Bifurcation analysis of vortex induced vibration of low

Noname manuscript No.(will be inserted by the editor)

Bifurcation analysis of vortex induced vibration of lowdimensional models of marine risers

Dan Wang · Zhifeng Hao · Ekaterina Pavlovskaia · Marian Wiercigroch

Received: date / Accepted: date

Abstract A low dimensional model of a top tensionedriser (TTR) under excitations from vortices and time-varying tension is proposed, where the van der Pol wake

oscillator is used to simulate the loading caused bythe vortex shedding. The governing partial differen-tial equations describing the fluid-structure interactions

are formulated and multi-mode approximations are ob-tained using the Galerkin projection method. The onemode approximation is applied in this study and two

different resonances are investigated by employing themethod of multiple scales. They are the 1:1 internal re-sonance between the structure and wake oscillator (al-

so known as ’lock-in’ phenomenon) and the combined1:1 internal and 1:2 parametric resonances. Bifurcationsunder the varying nondimensional shedding frequency

for different mass-damping parameters are investigatedand the results of multiple scale analysis are comparedwith direct numerical simulations. Analytical respons-

es are calculated using the continuation method andtheir stability is determined by examining the eigenval-ues of the corresponding characteristic equations. Ef-

fects of the system parameters including the ampli-tude of the tension variation, vortex shedding frequencyand mass-damping parameter on the system bifurca-

tions have been investigated. The analytical approachhas allowed to probe bifurcations occurring in the sys-tem and to identify stable and unstable responses. It isshown that the combined resonances can induce large-

Dan Wang · Zhifeng HaoSchool of Mathematical Sciences, University of Jinan, Ji-nan 250022, China, E-mail: [email protected],[email protected]

Ekaterina Pavlovskaia · Marian WiercigrochCentre for Applied Dynamics Research, School of Engineer-ing, University of Aberdeen, King’s College, Scotland, UK, E-mail: [email protected], [email protected]

amplitude vibration of the structure. Counter intuitive-ly, the amplitude of such responses increases rapidlyas the amplitude of the tension variation grows. Com-

parisons between the analytical and numerical resultsconfirm that the span of the system vibration can beaccurately predicted analytically with respect to the ob-

tained response amplitudes of responses. The proposedmulti-mode approximation and presented findings ofthis study can be used to enhance design process of

top tension risers.

Keywords Vortex-induced vibration · wake oscillator ·parametric resonance · bifurcation diagram · marine

riser model

1 Introduction

Long slender structures and their dynamic responsecharacteristics have attracted significant fundamentaland practical interests in the recent years. With the

development of technology, risers have been used foroil and gas exploration and production in deep water-s. The working conditions of risers are very compli-

cated in deep waters, as sea waves and currents canoccur simultaneously. Vortex-Induced Vibration (VIV)and parametric resonances are two main phenomena en-

countered in practice that can induce large-amplitudevibration of risers. Both phenomena have significanteffect on the lifetime and safety of the structures but

their mechanisms are still not entirely clear.VIV phenomena arise from the interactions between

the structures and fluid flow around them and forced

and self-excited vibration are typically encountered. Whenthe vortex shedding frequency approaches one of thenatural frequency of the riser, the ’lock-in’ phenomenon

can occur, which if it is persistent may induce fatigue

2 Dan Wang et al.

damage of the structure. A number of comprehensive

reviews [1–5] discuss earlier studies of VIV were fo-cussed on the multi-mode response, VIV response athigh Reynolds numbers, flow-induced vibrations between

multiple marine risers, VIV of inclined risers and inter-mittent VIV in oscillatory flows, impacts of the floatingplatforms and internal flows, and other effects.

Experimental [6–11] (e.g. the full-scale, the small-

scale, the model test) and numerical methods [12–19](e.g. Finite Element, Computer Fluid Dynamics, force-decomposition, time-domain simulation methods) are

used to analyse the dynamic responses of the risers.Effects of tensions on the in-line (IL) and cross-flow(CF) vibration amplitude and frequency responses, sup-pression of vibration, and hydrodynamic force coeffi-

cients were studied by Sanaati and Kato [20]. An ex-periment performed by Chen et al. [21] on the VIV ofa flexible cable in an oncoming shear flow has shown

that the cable model produced single-mode and multi-mode VIV under different oncoming velocities. They al-so found that amplitudes of displacement of the single

mode VIV were larger than those of multi-mode VIV,and the cross-flow (CF) response was larger than thatof in-line (IL) direction in both cases of single mode

and multi-mode approximations. The flow-induced vi-bration (FIV) of an elastically mounted circular cylin-der with different mass-damping parameters m∗ζ was

studied in a wind channel by Hu et al. [22]. These resultsindicated that the vibration amplitude and region weregradually decreasing for the increasing values ofm∗ζ. In

fact, effects of the mass-damping parameter m∗ζ on theresponses of risers have been studied previously bothusing experimental models [23,24] and test models [25,

26]. Effect of mass ratio on the vortex-induced vibra-tions of a top tensioned riser was investigated by Fu etal. [27] using a numerical method. An inverse method

to obtain the hydrodynamic forces in the CF and ILdirections of a flexible riser undergoing VIV based onmeasured strain was proposed by Song et al. [28]. A

cylinder with low mass and damping was considered byKonstantinidis et. al [29], where an instantaneous phasebetween the cylinder motion and the fluid forcing was

calculated.

The cost of the experimental and numerical analy-sis of the 3-D model of the risers coupled with a com-plex flows is high and often prohibitive. Hence some

reduced order models were introduced to simulate thetime-varying characteristics of fluid flow using wake os-cillators such as the van der Pol oscillator, the Rayleigh

oscillator, and others. The low-order models for thecoupling between an elastically supported cylinder andwake oscillator were analysed by Facchinetti et al. [30],

where displacement, velocity and acceleration couplings

between the structure and the wake oscillator were in-

troduced and analysed in detail. A semi-empirical the-oretical model consisting of structural nonlinear equa-tions of coupled motions in three directions (cross-flow,

in-line and axial) for VIV was proposed in [31], in whichthe vortex-induced hydrodynamic lift and drag forceswere simulated by two distributed and coupled wake

oscillators. Due to the fact that wake oscillator modelsare semi-experimental, calibrations of fluid (namely co-efficients of lift and drag force, mass-damping parame-

ter, strength of nonlinear terms) are essential to explainthe phenomena captured in the experiments. The cali-brations and comparisons of the VIV of rigid cylinders

on elastic supports were investigated by Postnikov et al.[32] and Kurushina et al. [33], and a similar approachwas employed for flexible structures by Kurushina et al.

[34]. Multi-modes responses of vortex-induced vibrationof a slender pipe with two configurations were investi-gated by Pavlovskaia et al. [35], where the time varyinglift force of vortices was modelled as the van der Pol

oscillator. A new single wake oscillator with frequencydependent coupling for the model of VIV was proposedby Ogink and Metrikine [36], which was able to mimic

observations from both the free and forced vibration ex-periments. Then the model was further improved as thenonlinear coupling wake oscillator by Qu and Metrikine

[37].

The external excitation resulting from the timevarying tension on the risers, especially for the top ten-sioned risers (TTRs) connecting the floating platform

and the wellhead, can produce parametric resonances.TTRs can be subjected to the waves as well as currentsin the real sea environments. Relative motions between

the surface waves and floating platform can induce thefluctuations of the tension for the risers, which thencan induce large-amplitude vibration of risers. Hsu [38]

first analysed the marine cable parametric resonance in1975. Some theoretical analyses at the early stage fo-cussed on the nonlinear parametric responses of risers

or tensioned cable legs. Nonlinear dynamic responsesin the transverse direction of vertical marine risers or atensioned cable legs subjected to parametric excitation

at the top of the structure were studied by Chatjigeor-giou and Mavrakos [39], and results indicated that thetransverse motion was dominated by the first mode of

vibration, which was a subharmonic with respect to theparametric excitation. The 1:2 parametric excitation ofa slender pipe conveying fluid for marine applications

was analysed in [40], and the results indicated that un-der specific conditions, the dynamics of the structurein transverse direction can be described by the coupled

Mathieu-Duffing oscillator.

Bifurcation analysis of vortex induced vibration of low dimensional models of marine risers 3

With the development of technologies, impacts of

the parametric excitation on the vibration of risersattract more and more attention as shown by the con-ducted experimental and numerical studies. The effect

of top tension on dynamic behaviour of deep-sea riser-s was investigated numerically and experimentally byZhang et al. [41], and results showed that the vibration

displacement for TTR increases and the bending stressdecreases as the top tension increases. The experimentof VIV for a vertical variable-tension riser was conduct-

ed by Li et al. [42], and it was shown that the dominantfrequencies of risers in IL and CF directions, decreasegradually as the top tension is increased for a fixed flow

velocity.

As experimental studies are expensive, various nu-merical simulations have been performed to analyse the

dynamic responses of a TTR subjected to the varyingtension during the VIV. Srinil [43] studied the VIV ofthe variable-tension vertical risers, and analysed effects

of shear and tensioned-beam parameters (tension versusbending). Yang and Xiao [44] investigated the effect-s of the key design parameters on the dynamic prop-

erties of a TTR under the combined parametric andvortex-induced excitations. The parametric instabilityof a top-tensioned riser (TTR) in irregular waves was

predicted for multi-frequency excitation in [45]. A cou-pled dynamic analysis of a marine riser under combinedforce excitation and parametric excitation was carried

out by Yang et al. [46]. Results showed that responses ofthe first vibration mode were dominant under these twocases. Effects of tension variations on the VIV response

were investigated in [47] by a numerical simulation, andresults showed the VIV responses contain several modesand the dominating mode can vary with time. Numeri-

cal simulations of VIV for a vertical riser which issinusoidally excited at its top end in both one and twodirections in still water were carried out in [48]. Results

revealed low-frequency oscillations at higher modes canbe induced when the current speed varies along thespan and the riser was excited at its top. The vibration

amplitude of riser bending moment under combinationaction of sea currents and waves was bigger than thatunder sea current condition. Moreover, the riser total

shear force has main relation with the first order modeunder the sea current condition. A numerical methodbased on the force-decomposition model was proposed

by Yuan et al. [49] to investigate the cross-flow VIVresponse characteristics with time-varying tension. AVIV dynamic model of a marine riser transporting fluid

subjected to top harmonic tension in the cross flow wasinvestigated in [50] by Finite Element Method. Resultsrevealed that the excitation frequency of the harmonic

tension near twice fundamental frequency of the ri-

ser had an important role in the displacement, and the

displacements and stresses can increase as the harmon-ic tension amplitude increase. The study of horizon-tal parametric vibration of a compliant vertical access

riser resulting from heave motion in the platform byLou et al. [51] showed that the parametric excitationcan occur mainly in first-order unstable regions. Mode

coupling may have caused parametric excitation in theleast stable regions. A parametrically excited top ten-sioned riser model subjected to simultaneous stochastic

waves and vortex excitations was proposed in [52], andone of results showed more complicated and differentdynamic responses can occur when the riser system is

subjected to multi-frequency excitations. An improvedtime domain prediction model was proposed to simu-late the riser subjected to axial parametric excitations

by Gao et al. [53], and the numerical methods were usedto study the responses for different cases. The resultsshowed that vibration displacements of the riser werelarger than those in the case without vessel heave mo-

tion. The stochastic response of a marine riser subjectedto parametrical and external excitations was studied byZhu et al. [54] with the path following method.

The cross-flow VIV response of a top tension riser

under different flow fields was comprehensively studiedin [55], where it was shown that the VIV responses ofthe riser exhibit obvious multi-modal characteristics.

In real life working conditions, risers can experiencea combined vortex shedding and parametric excitation

at the same time, which can has a significant impact onthe safety and lifetime of risers. However, there arestill many design challenges for top tensioned risers [56].

Studies on the combined resonances are limited, espe-cially the theoretical study that would help to under-stand the mechanism of interactions between fluid and

structure. In order to investigate the mechanisms of thecombined resonances of risers, the reduced theoreticalmodels of a TTR excited by the vortices and varying

tension are proposed. The dynamic response character-istics are analysed for different values of mass-dampingparameter.

The content of the paper is organized as follows.The mathematical models of a TTR under the com-

bination of vortex shedding and parametric excitationare constructed in Section 2, where the wake oscilla-tor is used to model the time-varying characteristic of

the fluid flow. The first-mode equations of motion arederived by the Galerkin projection. In Section 3, themultiple scale method is used to analyse the dynamic

system. The VIV phenomena and the combined vibra-tion resulting from vortex shedding and parametric ex-citation, where dynamic responses computed analyti-

cally and verified by direct numerical simulation, are

4 Dan Wang et al.

studied in Section 4 and 5, respectively. Finally, the

conclusions are drawn in Section 6.

2 Coupled model of structure and wake

oscillator

The nonlinear models of a TTR under combined fluid

flow and parametric tension variation excitation are p-resented in this section. Here, the wake oscillatorapproach is employed to simulate the time-varying cha-

racteristics of the vortex shedding. The partial differ-ential equations of motion for fluid-structure interac-tions are derived and discretized using the Galerkin

method. The reduced order coupled models of struc-ture and wake oscillator are then obtained.

2.1 Model of structural vibration

The working conditions for a TTR are complex because

currents and waves can occur simultaneously as shownschematically in Fig. 1(a), where the TTR is connectedto a floating platform and a wellhead. Herein, a TTR

is modeled as a uniform Euler-Bernoulli beam simplysupported at both ends as presented in Fig. 1(b), wherethe top pre-tension T0 +∆T cos(ωpt) varying harmoni-

cally is introduced. The pre-tension variation is chosento avoid bottom tension becoming negative in this s-tudy. The current velocity U is assumed to be constant

through the water column (along a riser), and only thetransverse vibration of a riser in cross flow direction isconsidered.

The equation of motion for the transverse cross flowvibration of a uniform Euler-Bernoulli beam with vary-ing tension can be presented as:

m∗∂2w

∂t2+ c

∂w

∂t+

∂2

∂z2

(EI

∂2w

∂z2

)− ∂

∂z

(T (z, t)

∂w

∂z

)= FF (z, t), (1)

where w(z, t) is the cross flow transverse displacement,

z is the axial position, m∗ = (µ+Ca)πρfD

2

4is the to-

tal mass per unit length including structural mass andadded mass of fluid, µ is the mass ratio (which can af-

fect on the range of ’lock-in’ and the peak amplitudeof the VIVs), Ca is the coefficient of fluid added mass,ρf is the density of the displaced fluid, D is the out-

er diameter of the riser, EI is the bending stiffness,c is the structural damping, and FF = FD + FL de-notes the fluid force being calculated as a sum of fluid

added damping and lift force due to the vortex-induced

vibration. T (z, t) is the varying axial tension which is

calculated as

T (z, t) = T0 − (L− z)Ww +∆T cosωpt

= Tb +Wwz +∆T cosωpt,

where T0 and Tb denote the top and bottom constant

tension of the riser, T0 = Tb + LWw, Ww is the wet(apparent) weight of the riser per unit length, L is thelength of the riser that is equal to the depth of water,

∆T is the amplitude of tension variation, and ωp is thefrequency of tension variation.

The lift fluid force acting on the structure are mo-

delled as FL(t, z) =1

2ρfU

2DCL, where CL is the lift co-

efficient. In order to simulate the interactions betweenthe structure and vortices, the time-varying variable

q =2CL

CL0is introduced [30,33,34,57–59], where CL0

is the reference lift coefficient. The lift force then can

be written as

FL(t, z) =1

4ρfU

2DCL0q.

The fluid added damping can be expressed as

FD = −1

2CDρfDU

∂w

∂t,

where CD is the coefficient of fluid added damping, U

is the flow velocity. Therefore, the transverse cross flowvibration of the structure can be described by the fo-llowing equation

m∗∂2w

∂t2+ c

∂w

∂t+ EI

∂4w

∂z4−Ww

∂w

∂z

− (Tb +Wwz +∆T cos(ωpt))∂2w

∂z2

= −1

2CDρfDU

∂w

∂t+

1

4ρfU

2DCL0q. (2)

By introducing the following non-dimensional variables

v =w

D, ζ =

z

L, τ = ω0t,

and parameters

Ωf = 2πStU

D,ΩR =

Ωf

ω0, ω =

ωp

ω0, ξ =

c

2m∗ω0,

β =∆T

Tb, a =

1

4

CDρfD2

m∗, b =

ρfD2CL0

16m∗, r =

Ww

m∗Lω20

,

Eq. (2) can be rewritten as

∂2v

∂τ2+ 2ξ

∂v

∂τ+

EI

m∗L4ω20

∂4v

∂ζ4− r

∂v

∂ζ

− Tb

m∗L2ω20

(1 + β cos(ωτ))∂2v

∂ζ2− rζ

∂2v

∂ζ2

= − 1

πStaΩR

∂v

∂τ+

1

π2St2bΩ2

Rq, (3)


Flexible Joint

Top Tensioned Riser

Floating Platform

Current

Wellhead

Wave

Seabed

w

z

x

y

q

( )a ( )b

T0+ Tcos( )ptw

Fig. 1 (a) Schematic diagram of a TTR in the currents and waves: a TTR connected by the floating platform and wellhead;(b) physical model of the riser under excitation from the currents with velocity U and waves resulting in top varying tension.

where v is the nondimensional cross flow transverse dis-

placement, ζ is the nondimensional axial position, ω0

denotes the reference frequency, ξ is the damping ra-tio, Ωf and ΩR denote the shedding frequency and

nondimensional shedding frequency of vortices, St isthe Strouhal number, β and ω are nondimensional am-plitude and nondimensional frequency of the tension

variation, and r is the wet weight ratio.

2.2 Wake oscillator model

The van der Pol oscillator is widely used to simulate

the lift force acting on the structure [30,59–62], andthe corresponding equation of the wake oscillator canbe written as

∂2q(t, z)

∂t2+ λΩf

(q2(t, z)− 1

) ∂q(t, z)∂t

+Ω2fq(t, z)

=A

D

∂2w(t, z)

∂t2, (4)

where λ is the van der Pol damping and A is the nondi-mensional coupling coefficient. Here the accelerationcoupling is introduced as suggested in [30].

Similarly, Eq. (4) can be nondimensionalized as

∂2q(τ, ζ)

∂τ2+ λΩR

(q2(τ, ζ)− 1

) ∂q(τ, ζ)∂τ

+Ω2Rq(τ, ζ)

= A∂2v(τ, ζ)

∂τ2. (5)

2.3 Galerkin projection of the coupled system

Following earlier studies [33,61,62], an approximate so-lution of the Eqs (3) and (5) is obtained employing the

Galerkin approach. Here the displacement and lift co-efficient are assumed in the following form

v(ζ, τ) =∞∑j=1

vj(τ)vj(ζ), (6)

and

q(ζ, τ) =∞∑j=1

qj(τ)qj(ζ), (7)

where

vj(ζ) = qj(ζ) = sin(jπζ). (8)

Therefore∫ 1

0

qj(ζ)qk(ζ)dζ =

∫ 1

0

vj(ζ)vk(ζ)dζ =

0, j = k12 , j = k

(9)

By substituting Eqs (6-8) into Eq. (3), multiplyingit by vk(ζ), integrating it along the length of the riser,and applying the orthogonality condition Eq. (9), we

obtain

vk(τ) + 2ξvk(τ) +aΩR

πStvk(τ) +

(EIk4π4

m∗L4ω20

+Tbk

2π2

m∗L2ω20

(1 + β cos(ωτ))

)vk(τ)− 2r

∞∑j=1

jπvj(τ)Φjk

+2r∞∑j=1

vj(τ)j2π2Φjk =

bΩ2R

π2St2qk(τ), (10)

6 Dan Wang et al.

where dot denotes differentiation with respect to time

τ , Φjk =∫ 1

0cos(jπζ) sin(kπζ)dζ and

Φjk =∫ 1

0ζ sin(jπζ) sin(kπζ)dζ.

Using the same approach Eq. (5) is transformed intothe following equation

qk(τ) + 2λΩR

∞∑j=1

∞∑m=1

∞∑n=1

qj(τ)qm(τ)qn(τ)Ψjmnk

−λΩRqk(τ) +Ω2Rqk(τ) = Avk(τ), (11)

where

Ψjmnk =

∫ 1

0

sin(jπζ) sin(mπζ) sin(nπζ) sin(kπζ)dζ.

According to previous studies [39–41], a parametricresonance often occurs at the low-order frequency of the

structure. Therefore, in order to analyse the dynamicsof the coupled system, the vibration of the first ordermodes for the riser and wake oscillator are investigated

as follows.

When k = j = 1, Φ11 = 0, Φ11 =1

4. By introducing

ω1 =

√EIπ4

m∗L4+

Tbπ2

m∗L2+

Wwπ2

2m∗L(the natural frequency

of the first-mode of the structure), ωR1 =ω1

ω0, η =

Tbπ2

m∗L2ω20

, a =CD

π(µ+ Ca)and b =

CL0

4π(µ+ Ca), Eq.

(10) becomes

v1(τ) + 2ξv1(τ) +aΩR

πStv1(τ) + ω2

R1v1(τ)

+ηβ cos(ωτ)v1(τ) =bΩ2

R

π2St2q1(τ), (12)

where ωR1 denotes the first nondimensional frequency

of the structure and the reference frequency ω0 is chosenas ω0 = ω1 in the following.

When j = m = n = k = 1, Ψ1111=3

8and the vi-

bration of the first mode equation for the van der Poloscillator can be obtained as

q1(τ) + λΩR

(3

4q21(τ)−1

)q1(τ)

+ Ω2Rq1(τ) = Av1(τ). (13)

Eqs (12) and (13) are one mode approximation ofthe model describing the interactions between the fluid

flows and structure. Although the analysis presented inthis paper is only conducted for this approximation, theoverall procedure of Galerkin discretization described

here allows to obtain the higher mode approximations.The subscripts of v1 and q1 are omitted in the followinganalysis, where variables v and q are used instead to

simplify derivations.

3 Application of multiple scale method

In this nonlinear system which is similar to the Mathieu–van der Pol type oscillators, a rich dynamic behaviour

can occur, especially under the resonance conditions.In order to investigate the qualitative features of thesystem, the multiple scale method [59,63] is used to

analyse possible resonances.By introducing the scaled parameters as CD → εCD

(indicating a → εa), CL0 → εCL0 (indicating b → εb),

λ → ελ, A → εA, β → εβ, ξ → εξ, Eqs (12) and (13)can be expressed as

v + ω2R1v =

εbΩ2R

π2St2q − ε

(2ξ +

aΩR

πSt

)v

− εηβ cos(ωτ)v, (14)

q +Ω2Rq = εAv − ελΩR

(3

4q2−1

)q. (15)

To construct approximate solutions, the method ofmultiple scales in employed in this study assuming two

time scales T0 and T1 (Tn = εnτ, n = 0, 1) so that

v(τ) = v0(T0, T1) + εv1(T0, T1),q(τ) = q0(T0, T1) + εq1(T0, T1).

(16)

Here only first order terms are retained so derivatives

with respect to time τ can be expressed asd

dτ= D0 +

εD1,d2

dτ2= D2

0 + 2εD0D1, Dn =∂

∂Tn(n = 0, 1) and

D20 =

∂2

∂T 20

.

Substituting (16) into Eqs (14) and (15), and col-

lecting terms in powers of ε, one can obtain that:

ε0 :

D20v0 + ω2

R1v0 = 0, (17)

D20q0 +Ω2

Rq0 = 0. (18)

ε1:

D20v1 + ω2

R1v1 = −(2ξ +

aΩR

πSt

)D0v0 − 2D0D1v0

+bΩ2

Rq0π2St2

− ηβeiωτ + e−iωτ

2v0, (19)

D20q1 + Ω2

Rq1 = AD20v0 − 2D0D1q0

− λΩR

(3

4q20−1

)D0q0, (20)

where cos(ωτ) from Eq. (14) is expressed in the complexform in Eq. (19) and i denotes the imaginary number.

The solutions of Eqs (17) and (18) can be expressedas

v0 = V (T1)eiωR1T0 + V (T1)e

−iωR1T0 , (21)

q0 = Q(T1)eiΩRT0 + Q(T1)e

−iΩRT0 , (22)


where V and Q denote the complex conjugates of func-

tions V,Q, which can be expressed in the complex formas

V =1

2a1(T1)e

iθ1(T1), (23)

Q =1

2a2(T1)e

iθ2(T1), (24)

and aj , θj (j=1, 2) denote the amplitudes and phaseangles, respectively.

Substituting the anticipated solutions (21) and (22)

into Eqs (19) and (20), we obtain

D20v1 + ω2

R1v1 = −iωR1

(2ξ +

aΩR

πSt

)V eiωR1T0

− 2iωR1D1V eiωR1T0 +bΩ2

R

π2St2QeiΩRT0

− 1

2ηβV [ei(ω+ωR1)T0 + ei(ωR1−ω)T0 ]

+ iωR1

(2ξ +

aΩR

πSt

)V e−iωR1T0

+ 2iωR1D1V e−iωR1T0 +bΩ2

R

π2St2Qe−iΩRT0

− 1

2ηβV [e−i(ω+ωR1)T0 + e−i(ωR1−ω)T0 ], (25)

D20q1 + Ω2

Rq1 = −ω2R1Ae

iωR1T0V − 2iΩReiΩRT0D1Q

− iλΩ2R

[3

4Q3e3iΩRT0 +

(3

4QQ− 1

)QeiΩRT0

]+ iλΩ2

R

[3

4Q3e−3iΩRT0 +

(3

4QQ− 1

)Qe−iΩRT0

]− ω2

R1Ae−iωR1T0 V + 2iΩRe−iΩRT0D1Q. (26)

Eqs (25) and (26) indicate possible resonances between

the frequencies ΩR, ωR1 and ω. Two types of reso-nances are investigated in the current study. First caseis the 1:1 internal resonance between the frequencies

ΩR and ωR1, and the other case is the combined reso-nance including the internal resonance and parametricresonance.

4 Internal resonance for the VIV

In riser systems, an important ’lock-in’ phenomenonis observed when the frequency of the vortex shed-ding is close to a natural frequency of the riser. There-

fore, the internal resonance between the structure andwake oscillator is investigated first. Here the frequency-amplitude response and bifurcation curves are derived,

hence the effects of parameters can be studied.

4.1 Solvability conditions

First of all we assume that nondimensional frequencies

of wake oscillator and structure are related as ΩR =ωR1 + εσ, where σ is the detuning parameter. Then ta-king into account that T1 = εT0, the secular terms can

be eliminated by satisfying the solvability conditions.Here, complex conjugate terms do not appear as theyare automatically eliminated if the conditions below are

fulfilled [64],

− iωR1

(2ξ +

aΩR

πSt

)V − 2iωR1D1V

+bΩ2

R

π2St2QeiσT1 = 0, (27)

− ω2R1Ae−iσT1V − 2iΩRD1Q

− iλΩ2R

(3

4QQ−1

)Q = 0. (28)

The derivatives of amplitudes V and Q with respectto T1 can be obtained as

D1V = −(ξ +

aΩR

2πSt

)V − ibΩ2

R

2ωR1π2St2QeiσT1 , (29)

D1Q =1

2

ω2R1

ΩRiAe−iσT1V − 1

2λΩR

(3

4QQ−1

)Q. (30)

By substituting (23) and (24) into Eqs (29) and (30),

the first-order differential equations can be obtained af-ter separating the real and imaginary parts:

a1′ = −

(ξ +

aΩR

2πSt

)a1 +

bΩ2Ra2

2ωR1π2St2sinΦ, (31)

θ1′ = − bΩ2

Ra22ωR1π2St2a1

cosΦ, (32)

a2′ =

1

2

ω2R1Aa1ΩR

sinΦ− 1

2λΩRa2(

3

16a22 − 1), (33)

θ2′ =

1

2

ω2R1

ΩRa2Aa1 cosΦ. (34)

where Φ = θ2 + σT1 − θ1 and ′ denotes derivative withrespect to T1. The first-order differential equation of Φcan be derived from Eqs (32) and (34) as

Φ′ = σ +1

2

ω2R1

ΩRa2Aa1 cosΦ+

bΩ2Ra2

2ωR1π2St2a1cosΦ. (35)

The steady-state solutions can be obtained whenletting aj

′ = 0 (j=1, 2), Φ′ = 0 in Eqs (31), (33) and(35). Then eliminating variable Φ from the obtained

equations and using standard equality cos2 Φ+sin2 Φ =

8 Dan Wang et al.

1, the following equations for unknown amplitude a1and a2 are derived:

F1(a1, a2) = 2Aω3R1π

2St2(ξ +

aΩR

2πSt

)a21

−bλΩ4Ra

22

(3

16a22−1

)= 0,

F2(a1, a2) = a21a22

[ξ +

aΩR

2πSt+

1

2λΩR

(3

16a22−1

)]2+σ2a21a

22 −

(1

2

ω2R1

ΩRAa21 +

bΩ2Ra

22

2ωR1π2St2

)2

= 0.

These two equations can be rewritten as scaled para-

meters change back to the original ones, namely

2Aω3R1π

2St2(ξ +

aΩR

2πSt

)a21 −

bλΩ4Ra

22

(3

16a22−1

)= 0, (36)

a21a22

[ξ +

aΩR

2πSt+

1

2λΩR

(3

16a22−1

)]2+

(ΩR − ωR1)2a21a

22 −(

1

2

ω2R1

ΩRAa21 +

bΩ2Ra

22

2ωR1π2St2

)2

= 0. (37)

Eqs (36) and (37) can be solved numerically to findsteady-state amplitudes.

The first-order equations of v1 and q1 can be sim-

plified to

D20v1 + ω2

R1v1 = −ηβ

4a1[e

i((ω+ωR1)T0+θ1)

+ei((ωR1−ω)T0+θ1)]

−ηβ

4a1[e

−i((ω+ωR1)T0+θ1) + e−i((ωR1−ω)T0+θ1)]. (38)

D20q1 +Ω2

R q1 = − 3

32iλΩ2

Ra32e

3i(ΩRT0+θ2)

+3

32iλΩ2

Ra32e

−3i(ΩRT0+θ2) . (39)

Solutions of v1 and q1 can be obtained from Eqs(38) and (39) as

v1 = −ηβa1[ei((ω+ωR1)T0+θ1) + e−i((ω+ωR1)T0+θ1)]

4[ω2R1 − (ω + ωR1)

2]

− ηβa1[ei((ωR1−ω)T0+θ1) + e−i((ωR1−ω)T0+θ1)]

4[ω2R1 − (ωR1 − ω)

2]

, (40)

q1 =3iλa32256

e3i(ΩRT0+θ2) − 3iλa32256

e−3i(ΩRT0+θ2). (41)

Thus the first order approximate solutions of v and

q can be obtained by combining the solutions for (v0,

q0) and (v1, q1)

v ≈ v0 + εv1 = a1 cos(ωR1T0 + θ1)

−εηβa12

[cos((ω + ωR1)T0 + θ1)

ω2R1 − (ω + ωR1)

2

+cos((ωR1 − ω)T0 + θ1)

ω2R1 − (ωR1 − ω)

2

], (42)

q ≈ q0 + εq1 = a2 cos(ΩRT0 + θ2)

−3ελa32128

sin(3(ΩRT0 + θ2)), (43)

where the amplitudes a1 and a2 satisfy Eqs (36) and

(37). As can be seen from Eq. (42), that the parametricexcitation only affects the first order solution v1, andalso a weak effect on the approximation of v. Corre-

spondingly, this has no obvious influence on the dyna-mic response q, representing the van der Pol oscillator.

4.2 Amplitude–frequency curves

As can be seen from Eqs (36) and (37), a1 = 0 anda2 = 0 are the trivial solutions giving zero amplituderesponses. Other solutions can be obtained numerical-

ly using the method presented in [64], and thereforeamplitude-frequency curves can be computed where thestabilities are determined by examining the eigenvalues

of the corresponding characteristic equations [65] forthe dynamical system described by Eqs (31), (33) and(35).

In order to investigate effects of the added mass co-

efficient µ and nondimensional damping coefficient ξ,four sets of system parameters shown in Table 1 are cho-sen to calculate the amplitude–frequency curves. The

other parameters are fixed as CD = 2, CL0 = 0.3, Ca =1, λ = 0.3, A = 12, St = 0.2 [30,33,62].

Figure 2 shows the analytical results obtained fromEqs (36) and (37) as solid curves and the results from adirect numerical integration of Eqs (12) and (13), where

the obtained maximum values of the displacements foreach selectedΩR are presented by circles under differen-t mass-damping parameters. The other parameters are

fixed at η = 0.8, β = 0.001, ω = 0.2, ωR1 = 1. As can beseen from Fig. 2, when increasing the nondimensionalshedding frequency ΩR, the amplitudes a1 and a2 in-

crease slowly at first to ΩR ≈ 0.5, and then grow rapid-ly to the maximum values near the resonance frequencyΩR ≈ 1. As ΩR is increased further, the amplitudes a1and a2 decrease rapidly at first to ΩR ≈ 1.5 and thencontinue to gradually decrease. It should be noted thatthe amplitude of vibration of the structure and wake os-

cillator at the resonance is smaller for larger values of


Table 1 Selected structural parameters with different mass ratios and damping ratios chosen from the previous studies

mass ratio µ damping ratio ξ mass-damping parameter µξ

Sun’s riser model (2014) [26] 1 0.003 0.0030Tsahalis and Jones (1981) [23] 2.45 0.00796 0.0195Yang et al.(2009)[24] 3.87 0.0152 0.0588He’s riser model(2010)[25] 2.9 0.045 0.1305

Am

plitu

de,

Am

plitu

de,

Shedding frequency, R

-0.3 0.0 0.3

-0.3

0.0

0.3

-4 0 4

-6

0

6

0.0 0.5 1.0 1.5 2.0 2.50.0

0.1

0.2

0.3

0.0 0.5 1.0 1.5 2.0 2.5

2.0

2.5

3.0

3.5

4.0

4.5

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

A1

B1

C1

A1

A2

B2

C2

B1C

1A

2B

2C2


Displacement,

Ve

loci t

y,

Wake strength,

Ra

teo

fw

ake

str

en

gth,

Fig. 2 Dynamic responses amplitudes for (a) the structure and (b) the wake oscillator as function of the nondimensionalshedding frequency ΩR obtained analytically (marked by solid curves) and numerically (marked by circles) for different valuesof mass-damping parameters and for fixed other parameters as η = 0.8, β = 0.001, ω = 0.2 and ωR1 = 1. (c) and (d) showphase portraits for the structure and the wake oscillator calculated using Eqs (12) and (13) for µ = 1, ξ = 0.003 and ΩR = 0.6(red), 1.1 (blue) and 1.6 (purple).

the mass-damping parameter. This is illustrated in Figs2(a) and 2(b) by points A1(0.6, 0.0645), B1(1.1, 0.2606),C1(1.6, 0.1993) and A2(0.6, 3.3851), B2(1.1, 4.0621),

C2(1.6, 3.0247), are marked with the black solid circles.

Figs 2(c) and 2(d) present the phase portraits ofsystem computed numerically using Eqs (12) and (13)

for µ = 1, ξ = 0.003 at ΩR = 0.6, 1.1 and 1.6. The ab-scissa of the points A′

i, B′i, and C ′

i, (i = 1, 2), are themaximum amplitude of responses of the structure and

van der Pol oscillator computed for ΩR = 0.6, 1.1 and1.6 respectively.The other parameters A′

1(0.0594, 0),B′

1(0.2709, 0), C′1 = (0.2309, 0) and A′

2 = (2.7645, 0),

B′2 = (4.1665, 0), C ′

2 = (3.7214, 0), show the evolution

of the amplitudes on phase planes (panels (c) and (d)of Fig. 2) used to construct the frequency-amplitudecurved (panels (a) and (b) of Fig. 2). Although the p-

resented comparison demonstrates some differences be-tween the analytical and numerical results, which aremore evident for the frequencies above the resonance,

the presented approach allows to determine the overallshape of the resonance curves and the maximum valuesof the displacement amplitudes reasonably well.

10 Dan Wang et al.

5 Combined resonances of the TTR

As can be seen from Eqs (31)-(34), the parametric ex-citation contributes to the system responses in the first

order approximate solution v1(τ). However, the para-metric resonance can induce large vibration of the struc-ture. In order to understand this phenomenon more

clearly, the internal resonance coupled with the para-metric resonance is investigated in this section. Thefrequency-amplitude response equations and bifurca-

tion curves are constructed and then the effects of pa-rameters on the system responses are studied.

5.1 Solvability conditions

We assume that the following relations between the fre-quencies are satisfed ΩR = ωR1 + εσ1, ω = 2ωR1 + εσ2,

where σ1 and σ2 are the detuning parameters. Then thesolvability conditions can be obtained from Eqs (25)and (26) in the same way as before,

− iωR1

(2ξ +

aΩR

πSt

)V − 2iωR1D1V

+bΩ2

R

π2St2Qeiσ1T1 − ηβ

2V eiσ2T1 = 0, (44)

− ω2R1Ae

−iσ1T1V − 2iΩRD1Q

− iλΩ2R

(3

4QQ−1

)Q = 0. (45)

By re-arranging these equations, one can obtain

D1V = −(ξ +

aΩR

2πSt

)V − ibΩ2

R

2ωR1π2St2Qeiσ1T1

+iηβ

4ωR1V eiσ2T1 , (46)

D1Q =1

2

ω2R1

ΩRiAV e−iσ1T1 − 1

2λΩR

(3

4QQ− 1

)Q. (47)

Here functions V and Q can also be expressed as beforeby Eqs (23) and (24) where the unknown amplitudes a1and a2 will now satisfy new set of the equations (Eqs

(54) and (55)) derived later in this section. Substitut-ing the complex variable expressions of Eqs (23) and(24) into Eqs (46) and (47), the first-order differential

equations can be obtained after separating the real andimaginary parts as

a1′ = −

(ξ +

aΩ

2πSt

)a1 +

bΩ2Ra2

2ωR1π2St2sinΦ1

− ηβa14ωR1

sinΦ2, (48)

θ1′ = − bΩ2

Ra22ωR1π2St2a1

cosΦ1 +ηβ

4ωcosΦ2, (49)

a′2 =1

2

ω2R1

ΩRAa1 sinΦ1 −

1

2λΩRa2

(3

16a22−1

), (50)

θ′2 =1

2

ω2R1

ΩRa2Aa1 cosΦ1, (51)

where Φ1 = θ2 + σ1T1 − θ1, Φ2 = σ2T1 − 2θ1.

Furthermore, the first order differential equation ofΦ1 and Φ2 with respect to time T1 can be derived fromEqs (49) and (51) as

Φ′1 = σ1 +

1

2

ω2R1

ΩRa2Aa1 cosΦ1

+bΩ2

Ra22ωR1π2St2a1

cosΦ1 −ηβ

4ωcosΦ2, (52)

Φ′2 = σ2 +

bΩ2Ra2

ωR1π2St2a1cosΦ1 −

ηβ

2ωR1cosΦ2. (53)

The steady-state solutions can be obtained assum-ing a1

′ = 0, a2′ = 0, Φ′

1 = 0, Φ′2 = 0 as follows

G1(a1, a2) =[bλΩ4

Ra22

ω3R1π

2St2A

(3

16a22 − 1

)− a21

(2ξ +

aΩR

πSt

)]2

+

[a21σ2 +

bΩ3R(σ2 − 2σ1)a

22

ω3R1π

2St2A

]2− η2β2

4ω2R1

a41 = 0,

G2(a1, a2) = λ2Ω2Ra

22

(3

16a22 − 1

)2

+(σ2 − 2σ1)2a22 −

ω4R1

Ω2R

A2a21 = 0.

The above equations are rewritten as the scaled para-meters change back to the original ones as follows.[

bλΩ4Ra

22

ω3R1π

2St2A

(3

16a22 − 1

)− a21

(2ξ +

aΩR

πSt

)]2+

[a21(ω − 2ωR1) +

bΩ3R(ω − 2ΩR)a

22

ω3R1π

2St2A

]2− η2β2

4ω2R1

a41 = 0, (54)

λ2Ω2Ra

22

(3

16a22 − 1

)2

+ (ω − 2ΩR)2a22

−ω4R1

Ω2R

A2a21 = 0. (55)

In order to analyse the effects of the nondimensional

shedding frequency ΩR on the responses, Eqs (54) and(55) are solved numerically. It should be noted that asbefore these equations have trivial solutions a1 = 0 and

a2 = 0.


Similarly, the first order solutions of v1 and q1 can

be obtained as

v1 = −ηβa1[ei((ω+ωR1)T0+θ1) + e−i((ω+ωR1)T0+θ1)]

4[ω2R1 − (ω + ωR1)2]

,

q1 =3iλa32[e

3i(ΩRT0+θ2) − e−3i(ΩRT0+θ2)]

256.

Thus the first order approximate solutions of v andq can be obtained by combining solutions (v0, q0) and

(v1, q1)

v ≈ v0 + εv1 = a1 cos(ωR1T0 + θ1)

− εηβa12

cos((ω + ωR1)T0 + θ1)

ω2R1 − (ω + ωR1)

2 ,

q ≈ q0 + εq1 = a2 cos(ΩRT0 + θ2)

− 3ελa32 sin(3(ΩRT0 + θ2))

128,

where amplitudes a1 and a2 can be found from Eqs

(54) and (55).

5.2 Analysis of the amplitude-frequency curves

Non-trivial solutions of Eqs (54) and (55) provide the

amplitudes of the vibration responses of the structureand the wake oscillator as functions of the nondimen-sional shedding frequency ΩR, and they are obtained in

this study by using the parameter continuation method.In order to analyse the influence of the parametric forceon the system responses, different values in a large range

of the amplitude ratio β are considered and bifurcationcurves are computed under the resonance condition.Their stabilities are determined by examining the eigen-

values of the corresponding characteristic equations for(48), (50), (52) and (53). The numerical bifurcationcurves are also calculated by using the Runge-Kutta

method to verify the analytical results. The fixed pa-rameters are chosen at CD = 2, CL0 = 0.3, Ca = 1, λ =0.3, A = 12, St = 0.2, η = 0.8, ωR1 = 1 and ω = 2.1,

respectively.Figure 3 presents bifurcation curves of the dynamic

responses of the structure computed for 4 different val-

ues of the amplitude ratio β at µ = 1, ξ = 0.003. Hereonly structural response a1(ΩR) is shown as the wakeoscillator response a2(ΩR) has the same behaviour. As

can be seen from Fig. 3(a) obtained for β = 0.8, on-ly trivial solution exists below ΩR ≈ 0.4216 where anunstable non-trivial solution is born via the Hopf bifur-

cation (labeled by letter H). As the frequency increases,amplitude of this unstable solution continue to increaseup to ΩR ≈ 1.3253, where a saddle-node bifurcation

(labeled as SN) is observed and it results in creation

of stable large amplitude responses. The amplitude of

this stable solution is increasing with a decrease of theshedding frequency and it is observed in the range ofΩR ∈ (0.6665, 1.3253). As the frequency increases be-

yond ΩR ≈ 1.3253, only trivial solution a1 = 0 is found.

When β is decreased to 0.5, two branches of solu-

tions occur as shown in Fig. 3(b). A large-amplitudesolution is observed at a narrow range of low valuesof the nondimensional shedding frequency and a small-

amplitude solution is found around main resonance fre-quency of ΩR = 1. For the large-amplitude solution,which has a similar varying trend as that of shown

in Fig. 3(a), the Hopf and SN bifurcation occur atΩR ≈ 0.2390 and 0.3853 respectively. As the nondi-mensional shedding frequency increases, a pair of sta-

ble and unstable small-amplitude solutions is born viathe SN bifurcation at ΩR ≈ 0.9690. They exist over asmall range of frequencies and disappear via the SN b-ifurcation at ΩR ≈ 1.2324. Only trivial solution a1 = 0

is observed in the range ΩR ∈ (0.3853, 0.9690) and forthe higher frequencies above ΩR ≈ 1.2324.

For smaller values of the parameter β such as β =0.3, only the small-amplitude solutions are found in therange of frequency near ΩR = 1 as shown in Fig. 3(c).

As the parameter β is decreased further to 0.05, the si-milar scenario as that of for β = 0.3 is observed with therange of frequencies, where this pair of small amplitude

solutions exist, is being reduced and lower values ofamplitude are obtained. As β is decreased below 0.3, theoccurrence of the first saddle node bifurcation is shifted

to the higher values of the nondimensional sheddingfrequency while the second saddle node bifurcation isobserved for lower values as can be seen from Figs 3(c)

and 3(d).

Next the influence of the mass-damping parame-

ter on the system responses was examined. Five setsof bifurcation curves for the amplitude ratio β varyingbetween 0.8 and 0.05 were calculated for four differ-

ent mass-damping parameters. Both amplitude of thestructure response a1(ΩR) and wake oscillator ampli-tude a2(ΩR) were determined and they are presented

in Figs 4 – 8 together with selected trajectories on thephase plane which were obtained by numerical integra-tion of Eqs (12) and (13).

The results for β = 0.8 are shown in Fig. 4, whereamplitudes a1(ΩR) and a2(ΩR) are presented in Figs.

4(a) and 4(b), respectively. Solid and dashed lines presentthe non-trivial analytical solutions of Eqs (54) and (55)and small circles demonstrate a direct numerical inte-

gration of Eqs (12) and (13) obtained for µ = 1, ξ =0.003. As can be seen from this figure, the same scenarioas the one presented in Fig. 3(a) and described earlier

is obtained for all considered mass-damping parameter

12 Dan Wang et al.

0.0 0.5 1.0 1.50

1

2

3

4

H

SN

SN

SNSNH

SN

Am

plitu

de

,

Am

plitu

de

,A

mp

litu

de

,

Am

plitu

de

,


=0.8 =0.5

=0.3

SN

SN

1.00 1.10 1.200.15

0.20

0.25

0.30

0.35

0.40

1.05 1.15

=0.05

SNSN

1.00 1.10 1.200.15

0.20

0.25

0.30

0.35

0.40

1.05 1.15

SN

0.0 0.5 1.0 1.50

1

2

3

4


Shedding frequency, R Shedding frequency, R

Fig. 3 Evolution of bifurcations of the amplitude-frequency curves for the structure obtained analytically (marked by solidand dashed curves) and numerically (marked by small circles) for different values of amplitude ratio β: (a) β = 0.8; (b) β = 0.5;(c) β = 0.3; (d) β = 0.05. As can be seen from the figures that the Hopf and SN bifurcation can occur under certain parameterconditions.

values. The large-amplitude oscillations are found in arelatively wide range of the nondimensional sheddingfrequency ΩR and they undergo the Hopf and saddle-

node bifurcations. With increasing the mass-dampingparameters from µ = 1, ξ = 0.003 to µ = 2.9, ξ =0.045, µ = 2.45, ξ = 0.00796 and µ = 3.87, ξ = 0.0152

successively, both Hopf and saddle-node bifurcationsare observed for higher values of the nondimensionalshedding frequency ΩR. This is illustrated in Figs 4(a)

and 4(b) by points D1(1.2, 0.7358), E1(1.2, 1.4875) andD2(1.2, 5.1159), E2(1.2, 6.3187), marked with the blacksolid circle.

Phase portraits of the original systems (12) and (13)are computed with ΩR chosen at 1.2 for two values

of mass-damping parameter – µ = 1, ξ = 0.003 andµ = 2.9, ξ = 0.045 and they are presented in Figs 4(c)and 4(d). Points in Figs 4(c) and 4(d) D′

1 = (0.7300, 0),

E′1 = (1.2781, 0) andD′

2 = (5.3661, 0),E′2 = (6.2267, 0),

show the evolution of the amplitudes on phase planes(panels (c) and (d) of Fig. 4) used to construct the

frequency-amplitude curved (panels (a) and (b) of Fig.

4). As can be seen, the span of the system vibration isestimated well by the amplitudes obtained analytically.

Figure 5 presents the obtained results for β = 0.7.Here small circles in Fig. 5(a) and (b) demonstratethe results of numerical integrations of Eqs (12) and

(13) obtained for µ = 3.87, ξ = 0.0152. In general, thesame behaviour is observed for this amplitude ratio val-ue as for β = 0.8 described earlier. Phase portraits of

the original systems (12) and (13) are computed forΩR = 1.1 are presented in Figs 5(c) and 5(d) for allconsidered mass-damping parameter values, and very

large-amplitude vibration are noticable for µ = 3.87and ξ = 0.0152.

As the amplitude ratio is decreased to β = 0.5, d-ifferent bifurcation scenarios are observed as shown in

Fig. 6. As can be seen from this figure for larger val-ues of mass-damping parameter µ = 2.45, ξ = 0.00796(cyan curve) and µ = 3.87, ξ = 0.0152 (black curve)

the system response is similar to the case shown inFig. 3(a) and also in Figs 4 and 5, while for small-er values of this parameter µ = 1, ξ = 0.003 (blue

curves) and µ = 2.9, ξ = 0.045 (green curves), the


0.2 0.6 1.0 1.4 1.80

1

2

3

4

SN

SNSN

SN

H

0.2 0.6 1.0 1.4 1.80

5

10

15

SN SNSNSN

H

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

-1.5 0.0 1.5-1.5

0.0

1.5

-8 0 8-15

0

15R=1.2

R=1.2

Am

plitu

de

,

Am

pl itu

de

,

D1

E1 D2

E2

D1 E1 D2 E2

Shedding frequency, RShedding frequency, R

Ve

locit

y,

Displacement, Wake strength,

Ra

t eo

fw

ake

str

en

gth,

Fig. 4 (a) and (b) Bifurcations of the frequency-amplitude curves for the structure a1 and the wake oscillator a2 obtainedanalytically (marked by solid and dashed curves) and numerically (marked by small circles) for β = 0.8. The solid lines denotethe stable solutions and the dashed lines denote the unstable solutions. Phase portraits of (c) the structure and (d) the wakeoscillator computed for mass-damping parameters µ = 1, ξ = 0.003 and µ = 2.9, ξ = 0.045 at ΩR = 1.2.

behaviour is similar to the case shown in Fig. 3(b).In the case of larger mass-damping ratio where only

one branch of solution exists, the Hopf bifurcations areobserved at ΩR = 0.3616 for µ = 2.45, ξ = 0.00796and at ΩR = 0.4418 for µ = 3.87, ξ = 0.0152, and

the saddle node bifurcations occur at ΩR = 1.3530for µ = 2.45, ξ = 0.00796 and at ΩR = 1.4524 forµ = 3.87, ξ = 0.0152. It should be noted that for a

fixed nondimensional shedding frequency ΩR, the am-plitude for µ = 3.87, ξ = 0.0152 is larger than thatof for µ = 2.45, ξ = 0.00796. In the case of smaller

mass-damping ratio, where two branches of the solu-tion exist, the unstable large amplitude solutions areborn via Hopf bifurcation at ΩR = 0.2390 and they be-

come stable at ΩR = 0.3851 for µ = 1, ξ = 0.003 andat ΩR = 0.2427, 0.3573 respectively for µ = 2.9, ξ =0.045. A small amplitude solution branch is observed

for ΩR ∈ (0.9690, 1.2325) for µ = 1, ξ = 0.003 andfor ΩR ∈ (0.9433, 1.2889) for µ = 2.9, ξ = 0.045. Inaddition, Figs 6(c) and 6(d) present phase portrait-

s of the original systems (12) and (13) computed forΩR = 1.1. As can be seen from these figures, the am-

plitudes a1 and a2 are almost same for µ = 1, ξ = 0.003and µ = 2.9, ξ = 0.045.

Fig. 7 presents the obtained results for β = 0.3.

Here small circles in Fig.7(a) and 7(b) demonstratethe results of numerical integrations of Eqs (12) and(13) obtained for µ = 3.87, ξ = 0.0152. As can be seen

from this figure, the same scenario as the one presentedin Fig. 3(c) and described earlier is obtained for al-l considered mass-damping parameter values. Only the

small-amplitude responses are found in the range of fre-quency near ΩR = 1 and they appear and disappearthrough the saddle-node bifurcations. With increasing

the mass-damping parameters from µ = 1, ξ = 0.003to µ = 2.9, ξ = 0.045, µ = 2.45, ξ = 0.00796 andµ = 3.87, ξ = 0.0152 successively, the first saddle-node

bifurcation is shifted to the lower values of the nondi-mensional shedding frequency while the second sad-dle node bifurcation is observed for higher values as

can be seen from Fig. 7(a) and 7(b). Phase portrait-s of the original systems (12) and (13) computed forΩR = 1.15 are presented in Figs 7(c) and 7(d) for all

considered mass-damping parameter values, and when

14 Dan Wang et al.

0.2 0.6 1.0 1.4 1.80

1

2

3

SNSN

SN

SN

H

0.2 0.6 1.0 1.4 1.80

5

10

15

SNSNSN SN

H

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Am

plitu

de

,

Am

plit u

de

,

-20 0 20-100

0

100

-40 0 40

-40

0

40R=1.1

-2 0 2-2

0

2

R=1.1


Ve

locit

y,


Ra

teo

fw

ake

st r

en

gt h,

Fig. 5 (a) and (b) Bifurcations of the frequency-amplitude curves for the structure a1 and the wake oscillator a2 obtainedanalytically (marked by solid and dashed curves) and numerically (marked by small circles) for β = 0.7. The solid lines denotethe stable solutions and the dashed lines denote the unstable solutions. Phase portraits of (c) the structure and (d) the wakeoscillator computed for four values of mass-damping parameters at ΩR = 1.1.

ΩR ∈ (1.0318, 1.0823), the amplitudes are very similarfor µ = 2.45, ξ = 0.00796 and µ = 3.87, ξ = 0.0152,which are generated owing to the combination of mass-

ratio and damping under the combined resonance con-dition.

Fig. 8 shows the obtained results for β = 0.05. Here

small circles in Fig. 8(a) and (b) demonstrate the re-sults of numerical integrations of Eqs (12) and (13) ob-tained for µ = 2.45, ξ = 0.00796. The same behaviour

is observed for this amplitude ratio value as for β = 0.3described earlier but with the difference is the rangeof frequency as ΩR ∈ (1.0989, 1.1219) for µ = 1, ξ =

0.003, ΩR ∈ (1.1042, 1.1305) for µ = 2.9, ξ = 0.045,ΩR ∈ (1.1231, 1.1572) for µ = 2.45, ξ = 0.00796, andΩR ∈ (1.1339, 1.1734) for µ = 3.87, ξ = 0.0152. Phase

portraits of the original systems (12) and (13) comput-ed for ΩR = 1.1 are presented in Figs 8(c) and 8(d)for ΩR = 1.1 when µ = 1, ξ = 0.003 and µ = 2.9,

ξ = 0.045.

The presented results demonstrate that for smal-l amplitude ratio β only small amplitude responses arefound in the narrow range of nondimensional shedding

frequency near ΩR = 1. As the amplitude ratio increas-es to β = 0.5, for lower mass-damping parameter valuestwo branches of the solution are observed, i.e. similar

small amplitude responses near Ω = 1 and large ampli-tude responses at the lower nondimensional sheddingfrequencies. Finally as β increases from 0.5 to 0.8, two

branches are merged together resulting in appearanceof stable large amplitude solution over wide range ofnondimensional shedding frequency for all considered

values of mass-damping parameter.

6 Conclusions

A low dimensional model of a top tensioned riser (T-TR) under excitations from vortices and time-varyingtension is investigated analytically by multiple scales

method, where the van der Pol wake oscillator is used


0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

1.6

2.0

SN

SN

SNSN

H

0.0 0.4 0.8 1.2 1.6 2.0

0

4

8

12SN

SN

SN

H

-6 0 6-15

0

15

-0.8 0.0 0.8-1

0

1

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

R=1.1R=1.1

Am

plitu

de

,

Am

plitu

de

,


0.24 0.400

4

8

12 SN

H

0.22 0.30 0.380.0

0.4

0.8

1.2

H H

SN

Ve

locit

y,


Ra

teo

fw

ake

str

en

gth,


to simulate the loading caused by the vortex shedding.The governing partial differential equations describingthe fluid-structure interactions are formulated and multi-

mode approximations are obtained using the Galerkinmethod. The one mode approximation is adopted inthis study and two different system resonances are in-

vestigated employing the multiple scale method. Theyare the 1:1 internal resonance between the structure andwake oscillator (also known as ’lock-in’ phenomenon)

and the combined 1:1 internal and 1:2 parametric re-sonance.

Bifurcations of the system responses under the vary-ing nondimensional shedding frequency for different mass-

damping parameters are investigated and the resultsof multiple scale analysis are compared with numeri-cal results obtained by the Runge-Kutta method. Ana-

lytical responses are calculated using the continuationmethod and their stability is determined by examiningthe eigenvalues of the corresponding characteristic e-

quations. Influence of the system parameters includingthe amplitude of the tension variation, vortex sheddingfrequency and mass-damping parameter on the system

bifurcations has been investigated.

First the 1:1 internal resonance between the struc-ture and wake oscillator was studied. The amplitude-frequency equations were derived from the modulated

equations, and bifurcation diagrams for the responseamplitude under varying nondimensional shedding fre-quency were constructed for four different values of the

mass-damping parameter. The presented results showthat increasing the mass-damping parameter can sup-press the oscillations of the structure and wake oscilla-

tor.

Then the combination of internal and parametricresonance was studied, where frequencies satisfy ΩR ≈ωR1, ω ≈ 2ωR1. Here the influence of the amplitude ra-

tio β on the dynamic responses was investigated andthe results are presented for the same four values of themass-damping parameter. A series of bifurcations in-

cluding Hopf and saddle-node types were found analyti-cally defining the range of existence of stable and unsta-ble responses. The presented results demonstrate that

for small amplitude ratio β only low amplitude respons-es are found in the narrow range of nondimensionalshedding frequency near ΩR = 1. As the amplitude ra-

tio increases to β = 0.5, for lower mass-damping param-

16 Dan Wang et al.

0.9 1.0 1.1 1.2 1.3 1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.9 1.0 1.1 1.2 1.3 1.43.0

3.6

4.2

4.8

SN

SNSN

SN

SN

SN SN

SN

SN

SN

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

Numerical results

= ,1= ,2.45= ,3.87

= ,2.9

-5 0 5-8

0

8

-0.4 0.0 0.4-0.4

0.0

0.4R=1.15

-0.02 0.10

-6.30

-6.05

R=1.15

Am

plitu

de

,

Am

plitu

de

,


Ve

locit

y,


Ra

teo

fw

ake

str

en

gth,


eter values two branches of the solution are observed,i.e. similar small amplitude responses near Ω = 1 and

large amplitude responses at the lower nondimensionalshedding frequencies. Finally as β increases from 0.5 to0.8, two branches are merged together resulting in ap-

pearance of stable large amplitude solution over widerange of nondimensional shedding frequency for all con-sidered values of mass-damping parameter.

In addition, results of analysis of the combined reso-nance comparing to that of only the internal resonanceshow that (i) amplitudes of responses of the structure

under the combined resonance condition are much larg-er than that for the case of VIV only and the amplitudeof responses increases rapidly as β grows; (ii) bifurca-

tions including the saddle-node and Hopf bifurcationsare observed in the considered frequency range; (iii) in-creasing the amplitude ratio β results in shift of the

observed bifurcations to higher values of frequency ΩR.

Acknowledgements The first two authors acknowledge thefinancial supports of the National Natural Science Founda-

tion of China (No.11702111, 11732014), the Natural ScienceFoundation of Shandong Province (No. ZR2017QA005) andthe State Scholarship Fund of CSC.

Declarations: The authors declare that they have noconflict of interest.

Data Availability Statements: The datasets generat-ed during and/or analysed during the current study are avail-able from the corresponding author on reasonable request.

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1.0 1.1 1.2 1.3

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Bifurcation analysis of vortex induced vibration of low