Slamming clustering on fast ships: From impact dynamics to global response analysis

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Ocean Engineering 62 (2013) 1 1 0 - 1 2 2

ELSEVIER

C o n t e n t s lists a v a i l a b l e at S c i V e r s e S c i e n c e D i r e c t

Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g

Slamming clustering on fast ships: From impact dynamics to global

response analysis

Daniele Dessi*, Elena Ciappi

CNR-INSEAN, Via di Vallerano 139, 00128 Rome, Italy

A R T I C L E I N F O

Article history:

Received 2 November 2011 Accepted 23 December 2012 Available online 28 February 2013 Keywords:

Slamming statistics Impact i d e n t i f i c a t i o n

W h i p p i n g induced bending m o m e n t Scaled physical models

A B S T R A C T

In this paper the statistical properties of the slamming impact process are analyzed w i t h the help o f experimental data acquired in the t o w i n g tank on a high speed ferry model. The physical model is a segmented-hull w i t h a flexible backbone-beam equipped, among other devices, w i t h sensors to measure the wetness of h u l l sections and strain gauges to estimate the induced vertical bending m o m e n t This setup allows us to analyze the slamming process not only on the basis of the detected slamming events but also on the basis of the w h i p p i n g response produced by the impacts. IVIoreover, due to the particular model selected for the present analysis, characterized by a V-shaped hull, bottom as well as bow flare slamming contributions are investigated. One of the major findings is the evidence that the impact statistics are largely affected by the grouping of slams into clusters thus violating the hypothesis of mutual independence between successive impacts that is at the basis of most of the statistical models. The dependence o f t h e w h i p p i n g response on the impact velocity is also investigated. Finally, the definition o f a new criterion for slamming identification based on the evaluation of the w h i p p i n g bending moment is discussed

1. Introduction

A s l a m m i n g e v e n t has g e n e r a l l y t w o d i s t i n c t e f f e c t s i n t h e t i m e - d o m a i n : a n i n s t a n t a n e o u s increase o f t h e f o r c e d l o a d i n g d u r i n g t h e w a t e r - e n t r y a n d w a t e r - e x i t phases, a n d a t r a n s i e n t v i b r a t i o n o f t h e s t r u c t u r e , also k n o w n as w h i p p i n g , t h a t s u p e r -i m p o s e s -i n e r t -i a l l o a d cycles w -i t h decreas-ing -i n t e n s -i t y o n t h e w a v e l o a d i n g . M o d e l - s c a l e a n d f u l l - s c a l e m e a s u r e m e n t s h a v e s h o w n t h a t t h e i m p u l s i v e l y i n d u c e d stresses a m i d s h i p due t o s l a m m i n g can be o f t h e same o r d e r o f m a g n i t u d e as t h o s e d u e t o t h e c o n d n u o u s w a v e l o a d i n g (e.g., see Ramos e t al. ( 2 0 0 0 ) ) . For t h i s reason, c l a s s i f i c a t i o n societies have considered the c o m b i n a t i o n o f s l a m m i n g - i n d u c e d loads w i t h t h e c o n t i n u o u s w a v e l o a d i n g as one o f the c r i t i c a l aspects i n establishing t h e ship design c r i t e r i a o r rules. The c o n s e q u e n t need t o assess t h e ship safety w i t h respect t o g i v e n e n v i r o n m e n t a l a n d o p e r a t i o n a l c o n d i t i o n s has led t o t h e search f o r t h e direct staristical c o r r e l a t i o n b e t w e e n a set o f para-meters, u s u a l l y d e s c r i b i n g t h e sea state a n d the ship course a n d speed, w i t h t h e r e l e v a n t response variables, e.g., t h e m a x i m u m stresses or t h e stress cycle d i a g r a m (see Junker, 2 0 0 9 ; Gao a n d M o a n , 2 0 0 8 ) . H o w e v e r , less a t t e n t i o n t h a n i n the past has been recently p a i d t o e x p l o r e t h e physical m e c h a n i s m s u n d e r i y i n g these

• C o r r e s p o n d i n g author. T e t : -F39 0650299254; fax; - f 3 9 065070619. E-mail address: daniele.dessi@cnr.it (D. Dessi).

c o r r e l a t i o n models. One o f t h e m o s t challenging, d u e t o possible h y d r o e l a s t i c c o u p l i n g a n d n o n l i n e a r i t i e s d e m a n d i n g f o r t i m e d o m a i n analysis, is t h e c o n n e c t i o n b e t w e e n s l a m m i n g a n d w h i p -p i n g , t h a t m a y still u n v e i l -p h e n o m e n a w o r t h f u r t h e r r e v i e w i n g o f t h e g l o b a l sea-state t o stress r e l a t i o n . T h e present p a p e r r e n e w s this insight, b y i n v e s t i g a t i n g the s l a m m i n g b e h a v i o r o f a fast f e r r y .

T h e i d e n t i f i c a t i o n o f w a t e r i m p a c t s , as w e l l as t h e e v a l u a t i o n o f t h e i r i n t e n s i t y , has c o n s t i t u t e d t h e p r e r e q u i s i t e f o r p r o b a b i l i s t i c t h e o r i e s o f t h e s l a m m i n g o c c u r r e n c e . Thus O c h i ( 1 9 6 4 ) p a v e d t h e w a y t o s l a m m i n g statistics b y s t a t i n g t w o c o n d i t i o n s f o r s l a m -m i n g t o o c c u r t h a t w e r e : ( i ) r e l a t i v e -m o t i o n -m u s t exceed t h e s e c t i o n a l d r a f t ; ( i i ) r e l a t i v e v e l o c i t y a t t h e i n s t a n t o f r e - e n t r y m u s t exceed a c e r t a i n m a g n i t u d e . These c o n d i t i o n s s u c c e s s f u l l y d e l i m -i t e d t h e t y p e o f w a t e r - e n t r y p h e n o m e n a f o r w h -i c h a p r o b a b -i l -i s t -i c analysis o f t h e s l a m m i n g i n d u c e d response w a s c a r r i e d o u t i n several papers, b e g i n n i n g w i t h t h e p r o b a b i l i t y d i s t r i b u t i o n s f o r t h e i m p a c t pressure a n d s l a m m i n g i n t e r a r r i v a l times t h a t w e r e o b t a i n e d b y f i t t i n g e x p e r i m e n t a l d a t a b y O c h i h i m s e l f ( 1 9 6 4 ) . F o l l o w i n g t h i s n o v e l a p p r o a c h , O c h i a n d M o t t e r ( 1 9 7 1 , 1 9 7 3 ) a n d M a n s o u r a n d L o z o w ( 1 9 8 2 ) a n d , f r o m t h e e x p e r i m e n t a l side. W h e a t o n a n d et al. ( 1 9 7 0 ) p o i n t e d o u t t h e r e l e v a n c e o f t h e s l a m m i n g i n d u c e d stresses alone as w e l l as t h e i r c r i t i c a l c o m b i -n a t i o -n w i t h t h e s t e a d y - s t a t e ( s t i l l - w a t e r ) , l o w - f r e q u e -n c y ( w a v e s p e c t r u m e x c i t a t i o n ) o r r e s o n a n t stresses ( s p r i n g i n g ) . F o l l o w i n g t h e a s s u m p t i o n t h a t t h e s l a m m i n g p h e n o m e n o n is s t a t i s t i c a l l y r e g a r d e d as a Poisson process, O c h i a n d M o t t e r ( 1 9 7 3 )

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D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110-122 111 e s t i m a t e d t i i e t i m e i n t e r v a l b e t w e e n successive s l a m i m p a c t s a n d s u g g e s t e d a t r u n c a t e d e x p o n e n t i a l p r o b a b i l i t y d i s t r i b u t i o n o f t h e i m p a c t pressure. T h e y e s t i m a t e d c o n s e r v a t i v e l y t h e s l a m m i n g i n d u c e d m o m e n t a m i d s h i p b y c a l c u l a t i n g t h e d y n a m i c response o f t h e s h i p - b e a m t h a t w a s supposed t o be loaded b y s e c t i o n a l i m p a c t forces g e n e r a t e d b y e x t r e m e s l a m m i n g pressures. M a n s o u r a n d L o z o w ( 1 9 8 2 ) r e f i n e d t h i s p r o b a b i l i s t i c m o d e l by a s s u m i n g f o r t h e s l a m m i n g i m p a c t s a Poisson pulse process w i t h i n d e p e n d e n t a m p l i t u d e s a n d i n t e r a r r i v a l t i m e s . Ferro a n d M a n s o u r ( 1 9 8 5 ) f o c u s e d o n t h e p r o b a b i l i s t i c c o m b i -n a t i o -n o f s l a m m i -n g a -n d l o w - f r e q u e -n c y stresses f o r w h i c h t h e y p r o p o s e d a m a t h e m a t i c a l m o d e l , a n d e s t i m a t e d t h e e x t r e m e v a l u e d i s t r i b u t i o n o f t h e c o m b i n e d w a v e - i n d u c e d a n d s l a m m i n g r e s p o n s e w i t h t h e T u r k s t r a ' s ( 1 9 7 0 ) r u l e .

Successively, some o f t h e basic a s s u m p t i o n s t h a t h a d served as s i m p l i f y i n g g u i d e l i n e s f o r a l l t h e p r e v i o u s studies w e r e r e v i e w e d a n d e v e n r e m o v e d i n o r d e r to achieve s a r i s f a c t o r y m o d e l i n g o f t h e s l a m m i n g occurrence. The seakeeping tests d e s c r i b e d l a t e r i n t h i s p a p e r w i l l f o l l o w t h i s d i r e c t i o n . N i k o l a i d i s a n d K a p l a n ( 1 9 9 2 ) c r i t i c i z e d t h e a s s u m p t i o n t h a t t h e t i m e s o f o c c u r r e n c e o f s l a m m i n g i m p a c t s are i n d e p e n d e n t , because o f t h e p e r i o d i c n a t u r e of. r e l a t i v e m o t i o n . T h e y f o u n d , as i m p l i c i t l y p o i n t e d o u t b y O c h i a n d M o t t e r , t h a t t h e rimes o f o c c u r r e n c e o f t h e s l a m m i n g or o f t h e w a v e i n d u c e d stress peaks are h i g h l y c o r r e l a t e d . Jiao ( 1 9 9 6 ) p r o c e e d e d f u r t h e r i n t h i s w a y a n d d e r i v e d t h e e x t r e m e p r o b a b i l i t y d i s t r i b u t i o n s f o r t h e c o m -b i n e d stresses, a c c o u n t i n g f o r t h e process n o n - s t a t i o n a r i t y a n d t h e m u t u a l d e p e n d e n c e b e t w e e n t h e s l a m m i n g a n d t h e w a v e -i n d u c e d stresses. A p r o b a b i l i s t i c m o d e l , capable t a k i n g i n t o a c c o u n t t h e presence o f clusters o f s l a m m i n g events, w a s f i n a l l y p r o p o s e d by H a n s e n ( 1 9 9 4 ) . H a n s e n drops t h e h y p o t h e s i s o f a Poisson p u l s e - t r a i n process, t h a t w a s c o n s i d e r e d t o be a s y m p t o t i c a l l y c o r r e c t o n l y as l o n g as the d r a f t approaches i n f i n i t y . He m o d e l e d t h e s l a m m i n g p h e n o m e n o n as a Slepian process, t h a t is a n o n -Gaussian a n d n o n - s t a t i o n a r y process. This p r o v i d e s a c o m p l e t e d e s c r i p r i o n o f t h e o r i g i n a l a n d ergodic Gaussian process a f t e r a n a r b i t r a r y u p c r o s s i n g i n t o a c r i t i c a l i n t e r v a l w h e r e a c l u s t e r i z a t i o n o f s l a m m i n g events occur. T h o u g h r e m o v i n g t h e h y p o t h e s i s o f m u t u a l i m p a c t i n d e p e n -d e n c e is w i -d e l y accepte-d i n -d e v e l o p i n g s t a t i s t i c a l m o -d e l s o f s l a m m i n g , t h e r e is s r i l l need f o r extensive e x p e r i m e n t a l v a l i d a -t i o n o f r e c e n -t l y i n -t r o d u c e d -theories. Evidence o f c l u s -t e r i n g o f s l a m s i n f u l l scale data w a s r e p o r t e d b y Fu e t a l . ( 2 0 0 9 ) w h i c h i n v e s t i g a t e d t h e o c c u r r e n c e o f w e t d e c k s l a m m i n g i n t h e case o f a h i g h s p e e d c a t a m a r a n . H o w e v e r , s a t i s f a c t o r y a v a i l a b i l i t y o f f u l l -scale m e a s u r e m e n t s i n e x t r e m e c o n d i t i o n s is o f t e n s u b j e c t e d t o o p e r a t i o n a l c o n s t r a i n t s . Recent a d v a n c e m e n t i n m e a s u r e m e n t t e c h n i q u e s a l l o w s us t o achieve a clearer p i c t u r e o f t h e w h o l e p h e n o m e n o n a t m o d e l scale, b e y o n d a m e r e c o u n t i n g o f t h e i m p a c t s , as w i l l be s h o w n i n t h e p r e s e n t paper. I n t h i s perspec-t i v e , perspec-t h e use o f an elasperspec-tic s e g m e n perspec-t e d m o d e l f o r perspec-t h e seakeeping tests p r o v i d e s a d d i t i o n a l chances f o r t h e m o n i t o r i n g o f t h e response i n t e r m s o f accelerations or b e n d i n g m o m e n t s a l o n g t h e s h i p .

I n t h i s paper, t h e seakeeping tests a c c o u n t i n g f o r t h e vessel s l a m m i n g b e h a v i o r at h i g h speeds c l e a r l y o f f e r e d t w o c h a l l e n g i n g aspects d e m a n d i n g a d e e p e r i n s i g h t : t h e p r e v a l e n t g r o u p i n g o f s l a m m i n g e v e n t s i n clusters - n o t j u s t a r e m o t e chance b u t a l m o s t t h e r u l e - a n d t h e presence o f w e d g e - s h a p e d sections i n t h e i m p a c t i n g b o w t h a t m a k e s t h e d i s t i n c t i o n b e t w e e n b o t t o m a n d f l a r e s l a m m i n g events less e v i d e n t . C l u s t e r i n g o f i m p a c t s s h o w s e x p e r i m e n t a l l y t h a t t h e i m p a c t v e l o c i t y loses p a r t i a l l y its e f f e c t i v e n e s s i n a c c o u n t i n g f o r t h e g l o b a l s l a m m i n g e x c i t a t i o n , l i k e t h e v e r t i c a l b e n d i n g m o m e n t s o r t h e s e c t i o n a l stresses, t h o u g h r e t a i n i n g i t s v a l i d i t y f o r t h e local

analysis. This l a c k o f c o r r e l a t i o n suggests a need t o r e v i e w t h e i m p a c t d y n a m i c s a n d t o i n t e r p r e t t h e e n e r g y d i s t r i b u t i o n w i t h i n a s l a m m i n g cluster. C o n c e r n i n g t h e d i s t i n c t i o n b e t w e e n b o t t o m a n d f l a r e s l a m -m i n g , t h e r e l e v a n t w h i p p i n g r e c o r d e d a f t e r b o w - f l a r e s l a -m -m i n g events c o n f i r m s t h e o b s e r v a t i o n t h a t t h e absence o f a f l a t b o t t o m f o r w e d g e - s h a p e d h u l l s reduces t h e d i f f e r e n c e i n t h e f o r c e i n t e n s i t y b e t w e e n v i o l e n t w a t e r - e n t r y p h e n o m e n a w i t h a n d w i t h o u t b o w e m e r g e n c e . This e v i d e n c e has m o t i v a t e d t h e i n c l u -s i o n o f the-se e v e n t -s i n t o t h e p r e -s e n t i n v e -s t i g a t i o n , d u e t o t h e i r relevance e s p e c i a l l y f o r t h e s l a m m i n g i n d u c e d g l o b a l response.

The use o f a n e l a s t i c a l l y scaled p h y s i c a l m o d e l a l l o w s us t o associate t h e peak v a l u e o f t h e t r a n s i e n t b e n d i n g m o m e n t response t o t h e s l a m m i n g l o a d r e s p o n s i b l e f o r t h i s v i b r a t i o n . Indeed, i f t h e r e l a t i v e v e r t i c a l v e l o c i t y o f t h e i m p a c t i n g s h i p s e c t i o n can g i v e a n a p p r o x i m a t e b u t acceptable e s t i m a t i o n o f t h e pressure a c t i n g o n t h e w e t t e d h u l l panels, i t is s h o w n t o be less c o r r e l a t e d t o t h e w h i p p i n g t r a n s i e n t l o a d i n g . T h i s o b s e r v a -t i o n leads -t o a r e v i s i o n o f -t h e p r o c e d u r e f o r -the i d e n -t i f i c a -t i o n o f t h e s l a m m i n g e v e n t s as w e l l as t h e e v a l u a t i o n o f t h e i r s t r e n g t h i f g l o b a l effects are t a k e n i n t o a c c o u n t . Thus, a n o v e l p r o c e d u r e f o c u s e d a n d based o n t h e g l o b a l s h i p response w i l l be p r o p o s e d a n d discussed i n t h e p a p e r a n d t h e n c o m p a r e d w i t h t h e w e l l -k n o w n Ochi c o n d i t i o n s . The c o m p a r i s o n w i l l cover b o t h t h e case o f b o w - f l a r e s l a m m i n g a n d t h e case o f b o t t o m i m p a c t s w i t h a v e r t i c a l e n t r y speed n o t s a t i s f y i n g t h e O c h i t h r e s h o l d .

The paper is d i v i d e d i n t o several sections. I n Section 2 t h e e x p e r i m e n t a l set-up is presented. The i m p l e m e n t a t i o n o f t h e Ochi conditions as a c r i t e r i o n t o i d e n t i f y s l a m m i n g events f r o m t h e collected e x p e r i m e n t a l d a t a is discussed i n Section 3, a n d a s i m i l a r c r i t e r i o n t o i d e n t i f y also t h e occurrence o f b o w - f l a r e i m p a c t s is i n t r o d u c e d . In Section 4, the a p p l i c a t i o n o f the Ochi's c r i t e r i o n f o r t h e s l a m m i n g analysis o f t h e scaled-model seakeeping tests is presented, h i g h l i g h t i n g the existence o f i m p a c t clusters; m o r e o v e r , clustering o f slams is s h o w n t o be responsible f o r deviations f r o m statistical p r e d i c t i o n s o b t a i n e d w i t h the Poisson m o d e l . I n Section 5, t h e transient b e n d i n g response peak, obtained f r o m strain-gage data, is d e f i n e d a n d chosen t o be i n d i c a t i v e o f t h e g l o b a l response i n d u c e d by s l a m m i n g ; therefore, these values are related t o t h e i m p a c t velocity t o analyze t h e s l a m m i n g - w h i p p i n g c o r r e l a t i o n . The inclusion o f b o w - f l a r e i m p a c t s i n the present analysis is carried o u t i n Section 6. I n Section 7 a procedure t o i d e n t i f y t h e s l a m m i n g impacts o n the basis o f t h e i n t e n s i t y o f the global response, t h u s o v e r c o m i n g t h e d i s t i n c t i o n b e t w e e n b o t t o m i m p a c t a n d b o w - f l a r e slams, is o u t i i n e d and m a y c o n s t i t u t e a s t a r t i n g p o i n t t o d e f i n e n e w criteria f o r s l a m m i n g i d e n t i f i c a t i o n .

2. Experimental set-up

The m o d e l e x p e r i m e n t s w e r e c a r r i e d o u t at t h e INSEAN t o w i n g t a n k basin. The b a s i n ( 2 2 0 m l o n g ) is e q u i p p e d w i t h a s i n g l e - f l a p w a v e - m a k e r capable o f g e n e r a t i n g r e g u l a r a n d i r r e g u l a r w a v e p a t t e r n s . A s e g m e n t e d m o d e l o f t h e f a s t f e r r y MDV300Q w a s c o n s t r u c t e d a c c o r d i n g t o t h e b a c k b o n e - m o d e l i n g t e c h n i q u e . T h e p r i n c i p a l m o d e l c h a r a c t e r i s r i c s are l i s t e d i n T a b l e 1, a n d i n Fig. 1 a s c h e m a t i c v i e w o f t h e e x p e r i m e n t a l s e t - u p is s h o w n . The v e r t i c a l b e n d i n g b e h a v i o r o f t h e s h i p w a s r e p r o d u c e d b y s h a p i n g p r o p e r l y t h e e l a s t i c b e a m t h a t f o r m e d t h e b a c k b o n e o f t h e s e g m e n t e d m o d e l . The elastic b e a m w a s m a d e o f a n a l u m i -n i u m a l l o y w i t h 2 0 e l e m e -n t s o f c o -n s t a -n t l e -n g t h a -n d v a r i a b l e transverse s e c t i o n , as r e p r e s e n t e d i n Fig. 2. I n o r d e r t o shape t h e b e a m secrions o f t h e b a c k b o n e , t h e b e n d i n g s t i f f n e s s a n d shear area d i s t r i b u t i o n s , o b t a i n e d b y c o l l a p s i n g t h e s t r u c t u r a l 3 D FE m o d e l o f t h e f u l l - s c a l e s h i p i n t o a n e q u i v a l e n t T i m o s h e n k o b e a m , w e r e used as r e f e r e n c e data. The m o d e l scale w a s 1:30, t h a t

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112 D, Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110-122 i m p l i e s a c c o r d i n g t o Froude scaling a m o d e l f r e q u e n c y o f t h e t w o -n o d e b e -n d i -n g m o d e i -n a i r equal t o 11.1 Hz. This r e f e r e -n c e v a l u e w a s c l o s e l y a p p r o x i m a t e d i n d r y - v i b r a t i o n tests o f t h e s e g m e n t e d m o d e l C^<';;,>,y^ 10.8 H z ) . T h i s s a t i s f a c t o r y r e s u l t w a s c o n f i r m e d b y C o p p o t e l l i et a l . ( 2 0 0 8 ) f o r t h e c o r r e s p o n d i n g w e t m o d e e s t i m a t e d w i t h v i b r a t i o n tests u s i n g w a v e e x c i t a t i o n ( / ' ^ j t 7.4 Hz w i t h r e s p e c t a r e f e r -ence f r e q u e n c y o f 7.3 H z ) . The m o d a l c o r r e s p o n d e n c e , u p o n w h i c h t h e h y d r o e l a s t i c s c a l i n g w a s based, i n v o l v e d a m o d e l d e s i g n w i t h a s l i g h t l y r e d u c e d c u r v e o f t h e s e c t i o n a l i n e r t i a ( i n o r d e r t o c o u n t e r b a l a n c e t h e l o w e r v a l u e a t t a i n e d b y t h e shear d e f o r m a b i l i t y ) a n d an o p t i m i z e d b a l l a s t p o s i t i o n . T h e ballasts w e r e placed b y a n o p t i m i z i n g a l g o r i t h m i n t h e n e i g h b o r h o o d o f t h e r e a r n o d e o f t h e f i r s t ( t w o - n o d e s ) b e n d i n g m o d e , i n o r d e r n o t t o decrease t o o m u c h its f r e q u e n c y a n d so t o keep i t close t o its r e f e r e n c e (scaled) v a l u e . The h u l l w a s d i v i d e d i n t o s i x s e g m e n t s , each o n e c o n n e c t e d t o t h e elastic b e a m w i t h a v e r t i c a l steel leg; t h e r e f o r e , a n a d e q u a t e s p a t i a l s a m p l i n g o f t h e t r u e f l u i d l o a d i n g a l o n g t h e h u l l w a s a c h i e v e d . The l o n g i t u d i n a l p o s i t i o n s o f t h e h u l l c u t s w e r e chosen i n o r d e r t o l o a d t h e legs i n a s i m i l a r w a y d u r i n g t h e s e a k e e p i n g tests (see Table 2 ) . These s e g m e n t s w e r e m a d e o f f i b e r - g l a s s , a n d t h e gaps b e t w e e n a d j a c e n t s e g m e n t s w e r e m a d e w a t e r - t i g h t b y u s i n g r u b b e r straps. The s e g m e n t s are n u m b e r e d f r o m t h e b o w ( N o . 1) t o t h e s t e r n ( N o . 6), as s h o w n i n Fig. 1. As a r i g i d - b o d y , t h e p h y s i c a l m o d e l w a s f r e e t o heave, t o p i t c h a n d , p a r t i a l l y , t o surge.

I n every test t h e f o l l o w i n g physical q u a n t i t i e s w e r e measured: ( i ) t h e absolute w a v e height, ( i i ) the d r a f t at t w o specific sections o n t h e h u l l , ( i i i ) the r i g i d - b o d y degrees o f f r e e d o m (dofs), ( i v ) the vertical force o n each segment and ( v ) the vertical b e n d i n g m o m e n t o n several b e a m sections. The i n c o m i n g , absolute w a v e h e i g h t was

Table 1

General characteristics o f the scaled model.

Scale ). 1 : 30

Length between perpendiculars ( m ) f-pp 4.280

Beam ( m ) B 0.6

Draft (mean) ( m ) d 0.137

Mass (displacement) ( k g ) M 144.74

Longitudinal position of center of gravity ( m ) XG 1.621

Vertical position of center of gravity ( m ) ZG 0.259

M o m e n t o f inertia about center of gravity ( k g m ^ ) Jyy 201.93

measured b y using t w o f i n g e r probes placed at f i x e d positions w i t h respect to the t o w i n g t a n k (see Figs. 1 and 3 ) . These positions, one i n f r o n t o f the m o d e l (FPQ), the o t h e r o n the m o d e l l e f t side (FP2), w e r e chosen so t h a t the i n c i d e n t w a v e was n o t y e t a f f e c t e d b y t h e presence o f the h u l l . The local d r a f t was d i r e c t l y m e a s u r e d a t t w o h u l l sections WP2 a n d WP3 (corresponding to the m i d d l e o f the segments 2 and 3 ) b y u s i n g capacitive w i r e probes, s h o w n i n Fig. 3. The heave, p i t c h a n d surge dofs w e r e measured w i t h a n o p t i c a l system, based o n cameras placed o n the carriage a n d o n f o u r LEDs g l u e d o n a plate carried o n b o a r d the m o d e l . The optical s y s t e m i n d i c a t e d d i r e c t l y t h e sensed m o t i o n w i t h respect to t h e m o d e l center o f g r a v i t y C. The legs connecting t h e segments to t h e elastic

Table 2

Mass and length at w a t e r line o f the segments.

Segment 1 2 3 4 5 6 Mass (kg) Length at W . L ( m ) 9.250 11.050 0.682 0.615 12.050 12.600 14.400 15.550 0.779 0.820 0.820 0.567

Fig. 3. Finger probes for absolute wave elevation ( o n the left) and wire-probes f o r local d r a f t measurement onboard ( o n the right).

r u b b e r strip

wire w a v e p r o b e _Fff,

strain g a u g e b a c k b o n e

f i n g e r w a v e p r o b e

Fig. 1. Schematic representation o f the segmented-hull and the wave probes layout.

II

_III

D11 D10 3 D 2 D 1 D I , i r (—_

P

u u 2 1

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D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110-122 113

Fig. 4. Strain-gages placed on the beam top-face.

b e a m had e m b e d d e d load cells to measure t h e vertical force. The use o f these l o a d cells needed some care. T h e f i r s t r e q u i r e m e n t t o f u l f i l l was t h a t t h e i r d e f o r m a t i o n under load induces s m a l l relative displacements o f one segment w i t h respect t o the o t h e r i n o r d e r t o avoid contact b e t w e e n t h e m . A second r e q u i r e m e n t w a s t h a t the f l u i d m o m e n t should n o t a f f e c t s i g n i f i c a n t l y the v e r t i c a l force m e a s u r e m e n t , at least w i t h m e d i u m and l o n g waves ( w i t h respect t o the s e g m e n t lengths), and d u r i n g the s l a m m i n g tests. Thus, the elastic d e f o r m a t i o n and m e a s u r e m e n t p e r f o r m a n c e o f t h e load-cells w e r e i n i t i a l l y tested r e p r o d u c i n g the above c o n d i t i o n s i n a c o n -t r o l l e d e n v i r o n m e n -t . The b e n d i n g m o m e n -t a c -t i n g u p o n -the b e a m was measured i n 12 points by using s t r a i n gauges g l u e d o n t h e t o p face o f the b e a m (see Fig. 4 ) . The c a l i b r a t i o n o f t h e s t r a i n gauges was p e r f o r m e d l o a d i n g statically t h e b e a m a n d c o m p a r i n g t h e t h e o r e -tical b e n d i n g m o m e n t s w i t h t h e voltage values. The a c q u i s i t i o n system based o n a N a t i o n a l I n s t r u m e n t s SCXl m o d u l e recorded globally 28 signals at a 5 0 0 H z s a m p l i n g rate whereas t h e w i r e probes had t o be s a m p l e d at 5 0 Hz due t o l i m i t a t i o n o f t h e i r o w n a c q u i s i t i o n cards. The highest s a m p l i n g rate a t w h i c h m o s t o f the signals w e r e recorded was selected to represent accurately t h e s l a m events (see Dessi and M a r i a n i , 2 0 0 8 ) . The w a t e r - e n t r y phase f o r the scaled f a s t - f e r r y m o d e l lasts generally about 0.2-0.3 s and t h e r e f o r e a b o u t 100 p o i n t s are used f o r d e s c r i b i n g t h e rising side o f a s l a m m i n g i m p u l s e . I t i m p l i e s t h a t i n the w o r s t case the e r r o r relative t o the peak value o f t h e curves is a b o u t 1% o f the o v e r a l l v a r i a t i o n (peak t o peak) o f t h e considered variable. The e r r o r is larger i n the case o f t h e signals acquired f r o m t h e w i r e probes b u t these data are considered j u s t f o r c o m p a r i s o n . A t r i g g e r w a s used t o synchronize b o t h the a c q u i s i t i o n systems a n d t h e w i r e signals w h i c h w e r e t o be n u m e r i c a l l y re-sampled u s i n g splines.

3. The kinematic criterion: application to experimental data

and related issues

3.J. Applicability ofthe Ochi-MoCCer criterion

The c r i t e r i o n f o r t h e o c c u r r e n c e o f s h i p s l a m m i n g has b e e n d e b a t e d f o r t w o decades since t h e p u b l i c a t i o n o f t h e o r i g i n a l Ochi p a p e r ( 1 9 6 4 ) . F o l l o w i n g O c h i , t h e necessary a n d s u f f i c i e n t c o n d i -t i o n s f o r -t h e o c c u r r e n c e o f a s l a m m i n g e v e n -t are: • t h e r e l a t i v e m o t i o n Wr at a r e f e r e n c e s e c r i o n m u s t be e q u a l t o t h e s e c t i o n a l d r a f t d a t i n s t a n t o f s l a m m i n g T J ; • t h e r e l a t i v e v e l o c i t y Wr at t h e same s e c t i o n m u s t be less t h a n o r e q u a l t o a g i v e n negative t h r e s h o l d -VBS, i.e., VVr|„^ = rf<-VB5, (1) w h e r e t h e s u b s c r i p t BS indicates t h a t t h i s t h r e s h o l d is r e l a t i v e t o b o t t o m s l a m m i n g . T h e c r i t e r i o n i n t r o d u c e d b y O c h i w a s e s t a b -l i s h e d c o n s i d e r i n g t h e m e a s u r e d pressure o n t h e k e e -l p -l a t e o f t h e MARINER, t h a t w a s t h e n c o r r e l a t e d w i t h t h e e x p e r i m e n t a l l y e s t i m a t e d i m p a c t v e l o c i t y . T h e t h r e s h o l d v e l o c i t y w a s d e f i n e d as t h e v a l u e b e l o w w h i c h n o i m p a c t pressure w a s o b s e r v e d a n d , f o r a d i f f e r e n t s h i p , t h i s v a l u e s h o u l d be m o d i f i e d a c c o r d i n g t o Froude s c a l i n g l a w regardless o f t h e shape o f t h e s e c t i o n close t o t h e b o t t o m , i.e., b e l o w 1 / l O t / i o f t h e n o m i n a l d r a f t . L o o k i n g at t h e o r i g i n a l p r e s s u r e / v e l o c i t y d i a g r a m t h a t i n s p i r e d Ochi's s t a t e m e n t , i t seems t h a t t h e v e l o c i t y t h r e s h o l d s i m p l y r e p r e s e n t s t h e l o w e r l i m i t o f v a l i d i t y o f t h e r e l a t i o n s h i p p = l / 2 p v v ? C p , (2) w h e r e Cp is t h e p r e s s u r e c o e f f i c i e n t , vvr is t h e r e l a t i v e v e r t i c a l v e l o c i t y a n d p is t h e fluid d e n s i t y . I n fact, e x p e r i m e n t a l p o i n t s a p p e a r t o d e v i a t e f r o m t h e regression l i n e o n l y as t h e y a p p r o a c h t h e area o f less i n t e n s e w a t e r i m p a c t s . A s i m i l a r a p p r o a c h f o r t h e choice o f t h e t h r e s h o l d v a l u e w a s t h e one i n t r o d u c e d b y C o n o l l y ( 1 9 7 4 ) a decade l a t e r . He f o c u s e d o n t h e d e f i n i t i o n o f a t h r e s h o l d as t h e l o w e r l i m i t o f s l a m m i n g p r e s s u r e peaks m e a s u r e d a l o n g t h e h u l l b o t t o m . F r o m t h i s , a l i m i t v e r t i c a l v e l o c i t y c a n be o b t a i n e d as w e l l . I t is w o r t h w h i l e t o recall t h a t O c h i a n d M o t t e r ( 1 9 7 3 ) s h i f t e d t h e i r a t t e n t i o n also t o t h e d e f l n i t i o n o f a s h i p f o r w a r d speed l i m i t based o n t h e c o n s i d e r a t i o n o f several s l a m m i n g - i n d u c e d e f f e c t s . I n p a r t i c u l a r , t h e y p o i n t e d o u t t h e l i n k b e t w e e n s l a m m i n g e v e n t s a n d the d a m a g e t h a t m i g h t o c c u r i n t h e b o t t o m k e e l plates, aspects t h a t h a v e since b e e n i n v e s t i g a t e d m o r e d e e p l y succes-s i v e l y . F r o m t h e analysucces-sisucces-s o f m a x i m u m succes-stresucces-ssucces-sesucces-s a n d / o r p l a t e d e f l e c t i o n , t h e y d e v e l o p e d a r u l e a l t e r n a t i v e f o r t h e v o l u n t a r y speed r e d u c t i o n o f t h e c a p t a i n t o select i f s l a m m i n g is t o b e a c c e p t a b l e f o r t h e s h i p . T h e O c h i c r i t e r i o n , a s s i i m e d as t h e basic r e f e r e n c e i n t h i s paper, is e s s e n t i a l l y b u i l t o n clear b u t s i m p l i f l e d k i n e m a t i c c o n s i d e r a t i o n s e s t a b l i s h e d o n t h e basis o f e x p e r i m e n t a l p r e s s u r e o b s e r v a t i o n s . D e s p i t e its u s e f u l n e s s , i t rises s o m e q u e s t i o n s a b o u t its p r a c t i c a l a p p l i c a t i o n a n d its p h y s i c a l m e a n i n g , t h a t are discussed i n t h e f o l l o w i n g sections.

3.2. Measurement of water entry kinematics

T h e e x p e r i m e n t a l p r o c e d u r e c a r r i e d o u t t o e v a l u a t e t h e O c h i p h y s i c a l c o n d i t i o n s necessary t o d i s c r i m i n a t e t h e s l a m m i n g o c c u r r e n c e is flrst i l l u s t r a t e d . T h e r e l a t i v e m o t i o n Wr(x,t) is c a l c u l a t e d as f o l l o w s : Wr{x,t} = Wc(t)+xOit}-h(x,t). (3) w h e r e h(x,t) is t h e a b s o l u t e w a v e e l e v a t i o n m e a s u r e d w i t h t h e w a v e flnger-probe, wdt) a n d 0(t) are t h e r e c o r d e d heave a n d p i t c h , r e s p e c t i v e l y , a n d x i n d i c a t e s t h e abscissa o f t h e c o n s i d e r e d s e c t i o n ( w i t h r e s p e c t t o t h e m o d e l c e n t e r o f g r a v i t y G).

D e s p i t e its s i m p l i c i t y , i n some cases t h i s p r o c e d u r e m a y b e f a u l t y , l e a d i n g t o possible inaccuracies i n s l a m m i n g i d e n t i f i c a t i o n t h a t need t o be c o n s i d e r e d . I f t h e w a v e s u r f a c e b e l o w t h e i m p a c t i n g b o t t o m is a f f e c t e d b y t h e s h i p r a d i a t e d w a v e field, Eq. ( 3 ) is n o l o n g e r v a l i d d u e t o these u n a c c o u n t e d d i s t u r b a n c e s a n d j u s t p r o v i d e s a n a p p r o x i m a t i o n o f t h e real s i t u a t i o n . F u r t h e r -m o r e , i t -m a y also o c c u r t h a t t h e finger-probe -m e a s u r e -m e n t p o i n t is r e a c h e d b y t h e w a v e s g e n e r a t e d b y t h e h u l l , so t h a t t h e u n d i s t u r b e d w a v e e l e v a t i o n can n o t be m e a s u r e d i n t h a t p o s i t i o n .

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114 D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110-122 T h e r e f o r e , a s e c o n d m e a s u r e m e n t t e c h n i q u e , b a s e d o n t h e d i r e c t m e a s u r e m e n t o f t h e l o c a l d r a f t d(x,t), w a s e x p l o i t e d i n t h i s e x p e r i m e n t a l p r o g r a m t o c h e c k i f t h e c a l c u l a t i o n o f t h e r e l a t i v e m o t i o n y i e l d s a g o o d e s t i m a t i o n o f t h e i m p a c t t i m e a n d v e l o c i t y . As a n t i c i p a t e d i n S e c t i o n 2 , t w o w i r e p r o b e s w e r e i n s t a l l e d o n b o a r d i n t h e m i d d l e o f t h e second a n d t h i r d s e g m e n t s ( n u m -b e r e d f r o m t h e -b o w ) , w h e r e t h e h u l l w a s l i k e l y t o e x p e r i e n c e s l a m m i n g . I n d e e d , t h i s e x p e r i m e n t a l t e c h n i q u e a l s o d e m a n d s s u i t a b l e p r o c e s s i n g o f t h e r e c o r d e d signals. B e y o n d t h e s e n s o r c a l i b r a t i o n t h a t a l l o w s us t o t r a n s f o r m t h e v o l t a g e o u t p u t i n t o t h e w i r e w e t t e d l e n g t h , t w o o t h e r r e l e v a n t issues n e e d t o be c o n s i d e r e d : t h e first c o n c e r n s t h e o r t h o g o n a l p r o j e c t i o n o f t h e w e t t e d l e n g t h o v e r t h e v e r t i c a l p l a n e o f t h e s h i p t o o b t a i n t h e s e c t i o n a l d r a f t , w h e r e a s t h e s e c o n d relates t o t h e l a c k o f s m o o t h n e s s t h a t t h e r e c o r d e d signals m a y e x h i b i t . T h e m a i n findings o f t h i s a n a l y s i s are s u m m a r i z e d b y t h e c o m p a r i s o n s h o w n i n Fig. 5 r e l a t i v e t o r e g u l a r w a v e tests.. To e n h a n c e t h e c o m p a r a b i l i t y , t h e r e c o r d e d l o c a l d r a f t is s h i f t e d d o w n w a r d b y a n a m o u n t e q u a l t o t h e n o m i n a l d r a f t , p r o v i d i n g t h e s o - c a l l e d r e l a t i v e d r a f t d r ( x , f ) = d ( x , t ) - d , t h a t is p l o t t e d t o g e t h e r w i t h t h e ( u n d i s t u r b e d ) r e l a t i v e w a v e e l e v a t i o n C(x,t) = -Wr(x,t) ( i n Fig. 6 , t h e m e a n i n g o f dr is c l a r i f i e d ) . As e x p e c t e d , t h e r e l a t i v e w a v e e l e v a t i o n ( d a s h e d l i n e i n Fig. 5 ) has a r a t h e r d i f f e r e n t b e h a v i o r w i t h r e s p e c t t o t h e r e l a t i v e d r a f t ( s o l i d l i n e ) , s i n c e t h e l a s t one a c c o u n t s a l s o f o r w a t e r , u p r i s e g e n e r a t e d b y s l a m m i n g e v e n t s a n d n e a r - f i e l d e f f e c t s d u r i n g t h e e x i t phase. H o w e v e r , i t is i n t e r e s t i n g t o n o t e t h a t t h e t i m e d e r i v a t i v e o f t h e r e l a t i v e w a v e e l e v a t i o n , a t i m p a c t t i m e ( m a r k e d w i t h a v e r t i c a l l i n e ) , p r o v i d e s r a t h e r t h e s a m e e s t i m a t i o n o f t h e i m p a c t v e r t i c a l v e l o c i t y . T h i s e x p e r i m e n t a l e v i d e n c e i n d i c a t e s t h a t b e f o r e t h e i m p a c t t i m e t h e 0.6

Relative Draft (Wire Probes) Draft Derivative (Wire Probes)

Rel. Wave Elevation (Abs. Wave - RB Displ.) o-— Rel. Wave Elev. Derivative (Abs. Wave - RB Displ.)

Nominal Water Exit/ Entry Threshold

w a t e r s u r f a c e u n d e r t h e i m p a c t i n g b o w is w e l l r e p r e s e n t e d b y t h e u n d i s t u r b e d w a v e e l e v a t i o n , because t h e h i g h f o r w a r d speed o f t h e s h i p m a k e s n e g l i g i b l e t h e d i s t u r b a n c e s p r o d u c e d b y t h e d i f f r a c t e d a n d r a d i a t e d w a v e fields. T h e s a m e r e m a r k a p p l i e s also t o t h e flnger-probe p o s i t i o n . T h e r e f o r e , f o r t h e p u r p o s e o f t h e p r e s e n t i n v e s t i g a t i o n , t h e e x p e r i m e n t a l l y e v a l u a t e d r e l a t i v e m o t i o n can be a s s u m e d t o i m p l e m e n t t h e O c h i c r i t e r i o n .

3.3. Extension ofthe Octii-Motter criterion to tlie bow-flare slamming case The h u l l i m m e r s i o n can be e f f i c i e n t l y a n d a c c u r a t e l y d e t e r -m i n e d b y t h e w i r e p r o b e -m e a s u r e -m e n t a n d p r o v i d e s a g e n e r a l w a y t o d e t e r m i n e t h e w e t n e s s o f t h e s h i p b o t t o m . H o w e v e r , since t h e s h i p f o r w a r d speed is s u f f i c i e n t l y h i g h i n the p r e s e n t e x p e r i -m e n t a l analysis, i t is r e a s o n a b l e t o set Wr(x,t) = -C(x,t) -dr(x,t) a n d t h e r e l a t i v e m o t i o n can s a t i s f a c t o r i l y describe the w a t e r -e n t r y o f t h -e b o w . T h -e a d v a n t a g -e o f t h i s choic-e li-es also i n t h -e p o s s i b i l i t y to e x t e n d t h i s analysis i n t i m e i n s t a n t s a f t e r t o t h e i m p a c t occurrence.

T h e m i n i m u m e n t r y v e l o c i t y , u s u a l l y r e a c h e d a r o u n d t h e m i d d l e o f the e n t r y phase, is r e l e v a n t i n t h e p r e s e n t analysis because i t is a s s u m e d i n t h e f o l l o w i n g t o be s y m p t o m a t i c o f t h e i n t e n s i t y o f b o w - f l a r e s l a m m i n g events.^ T h e k i n e m a t i c c o n d i t i o n f o r b o w - f l a r e s l a m m i n g is t h e n d e f i n e d as M i n [ W r ] < - V p s . ^ ( 4 ) w h e r e Vfs is a n e w ( p o s i t i v e ) t h r e s h o l d g r e a t e r or a t least e q u a l t o t h e O c h i t h r e s h o l d VBS, since f o r b o t t o m s l a m m i n g cases Mm[Wr]<Wr\^^^^ h o l d s s t r i c t l y , as t h e e x p e r i m e n t a l e v i d e n c e s h o w s . Since t h e m i n i m u m v e l o c i t y is d e t e r m i n e d i n a w a y c o n s i s t e n t w i t h t h e d e t e r m i n a t i o n o f t h e i m p a c t v e l o c i t y , a n e x p e r i m e n t a l c o r r e l a t i o n b e t w e e n t h e i m p a c t v e l o c i t y a n d t h e m i n i m u m w a t e r - e n t r y v e l o c i t y m a k e s sense, a n d i n p a r t i c u l a r M i n [ W r ] 1.5 vvrljy^ ^ t h u s s u g g e s t i n g t h a t w e s h o u l d select V K = 3 / 2 VBS. O n t h e o t h e r h a n d , t h e use o f t h e t i m e d e r i v a t i v e o f t h e l o c a l d r a f t i n Eq. ( 4 ) ( i n s t e a d o f W r ) m a y lead t o r e s u l t s h e a v i l y a f f e c t e d b y t h e h u l l s h a p e d u e t o t h e flow a c c e l e r a t i o n as i t rises a l o n g t h e s e c t i o n sides. T h i s f e a t u r e w o u l d be n o t d e s i r a b l e w h e n c o m p a r i n g b o t t o m s l a m m i n g a n d flare s l a m m i n g e v e n t s .

4. Cluster analysis

It is w o r t h r e c a l l i n g t h a t t h e g e n e r i c s l a m m i n g process Z c a n be r e p r e s e n t e d , as a l r e a d y p r o p o s e d b y M a n s o u r a n d L o z o w ( 1 9 8 2 ) a n d Ferro a n d M a n s o u r ( 1 9 8 5 ) , b y a pulse process as

Fig. 5. Evaluation o f the d r a f t obtained by direct w i r e - p r o b e measurement and by reconstruction f r o m rigid-body and absolute wave elevation measurement (x position at the m i d d l e o f segment 2).

' /

W a t e r s u r f a c e , / N'" A b s o l u t e w a v e Relative d r a f t e l e v a t i o n /; c/,/x,t) ^ Local d r a f t d f x j j M e a n v^ater s u r f a c e -7' W . L . N o m i n a l d r a f t d ( 5 ) w h e r e t h e s u p e r s c r i p t r d e n o t e s a s p e c i f l c t o w i n g t a n k r u n ( i n t h e p r e s e n t case, r = l 1 1 ) , N"^' is t h e Poisson c o u n t i n g process ( p e r t a i n i n g to r u n r ) , 5 ( - ) is t h e D i r a c delta f u n c t i o n , y^P a n d x^p are t h e i n t e n s i t y a n d t h e i m p a c t t i m e , r e s p e c t i v e l y , a s s o c i a t e d w i t h t h e i t h s l a m m i n g e v e n t o c c u r r e d d u r i n g r u n r ( t h e s u b s c r i p t s is d r o p p e d f o r sake o f conciseness i n t h e i m p a c t t i m e x w h e n e v e r i t does n o t g e n e r a t e a m b i g u i t y ) . To c l a r i f y t h e p h y s i c a l m e a n i n g o f Eq. ( 5 ) , i n t h e t o p o f Fig. 7 t h e process Z ' ^ ' is s y n t h e t i c a l l y r e p r e s e n t e d as a sequence o f v e r t i c a l s e g m e n t s i n d i c a t i n g t h e i m p a c t v e l o c i t y , n a m e d as V j f ^ , at t h e c o r r e s p o n d e n t i m p a c t t i m e s TJ^*. The i m p a c t t i m e s T P ' are o b t a i n e d f r o m t h e t i m e -h i s t o r y o f t -h e v e r t i c a l r e l a t i v e v e l o c i t y Wr(t) ( s l a m m i n g events are

Fig. 6. Physical meaning o f variables related to wave elevation.

' The absence of the bow-emergence prevents to i d e n t i f y a particular t i m e at w h i c h the water-entry velocity has to be evaluated.

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D. Dessi, E. Ciappi/Ocean Engineering 62 (2013) 110-122 115 m a r k e d w i t h circles i n t h e m i d d l e c u r v e ) , w h e r e a s t h e i m p a c t v e l o c i t i e s can be e v a l u a t e d f r o m t h e t i m e - h i s t o r y o f t h e v e r t i c a l r e l a t i v e v e l o c i t y vvr(t), p l o t t e d at t h e b o t t o m o f Fig. 7. W i t h r e s p e c t t o Ferro a n d M a n s o u r ( 1 9 8 5 ) , t h e s l a m m i n g i n t e n s i t i e s x f \ r e p r e s e n t e d by t h e s l a m m i n g i m p a c t v e l o c i t i e s , w i l l be s h o w n n o t t o be i n d e p e n d e n t a n d e q u a l l y d i s t r i b u t e d r a n d o m variables.

The Ochi k i n e m a t i c c r i t e r i o n was t h e n a p p l i e d t o t h e e x p e r i -m e n t a l data collected i n t h e 11 t o w i n g - t a n k -m n s t o i d e n t i f y the r e l e v a n t statistics. The tests w e r e c a r r i e d o u t i n i r r e g u l a r sea a c c o r d i n g t o a J o n s w a p s p e c t r u m w i t h H 1 / 3 = 5 m a n d T] = 7 . 5 s a t f u l l - s c a l e ( H , / 3 ~ 0.016 m a n d T, ~ 1.30s a t m o d e l scale). The m e a s u r e d sea s p e c t r u m is s h o w n i n Fig. 8 a n d c o m p a r e d w i t h the t h e o r e t i c a l s p e c t r u m o f the w a v e m a k e r . Each v a l i d a c q u i s i t i o n lasted a b o u t 3 0 s, a n d overall t i m e at m o d e l scale o f a b o u t 3 6 0 s is e q u i v a l e n t t o nearly 3 0 m i n at f u l l - s c a l e . I n Fig. 9 t h e i m p a c t t i m e z'P f o r each i d e n t i f i e d s l a m s a t i s f y i n g the Ochi c o n d i t i o n s (see Eq. ( 1 ) ) is d r a w n as a p o i n t w i t h the abscissa i n d i c a t i n g t h e c o r r e -s p o n d i n g t o w i n g - t a n k r u n . Thu-s, a l i g n e d dot-s o n v e r t i c a l line-s i n d i c a t e s l a m m i n g i m p a c t s o f t h e same test. The s l a m m i n g i m p a c t s s h o w t h e t e n d e n c y t o appear g r o u p e d i n t o s h o r t sequences, n a m e d as s l a m clusters (one o f t h e m is c i r c l e d ) . This e x p e r i m e n t a l o b s e r v a t i o n ' c o n t r a d i c t s t h e h y p o t h e s i s o f i m p a c t i n d e p e n d e n c e u p o n w h i c h several early theories o f s l a m m i n g statistics w e r e b u i l t o n . The sequence o f repeated s l a m m i n g events seems o n l y p a r t i a l l y r e l a t e d t o g r o u p i n g o f waves w i t h r e l e v a n t heights. I n Fig. 10 the r e l a t i v e w a v e e l e v a t i o n ( c o n t i n u o u s l i n e ) is p l o t t e d t o g e t h e r w i t h

20 25 30 Time [s]

Fig. 7. S l a m m i n g process Z'^' obtained f r o m the seakeeping tests ( t o w i n g - t a n k r u n r-2). Dashed lines indicate the relative m o t i o n and velocity thresholds of the Ochi

criterion, respectively. 0.0015 0.001 • • Q. 0.0005 0.5 1 Frequency [Hz]

Fig. 8. Measured and theoretical sea spectrum.

1.5 40 35 30 25 20 E-15 l o t 5 0

0

I 1 I I -0 1 2 3 4 5 6 7 Run 9 10 11 12

Fig. 9. Impact times f o r each r u n recorded d u r i n g t o w i n g - t a n k tests.

25 30 Time [s]

Fig. 10. Encountered w a v e elevation (line and points) and relative m o t i o n

(solid line).

t h e absolute w a v e e l e v a t i o n o f t h e e n c o u n t e r e d w a v e s ( l i n e and p o i n t s ) relative to t h e same s h i p section. There is n o t a s t r i c t c o r r e l a t i o n b e t w e e n t h e e l e v a t i o n o f t h e w a v e s a n d t h e r e l a t i v e w a v e e l e v a t i o n ( o r r e l a t i v e m o t i o n ) o c c u r r i n g s h o r t l y a f t e r t h e w a v e crest has passed. M o r e o v e r , i t m a y also h a p p e n t h a t t h e second h i g h w a v e crest f o l l o w i n g a n d a n t i c i p a t i n g a w a t e r i m p a c t (see g r o u p e d slams at T 4 - T 5 a n d at T S - T ? ) e x h i b i t s an e l e v a t i o n l o w e r t h a n t h e f i r s t crest. A c c o r d i n g to O c h i ( 1 9 6 4 ) , the p r o b a b i l i t y o f s l a m m i n g e v e n t s ( s a t i s f y i n g t h e O c h i c o n d i t i o n s ) s h o u l d c o m p l y t h e f o l l o w i n g e x p o n e n t i a l l a w : Pr{Slam) = e^e (6) w h e r e a = - d /(lal;) a n d /( = V ^ s / ( 2 < ) . w i t h < a n d t h e v a r i a n c e o f t h e r e l a t i v e m o t i o n a n d r e l a t i v e m o t i o n v e l o c i t y a t t h e c o n s i d e r e d s h i p s e c t i o n f o r a l l t h e r u n s . T h e n u m b e r o f i m p a c t s p e r u n i t time, d e f i n e d as /. = N^/T w i t h Ns the o v e r a l l n u m b e r o f o b s e r v e d s l a m m i n g e v e n t s i n t h e t i m e i n t e r v a l T, can be e v a l u a t e d as (7) w h e r e /. is also t e r m e d u n c o n d i t i o n a l m e a n s l a m m i n g rate. I n Table 3 t h e p r e d i c t e d a n d e x p e r i m e n t a l value o f t h e m e a n s l a m m i n g r a t e are r e p o r t e d , s h o w i n g t h a t Ochi t h e o r y s i g n i f i -c a n t l y u n d e r e s t i m a t e s t h e n u m b e r o f e x p e -c t e d s l a m m i n g e v e n t s . A m e a s u r e o f t h e t e n d e n c y o f t h e slams t o a p p e a r i n c l u s t e r s can be e v a l u a t e d w i t h d i f f e r e n t s t a t i s t i c a l indices. T h e f i r s t one c o n c e r n s t h e e s t i m a t i o n o f t h e t i m e i n t e r v a l A T J b e t w e e n

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116 D. Dessi, E. Ciappi / Ocean Engineenng 62 (2013) 110-122

Table 3

Predicted and experimental number of s l a m m i n g impacts per unit t i m e (unconditional mean slam-m i n g rate at slam-model scale).

Type o f estimation / .

Predicted (Eq. ( 7 ) ) 0.094

Experimental 0.300

•a 2

Theoretical (Poisson Law)

, = = Mixed (Poisson Law with exp. slamming rate) Experimental

,1ltir

:flïïf-:mn-n

-.-n

',fl-,- , , n

0 A To 2 4 6 8 T i m e b e t w e e n s l a m s [s]

Fig. 11. Comparison between theoretical (solid and dashed lines) and experi-mental (bars) probability density functions relative to the t i m e interval between successive slams (at model scale).

successive i m p a c t s , t h a t f o r a Poisson process is g i v e n b y t h e f o l l o w i n g p r o b a b i l i t y d e n s i t y f u n c t i o n ( p d f ) (8) w h e r e Tpicch is t h e n a t u r a l p i t c h i n g p e r i o d . T h e c o m p a r i s o n b e t w e e n t h e t h e o r e t i c a l a n d e x p e r i m e n t a l t i m e i n t e r v a l s b e t w e e n successive s l a m s is s h o w n i n Fig. 1 1 . I t is w o r t h n o t i n g t h a t m o s t o f t h e o b s e r v e d t i m e i n t e r v a l s f a l l i n a n a r r o w b a n d c e n t e r e d o n t h e v a l u e d e n o t e d as A T Q , w h i c h is c l o s e l y r e l a t e d to t h e n a t u r a l p i t c h i n g p e r i o d , i.e., A T Q ~ Tpitch = 1 s at m o d e l scale, a n d also t o t h e f r e q u e n c y o f t h e e n c o u n t e r e d sea s p e c t r u m peak, since A T O 1 / / ^ ' ' " ' ' " = 0 . 8 4 s. As also a p p a r e n t i n Fig. 9, t h e s l a m m i n g e v e n t s i n s i d e t h e c l u s t e r sequences are o f t e n s e p a r a t e d i n t i m e b y n e a r m u l t i p l e s o f A T Q . T h i s e x p e r i m e n t a l e v i d e n c e c o n t r a d i c t s t h e e x p e c t a t i o n o f a s m o o t h p r o b a b i l i t y d i s t r i b u t i o n o v e r t h e v a l u e s A T > A T O , as s h o u l d h o l d i n t h e case o f i n d e p e n d e n t l y o c c u r r i n g i m p a c t s . B o t h t h e s o l i d a n d t h e d a s h e d l i n e s r e p r e s e n t t h e values g i v e n b y Eq. ( 8 ) , b u t u s i n g d i f f e r e n t u n c o n d i t i o n a l s l a m m i n g rates 2: f o r t h e s o l i d l i n e t h e v a l u e o f X is p r o v i d e d b y Eq. ( 7 ) , w h e r e a s f o r t h e d a s h e d l i n e t h e e x p e r i m e n t a l v a l u e o f / r e p o r t e d i n T a b l e 3 is used. I t is e v i d e n t t h a t t h e p o o r p r e d i c t i o n g i v e n b y Eq. ( 8 ) is a c o n s e q u e n c e o f a t o o l o w e s t i m a t i o n o f t h e u n c o n d i t i o n a l m e a n s l a m m i n g r a t e . A second analysis c o n c e r n s t h e e x p e c t e d n u m b e r o f i m p a c t s i n a g i v e n t i m e w i n d o w A f . Its average v a l u e is f o r m a l l y d e f i n e d as Hj = A t • / . IVIoreover, t h e p r o b a b i l i t y t h a t ris = k e v e n t s w i l l o c c u r i n A t is e v a l u a t e d f o r a Poisson process b y t h e f o l l o w i n g p r o b a b i l i t y f u n c t i o n d i v i d e d i n t o L c o n s e c u t i v e t i m e i n t e r v a l s o f l e n g t h A t = Tr/L T h u s , A t d e n o t e s t h e c o n s t a n t l e n g t h o f t h e e l e m e n t a r y t i m e w i n d o w t o be used f o r s l a m m i n g c o u n t i n g . I n each i t h t i m e i n t e r v a l , i t m a y h a p p e n t h a t one b r m o r e s l a m m i n g events, say k{i) events, are o b s e r v e d . T h e n , l e t us d e n o t e w i t h m(/<) t h e n u m b e r o f t i m e s t h a t k e v e n t s o c c u r i n a single o b s e r v a t i o n w i n d o w ( f o r i n s t a n c e , m ( 3 ) = 2 m e a n s t h a t t w i c e t h r e e d i s t i n c t s l a m m i n g e v e n t s o c c u r r e d i n d i s t i n c t o b s e r v a t i o n t i m e w i n d o w s ) . T h i s a l l o w s us to m a k e a h i s t o g r a m a p p r o x i m a t i n g , a f t e r n o r m a l i z a t i o n , t h e e x p e r i m e n t a l p d f t h a t a c c o u n t s f o r the p r o b a b i l i t y t h a t k s l a m -m i n g e v e n t s are o b s e r v e d i n a t i -m e w i n d o w o f l e n g t h A t . T h e c h o i c e o f A f i n d e e d a f f e c t s t h i s analysis. T h e s h o r t e s t u s e f u l o b s e r v a t i o n w i n d o w f o r t h e analysis c a n be i d e n t i f i e d w i t h r e s p e c t t o t h e m i n i m u m A T , say A T „ „ n , c o n s i d e r e d i n t h e e x p e r i -m e n t a l data o f Fig. 11. Since n o -m o r e t h a n one e v e n t can o c c u r i n a n o b s e r v a t i o n t i m e w i n d o w s u c h as A f < A T , , , , , , , i t is t r i v i a l t o set A f > A T „ „ „ . O n t h e o t h e r h a n d , as l o n g as A t - ^ T r , t h e o v e r a l l a c q u i s i t i o n t i m e , a l l t h e events w i l l be t r i v i a l l y i n c l u d e d i n t h i s rime w i n d o w . T h e r e f o r e , f o r sake o f conciseness, w e w i l l f o c u s o n m u l t i p l e s o f a basic i n t e r v a l n e a r l y e q u a l to t h e p i t c h i n g p e r i o d Tpicch, t h u s l e a d i n g t o A f ~ l . l s , 2.2 s, 3.3 s, 4 . 4 s , 5.5 s. T h i s ensures h i g h l i g h t i n g o f t h e p o t e n t i a l i m p a c t c l u s t e r i z a t i o n . I n Fig. 12 t h e p d f o b t a i n e d b o t h e x p e r i m e n t a l l y ( u s i n g t h e h i s t o g r a m m e t h o d ) a n d t h e o r e t i c a l l y ( p r o v i d e d b y t h e Poisson m o d e l ) are d e p i c t e d t o g e t h e r f o r t h e c h o s e n d i f f e r e n t t i m e w i n d o w s . For A f = l . l ^Tpitch< i t begins t o be e v i d e n t t h a t t h e Poisson m o d e l u n d e r p r e d i c t s t h e p o s s i b i l i t y t h a t m o r e t h a n o n e e v e n t m a y o c c u r i n t h e g i v e n time w i n d o w . T h i s lack o f a g r e e m e n t is a c o n s e -q u e n c e o f t h e c l u s t e r i n g e f f e c t n o t a c c o u n t e d f o r b y t h e Poisson m o d e l . T h i s r e m a r k is also c o n f i r m e d i n t h e c o m p a r i s o n f o r A t = 2.2 =;2rp,tch. As l o n g as A t is increased, t h e p e a k i n t h e o b s e r v e d s l a m m i n g d i s t r i b u t i o n is m o v e d f o r w a r d t o h i g h e r s l a m m i n g rates a n d , w h e n A f is decreased, t h e r e is a r e d u c e d p r o b a b i l i t y t h a t a n y e v e n t s are r e c o r d e d at a l l i n t h e same t i m e w i n d o w .

5. Correlation between impact dynamics and global effects

5.1. Impact severity: local analysis

I t is w e l l k n o w n t h a t t h e r e is a r e l a t i o n s h i p b e t w e e n t h e n u m b e r o f s l a m s p e r u n i t rime a n d t h e i r s t r e n g t h , r o u g h l y o b e y i n g t o t h e p r i n c i p l e t h a t m o r e slams are o b s e r v e d , m o r e i n t e n s e t h e y are s u p p o s e d t o be. For instance, F e r r o a n d M a n s o u r ( 1 9 8 5 ) s t a t e d t h a t

/<f"" = 2;.c24^

1 +

vi

(10) w h e r e iJ.f'^'"^ is t h e m e a n v a l u e o f t h e s l a m m i n g e x c i t a t i o n , c is t h e c o n s t a n t r e l a t i n g t h e square o f the i m p a c t v e l o c i t y t o t h e pressure.^ I f t h e s q u a r e o f t h e e n t r y v e l o c i t y is a s s u m e d as t h e r e f e r e n c e s l a m m i n g e x c i t a t i o n , t h u s s e t t i n g Xi = Wf i n Eq. ( 5 ) a n d c = l i n Eq. ( 1 0 ) , i t s m e a n v a l u e c a n be e x p e r i m e n t a l l y e v a l u a t e d as " ^ 1 = 1

(11)

Pr{k,At]=^-^^e-'^'\ (9) Its e x p e r i m e n t a l c o u n t e r p a r t can be d e f i n e d a n d e v a l u a t e d as e x p l a i n e d as f o l l o w s . The a c q u i s i t i o n t i m e f o r each r u n , Tr, is f i r s t I n T a b l e 4 t h e p r e d i c t e d a n d e x p e r i m e n t a l m e a n s l a m m i n g e x c i t a t i o n are c o m p a r e d t o each o t h e r . I t is a p p a r e n t t h a t ,

(8)

D. Dessi. E. Ciappi / Ocean Engineering 62 (2013) 110-122 117 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 e. 1 r 0.8 0.6 0.4 0.2 0 •D a 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 o • Experimental G T h e o r y (Poisson Distribution) At = 1.1s 1 2 3 4 5 N u m b e r of slams / At

•

Experimental

- O 0 Theory (Poisson Distribution)

At = 2.2s 1 0 1 2 3 4 5 6 N u m b e r o f s l a m s / A t I I Experimental e T h e o r y (Poisson D i s t r i b u t i o n ) At = 3.3s 1 2 3 4 5 N u m b e r o f s l a m s / A t • Experimental • T h e o r y (Poisson D i s t r i b u t i o n ) At = 4.4s 1 2 3 4 5 N u m b e r o f slams / At I I E x p e r i m e n t a l o T h e o r y (Poisson D i s t r i b u t i o n ) At = 5.5s 0 1 2 3 4 5 6 N u m b e r o f slams / At

Fig. 12. Comparison between theoretical and experimental probability density func-tions relative to the number of slam impacts observed in given time windows (at model scale).

Table 4

Predicted and experimental mean s l a m m i n g excitation (at model scale).

Type of estimation /'z Predicted 0.440 Experimental 1.595 Experimental Poisson 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.1 Impact v e l o c i t y [m/s]

Fig. 13. Comparison between theoretical and experimental p r o b a b i l i t y density functions relative to the impact velocity.

T h i s d i s c r e p a n c y is f u r t h e r e v i d e n t i f t h e p d f o f t h e e n t r y v e l o c i t y is t a k e n i n t o account. A c c o r d i n g t o O c h i , i t is expressed as

2a}, ( 1 2 )

I n Fig. 13 t h e p r e d i c t e d (Eq. ( 1 2 ) ) a n d e x p e r i m e n t a l p d f s are s h o w n . First, w e see t h a t t h e e x p e r i m e n t a l c u r v e s h o w s a p e a k b e y o n d t h e i m p a c t v e l o c i t y t h r e s h o l d , w h e r e a s t h e t h e o r e t i c a l c u r v e decreases m o n o t o n i c a l l y . IVIoreover, e x p e r i m e n t s s h o w t h a t i m p a c t s are l i k e l y to o c c u r also a t h i g h v e l o c i t i e s at w h i c h O c h i t h e o r y p r e d i c t s less o r even n e g l i g i b l e s l a m m i n g events.

As a f i n a l r e m a r k o n t h e discrepancies b e t w e e n p r e d i c t i o n s a n d e x p e r i m e n t s , i t is w o r t h w h i l e t o stress t h a t t h e c o m b i n a t i o n o f h i g h e r s l a m m i n g rates w i t h l a r g e r i m p a c t p r e s s u r e s is r e s p o n sible f o r r e l e v a n t local effects t h a t , u n f o r t u n a t e l y , are u n d e r -e s t i m a t -e d by Ochi t h -e o r y a n d succ-essiv-e i m p r o v -e m -e n t s . To h i g h l i g h t t h i s aspect, i t is necessary t o evaluate h o w t h e i n t e n s i t y o f these s l a m m i n g events, p r o p o r t i o n a l t o W f , is d i s t r i b u t e d i n t i m e . For t h i s p u r p o s e , a s e v e r i t y i n d e x I , d e f i n e d as t h e t i m e average o f t h e s l a m m i n g e x c i t a t i o n , is i n t r o d u c e d , i.e., 1 ( 1 3 ) t h a t , r e c a l l i n g T = Ns//., leads t o a s t r o n g e r d e p e n d e n c e o n t h e m e a n s l a m m i n g r a t e / as s h o w n also b y t h e Ferro a n d M a n s o u r f o r m u l a

Vi

(14) ( 1 5 ) a s s u m i n g t h e e n t r y v e l o c i t y as a n i n d i c a t o r o f t h e e x t e n t o f t h e s l a m m i n g e x c i t a t i o n , t h e e x p e r i m e n t a l v a l u e o f t h e e x c i t a t i o n level is a b o u t f o u r t i m e s t h e p r e d i c t e d one. T h e u n d e r e s t i m a t i o n o f /. (Table 3 ) a n d o f t h e m e a n s l a m m i n g e x c i t a t i o n (Table 4 ) s t r o n g l y a f f e c t s t h e f i n a l v a l u e o f t h e p r e d i c t e d s e v e r i t y i n d e x I a c c o r d i n g t o Ochi t h e o r y .

(9)

D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110-122

l i s

5.2. Impact severity: global effects

I t has b e e n s h o w n u p t o t h i s p o i n t t h a t t h e analysis o f s l a m m i n g p h e n o m e n a c h a r a c t e r i z e d b y a r e l e v a n t presence o f i m p a c t c l u s t e r s requires a d i f f e r e n t t h e o r e t i c a l f r a m e w o r k t o d e t e r m i n e t h e i m p a c t statistics a n d , c o n s e q u e n t l y , t h e e v a l u a t i o n o f l o c a l e f f e c t s . This lack o f a c c u r a c y r e l a t i v e t o t h e m o d e l s based o n O c h i t h e o r y is increased i f g l o b a l e f f e c t s are c o n s i d e r e d . I n t h e p a p e r o f Ferro a n d M a n s o u r ( 1 9 8 5 ) , t h e presence o f a t r a n s i e n t response d u e t o s l a m m i n g , i.e., w h i p p i n g , is t a k e n i n t o a c c o u n t b y d e c o m p o s i n g t h e t o t a l s e c t i o n a l l o a d R{t) as t h e s u m o f t w o s t a t i o n a r y stochastic processes. T h u s , a s s u m i n g R(t) = M y ( t ) , t h e v e r t i c a l b e n d i n g m o m e n t M y ( t ) is expressed as My(t) = M f \ t ) + M f \ t ) , ( 1 6 ) w h e r e M'-y''\t) is t h e s u m o f t h e l o w - f r e q u e n c y c o n t r i b u t i o n (LF), d i r e c t l y r e l a t e d v i a t h e D u h a m e l ' s i n t e g r a l t o t h e local w a v e e l e v a t i o n f i ( f ) , and Mf"'\ t h e h i g h - f r e q u e n c y ( H F ) c o n t r i b u t i o n , g i v e n as M f ' \ t ) = r h^'^\t-x)Z(x)dx, ( 1 7 ) w h e r e / i " ^ ' ( t ) is t h e i m p u l s e response f u n c t i o n f o r t h e chosen HF

l o a d . Since Z ( t ) is n o t a z e r o - m e a n process, N l f X t ) has a n o n - z e r o m e a n , g i v e n b y ( c f r . Ferro a n d M a n s o u r , 1 9 8 5 ) ^ ( M = f e H ' ^ ' ( 0 ) , ( 1 8 ) w h e r e H"^\CO) is t h e F o u r i e r t r a n s f o r m o f t h e i m p u l s e response f u n c t i o n . A c c o r d i n g t o Eq. ( 1 8 ) , t h e e v a l u a t i o n o f t h e average o f t h e g l o b a l response IA,^ w i l l s i g n i f i c a n t l y s u f f e r o f t h e a p p r o x i m a t i o n i n t r o d u c e d i n t h e p r e d i c t i o n o f fi^. H o w e v e r , i n t h e p r e s e n t case, i t is w o r t h n o t i n g also t h a t w i t h i n a c l u s t e r t h e t r a n s i e n t o s c i l l a t i o n s m i g h t have n o t been e x t i n g u i s h e d b e f o r e t h e n e x t i m p a c t , m a k i n g t h e p r e d i c t i o n w o r s e . For t h i s reason, t h e c o r r e l a t i o n b e t w e e n t h e i m p a c t k i n e m a t i c s . a n d t h e i n d u c e d response needs t o be i n v e s t i g a t e d i n m o r e d e t a i l b e f o r e t h e averaged q u a n t i t i e s are c o n s i d e r e d . I n p a r t i c u l a r , t h e f i r s t peak o f M'"''\t), o c c u r r i n g i m m e d i a t e l y a f t e r t h e s l a m m i n g e v e n t , can be a s s u m e d t o be r e p r e s e n t a t i v e o f t h e i n t e n s i t y o f t h e s l a m m i n g i n d u c e d response a n d w i l l be c o r r e l a t e d t o t h e i m p a c t v e l o c i t y Wi. The peak o f t h e w h i p p i n g b e n d i n g m o m e n t is f o r m a l l y d e f i n e d as M i = max[M<"'''(t)] w i t h t - t , - = m i n > 0, ( 1 9 ) w h e r e Mf"'\t) is o b t a i n e d t h r o u g h t h e analysis o f e x p e r i m e n t a l d a t a a n d T,- r e p r e s e n t s t h e c o r r e s p o n d i n g i m p a c t t i m e . I n o r d e r t o extract, f r o m t h e b e n d i n g m o m e n t t i m e h i s t o r y M y ( t ) , its HF p a r t M f ' ^ t ) , t h e c o n t i n u o u s w a v e l e t t r a n s f o r m ( C W T ) has b e e n used. The C o n t i n u o u s W a v e l e t T r a n s f o r m T*"'™' o f a generic s i g n a l s ( t ) is d e f i n e d as

p^^"%(tmo = - J = s(t)^ ( ^ ^ ) dt, (20)

w h e r e t h e f u n c t i o n i/'(t) is t e r m e d t h e m o t h e r w a v e l e t . T h e p a r a m e t e r s a a n d c are d e n o t e d as t h e s c a l i n g a n d t h e s h i f t p a r a m e t e r , r e s p e c t i v e l y . The s c a l i n g p a r a m e t e r plays t h e r o l e o f s e t t i n g t h e t i m e scale o r t h e f r e q u e n c y a t w h i c h t h e s i g n a l is a n a l y z e d . A t y p i c a l choice f o r t h e ' m o t h e r w a v e l e t ' is t h e so-called M o r l e t W a v e l e t , w h o s e e x p r e s s i o n is il/{t) = e-'''^é"">'. ( 2 1 ) w i t h coo = 2n. The w a v e l e t t r a n s f o r m o f t h e o v e r a l l V B M a m i d s h i p (s(t) = M y ( t ) i n t h e p r e s e n t case) is s h o w n as a c o n t o u r p l o t i n Fig. 14, w h e r e t h e d i f f e r e n t c o n t o u r s are t h e C W T levels at m o d e l scale f o r t h e t e s t r u n r = 2 . F r o m t h e analysis o f Fig. 14, i t appears

^2 r

t 1 C W T

10 15 20 26 30 35 40 Time [s]

Fig. 14. Continuous wavelet transform of the vertical bending m o m e n t a m i d s h i p (model scale data).

Time [s]

Fig. 15. Slice o f the continuous wavelet transform at ƒ = 6,8 Hz: envelope o f the HF-VBM. t h a t t h e w h i p p i n g C W T peaks lie a t a f r e q u e n c y o f a b o u t 6.6 Hz, s l i g h t l y s m a l l e r t h a n t h e f i r s t v e r t i c a l b e n d i n g m o d e f r e q u e n c y , t h a t w a s e v a l u a t e d t o be a b o u t 7.3 Hz i n c a l m w a t e r w i t h n o -f o r w a r d speed ( C o p p o t e l l i e t al., 2 0 0 8 ) . I t is w o r t h n o t i n g t h a t , since m o r e t h a n 97% o f t h e e n e r g y is u s u a l l y c o n t a i n e d i n t h e f i r s t b e n d i n g m o d e f o r t h i s s h i p ( c f r . M a r i a n i a n d Dessi, 2 0 1 2 ) , i t is r e a s o n a b l e t o d e n o t e t h e c o r r e s p o n d i n g f r e q u e n c y as t h e w h i p -p i n g f r e q u e n c y . T h e r e f o r e , t h e slice o f t h e C W T a t t h e res-ponse peak f r e q u e n c y w a s a s s u m e d t o be r e p r e s e n t a t i v e o f t h e h i g h -f r e q u e n c y V B M response a m p l i t u d e ( H F - V B M -f o r s h o r t i n t h e f o l l o w i n g ) a t m i d s h i p . This c u r v e is p l o t t e d i n Fig. 15 f o r t h e t o w i n g - t a n k r u n r = 2 .

Several H F V B M response peaks M, a p p e a r c l e a r l y d i s t i n g u i s h -able i n Fig. 15 a n d t h e i r c o r r e l a t i o n w i t h the associated s l a m m i n g e v e n t is s h o w n i n Fig. 16, w h e r e t h e H F - V B M ( t h i c k s o l i d l i n e ) is p l o t t e d t o g e t h e r w i t h t h e r e l a t i v e m o t i o n ( t h i c k d a s h e d l i n e ) a n d t h e r e l a t i v e v e l o c i t y at t h e p o s i t i o n W P l ( t h i n s o l i d l i n e ) .

F r o m t h e b o t t o m s l a m m i n g analysis, several e v e n t s are i d e n -t i f i e d a n d a d e l -t a s y m b o l is c h o s e n -t o i n d i c a -t e -t h e e n -t r y -t i m e i n s t a n t T,-. I n t h e same f i g u r e , t h e v e r t i c a l a r r o w s d e n o t e , at these t i m e i n s t a n t s , t h e e n t r y v e l o c i t y e q u a l t o Wj = W r ( T , ) . I t is e v i d e n t t h a t t h e H F - V B M peaks are l i k e l y t o o c c u r a f t e r t h e s l a m m i n g events a n d , f r o m t h e analysis o f t h i s flgure, t h e i r v a l u e s M , a p p e a r to be d e p e n d e n t o n t h e W,. H o w e v e r , t h e r e are also s o m e w a t e r -e x i t -ev-ents (s-e-e f o r instanc-e b -e t w -e -e n 23 s a n d 25 s) w h i c h d o n o t cause a n y a p p r e c i a b l e w h i p p i n g a t a l l o r s o m e s l a m m i n g e v e n t s ( l o o k a r o u n d 26.5 s) w i t h s u b s e q u e n t r e l a t i v e l y s m a l l w h i p p i n g . Also i t m a y h a p p e n t h a t a s i g n i f l c a n t H F - V B M p e a k is n o t r e l a t e d t o a n y w a t e r - e x i t event. F o c u s i n g o n t h e flrst i m p a c t i n t h e s l a m m i n g sequence o r o n i s o l a t e d i m p a c t s , i t appears t h a t t h e