r
A n a l y s i s o f o f a p p a r e n t l y apontarwoua yawing o s c l l l a t l o n a f o r a a h i p undar way.Dr. C, de W i t , D e l f t U n i v . o f Techn., Dept o f Hathenatloa & I n f o r a a t l c s .
0. Suomary.
I n t h i s paper a o a t h e q a t l o a l nodel l a presented t o e x p l a i n t h e phenoaenon o f course o s c l l l a t l o n a o f a a h i p , s a i l i n g I n a nean a t r a l g h t eourae w i t h a c o n s t a n t oean speed.
W i t h t h i s model, p e r i o d i c course changing can be s i m u l a t e d aa a r e s u l t o f s i m u l t a n e o u s p e r i o d i c r o l l i n g and p i t c h i n g and e m p h a t i c a l l y w i t h o u t t h e i n p u t o f any rudder a c t i o n .
The f r e q u e n c y spectrum o f t h i s course e r r o r appears t o c o n s i s t o f two narrow bands around tho f r e q u e n c i e s f • f and f - f .
p r p r ' where f p and r^, a r e t h e s h i p ' s n a t u r a l p i t c h i n g and r o l l i n g f r e q u e n c i e s . On t o p o f t h i s , the course e r r o r I s found t o have a non z e r o t i m e average.
I
I . I n t r o d u c t i o n .
AaauBc a a h i p o f aoae 20,000 tona t o be s a i l i n g I n a choppy aea. The aea waves have a s i g n i f i c a n t h e i g h t o f 1 t o 5 meters and
the mean wave d i r e c t i o n I s n o t c o i n c i d i n g w i t h t h e l o n g s h i p ' s f or t h w a r t s h l p ' s d i r e c t i o n s . .
The mean s t a t e o f t h e s h i p l a assumed t o be s t a t i o n a r y , meaning
t h a t t h e mean apeed and t h e mean course a r e conatant i n t i m e . I T I The wind waves a r e aupposed t o a c t as d l s t u r b a n e e a o f t h i s mean " T f steady a t a t e , r e s u l t i n g i n p e r i o d i c movements o f t h e s h i p ' s
g r a v i t y o e n t r e G , U k e h e a v i n g , i . e . movementa i n a v e r t i c a l d i r e c t i o n , and awaylng, i . e . i r r e g u l a r movements I n t h e beam d i r e c t i o n .
As a lumped naas, t h e s h i p l a assumed t o be o s c i l l a t i n g about the l o n g a h i p ' a and t h w a r t s h l p ' s axes through G . These o a c l l -- l a t l o n a a r e commonly known as r o l l i n g and p i t c h i n g .
A l a a t phenomenon t h a t i s known t o occur i a p e r i o d i c yawing, i . e . p e r i o d i c movementa o f t h e compaaa l u b b e r mark w i t h reapect t o t h a s h i p ' s compaaa, which l a assumed t o m a i n t a i n I t s h o r i z o n t a l and a z i m u t h a l p o a l t i o n .
T h i s yawing i s o f t e n caused by a d i s t u r b i n g couple around t h e s h i p ' s V^.«xla, e x e r t e d by t h e J o i n t d i s t u r b i n g a c t i o n o f wind and waves upon t h e s h i p ' s h u l l .
T h i s d i s t u r b a n c e may r e a u l t i n a d e v i a t i o n o f t h a s h i p ' s course from i t s mean v a l u e . I n t h a t case i t i a u s u a l l y c o u n t e r a c t e d by a rudder a n g l e .
I f the s h i p l a equipped w i t h an a u t o m a t i c s t e e r i n g d e v i c e , t h e
r e s u l t o f these time changing v e r t i c a l d i s t u r b a n c e s and t h e o o r - g j j " -responding rudder c o u n t e r a c t i o n s w i l l be some k i n d o f q u a s i - " H I OT*
ro •
- p e r i o d i c eourae changing about a mean courae. O
There l a however a second p e r i o d i c yawing movement, c o n s i s t i n g 3 " ^ o f p e r i o d i c eourae changing w i t h a r a t h e r h i g h frequency o f 5 vi'
t o 10 p e r i o d s per m i n u t e . These o s c l l l a t l o n a have been expe- 3— r i e n c e d t o occur i n t h e a b s e n c e o f a n y r u d
-- d e r a c t 1 0 n . I t i s t h i s p a r t i c u l a r movement, t h a t t h i s Sg! " S paper l a concerned w i t h . ^ J j -The s p e c i a l and e x c e p t i o n a l n a t u r e o f t h i a phenomenon i s t h a t , ^ § a l t h o u g h i t hae been n e t w i t h i n p r a c t i c e , t h e r e I s no p l a u s i b l e €L.
§ - r
a-.
hydrodynamic e x p l a n a t i o n f o r I t , l i k e t h e r e I s f o r p e r i o d i c r o l l i n g and p i t c h i n g .
A c e r t a i n r o l l o r p i t c h angle l a always c o u n t e r a c t e d by some r e a t o r l n g raonent, a r i s i n g from t h e f a c t t h a t t h e c e n t r e o f buoy--ancy B i s beyond t h e v e r t i c a l l i n a through t h e c e n t r e o f g r a v i t y 0 .
However, i f a course e r r o r i s n o t being c o u n t e r a c t e d by a c o u p l e around t h e v e r t i c a l a x i s , generated by a c e r t a i n rudder a n g l e , t h e r e i s no f i r s t o r d e r e x p l a n a t i o n f o r t h e f a c t , t h a t t h e s h i p t u r n s back t o h e r p r e v i o u s courae.
I n t h e f o l l o w i n g paragraph t h i s apparent spontaneous yawing i s modelled as a r e a u l t o f combining t h e o a c l l l a t l n g r o l l i n g and p i t c h i n g movements.
2. A m a t h e m a t i c a l model f o r spontaneous p e r i o d i c yawing. We f i r s t I n t r o d u c e some n o t a t i o n s and c o n v e n t i o n s .
With t h e o r i g i n i n G, we work w i t h a r i g h t handed c o - o r d i n a t e system, a t t a c h e d t o t h e s h i p .
Por t h e s h i p I n a p u r e l y steady s t a t e . I . e . s a i l i n g on a plane w a t e r - s u r f a c e w i t h a c o n s t a n t speed, t h e X * - d l r e c t i o n I s adopted
s
as t h e h o r i z o n t a l d i r e c t i o n o f the ahlp's stem . t h e ï*-direc-- t l o n I s t h e h o r i z o n t a l d i r e c t i o n o f t h e s t a r b o a r d beam and t h e V ^ . d i r e c t i o n l a p o i n t i n g v e r t i c a l l y downward.
I n t h i s c o - o r d l n a t 8 aystea a p o a l t l v s a n g u l a r v e l o c i t y v e c t o r p o i n t i n g I n t o t h e X * - d l r e c t l o n , correaponds w i t h a r o l l i n g t o
a
s t a r b o a r d . Analogously, a p o s i t i v e t h w a r t s h l p ' s a n g u l a r v e l o c i t y Wp, p o i n t i n g I n t o t h e ï^-directlon, corresponds w i t h an upward
t u r n o r t h e s h i p ' s s t e a .
The r o l l Angle f and the p i t c h a n g l e 6 a r e d e f i n e d aa t h e anglea between t h e ï*axla f o r f and t h e X*axls f o r 8
-s -s w i t h t h e h o r i z o n t a l p l a n e . I n consequence w i t h t h e above n e n t i o n e d s i g n c o n v e n t i o n s , a r o l l angle i s p o s i t i v e . I f t h e s h i p h e e l s t o s t a r b o a r d and a p i t c h a n g l e i s p o s i t i v e . I f t h e s t e n d i r e c t i o n i s above t h e h o r i z o n .
L e t us now c q p s i d e r f i g u r e 2.a on page 5, showing a sphere w i t h r a d l u a R . The sphere's c e n t r e c o i n c i d e s w i t h t h e s h i p ' s g r a v i t y o e n t r e C . TV i a t h e v e r t i c a l d i a m e t e r .
The h o r i z o n on t h i s sphere i s t h e g r e a t c i r c l e NESW. CH p o i n t s i n t o t h e t r u e N o r t h d i r e c t i o n e t c .
The s h i p haa a p o a i t l v e p i t c h Ö and a p o s i t i v e r o l l a n g l e f . A and B a r e t h e i n t e r s e c t i o n s o f t h e sphere w i t h t h e s h i p ' s momentary X*- and Y*-axea. We thua have
s a
arc TA s
«/2 - 0
, a r c AD a0 ,
a r o TB »
«/2
• v , a r c FB = • , a r c AB s i / 2.
I n t h i s skew a t a t e , t h e s h i p ' s course can be d e f i n e d I n two ways. The most common way i s t o d e f i n e i t aa t h e a z i m u t h a l d i r e c t i o n o f t h e l o n g s h i p ' s a x i s CX* . T h i s course I s denoted aa T .s a A c c o r d i n g t o i n t e r n a t i o n a l s t a n d a r d r u l e s t h e a h l p ' s
compass bowl should be mounted w i t h t h e o u t e r gimbal a x i s p a r a l l e l t o t h e CX*-axl3. T h i s meane, t h a t t h e a z i m u t h a l course i s t h e course, i n d i c a t e d on tha - asaumed t o be h o r i z o n t a l - compass c a r d by t h e l o n g s h i p ' s l u b b e r mark.
Prom a hydrodynamical p o i n t o f view however, i t seems more l o g l -- c a l t o conclude t h a t , i n the absence o f any r u d d e r a c t i o n , t h e h o r i z o n t a l d i r e c t i o n i n t o which t h e s h i p moves, i s t h e l i n e o f i n t e r s e c t i o n o f t h e l o n g s h i p ' s plane X* G V* w i t h the h o r i z o n ¬ - t a l p l a n e . I n f i g u r e 2.a t h i s i s t h e l i n e OH .
I n t h e s e q u e l these courses w i l l be c a l l e d t h e a z i m u t h a l and the l o n g a h i p ' a courae r e s p e c t i v e l y , denoted aa and . I n f i g u r e 2.a we see t h a t T > a r c ND and T, « a r c HH .
s i n e s B i s a p o l e t o the g r e a t c i r c l e t h r o g h A and H , we oan conclude t h a t arc BH = t / 2 .
With BP _[ PH we now see t h a t arc HF = V 2 , ao t h a t arc HF = • 1/2 .
T h i s l o p l l e s t h a t , t h e time d e r i v a t i v e o f t h e l o n g s h i p ' s courae Tj^ i s equal t o t h e a z i m u t h a l v e l o c i t y o f B .
We are how I n a p o s i t i o n t o e s t a b l i s h d i f f e r e n t i a l e q u a t i o n s f o r the change i n t i m e o f t h e angles 9 , * , t and . The s h i p i s aqsumed t o ba r o t a t i n g about t h e X*-axla w i t h an a n g u l a r v e l o c i t y and about t h e Y * - a x i 3 w i t h an a n g u l a r v e l o c i t y w .
P
F i g u r e 2.b shows, how t h e s h i p I s momentarily r o t a t i n g about the l i n e CC i n the ahlp's G Y^plane w i t h t h e a n g u l a r v e l o -- c l t y
w. = (w^ + w ^ ) J .
t r p
The d i s t a n c e from A t o t h i s a x i s GC l a R a i n G, , ao A haa an upward l i n e a r v e l o c i t y
= W|.R a i n 0, = R Wp .
For t h e a r c v e l o c i t y o f A along t h e sphere we f i n d "A = "A'" ' "p •
A I s moving upward I n a d i r e c t i o n , p e r p e n d i c u l a r t o AB. A n a l o g o u s l y , we f i n d t h a t B haa a downward a r c v e l o c i t y w ,
d i r e c t e d 90° from BA. '' Denoting the t i m e d e r i v a t i v e s o f e , v. e t c . aa Ó . ' ^ e t c .
and p u t t i n g ^ BAT = a , / AST = B , we see t h a t é = WpOoaCo . 1 / 2 ) = w^gin a , f = w c o s ( t / 2 _ B) = W s i n B , r Por t h e t i m e d e r i v a t i a o f T and », we f i n d •^cos e = Wp3in(o - «/2) , • ^003 f = - w^ain(«/2 - B) , '° 'a ° " a/coa 0 • j ^ = - w^ooa B/coa » .
A p p l y i n g t h e r u l e o f ooalnes I n t r i a n g l e TAB, ue f i n d ooa( i / 2 + f ) J s i n ( i / 2 - 9 ) e o 3 « , ooa( i / 2 - 9 ) 5 s l n ( «/2 * t ) c o 3 ft , We t h u s come t o t h e d i f f e r e n t i a l e q u a t i o n s è = " p " - ^ i " ^ * / c o 8 ^ 9 ) i i = w ^ { l - 3 l n ^ 9 /ooa^» >^ i = u ain f / 008^9 , a p J = - H ^ s l r 8/coc ^
Given t h e i n i t i a l v a l u e s o f e , » , and , we oan now aee, what happens t o these q u a n t i t i e s . I f w and w a r e a i v e n as
p r = f u n c t i o n s o f t i m e .
S i m u l a t i o n o f t h e t i m e b e h a v i o u r o f t h e a z i m u t h a l and t h e l o n g -- 3 h i p ' 3 c o u r s e s on a r o l l i n g and p i t c h i n g s h i p .
For t h e r o l l i n g and p i t c h i n g v e l o c i t i e s we adopt t h e t i m e f u n c t i o n s = Ap co3( 2 • f p t ) ,
w = A 0O3( 2 « f t ) . P P P
These t i m e f u n c t i o n s can be aeen as the reaponaea o f t h e s h i p t o d i s t u r b i n g I n p u t s , caused by wind and s w e l l waves. The r e a p o n d l n g f r e q u e n c i e s f ^ and f ^ a r e known t o be r a t h e r c l o a e t o the s h i p ' s own r o l l i n g a n d p i t c h i n g f r e q u e n c l e a f and f . The s m a l l
r n pn d i f f e r e n c e s between f and f and between f and f
r r n p pn a r e m a i n l y due t o t h e f a c t , t h a t the b a s i c second o r d e r d i f f e r e n -- t i a l e q u a t i o n s u s u a l l y have a s m a l l damping coëfficiënt. The s i m u l t a n e o u s s e t ( 2 . 1 ) was i n t e g r a t e d n u m e r i c a l l y w i t h Heun's p r e d i c t o r - c o r r e c t o r method w i t h t h e i n i t i a l o o n d l t i o n a 9 ( 0 ) = , ( 0 ) = r ( 0 ) = T,<0) = 0 . a X I n a f i r s t example t h e a u t h o r s e l e c t e d Ap = Ap = «^/108 r a d i a n s per second, f p = 1/12 sec"' , r = 1/6 aec"' . ( 2 . I . a ) (2.1.b) ( 2 . l . C ) ( 2 . I . d )
These v a l u e s c o r r e s p o n d w i t h - d l s c o u p l e d - r o l l i n g and p i t c h i n g a n p l l t u d e a o f 10° and 5° and w i t h r o l l i n g and p i t c h i n g p e r i o d s o f 12^^° and 6^°° r e s p e c t i v e l y .
The r e s u l t i n g graphs o f »^ and a r e shown i n f i g y r e s 3.a4b. The second example shows t h e e f f e c t o f p e r i o d i c p i t c h i n g w i t h 5° a m p l i t u d e and 5'®° p e r i o d w i t h r o l l i n g w i t h 12° a m p l i t u d e and
12'®° p e r i o d . The r e s u l t i n g graphs a r e shown i n f i g u r e s l . a i b .
A n a l y s i s o f t h e l i n e a r i z e d model. W i t h l f l 5 0.2 r a d . and |e| = 0.1 r a d . t h e d i f f e r e n t i a l e q u a t i o n s (2.1.a,b.o&d) oan be l i n e a r i z e d w i t h o u t a s e r i o u s l o s s o f a c c u r a c y . P u t t i n g 3 l n • s » + 0( if^) , s i n 0 Ï 8 + 0< n^) COS » = 1 > 0( 9^) , Cos «) 5 1 • 0 ( 8^)
and o m i t t i n g terms o f 0( 9^) and o f 0 ( 8 ^ ) , t h e s e t <2.1) can be reduced t o 8 = w , ( 1 . 1 ) P » = Wp , ( I t . 2 ) * = »„<fi . ("-3) •V = - W p ö . (11.D) A d o p t i n g Wp = ApCos(2 « f p t ) , ( 1 . 5 ) Wp = ApCoa(2 I f p t ) , and t a k i n g 8 ( 0 ) = q)(0) = 0 , we f i n d ( I 1 . 6 ) 9 ( t ) = Ap 3 l n ( 2 I r p t ) / ( 2 « f p ) , ( 1 . 7 ) ( t ) = Ap 3 l n ( 2 f f p t ) / ( 2 I f p ) ( I t . 8 ) Assuming z e r o i n i t i a l c o n d i t i o n s f o r V^ and , we have,
as a consequence o f (".3) t o (U.8) T ( t ) = A A (-cos(2 i ( f + f ) t ) / ( f + f ) + a p r P r p r • c o s ( 2 i ( r - f ^ ) t ) / ( r - f ) ) / ( 8 K ^ r ) p r P r r - A„A^/(lt , 2 ( f ^ - f ^ ) ) . p r P r
T j ( t ) T A p A p ( c o s ( 2 i ( r p * f p ) t ) / ( f p + r p ) + c o 3 ( 2 , ( r p - r p ) t ) / ( r p - f p ) ) / 8 , ' r p )
I n the f i r s t example t h i s would amount t o
f g C t ) = - 0.29°«oo8(90°''t) + O.87°'co3(30°»t) - 0.58° , * j ( t ) = 0.1il°»oos(90°«t) + O.')li°»oo3(30°'t) - 0,58° , and i n t h e second example t h l S would leSd t o
• g ( t ) = - 0.37°*cos(102°»t) • 0.90°'cos(ft2°'t) - 0.53° , ' ' j ( t ) = 0.15°*cos(102°«t) + O.37°'cos(lt2°«t) - 0.53° . These e x p r e s s i o n s g i v e r i s e t o t h e f o l l o w i n g c o n c l u s i o n s : ( I ) Both t h e a z i m u t h a l and t h e l o n g s h i p ' s courses have
double p e r i o d i c o a c U l a t l o n a w i t h f r e q u e n c i e s equal t o t h e 3um and t h e d i f f e r e n c e o f t h e p i t c h i n g and r o l l i n g f r e q u e n c i e s .
( I I ) Both yawing movements have a non zero t i m e average. I n t h e asaumed examples t h i s t i m e average o f t h e course d e v i a t i o n from t h e steady course amounts t o
