Dynamic behaviour of suction caissons

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x Contents 3 Dynamic stiffness of suction caissons—torsion, sliding and rocking 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Static and dynamic stiffness formulation . . . . . . . . . . . . . . . . . . . 32 3.4 Dynamic stiffness for torsional vibrations . . . . . . . . . . . . . . . . . . 33 3.4.1 Boundary Element/Finite Element model . . . . . . . . . . . . . . 34 3.4.2 Static stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.3 Dynamic stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.4 High-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Dynamic stiffness for coupled sliding–rocking vibrations . . . . . . . . . . 38 3.5.1 Boundary Element/Finite Element model . . . . . . . . . . . . . . 38 3.5.2 Static stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.3 Dynamic stiffness—variation of Poisson’s ratio . . . . . . . . . . . 39 3.5.4 Dynamic stiffness—variation of skirt length . . . . . . . . . . . . . 43 3.5.5 High-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6.1 Torsional vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6.2 Coupled sliding–rocking vibrations . . . . . . . . . . . . . . . . . . 47 4 Prototype bucket foundation for wind turbines—natural frequency estimation 49 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 The prototype in Frederikshavn . . . . . . . . . . . . . . . . . . . . 50 4.2 Experimental estimation of natural frequencies . . . . . . . . . . . . . . . 52 4.2.1 Modal identification technique . . . . . . . . . . . . . . . . . . . . 52 4.2.2 Natural frequencies for idle conditions . . . . . . . . . . . . . . . . 55 4.2.3 Natural frequencies for wind turbine without wings . . . . . . . . . 55 4.2.4 Natural frequencies for wind turbine without wings and nacelle . . 57 4.3 Numerical estimation of natural frequencies . . . . . . . . . . . . . . . . . 59 4.3.1 Finite Element model of the wind turbine . . . . . . . . . . . . . . 59 4.3.2 Foundation models . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.3 Numerical analysis of steady state response . . . . . . . . . . . . . 62 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.1 Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.3 Recommendations for future work: . . . . . . . . . . . . . . . . . . 68 5 Concluding remarks 69 5.1 Conclusion—main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.1 Evaluation of the frequency dependent stiffness of suction caissons 69 5.1.2 Experimental estimation of resonance frequencies . . . . . . . . . . 71 5.1.3 Formulation and implementation of lumped-parameter models . . 72 5.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1 Experimental approaches . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.2 Numerical approaches . . . . . . . . . . . . . . . . . . . . . . . . . 74 Morten Liingaard
Contents xi References 75 A Solution for an infinite cylinder 83 B Experimental modal analysis 87 B.1 Output-only Modal Analysis Software . . . . . . . . . . . . . . . . . . . . 87 B.1.1 Output-only Modal Identification . . . . . . . . . . . . . . . . . . . 87 B.1.2 ARTeMIS Testor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 B.1.3 ARTeMIS Extractor . . . . . . . . . . . . . . . . . . . . . . . . . . 90 B.2 General digital data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 96 B.2.1 Data Sampling and aliasing . . . . . . . . . . . . . . . . . . . . . . 96 B.2.2 Structure of measured data . . . . . . . . . . . . . . . . . . . . . . 96 B.2.3 Nyquist frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 B.2.4 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 B.2.5 Signal processing (digital data analysis) . . . . . . . . . . . . . . . 100 B.3 Basics of structural dynamics and modal analysis . . . . . . . . . . . . . . 106 B.3.1 Dynamic model of second-order structural system . . . . . . . . . 106 B.3.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3.3 Spectral analysis of dynamic excited system . . . . . . . . . . . . . 109 B.4 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.4.1 ID by Frequency Domain Decomposition (FDD) . . . . . . . . . . 111 B.4.2 ID by Stochastic Subspace Iteration (SSI) . . . . . . . . . . . . . . 112 C Lumped-parameter models 115 C.1 Static and Dynamic Stiffness Formulation . . . . . . . . . . . . . . . . . . 115 C.1.1 Static stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 C.1.2 Dynamic stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 C.2 Simple lumped-parameter models . . . . . . . . . . . . . . . . . . . . . . . 118 C.2.1 Standard lumped-parameter model . . . . . . . . . . . . . . . . . . 118 C.2.2 Fundamental lumped-parameter models . . . . . . . . . . . . . . . 120 C.3 Advanced lumped-parameter models . . . . . . . . . . . . . . . . . . . . . 124 C.3.1 Dynamic stiffness obtained from rigorous methods . . . . . . . . . 124 C.3.2 Decomposition of the dynamic stiffness . . . . . . . . . . . . . . . 125 C.3.3 Polynomial-fraction approximation . . . . . . . . . . . . . . . . . . 126 C.3.4 Discrete models for partial-fraction expansions . . . . . . . . . . . 127 C.3.5 Example — vertical dynamic stiffness of a suction caisson . . . . . 132 D Application of lumped-parameter models 137 D.1 Lumped-parameter models for the suction caisson . . . . . . . . . . . . . . 137 D.1.1 Determination of the exact solution for the dynamic stiffness . . . 137 D.1.2 Lumped-parameter models for vertical vibrations . . . . . . . . . . 140 D.1.3 Lumped-parameter models for sliding vibrations . . . . . . . . . . 144 D.1.4 Lumped-parameter models for rocking vibrations . . . . . . . . . . 148 D.1.5 Lumped-parameter models for the coupling term . . . . . . . . . . 151 D.1.6 Lumped-parameter models for the torsional term . . . . . . . . . . 154 D.2 Assembly of the global dynamic stiffness matrix . . . . . . . . . . . . . . . 157 D.2.1 Structure of the local dynamic stiffness matrices . . . . . . . . . . 157 December 4, 2006
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76 BIBLIOGRAPHY Brincker, R. and P.
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78 BIBLIOGRAPHY Kitahara, M. (1984)
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80 BIBLIOGRAPHY Morten Liingaard
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82 BIBLIOGRAPHY Morten Liingaard
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Dynamic behaviour of suction caissons

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