Theoretical, experimental and numerical investigations into nonlinear motion of a tethered-buoy system

J Mar Sci Technol (2016) 21:396-415 D O I 10.1007/S00773-015-0362-X

O R I G I N A L A R T I C L E

Theoretical, experimental and numerical investigations

into nonlinear motion of a tethered-buoy system

Chong Ma^ • K a z u h i r o lijima^ • Y a s u n o r i NiheP • M a s a h i k o Fujilmbo^

Received: 12 January 2015/Accepted: 21 December 2015/Published online: 22 January 2016 © J A S N A O E 2016

Abstract I n this paper, we discuss nonlinear m o t i o n o f a buoy connected vertically to the seabed via a tensioned tether (tethered-buoy). A series o f scaled model tests has been conducted and a significant nonlinear behavior o f the buoy motion, sub-harmonic m o t i o n i n particular, is observed. T a k i n g account o f the influence o f time-varying tether tension on the buoy motion, theoretical explanation is made f o r the sub-harmonic response. The stability o f the tethered-buoy system is focused based on Mathieu insta-b i l i t y theory. A strongly coupled numerical model insta-between the buoy m o t i o n and the tether behavior is established to c l a r i f y the mechanism o f the nonlinear m o t i o n o f the tethered-buoy system. A comparison between the experi-ment data and simulation results is presented not only f o r the linear but also f o r the sub-harmonic components. Influential factors f o r the sub-harmonic m o t i o n are dis-cussed i n detail. It turned out that the sub-harmonic m o t i o n is dominated by the nonlinear coupling effect o f time-varying tension i n the tether w i t h the buoy motion. Finally, the influential factors to the sub-harmonic m o t i o n are indicated throughout the comparison between t w o different buoy models.

K e y w o r d s Tethered-buoy • Single-point-mooring • Stability analysis • Nonlinear simulation • Strongly coupling • Sub-harmonic motion

E l Chong M a

ma_chong@naoe.eng.osaka-u.ac.jp

' Department o f Naval Ai'chitecture and Ocean Engineering, Osaka tJniversity, Osaka, Japan

- Department o f Marine System Engineering, Osaka Prefecture University, Osaka, Japan

1 I n t r o d u c t i o n

I n this research, we consider a buoy connected w i t h a vertical pre-tensioned tether to the sea bed. The tethered-buoy system is f o u n d i n various marine applications. F o r example, a buoy system i n single-point-mooring (SPM) f o r Floating, Production, Storage and O f f l o a d i n g systems (FPSOs) is w i d e l y adopted. As the S P M system needs less amount o f material and less number o f anchors than the conventional catenary system, lower installation cost can be attained. A l s o , by introducing the S P M system, the installation space i n the sea bottom can be saved s i g n i f i -cantly compared w i t h the catenary system. The system may be utilized i n other applications such as Floating O f f s h o r e W i n d Turbines ( F O W T s ) . The authors proposed an F O W T system consisting o f a semisubmersible p l a t f o r m and an S P M so that the F O W T can weathervane to the change o f the w i n d , wave and current directions. T h e weather-vane behavior was verified i n a series o f scaled m o d e l tests f o r the S P M - F O W T scaled model [ 1 ] .

M e a n w h i l e , a severe nonlinear response o f the tethered buoy was observed w h i c h was totally d i f f e r e n t f r o m the normal harmonic motion. A similar nonlinear response was reported i n Otaka and N i h e i [ 2 ] . A large nonlinear pitch m o t i o n occurred during the model test f o r the F O W T w h i c h consists o f spar-type p l a t f o r m and single-tension-leg m o o r i n g system. I t turned out that the severe m o t i o n was partly due to subharmonic excitation whose m o t i o n f r e -quency is h a l f o f the incident wave fre-quency w h i c h may be interpreted as M a t h i e u Type problem [ 3 ] .

M a n y researchers have studied the M a t h i e u instability o f m o t i o n o f spar p l a t f o r m w i t h catenary m o o r i n g system. Amongst others, Haslum and Faltinsen [4] discussed the Mathieu instability problem by taking into account the heave-pitch coupling and instantaneous water-plane area.

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A similar heave-pitch coupling problem was reported i n Rho et al. [5] i n w h i c h more complicated spar was modeled and studied. He also checked the influence o f catenary mooring system on the Mathieu instability o f spar by a series o f model tests [ 6 ] . A l l o f the above researches pointed out that, f o r the spar p l a t f o r m w i t h catenary m o o r i n g system, i f the natural frequency ( f o r convenience, all the frequency i n this paper means the circular frequency w i t h unit rad/s) f o r heave is twice the natural frequency f o r pitch m o t i o n , the heave motion can undergo 10 times as large as the incident wave amplitude at resonance and the Mathieu instability problem occurs. Parametric r o l l o f ships is a well-studied Mathieu instability problem. The para-metric r o l l motion occurs even i n head seas. Hashimoto and U m e d a [ 7 ] recently proposed a coupled heave-roll-pitch mathematical model to investigate the coupling influence between r o l l , pitch and heave when the Mathieu instability occurs. For spar type o f p l a t f o r m , a coupled heave-roll-pitch numerical tool was also established by Rodriguez and Neves [8] w i t h considering the influence o f nonlinear Froude-Kj-ylov force.

The tethered buoy considered i n the present study is connected by one pre-tensioned tether instead o f the cate-nary m o o r i n g system. I f we assume the axial stiffness o f the tether is large enough, the heave m o t i o n o f the buoy is completely associated w i t h the pitch and surge motions by the tether and i t is given i n a quadratic and even higher order f o r m o f the pitch and surge motions. The heave motion gets the peak value when the surge or pitch m o t i o n become the largest and the smallest. Thus, the heave m o t i o n naturally includes twice the m o t i o n frequency o f pitch or surge motions, w h i c h may cause the Mathieu instability problem as discussed i n the above references.

The nonlinear coupling effect o f time-varying tether tension may also cause the sub-harmonic motion f o r the tetheredbuoy system. A similar e f f e c t is studied f o r T L P -type p l a t f o r m . Minematsu and K a j i t a [9] discussed the Mathieu instability problem f o r the surge m o t i o n o f T L P by considering the time-varying tether tension theoretically.

For the pitch motion, however, there is still less research w h i c h covers the nonlinear coupling effect o f tether ten-sion, especially, f o r the tethered buoy system as far as the present authors know. A nonlinear numerical model f o r the pitch m o t i o n o f a tethered buoy has been researched b y Sao et al. [10] i n time domain. He pointed out that, f o r the tethered buoy system, the sub-harmonic m o t i o n occurs and may be mainly caused by the high order hydrodynamic forces under waves w i t h the sufficiently large wave amptitirde. B u t he did not discuss it f r o m the viewpoint o f the M a t h i e u instability problem. The clarification o f the mechanism o f the sub-harmonic m o t i o n is yet to be made. In the present research, a series o f m o d e l tests is con-ducted solely on the tethered buoy f o r various wave

conditions to c l a r i f y the mechanism o f the sub-harmonic motion. A Mathieu type analytical model taking account o f the coupling between the buoy and the tether is proposed. To s i m p l i f y the problem and focus on the influence o f mooring tension variation, only the coupling between the motion o f surge, heave, pitch and m o o r i n g tension is taken into account i n this paper.

Y h e model is shown to explain the tank test results qualitatively. Besides, a time-domain numerical model w h i c h accounts f o r various coupling effects including the one between the buoy motion and the tension variation i n the tether, and that between the heave and pitch motions, is also developed. A comparison between the experiment and numerical simulation is made. A good qualitative and quantitative correlation is obtained f o r both linear and nonlinear response components. The numerical model is further utilized to discuss w h i c h coupling term contributes the most to the sub-harmonic motion. I t turned out that the sub-harmonic motion is dominated by the nonlinear cou-p l i n g effect o f time-varying tension w i t h the buoy motion.

2 A n a l y t i c a l m o d e l

2.1 N a t u r a l frequency and the associated mode

The tethered-buoy system considered i n this research consists o f a buoy and a pre-tensioned tether w h i c h are connected b y a p i n hinge. The tether is also hinged on the seabed. So, only tension force along the tether is acted on the buoy bottom. T h e wave is assumed to propagate along the X-axis and the tethered-buoy system is assumed to be symmetric along x-axis as w e l l .

I f we confine the motions w i t h i n x-z plane, the equa-tions o f motion f o r the buoy is established f o r surge, heave and pitch m o t i o n as f o l l o w s (Fig. 1):

(OT -f- ;n.„.v)X + C,X = -Tsiu{Oi) + F,,

{m + m„,)Z + C,Z = -mg - Tcos{e^) + + F, (I + D'Oi + CgÓ2 = -T • L2sm{e2 - 0 i ) -

FB-G M J O T ( 0 2 ) - F M (1)

where, X: horizontal displacement o f the buoy (surge), Z: vertical displacement o f the buoy (heave), dy. tether angle (angle between the z-axis and the inclined tether), 62-angular displacement o f the buoy (pitch), OT.: the mass o f the buoy, ;«.„,., uia.: the added mass f o r surge (x) and heave (z) motions, respectively, / : the moment o f inertia o f the buoy f o r pitch motion, ƒ„: the added moment o f inertia f o r pitch m o t i o n , Q , CQ: the hydrodynamic damping f o r surge, heave and pitch motions, respectively, T: tension o f tether i n c l u d i n g the static and dynamic components, Li. length o f tether, Ly. the height o f the center o f gravity f r o m

398 J Mar Sci Teclinol (2016) 21:396-415

s i n k i n g d o w n ( - Z

(2)

F i g . 1 Definition o f the tethered buoy and coordinate system

the keel o f the buoy, G M : the metacentric height measured f r o m the center o f gravity, Fg. buoyancy o f the buoy, F.,-, F^: hydrodynamic external force about x-axis and z-axis, respectively, M: hydrodynamic external moment about y-axis.

Geometrically, the surge and heave f o r the buoy are expressed by E q . 2:

X = L l s i n ( 0 i ) - f L2sin(fl2)

Z = L i [ c o s ( 0 i ) - 1] + L 2 [ c o s ( 0 2 ) - 1]

B y taking up to the second order w i t h respect to the angular displacement 0 i and 02, E q . 2 is s i m p l i f i e d to,

Z = L i 0 i - f L 2 0 2

1 2 2

It can be understood f r o m E q . 3 that the heave m o t i o n o f a quadratic order naturally exists i n the tethered-buoy system. The second equation may be approximated as z = 0 to the first order. Here, we only discuss the equation o f motion i n the first order so that the equation o f heave

motion can be neglected. Also, the varying tension T, the varying buoyancy F^ and the varying metacentric height G M due to the heave motion are replaced by the pre-ten-sion To, i n i t i a l FBO and initial metacentric height G M Q i n Eq. 1. T o discuss the natural mode o f tethered-buoy sys-tem, all the hydrodynamic external force (or moment) and damping i n Eq. 1 are neglected. Then, substituting E q . 3 into E q . 1, we obtain Eq. 4.

(4)

{m + /H„.v)Li0i + r o 0 | + {m + « w ) L 2 0 2 = 0 (/ + 4 ) 0 2 + (roL2 + FBQ • GMo)02 - TGLOBI = 0

Assuming the angular displacement 0i 02 o f the f o r m

01 = 0 1 6 _{92 = (1)2^"} respectively (cpi, 02 is

amplitude o f 0 i and 02), Eq. 4 can be transformed to the f o l l o w i n g matrix f o r m :

- f f l ^ { i n + max)L\ +TQ - C Ü ^ { m + in„^)L2

-T0L2 TOL2+FBO-GMO-CO\I + I„)

X (5)

To obtain a nontrivial solution, the determinant o f the above matrix has to be set zero:

-OJ^{m + inax)L\ + To -T0L2

-ap-{in + inax)T2 ToLi + F BO • GMo - 0? {1 + la)

= 0

(6) The E q . 6 is simplified as E q . 7 by introducing the coefficients A, B, C: Am'^ - Boj^ + C =

0

C ^ To{ToL2 + FBO • GMo)

It is easily k n o w n that when C < 0, Eq. 7 w i U have an imaginary root w h i c h means that the system becomes unstable. Therefore, f o r the system to be stable, the i n i t i a l metacentric height G M Q has to satisfy the f o l l o w i n g condition.

GMo > -T0L2 FBO

(8)

A t last, the t w o natural frequencies can be obtained as roots o f E q . 7 as f o l l o w s :

- , » 2 = \ / (9)

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where,

S- - 4/iC : _{T^^j^Q + + "^oLl + Toil + /„)}

To{I + Q + + ' " - ) r o L ^ ~ 7-o(/ + 4 ) + A{m + m„,)TlÜi{I + Q>Q

-AAC

Substituting the natural frequency m into Eq. 5, the motion ratio 1 which is defined by the ratio o f amplitude f o r 01, 02 is obtained:

02 - c u ^ ( m + m„,,-)Li + 7b (10) It can be proven that f o r the natural frequency cüi, the angular displacement 0 i , and 02 have the opposite phase. W h i l e , they are in-phase f o r the natural frequency <xi2. The detailed derivation is given i n " A p p e n d i x 1 " . T w o d i f f e r -ent modes associated w i t h the t w o natural frequencies are shown i n F i g . 2.

Especially f o r natural frequency o j i , the motion rado 1 can be easily estimated by E q . I I . See the detail derivation in " A p p e n d i x 2 " .

k

K m ₍₁₁₎

I f we substitute Eq. 11 into the Eq. 3, it is k n o w n that, the surge motion measured at the center o f gravity o f buoy w i l l be close to zero when the motion mode w i t h the nat-ural frequency oji occurs. I f we assume that the heave motion is small enough as well, the natural mode associ-ated w i t h the natural frequency co^ is regarded as pitch dominant.

2.2 Mathieu instability

W h e n we consider the hydrodynamic external force (or moment) f o r the tethered-buoy system, the M a t h i e u insta-b i l i t y may occur w h i c h is accounted f o r insta-by the nonlinear component existing i n the Eq. 1 which includes the varying tension T{t), varying buoyancy pgA^ • ( D r a f t - Z ( f ) ) and v a r y i n g metacentric height G M ( / ) . Equation 12 rewrites the equation o f pitch m o t i o n i n Eq. 1 by marking out a l l the time varying components. Tether angle 0 i is substituted by the m o t i o n ratio A.

( / + /«)02 + Co02 + [pgA, • ( D r a f t - Z{t)) • G M ( r )

+ ( l - ; . ) L 2 r ( O ] 0 2 = M ( O (12) where, A^. waterplane cross-section area. GM(?) =

GMo — dynamic metacentric height due to heave.

F i g . 2 The motion modes associated with the natural frequency cui and oJo- a (Ui*, b 0)2**. *pitch dominant mode, ••"•'surge dominant mode

0 1

(a)

* pitch dominant mode ** surge dominant mode

I I ! 1 02 1 ! I I I (b) Ö Springer

400 J Mar Sci Technol (2016) 21:396-415

Draft: the initial draft.

The first term, pgAc • ( D r a f t - Z ( / ) ) • G M ( / ) , i n restoring stiffness has been discussed i n many researches as the cou-pling effect o f heave and pitch. Haslum [4] pointed out that, the dynamic metacentric height plays a dominant role i n the coupling behavior and the equation o f pitch motion (the influence o f tether tension on the body was not considered) can be simplified as a typical f o r m o f Mathieu type equation. The system w i l l become unstable when the frequency o f heave m o t i o n is twice the natural frequency f o r pitch m o t i o n . The second term, (1 - X)LaT{t), i n restoring stiffness represents the nonlinear influence o f time-varying tether tension. I f we do not consider the heave-pitch coupling and focus on the mechanism o f tether influence, the E q . 12 reads:

(/ + Ia)h + Ce02 + [pgAc • D r a f t • G M Q (1 - XWit)]

0 2 = M ( O (13) The equation o f heave m o t i o n can be written as

(m + ff!„,)Z + Q Z + pgA,Z = F,{t) - {T{t) - To) c o s ( 0 | ) (14) The heave m o d o n is regarded second order as discussed previously. T o account f o r the nonlinear contributions i n Eq. 13 up to the second order, we may neglect the second and even higher order components i n E q . 14 to obtain the f o l l o w i n g linear approximation.

T{t) = To + F,{t) (15)

The second term F^(t) represents the linear contribution o f the vertical wave e x c i t i n g force on the buoy. For the reg-ular wave w i t h the circreg-ular frequency co, the F^(t) is obtained by applying the linear potential theory as:

F,{t)=af,cos{cot + y) (16)

where, a: the wave amplitude, f^: the vertical force amplitude per unit wave amplitude, y: the d i f f e r e n t phase existing i n the heave exciting force and wave profile.

Substituting Eqs. 15 and 16 into Eq. 13, E q . 17 is obtained:

(17)

(/ + 4 ) 0 2 + Q 0 2 + [Ko + Kl co&{cot + y)]02 = M(t)

where,

Ko = pgAc • D r a f t • G M Q + {\ - X)L2TO

Kl = f l ( l

The homogeneous equation o f E q . 17 may be rewritten as f o l l o w s . 02 - I - ;;ct),-02 + [co^ + E C 0 s( a ) f -|- y)] 02 = 0 (18) where. (Oi CO, 0J2 Co COi{I + I„) Kl ' I + Ia

I f the relationship x = cot + y is introduced, Eq. 18 may be deduced to the standard Mathieu equation:

, -02 + /( — 0 2 + [5 + £ ' C O S ( T ) ] 0 2 = O (19)

Ë =77f w h e r e / ( = ^ f ' , ,5=

The stability f o r the Mathieu equation has been w e l l studied. Figure 3 shows the stability diagram f o r the damped Mathieu's equation [11] f r o m w h i c h i t is f o u n d that, when the 5 — 0.25 or 1 (OJ = 2(W; or OJ,), the system becomes unstable. Especially when ö locates around 0.25, the stable criterion can be written i n the f o l l o w i n g inequality [ 1 2 ] .

:{-\{e''-n')>0 (20)

Substituting 3, Ë', p i n Eq. 20 by a,f^, X, E q . 20 can be written as a f u n c t i o n o f wave amplitude and wave f r e -quency as given b y E q . 2 1 . a< 1 (1 - ^ O W . 4 ) ' ( 4 c CO' _•2^n2 Cico ₍₂₁₎

It can be interpreted that larger vertical hydrodynamic f o r c e / j , larger absolute value o f m o t i o n ratio (when m o t i o n ratio is minus) |.^.| and higher height o f the center o f gravity La tend to cause the instability o f the tethered buoy system

2.5 1.5 1 0.5 \ ; Uns&ble \ . '\ . k \ . Unstable \ . V" " • . \ . :\ \ . • \ V \ . . » / / \ J > /Jr Stable' S t a b l e \ / / • stable \ '• *• \ ., \ . ^ ' . / 1 1 — 0% darpping 3% damping 5% damping 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 5

F i g . 3 Stability diagram for damped Mathieu's equation

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r & Load Transducer F i g . 4 Photos o f SPM model test

Table 1 Principal particulars o f SPM model

Item Value Unit

Scale Ratio 1/200

Mass 1.035 kg

Moment o f Inertia (ly^l) 0.00786 l<:g*m^ The height o f the center o f gravity ( L j ) 0.073 m

Height 0.240 m

Diameter 0.100 ni

Draft 0.150 111

Tether Length ( L j ) 0.270 ni

Pretension 1.400 N

for a given amplitude o f wave. Also, sinaller damping coefficient Cg may cause the instability especially f o r the waves w i t h wave frequency twice the natural frequency. The influence o f pre-tension is reflected i n terms o f the natural frequency.

3 S c a l e d m o d e l test

3.1 S P M model test

T o better understand the behavior o f the tethered-buoy system, a series o f model tests was conducted by the authors i n the water tank o f Osaka University. A scaled model f o r the buoy part o f the S P M system w i t h the scale ratio 1/200 (See Fig. 4) was fabricated w i t h GFRP. Table 1 shows its principal particulars. The model is called S P M model herein.

T o c l a r i f y the behavior more i n an analytical manner, a series o f tank tests under regular waves was conducted and the wave condition is given i n Table 2.

The side-view and top-view f o r the arrangement o f the model test is shown i n F i g . 5. T w o L E D s were attached along the vertical r i g i d bar on the buoy so that the surge, heave and p i t c h motions o f the buoy can be measured by optical-video-analysis o f the m o t i o n o f the L E D s . The tension o f tether was also measured by the load transducer w h i c h located on the bottom.

As the value o f heave m o t i o n is second order compared w i t h the surge and p i t c h [see Eq. 3] due to the restriction o f the tether, f o r S P M model, the heave motion is always smaller than 5 m m w h i c h cannot be accurately measured by the utilized optical-video-analysis system. For the optical-video-analysis system, the resolution is 600 x 400 pixels and the samphng frequency is óOfps. Considering

Table 2 Tested wave conditions, SPM Case I D Wave frequency

(rad/s) Wave period (s) Wave amplitude C (cm) Wave steepness (wave height/wave length) T SPM_01 12.6 0.5 1.793 0.092 2 SPM_02 10.5 0.6 2.219 0.079 2 SPM_03 9.0 0.7 3.440 0.090 2 SPM_04 7.9 0.8 2.577 0.052 2 SPM_05 7.0 0.9 3.169 0.051 2 SPM_06 5.2 1.2 3.173 0.032 2 SPM_07 4.8 1.3 1.804 0.016 2 SPM_08 4.5 1 4 2.717 0.022 2 SPM_09 4.2 1.5 1.619 0.012 2 Test times < ^ Springer

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F i g . 5 Anangement of SPM model test, a Side-view, b top-view W a v e G E N E

-/

*

^ 1

1

the physical length o f the model, the accuracy of the optical-video-analysis system is about 0.5-1.5 m m .

3.2 T L S P A R model test

The sub-harmonic behavior was also observed i n the F O W T model test [2] w i t h the special designed p l a t f o r m model ( T L S P A R , Fig. 6) w h i c h consists o f SPAR type

p l a t f o r m and single pre-tensioned mooring tether (a similar concept may be f o u n d i n " S W A Y " [13]). The principal particulars f o r the T L S P A R model are shown i n Table 3. I f neglecting the w i n d influence, the mechanism of T L S P A R m o t i o n totally coincides w i t h the tethered-buoy system.

A series o f regular only wave tests (see Table 4 ) was earned out f o r T L S P A R model i n the Ocean Engineering Basin o f Institute o f Industrial Science, the University o f T o k y o . T o obtain the nonlinear response, the wave f r e -quency ( 5 - 1 1 rad/s) is focused around the twice o f natural frequency o f pitch m o t i o n f o r T L S P A R model. The experimental setup and the definition o f coordinate system f o r T L S P A R model test is shown i n F i g . 7.

4 N u m e r i c a l m o d e l

T o simulate the couphng influence between floater (Buoy or SPAR) and mooring tether, two numerical models are established f o r the floater and the tether, respectively. A n d then, these t w o models are strongly coupled i n the time domain. A more detailed discussion about the simulation can be f o u n d i n the previous papers by the present authors [14, 15]. Herein, a brief explanation is given to the numerical models and the coupling procedure.

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Table 3 Principal particulars o f T L S P A R model

Item Value Unit

Scale ratio 1/100

-Mass 7.91 kg

Moment o f inertia (Z^,,) 2.28 k g X m^ The height o f the center of gravity (Z^) 0.62 111

Height o f tower 0.75 m

Height of spar 1.19 m

Diameter (minimum) 0.07 m

Diameter (maximum) 0.25 m

Tether length (Li) 1.70 m

Draft 1.09 111

Pretension 14.71 N

4.1 F l o a t e r

It is assumed that the m o t i o n amphtude o f the floater is small. The structure is assumed to be either flexible or r i g i d . I f the structure is modeled to be flexible, the floater is modeled by linear beam elements so that 6DOFs equation of m o t i o n can be established f o r the respective nodes [16]. A n in-house code has been developed to this end. B y solving a system o f equations o f m o t i o n , the nodal dis-placements are obtained. Then, the sectional forces can be obtained f o r each element. I n the present analysis, the r i g i d body approach is taken f o r modeling the buoy to account only f o r the r i g i d body motions i n the present simulation.

The hydrodynamic coefficients are calculated based on the linear singularity distribution method. The viscous drag f o r c e is also taken into account based on the drag force term i n Morison's f o r m u l a [ 1 7 ] . As the response o f the floater may include the transient motions arising f r o m the nonlinear coupling, the floater-motion induced influence on the free surface should be taken into account by applying the retardation f u n c t i o n [ 1 8 ] . T h e nonlinear influence o f the varying tether tension is included as a nonlinear external force. The nonlinear coupling effect between heave and

pitch motions as in Eq. 12 may be accounted f o r i n terms o f the hydrostatic restoring stiffness. Finally, a system o f equations o f motion f o r the floater can be expressed as in Eq. 22. t [M + A(oo)]x + Dx +{K + iQx + j k{t - x)x{x)dx 0 = F,, + + Fl, + F,„ + Fd (22) where, M: the mass matrix, A ( o o ) : the added mass matrix con-esponding to the high-frequency l i m i t , D: Rayleigh's structural damping matrix, K: the hydrostatic restoring stiffness matrix w h i c h considered the nonlinear coupling between heave and pitch, see E q . 12. K^. linear structural stiffness matrix (neglected when the model is modeled as a r i g i d body), k: the retardation f u n c t i o n vector, F,,,: the time-dependent first-order wave excitation load vector. Fg. the gravity load vector, F;,: the buoyancy load vector, F,„: the nonlinear m o o r i n g load vector, F^: the nonlinear viscous drag load vector according to M o r i s o n ' s f o r m u l a .

4.2 Mooring

The tether is modeled as a slender element w i t h large deflections and w i t h small strains. Nonlinear structural stiffness may affect a lot on the behavior o f the tether. U n l i k e catenary system w h i c h can be regarded as a quasi-static problem, the inertial loads on the tether may play an important role and they need to be considered to account f o r the dynamic and nonlinear behaviors. Besides, the tether system may cause nonlinear influence when considering the tension variation and tension direction. The other in-house code is developed by u t i l i z i n g beam element ( 6 D 0 F ) or bar element ( 3 D 0 F ) w i t h a consideration o f the geometrical nonlinear stiffness [ 1 4 ] . The hydrodynamic force on the tether is calculated based on M o r i s o n ' s Formula. T o v e r i f y the nonlinear influence o f time-varying tether tension, the linear m o o r i n g model is also derived according to [ 1 6 ] .

Table 4 Tested wave conditions, T L S P A R

Case I D Vv'ave frequency Wave Wave amplitude Wave steepness Test times

(rad/s) period (s) C (cm) (wave height/wave

length) TLSPAR_01 104 0.60 1.215 0.043 1 TLSPAR_02 9.2 0.68 1.267 0.033 I TLSPAR_03 8.0 0.79 2.263 0.045 1 TLSPAR_04 6.7 0.93 1.817 0.029 1 TLSPAR_05 6.2 1.00 2.501 0.032 1 TLSPAR_06 5.5 1.14 3.025 0.032 ₁ Springer

404 J Mar Sci Technol (2016) 21:396-415

F i g . 7 Anangement o f T L S P A R model test, a Side-view, b top-view W i n d W i n d B l o w e r —••• W a v e L E D W a v e G a u g e T o w e r W i n d T u r b i n e T e t h e r L o a d C e l l (a) Wave Generator

1

_Hi'

.[ M a r Sci Teclinol (2016) 21:396-415 405

4.3 Coupling technique and numerical solver

The two mimerical models introduced above are coupled together to account f o r the mutual interactions. The cou-p l i n g is required since the tension variation i n the tether affects much on the motion o f the floating body, i.e., the m o t i o n of the buoy is strongly constrained by the tether. There are i n general two strategies in the coupling proce-dure. One is k n o w n as weakly coupling while the other is k n o w n as strongly coupling.

T h e weakly coupling means the simple displacement-force coupling i n different time step. D u r i n g the weakly coupling, the two numerical models (here, floater and mooring) can be kept relatively independent and only m o d i f y i n g the interface o f two models is necessary. Therefore, the weakly coupling is much easier to be adopted, especially when the numerical model is c o m p l i -cated. However, as the weakly coupling does not solve the t w o numerical models simultaneously, i t cannot give the exact solutions unless the convergence is confirmed at each time step. Especially, when the stiffness o f numerical models is high, the solution o f weakly coupling w i U be easy to diverge. As the m o o r i n g model has high stiffness, the wealsly coupling is insufficient i n this research. Instead, the strongly coupling is utilized here by coupling the two numerical models i n matrix f o r m . U n l i k e weakly couphng, strongly coupling can solve aU the exact displacement and f o r c e f o r fioater and m o o r i n g simultaneously. But the strongly coupling is more d i f f i c u l t to conduct as it needs m o d i f y i n g the numerical models much more than the weakly couphng. The m a i n process of strongly coupling u t i l i z e d i n this research is discussed as f o l l o w i n g .

Generally, the equation o f m o t i o n f o r both floater and m o o r i n g can be derived as given i n the f o l l o w i n g f o r m :

MfXf + CfXj + KfXf = Ff + F* (23) M,„X,„ + C„,X,n + K,„X,„ = F„, - F* (24)

where the subscripts ƒ and m mean the relevant matrix or vector f o r floater or mooring, respectively. F * is the con-nection force between the floater and the m o o r i n g . A f t e r transformation according to Newmark-Beta method, the w h o l e equation f o r t w o numerical models can be shown i n matrix manner as F i g . 8.

where Parts 1 and 2 are derived based on Eqs. (23) and (24). Part 3 can be obtained according to the boundary condition i n the j o i n t o f floater and m o o r i n g . MATRIX represents the transformed matrix according N e w m a r k -Beta method. I means the coupling m a t r i x w h i c h has the identical value on the coupling D O F (other values are zero). Finally, the size o f the whole m a t r i x is iif + n,„ + n^, where Uf, «.,„, are the number o f D O F f o r floater, mooring and the coupling j o i n t , respectively.

As the simulation is designed f o r dealing w i t h the nonlinear and coupled problem, the time domain analysis is performed by the h a l f - i m p h c i t Newmark-Beta method [19]. For each time step, the predictor-corrector method is conducted to get the cun-ent nonhnear stiffness matrix.

5 R e s u l t s a n d d i s c u s s i o n

A l l the f o l l o w i n g results are presented i n model scale.

5.1 SPiVI model

5.7.7 Free decay test

T o get the natural frequency o f S P M model, a series o f free decay tests is conducted. The floater and tether are inclined manually at the same time w i t h opposite direction. By using opposite inclination, the natural mode corresponding to CO I is desired to occur easily. The time domain results and corresponding F F T analysis results i n frequency domain are shown i n F i g . 9. T w o m o t i o n modes are observed as discussed i n the theoretical analysis. B y picking up the frequency at w h i c h the F F T result takes the peak i n F i g . 9(b), the natural frequency is obtained. Table 5 gives a comparison of the natural frequency among the decay tests, theoretical analysis, i.e., Eq. 9, and numerical simulation. For the higher natural frequency c/Ji, the results o f theoretical analysis and numerical simulation can explain the test results to some extent. The test value 0J2 may have larger uncertainty due to the small number o f measured cycles. B u t i t w i f l give less influence as only the sub-harmonic m o t i o n associated w i t h cox is discussed i n this paper. T o i m p r o v e the reliability o f the decay test, the decay test was carried out by three times and the discrep-ancy o f natural frequency among the m u l t i p l e tests is less than 5 % f o r the natural frequency o j i .

The damping coefficient associated w i t h coi is obtained after conducting the I F F T analysis by ehminating the response w i t h the frequency smaller than 1.5 rad/s. The ratio between the damping coefficient and critical damping is between 3 and 4 % w h i c h is utilized i n the f o l l o w i n g simulation.

5.1.2 RAO comparison

The tethered-buoy system may have large response or even unstable behavior when the wave frequency is the same as or twice the natural frequency due to the M a t h i e u insta-bihty as discussed f o r Eq. 12. For instance, F i g . 10 shows a sample o f the model test results i n the time d o m a i n and frequency domain under regular waves w i t h wave f r e -quency i n 9 rad/s or 0.7 s i n period.

406 J Mar Sci Technol (2016) 21:396-415

F i g . 8 The coupling matrix and vector for the whole equation of motion

1/

Aim

MATRIX/

MATRIX^

T a b l e 5 Comparison o f natural frequency among model test, theo-retical analysis and numerical simulation f o r SPM model

0 1 0 2 0 3 0 4 0 Circular frequency [rad/s]

( b )

F i g . 9 Free decay test results f o r SPM model, a Time domain, b frequency domain

F r o m F i g . 10a, it can be clearly observed that, the pitch m o t i o n is not harmonic any more after 60 s. F F T analysis is carried out f o r the data f r o m 60 to 100 s and the results i n frequency domain is given by F i g . 10b where two signifi-cant peaks are observed. The peak marked by the circle coiTesponds to the wave frequency w h i c h means the

Item Test Theoretical analysis Simulation Unit

5.14 4.57 4.83 rad/s

CO2** 1.46 2.0 1.81 rad/s

* coi is natural frequency f o r the pitch dominant mode

** 0J2 is natural frequency f o r the surge dominant mode

harmonic component and the peak marked by the triangle represents the sub-harmonic component w h i c h has the m o t i o n frequency as half o f the wave frequency. I t is f o u n d that, f o r this case, the amplitude of the sub-harmonic response is even larger than the harmonic response.

Figure 11 shows the comparison o f R A O s f o r harmonic response between the model tests and numerical simula-tions results f o r surge, heave and pitch. The horizontal axis shows the incident wave frequency. The value o f the ver-tical axis is obtained by taking the ratio between the amplitude o f the first order of the response and the amplitude o f the first order of the wave amplitude (for surge and heave) or wave slope ( f o r pitch). The tendency can be reproduced by the simulation except the resonant frequency range around wave frequency 5 rad/s. The heave m o t i o n is relatively small due to the restriction o f tether. Figure 12 shows the same results o f F i g . 11 but w i t h the comparison o f sub-harmonic response w h i c h is marked by suffix "sub". For the sub-harmonic response, the horizontal axis means the wave frequency w h i c h is same w i t h F i g . I I . The pseudo-RAO is obtained by non-dimensionalizing the sub-harmonic response (e.g., the triangle i n F i g . 10b) by

J Mai- Sci Technol (2016) 21:396-415 407

•a 3

•— _\

A

__ ( 3 Harmonic response

jy^ Sub-harmonic response

-J-

u

V—

-0 5 1 -0 1 5 2 -0 2 5 3 -0 Circular frequency [rad/s]

( b )

Fig. 10 Measured pitch motion o f SPM model at wave frequency 9 rad/s. a Time domain, b frequency domain

the amphtude o f the first order o f the wave (amplitude or slope) as well so that the value o f harmonic and sub-har-monic response can be compared directly. The onset o f sub-harmonic response is observed f r o m the model tests when the wave frequency is around twice the natural f r e -quency (marked by the circle) and the value o f sub-har-monic response is significant compared w i t h the value o f harmonic response. A similar behavior can be predicted w e l l b y the simulation.

The respecfive test cases are repeated twice as indicated i n Table 2. F r o m the comparison between the t w o test cases, we can c o n f i r m the small scatter except f o r the resonant f r e -quency range. I t is noted that even a slight difference o f configuration/installation o f the model may induce a large influence on the test results due to the small scale ratio. For example, a difference o f 5 m m i n draft may change the pre-tension value by about 30 % w h i c h further results i n the shift o f the natural frequency b y about 14 % . Careful attention was paid to the installation work. A s the series o f model tests were completed i n different days and the model was installed several times, i t was impossible to keep the identical draft during the whole model tests. This may contribute some experimental uncertainty w h i c h influences the results i n l o w frequency range (close to natural frequency) to some extent.

Figure 13 shows the frequency characteristics o f the tension variation w h i c h includes the R A O f o r harmonic

4 6 S 1 0 1 2 1 4 16 Wave frequency [rad/s]

(a) 18 2 . 5 2 1.5 1 0 . 5 —1— Sim Exp 1-1 j -Kl t 1 t 1 t t 1 ( 1-4 6 8 1 0 1 2 11-4 16 Wave frequency [rad/s]

( b ) 5 4 O < 3 CC 2 1 ! Sim 1 Exp

\

Wave frequency [rad/s] (c)

F i g . 11 RAOs o f motions o f SPM model, a Surge, b heave, c pitch

component and pseudo-RAO f o r sub-harmonic component. The vertical value of R A O is calculated by taking the ratio between the amplitude o f the first order o f the tension and the first order o f buoyancy variation (pAgi], p: water den-sity, A: cross-section area, g: gravity acceleration, )/.• first order o f wave amplitude). The pseudo-RAO f o r sub-har-monic components is obtained by non-dimensionalizing the corresponding order tension by the first order o f buoyancy variation as w e l l . A n acceptable correlation between the tests and simulation is obtained f o r both the R A O and pseudo-RAO. For the harmonic R A O results, a small discrepancy is observed f r o m wave frequency 4.5-5.0 rad/s where the linear resonant response o f pitch occurs. The sub-harmonic tensions are usually small compared w i t h the harmonic component.

408 J M a r Sci Technol (2016) 21:396-^115 — 1 — Sim - - K — Sim_sub • Exp n Pvn c;(ih .. — 1 — Sim - - K — Sim_sub • Exp n Pvn c;(ih .. , • X ><0 r>x^X)^|g)<0 X X-X--• • X ><0 r>x^X)^|g)<0 _{X X-X-- ^ — X E )• '} 6 8 1 0 1 2 1 4 Wave frequency [rad/s]

(a) 2.5 o 2 < cc 6 1.5 <U V) a 1 oa o R A 0.5 0 — ) — Sim - X - - Sim_sub • Exp • Exp_sub -a-6 8 1 0 1 2 1 4 Wave frequency [rad/s]

(b) 16 18 —1—s ~ X — S • E m m_sub ~ <P (p__sub —1—s ~ X — S • E m m_sub ~ <P (p__sub • E m m_sub ~ <P (p__sub

1

j Res ults of Fig. 10

\

ft

6 8 1 0 - 1 2 1 4 Wave frequency [rad/s]

(c)

F i g . 12 Comparison between harmonic and sub-harmonic compo-nents o f motions of SPM model, a Surge, b heave, c pitch

g

1

Wave frequency [rad/s]

F i g . 13 Comparison between harmonic and sub-harmonic compo-nents o f mooring tension o f SPM model

3.5 3 ^ 2.5 OJ 2, J: 1.5 u °- 1 0.5 0 0 5 1 0 1 5 2 0 2 5 Circular frequency [rad/s]

(b)

Fig. 14 Free decay test results f o r T L S P A R model, a Time domain, b frequency domain

5.2 T L S P A R model

5.2.1 Free decay test

T o obtain the natural frequency o f pitch f o r T L S P A R model, a free decay test is also caiTied out and the time domain and frequency domain results are given i n F i g . 14. Besides the higher natural frequency coi, the m o t i o n mode w i t h l o w e r natural frequency 0)2 is also evidenced as dis-cussed i n the theoretical analysis. Table 6 shows the nat-ural frequency o f T L S P A R model w h i c h is obtained by the f r e e decay test, theoretical analysis and numerical simulation. A good conelation exists f o r the higher natural f r e -quency CO I. Due to the large damping, the lower natural frequency CO2 cannot be obtained b y the model test so clearly. A f t e r I F F T analysis focusing on the frequency range around natural frequency oJi, the damping coefficient is obtained w h i c h is about 3 % o f critical damping and i t is utilized i n the f o l l o w i n g simulation. The free decay test f o r the m o t i o n mode i n r o l l direction was also conducted and the measured natural frequency corresponding to the r o l l m o t i o n locates around 3.4 rad/s.

J M a r Sci Technol (2016) 21:396-415 ₄₀₉

T a b l e 6 Comparison o f natural frequency among model test, theo-retical analysis and numerical simulation f o r T L S P A R model Item Test Theoretical analysis Simulation Unit

3,07 3,16 2.92 rad/s

« 2 * *

_-

* CÜ, is natural frequency for the pitch dominant mode ** £02 is natural frequency f o r the surge dominant mode

(_") Hannonic response ^ \ Sub-harmonic response 7 6 5 4 3 2 1 0 10 15 20 Circular frequency [rad/s]

( b )

Fig. I S Measured pitch motion of T L S P A R model at wave fre-quency 6.2 rad/s. a Time domain, b frefre-quency domain

5.2.2 RAO comparison

As mentioned previously, the obvious sub-harmonic m o t i o n was observed i n the tank test f o r the T L S P A R model. Specially, one typical case w i t h wave frequency 6.2 rad/s (wave period 1.0 s) is selected and the cone-sponding pitch motion is shown i n F i g . 15. Figure 15a shows the time domain results f o r the pitch m o t i o n where we can find that, the harmonic pitch occurs i n i t i a l l y and the pitch m o t i o n gradually transforms f r o m harmonic to sub-harmonic response after several cycles. F i n a l l y , the pitch m o t i o n becomes prominently sub-harmonic. I f we conduct the F F T analysis f o r the time domain data between 25 s to 40 s, the sub-harmonic response w i l l be more clearly checked w h i c h is shown i n F i g . 15b. 1.5 1.25 1 o < 0.75 0.5 0.25 0 _ —I—Sim i 1 • Exp 6 7 8 9 10 Wave frequency [rad/s]

(a) 12 1.5 1.25 1 O < 0.75 I t : 0.75 0.5 0.25 0 -__ Sim Exp -__ Sim Exp -r -__ Sim Exp 4 — 6 7 8 9 10

Wave frequency [rad/s] (b) 11 12 0.5 0.4 0.3 0.2 0.1 0 —1—Sim • exp 1

^^^^

(c)

Fig. 16 RAOs of motions o f T L S P A R model, a Surge, b heave, c pitch

Figure 16 shows a comparison o f harmonic component f o r the m o t i o n o f T L S P A R between model test and simu-lation. It turned out that, the simulation can predict the harmonic response o f model test w i t h good accuracy. As the natural frequency f o r the pitch dominant mode locates around 3 rad/s, the surge and pitch increases w i t h the decrease o f wave frequency. For heave m o t i o n , it has a very l i m i t e d value f o r most wave frequency. However, f o r wave frequency f r o m 5.5 to 6.5 rad/s w h i c h is around the twice o f the natural frequency, the harmonic component o f heave can have a significant value w h i c h is caused by the large sub-harmonic response existing i n surge and pitch. This behavior is not observed i n S P M model as the external

410 J Mai- Sci Teclinol (2016) 2 1 : 3 9 6 ^ 1 5 1.5 S 0-75 Q. n.5 0.25 .... — \ — Sim - • X —- Sim_sub • Exp • Exp_sub • " % / — k o -— X — « < - X — X — X X 5 5 7 8 9 10 Wave frequency [rad/s]

(a) 11 0 1.25 Ó 1 1 0.75 Q. «3 0.5

s

Wave frequency [rad/s] (b) 11 12 o 3.5 3 2.5 2 1.5 1 0.5 0 I I • I * I — I— S i m - ^ - - S i m _ s u b " • Exp • Exp_sijb

Response with motioü frequeocy 3.1 rad/s under wave frequency 6.2 rad/s

I I

Response with motion frequency 6.2 rad/s ^uiider wave frequency 6.2 rad/s

4 5 5) 7 8 9 10 11 12 / Wave frequency [rad/s]

Results o f Fig. 15

(C)

Fig. 17 Comparison between harmonic and sub-harmonic compo-nents o f motions o f T L S P A R model, a Surge, b heave, c pitch

4 5 6 7 8 9 10 Wave frequency [rad/s]

Fig. 18 Comparison between harmonic and sub-harmonic compo-nents o f the mooring tension o f TLSPAR model

the sub-harmonic heave motion, the measured value is larger than the prediction by the numerical simulation when the wave frequency is equal to 6.7 rad/s. I n fact, due to the existence o f w i n d turbine, the moment o f inertia f o r T L S P A R model around x (roll) and y (pitch) is slightly different and the natural frequency f o r r o l l is slightly larger than the natural frequency f o r pitch (see F i g . 7 ) . W a v e frequency 6.7 rad/s is just twice the r o l l natural frequency and a larger sub-harmonic roll can easily happen even when the wave propagates along x-axis due to the unsymmetrical influence of disturbance. T h i s influence exceeds the scope o f this paper and i t w i l l not be discussed here.

A comparison f o r the mooring tension is shown i n Fig. 18. A good agreement f o r both the harmonic and sub-harmonic components o f the m o o r i n g tension can be obtained f o r T L S P A R model. The sub-harmonic response is relatively large when the sub-harmonic m o t i o n occurs (wave frequency f r o m 5.5 to 6.5 rad/s).

5.3 Discussion on sub-harmonic motion

5.3.1 Source of sub-hcinnonic motion

force i n heave direction is relatively small compared w i t h the T L S P A R (see Sect. " 5 . 3 . 4 " ) .

A comparison of R A O s and pseudo-RAOs (non-di-mensionalized b y the amplitude o f the first order o f the wave, same manner w i t h S P M model) f o r both harmonic and sub-harmonic response between model test and simu-lation is given i n F i g . 17. The data name w i t h a suffix

"sub" represents the corresponding sub-harmonic response. I t can be f o u n d that, an obvious sub-harmonic response occurs when the wave frequency are ranged f r o m 5.5 to 6.5 rad/s. I t is noted that the amplitude o f the sub-harmonic m o t i o n is even larger than the m a x i m u m amplitude o f the linear response.

The correlation between the model tests and simulation results f o r the sub-harmonic m o t i o n is also acceptable. For

T o c l a r i f y the mechanism o f the nonlinear behavior o f the tethered-buoy system, a comparison f o r sub-harmonic pitch w i t h f o u r d i f f e r e n t simulation models f o r T L S P A R is presented i n F i g . 19. W i t h considering the nonlinear influence o f tether tension, Sim_sub and Sim(NC)_sub show the simulation results w i t h and w i t h o u t coupling between heave and pitch f o r T L S P A R model, respectively (See, E q . 12). S i m ( L M ) _ s u b and S i m ( L M N C ) _ s u b gives the same comparison without all the nonlinear influence o f tether [16]. The figure shows that, f o r the sub-harmonic response, only the models w i t h the coupling e f f e c t between the buoy and m o o r i n g can reproduce the same behavior when wave frequency is twice as large as the natural f r e -quency o f T L S P A R model. It means that, the sub-harmonic behavior is m a i n l y caused by the nonlinear influence o f

J Mar Sci Tectinol (2016) 21:396-415 411 4 3.5 O Sirr _sub - _{— .} - - Sim(NC)_5ub -T4--Sim(LM)_sub --K--Sim(LMNC)_5ub -O Exp_sub 1 — - - Sim(NC)_5ub -T4--Sim(LM)_sub --K--Sim(LMNC)_5ub -O Exp_sub '1 -4 5 6 7 8 9 10 11 12 Wave frequency [rad/s]

F i g . 19 Comparison among the sub-harmonic pseudo-RAOs o f pitch motion f o r T L S P A R obtained by using different numerical models tether tension. Meanwhile, by comparing the model "Sim_sub" and " S i m ( N C ) _ s u b " , i t can be k n o w n , due to the constrained heave m o d o n , that the nonlinear coupling between heave and pitch has very h m i t e d influence on the subharmonic motion compared w i t h the nonlinear i n f l u -ence o f v a r y i n g tether tension.

5.3.2 Motion ratio between pitch of buoy and tether angle

The m o t i o n ratio X affects the Mathieu instability o f teth-ered-buoy system a lot as it can amply the influence of the variation o f tether tension (see Eq. 21). T w o typical cases w i t h T L S P A R model are selected and the time domain results obtained f o r the cases f r o m model test and simu-lation are presented i n Figs. 20 and 2 1 . They include two series of data, the pitch m o t i o n o f T L S P A R model (marked by " P i t c h " ) and tether angle (marked by "Tether angle") w h i c h is defined by öi i n Fig. 1. I t turned out that, f o r the both cases, the simulation can w e l l reproduce the model test not only f o r the pitch o f T L S P A R model but also the tether angle.

For the case w i t h wave frequency 9.0 rad/s (wave per-iod: 0.7 s) when the sub-harmonic m o t i o n does not occur, we can find f r o m Fig. 20 that, the tether angle is relatively small compared w i t h the pitch m o t i o n of T L S P A R model. Figure 21 shows the case w i t h wave frequency 6.2 rad/s (wave period: 1 s) when the sub-harmonic motion occurs. It can be observed that, the tether angle has the opposite phase compared w i t h the pitch o f T L S P A R model, which coincides w i t h the theoretical analysis. The motion ratio X obtained f r o m the model test, theoretical analysis and simulation are listed i n Table 7. The theoretical analysis turns out to be reasonable by comparing w i t h model test and simulation results.

5.3.3 Stability diagram

The stability diagram f o r the tethered-buoy system can be easily drawn according to Eq, 2 1 . A d d e d mass and

•n -0.25 Pitch T e t h e r ar gle 52 163/ 64 Time [s] (a) Model Test

0.25 S 0

A A

1

1

1

F i g . 2 0 T i m e domain results o f pitch motion o f T L S P A R model at wave frequency 9.0 rad/s. a Model test, b simulation

Fig. 21 Time domain results o f pitch motion o f T L S P A R model at wave frequency 6.2 rad/s. a Model test, b simulation

damping coefficients i n Eq. 21 are obtained by the poten-tial theory. The natural frequency and m o t i o n ratio X can be calculated based on Eqs. 9 and 10, respectively. Figure 22

412 J Mar Sci Technol (2016) 2 1 : 3 9 6 ^ 1 5

T a b l e 7 M o t i o n ratio for TLSPAR model

Item Test Analysis Simulation

X - 0 . 4 2 8 - 0 . 3 9 4 - 0 4 1 6

6 6.5 7 7.5 a 8.5 Wave frequency [rad/s]

F i g . 22 Stability diagram f o r T L S P A R model

shows the stabiUty diagram f o r the T L S P A R model. The horizontal axis represents the wave frequency and the vertical axis gives the wave amplitude. The solid line marked by "Mathieu amp" shows the results by substi-tuting vertical hydrodynamic force obtained f r o m potential theory into Eq. 2 1 . The dashed line represents the results b y regarding the tether tension amplitude obtained f r o m simulation as the i n Eq. 2 1 . "Break amp" line shows the wave breaking criterion ( f ) ^ ^ ^ = itanh(2f^^) [20]. The figure indicates that when the wave amplitude locates higher than the solid line or dashed line, the system w i l l become unstable. It is f o u n d that, f o r the case w i t h wave frequency f r o m 5.5 to 6.5 rad/s w h i c h is around the twice the natural frequency (3.07 rad/s), the system turns to be unstable. Then, the sub-harmonic m o t i o n should be observed i n both model test and numerical simulation. The wave amplitude used i n the model tests is shown by the triangles. The filled triangles represent the sub-harmonic cases w h i c h have been detected by the model test and simulation. A good coiTelation can be f o u n d among the theoretical analysis, model test and simulation f o r the prediction o f sub-harmonic response.

5.3.4 Different mechanism between SPM and TLSPAR models

W h e n we revisit the model test results i n F i g . 10a and i n F i g . 2 I a , i t is f o u n d out that the S P M and T L S P A R models shows slightly different property when the sub-harmonic m o t i o n occurs. For SPM model ( F i g . lOa), both harmonic and sub-harmonic pitch exists w i t h slow beating. On the other hand, f o r T L S P A R model ( F i g . 21a), a more "pure" and "prominent" sub-harmonic pitch m o t i o n is observed.

£ -15

I oyyyywwwwvw

ar 85 90 9

Time [s] (a)

Fig. 23 Simulation results f o r the pitch motion o f SPM and T L S P A R models at the frequency when the sub-harmonic motion prominently occurs, a Typical pitch motion o f SPM model at wave frequency 9.6 rad/s, b typical pitch motion o f T L S P A R model at wave frequency 6.2 rad/s T a b l e 8 Non-dimensional acceleration (a/a^) SPM T L S P A R Surge 5.08 1.19 Heave 1.00 1.00 Pitch 36.48 1.44

w h i c h includes much less harmonic response. A similar phenomenon can be also f o u n d i n the simulation results i n Fig. 23a and b . • Figure 23 shows a typical nonhnear response f o r S P M model (wave frequency 9.6 rad/s) and T L S P A R model (wave frequency 6.2 rad/s) w h i c h is obtained by the simulation.

These discrepancies can be interpreted by the d i f f e r e n t hydrodynamic properties. Table 8 compares the hydrody-namic forces i n terms o f the relative acceleration w h i c h is non-dimensionalized by the heave acceleration. A l l the acceleration is obtained by taking the ratio between the respective hydrodynamic force and mass (including added mass). Since the acceleration is obtained by subdividing the force b y the mass, the values i n the table may be regarded as non-dimensionalized hydrodynamic force. I t is k n o w n f r o m the table that, f o r SPM model, the force i n surge and pitch are m u c h larger than the T L S P A R model w h i c h means that the external force about surge and pitch f o r S P M model gives more infiuence on the m o t i o n compared w i t h the T L S P A R model. As the external force about surge and pitch can be regarded as the excitation f o r the har-monic response, the harhar-monic response of S P M model is more significant than the T L S P A R model. The slow beat-ing is always observed i n the M a t h i e u type problems as a

J Mar Sci Technol (2016) 2 1 : 3 9 6 ^ 1 5 413 30 -30 I Time [s] (a) 30 -30 I 1 Time [s] (b)

Fig. 24 Simulation results for pitch motion of SPM model with different hydrodynamic coefficients, a Pitch motion of SPM model with the hydrodynamic external force in surge and pitch decreased to 10 % o f the original values, b pitch motion o f SPM model with the hydrodynamic external force in surge and pitch decreased to 10 % o f the original values while increasing the hydrodynamic damping coefficient increased by 1.2 times

transient behavior [21] and it vanishes after a sufficiently long time depending on the system damping; The slow beadng can be suppressed for the large damping.

T o show the influence of external force and damping, more simulation f o r SPIVI model w i t h same wave condition as F i g . 23a is conducted with 9 0 % smaller external force about surge-pitch direction and w i t h smaller external force (same w i t h before) and 1.2 times larger damping. The simulation results are shown b y F i g . 24.

It is f o u n d f r o m Fig. 24a that the harmonic response vanishes as the hydrodynamic force i n surge and pitch m o d o n decreases. Based on the small force i n Fig. 24a, through manually increasing the damping, the slow beating w i l l also disappear and the response o f S P M model can be regarded as same behavior as T L S P A R model.

I n the same manner, we can obtain results similar to S P M model f o r T L S P A R model i f we m o d i f y the hydro-dynamic coefficients by decreasing the damping and increase the external force i n surge and pitch direction. Figure 25 shows the simulation pitch m o t i o n under same wave condidon as Fig. 23b w i t h 1 % o f o r i g i n a l damping and w i t h smaller damping (same as before) and 5 times larger external force i n surge and pitch direction. It is seen that, the conclusion derived f r o m F i g . 24 can be proven again by F i g . 25.

The sub-harmonic motion may pose technical chal-lenges i n the operability, maintainability, structural safety, etc. of the system. I n this regard, it is important to control the sub-harmonic motion. Equation 21 indicates that, to get

30

- s o l 1

Time [s] (b)

F i g . 25 Simulation results f o r pitch motion of T L S P A R model with different hydrodynamic coefficients, a Pitch motion of TLSPAR with the hydrodynamic damping coefficient decreased to 1 % o f the original value, b Pitch motion o f T L S P A R w i t h the hydrodynamic damping coefficient decreased to I % of the original value which increasing the hydrodynamic external force i n surge and pitch increased by 5 times

a prominent sub-harmonic m o d o n , the nonlinear influence o f varying tether tension should be increased by increasing vertical hydrodynamic force f^. Increasing the m o t i o n ratio X between L2 and L ] i n E q . 11 can a m p l i f y the influence o f vertical hydrodynamic f o r c e F o r T L S P A R model, due to the existence o f the large diameter part w h i c h is near to the free-water-suiface, the vertical hydrodynamic force is rel-atively larger compared w i t h the normal c o l u m n design as shown by the S P M model. Thus, the floater shape such as T L S P A R model is advantageous i n obtaining sub-harmonic motion.

Meanwhile, we cannot expect a prominent sub-harmonic pitch m o t i o n by simply increasing the wave amplitude to obtain the larger hydrodynamic vertical force even when the sub-harmonic frequency o f the external force hits the natural frequency. As is the case w i t h S P M model, a combined harmonic and sub-harmonic m o t i o n may appear when the hydrodynamic force i n horizontal direction and hydrody-namic pitching moment is relatively large. They should be kept sufficientiy small to decrease the harmonic response i n the background o f the sub-harmonic component.

6 C o n c l u s i o n

I n this research, the mechanism o f the nonlinear m o t i o n o f t w o tethered-buoy systems has been studied. A theoretical analysis is developed to predict the natural frequency, and hydrostatic and hydrodynamic stability of the tethered-buoy systems. M o d e l test is carried out to observe the

414 .1 Mar Sci Technol (2016) 21:396-415

nonlinear behavior o f buoy motion. Nonlinear numerical model has been established and validated against the model test i n terms o f both buoy motions and tether tension variations. The different sub-harmonic behavior between S P M and T L S P A R models has been discussed. W e may conclude as f o l l o w s .

• A sub-harmonic pitch motion is observed f o r the tethered buoy system. The behavior may be described by M a t h i e u instability. I t occurs when the Mathieu instabihty condition is satisfied, i.e., the frequency o f the external force is twice the natural frequency o f the pitch m o t i o n o f the buoy.

• The nonlinear influence o f time-varying tether tension gives a dominant influence on the sub-harmonic pitch motion o f the buoy.

• The coupling between heave and pitch f o r the buoy gives only a l i m i t e d effect to the sub-harmonic motion as the small heave motion w h i c h is constrained by the tether.

• I n the design, we can suppress the sub-harmonic motion by decreasing the vertical hydrodynamic force at the frequency where the sub-harmonic motion easily occurs. I t can be also mitigated by adopting the smaller motion ratio X w h i c h can be approximated as a ratio o f the center o f gravity height La to the tether length L j . • Large external force i n horizontal direction and pitch-ing moment can induce a harmonic m o t i o n i n a combined manner when the sub-harmonic motion occurs. Then, the slow beating m o t i o n is observed w i t h the sub-harmonic motion. I t can be decreased or eliminated by increasing the system damping.

The present research results indicate that the Mathieu instability due to floater-mooring coupling may occur even f o r a catenary-moored floater system when the mooring load undergoes the wave-induced variation and the moor-ing affects the floater motion. Such an example may be f o u n d , e.g., i n the yaw m o t i o n o f a floating structure moored by catenaries. The mooring load has a variation w i t h the waves and the tensile load i n the m o o r i n g hnes gives a vaiiable restoring moment around the yaw axis [22]. The discussion made i n the present research may hold to such a problem, too.

Aclinowledgments It is acknowledged that the present research was financially supported by JSPS K A K E N H I , Grant Number 26630453.

A p p e n d i x 1

The m o t i o n ratio X(oi) between the amplitude o f ö i , Ö2 is given i n Eq. 10. I n this appendix, i t w i l l be proven that the f o l l o w i n g inequahties hold.

A ( f f l i ) < 0

X{(Ü2) > 0

(25) (26)

Inequality (25) can be proved by contradiction.

First, it is assumed that /'.(coi) > 0 , so we can get (to be convenient, the added mass and added inertia moment are included i n the in and / i n the f o h o w i n g derivation):

To > ojjniLi =

oj]A _ fi+^/^;^-'^^^A _ g + - 4AC

T~

(27)

where, the coefficients / I , fi, C are defined as Eq. 7 w h i c h is shown below,

A = inlLi

B = [m{Ll + Lila) +L]-TQ+ niGMo • U • Fgo C = TO(TOL2 + FBO-GMO)

Coefficient B can be written as a f u n c t i o n as coefficients A and C:

B ^ ^ ^ + mToLl + ToI (28)

Inequality (27) can be transformed to (29)

2ToI -B> VB^ -4AC (29) Take the square value f o r both sides o f (29)

{2ToI-Bf>B^-4AC (30) Substituting the Eq. 28 into inequality (30), inequality

(31) is derived:

1,1^11 <0 (31)

I t is obviously knows that, inequality (31) cannot h o l d w h i c h means that the assumption X{coi) > 0 , is wrong.

Therefore, the inequality (25) is proved.

Inequality (26) can be proved i n the same manner.

A p p e n d i x 2

The equation o f m o t i o n ratio X Eq. 10 can be written i n inverse f o r m : X{o,) 4>i where 5 = —

ml

© 0 = J m natural frequency i f the buoy is regarded as a mass m attached and suspended to a pendulum by the tether w i t h pre-tension TQ.