Bert Sweetman

The Hermite Moment Model for Highly Skewed Response With Application to Tension Leg Platforms

A new addition to the statistical Hermite moment model of extremes is introduced for use on processes with high skewness and near-Gaussian kurtosis. The monotone limits of the existing model are expressed as ellipses in response moment space and a new methodology is introduced that combines hardening and softening models to overcome these limits. The result is that any fractile of a distribution described by its first four statistical moments can be transformed to or from the Gaussian. An example application to a tension leg platform is presented.

Efficient Calculation of Statistical Moments for Structural Health Monitoring

Wireless networks of smart sensors with computations distributed over multiple sensor packages have shown considerable promise in providing low-cost structural health monitoring. In these networks, microprocessors are typically embedded in individual smart sensor packages. The efficiency of embedded computational algorithms is of critical importance because the size, cost, and power requirements of the sensor arrays are central concerns. Here, very efficient methodologies are presented to compute statistical moments of a measured response time-history. These moments: the mean, standard deviation, skewness, and kurtosis are often used to characterize a measured irregular response. Two alternative approaches are presented, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach reconsiders the computational benefits of computing statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of the sensor hardware. The second approach is a new analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used to allow for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times. A worked example is presented comparing two implementations of the new methodologies with conventional calculations in monitoring the global performance of an offshore tension leg platform. Accuracy, efficiency, and storage requirements of the calculation methods are compared with those of conventional methods. The results show that substantial CPU and memory savings can be attained with no loss in accuracy and that more dramatic savings can be attained if a slight reduction in accuracy is acceptable.

Practical Airgap Prediction for Offshore Structures

Two new methods are proposed to predict airgap demand. Airgap demand is the maximum expected increase in the water surface elevation caused incident waves interacting with an offshore structure. The first new method enables inclusion of some second-order ef- fects, though it is based on only first-order diffraction results. The method is simple enough to be practical for use as a hand-calculation in the early stages of design. Two existing methods of predicting airgap demand based on first-order diffraction are also briefly presented and results from the three methods are compared with model test results. All three methods yield results superior to those based on conventional post-processing of first-order diffraction results, and comparable to optimal post-processing of second-order diffraction results. A second new method is also presented; it combines extreme value theory with statistical regression to predict extreme airgap events using model test data. Estimates of extreme airgap events based on this method are found to be more reliable than estimates based on extreme observations from a single model test. This second new method is suitable for use in the final stages of design.@DOI: 10.1115/1.1710870# Introduction and Background Present airgap design methodologies for floating structures are not standard and rely heavily on empirical knowledge and model tests. Large production semi-submersibles and novel large-volume offshore structures provide design challenges that are well outside the present design experience base. Unlike drilling semi- submersibles, these vessels are generally required to remain on station throughout the most severe weather. Urgency is added by the fact that airgap design problems ~wave impacts! have been encountered on large North Sea semi-submersibles, including the Veslefrikk B platform in the Norwegian sector of the North Sea, the vessel which is the subject of the analysis and model tests presented in this paper. High volume structures, whether fixed or floating, complicate the airgap calculation by significantly diffracting the incident waves. For these structures, ignoring diffraction effects is non- conservative in that diffraction effects generally worsen the airgap demand. Large-volume floating structures, including semi- submersibles and floating production, storage and offloading ~FP- SOs !, vessels, offer the most significant challenge. Two distinct hydrodynamic effects are observed: ~1! global forces and resulting motions are significantly affected by diffraction; and ~2! the local wave elevation, h(t), is also significantly influenced by diffrac- tion. For semi-submersibles, these wave amplification effects are most extreme at locations above a pontoon and/or near a major column. Here, four methods for prediction of airgap demand without use of second-order diffraction are presented and compared with model test results. First, a theoretical overview and motivation section outlines which physical terms are included in conventional first-and second-order diffraction analysis and the implications of the assumptions implicit to various post-processing methods. Some background is then presented which is relevant to all of the methods presented here. This background includes airgap nota- tion, model testing issues and methodologies and some statistical background. Two existing and one new method of predicting air- gap demand using results from only first-order diffraction are then presented and critically compared with model test data. It is con- cluded that second-order effects should be included in airgap pre- diction throughout the early stages of design, but that these meth- ods are inadequate for final design. Another new methodology is then presented which is useful for final design. In the new meth- odology, extreme value theory and regression techniques are com- bined to predict extremes from model test results. Theoretical Overview and Motivation The disturbed water surface in the presence of the vessel, h(t), is assumed to be a sum of incident and diffracted waves, h i and h d , each of which is a sum of first- and second-order components:

Floating Offshore Wind Turbine Dynamics: Large-Angle Motions in Euler-Space

Floating structures have been proposed to support offshore wind turbines in deep water, where environmental forcing could subject the rotor to meaningful angular displacements in both precession and nutation, offering design challenges beyond conventional bottom-founded structures. This paper offers theoretical developments underlying an efficient methodology to compute large-angle rigid body rotations of a floating wind turbine in the time domain. The tower and rotor-nacelle assembly (RNA) are considered as two rotational bodies in the space, for which two sets of Euler angles are defined and used to develop two systems of Euler dynamic equations of motion. Transformations between the various coordinate systems are derived to enable solution for motion of the tower with gyroscopic, environmental and restoring effects applied as external moments. An example is presented in which the methodology is implemented in MATLAB to simulate time-histories of a floating tower with RNA. Nomenclature (X ,Y,Z) Translating non-rotating coordinate system with respect to the earth, with the origin O fixed at the center of mass of the tower (x,y,z) Translating rotating coordinate system with the origin O fixed at the center of mass of the tower; the z-axis is fixed on the centerline of the tower while the xand y-axes rotate about the z-axis independent of body rotations about that axis. (Xn,Yn,Zn) Translating non-rotating coordinate system with respect to the earth, with the origin O′ fixed at the center of mass of the RNA (A,B,C) Translating rotating coordinate system with the origin O′ fixed at the center of mass of the RNA B-axis Spinning axis of the blades Bp-axis Horizontal projection of the B-axis on the XOY plane yp-axis Horizontal projection of the y-axis on the XOY plane (θ1,θ2,η) Three 3-1-3 sequence Euler angles used to describe the rotation of the tower, where θ1 is measured from the Z-axis to the z-axis with the positive direction right-handed along the x-axis; θ2 is measured from the X-axis to the x-axis with positive right-handed along the Z-axis; η is rotation along the z-axis with positive right-handed along the z-axis. (φ,θ,ψ) Three 3-1-3 sequence Euler angles used to describe the rotation of the RNA, also called precession, nutation and spin angles, where φ is measured from the Y -axis to the Bp-axis with the positive direction along the Z-axis; θ is measured from the Z-axis to the B-axis with the positive direction along the C-axis; ψ is the spinning around the B-axis with the positive direction along the B-axis. λ Measured from the Bp-axis to the B-axis with positive right-handed along the C-axis α Measured from the yp-axis to the Bp-axis with positive right-handed along the Z-axis ~Ω The angular velocity of the (A,B,C) coordinate system relative to (X ,Y,Z) ~ω The angular velocity of the RNA relative to (X ,Y,Z) CT Thrust coefficient, used to calculate wind force on the blade swept area

Analytical Predictions of the Air Gap Response of Floating Structures

Two separate studies are presented here that deal with analytical predictions of the air gap for floating structures. (1) To obtain an understanding of the importance of firstand second-order incident and diffracted wave effects as well as to determine the influence of the structure’s motions on the instantaneous air gap, statistics of the air gap response are studied under various modeling assumptions. For these detailed studies, a single field point is studied here – one at the geometric center (in plan) of the Troll semi-submersible. (2) A comparison of the air gap at different locations is studied by examining response statistics at different field points for the semi-submersible. These include locations close to columns of the four-columned semi-submersible. Analytical predictions, including firstand second-order diffracted wave effects, are compared with wave tank measurements at several locations. In particular, the gross root-mean-square response and the 3-hour extreme response are compared. BACKGROUND The air gap response, and potential deck impact, of ocean structures under random waves is generally of considerable interest. While air gap modeling is of interest both for fixed and floating structures, it is particularly complicated in the case of floaters because of their large volume, and the resulting effects of wave diffraction and radiation. These give rise to two distinct effects: (1) global forces and resulting motions are significantly affected by diffraction effects; and (2) local wave elevation modeling can also be considerably influenced by diffraction, particularly at locations above a pontoon and/or near a major column. Both effects are important in air gap prediction: we need to know how high the wave rise (step 2), and how low the deck translates vertically (due to net heave and pitch) at a given point to meet the waves. Moreover, effects (1) and (2) are correlated in time, as they result from the same underlying incident wave excitation process. We focus here on analytical diffraction models of air gap response, and its resulting stochastic nature and numerical predictions under random wave excitation. Attention is focused on a semi-submersible platform, for which both slow-drift motions (heave/pitch) and diffraction effects are potentially significant. This air gap response presents several new and interesting challenges. It is the first response limit state where we need to simultaneously include both second-order sumand difference-frequency effects (on the wave surface), and second-order differencefrequency effects (on slow drift motions and generally, on the wave surface as well). The sumand differencefrequency waves and the difference-frequency heave and pitch motions can both influence the air gap. The air gap response is further complicated because the heave, pitch, and roll motions of the floating structure are generally coupled. Moreover, the motions and the net wave elevation, both of which affect the air gap, are correlated in time as they result from the same underlying incident wave excitation process. Note that air gap modeling has been the subject of previous work within the Reliability of Marine Structures Program at Stanford University. For example, Winterstein and Sweetman (1999) apply a fractile-based approach to develop a scaling factor between the statistics of the incident waves and those of the associated air gap demand. Results are shown here from frequency-domain analyses which permit careful study and isolation of various effects: e.g., wave forces on a fixed (locked-down) structure, the effect of structural motions on air gap response, and finally, the effect of different local wave elevation models. For reference, a complete second-order diffraction model is formulated and studied. Compared with this complete model, various simplified models are imposed and evaluated: (a) second-order wave elevation effects are neglected completely, or (b) these second-order effects are approximated by analytical Stokes theory, which retains second-order effects on the incident wave but neglects second-order diffraction. Use of (a) and (b) would significantly simplify the analysis, avoiding the costly step of second-order diffraction. The local wave modeling in step (b) is found to be quite important in predicting the air gap response of two semi-submersibles studied here. THEORY Volterra Series and Transfer Functions In modeling nonlinear systems, such as floating structures, it is common to employ Volterra series that permit one to describe the response (output) of such systems. The nonlinear system is defined in terms of firstand second-order transfer functions. For floating structures, these transfer functions are obtained from firstand second-order wave diffraction analysis programs (e.g., WAMIT, 1995). In order to study the response of a floating structure to random seas, we start by defining an irregular sea surface elevation, η(t), as a sum of sinusoidal components at N distinct frequencies: ) exp( ; ) ( 2 where ; ) exp( Re ) cos( ) (

Air Gap Response of Floating Structures: Statistical Predictions Versus Observed Behavior

Summary and Conclusions • A new, fractile-based approach has been proposed to definean amplification factor,b p , on extreme wave crest levels due todiffraction. This is calculated as the rms-ratio between input andoutput peaks in all wave cycles ~Eq. ~8!!. The scaling factor re-sulting from this new approach is found to describe the outputfractile results quite well, even at extreme levels within the 18 hof seastate test results, e.g. Figs. 5–7.• Figure 8 shows that the largest observed amplification,roughly 1.4, is found at location 1 ~in front of the up-wave col-umn; see Fig. 1!. Other near-column locations ~4, 5, 6, and 9!show amplification factors of roughly 1.2. As might be expected,somewhat greater amplifications generally occur for the smallerT P case ~with shorter wavelengths, hence larger relative effect ofthe structure!. The largest amplifications, however, at the near-column locations are relatively constant for the two T P cases con-sidered.• An analogous amplification factor, b