A Mathieu equation for ships rolling among waves I-II

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DET KONGELIGE NORSKE VLDENSKABERS SELSKAB FORHANDLINGER BD XXII, NR 25

For a trochoidal wave the slope is tan

i =

2771 h2 . 4711

h sin

_{T -}

Sin

T,

+

b =

the maximum slope.

ination p = tan ip = sin

no difficulty in of equation (1) will then be (2) F Sill 532.593

A Mathieu Equation for Ships Rolling

Among Waves I

By

GEORG VEDELER

(Framlagt i Fellesmøte 6. oktober 1949)

For describing the rolling of ships among waves naval archi-tecEs have up to this date been using the equation

(1) d2 p± k2 (p - ç") = O

where is the angle of roll, p the wave slope, k = ,

T, =

27t _{being tI'e natural period of the ship.}

2nt T 2711 R H- r cos

'w

where T,,, is the wave period and Throughout this Ilote the

approxi-271f .

-b sin _T,, viIl he use(l, there being adding additional terms if wanted. The solution with initial condihions

=

O when t = O

b . 2n1 T,

(T, 2

(sn__

f

shy-This solution was given by \ViLLiAI FROUDE in 1861. [1J (see

NoLe Il).

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114 D. K. N. V. S. Forhajidlinger Bd XXII, Nr 25

For comparison wiLh the results of hie following it may he

convenient to substitute the lime t by a new variable s =

and wrìle equation (1)

d2

( _dX2ma =a2b sin 2x

with solution corresponding Lo (2) ab

q=24(a sin 2x-2 sin ox).

2T,,

Here a=

_T5

Froude's solution was based ou several assumptions, which I have discussed ou a I)res'ious occasion [2] . Model experiments

by professor SUYE-ulno [3] and others seem to justify the assump. tion that the centre of gravity of the ship moves in a circular

path and that the resultant of weight and centrifugal

force

must act along the normal to the wave surface, or to Ihe trocli-oid through the cenLre of gravity. Froude, however, assumed that this resultant force had a constant value equal to

the weight W of the

ship, while in reality it must equal

W . b2 - 2b cos , which is not constant. For a

troch-oid of height one-twentieth of the length b = = 0.157, which

means that in this case the resultant force varies between 0.843

W and 1.151W.

On the assumption that the forces act as shown in Fig. i

the equation of motion can be written

Here GM' sin p = GM sin

- BM sin

= GM. BM.

¿27't . ¡2x1

approximately, and GX = BM sin

-- ;-

GM sin

¡2n1 . 2i/ b .

4t

where approximately sun = sin sin

T and sin I2it i . 2i1

-

2711

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Geoig Vedeler: A Malhieu Equation for Ships Rolling Among Waves 1 115

The equation of

mo-tion (5) can then he

written d2çp + (6) dt2 + k2(i + b cos =k'2b (sie f!+ lv i 4'ii

+bnts1n-BM where ¡n should be a constant for

small angles of roll

when GM is not very small. The assumption that the mela-centric height GM is not very small applies also Lo the expres.

sion used for T.

Substituting x = , this equation cati he written

H- 02 (1 + b cos 2x1 ,c = a2 b 2x + b ni sin 4x)

The equation is of the Mathieu type, the usual attempts at

solving of which actually give ito information of value to the physical interpretation.

To get at least some information about the influence of the second term on the right hand side it is natural to start with the approximation

+ar =

(12b(Sifl _2x+ bin sin 4x)

With the same initial conditions as used before the solution is now

rasin2x-2sinax

1

asin4x-4sinax

p=abL _{02_4}

+-bm

_a2-16

_i.

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116 D. K. N. i'. S. Forhandlinger Bd XX!!, Nr 25

will now be made Io

solve the Mathieu equation (7). As a first approximation the function a2 (1 + b cos 2x) will he sub-stituted by _(c1s1) , where the

constants A and c can be

chosen so as to

fit the original function in the best possible

way. The homogeneous part of equation (7) will then read

(f2 _A2

(10)

The new function is given a period of ir by shifting the

ori-gin to new positions at intervals of . Starting with zero, x

is increased regularly to -J- , then the origin is shifted to ir so

that x jumps to -

, increases regularly through zero to +

when the origin is again shifted, now to 2ir, and x again

jumps to -

etc. If one wants to have the same values as

the original function at x = O and x = *

-, A2 = 184 000 a2

and c=20 for b=0.15.

Equation (10) has the general solution

(Il)

(c+ x) C ( cos str sin s) + D sin S

Ax

where S_{= c(c-- Jx)} and C and D are constants of integration. The solution has been given in this form so as lo make p as

d . . dx

well as - continuous at x_dx 0. Rememberiii that _dx

± i

positive when s is positive and negative when s is negative, i.e. discontinuous at s 0, it will be seen that the upper sign

in front of the term sin S must be used when s is positive, the lower sign when x is negative. In our case this extra term is a refinement of small practical value because will be very small. Substituting the numerical values of A and c given above

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Georg Vede/cr: A Mathieu Equationfor ShipsRolling Among Waves 1117

± 1.57 only. In Fig. 2 the function a2 (1 + b cos 2x) has been plotted by a full line and its substitution

_(c±x)4

by a dotted line.

o &i*0C*2,) t P0SIflV r SFsAt!Ve r B Q+ -bmsin2Q) li i.r..!i

tN

r Po,vr,vE i ZNEG&Tv* T

CUVWS OPAWN POR CASt

-- 94000u b .0-IS u . 20-0 g, .0055. 1, t. r P05IriVE ) r

To be able to find an approximate particular solution

of the non-homogeneous differential equation the right hand

side of equation

(8) may be substituted

by the function

, where B and Q can he chosen so as to (c + x

approximate the original function as well as possible. I have

found it practical to choose B = a2b

(± )

= 1347 a2 and

qx

2 21.57

+ x

with q = 43 .

14. The function b

sin 2x has been plotted by a

full

line and its

substitution

B sin Q

a2 (c ± )5 by a dotted line in the upper part of Fig. 3.

Simi-larly the function b sin 4x has been shown by a full line and

B sin 2Q

its substitution I by a dotted line in the lower part of Fig. 3.

The equation of motion will now be

(12)

Q±bm sin 2Q)

(6)

118 D. K. N. V. S. Forhandliiìger Bd XXII Nr 25 PO i*;i;ii.' (:fr) t E..

=(c+Ix()[C(cosSisinS)-i-DsinS+

j sinQ i sin2Q

BA.q2c2+bmAt_4q2ci)

Adjusting the constants of integration and substituting numeri-cal values we get

2145

a sin i4x 21.57

) 4 02 sin (ax.20.

+bm

20+1x1

a5 I 6 184

The slight diffèrence in the positions of synchronism shown by the numerators in (14) compared with (9) is due to the choice of Q, the number 21 . 57 appearing in the denominator compa-red with 21 . 45 in S . It would have made no perceptible diffe-rence in the curve of substitution in Fig. 3 if 21 . 45 had been chosen also for Q

Trykt lOde februar 1950

I kommsjon hos F. Bruns Bokh15ndel .&ktietrykkeriet i Trundlijem B l347a. 0.15 O - 20 o. -0-07i. B - 347E OIE. 20.0. -0.10. (14) 20+1x1 r i _21.57 12x. J 2 01sin(ax 21 . 45 t 20-i-ix!, Fa sin 20+!x!.J±

q=ab.

_20.49

_a2-4046

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DET KONGELIGE NORSKE VIDENSEABERS SELSKAB

FORHANDLINGER BD XXII, NR 26

532.593

A Mathieu Equation for Ships Rolling

Among Waves II

By

GEORG VEDELER

(Framlagt i Fellesmøte 6. oktober 1949)

The equation of motion (7) of Mathieu type given in note I

of this paper was substituted by an approximate expression equation (12), of which the general solution is as given by

equation (13). Finally equation (14') of noie I gave the solu

tion with the constants of integration adjusted to the initial d';

conditions

=

O for

I = 0.

Equation (14) is valid only for the interval x = O to x

qcBI

i

with constants of integration C0 = O. D0

= -

_{A ¶A2}

-q2c2

+

bm 2OIab/

i

bm

+

_{B2 _4q2c2!}

-

_20.49

02 4046

+

a2 -16.1841 When pas. sing x =-the origin has to be changed to ir, so that x jumps

to -

and new constants of integration must be chosen so as to make and continuous. If the new constants are called C1 and D1 , they will have the values

(15) C1

=

kC0 + mD( and D1 = /C + kD0

¡ 7

dc +

where k = i + sin Si cos S - 2 Ii + _A2 I sin2S1

7t

C + -m=2 sin S1 cos S1 -

ASi

_S1

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120 D. K. N. V. S. Forhandlinqer Bd XXII, Nr 26

/=_2[l+fl sinS1

cossi+2c[+

)j

s,,

S1 being the value of S for x

=

A similar substitution must be made every time the origin

is shifted. We have generally for the n'th shifting of origin

C=AC_1-i-mD,_1 and D=1C_1+kD_1

or written in matrix form

(1g) _{( C,,\} k m\n (C

I k J

The approximation used to solve equation (7) may seem rather

rough. It therefore seems worth while to try the method of successive approximations suggested by BRILLOUIN [1]. The

equa-tion of moequa-tion may be written

d2 A2

dx2 (1 ±Ejsin 2Q ±E2sinQ)q=

B

where the upper sign applies for positive x, the lower sign for negative x and the small quantities , , can be

cho-sen so as to give the best possible approximation to the

func-91.

1 .

lions a2(1 +b cos 2x) and absin 2x+-bm sin 4x),

respec-tively. In the case dealt with here is is possible to get a very good approximation by choosing

=

0.033,

=

0.013,

=

0.071, F4=O.lO.

If the solution of the

first approximation equation (11) is

called qì,, a second and better approximation may he obtained by solving the equation

d2 A2 A2

f18)

dx2 + + 2sinQ)0 +

B

[.

11 .

..1

+

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Georg Vede/cr: A Math/eu Equation for Ships Rolling Among Waves II 121

The improved solL1tion will then be

q =(c+ x)C[cosSTsinS±

+ ,

A2 qc sin 2Q s ksins)

-

A 2Q (sins ± cos s)

-_

q2c2A2

qcsinQ(cosS _--sin

s)

2AcosQ (sins ± coss)]

1+

±

_qc

_-

_qc2-4A2 +D[sins±1A2 qcsin'2QsinS+Acos2QcosS 4qc q2c2A2 csinQsinS+2AcosQcosS] qe qc2 - 4A2 + B [sin Q + ( brnT8) sin 2Q - 1 sin 3Q i A2q2c2 = A4 4q2c2 ± - bme4A2 _9q2c2j To illustrate the difference between the solutions, curves for

the angle of roll q have been drawn in Fig. 4 for two

exam-p es, the first (tipper diagram) for a = 1.5 and rn = 16.0, the

I/

\\ /

'y,

_-_ .4(X - 2)

+ ab(-o :.r'. -À - O-II. . 20-O- -A'- I4oOOj' B -I547'

Ao

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12 D. K. N. V. S. Forhandlinger Bd XXII. Nr 26

second (lower diagram) for a = 3.0 and m = 4.0. The dash. and-dot lines give the Froude solution equation (4) where no m enters, the full lines the solution according to equation (9)

and the dotted lines the solution according to equations (13)

and (14). In all cases b = 0.15 has been used.

It will he seen that the two last mentioned solutions, equa-tions (9) and (13), give larger maximum angles of roll than

the Froude solution when a = 1.5, but less maximum angles when a = 3.0. The difference between the maximum

amplitu-des is proportional to ni . When a = 1.5 the ship period is larger than the wave period, when a = 3.0 it is less.

It is remarkable how close the two solutions equations (9) and (13) follow each other, although they are based on very

different equations, the difference in the factor in front of the second term of the differential equations being well illustrated

in Fig. 2. When so different coefficients make such a small

difference in the result, the refinement in the solution equation

(17) can be expected to make only a very small alteration Io

the curves drawn.

The solutions equations (4) and (9) are seen to repeat them-selves after cerlain intervals, for a = 1.5 every 42-r, for a = 3.0

every 22-r

of x.

For such a repeat to occur for the solution

equation (3) the constants of integration must also repeat

themselves. It can be seen from equation (16) that this will

happen allei- n intervals of 21 ii

(20)

(k rnn(1 O

_\lk!

0 i

If for simplicity we consider oniy the most important terms,

we can write approximately k= cos 2S1 and in = sin 2S1 = - 1. 1f we also use the approximation S1 = a , the condition of

repeat equation (20) will be fulfilled by n = 2 if a is an integer,

2 4 8 10 14

by n=3 if

...,

b n=4 if a=0.5, 1..5,

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Georg Vedeler: A Mat hieu Equation for Ships Rolling Among Waves II 123

No friction being taken into account in the equations of

mo-tion il has been considered outside the scope of this note to

discuss the cases of synchronism.

REFERENCES:

FROUDE 1LLIAM. ,,The Rolling of Ships", Trans. Inst. Naval

Archi-tects 1861, p. 180 if.

VEI)ELER GEORG'. ,,Notes on the Rolling of Ships", Trans. Inst. Nay.

Arch. 1925, p. i66 if.

SUYEHIRO K. :,,The Drift of Ships Caused by Rolling Among Waves",

Trans. Inst. Nay. Arch. 1924, p. 60 if.

BRILL0uIN L.: ,,A Practical Method for Solving Hill's Equation".

Quarterly Appl. Mathematics, .Julv 1948, p. 167 if.

Trykt lOde februar 19S0

I koinrnisjon ho F. Bruits &khande1