Computer aided simulation for non-linear model of ship motion

The ship motion is simulated on computer by non-linear model of small vibration and strong non-linear model. The results are compared between two models. The influences of different parameters to the ship motion are considered.

Journal of Mechanics, NCNST of Vietnam T XVI, 1994, No 2 (33 - 37)

COMPUTER AIDED SIMULATION FOR

NON-LINEAR MODEL OF SHIP MOTION

NGUYEN NHAT LE

SUMMARY The ship motion is simulated on computer by non-linear model of small vibration and strong non-linear model The results are compared between two models The influences of different parameters to the ship motion are considered

The non-linear model simulation of ship motion has been studied by Nguyen Van Dao [1] proposed model consists of a mass M restrained by a non-linear elastic spring and a mass m

,ched to a hinged, weightless rod {Fig 1) This sfstem has the vertical and angular oscillations

§2 SIMULATION FOR NON-LINEAR MODEL

OF SMALL VIBRATION The differential equations of motion for the system represented in Fig 1 are writen as follows

(M + m)(Z + u) + k 0 Z + fJ 0 Z 3 + h 0Z + ml(9sin 'P +<{?cos 'P) = o

m£29 + Co<fo + ml(g + Z + ii) sin<p = 0 ( 2.1)

~e Z = x - u is the relative vertical displacement of the mass M; x is the vertical displacement

te mass M from its static position of equilibrium; u = q cos wt is the vertical displacement of base of the system; rp is the angular displacement of the pendulum; k 0 and {3 0 are the linear non-linear characteristics of the spring, respectively; ho and Co are the damping coefficients .e vertical and angular motions, respectively

Supposing that the damping forces and the ratios"= qjl; p = mj(M + m) are small, the rential equations of small vibrations are written as follows [1]:

e

W" + k 2 W = -«[-ary 2 cosryr + hW' + fJW 3 + P.('P'P" + <p'2)) + e2

3

<p 11 + <p = e[-Ctp 1

+ ~ - <pW 11 + at] 2

rpcosryr] + e2

••

(2.2)

z

W=-· t , ' J = -w j

Wo

h = ho

Wo =If;

m

t

(3=w5(M+m)

Computer aided simulation for differential equations of small vibrations {2.2) with the different data in ]1] with:

Jl = 0.05; k = 2; h = C = 0.10; "= 4.5 · w-2

; (3 = 0 and (3 = 1

we obtain the behaviours of so(t) and W(t) as shown in Fig 2 Where so(r): damped vibration,

W ( r): periodic vibration affter a timẹ

The change of coefficient (3 ((3 = 0 or (3 = 1) and the change of coefficient ry have a very small

influence to the motions

m

-Fig 1

, I

'

I

IIIIIIÃ~~~~~~Ặ·~-~~~~~~ vlf v

v

f : 'fJ(C}

2 W(r}

I

'

fig 2

§3 SIMULATION FOR STRONG NON-LINEAR MODEL

For general strong non~linear model of ship motion, the differential equations of motion has

the form {2.1) This form has been studied by means of the asymptotic method of non-linear

mechanics [1) It is can bẹsimulated on computer as shown in Fig 3:

With this computer aided simulation ~e can observe all motions correspondịng to the different real parameters-

1 Influence of the non-linear damping coefficients:

Making change the non-linear damping coefficients, we see that the term f3o in the non-linear characteristic of the spring has a very small influence to the vertical motiõ, while the damping

term Co of the angular motion has a considerable influence to this motion

2 Influence of the length of the rod:

Taking the ratios

Jl = (M: m) = 0.05; u = ~ = 0.045; ry= w 1.8

wo

the change of q and l has a influence to the behaviours of motions

- _with l = 2.2, q = 0.1: The response curves of displacements Z and rp are plated in Fig 4

Fig 9

.t if! It) e It)

0' 0000 O.Jaooooo a ~oaaoco

1) 1000006 a Jtl&oooo • .1oaoaoa

0 2000000 0 2931420 0 0952.1:113

0 JOOOOOG 0.25fi51i'OO 0 085i727

0.4ooaooo ~-2J8621C o 06644Ji

o 5000000 ~-nssago 0 OJ6HSS'

0.6000000 a lZ1725D ·0 00662li!;

q roooooo () 0650743 ·0 0:1<!Z95J

(), 8000000 a aosaBeo ·0 H568S.O

11

0-9GOOO(f(} -o 05i€1155 -0 1665080

t.oooa -' -lo7J970 -0 200<11/D ;,

I 20 00 -o IB93S'Iio - o 4834020

j 2

; .300 0 -0 2~ U860 · 0 HJ50fa

1.4000 -O.t20Jf50 -o 0347029

1 6a a~ ·0 .20593'f.O 0 175!t800 2 2

1- ?000 -0 1.;~•~~o 0 2.61'18~0

~

1-8000 -·0 (67J940 o 3101000

-1' ~0{] 0 _, ~"4 It 50

' Ji!:lH~D 2

2' 10 ~0 - 0 ~'11 G46t o 154846'0

'

2-lOOO -~- 06-17130 0.62~l57+

2 JO Ofl -O.a2.9-1Bsl -D- -12!55060 rp ( t)

2 -4JOO {] 00-499<98 -o 25rs7'0a - ~'

2 sooo o.a::~aurs -0.352.12'70

2 600 0 0.06!12~21 -O.J>~a5460

Fig 4

Where, the angular displacement IP is damped, while the vertical displacement Z has a beat phenomenon

Thus, corresponding with the observation in [1], for the cas.e considered this is only the purely vertical motion

- with l = 1; q = 0.'045: the shape of the response curves and the coupling between <p and Z

are shown in Fig 5 Where, both the behaviours p(t) and Z(t) have the beat phenomenon

2

1 ' <p{t)

2 l(f)

Fig 5

3 Coupling between vertical and angular motions:

The coefficient '1 = .w fwo has a essential influence on the coupling between vertical and angular motions However the coupling b.etween two displacements ~ and Z is small with

1.65 :5 ry < 2 where p(t) is damping oscillation, Z(t) has the beat phenomenon

Its behaviours are analogous with those in Fig 4

The coupling strongly occurs in the resonant regions, i.e

2 = ry :5 2.12

The coupling at the resonance (ry = 2) is ploted in Fig 6

/ 'P{I)

-2 i!(t)

0 0000

iiJWVWfM!iMJ,

Fig 6

50.0000

With YJ 2 2.5, the shape of the response curves is analogous with that in Fig 5

4 Others influence of 'the parameters:

Making change others parameters of the different blocs in the flow diagram on Fig 3 we can see on computer screen the direct displays of different motions considered

Basing on the study in [1] and the computer aided simulation we can observe others phe-nomenons occured by t.he strong non-linearity of the system (2.1) Also, we can select the conve-nient parameters to occur the desirous motions

CONCLUSION

The ship motion is simulated on computer by two models of small vibration and of strong non-linearity

The influences of different parameters are considered and corhpared The experimental results are in accordance with theoretical ones [1] With this computer aided simulation we can select the reel parameters to find the convenient motions

This publication is completed with financial support from the National Basic Research Pro-gram· in Nat ural Sciences

REFERENCES

1 Nguyen Van Dao Non-linear model simulation of ship motion Journal of Mechanics, T XIV,

No 2, 1992

Received May 3 1 1993

MO PRONG TREN MAY VI TiNH MO HINH PHI TUYEN

CUACHUYENDQNGTAUTHUY

MO hlnh phi tuyen yeu cho dao d9ng lktc ngang_ va th~ng dU.ng ella tau thdy di duyc nghien cU:u b~ng plnrang phap ti~m c~n ella dao d9ng phi tuyen [1]

Trang bai nay, mO hlnh phi tuye'n tren di drrq'c mO phdng tr&n may tinh, v&i hai mO hlnh: dao di?ng b€ va phi tuye'n m<_tnh Da xet inh hrr&ng cUa d.c thOng sO den dang di~u cUa chuy~n di?ng V&i sv mO ph6ng nay, chUng ta c6 th~ d~ dang thay d&i c&c tham sO va quan sat trlfc tiep tren man hinh c<i.c d<;tng chuy~n d9ng dOng th&i th&ng dU·ng va l£c ngang c-da tau

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