Ship’s spacious motion mathematical modelling by PC

ABSTRACT

The matematical model for solving different problees. so as a motion of submerged crafts and surface ships in a sea environment, and according implementation by PC IBM type, are presented

Mation and control process simulation executes in PC by the interactive performance, so that simulation (modelling) may be interrupted by operator for introduction a new data.

MatematicaJ model includes a number of nonlinear

differential equations for the spacious craft motion with various craft mass, depending of time; the craft is influenced by: control forces initiazed by a water ballast tank, control forces due to variation of plane or rudder angles, variation of propeller thrust. external forces due to environment perturbations. Complete Mathematical model contains particular model of processes in ballast tanks during transfer motion of water ballast. umpelling by pumps or by press air. Particular mathematical model uncludes also simulation (by subrutine) of the effects of switching on-off valves, controlling over pumps, press air. etc.

Various pcsitions of the craft represents on colour monitor screen together with positions of controlable

valves, and a water quantity in ballast tanks.

Matematical model and software package are avai labbe for simulation a qreat number of tasks in a different environment, including situation in case of ship's wreck.

Introducing of results of full scale ships test is possible (due to interactive performance) during the ail simulation, so the presented suitable software package is available in engineering, design, science, education process control and monitoring and for operators

informative support.

NOMENCLATURE

Crafts motion equations M - crafts mass, kg M4o - starting mass, kg

1 Leading engineer 2 Leading engineer, Ph.D.

SHIP'S SPACIOUS *3TION MATh)1ATICAL MODLING BY PC

1 2

P.P.Tolkachev S.L.Kariinsky

Central Design Bureau of Marine Engineering 'Rubin" 191126. S-Petersburg. Mai-at str, 90, USSR

Ñthlef

Mulcelweg 2,2628 CD Deift

T

015- ThN,. Fac 015. 781838

n

M = (E Fj)/g - mass variation (due to the variation of

j=l buoancy). kg

Vx.Vy,Vz - components of the craft's velocity vector rn/sec

components of the angular velocity vector. rad/sec

's

-

water density. kg/rn

2 g - gravity constant. a/sec >, - additional water masse.. kg

V - craft.s displacement (hull volume), Vo - starting displacement. m

V = f(«t) - volume variatic'n lue to a compression of the hull, m

Ix.Iy.Iz - craft's moments of gyration. kg*rn Ixo.Iyo,Izc' - starting moments of gyration, kgm

n 2 n 2 n

Ix= (E Fj xj) /g, 1y (Z Fj yj) /g. lz= (E Fj zi /g

.i=1 j=1

yj.

zj -

the distance from the craf's mass to j-th tanks centre. ni

Fxh . Fyh.. Mzh - components of h idrod i nam i c forces. N

Tx,Ty.Th - components rif propeller's thrust. k - rotat ive velocity of the pr«4l 1er

shaft (RPM), rot/sec

Fxp, . .Mzp - components

cf

I:he craft 's buoyancy,kN

Fxe, . .We - components of Lhe external influence. kN

'rj,'Ç,l - crdft's depth, drïftag'. way, s

- roil, pitch, heading, rad - incidence and drift anglos, rad - plane's or rudder's (flaps) angle, rad Pj - anr precsure in j -tò tank, Timt

Pi - external pressure near valve. T/m Vrj - air volume in j -th tank, m's

Vo - tank volume. m's

qPH.KB.Wfl

-

expense coefticient. Q - air expense. T/sec

Hj - water level height in the tank, a Yg - mass centre height. m

hj - the top of tank height. ni

*

xj.xj - distance from the mass centre t.o the valve, ni

V6 - air volume in reeervuars,

- gas

constant

for air, R= 29,3 rn/degree T - &ir temperature in reservuars, T= 278 K P6 - air pressure in reservuars. T/mm Fee. - eqiva lent hole area m

Zo - air compressing coefficient. c/rn k - a number of tanks for j -th group of

reservuars,

di - air tube dameter. mm i - resistance coefficient,

ki - opening coefficient for j-th air valve. Du - valve diameter. mm

fsn - spigat hole area, a

(i -

air expense from i -th resrvuar group, T/sec

Fsj - equalent hole area for tube to j -th tank, mm

Fsij - equalent hole area for tubes from j -th reservuar 's group, mm

1. CIAfl'S SPACIOUS MOTION EQUATIONS

It was proposed for the equations, describing the crafts motion, the following (add to a usual form):

various crafts mass and moments of gyration, depending upon a transfer motion of the water

be-(last trough the ballast tanks are considered; various craft's buoyancy, due to the water density depth variation (internal waves also).

hull compression, surface waves. etc. are con-sidered;

it is no limitations regarding to incidence or drift angles are. aiment;

the complete metematical model includes also the particular model, describing the procceses into ballast tanks during the transfer motion of a water ballast impelling by pumps or by a press air.

These assumtions allows executing calculation and time modelling for a great number of tasks. including strenuous process, so as an emergency surfacing, the propeller thrust loes, circulation at a high craft's speed, etc.

The craft's spacious motion equations under the noted assumptions may be written (by connected with a craft's hull coordinate system OXYZ, FiglI as:

1. (M + >,11) Vx + M Zg Wy - M Yg Wz - (M + >,22( Vy Wz +

2 2

2 2

(>,35 - M Xg) Wx Wy M Zg Wy Wz - M 1g (Wz + Mx) =

= (ç V - Mo)

g

Con 6 C0s4)+ Ty +

Fyr + Fye + Fyss

3. (M + >,33) Vz + (>,34 + M 1g) Wx + (>,35 - M Xg) Wy j (M + >41) _IVxI Wy + (M + >,22) Vy Mx + (>,26 + M Xg) Wx Wy 2 2

+MZg Wy Wz- m(Wx+Wx) =- (V-Mol g SineCoep

+ Tz + Fzr + Fze + FZBH (1)

(Jx +

)44) Mx - M Zg Vy + (>,34 + M Yg)Vz -(>,26 + >,35) Vz Wz -

(Jy +

>,55 -

Jz +

>,66) Wy Wz --)34 Vy Wx + M 1g (IVxI

Wz - Vy Mx) -

M Zg

_(IVx

Wz -- Vz Mx) + (>,26 + >,35) Vy Wy = Mo g (1g Sin G +

+ Zg Cas4)) Cas₄+ Mxr + Mxe + Mxss

(Jy + >,55) Wy + M

Zg Vx + (>,35 - M Xg)

1z

-(>,26 + M Xg) Vy Wx - (>,35 - M Xg) IVXI Wy - >,34 Vx Mx +

+ >,35 Vz Wz - (Jz +

>,66 -

Jx -

>,44) Mx Wz

-M Zg (Vy Wz - Vz Wy) = -Mo g (Xg Sine Cosq.i + Zg Sin*) +

+ Myr + Mye + Myse

(Jz +

>,66) Wz - M 1g Vx + (>,26 + M Xg) Vy

-- (>,26 + M Xg) IVxI Wz -- (M Xg -- >,35) Vz

Mx

-- (Jx +

>,44 -

Jy - >,55) Wx Wy - M

1g (Vz My - Vy WZ) =

=Mog(XgCasGCos4) -YgSin'f( +Mzr+Mze+MzSH

+ Vz (Con G Cos - Sin & Siri

Sin)

Mx -

(Wy Cas G - Wz Sin 8) SinY/Cos'j

t (Wy Cas 6 - Wz Sin 8) l/Con'' j.)= Wy Sin C + Wz

Con 8

+ (>,34 + M Yg) Wx Wy -(>,26 + M Xg) Wz - (M Xg - >,35) Wy + It must be added to these equations (1) a number of formulations (2) for the correspondence upon

coordi-nate systems Oand Oxyz:

+ M Zg Mx Wz = (V - Mo) g Sin$l+ Tx Fxr + Fxe + FXBH

Vx Cos'Cosjd+ Vy (Sin 6 Sine Cas 6 Coe'Sin)) + + Vz (Con G Sing + Sin G CoeSin')

2. (M + >,22) Vy - M Zg Mx + 26 + M Xg) Wz +

Vx Sin'+ (Vy Con 9- Vz Sin G) os4' (2) 2

nos sp 2 2/3 (3)

Mir = [

Ma + Ma + Mi

(ôj)]/2 y

V

2 PARTIAL MA4ATICAL MODELS 2.1.Plane's (rudder's) control

Matematical model of the plane's (rudder's) control system includes description of a movement for a num-ber (up to eight) of independed splat flaps. These flapes say de also used in pairs. The all flap's servomes are Identical; theirs cinematic hôracteris-tics are shown in fig.2, where -a control signal for n-th flap (in degrees), -flap's angular velocity. (degree/eec).

Either a remote control or an automatic control may be implemented un the following rules:

for remote control

= f (i),

I'I

imax (4)

where -target control signal, n - the number of the flap

for automatic control (as pitching control, for example):

'3,:,

K**t'44'L

,

W-4'

(5)

where ₉ -target pitch.

It as possible, however, the complex implementation. i.e. the depth and the pitch simultaneously, for example:

' (6)

The forms (4).(5).(6) allows to introduce the flap angles as the array = f(t). and so this matematical model may be used for an analisas of a ship trials, also.

2.2.Propeller's control

The variation of the propeller's thrust may be is

expressed as:

_I

2

Tx=An +BnVx

where A.B -coefficients determined as function of propu-isional propeller's characteristics.

Determination the function n=f(t) by an array allows to implement in subruitin characteristics of any engine

(Diesel, turbine, electric motor, etc'.) based on the results of the ship's trials.

2.3.Buoyancy control

Matematica) model of buoyancy control conoists of two parts: 1) formulations of forces and moments in the right parts of equations (1), and 2) equations descri-bing the dinamic processes in ballast tanks.

These equations are intended to determine various values of the air and water volumes ici the tanks during a water transfer motion in-or-out the tanks.

The ballast tank scheme is depicted in fig.3. where are also the different valves, described by the partial mathematical model.

The following processes occurs in the ballast tank: - transfer water motion trough the kingston or the

spigat without a ventilation, connecting with air compression or decompression in the tank.

- the tank's filling trough the kingston or the spi-gat with a ventilation,

85-3

valve are closed. A few logical conditions may be introduced in the model, so as the blocking c'f any valve opening without opening the definite other valves, for example.

All processes, mentioned abave are described under foiiowing equations:

1) for the air weight variation in j -tb tank: d/dt (Vrj''rj)

=!

.j.1 qx f2

JO. (VrjVocr) or (Pj Psa6j)

:1511 f2 n

_f

O. Pj

-E

j=i

[qB

f2 where n

-E

j=i dF/dt = - (R T )/ V6

c, y<yicp

*

[(y - yxp)/(l -

yxp)] },

y)

(12)

yPy/P

where

Py = E Pj /

n

i=i

s

n j O. (Vrj< Voc'r) or (Pj < Psa6j) j=J. qr.H fi - kingston (8) n [ O, (Vrjc Voc'i) ox (Pj PaGj)

-E

i=i qsn fi - opigat n D. Pj

Psa6j

-

vent.velve

-E

fi + q; Kfl j

-

air voVC 1/2 fi [ 2g

'eri

(Pj - PsaGj)j (10) 1/2 f2 = ( 2/

Pj -

Psaf I Sign (Pj

-ç g ['- +1O+Yg Cosq'- X Sins)

-Hj Cc's

_J

* .) * ₍₁₁₎

= [-1'+lO+Yg Costi'- X Sin -hj Cos1) j

3) air pressure variation in press air reservuare: 2) for the air volume variation an -th tank,

d/dt (Vr,) = ri

f

û. (VrjVoci',

or (Pj kingston (9) spagat v-rit valve

1/2

t=(3.38F3oP)/(RTZo)

(13)

2

21/2

F3o=(F3e+F35)/[(F3e)+(F6)

where

k n

F3e=E

E

F3ij

(14)

1=1 j=1

2

-6

1/2

F3j = ( k'iídj 10

) / { 4 [

2

( 0.55j + 1 )]

4) valve expense coefficient determination:

-3 2

qnLlcs sTr C OrlO

)

/4

- kingston (vent.valve)

qsri = fia

,fn

- spigat

₍₁₅₎

n

qsrl = L (

G6j F3j

)

/F3ej

- air valve

.i=1

2.4. Environment

Environmental conditions, such as a water temperature

and a density, and also a depth(trough the variation

of the hull compression) influences upon the

crafts buoyancy and the moment. under the following

formulations:

-(16)

These external forces are the functions of s depth and,

so that the dependences

_{? (1?). V()}

shownedin

fig.4.5 as a typical, used in the following example.

3. I).LDTATI0N BY PC AND AN EMPLOIMENT OF THE METHOD

3.1. Implementation

Matematica) mcxiel and the software package have been

adapted to PC IBM type. Control process modelling

executes in PC by interactive performance, so that

ope-rator is enable to carry out (simultaniously):

plane (rudders) control, propeller's control and

bu-oyancy control. Various ship positions during her

maneuring represents at colour monitor screen in

vertikal or horizontal plane (by turns). In the same

time the picture shows the changeable information in

tables. including following parameters (table):

Table

Plane's and rudders angles are represented at dials

(fig.6); the running information regarding to the crafts

buoyancy, positions of control valves and the water

qu-antity in every ballast tank occupies the right part

of the screen by the operator's desire. Hear is. also.

the information about buoyancy siimmmry value

and the moment; in the centre of the screen places

the moving graph, according to craft's movement in time

in one of two planes. The most important parameters

present at the screen constantly and others may be calls

to the screen by operator's desire.

Service si.thruatines allows to depict a time graphs

for up to 168 parameters) after modelling

and the time record about operator's or automatic

cont-rolling manipulations, also. Files, containing the

gra-phics information and the time record. may

be

viewed.

saved in PC memory and

be

plotted by a printer or a

graph plotter. The monitor screen picture is depicted

in fig.6 (vertical plane) and fig.7 (horizontal plane).

3.2.THE EXAMPLE

Computers simulation results for the emergency subs

surfacing is considered

as

a representative example.

It is supposed. that the emergency situation occurs due

to a damage of the pressure hull; that can be due to

spoiling of the board valve.

At the moment of a damage the sub's

depth was

200 s

and the speed was 3.05 m/sec (6 knots). For the

emer-gency sub's surfacing used,as usual, following actions:

(1)forcing the water ballast out of tanks by a press air.

(2)the sub's speed increasing. (3)mounting the planes to

lifting angles. These actions may

be

carried out by

ope-rator on purpose of his educating or research (by his

desire). The graphs ot the most important parameters of

of the mentioned manoeuvre and the protokol of the

operator's actions are depicted at fig.8.9,1D.11.

3.3. THE E1PLOYMENT OF THE METHOD

The complete description of the f orces.( by which the

craft is influenced) and of the according technical

means allows the employment of this matematical model

for solving of a great nurnh.r tasks

as

following:

Researh

- determination of the nomenclature and parameters of

technical means for the buoyancy forces;

- rudders and buoyancy control system alhortms

deve-lopment;

- information sistem requirments development.

Ship's test data examination and parameter's

identification

- matematica) model identification,

- control system alhoritms check,

- information means examination.

- external forces identification.

Dinamic modelling

- craft's controlling check and education, including

emergency situations.

- action's solution education and examination.

Naine

Depth Pitch

Heading

Roll

Speed

Vert.

veloc.

Mark . W Vx

Measure

s

degree degree degree m/sec

rn/sec

Naine

Angul.

veloc.

Drift

veloc.

Heading

veloc.

Thrust Time

Mark Vz (t) Tx T'

Fig.2

s

syst

Fig.3

1020

1025

1030

-2

-1

Pre

air

rese'uar

Air valve

p

kgiui3

Fig.4

Flg.5

Water

Xinston

Siat

Air valve

Ai r loo

200

300

v,

loo

200

300

7).m

0 o 33

1J

13

1J

210 240 270

1)39Ii1fl3

V=8i

P=173.4

a N.Ø.T. 2

V: 33.0

P1%.i

aLB.T. 3

V: 8.1

P= 1.1

4. Ca'tparthent

V=5000

P=1ft10

Fig.6

Fy

1000

Mz= 133JX P P O

V

' ú)

Vz

Ó)

Ix

Time

Buoyancy [Pj

1.

N.B. T.

i

V 8.1

P= 173.4

ZN.9.T.2

V=33.OP:1%.1

V=8.1 P; 190.1

4CompartAent

Fig.7

Fy 00000 MZr 0.0000 300.0 -1.00 30.00 S000 4.972 -0.00 -0.00 OE000 OE000 1.402 120

1)

210 240 270 300

0104006007008009001000

-Rudder L R I

o

-3020100102030

CommAnd.

Fi -Help

P-R - Controt

N-Z-pRornet2r

y- optrtng p1ne

Esc

directored

[NI

o'i"z

Corniwuid:

Fi -Help

P-R Control

N-Z-pmetr

V- Operiutin

piule

d ¿T ¿

I Ifll

I [f) L L S S

'I

'

01

0

j

öl o

rj

i

0.0

50.0

100.9

250.0

Fig. 9

200.0

'J

o

o

250.0

hue,

eo

00

50.0

100.0

250.0

200.0

250.0

Fig. 8

Time,

ec

i 0- i

O

Tr TD

S

o,

L

C.

000

_{,- s O} fl O

(l

01

0

O si O

-'I

4

I')_

iI

I

valve's switching protocol

Fig. il 85-9 Time iL VALVE CONTROL

Tank's name Valve's name

Control signal

5.

Nater pouring in the compartment

p

16.

TANK 2

KINGSTON OPEN

19.

TANK 2

AIR VALVE

OPEN

26.

TAtI( i

KINGSTON OPEN

20.

TANK i

AIR VALVE

OPEN

36.

TANK i

AIR VALVE

CLOSE

98.

TANK 3

AIR VALVE

CLOSE

50.0 100.0 150.0 200.0 250.0

Fig. 10 Time, sec

I L.

o

o

.0'' C e E 4' L.

I

t

£ o

o

o

o

o

o