821. Study on analysis method for deepwater TTR coupled vibration of parameter vibration and vortex-induced vibration

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1 81. Study on analysis ethod for deepwater TTR coupled vibration of paraeter vibration and vortex-induced vibration Wu Xue-Min 1, Huang Wei-Ping Shandong Key aboratory of Ocean Engineering, Ocean University of China, Qingdao 66100, China E-ail: 1 wuxuein@yahoo.cn, wphuang@ouc.edu.cn (Received 13 April 01; accepted Septeber 01) Abstract. Considering the vertical vibration, caused by floating platfor, of top tensioned riser (TTR), an analysis ethod for the coupled vibration of paraeter excited vibration and vortex-induced vibration is presented in this paper. With the section rotation and shear deforation due to the bending large displaceent of TTR, a coupled vibration odel of paraeter excited vibration and vortex-induced vibration is put forward. And the vortex-induced vibration (VIV) of a TTR for 1500 water depth is analyzed based on the odel. The results show that the vertical vibration caused by floating platfor increases transverse vibration displaceent of TTR. Keywords: deepwater riser, TTR, vortex-induced vibration (VIV), fluid-structure interaction, paraeter excited vibration. 1. Introduction Riser syste is the ain constituent of ocean infrastructure, and its safety becoes ore and ore iportant. However, with the depth increasing, the design analysis of riser syste faces great challenges. Now long flexible deepwater riser vortex-induced odels can be divided into: epirical odels, ixed odels and CF odels. A coplete description is conducted [1]. Hartlen and Currie [] originally established vortex-induced transverse vibrations odel and wake oscillator odel. After the, any scholars have it aended and iproved. For exaple, Skop-Griffin odel, Iwan-Blevins odel, andl odel, Iwan wake oscillator odel, Skop-Griffin wake oscillator odel, Krent-Nielsen two oscillator odel and so on. The developent of coputer provides favorable conditions for the research and application of CF, and akes the nuerical ethod be widely used [3, ]. The ephasis on vortex-induced vibration of deepwater riser with two degrees of freedo (in-line and cross flow) is increasing [5]. When transverse bending vibration happened under wave load, the vertical vibration caused by floating platfor does not only affect the vertical paraetrically excited vibrations, but also affect the transverse bending vibration, thus there is a coupled vibration of paraeter excited vibration and vortex-induced vibration.. Matheatical odel ong flexible deepwater riser will have paraeter excited vibration caused by floating platfor heave [6], so when analysis the dynaical response of deepwater riser, not only consider the transverse excitation, but also consider axial excitation, because axial excitation caused by floating platfor heave increases transverse vibration aplitude of deepwater riser. Fig. 1 displays the principle of paraeter excited vibration and 1, denote transverse load and axial load. Existing deepwater riser bending vibration analysis ethods generally doesn t consider transverse displaceent caused by vertical displaceent; only calculate the bending 977

2 displaceent caused by transverse loads. And the bending vibration equation of deepwater drilling riser is as follows: y y y y EI ( T ) + + c = q( x, t) x x x t t where y is the transverse displaceent, x - the riser axial coordination, t - the tie, EI - the transverse flexural rigidity of TTR, T - the tension, it is the function of x and t, naely T = T(x, t), - the riser ass per unit length, c - the daping coefficient, q(x, t) - the fluid force in the transverse direction. (1) a) b) Fig. 1. a) ateral vibration, b) Paraetric vibration Now, with the section rotation and shear deforation due to the bending large displaceent of TTR, a coupled vibration odel of paraeter excited vibration and vortex-induced vibration is put forward. Its derivation process is as follows. Select a icro-eleent of riser dx (Fig. ). According to geoetry relation, there is: du dx= ρdθ () where ρ is the curvature radius of selected infinitesial section, θ the section corner, u the axial displaceent of riser, the top of riser is influenced by platfor otion, in this paper, we assue the platfor on the top of the riser as a particle, and only consider its heave otion. du is the axial displaceent of selected infinitesial section. Fig.. Riser bending scheatic diagra Fig. 3. Geoetric description of bea deforation with the consideration of the effects of shearing Take shear deforation (Fig. 3) into account: 978

3 ( dy dx) θ = γ (3) where γ is the shear strain and Eq. () is expressed as: γ = N / GA. ( du / dx) 1=ρ( d y / dx dγ / dx) () We know: 1/ρ = d y/dx, thus Eq. () is expressed as: 1 du / dx= 1 ρdγ / dx (5) du / ρ dx= dn / GAdx (6) 3 3 N = EId y / dx, and we assue κ= 1/ ρ, we can get Eq. (7): du d y GA κ = EI (7) dx dx As known the bea bending equation under transverse loads is as follows: d y EI = q( x, t) (8) dx GAκdu/dx is equivalent to transverse loads, we can add it to the right of Eq. (1) directly, and we can get a coupled vibration odel of paraeter excited vibration and vortex-induced vibration: y y y y du ( ) + + = (, ) + κ x x x t t dx EI T c q x t GA GA is the riser s shear rigidity, other paraeters are sae with Eq. (1). With the section rotation and shear deforation due to the bending large displaceent of TTR, the new coupled vibration odel is put forward. In the study, riser vibration analysis takes the effects of fluid within the riser and riser inner tension effects into account. 3. Fluid-structure interaction vortex-induced forces The riser under the fluid loads will vibrate, and the vibration of the riser will affect the flow field in turn, vice versa, naely fluid-structure interaction. When considering the effects of fluid-structure interaction, the structure will have nonlinear daping force and inertia force in transverse direction, as the Morison equation [7] expresses: f = C y ρ vɺ vɺ / + C ρπ vɺ / Cross-flow force expression as follows: (9) (10) f y = C ρ ( u wɺ ) cosω t / + C ρvɺ vɺ / + C ρπ ɺ / (11) 0 s v f z In-line force expression as follows: = C ρ ( u wɺ ) cosω t / + C ρ( u wɺ ) u wɺ / + C ρπ ɺ / (1) 0 s 0 0 w where C is the cross-flow lift coefficient, C - the in-line lift coefficient, - the riser 979

4 diaeter, ω - the vortex shedding frequency, s C - the drag coefficient, C - the added ass coefficient, vɺ - the vibration speed of riser in cross-flow direction, vɺɺ - the vibration acceleration of riser in cross-flow direction, wɺ - the vibration speed of riser in in-line direction, wɺɺ - the vibration acceleration of riser in cross-flow direction.. Nuerical analysis Put forula (11) and forula (1) into Eq. (9), and use Newark β ethod increental for to analysis of dynaic response of riser. Newark β ethod is a direct integration ethod, it s unconditionally stable. The full variable for of direct integration ethod can be expressed as follows: [ M] { vɺ e t [ ] e t t [ ] e i+ t + C { vɺ i+ + K { v ti+ t = { Fy t + t ɺ (13) [ M] { wɺ e t [ ] e t t [ ] e i+ t + C { wɺ i+ + K { w ti+ t = { Fz t + t i ɺ (1) Increental equations of otion of the syste can be expressed as: [ M ] t { vɺ [ ] [ ] + C { vɺ + K { v = { F i y ti i ɺ (15) [ M ] t { wɺ [ ] [ ] + C { wɺ + K { w = { F i z ti ɺ (16) where [M] is the ass atrix of syste, [C] - the daping atrix of syste, we usually use Rayleigh daping [C u ] = α[m u ] + β[k u ], in which α and β are Rayleigh daping coefficients, their unit are s -1 and s, and their value according to: α ξ ϖϖ n β = ϖ+ ϖn 1 and [16] ξ = 0.05, ϖ is fundaental frequency, ϖ n is the third order natural frequency. [K] - the stiffness atrix of syste, {v e, {w e - the node displaceent of eleent, { vɺ e, { wɺ e - the node speed of eleent, { vɺɺ e, { wɺɺ e - the node acceleration of eleent. ue to introducing large deforation, the ass atrix, daping atrix and stiffness atrix change with tie. So it can be used to solve geoetrical and physical non-linear proble. In order to iprove calculation accuracy, iteration is carried out in every step. Tie step is dt = 0.01 s. 5. Results and discussion Based on the new above progra, the features of a top tension riser considering large deforation are further investigated. The selected TTR is a double casing production riser, its external diaeter of external pipe is 3, diaeter of inner pipe is, diaeter of internal pipe is 11. Based on criterions of bending stiffness equivalent this double-layered pipe is equivalent to single-layered pipe. The equivalent inner diaeter is 9. Top tension coefficient is 1.. Boundary conditions are one fixed end and one articulated end. 980

5 Paraeters of odel riser are given in Table 1. Table 1. Paraeters of odel riser Riser length 1500 Outer diaeter Inner diaeter 0.85 Elastic odulus 07 GPa Shear odulus Pa Material density 7850 kg/ 3 Table. VIV response paraeters V (/s) C C C ' C 0.18 / According to Fig., the relation curves of reduced aplitude and reduced speed, TTR cross-flow vibration lock-in range is: < U < 6.3, thus in this paper, velocity 0.18 /s, which r is in lock-in range, and velocity 0. /s which is outside lock-in range are selected as siulation speeds. Fig.. Cross-flow reduced aplitude and reduced velocity As that different velocity lead to different odes being excited by vortex-induced vibration of riser, this paper selects the points where axiu vibration displaceent happened at 885 water depth (velocity 0.18 /s) and 1065 water depth (velocity 0. /s) as analysis points. In-line direction and cross-flow direction displaceent tie curve of analysis points are given in Fig. 5 - Fig. 6 and Fig. 7 - Fig. 8 under the two vortex-induced force odels (considering paraeter excited vibration and not considering paraeter excited vibration odel). By running it was deterined that the progra of the TTR is stable, so in this paper 0 50 s tie history is selected. Response results of vortex-induced vibration are given in Table 3 and Table. Fro Fig. 5-8 displaceent tie histories, we could see that: the riser displaceent increase whatever in cross-flow direction or in in-line direction if we consider the paraeter excited vibration caused by floating platfor. When velocity is 0.18 /s, at 1065 water depth, the axiu vibration displaceent of TTR in in-line direction increases 0.015, the axiu vibration displaceent of TTR in cross-flow direction increases When velocity is 0. /s, at 885 water depth, the axiu vibration displaceent of TTR in in-line direction increases 0.091, the axiu vibration displaceent of TTR in cross-flow direction increases According to above analysis, the paraeter excited vibration caused by floating platfor increases transverse vibration displaceent of TTR. 981

6 Fig. 5. isplaceent tie histories of in-line direction riser vibration at 1065 water depth. otted line is on behalf of not considering paraeter excited vibration, the solid line is on behalf of considering paraeter excited vibration Fig. 6. isplaceent tie histories of riser vibration at 1065 water depth in cross-flow direction. otted line is on behalf of not considering paraeter excited vibration, the solid line is on behalf of considering paraeter excited vibration Fig. 7. isplaceent tie histories of in-line direction riser vibration at 885 water depth. otted line is on behalf of not considering paraeter excited vibration, the solid line is on behalf of considering paraeter excited vibration Fig. 8. isplaceent tie histories of cross-flow direction riser vibration at 885 water depth. otted line is on behalf of not considering paraeter excited vibration, the solid line is on behalf of considering paraeter excited vibration 98

7 Table 3. Response results of vortex-induced vibration at 1065 water depth irection Maxiu displaceent No coupled vibration odel The coupled vibration odel Increasing rate In-line % Cross-flow % Table. Response results of vortex-induced vibration at 885 water depth irection Maxiu displaceent No coupled vibration odel The coupled vibration odel Increasing rate In-line % Cross-flow % 6. Conclusions The vertical vibration caused by floating platfor increases transverse vibration displaceent of TTR. So, when forecasting dynaic response of deepwater riser, the paraeter excited vibration caused by floating platfor shouldn t be ignored, instead, it should be taken into account. The coupled vibration odel of paraeter excited vibration and vortex-induced vibration of TTR can provide reference for riser design and analysis. Acknowledgents This study was supported financially by the Natural Science Foundation of China (NSFC) (No ; No ). The authors would like to express profound thanks for the. References [1] Wu Y. X., Xue W. H. et al. The ethods for predicting vortex-induced vibration of arine risers. Shipbuilding of China, Vol. 51, 010, p. 0-. [] Hartlen R. T., Currie I. G. ift-oscillator odel of vortex-induced vibration. ASCE Journal of the Engineering Mechanics, Vol. 96, 1970, p [3] Tang S. Z., Haung W. P. et al. Nuerical analysis of deepwater riser vortex-induced vibration with two degrees of freedo. Journal of Vibration and Shock, Vol. 9, 010, p [] Huang Z. Y., Pan Z. Y. et al. Nuerical siulation of VIV of a circular cylinder with two degrees of freedo and low ass-ratio. Journal of Ship Mechanics, Vol. 11, 007, p [5] Tang G. Q., v. et al. aboratory easureent of vortex-induced vibration of long flexible riser. Ocean Engineering, Vol. 9, 010, p [6] Nie W., iu Y. Q. Ocean Engineering Structure ynaic Analysis. Haerbin Engineering University Press, China, 00. [7] ong Y. Q. Wave oads and Response of the Oil-Extraction Platfor in eep Ocean. Tianjin University Press,