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Applied Ocean Research
j o u rnaI h omepagerwww.etseviBr.coTn/lorate/apor"O C E A N R E S E A R C H
Uncertainty of full-scale manoeuvring trial results estimated using a
simulation model
Sergey Gavrilin*, Sverre Steen
Jhe Department of Marine Technology, The Norwegian University of Science and Technology (tmtU), Otto Nielsens veg. JO, Trondheim, 7052, Norway
jCrossMaik
A R T I C L E I N F O
Article history: Received 14 April 2016
Received i n revised f o r m 22 July 2016 Accepted 21 March 2017
Available online 29 March 2017
Keywords: Uncertainty Monte Carlo Sensitivity Manoeuvring A B S T R A C T
This paper describes how to estimate the uncertainty of manoeuvring sea trial results without perform-ing repeated tests usperform-ing only a simulation model. The approach is based on the Monte Carlo method of uncertainty propagation. Moreover, the global sensitivity analysis procedure based on variance decom-position is described. As an example, the method is applied to estimate the uncertainty of 10°/l 0° zigzag overshoot angles and a 20° turning circle advance and tactical diameter for a small research vessel. The estimated uncertainty is compared w i t h corresponding experimental uncertainty assessed from repeated tests. The method can be useful for validation studies and other studies that involve the uncertainty of sea trial results.
® 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Currently, ship m a n o e u v r i n g s i m u l a t i o n models are w i d e l y a p p l i e d f o r t r a i n i n g , standardization a n d engineering purposes. H o w e v e r , recent studies [1) have s h o w n a large scatter i n the results predicted by models developed b y d i f f e r e n t organizations. This indicates h i g h demand f o r the d e v e l o p m e n t of v a l i d a t i o n t e c h -niques, as w e l l as f o r the i m p r o v e m e n t o f ship s i m u l a U o n models. The ITTC M a n o e u v r i n g C o m m i t t e e indicates the i m p o r t a n c e and a lack o f v a l i d a t i o n activities f o r ship s i m u l a t o r s [2,3]. V a l i d a t i o n is p e r f o r m e d via comparison o f p r e d i c t i o n s m a d e b y a s i m u l a t o r w i t h e x p e r i m e n t a l results f o r idenrical trials. Full-scale or model-scale experiments are used to o b t a i n a b e n c h m a r k f o r v a l i d a t i o n . H o w -ever, model-scale experiments are prone t o scale effects; therefore, full-scale experiments are preferable. Evaluation of the u n c e r t a i n t y of e x p e r i m e n t a l results is an i m p o r t a n t p a r t o f v a l i d a t i o n . The ITTC issued a r e c o m m e n d e d procedure f o r t h e u n c e r t a i n t y analysis o f f r e e r u n n i n g m o d e l tests [4] t h a t outlines t l i e m a i n sources of uncer-t a i n uncer-t y i n m a n o e u v r i n g experimenuncer-ts. They use a c o m b i n a uncer-t i o n o f t h r e e approaches t o o b t a i n the c o m b i n e d u n c e r t a i n t y : measure-m e n t u n c e r t a i n t y analysis, r e p e a t a b i l i t y analysis and u n c e r t a i n t y p r o p a g a d o n analysis. The u n c e r t a i n t y p r o p a g a r i o n analysis is based on the Taylor series m e t h o d . A c c o r d i n g t o the m e t h o d , the uncer-t a i n uncer-t y o f uncer-the e x p e r i m e n uncer-t a l resuluncer-t due uncer-t o some i n p u uncer-t facuncer-tor is equal
* Corresponding author.
E-mail addresses: sergey.gavrilin@ntnu.no (S. Gavrilin), s v e r r e . s t e e n ® n t n u . n o (S. Steen).
http://dx.doi.Org/10.1016/j.apor.2017.03.011 0 1 4 1 - 1 1 8 7 / ® 2017 Elsevier Ltd. All rights reserved.
to the p r o d u c t o f t h e u n c e r t a i n t y o f this f a c t o r and the so-called u n c e r t a i n t y m a g n i f i c a t i o n f a c t o r (UMF). The UMF is a linear local absolute s e n s i t i v i t y c o e f f i c i e n t and can be n u m e r i c a l l y estimated using a s i m u l a t i o n m o d e l . The t o t a l c o m b i n e d u n c e r t a i n t y is cal-culated as the r o o t s u m m e d squared o f t h e i n d i v i d u a l u n c e r t a i n t y c o n t r i b u t i o n s . A l t h o u g h this approach can be p a r t l y a p p l i e d t o f u l l -scale tests, some features cause a s i g n i f i c a n t difference b e t w e e n full-scale tests and model-scale tests. The price o f r e p e t i r i o n s is v e r y h i g h f o r full-scale tests; therefore, the r e p e t i t i o n s are rarely p e r f o r m e d . Moreover, sea t r i a l results are i n f l u e n c e d by e n v i r o n -m e n t a l effects, w h o s e c o n t r i b u t i o n t o the r e s u l t i n g u n c e r t a i n t y is sometimes d o m i n a t i n g . These effects are represented b y t w o or m o r e i n d e p e n d e n t factors (such as c u r r e n t speed and d i r e c t i o n , or w a v e height, p e r i o d and d i r e c t i o n ) w i t h s t r o n g i n t e r a c t i o n . There-fore, t h e c o m b i n e d result u n c e r t a i n t y cannot be e s t i m a t e d using the UMFs.
In t h i s paper, w e consider an alternative approach t o uncer-t a i n uncer-t y p r o p a g a uncer-t i o n based o n uncer-the M o n uncer-t e Carlo m e uncer-t h o d [5,6]. The M o n t e Carlo m e t h o d is m o r e f l e x i b l e and suitable f o r h i g h l y n o n -linear systems and i n t e r c o n n e c t e d i n p u t factors. The M o n t e Carlo m e t h o d was p r e v i o u s l y used i n m a n o e u v r i n g to propagate the u n c e r t a i n t y o f f o r c e measurements i n captive tests t o t h e f i n a l u n c e r t a i n t y o f m a n o e u v r i n g indices (overshoot angles) [7,8]. W e a p p l y t h e m e t h o d t o estimate the u n c e r t a i n t y o f repeated tests and compare i t w i t h t h e e x p e r i m e n t a l l y d e t e r m i n e d u n c e r t a i n t y . W e also describe t h e global s e n s i t i v i t y analysis based o n variance d e c o m p o s i t i o n and a p p l y i t to estimate t h e c o n t r i b u t i o n of t h e i n p u t factors t o the t o t a l u n c e r t a i n t y . Thus, the m a i n goal o f t h e paper is to d e m o n s t r a t e h o w t o estimate u n c e r t a i n t y o f fullscale m a n o e u
-2V,7. S. Gavrilin, S. Steen /Applied Ocean Researcli 64 (2017) 281-289
Nomenclature
CT Standard d e v i a t i o n
CDF Cumulative d i s t r i b u t i o n f u n c t i o n
N(|x, CT) N o r m a l d i s t r i b u t i o n w i t h the m e a n and the s t a n -dard d e v i a t i o n a —
OA Overshoot angle
PDF Probability d i s t r i b u t i o n f u n c t i o n
•-1(^1. ^2) U n i f o r m d i s t r i b u t i o n , w i t h the l o w e r and u p p e r borders ^1 and
Ugs Expanded uncertainty, w i t h 95% confidence U M F U n c e r t a i n t y m a g n i f l c a d o n factor
v r i n g tests i n practice, w i t h o u t actually p e r f o r m i n g repeated tests. Therefore, w e do n o t consider some possible sources o f u n c e r t a i n t y w h i c h are negligible, w h i l e f o c u s i n g o n m o r e i m p o r t a n t ones. W e emphasize also t h a t a l l the s e n s i t i v i t y coefficients obtained are spe-cific f o r the case vessel and prescribed conditions. H o w e v e r , the same a l g o r i t h m can be applied to any other vessel.
The paper is organized as f o l l o w s . Section 2 describes the M o n t e Carlo m e t h o d o f u n c e r t a i n t y p r o p a g a t i o n . Section 3 describes h o w t o estimate the c o n t r i b u t i o n o f the i n p u t factors t o the total u n c e r t a i n t y o f the result using graphical analysis and variance d e c o m p o s i t i o n . Section 4 describes h o w t o use the M o n t e Cario m e t h o d t o estimate the u n c e r t a i n t y o f m a n o e u v r i n g t r i a l results. Section 5 presents an example o f t h e a p p l i c a t i o n o f t h e analysis to estimate t h e u n c e r t a i n t y o f 1 0 ° / 1 0 ° zigzag and 2 0 ° t u r n i n g circle test results. Section 6 contains discussion and conclusions.
2 . Monte Carlo m e t h o d
Consider the result V p r e d i c t e d b y a s i m u l a t i o n m o d e l . The result depends o n the set o f uncertain i n p u t factors X :
Y = / ( X ) (1)
Each o f the i n p u t factors i n X has an associated k n o w n p r o b a b i l -i t y d -i s t r -i b u t -i o n . The goal o f t h e M o n t e Carlo p r o p a g a t -i o n m e t h o d is to estimate the u n c e r t a i n t y o f Y due to the u n c e r t a i n t y o f X . The f o l l o w i n g a l g o r i t h m describes t h e procedure:
1. ) Generate a m a t r i x A w i t h n u m b e r s d i s t r i b u t e d r a n d o m l y on [0,1 ], w i t h N r o w s and k columns, w h e r e JV is a s u f f i c i e n t l y large n u m b e r , k is the n u m b e r o f i n p u t factors i n X . N defines the total n u m b e r o f s i m u l a t i o n s .
2. ) A p p l y t h e c o r r e s p o n d i n g inverse c u m u l a t i v e d i s t r i b u t i o n f u n c -tion t o the samples o f each c o l u m n and compose a n e w m a t r i x X f r o m the r e s u l t i n g n u m b e r s :
(2)
3. ) N o w , each r o w o f t h e m a t r i x X c o n t a i n s a set o f i n p u t parameters f o r (1). P e r f o r m t h e s i m u l a t i o n s f o r each r o w o f X and calculate the array o f results Y according to (1).
4. ) A c c o r d i n g t o the central l i m i t t h e o r e m [ 9 ], t h e result V i s dis-t r i b u dis-t e d a p p r o x i m a dis-t e l y n o r m a l l y i f i dis-t is n o dis-t d o m i n a dis-t e d by a single i n p u t factor. Therefore, calculate the standard d e v i a t i o n CT o f y and t h e n calculate expanded u n c e r t a i n t y U95 b y m u l t i -p l y i n g CT b y the coverage f a c t o r 2. I n some cases, t h e r e s u l t i n g d i s t r i b u t i o n o f y is n o t close t o a n o r m a l d i s t r i b u t i o n . Then, to f i n d t h e 95% confidence i n t e r v a l , b u i l d the e m p i r i c a l CDF o f Y and f i n d the l o w e r and the u p p e r border o f t h e confidence i n t e r -val as the a r g u m e n t o f the f u n c t i o n , w h e r e i t equals 0.025 and 0.975, respectively.
Thus, the M o n t e Carlo m e t h o d provides the u n c e r t a i n t y o f the m o d e l o u t p u t due to the u n c e r t a i n t y o f the i n p u t factors X . However, the m e t h o d does n o t p r o v i d e i n f o r m a t i o n r e g a r d i n g the c o n t r i b u t i o n o f each i n d i v i d u a l i n p u t factor to the t o t a l u n c e r t a i n t y .
3 . Global sensitivity analysis
I n the Taylor series m e t h o d o f u n c e r t a i n t y propagadon, the i n d i -v i d u a l c o n t r i b u t i o n s o f the i n p u t factors' u n c e r t a i n t y are calculated as a p a r t of the analysis. In the M o n t e Cario m e t h o d , i t is n o t possible to say d i r e c t l y w h i c h i n p u t parameters make the m a i n c o n t r i b u t i o n to result u n c e r t a i n t y . H o w e v e r , t h i s k n o w l e d g e is v e r y u s e f u l . It increases confidence i n the u n c e r t a i n t y analysis and helps to detect faults or to i m p r o v e the e x p e r i m e n t .
W e consider t w o methods t o assess the relative i m p o r t a n c e o f the i n p u t factors. The f i r s t m e t h o d is graphical. A c c o r d i n g t o the m e t h o d , one s h o u l d p l o t the r e s u l t Y versus the i n p u t f a c t o r X , (this w i l l be i l l u s t r a t e d i n subsection 5.5, see Fig. 6). I f the result changes f o r d i f f e r e n t values o f the i n p u t factor, or, i n other w o r d s , a p a t t e r n is observed, the p a r a m e t e r is i m p o r t a n t . The stronger t h e p a t t e r n is, the m o r e i m p o r t a n t the p a r a m e t e r is. The m e t h o d is s i m p l e and does n o t d e m a n d a d d i t i o n a l s i m u l a t i o n s . However, i t does n o t give any objective q u a n t i t a t i v e measure o f the factor's i m p o r t a n c e , and i t is n o t suitable f o r s t u d y i n g t h e j o i n t effects o f several i n t e r a c t i n g factors.
The second m e t h o d is based o n variance d e c o m p o s i t i o n . The f u r -t h e r d e s c r i p -t i o n o f -the m e -t h o d closely f o l l o w s [ 10,11 ]. The m e t h o d uses the concept o f a s e n s i t i v i t y i n d e x as the measure o f factors' i m p o r t a n c e . There are f i r s t - o r d e r , j o i n t and t o t a l effect s e n s i t i v i t y indices. The f i r s t - o r d e r s e n s i t i v i t y index S, and the t o t a l e f f e c t S^- o f the factor Xj are d e f i n e d c o r r e s p o n d i n g l y as:
Vx,{Exjy\Xi)) V{Y) ViY) = 1 VxJEx.{Y\X^i)) V{Y) (3) (4)
w h e r e Vxif- • J . % f - • . j is t h e variance and t h e mean, respectively, o f the a r g u m e n t ( . . . ) taken overX,-; Vx~i(- • •)• Ex~i(- • •) is the variance or m e a n o f the a r g u m e n t ( . . . ) t a k e n over a l l factors b u t X j ; V(Y) is u n c o n d i t i o n a l variance. Thus, Sj is t h e expected relative r e d u c t i o n i n variance V(Y) t h a t w o u l d be obtained i f X ; c o u l d be f i x e d ; Sn is the expected r e l a t i v e variance t h a t w o u l d be l e f t i f all factors butX,-c o u l d be fixed. The j o i n t s e n s i t i v i t y indibutX,-ces are d e f i n e d b y analogy, f o r instance, f o r t w o factors:
Vx,{ExJY\Xy))
'J V{Y) (5)
The h i g h e r o r d e r effects (interactions) are d e f i n e d as t h e r e s i d -ual c o m p o n e n t as:
Sii = S^j-Si-Sj (6)
I n fact, the u n c o n d i t i o n a l variance can be decomposed as the s u m o f the first-order and h i g h e r - o r d e r effects:
1 V{Y)
V{Y) E ^ ' + E E ^ " + E E E % + - + ^ - ^ (7)
i i j i i li Ij
The t o t a l e f f e c t o f the factor X,- contains a l l t e r m s i n (7) t h a t i n v o l v e this factor, f o r instance:
S „ = S I + S I 2 - F S , 3 4-5,23 + . . .
(8)
To s u m up t h e m e t h o d , w e list i m p o r t a n t properties o f the sen-s i t i v i t y indicesen-s:
283 • Sj and ST-,' are n u m b e r s b e t w e e n 0 and 1. Higher value indicates
higher i n f l u e n c e .
• Sji = 0 means t h a t X ; does n o t i n f l u e n c e V and can be f i x e d . • Sj shows the p r i m a r y e f f e c t of t h e f a c t o r ^ , . The difference f S „ - 5 ; j
is the measure o f the i n t e r a c t i o n b e t w e e n Xj and the other i n p u t factors. ST-, =5, indicates the absence o f the i n t e r a c t i o n .
One can use the f o l l o w i n g a l g o r i t h m to estimate S,- and Sn n u m e r i c a l l y :
1 Generate a 2fc-dimensional Sobol sequence o f quasi r a n d o m n u m b e r s o f l e n g t h N{k n u m b e r o f i n p u t parameters, N s u f -f i c i e n t l y large n u m b e r ) . The -f u n c t i o n 'sobolse-f can be used i n IVlatlab. Split the sequence i n t o t w o matrices: A, c o n t a i n i n g f i r s t k columns o f the sequence, and B, c o n t a i n i n g the r e m a i n i n g k columns.
2 Compose a u x i l i a r y matrices Ab'''\ w h e r e all columns come f r o m A except f o r the i'"" c o l u m n , w h i c h comes f r o m B.
3 For each c o l u m n , use the m a p p i n g f u n c t i o n CDF,-' t h a t maps, u n i f o r m l y d i s t r i b u t e d on [0,1], n u m b e r s t o the custom d i s t r i b u -tions according to (2). For s i m p l i c i t y , w e keep the n o t a t i o n o f the matrices.
4 Run s i m u l a t i o n s using each r o w f r o m the matrices A, B and ylflC' as the i n p u t o f m o d e l (1) and calculate corresponding o u t p u t s / W . / W a n d / f A f l O j .
5 Calculate variance V(Y) u s i n g / ( A j at\d f(B).
6 Estimate c o n d i t i o n a l variances and expectations using f o r m u l a s :
^- {Ex^Ay\>^i)) = v m - ~ f ^ ( f ( B ) j - f ( A ^ ^ ) ) l ' (9)
7 Calculate S,- using (3) and (9) a n d Sj-,- u s i n g (4) and (10).
Instead o f the Sobol sequence, one can use r a n d o m n u m b e r s t o compose the design matrices A a n d B. However, m o r e s i m u l a t i o n s can be necessary t o achieve convergence. For alternative s a m p l i n g techniques and f o r m u l a s f o r c o n d i t i o n a l variance and mean, c o n -s u l t [10]. One needs N(k+2) s i m u l a t i o n s t o estimate S,- and Sj-,- f o r each i n p u t factor. The u n c e r t a i n t y o f the result is obtained as p a r t of the analysis ( i n step 5); t h e r e f o r e , no separate M o n t e Cario s i m -u l a t i o n s are needed. To calc-ulate s e n s i t i v i t y indices f o r the j o i n t effects, one needs to compose a d d i t i o n a l a u x i l i a r y matrices i n step 2, s u b s t i t u t i n g several c o l u m n s instead o f one c o l u m n f r o m B.
The analysis based o n variance d e c o m p o s i t i o n is m o r e objec-t i v e compared objec-to objec-the graphical analysis. I objec-t provides objec-the q u a n objec-t i objec-t a objec-t i v e measures o f the relative i m p o r t a n c e o f the i n p u t factors and a l l o w s f o r i n v e s t i g a t i n g i n t e r a c t i o n effects. H o w e v e r , i t demands m o r e s i m u l a t i o n s t h a n one needs f o r t h e M o n t e Cario analysis itself.
4. Application to m a n o e u v r i n g t r i a l results
I t is v e r y expensive (and sometimes impossible) to p e r f o r m m u l t i p l e r e p e t i t i o n s o f the t r i a l t o estimate u n c e r t a i n t y statis-tically. Instead, the s i m u l a t i o n m o d e l can be used. Thus, t h e M o n t e Carlo s i m u l a t i o n s represent v i r t u a l e x p e r i m e n t s w i t h v a r y -i n g -i n p u t factors. System-based m a n o e u v r -i n g models t y p -i c a l l y a l l o w a c c o u n t i n g f o r : uncertainties i n c o n t r o l parameters, such as r u d d e r angles, r p m and execute time; k i n e m a t i c parameters d u r i n g approach and i n i t i a l c o n d i t i o n s ; c u r r e n t speed and d i r e c t i o n ; w i n d speed and d i r e c t i o n ; w a v e parameters ( f o r c o m b i n e d m a n o e u v r i n g
Fig. 1. Research vessel "Gunnerus".
Table 1
Main dimensions of research vessel "Gunnerus".
Length overall [m] 31.25
Length between perp. [mj 28.90
Length in wateriine [m] 29.90
Breadth midships [m] 9.60
Mast height/antetma [m] 14.85/19.70
Dead weight | t j 107
a n d seakeeping m o d e l s ) ; mass a n d m o m e n t s o f i n e r t i a ; measure-m e n t u n c e r t a i n t y . The d i s t r i b u t i o n s f o r t h e factors s h o u l d be based o n t h e observations d u r i n g t h e e x p e r i m e n t and t h e p r o b l e m state-m e n t (an exastate-mple is g i v e n b e l o w ) . Sostate-metistate-mes, a single physical p a r a m e t e r can have m u l t i p l e associated uncertainties, w h i c h are considered as separate i n p u t factors w i t h t h e i r o w n p r o b a b i l i t y d i s t r i b u t i o n s . For instance, r u d d e r angle can have associated bias u n c e r t a i n t y (non-zero n e u t r a l r u d d e r angle) a n d r a n d o m uncer-t a i n uncer-t y ( d e v i a uncer-t i o n from uncer-t h e c o m m a n d e d value, c h a n g i n g each r u d d e r execution). I m p o r t a n t factors t h a t cannot be i n c l u d e d i n t h e M o n t e Cario analysis are l o a d i n g c o n d i t i o n s a n d t h e c o r r e s p o n d i n g change o f h y d r o d y n a m i c effects ( t h i s w o u l d d e m a n d r e c a l c u l a t i o n o f a l l h u l l h y d r o d y n a m i c coelHcients at each i t e r a t i o n o f M o n t e Cario s i m u l a t i o n s ) . These factors are n o t revealed i n repeated tests (unless t h e l o a d i n g c o n d i t i o n s change). H o w e v e r , t h e y are i m p o r -t a n -t f o r v a l i d a -t i o n analysis. U n c e r -t a i n -t y due -t o l o a d i n g c o n d i -t i o n s can be e s t i m a t e d u s i n g CFD o r a n o t h e r n u m e r i c a l m e t h o d , and t h e T a y l o r series m e t h o d o f t m c e r t a i n t y p r o p a g a t i o n can be a p p l i e d as e x p l a i n e d in [4]. Thus, t h e t o t a l c o m b i n e d u n c e r t a i n t y o f the result is calculated as:
"c ( y ) = V ( u c m . „ (Y))' + (uc„™,^„^ i Y ) f + (uc„„^^„^ iY)Y (11)
w h e r e UcjMeasfVj is the m e a s u r e m e n t u n c e r t a i n t y o f Y, ^cMCM.prop(y) is the p r o p a g a t i o n u n c e r t a i n t y f o r t h e factors t h a t can be v a r i e d at each s i m u l a t i o n and are e s t i m a t e d using the M o n t e Cario m e t h o d , and UcjsM.prop(y) is the p r o p a g a t i o n u n c e r t a i n t y f o r the factors t h a t cannot be v a r i e d at each s i m u l a t i o n and are estimated using t h e Taylor series m e t h o d .
5. Case study: u n c e r t a i n t y of repeated tests
5.1. Case vessel destription
I n this study, w e use a r e l a t i v e l y s m a l l (31.25 m overall l e n g t h ) research vessel, i.e., "Gunnerus", as a case vessel (Fig. 1). The ship is o w n e d and operated b y the N o r w e g i a n U n i v e r s i t y o f Science and Technology (NTNU), w h i c h makes i t cheaper and easier t o access f o r full-scale trials f o r research purposes. Due to this fact, i t is possible t o p e r f o r m repeated tests. Table 1 lists t h e m a i n parameters o f the vessel. "Gunnerus" is e q u i p p e d w i t h two d u c t e d propellers and t w o
284 S. Gavrilin, S. Steen/Applied Ocean Researcli 64 (2017)281-289 Table 2 Table 3
Accuracy of the Seapath 330+ measurements, i,e„ RMS, according to the manufac- Results o f l 0 ° ; i 0 ° zigzag tests.
turer (position depends on the distance to the closest correction station). 2nd OA
Heading 0.04- mean|deg| 8 5 ^ 7-9 Position (X and Y ) I c m + l p p m aldeg] 03 0.4 Velocity 0.03m/s y j ^ j , 6 ^ 13^1
flapped rudders. The diesel-electric propulsion system comprises main electric propulsion ( 2 x 5 0 0 k W ) a n d the b o w tunnel thnister (200 k W ) , powered by generators (3 x 5 0 0 kW)- The cruising speed of the vessel is 10.5 knots.
5.2. Measurement equipment
To register ship motions, two integrated orientation and p o s i -tion sensors produced by Kongsberg Seatex were used: the permanently mounted Seapath 3 3 0 + w i t h correction signals and the temporarily instaUed Seapath 330. The sensors combine GNSS signals and inertial measurements. Table 2 demonstrates some accuracy l i m i t s o f the m e a s u r e m e n t system. Further i n f o r m a t i o n can be f o u n d o n the webpage o f the s u p p l i e r (h t t p : / / w w w . k m . kongsberg.com).
Propeller RPMs w e r e measured v i a o p t i c a l measurement o f pulses f r o m the shaft. W i r e - o v e r - p o t e n t i o m e t e r distance sensors attached to each r u d d e r stock w e r e used f o r r u d d e r angle measure-ments. The sensors w e r e calibrated u s i n g the mechanical indicators on top o f the r u d d e r stocks. Rudder measurements b y the DP system of the vessel w e r e also recorded, w h e r e available. W i n d d i r e c t i o n and speed w e r e measured b y the w i n d sensor m o u n t e d o n the m a s t of the ship, a p p r o x i m a t e l y 10 m above t h e w a t e r surface.
A p a r t f r o m the m e a s u r e m e n t e q u i p m e n t on t h e ship, t w o w a v e buoys w e r e used. A n advanced Fugro Oceanor Wavescan b u o y w a s anchored near t h e t r i a l site d u r i n g the f i r s t series o f trials ( f u r t h e r referred to as "the f j o r d trials"). The b u o y measures the d i r e c t i o n a l wave s p e c t r u m , c u r r e n t speed a n d d i r e c t i o n at 1.5 m d e p t h , w i n d speed and d i r e c t i o n , a n d some other parameters. For the second series o f trials ( f u r t h e r r e f e r r e d to as "the ocean trials"), a d r i f t i n g Datawell D W R - G 4 40 c m buoy w a s used. The b u o y measures o n l y its o w n p o s i t i o n and a d i r e c t i o n a l w a v e s p e c t r u m .
5.3. Experimental program and conditions
Results f r o m t w o d i f f e r e n t series o f trials are used i n the paper. Both w e r e p e r f o r m e d near T r o n d h e i m , N o r w a y . The f i r s t series o f trials ( t h e f j o r d trials) w a s c o n d u c t e d i n August 2013 i n the T r o n d -h e i m f j o r d ( N 6 3 ° 29.717', E 1 0 ° 27.951'). T-he p r o g r a m i n c l u d e d 1 0 ° / 1 0 ° and 2 0 ° / 2 0 ° zigzag tests w i t h d i f f e r e n t approach speeds, executed b o t h m a n u a l l y and i n the a u t o m a t i c mode, 2 0 ° and 3 5 ° t u r n i n g circles, speed tests, and some l o w speed and DP tests. I n t h e automatic mode, the r u d d e r is c o n t r o l l e d b y a special p r o g r a m i n the DP system, w h i c h is made t o a u t o m a t i c a l l y execute 1 0 ° / l 0° a n d 2 0 ° / 2 0 ° zigzag tests. The f j o r d area is w e l l sheltered, so despite t h e relatively s t r o n g w i n d ( s o m e t i m e s u p t o 8 - 1 0 m/s), the s i g n i f i c a n t wave h e i g h t w a s t y p i c a l l y 0.2-0.3 m . Current speed measured b y the b u o y was u p to 0.4 m/s; t h e d i r e c t i o n o f the c u r r e n t varied due to tides and t w o rivers f l o w i n g i n t o the f j o r d . The second series ( t h e ocean t r i a l s ) w a s c o n d u c t e d i n the open ocean ( N 63° 35.334', E 8° 5.980') i n N o v e m b e r 2013, w i t h an emphasis o n t u r n i n g circles. Sev-eral tests w e r e executed i n a sheltered area b e h i n d islands, w h e r e no s i g n i f i c a n t waves w e r e observed. W i n d speed was u p to 6 m/s. Current was e s t i m a t e d b y a p p l y i n g t h e IMO c o r r e c t i o n procedure [12] f o r each t u r n i n g m a n o e u v r e :
i f r ( ^ , + 2 ^ ) - r ( V r , ) ^ - N ^ t { r l t , + 2 7 t ) - t { f , )
hcadmg SB radder PT radder
O 10 20 30 40 t[sec]
F i g . Z Results of 10°/10= zigzag trials.
Table 4
Results of the turning drcle tests.
Advance Tactical diam.
not cor. cor. not cor. cor.
mean (ml ff[ml U35 [%l 92.8 7.2 21.7 91.0 1 5 4.5 118.0 7.7 18.2 1153 1.4 3.4
w h e r e points ^ , are chosen during steady tummg, then equal points +2n are found on the next t u m . and then corresponding differ-ences of position r and time t are calculated. Note that in general, this estimate includes the actions of other factors, such as w i n d a n d waves. Current speed estimated using ( 1 2 ) was u p to 0.5 m/s.
5.4. Kesuta 0 / t r i a l s
For d e m o n s t r a t i o n , w e focus o n l y o n t w o types o f t r i a l s : 1 0 ° / l 0° zigzag (1st overshoot a n d 2 n d overshoot angles) and 2 0 ° t u r n i n g circle (advance a n d tactical d i a m e t e r ) . The trials are executed i n a u t o m a t i c m o d e . Five r e p e t i t i o n s o f each t r i a l are used. Table 3 contains the m e a n values o f f i r s t and second overshoot angles f r o m t h e zigzag tests, standard u n c e r t a i n t y and r e l a t i v e e x p a n d e d u n c e r t a i n t y , w i t h 95% confidence. The l a t t e r one is calculated u s i n g a coverage factor o f 2.776, taken f r o m t w o t a i l Student's t - d i s t r i b u t i o n , w i t h 95% confidence f o r f o u r degrees o f f r e e d o m (equal t o the n u m b e r o f e x p e r i m e n t s m i n u s one). Fig. 2 d e m o n -strates t h e t i m e series o f the heading a n d b o t h r u d d e r angles f o r t h e zigzag tests. It is easy t o see some reasons f o r t h e scatter o f o v e r s h o o t angles. The r u d d e r angles d i f f e r f r o m the c o m m a n d e d 1 0 ° or - 1 0 ° (the deviations are w i t h i n 1°). The 2 n d and 3 r d r u d d e r executions occur w i t h some delay a f t e r v a r i a t i o n o f t h e heading reaches 1 0 ° (the m e a n o f the delays is 1.6° and 2.1 °, w i t h a standard u n c e r t a i n t y o f 0.3° and 0.5° f o r t h e 2 n d a n d 3 r d r u d d e r executions, c o r r e s p o n d i n g l y ) . The rudders deviate l ° - 2 ° d u r i n g approach t o keep t h e vessel o n a s t r a i g h t course ( w h i c h also causes deviations o f y a w rate d u r i n g the 1 st e x e c u t i o n ) .
Table 4 contains the m e a n values o f advance a n d tactical d i a m -eter f r o m the t u r n i n g circle tests. Corrected values a f t e r a p p l y i n g ( 1 2 ) are also i n c l u d e d .
285 Table 5
Input factors w i t h distributions for turning circle simulations.
ID Input factor Distribution
Wind direction |deg| Wind speed |m/sl Current direction [deg] Current speed |m/s] taitial rudder dev. [deg] SB rudder dev. [deg] FT rudder dev. [deg]
U(0.36O) N ( 5 . 2 ) U(0.360) N(0.25.0.07) U ( - 1 . 1 ) U ( - 0 5 . 0 . 5 ) U ( - 0 3 . 0 . 5 ) 600 84 86 88 90 92 94 Advance [ m ]
Fig. 3. Results of Monte Carlo simulations of turning circles witii correction: advance.
Table 6
Results of Monte Carlo simulations of turning circles.
Advance Advance (corr) Tact diam. T a c t diam. (corr)
mean [deg] 88.1 88.1 113.7 113.7
a[deg] 4.7 1 3 8.6 1.1 Ugs (%] 10.7 3.1 15.2 IS
5.5. Uncertainty anil sensitivity analysis
In this section, w e demonstrate h o w to estimate the uncer-tainty of repeated tests using a simulation model. The model is described i n the A p p e n d i x A. For s i m p l i c i t y , w e neglect the m e a -s u r e m e n t u n c e r t a i n t y , a-s i t i-s negligible compared to u n c e r t a i n t y due to e n v i r o n m e n t a l effects and c o n t r o l factors. W e also neglect all bias sources o f u n c e r t a i n t y as t h e y do n o t change d u r i n g the tests series. Thus, the analysis answers t h e question " w h a t w o u l d the u n c e r t a i n t y be i f the test was repeated m u l t i p l e t i m e s u n d e r p a r -t i c u l a r condi-tions". These condi-tions are based on -t h e observa-tions m a d e d u r i n g the sea trials.
Table 5 hsts i n p u t factors t h a t are treated as u n c e r t a i n i n s i m u -lations o f t h e t u r n i n g circle test, w i t h corresponding PDFs. A l l o t h e r m o d e l parameters are fixed and t h e r e f o r e do n o t a f f e c t the r a n d o m u n c e r t a i n t y o f the result. As n o a d j u s t m e n t w i t h respect to w i n d a n d c u r r e n t directions was made d u r i n g t h e trials, every d i r e c t i o n is e q u a l l y possible. The u n c e r t a i n t y o f i n i t i a l c o n d i t i o n s is presented b y a single factor the i n i t i a l r u d d e r deviations. The t o t a l n u m -ber o f s i m u l a t i o n s p e r f o r m e d is 4000. Figs. 3 and 4 d e m o n s t r a t e d i s t r i b u t i o n s o f t h e values o f advance and tactical d i a m e t e r f o r a corrected t u r n i n g circle r e s u l t i n g f r o m the M o n t e Carlo s i m u l a -t i o n s . Table 6 presents the m e a n values, standard deviations and expanded u n c e r t a i n t y (a coverage factor o f 2 is used) o f advance a n d tactical diameter b o t h f o r corrected and n o n - c o r r e c t e d t u r n i n g circles. F r o m the c o m p a r i s o n o f Tables 4 a n d 6, i t f o l l o w s t h a t t h e M o n t e Carlo analysis gives reasonable estimates o f t h e u n c e r t a i n
-112 114 116 Tactical diam. [ m ]
l i s
Fig. 4. Results of Monte Carlo simulations of turning circles w i t h correction: tactical diameter.
: First order SI Total SI
1
0.8
0.6
0.4
0.2
1 1 1 1 1 1 1m _ It
1
1 ^ i
Fig. 5. Sensitivity indices for advance ( w i t h o u t correction).
ties, a l t h o u g h f o r the advance, t h e value o f u n c e r t a i n t y is l o w e r i n t h e s i m u l a t i o n s . The m e a n values are also similar.
To i d e n t i f y w h i c h o f t h e factors f r o m Table 5 make the largest c o n t r i b u t i o n t o the u n c e r t a i n t y o f the results, w e p e r f o r m a sensi-t i v i sensi-t y analysis. Fig. 5 demonstrates the first-order s e n s i t i v i t y indices and the t o t a l effects f o r advance. For the c u r r e n t d i r e c t i o n and cur-r e n t speed, the t o t a l e f f e c t is h i g h e cur-r t h a n t h e ficur-rst-ocur-rdecur-r s e n s i t i v i t y index. The same holds f o r t h e w i n d d i r e c t i o n and speed, a l t h o u g h less obviously. This d i f f e r e n c e indicates the presence o f interac-t i o n b e interac-t w e e n interac-the i n p u interac-t facinterac-tors. The alinterac-ternarive w a y interac-to see w h e interac-t h e r t h e i n p u t f a c t o r is i m p o r t a n t is graphical analysis. Fig. 6 d e m o n -strates advance r e s u l t i n g f r o m M o n t e Cario s i m u l a d o n s p l o t t e d versus some i n p u t factors. A s t r o n g effect is observed f o r the c u r r e n t d i r e c t i o n . There is also some p a t t e r n f o r c u r r e n t speed. However, t h e m e a n values o f advance f o r each c u r r e n t speed are a p p r o x i -m a t e l y the sa-me. Therefore, t h e r e is s t r o n g i n t e r a c t i o n o f c u r r e n t speed w i t h o t h e r factors. A w e a k e r effect is observed f o r w i n d direc-t i o n , w h i l e a l m o s direc-t n o e f f e c direc-t is observed f o r r u d d e r d e v i a direc-t i o n . Thus, graphical analysis leads t o the same conclusions as variance-based analysis.
Figs. 7 a n d 8 present the s e n s i t i v i t y indices o f t h e advance and t h e tactical d i a m e t e r f o r n o n - c o r r e c t e d a n d corrected t u r n i n g cir-cles, w h e r e related i n p u t factors are c o m b i n e d . I n b o t h cases, the c u r r e n t e f f e c t is the m a i n c o n t r i b u t i o n to t h e c o m b i n e d u n c e r t a i n t y o f the results f o r the n o n - c o r r e c t e d t r a j e c t o r y . The o t h e r factors are m u c h less i m p o r t a n t . H o w e v e r , the e f f e c t o f c u r r e n t is negligible i f t h e correction is applied. The e f f e c t o f w i n d is also p a r t l y reduced. Nevertheless, i t becomes an i m p o r t a n t c o n t r i b u t i o n t o t h e t o t a l u n c e r t a i n t y o f b o t h results. A s i g n i f i c a n t p a r t o f t h e t o t a l
uncer-2S6 S. Gavrilin, S, Steen/Applied Ocean Research 64 (2017) 281-289
0 90 180 270 360
Wind dir. [deg]
0 90 180 270 360
Current dir. [deg]
Table 7
Input factors w i t h distributions for zigzag simulations.
5
fi
7 8 9 10 11 Input factor Wind direction jdeg] Wind speed [m/s] Initial rudder dev. [deg] Heading dev. 2nd exec [deg] Heading dev. 3 n i exec [deg] SB rudder dev. I s t exec, [deg] FT rudder dev. 1st exec [deg] SB rudder dev. 2nd exec [deg] FT rudder dev. 2nd exec [deg] SB rudder dev. 3rd exec [deg] FT rudder dev. 3rd exec, [deg]Distribution N(0.15) N(5.2) U ( - 1 . 1 ) N ( 0 . 0 3 ) N ( 0 . 0 3 ) U ( - 0 3 . 0 . 5 ) ü ( - 0 5 . 0 . 5 ) U ( - 0 5 , 0 . 5 ) U ( - 0 5 . 0 . 5 ) U ( - 0 . 5 . 0 5 ) U(-0.5.0.5) 1400
0 0.2 0.4
Current speed [m/s]
-0.5 0 0.5
SB rudder dev. [deg]
Fig. 6. Advance [m] vs i n p u t parameters.First order SI Total SI 1 0.8 0.6 0.4 0.2 0 n o. Without correction _na_ Will, correction - 0 0 .
I B I
1 1 1 1
Fig. 7. Sensitivity indices for the advance (without and w i t h correction).
First order SI Total SI
0.8 0.6 0.4 0.2 Witliout correction Willi correction
Fig. 8. Sensitivity indices for the tactical diameter ( w i t h o u t and w i t h correction).
7 8 9 10 n overshoot angle [deg]
Fig. 9. Results ofMonte Carlo simulations ofthe zigzag trial: 1st overshoot angle.
tainty of advance after correction is due to uncertainty in the initial conditions. The rudder deviation from the commanded value gives the main contribution to the advance for the corrected turning cir-cle. These results are reasonable: initial conditions affect time w h e n the heading changes by 9 0 ° or 1 8 0 ° and therefore have high influ-ence on the distance travelled along the approach direction (but not in the transverse direction), while small rudder deviations affect the curvature of the track.
Table 7 lists i n p u t factors t h a t are treated as u n c e r t a i n i n t h e simulations o f zigzag tests. The i n i t i a l r u d d e r d e v i a t i o n accounts f o r u n c e r t a i n t y i n the i n i t i a l conditions d u r i n g the f i r s t execution. The heading deviations d u r i n g the 2 n d and 3 r d execution account f o r t h e delay i n the DP system c o n t r o l l i n g the a u t o m a t i c a l l y exe-cuted zigzag trials. Because t h e ship is s m a l l a n d responds q u i d d y , t h e 2 - to 4-s delay f r o m d e t e c t i n g the t h r e s h o l d heading change t o executing the r u d d e r w a s a m a j o r p r o b l e m , w h i c h was o n l y p a r t l y compensated f o r by means o f t r i a l and e r r o r l o w e r i n g o f t h e actuation t h r e s h o l d angle f r o m 10° o r 2 0 ° to l o w e r angles. In fact, the m e a n value o f t h e heading t h a t activates t h e r u d d e r execution i n the simulations is 1 2 ° instead o f 1 0 ° to increase t h e s i m i l a r -i t y o f the s -i m u l a t -i o n s to the e x p e r -i m e n t . Rudder dev-iat-ions f r o m c o m m a n d e d values are i n d e p e n d e n t f o r each r u d d e r and each exe-c u t i o n . The effeexe-ct o f the exe-c u r r e n t is n o t i n exe-c l u d e d as i t does n o t a f f e exe-c t t h e heading.
Figs. 9 and 10 demonstrate the h i s t o g r a m of the first and sec-o n d sec-overshsec-osec-ot angles r e s u l t i n g f r sec-o m M sec-o n t e Carlsec-o s i m u l a t i sec-o n s sec-o f the zigzag test and corresponding n o r m a l d i s t r i b u t i o n s . The t o t a l n u m b e r o f simulations is 4 0 0 0 . Table 8 contains the m e a n values, t h e standard deviations and t h e relative expanded uncertainties o f t h e overshoot angles. For the expanded uncertainties, a coverage factor o f 2 is used. C o m p a r i n g Tables 3 and 8, w e see t h a t b o t h t h e m e a n values and t h e expanded uncertainties are s i m i l a r .
287
1400
7 8 9 10 11 12 overshoot angle [deg]
Fig. 10. Results o f M o n t e Carlo simulations of tlie zigzag trial: 2nd overshoot angle.
Table 8
Results o f M o n t e Carlo simulations of the zigzag test.
1st OA 2nd OA
mean [deg] 9.5 9.4 o-Idegl 0.6 0.6 Uss l%I 12.5 12.1
First order SI [ J ^ H B Total SI
!
0.8
( 1 Thefirst overshoot angle0.6
• \ •0.4
•\
•0.2
0 r-.-1.
m
. .
1 The second 0.8 overshoot angle 0.6 0.4 • 0.2 • -0. m
.
I I | ö |ö' I d | ö .|d
g c :gï
rg-j
I 1^ ^';!
" :ë :ë I I I
Fig. 11. Sensitivity indices for the first (top) and second (bottom) overshoot angles.To evaluate t h e c o n t r i b u t i o n o f each o f t h e factors t o t h e t o t a l u n c e r t a i n t y o f t h e result, w e p e r f o r m g l o b a l s e n s i t i v i t y analy-sis. Fig. 11 presents t h e s e n s i t i v i t y indices. For convenience, w e g r o u p e d related factors such as w i n d speed a n d d i r e c t i o n a n d star-board a n d p o r t r u d d e r deviations f o r the same execution. W e n o w see t h a t t h e u n c e r t a i n t y o f the heading at t h e t i m e o f change o f t h e rudders p r e c e d i n g the corresponding overshoot angle is t h e largest c o n t r i b u t i o n t o the t o t a l u n c e r t a i n t y . The r u d d e r d e v i a t i o n ( p r e v i -ously t h e overshoot) a n d w i n d also have some i n f l u e n c e . The rest o f t h e i n p u t factors are i n s i g n i f i c a n t . The i m p o r t a n c e o f t h e head-ing d e v i a t i o n is easy t o u n d e r s t a n d : i f h e a d i n g d u r i n g t h e r u d d e r
execution is d i f f e r e n t f r o m the "planned" value ( f o r example, 1 1 ° instead o f 10°), this difference w i l l d i r e c t l y a f f e c t the result ( t h a t is, t h a t 1° w i l l be added t o the corresponding overshoot angle).
6. Discussion a n d conclusions
I n t h e paper, w e have d e m o n s t r a t e d h o w t o estimate t h e u n c e r t a i n t y o f t h e results o f m a n o e u v r i n g trials due t o e n v i r o n -m e n t a l effects and other factors w i t h o u t p e r f o r -m i n g repeated tests. A l t h o u g h n o t a l l factors a f f e c t i n g t h e results can be i n c l u d e d i n the analysis, the approach m a y give a reasonable estimate o f t h e u n c e r t a i n t y o f t h e result and s i g n i f i c a n t l y reduce t h e cost o f anal-ysis. Contrary t o t h e Taylor series p r o p a g a t i o n m e t h o d , the M o n t e Carlo m e t h o d considered i n the paper is suitable f o r the cases w h e n t h e r e l a t i o n b e t w e e n a result a n d i n p u t factors is nonlinear and i n t e r a c t i o n effects b e t w e e n d i f f e r e n t i n p u t factors are i m p o r t a n t . It is easy t o apply the m e t h o d a n d i n t e r p r e t t h e results. However, t h e m e t h o d demands m u l t i p l e s i m u l a t i o n s ; thus, i t is applicable t o system-based models w i t h i n p u t factors t h a t can easily be varied. The choice o f t h e d i s t r i b u t i o n s o f t h e i n p u t factors is v e r y i m p o r t a n t f o r the estimated u n c e r t a i n t y o f t h e result. Therefore, one should m a k e this choice w i t h care, f o r instance, based on observations d u r -i n g the sea tr-ials. H o w e v e r , w e w o u l d l -i k e t o emphas-ise t h a t -i t -is m u c h easier to m a k e good engineering estimates o f the u n c e r t a i n t y of i n p u t variables such as w i n d and c u r r e n t t h a n t o d i r e c t l y estimate the u n c e r t a i n t y o f the o u t p u t results trials. This is actually t h e k e y c o n t r i b u t i o n o f t h i s paper.
W e have also d e m o n s t r a t e d h o w t o assess t h e c o n t r i b u t i o n o f i n d i v i d u a l i n p u t factors t o t h e t o t a l u n c e r t a i n t y . The graphical m e t h o d does n o t d e m a n d a d d i t i o n a l simulations. However, o n l y the global s e n s i t i v i t y analysis based o n the variance d e c o m p o s i t i o n provides q u a n t i t a t i v e measures o f the i m p o r t a n c e o f the d i f f e r e n t factors and shows j o i n t effects. P e r f o r m i n g t h e global s e n s i t i v i t y analysis reduces t h e risk o f a f a u l t i n t h e u n c e r t a i n t y analysis. The u n c e r t a i n t y estimates and t h e s e n s i t i v i t y o f factors h o l d o n l y f o r a specific vessel a n d specific i n p u t parameters. For instance, i f the same s i m u l a t i o n s w e r e p e r f o r m e d w i t h r a n d o m w i n d d i r e c t i o n , the c o m b i n e d u n c e r t a i n t y o f overshoot angles a n d the sensitiv-i t y t o w sensitiv-i n d w o u l d be bsensitiv-igger. S sensitiv-i m sensitiv-i l a r l y , the delays o f reactsensitiv-ion a n d a c t u a t i o n o f t h e c o n t r o l system o r h e l m s m a n are i m p o r t a n t f o r this vessel, as i t has a p a r t i c u l a r l y fast reaction; however, t h e y are m o s t l y i n s i g n i f i c a n t f o r s l o w r e a c t i n g vessels. The analysis m a y be u s e f u l b o t h i n t h e design stage o f t h e e x p e r i m e n t t o assess t h e pos-sible u n c e r t a i n t y o f t h e results and i m p o r t a n c e o f d i f f e r e n t factors and possibly m o d i f y t h e e x p e r i m e n t t o reduce the u n c e r t a i n t y o f the result and a f t e r t h e e x p e r i m e n t t o estimate t h e actual uncer-t a i n uncer-t y .
A c l a i o w l e d g e m e n t s
The research, as a p a r t o f t h e SimVal project, is sponsored by t h e N o r w e g i a n Research Council ( 2 2 5 1 4 1 / 0 7 0 ) ; t h e s u p p o r t is greatly a c k n o w l e d g e d b y t h e authors. W e w o u l d also like t o express o u r g r a t i t u d e t o MARINTEK f o r technical a n d professional help, as w e l l as t o the captain a n d c r e w o f R/V "Gunnerus" for t h e i r great patience and good seamanship d u r i n g t h e trials.
A p p e n d w A. Description of s i m u l a t i o n m o d e l
In this study, w e use a r e l a t i v e l y s i m p l e 3 ° - o f - f r e e d o m m o d u l a r m a t h e m a t i c a l m o d e l . The m o d e l is m a i n l y based on [ 1 3 - 1 5 ]. Fig. A l illustrates reference systems and some corresponding notations. Coordinates o f t h e s h i p j / = / X Y are expressed i n t h e i n e r t i a l reference f r a m e {0-X-Y}, o r [i}. Velocities o f t h e ship V=luv rf are expressed i n t h e reference f r a m e t h a t moves w i t h t h e ship {C-x-y],
288 S. Gavrilin, S. Steen/Applied Ocean Research 64 (2017) 281-289
hi
O Y
F i g . A l . Reference systems: inertial (O-X-Y) and body-fixed {C-x-y).
or {fa}, w i t h the o r i g i n at Lpp/2 and axes x and y p o i n t i n g f o r w a r d a n d t o w a r d starboard, respectively.
The velocities i n [b] and [i] are related as f o l l o w s :
(A.1) w h e r e R(\j/) is the r o t a t i o n m a t r i x f r o m {b} to { i } . The m a n o e u v r i n g m o d e l is expressed i n {b} i n general f o r m as (see, f o r instance, [13]):
m-Xi, 0 0 0 m - Vv ^"c - Y r 0 mxc - Nv fzz - Nj. Ü V r
The subscripts ' H ' , 'P, 'R' a n d ' W r e f e r to t h e h u l l , propeller, r u d -der and w i n d forces, respectively. I n the presence o f a constant i r r o t a t i o n a l c u r r e n t w i t h speed Vc a n d d i r e c t i o n ac, (A.2) holds f o r r e l a t i v e velocities (see [13]), d e f i n e d as
(A.3) Ur = u-Vc COS (ac - f )
vr = v-Vc s i n (ac - ^ )
Let us n o w consider t h e components o f forces and m o m e n t o n t h e r i g h t - h a n d side o f (A.2). W e use the f o l l o w i n g representation o f t h e h u l l h y d r o d y n a m i c v e l o c i t y - d e p e n d e n t forces:
X H = X ( u ) - Y v \ r - Y f r 2
YH =X|,ur + Y^v+Y,r-^Y,^,,MM + yn^r\M + Y^mMr\ + Y,„rlrl (A.4) NH = YfUr + ( Y v - X i , ) UV+ NvV-^ N^r + N^MV\M + NriMr|v| + iV„riMr| + N,m'^\r\
w h e r e X(u) is ship resistance. The p r o p u l s i o n forces are calculated u s i n g a propeller open w a t e r characteristic KT(}):
Xp = (1 - t)pD^p {n]KTUt) + nlKTU2)] Np = - (1 - t ) p D 4 { y p i n f K j - (Ji) +yp2njKT (J2)} w h e r e
(1 -W{)U
= { 1 , 2 }
(A6)
Here and i n the f o l l o w i n g expressions, t h e subscripts '1' and '2' r e f e r to the starboard and p o r t sides, respectively. Forces acting o n
Fig. A2. Rudder inflow velocity and forces.
a r u d d e r are based o n l i f t and drag coefficients (see Fig. A2 f o r the angles d e f i n i t i o n s ) : 2 Ym = IpV^M^LCOS&i - Cusinêi) w h e r e
(A7)
(A.8) W e assume t h a t the w h o l e r u d d e r is situated w i t h i n the p r o -peller slipstream. L o n g i t u d i n a l i n f l o w v e l o c i t y is calculated as f o l l o w s : "Ri =JiniDpE 1 + KR - - ^ - X c t ^ ' XH +Xp +XR -l-Xyv w h e r e + m ur YH + YR + YW ( ^ • 2 ) KR = 0.5 -1-_NH+NP + NR + NW. 0.5 1 - ^ 0 . 1 5 / ({XP-XR)/DP) ^ i = y i ( v + r x R ) Thus, &i = atan V U R , / (A.9)(AlO)
ng:( A l l )
(A.12) and Q!j = Si - i?i (A.13) The t o t a l r u d d e r lateral force and m o m e n t i n c l u d e the r u d d e r s-h u l l i n t e r a c t i o n v i a t-he r u d d e r - -h u l l i n t e r a c t i o n coefficients O H , tR a n d x w :
XR = ( l - t R ) ( X R , - f X R 2 )
yR = ( l + a H ) ( y R i + y R 2 ) (A.14) N R = ( X R - | - X H a H ) ( V ' R l + V R 2 )
Finally, the w i n d forces a n d m o m e n t are calculated according t o [15]:
(A.5) 2^'''''^Vir.a^F^w,x
( A 1 5 ) Yw = 2PaiTVw,a^LCw,Y
Nw = ^PairVw.o^iiCw.N
w h e r e P a i r is air density, Vw,a is the apparent speed o f w i n d , Ap and AL are t h e f r o n t a l and lateral p r o j e c t e d areas o f the ship, and L is the
Table A l
Hull hydrodynamic coefficients in non-dimensional f o r m , 1 0 ^ .
Xu' -0.89 Yr 19.09 Xya' -0.79 -19.68 X„aii' -0.90 y,^' 2.46 x ^ ' -1.10 y^rt' 1.15 X ^ ' - J ) £ 3 ^ r t i ' 334 Xi' -6.12 N,' 2035 Yi -3734 N / - 6 i 3
n'
0.47 N„„' 15.85 Ni 0.47 Nr,,,' -2.19 N / -1.12 N„n' 2.18 y,' -66.05 N„„" -6.16The hydrodynamic coefficients of the hull in (A.4) are estimated f r o m PIVIM tests. Table A l lists the n o n - d i m e n s i o n a l coefficients (terms pLpp^lPlJ and pLpp'^lflft are used as d i m e n s i o n a l factors f o r forces and m o m e n t , respectively; 17 = 5.28 m/s).
Coefficients Cj, and Cp, f o r the f l a p p e d r u d d e r are evaluated based on the results o f open w a t e r tests f o u n d i n [ 16]. The h u l l p r o p e l l e r -r u d d e -r i n t e -r a c t i o n coefficients a-re e s t i m a t e d f -r o m the f-ree -r u n n i n g m o d e l tests. The w i n d coefficients Cw,x, Cw,y and CW,N f o r the research vessel are f o u n d i n [15].
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