New insight into the wave-induced nonlinear vertical load effects of ultra-large container ships based on experiments

J Mar Sci Teclinol (2013) 18:87-114 D O I 10.1007/S00773-012-0186-X

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

New insight into the wave-induced nonhnear vertical load effects

of ultra-large container ships based on experiments

S u j i Z h u • Torgeir M o a n

Received: 14 October 2011/Accepted: 27 June 2012/Published online: 4 August 2012 © J A S N A O E 2012

Abstract Accurate estimation o f the wave-induced extreme hogging vertical bending moment ( V B M ) is o f v i t a l importance f o r the design o f container ships because container ships are n o r m a l l y under hogging conditions i n still water. A c c o r d i n g to the empirical formulas proposed by the classification society rules, the design hogging V B M can be approximately 20 % smaller than the design sagging V B M f o r vessels w i t h small b l o c k coefficients. High-order harmonic components i n the vertical load effects, w h i c h are induced by the nonlinearities i n the hydrodynamic forces and ship h u l l geometry, contribute to the asymmetry. Pre-vious studies have shown that the nonlinear hydrostatic and F r o u d e & y l o v forces increase the sagging V B M s i g n i f i -cantly. CuiTent numerical tools are able to reveal this asymmetry to a certain extent. There is, however, little focus on the nonlinear pressure under the bow bottom, w h i c h is a more l i k e l y contributor to the hogging V B M . Several unexpected phenomena have been observed f o r large container ships. The wave-frequency sagging and hogging V B M s f o l l o w e d each other closely, and hence d i d not reflect the significant nonlinear factors as expected. I n this paper, the test data o f t w o ( 8 6 0 0 - T E U and 13000-TEU) ultralarge containership models i n both regular and i i r e g -ular head waves are systematically studied. I n reg-ular waves, the influence o f the second and third harmonics o n the fundamental hogging peaks and sagging troughs is estimated b y comparing both the amplitude and phase d i f -ference relative to the first harmonic peaks. I n irregular waves, the focus is on the statistical characteristics o f the

S. Z h u ( E l ) • T. Moan

Centre f o r Ships and Ocean Structures, Norwegian University of Science and Technology, Otto Nielsens veg 10, 7491 Trondheim, Norway

e-mail: suji.zhu@ntnu.no

wave-induced nonlinear vertical load effects. T o achieve a balance between results i n regular and hregular waves, the influence o f the second harmonics is evaluated through bispectral analysis.

K e y w o r d s H o g g i n g • Vertical bending moment • Ultra-large container ship • M o d e l test • Head waves

1 Introduction

The wave-induced vertical bending moment ( V B M ) nor-m a l l y shows a strongly nonlinear behaviour f o r ships w i t h small b l o c k coefficients. Experimental data and full-scale measurements [ 1 - 6 ] revealed that the sagging V B M was significantly larger than the hogging V B M . The classifica-t i o n socieclassifica-ty rules, classifica-thus, have proposed alclassifica-ternaclassifica-tive e m p i r i c a l formulas to determine the wave-induced design sagging and hogging V B M s [ 7 ] . Current time-domain simulation tools [ 8 1 0 ] that consider the nonlinear hydrostatic and F r o u d e -Ki-ylov forces can reflect the strong asymmetry o f the ver-tical loads to some extent. I n regular waves, the asymmetry o f the vertical loads was essentially caused b y the mean s h i f t value and the second harmonic response [ 1 1 ] . The fast Fourier transformations (FFTs) o f the heave and p i t c h motions o f t w o W i g l e y h u l l f o r m s [12] showed m i n o r h i g h -order harmonic peaks, while the second and t h i r d harmonic peaks o f the V B M s were clearly distinguished. E x p e r i -mental results o f a container vessel (length between per-pendiculars Lpp = 240 m ) w i t h a b l o c k coefficient o f 0.6 i n steep, regular waves [13] showed that the second, third, and f o u r t h hannonics increased the absolute values o f the sag-g i n sag-g trousag-ghs step by step w h i l e the absolute values o f the hogging peaks d i d not change much. I n in^egular waves, the Rayleigh distribution underestimated the sagging response

88 J Mar Sci Teclmol (2013) 18:87-114

and overestimated tlie hogging response as l o n g as the ship h u l l was considered r i g i d [ 1 4 - 1 7 ] .

The above observations are mainly drawn based on conventional ships w i t h a length below 250 m . The non-linear vertical loads may show d i f f e r e n t behaviour when the ship length nears 300 m . The test data o f a container-ship model w i t h a full-scale length between perpendiculars (Lpp) o f 281 m [18] showed that the measured hogging V B M was slightly larger than the sagging V B M f o r both the rigid-body and flexible-body responses. The numerical simulations based on a nonlinear hydroelastic program, however, indicated that the sagging V B M was noticeably larger than the hogging V B M . The full-scale measurements o f a 4 4 0 0 - T E U container ship [19] showed that the wave-frequency sagging and hogging stresses f o l l o w e d each other closely, w h i c h did not reflect the significant nonlinear factors as expected. The observed hogging stress was even larger than the sagging stress when the mean period o f the wave-frequency V B M was smaller than 8.6 s. The extreme hogging stress was observed to be above the rule values i n f o u r irregular wave conditions, although the significant wave height was still below 7 m . M a o et al. [20] f o u n d that the r i g i d - b o d y V B M response can be w e l l approximated by Gaussian assumption. The full-scale measurements o f an 8 6 0 0 - T E U Post-Panamax container vessel over 9 months showed that the measured m a x i m u m hogging V B M was f o u n d to be approximately 18 % larger than the measured m a x i m u m sagging V B M [ 2 1 ] . Stoihaug et al. [22, 23] presented the measured m a x i m u m hogging and sagging values f o r t w o ultra-large container ships i n sixteen to eighteen d i f f e r e n t UTegular head waves. Under certain iiTcgular wave conditions, the i n a x i m u m hogging V B M s amidships were even larger than the m a x i m u m sagging V B M s amidships. A l t h o u g h these unexpected phenomena mentioned above have been w i d e l y reported, there is no appropriate explanation w h y the asymmetry between sag-ging and hogsag-ging V B M s no longer exists. I n other words, most researchers still doubt that hogging V B M can be equal to or larger than sagging V B M . Systematic validation and appropriate explanation are i n urgent need to prove that the current expectations for behaviour are no longer applicable f o r ultra-large container ships so that further investigations can be continued.

The nonlinear pressure under the b o w b o t t o m is believed to be the m a i n reason f o r the increase i n the hogging V B M s . Z h u et al. [24] presented the experimental results o f a backbone m o d e l i n steep, regular waves. Due to the large, flat bottom o f the backbone model, the change i n the added mass induced a large suction pressure, and the flat bottom further increased the integration area of the suction force. Consequently, the hogging V B M was approximately 30 % larger than the sagging V B M . Z h u [25] indicated that the change i n the added mass contributed significandy to the

substantial negative pressure. The nonlinear pressure near the ship b o w was investigated f o r a high-speed vessel [ 2 6 - 2 8 ] . The sensor under the b o t t o m revealed a large negative pressure d u r i n g the b o w - u p stage. M o r e o v e r , the nonlinearity o f the pressure under the bottom was even stronger than that at the waterline. Furthermore, the nonlinear suction forces under the b o w b o t t o m were also reported to induce the vertical vibrations f o r a catamaran and a large ocean-going vessel [29, 3 0 ] . D r u m m e n et a l . [31] f o u n d that the vibrations started i n the h o g g i n g process f r o m the f u l l - s c a l e measurements o f a container vessel (block c o e f f i c i e n t o f 0.672). A l t h o u g h no pressure sensors were installed under the b o w o f the realistic container vessel, the suction force was the most probable excitation source. S i m i l a r phenomena, however, have not been observed d u r i n g the m o d e l tests o f a similar c o n -tainer m o d e l ( b l o c k c o e f f i c i e n t o f 0.707). This i m p l i e s that the nonlinear suction f o r c e m a y be more significant i n realistic vessels.

The numerical solution o f the exit-water p r o b l e m is complex, especially i n the presence o f a f o r w a r d speed and severe waves. Therefore, i t is a compromise to investigate the influence o f high-order harmonic responses instead. A l t h o u g h the amplitudes o f the second harmonics o f the V B M s were provided [2, 4 ] , the phase differences o f the second-order haimonics relative to the first harmonic peaks have rarely been reported. I n in-egular waves, the responses o f the vertical load effects can be regarded as non-Gaussian random processes. The bispectrum, w h i c h is an ensemble average o f a product o f thi-ee spectral components, has proved to be a p o w e r f u l diagnostic tool i n experimental studies of nonlinear wave interactions [ 3 2 ] . O c h i and A h n [33] adopted the bispectral analysis to separate the n o n -linear energy components f r o m the energy spectrum. Based on the bispectral analysis, the asymmetry o f the w i n d waves i n a laboratory tank was successfully explained [ 3 4 ] . I n general, there is little focus on the application o f the bispectral analysis to the nonlinear vertical loads that ships experience.

The above phenomena have not been accounted f o r properly i n the numerical predictions o f the vertical loads f o r ultra-large container ships. N u m e r i c a l results o f an ultra-large container ship whose length was 320 m showed that the extreme sagging V B M amidships was more than twice the hogging V B M at the f o r w a r d speed o f both 0 knots and 5 knots [ 3 5 ] . M o a n et al. [36] noted that the container vessels i n general were under hogging conditions i n still water. The m a x i m u m still water hogging V B M during the 3,475 studied voyages was 123 % o f the max-i m u m allowed value accordmax-ing to l A C S rules, w h max-i l e the m a x i m u m still water sagging V B M was less than 10 % o f the prescribed value. Similar trends can also be f o u n d i n Huang and M o a n [ 3 7 ] . The possible underestimation o f the

J M a r Sci Teclinol (2013) 18:87-114 89

wave-induced liogging V B M is a great concern during tlie design process.

This paper provides some new insights into the wave-induced nonlinear vertical load effects f o r ultra-large container ships. The hogging V B M s are compared w i t h con'esponding sagging V B M s under d i f f e r e n t conditions. The comparisons are illustrative to prove the underesti-mation o f the wave-induced extreme hogging V B M . First, test results o f the 8 6 0 0 - T E U and 13000-TEU ultra-large container ships i n steep, regular waves at t w o f o r w a r d speeds are investigated. The contributions o f the second and third harmonics o f the V B M s to the fundamental hogging peaks and sagging troughs are evaluated i n terms o f the amplitude and phase difference relative to the first harmonic peaks. Secondly, the stochastic characteristics i n severe hregular waves are presented. The f o r w a r d speeds used are realistic, and are based on s i m p l i f i e d calculations o f speed reduction and captains' experience. Next, b i -spectral analysis is adopted to explain the asymmetry o f the wave-frequency V B M i n order to achieve a balance between results i n regular and iiTegular waves. Then, the factors influencing the rigid-body hog/sag ratio are listed, and the m a x i m u m hogging and sagging V B M s that ultra-large container ships may experience are discussed. F i n a l l y , uncertainty analysis is conducted, and the con-clusions are provided.

2 E x p e r i m e n t a l setup

A n international joint-industry project (JIP), w i t h hyundai heavy industries ( H H I ) , centre f o r ships and ocean structures (CeSOS), det norske Veritas ( D N V ) , bureau Veritas ( B V ) , and marintek ( h t t p : / / w w w . s i n t e f . n o / H o m e / M A R I N T E K y ) as partners, was established to investigate the influence o f w h i p p i n g and springing on the load effects i n ultra-large container ships, namely the 8 6 0 0 - T E U and 1 3 0 0 0 - T E U

Table 1 M a i n particulars o f the 8600-TEU and 13000-TEU

ultra-large container ships

Vessel type 8600-TEU 13000-TEU Length between peipendiculars, Lpp (m) 322.6 350.0

Breadth moulded, B (m) 45.6 48.2

Depth moulded, D (m) 24.6 29.85

Draught s c a n ü i n g s (m) 14.5 14.5

Mass (ton) 128307 169692

2-Node vertical natural frequency (Hz) 0.48 0.48 3-Node vertical natural frequency (Hz) 0.99 0.91

Scale o f model 1:45 1:45

Damping ratio to critical damping (%) 0.9 0.9 B l o c k coefficient 0.587 0.652

container ships designed by H H I . B o t h o f the m o d e l tests were carried out i n the t o w i n g tank at the M a r i n e Technol-ogy Centre i n Trondheim. The full-scale main particulars o f the two vessels are listed i n Table 1. For both ship models, the load condition during model test represents a realistic operational condition.

B o t h o f the models are made o f D i v i n y c e l l f o a m and w o o d w i t h coating on the surface. A s illustrated i n F i g . 1, the 13000-TEU model is composed o f f o u r r i g i d segments connected by three flexible joints. The V B M s were measured at 0.28 Lpp, 0.48 Lpp, and 0.75 Lpp before A P f o r the 8600-T E U vessel. 8600-The measured locations were 0.30 Lpp, 0.51 Lpp, and 0.72 Lpp before A P f o r the 13000-TEU vessel. Hereafter, f o r convenience, the measured locations are approximately described as 1/4 Lpp, 1/2 Lpp, and 3/4 Lpp before A P f o r b o t h models. The t w o models were both towed under the t o w i n g caniage through a transverse beam, w h i c h was fixed to the a f t segments. W u et al. [38] presented some comparisons between the experimental data and the numerical calcula-tions i n both regular and in-egular waves f o r the 1 3 0 0 0 - T E U vessel. A more detailed description o f these two models can be f o u n d i n reports b y Storhaug et al [22, 2 3 ] .

Selected test results i n steep, regular waves are g i v e n i n Table 2. Because b o t h o f the models were tested i n steep waves, the green water on deck is a concern. I n order to prevent water f r o m entering the models, the upper deck o f the 8 6 0 0 T E U m o d e l was wrapped w i t h a plastic m e m -brane and steep plates were installed near the b o w f o r the 13000-TEU vessel. There is only slight green water at 15 knots f o r b o t h models. D u et al. [11] indicated that green water can increase the hogging V B M and b o w flare slamming can increase the sagging V B M . Ships usually suffer speed reduction under high-wave conditions, either voluntarily or involuntarily. Therefore, the test results at 15 knots are addressed. The test data at 25 knots can be used to evaluate the influence o f the f o r w a r d speed on the nonlinear vertical load effects. The wave lengths used were 225.8, 258.1, 290.3, 322.6, and 354.9 m , respectively. The nonlinear characteristics o f the vertical load effects i n these waves deserve special attention. The m a x i m u m wave steepness is 0.065 at a wave height o f 15 m , w h i c h is s t i l l w i t h i n an acceptable range. The amplitudes o f the second and third harmonics o f the encounter waves are approxi-mately 4 and 2 % o f the first harmonic amplitude.

The wave spectra and f o r w a r d speed adopted i n iiTCgular waves are realistic considering the effects o f real operation to some extent. The 8 6 0 0 - T E U model was tested under a total o f 16 irregular wave conditions, and the 1 3 0 0 0 - T E U model was tested under 18 iiTegular wave conditions [22, 2 3 ] . This paper emphasises the results under the most severe wave conditions. As shown i n Table 3, f o u r irreg-ular wave conditions were chosen f o r the 8 6 0 0 - T E U vessel.

90 J Mar Sci Teciinol (2013) 18:87-114

Table 2 Test program i n steep, regular waves Test Test Wave Speed

state model height (knots) (m) A 8600-TEU 10 15 0.70, 0.80, 0.90, 1.00, 1.10 B 8600-TEU 10 25 0.70, 0.80, 0.90, 1.00, 1.10 C 13000-TEU 10 15 0.65, 0.74, 0.83, 0.92, 1.01 D 13000-TEU 15 15 0.65, 0.74, 0.83, 0.92, 1.01 E 13000-TEU 10 25 0.65, 0.74, 0.83, 0.92, 1.01

X denotes the incident wave length

Table 3 Selected irrei gular wave conditions f o r the 8600-TEU vessel

Condition A _ l A _ 2 A „ 3 A _ 4

Duration (min) 46.3 44.7 45.2 44.5

7-.(s) 7.5 (8.3) 9.5 (9.2) 11.5(11.1) 13.5(12.5) (ra) 9.5 (7.7) 9.5 (8.3) 9.5 (8.7) 9.5 (9.0) The forward speed used is 15 knots. The measured average wave period and significant wave height are shown i n parentheses

For each hregular wave condition, full-scale time records lasting approximately 45 m i n were obtained. The measured average wave periods are given i n parentheses, and they are slightly different f r o m the target values. The measured aver-age wave period is shghtly larger than the target value o f 7.5 s under condition A _ l , while the measured average wave period is shghfly smaller than the target value o f 13.5 s under con-dition A _ 4 . D u e to the decay o f the waves, the encountered significant wave heights are a l l lower than the target values.

Table 4 Selected irregular wave conditions f o r the 13000-TEU vessel Condition B _ l B_2 B_3 B_4 B_5 B_6 Duration 43.0 83.9 42.6 41.8 93.9 193.2 (min) r . ( s ) 7.5 9.5 11.5 13.5 9.5 11.5 (8.2) (9.7) (11.5) (13.2) (9.7) (11.4) f f , ( m ) 9.5 9.5 9.5 9.5 11.5 11.5 (7.9) (8.3) (8.4) (9.0) (10.2) (11.3) The forward speed used is 15 knots under conditions B _ l to B _ 4 and 10 knots under conditions B_5 and B_6. The measured average wave period and significant wave height are shown i n parentheses

Because shoiter waves decay more quickly, the measured significant wave height is the smallest under condition A _ l . Table 4 presents six iiregulai- wave conditions f o r the

13000-TEU vessel. The measured average wave periods and significant wave heights are shown i n parentheses. Similar comparisons between the target values and the measured values are observed f o r both vessels.

3 E x p e r i m e n t a l results in regular waves

3.1 General

The FFTs o f the vertical load effects i n regular waves show that the energies are concentrated i n several harmonic peaks. The vertical motions or the V B M at any cross sec-tion, thus, could be described as f o l l o w s :

J M a r Sci Teclinol (2013) 18:87-114 91

y{t) ^Ro + Ri cos(ffle? -ei)+R2 cos(2(cüef - £ 2 ) ) 00

+ i?3 C0s(3(a)e/ - £3)) +

J2^l<

COs{k{c0^t - Sk)) k=4

(1) where is the absolute amplitude o f the A:-th harmonic component. The first harmonic response Ri cos{cOgt — Si) is normally the fundamental response. The asymmetry between the crest height and the trough depth is m a i n l y caused by the mean value RQ and the second and t h i r d harmonics. The mean shift value RQ can be induced by the sinkage and t r i m o f the ship h u l l due to the f o r w a r d speed effects and even-order harmonic components. The mean values are obtained b y averaging the V B M responses that are filtered b y the band-pass frequency f r o m 0 to 0.1 rad/s. I n general, the filtered mean V B M responses are quite stable.

T h e convention f o r V B M is such that the hogging V B M is positive. A U o f the amplitudes, the hogging peaks, the steady values, the mean values, and the sagging troughs are given i n a nondimensional f o i i n a t , divided by pgL^^Bq^. p and g represent the density o f water and the acceleration o f gravity, respectively.

3.2 V e r t i c a l motions

Figure 2 presents the nondimensional heave amplitudes f o r the 8 6 0 0 - T E U vessel i n the l e f t plot and f o r the 1 3 0 0 0 - T E U vessel i n the right plot. A t 15 knots, the amplitudes change slightly f o r both vessels and are approximately 0.3 under all incident wave periods. For the 13000-TEU vessel, the slight reduction i n the nondimensional heave amplitudes due to the increase i n the wave height is observed. A t 25 knots, the amplitudes increase f r o m 0.2 to approximately 0.55 f o r b o t h vessels. The FFTs o f the heave motions show that the peaks o f the second harmonic responses can hardly be i d e n t i f i e d .

T h e p i t c h amplitudes are shown i n F i g . 3. denotes the wave number. The amplitudes increase approximately linearly w i t h the incident wave period f o r both vessels. T h e change i n the f o r w a r d speed has httle influence on the p i t c h ampUmdes. The pitch amplitudes of the 13000-TEU vessel are generally smaller than the pitch amphtudes o f the 8600-TEU vessel. The FFTs o f the pitch motions show that the third hai'monic pitch responses can hardly be distinguished. W i t h regard to the amphtudes o f the second harmonics, they are approximately 3 % o f the first harmonic amphtudes.

A s depicted i n F i g . 4, the amplitudes o f the relative motions at station 17 are pronounced f o r both vessels. For the 8 6 0 0 - T E U vessel, the relation m o t i o n increases sig-nificantly as the incident wave period increases, and a shght increase i n the amplitude is observed at 25 knots.

W i t h regard to the 1 3 0 0 0 - T E U vessel, the relative m o t i o n changes slightiy and keeps approximately 2.0 at 15 knots. W h e n the f o r w a r d speed changes f r o m 15 knots to 25 knots, the relative m o t i o n increases significantiy at a higher incident wave period. The slight reduction i n the nondi-mensional relative m o t i o n amplitude due to the increase i n the wave height at 15 knots is observed i n the right plot. I n general, the amplitudes o f the relative motions f o r the 8 6 0 0 - T E U vessel are larger than the amplitudes f o r the 13000-TEU vessel. The amplitude o f the second harmonic is approximately 7 % o f the first harmonic amplitude. Due to the second harmonic, the wave moves more up than d o w n f o r the 8 6 0 0 - T E U vessel and the wave moves more d o w n than up f o r the 1 3 0 0 0 T E U vessel. The other h i g h -order harmonic responses o f the relative motions can be neglected.

I n general, the experimental data o f the heave, pitch, and relative motions at station 17 show that the influence o f the high-order harmonics are quite l i m i t e d .

3.3 V B M results at 15-knot f o r w a r d speed

3.3.1 Second harmonic responses at 15 knots

The harmonic transfer functions o f the second harmonics o f the V B M s at three cuts are iUustrated i n terms o f the amplitude and phase difference relative to the first har-monic peaks f o r both vessels i n F i g . 5. Similar trends w i t h respect to the amplitudes can be observed f o r both vessels. The amplitude decreases as the incident wave period increases. The amphtude at 3/4 Lpp before A P is close to the amplitude at 1/4 Lpp before A P w h e n the incident wave period is 12.0 s. The amplitude at 3/4 Lpp before A P approaches the amplitude at 1/2 Lpp before A P w h e n the incident wave p e i i o d increases to 15.1 s. For the 13000-T E U vessel, the amplitude increases noticeably w h e n the wave height changes f r o m 10 to 15 m . The amplitudes o f the second harmonic seem to be more sensitive to the encounter frequency than to the slamming impacts.

The phase difference (22 E I ) has a more direct i n f l u -ence on the combined amplitudes between the first and second harmonic responses. W h e n the absolute value o f the phase difference (£2 - e i ) is w i t h i n 30°, the second hai-monic w i l l increase the hogging peaks and decrease the absolute values o f the sagging troughs. A n d the second harmonic plays an opposite role when the absolute value o f (£2 — e i ) is larger than 6 0 ° . The phase difference (£2 - £1) presents a similar trend but shghtiy different values f o r t w o vessels.

For the V B M s at 3/4 Lpp before A P , the phase difference (£2 - £1) f o r the 8 6 0 0 - T E U vessel is lai-ger than that f o r the 13000-TEU vessel. This implies that the second harmonic w i l l increase the absolute values o f the sagging troughs

92 _{J Mar Sci Teclinol (2013) 18:87-114}

T r a n s f e r f u n c t i o n s o f h e a v e for 8 6 0 0 - T E U v e s s e l . H = 1 0 m .

T r a n s f e r f u n c t i o n s of h e a v e for 1 3 0 0 0 -T E L; v e s s e l .

12.5 13 13.5 14 14.5 Incident wave period 7 [s]

15 15.5 1 0.9 0.8

1=1

0.7 0.6 N 0.5 0.5 > 0.4 0,3 0.2 0.1 0 1 1 — B - = 1 5 k n o t s , H = 1 0 m . — U = 1 5 k n o t s , H = 1 5 m . ' vv ^ A - U = 2 5 k n o t s , H = 10 m. ' — B - = 1 5 k n o t s , H = 1 0 m . — U = 1 5 k n o t s , H = 1 5 m . ' vv ^ A - U = 2 5 k n o t s , H = 10 m. ' — B - = 1 5 k n o t s , H = 1 0 m . — U = 1 5 k n o t s , H = 1 5 m . ' vv ^ A - U = 2 5 k n o t s , H = 10 m. ' i l l : ' ; i i i Ia^ --<• A - ^ ^ ^ — ' i i i i i i i i i i i 12 12.5 13 13.5 14 14.5 15 incident wave period T [s]

F i g . 2 Transfer functions o f tlie heave amplitudes. The left plot coiTesponds to the 8600-TEU vessel, and the right plot coiTesponds to the 13000-TEU vessel. Test results at 15-knot and 25-knot forward speeds are compai-ed

T r a n s f e r f u n c t i o n s of p i t c h for 8 6 0 0 - T E U v e s s e l . H =10 m.

•as

O a.

12.5 13 13.5 14 14.5 incident wave period T [s]

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T r a n s f e r f u n c t i o n s o f p i t c h f o r 1 3 0 0 0 - T E U v e s s e l . 1 1 i

1

1 I 1 - e - L / = 1 5 k n o t s , H = l 0 m . U = 1 5 k n o t s , H = 1 5 m . ^ A , - U = 2 5 k n o t s , H = 10 m. - e - L / = 1 5 k n o t s , H = l 0 m . U = 1 5 k n o t s , H = 1 5 m . ^ A , - U = 2 5 k n o t s , H = 10 m. - e - L / = 1 5 k n o t s , H = l 0 m . U = 1 5 k n o t s , H = 1 5 m . ^ A , - U = 2 5 k n o t s , H = 10 m. i _{i 1 1} 12 12.5 13 13.5 14 14.5 Incident wave period T [s]

15 15.5

F i g . 3 Transfer functions o f the pitch motions. The lefi plot coiTesponds to the 8600-TEU vessel, and the riglit plot corresponds to the 13000-TEU vessel. Test results at 15-kiiot and 25-knot forward speeds are compared

more significantly f o r the 8 6 0 0 - T E U vessel. W i t h regard to the V B M s at 1/2 Lpp before A P , the phase difference (s2 - £i) is b e l o w 30° f o r both vessels and it is closer to 0 ° f o r the 13000-TEU vessel. As f o r the V B M s at 1/4 Lpp before A P , the phase difference (e2 - £i) changes f r o m - 2 7 . 2 ° to 14.0° f o r the 8 6 0 0 - T E U vessel, w h i l e i t is approximately - 4 0 ° at a wave height o f 10 m and f r o m —33.0° to - 8 . 9 ° at a wave height o f 15 m f o r the 13000-TEU vessel.

3.3.2 Third imrmonic responses at 15 knots

The results o f the third hai'monic responses are provided i n Eig. 6. The ampliUides o f the third harmonics show a signif-icant decrease as the incident wave period increases f o r both vessels. The m a x i m u m amplitudes o f the tliird hai-monics are

approximately 50 % lai'ger than the m a x i m u m amphtudes o f the second harmonics. The amphtudes at 3/4 Lpp before A P are close to the amphtudes at 1/4 Lpp before A P when the incident wave period is 12.0 s. The amphtudes at 3/4 Lpp before A P approaches the amplitudes at 1/2 Lpp before A P when the incident wave period increases to 15.1 s. The third hannonic also seems sensitive to the encounter frequency.

The trends o f the phase difference (£3 - £1) are also similar f o r both vessels. The phase difference (£3 - £1) at 1/2 Lpp before A P is approximately - 6 0 ° f o r both vessels. The combined hogging peaks and sagging troughs o f the V B M s amidships do not change much f o r both vessels when the phase difference (£2 - £ i ) is close to 0 ° . W i t h regard to the V B M s at 1/4 Lpp and 3/4 Lpp before A P , the third harmonic seems to cause more o f an increase i n the

J M a r Sci Teclinol (2013) 18:87-114 93 T r a n s f e r f u n c t i o n s of relative m o t i o n s f o r 8 6 0 0 - T E U v e s s e l . 1—1 _{u = 15 k n o t s} L—l u = 15 k n o t s A _{u u = 2 5 k n o t s = 2 5 k n o t s} 12.5 13 13.5 14 14.5 Incident w a v e period T [s] 15.5

F i g . 4 Transfer functions o f the relative motions at station 17. The

left plot coiTesponds to the 8600-TEU vessel, and the right plot

conesponds to the 13000-TEU vessel. Test results at 15-knot and

T r a n s f e r f u n c t i o n s of relative m o t i o n s for 1 3 0 0 0 - T E U v e s s e l 4 3.5 i j » 3 2.5 O E a> 1.5 0.5 - B - U = 1 5 k n o t s , = 1 0 m . U = 1 5 k n o t s , = 1 5 m . ^ A , - L / = 2 6 k n o t s , H = 1 0 m . 12 12.5 13 13.5 14 14.5 Incident wave period T [s]

15

25-knot f o r w a r d speeds are compared. The gauge f a i l e d to measure the relative motions at the incident wave period o f 15.1 s f o r the 8600-TEU vessel 2"'' h a r m o n i c a m p l i t u d e s of V B M for 8 6 0 0 - T E U v e s s e l . P h a s e d i f f e r e n c e s ( - ff,) for 8 6 0 0 T E U v e s s e l . 12.S 13 13.5 14 14.5 I n c i d e n t w a v e p e r i o d T [ s ] n V B M at 3/4 L b e f o r e A P - O - V B M at 1/2 b e f o r e A P ^ a s - V B M a t l / 4 L b e f o r e A P 1— a - - H

1

—

1

IE:

-G

©

i—®~

A i — ^ 12.5 13 13.S 14 14.5 I n c i d e n t w a v e p e r i o d T - [ s ] 2"" h a r m o n i c amplitudes of V B M for 1 3 0 0 0 - T E U v e s s e l . P h a s e d i f f e r e n c e s ( e-^ - ) f o r 1 3 0 0 0 - T E U v e s s e l . 0.01 • .009 0.008 0.008 Q.oor cn O.OOB O.O05 •5 0.004 s 0.003 m

>

0.002 0.001 - V B M at 3/4 i. before A P - 1 5 m - V B M at 1/2/.pp before AP - 1 5 m - V B M a t 1 / 4 L b e f o r e A P - 1 5 m pp - V B M at 3 / 4 / . before A P - 1 0 m pp - e - V B M at 1/2 Lpp before A P - 1 0 m V B M at 1/4 L before A P - 1 0 m pp 12.5 13 13.5 14 14.5 I n c i d e n t w a v e p e r i o d T [ s ]

fc

TJ CC .c Q. - V B M at 3/4 L b e f o r e A P - 1 5 m pp - V B M a t 1 / 2 L b e f o r e A P - 1 5 m pp - V B M a t 1 / 4 L b e f o r e A P - 1 5 m pp - V B M at 3/4 L b e f o r e A P - 1 0 m pp - V B M a t 1 / 2 L b e f o r e A P - I O m pp - V B M at 1/4 L b e f o r e A P - I O m pp 12.5 13 13.5 14 14.5 I n c i d e n t w a v e p e r i o d T [ s ] F i g . 5 Haimonic transfer functions o f the second harmonics o f the

V B M s at three cuts as a function o f the incident wave period f o r the 8600-TEU and 13000-TEU vessels. The amphtudes are illustrated i n

the left plots, while the corresponding phase differences (B2 — si) are given i n the right plots. The forward speed used is 15 Icnots. F o r the 13000-TEU vessel, wave heights o f 10 and 15 m are used

94 _{J M a r Sci Technol (2013) 18:87-114} 5.5 I n c i d e n t w a v e p e r i o d T [ s ] I n c i d e n t w a v e p e r i o d T [ s ] 0.01 0.009 o.Goa 0.007 00 =3. Q) O.OOB 0.005 0.004 2 0.003 tn > 0.002 0.001 0 3 " h a r m o n i c a m p l i t u d e s of V B M for 13000-TELJ v e s s e l . P h a s e d i f f e r e n c e s ( for 1 3 0 0 0 - T E U v e s s e l . H i - V B M at 3/4 L^^ before A P - 1 5 m - # — V B M a t 1 / 2 L before A P - 1 5 m pp ^ A — V B M a t 1 / 4 L b e f o r e A P - 1 5 m pp - B —V B M at 3/4 L^^ before A P - 1 0 m - e - V B M at 1/2 L b e f o r e A P - I O m - A - V B M at 1/4 before A P - 1 0 m 12 12.5 13 13.5 14 14.5 15 15.5 I n c i d e n t w a v e p e r i o d T [ s ]

F i g . 6 Haimonic tansfer functions o f the third harmonics o f the V B M s at thi-ee cuts as a function o f the incident wave period f o r the 8600-TEU and 13000-TEU vessels. The amplitudes are illustrated i n

150 TJ a. - » - V B M a t 3 / 4 L before A P - 1 5 m pp V B M at 1/2 Lpp before A P - 1 5 m - ^ V B M at 1/4 Lpp before A P - 1 5 m - B -V B M at 3/4 Lpp before A P - 1 0 m - e - V B M at 1/2 Lpp before A P - 1 0 m -^VBM at 1/4 L„„ b e f o r e A P - 1 0 m pp 12.5 13 13.5 14 14.5 I n c i d e n t w a v e p e r i o d T [ s ] 15.5

the left plots, while the con-esponding phase differences ( 6 3 - E I ) are given i n the right plots. The forward speed used is 15 linots. For the

13000-TEU vessel, wave heights o f 10 and 15 m are used

absolute values o f the sagging troughs. For the 13000-TEU vessel, the increase i n the wave height has little infiuence on the phase difference (£3 - £1).

3.3.3 Combined VBM amplitudes at 15 knots

Figure 7 presents the hogging peaks, the mean values, the steady values, and the sagging troughs at three cuts f o r the 8 6 0 0 - T E U vessel. The mean values represent the average values o f the V B M s , i n w h i c h the wave-frequency and high-frequency components are completely filtered. Between two consecutive zero up-crossings i n the band-pass-filtered (0.0-1.5 cUe) V B M s , one highest peak and one lowest trough are recorded f o r up to the first, second, and t h i r d harmonic responses as w e l l as the total response. The average values o f the positive peaks and the negative troughs are obtained. The mean values are exactly i n the m i d d l e o f the circle peaks and the circle troughs because

the first harmonic responses are themselves absolutely sinusoidal. The steady V B M s at three cuts are a l l sagging as expected. The absolute values o f the mean V B M s are larger than the absolute values o f the steady V B M s . The odd harmonic responses increase the sagging V B M s at three cuts, as shown f r o m the differences between the thin lines and the plus points.

For the V B M at 3/4 Lpp before A P , the second and third harmonics increase the absolute values o f the sagging troughs significantly and do not notably change the hog-g i n hog-g peaks. Consequently, the diamond peaks and trouhog-ghs are highly asymmetric suirounding the mean values. W i t h regard to the V B M at 1/2 Lpp before A P , the second har-monic generally increases the absolute values o f the hog-g i n hog-g peaks and decreases the absolute values o f the sahog-ghog-ginhog-g troughs, especially at relatively l o w incident wave periods. The third harmonie plays the opposite role. The diamond peaks and troughs are, thus, symmetric suiTOunding the

J Mai- Sci Teclinol (2013) 18:87-114 ₉₅ V B M at 3/4 L^^ before A P for 3 6 0 0 - T E U v e s s e l . -0 03 L i i i ^ i i i i i i i i 12 12.5 13 13.5 14 14.5 15 15.5 I B 16.5 17 17.5 I n c i d e n t w a v e p e r i o d T [ s ] V B M a m i d s i i i p s for 8 6 0 0 - T E U v e s s e l . O 0 B | - ] 1 1 1 r 1 1 1 I -0 06 L i i i i i i i i I i i i 12 12.5 13 13.5 14 14.6 15 15.5 16 16.5 17 17.5 I n c i d e n t w a v e p e r i o d T [ s ] V B M at 1/4 L„„ before A P for 8 6 0 0 - T E U v e s s e l . -0 04 L i i i i i i i i i \ i I 12 12.5 13 13.5 U 14.5 15 15.5 I B 1B.5 17 17.5 I n c i d e n t w a v e p e r i o d T [ s ]

mean values. For the V B M at 1/4 Lpp before A P , the second harmonic also increases the absolute values o f the hogging peaks and the third harmonic increases the absolute values

Fig. 7 Measured hogging pealcs, mean values, steady values, and sagging troughs at three cuts as a function o f the incident wave period for the 8600-TEU container ship. The steady values were measured using the model advancing i n still water. U p to 1st, 2nd, and 3rd, total mean that the upper band-pass filter frequency is 1.5 m^, 2.5 cOe, 3.5 oje, and 20.0 rad/s, respectively. Head seas, { / = 1 5 Icnots, //,„ = 10 m

of the sagging troughs more significantly than the absolute values o f the hogging peaks.

The difference between the diamond points and the triangle points is even more noticeable because the h u l l flexibility plays an important role. The natural frequency o f the 2-node vertical bending mode is 3.01 rad/s. W h e n the incident wave period is 12.0 s, the encounter frequency is approximately 0.74 rad/s and the fourth-order resonance is greatly excited. W h e n the incident wave period is 14.4 s, the encounter frequency nears 0.60 rad/s and the fifth-order resonance is excited to some extent.

The V B M results f o r the 13000-TEU vessel are pre-sented i n F i g . 8. The steady and mean V B M s at three cuts are all sagging as expected, and their amplitudes are smaller than the amplitudes o f the 8 6 0 0 - T E U vessel. For the V B M s at 3/4 Lpp before A P , the asymmetry c o u l d hardly be identified at 10-m wave height compared w i t h the corresponding results i n Fig. 7. W i t h regard to the V B M s at 1/2 Lpp before A P , the influence o f the second harmonic seems more significant especiaUy at relatively l o w incident wave period w h i l e the third harmonic plays the opposite role. Even though the mean values are negative i n sagging, the absolute values o f the triangle peaks are larger than the absolute values o f the triangle troughs at a wave height o f 10 m w h e n the incident wave period equals 12.9 and 13.6 s.

I n general, the combined amplitudes at 15-knot speed can shed some light on the determination o f design loads for ultra-large container ships. The wave length used is quite close to the ship length, and the speed reduction effects are also considered.

3.4 V B M results at 25-knot f o r w a r d speed

3.4.1 Second harmonic responses at 25 knots

The harmonic transfer functions o f the second harmonics of the V B M s at 25 knots are presented i n F i g . 9. A s shown i n the l e f t plots between Figs. 5 and 9, the trends o f the second harmonic amplitudes are quite different. W h e n the f o r w a r d speed changes f r o m 15 knots to 25 knots, the amplitudes increase slightly at a smaller incident wave period and increase significantly at a larger incident wave period. The amplitude reaches its m a x i m u m at an incident wave period o f 13.6 s f o r the 8 6 0 0 - T E U vessel and its m a x i m u m at an incident wave period o f 14.4 s f o r the

96 J Mar Sci Teclinol (2013) 18:87-114 V B M at 3/4 L before A P for 1 3 0 0 0 - T E U v e s s e l . H = 10 i pp w 0,03 13 14 15 16 I n c i d e n t w a v e period 7 [ s ] V B M a m i d s h i p s for 1 3 0 a O - T E U v e s s e l . H = 10 m 13 14 15 16 I n c i d e n t w a v e period 7 [ s ] V B M at M i . before A P for 1 3 0 0 0 - T E U v e s s e l . H = 10 m 0,03 0,02 V P ta 0,01 g -0.01 CD > •0.02 • 0 . 0 3 —B— Up to

a""*

Up to S"* Total —!— Mean S t e a d y e f f e c t s 13 14 15 16 I n c i d e n t w a v e period 7 [ s ] V B M at 3/4 L before A P for 1 3 0 0 0 - T E U v e s s e l . H = 15 m pp w 0.03 I - . . : 0.02 « 0.01 5 -0.01 CO

>

-0.02 -0.03 - e - u p t o i " s - Up to 2"" U p to 3"' Total H —M e a n S t e a d y e f f e c t s i h 12 13 14 15 16 17 Incident w a v e period 7* [ s ] V B M a m i d s h i p s for 1 3 0 0 0 - T E U v e s s e l . H = 1 5 m 13 14 15 16 I n c i d e n t w a v e p e r i o d T [ s ] V B M at 1/4 L before A P for 1 3 0 0 0 - T E U v e s s e l . H = 1 5 m pp w

>

0.04 0.03 0.02 0.01 0 -0.01 •0.02 -0.03 •0.04

- e -

U p to l " U p to 2"" U p to 3"* T o t a l M e a n S t e a d y e f f e c t s -13 14 15 16 Incident w a v e period 7 [ s ] 17 Springer

J M a r Sci Teclinol (2013) 18:87-114 97

•4 F i g . 8 Measured hogging peaks, mean values, steady values, and sagging troughs at three cuts as a function of the incident wave period for 13000-TEU vessel. The steady values were measured using the model advancing i n still water. Up to 1st, 2nd, and 3rd, total mean that the upper band-pass filter frequency is 1.5 co^, 2,5 cOe, 3.5 m^, and 20.0 rad/s, respectively. Head seas, U = 15 knots. The left plots coiTespond to a wave height o f 10 m, and the right plots are f o r a wave height o f 15 m

13000-TEU vessel. A t these t w o incident wave periods, the ratio between the wave length and the ship length is approximately 0.9 f o r both vessels. I t seems that the slamming impact contributes most to the second harmonics at a 25-knot f o r w a r d speed due to the large relative motions at higher incident wave periods.

Compared w i t h the coiTesponding phase difference results i n F i g . 5, the phase differences (£2 - £1) at three cuts increase noticeably at 25 knots. For both vessel models, the phase differences (£2 — £ 1 ) increase b y

approximately 15° at an incident wave period o f 12.0 s and increase by approximately 2 5 ° at other incident wave periods.

3.4.2 Third imrmonic responses at 25 Icnots

Figure 10 presents the t h i r d harmonics o f the V B M s at three cuts f o r both vessels. The trends o f the third harmonic amplitudes are somewhat similar to the trends o f the sec-ond harmonic at the same f o r w a r d speed o f 25 knots. The m a x i m u m amplitudes o f the third harmonics are approxi-mately 50 % larger than the m a x i m u m amplitudes o f the second harmonics f o r both vessels.

For the V B M s at 3/4 Lpp before A P , the phase difference (£3 - £1) is slightly larger than 0° f o r the 8 6 0 0 - T E U vessel and shghtly below 0 ° f o r the 13000-TEU vessel. Consid-ering the phase difference (£2 — £1) at 3/4 Lpp before A P , the asymmetries f o r the V B M s at 3/4 Lpp before A P w i l l be

0.02 0.018 0.016 0.014 • 0.012 2"° h a r m o n i c a m p l i t u d e s of V B M for 8 6 0 0 - T E U v e s s e l . P h a s e d i f f e r e n c e s ( - ) for 8 6 0 0 - T E U v e s s e l . S 0.008 m > 5 " 0.006 0.004 0.002 002 0.018 0.016 0.014 0.012 0.01 0.008 - a - V B M a t 3/4 L^^ before A P - e- V B M a t 1 / 2 / . before A P V B M a t 1/4 i. b e f o r e A P pp 12.5 13 13.5 14 14.5 i n c i d e n t w a v e p e r i o d T [ s ] 2"" h a r m o n i c a m p l i t u d e s of V B M for 1 3 0 0 0 - T E U v e s s e l . f 0.006 > 0.004 0.002 0 - B - V B M at 3/4 L before A P - e - V B M at 1/2 i. before A P V B M at 1/4 L before A P pp 12.5 13 13.5 14 14.5 I n c i d e n t w a v e period T [s] 100 40 100 40 E 20 - a -V B M at 3 / 4 i. b e f o r e A P '-' PP - e - V B M a t 1/2 L b e f o r e A P V B M at 1 / 4 / . b e f o r e A P pp 2 12.5 13 13,6 14 14 5 15 I n c i d e n t w a v e period / [s] P h a s e d i f f e r e n c e s ( - E, ) for 1 3 0 0 0 - T E U v e s s e l . - B - V B M a t 3/4 L^^ b e f o r e A P V B M a t 1/2 i. b e f o r e A P ^ pp ^ A -V B M at 1/4 Z. before A P pp 12,5 13 13,6 14 14 6 I n c i d e n t w a v e p e r i o d T [ s ]

F i g . 9 Harmonic transfer funcdons o f the second harmonics o f the V B M s at three cuts as a function o f the incident wave period f o r 8600-TEU and 13000-TEU vessels. The amplitudes are illustrated i n

the left plots, and the corresponding phase differences are given i n the

right plots. The forward speed used is 25 knots, and the wave height

used is 10 m

98 J M a r Sci Technol (2013) 18:87-114 0.02 0.018 0.016 ^ 0.014 0.012 0.01 5 0.008 0.006 0.004 0.002 0

3 h a r m o n i c a m p l i t u d e s of V B M for 8 6 0 0 - T E U v e s s e l . P h a s e d i f f e r e n c e s ( e^- ) for 8 6 0 0 - T E U v e s s e l .

> n V B M a t 3/4 L b e f o r e A P - e - V B M at 1/2 L before A P pp A V B M a t 1/4 i. before A P pp 12.5 13 13.5 14 14.5 i n c i d e n t w a v e p e r i o d T [ s ] 15 _15.5

.5

0.02 0.018 0.016 0.014 0.012 0.01 3™ h a r m o n i c a m p l i t u d e s of V B M for 1 3 0 0 0 - T E U v e s s e l . 5 0.008 0,006 • V B M a t 3/4 L before A P - 1 = 1 — pp - e - V B M at 1/2 L b e f o r e A P ^ pp V B M at 1/4 L b e f o r e A P pp 12.5 13 13.5 14 14.5 i n c i d e n t w a v e p e r i o d T [ s ] 15 15.5

F i g . 10 Hai'monic transfer functions o f the third harmonics of the V B M s at three cuts as a function of the incident wave period f o r the 8600-TEU and 13000-TEU container ships. The amplitudes are quite pronounced f o r both vessels. W i t h regard to the V B M s at 1/2 Lpp before A P , the phase difference (£3 - EJ ) is s t i l l close to - 6 0 ° at 12.0-s incident wave period and approaches —20° as the incident wave period increases.

3.4.3 Combined VBM amplitudes at 25 knots

The hogging peaks, the mean values, the steady values, and the sagging troughs at three cuts are presented f o r the 8 6 0 0 - T E U vessel i n F i g . 11. Similar results are provided f o r the 13000-TEU vessel i n F i g . 12. The absolute values o f the mean V B M s are larger than the absolute values o f the steady V B M s . The absolute mean values generally increase as the incident wave period increases. A s the f o r w a r d speed increases f r o m 15 loiots to 25 knots, both the steady and mean values increase shghtly f o r b o t h vessels. For the V B M s at 3/4 Lpp before A P f o r both vessels, the second and third harmonics increase the absolute values o f the sagging troughs significantly, w h i l e the absolute values

'•5 60 40 20 0 -20 -40 -60 -80 - a - V B M at 3/4 L b e f o r e A P ' - ' pp ^ 0 - V B M at 1 /2 L b e f o r e A P ^ pp - A - V B M a t 1/4 L b e f o r e A P " pp 12 12.5 13 13,5 14 14,5 15 I n c i d e n t w a v e period T [ s ] P h a s e d i f f e r e n c e s ( ffj - ) for 1 3 0 0 0 - T E U v e s s e l . 40 20 0 -20 -40 -60 -80 - e -V B M a t 3 / 4 / . b e f o r e A P '-' PP n V B M a t 1/2 Z. b e f o r e A P ^ pp V B M a t 1 / 4 / . b e f o r e A P pp 12,5 13 13.6 14 14.5 I n c i d e n t w a v e p e r i o d T [s] 15 15.5

presented i n the left plots, w h i l e the phase differences are given i n the

right plots. The forward speed used is 25 knots, and the wave height

used is 10 m

o f the hogging peaks are o n l y increased slightly. Pro-nounced asymmetries between the hogging peaks and the sagging troughs can be identified. The m a x i m u m absolute values o f the d i a m o n d sagging troughs change f r o m approximately 0.017 at 15 knots to 0.031 at 25 knots f o r both vessels. The m a x i m u m absolute values o f the trian-gular sagging troughs change f r o m approximately 0.028 at 15 knots to 0.047 at 25 knots f o r the 8 6 0 0 - T E U vessel, and change f r o m 0.025 to 0.042 f o r the 13000-TEU vessel. The increase i n the hogging peaks, however, is m u c h smaller.

Obvious asymmetries o f the V B M amidships can be observed although they are not as pronounced as those at 3/4 Lpp before A P f o r both vessel models. The i n a x i m u m absolute value o f the diamond trough can be 2.5 times the m a x i m u m absolute value o f the diamond peak f o r the 8 6 0 0 - T E U vessel, and the ratio is approximately 1.5 f o r the 13000-TEU vessel. The second harmonic increases the absolute values o f the sagging troughs f o r the 8 6 0 0 - T E U vessel and does not change the absolute values o f the

J M a r Sci Teclinol (2013) 18:87-114 99 V B M at m L b e f o r e A P for 8 6 0 0 - T E U v e s s e l , pp 0.05 0.04 0.02 0 -0.02 -0.04 -0.08 -0.08 •0.1 13 14 15 16 I n c i d e n t w a v e period T [S] V B M a m i d s h i p s f o r 8 6 0 0 - T E U v e s s e l . ^ > - U p t o l " - B - U p to 2""' Up to 3'" A T o U l — I—M e a n — S t e a d y e f f e c t s 12 13 14 15 16 11 I n c i d e n t w a v e p e r i o d T [ s ] V B M at 1/4 L b e f o r e A P for 8 6 0 0 - T E U v e s s e l . pp © - U p t o l " B - U p to 2"" U p to 3"* A - T o t a l Mean S t e a d y e f f e c t s . 5 > 0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 V B M a t 3/4 L before A P for 1 3 0 0 0 - T E U v e s s e l , pp - e - u p t o i ^ ' - B — U p to 2 " " ^ U p to 3'" T o t a l H—M e a n S t e a d y e f f e c t s 13 14 15 16 Incident w a v e p e r i o d T [ s ] V B M a m i d s h i p s for 1 3 0 0 0 - T E U v e s s e l . 0.02 > 13 14 15 16 Incident w a v e p e r i o d 7 [ s ] V B M a t 1/4 L before A P for 1 3 0 0 0 - T E U v e s s e l , pp 13 14 15 16 I n c i d e n t w a v e p e r i o d 7 [ s ] 13 14 15 16 Incident w a v e p e r i o d T [ s ]

F i g . 11 Measured iiogging pealcs, mean values, steady values, and sagging troughs at three cuts as a f u n c t i o n of the incident wave period for 8600-TEU vessel. The forward speed used is 25 knots, and the wave height used is 10 m . U p to 1st, 2nd, and 3rd, total mean that the upper band-pass filter frequency is 1.5 co^, 2.5 w^, 3.5 We> and 20.0 rad/s, respectively

F i g . 12 Measured hogging peaks, mean values, steady values, and sagging troughs at three cuts as a function o f the incident wave period for 13000-TEU yessel. The forward speed used is 25 knots, and the wave height used is 10 m . Up to 1st, 2nd, and 3rd, total mean that the upper band-pass filter frequency is 1.5 We, 2.5 cOe, 3.5 co^, and 20.0 rad/s, respectively

100 J Mar Sci Technol (2013) 18:87-114

sagging trouglis f o r the 1 3 0 0 0 - T E U vessel. The t h i r d harmonic increases the absolute values o f the sagging troughs significantly. For the 8 6 0 0 T E U vessel, the m i n -i m u m values o f the sagg-ing V B M am-idsh-ips up to the t h i r d harmonic can be as l o w as - 0 . 0 5 5 and can reach —0.095 i f the total response is considered. F r o m a prac-tical p o i n t o f v i e w , these values are too large under extreme sea conditions.

W i t h regard to the V B M s at 1/4 Lpp before A P , the asymmetries between the hogging peaks and sagging troughs are shght and mainly caused by the mean values. The second harmonics increase the absolute values o f the hogging peaks f o r both vessels. For the 13000-TEU vessel, the second harmonic decreases the absolute values o f the sagging troughs. The third haimonics generally increase the absolute values o f the sagging troughs and do not change the absolute values o f the hogging peaks.

I n general, the results i n Figs. 11 and 12 are based on an um-ealistic f o r w a r d speed o f 25 knots at the wave height o f 10 m , w h i c h are j u s t presented f o r comparative purposes. As shown f r o m the comparisons between results i n Figs. 11 and 12, the nondimensional wave-frequency hogging V B M s are slightly larger f o r the 13000-TEU vessel, although the nondimensional sagging V B M s are much larger f o r the 8 6 0 0 - T E U vessel.

3.5 Discussion about the second and third harmonic amplitudes

The h a r m o n i c transfer f u n c t i o n s o f the second and t h i r d h a r m o n i c amplitudes f o r the S-175 containership have been reported i n [ 4 , 6 ] . B o t h the second and t h i r d har-m o n i c ahar-mplitudes reach har-m a x i har-m u har-m when the ratio between wave l e n g t h and ship length is close to 1.2. W h e n the ratio between wave length and ship length is less than 0.8, the second and t h i r d harmonic amplitudes are quite l i m i t e d . The second harmonic amplitudes f o r the V B M amidships have m a x i m u m values that vary between 40 and 60 % o f the first harmonic amphtudes. T h e t h i r d harmonics reach values o f a p p r o x i m a t e l y 15 % o f the first harmonics.

The second and third harmonics f o r the 8 6 0 0 - T E U and 13000-TEU vessels show quite d i f f e r e n t behaviour. First, the t h i r d harmonic amplitudes are approximately 50 % larger than the second harmonic amplitudes. Second, the second and third harmonic amplitudes are noticeable when the ratio between wave length and ship length is less than 0.8. T h i r d , the second and third harmonic amplitudes are insignificant at a 15-knot f o r w a r d speed when the ratio between wave length and ship length is close to 1.0. This implies that the nonlinearity i n V B M is not as large as expected when relative m o t i o n is the largest at 15 knots.

Fourth, the trends f o r the second harmonic observed f r o m the S-175 containership is only similar to the trends observed at a 25-knot f o r w a r d speed f o r t w o models.

4 Experiments in irregular waves

4.1 Statistical characteristics o f the vertical motions

T h e statistical characteristics o f the heave, pitch, and rel-ative m o t i o n at station 17 under conditions A _ l to A _ 4 are shown i n Table 5 f o r the 8 6 0 0 - T E U vessel. The upward direction o f the ship gravity is defined as positive f o r the heave motion. The convention f o r the p i t c h m o t i o n is such that the bow moves up i n the positive direction. W i t h regard to the relative m o t i o n at station 17, the positive direction means that the wave moves up. The steady values o f the heave, pitch, and relative m o t i o n at station 17 are - 0 . 4 2 m , - 0 . 0 2 5 ° , and 0.11 m , respectively. The noise i n the measured records is removed by cutting o f f a l l o f the components whose frequencies are above 3.0 rad/s. The number o f cycles between consecutive up-crossing points is also provided.

The mean value o f the heave m o t i o n is negative under all conditions. Compared w i t h the steady value i n still water, the vessel submerges f u r t h e r i n h i g h waves. The standard deviation cr increases f r o m 0.420 to 1.287 m as expected when the average wave period increases f r o m 7.5 to 13.5 s. The skewness is positive under a l l conditions. This implies that the vessel emerges more than i t sub-merges. Furthermore, a two-parameter W e i b u l l distribution is used to fit the measured peaks and troughs. Cpeak and Ctiough represent the shape parameters o f the peak and trough distributions, respectively. The exponential distri-b u t i o n is a special case o f the W e i distri-b u l l distridistri-bution when the shape parameter equals 1.0. Another special case o f the W e i b u l l distribution is the Rayleigh distribution w i t h the shape parameter o f 2.0, w h i c h is usually used to esti-mate the probability distribution o f linear systems. Thus, the value o f the shape parameter can reflect the nonlin-earity to some extent.

For the p i t c h motion, the mean value is slightly positive under a l l conditions compared w i t h the steady value i n still water. This means that the ship b o w moves slightly. The standard deviation increases f r o m 0 . 3 3 5 ° to 1.287° under conditions A _ l to A _ 4 . The skewness is positive under conditions A _ l and A _ 2 and negative under conditions A _ 3 and A _ 4 . I n short waves, the b o w moves more up than d o w n . Conversely, the b o w moves slightly more d o w n than up. The values o f the fitted shape parameters are close to 2.0 f o r both the peak and trough distributions. This implies that the pitch response is nearly linear.

J Mar Sci Teclinol (2013) 18:87-114 101

Table 5 Statistical characteristics o f the heave, pitch, and relative motion at station 17 under conditions A _ l to A _ 4

Condition Cycles Mean (m) (T (m) Skewness •^peak •-tough ^ trough

Heave

A ^ l 319 - 0 . 7 8 1 0.420 0.429 2.026 0.597 1.986 0.565

A _ 2 255 - 0 . 7 5 2 0.655 0.353 1.971 0.945 2.140 0.950

A_3 224 - 0 . 6 6 6 0.951 0.288 2.026 1.373 2.226 1.333

A _ 4 189 - 0 . 7 4 8 1.287 0.280 1.612 1.736 1703 1.753

Condition Cycles Mean (°) Skewness ^peak •^peak Qough •^trough

Pitch

A _ l 294 0.0022 0.335 0.359 2.024 0.489 2.058 0.479

A _ 2 250 0.0123 0.653 0.145 2.143 0.937 2.168 0.932

AJ, 224 0.0219 0.944 - 0 . 1 3 5 2.242 1.360 2.402 1.385

A _ 4 211 0.0150 1.042 - 0 . 1 3 2 1.800 1.439 1.819 1.451

Condition Cycles Mean (m) a (va) Skewness Cpeak •^peak ^tough '^trough

Relative motion

A _ ] 369 - 0 , 2 2 7 3.024 0.413 1.854 4.648 1.997 4.167

AJ. 308 - 0 . 3 0 9 4.177 0.133 1.895 6.040 1.941 5.817

AJ 269 - 0 . 3 1 0 4.802 0.400 2.093 7.230 2,167 6.676

A _ 4 249 - 0 . 2 1 8 4.622 0.378 2.105 6.945 2.104 6.370

Cpeak and ipeak represent the shape and scaling parameters o f the pealc distributions. Ctough and Ciough denote the shape and scaling parameters o f the trough distributions

W i t h regard to the relative motion at station 17, more cycles are observed compared w i t h the cycle numbers o f the heave and pitch motions. The mean value is negative under a l l conditions. The standard deviation does not change m u c h under different conditions. I t reaches a m a x i m u m under condition A _ 3 . The skewness is positive under aU conditions. This implies that the wave moves more up than d o w n at station 17.

Table 6 presents the statistical characteristics o f the vertical motions under conditions B _ l to B _ 6 f o r the 1 3 0 0 0 - T E U vessel. The trends o f the heave m o t i o n are similar to those f o r the 8600-TEU vessel. For the pitch m o t i o n , the standard deviation increases as expected f r o m 0 . 2 6 7 ° to 0 . 9 7 8 ° under condition B _ l to B _ 4 . The values o f the fitted shape parameter reveal that the pitch response is close to the Rayleigh distribution under all conditions. W i t h regard to the relative m o t i o n at station 17, the stan-dard deviation changes slightly among the d i f f e r e n t con-ditions. The skewness is negative under a l l conditions, w h i c h is different f r o m the results observed f o r the 8600-T E U vessel. 8600-This difference reflected i n ii8600-Tegular waves is inconsistent w i t h that i n regular waves.

I n general, the peak and trough distributions o f the vertical motions agree w e l l w i t h the Rayleigh distribution f o r b o t h vessels, and the asymmetries between the positive peaks and the negative troughs are l i m i t e d . T h e heave

positive peaks tend to be larger than the negative troughs. For the pitch m o t i o n , the asymmetry depends on the average wave period. W i t h regard to the relative m o t i o n , the skewness is positive f o r the 8 6 0 0 - T E U vessel and negative f o r the 13000-TEU vessel.

4.2 Statistical characteristics o f the V B M s

4.2.1 General

A band-pass filter is used to extract the rigid-body (0.1-2.0 rad/s) and flexible-body (0.1-20.0 rad/s) respon-ses f r o m the measured time records. A frequency o f 2.0 rad/s is w e l l above the wave-frequency region but much lower than the wet natural frequency o f the first flexible mode (3.01 rad/s). The rigid-body response can be regarded as a narrow-banded process, w h i c h can also be described as the wave-frequency response. Furthermore, the mean values can be obtained by averaging the V B M responses that are filtered by the band-pass frequency f r o m 0 to 0.1 rad/s. Because there are certain amounts o f wave components i n generated iiTegular waves, the mean values are not as stable as those i n regular waves. This could b r i n g certain uncertainties to the measured values. However, the influence o f mean values is l i m i t e d and w i f l not influence the general trend.

102 _{J M a r Sci Technol (2013) 18:87-114}

Table 6 Statistical characteristics o f the heave, pitch, and relative motion at station 17 under conditions B _ l to B_6. Cpeak and Speji; represent the shape and scaling parameters o f the pealc distributions

Condition Cycles Mean (m) a (m) Skewness _{•^peak Ctough Ctough}

Heave B ^ l 279 - 1 . 0 4 0 0.452 0.396 1.778 0.630 1.863 0.617 B_2 494 - 0 , 9 2 3 0.660 0.302 1.878 0.922 2.056 0.920 B_3 214 - 0 . 7 7 8 0.871 0.214 2.078 1.164 1.951 1.137 B_4 177 - 1 . 1 0 7 1.212 0.208 1.954 1,733 2.002 1.742 B_5 521 - 0 . 9 3 0 0.792 0.293 2.054 1.148 2.075 1.119 B_6 912 - 0 . 7 3 7 1.096 0.375 1.889 1.542 2.027 1.514

Condition Cycles Mean (°) _<T(°) Skewness Cpeak -^peak Ctough •^trough

Pitch B _ l 277 - 0 . 0 5 2 0.267 0.091 1.956 0.384 2.160 0.384 B_2 452 - 0 . 0 8 0 0.570 0.199 1.850 0.792 1.916 0.791 B_3 210 - 0 . 0 8 0 0.819 - 0 . 2 4 4 2.015 1.166 1.965 1.160 B_4 186 - 0 . 0 5 7 0.978 - 0 . 3 2 8 2.079 1.374 1.915 1.402 B_5 487 - 0 . 0 3 5 0.733 - 0 . 2 0 8 1.998 1.031 1.990 1.034 B_6 903 - 0 . 0 8 2 1.040 - 0 . 3 5 7 1.957 1.429 1.835 1.448

Condition Cycles Mean (m) cr (m) Skewness Cpeak •^peak Ctough •^trough

Relative motion B _ l 385 - 0 . 0 5 6 3.596 - 0 . 4 9 8 2.138 5.462 1.911 4.975 B_2 665 - 0 . 0 8 7 3.757 - 0 . 5 4 6 2.283 5.702 2.197 5.496 B J 309 - 0 . 1 3 6 3.503 - 0 . 6 7 7 2.460 5.107 2.014 5.240 B_4 299 - 0 . 0 1 1 3.027 - 0 . 7 1 4 2.337 4.337 1.769 4.277 B_5 687 0.118 3.649 - 0 . 5 5 4 2.163 5.391 1.899 5.283 B_6 1359 - 0 . 0 7 1 3.572 - 0 . 6 0 6 2.037 5.118 1.820 5.132

Ctough and Ctougii denote the shape and scaling parameters o f the trough distributions

C o r r e l a t i o n c o e f f i c i e n t s for 8 6 0 0 - T E U v e s s e l -6000 S a g g i n g ^ 0 0 0 -2000 0 2000 4000 6000 W a v e - f r e q u e n c y m a x i m a [MNm] H o g g i n g z 10000 9000 8000 7000 E X 6000 ra 5000 u ffl 4000 cr ,S 3000 S 2000 1000 0 C o r r e l a t i o n c o e f f i c i e n t s for 1 3 0 0 0 - T E U v e s s e l o o S a g g i n g -05 0 0.5 1 W a v e - f r e q u e n c y m a x i m a [MNm] Hogging<

F i g . 13 Correlations between high-frequency maxima and wave-induced sagging and hogging V B M s . The left plot con-esponds to resuhs under condition Aji f o r the 8600-TEU vessel and the right plot corresponds to the results under condition B_6 the 13000-TEU vessel

J M a r Sci Teciinol (2013) 18:87-114 103

To study the influence o f the high-frequency vibrations on the hogging and sagging V B M s , the coiTclation between high-frequency maxima and wave-induced maxima is studied i n this paper. A band-pass filter is used to extract the wave-frequency (0.1-2.0 rad/s) and high-frequency (2.0-20.0 rad/s) responses. Between consecutive zero up-crossing and zero down-up-crossing points i n the wave-fre-quency response, one m a x i m u m peak is recorded as the hogging V B M f o r the wave-frequency response as w e l l as the high-frequency response. Between consecutive zero down-crossing and zero up-crossing points i n the wave-frequency response, one m i n i m u m trough is recorded as the sagging V B M f o r the wave-frequency response as w e l l as the high-frequency response. For instance, the coirelation between the high-frequency maxima V B M and wave-induced sagging and hogging V B M s is shown i n F i g . 13. Baarholm and Jensen [39] showed that the coirelation coefficient is smaller f o r hogging V B M s than f o r the sag-ging V B M s based on numerical simulation results. There are, however, f e w discussions about the correlation coefficients f o r the measured results f r o m model tests or f u l l -scale measurements.

4.2.2 8600-TEU vessel

Figure 14 presents the exceedance probability distributions of the V B M s at 3/4 Lpp and 1/2 Lpp before A P under conditions A _ l to A _ 4 f o r the 8 6 0 0 T E U vessel. The t w o -parameter W e i b u l l distribution is used to fit the hogging peaks and sagging troughs based on the least-square method.

For the V B M at 3/4 Lpp before A P , the sagging V B M s are noticeably larger than the hogging V B M s f o r both the rigid-body and flexible-body responses under a l l condi-tions. Under condition A _ 4 , the rigid-body sagging V B M s are even larger than the flexible hogging V B M s at the exceedance probabilities o f 10~^ and 10~^. The asymmetry between the hogging peaks and sagging troughs seems to be m a i n l y caused by the wave-frequency response. The shape parameter f o r the hogging peaks is above 1.38 and below 1.73, w h i l e the shape parameter f o r the sagging troughs is always below 1.38. This implies that the V B M response at 3/4 Lpp before A P is highly nonlinear.

W i t h regard to the V B M at 1/2 Lpp before A P , the r i g i d -body hogging V B M s are slightly larger than the rigid--body sagging V B M s , especially under conditions A _ l and A _ 2 . It seems that certain nonlinear effects contribute to the wave-frequency hogging peaks i n relatively short waves. The high-frequency vibrations, however, increase the sagging V B M s shghtly more than the hogsagging V B M s . C o n -sequently, the flexible-body hogging V B M s are almost identical to the flexible-body sagging V B M s under

condition A _ l and the flexible-body hogging V B M s are slightly smaller than the flexible-body sagging V B M s under conditions A _ 2 to A „ 4 . The shape parameter f o r the hogging peaks varies f r o m 1.56 to 1.95, and the shape parameter f o r the sagging troughs varies f r o m 1.48 to 1.89. In most cases, the tails o f the peak and trough distributions are both slightly heavier than the Rayleigh distribution.

To provide better insight into the experimental results, the statistical characteristics o f the V B M s at 3/4 Lpp and 1/2 Lpp before A P are listed i n Tables 7 and 8, respectively. The steady V B M at 3/4 Lpp before A P advancing i n still water at 15 knots is - 1 0 0 . 9 M N m . The mean values are approximately twice the steady value. This implies that the mean sagging V B M s increase i n high waves. The standard deviation o f the wave-frequency response CTWF is the largest under condition A _ 3 . The skewness is estimated based on the rigid-body response, because the total response can not be regarded as narrow-banded. The value o f the skewness is approximately - 1 . 0 under afl conditions. Hog^yp and SagwF are the rigid-body hogging and sagging V B M s at the exceedance probability o f 10~^ based on the fitted curves. SagwF is more than 50 % larger than Hog^vp- The standard deviation of the high-frequency vibrations C J H F reaches a m a x i m u m under condition A _ 2 . The correlation coefficient is above 0.50 f o r the sagging V B M s , w h i l e i t is smaller than 0.41 f o r the hogging V B M s . Especially under condi-t i o n A _ 4 , condi-the coirelacondi-tion coefficiencondi-t is 0.269 f o r condi-the hog-ging V B M s . This implies the high-frequency vibrations have l i m i t e d influence on the hogging V B M s i n l o n g waves. H o g j . and Sagj- denote the flexible-body hogging and sagging V B M s at the exceedance probability o f 10"^ based on the fitted curves. A l t h o u g h Sag^p is the largest under condition A _ 3 , Sagy reaches a m a x i m u m under condition A _ 2 due to the contribution o f the h i g h f r e -quency vibrations.

W i t h regard to the statistical characteristics o f the V B M at 1/2 Lpp before A P , the steady V B M advancing i n still water at 15 knots is - 2 7 8 . 9 M N m . The mean values are approximately 60 % larger than the steady value. The values o f the skewness also c o m p l y w e l l w i t h the right plots i n F i g . 14. The skewness shows a clear decrease trend when the average wave period increases. H o g ^ p is the largest under condition A _ 2 , although the standard deviat i o n o f deviathe wavefrequency response awF reaches a m a x i -m u -m under condition A _ 3 . This is because the nonhnearity is more pronounced Under condition A _ 2 . The standard deviation o f the high-frequency vibrations O H F is notice-ably the lai'gest under condition A _ 2 . The coireiation coefficients Csag show a clear decrease trend when the average wave period increases, w h i l e Chog o n l y decreases slightly. Consequentiy, Sagj- is slightly larger than H o g j . under a l l conditions. Under condition A _ 2 , the m a x i m u m