IMPROVED APPROXIMATION OF EXTREME TENSIONS FOR FREQUENCY DOMAIN ANALYSIS OF MOORING CABLES

Transcription

1 7 th IBC Deepwater Risers, Moorings and Anchorings conference, Oct 2002, London. IMPROVED APPROXIMATION OF EXTREME TENSIONS FOR FREQUENCY DOMAIN ANALYSIS OF MOORING CABLES P. P. A. ONG Department of Engineering, University of Cambridge S. PELLEGRINO Department of Engineering, University of Cambridge X. M. CHENG Noble Denton Europe Ltd Abstract This paper presents a simple method of improving the initial approximation of the mean tension of a mooring cable, such that when the cable is analysed in the frequency domain, an extreme tension closer to that provided by the time domain solution is obtained. The cable is first analysed quasi-statically beyond the mean offset position to the positions of maximum excitation, yielding a non-linear relationship between the fairlead tension and the amplitude of excitation. This non-linear relationship is then linearised such that the work done by the forces applied at the fairlead over a complete quasi-static oscillation cycle is equal to that of the non-linear system. The outcome is a higher mean tension at the offset position. When the cable is analysed at this higher mean tension, the resulting dynamic tension is also higher, converging towards that obtained from time domain solutions. The proposed method is shown to work well and to increase the accuracy of frequency domain analyses, especially for cables with low pretensions. Keywords: catenary mooring, quasi-static analysis, frequency domain analysis.

2 Introduction Mooring structures are becoming increasingly important to the offshore oil and gas industry. There is a continuing need to improve the designs of such structures through better understanding and hence prediction of their non-linear behaviour. Many time domain codes exist to fulfil this need but many fall short of a fast and accurate prediction. In general, even with the fastest codes, the time taken for a full simulation is almost equal to "real time". The alternative is a frequency domain analysis. However, this requires the linearisation of the various non-linear aspects of the cable system and thus, this type of analysis is usually accurate for small excitation. In general, it has been shown that frequency domain analyses often lead to under-prediction of the maximum tensions when compared to time domain analyses, and even more so when the excitation is large (Brown and Mavrakos, 1999). It is this shortcoming that this study endeavours to resolve. The study is based primarily on a single two-dimensional (2D) catenary cable as opposed to a taut cable or a threedimensional system of cables. The proposed method is a simple improvement to the initial approximation of the static tension such that when the cable is analysed in the frequency domain, an extreme tension closer to that of time domain solutions is obtained. In conventional analysis, the mooring cable is first displaced quasi-statically from its initial position to its maximum offset position, where it is then analysed dynamically. In the new method, however, the cable is analysed quasi-statically beyond the mean offset position to the positions of maximum excitation, yielding a non-linear relationship between the fairlead tension and the amplitude of excitation. This non-linear relationship is then linearised such that the work done by the forces applied at the fairlead over a complete quasi-static oscillation cycle is equal to that of the non-linear system. The outcome is a higher mean tension at the offset position. When the cable is analysed at this higher mean tension, the resulting dynamic tensions is also higher, converging towards the value provided by the time domain solutions. The new method can be easily implemented into any existing frequency domain codes and is shown to work well when compared to time domain solutions. Improved accuracy in the prediction of maximum tensions is observed for cables with low pretensions. However, the improvement is found to decrease as the cable pretension increases, in which case the standard frequency domain solutions are adequate.

3 Frequency Domain Formulation Figure 1 Schematic of mooring cable The governing equations for the dynamic behaviour of the 2D mooring cable depicted in Figure 1 are as follows (Goodman and Breslin, 1976): Equations of Motion sv sg sg m0m m0umv cosg T wcosgf st st ss a a c Dn u G T m s s v s ž w G F žÿ st st ss 0 sin Dt (1) (2) Kinematic Relations su sg sf v ss ss st sv sg sg u (1 F) ss ss st (3) (4) Constitutive Relations T F (5) AE Cartesian Relations sz vcosg using st (6)

4 sx ucosg vsing st (7) where u and v are the tangential and normal velocities; z and x, the vertical and horizontal Cartesian displacements; T, the effective tension; φ, the inclination of the cable; ω, the submerged unit weight; m 0 and m a, the dry and added unit mass; and V c, the velocity of wave current. F Dt and F Dn are the linearised tangential and normal drag forces derived by Krolikowski and Gay (1980). For purpose of frequency domain analysis, the above non-linear equations must be linearised. This is achieved by assuming the cable is undergoing simple harmonic motion, whereby we substitute QQ Q (8) i t e X 0 1 to remove the time dependent terms. Here, < u v T G> T Q ; the subscripts 0 and 1 denote the static and dynamic values of the cable, respectively; and ω is the frequency of excitation. Thus, we obtain the following first-order ordinary differential equation dq ds 1 A( s, Q, X) Q B( s, Q, X) (9) which can be solved numerically. It is normally assumed that the cable is horizontal and pinned at touchdown, and it is excited at the fairlead where the maximum tensions will also be found. The static values of Q 0 can be easily obtained from standard catenary equations (Ahilan, 1998).

5 Improved Approximation Conventionally, the mooring cable is displaced quasi-statically to its offset position and then analysed with the properties found at that same offset position. In other words, the static values Q 0 at the offset position are used to determine the dynamic amplitudes Q 1. This is better illustrated in the left-hand plot of Figure 2. Here, TD represents time domain; FD, frequency domain; and QS, quasi-static. For low frequency excitation, it is valid to assume the time domain and quasi-static solutions are similar and so the TD and QS solutions are represented by a single curve in the plot. The straight line tangent to the curve at the point (X 0, Q 0 ), where X 0 indicates the mean offset position, represents the linearised frequency domain analysis carried out with the static values of Q 0. While this may be acceptable for small excitation, it can be seen that the frequency domain analysis will always under-predict the maximum tension, represented by (Q 1 + Q 0 ), when the excitation is large and/or the actual solutions are highly non-linear. The improved method illustrated in the right-hand plot will however take into account the non-linear behaviour of the cable beyond the offset position X 0, up to the position of maximum excitation (X 0 + X 1 ). The cable is first analysed quasi-statically from the point of minimum excitation to the point of maximum excitation, producing a standard tensionexcursion curve similar to the TD, QS curve in the plot. This tension-excursion curve is then linearised either by the method of least squares or based on a work done argument. The outcome is the straight line shown bold in the righthand plot, where the work done, given by the areas under the curve and the bold line, are equal. In most cases, it can be seen that the straight line will lie above the curve at the offset position X 0, producing a new Q * 0 which is higher than the original Q 0. Previously, the dynamic properties of Q 1 were obtained with the static properties of Q 0. With the improved method, Q 1 is obtained using the new properties of Q * 0. Hence, we now have dq ds 1 A( s, Q *, X) Q B( s, Q *, X) (10) where the frequency domain analysis in the right-hand plot is now carried out at the new point (X * 0, Q * 0). The tangent stiffness at this new position is higher and this also tends to yield higher values of Q 1. For the final solution of Q, we add Q 1 obtained with the new Q * 0 to the original Q 0. At higher frequency of excitation, the principle of the improved method remains the same. However, the time domain and quasi-static solutions are now different, as illustrated in Figure 2. At the mean offset position X 0, the cable is stationary and hence the TD and QS curves intercept. At other positions, the time domain solutions deviate from the quasistatic as a result of dynamic effects. Nevertheless, it can be seen that the improved method will still yield a higher value of Q 1.

6 Figure 2 Schematic of dynamic analysis at low frequency Figure 3 Schematic of dynamic analysis at high frequency

7 Results The above formulation has been programmed in MATLAB 6 (MathWorks, 2001) and the improved method tested on two mooring cables of which details are given in Table 1. These cables were taken from a comparative study on the dynamic analysis of moorings initiated by the International Ship and Offshore Structures Congress Committee I2 (Brown and Mavrakos, 1999). Cable 6 represents a chain mooring in shallow waters and Cable 7 a wire mooring in somewhat deep waters. These conditions generally reflect the usage of the two line types in different water depths. The results are then validated against time domain simulations carried out using Visual Orcaflex 7.2 (Orcina, 2001). Description Units Cable 6 Cable 7 Line Type - Chain Wire Cable Diameter m Submerged Unit Weight kn/m Axial Stiffness kn 1.69 x x 10 6 Dry Unit Mass kg/m Added Unit Mass kg/m Normal Drag Coefficient Tangential Drag Coefficient Water Depth m Total Cable Length m Mean Offset Tension kn Current Velocity m/s Seabed Inclination - Horizontal Horizontal Seabed Friction Coefficient Figures 4 and 5 show the quasi-static tension-excursion plots of Cable 6 and 7 obtained over a surge excursion of amplitudes 10 m and 20 m respectively. It can be seen that the static tension variation for Cable 6 is more non-linear than that of Cable 7. The non-linear and linear quasi-static solutions are almost identical for Cable 7. As such, the linearised solution yields a higher mean tension for Cable 6 than 7 at the mean offset position. The resulting effect can be seen in Figures 6 and 7, where the maximum tensions of Cable 6 and 7 under a 10 s surge excitation of up to 10 m and 20 m are shown, respectively. There is an increased accuracy in the predictions of maximum tensions for Cable 6 with the new method. For Cable 7, however, there is hardly any benefit. This is caused by the fact that the new tension found with the linearised quasi-static solution is almost identical to that found originally.

8 Conclusions The improved method has shown to work well when compared against time domain solutions. In general, for a cable with low pretensions, the improved method can increase the accuracy of frequency domain solutions by up to 25% in one case. It is however less beneficial for a cable of higher pretensions. Although the accuracy of prediction is improved, the frequency domain analysis continues to under-predict the maximum tensions when the excitation is large. This may be due to several reasons, but in particular, note that the interaction of the cable with the seabed has not been modelled in this study. Even though this seabed interaction can be modelled fully in the time domain, a new method of modelling this interaction in the frequency domain is currently underway. Acknowledgements The authors wish to thank Dr R. V. Ahilan of Noble Denton Europe, Mr Brian Corr of BP Amoco, and Prof R. S. Langley of University of Cambridge for their contributions and advice. Financial support from the Department of Trade and Industry and BP Amoco is gratefully acknowledged.

9 References Ahilan, R. V. (1998). Mooring Systems. Chapter in Floating Structures: a guide for design and analysis. CMPT Publication 101/98, London, First edition. Brown, D. T. and Mavrakos, S. (1999). Comparative study on mooring line dynamic loading. Marine Structures, 12: Goodman, T. R. and Breslin, J. P. (1976). Statics and dynamics of anchoring cables in waves. J Hydronautics, 10(4): Krolikowski, L. P. and Gay, T. A. (1980). An improved linearization technique for frequency domain riser analysis. Offshore Tech Conf, (3777): MathWorks (2001). MATLAB Student Version Release 12 learning MATLAB 6. The MathWorks Inc. Orcina (2001). Visual Orcaflex 7.5 User Manual. Orcina Ltd, Cumbria, First edition.

10 Figure 4 Quasi-static tension-excursion plot of Cable 6 Figure 5 Quasi-static tension-excursion plot of Cable 7

11 Figure 6 Maximum tensions of Cable 6 at 10 s excitation Figure 7 Maximum tensions of Cable 7 at 10 s excitation