A preliminary analysis of nonlinear coupled pitching and heaving motions in regular waves

A. Preliminary Analysis of Nonlinear

Coupled Pitdiing

and Heaving Motions in Regular

The strip method is employed to approximate the motions of a long ship in a regular sea. The amplitude of the motion and the slope of wave components are assumed to be only moderately small, and bow emergence

is allowed. Basically, only elementary hydrodynamic theory is employed, including some approximate ref

me-ments. Graphical or numerical integration of the resultant dynamic equations' in simultaneous phase planes

is described.

Introduction

In this atomic age, power. resources are practically

un-Jted, and thus both the size of ships and their cruising ds will tend to pass beyond the extrapolable range of past experience, so that the performance of a ship in a seaway has

become a problem of increased concern and researd in re-cent years. For example, the amplitude of the ship motion determines whether there will be a dry deck, the maximum

acceleration affects the comfort of the crew or passengers, and

the hydrodynamic slamming forces require a ship hull of

sufficient strength.

-Approximate theories for prediction of slamming forces

and pressures of a two-dimensional object entering water have

been reviewed in (1), (2), and (3). An experimental study

of pure heaving oscillations was given by Golovato (4); it was

found that the nonlinear effect on magnification factor, in

some cases, can be as muth as fifty percent. The simple

theory, using the added apparent mass coefficient for a

re-flected body (5), is shown to be a reasonable approximation at

high frequencies. The additional damping due to generation of surface waves for an oscillatory body was investigaredby

Ursell (6) and others [c. f. (7), (8)1. Pure pitdiing motion with nonlinear hydrodynamic forces was analyzed theoretically by

T-.td (9), but the contribution of surface waves in. a regular

was, however, not included.

The linearized theory of coupled heaving and pitdsing ship

motion is given in (7) and (8), taking into account also the effect of surface waves generated by oscillations. Some ex-perimental information on pitdiing and heaving motion in a

single wave is given by (10) in nondimensionalized plots. The

effect of surface wave on wetted width, and thus on the apparent mass forces for pitdiing and heaving motions, are in. -cluded in (11) by a graphical method; however, no analytical

expressions were given to show its application to larger

ampli-tude oscillations. In (12), nonlinear coupling effects in pitth-roll, pitth-heave and roll-heave are investigated. For

longi-tudinally' symmetric ships, regions of frequency ratio against

nonlinear coupling coefficient, for whith the coupled motion

is. unstable, were found.

It is the purpose of this paper to include nonlinear damp-ing terms as well as nonlinear coupled terms in the analysis. Methods of solving the system of dynamic equations and

in-vestigating slamming conditions are discussed.

"-3) The rsiuZta presented in this 'paper were obtained during

support of David Taylor Model Basin, Department of the Navy.

' ashington 7, D. C.. under Contract Nonr-2?39(OO) of the Fursda-mental Hydromechantce it cf-Pi-o gram.

SenIor kesearcft Engineer, Department of Mechanical

Scien-ces.

Numbers in parentheses indicate references listed for this

section in the Bibliography at the end of this dissertation.

,'1

Wen-Hwa Chu2)

Southwest Researdi Institute, San Antonio, Texas

Techscuie HogcdoI

Formulatiofl of the Problem

We shall assume that the seaway is composed of regular surface waves, the displacement of whith, in, a coordinate

moving with forward velocity U, takes the following form

2t

2nx

il = I a sin

(x - it1 t) I a, sin

-I xi

i,

where

g ?

c=

(ancI)1/2

and the cot-responding potential is, for a coordinate system

moving with the ship

tSZ 2it

=Ia1c1ej,coa

(xujt)

-Let the Cartesian coordinates it, y, z pass' through the c g. of the ship translating with an i-component of velocity U. The it', z' axes are rotating with the ship (Fig. 1). Consider two degrees of freedom, pitthing and heaving of the ship, so

that M0z0

= F

(1) 10a = M (2, where FB + F,., + Fm + F + F1 + Fr (3) M= M8 + M,., + M + M,+ MT + MD . (4)

(a) The buoyancy force and moment FB and MA')

-- F

g A. (Z) dx' - g f

t (x, t)b1dx

= pg$A(Z) 1(Z) dx'Qgfb5dx

(5)

5f.e.

where the effective draft

Z = z sec a + x' tan a + (it') +ij(it', z0, a, t) sec a (6a)

and

= x'cosa + [Z(xj')t (xi')] sin a;

x2 = x2' cos a + [Z(x2') (it2)] sin a (6b)

b1

=

(b1. (Z)].

_xcosa(z0

+ )sifla- (6c) The elevation of the free surface in the strip (x', x' + dx'),

ij (x,z0,a, t) is determined by the two equations

it = it' sec a +10tan a + i tan ci (7)

and

I) Use PB g (z + 1) for the origin at the C. g. and trans-form the surface integral by the divergence theorem; th, second

term in Equation (SJ ii due to the fact that the bodV is not

fully submerged.

Deift

'vu

Mea, ,ltwfl of_I'.. l,rFICt where -.'/ b,' 'C (2az,,(Y)co5o1 F1(x',a)

= J

e I,

Fig. 1 illustratIon of semis Definitions

iasan

2,t 1

12n

(8)

j I

from whith the solution can be obtained graphically by the method of false position (interpolation), or by iteration, for given x',z0and a. Then

(9) where the effective relative velocity w is approximately

wUslna+x'à+iocoSU+W,o

(15)

The moment due to buoyancy force is, similarly

_2t

2atx'5lfla+(1')COSU+ZoJ

'2 12

- a C1 -

e

M5

Qgf(coen)A,.(Z) x'dx'Qgsinuf S,.(Z) dx'

I I

11'

2t

(")

-cos

(it cosa(x)sinau1t) +a

(16)

- g 5 i (x, t)

b, x dx - Qg $ t (it, t) b,

(z + ) ---- dx

Al

ax w -is the negative z' component of the velocity of the wave

where at the keel. z' = (x') specifies the shape of the ship bottom.

S,. (Z) =

j5

' ' (11) The correction due to momentum transfer through she control

submerged area with local draft surfaces are included in Equation [141.

The displacement effect of the regular waves isincluded From Equations [141 and [15]

in the above expressions.

(b) The effect of 'impulsive"5) pressure due to wave poten-

Fm = -

cos'a $ rn,. (Z) dx' + ö cos a $m,. (Z) it' dx' + tial: The force produced is

X1

- a1c112x

12

21(X'Btfla+z()

+ (Ucos'a) à$m,. (Z) dx' _0àsinacosa$m,.

(Z) dx' +

- F,.,

p

_{f e}

.. { sin A1

j

Xe XC C

dm,./aZ\

+ cos a m (Z) dx + cos a w I

dx +

J

dZ \3t/'

Xt' xl' ± U c085a [w,2. m,a' - w,1' (17) where

iaz\

-= ±0seca + z, (seztana)U +

\

t /x'

- (17a)°)

+

_(----

sec a + (x', a0, a, t) (sec u tan a) a + x' sec2 ci

\

t Jx'

Neglecting the effect of "piled up" water, the added

ap-parent mass is approximately

=

A,. (Z) _{= 55}dy' dz'

-submerged area with local draft

xi'

(it' cos a - u t)jF1 (x', a)

-

[-(it' cos a - u1t)} F2 (x',a)} dx' cosu. (12)

2t

cos -- --z,.(y)slnu dy A1 (12a) - 'I. b,' 'I. (2nz,,(y)cOSai _2'r F2 (x',a)

=

J --- --- -sin

--- z, (y) sina] dy - 'I, b,' (12b)

5) Here, 'impulsive pressure" is used to describe the pressure

due tou --, C being the veIocitV potentiaL, and

6j ( can be determined from an iterative prOcedVI

e-3)

(0)

pLoVtnp the time derivatives of EquatiOns (7) and (51 wtUi the

---

_xUt.l

s(-_--)

_{ot ,y}

-

_,t variables x, a, i, tj.

Sdlffstednlk Bd. 10 1033 Hert 54 188

-z',' (y') is the contour of the ship at x', a',' (0) being The moment produced is, by strip theory

XC a. c 2 C 2* (X' sin a -4- 1,,)

---e

A1

J

Xl'

(.

2r sin ---- (x cos a - u1 t) F1 (x , a) -xi 2it , ,

1.

- cos -- (it cos a - u t)

F4 (x , a) it dx . (13)

The effect of dynamic pressure due to wave potential is

neg-lected.

(c) The apparent mass forces according to strip theory:

The downward apparent mass force acting on the ship ii approximately

Xe'

Fm

=

(wmx.)dx'jcoscI

-

{J

_xl, (wm,) x' dx'

-4- U cosu w' m,., x.,' - w11' m11.

- Ucosu

_[Jwmt'dx'l}.

xl,

From Equations [15) and [19]

12 12 Mm = - z cos a m1. x' dx' - J m1' x" dx' XI' x..,' 1' ,dm1,3Z

- I wx ---dx Ucosa

J

dZ at

xl, 12' 12'

+ Ucosa[(z0cosa + Usina)fm1'dx' + fwm1'dx'].

x1' x1' (20)

The effect of friction force in vertical flow be neglected

as it is generally considered as small and cannot be easily

obtained.

The downward force of weight

FM0g.

(21)

(1) The effect of waves generated by the motion of the ship is neglected.

;) Force and moment produced by thrust: The effect of

Surging on the motion is neglected, and the thrust is assumed

to be maintained at a constant value parallel to the x'-axis.

The magnitude of the thrust is

T j, (z,, 0) è U (22)

wiere the drag coefficient D is determined experimentally or is obtained by integration of the profile drag per unit draft

in calm water. The downward force due to the thrust is

FT = Tsina.

(23)

The moment due to thrust is

MT

TtT.

(h) Force and moment du; to profile drag in forward

velo-city: The profile drag is approximately parallel to x. The moment produced by profile drag is approximately

Z (Z0,it)

M1, =

- f

C (z, z,, a) I U' £ (a, z0, a) a dz (25)

- Zo

(Note z0 positive from unperturbed free surface) For preliminary analysis, assume CD = constant

Zmx(Zo,a)

MD - CD I p U5 $ £ (a, a0, a) z dz (25a)

- Zn

where the local wetted length £ (x,z0, a) and the maximum value of z, La., z., can be determined graphically or

analy-tically in the plane of symmetry, neglecting surface wave.

'x'dx' +

asinaJmxix'dx'_

[Wt2' m1.. 12' - w11 m11. Ii'I I EM,, + cos a) M

[M0 + s0cos'a] [I + 2J-1' cos2a

- (s1 cosa) F] + K22a (29) (19) s0 (t) = m1' (Z) dx' (30) Xj' 12' s (t) = Jm1. (Z) x' dx' (31) xl' 12 = $ m1. (Z) x' dx' (32) xl,

M=M(z,ó,z0,a,t)=M+s2ö+s1cosa

(33)

F = F (1, a., z0, a, t) = F + (s coe' a) + (s cos a) ii.

(34)

Reasonable initial conditions may be assumed as desired.

Methods of Solution

Numerical Method

The equations of motion can be represented by a system of first order differential equations with ±, z0, a., a as de-pendent variables. Methods given in Reference (14) can be

employed for numerical solution of these equatioos. Extension o/ Jacobsen', Graphical Metho&)

Jacobsen's graphical method can be easily extended to

multidegree of freedom vibrations described by a system of

second order differential equations of the form

H0 (q1, q2.. .

- q;

q2. .

.

t) n = 1,... N. N phase

planes (q0 versus q5) are employed. The extension is

demon-strated by the present problem of two degrees of freedom

coupled motion.

Writing a for 10, one has

dx = K' a + C1 (z, a, a, a, t) dx - dà

a - = - K2' a + G2 (t, a, a, a, 1)

da

where K1, K2 may be fictitious spring constants. The equations are equivalent to

(t/K1)

G1

1--

_K1'

-

- (a/K0)

a

-K02

7) Thts method t ruggested to the author by Dr. H. N.

Abram-son. cf. Abramson, H. N. & Chu. W. H. Appucatton of the Gene-raLized Phase Plane b-method to Multi-degree of Freedom

Vibra-tton Systems" J. Appi. Mech. 29. S pp. 580U2, Sept. 1962. dz

K1

da

e apparent mass coefficient K. is that of the reflected The ..quations of motion can therefore now be written in x' when the local draft is Z. In many applications the form

and mx2, are zero, since the wetted widths and _{= G1 (z0,à,z0,a,t)--K12z0 = H1(±0,à,z0,a,t)}

(26)

are zero.

ii = G2 (*.,, a, z, a, t) - 1(2' a = H0 a, z0, a, t) (27)

Similarly, the moment due to apparent mass is

where

M =

-X:t'

ía

.

r

a - --- _{(wm1,) x dx + U C08 (wmf) dx} I 3x 1

[M0 + s cos' a) [I. + 521 - SI' costa

-xl' x1'

(Sj cosa) Mj ± K12z (28)

X2

Fig. 3 GrapbICaI Construction in Bimultaneous Phase Planes

dz dci

d (*1K1) ' d (à1K2)

to O P1 and 02P2, respectively (Fig. 2). Assuming the

ra-dius of curvature remains constant and the center of

cur-vature, fixed during the small time interval [t, t + bt], the points Q1, Q2 are determined (Fig. 2). The small angular

dis-placements, ip = 6zI(-'

)Ki5t

and

K2 8t. For a selected 6*, and lpr can be calculated. If 'l'i. say, is drawn, then 6*, ip2 can be calculated. It is more

con-venient to select K1 = K2 so that

=

whidi can be

athieved graphically with ease5). At the next time interval, (t + 5t,

t + bt +

t1,

the centers 01, 02 are shifted

to

(G1).t,

oJ and

K2' (G2).at, 0

, respectively.

New arcs can then be drawn to determine the motion at the end of the new time increment. The entire procedure can, of course, also be carried through entirely on a numerical basis.

Slamming ConditiOns

Instantaneous Loading per Unit Length along the Ship The loading force per unit length in the z' (downward)

direction is approximately the sum of the following, whidi

may be used for estimating bending stresses: (a) Buoyancy per unit length is

-- g cos ci A1. (Z) + (x', z0, a, t) b1. (Z) . (36a)

a) For hIgher accuracy, one may first estimate the values of

a. , z. z at t + 61 ci described, then recalculate these values by

evaluating G1 and at the eattmated average vat,es pJ.i. a. 2,

z in ft + StJ and t + '/i St

-are found to be in the direction normal

dFR'

dx'

Sdi1ftstedn1kBd. 1963Heft 54 ₁₉₀

-"Impulsive pressure" loading per unit length due to

wave potential is

ac22n

2(x'Stha+Z(,) f

--

I

e---1TFi(x,a)sin

A, I (36b)

{!.(x'cosa_uit)] ___F2(xt'a)cos[(x' cosa_uit)]]

Loading per unit length due to apparent mass force,

given by dF01' dx'

ía

a =

-

(wm.) + U cosa - (wm2.)

= -

{[(uCOSfl)à + x'

+0c:a*0a8ina +

I 3w,, \

+ (-

-, -)

m.(Z) +

[z0sec a + z (sec a tan a) a +

\

Ot /'

I afj \

+x'asec'a+(----i seca+

\ 3t /'

dm ' + i(x',z0,ct,t) (seers tanri) a]

_{w ---- +}

+

[tanci + tan ci + dt (x')1 ax'

.---1(Uco8a)w

3w.

+ a + -- (U cosQ) m1(Z)

. (36c) a1

Loading per unit length due to friction in the velocity w is

dF'

.

= - c1

(Z, U, w, v, x , t) - (sgn w)

dx 2

whidi is likely negligible in the calculations. Effective inertia loading per unit length is

dF1'

_{=mf(x) g0cosa+ x}

iizasinagcoscL-dx

.L

sin a) (36d)

dt

where d/dt (ii, sin a) is determined by the equilibrium

condi-tion

F' + F.' '+ F11' + F' + F1' = 0.

(36e) Loading per unit length due to wave generation is neglected.

Thrust force is assumed to be parallel to the x'-axis.

hence dFT'

dm.

dZ

=0.

(36f) dx'

Load distribution per unit length due to drag in the velocity U is difficult to obtain and possibly can be neglected

for the prediction of stress.

-The total loadings may be calculated at selected time

inter-vals, for purposes of stress estimation in whith the ship hull

structure is replaced by a nonuniform beam. Instantaneous Local Pressure Distribution

Two different approadies may be used for predicting

pres-sure distributions. One is to isolate the slamming problem by

assuming a predicted entry velocity and neglecting the effect

of the surface waves (1). The second, presented here. aasuñes

some reasonable prediction of the ship motion. To simplify the calculations, the body is replaced by a flat plate having the proper instantaneous wetted width. The pressure along

e yields the pressure at the corresponding point on the r. The pressure is approximately

(3!

₊

)

(y0' - y)

P =

\ 3t

where y0 is determined graphically by

3!

y2

=0

3t

3!

b5. db5, (Z) / 3Z' / 3w \

--

_-

--

( ---) + I

_; Vb -. y2 3t

j/by

dZ

_{\ 3t ,,}

3t _ix' (38)

z,, seco + z, (seca lana) a + (sec2a) a +

-+

sec a + i (sec a tan a) a

\

t /x

(40) (37)

(39)

(U cos a) a + x' ii + L cos a + ± a

_{- sin a) +}

(" '

\ 3 /1

The interference of the wave is approximately taken care of

but the effect of piled up water is neglected.

Discussion

Although the present analysis, based on a reflected body ipparent mass coefficient, is approximate, it is hoped that it s'ill yield useful information for designers. In view of the

umerous integrals entering the calculations, a combination ii numerical and graphical methods may prove to be time-aving, if the computer program exceeds the capacity of the omputer but a large amount of time may still be requiredto

arry out a practical example. In some cases, the motion might

un into a stable limit cycle, thus determining the maximum mplitude and the other desired information. It is also pos-ibi at useful estimates or upper bounds might be obtained

ur _{a sufficiently long period of transient motion, even} Fiough a limit cycle may not exist_{or has not been reathed. An}

ppropriate thoice of the initial conditions may reduce the

umber of integration steps required.

The difference between the present analysis and Korvin-roukovsky's analysis (7) (8) is principally in the nonlinear Tect of the apparent mass force and the buoyancy force for

rger amplitude oscillations; the interaction between sur-ce waves and the ship is approximately taken care of in _a

mewhat different manner. For a ship with a circular bottom zero forward speed, Korvin-Kroukovsky's result should be

ightly more accurate than the present analysis, whith in.

udes the effect to the wave velocity _{at the fore part of the}

Ittom; however, the difference is of second order for small 4th-wave length ratios. On the other hand, _{the forward} eed effect seems to be more reasonable, _{as included in the}

esent analysis9). Another difference between the _two

ana-tea is the inclusion, in Korvin-Kroukovsky's paper, of damp. due to waves generated by steady oscillations of the ship.

is damping was also neglected in (11), since during the

n t period the damping due to wave generation is

)L_y muds smaller than the steady value, and _this

ap-)ximation is therefore conservative in estimating the

criti-5) For cfrcuiar ship bottoms, tho two theories are approxlmgtet

tvaLent, while the present theorij may be more accurate for

er ship bottom,.

191

-cal conditions. Modifications _{may be added if the theory is}

overly conservative.

Obviously, the present analysis is only preliminary in nature, and considerable work yet remains to be done. An

example of an idealized ship form is now being carried

through by the author's collegue, employing the present

ana-lysis for comparison with other available _theories.

Program-ming for an actual ship is also under consideration.

Acknowledgment

The author wishes to express his appreciation to Mr.

Wil-liam Squire for valuable discussions during the course of this

study and to Dr. H. N. Abramson for his _{overall supervision.}

(Received 4th July 1963)

Nomenclature

a _{maximum wave height of the Ith surface wave}

A1, wetted area at x' in the strip tx', x' + dx'J Ami maximum frontal area

b1, _{wetted width at x' In the strip tx', x' + dx'J}

c1 _{propagation velocity of the liii surface wave}

Cf (Z. w, x', t) coefficient of friction

I' vertical force (positive down)

K1 K2 1(1,(Z)

gravitational acceleration

a

10 moment of Inertia of the ship about the y' axis through c. g. of the ship

fietltlou, spring constants

apparent mass coefficient at station x' with draft Z

l(z, Z, a) local wetted width measured alongx axis

ml' added apparent mass per unit length at x'

Tfl)) Cx') mass per unit length Of ship at x'

M moment about C. g. (positive bow up)

M0 ma,, of the .hlp

9, s, s

_{defined by uquatlons (301, (3lJ, [32J, respectively}

S1. first moment about y axi, of the wetted area A1, t time

u1 _{relative propagation velocity of the ith surface}

wave

U forward velocity of ship In the negative direction

w relativ, velocity at infinity In the strip ,c1' (t) effective fore end

x2' (t) effective aft end

acceleration of the ahip parallel to x axis Cartesian coordinates rotating with the ship

Cc. 1., FIg. 1)

positive downward coordinate measured from horizontal ixi,

distance of c. g. from mean position of the

un-perturbed free surface (LWL); posItive z cor-responds to C. g. below LWL

draft normal to x' axis Cc. f., Fig. 1)

angle of pitch from mean position of unperturbed free surface

contour of keel from x' axis displacement of the surface wave

vertical height of free water surface aSsociated

with ship section x' (see Fig. 1) wave length of the Ith surface wave kinematic viscosity of the fluid density of the fluid

defined in Figure 2 unit step function

Subseripta

() related to weight

m () related to apparent mass )f ( ) related to friction In vertical flow

I.) ()

related to buoyancy related to wave related to the plane x' differentiation with respect to t, holding x' constant

Superscripts

first time derivative of second time derivative of

References

(1) Szebehely, V. 0., ,,HydrodyflamiCS of Slamming of Ships", DTMB Report 823, July 1952.

Fabula, A. G., Ellipse Fitting Approximation of Two-Dimensional Normal Symmetric Impact of Rigid Bodies

in Water", Fifth Midwestern Con!. on FluidMeth., Univ.

of Mith. Press, Ann Arbor, 1957.

C h u, W. H.. and Abraznson, H. N., HydrodynamiC Theory

of Ship Slamming - Review and Extension, Southwest Researdi Institute, Teds. Rept. No. 1, Contract No.

Nonr-2729(00), SwRI Project No. 23-834-2, November 1959,

pub-lIshed in the Journal of Ship Researcti, MardI 1961.

(41 Golovato, P., .A Study of the Forcesand Moments on a Heaving Surface Ship", DTMB Rept. No. 1974, September 1951.

(5] Lewis, F. M., The Inertia of the Water Surrounding a Vibrating Ship", Trans. SNAME, Vol. 37, 1929.

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Schiftstethnikfld.10-1983Heft54 192

-[61 Ursell, F., ,,Short Surface Waves Due to anOscillating

Immersed Body". Proc. of the Royal Soc., Ser. A., Vol.

220, No. 1140, pp. 90-103, October 1953.

[7] Korvin_KroUkOVskY, B. V.. and

Jacobs, W. R.,

,,Pitthing and Heaving Motion of a Ship in Regular Waves", Experimental Towing Tank, Stevens Institute of

Tedinology, Hoboken, New Jersey, Rept. No. 659, May

1957.

[8) Korvin-KrOUkOVskY, B. V., ,,Investigation of Ship

Motions in Regular Waves", Trans. SNAME, Vol. 63, pp.

385-418, 1955.

(9] Todd, M. A., ,,Slarnming Due to Pure Pitthing Motion",

DTMB Rept. No. 883, January1955.

[101 Sims, A. J., and Williams, A. J., The PItthing and Heaving of Ships", Quarterly Trans. INA, Vol. 98, No. 2, pp. 113-128, April 1956.

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