The purely vertical motion and vertical - angular motion aưid their stability are studied.
Trang 1I Ệ N K H O A H Ọ C V I Ệ T N A M• • •
Tạp chi'
J o u r n a l o f M ec h a n ic s, N C S R o f V ie tn a m
( T Ậ P X I V )
1 9 9 2
Trang 2T ập chí C ơ H ọc J ou rn al o f Mechaưiics, NCSR of V ietnam T x r v , 1992, No 2 (7 — 12)
N O N L I N E A R M O D E L S I M U L A T I O N O F S H I P M O T I O N
N G U Y E N VAN DAO
I nst it ute o f Mechanics N C S R Vietnam
SU M M A R Y Nonlinear m odel simulation of coupling between heave - roll and pitch -
roll ship m o t i o n s ia considered b y m e a n s of the M y m p t o t i c m e t h o d of nonlinear mechanics.
The purely vertical motion and vertical - angular motion aưid their stability are studied.
1 IN T R O D U C T IO N
T he s im u la tio n o f th e c o u p lin g b etw een heave roll and p itch roll m o tio n s of a ship running
in a regular lo n g itu d in a l or o b liq u e sea has been stu d ied by T ondl A and N abergoj R | l | T h e
p roposed m o d e l c o n s is t s o f a m ass M restrain ed by a linear e la stic spring, w hich, in turn, carries
a sim p le free o s c illa t in g p e n d u lu m m ad e up o f a m ass m a tta ch e d to a h in ged , w eightless rod (F ig 1) T h e sy s te m b fo r ced to o sc illa te sin u soid ally ill the vertical d ừ ectio n by m eans o f an externa] driver w ith c o n s ta n t a m p litu d e q an d frequency CJ. T he co u p lin g b etw een th e vertical and angular
o scilla tio n s is a c c o m p lish e d by co n n ectin g th e tw o m asses and th e effect o f th e w aves is sim u lated
by m ean s o f e x te r n a l forcin g.
In th e p r e se n t p a p e r so m e resu lts o b ta in ed by T ondl A and N abergoj R [ l , 2) w ill be exten d ed for th e case o f a n o n lin e a r ela stic sp rin g and nonlinear exp a n sio n s o f trigonom etric functions.
1
2 M O T IO N E Q U A T IO N S
U sing th e L a g ra n g e e q u a tio n s for th e sy stem represented in F ig l w e have th e follow ing dif feren tial e q u a tio n s o f m o tio n :
( m + M ) ( Z + Ũ) + k q Z + + h o Z + m t [ < p s i i \ <0 + tp 2 COS <p) = 0, ^ ^
m l 2 ip + Co¥? + + z + ũ) 9ÌII <p = 0>
w here z = X — u is th e rela tiv e v e r tic a l d isp lacem en t o f th e m ass My X is th e vertical disp lacem en t
o f th e m a st M fro m it s s ta tic p o sitio n o f equilibrium , u = q c o i w t is th e v ertica l disp lacem en t o f
th e b ase o f th e sp r in g - m a ss sy ste m , <p is th e angular d isp la cem en t o f th e p en d u lu m , kị) and p it
are th e lin ea r an d n o n lin e a r c h a ra cteristics o f the spring resp ectively, I is the len gth o f the rod,
7
Trang 3ho and Co * rt th e d a m p in g coefficien ts o f th t lin ear and angular m o tio n s, resp ectiv ely , g is the gravity accelera tio n and an overdot d en o tes a d eriv a tiv e w ith respect to tim e t.
B y in tro d u cin g the n o ta tio n s
ttf = -7 , n = — , Uo s \ / 7 I c = — -77^ ,
^ = — TT7 » m + M ơ ~ i£ » r = "<>*»
( 2 2 )
and su p p o sin g th a t the d a m p in g forces and the ra tio s • = CT, — - — = /i are sm all, and lim itin g
by con sid erin g sm a ll v ib ra tio n s of co o rd in a tes, 80 th a t lơ3 , £>3 , p<p"} <p'7y <pw" are sm all, w e have
th e follow in g eq u a tio n s o f m otion:
w " + k 2 w = - e ỉ { w ì w \ <pt <p\ <p") + Í 3 ,
<p" + <p = £ > , * > ' ) + e 2 ,
where a p rim e d en o tes a d erivative w ith respect to th e dim en sion less tim e r,
/ = -ơ ry 2 cos ĨỊT + + /9 u ; 3 + ậi(<pp" + v^, 2 )»
$ = - c ^ ' + - - -p u /' + ƠTỊ <pcOB TỊT
6
(2.4)
and £ is a sm a ll d im en sio n less p o sitiv e param eter th a t is used as a book keeping d evice and w ill b« set eq u al to u n ity in th e final resu lts T he case p = 0, sin <p23 <p, COS <p ss 1 haa been exam in ed
in [1, 2|
3 A P P R O X IM A T E S O L U T IO N
Let Ufl con sid er the resonant region determ in ed by
(3 1 )
Tl - 4(1 + « A ),
w here r a n d A are d e tu n in g p aram eters U sing in eq u a tio n s (2 3 ) th e tran sform ation in to new
am p litu d e an d phaae va ria b les R t 0, a , iỊ> by m ean s o f th e form ulae
w = R COS $, t i/ = -/? r ; sin £, t =z T)T + 0,
<p = a c o s a , ip = z - - a r f Bin a , a = -» jr + 0 ,
we have:
I|iỉ# = c ( / - ru;)sin rja1 = - 2 e ( # + A^>) sin a,
= « (/ - ru/)cos£; = — 2e(Q + Apjcosa.
T h ese e q u a tio n s are in th e stan d ard form for w hich th e averaging tech n iq u e o f n on lin ear m ech an ics (3] can b e u sed S o, in th e first ap p roxim ation on e can replace (3 3 ) b y th e follow in g evaraged
Trang 4R ' * - J [fc/ĩ + <x»í sin 0 + - ịir ịá * *in(0 - 2ự»)],
Rr)0' = - - [ r / ỉ - -/9 Ì Ĩ 3 + ơrj3 c o s 0 + “ M1?3**3 c° s (0 ~
à — - - [ca + ơ ^ a sin - R tjq sin (0 - 2 0 ) ] ,
ù arjrịỉ' = — - [2A a + - a 3 + ơrj7a COS 2 0 + /?r;2a CO3(0 — 2 0 )].
4 P U R E L Y V E R T I C A L S H I P M O T I O N
A s ta tio n a r y se m i - triv ia l solution o f eq u ation s (3-4) is
/? = ft), Ớ = Í0 , where 01) ifl an arb itrary co n sta n t and # ,), 0<) are c o n sta n ts sa tisfy in g the relations
(3.4)
(4.1)
hRo + ƠTỊ sin 00 = 0,
T his so lu tio n co r resp o n d s to the vertical m otion o f th e sh ip , w hile its angular m otion rem ain s
u n ex cited
E lim in a tin g th e phase 00 from (4.2) we o b ta in an e q u a tio n which defines the adm issible values
o f the a m p litu d e Ri) a* a fu n ction of the e x c ita tio n freq u en cy Tf:
here
«r = T)* — k7.
T h is r e la tio n sh ip c a n be expressed approxim ately aa
I Z ĩ ~t*~
(4 5 )
1» — 9 Jk i n * i /T
and = 1 (c u r v e 2 ) W ith very sm all v a lu es o f a , th e a m p litu d e Hi) is alm ost a sm all co n sta n t:
R * ~ k 2* * / h 2.
To s tu d y th e s ta b ility o f the sem i - tr iv ia l so lu tio n (4 1 ) on e lets
a = 6at 0 — ipo ■+■ ^ = Ao + 0 5=5 00 +
T he follow ing v ariation al equations will b e obtained
j [ 6 R ) = - ị ( h 6 R + ơTỊCo»0qS ê ) t
~ (f a ) = - 1 [c + ƠVỊ sin 2^0 —• »in(0o * 2^o)] £<*»
0 = [2A + (Try2 COS 2 0 0 + *J2 /Ỉo c o s(0 o — 2ự»o)]ía*
Trang 5FYom th e first tw o eq u a tio n s o f (4 6 ) and from (4 2 ) and (4 3 ) one can find after a short ca lcu la tio n
th e sta b ility co n d itio n
£ > •
w hich is im posed upon the reso n a n t curve (see h eavy lines m F ig 2) of the vertical m o tio n o f th e
T h e b ou n d a ry o f the in sta b ility region for the appearance of a param etric reson an ce o f a n gu lar ship m o tio n is d eterm in ed from th e last tw o e q u ation s o f (4.6) a£
c + or\ sin 2 0 0 — v f y 8Ìn(0o — 2V>o) = 0,
2 A + ƠĨỊ 2 COS 2 0 0 + V2Ro cos(0o - 2 0 0 ) = 0.
E lim in a tin g th e p h ases ĩpo, 00 gives
+ 4 A 3 = r,*{a* + R*) + 2ri 7 R l { - B R l - r ) ,
4
o r
TỊ 7 * 4 ± ik€y/k*l<r* + R i ) - c’ + + I f i B Z ) (4 9 )
w here Ro satisfies eq u a tio n (4 3 ) T h e relation (4 9 ) is p lo tte d in F ig 3 for th e case / 9 = 1 ,
c = 1 0 " 1, h = 10“ \ and k = 1.9 (curvc 1), k = 2 (curve 2), k = 2.1 (curve 3 ) T h ese curves are a p p ro x im a te ly parabolic and th e in sta b ility region is located a b o v e the p arab ola (see sh a d ed region in F ig 3 ) For su fficen tly sm a ll v a lu es o f th e e x c ita tio n ( a ) , th e se m i - trivial so lu tio n rem ain s
ex c eed ed b efore in sta b ility can occu r.
Taking into account curve 2 in F ig.3, one can s e t th a t for Ơ = 4-5 • 10~3 and 2.75 < TỊ2 < 4.9 the param etric vibration of angular m otion may occur So, in th e interval 2.75 < TỊ2 < 4.9 th e
sh ip m o tio n w ill n o t be characterised o n ly b y th e reson an t curve in F ig 2 O u tsid e th e m en tio n ed
in terv a l th e an gu lar m otion o f th e sh ip w ill b e sero and th e sh ip m o tio n b characterised o n ly b y
Trang 6th e reson an t curve in F ig 2 In th e la*t case the a m p litu d e o f ship v ib ra tio n w ill be sm a ll and
a lm o st c o n sta n t T herefore, for th e case considered, th e purely vertical m o tio n (w ith o u t angular
m o tio n ) occu rs only w ith sm all a m p litu d e T h e stron g v ertica l m otion w ill be accom p an ied by th e
a n g u l a r o n e
6 *
Fig 3
5 C O U P L IN G B E T W E E N V E R T IC A L A N D A N G U L A R M O T IO N S
T he non - trival sta tio n a r y so lu tio n of eq u a tio n s (3.4) w ith a Ỷ 0» R Ỷ 0 is d ete rm in ed from
t h e r e l a t i o n :
ƠTỊ2 s i n 9 + \ i i r Ị 2 CL2 8 Ì n ( 0 — 2rp) = — h r ) R y
4
ƠĨỊ 2 sin 2ip — Rrj2 8Ìn(0 — 2ip) = —cry,
ƠĨ ) 2 cos + Rrj 2 cos(0 — 2ĩp) = —( 2 A + 7 a2)
T h is s o lu tio n corresp on d s to th e sim u lta n eo u s v er tic a l and an gu lar m otion s o f th e sh ip E lim in a tin g
th e p h ase v ariab les in th ese eq u a tio n s w e o b ta in th e follow in g exp ression g iv in g th e d ep en d en ce o f
th e a m p litu d e s o f v ib r a tio n a, R on th e frequency rj o f th e e x c itin g force:
* v - h 9ữ ) + [2A + \ a ' ~ i / w i V ] * - - r + \ p R 2}2. , (5,2)
T h is r e la tio n sh ip is p lo tte d in F ig 4 for th e case p = 1, k = 2, k = c = 0 1 and ịẰ = 0 0 5 FYom F ig 4
it is seen t h a t , in creasin g rj from th e reson an t value (r; = 2) and keeping c o n sta n t th e a m p litu d e o f vertical m o tio n , the am plitude of angular ship vibration (a) decreases, and at a con stan t value of
or decrease sim ultaneously.
11
Trang 7Fig.4
C O N C L U S I O N
T he n o n lin e a r m o d el sim u lation of coupling betw een v e rtic a l and an gu lar ship m o tio n s has been co n sid ered T h e n on lin ear term s in the m otion eq u a tio n s have esse n tia l influence on b o th th e shape o f th e r e sp o n se curve and th e in stab ility region in com p arison w ith th e linear e q u a tio n s ( l , 2Ị T he c o n d itio n for th e appearance of a purely vertica l v ib ra tio n o f th e sh ip has b«en d eriv ed
In the case co n sid ere d th is vib ration occurs only w ith sm all am p litu d e and th e strong v e rtic a l sh ip
angular m o tio n s o f th e sh ip have been studied too.
R E F E R E N C E S
1 Tondl A , N a b erg o j R M odel S im ulation o f p aram etrically ex c ited sh ip rollin g J N o n lin ea r
D yn am ics 1 (1 3 1 - 1 4 1 ), 1990 Kluwer A cadem ic P u b lish ers N eth erlan d s.
2 N abergoj R P a ra m etric resonance o f a spring - p en d u lu m sy ste m E igh th world con gress on the th eory o f m a ch in e and m echanism s, Prague, Czechoslovakia, A u g u st Ỉ991 (2 9 9 - 3 0 2 ).
3 B o g o liu b o v N N , M itro p o lx k i Yu A A sy m p to tic m eth o d s o f th e th eory o f n on lin ear v ib r a tio n s, M o sc o w , 1974.
Re cei ved Ma r c h 28, 1 9 9 2
TYong b à o b á o n à y m ô hình tu y ến tín h củ a T ondl A v à N ab ergoj R [1, 2] ve s ự lắc n g a n g v à chuyển động t h i n g đ ứ n g của tàu thủy được mír rộng cho tn rờ n g h ọp phi tu yến , tạo nền do đặc
t n m g dàn h ồi v à k ề đ ế n c á c so h ạn g bậc cao trong k h ai triển các h à m lư ợ n g giác Đ ã x é t đ ến k h ả nẵng x ẩ y ra d ồ n g thòri cộ n g hưdrng cư ỡ ng bức (doi v ớ i chuyền d ộ n g th ẳ n g d ứ n g ) v à cộn g hưỏrng thông số (đối v ó i chuyển dộng lắc ngang) của tầu thủy; cũng như đ ã nghiền cứu chi riêng chuycn dộng thẳng đ ứ n g m ì không có chuyen động lắc ngang của tàu.
C huyên đ ộ n g d ồ n g thòri th ẳn g dử ng và l í c ngang c d a tà u có d ặc điềm lằ ò cùng m ột tầ n số kích dộng các biền đ ộ dao động dứng và dao dộng lắc ngang cùng tăn g hoặc cùng giảm Khi tăng tần sổ kích d ộ n g t ừ g iá trị cộng hưárng và giử cho b iỉn đ ộ d a o đ ộ n g đ ứ n g k h ồn g đổi th ì b iỉn đ ộ dao dộng lắc n gan g giảm.