IỆN K H O A H Ọ C V I Ệ T N AM • • • Tạp ch i' Journal of Mechanics, NCSR of Vietnam (T Ậ P X I V ) 1 99 2 HÀ NỘI Tập chí Cơ Học Journal of Mechaưiics, NCSR of Vietnam T. x rv , 1992, No 2 (7 — 12) N O N L IN E A R M O D EL SIM ULA TION O F SH IP MOTION N G U Y EN VAN DAO Institute of Mechanics NCSR Vietnam SUMM ARY. Nonlinear model simulation of coupling between heave - roll and pitch - roll ship mo t i o n s ia considered by me a n s of the M ym p to t ic m e t h o d of nonlinear mechanics. The purely vertical motion and vertical - angular motion aưid their stability are studied. 1. INTRODUCTION The sim u lation o f the coupling between heave roll and pitch roll m otions of a ship running in a regular lon gitu d inal or oblique sea has been studied by Tondl A. and Nabergoj R. |l|. The proposed m odel con sists of a mass M restrained by a linear elastic spring, which, in turn, carries a sim ple free o scillatin g pendulum m ade up of a mass m attached to a hinged, weightless rod (Fig. 1). T he system b forced to oscillate sinusoidally ill the vertical dừection by means of an externa] driver w ith con stan t am plitud e q and frequency CJ. The coupling between the vertical and angular oscillations is accom plish ed by connecting the two masses and the effect of the waves is sim ulated by means of ex tern a l forcing. In the present p aper som e results obtained by Tondl A. and Nabergoj R. [l, 2) will be extended for the case of a non linear elastic spring and nonlinear expansions of trigonometric functions. .1 t • > : r .iij 2. MOTION EQUATIONS Using the Lagrange eq uations for the system represented in F ig .l we have the following dif ferential equa tions o f m otion: ( m + M )(Z + Ũ) + k q Z + + h o Z + mt[<psii\ <0 + tp2 COS <p) = 0, ^ ^ m l2 ip + Co¥? + + z + ũ) 9ÌII <p = 0> where z = X — u is th e relative vertical displacem ent o f the mass My X is the vertical displacement o f th e m ast M from its static position of equilibrium , u = qcoiwt is the vertical displacem ent of the base of the spring - m ass system , <p is the angular displacem ent of the pendulum , kị) and pit are the linear and nonlinear characteristics of the spring respectively, I is the length of the rod, 7 ho and Co *rt the dam ping coefficients of th t linear and angular m otions, respectively, g is the gravity acceleration and an overdot denotes a derivative w ith respect to tim e t. By introducing the n otations z _ LJ fg Co ttf = -7 , n = — , Uo s \ / 7 I c = — - 77^ , i UJQ \ t (Ji)mi2 1 __ ____ ^0 ____ 12 ____ ^0 ứ _ ^ Wi)(m Ỷ M ) ’ u*(m ■+• M ) * wị;(m + AÍ) m g ^ = — TT7 » ơ ~ i » r = "<>*» m + M £ ( 2 .2 ) and supposing that the dam ping forces and the ratios • = CT, — — = /i are sm all, and lim iting t m + M by considering sm all vibrations of coordinates, 80 that lơ3, £>3 , p<p"} <p'7y <pw" are sm all, we have the following equations of motion: w" + k2w = -eỉ{w ì w\ <pt <p\ <p") + Í 3 . , < p " + <p = £ > , * > ') + e 2 , where a prim e denotes a derivative with respect to the dim ensionless tim e r, / = -ơry2 cos ĨỊT + + /9u;3 + ậi(<pp" + v^,2 )» $ = - c ^ ' + p u /' + ƠTỊ <pcOB TỊT 6 (2.4) and £ is a sm all dim ensionless positive param eter that is used as a book keeping device and will b« set equal to unity in the final results. The case p = 0, sin <p 23 <p, COS <p ss 1 haa been exam ined in [1, 2| . 3. APPROXIMATE SOLUTION Let Ufl consider the resonant region determ ined by (31) Tl - 4(1 + « A ), where r and A are detuning param eters. Using in equations (2.3) the transform ation into new am plitude and phaae variables Rt 0, a, iỊ> by m eans of the formulae w = R COS $, ti/ = -/?r; sin £, t =z T)T + 0, , 1 ; 1 . (32) <p = acosa, ip =z arfBin a , a = -»jr + 0 , • A we have: I|iỉ# = c ( /- ru;)sin rja1 = -2e(# + A^>) sin a, = «(/ - ru/)cos£; = — 2e(Q + Apjcosa. These equations are in th e standard form for which the averaging technique of nonlinear mechanics (3] can b e used. So, in the first approxim ation one can replace (3.3) by the following evaraged equations: R ' * - J [fc/ĩ + <x»í sin 0 + -ịirịá* *in(0 - 2ự»)], 2 4 Rr)0' = - - [ r / ỉ - -/9ÌĨ3 + ơrj3 co s0 + “ M1?3**3 c°s(0 ~ 2 4 4 à — - - [ca + ơ^a sin - Rtjq sin(0 - 2 0)], ù arjrịỉ' = — - [2Aa + - a 3 + ơrj7a COS 2 0 + /?r;2a CO3(0 — 2 0)]. 2 4 4. P U R E L Y V E R T IC A L S H I P M O T IO N A sta tio nary sem i - trivial solution of equations (3-4) is a = 0, 0 = 00» /? = ft), Ớ = Í0, where 01) ifl an arbitrary constant and #,), 0<) are constants satisfying the relations (3.4) (4.1) hRo + ƠTỊ sin 00 = 0, 3 (4.2) r /?0 “ ■“)WĨỖ + ơr72 cos ^0 = 0- 4 This solution corresponds to the vertical m otion of the ship, while its angular m otion remains unexcited. E lim inating th e phase 00 from (4.2) we obtain an equation which defines the admissible values of the am plitude Ri) a* a function of the excitation frequency Tf: W (R l V*) = 0, (4.3) here = J $ [ * v + (r - \ p r ĩ Ý \ - * v , {4 4) «r = T)* — k7. This relationship can be expressed approxim ately aa IZĩ~t*~ (4.5) r2 = k7 + ± e* y — - - h2 , 1» — 9 Jk - in * i /T - and is p lotted in F ig. 2 for the parameters k = 2, h = 10” A, Ơ = 4.5 • 10~2, /9 = 0 (curve 1) and = 1 (curve 2). W ith very sm all values of a, the am plitude Hi) is almost a sm all constant: R * ~ k 2**/h2. To study th e stab ility of the semi - trivial solution (4.1) one lets a = 6at 0 — ipo ■+■ ^ = Ao + 0 5=5 00 + The following variational equations will be obtained j [ 6 R) = - ị ( h 6 R + ơTỊCo»0qSê)t ~ (fa) = - 1 [c + ƠVỊ sin 2 ^ 0 —• »in(0o * 2^o)] £<*» 0 = [2A + (Try2 COS 200 + *J2/Ỉocos(0o — 2ự»o)]ía* FYom the first two equations of (4.6) and from (4.2) and (4.3) one can find after a short calculation the stab ility condition £ > • which is im posed upon the resonant curve (see heavy lines m F ig.2) of the vertical m otion of the T he boundary of the instability region for the appearance of a parametric resonance of angular ship m otion is determ ined from the last two equations of (4.6) a£ c + or\ sin 200 — v fy 8Ìn(0o — 2V>o) = 0, 2A + ƠĨỊ2 COS 200 + V2Ro cos(0o - 200) = 0. Elim inating the phases ĩpo, 00 gives + 4 A 3 = r,*{a* + R*) + 2ri7 Rl{-BR l - r ) , 4 (4 8) eA = - 1, 4 or TỊ7 * 4 ± ik€y/k*l<r* + Ri) - c’ + + IfiBZ) (4.9) where Ro satisfies equation (4.3). T he relation (4.9) is plotted in Fig.3 for the case / 9 = 1 , c = 10"1, h = 10“ \ and k = 1.9 (curvc 1), k = 2 (curve 2), k = 2.1 (curve 3). These curves are approxim ately parabolic and the instability region is located above the parabola (see shaded region in F ig.3). For sufficently sm all values of the excitation (a ), the sem i - trivial solution rem ains stable. Corresponding to every instability region there exists a “theshold* Ơ — ơ0 which must be exceeded before instability can occur. Taking into account curve 2 in Fig.3, one can set that for Ơ = 4-5 • 10~3 and 2.75 < TỊ2 < 4.9 the parametric vibration of angular motion may occur. So, in the interval 2.75 < TỊ2 < 4.9 the ship m otion w ill not be characterised only by the resonant curve in F ig.2. O utside the m entioned interval the angular m otion of the ship w ill be sero and the ship m otion b characterised only by 10 the resonant curve in F ig.2. In the la*t case the am plitude of ship vibration w ill be sm all and alm ost constant. Therefore, for the case considered, the purely vertical m otion (w ithou t angular m otion) occurs only with sm all am plitude. The strong vertical m otion will be accom panied by the a n g u la r o ne. 6 * Fig. 3 5. COUPLING BETWEEN VERTICAL AND ANGULAR MOTIONS The non - trival stationary solution of equations (3.4) w ith a Ỷ 0» R Ỷ 0 is determ ined from t h e re l a tio n : ƠTỊ2 sin 9 + \ i i r Ị 2 CL2 8Ìn(0 — 2rp) = — h r ) R y 4 ơĩị2 COS 6 + Ị/xT/3a2 cos(0 - 2V>) = - r) ^ ƠĨỊ2 sin 2ip — Rrj2 8Ìn(0 — 2ip) = —cry, ƠĨ)2 cos + Rrj2 cos(0 — 2ĩp) = —(2 A + 7 a2). . /: * ' t ■? 4 Í* • T his solution corresponds to the sim ultaneous vertical and angular motions of the ship. Elim inating the phase variables in these equations we obtain the following expression giving the dependence of the am plitudes of vibration a, R on the frequency rj of the exciting force: * v - h 9ữ) + [2A + \a'~ i/wiV]* - - r + \pR2}2. . , (5,2) This relationship is plotted in F ig .4 for the case p = 1, k = 2, k = c = 0.1 and ịẰ = 0.05. FYom Fig.4 it is seen th a t, increasing rj from the resonant value (r; = 2) and keeping constant the amplitude o f vertical m otion, the amplitude of angular ship vibration (a) decreases, and at a constant value of the excitation frequency (77) the amplitudes of vertical and angular ship vibrations either increase or decrease sim ultaneously. 11 Fig.4 CONCLUSION The nonlinear m odel simulation of coupling between vertical and angular ship m otions has been considered. The nonlinear terms in the m otion equations have essential influence on both the shape of the respon se curve and the instability region in com parison with the linear equations (l, 2Ị. The condition for the appearance of a purely vertical vibration of the ship has b«en derived. In the case considered th is vibration occurs only w ith sm all am plitude and the strong vertical ship vibration is alw ays accom panied by angular m otion. The stationary' sim ultaneous vertical and angular m otions of the ship have been studied too. REFERE NC ES 1. Tondl A ., N abergoj R. M odel Simulation of parametrically excited ship rolling. J. N onlinear Dynam ics 1 (131 - 141), 1990. Kluwer Academ ic Publishers Netherlands. 2. Nabergoj R. Param etric resonance of a spring - pendulum system . Eighth world congress on the theory o f m achine and mechanism s, Prague, Czechoslovakia, A ugust Ỉ991 (299 - 302). 3. B ogoliubov N. N ., M itropolxki Yu. A. Asym ptotic m ethods of the theory of nonlinear vibra tions, M oscow , 1974. Received March 28, 1992 M Ô HÌNH PHI TU Y ẾN MÔ PHỔNG CH U Y Ể N đ ộ n g t à u t h ủ y TYong bào b áo này mô hình tuyến tính của Tondl A. và Nabergoj R. [1, 2] ve sự lắc ngang và chuyển động th in g đứng của tàu thủy được mír rộng cho tnrờng họp phi tuyến, tạo nền do đặc tn m g dàn hồi v à kề đến các so hạng bậc cao trong khai triển các hàm lượng giác. Đ ã xét đến khả nẵng xẩy ra d ồng thòri cộng hưdrng cưỡng bức (doi với chuyền dộn g thẳng dứng) và cộng hưỏrng thông số (đối vó i chuyển dộng lắc ngang) của tầu thủy; cũng như đã nghiền cứu chi riêng chuycn dộng thẳng đứng m ì không có chuyen động lắc ngang của tàu. Chuyên độn g dồng thòri thẳng dửng và líc ngang cda tàu có dặc điềm lằ ò cùng m ột tần số kích dộng các biền độ dao động dứng và dao dộng lắc ngang cùng tăng hoặc cùng giảm. Khi tăng tần sổ kích dộng từ giá trị cộng hưárng và giử cho biỉn độ dao động đứng khồng đổi th ì b iỉn độ dao dộng lắc ngang giảm. 12 . SIM ULA TION O F SH IP MOTION N G U Y EN VAN DAO Institute of Mechanics NCSR Vietnam SUMM ARY. Nonlinear model simulation of coupling between heave - roll and pitch - roll ship mo t i o n s ia. vertical m otion of the T he boundary of the instability region for the appearance of a parametric resonance of angular ship m otion is determ ined from the last two equations of (4.6) a£ c +. angular motions of the ship. Elim inating the phase variables in these equations we obtain the following expression giving the dependence of the am plitudes of vibration a, R on the frequency rj of