Riser Dynamic Analysis Using WKB-Based Dynamic Stiffness Method

Size: px

Start display at page:

Download "Riser Dynamic Analysis Using WKB-Based Dynamic Stiffness Method"

Annis Kelly
5 years ago
Views:

1 Rier ynamic Anayi Uing WKB-Baed ynamic Stiffne Method The MIT Facuty ha made thi artice openy avaiabe. Peae hare how thi acce benefit you. Your tory matter. Citation A Pubihed Pubiher Cheng, Yongming, and J. Kim Vandiver. Rier ynamic Anayi Uing WKB-Baed ynamic Stiffne Method. Voume : Pipeine anier Technoogy (Juy, ). American Society of Mechanica Engineer Verion Fina pubihed verion Acceed Thu Ju 6 :8:5 ET 8 Citabe Link Term of Ue etaied Term Artice i made avaiabe in accordance with the pubiher' poicy and may be ubject to US copyright aw. Peae refer to the pubiher' ite for term of ue.

2 Proceeding of the ASME t Internationa Conference on Ocean, Offhore and Arctic Engineering OMAE Juy -6,, Rio de Janeiro, Brazi OMAE-8 RISER YNAMIC ANALYSIS USING WKB-BASE YNAMIC STIFFNESS METHO Yongming Cheng *, J. Kim Vandiver ** * Richtech Int Engineering, Inc, Houton, TX, USA ** Maachuett Intitute of Technoogy, Cambridge, MA, USA ABSTRACT Rier are fuid conduit from ubea equipment to urface foating production patform. The integrity of a rier ytem pay a very important roe in deepwater deveopment. Rier dynamic anayi i an important part to the ytem deign. Thi paper invetigate rier dynamic anayi uing the WKB-Baed dynamic tiffne method. Thi paper firt preent a theoretica formuation of the dynamic tiffne method. It then combine the dynamic tiffne method with the WKB theory, which aume that the coefficient in the differentia equation of motion are owy varying. The WKB-baed dynamic tiffne method i derived and a frequency dependent hape function i expreed impicity. The Wittrick and Wiiam (W-W) agorithm i further extended to ove eigen vaue probem for a genera non-uniform marine rier. Exampe of non-uniform rier are anayzed and the reut how the efficiency of thi method. In addition, a pipe-in-pipe rier ytem i anayzed for natura frequencie and mode hape uing the WKB-baed dynamic tiffne method with the W-W agorithm. The characteritic of the mode hape i decribed for uch a rier ytem. INTROUCTION A rier i a fuid conduit from ubea equipment to the urface foating production ytem uch a a Spar or TLP. It i a key component in a deepwater driing and production ytem. It dynamic deign and anayi i very important in the deepwater appication. Hitoricay, Koouek firt preented the idea of ynamic Stiffne Method (SM) in the eary 9 [], and gave an eaborate formuation of thi method in 95 []. Since then the SM ha been widey ued in the vibration anayi of beam tructure. Improvement on cacuating natura frequencie have been made by the Wiiam and Wittrick (W-W) agorithm [, ]. The SM ha a great appea for an exact dynamic anayi of a uniform beam tructure, a it i baed on the exact dynamic tiffne matrix derived from the free vibration anayi. The SM perform free and forced vibration anayi within the differentia equation theory of beam, thu avoiding aumed mode and umped mae. Thi method enabe one to anayze an infinite number of natura frequencie and mode accuratey by mean of fewer degree of freedom, compared with a traditiona finite eement method by uing a poynomia hape function. Thi paper deveop the WKB-baed dynamic tiffne formuation for a rier ytem, which aume that the coefficient in the differentia equation of motion are owy varying. The WKB-baed dynamic tiffne method i derived and a frequency dependent hape function i expreed impicity. The Wittrick and Wiiam (W-W) agorithm i extended to ove eigen vaue probem for a genera non-uniform marine rier. Exampe are anayzed and the reut how the efficiency of thi method. In addition, a pipe-in-pipe rier ytem i anayzed for natura frequencie and mode hape uing the WKB-baed dynamic tiffne method with the W-W agorithm. Copyright by ASME ownoaded From: on /5/7 Term of Ue:

3 The characteritic of the mode hape i decribed for uch a rier ytem. THEORETICAL FORMULATION OF A RISER SYSTEM WKB-baed dynamic tiffne matrix A great dea of reearch ha focued on vibration anayi of a beam tructure. For a uniform Euer beam under a contant axia oad, the effect of the axia oad on the natura frequencie ha been found by conidering the natura frequencie to be function of a non-dimeniona oad parameter and boundary condition. Uing a power erie expanion, areing and Huang [5] found the natura frequencie of a uniform marine driing rier. A dynamic rier mode i needed which i abe to account for non-uniform propertie uch a ma denity, bending rigidity and tenion ditribution, and dicontinuitie uch a intermediate upport. A coed form oution to uch a ytem i not generay poibe. An approximation to the vibration anayi of uch a rier may be accompihed by repacing the variabe parameter with contant one. For exampe, a variabe axia oad i often approximated by a tenion that i contant over each eement. However, many degree of freedom in the approximation are required in order to obtain accurate reut. Thi paper invetigate the vibration anayi of marine rier by combining the dynamic tiffne method with the WKB theory, which aume that the coefficient in the differentia equation of motion are owy varying. A genera marine rier i a ong ender beam ytem with variabe tenion ditribution, bending rigidity and ma denity. The ma/ength change are often dicontinuou. Such a rier can be dicretized into eement having continuouy varying propertie within the eement and aowing dicontinuitie to occur between eement. The equation of motion of a rier i written a: x w w w EI ( x) ] [ T ( x) ] m( x) f ( x, t) x x x t [ where w i the tranvere dipacement of rier, E i the Young' moduu of the materia, I (x) i the area moment inertia of the beam, T (x) i the tenion of the rier, m (x) i ma per unit ength, and f ( x, t) i externa force per unit ength. The dimenione parameter are defined a foow: x /, E I w Y,, t, m () Where the ubcript '' repreent the vaue at a reference cro ection, i ength of a rier eement and i a reference diameter for the rier. Eq. () i thu written into the foowing non-dimeniona form: [ Y Y Y P( ) ] [ Q( ) ] U( ) f (, ) i Auming Y (, ) R( ) e and ubtituting it into Eq. () reut in the foowing equation of motion of a free vibration: d d [ P( ) ] d d d dr [ Q( ) ] U( ) R d where i a dimenione frequency, /. (), () Auming that P (), Q () and U () in Eq. () vary owy with repect to, compared with variation of, rewrite Eq. () a: R (), R () an () dr P( z) P( z) [ P ( z) Q( z)] Q( z) U( z) R dz dz dz dz () Where z, i a ma parameter. The forma WKB expanion i written a: R(z) exp n [ Sn ( z)],. n (5) The foowing aymptotic oution can be found by ubtituting Eq. (5) into (), identifying the ame order term, truncating the erie and eecting R( ) T ( )[ C T ( )[ C in( inh( h ( ) d ) C co( h ( ) d ) C. h ( ) d )] coh( h ( ) d )] Where, C i ( i to ) are contant of integration; Ti ( ), hi ( ) ( i to ) are function of P (), () U() and. Note that: dw dy d (6) Q,, d w d Y d w d Y, and. Negecting higher order term, then the eement noda dipacement vector, V e, can be formuated in the foowing matrix form: d d Copyright by ASME ownoaded From: on /5/7 Term of Ue:

4 v y T () h () v B () T () y B () T () h () In which, T () B () T () B () T () h () B ( ) in h ( ) d T () h () B () T () B () T () h () B ( ) inh h ( ) d, B ( ) coh h ( ) d. T () C C B () T () C B () T () h () C (7), B ( ) co h ( ) d, The Eq. (7) can be written in abbreviated form a: V e = G C. (8) The noda force, F e, for an eement with changing propertie are formuated a: m m y y EI ( x) EI ( x) EI ( x) EI ( ) EI ( ) EI ( x) d d x x x x EI ( ) T ( ) T ( ) EI ( ) T ( x) dr d T ( x) dr d d d dr dr x x Subtituting R() from Eq. (6), Eq. (9) can be written a the foowing matrix form: F e = H C. () The reationhip between the eement noda force and dipacement can be etabihed by combining Eq. (8) with Eq. (): F e = K e (ω)v e, () In which K e (ω) = H G -, i the WKB-baed dynamic eement tiffne matrix, whoe eement were derived by uing Mape V. Frequency ependent Shape Function In order to derive the frequency dependent hape function, rewrite Eq. (6) a: (9) T ( )in h ( ) d C T ( ) co h ( ) d C R( ) C T ( )inh h ( ) d C T ( ) coh h ( ) d T. () The contant of integration C i ( i to ) are oved from Eq. (8) a: C = / G - V e. () Subtituting Eq. () into Eq. () reut in: R() = Φ V e, () Where Φ i the frequency dependent hape function. Goba ynamic Stiffne Matrix Formuation The dynamic tiffne formuation of a rier ytem i obtained by etabihing a weak form of the equation of motion uing the Gaerkin procedure. Integrating over the domain of interet and tranforming to ower the order of the derivative and incorporate the boundary condition a forcing term give the variationa equation to be dicretized by finite eement interpoation. The formuation of the pectrum eement method for a rier ytem i thu deveoped by foowing the procedure of the conventiona finite eement method [8], in which oca eement are cat into a goba form by coordinate tranformation. The equation of motion of free vibration in the retrained goba dynamic tiffne form can be written a: K G (ω) X =. (5) The eement of goba tiffne matrix, K G (ω), are generay trancendenta function of circuar frequency ω. Natura frequencie can be found by equating to zero the determinant of the goba dynamic tiffne matrix, K G (ω). The eigenvaue, or the natura frequencie, are obtained by potting det (K G (ω)) and finding the root. For a uniform beam member, Wittrick and Wiiam (W-W) [] preented an automatic computation of natura frequencie. For a tapered beam whoe ection propertie vary reguary, Banerjee and Wiiam [] gave a procedure to cacuate natura frequencie. However, a typica marine rier ha non-uniform propertie incuding ma ditribution, bending rigidity and tenion. The procedure in [, ] can t be directy ued for the Copyright by ASME ownoaded From: on /5/7 Term of Ue:

5 appication in thi paper. The W-W agorithm i thu extended to a genera non-uniform rier ytem for an automatic computation of natura frequencie [6]. Once the natura frequencie are found, one can ue Eq. (5) to ove for a pecific mode hape. An effective way i to ue a trianguar decompoition. Simiary, the equation of motion for a rier ytem under a forced excitation can be formuated a: K G (ω) X = F (ω). (6) The rier frequency repone can be oved by uing the agorithm baed on Gau eimination [7]. The kyine reduction method i ued in the computer impementation of the Gau eimination. A computationa program ha been deveoped to ove for the natura frequencie and mode hape, and the frequency repone for a generay rier ytem. In addition, the above WKB-baed dynamic tiffne formuation ha been extended to a couped pipe-in-pipe rier ytem [9]. RESULTS Uniform riing Rier under Lineary Varying Tenion [5] The parameter of a impy upported rier are: Length L 5 feet; Outer diameter d inche; Wa Thickne t. 65 inche; Young' moduu 6 E bf/in ; Ma per unit ength m. 8 ug/ft (incude ma of driing mud and ea water); Tenion at the bottom ba joint T =86, b; and Net weight of rier per unit ength in ea water w b/ft (incuding 8 b/ft for choke and ki ine). Figure how the determinant of the dynamic tiffne matrix of the 5-ft rier veru frequency. Tabe it the firt five natura frequencie found from Figure. In order to verify the reut, a finite eement procedure which aumed contant tenion over each beam eement wa deveoped. Converged vaue for natura frequencie were found empoying 6 eement in the FEM. The approximation reut [5] obtained by mean of a power erie expanion i ao incuded for comparion. It i oberved from Tabe that the natura frequencie acquired by the WKB-baed dynamic tiffne method uing ony five eement are accurate. Figure depict the firt three mode hape. Tabe indicate that the natura frequencie obtained by areing and Huang [5] are ao accurate, compared with thoe obtained by uing the FEM and the WKB-baed dynamic tiffne method. However, their finding of ``point of infection'' in mode hape i not correct. In addition, ony one eement i needed to obtain accurate natura frequencie by uing the W-W agorithm. Tabe Comparion of Circuar Natura Frequencie areing and FEM WKB-SM Order Huang [5] (6 eement) (5 eement) Non-Uniform Rier under Lineary Varying Tenion ue to attachment uch a buoyancy modue, a typica marine rier i a ytem with variabe propertie incuding tenion and ma denity. Such a rier ytem, impy upported, ha the foowing propertie: Length L m; Outer tee diameter d. inche; Wa Thickne t. 65 inche; and Buoyancy diameter d. 5 inche. b o Figure and how the variation of the ma and tenion at the meaured point which are marked repectivey. The poition i meaured from the bottom. Thee figure demontrate that the ma denity doe not change continuouy, and tenion doe not vary ineary. There are eeven () egment in Figure and, each of which ha continuou variation of ma and tenion. Figure 5 how the firt natura frequencie found by uing the WKBbaed dynamic tiffne anayi with eement. The approximate reut uing Shear7, which aumed the rier to be an equivaent uniform beam with an average ineary varying tenion aong the rier, are incuded for comparion. The Shear7 reut are accurate ony for ower order natura frequencie. Figure 6 depict the th mode hape, ope and curvature. The moda information i important to predict VIV fatigue damage to the rier. The ocation of the antinode are not eveny paced. Therefore, the mode differ from trigonometric one. Copyright by ASME ownoaded From: on /5/7 Term of Ue:

6 It i found that 8 eement are needed for the tandard finite eement method to obtain a good th mode hape and a converged natura frequency of.695 Hz. Thi i coe to.6955 Hz by the WKB-baed dynamic tiffne method with ony eement. Very few eement are neceary if they are choen wiey. Within each eement, propertie mut vary owy o a to atify the WKB aumption. icontinuitie houd occur at the junction of eement. In thi exampe, the ma/ength change abrupty ten time requiring a tota of eeven eement to adequatey mode the ytem. Couped Pipe-In-Pipe Rier Sytem A couped pipe-in-pipe rier ytem, hown in Figure 7, i ued for demontrating the appication. Both the externa and interna caing are impy upported and their pecification are a foow: Outer diameter of externa pipe =.75 inche; Wa thickne of externa pipe =.8 inche; Added ma coefficient for externa pipe=.; Outer diameter of interna pipe =9.75 inche; Wa thickne of interna pipe =.975 inche; Young' moduu E = ki; Length of both cyinder L =9 ft; Minimum tenion on externa pipe T.5 b; Tenion varying factor of externa pipe = 7. bf/ft; Minimum tenion on interna pipe T. b; Tenion varying factor of interna pipe =.6 bf/ft; Number of eveny ditributed identica centraizer=9; and The ditance between centraizer = 97. ft. Each rier i dicretized into eveny ditributed eement. The centraizer are eveny ditributed aong the rier. The * foowing dimenione tiffne k i ued to decribe the reative tiffne of centraizer. k * k / 8E I, n where the ubcript denote the tandard reference vaue for the pipe and i the ditance between centraizer. The pipe-in-pipe rier ytem i couped by centraizer and idea fuid in the annuu. Tabe it the natura frequencie and incude thoe couping cae for pring and fuid ony for comparion. Thi tabe demontrate that the fuid ower the natura frequencie. The cae couped by fuid ony generate the owet natura frequencie whie that couped by centraizer ony generate the highet natura frequencie. The natura frequencie for a genera cae couped by fuid and centraizer ie in between the two other cae. Figure 8 iutrate the firt 8 mode hape i, ( i,8) of the couped rier ytem. It indicate that when the ytem i weaky couped by centraizer, it mode hape are either in-phae or out-of-phae. The difference in the firt two mode, and, i that the deformation of the interna rier i arger in mode one,, whie that of the externa rier i arger in mode,. The natura frequencie of the couped ytem increae with the tiffne of centraizer. Tabe Natura frequencie (Hz) of a couped rier ytem Order Spring ony Fuid ony Spring/Fuid CONCLUSIONS Thi paper invetigate the rier dynamic anayi uing WKBbaed dynamic tiffne method. The concuion that can be drawn from the work in thi paper are: () The theoretica formuation for a genera non-uniform rier ytem i contructed by uing a pectrum eement method and WKB-baed frequency-dependent hape function. The theoretica formuation can be extended to a couped pipe-inpipe rier ytem. () The W-W agorithm i extended to the WKB-baed dynamic tiffne method for an automatic computation of natura frequencie for a non-uniform rier. The minimum eement are needed to accuratey compute natura frequencie and moda information. The advantage of thi approach i evident for oving high order natura frequencie. 5 Copyright by ASME ownoaded From: on /5/7 Term of Ue:

7 () The fuid/rier couping can be deigned to uppre the vibration of an externa caing caued by VIV [9]. The couping can be optimized to provide damping to the externa caing. ACKNOWLEGEMENT Thi paper repreent part of the reearch work when the firt author tudied at MIT for hi Ph.. with the econd author a hi upervior. Thank to SHEAR7 JIP member who upported the work. REFERENCES. V. Koouek, Anwendung de Geetze der virtueen und de reziprozitatatze in der tabwerkdynamik, Ingenieur Archiv, : 6-7, 9. V. Koouek, Structura ynamic of Beam and Frame Sytem, Prague, (in Czech), 95. W. H. Wittrick and F. W. Wiiam, A Genera Agorithm for Computing Natura Frequencie of Eatic Structure, Vo. xx iv, pt., 6-8, Quart. Journ. Mech. and Appied Math., 97. J. R. Banerjee and F. W. Wiiam, Exact Bernoue-Euer ynamic Stiffne Matrix for a range of Tapered Beam, Vo., pp. 89-, Internation Journa for Numerica Method in Engineering, W. areing and T. Huang, Natura Frequencie of Marine riing Rier, PP8-88, Journa of Petroeum Technoogy, Y. Cheng, ynamic Stiffne and Tranfer Matrix Anayi of Marine Rier Vibration (Ph.. thei), MIT, Cambridge, 7. Y. Cheng, J. Kim Vandiver, and G. Moe, The Linear Vibration Anayi of Marine Rier Uing the WKB-Baed ynamic Stiffne Method, 5(), 75-76, Journa of Sound and Vibration, 8. K. J. Bathe, Finite Eement Procedure, Prentice-Ha, Inc., Y. Cheng and J. Kim Vandiver, ynamic Anayi for an Internay Couped Fuid/Rier Sytem, 9 th Internationa Conference on Ocean, Offhore and Artic Engineering, OMAE-9, 6- June,, Shanghai, China. G. Moe, Y. Cheng, and J. Kim Vandiver, Rier Anayiby Mean of Some Finite Eement Approache, Proceeding of ETCE/OMAE, 6 Copyright by ASME ownoaded From: on /5/7 Term of Ue:

8 Figure eterminant of the ynamic Stiffne Matrix Figure Firt Three Natura Mode Shape 7 Copyright by ASME ownoaded From: on /5/7 Term of Ue:

9 Figure Ma Variation aong the Rier Figure Tenion Variation aong the Rier 8 Copyright by ASME ownoaded From: on /5/7 Term of Ue:

10 Figure 5 Natura Frequencie of a Non-Uniform Rier Figure 6 th Moda Shape, Sope and Curvature of the Rier 9 Copyright by ASME ownoaded From: on /5/7 Term of Ue:

11 Figure 7 Schematic of Two Concentric Pipe Containing Vicou Fuid Figure 8 Firt 8 Mode Shape of the Coupeier Sytem * ( k =., oid ine: externa rier; dah-dot ine: interna rier) Copyright by ASME ownoaded From: on /5/7 Term of Ue:

DESIGN SPECTRA FOR BURIED PIPELINES

DESIGN SPECTRA FOR BURIED PIPELINES th Word Conference on Earthquae Engineering Vancouver, B.C., Canada Augut -6, 4 Paper o. 94 DEIG PECTRA FOR BURIED PIPEIE i-ing HOG and Tzuchien CHE UMMARY For a buried pipeine ytem, the imum repone aong