226 Marine, Offshore and Ice Technology - WIT Press€¦ · Marine, Offshore and Ice Technology 229...

Time domain analysis of vertical ship

motions

N. Fonseca & C. Guedes Soares

Instituto Superior Tecnico, Universidade Tecnica

de Lisboa, Av. Rovisco Pais, 1096 Lisboa, Portugal

Abstract

The time domain description of the vertical motions of a ship travelling atconstant forward speed is studied. The hydrodynamic forces are represented byinfinite frequency added mass, convolution integrals and restoring forcecoefficients which are obtained from Fourier transforms of the frequencydomain solutions. The exciting forces are calculated in the frequency domain.The hydrostatic forces are evaluated both around the mean equilibrium positionand over the instantaneous wetted surface in order to assess the effect of thenon-linearity of these forces. A strip theory is used to obtain a numericalsolution to the problem. Results of heave and pitch motions are presented for alarge containership and a tanker in head waves, and comparisons are madebetween the solutions for linear and for non-linear hydrostatic forces.

1 Introduction

Time domain solutions of the ship motions are important to study variousphenomena. Of special importance are the different non-linear responses thatcan be associated with the motions. A specific example are the stresses thatresult from the impact of ships with the sea surface. Often in large ships themotions can still be described adequately by linear theory but the forcesassociated with the impact phenomena will be non-linear. For small ships or forships with large bow-flare the magnitude of the non-linear forces can be such asto induce non-linear motions. Therefore, it is clear that at least for the problem

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226 Marine, Offshore and Ice Technology

mentioned it is important to be able to describe the ship motions in the timedomain so that non-linear forces can be calculated in an irregular sea state.

The interest for time domain representations of ship motions is relatively recentas compared with the frequency domain formulations that were developed in thelate 60's and early 70's. After the development of a two-dimensional harmonicflow solution by Ursell [1] for the heaving motion of a half-immersed circularcylinder and the generalisation to arbitrary shapes using conformal mappingtechniques, these results were applied in frequency domain strip theories, firstlyby Korvin-Kroukovsky [2] and then by many other researchers who improvedthe theory.

A more recent development in the linear frequency domain solutions of shipmotions was achieved by Chang [3] who represented the ship hull by adistribution of singularities, and solved the fluid domain and free surfaceequation using a panel discretization of the mean hull position. This is a veryacurate method to calculate the linear wave excitation load and response ofships, but the numerical evaluation of the speed dependent Green function isvery difficult and it requires powerful computers.

The time domain simulations are often used to study non-linear ship motionsand two main approaches have been used. In one of them only the equations ofmotion are solved in the time domain, and the hydrodynamic forces arerepresented by frequency dependent coefficients. This is correct if the motion isstricly sinusoidal in time, but not in the case of strongly non-linear motions or ofirregular motions.

Another approach solves the hydrodynamic problem in the time domaintogether with the equations of motion. Finkelstein [4] and Stoker [5] were thefirst authors who discussed the time-domain direct solution of hydrodynamicproblems. They applied the formulation to the problem of water waves. The useof time domain analysis to solve the unsteady ship motion problem was initiatedby Cummins [6]. Since the instantaneous hydrodynamic forces depend of theunknown body motion, Cummins decomposed the motion and the associatedvelocity potential in a sequence of impulses. This decomposition allows thehydrodynamic forces for arbitrary motions to be evaluated in terms ofconvolution integrals of impulse response functions, which are independent ofthe body motion. Ogilvie [7] studied the problem by the same approach, andrelated the time domain motion equations with the classical frequency domainequations using Fourier analysis. However, Cummins and Ogilvie did not obtaina numerical solution to the time domain ship motion problem.

A linear time domain formulation for ship motions was developed and appliedby Chapman [8]. This theory is in some aspects similar to Chang's panelmethod, but Chapman did not use Green's functions representing solutions of


Marine, Offshore and Ice Technology 227

the free surface equation as Chang did. A panel discretization is used and thesolution is generated in terms of a source distribution representation of the bodyand a spectral representation of the free-surface. The use of a time dependenthull boundary to evaluate the hydrodynamic forces was proposed, dealing thuswith some effects of large-amplitude motions, but the computational effort ofsuch an extension is enormous.

Beck and co-workers [9-11] formulated the time domain hydrodynamicproblem with the same basic assumptions as Chang [3]. The linear potentialflow theory and the linear free surface condition are used together with a paneldiscretization of the mean hull boundary under the still water line. Again thesolution is obtained by applying Green's theorem to the fluid domain.Comparing with the frequency domain Chang's panel method the proper timedomain Green functions are easier to derive and to compute regardless of theship speed. Following the work described, Beck and Magee [12] generalized themethod to include a time dependent hull shape, but numerical solutions for shipstravelling on the free surface were not available yet.

The main disadvantage of the time domain panel methods is the need ofpowerful! computers to solve the complicated integro-differential equations ateach time step. Basically the idea of the present work is to develop and apply atime domain method, where the hydrodynamic forces are treated with realismfor non linear and irregular motions, and on the other hand the computationaleffort is reduced to a point where the simulations can be made by a desk topcomputer in a few minutes.

In this work, the formulation developed in [13] is briefly described, and it isapplied to a containerhsip and a tanker in order to illustrate the differentcharacteristics of the response of these ships hulls. The linear time domainsolutions were found to reproduce almost exactly the frequency domainsolution. The influence of including a non-linear description of the hydrostaticterm was also investigated and these results were compared with the linearones.

2 Present Theory

A time domain strip theory is developed, based on the linear potential flowformulation, and on the linearized free surface and body boundary conditions. Inorder to linearize the velocity potential and the boundary conditions, the shipmust be slender, her forward speed low, and both the amplitudes of theincoming waves and the oscillatory motions must be small. The radiation forcesare formulated in the time domain and they are related with the frequencydomain solutions by means of Fourier transforms. The exciting forces areevaluated assuming sinusoidal incoming waves acting upon the restrained ship.



The hydrostatic forces have a non-linear characteristic since they are evaluatedup to the instantaneous wetted surface.

The time domain method followed for the radiation forces is based on theformulation presented by Cummins [6] where the basic assumption is thelinearity of the forces. The formula for the potential obtained is anhydrodynamic analogue of the impulse response function, which Cumminsassumed that exists but he did not calculate it. The equations of motion are validfor any excitation, as long as they result in small amplitude motions. The inertialproperties of the fluid were represented by coefficients which are independent ofthe frequency, of the previous history of the motion and of the ship's forwardspeed. Cummins called them "legitimate added masses". The effect of pasthistory is represented by a convolution integral over the oscillatory velocity.

The method used here applies a strip theory to obtain a solution to theformulation proposed by Cummins. This method works with truly time domainradiation forces, and on the other hand asks for much less computer effort thanthe existing panel methods since the boundary value problem is solved onlyonce rather than at every time step. Only the final equations are indicated in thetext while the detailed derivation of the expressions can be found in Fonsecaand Guedes Soares [13].

A coordinate system fixed with respect to the mean position of the ship isdefined, X = (x , y , z), with z in the vertical upward direction and passingthrough the centre of gravity of the ship, x along the longitudinal direction ofthe ship, and y perpendicular to the latter and in the port direction. The origin isin the plane of the undisturbed free-surface.

t» (h«ve)

S, (surge)

$4 (roll)

Figure 1: The coordinate system and six modes of ship motion

Considering a ship advancing in waves and oscillating as a unrestrained rigidbody, the oscillatory motions will consit of three translations and threerotations. The translatory displacements in the x, y, and z directions arerespectively the surge (£, ), the sway (%% ), and the heave ( ). The rotationaldisplacements about the x, y, and z axes are respectively the roll (£4), the pitch (



£5 ), and the yaw ( ). The coordinate system and the linear and angular

displacements are shown in figure 1.

2.1 Radiation ForcesIn order to implement a time domain method all the forces in the equations ofmotion must be represented in the time-domain. This raises no major difficultiesin the evaluation of the exciting and restoring forces since, for the presentlinearized model, these forces do not have the time dependency of the previoushistory of the fluid motion. However the radiation forces behave in a differentmanner. The existence of radiated waves implies a complicated time dependenceof the fluid motion and hence of the pressure forces. Waves generated by thebody at time t will persist, in principle, for an infinite time thereafter, as well asthe associated pressure force on the body surface.

The time domain formulation derived by Cummins [6] is used to represent theradiation forces in terms of unknown velocity potentials. The basic assumptionis the linearity of the radiation forces. The equations representing the radiationforces are given by:

where the dots over the symbols represent differentiation with respect to thetime variable, and the indices 'kj' are associated with forces in the k-directiondue to an oscillatory motion in the j-mode. In this study consideration is givenonly to vertical motions i.e. k, j = 3,5.

In this expression Aj° represents the infinite frequency added mass coefficient,

which is dependent only on the ship geometry. The coefficient B^ is a constant

which depends on the ship geometry and on the forward speed. It was provenby King [14] that B^ is zero. The coefficient C^ is also a constant which

depends on ship geometry and forward speed. The force proportional to thiscoefficient is a "radiation restoring force", and it represents a correction to thehydrodynamic steady forces acting on the ship due to the steady flow. Thecorrection arises because the steady force is evaluated assuming the ship in theequilibrium position, and in fact the ship's position changes with time. In manyforward speed formulations this term does not appear, at least explicitly. Thisconstant can be obtained from the following expression:



q, = <O*[A; - A,(*>)]-<»}[ (?)sin**]a (2)0

where A (a>) are frequency dependent added mass, and the function K (t) is

dependent on time, geometry, and forward speed. This quantity contains all thememory of the fluid response. It is interesting to note that K (t) is equivalent

to the impulse response function of any stable linear dynamic system. Thesefunctions are obtained from the frequency dependent damping coefficientsBQ(G)) by means of Fourier transforms.

(3)

The last term in equation (1) is a convolution integral which involves the effectsof the whole past history of the motion, representing the memory effects due tothe radiated waves.

It is important to stress that none of the quantities described above (AJj°, B^,

Cjy, and Kq(t) is dependent of the past history of the unsteady motions. This

means that they need only to be calculated once for a given vessel, and then theradiation forces can be evaluated for any arbitrary motion using equation (1).

The relations obtained for the radiation forces do not have coefficientsdependent on the frequency, and they are valid to evaluate the radiation forcesassociated with non-sinusoidal motions, for example irregular motions. The onlycondition necessary to apply this method is the linearity of the response, and thismeans that the unsteady motions must be of small amplitude.

In spite of the fact that the forces obtained are truly time domain forces, theyare not evaluated directly in the time domain. This means that the boundaryvalue problem is not solved in the time domain, but instead infinite frequencyadded masses, AJ°, and frequency dependent damping coefficients, B .(co), are

used. A frequency domain strip theory very similar to the one presented bySalvesen et al. [15] is used to obtain these hydrodynamic coefficients in terms ofsectional coefficients.

The sectional damping coefficients are computed by the Frank's close-fitmethod [16], while the added masses corresponding to the infinite frequencylimit are computed by a modified Frank's method which takes into account thezero potential condition on the free surface.



2.2 Exciting ForcesThe calculation of the wave exciting forces is treated as an independent problemof a vessel restrained at its static equilibrium position and being subjected toincoming waves. Again the strip theory approach is used to evaluate the totalforce on the ship in terms of sectional forces but neglecting interferencesbetween adjacent cross sections.

The exciting force is divided in two components, F^ = F* +F°, where the first

part, the Froude-Krilov force, represents the effect of the wave incidentpotential. The second part is the diffraction force which represents the effects ofthe disturbance in the incident potential caused by the presence of the ship. Thislast force is calculated with the formulation presented by Salvesen et al [15].

The amplitudes of the Froude-Krilov force associated with the heave and pitchmotions due to sinusoidal incoming waves travelling on the negative x-directions are:

(4)

(5)

Here the integrations are performed over the ship's length L, £* is the incident

wave amplitude, co^ and k^ =co//g are the wave frequency and the wave

number, respectively and // is the sectional heave Froude-Krilov force due toan unit amplitude incident wave which is given by:

(6)

where the integration is performed over the cross section contour C%, n, is thevertical component of the two-dimensional unit vector contained in y-z planesand normal to the hull surface at the mean position. q>*(z) is the amplitude of

the potential of an unit wave acting on a cross section, which in accordancewith the linear gravity wave theory is given by:

/W = -/°' (7)

The amplitudes of the diffraction force, associated with the incident sinusoidalwaves, for the heave and pitch motions are:



where the ff is the sectional heave diffraction force associated with an incidentwave of unit amplitude. This force can be evaluated using the Haskind-Newmanrelations, which means that instead of solving directly the diffraction problem,the solutions of the forced oscillations problem are used to represent thediffraction force acting on the ship restrained at mean position. Haskind [17]derived the relations for the ship's zero speed condition, and some years laterNewman [18] showed that an analogous result can be obtained for the case of aship with forward speed. Salvesen et al [15] used these relations together withthe strip theory simplifications to obtain the expressions for the sectionaldiffraction forces. The sectional heave diffraction force becomes:

<; (10)C,

where ¥ (y,z) is the two-dimensional radiation potential associated with the

forced heave oscillations of the cylinder with the cross section C% Thispotential is calculated by the Frank close fit method.

Finally the time description of the total exciting forces is given by the real partof the sum of incomming and difracted forces:

k = 3,5 (11)

2.3 Hydrostatic ForcesThe hydrostatic forces are the result of the hydrostatic pressure acting on thehull. The formulation for the hydrostatic forces used in the present model doesnot impose any kind of restriction on the body shape or motion amplitudes. Infact these forces are evaluated over the "exact" wetted surface at each timeinstant, taking into account the free surface elevation due to the incident waveIt is assumed that the contribution to the free surface elevation from theradiated and diffracted waves can be neglected. This hypothesis is reasonablefor wave lengths of the same order of the ship length, which is the wave lengthrange of interest.

The hydrostatic forces associated with the heave and pitch motions are givenby:



F"(t] = Pg V*(0 - pg\ \(z ntyg dt (12)1C,

= -PS V*(/) x,(t) + pgj Jz(x n?)dg dt (13)LC,

where n^' is the vertical component of the two-dimensional unit vector normalto the intersection of the free surface of the fluid with the hull at each timeinstant. The instantaneous immersed volume under the intersection of the freesurface with the hull, V „,(/), and the coordinates of the centre this volume are

given by:

X, = — Mxdv, y,=-—\\\ydv (15a,b)

The free surface elevation at any point represented on the reference systemadvancing with the ship's speed is given by:

£ (x,t) = C cos[*o* cos(&)/)] - sin[*ojc sin( )] (16)

With the knowledge of the positions in space of the ship's surface and the fluidfree-surface, the intersection between both surfaces can be determined. It isassumed that this intersection is given by a plane parallel to the x-axis, whichmeans that the hydrostatic force in the x-direction can be neglected.

2.4 Equations of MotionThe equations of motion for heave and pitch are obtained applying Newton'ssecond law:

(17)

+-oo



(18)

where x<,, y^ are the coordinates of the centre gravity and the coefficients withindices 35 represent the coupling between the motions.

The method chosen to solve the problem is the fourth-order Runge-Kutta,which processes the integration of the coupled first order ordinary differentialequations.

3 Numerical results

Results are presented for the motion of a containership and a tanker in order toidentify the different behaviour when the non-linear hydrostatic term is includedin the formulation. The non-linearity of this term is associated with non-verticalship sides which are more pronounced for the containership.

The containership hull is the one used in the experiments of Gie [19] and usedby Flokstra [20] to compare with calculated values from strip theories. Themain characteristics of the ships are listed in table 1.

Characteristics of shipsLength between perpendiculares(Lpp)BreadthDraught even keelDisplaced volumeBlock coefficientLCG relative to section 10 (midship)Centre of gravity above base lineLongitudinal radius of gyrationShip's speed

Contair270,032,210,85560970,59810,1213,4924,824,5

icrshipmmmm'

maft.m%LppKnots

Tan168,5628,010,9

425960,8281,66

8,95627,214,0

kermmmm>

mfwd.m%LppKnots

Table 1 - Characteristics of the Ships Analysed

Before the time domain computations are performed it is necessary to computethe retardation functions given by expression 3. These functions include theeffect of frequency dependent damping coefficients but they are frequencyindependent. Figures 1 and 2 show the damping coefficients as a function of thefrequency for the tanker hull and figures 3 and 4 show the correspondingretardation functions or the impulse response functions.



0.000.0 1.1 i.b ,2:0'' *s.b'' '4/0W(nod/s)

Figure 2: Heave (#33 / M^g I L^ \ and pitch (B^ I Mll^^gl L\

damping coefficient for the tanker

0.75

0,50

0.25

W(md/m)

853

-0.00-

-0.10-

-0.20

-0.30

-0.40

non-dimensional

W(rad/s)3.0 4.0

Figure 3: Heave (#351 ML^^gl L^ and pitch (B^ I ML^^gl L^ ) non-dimensional

coupling coefficients B% and 3 for the tanker

1 00 •

J0.50 •

0 00 •

A V) >0.50

I

rv/

0' ' 10.0 ' '20Time&*.0" ' 40

0 10 •

8 0.05 -

0.00 •

—0 OS •o o.

1

r* — H

0 10

—

0 20Time 0 'JO(sec) .0 " "40

Figure 4: Non dimensional impulse response function for heave and pitch for the tanker



).0 10.0 20.0 30.0 400Time (sec)

10.0 20.0 30.0 400Time (sec)

Figure 5: Non dimensional impulse response function for the coupling between heave andpitch for the tanker

The corresponding results for the containership showed the same general aspectbut had different numerical values

An important conclusion from the observation of these retardations functions isthat they can assume significant values for the first two seconds and that theyvanish after 20 to 30 seconds. This shows the duration of the memory effects inthe response.

To verify the method of evaluating the radiation forces in the time domain, theresponses obtained from the frequency solution and from the time domainresponse with linear restoring coefficients have been compared. The predictionsobtained for both motions were exactly the same, which was reassuring.

Several time histories have been obtained for the non-linear restoring forces anda representative sample of them is shown in figures 6 to 12 for the tanker andthe containership. Since the motions are not initiated from the rest position ofthe ship, the time histories of the motions before the first instant (t=0) are givenby the frequency domain solutions. It can be observed that a transient phaseexists during the first few seconds of the run and afterwards the responsesbecome periodic.

8.004.00 -0.00

S -4.00 --8.00

A A A

7 V V V V

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0Time (sec)

Figure 6: Linear and non-linear heave motion of the tanker in head waves (l IL - 2.4J



"1 i 1 1 1 1 1 1 1 r10.0 20.0 30.0 40.0 50.0 60.0 700

Time (sec)

Figure 7: Linear and non-linear pitch motion of the tanker in head waves (z, / = 2.4)

6.00

^ 2.00 -

I -2.00

A A

\j y y y

1 : 1 1 1 1 1—10.0 20.0 30.0 40.0

Time (sec)

Figure 8: Linear and non-linear heave motion of the tanker in head waves (L^ I Lpp = 1.4)

"-6.00.0 10.0

1 1 1 1 r20.0 30.0 40.0

Time (sec)50.0 60.0

Figure 9: Linear and non-linear pitch motion of the tanker in head waves (L^ I Lpp - 1.4)

12.00

0.0 15.01 1 1 1 1 T

30.0 45.0 60.0Time (sec)

75.0 90.0

Figure 10: Linear and non-linear heave motion of the containership in head waves



9.0

sfT30.0

-9.00.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 BO.O 90.0

Time (sec)

Figure 11: Linear and non-linear pitch motion of the containership in head waves

o.uu -

\w/

oQ)-C

_A nn -

A /

J V

\ /

V

\ /\j

\ If

V

0.0 10.0 20.0 30.0 40.0 50.0 60.0Time (sec)

Figure 12: Linear and non-linear heave motion of the containership in head waves

8.0

0.0

IQ.-8.00.0

A A A A

10.0 20.0 30.0 40.0Time (sec)

50.0 600

Figure 13: Linear and non-linear pitch motion of the containership in head waves

The wave steepness defined as the ratio of wave height to ship length (2£* / L)was kept constant and equal to 1/40.

For waves with length equal to the ships length both responses were basicallylinear and they therefore are not shown The non-linear time domain responsewhen compared with the linear one, shows different positive and negativeamplitudes. For the heave motion the positive amplitudes are very similar forboth models but this is not the case for the negative ones.

To summarise the response as a function of the ratio between the length of thewave and the ship, linear transfer functions are shown in Fig. 14 for the tanker



and 15 for the container ship. In view of the asymetry of the non-linearresponses as concerns the positive and negative peaks of the response, pseudo-transfer functions have been determined for the non-linear responses as shown.

1.60 -.

1.20 -

oQ.80 -

0.00

1.35 i

0.90 -

0.45 -

0.00 1.00 2.00 3.00Lw/Lpp

0.00 0.00

***** linear•n linear ( +in linear (—

1.00' ' ' 2.60 ' ' 3.00Lw/Lpp

Figure 14: Heave and pitch transfer functions for the tanker

1.60 -| 2.00 -i

0.00

***** linearnon linear (•+non linear (—

i .60' ' ' 2.60Lw/Lpp

0.00 1.00' ' ' 2.60Lw/Lp

3.00

Figure 15. Heave and pitch transfer function for the containership

Observation of those figures makes it clear that the non-linear results showdifference with respect to the linear responses and the method differenciatesbetween the amplitudes of the positive and negative peaks. Furthermore, theresults also show that the differences are much larger for the containership thanfor the tanker, as expected.

These results demonstrate that the present theory is capable of predicting theeffects of the non-linearities associated with the non-linear description of therestoring forces. Although it is felt that this effect is one of the majorcontributors to the non-linearity of the response [21,22] it is still necessary toinvestigate the influence of the non-linear formulation of other terms of themotion equations.



4 Conclusions

The comparisons of time domain solutions with frequency solutions for linearhydrostatic forces has shown that this method of calculating the radiation forcesis accurate.

The time domain model shows that the depicted non-linear effects, althoughrelatively small for the tanker they are large for the containership. The non-linear behaviour is more pronounced in pitch than in heave.

Although the present results concern the response to regular seas, it is believedthat the main advantages of the time domain solution is on calculating themotion response to irregular seas. However this can be done whenever theresponses to regular seas have been validated.

5 Acknowledgements

This work has been made within the scope of the research project "WaveClimate and Ship Behaviour in the Portuguese Exclusive Economic Zone",which has been financed by JNICT: Junta Nacional de Investigate Cientifica eTecnologica under contract PMCT/MAR/931/90.

6 References

1. Ursell, F. "On the heaving motion of a circular cylinder on the surface of afluid", Q. J. Mech. Appl. Math., Vol. 2, (1949), pp. 218-231.

2. Korvin-Kroukovsky, B V., "Investigation of ship motions in regularwaves", Trans., Soc. NavalArchit. Mar. Eng., Vol. 63, 1955, pp. 385-435.

3. Chang, M.S., "Computations of three-dimensional ship motions withforward speed", Proc. 2nd. Int. Numer. Ship Hydrodyn, University ofCalifornia, Berkeley, 1977, pp. 124-135.

4. Finkelstein, A. B., "The initial value problem for transient water waves",Commun. on Pure andAplliedMathem, Vol. 10, 1957, pp. 511-522.

5. Stoker, J. J. Water waves, Interscience Publishers, Inc., New York, 1957.

6. Cummins, W. E, "The impulse response function and ship motions",Schiffstechnik, Vol. 9, 1962, pp. 101-109.

7. Ogilvie, T F., "Recent progress toward the understanding and prediction ofship motions", Proc. Symp. 5th on Naval Hydrodyn, Washington DC,1964, pp. 3-12.



8. Chapman, R, "Time Domain Method for Computing Forces and MomentsActing on Three-Dimensional Surface Piercing Ship Hulls with ForwardSpeed", 3rd International Conference on Numerical Ship Hydrodynamics,Vol.2, 1981.

9. Liapis, S. J and Beck, R. F, "Seakeeping Computations Using Time-Domain Analysis", Proc. 4th Int. Symp. on Numerical Hydrodyn, NationalAcademy of Sciences, Washington, DC, 1985, pp. 34-54.

10. Beck, R.F. and Liapis, S. "Transient Motions of Floating Body at zeroForward Speed", Journal of Ship Research, Vol. 31, 1987, No. 3, pp. 164-176.

11. King, B K, Beck, R F and Magee, A. R, "Seakeeping Calculations withForward Speed Using Time-Domain Analysis", Proc. 18th Symp. on NavalHydrodyn., Delft, Netherlands, 1988, pp. 577-596.

12. Beck, R F and Magee, A. R., "Time-Domain Analysis for Predicting ShipMotions", Dynamics of Marine Vehicles and Structures in Waves, W.H.Price et. el. (Eds), Elsevier Science Publishers B V. , 1991, pp. 49-64.

13. Fonseca, N. and Guedes Soares, C, "Time Domain Formulation forVertical Ship Motion", to be published.

14. King, B K., "Time-Domain analysis of Wave Exciting Forces on Ships andBodies", Report No. 306, Dept of Naval Arch, and Marine Engineering,University of Michigan, 1987.

15. Salvesen, N., Tuck, E. O , and Faltisen, O., "Ship motions and sea loads",Trans. Soc. Naval Archit. Mar. Eng., Vol. 78, 1970, pp 250-287.

16. Frank, W, "Oscillation of Cylinders in or Below the Free-Surface of DeepFluids", Report 2375, Naval Ship Research and Development Center,Washington D.C., 1967.

17. Haskind, M. D, "The Exiting Forces and Wetting of Ships", TranslationNo. 307, David Taylor Model Basin, 1962.

18. Newman, J N., "The Exciting Forces on a Moving Body in Waves", J. ofShip Research, Vol. 9, 1964, pp. 190-199.

19 Gie, T S., "Wave load measurements on a model of a large container ship",Report No. 173s, Netherlands Ship Research Centre, 1972.

20. Flokstra, C , "Comparison of ship motion theories with experiments for acontainer ship", Int. Shipbuilding Progress, Vol 21, 1974, pp. 168-189.



21. Guedes Scares, C , "Effect of Transfer Function Uncertainty on ShortTerm Ship Responses", Ocean Engineering, Vol. 18, N° 4, 1991, pp. 329-362.

22. Guedes Scares, C , "Long-Term Distribution of Non-Linear Wave InducedVertical Bending Moments" Marine Structures, Vol. 6, 1993, pp. 475-483.


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