Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation

Journal of Natural Gas Science and Engineering xxx (2015) 1e9

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Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation Yanbin Wang*, Deli Gao**, Jun Fang MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 July 2015 Received in revised form 27 October 2015 Accepted 29 October 2015 Available online xxx

This paper presents a coupled dynamic analysis of a marine riser under combined forcing excitation and parametric excitation. The forcing and parametric excitation are induced by lateral sea wave and current, and time-varying heave motion of a floating vessel respectively. Firstly, the mechanical model and control equations, which can be solved by Runge-Kutta method, have been established. Then, the Mathieu instability region has been figured out by perturbation method through transforming the control equation into Mathieu equation. On this basis, riser vibration response, bending moment and shear force under the two kinds of actions, sea current (Case I) and combination action of sea current and sea wave (Case II), have been discussed respectively. Results show that, under these two cases, riser vibration responses are both given priority to the first order of vibration mode. The vibration amplitude of riser bending moment under Case II is bigger than that under Case I. Moreover, riser total shear force is mainly determined by the first order mode under Case I, while under Case II, the influences of the first several order vibration modes on the total shear force are non-ignorable. © 2015 Elsevier B.V. All rights reserved.

Keywords: Deepwater drilling riser Forcing excitation Parametric excitation Vibration response

1. Introduction With the marked development of deepwater drilling technology, riser which connects the float drilling vessel and the subsea wellhead has been applied more and more extensively. Riser is the key equipment to provide a roundtrip channel for drilling mud and takes an important role of supporting auxiliary lines, guiding drilling tools, installing and recycling subsea blowout preventers (BOP). Normally, as shown in Fig. 1-a, riser top and bottom end are connected with the floating vessel and low marine riser package (LMRP) and BOP through top tension system and lower ball joint respectively. Under normal conditions, there are two dynamic excitation sources affecting riser vibration response: the first one, which called forcing excitation, is induced by horizontal sea wave and current forces, and the second one, which called parametric excitation, is due to changes in axial tension resulting from heave motion of a drilling vessel. However, the dynamic behavior of marine riser becomes a combined forcing and parametric excitation problem when the two kind of dynamic excitations are considered

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Y. Wang), [email protected] (D. Gao).

simultaneously. At present, the forcing excitation problem has been widely studied. Robert M. Sexton and L. K. Agbezuge (1976) have developed a computer model to make a dynamic analysis of the riser by calculating the riser stress, deflections and lower ball joint angle. Egeland et al. (1982) have presented some common research methods for riser dynamic analysis. Simmonds D.G (1980) has established a nonlinear equation to analyze riser dynamical response and the equation has been solved by finite difference method. Geir Moe and Bjern Larsen (1997) have developed a differential equation describing the motions of marine riser with asymptotic solution. A. Ertas (2006) has proposed a riser dynamic differential equation solved by the finite difference method. Khan R. A. (2006) has made some dynamic analysis of risers subjected to regular or irregular wave with ABAQUS software and the variation of riser bending stress with low frequency drilling ship movement and wave motion and current velocity have also been analyzed. Wang et al. (2015) have established the mechanical model and control equation for analyzing lateral vibration of marine riser in installation via variational approach. For the parametric excitation problem, early research mainly focuses on marine cable and tension leg of tension leg platform (TLP). Hsu (1975) has studied the parametric excitation of long slender marine structure and analyzed the vibration response for the first time. Chang (1986) have analyzed the vibration response of

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Please cite this article in press as: Wang, Y., et al., Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.038

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Fig. 1. Marine riser system and mechanical model.

tension leg through taking the heave and surge motion of floating body as parametric excitation, and the Mathieu equation has been figured out by Galerkin method. Patel and Park (1995) have obtained the dynamic response of a tension leg under low axial force and the influences of different strength excitation parameters on instability region have been discussed. Han and Benaroya (2000) have established the coupled axial and horizontal dynamic control equation according to the Hamilton principle and Kirchhoff assumption and the equations have been solved by finite difference method. Park and Jung (2002) have presented a 3-D finite element analysis method to study the coupled dynamic vibration of long slender marine structure. Chatjigeorgiou et al. (2004); Chatjigeorgiou and Mavrakos. (2005) have discussed the influence of damping effect on riser stability of parametric excitation through simplifying the heave as harmonic motion. Kuiper et al. (2008) have solved the control equation using Galerkin method to analyze the fluctuation of riser tension force induced by heave motion. Dong et al. (1992); Dong, (1994) have established the Mathieu equation with consideration of the heave motion and analyzed the Morison hydrodynamic, the vortex-induced response and its stability. Lou et al. (2010) have figured out the time history response of riser under parametric excitation and forcing excitation using finite difference method. Yang and Li (2009) have studied the parametric excitation response of riser through simplifying the heave as harmonic motion, but the hydrodynamic load induced by current is his study is neglected. As shown in the published papers, the dynamic response about long slender marine structures mainly focuses on marine cable and tension leg, and the coupled dynamic analysis of marine riser is rare until now. What's more, the axial tension force is regarded as

constant in the above research which is inconsistent with the actual situation. So, in this paper, the mechanical model and control equations have been established to take coupled dynamic analysis of marine riser under combined forcing and parametric excitation with consideration of changes of the axial tension force, which is different from the published papers and more consistent with the actual situation. The remainder of the paper is organized as follows. The overall mechanical model and control equations are described in Section 2. The Mathieu equation and stability analysis of parametric excitation are shown in Section 3. The solution of the control equation is given in Section 4. A case study and corresponding discussion are presented in Section 5 and the conclusions are drawn in Section 6.

2. Mechanical model and control equation As mentioned above, riser top end is connected with the floating vessel through top tension system. The top tension system, which consists of tensioner lines, tensioner rings and telescopic joints, has two impacts: one is to provide top tension force to avoid riser buckling, the other is to avoid large riser axial force. In fact, the top tension system is a spring between riser and the floating vessel, and the spring stiffness is far less than riser axial stiffness. As shown in Fig. 1-b, riser upper and lower boundary conditions can be regarded as hinge constraint respectively. In this paper, the following assumptions are applied during equation deduction: (1) The motion direction of sea wave is the same with sea current.

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(2) Riser geometry characteristic and material properties are kept as constant. (3) Riser curvature and torsion are small and the geometric nonlinearity is neglected. As shown in Fig. 1-b, the origin of coordinate locates at the mud line, the positive direction of y axis is the same with lateral force and the positive directive of axis is perpendicular upward to they axis. Based on the above assumptions and coordinate system, the control equation of riser coupled dynamic can be written as:

EI

v4 yðz; tÞ v2 yðz; tÞ vyðz; tÞ v2 yðy; tÞ þm  ½TðzÞ þ S cos Ut þc vt vz4 vz2 vt 2 ¼ Fy ðz; tÞ

Fr ðz; tÞ ¼

3

  · · ·· ·· 1 pD2 ·· rw CD Do yy þ CA rw o y ¼ Kd CD sgn y y2 þ mi y 2 4 (6)

where, Kd ¼ rwDo/2; CD and CA are damping coefficient and addi· tional mass coefficient respectively, dimensionless; sgnðyÞ equals to · 1 or 1, which is depended ony. Both CD and CL are related with Reynolds number Re and the Keulegan-Car-Penter number K. What's more, CA is related with riser vibration frequency. If riser natural frequency closes to the vortex discharge frequency (under resonance condition), CL ranges from 0.6 to 2.4, CD ranges from 0.4 to 2.0 and CA equals to 1.0. 3. Stability analysis of parametric excitation

(1) where, EI is bending stiffness, N m2; T(z) is riser axial force along water depth, N; S is vibration amplitude of dynamic tension force, N; U is frequency of parametric excitation, rad/s; m ¼ ms þ mi þ ma is vibration mass of riser per length, kg; ms ¼ rs pðD2o  D2i Þ=4 is mass of riser per length, kg; mi ¼ ri pD2i =4 is mass of fluid in riser per length, kg;ma ¼ CA rw pD2o =4 is mass of additional water outside riser per length, kg; rs is density of riser, kg/m3; ri is density of fluid in riser, kg/m3; rw is density of sea water, kg/m3; Do is riser outside diameter, m; Di is riser inner diameter, m. Since riser upper and lower boundary conditions are regarded as hinge constraint, the boundary conditions of Eq. (1) can be written as:

 8 v2 yðz; tÞ > ¼ 0; ¼0 yðz; tÞj  > z¼0 > vz2  < z¼0  >  2 > v yðz; tÞ >  : yðz; tÞj ¼0  z¼L ¼ 0; 2  vz

3.1. Mathieu equation According to vibration mechanics, the solution of Eq. (1) can be written as (Cheng, 1990):

yðz; tÞ ¼

∞ X

fn ðzÞ$qn ðtÞ

(7)

n¼1

where, fn(z) is the vibration mode function, and qn(t) is the time function. After mathematical deduction, the Mathieu equation can be presented by Eq. (8), and the detailed derivation process is shown in Appendix A. ··

qn ðtÞ þ u2n ð1 þ b cos UtÞqn ðtÞ ¼ 0

(2)

z¼L

(8)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 p4 n2 p2 is riser vibration frequency of the n'th Where, un ¼ EIn þ TemL 2 mL4 order, rad/s; Te is riser effective axial tension and can be represented by the tension force at riser middle section, N; b is the strength of parametric excitation, which can be calculated by:

Z

where, L is riser total length, m. The combined hydrodynamic force deduced by sea wave and current can be described as two parts: one is FL(z,t), the vortexinduced lift force due to vortex discharge, the other is Fr(z,t), the fluid damping force due to riser motion toward y axis, which is:

S b ¼  2 $ 0Z un $m

Fy ðz; tÞ ¼ FL ðz; tÞ  Fr ðz; tÞ

According to Park et al. (1999), the single degree of freedom analysis method can be employed to analyze this problem in spite of the multiple degrees of freedom.

(3)

where, FL(z,t) is the harmonic function of vortex discharge frequency, and vortex-induced lift force can be calculated by:

FL ðz; tÞ ¼

1 r Do ðvc þ uÞ2 CL cos us t ¼ KL ðzÞCL cos us t 2 w

(4)

where, KL ðzÞ ¼ 12 rw Do ðvc þ uÞ2 ; vc is the horizontal current velocity, m/s; CL the coefficient of lift force, dimensionless; us is vortex discharge frequency, rad/s; u is the horizontal wave velocity, m/s. According to Zhu (1991), u can be estimated by linear wave theory, which is:

u¼

pH kðzLÞ e cosðky þ uw tÞ Tw

(5)

where, H is wave height, m; Tw is wave period, s; uw is wave frequency, rad/s; k ¼ 2p/Lw is wave number, rad/m; Lw is wave length, m. The fluid damping force can be calculated by Morison equation (Zhu, 1991), which is:

L

00

fn fm dz L

0

f2n dz

¼

n2 p2 S u2n mL

(9)

3.2. Stability analysis The perturbation method can be used to analyzed instability region of Mathieu equation when the time-varying tension is relatively small (b is far less than 1 in Mathieu equation). Generally, the basic principle of perturbation method is to describe q(t) with the power series of b (Joost Brugmans, 2007), which is:

qðtÞ ¼ q0 ðtÞ þ bq1 ðtÞ þ b2 q2 ðtÞ þ b3 q3 ðtÞ

(10)

Substituting Eq. (10) into Eq. (8) and letting the coefficients of the same power of b equal to 0, one obtains:

8 ·· 2 > < q0 ðtÞ þ u q0 ðtÞ ¼ 0 ·· 2 q ðtÞ þ u q1 ðtÞ ¼ u2 cos Ut$q0 ðtÞ 1 > : ·· q2 ðtÞ þ u2 q2 ðtÞ ¼ u2 cos Ut$q1 ðtÞ

(11)

where, u is the natural frequency of heave motion. After solving Eq. (11), the first and second order parametric resonance vibration

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occurs whenU ¼ 2u, and U ¼ u respectively. The specific solving process of Eq. (11) is presented in Appendix B. However, in the actual situation, the first unstable order has the dominant influence on the parametric excitation characteristic. The instability region of Mathieu equation, shown in Fig. 2, can be analyzed according to the method introduced by Feng (Feng, 2010) since the resonant frequency of parametric excitation has been figured out. 4. Solution method According to the boundary condition of Eq. (1), riser vibration response can be written as (Jia and Fang, 1995): ∞ X

npz yðz; tÞ ¼ yn ðtÞsin L n¼1

(12)

Rðz; tÞsin

ipz dz ¼ 0 L

ðj ¼ 1; 2…Þ

(13)

0

The detailed expression of R(z,t) is presented in Appendix C. Eq. (13) is a system of ordinary differential equations. Due to the complexity of damping force in the nonlinear equation, the number of n should be less than 3. In this paper, the following notation is employed:

ZL Dj ¼ Kd CD

· · ipz dz sgn y y2 sin L

 2 np L

T0 m

is the natural frequency of riser axial

vibration, rad/s; T0 is the average value of riser axial tension force, N; a ¼ S/T0; Cn ¼ 2mðl2Bn þ l2Cn Þ1=2 xn is the viscous damping coefficient; xn is the damping ratio. The sea current velocity can be calculated by Eq. (16), which is:

vc ðzÞ ¼ a þ bz

(16)

Substituting Eq. (4), Eq. (5) and Eq. (16) into the right item of Eq. (15), one obtains:

2CL cos us t mL

ZL KL ðzÞsin

npz dz ¼ Ac cos us t þ Aw cos 2uw t cos us t L

0

þ 2Acw cos uw t cos us t ðn ¼ 1; 2; 3Þ

Substituting Eq. (3), Eq. (4), Eq. (6) and Eq. (12) into Eq. (1), one obtains:

ZL

vibration, rad/s; l2Cn ¼

ðj ¼ 1; 2; …; nÞ

(14)

0

P · · · Where, y ¼ yðz; tÞ ¼ ni¼1 yi ðtÞsin ipz L , and Eq. (14) can be solved by numerical integration method. If n ¼ 3, the discretization format of Eq. (13) can be written as:

h i ·· Cn · 2K C yn þ d D Dn yn þ l2Bn þ l2Cn ð1 þ a cos utÞ yn þ m pm ZL 2C npz dz ðn ¼ 1; 2; 3Þ ¼ L cos us t KL ðzÞsin L mL

(15)

(17) In Eq. (17), the first right item stands for vortex excitation force generated by sea current, the second item is caused by sea wave and the third item represents coupling vortex-induced force resulted from the combination action of sea wave and current, where Ac, Aw and Acw are presented in Appendix C. 5. Case study and discussion After solving Eq. (15) with Runge-Kutta method, two different cases have been discussed in this paper: one is the vibration response caused only by sea current (Case I), and the other is the vibration response due to the combination action of sea wave and current (Case II). According to the results of Section 3.2, we have U ¼ 2u when the first order of parametric excitation occurs. Other calculation parameters are shown in Table .1. When the first order resonance occurs, the first three orders of riser dynamic response, y1(t), y2(t) and y3(t), are shown in Fig. 3. As shown in Fig. 3, the maximum value of y1(t) is far bigger than that of y2(t) and y3(t), which indicates that the first order vibration mode has the dominant influence on riser response and the impacts of higher order modes are slight. According to Eq. (12), riser vibration bending moment can be calculated by:

0

Where, l2Bn ¼

 2 np L

EI m

is the natural frequency of riser flexural

v2 y vz2   2 EIp pz 2pz 3pz þ 4y2 ðtÞsin þ 9y3 ðtÞsin ¼ 2 y1 ðtÞsin L L L L

Mðz; tÞ ¼ EI

(18) Eq. (19) is satisfied at the top and bottom end of riser, which is:

z ¼ 0; z ¼ L

(19)

Substituting Eq. (19) into Eq. (18), Mjz¼0 ¼ 0 and Mjz¼L ¼ 0 are satisfied. Whereas, in the middle of the riser (z ¼ L/2), one obtains:

Mðz; tÞ ¼

Fig. 2. Mathieu instability chart.

EIp2 ½y1 ðtÞ  9y3 ðtÞ L2

(20)

The fluctuation of riser dynamic bending moment is shown in Fig. 4, which shows that the bending moment at z ¼ L/2 is similar with the first dynamic response mode. Fig. 5 is the shear force distribution, which also shows the total shear force is mainly caused by the first order mode. When the wave height is 1/20e1/10 of the wave length, Aw and Acw is proportional to Ac. If the wave frequency equals to vortex induced frequency (Lw ¼ 85 m), the order of magnitudes of is Aw

Please cite this article in press as: Wang, Y., et al., Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.038

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Table 1 Calculation parameters. Sea current Sea wave Riser Density Heave motion Tension force Damping coefficient

a ¼ 0.4m/s uw ¼ 0.5938rad/s L ¼ 1500m rs ¼ 7850kg/m3 l1 ¼ 0.869rad/s S¼1.693MN xn ¼ 1.8  103

b ¼ 1.2857  103m/s H ¼ 10m Do ¼ 0.5334m rw ¼ 1025kg/m3 l2 ¼ 1.817rad/s a ¼ 0.4

Lw ¼ 175m Di ¼ 0.4826m rm ¼ 1200kg/m3 l3 ¼ 2.910rad/s

Fig. 3. The first three orders of modal dynamic response when CL ¼ 2.4, CD ¼ 0.6, CA ¼ 1.0 and U ¼ 1.74 rad/s.

Fig. 4. Dynamic bending moment at z ¼ L/2 under Case I.

Fig. 5. Shear force distribution under Case I.

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and Acw is about 105 ~ 103, which is far less than Ac and can be ignored. The first three orders of riser dynamic response mode under the combined excitation is shown in Fig. 6. As shown in Fig. 6, the vibration amplitude, phase angle and vibration frequency changes complexly, which indicates the nonlinear characteristic of riser vibration response is more significant under the combined excitation comparing to Fig. 5. The specific reason for riser nonlinear characteristic under combined excitation is shown in Appendix C. The dynamic bending moment distribution of riser under the combined excitation is shown in Fig. 7, which shows that the dynamic bending moment is also similar with the first dynamic response mode. However, the bending moment under Case II is bigger than that under Case I. However, all of the first three dynamic response modes have significant influence on the total shear force, as shown in Fig. 8. So, more vibration modes should be considered when calculate the shear force. Fig. 7. Dynamic bending moment at z ¼ L/2 under Case II.

6. Conclusions (1) The mechanical model and corresponding control equations analyzing the dynamic behavior of marine riser under combined forcing and parametric excitation have been established in this paper. (2) The Mathieu instability region has been figured out through transformation of the control equation into Mathieu equation using perturbation method, and the control equations have been solved by Runge-Kutta method.

(3) Riser vibration responses are given priority to the first order vibration mode, and the vibration amplitude of dynamic bending moment under the combination action of sea wave and current is bigger than that under the single action of sea current. (4) The total shear force of riser under the single action of sea current is mainly caused by the first order mode, while under the combination action of sea wave and current, the first

Fig. 6. The first three orders of modal dynamic response or riser under the combined excitation when uw ¼ 0.4345 rad/s, H ¼ 20 m and Lw ¼ 327 m.

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Y. Wang et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e9 ∞  X n¼1

7

  ·· cos Ut 00 Sf ðzÞqn ðtÞ ¼ 0 qn ðtÞ þ u2n qn ðtÞ fn ðzÞ  m

(A-3)

Multiplying Eq. (3) by fm(z) and doing integral from z ¼ 0 to z ¼ L with consideration of the orthogonality of vibration mode, one obtains:

 ··  ZL qm ðtÞ þ u2m qm ðtÞ f2m ðzÞdz 0

2 L 3 Z ∞ cos Ut X 00 4 Sf ðzÞfm ðzÞdz5qn ðtÞ ¼ 0  n m n¼1

(A-4)

0

Z Divided Eq. (A-4) by

L 0

·· qm

Fig. 8. Shear force distribution under Case II.

f2m ðzÞdz, one obtains:

S cos Ut ðtÞ þ u2m qm ðtÞ  m

Z

L

0

Z

L 0

several order vibration modes have non-ignorable influence on the total shear force. Acknowledgments The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (NSFC, 51221003, U1262201). This research is also supported by other projects (Grant numbers: 2013AA064803, 2011ZX05009-005).

According to the boundary condition of Eq. (2), the vibration mode function can be written as (Tang, 2008):

np  z fn ¼ sin L

(A-6)

Substituting Eq. (A-6) into Eq. (A-5), one obtains: ·· qn

ðtÞ þ u2n ð1 þ b cos UtÞqn ðtÞ ¼ 0

(A-7)

Eq. (A-7) is the Mathieu equation.

The derivation process of Mathieu equation is shown as follows: The stability of parametric excitation is depended on the system vibration characteristic and has nothing with damping effect (Feng, 2010). Substituting Eq. (7) into the riser vibration equation and neglecting the influence of the axial tension force changes, one obtains:



¼ 0; n ¼ 0; 1; 2; 3…

f2n dz (A-5)

Appendix A

d4 fn ðzÞ EI dz4

00

fn fm dz

d dfn ðzÞ T  mu2n fn ðzÞ ¼ 0 dy dz

(A-1)

Substituting Eq. (7) into Eq. (1) and neglecting the damping effect, one obtains:

Appendix B The resonance frequency of parameter excitation is solved by the following derivation: The general solution of the first equation of Eq. (11)is:

q0 ðtÞ ¼ A0 sin ut þ B0 cos ut

(B-1)

Substituting Eq. (B-1) into the second equation of Eq. (11), one obtains: ··

q1 ðtÞ þ u2 q1 ðtÞ ¼ u2 cos UtðA0 sin ut þ B0 cos utÞ

(B-2)

The general solution of Eq. (B-2) is:

q1 ðtÞ ¼ A1 sin ut þ B1 cos ut þ

∞  X

u2 ðB0 ð2u þ UÞcosðUt  utÞ  A0 ð2u þ UÞsinðUt  utÞ þ B0 cosðUt þ utÞÞ

 0 v  ðTðzÞ þ S cos UtÞfn ðzÞ qn ðtÞ vz n¼1  ·· þ mqn ðtÞfn ðzÞ ¼ 0

8Uu2 þ 2U3

0000

EIf ðzÞqn ðtÞ 

(A-2)

Substituting Eq. (A-2) into Eq. (A-1), after complex mathematical deduction, one obtains:

(B-3)

In Eq. (B-3), if 8Uu2 þ 2U3 ¼ 0 (U ¼ 2u), the riser vibration displacement (q1(t)) is infinite, which indicates that the first order resonance occurs. Substituting q0(t) and q1(t) into the third equation of Eq. (11) and solving the equation, one obtains:

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q2 ðtÞ ¼ A2 sin ut þ B2 cos ut þ þ

B0

u5 ððu

u2 ðB1 ð2u þ UÞcosðUt  utÞ  A1 ð2u þ UÞsinðUt  utÞ þ ðU  2uÞcosðUt þ utÞÞ 8Uu2 þ 2U3

þ UÞcosð2Ut  utÞ  ðU  uÞcosð2Ut þ utÞÞ

(B-4)

32U2 u4  40U4 u2 þ 8U6

Similarly, in Eq. (B-4), if 32U2u4  40U4u2 þ 8U6 ¼ 0 (U ¼ u), the second order resonance occurs. Appendix C a: The detailed expression of parameters in Eq. (13) and Eq. (17) are:

Rðz; tÞ ¼

∞   np2 X np2 EI yn ðtÞ þ ðTðzÞ þ S cos UtÞ yn ðtÞ L L n¼1  · ·· npz þ cyn ðtÞ þ myn ðtÞ sin L " " #2 # ∞ ∞ X X · · npz npz þ KD CD yn ðtÞsin $sgn yn ðtÞsin L L n¼1 n¼1

Bn ¼

np L

np

L  2 ; B n ¼ h  2 i 2 np 2 2 k þ np k þ L L

(C-5)

b: The reason why the nonlinear characteristic of the vibration response is more significant under Case II is explained as follows: The right end of Eq. (17) can be expressed with trigonometric function, which is:

 KL ðzÞCL cos us t (C-1)

(  " 8  2  3 #) 3 rw Do CL L L > 2 2L 2 L > > þ 2ab a þ b  4 > < mL p p p p Ac ¼   2  3  > > > r D C L L > :  w o L ab þ b2 mL p 2p

ðn ¼ 1; 3Þ (C-2) ðn ¼ 2Þ

Ac cos us t þ ðAw cos 2uw t þ 2Acw cos uw tÞcos us t np 8  h i > r D C pH 2 > 2kH 2kL L < w o L e þ e   Tw mL np 2 Aw ¼ 4k2 þ > > L : 0

ðn ¼ 1; 3Þ

(C-6) ðn ¼ 2Þ (C-3)

Acw

1 ¼ Ac cos us t þ Aw ½cosð2uw þ us Þt þ cosð2uw  us Þt 2 þ Acw ½cosðuw þ us Þt þ cosðuw  us Þt As can be seen from Eq. (C-6), excitation frequency consists of three parts: us, uw þ us and uw  us.

8  h i h  io r D C pH 2 n > > > w o L aBn ekH þ ekL þ b Bn LekH  2kBn ekH þ ekL < Tw mL ¼ > > rw Do CL pH > : Bn lekH mL Tw

ðn ¼ 1; 3Þ (C-4) ðn ¼ 2Þ

Please cite this article in press as: Wang, Y., et al., Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.038

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References Chang, Yi Long, 1986. Nonlinear Analysis of Tension Leg Platform. University of California, Berkeley, pp. 32e56. Chatjigeorgiou, L.K., 2004. On the parametric excitation of vertical elastic slender structures and the effect of damping in marine applications. Appl. Ocean Res. 26, 23e33. Chatjigeorgiou, L.K., Mavrakos, A., 2005. Nonlinear resonances of parametrically excited risers numerical and analytical investigation for U¼2u1. Comput. Struct. 83 (8e9), 560e573. Cheng, Yaodong, 1990. Mechanical Vibration. Zhejiang University Press, Hangzhou, pp. 212e240. Dong, Yanqiu, 1994. Vortex-induced nonlinear vibration analysis of tension leg subjected to sea wave and current. ACTA Oceanol. SIN. 16 (3), 121e129. Dong, Yanqiu, Jack, Y., Loa, K., 1992. A study on stability of tension leg platform nonlinear oscillation induced by vortex shedding. Shipbuild. China 4, 85e97. Egeland, O., Wiik, T., Natvig, B.J., 1982. Dynamic analysis of marine riser. In: Offshore South East Asia 82 Conference. SPE-10420-MS. Ertas, A., 2006. Numeric solution techniques for dynamic analysis of marine riser. J. Energy Resour. Technol. 109e111. Feng, Limei, 2010. The Study on the Parametric Vibration for Top-tension Riser in Deep-sea. Tianjin University, Tianjin, pp. 28e42. Han, S.M., Benaroya, H., 2000. Nonlinear coupled transverse and axial vibration of a compliant structure 1: formulation and free vibration. J. Sound Vib. 237 (5), 837e873. Hsu, C.S., 1975. The response of a parametrically excited hanging string in fluid. J. Sound Vib. 39, 305e316. Jia, Xinlan, Fang, Huacan, 1995. Dynamic response of riser system in marine drilling. CPM 23 (8), 18e22. Joost, Brugmans, 2007. Parametric Instability of Deep-water Risers. The Master's Thesis of Delft University of Technology, pp. 35e78.

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Khan, R.A., 2006. Nonlinear dynamic analysis of deep water marine risers. In: Proceedings of the 2nd International Congress on Computational Mechanics and Simulation, Cuwahati, India. Kuiper, G.L., Brugmans, J., Metrikine, A.V., 2008. Destabilization of deep-water risers by a heaving platform. J. Sound Vib. 310, 541e557. Lou, min, Dong, Wenyi, Wang, Teng, 2010. Study on dynamic response in floating system with pitching of Marine and sway riser motions. China Offshore Platf. 25, 14e18. Moe, Geir, Larsen, Bjern, 1997. Dynamics of deep water marine risers-asymptotic solutions. In: International Offshore and Polar Engineering Conference, Honolulu, USA. Park, H.I., Jung, D.H., 2002. A finite element method for dynamic analysis of long slender marine structures under combined parametric and forcing excitations. Ocean. Eng. 29, 1313e1325. Park, H.I., Jung, Dong-ho, Piao, Chun-jun, 1999. 3-D Numerical Analysis of a Long Slender Marine Structure under Combined Axial and Lateral Excitation. International society of offshore and polar engineering. ISOPE-I-99e159. Patel, M.H., Park, H.I., 1995. Combined axial and lateral responses of tensioned buoyant platform tethers. Eng. Struct. 17, 687e695. Sexton, Robert M., Agbezuge, L.K., 1976. Random wave and vessel motion effects on drilling riser dynamics. OTC 2650, 391e398. Simmonds, D.G., 1980. Dynamic Analysis of the Marine Riser. SPE 9735-MS. Tang, yougang, 2008. Dynamics of Ocean Engineering Structure. Tianjin University Press, Tianjin, pp. 160e169. Wang, Yanbin, Deli, Gao, Fang, Jun, 2015. Study on lateral vibration analysis of marine riser in installation-via variational approach. J. Nat. Gas Sci. Eng. 22, 523e529. Yang, Hezhen, Li, Huajun, 2009. Vibration analysis of deep-sea risers under parametric excitations. J. Vib. Shock 28 (9), 65e69. Zhu, Yanrong, 1991. Ocean Engineering Wave Mechanics. Tianjin University press, pp. 56e89.

Please cite this article in press as: Wang, Y., et al., Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.038