Free Vibrations of Catenary Risers with Internal Fluid

Transcription

1 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, 216. Trabalh apresentad n DINCON, Natal - RN, 215. Prceeding Series f the Brazilian Sciety f Cmputatinal and Applied Mathematics 1 Free Vibratins f Catenary Risers with Internal Fluid Jseph Arthur Meléndez Vásquez 1 Departament de Engenharia Mecânica, UFABC, Sant André, SP, Juan Pabl Julca Avila 2 Departament de Engenharia Aerespacial, UFABC, Sant André, SP, Abstract. An investigatin emphasizing n free vibratins f steel catenary risers t determine the natural frequencies is numerically apprached in this wrk. A mdel frmulatin is develped using the variatinal apprach. Nnlinear equatins f mtin cupled in axial and transverse displacements are derived thrugh the principle f the statinary ptential energy. Keywrds. Steel catenary risers, Free vibratin, Finite element methd, Eigenvalue prblem. 1 Intrductin Fluid traveling thrugh a steel catenary riser (SCR) is subjected t centrifugal and Crilis acceleratins due t the curvature and angular mvement f the riser, respectively. The dynamics f the internal fluid flw affects the natural vibratin parameters f an SCR, reducing even mre its useful life in relatin t the useful life withut cnsidering the internal fluid. Few investigatins have been cnducted t study the effect f internal fluid dynamics n the glbal dynamic behavir f the SCRs. The prblem f SCRs naturally vibrating n the effects f internal fluid dynamics is still an pen research field. The present wrk appraches numerically the free vibratin prblem f SCRs. 2 Mathematical Frmulatin The SCR is fully immersed in sea water f density ρ w and has an initial arc length equal t S. The external and internal diameters are D e and D i, respectively, and the density f the riser material is ρ r. The SCR is cnnected t the platfrm and cnnected t the well head thrugh hinged supprts in bth ends, see Figure 1. This transprts an incmpressible fluid f density ρ f with cnstant velcity U. We cnsider three cnfiguratins in the mathematic mdeling f the riser. The cnfiguratin 1-1 represents the riser withut any lad applied. The cnfiguratin 2-2 is the static equilibrium cnfiguratin f the riser when subjected t time-independent lads, and the cnfiguratin 3-3 when subjected t the actin f time-dependent lads. 1 j.melendez.v@htmail.cm 2 juan.avila@ufabc.edu.br DOI: 1.554/ SBMAC

2 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, Figure 1: Risers cnfiguratins. The equatins f mtin f the riser are deduced using the principle f statinary ptential energy that is expressed by δπ = δu a + δu b (δw w + δw + δw I ) = (1) where δu a and δu b are the virtual strain energies due t axial stretching and bending mment, respectively; δw w, δw and δw I are the virtual wrk dne by the effective weight, hydrdynamic frces, and inertial frces, respectively. 2.1 Virtual strain energy due t axial stretching The strain energy stred at the riser due t the axial defrmatin is due t the axial tensin n the linear elastic riser and due t the axial stress resulting frm enclsing external and internal hydrstatic pressures [6]. The axial tensin is defined by T and the axial stress due t the enclsing hydrstatic pressures is defined by 2 ( pe Ae pi Ai ) / Ar where p e and p i are the external and internal pressures, respectively, A r is the crss-sectinal area f the riser, A e and A i are the external and internal crss-sectinal areas f the riser, respectively, and ν is the Pissn s rati. Thus, the strain energy due t axial defrmatin is given by 1 U EA ds A ds S S 2 a r r 2 (2) where E is the mdulus f elasticity. The axial strain ε is defined as 2 2 ux vy 1 u v s s 2 s (3) DOI: 1.554/ SBMAC

3 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, /2 where s ( x y ) and the peratr ( ) express the derivative f ( ) with respect t y. By intrducing the Eq. (3) int the Eq. (2) and applying the variatinal peratr we btain EAr T a ( x u) ( y v ) Ua u x 3 v y u v u v dy 2 s (1 ) (1 ) s (4) where T a = EA r ε + σa r, is the apparent axial frce at the static equilibrium cnfiguratin Virtual strain energy due t bending Cnsidering that the initial cnfiguratin f the riser is a straight line, defined by linking bth ends f riser, the virtual strain energy due t the bending mment M is by definitin 3 U M ' dy (5) b where M EIκ(1 ), M is detailed in [2], I is the mment f inertia and κ is the curvature f the riser at the displaced cnfiguratin. This last term is defined by: 3 x'' y ' x' y '' s ' (6) where s is the length f a riser element at the displaced cnfiguratin. The variable δθ f Eq. (5) is btained by applying the variatinal peratr t θ = κs, the definitin f curvature. Finally, by substituting the equatin (6) in the Eq. (5), and using the fllwing relatins, valid at the displaced cnfiguratins, δx = δu, δy = δv, we btain U Virtual wrk dne by the apparent weight The apparent weight by unit f length, w e, is defined as w e = (ρ r A r + ρ f A i ρ w A e )g. Then the virtual wrk dne by the apparent weight is expressed by S δw W = w e δv ds = w e ( s ) δv dy 1 + ε (7) 2.4. Virtual wrk dne by the hydrdynamic frces Figure 2 shws a riser element mving submerse in an ceanic current that fllws the psitive directin f the x axis. V n and V t are the cmpnents f velcity f the fluid incident n the riser at the nrmal and tangential directins, respectively. The hydrdynamic frce by unit f length acting n a riser can be determined using the Mrisn equatin expressed by: D De m Dw n n n n e Aw n n w n e f C V u V u D C V u V (8) where C D and C A are the drag and added mass cefficients, respectively. The first term n the right hand f the Eq. (8) is the drag frce, the secnd term is the added mass frce and the last term is the Frude-Krylv frce. Then, the virtual wrk dne by the hydrdynamic frces is expressed by δw = f m y δu dy f m x δv dy (9) b DOI: 1.554/ SBMAC

4 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, Figure 2. Nrmal hydrdynamic frce acting n inclined riser and the displacement cmpnents f a pint f riser Virtual wrk dne by the inertial frces The riser is subject t inertial frces; due t the rigid bdy dynamic and due t the dynamics f the internal fluid. The virtual wrk dne by the inertial frces is expressed as δw I = [(m r a r + m f a f ) (δui + δvj)] s dy (1) where a r is the acceleratin f the riser, a f is the acceleratin f the internal fluid, m r and m f are the riser mass and the internal fluid mass per unit length, respectively. The acceleratin f the riser in Cartesian crdinates is given by a r = u i + v j (11) and the internal fluid acceleratin is expressed by the fllwing expressin, detailed in [4], a f = [u + ( 1 s 2 x 2 s 4) (x + u )U 2 ( x y s 4 ) (y + v )U 2 + 2U s u ] i + (12) [v + ( 1 s 2 y 2 s 4) (y + v )U 2 ( x y s 4 ) (x + u )U 2 + 2U v ] j s By substituting equatins (4), (5), (7), (9) and (1) in Eq. (1) and integrating by parts the resulting equatin, we btain tw equatins f mtin: ne in the x-directin and anther in the y-directin. Then, thse tw equatins can be expressed in matrix frm as: [A]{x } + [B]{x } + [C]{x } + ([k t1 ]{x }) + ([k t2 ]{x }) + ([k b1 ]{x }) + ([k b2 ]{x }) = {F} (13) In this wrk an analysis f free vibratin f a riser was cnducted cnsidering the fllwing external frces: weight f the riser, buyancy, frces due t internal fluid, and Mrisn s frce. The last frce is mdelled by the secnd term n the right hand side f Eq. (8), and then the frce vectr {F} becmes: {F} = C A ρ m A e ( 1 s ) [ 1 x x x 2 ] { x } (14) y Substituting the Eq. (14) in Eq. (13), we btain the equatin f mtin fr free vibratin [A ]{x } + [B]{x } + [C]{x} + ([k t1 ]{x} ) + ([k t2 ]{x} ) + ([k b1 ]{x} ) + ([k b2 ]{x} ) = {} (15) In ur wrk, we cnsider that the static equilibrium cnfiguratin f the riser is knwn. DOI: 1.554/ SBMAC

5 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, Numerical Methd This sectin presents the prcedure used t determine the riser s linear natural frequencies with internal fluid. Frm Figure 1: {x} = { x y } = {x } + {u} = {x y } + { u v } (16) Substituting Eq. (16) in Eq. (15), we btain the fllwing equatin f mtin [A ]{u } + [B]{u } + [C]{u} + ([k t1 ]{u} ) + ([k t2 ]{u} ) + ([k b1 ]{u} ) + ([k b2 ]{u} ) = {} (17) The dynamic displacement f the riser in its bttm and tp ends is cnstrained using a hinged supprt. In dynamic analysis the displacement field {u} is interplated as {u} = [N]{d} (18) [N] = [ N 1 N 2 N 3 N 4 N 5 N 6 ] N 1 N 2 N 3 N 4 N 5 N 6 (19) {d} = {u 1 u 1 u 1 v 1 v 1 v 1 u 2 u 2 u 2 v 2 v 2 v 2 } T (2) where {d} is the vectr f ndal displacements f riser s element frm the equilibrium psitin, [N] is the matrix f interplatin functins, where N 1, N 2, N 3, N 4, N 5 and N 6 are the interplatin functins f fifth degree. Using the Galerkin finite element methd [3] n the Eq. (17), the equatin f mtin f a riser s element is btained, and then, by assembling the matrices f all the finite elements, we get the glbal equatin f mtin [M]{D } + [G]{D } + [K]{D} = {} (21) where {D}, {D } and {D } are the vectrs f ndal displacements, ndal velcities and ndal acceleratins f the riser, respectively, and the matrices [M], [G] and [K] express the Mass, Gyrscpic and Stiffness matrix, respectively. 3.1 Linear Free Vibratin Fr the study f linear free vibratin, the nn-linear terms f the stiffness matrix [K] are neglected. The nn-linear terms are thse that cntaining the variables u and v in their frmulatin. Then, the Eq. (21) takes the fllwing frm: [M]{D } + [G]{D } + [K L ]{D} = {} (22) where [K L ] represents the linear stiffness matrix. Equatin (22) has harmnic slutin fr cmplex eigenvalues λ i = α i ± iω i in the frm {D} = {D }e λt. Then the quadratic eigenvalue prblem is btained (λ 2 [M] + λ[g] + [K L ]){D } = {} (23) Slving the Eq. (23), the linear frequencies and mdal frms f SCR are btained. 5 DOI: 1.554/ SBMAC

6 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, Results The riser prperties and ther parameters used in the simulatins are De=.26 m, Di=.2 m, =3 m, ρ r =785 kg/m3, ρ w =125 kg/m3, ρ f =998 kg/m3, E=2.7x1 11 N/m2, ν=.3, Nrmal drag cefficient (C Dn ) and add mass cefficient (C A ) are.7 and 1, respectively. The Axial frce applied at the tp is T =476.2 kn. Table 1 shws the linear fundamental frequencies f the vertical riser btained fr different internal flw speed with current velcity equal t m/s. The results btained in this study are cmpared with the results btained in [1] and [5]. The results are in gd agreement. Table 1: Fundamental frequencies f the vertical riser, ω (rad/s). U (m/s) Me, et al., (1988) Chucheepsakul, et al., (1999) This study Analytical slutin 2-finite elements 2-finite elements In additin, it was studied the effect f the gyrscpic matrix n the values f linear natural frequencies. The quadratic eigenvalue prblem expressed by Eq. (23) has been slved with and withut the gyrscpic matrix. Table 2 present the results btained. Table 2: Effect f the gyrscpic matrix n the fundamental frequencies, ω (rad/s). U (m/s) With gyrscpic matrix Withut gyrscpic matrix [G] [] [G]=[] % errr Fllwing, a SCR has been simulated in rder t study the effects f the bending stiffness n the linear natural frequencies. The characteristics f the simulated riser are the same used in the previus simulatins with the unique additin f that the hrizntal displacement f platfrm is 7 m. Table 3 shws the effects f the elasticity mdulus E and internal flw velcity n the linear fundamental frequencies f SCR. Table 3: Fundamental frequencies f a catenary riser, ω (rad/s). E (kn/m 2 U (m/s) ) x DOI: 1.554/ SBMAC

7 Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, Cnclusins 2.7 x x x x x The present wrk deals with the prblem f free vibratin f catenary risers with internal flw. Results are in gd agreement with results btained by anther wrks. 1. In the analysis f free linear vibratin f catenary risers, with a fixed value fr the axial frce at the tp and withut current, it is cncluded that the natural frequencies decreases with increasing the velcity f the internal fluid. Simulatins were als cnducted t study the effects f the mdulus f elasticity ver the natural frequencies f catenary risers, cncluding that fr a cnstant velcity f the internal fluid, the natural frequency increases with the mdulus f elasticity E. 2. Simulatins were cnducted in linear vibratins withut cnsidering the effect f the gyrscpic matrix. By cmparing the natural frequencies btained with and withut the gyrscpic matrix, we can cnclude that fr lw values f internal fluid velcity, the effects f the gyrscpic matrix are negligible. Acknwledgements The authrs gratefully acknwledge the financial supprts by CAPES Crdenaçã de Aperfeiçament de Pessal de Nível Superir and UFABC University Federal f the ABC. References [1] S. Chucheepsakul, T. uang and T. Mnprapussrn, Influence f transprted fluid n behavir f an extensible flexible riser/pipe, Prceedings f the 9th Internatinal Offshre and Plar Engineering Cnference, Brest, France (1999). [2] S. Chucheepsakul, T. Mnprapussrn and T. uang, Large strain frmulatins f extensible flexible marine pipes transprting fluid, Jurnal Fluids and Structures 17, Arlingtn, USA (23). [3] D. Ck, Cncepts and applicatins f finite element analysis, 4th Ed., Jhn Wiley & Sns, New Yrk, USA (22). [4] T. uang, Kinematics f transprted mass inside risers and pipes, Prceedings f the 3rd Internatinal Offshre and Plar Engineering Cnference, Singapre (1993). [5] G. Me and S. Chucheepsakul, The effect f internal flw n marine risers, Prceedings f the 7th Internatinal Offshre Mechanics and Artic Engineering Cnference, ustn, USA (1988). [6] C. Sparks, The influence f tensin, pressure and weight n pipe and riser defrmatin stresses, ASME Jurnal f Engineering Mechanics 11, France (1984). 7 DOI: 1.554/ SBMAC