed for the 2DOF system, which is a function of the structural, as well as the aerodynamic, parameters. In general a numerical solution is required, but for the special case of equal mass (m), structural damping coefficient (c) and natural frequency (ω x = ω y = ω n ) for both DOFs, a closed form result is derived as below. The equations of motion m¨ x + (c +c xxa ) ˙ x +mω 2 x x = −c xya ˙ y , m¨ y + (c +c yya ) ˙ y +mω 2 y y = −c yxa ˙ x , (4.15) are assumed to possess a solution of the form x = X exp(λt) , y = Y exp(λt) , (4.16) where the eigenvalues λ and the amplitudes X, Y are in general complex valued. Eqs.(4.15&4.16) yield Y X = − λ 2 + c xx m λ +ω 2 x λ c xya m = − λ c yxa m λ 2 + c yy m λ +ω 2 y , (4.17) λ 4 + _ c xx +c yy m _ λ 3 + _ c xx c yy −c xya c yxa m 2 +ω 2 x +ω 2 y _ λ 2 + _ c xx ω 2 y +c yy ω 2 x m _ λ +ω 2 x ω 2 y = 0 , (4.18) where c xx = c + c xxa and c yy = c + c yya . For ω x = ω y = ω n , solving the biquadratic Eq.(4.18) for the stability boundary (i.e. purely imaginary eigenvalues, λ=iω) results in ω = ω n , together with c xx c yy −c xya c yxa = 0 , (4.19) or c xx +c yy = 0 (with ω not restricted to equal ω n ) . (4.20) 74 Chapter 4. Quasi-steady galloping analysis revisited Eq.(4.19) translates, by analogy with Eqs.(4.2, 4.8&4.14a&d), to the criterion for cou- pled galloping (for no structural damping): S 2D = 1 2 _ 3C D +C ′ L ± _ _ C D −C ′ L _ 2 + 8C L _ C ′ D −C L _ _ < 0 , (4.21) where S 2D denotes the non-dimensional effective aerodynamic damping coefficient of the coupled motion (equivalent to S xx and S yy for uncoupled motion) and the nega- tive square root obviously gives the critical case. Here Y/X is real, indicating planar trajectories. As expected Eq.(4.21) does not explicitly include α 0 . The solution in Eq.(4.20) corresponds to the so-called complex response [134, 191], which arises when the term under the square root in Eq.(4.21) is negative. Then the criterion for coupled galloping becomes S 2D = 1 2 _ 3C D +C ′ L _ < 0, (4.22) which coincides with the real part in Eq.(4.21), but in addition, the frequencies of the resulting two in-wind modes are released from being equal. This solution is often missed (as in [192,193]) by constraining X and Y to be real. From Eq.(4.17), for λ = iω with ω = ω n , Y/X is complex, indicating elliptical trajectories. Since the two modal responses occur simultaneously at different frequencies, a 2D beating-type behaviour occurs, as described in [134, 191, 195]. More generally, in the presence of structural damping (the same in both planes), the right hand side of Eqs.(4.21&4.22) becomes −2c/ρBU (equivalent to Eq.(4.7)). Also of interest is the case where the initial natural frequencies in the two DOFs (ω x and ω y ) are not equal. Then for the stability boundary, similar to Eqs.(4.17&4.18), is given by [135]: (c xx c yy −c xya c yxa )(c xx +c yy )(κ 2 c xx +c yy ) +c xx c yy (1 −κ 2 ) 2 m 2 ω 2 x = 0, (4.23) where κ = ω y /ω x . This can generally only be solved numerically. For all detuned cases, the trajectories become elliptical, similarly to the complex response for the perfectly tuned system and to actual occurrences of galloping in the field. 4.2.1 Relevance to uniform continuous systems It is worth noting that all the derived aerodynamic damping estimates (and hence the galloping criteria), although referring explicitly to a unit length section, are often also 4.3. Application: quantifying differences 75 applicable for a uniform continuous system allowing motion in two orthogonal planes, in a uniform flow. This can be easily proved by applying in Eqs.(4.15) the standard separation of time and space variables, x(s, t) = n φ xn (s)q xn (t) , y(s, t) = n φ yn (s)q yn (t), (4.24) where s is the distance along the continuous system, φ xn (s) and φ yn (s) are the n th undamped mode shapes in the x, y planes and q xn (t), q yn (t) the corresponding gener- alised displacements. Employing apart from the standard orthogonality conditions for same plane modes _ φ xn 1 (s)φ xn 2 (s)ds = 0 for n 1 = n 2 , _ φ yn 1 (s)φ yn 2 (s)ds = 0 for n 1 = n 2 , (4.25) the condition that the mode shapes in the two planes are the same (i.e. φ xn (s) = φ yn (s) = φ n (s)), for any pair of modes in the two planes, Eqs.(4.15) transforms to _ mφ(s)φ(s)¨ q x (t)ds + _ c xx φ(s)φ(s) ˙ q x (t)ds +ω 2 x _ mφ(s)φ(s)q x (t)ds = − _ c xya φ(s)φ(s) ˙ q y (t)ds , _ mφ(s)φ(s)¨ q y (t)ds + _ c yy φ(s)φ(s) ˙ q y (t)ds +ω 2 y _ mφ(s)φ(s)q y (t)ds = − _ c yxa φ(s)φ(s) ˙ q x (t)ds . (4.26) Since for a uniform section in a uniform wind, the generalised coordinates, mass per unit length and damping coefficients are not functions of s, the integral term _ φ(s)φ(s)ds cancels out, yielding back again Eqs.(4.15) and thus rendering the deduced instability thresholds in Eqs.(4.14a&d, 4.21&4.22) still valid. It is noted that when the aero- dynamic damping coefficients cannot be deemed to be constants over the structural length, as for instance for high rise bridge towers where the wind velocity profile is significant, or for a varying section, then the integrals in Eq.(4.26) should be calculated explicitly. However, in many cases the simplifying approach of uniform wind velocity and section is adequate, and in the present situation it allows comparison between the relatively simple different criteria presented above. 4.3 Application: quantifying differences The differences in the results arising from the different galloping criteria are quantified by utilising data of aerodynamic force coefficients for a variety of cross-sectional shapes. 76 Chapter 4. Quasi-steady galloping analysis revisited Fortunately such data are available in the literature for many shapes (e.g. see [190,191, 202–207]), although they have almost exclusively been used in the Den Hartog criterion only, which, as previously stressed, is not always the case in hand. For the current study a number of representative shapes, as illustrated in Fig.4.2, were chosen to span a whole range of possible relative differences between the different galloping criteria. The last three iced cable shapes (Figs.4.2(j,k,&l)), although only specific examples of the infinite number of possible iced geometries, were chosen for direct comparison with the work of Jones [191], since, although she attempted to define the worst case for 1DOF or perfectly tuned coupled galloping, she chose a presentation method that did not make the actual differences in the results clear. [190] [204] [204] [203] [205] [206] [203] [190] [207] [191] [191] [191] Figure 4.2. Sections with aerodynamic coefficients provided in the literature [190, 191, 203– 207], used in the galloping analysis herein. Fig.4.3, presents the non-dimensional aerodynamic damping coefficients for each galloping criterion, for each section, for the whole angle range of angles of attack that are available. Negative values identify aerodynamically unstable regions, where galloping would occur in the absence of structural damping. More generally, with structural damping, galloping occurs when the non-dimensional aerodynamic damping coefficient is below −2c/ρBU. For each shape, three lines are plotted, corresponding to the aerodynamic damping contributions from: i) the classical Den Hartog summation, S DH , in Eq.(4.8); ii) the more adverse of the two rotated 1DOF cases, S xx or S yy , as given in Eqs.(4.14a&d); and iii) the perfectly tuned 2DOF coupled motion case, S 2D , in Eqs.(4.21&4.22). It is pointed out that these correspond to three conceptually different motion scenarios: i) applies to different aerodynamic angles of attack of the cross- section, α, but with the motion always constrained to be purely across-wind; ii) applies to the instance where the principal structural axes and cross-sectional shape are fixed to each other and rotate together with respect to the wind (or the wind rotates relative to the structural axes and shape) as in Fig.4.1(a), with α 0 becoming the variable; and iii) applies to combined 2D motion with perfect tuning of the structural natural frequencies in the two planes, in which case the orientation of the structural axes is arbitrary, 4.3. Application: quantifying differences 77 reflected by Eqs.(4.21&4.22) being independent of α 0 , and only the orientation of the cross-sectional shape with respect to the wind direction is then relevant. The sub-case of the 2DOF complex response is distinguished in Fig.4.3 by plotting open circles. It is notable that there is no case of instability linked to this scenario (i.e. S 2D from Eq.(4.22), when it applies, is never negative). This is in keeping with the suggestion by Macdonald et al. [132], for galloping of a skewed stranded cable in the critical Reynolds number range, that the combination of parameters required for galloping of a complex response is unlikely to occur in practice. In addition, comparing Eqs.(4.8&4.22), since C D is always positive, the condition for 2DOF complex galloping is less onerous than the condition for pure across-wind galloping. Hence, in contrast to Jones’ [191] suggestion that observations of elliptical galloping trajectories in the field can be attributed to complex galloping, here it seems most probable that this is not the actual case. The results presented here are for the cases of the wind and motion direction fixed at right angles to each other or for the principal structural axes and cross-sectional shape fixed to each other (or for perfect tuning in 2DOF). The full generalisation allows the wind direction, principal structural axes and the orientation of the cross-sectional shape to all be independent. This could occur, for example, for a transmission line, where the wind is close to horizontal, the structural axes are given by the inclination of the cable catenary in the mean wind, and the cross-sectional shape can rotate due to the influence of gravity on any accreted ice. Such a situation is still covered by Eqs.(4.13- 4.15), where in Eq.(4.14) α 0 is the angle between the wind direction and the structural x-axis (Fig.4.1(a)), but C D , C L and their derivatives should be evaluated at the angle of attack between the wind direction and the reference direction of the cross-sectional shape (not necessarily equal to α 0 due to the cross-section rotating). Commenting on the individual plots in Fig.4.3, the first impression is that in most cases all the values follow roughly similar trends and predict close instability zones with respect to the angle of attack. Especially for sections being or resembling rectangles, in- cluding the square in Fig.4.3(d), the rectangle with side ratio 3:1 in Fig.4.3(g), and the rectangle with rounded ends in Fig.4.3(h), the instability zones from the three different criteria are almost indistinguishable, showing some insensitivity of the susceptibility to galloping for the different cases. The square and rectangle were actually chosen for this study for exhibiting different characteristics, with the square galloping for zero angle of attack and the 3:1 rectangular not (for a classical treatise on the instabilities of rectangular sections with different side ratios see Nakamura and Hirata [57]), although the most severe zone of instability is for an angle of attack near 10 ◦ in both cases. This is the case for the section in Fig.4.3(h) also. A similar connection exists between the rectangle in Fig.4.3(g) and the ellipsoid in Fig.4.3(c) with a strong instability near 70 ◦ for both, showing that sections very close to circular can still exhibit negative aerody- 78 Chapter 4. Quasi-steady galloping analysis revisited Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (a) 0 20 40 60 80 100 -3 -2 -1 0 1 2 3 4 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (b) 0 20 40 60 80 10 30 50 70 90 -3 -2 -1 0 1 2 3 4 5 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (c) 0 20 40 60 80 10 30 50 70 90 -20 -15 -10 -5 0 5 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (d) 0 20 40 60 80 90 10 30 50 70 -8 -6 -4 -2 0 2 4 6 8 10 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (e) 0 10 20 30 40 50 60 -4 -2 0 2 4 6 8 10 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (f) 0 20 40 60 80 100 120 140 160 180 -2 -1 0 1 2 3 4 5 6 Figure 4.3. Non-dimensional aerodynamic damping coefficients (S DH , min(S xx , S yy ), S 2D ) as a function of angle of attack for the cross-sectional shapes given in Fig.4.2 (inset letters link the two figures). Negative values indicate unstable behaviour (in the absence of structural damping). 4.3. Application: quantifying differences 79 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (g) 0 20 40 60 80 10 30 50 70 90 -5 0 5 10 15 20 25 30 35 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (h) 0 20 40 60 80 100 120 140 160 180 -25 -20 -15 -10 -5 0 5 10 15 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (i) 0 20 40 60 80 100 120 140 160 180 -4 -2 0 2 4 6 8 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (j) 100 120 140 160 180 200 220 240 260 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (k) 0 20 40 60 80 100 120 140 160 180 -8 -6 -4 -2 0 2 4 6 8 Angle of attack α[ ◦ ] S D H , m i n ( S x x , S y y ) , S 2 D (l) 0 50 100 150 200 250 300 350 -0.5 0 0.5 1 1.5 2 2.5 Figure 4.3 (continued) 80 Chapter 4. Quasi-steady galloping analysis revisited namic damping values and consequently unstable behaviour, in this case because, after separation at the sharp corner, the flow then does not reattach, causing a rapid drop in lift. In any case, for all the figures referenced so far, the differences are sufficiently small to consider that any of the above instability bounds works quite well in any ac- tual case. Indeed the previous lack of a study to quantify the differences arising from the use of different criteria can probably be linked to the fact that benchmark studies of galloping analysis have often been performed on rectangles or rectangle-like shapes, where the differences are unimportant. On the other hand, another section, equally widely studied in wind tunnel tests and historically connected with galloping, the D-section in Fig.4.3(a), shows more diverse behaviour. For zero angle of attack, the Den Hartog summation (S DH ) predicts zero aerodynamic damping (the D-section is a hard oscillator [208] that, for motion exceeding a certain amplitude, will gallop even for this angle of attack, but this is beyond the scope of the current presentation). Evidently the lesser of S xx and S yy (in this case S yy ) from Eqs.(4.14a&d) (referred to as the 1D rotated case hereafter) becomes the same as S DH when α 0 = 0, and slightly more unexpectedly the 2DOF solution (S 2D ) also falls on the same value giving a common start for all three. As the angle of attack increases, S DH departs from the other two, which have a negative peak near 40 ◦ , representing almost twice the negative aerodynamic damping as for S DH . This is a significant difference and it clearly demonstrates that the appropriate galloping criterion should always be applied carefully to the actual the problem in hand. Increasing the angle of attack further, a smaller instability zone is expected near 100 ◦ where now the most severe condition is for 2D motion and the Den Hartog summation estimates slightly more negative aerodynamic damping than for the 1D rotated case. Near 80 ◦ it is seen that S DH gives extremely positive aerodynamic damping. Looking more broadly it is seen that actually for nearly all sections the Den Hartog summation gives the highest positive aerodynamic damping value. Such extreme values are often very close to the ones coming from the alternate 1D rotated case (the greater of S xx or S yy ), which is not shown in Fig.4.3 that presents only the worse case. This also explains why in Figs.4.3(f,g&h) the Den Hartog summation does not match the 1D rotated case for zero angle of attack - the aerodynamic damping of along-wind vibrations is lower than for across-wind, although both are positive. Other sections considered also show notable differences between the outputs for the different cases. The results for the triangle with a vertex angle 30 ◦ as shown in Fig.4.3(f) show that near 20 ◦ the 1D rotated case is close to being stable while the other two cases are clearly unstable, and around 30 ◦ the 2D case is unstable whereas the other two are not. The same section near 180 ◦ (presenting a flat face to wind) on the other hand shows all the three lines in Fig.4.3(f) coinciding. Similarly the 4.3. Application: quantifying differences 81 Angle of attack α[ ◦ ] S s c , S y y Ssc, Eq.(4.2) Syy, 1D rotated, Eq.(4.14d) (a) 0 20 40 60 80 90 10 30 50 70 -8 -6 -4 -2 0 2 4 6 8 10 Angle of attack α[ ◦ ] S s c , S y y Ssc, Eq.(4.2) Syy, 1D rotated, Eq.(4.14d) (b) 0 20 40 60 80 100 120 140 160 180 -4 -2 0 2 4 6 Figure 4.4. Comparison between the erroneous S sc (Eq.(4.2)) and the correct value for the 1D rotated y-axis case, S yy (Eq.(4.14d)), for (a) the square in Fig.4.2(d) and (b) the triangle in Fig.4.2(f). equilateral triangle (Fig.4.3(e)) exhibits significant differences, although the 1D and 2D rotated cases generally give close results. In addition it is of interest to note that the two triangles behave quite differently (Fig.4.3(e&f)) when presenting their flat faces to the wind although only a small vertex angle change has occured. Differences are also apparent in Figs.4.3(b,j&k), with the 1D rotated case giving the least unstable results, thus rendering the simple Den Hartog calculation to be unnecessarily conservative if the structural axes rotate with the section. Conversely for the L-section in Fig.4.3(i), in the most critical region near 60 ◦ , the Den Hartog summation is unconservative. Drawing some general conclusions, although in most cases the broad picture from the three criteria is similar, the absolute aerodynamic damping values at certain angles can be quite different. There are many instances where a section stable according to one criterion can be unstable according to another, and there is no set sign for the relative differences, with changes being possible even for the same shape in a different range of angle of attack. Still, in almost all the examples interestingly the worst case occurs for the 2DOF criterion. It should be noted that the accuracy of the results is of course limited by the accuracy of the available data. But additionally there is the need to convert the discrete point measurements of static force coefficients into a continuous or piecewise continuous function in order to determine their derivatives. Many options were pursued towards this goal, including polynomial and harmonic curve fits of different orders. In any case there is a need for a very high order for any function to accurately fit the measured points, which was recognised early by Blevins [199]. In Blevins’ analysis the nonlinear 82 Chapter 4. Quasi-steady galloping analysis revisited terms enter the equation of motion and subsequently solutions are sought to yield the steady state amplitudes, thus it is detrimental that the introduction of different non- linearity, from different fitted functions, strongly influences the results. However for the purposes of the present analysis, different choices make little change to the results of the comparison (except making the plots smoother). For this reason the simplest possible piecewise linear assumption was picked for estimation of the derivatives in Fig.4.3. One of the main initiatives of this section was to correct Eq.(4.2), but it is equally important to show the error arising from its use. At first sight it is clear that for 180 ◦ rotation it is the opposite of the Den Hartog criterion, while for 90 ◦ any correlation with the Den Hartog criterion should be deemed as fortuitous. Judging from the general similarity that was earlier found between the Den Hartog case and the 1D rotated case, it is expected that great differences from Eq.(4.2) can emerge. Actually it should be noted that the correct direct equivalent of Eq.(4.2) is not the worse of Eqs.(4.14a&d), as considered earlier, but only Eq.(4.14d), for motion in the y direction. As can be seen in Fig.4.4(a), for plotted results for the square section, although Eq.(4.2) inaccurately estimates a weak instability zone near 80 ◦ it otherwise predicts instabilities in agreement with the correct result. This is only because a square’s critical zone occurs for small angles of attack, where Eq.(4.2) should be close to the Den Hartog criterion. On the other hand, for the triangular section in Fig.4.2(f) the errors are alarming (Fig.4.4(b)). Apart from the initial coalescence of the two curves, Eq.(4.2) is consistently unable to capture not only the value of the aerodynamic damping but even its correct sign. This is true for most of the section shapes considered (Fig.4.2). 4.4 The detuning effect The 2DOF solution presented earlier is restricted to perfectly tuned natural frequencies in the two motion planes. It is evident that in a great number of situations different natural frequencies exist for the different directions of motion. Such a case is found for instance on cables, where, due to sag, the frequencies of odd in-plane modes are higher than for the corresponding out-of-plane ones. For any detuning the coupled Eqs.(4.15) still apply, but for increasing detuning the coupling terms on the right hand side move further from the relevant natural frequency and hence have a reduced effect on the behaviour. Eventually, for greatly detuned systems the coupling becomes irrelevant and the system behaves like two uncoupled 1DOF systems in the orthogonal planes. This poses the interesting questions of: i) what happens for close but not equal natural frequencies and ii) what will the behaviour be for quite large detuning values. Utilising Eq.(4.23) for all the different sections in hand it is found, as intuitively expected, that the two 1DOF and the tuned 2DOF solutions define an envelope within 4.4. The detuning effect 83 Frequency ratio, κ = ωy ωx S d e t u n e d , S x x , S y y , S 2 D Sdetuned, 2D detuned solution, Eq.(4.23) Sxx, Syy, 1D solutions, Eqs.(4.14a&d) S2D, 2D tuned solutions, Eq.(4.21) (a) 1D y 1D x 0.8 0.9 1 1.1 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Frequency ratio, κ = ωy ωx S d e t u n e d , S x x , S y y , S 2 D Sdetuned, 2D detuned solution, Eq.(4.23) Sxx, Syy, 1D solutions, Eqs.(4.14a&d) S2D, 2D tuned solutions, Eq.(4.21) 1D y 1D x (b) 0.8 0.9 1 1.1 -1 -0.5 0 0.5 1 1.5 2 Figure 4.5. Evolution of the aerodynamic damping solution for different values of detuning, κ, for (a) the section in Fig.4.2(j) and α = 123 ◦ and (b) the section in Fig.4.2(k) and α = 30 ◦ . The lower branch is the important one. In (a) for perfect tuning the solution is unstable (negative aerodynamic damping) and for detuning above about 7% it approaches the 1D solution, which in this case is stable. In (b), on the other hand, for perfect tuning the solution is stable and for detuning above 1% it becomes unstable while moving towards the 1D solution. which the detuned coupled solutions always fall. The actual evolution with detuning consists of starting from the tuned 2DOF solutions (using ± in Eq.(4.21)) and progres- sively converging towards the 1DOF ones, as presented, for example, in Fig.4.5(a) for the iced cable section of Fig.4.2(j) for an angle of attack of 123 ◦ (other parameters (m, ω x , ρ, B, U) were taken from Jones [191]). The alternate path is given in Fig.4.5(b) for the iced cable section of Fig.4.2(k) for an angle of attack of 30 ◦ , where the 1D solution is more onerous than the 2D tuned one. The actual rate of convergence is found to be strongly dependent on the force coefficients and hence the angle of attack. The smaller the initial distance between the coupled and uncoupled solutions in Fig.4.3, the slower the rate of convergence seems to become. As clearly shown in Fig.4.5(a), for detuning as low as 7% the 2D solution, which is unstable when tuned, becomes stable and ap- proaches the 1D solution (see the lower critical branch in the figure and also Fig.4.3(j) at 123 ◦ ). Incidentally, this particular example gives the opportunity to correct an erroneous conclusion reached by Jones [191]. Results from tests on this section by Nigol and Buchan [209], where no galloping occurred, were taken as support for an assertion that in general when along-wind motion is restrained then no instability can ever occur. However, the reasoning was only a result of forgetting the associated external restraining force in the balance of forces in the equation of motion. As seen in Fig.4.3(j) at 123 ◦ , picked as being the most critical orientation, the tuned 2D solutions and Den Hartog 84 Chapter 4. Quasi-steady galloping analysis revisited criteria produce indistinguishable values. Thus in this case, with allowance for across- wind motion, presence or absence of along-wind motion, whatever the frequency ratio between the two, hardly changes the galloping threshold. The true reason for lack of observed galloping in this case, which is theoretically only slightly unstable, is likely related to the level of structural damping and/or slight inaccuracies in quasi-steady theory, as discussed by Bearman et al. [202] or Luo et al. [210]. Bearman et al. , while studying the square sections’ aerodynamic behaviour, noted interaction effects between galloping and vortex shedding that could delay galloping. Quite unexpectedly the structural damping values during limit cycle oscillations were at least 13% higher than the wind-off measured values. This was not a mere amplitude effect, and showed strong interplay with turbulence conditions. Further the transition to high U r , where quasi-steady theory assumes wind forcing in quadrature with motion was not recorded. Even for the highest attained U r the phase advance of the aerodynamic force was approximatelly 80 ◦ ahead of the motion. Luo et al. had a similar finding with unstable galloping sections only slowly reaching the theoretical phase advance of 90 ◦ . Closing this short parenthesis there should be a note on the response trajectories, which as already mentioned can range from planar to elliptical. As discussed above and shown in Fig.4.5, the detuned solutions quickly approach the aerodynamic damping values that correspond to the 1D solutions. However, they still remain qualitatively different from them in terms of the trajectories, which are described by Eq.(4.17). Such differences are presented for the case of Fig.4.5(a) in Fig.4.6. When perfectly tuned (Fig.4.6(a)) two planar modes occur (not necessarily in orthogonal planes). For small detuning values, the planar motion of the modes turns into ellipses with growing magnitudes of their minor axes as the detuning increases. This is shown for κ = 1.005 in Fig.4.6(b). For larger detuning the axes of the elliptical modes also rotate as in Fig.4.6(c) where κ = 1.05. This rotation continues until the principal structural axes are reached, and when that is virtually accomplished (for detuning of the order of 10% as in Fig.4.6(d)) the width of the trajectories reduces as they converge on the uncoupled planar solutions. The detuning values required to produce essentially planar responses are in fact much larger (of the order of 200% in this case). The relevant aerodynamic damping estimates are also shown in Fig.4.6 to indicate instabilities and to establish the link to Fig.4.5(a). It is a noteworthy conclusion that the elliptical galloping paths observed in the field (see discussion in [191]), are almost certainly due to a detuning effect between the structural axes. 4.4. The detuning effect 85 x y S2D = −1.24 S2D = 1.76 (a) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x y Sdetuned = −1.22 Sdetuned = 1.74 (b) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x y (c) Sdetuned = −0.13 Sdetuned = 0.65 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x y (d) Sdetuned = 0.48 Sdetuned = 0.04 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Figure 4.6. Modal trajectories corresponding to Fig.4.5(a) for (a) κ = 1, (b) κ = 1.005, (c) κ = 1.05 and (d) κ = 1.1. The applicable S detuned value is also indicated. Unstable modes are plotted with solid lines while stable ones are dotted. Note for comparison that S xx = 0.45, S yy = 0.06. For all plots the structural damping value was c = 0. 86 Chapter 4. Quasi-steady galloping analysis revisited 4.5 Concluding Remarks Quasi-steady aerodynamic theory, which lies at the core of the current analysis, has known limitations and it has long been recognised that in certain operational regimes (only approximately identifiable) it breaks down. Still, for a broad range of conditions for low frequency modes, quasi-steady analysis has a proven ability to predict aerody- namic damping and galloping behaviour. The current chapter identifies cases where limitations are introduced only because of shortcomings in the actual application of the method. In particular, the case sketched in Fig.4.1(a), with the structural axes inclined to the wind direction, has hardly been properly considered before and the differences in the behaviour from classical across-wind galloping had not been quantified. The Den Hartog criterion is correct for pure across-wind vibrations, but not otherwise. Its ap- plication, or even worse a faulted extension of it presented in Eq.(4.2), for the rotated system or wind, can yield solutions that can range from close to even the opposite of the correct ones. Although the Den Hartog summation often gives reasonable estimates of the aerodynamic damping, it can in some circumstances give negative estimates of only around half the magnitude of the real values, which is potentially unsafe. Conversely in some other cases it can be unnecessarily conservative. Furthermore, the dynamic stability of a section can be determined not only by its shape, the aerodynamic ori- entation and the orientation of the principal structural axes (which may or may not follow the aerodynamic orientation), but also by the proximity of the structural natural frequencies in the two planes. The correct equations for the non-dimensional aerodynamic damping coefficients, and hence the instability criteria, have been devised for arbitrary relative orientations of the system with respect to wind, and for perfectly tuned and well detuned natu- ral frequencies. Such a presentation is essential in realising the way galloping theory embodies geometric arrangement details and detuning. In particular detuning is a parameter almost entirely neglected in the existing literature. Here, detuned results, numerically obtained, always fall between the solutions for the 2D perfectly tuned case and the more adverse of the two uncoupled 1D cases, so the more critical of the 1D or 2D cases can be used conservatively. The equations provided are almost as easy to apply as the classical Den Hartog equation, yet they avoid potential errors and give accurate estimations of the aerodynamic damping and the propensity of a cylinder to gallop. But it is important to use the particular equation relevant to the specific problem being addressed. A main application of the above design tool in bridges are the bridge cables. Being of relatively small sectional dimension, having low motion frequencies due to their long 4.5. Concluding Remarks 87 lengths, and moving both in and out of the bridge-cable plane they seem an excellent practice field for the above analysis. An interesting illustration of a case where the above description becomes relevant is the lightning protection system of a cable-stayed bridge. This typically consists of ordinary stranded cables as in transmission lines. The Rion Antirion Bridge, which suffered a cable stay failure due to a lightning strike, had several issues with its lightning protection cables [211]. A simple galloping analysis as above would be able to efficiently mitigate the problem. On the other hand, in view of Eqs(4.14a,d&4.21) a perfectly circular cable with C L =C ′ L =C ′ D =0, C D >0 should not gallop. The reasons why bridge stays do seem to gallop in reality are subsequently pursued. Chapter 5 Experiments on galloping vibration of a circular cylinder This chapter addresses large galloping-like vibrations of bridge cables, generically in- clined and yawed to the flow. To this goal, wind tunnel experiments were performed for various geometric arrangements of a rigid circular cylinder covered with a smooth high density polyethylene (HDPE) duct as in real bridge stays. Both static and dynamic configurations of the cylinder model were tested, while Reynolds numbers spanned the estimated critical range 10 5 –4.5×10 5 . In what follows, initially the experimental appa- ratus is presented and subsequently the flow features and excessive motions observed are discussed. For motion frequencies far from K´ arm´ an vortex resonance, unsteady pressure measurements are utilised in order to uncover the aeroelastic forcing func- tion. It is shown that a fundamental difference between the inclined and non-inclined cylinder aerodynamics exists producing different pressure distributions and resulting in alternate dynamic behaviour. Reynolds number-induced transitional behaviour seems to be crucial in the recorded instability phenomenon in a way that was never suggested before in the literature. Intermittent ‘jumps’ in overall sectional forces, wake discon- tinuities between cell structures along the cylinder span-wise direction and axial flow are conjectured to be key elements towards realising the complex mechanism of dry galloping. 5.1 Introduction As it was shown in §2.2, there are many conflicting theories on the explanation of dry galloping. Most of them are qualitative and schematically demonstrate complex flow structure interaction mechanisms that could instigate response. However, they are un- able to seize the magnitude of the involved aerodynamic actions that is essential in 89 90 Chapter 5. Experiments on galloping vibration of a circular cylinder designing mitigation measures. One exception to this rule, that allows for quantifying damping thresholds, analogously to Chapter 4, is the Reynolds number based gener- alised quasi-steady model initially proposed by Macdonald and Larose [133]. Circular cylinders on different inclination-yaw configurations, could have a varying Reynolds number dependence of their mean static force coefficients. This renders the existence of dedicated sectional asymmetry terms, a sufficient but not necessary condition for galloping. The first wind tunnel tests, specifically designed to cover the range where Reynolds number effects on a dry circular cylinder’s mean forcing become dominant (i.e. critical Reynolds numbers), took place in 2001. Previous tests on moving cylinders in high Reynolds numbers are extremely limited, mostly refer to marine applications and focus on ordinary self-limited K´ arm´ an vortex shedding motions e.g. [110, 212]. Most impor- tantly they do not assess the influence of a cylinder’s inclination. The series of the 2001 tests, hereafter referred as Phase 1, was conducted on a 6.687m long inclined rigid ca- ble model, spring mounted (in two perpendicular directions) in the National Research Council Canada (NRC) open circuit propulsion wind tunnel. The tests produced data of significant vibrations, including a record that was quoted being of diverging character, see Cheng et al. [213, 214]. This case, which appeared to be dry galloping, proved to be non-reproducible in a single repeated run. The main reason for the non-repeatability, was suspected to lie on ambient weather conditions change. The only explicit infor- mation reporting on this is that during the unstable test it was raining outside while the unsuccessful run arose on a dry sunny day. Yet it could have also been a system- atic behaviour of a complex coupled system that even under identical conditions can produce alternate results. The Phase 1 tests applied a range of parametric scenarios, varying the cable inclination angle, critical structural damping ratio, support spring orientation and surface roughness, with wind speeds covering the full nominal critical Reynolds number range. Unfortunately there is no direct evidence of the actual flow regimes encountered. Various attempts have been made to explain the significant vi- brations observed. To complement these unique Phase 1 dynamic tests, a new test series of static tests, referred to as Phase 2, was undertaken in the NRC 2m×3m closed circuit wind tunnel. The cylinder this time was much smaller and testing parameters such as end conditions, flow smoothness, aspect ratio and Mach numbers also differed. The aim was to gain further insight into the cable forcing by testing a similarly inclined static cylinder equipped with pressure measurement taps [215, 216]. The mean static force coefficients subsequently deduced were used by Macdonald and Larose [133–135] in their earlier referenced galloping framework to identify insta- bility regions. The largest response incident from Phase 1 was adequately predicted but obviously not also its non-repeatability. Other Phase 1 large response cases, of smaller 5.2. Wind tunnel tests 91 amplitude though, could not be traced by the galloping analysis and were thus waived the galloping characterisation. Another very similar variant of quasi-steady analysis was independently put forward by Carassale et al. [137]. In contrary to Macdonald and Larose they consider the Reynolds number to have a ‘slow’ effect on the aerodynamic forcing. This translates in disregarding all derivatives with respect to Re when form- ing aerodynamic damping expressions (a process as in Eqs.(4.10&4.11)). The method was still found capable of reproducing the observed dry galloping incident. Cheng et al. [217] attempted to formulate a simplified Den Hartog-like criterion, different from the above two, for studying the instability attributes. As a matter of fact the plausible expression acquired is only product of mishandling derivatives’ estimates [217, p2272 Eqs.(A.11&A.12)]. When the correct derivation is performed results revert to the orig- inal Macdonald and Larose, Carassale relations. In a distinct approach, Jakobsen et al. [218] utilised the unsteady force components contained in the static measurements to show that they could also cause large responses. And this is without employing any force lock-in or amplitude dependent action. The ultimate decisive test for all these treatises would be a comparison with the real dynamic forcing, which has never been measured before. In this chapter a newer series of wind tunnel tests, Phase 3, carried out under a collaborative project between NRC, Canada, the University of Stavanger, Norway and the University of Bristol, UK is described. It used the same large-scale aeroelastic model as in Phase 1, but additionally instrumented with pressure taps and covering a different range of parameters, strategically selected for large response to emerge. The main objectives pursued herein are to identify the local characteristics of the aerodynamic forcing causing the vibrations, and to highlight the true effects of Reynolds number. 5.2 Wind tunnel tests 5.2.1 Preliminary Imagine the case of a real inclined bridge cable lying on a vertical plane subject to horizontal arbitrary yawed wind as in Fig. 5.1. The cable orientation against the wind can be uniquely described by the set of cable inclination angle ϑ and yaw angle β. When modelling cable wind induced vibrations three directions are of major importance: the mean wind direction, the cable orientation and the motion direction. Preserving the way the wind sees the cable should leave unaltered the relative angles between them. In wind tunnels, space restrictions often put limitations in the range of angle sets ϑ, β that can be reproduced. This is also the current case. Since for the aerodynamic problem in hand gravity is not relevant, as in rain-wind induced vibrations for instance, a rotational 92 Chapter 5. Experiments on galloping vibration of a circular cylinder Figure 5.1. Transformation from real cable to wind tunnel model. 5.2. Wind tunnel tests 93 transformation that will best fit the testing facility is performed, while keeping relative angles constant. Thus ϑ and β convert to ϕ, which is the angle between the wind and cable axis vectors, and α, which is the angle between the out-of plane motion direction x 1 and the wind-cable plane normal x 2 . It is straightforward to prove that the mapping operation yields the relations cos φ = sin β cos θ , tan α = cot β/ sin θ . (5.1) In the setup of both Phase 1&3 series, ϕ was implemented as the vertical angle inclina- tion i.e. its horizontal projection is parallel to the wind. This corresponds to rotating the real cable in Fig. 5.1 around the AB axis until the cable-wind plane coincides with the vertical plane. Evidently α will be realised as the angle between the spring support axes and the across-wind direction. Every geometric configuration ever tested with its analogy to prototype set of angles and its naming convention as introduced in [146] is given in Table 5.1. Table 5.1. Orientation angles for studied cases. P h a s e 1 Setup ϑ β ϕ α P h a s e 3 Setup ϑ β ϕ α ( ◦ ) ( ◦ ) ( ◦ ) ( ◦ ) ( ◦ ) ( ◦ ) ( ◦ ) ( ◦ ) 1B 45 90 45 0 2A 60 90 60 0 1C 30 54.7 45 54.7 2C 45 45 60 54.7 2A 60 90 60 0 4A 90 90 90 0 2C 45 45 60 54.7 4B 90 60 90 30 3A 35 90 35 0 5B 75.1 61 77 30 3B 20 60.6 35 58.7 5A 77 90 77 0 The (NRC) open circuit propulsion wind tunnel is an open loop wind tunnel blowing air from the outside environment. Air is sucked via a conical intake from an electrically powered fan with diameter 7.9m. The screens available for the inlet entry were not in place during the time of Phase 3 tests. Before entering into the settling chamber the flow passes additional screens, various stators and straightening vanes to counteract the fan-induced swirl. Following, the air goes through a 1:6 contraction prior to entering the test section. The test section is 12.2m long, 6.1m high and 3.1m wide. The model was positioned within the second third of the tunnel after the end of the contraction. A view of the facility and its inner section with the model in place is given in Fig. 5.2. Right after completion of Phase 3 a calibration operation was performed by NRC staff, acquiring local flow characteristics at 105 stations across the tested section area. Results from data acquired, are illustrated in Fig.5.3. Turbulence intensities, referring to the 94 Chapter 5. Experiments on galloping vibration of a circular cylinder Figure 5.2. View of the NRC wind tunnel facility and its test section with the model in place. total wind vector, were of the order of 0.5%. As it can be seen a vertical asymmetry exists in the tunnel. This was quoted to be mainly due to the removal of the lower fillets. 5.2.2 Setup details In an attempt to reduce any potential scaling artefacts, the cable model had typical properties of a full-scale bridge cable. The actual cylinder was a rigid steel pipe with diameter d=140mm and thickness 16mm. It was covered with a smooth close fit HDPE pipe of the same type that is used for covering cable strands on site. The duct had a nominal outside diameter of 159mm. The final model was measured to have d=161.7mm (though quoted as 160mm in all later discussion) and a total mass of 60.3kg/m. For simply supported ends, these numbers give a natural bending frequency estimate of 7Hz. The aspect ratio (l/d) was approaching 40, which according to West and Apelt [219] is sufficient to minimise contamination of pressure data due to end effects. Still the tested models of West and Apelt were horizontal and fitted with end plates, which is not the case here. The wind tunnel blockage ratio is calculated at 5.3%. In line with Blackburn [220], this value is sufficiently low not to cause added artificial span-wise coherence. The wind speed (U) ranged for most tests between 10-40m/s, corresponding roughly to Reynolds numbers of 10 5 to 4.5×10 5 , fully covering the expected critical range. The maximum Mach number tested was less than 0.12. 5.2. Wind tunnel tests 95 Figure 5.3. Turbulence intensity (U rms /U) and mean wind velocity variation with height across the wind tunnel section. The cable assembly was mounted on spring supports at both top and bottom, allowing motion in the ‘sway’ and ‘heave’ directions normal to the cable axis as shown in Fig. 5.4. A cable support was additionally provided at the top to carry the axial component of the cable weight. The natural frequency for translational motions (f) was set at approximately 1.4Hz. An end to end rotation also existed with a frequency around 2.21Hz. For practical reasons, the lower (down-wind) end of the cable support was positioned inside the wind tunnel, whereas the upper one was outside the section (see Fig. 5.4). The cable passed through a rectangular opening in the roof (approximately 0.43m along-wind × 0.42m across-wind) to allow motion. The orthogonal sets of springs at both ends were connected to the cable body with spacer plates that indented to inhibit any rotational motion. Structural damping ratios were attempted to be very low to allow for aeroelastic effects to be distinguished. Values ranged between ζ=0.06-0.33% for low amplitude response. It should be noted that the quoted numbers significantly increased during high levels of response. For identical natural frequencies in the two vibration planes, the orientation of the principal axes is arbitrary. However, real cables have slight detuning of their natural frequencies in the vertical and horizontal directions due to the cable sag. This was modelled with slightly different spring stiffness in the two perpendicular directions. With detuning, the orientation of the principal axes becomes significant. Former experience from Phase 1, indicated the importance of the 96 Chapter 5. Experiments on galloping vibration of a circular cylinder detuning parameter. The range of detuning between the natural frequencies in the two planes was varied up to approximately 3%. To cover a variety of possible geometric arrangements, the cable was tested at three different inclination angles attempting to expand the previous Phase 1 range. As presented in Table 5.1, values of ϕ were 60 ◦ , 77 ◦ and 90 ◦ , with 60 ◦ being tested again to cover the previously quoted divergent response record [214]. The support spring rotation angles were chosen as 0 ◦ and 54.7 ◦ for ϕ= 60 ◦ , and 0 ◦ and 30 ◦ for the other inclinations. The rationale behind angle selections is mostly founded on analysis from previous Phase 2 results [135, 137]. Further the vertical cylinder scenario is a benchmark case that can allow for comparisons with existing literature. It was identified that the principal axes in cases with α=54.7 ◦ did not coincide with the rotated spring axes, indicating the existence of structural coupling. For each of the inclination angles, a set of tests was also taken with the model fixed in position, to measure the pressures on the stationary cylinder and compare with the corresponding dynamic cases. The static tests for the inclinations of 60 ◦ and 90 ◦ were conducted with the hole around the top end of the cable both open and sealed in order to assess the influence of the top end condition. Judging from the attained mean pressure profiles, very small differences between the two alternatives established that the top end effects were not significant, at least locally in the vicinity where the pressures were measured. The pressure measurement instrumentation was arranged in four rings of 32 pressure taps each, more densely spaced on the leeward side, as illustrated in Fig. 5.4. Additional lines of pressure taps were positioned close to the expected separation and back pressure points at 100 ◦ and 150 ◦ from stagnation (measurements for when α=0) at one diameter span-wise intervals. Taps were connected to four electronic pressure scanners embedded in the model via 1mm diameter urethane tubing of varying length. All data had to be corrected for the tubing frequency response. In short, the frequency response function of each tap was measured prior to the experiments. Typical examples of them shown in Fig. 5.5, look very similar to previous results obtained by Irwin et al. [221]. Using these transfer functions the recorded measurements were corrected for magnitude and phase distortions in the frequency domain. Details about the pressure measuring technique on a similar application are given by Gatto et al. [222]. Both the displacements and accelerations of each end of the model were measured in the two spring directions (termed heave and sway throughout, defined in Fig. 5.4). Data sampling was performed at two different frequencies, 500Hz and 1250Hz, to assess the significance of very high frequency components. It was found that the higher sampling frequency was redundant. Being interested in fine details of the flow, the turbulence intensity was measured with two dynamic three-component cobra probe instruments, upstream and downstream of the cable model. Most of the upstream 5.2. Wind tunnel tests 97 cobra measurements were later found to be corrupted by noise and were discarded. An additional effect that could influence the studied flow transitions was ‘dirt’ accumulation on the cable surface particularly on the windward face. This was due to the open return design of the wind tunnel, driving in air from outside. The size of these random roughness anomalies (up to 1mm high) could be large enough (according to Achenbach and Heinecke [223] and Shih et al. [224]) to alter the flow transition behaviour. To minimise this effect the cable was regularly cleaned. The pattern established was that an unclean cable would not present large motion. All the tests described below are believed to not have been exposed to the influence of this uncontrolled parameter. On a clean cable, surface roughness corresponded to a roughness-to-diameter ratio (ε/d) of 6×10 −6 . Figure 5.4. Elevation of cable model showing instrumentation arrangement. Some distances are not in scale. For accurate positioning details consult Table 5.2. 98 Chapter 5. Experiments on galloping vibration of a circular cylinder Table 5.2. Position details for the model. For rings and lowest cable end ‘distance from floor’ refers to stagnation points, while for cobra probes ‘distance from model’ refers to along-wind distance. ϕ ( ◦ ) Distance from floor (m) Distance from model (m) target/actual Ring1 Ring2 Ring3 Ring4 Low end Cobra1 Cobra2 Cobra1 Cobra2 60 / 59.4 3.87 3.594 3.043 2.63 0.859 4.34 3.19 3.018 0.133 77 / 76.7 4.231 3.919 3.296 2.829 0.827 4.34 3.24 2.381 0.305 90 / 90.9 4.152 3.832 3.192 2.712 0.655 4.34 3.19 1.702 0.304 Figure 5.5. Typical frequency response curves for three pressure taps. 5.3. Results 99 5.3 Results 5.3.1 Overview and large responses Large responses of primarily across-wind character were observed, only for the incli- nation of ϕ=60 ◦ , for both cases of spring rotations examined, α= 0 ◦ and 54.7 ◦ , and for various combinations of spring tunings in the two vibration planes. All events refer to the lowest translational modes with end to end motion always yielding much less significant amplitudes. Results are consistent with previous findings by Matsumoto et al. [120], who, for a horizontal cylinder restricted to planar across-wind motion, identi- fied the range of yaw angles for which large cable vibrations occurred to be β ∈[22.5 ◦ , 45 ◦ ]. Note that cable orientations relatively close to parallel to the wind direction were not considered and that the test wind speeds were intended to be in the subcritical Reynolds number range. In the current study, the large responses for a cable-wind angle of ϕ=60 ◦ fell in the unstable region, as previously identified in Phase 1, while inclinations of ϕ=77 ◦ , 90 ◦ did not produce any similarly large responses. All large vibrations in the Phase 1&3 tests only occurred within a limited range of wind speeds. This feature is reminiscent of typical K´ arm´ an vortex shedding, although the frequency content was very far from K´ arm´ an vortex resonance (cf. K´ arm´ an vortex shedding corresponds to a reduced velocity U r =U/fd ≈5 or lower, while in the tests exhibiting large vibrations it was over 100). The feature also distinguishes the observed response from classical Den Hartog galloping, which occurs for all wind speeds exceeding a certain threshold (see Fig. 2.5). However such behaviour does not necessarily mean that the term galloping is inappropriate. Galloping response of such a transient type, though of different attributes, has previously been presented in Fig.2.6. Conjecturing that instability may be triggered by some sustained boundary layer asymmetry in the critical Reynolds number range, the excitation mechanism may be similar to normal galloping but the restoration of symmetry in the supercritical Reynolds number range for increased wind speeds may bring it to an end. This is exactly the underlying mechanism implication of the quasi-steady analysis proposed by Macdonald and Larose [133–135]. Their predicted unstable vibrations are limited to the critical Reynolds number range, as observed, in contrast to classical galloping and to other proposed mechanisms that are not limited to a specific wind speed range. However it should be pointed out that the vibrations expected from the quasi-steady analysis for the setup 2A (see Table 5.1) were more along-wind, rather than across-wind as observed in Phase 3. The quasi-steady implementation of Carassale et al. [137] cannot predict any large amplitude response for this case. The maximum amplitude of vibrations observed was around 0.75d, which reached the maximum available clearance of the hole through the wind tunnel roof and corre- 100 Chapter 5. Experiments on galloping vibration of a circular cylinder sponded to significantly increased structural damping values. The aerodynamic limit state, if one exists, would occur at higher amplitude. In every case the trajectories of the cable for large vibration events were elliptical, generally with the major axis at an angle to the spring directions, suggesting coupling action induced by aeroelastic forces. The motion trajectories evolution with detuning shown in Fig. 5.6 is qualitatively in excellent agreement with theoretical quasi-steady analysis of coupled translational gal- loping oscillations as presented earlier in Chapter 4. Relatively large detuning values (in this case of only 2%) produce an ellipse aligned with one of the uncoupled degrees of freedom (the more excited one) as in Fig. 5.6a. Closer tuning of the system (in this case roughly 1% detuning) rotates the ellipse as in Fig. 5.6b&c, and perfect tuning would lead to planar motion in the direction of the divergent coupled mode. The width of the ellipse is controlled by the coupling action induced by the wind. Since in all large amplitude cases the motion was predominantly across-wind, and in some cases almost exclusively so (e.g. Fig. 5.6a), it seems that the dominant aerodynamic forcing is across-wind and it is likely that the along-wind component arises as a secondary ef- fect from coupling, giving a significant response in that direction for very close tuning. 5.3.2 Pressure data When correlating large responses to Reynolds number it is observed that major events were grouped in two distinct regions at approximately Re=2.5×10 5 and 3.5×10 5 , falling inside the boundaries of the drag crisis. Proper Orthogonal Decomposition (POD) was performed on the pressure tap data and interestingly it was found that some consis- tency of loading exists in the aforementioned regions. As seen in Fig. 5.7, which presents the cumulative variance explained by a relatively small number of Proper Orthogonal Modes (POMs), two peaks emerge, approximately coinciding with the Reynolds num- bers for large responses. The number of POMs selected was such that the cumulative variance could reach a value of around 90%, which is routinely selected when POD is employed for filtering purposes. The trends presented remained even with alternate choices of POM numbers. Very noticeably, the behaviour is similar for both the static and dynamic tests indicating organisation of the loading even for the static cylinder, although the actual POMs themselves were found to differ in the two cases. An actual set of relevant modeshapes in both the static and dynamic tests is illustrated in Fig. 5.8. The POMs selected are the most energetic ones and correspond to the points indicated by arrows in Fig. 5.7. Interestingly modeshape coordinates for Ring 3 are quite similar in both dynamic and static cases. The greatest variance contributions are located at what seems to be the separation region on the left side. Actually for the dynamic tests 5.3. Results 101 −0.25 0 0.25 −1 0 1 A c r o s s w i n d d i s p l a c e m e n t ( / d ) −0.25 0 0.25 −1 0 1 −0.25 0 0.25 −1 0 1 heave sway a heave sway b heave sway c Along wind displacement (/d) U=30m/s Re=3.1×10 5 U=32m/s Re=3.4×10 5 U=35m/s Re=3.7×10 5 Figure 5.6. Motion traces for cases a) ϕ=60 ◦ , α=0 ◦ , with frequency ratio in heave/sway f h /f s =0.979 and structural damping ratio in heave/sway ζ h /ζ s =1.91; b) ϕ=60 ◦ , α=0 ◦ , f h /f s =1.007, ζ h /ζ s =1.18 and c) ϕ=60 ◦ , α=54.7 ◦ , f h /f s =1.014, ζ h /ζ s = 0.94. The heave and sway spring axes are also presented. ‘Along-wind’ really means normal to the cable in the cable-wind plane rather than along the wind itself. 102 Chapter 5. Experiments on galloping vibration of a circular cylinder Figure 5.7. Proportion of total variance from 20 POMs (from all pressure tap data) against Reynolds number. Model setup: ϕ=60 ◦ and, for dynamics tests, α=0 ◦ . all the Rings peak at around this location. On the right side there is a much broader contribution, which seems to attain its maximum near the across-wind pressure tap. The POD analysis indicated the existence of both localised and widespread energetic modes to explain data variances. There were cases where time coefficients of certain POMs from the static tests were primarily harmonic with a frequency close (but not equal) to the structural frequency, raising the question as to whether this component has the ability to lock in with motion to cause large response in the later dynamic tests. The subcritical Reynolds number behaviour is dominated by periodic vortex shedding and accordingly the possibility for fewer periodic modes to relatively accurately describe the pressure tap measurements. Increasing Reynolds number into the critical range it seems to destroy coherent structures and cause an increase in the dimensionality of the underlying dynamics. Still, there are two breaks in the expected monotonic decrease in the variance, which provide the system with energetic mechanisms that most probably accommodate the large responses. The differences presented in Fig. 5.7 may seem small but they are consistent and moreover seem to be amplified under the influence of large scale motion, thus giving an indication of a possible lock-in action. Power Spectral Densities (PSDs) of the fluctuating lift coefficient (C L ), averaged over the four pressure rings, for dynamic tests in different wind speeds are presented in Fig. 5.9. The PSDs are normalised by multiplying by U/d, the inverse of the nor- malisation of the frequency axis, to preserve the variance magnitude. The plot shows the evolution of the wind forcing in four characteristic behavioural cases for different 5.3. Results 103 Figure 5.8. First Proper Orthogonal modeshapes for a set of dynamic and static tests. Tests correspond to the cases indicated by arrows in Fig. 5.7. Interestingly the greatest variance contributions originate from the near separation regions. Figure 5.9. Spectra of the lift coefficient, averaged over all four pressure rings. Star-marked points, 32m/s case, show the resonant peaks that correspond to the structural frequency of 1.4Hz and twice this value. Model setup 2A. 104 Chapter 5. Experiments on galloping vibration of a circular cylinder Reynolds numbers. For the subcritical range, represented by Re=1.2×10 5 (U=11m/s), clear vortex shedding can be identified at a Strouhal (Sr) number of 0.17 corresponding to 12Hz (cf. the natural frequency of 1.4Hz). When applying the independence prin- cipal using the component of wind normal to the cable (Usin ϕ) the estimate matches the expected value for a static cylinder normal to the wind, Sr=0.2. Also the cable inclination broadens the frequency range of the forcing, relative to the normal wind case, which looks very similar to the effect of added turbulence [225]. Increasing the Reynolds number, e.g. Re=2.5×10 5 (U=23m/s), leads to the vortices becoming inco- herent, thus reducing the spectrum in the Strouhal reduced frequency region. At the same time some very low frequency components emerge fd/U <0.05) and the spec- trum becomes flat for a range of reduced frequencies from 0.1 to 0.2. This regime was reached for much higher Reynolds numbers for the cable normal to the flow. For the record for which there was a large cable response, at Re=3.4×10 5 (U=32m/s), there are large sharp peaks in the spectrum at reduced frequencies corresponding to the mo- tion frequency and twice this value, indicated by stars in Fig. 5.9. It seems that the energy in the broad low frequency band locks in to the structural frequency and large motion builds up, while some broader band low frequency forcing, probably related to weak vortex shapes, still survives for fd/U <0.1. In the supercritical Reynolds number regime, e.g. Re=4.3×10 5 (U= 40m/s), the low frequency components have vanished and only low-level broadband excitation remains for fd/U <0.3 not having consistent uniform periodic components. On the contrary, as expected according to Roshko’s (normal cylinder) tests [108], the static equivalents did show up a narrow band process near fd/U=0.2. It is important to note that the lift coefficients above were calculated as the average from all four rings, whist there were marked differences between the rings, particularly for the ϕ=60 ◦ inclination case. It is believed that this should not be a consequence of end-effects. As a matter of fact there was some consistency between rings for the two different sets of end conditions, which possibly implies that this be- haviour is mainly sourced by the way the 3D flow pattern establishes itself regardless of boundaries. To investigate the relationship between the aerodynamic forces at the different rings, lift and drag cross-correlation functions were estimated. Fig. 5.10 gives an illustration of such functions showing the systematic force delays along the cable, indicated by the asymmetry between the top right and bottom left of the figure. Such delays denote some propagation in the axial direction only along with the flow. When the time lags of the peak absolute values of the cross-correlation functions are transformed into a propagation velocity, they compare quite well with the axial component of the free- stream wind, although they are not constant along the length between different rings but show a variation. This result is in excellent agreement with the numerical findings 5.3. Results 105 Figure 5.10. Correlation functions (R) of lift coefficients (C Li C Lj ) between rings (i, j) for the subcritical dynamic test case of U=13m/s, Re=1.4×10 5 . Propagation is evidently one-sided. Model setup, ϕ=60 ◦ and α=0 ◦ . of Yeo and Jones [117]. Indications of axial propagation were clear in both the static and dynamic tests for the cable inclined at ϕ=60 ◦ and, as possibly expected, ceased (or almost ceased) for higher inclination values. Fig. 5.10 intentionally displays a subcritical case with apparent vortex shedding, and was picked in order to match the Reynolds number of the numerical simulations by Yeo and Jones [116, 117]. To continue on the inhomogeneity point raised earlier, it is indicative from the auto-correlation functions, along the diagonal of Fig. 5.10, that the state of shedding for each ring is very different. For Rings 1-3 vortex shedding is highly damped in contrast to the strong periodic phenomenon that is well known to cause large responses of cylinders normal to the flow. The periodicity translates typically to Sr=0.17 which is around 20% lower than the value recovered by Yeo and Jones. At Ring 4, most interestingly, vortex shedding is almost entirely masked (but not vanished) by a low-frequency broadband process. The case is reminiscent of so-called swirling structures and their associated pressure distributions [117]. In any case, signature indications of some axial correlation persisted throughout the wind speed range along the whole cable. Another possible effect of this apparent secondary flow can be conjectured in view of static pressure (C p ) profiles such as the ones given in Fig. 5.11. Ring 3, presents near 120 ◦ what looks as a sustained attached flow or an axial-vortex type disturbance. Its pressure contribution is reaching only up to the base pressure point (180 ◦ ). There the behaviour changes abruptly, which 106 Chapter 5. Experiments on galloping vibration of a circular cylinder appears to be consistent with the suggestion by Matsumoto et al. [120] that axial flow on the lee side of the cable acts as a splitter plate. Still it should be highlighted that only Ring 3 receives locally a splitting action if truly this is the case. 5.4 Discussion 5.4.1 Symmetry considerations The force measuring technique of pressure taps employed (integrating pressures over tributary areas) has the feature of allowing the assessment of the contribution of par- ticular segments of the cross section to the total drag 1 and lift forces. Dealing with an instability that is fundamentally fed by asymmetries, it would be useful to identify where and how these arise. As indicated in §2.2, which meticulously described the evolution of the flow characteristics for smooth flow past a normal circular cylinder over the whole Reynolds number range, asymmetries can occur during the initiation of critical transition. There the drag suddenly drops and considerable mean lift appears, since half the section contains a laminar separation bubble while the other half does not. The modified treatise, also allowing for the effects of turbulence and roughness on flow ranges, is much more complex. For the current case, to assess the transitional flow symmetry, the section illustrated in Fig.5.4 (for α=0 ◦ ) is split into a ‘right’ part containing taps 4-3-2. . . -20 (with only half the 4 and 20 contributions) and a ‘left’ part containing taps 4-5-6. . . -20 (again with half the 4 and 20 contributions). There is some conceptual difference in the behaviour of individual rings while vary- ing the inclination angle, ϕ. As presented in Fig.5.12, for the inclination of 60 ◦ , the drag crisis zone during static tests has very distinct features for different rings. For Ring 2 the half-perimeter drag contributions (C D 1/2 ) from the ‘right’ and ‘left’ parts almost coincide while for Ring 3 there is consistent spacing during the drag drops. These are signs of two bubbles forming simultaneously or sequentially for increasing Reynolds number. Thus for a given Reynolds number inside the drag crisis region, two states exist together along the cylinder length (Rings 2 and 3 are at a spacing of 4 diameters). But according to previous studies on smooth cylinders in smooth flow, the simultaneous formation of two bubbles, as for Ring 2, does not occur in the critical Reynolds number range but only in the supercritical or transcritical range. A similar situation to the above observation was described by Zdravkovich [109], who found that roughness of the order of only ε/d ∼ =0.003, or alternatively turbulence, not only shifts the drag crisis to lower Reynolds numbers but actually obliterates the 1 Drag is here taken to be normal to the cable in the cable-wind plane 5.4. Discussion 107 Figure 5.11. Mean pressure coefficient distribution around cylinder for large response case of U=32m/s, Re=3.4×10 5 (Fig.5.6b). Model setup ϕ=60 ◦ , α=0 ◦ . critical state and causes a transition directly from the subcritical to the supercritical state. More strikingly, in the presented results the two conditions are found to co- exist stably (i.e. pressure profiles locally did not change state) along the cylinder length. Neighbouring sections, with one and two bubbles respectively, will shed wake vortices at different frequencies, thus becoming the source of vortex dislocations similar to the ones identified by Bearman and Owen [40] in the wake of rectangular sections with sinusoidal variation of the along width dimension (i.e. by introduction of wavy front additions). Furthermore, the observation that a beyond-critical state emerges very early is consistent with the acquired forcing spectra. As shown by Schewe [106], the unstable supercritical state (supercritical to transcritical transition specifically) is characterised by lift fluctuations at reduced frequencies in a broad band around 0.2, along with stronger low frequency peaks, similar to the spectra found here (Fig. 5.9). Note that differences between the two bubbles (or even inconsistency in the timing of their existence) during the two bubble state can still give the opportunity for mean lift to arise. Finally it should be stressed that what would appear as a second bubble can yet be a non-conventional flow structure as earlier conjectured. Considering the 77 ◦ and 90 ◦ inclination angles, different behaviour is found. All rings seem to exhibit very similar behaviour between them for the mean drag force coefficients. This is consistent with a previous finding of closer agreement for the higher inclinations, between the mean coefficients from the four pressure tap rings and those back-calculated from mean static displacements during the dynamic tests, 108 Chapter 5. Experiments on galloping vibration of a circular cylinder 1 1.5 2 2.5 3 3.5 4 4.5 x 10 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Re C D 1 / 2 Ring 2− right Ring 2− left 1 1.5 2 2.5 3 3.5 4 4.5 x 10 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Re C D 1 / 2 Ring 3− right Ring 3− left Figure 5.12. Drag evolution acquired for ‘right’ (taps 4-3-. . . 20) and ‘left’ (taps 4-5-. . . 20) parts of Rings 2 and 3, during static tests. Model setup ϕ=60 ◦ . Double points correspond to increasing and decreasing wind speed runs. 5.4. Discussion 109 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 5 0 0.1 0.2 0.3 0.4 0.5 0.6 Re C D 1 / 2 Ring 3− right Ring 3− left Figure 5.13. Drag evolution acquired for ‘right’ (taps 4-3-..20) and ‘left’ (taps 4-5-..20) parts of Ring 3, during static tests. Model setup, ϕ=90 ◦ . which give a measure of the mean force over the whole cylinder length [226]. This could well be due to the conjecture of increased inclination angles having a consistent characteristic behaviour for the transitions of individual cylinder sections. Thus a simple non-weighted averaging can be performed that closely matches the global forcing coefficients. On the other hand, for ϕ=60 ◦ , since different states exist along the cylinder length, averaging over the four pressure tap rings with predefined weightings does not so accurately represent the global forcing. The drag crisis for Ring 3 for ϕ=90 ◦ is illustrated in Fig. 5.13. Similar one-bubble formations exist on all rings in a (closely spaced) narrow defined range of Reynolds number. It is significant that the Reynolds number corresponding to minimum drag is increased relative to the previous inclined cable case (Fig. 5.12) as e.g. also in [111]. The difference is amplified further when the modification of calculating Re based on the wind component normal to the cable, as successfully performed earlier for Sr, is used. Logically some additional mechanism should be operating for the ‘early’ drag reduction for the inclined cable. Apart from the evident curvature change of streamlines on the cable surface, which translates to a higher effective Reynolds number, this could also be due to the action of the quoted vortex dislocations, which when steadily located along the length can act effectively towards drag reduction [40]. An interesting case of enhanced drag reduction connected with stable flow patterns in the critical Reynolds number range was previously given by Humphreys [103] when he performed static tests tying fine silk threads at a cylinder’s stagnation points. One last feature not previously explicitly reported was that of sudden avalanche-like (intermittency for ‘quiescent’ flow 110 Chapter 5. Experiments on galloping vibration of a circular cylinder Figure 5.14. Ring 1 C L and C D transitional avalanche-like behaviour. Setup ϕ=90 ◦ , Re=3.2×10 5 , U=32m/s. Pressure distributions for time instants a, b, c, d are provided in Fig.5.15. conditions lends them this definition) jumps in the lift (and drag) coefficient, denoting alternating transitions between symmetric and asymmetric states. Such a case is shown in Fig.5.14, where Re=3.2×10 5 , U=32m/s. These jumps, reminiscent of driven chaotic vibrations (cc. the magnetoelastic problem [227]), could occur independently on all rings, during both static and dynamic tests. Interestingly they were not recorded as such for ϕ=60 ◦ . Clearly due to this finding, when high-angle arrangements are concerned, averaging of force coefficients should be performed cautiously. Analysis of the pressure distributions around the cylinder confirmed that they were due to real changes in the flow state. To illustrate this, four points (a,b,c,d) were selected in the record of Fig.5.14 and their instantaneous pressures were drawn in Fig.5.15. As it can be evidenced, all possible lift signs were realised. Time steps a and d do not only produce opposite C L , but also quite distinct C D . This could well be due to a structural asymmetry of the cylinder, however the fact that the lift flips sides during steady flow seems as an indication that an unbiased circular behaviour prevails. It could thus be a case where an ensemble of differently sized laminar separation bubbles (see §2.2) are possible. The sequence b→c→d was picked to surround a sudden drop in C L . Noticeably during the prior to the jump instant b, a small one-sided ‘bump’ develops in the back pressure region. When the intermittent process is midway through, at c, two bumps can be seen in the same region in both sides. Strikingly this yields an instantaneous C p profile exactly as in Ring 3 in Fig.5.11. 5.4. Discussion 111 Figure 5.15. Pressure distributions during the state ‘vibration’ indicated in Fig.5.14. Steps b, c, d describe the actual dynamic changes during a state jump. In the plots the radius corresponds to C p =1. 5.4.2 Mechanism implications With all these observations in hand it is possible to speculate on possible flow mecha- nisms underlying the dynamic excitation and on the influence of the cable inclination angle. Some secondary flow that was identified in the case of ϕ=60 ◦ is evidently propa- gating disturbances axially along the cable. Taking into account that no extreme state jump, as in Fig. 5.14, was observed for the numerous recorded cases it seems reasonable to suggest that, for this inclination, some reasonably ‘stable’ dynamic balance builds up between sections along the cylinder. Still, the dynamic balance is built from cell components that conceptually differ from each other thus becoming sources for vortex dislocations. Each section is prone to the effects of turbulence and surface roughness, as demonstrated by the frequent omission of the critical state in the transition from subcritical to transcritical. Interaction between zones along the cylinder could possi- bly allow for stable local formations of high lift to develop in the system (being an indication of a structured asymmetry). On the other hand, for ϕ=90 ◦ and 77 ◦ the sections appear to be less influenced by the similar turbulence intensity and surface roughness conditions and all transition states can clearly be distinguished. The existence of ‘delicate’ single bubble states is probably proof of the good flow conditions during the whole series of tests. It is suggested that the lack of significant secondary flow, providing the means for coupling, inhibits the formation of a stable dynamic balance along the cylinder, allowing each section to behave independently from its not too distant neighbours. The fact that 112 Chapter 5. Experiments on galloping vibration of a circular cylinder there seems to be some characteristic behaviour for the drag evolution, although data are only available from a few sections, is not inconsistent with this conjecture. Although considerable lift could occur locally, if the sign of lift is random, the expected total lift on the cable could be zero, and in any case have unstable features. Vortex dislocations if occurring (due to intermittent response) would also be unstable. Of course non-zero total mean lift can stably occur due to the finite length of the cylinder or some exogenous ‘driving’ (e.g. blockage) or evidently if the lift sign statistics are not Gaussian. Yet the above scenario could possibly explain the lack of large vibration incidents for high inclination values in the Phase 3 tests. For lower cable-wind angles, the suggested coherent dynamic behaviour over the length of the cable is more likely to result in stable structured asymmetry (e.g. non-zero lift), which could be a significant factor in a galloping mechanism. 5.5 Concluding remarks Large amplitude vibrations of dry inclined cables are still an unravelled mystery with ongoing research trying to establish connections and reasoning about the different pa- rameters that may trigger instabilities. This chapter has attempted to add to the previous observations of the behaviour, using the latest findings from a series of static and dynamic tests on a realistic cable model equipped with pressure measurement taps and identifying features not existing in the current literature. An interesting finding was the identification of behaviour indicating two separation bubbles existing very close to the subcritical Reynolds number range. This was only found for the lowest cable-wind angle of ϕ=60 ◦ and ceased for ϕ=77 ◦ and 90 ◦ . It was also established that the cable at ϕ=60 ◦ could retain different flow states along its length in a very stable way, while at the greater cable-wind angles the flow often intermittently jumped between states, altering the lift sign and/or value. Moreover, the Reynolds number value corresponding to minimum drag, nominally designating the end of the critical state, was found to be significantly lower for the lowest cable-wind angle. Accepting the limitation that only a few measurement sections were used, it is suggested that some stable dynamic balance of different states builds up along the cable for some inclination angles, probably connected with a specific range of along- cylinder wind component values. Introduction of structures such as vortex dislocations, resulting from a sustained asymmetry (e.g. as a distribution of pressure profiles similar to Ring 3 in Fig. 5.11) along the cable, could be an important factor in the excitation mechanism. On the other hand, it is thought that weak coupling, as appears to occur for flow normal to the cylinder and more importantly avalache-like disrupting behaviour, may tend to inhibit galloping. 5.5. Concluding remarks 113 Simultaneous existence of different flow states was suggested to be a case of different sensitivity of inclination angles to (even low) turbulence intensity and roughness, but it could alternatively be an effect of the inclination angle itself in smooth flow. The mean features identified are indicative of the flow characteristics. In any case the critical Reynolds number regime has been shown able to provide flow structures that could well be responsible for dynamic instabilities of inclined cables. Ongoing work is aiming to shed light on the links between these structures and the actual instability mechanism. Chapter 6 Conclusion and outlook The present work is a collection of aerodynamic studies concerning different parts of flexible long-span bridges. In all cases an attempt was made to elucidate and understand a number of distinct features that arise due to the complex structural interplay with the wind. Specifically threatening self-excitation phenomena that affect bridges were addressed. This concluding chapter aims to summarise the main findings and suggest areas for further research. Initially the flutter potential of the Clifton Suspension Bridge (CSB) was considered. It is well accepted that plate girder sections as on the CSB are extremely vulnerable to aerodynamic effects, yet in a modernised variant they are still in use. Utilising ambient vibration measurements from a long-term monitoring campaign, it was possible to per- form the conventional flutter analysis of Scanlan [12, 13] in an inverse way and deduce the flutter derivative description of the experienced aeroelastic loading. Records in- cluding a quite wide variety of wind conditions gave a good range of reduced velocities making possible further aeroelastic assessments. The results, obtained under uncon- trolled conditions, seem to reliably follow values from wind tunnel sectional models of similar sections. This is despite the fact that scaled tests are performed under homo- geneous flow without the variations of natural wind. Interestingly it was found that there is some similarity to the Tacoma Narrows Bridge, with single-degree-of-freedom torsional instability being a possibility for the bridge for wind speeds not far beyond the range experienced. For the highest obtained reduced velocities, flutter derivative A ∗ 2 becomes positive (contributing negative torsional aerodynamic damping), reducing the total available torsional damping. The actual A ∗ 2 function may be increasing even more rapidly than estimated, since the possible amplitude-dependent increase of struc- tural damping was not accounted for. Notably the analysis was successful in finding an estimate of the critical flutter speed, based solely on full-scale measurements, for the first time. The estimated value of approximately 21m/s is only slightly above the 115 116 Chapter 6. Conclusion and outlook maximum recorded wind speed on the bridge. The findings also raised an interest- ing question regarding the H ∗ 1 flutter derivative (responsible for vertical aerodynamic damping). On the Tacoma Narrows Bridge the sign reversal of H ∗ 1 , similar to A ∗ 2 , was assigned to vortex shedding. However, it occurred far from the expected resonant vortex shedding condition, which raises doubts about this explanation. On the CSB there was no recorded vortex-induced response throughout the testing period, but it shows a similar trend for H ∗ 1 (also at wind speeds far from the expected critical speed for vortex shedding), suggesting that indeed it is due to some other cause. Aeroelastic coupling action was also found but it was not a serious concern, at least for the range of reduced velocities considered. This study essentially adds to the experience of aeroelas- tic identification of bridges, which is very limited for full-scale structures, and provides a practical example for any similar future study. In any case the main achievement is that the best aeroelastic data of a real-scale bridge near critical behaviour to date were clearly identified. The obvious means of complementing the analysis would be to perform scaled wind tunnel tests or numerical simulations for comparison. There has been no previous full-scale validation of bridge flutter analysis. This could substantiate the empirical flutter framework currently in use and reassure engineering practice that the safety margins that modern bridges are designed for are realistic. An added benefit that could surface from the current analysis in the near future is putting forward a convincing response to the long-standing question of how small localised changes can improve the overall aerodynamic performance. This could be beneficial to assess the wind risk of many existing bridges that were not designed using the recently devised flutter-resisting framework. Next a topic that has often been mishandled in the existing literature was consid- ered;the generalisation of galloping in two dimensions. When the structural axes are inclined to the wind direction, the original Den Hartog derivation for galloping motion is not valid. The root of the error in some previous treatments (e.g. [189]) was iden- tified and succinct expressions for generalised galloping modelling were devised. The study was extended to cover two-degree-of-freedom motion with allowance for arbitrary detuning between vibrations along the two principal axes. Although the foundations for generalised 2DOF galloping were laid by Richardson and Martuccelli [190] in 1965 it has not previously been correctly and clearly presented and the implications quan- tified. The presentation clarifies the way quasi-steady theory incorporates geometric and structural details. Turning the derived force descriptions into instability criteria, three benchmark galloping scenarios were considered: single-degree-of-freedom motion normal to the wind and inclined to the wind and perfectly tuned two-degree-of-freedom motion. These cover the range of possible instability boundaries. Employing published data of static force coefficients it was possible to quantify the differences in the effective 117 aerodynamic damping (positive or negative) in the different cases. It was apparent that large differences occur. Furthermore the influence of detuning on the evolution of the in- stability has rarely been considered before. However it was shown that it is an essential parameter in defining the true stability boundary. The investigation was able to refute the suggestion that the introduction of an along-wind degree-of-freedom [137, 191] will necessarily stabilise a purely across-wind unstable motion. As presented, the potential for the opposite behaviour also exists. A number of similar shortcomings in previous analyses were also addressed, making the suggested updated galloping framework a potentially valuable tool for wind studies of slender elements such as cables and bridge towers. For both the preceding sections, although Reynolds number (Re) was accepted as a potentially influential variable, it was excluded from the analysis per se. The final piece of this thesis attempted to tackle the controversial aerodynamic problem of dry galloping of circular sections, in particular of inclined cables on cable-stayed bridges, where the development of Reynolds-induced lift has been suspected of having a key role. Therefore the study of Re effects was the focus of the final part of the investigation. An experimental approach was adopted, testing both dynamic and static models of cables with sectional dimensions as on real cable-stayed bridges. Observed large amplitude responses were primarily of across-wind character and occurred close to the critical Re and nowhere else. Quasi-steady theory, even in its most elaborate form [133], is unable to fully interpret the details of most of the responses recorded. The results indicate that only cylinders inclined at a limited range of angles exhibited large amplitude vi- brations, while cylinders close to normal to the flow only experienced limited responses. Strikingly it was recorded that for the near normal cable, aerodynamic forces vary in a discontinuous non-stationary way, jumping between different laminar separation bubble flow states and resulting in intermittent abrupt steps in the lift and drag time series. This so-called avalanche-like behaviour was apparent during both static and dynamic tests. The observed intermittent state jumps could not be identified on the cable in- clined at angles that produced galloping-like responses, so they were conjectured to have the function of a quenching disorder that effectively mitigates vibrations. In his seminal work addressing the transitional Re behaviour of a smooth circular cylinder normal to the flow, Schewe [228] was in search of period doubling phenomena to ac- company the turbulent transitions on the cylinder’s surface. In the context of the new findings it seems no surprise that he did not recover any. The current results confirm that the route to the chaotic turbulent state in the boundary layers follow the alterna- tive path of intermittency. Interestingly Schewe also found individual state jumps, but these were global and were not in an ‘oscillating’ mode (i.e. once a jump occurred it did not reverse) as in the present study. It would be most intriguing to discover how 118 Chapter 6. Conclusion and outlook and why locally erratic behaviour with numerous asynchronous jumps can combine over the cable length to result in the global spatio-temporal stable structure recorded by Schewe. Due to a lack of global force measuring equipment in the experiments this is not feasible from the current study. Intriguingly in §2.1.3 something similar was il- lustrated for classical vortex shedding past the lock-in region(Fig. 2.3). Local pressure fluctuations and global motion did not agree in terms of frequency content. During large response events non-conventional flow structures near the back pres- sure region that induce asymmetries were identified. Similarities were established be- tween the current tests and the numerical simulations of Yeo and Jones [116, 117], though no significant dynamic response occurred in the experiments near their simu- lation region as they anticipated. Finally it was summarised how correlation changes along the length of the cable, for cables inclined at various angles, influence the global dynamics. It was surmised that the dry galloping region overlaps with the critical Re regime not only because of an expected appearance of lift but due to the multitude of complex flow features that emerge. If such features are possible to appear in other flow regimes, which seems highly unlikely, then these would also be potential regions for dry galloping. In a study with similar aims to this one, Symes [229] conducted static wind tunnel tests for a smooth circular cylinder normal to the flow, in the critical Re range. He concluded that although nominally circular cylinders are normally treated as perfectly symmetric, they could have a consistent rotational asymmetry that may lead to a clas- sical galloping response similar to the one addressed in Chapter 4. It is uncertain how such a behaviour would combine with the more complex picture captured here to form a mixed origin instability. Additionally the range of Reynolds numbers where the critical regime was found was unusually low, raising questions as to whether nominally precrit- ical wind speeds on bridges could actually be critical. Future wind tunnel tests should also verify the circularity of the model cylinder, which would enable consideration of the role of the different possible galloping-like mechanisms involved. The intermittent and noisy aerodynamic force fluctuations with the distinct abrupt flow state jumps had a disturbance role since they did not allow motion-induced load- ing to set in. Yet is is questionable whether this is always true. Potentially the self- excitation mechanism, if existing, can carry on despite the appearance of noise or even use the energy content in the noise spectrum to develop. Actually non-harmonic forces with sharp noisy peaks have been seen during large rain-wind vibrations [97] (Chap- ter 2, Fig. 2.10). Preliminary results from circular cylinders forced to vibrate with large amplitudes indicate that there may well be such a different function even under dry conditions. It should be pointed out that as Parkinson comments [23] (based on the unique studies of Staubli [230] and Bearman [202]), in regions such as the critical 119 Re one, the equivalence between forced and free vibration tests could be invalid. Thus it would be useful to complement the series of free vibration tests studied here with equivalent forced ones and compare the relevant aerodynamic force characteristics. In the spirit of Bishop and Hassan [32], who first suggested modelling the fluid-structure interaction during vortex-shedding with a nonlinear Van der Poll oscillator, this study wishes to conclude by proposing an alternative modelling for dry galloping vibrations. Combining the actual ‘noisy’ intermittent state jumps with the suspected contribution when ‘driving’ the cylinder during forced vibration tests, it is conceivable that a formu- lation based on stochastic resonance could reproduce the dry galloping phenomenon. The existence of a second harmonic in the motion-induced lift presented in Fig. 5.9, is an additional detail that can also justify this choice. Publications Author’s publications related to the present thesis. Journal papers Nikitas N, Macdonald JHG, and Jakobsen JB. Wind induced Vibrations of the Clifton Suspension Bridge. Wind Struct., 14(3):221-238, 2011. Nikitas N, Macdonald JHG, and Jakobsen JB, Andersen TL. Critical Reynolds number and galloping instabilities – Experiments on circular cylinders. Accepted in Exp. Fluids. Nikitas N, and Macdonald JHG. Misconceptions and generalisations of the Den Hartog galloping criterion. Submitted for publication in J. Eng. Mech.-ASCE. Conference Proceedings Nikitas N, Macdonald JHG, and Jakobsen JB. Full Scale Identification of Modal and Aeroelastic Parameters of the Clifton Suspension Bridge. In 6th International Col- loquium on Bluff Bodies Aerodynamics & Applications (BBAA VI), pages 135–138, Milano, Italy, 2008. Macdonald JHG, Nikitas N, Symes JA, Jakobsen JB, Andersen TL, Savage MG, and McAuliffe BR. Large-scale wind tunnel tests of inclined cable vibrations- Preliminary findings. In 8th UK Conference on Wind Engineering, Guildford, UK, 2008. Nikitas N, Macdonald JHG, and Jakobsen JB., Andersen TL, Savage MG, McAuliffe BR. Wind Tunnel testing of an inclined cable model-Pressure and motion characteris- tics, Part I. In 5th European & African Conference on Wind Engineering (EACWE5), Florence, Italy, 2009. Jakobsen JB., Andersen TL, Macdonald JHG, Nikitas N, Savage MG, and McAuliffe BR. Wind Tunnel testing of an inclined cable model-Pressure and motion characteris- tics, Part II. In 5th European & African Conference on Wind Engineering (EACWE5), Florence, Italy, 2009. 121 122 PUBLICATIONS Andersen TL, Jakobsen JB, Macdonald JHG, Nikitas N, Larose GL, Savage MG,and McAuliffe BR. Drag-crisis response of elastic cable-model. In 8th International Sym- posium on Cable Dynamics, Paris, France, 2009. Nikitas N, and Macdonald JHG. The Den Hartog galloping criterion revisited: a non- classical case. In 9th UK Conference on Wind Engineering (WES-2010), Bristol, UK, 2010. Zhang J, Au FTK, Li J, Nikitas N, Macdonald JHG, and Jacobsen JB. Identifying bridge aeroelastic parameters from full-scale ambient vibration data. In 9th UK Con- ference on Wind Engineering (WES-2010), Bristol, UK, 2010. References [1] Blevins RD. Flow-Induced Vibrations. Van Nostrand, New York, 1st edition, 1977. [2] Simiu E and Scanlan RH. Wind Effects on Structures. John Wiley & Sons, New York, 1st edition, 1978. [3] Zdravkovich MM. Flow around circular cylinders Vol 1: Fundamentals. 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