MATHIEU STABILITY IN THE DYNAMICS OF TLP's TETHERS CONSIDERING VARIABLE TENSION ALONG THE LENGTH

Transcription

1 MATHIEU STABILITY IN THE DYNAMICS OF TLP's TETHERS CONSIDERING VARIABLE TENSION ALONG THE LENGTH Simos, A.M.' & Pesce, C.P. Escola Politecnica, USP, CP61548, S.P., Brazil * Dep. of Naval Architecture and Ocean Engineering / FAPESP scholarship ^ Dep. of Mechanical Engineering Abstract The trend in applying TLPs (Tension Leg Platforms) in deep waters and the necessity of reduction of the usually high values of pretension make the effect of variable tension in tethers dynamics more significant. This work presents a dynamic modeling of TLP's tethers considering the tension variation along the length due to the submerged weigh. The modal analysis considers a linear cable equation for tether modeling submitted to tension which varies linearly along its length. The standard Sturm-Liouville problem is solved by transforming it into a modified Bessel equation form. A Mathieu stability analysis is then performed so one can obtain the amplitudes of tethers vibrations and verify possibly unstable modes. The results are compared to those obtained by the model proposed by Patel' which considers constant tension along the length of the tethers, showing that the submerged weigh effect is indeed significant. INTRODUCTION As the use of TLP platforms moves in the direction of deeper waters the economic viability of their projects has been leading to huge platforms with high production levels. These characteristics emphasise the necessity of reduction of the usually high values of static pretension in order to allow higher values of payload and, consequently, enhancing the system's production. The operation in deeper waters with reduced pretensions makes

2 176 Offshore Engineering the effects of tension variation along the tethers more relevant in their dynamics. The investigation of such effects represents the main objective of the present work. An analysis on the effects of pretension reduction was previously made by Patel*. The dynamic model of the tethers lateral vibration considered in his work assumes constant value of static tension along the length of the tethers. The excitation comes from the linear heave motion of the hull induced by an harmonic wave. The tether's dynamic is thus represented by a non-linear Mathieu equation. Due to the consideration of constant tension along the tether's length, the modal forms of vibration are harmonic in his model. The global dynamics representation of the lateral motion governed by a parametric Mathieu equation derived in Patel's work is also assumed here: a linear cable equation forced by harmonically varying dynamic tension and limited in amplitude due to viscous damping consideration. However, the static tension linear variation along the tether's length due to its submerged weigh is here considered leading to a standard Sturm-Liouvilie problem which is analytically treated and solved in terms of modified BesseI equation. Therefore the modal forms in this model result non harmonic and are given in terms ofbessel functions^. Both models described above are then applied to determine the dynamic response of the tethers of two real TLP platforms: the Hutton TLP (example considered in Patel's work) and the Auger platform. The former platform was the first TLP designed and is operating in the Gulf of Mexico in a very low water depth. The Auger TLP, instead, is a more recent design and follows the modern trends. In order to simplify comparison, the main nomenclature used in the present text follows the one presented in Patel's work. THE DYNAMIC MODEL The dynamic model for the tether lateral vibration adopts the main hypotheses described bellow: linear cable model, thus the tether flexuralrigidityis assumed to be negligible since the element is very slender; static tension varying linearly along the tether's length due to the effect of its own submerged weigh; dynamic tension imposed by linear heave motion induced by a monochromatic wave; extremities simply supported; absence of current effects.

3 Offshore Engineering 177 The proposed global dynamic representation is based on Patel's work which considers the same hypotheses mentioned above but assumes the static tension to be constant along the tether's length. Figure 1 - Illustration of tether's dynamic model The Dynamic Equation Once neglected the tether's flexural rigidity, its lateral movement is governed by the following dynamic equation: where: M: tether's mass per unit length (physical plus added masses); By: quadratic fluid damping coefficient (viscous); T(x): total tension; The total tension acting along the tether is composed by: the static tension (T^x)), resulting from the platform pretension (P) and varying along the tether's length due to the action of its submerged weigh: being: i: physical mass per unit length; L: tether's length. the dynamic tension (Td(x)), imposed by the hull's first-order heave motion induced by a monochromatic wave with frequency (CD): (2)

4 1 78 Offshore Engineering where: (3) S: wave induced tension amplitude; leading to a total tension given by: T(x) = f + pg(l - x) -.S'. cos(<s>0 (4) Solution of the Eigenvalue Problem The dynamic equation for the tether's free lateral vibration is given by: f-o (5) a & Assuming the lateral motion of the nth. natural mode as: and substituting Equation (6) into Equation (1), one obtains: f + ^^M + «^>0 (?) \\ M M IL ^ " J J being: t3r.: natural frequency of the /?th. mode. As indicated by Equation (7) the dynamic model adopted leads to a classical Sturm-Liouville problem. By the introduction of the new variable (rj): " = V'+^7-^ (8) and after some algebraic work, Equation (7) can be rewritten in the form (see, e.g., [6]): 8' = or (9)

5 Offshore Engineering 179 The equation above is identified as a Modified Besse! Equation, being its solution given in terms of the Bessel functions (Jo,Yo): The constants (C^C:) are determined by the consideration of appropriate boundary conditions, such as: The resulting modal forms for the tether's dynamic model are then obtained as: with frequencies given by: The values of (3,, are determined numerically as solutions of the characteristic equation: Dynamic Solution in Terms of Natural Modes Following Pat el \ substituting Equation (12) into the dynamic Equation (1), applying Galerkhi's variational method (multiplying it by X,,(x) and integrating along the tether's length) and considering the following coefficients:

6 180 Offshore Engineering (15) the dynamic Equation (I) can be expressed in the form: with: i: dimensionless time variable (2r = cot); The tether's dynamic is hence represented by a Mathieu parametric equation with parameters given by: The stability analysis to be performed can be illustrated graphically by the Mathieu stability chart obtained by Patel^ (shade regions are unstable):

7 Offshore Engineering O.O 6 Figure 2 - Mathieu Stability Chart up to Large Parameters (extracted from [1]) THE BUTTON TLP EXAMPLE The Hutton TLP^ was the first platform of its kind to operate and due to its originality its design has some particularities when compared to more recent projects. Operating in the Gulf of Mexico at a low water depth (about 150 m), its tethers are subjected to a very high pretension. Data for the Hutton TLP tethers are: Number of tethers: Length: External diameter: Thickness: Static tension per tether: 16 ~ 115 in 0.26m 8.7 mm (estimated) 800 tonf Considering an wave exciting period of 6.8 seconds and the relation (S/P) equal to 2/3, one obtains the parameters presented by Patel for the Hutton operating condition (8=255;q=85). These values correspond to the tether's first mode of vibration. Patel also considers a damping coefficient (c) of 0.3 in his analysis. The dynamic analysis was then performed using both models (constant static tension model (Patel's) and the proposed variable static tension model) in order to obtain the phase plane and the time history for the generalised displacement amplitude f(t) (see Equation 6):

8 182 Offshore Engineering Constant Static Tension Model (Patel) 6 = 255;q = 85;c = f (t) Variable Static Tension Model f (t) Figure 3 - Generalised Displacement Amplitude for the Hutton TLP Tethers Considerations About the Results Obtained for Hutton TLP As can be observed in figure 3, the results obtained by the two distinct dynamic models are in this case indistinguishable. Nevertheless, this fact was already expected since the static tension for each tether of Hutton TLP is much greater than its submerged weigh. Considering the tension variation caused by such a weigh is, of course, of no relevance. It must be emphasised, however, that the Hutton TLP characteristics are very particular, mainly because the low water depth and high value of pretension. The present tendencies of oil exploitation are clearly leading to the use of TLPs in deeper waters. The reduction of pretension values is also desired in order to increase the payload and thus adequate the economic viability of such kind of project. Those tendencies may be verified observing the TLPs designed more recently. Besides, as already concluded in Patel's work, the deep water condition is critical for the tether's dynamic behaviour. Therefore the Hutton TLP is not an adequate example either to verify the influence of tension variation along the tether or to extract conclusions about its dynamics in deeper waters.

9 THE AUGER TLP EXAMPLE Offshore Engineering 183 The Auger TLP represents a more up-to-date design and its main characteristics are closer to those usually observed nowadays. It is also located in The Gulf of Mexico at a nominal water depth of 872 meters The platform presents mooring lines which contribute to control its offset, reducing the tethers pretension requirements. The main data for Auger tethers are presented bellow: Number of tethers: 12 Length: m External diameter: 0.66 m Thickness: 33 mm Static tension per tether: 903 tonf The dynamic analysis was then performed assuming a wave exciting period of 6 seconds and an induced dynamic tension amplitude (S) equivalent to 60% of the nominal static tension. Once again the phase plane and time history for the generalised displacement amplitude was determined considering both dynamic models. For the constant static tension model it was considered a mean static tension value once the weigh effect in this case is surely relevant. It can be proved that the mean-tension corresponds to the stationary point for sinusoidal representation if the Rayleigh-Ritz method is employed. The results are presented bellow for the first and the second mode of vibration: Auger TLP - First Mode Constant Static Tension Model (Patel) 6 = ; q = 250 ;c = O.6 f> 1.4 f.20 f'i A A A A A A f\ fi M ' \ M/ n ^ "<.2 I \/ \i\' \ \i ' I-' ^.4 J f V V U U V V V V ;.6 q f (t) f (t) I I I! IDOOO OOOC Time

10 184 Offshore Engineering Variable Static Tension Model 5," = ; q%* = 270 ;ci* = f (t) Time Figure 4 - Generalised Displacement Amplitude for the Auger TLP Tethers - First Mode Auger TLP - Second Mode Constant Static Tension Model (Patel) 5 = ; q= ; c = f (t) Time

11 Variable Static Tension Model 5," = ; q,* = ; c,* = Offshore Engineering _ 0.2 ii 0 " Time Figure 5 - Generalised Displacement Amplitude for the Auger TLP Tethers - Second Mode Considerations About the Results Obtained for Auger TLP The results above presented clearly demonstrate that, in this case, the influence of tension variation along tether's length performs an important role in its dynamics. The Auger's operation condition leads to an unstable response for tether's first mode of vibration, as shown in figure 4. The amplitude of generalised lateral displacement of 0.5 meters obtained considering tension variation is approximately 30% greater than the one calculated by the constant static tension model. The influence of submerged weigh can also be verified in the second mode case although with minor relevance, since the response in this case is stable and the amplitude decreases very fast. The analysis of Auger's tethers outlines some remarkable features concerning tethers dynamics in deep waters. A first aspect that must be pointed out is that for deep waters very small values of the parameters (5) and (q) control the dynamics of tether's lateral vibration. As the tether's length increases, the operation point in the Mathieu stability chart (figure 2) tends to the region near its origin. As can be observed form this figure, the dynamic behaviour is very sensitive to small variations in the parameters (6) and (q), in this region. The deep water case is then critical concerning tethers dynamics, not only due to the common sense fact of being larger the vibration amplitudes as larger is the span, but also because its inherent response instability. Another interesting feature tjiat results from this analysis is that the higher modes of vibrations tend to be more stable once the parameter (6) increases more rapidly than the parameter (q) with the modal number. This indication should be further investigated since higher modes have significant importance in the coupled platform-tethers dynamics.

12 186 Offshore Engineering CONCLUSIONS A dynamic model for TLP tethers lateral vibration considering the static tension variation along the length was derived. The global equation is governed by a parametric Mathieu equation and the natural modes of lateral vibration are obtained in terms of Bessel functions. A comparison between the proposed dynamic model and the one presented by Patel* was performed. The examples analysed show that tension variation plays an important role and its consideration is indeed necessary as the water depth increases and the usually high values of pretension are aimed to be reduced. This fact is still emphasised since the deep water TLP's design face critical conditions in tethers dynamics, as demonstrated. More accurate results are then requested in this case and the static tension variation assumption implies not only in more accurate amplitudes of vibration but also in a more precise stability analysis for each natural mode of vibration. ACKNOWLEDGEMENTS This work was supported by FAPESP - The State of Sao Paulo Research Foundation. REFERENCES [l]patel, M.H. & Park, H.I., "Dynamics of Tension Leg Platform Tethers at Low Tension. Part I - Mathieu Stability at Large Parameters", Marine Structures 4(3), pp , [2]Tetlow, J. & Leece, M, "Mutton TLP Mooring System", OTC 4428 (1982). [3]Mercier, J.A., Leverette, S.J. & Bliult, A.L., "Evaluation of Hutton TLP Response to Environmental Loads", OTC 4429 (1982). [4]Lohr, C.J. & Bo wen, K.G., "Design, Fabrication and Installation of Auger LMS and Tendons", OTC 7622 (1994). [5]Bender, C.M & Orzag, S., "Advanced Mathematical Methods for Scientists and Engineers", McGraw-Hill, New York, [6]Bowman, F, "Introduction to Bessel Functions", Dover, New York, 1958.