On unstable ship motions resulting from strong non-linear coupling

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Ocean Engineering 33 (2006) 1853–1883 www.elsevier.com/locate/oceaneng

On unstable ship motions resulting from strong non-linear coupling Marcelo A.S. Neves, Claudio A. Rodrı´ guez Department of Naval Architecture and Ocean Engineering, LabOceano/COPPE, Federal University of Rio de Janeiro, C.P. 68.508, Rio de Janeiro, 21.945-970, Brazil Received 4 April 2005; accepted 23 November 2005 Available online 27 March 2006

Abstract In this paper, the modelling of strong parametric resonance in head seas is investigated. Non-linear equations of ship motions in waves describing the couplings between heave, roll and pitch are contemplated. A third-order mathematical model is introduced, aimed at describing strong parametric excitation associated with cyclic changes of the ship restoring characteristics. A derivative model is employed to describe the coupled restoring actions up to third order. Non-linear coupling coefficients are analytically derived in terms of hull form characteristics. The main theoretical aspects of the new model are discussed. Numerical simulations obtained from the derived third-order non-linear mathematical model are compared to experimental results, corresponding to excessive motions of the model of a transom stern fishing vessel in head seas. It is shown that this enhanced model gives very realistic results and a much better comparison with the experiments than a second-order model. r 2006 Elsevier Ltd. All rights reserved. Keywords: Ship stability; Parametric resonance; Non-linear equations; Ship motions; Roll motion; Hill equation

Corresponding author. Tel.: +55 21 2562 8715; fax: +55 21 2562 8715.

E-mail address: [email protected] (M.A.S. Neves). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.11.009

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Nomenclature r r f y r0 r1 A0 Aw Fn g hi Ixx0 Iyy0 Jxx Jyy k m U xf0 z l z w oe ow ^ J; ^ K^ I; i;^ j;^ k^ x¯ y¯ z¯ G z¯ B0

incremental volume for displaced hull density of water roll angular displacement pitch angular displacement volume at average hull position volume at instantaneous hull position waterplane area at average hull position wave amplitude Froude number acceleration of gravity height of elemental prisms transversal second moment of waterplane area longitudinal second moment of waterplane area transversal mass moment of inertia longitudinal mass moment of inertia wave number ship mass ship speed of advance longitudinal co-ordinate of centroid of waterplane heave displacement of the ship wavelength wave elevation wave incidence encounter frequency wave frequency unit vectors along axes of inertial frame unit vectors along axes fixed in the ship longitudinal position of a transversal station half-beam of a transversal station vertical position of the ship’s centre of gravity vertical position of hull volume centroid

1. Introduction Any comprehensive investigation on the safety of intact vessels in waves must take into consideration the possible occurrence of parametric resonance. In fact, parametric rolling of ships has continuously received wide attention of researchers and designers, since it is a relevant instabilizing mechanism, see Kerwin (1955), Paulling and Rosenberg (1959), Blocki (1980), De Kat and Paulling (1989), Munif and Umeda (2000). Much of such attention has been devoted to the particular configuration of longitudinal regular waves, either with or without speed, bow or

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stern waves. Roll motion in such conditions has usually been modelled as an uncoupled Mathieu type equation. Considering the well-known existence of the Mathieu resonant frequencies, focus has been concentrated on the first region of instability, the one defined by the proximity of encounter frequency to twice the roll natural frequency. For many years, more attention was given to parametric rolling in astern seas, Oakley et al. (1974), De Kat and Paulling (1989), Umeda et al. (1995), Spyrou (2000). Yet, more recently, some authors and accidents have called attention to the problem of parametric excitation in head seas. France et al. (2003) reported on strong roll amplification in the case of a large container ship, which suffered severe damage to containers and structure due to excessive accelerations. Dallinga et al. (1998), Luth and Dallinga (1999) reported on the development of head seas parametric resonance in cruise vessels. Levadou and Palazzi (2003), recognizing the potentially dangerous situation, attempted to evaluate the operational risks associated with head seas parametric resonance. Another example of parametric resonance in head seas has been recently reported by Palmquist and Nygren (2004). A large experimental program with two similar models of fishing vessels on parametric resonance in head seas at different speeds and loading conditions was conducted some years ago, see Neves et al. (2002). The experiments showed that one of the tested vessels, a transom stern hull, is very sensitive to parametric excitation in the first region of instability. In particular, when tested in high waves and low metacentric height, the transom stern hull tended to undergo very strong roll amplifications at different speeds. Employing Taylor series expansions up to second order, Neves and Valerio (2000), Neves (2002) expressed restoring actions in the heave, roll and pitch modes in a coupled way. Wave action was taken into consideration not only in the FroudeKrilov plus diffraction first-order forcing functions, but also in second-order terms resulting from volumetric changes of the submerged hull due to wave passage effects. It can be shown that a system of second-order non-linear equations as defined above may be reduced to a set of coupled Mathieu equations when the linear variational equation is taken, Neves (2002). The set of coupled Mathieu equations then describes the essential aspects of the stability of the dynamic system when small perturbations are imposed on the basic linear periodic motions. Neves et al. (2003) suggested that a Hill equation should be employed, instead of the Mathieu equation, in order to describe the stability of parametrically excited motions. For many ship designs the simulation models available are capable of reproducing with confidence the roll amplifications resulting from parametric resonance. But, unfortunately, there are some known cases in which the numerical models tend to over-predict the resonant rolling motions observed in experiments, as pointed out by Umeda et al. (2003). These strong amplifications are associated with specific values of metacentric height, ship speed and stern shape (and, in general, other design parameters). In these cases, the classical Mathieu type modelling, in which parametric excitation is assessed considering terms up to the second order, tends to predict excessive excitation. The Authors attempts to reproduce the strong amplifications observed in the tests with the transom stern hull employing a

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second-order non-linear mathematical model also indicated excessive amplifications in the simulations. Bulian et al. (2003), Umeda et al. (2003), Matuziak (2003), are recent examples of investigations in which some form of non-linearities have been heuristically introduced as a possible remedy to the deficiencies encountered in matching experimental results with large amplifications. Skomedal (1982), Shin et al. (2003) report on numerical assessments of strong parametric amplifications, but with no comparisons with experiments. In this paper, the modelling of strong parametric resonance in head seas is investigated. In an attempt to give a robust answer to this question and improve the quality of the simulations, a new third-order mathematical model is introduced. The model describes the heave, roll and pitch motions in a coupled way—up to third order, thus establishing a mathematically congruent set of equations. Parametric excitation is associated with cyclic changes in the ship restoring characteristics, corresponding to variations in the submerged part of the hull due to the relative body motions with respect to the wave motion along the hull. A holistic derivative model is employed to describe the coupled restoring actions up to third order. Given the extended complexities of the coupled non-linear systems, some effort is devoted to interpreting the essential dynamic characteristics of the new mathematical model. The appearance of super-harmonics and increased stiffness proportional to wave amplitude squared due to third-order terms is highlighted. Non-linear coupling coefficients are analytically derived in terms of hull form characteristics. In particular, the longitudinal distributions of breadth and flare are derived as essential constituents of the coefficients defining the complex dynamics. This aspect may be of great interest and relevance in the ship design practice. In the present investigation, numerical simulations obtained from the derived third-order non-linear mathematical model are compared to experimental results corresponding to excessive motions of a transom stern fishing vessel in head seas. It is shown that this enhanced model more closely matches results from experiments than a second-order model.

2. Equations of motion Two right-handed co-ordinate systems are employed to describe the motions. An inertial reference frame (C,x,y,z) is assumed to be fixed at the mean ship motion, defined by the ship speed U. Regular waves are assumed to travel forming an angle w with ship course. Another reference frame ðO; x; ¯ y¯ ; z¯ Þ is fixed at the ship having the xy plane coinciding, for the ship at rest, with the undisturbed sea surface, z¯ -axis passing through the vertical that contains the centre of gravity. The two systems coincide when excitations are absent. See Fig. 1. Non-linear equations of motion considering the three restoring degrees of freedom may be expressed in matrix form using a displacement vector:
T ~ sðtÞ ¼ zðtÞ fðtÞ yðtÞ

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Fig. 1. Co-ordinate axis and definition of the six degrees of motions.

defining the heave translational motion and the roll and pitch angular modes, indicated in Fig. 1: _ zÞ. € _~ ~ext ðz; z; ~~ ~res ð~ ~ þ AÞ ~ fÞ s; zÞ ¼ C (1) ðM s€ þ Bð s_ þ C ~ Hull inertia M is a diagonal 3
3 matrix. Its elements are: m, the ship mass, Jxx, Jyy the mass moments of inertia in the roll and pitch modes, respectively, taken with reference to the chosen origin. Elements in matrix A~ represent hydrodynamic generalized added masses. Following the reasoning of Abkowitz (1969), these _ describe ~ fÞ hydrodynamic reactions will be taken as linear. Damping terms Bð hydrodynamic reactions dependent on ship velocities, and may incorporate non~res ð~ linear terms in the roll equation. Vector C s; zÞ describes non-linear restoring forces and moments dependent on the relative motions between ship hull and wave _ zÞ € ~ext ðz; z; elevation z(t). On the right-hand side of Eq. (1), the generalized vector C represents wave external excitation, usually referred to in the literature as the Froude-Krilov plus diffraction wave forcing terms, dependent on wave heading w, encounter frequency oe, wave amplitude Aw and time t. In component form, the heave force and roll and pitch moments will be defined as _ zÞ € ¼ ½Z W ðtÞ K W ðtÞ ~ext ðz; z; C

M W ðtÞ T .

It is observed that for longitudinal waves, there is no roll external excitation: K W ðtÞ ¼ 0.

3. Hydrodynamic coefficients associated with potential theory The hydrodynamic inertia and damping matrices are expressed respectively, as 2 3 2 3 0 Z y_ Z z_ Z z€ 0 Z y€ 6 6 _ 0 7 0 7 A~ ¼ 4 0 K f€ 5; B~ ¼ 4 0 K f_ ðfÞ 5. M z€ 0 M y€ M z_ 0 M y_

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_ all the other terms in It is generally accepted that, with the exception of K f_ ðfÞ, ~ matrices A and B~ may be evaluated by means of potential theory, as described, for instance, in Salvesen et al. (1970), Meyers et al. (1975). The uncoupled nature of hydrodynamic reactions in the roll, on one side, and the heave and pitch modes on the other, is evident from the structure of the A~ and B~ matrices. 4. Non-linear roll damping Due to the occurrence of strong viscous effects, the roll damping moment can not be satisfactorily computed by means of potential theory. The semi-empirical model of roll damping moment due to Ikeda, see Himeno (1981), may be used to estimate the main linear and non-linear viscous effects. In a modular way, the model assumes that the total roll damping may be subdivided into five main components, each one being computed separately: _ f_ ¼ BW þ BF þ BE þ BL þ BBK , K f_ ðfÞ where BW, BF, BE, BL and BBK correspond to wave, friction, eddy, lift and bilge keel damping, respectively. Damping due to the bilge keels may also be split into components: BBK ¼ BBKN þ BBKH þ BBKW , where BBKN, BBKH and BBKW are contributions due to normal force, interaction between hull and bilge keel and wave system generated by the bilge keels, respectively. Mathematically, the roll damping moment may be represented to third order as 3

_ _ _ f_ ¼ K _ f K f_ ðfÞ _f _ f_ f f þ Kf

or, in a more convenient (and equivalent) form, see Bass and Haddara (1988): _ _ _ _ f_ ¼ K _ f K f_ ðfÞ _ jf_ j fjfj. f þ Kf

(2)

Coefficients K f_ and K fj _ fj _ may be computed using expressions given in Himeno (1981). 5. Non-linear restoring actions We consider now the force and moments dependent on position. To third order, positional terms due to the combined actions of ship motions in calm water and wave elevation along the hull may be formally expressed in terms of a Taylor series expansion about the average position. For this purpose, it is convenient to define a generalized vector ~ q ¼ ½~ s; zT such that the positional actions are:      4 4 X 4 4 X 4 X 4 2~ 3~ X X X ~pos  q C 1 q 1 q C C   pos pos ~pos ¼ C  qi þ  qi qj þ  qi qj qk :   2 6 qq qq qq @q qq qq i i j i j k i¼1 i¼1 j¼1 i¼1 j¼1 k¼1 0

0

0

(3)

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Wave elevation, according to the Airy linear theory, see Newman (1977), may be defined as zðx; y; t; wÞ ¼ Aw cos½kx cosðwÞ þ ky sinðwÞ  oe t, where Aw is the wave amplitude, k the wave number, defined as k ¼ o2w =g ¼ 2p=l, ow the wave frequency, l the wave length, g the acceleration of gravity, w the wave heading (incidence), oe the encounter frequency, defined as oe ¼ ow  kU cos w. In longitudinal waves, head seas, the equation of wave surface elevation is zðx; tÞ ¼ Aw cos½kx þ oe t.

(4)

It is important to note in Eq. (3) that the first-, second- and third-order terms independent of ~ s ¼ ½ z f y T may be more appropriately accommodated on the right-hand side of Eq. (1). Following the formalism of Eq. (3):      2~ 3~ ~pos  q C 1 q 1 q C C   3 pos pos 2 ~extðFKÞ ðzÞ ¼ C z þ  zþ   z . 2 qz2  6 qz3  qz  0 0 0 These terms represent linear and non-linear Froude-Krilov force/moments _ zÞ € appearing on the right~ext ðz; z; excitations, to be included in the forcing vector C hand side of Eq. (1), in which are cast all the terms independent of the ship motions vector ~ s. ~res , representing non-linear restoring actions, will be expressed as The vector C ~res ¼ ½ZðtÞ C

~pos ðz; f; y; zÞ  C ~extðFKÞ ðzÞ. KðtÞ MðtÞ T ¼ C

(5)

It is clear that the heave, roll and pitch motions will be completely coupled to each other. We start to introduce the derivative nomenclature to be used onwards. In the Taylor expansions, as an example, below we show all the heave coefficients, where the notation ‘‘j0’’ used to represent derivatives of forces and moments at average equilibrium position was abandoned: First order: 

qZ ¼ Zz ; qz



qZ ¼ Zf ; qf



qZ ¼ Zy . qy

(6)

Second order: 

q2 Z ¼ Z zz ; qz2



q2 Z q2 Z ¼ ¼ Zzf ; qzqf qfqz



q2 Z q2 Z ¼ ¼ Zfy . qfqy qyqf



q2 Z ¼ Z ff ; qf2 



q2 Z ¼ Zyy , qy2

q2 Z q2 Z ¼ ¼ Zzy , qzqy qyqz ð7Þ

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Third order:  

q3 Z ¼ Z zzz ; qz3



q3 Z ¼ Z fff ; qf3



q3 Z ¼ Zyyy , qy3

q3 Z q3 Z q3 Z ¼ Zzzf , ¼  ¼  qz2 qf qf qz2 qz qfqz

q3 Z q3 Z q3 Z ¼ Z zzy , ¼  ¼  qz2 qy qy qz2 qz qy qz q3 Z q3 Z q3 Z ¼ Z ffz ,  2 ¼ ¼  qf qz qf qf qz qz qf2 



q3 Z q3 Z q3 Z ¼ Z ffy , ¼ ¼ 2 2 qf qy qf qf qy qy qf

q3 Z q3 Z q3 Z ¼ Z yyz , ¼  ¼  qy qz qy qy2 qz qz qy2 q3 Z q3 Z q3 Z ¼ Zyyf ,  2 ¼ ¼  qy qf qy qy qf qf qy2 



q3 Z q3 Z q3 Z q3 Z ¼ ¼ ¼ qz qf qy qf qz qy qy qz qf qf qy qz ¼

q3 Z q3 Z ¼ ¼ Z zfy . qy qf qz qz qy qf

ð8Þ

6. Restoring coefficients in calm water Considering the physics of the fluid-body interaction, see Paulling and Rosenberg (1959), it is evident that the dependencies of the heave and pitch coefficients on roll are even; and the coefficients in roll due to heave and pitch are odd; it follows that some of the coefficients in the Taylor expansions are nil: Zf ¼ Zzf ¼ Z fy ¼ Z fff ¼ Z zzf ¼ Z yyf ¼ Z zfy ¼ 0, K z ¼ K y ¼ K zz ¼ K ff ¼ K yy ¼ K zy ¼ K zzz ¼ K yyy ¼ K zzy ¼ K ffz ¼ K ffy ¼ K yyz ¼ 0, M f ¼ M zf ¼ M fy ¼ M fff ¼ M zzf ¼ M yyf ¼ M zfy ¼ 0.

ð9Þ

A general derivation (up to the third order) of all restoring terms is given in Appendix A. Tables 1a–1c shows all the calm water geometric derivatives given in terms of relevant hull geometric characteristics. 7. Interactions between non-linear motions and incident wave Wave effects due to incident waves of arbitrary direction along the hull are modelled as a change of the hull average submerged shape defined by the

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instantaneous position of the wave. Contributions independent of the ship motions, _ zÞ € forcing term, as is usually adopted, appear on the right~ext ðz; z; contained in the C hand side of Eq. (1), such that: _ zÞ € ¼C _ zÞ € ¼ ½Z W ðtÞ ~extðFKÞ ðzÞ þ C ~extðDifÞ ðz; ~ext ðz; z; C

K W ðtÞ M W ðtÞ T .

Higher order diffraction force and moments are assumed not to be relevant; for this reason, in the present mathematical model, diffraction forces/moments are defined as being proportional to the first-order wave motion. Non-linear effects due to body–wave interactions proportional to second and third-order hull-wave relative displacements, due to their mathematical affinity with the purely hydrostatic terms, are considered on the left-hand side of the equations of motion, Eq. (5), being named wave passage effects (on the parametric excitation). Employing similar nomenclature and procedures to those used for the establishment of the hydrostatic terms, and considering that the coefficients corresponding to body/incident-wave relative motion follow the same logics of Eq. (9), the following interaction derivatives are found to be zero: Z zf ¼ Zzzz ¼ Z zzz ¼ Zzzf ¼ Z zzf ¼ Z yyz ¼ Z zzy ¼ Z zzy ¼ Z zfy ¼ 0, K zz ¼ K zy ¼ K zzz ¼ K zzz ¼ K zff ¼ K zyy ¼ K zzy ¼ K zzy ¼ 0, M zf ¼ M zzz ¼ M zzz ¼ M zzf ¼ M zzf ¼ M yyz ¼ M zzy ¼ M zzy ¼ M zfy ¼ 0. ð10Þ Consequently, the non-zero derivatives are obtained as defined in Tables 2a and b in terms of hull and incident wave characteristics. It may be observed that, up to Table 1a Hydrostatic restoring coefficients (calm water)—Linear Heave

Roll

Pitch

Zz ¼ rgA0 Zf ¼ 0 Zy ¼ rgA0 xf 0

Kz ¼ 0 K f ¼ rg½r 0 ð¯zB0  z¯ G Þ þ I xx0  Ky ¼ 0

M z ¼ rgA0 xf 0 Mf ¼ 0 M y ¼ rg½r 0 ð¯zB0  z¯ G Þ þ I yy0 

Table 1b Hydrostatic restoring coefficients (calm water)—Second order Heave R y Zzz ¼ 2rg L q¯ q¯z dx Zzf ¼ 0 R y Zzy ¼ 2rg L x¯ q¯ dx R q¯z2 q¯y Zff ¼ 2rg L y¯ q¯z dx Zfy ¼ 0 R y Zyy ¼ 2rg L x¯ 2 q¯ q¯z dx

Roll

Pitch

K zz ¼ 0 K zf ¼ 2rg K zy ¼ 0 K ff ¼ 0 K fy ¼ 2rg K yy ¼ 0

R

R L

y ¯ 2 q¯ Ly q¯z

dx

q¯y q¯z

dx

x¯y2

M zz ¼ 2rg M zf ¼ 0

R L

x¯

q¯y q¯z

dx

R y M zy ¼ 2rg L x¯ 2 q¯ q¯z dx R 2 q¯y M ff ¼ 2rg L x¯ ¯ y q¯z dx M fy ¼ 0 R y M yy ¼ 2rg L x¯ 3 q¯ q¯z dx

a

   2 R y dx þ A x K zfy ¼ rg 4 L x¯ ¯ y q¯ 0 f 0 q¯z

M yyf ¼ 0

M zzf ¼ 0 M fff ¼ 0

  R  y2 dx þ A K zzf ¼ rg 4 L y¯ q¯ 0 q¯z   R 3 q¯y2 K fff ¼ rg 8 L y¯ q¯z dx þ 2I xx0    2 R y dx þ I yy0 K yyf ¼ rg 4 L x¯ 2 y¯ q¯ q¯z

Z yyf ¼ 0

Z zzf ¼ 0 Z fff ¼ 0

a

a

a

M zfy ¼ 0

M yyy ¼ 0

a

   2 R y dx þ I M ffy ¼ rg 4 L x¯ 2 y¯ q¯ yy0 q¯z

M zzy ¼ 0

K yyy ¼ 0

K ffy ¼ 0

K zzy ¼ 0

Zyyy ¼ 0

   2 R y dx þ A x Zffy ¼ rg 4 L x¯ ¯ y q¯ 0 f 0 q¯z

Zzzy ¼ 0

a These expressions were obtained analytically for the case of a ship with inclined wall side, corresponding to a good approximation in the case of ships of conventional forms, small displacements and smooth transversal curvatures ðq2 y¯ =q¯z2 ! 0Þ at the considered water-line.

Heave-roll-pitch coupling Zzfy ¼ 0

M yyz ¼ 0

M ffz

a

a

   2 R y ¼ rg 4 L x¯ dx þ A x ¯ y q¯ 0 f0 q¯z

Pitch M zzz ¼ 0

K yyz ¼ 0

K ffz ¼ 0

Roll K zzz ¼ 0

Zyyz ¼ 0

Zffz

  R  y2 ¼ rg 4 L y¯ q¯ dx þ A 0 q¯z

a

1862

Heave Zzzz ¼ 0

Table 1c Hydrostatic restoring coefficients (calm water)—Third order

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Table 2a Derivatives due to wave passage—Second order Heave

Roll

Zzz ðtÞ ¼ 2rg Zzf ðtÞ ¼ 0

R

q¯y L q¯z

Zzy ðtÞ ¼ 2rg

R

K zz ðtÞ ¼ 0

z dx

¯ Lx

q¯y q¯z

z dx

K zf ðtÞ ¼ 2rg K zy ðtÞ ¼ 0

Pitch R

y ¯ 2 q¯ Ly q¯z

z dx

R y M zz ðtÞ ¼ 2rg L x¯ q¯ q¯z z dx M zf ðtÞ ¼ 0 R y M zy ðtÞ ¼ 2rg L x¯ 2 q¯ q¯z z dx

second order, the coefficients in heave, roll and pitch are identical to those employed in Neves and Valerio (2000), see Table 2a. It is clear that the above coefficients are dependent on hull characteristics and on wave amplitude, frequency and time. These coefficients may be expressed in terms of their cosine and sine terms, such that their dependence on wave amplitude becomes explicit. Thus, for example, in the case of roll motion: K zf ðtÞ ¼ Aw K zfc cosðoe tÞ þ Aw K zfs sinðoe tÞ, K zzf ðtÞ ¼ A2w K zzfc cosð2oe tÞ þ A2w K zzfs sinð2oe tÞ þ A2w K zzf0 , K zzf ðtÞ ¼ Aw K zzfc cosðoe tÞ þ Aw K zzfs sinðoe tÞ, K zfy ðtÞ ¼ Aw K zfyc cosðoe tÞ þ Aw K zfys sinðoe tÞ,

ð11Þ

where the newly defined coefficients, independent of wave amplitude and time, are as given in Table 3. These coefficients are dependent on wave frequency and hull geometry. Similar derivations were presented by Paulling (1961), but with some simplifications in the third-order terms. The present derivation, based on the instantaneous wave surface intercepting the hull, aggregates the various integral terms involving the longitudinal distribution of y¯ ðq¯y=q¯zÞ2 in Table 3.

8. Non-linear heave–roll–pitch equations of motions Taking into account the developments of sessions 3 to 7, Eq. (1) may be expressed in the form of the following non-linear set of coupled equations for the heave, roll and pitch modes, valid for longitudinal waves: 1 1 1 ðm þ Z z€ Þ€z þ Zz_ z_ þ Z y€ y€ þ Zy_ y_ þ Zz z þ Zy y þ Z zz z2 þ Zff f2 þ Z yy y2 2 2 2 1 1 1 1 1 2 2 3 2 þ Z zy zy þ Z zzz z þ Z zzy z y þ Z ffz f z þ Z ffy f y þ Z yyz y2 z 6 2 2 2 2 1 3 þ Z yyy y þ Z zz ðtÞz þ Z zy ðtÞy þ Z zzz ðtÞz þ Z zzz ðtÞz2 þ Z zzy ðtÞy 6 þ Z zzy ðtÞzy þ Z ffz ðtÞf2 þ Z yyz ðtÞy2 ¼ Zw ðtÞ,

a

a

a

L

R

R

    2 y 2¯y q¯ þ y¯ z dx q¯z

   2 y 2x¯ þ x¯ ¯ y q¯ ¯ y z dx q¯z

L

R

M zzf ðtÞ ¼ 0 M zzf ðtÞ ¼ 0 M yyz ðtÞ ¼ 0 a

K yyz ðtÞ ¼ 0

L

    2 y 2¯y q¯ þ y¯ z2 dx q¯z     2 R y þ 2¯y z dx K zzf ðtÞ ¼ rg L 4¯y q¯ q¯z K zzf ðtÞ ¼ rg

Zzzf ðtÞ ¼ 0 Zzzf ðtÞ ¼ 0 Zyyz ðtÞ ¼ 0 a

Same hypothesis are applied, as in the case of geometric derivatives in calm water.

M ffz ðtÞ ¼ rg

Pitch M zzz ðtÞ ¼ 0 M zzz ðtÞ ¼ 0

K ffz ðtÞ ¼ 0

K zzz ðtÞ ¼ 0

Roll K zzz ðtÞ ¼ 0

Zffz ðtÞ ¼ rg

Heave Zzzz ðtÞ ¼ 0a Zzzz ðtÞ ¼ 0 a

   2 q¯y 4 x¯ y þ 2 x¯ y z dx ¯ ¯ L q¯z R M zzy ðtÞ ¼ 0 a M zzy ðtÞ ¼ 0 a M zfy ðtÞ ¼ 0

K zfy ðtÞ ¼ rg

K zzy ðtÞ ¼ 0

K zzy ðtÞ ¼ 0

Zzzy ðtÞ ¼ 0 a Zzzy ðtÞ ¼ 0 a Zzfy ðtÞ ¼ 0

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Table 2b Derivatives due to wave passage—Third order

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Table 3 Coefficients of wave passage in the roll equation R y K zfc ¼ 2rg L y¯ 2 q¯ cosðkxÞ dx  q¯z   2 R q¯y K zzfc ¼ rg L y¯ q¯z þ 12 y¯ cosð2kxÞ dx     2 R y 1 þ y dx K zzf0 ¼ rg L y¯ q¯ ¯ 2 q¯z     2 R y þ 2¯ y cosðkxÞ dx K zzfc ¼ rg L 4¯y q¯ q¯z     2 R y þ 2x¯ K zfyc ¼ rg L 4x¯ ¯ y q¯ ¯ y cosðkxÞ dx q¯z

R y K zfs ¼ 2rg L y¯ 2 q¯ sinðkxÞ dx  q¯z   2 R q¯y K zzfs ¼ rg L y¯ q¯z þ 12 y¯ sinð2kxÞ dx

    2 q¯y 4¯ y þ 2¯ y sinðkxÞ dx L q¯z i @¯y 2 R h ¼ rg L 4x¯ ¯ y @¯z þ 2x¯ ¯ y sinðkxÞ dx

K zzfs ¼ rg K zfys

R

1 2 _ þ K _ _ fj _ _ ðJ xx þ K f€ Þf€ þ K f_ f fjfj fj þ K f f þ K zf zf þ K fy fy þ K zzf z f 2 1 1 þ K fff f3 þ K yyf y2 f þ K zfy zfy þ K zf ðtÞf 6 2 þ K zzf ðtÞf þ K zzf ðtÞzf þ K zfy ðtÞfy ¼ 0; 1 1 ðJ yy þ M y€ Þy€ þ M y_ y_ þ M z€ z€ þ M z_ z_ þ M z z þ M y y þ M zz z2 þ M ff f2 2 2 1 1 1 1 1 þ M yy y2 þ M zy zy þ M zzz z3 þ M zzy z2 y þ M ffz f2 z þ M ffy f2 y 2 6 2 2 2 1 1 þ M yyz y2 z þ M yyy y3 þ M zz ðtÞz þ M zy ðtÞy þ M zzz ðtÞz 2 6 þ M zzz ðtÞz2 þ M zzy ðtÞy þ M zzy ðtÞzy þ M ffz ðtÞf2 þ M yyz ðtÞy2 ¼ M w ðtÞ.

ð12Þ

Analytic expressions for the determination of all—linear and non-linear–derivatives have been given in Tables 1–3. It is pointed out that in general parametric resonance is modelled in the literature considering uncoupled versions of the roll equation, with: (a) vertical motions, z(t) and y(t), assumed to be purely harmonic, (b) roll parametric excitation defined with non-linearities up to the second order, and (c) different levels of non-linearities in the calm water GZðfÞ curve, as discussed in Blocki (1980), Neves et al. (1999), Spyrou (2000), Umeda et al. (2003), Bulian et al. (2003). In the nomenclature adopted in Eq. (12), such a simplified model of parametric rolling would read as _ þ K _ _ fj _ _ ðJ xx þ K f€ Þf€ þ K f_ f fjfj fj þ K f f þ ½K zf z þ K fy y þ K zf ðtÞf 1 þ K fff f3 ¼ 0, 6

ð13Þ

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where the three terms within brackets represent contributions to parametric excitation (to second order) due to the heave and pitch motions and to volumetric changes in the submerged hull produced by the wave passage, respectively. Clearly, these contributions are proportional to the wave amplitude. Eq. (13) is sometimes referred as Mathieu–Duffing equation.

9. Theoretical analysis of perturbed roll motions Stability of motion of the non-linear set of equations may be assessed by means of the variational system, Cesari (1971). In its linear form it may be derived under the assumption that the non-linear motions may be decomposed as the sum of steady oscillatory solutions plus some small perturbations: zðtÞ ¼ z^ðtÞ þ xðtÞ ¼ Aw Z3 cosðoe t þ az Þ þ xðtÞ, ^ þ jðtÞ ¼ Aw Z cosðoe t þ af Þ þ jðtÞ, fðtÞ ¼ fðtÞ 4

^ þ WðtÞ ¼ Aw Z cosðoe t þ ay Þ þ WðtÞ, yðtÞ ¼ yðtÞ 5 ^ ^ where z^ðtÞ, fðtÞ and yðtÞ correspond to the heave, roll and pitch well known linear solutions, and Z3 ,Z4 ,Z5 are the corresponding transfer functions. Perturbations in the heave, roll and pitch modes are defined as xðtÞ, jðtÞ, and WðtÞ, respectively. The linear variational equation of the roll motion associated with Eq. (12) is then derived as ^ yÞx ^ _ þ K f j þ ðK zf f^ þ K zzf z^f^ þ K zfy f ðJ xx þ K f€ Þj€ þ K f_ j

 2 2 1 1 1 2 ^ ^ ^ ^ þ K zf z^ þ K fy y þ K fff f þ K zzf z^ þ K yyf y þ K zfy z^y j 2 2 2 ^ ^ ^ ^ þ K zzf ðtÞ^zj ^ þ ðK fy f þ K yyf fy þ K zfy z^fÞW þ K zf ðtÞj þ K zzf ðtÞfx ^ þ K zfy ðtÞfW ^ þ K zzf ðtÞj ¼ 0. þ K zfy ðtÞyj

ð14Þ

In the particular case of longitudinal waves, the roll linear solution is zero. That is, ^  0. Hence: f _ þ K f j þ ½K zf z^ þ K fy y^ þ K zf ðtÞj ðJ xx þ K f€ Þj€ þ K f_ j   2 1 1 K zzf z^2 þ K yyf y^ þ K zfy z^y^ j þ 2 2 þ ½K zzf ðtÞ^z þ K zfy ðtÞy^ þ K zzf ðtÞj ¼ 0,

ð15Þ

in which it is possible to identify the distinct third-order contributions to parametric excitation, from purely hydrostatic actions and wave interaction effects, respectively. The three terms inside the third bracket, corresponding to third-order Froude-Krilov moments, are given in Eq. (11) and Table 3.

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For the sake of comparison, it is noted here that the linear variational equation derived from Eq. (13) is in the form of a Mathieu equation: _ þ K f j þ ½K zf z þ K fy y þ K zf ðtÞj ¼ 0. ðJ xx þ K f€ Þj€ þ K f_ j

(16)

Returning to the examination of Eq. (15): using a very well known trigonometric relationship, it is possible to relate squared oscillatory functions to the sum of a constant term plus another oscillatory function with twice the frequency. For instance (from Eq. (11)): K zzf ðtÞ ¼ A2w K zzf0 þ A2w K zzfc cosð2oe tÞ þ A2w K zzfs sinð2oe tÞ. The same reasoning may be extended to all terms inside the second bracket in Eq. (15). Then it becomes apparent that each of these terms will introduce superharmonics proportional to wave amplitude squared, together with constant terms, again proportional to the wave amplitude squared. The occurrence of super-harmonics in the parametric excitation means that the essential dynamics behind the non-linear motions described by Eq. (12) is not in the form of coupled Mathieu equations; it is in fact a set of Hill equations. On the other hand, the constant terms may be interpreted as additional stiffness (proportional to wave amplitude squared) incorporated into the dynamical system through third-order terms. Thus, it may be concluded that the extension of the mathematical model to third-order brings new dynamic characteristics into the resulting motions.

10. Numerical results for the heave and pitch motions Numerical integration of Eq. (12) has been performed for a transom stern fishing vessel, here denominated TS, which is very sensitive to parametric excitation. Hull form and main characteristics of the vessel are given in Appendix B. First, the results for the vertical motions are discussed. In head seas, the heave and pitch motions are very important in favouring the occurrence of roll parametric amplification. For small wave amplitudes, these vertical motions are typically linear. Yet, for larger waves, these motions may lose this characteristic, and asymmetries may be noticed and become remarkable. A very complex set of coupling effects may now intervene in the resulting ship dynamics. Fig. 2 shows simulations of the heave and pitch motions for the considered ship in head seas, F n ¼ 0:11, wave amplitude Aw ¼ 0:5 m, oe ¼ 2on4 . Results from the third-order model are plotted against the linear response. It is observed that both third-order heave and pitch motions are comparable to the linear displacements. In Fig. 3 the same curves are obtained for the same speed and encounter frequency, but for a higher wave, Aw ¼ 01:0 m. Now, some asymmetries are observed in the responses of both modes, making it clear that the heave and pitch motions are now far from the linear range. This result points out to the fact that the exchange of energy between the vertical modes and the roll motion, which is an

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Heave - Ship TS: GM = 0.50 m, Fn = 0.11 We = 2Wn4, Aw = 0.50 m, ksi = 180 °

1.00

Heave (Linear) Heave (3rd order) Heave [m]

0.50 0.00 -0.50 -1.00

0

25

50 time [s]

75

Pitch - Ship TS: GM = 0.50 m, Fn = 0.11 We = 2Wn4, Aw = 0.50 m, ksi = 180 °

0.525

Pitch (Linear) Pitch (3rd order)

0.350 Pitch [rad]

100

0.175 0.000 -0.175 -0.350 -0.525

0

25

50 time [s]

75

100

Fig. 2. Linear and non-linear responses in heave and pitch (Aw ¼ 0:50 m).

essential aspect of the dynamics of parametric resonance in head seas, must be described as a set of coupled equations if all the complexities of excessive motions are to be taken into consideration.

11. Results for the roll parametric amplifications Figs. 4–9 show the roll motion obtained from numerical integration of the thirdorder mathematical model described above. These simulations are given for conditions that had been tested previously, Neves et al. (2002). In each of the figures, the results for the third-order model are compared to the experimental results and to numerical integration of the second-order counter part of the present model. In the numerical integrations the restoring curves have been approximated by seventh-order polynomials. Fig. B2 in Appendix B shows the static restoring curves (GZ curves) for two distinct metacentric heights, where it may be observed that the polynomials give a fair description of the calm water restoring moment. It was decided to retain the seventh-order modelling of the roll-roll restoring curve in the numerical simulations, yet describing all the coupling effects of the parametric excitation with terms up to the third order.

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Heave - Ship TS: GM = 0.50 m, Fn = 0.11 We = 2Wn4, Aw = 1.00 m, ksi = 180˚ 1.00

Heave (Linear) Heave (3rd order)

Heave [m]

0.50 0.00 -0.50 -1.00

0

25

50 time [s]

75

100

Pitch - Ship TS: GM = 0.50 m, Fn = 0.11 We = 2Wn4, Aw = 1.00 m, ksi = 180˚ 0.525 Pitch (Linear) Pitch (3rd order)

Pitch [rad]

0.350 0.175 0.000 -0.175 -0.350 -0.525

0

25

50 time [s]

75

100

Fig. 3. Linear and non-linear responses in heave and pitch, Aw ¼ 1:00 m.

Figs. 4 and 5 show the roll motion for GM ¼ 0:37 m, F n ¼ 0:15, oe ¼ 2on4 ¼ 1:717 rd=s. In Fig. 4, for a low wave amplitude, Aw ¼ 0:45 m, there is excellent agreement between the third-order model and the experimental results, whereas the second-order model gives responses well in excess of the other curves. For the same conditions, but for a higher wave amplitude, Fig. 5, the third-order model gives results slightly lower than the experiments, but the comparison may be said to be quite good. The second-order model is completely incapable of producing results comparable to the time series obtained from the experiments. Very similar interpretations are applicable to the cases shown in Figs. 6 and 7, corresponding to GM ¼ 0:37 m, F n ¼ 0:20 and F n ¼ 0:30, oe ¼ 2on4 ¼ 1:717 rd=s. In both conditions, agreement between the third-order model and experiments is quite good, whereas the second-order model fails to reproduce the experiments. Important to say, the condition given in Fig. 7 is one of quite excessive roll motions, of the order of 38 degrees. It should be observed that for the high GM case (GM ¼ 0:50 m, oe ¼ 2on4 ¼ 1:968 rd=s), Figs. 8 and 9, possibly due to the higher GM, the roll parametric amplification is not so intense. In particular, in the case of Fig. 9, F n ¼ 0:15, Aw ¼ 0:60 m, the second and third-order models give good descriptions of the time evolution of the roll motion.

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Ship TS: GM = 0.37 m, Fn =0.15 We = 2Wn4, Aw = 0.45 m, ksi=180˚

Roll Angle [deg]

60 3rd.order 2nd.order Experimental

40 20 0 -20 -40 -60

0

25

50

75 time [s]

100

125

150

Fig. 4. Roll motion, GM ¼ 0:37 m, F n ¼ 0:15, Aw ¼ 0:45 m.

Ship TS: GM = 0.37 m, Fn =0.15 We = 2Wn4, Aw = 1.02 m, ksi = 180°

Roll Angle [deg]

60

3rd. order 2nd. order Experimental

40 20 0 -20 -40 -60

0

25

50

75 time [s]

100

125

150

125

150

Fig. 5. Roll motion, GM ¼ 0:37 m, F n ¼ 0:15, Aw ¼ 1:02 m.

Roll Angle [deg]

60 40 20

Ship TS: GM = 0.37 m, Fn =0.20 We = 2Wn4, Aw = 0.60 m, ksi = 180° 3rd. order 2nd. order Experimental

0 -20 -40 -60 0

25

50

75 time [s]

100

Fig. 6. Roll motion, GM ¼ 0:37 m, F n ¼ 0:20, Aw ¼ 0:60 m.

ARTICLE IN PRESS M.A.S. Neves, C.A. Rodrı´guez / Ocean Engineering 33 (2006) 1853–1883 Ship TS : GM = 0.37 m, Fn =0.30 We = 2Wn4, Aw = 0.78 m, ksi = 180°

60 Roll Angle [deg]

1871

3rd. order 2nd. order Experimental

40 20 0 -20 -40 -60

0

25

50

75 time [s]

100

125

150

125

150

125

150

Fig. 7. Roll motion, GM ¼ 0:37 m, F n ¼ 0:30, Aw ¼ 0:78 m.

Ship TS: GM = 0.50 m, Fn =0.11 We = 2Wn4, Aw = 0.63 m, ksi = 180°

Roll Angle [deg]

60

3rd. order 2nd. order Experimental

40 20 0 -20 -40 -60

0

25

50

75 time [s]

100

Fig. 8. Roll motion, GM ¼ 0:50 m, F n ¼ 0:11, Aw ¼ 0:63 m.

Ship TS: GM = 0.50 m, Fn =0.15 We = 2Wn4, Aw = 0.60 m, ksi = 180°

Roll Angle [deg]

60

3rd. order 2nd. order Experimental

40 20 0 -20 -40 -60

0

25

50

75 time [s]

100

Fig. 9. Roll motion, GM ¼ 0:50 m, F n ¼ 0:15, Aw ¼ 0:60 m.

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12. Conclusions A derivative mathematical model was introduced, in which the heave, roll and pitch motions and wave passage effects were described with coupling terms up to the third order. Analytical derivation of all the coupling coefficients due to heave, roll and pitch, and wave passage has been given. The derivation is structured such that all coupling coefficients are expressed in terms of simple hull-wave characteristics. This aspect should be recognized as relevant from the practical point of view of understanding and correlating the complex non-linear responses to practical hull configurations. A qualitative analysis of the coupled non-linear system was performed. The analysis points out to the possible appearance of super-harmonics and additional stiffness due to the heave and pitch motions and wave passage. The essential dynamic characteristics are those of a Hill equation with a hardening term proportional to wave amplitude squared. It is shown that, depending on the level of excitation, the heave and pitch motions may display significant asymmetries. The appearance of these non-linear effects is indicative of the importance of taking the ship motions in a coupled way, whenever strong parametric excitation may occur. Roll amplifications for low metacentric height tended to be strong. In these cases the third-order numerical simulations based on the proposed model give quite good comparisons with the experimental results. The increased stiffness, proportional to wave amplitude squared, is thought to be responsible for giving to the third-order model such a realistic description of the parametric resonance. In the tested conditions with high metacentric height the roll amplification is not excessive. Second and third-order models do not give very distinct responses in these cases.

Acknowledgments The present investigation is supported by CNPq within the STAB project (Nonlinear Stability of Ships). The authors also acknowledge financial support from CAPES, FAPERJ and LabOceano.

Appendix A. Derivation of restoring coefficients A.1. Restoring force For the derivation of the hydrostatic heave force and restoring moments in roll and pitch, a methodology analogous to the one presented in Neves (2002) is employed. Vectors defined with respect to the coordinate systems of Fig. 1 of the ^ J; ^ K^ for the body fixed main text are: for the inertial system, unit vectors are I; ^ ^ ^ system, unit vectors are i; j; k.

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Z

z y

G

φ hi

O

bi

∆i dAi

C

Y

Bo

W B1

E1

Fig. A1. Weight and hydrostatic forces in a generically displaced condition.

By definition, the restoring force is the resultant between weight and instantaneous hydrostatic force: ~ þE ~1 , ~H ¼ W F where the weight is ^ ~ ¼ rgr 0 K. W The instantaneous hydrostatic force is ^ ~1 ¼ rgr 1 K^ ¼ rgðr 0 þ r ÞK. E Thus: ^ ~H ¼ rgr K. F In the above expressions, r is the density, r 0 is the submerged volume in the static equilibrium of the ship, r 1 is the instantaneous submerged volume, and r is the instantaneous incremental volume (variation of submerged volume) due to the motions of the ship in heave, roll and pitch, see Fig. A1. A.2. Restoring moments By definition, the restoring moment is the resultant between the moments of weight and instantaneous buoyancy: ~H ¼ M ~W þ M ~E M 1

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where   ~ W ¼ ! ~ ¼ rg ! M OG
W OG
r 0 K^ ,  !  h ! i X ! ! ~ ~ E1 ¼ OB ^ ^ M Ob i
dr i K^ 1
E 1 ¼ rg OB1
r 1 K ¼ rg OB 0
r 0 K þ It is clear that the restoring moment may be rewritten as " #  ! !  X ! ~ ^ ^ M H ¼ rg r 0 K
OG  OB 0 þ Ob i
dr i K . i

Referring again to Fig. A1, the following expressions may be derived: ! ^ OG ¼ z¯ G k, ! ^ OB0 ¼ z¯ B0 k, X X r ¼ dr i ¼ hi dAi , i

i

! hi ^ Ob i  xAi I^ þ yAi J^ þ K, 2 ^ k^ ¼ yI^  fJ^ þ K, where z¯ G is the vertical coordinate of the centre of gravity, z¯ B0 is the vertical coordinate of the centroid of the submerged volume, hi is the height of the elemental prism, and (xAi, yAi) are the coordinates of the elemental prism in the instantaneous plane of flotation, referred to system CXYZ. With substitutions and regrouping, the following vector expression is obtained for the restoring moment: " # X ~ H ¼  Irg ^ M r 0 ð¯zB0  z¯ G Þf  yAi hi dAi " ^  Jrg r 0 ð¯zB0  z¯ G Þy þ

i

X

# xAi hi dAi .

i

In order to determine the restoring force and the moments, an analytical expression will be derived to compute the height hi of each elemental prism. This expression, it is clear, is a function of the relevant variables z, f, and y, that is, the displacements in heave, roll and pitch (Neves, 2002). A.3. Generalized hydrostatic actions An approximation to obtain the height hi of each elemental prism (see Fig. A1) caused by a generic displacement of the ship (z, f, y) may be obtained assuming that

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each hi is composed of a linear superposition of three contributions: hi ¼ zr  yAi f þ xAi y, where (xAi, yAi), previously defined, is a point of a generic plane of flotation, and zr is the relative vertical displacement relative to the instantaneous elevation of the sea surface, defined by zr ¼ z  zðxAi ; yAi ; tÞ. In this section we will consider the free surface as un-deformed (z ¼ 0), such that zr ¼ z, but it is noted that wave passage effects may be obtained following the same procedure described in this Appendix. For zr ¼ z the volumetric variation will be X X X r ¼ dr i ¼ hi dAi ¼ ðz  yAi f þ xAi yÞdAi i

¼ z

X i

i

dAi  f

X

i

yAi dAi þ y

i

X

xAi dAi ;

i

where the summations represent known geometric properties of the plane of flotation: X dAi ¼ Aðz; f; yÞ area of instantaneous plane of flotation; i

X

yAi dAi ¼ Ayf ðz; f; yÞ

first static transversal moment of the area;

i

X

xAi dAi ¼ Axf ðz; f; yÞ first static longitudinal moment of the area;

i

such that:

r ¼ zAðz; f; yÞ  fAyf ðz; f; yÞ þ yAxf ðz; f; yÞ and the restoring force will be given by ^ ~H ¼ rg½zA þ fAyf  yAxf K. F For the restoring moment, again taking zr ¼ z and considering the following known geometric properties of the plane of flotation: X y2Ai dAi ¼ I xx ðz; f; yÞ transversal moment of inertia of the area; i

X

x2Ai dAi ¼ I yy ðz; f; yÞ longitudinal moment of inertia of the area;

i

X i

xAi yAi dAi ¼ I xy ðz; f; yÞ

product of inertia of the area;

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the following vector expression may be obtained for the restoring moment: ~ H ¼  rgf½Ayf z þ r 0 ð¯zB0  z¯ G Þf þ I xx f  I xy yI^ M ^ þ ½Axf z  I xy f þ r 0 ð¯zB0  z¯ G Þy þ I yy ygJ. Finally, the restoring force and moments may be written as Z H ¼ rgðAz þ Ayf f  Axf yÞ, K H ¼ rg½Ayf z þ r 0 ð¯zB0  z¯ G Þf þ I xx f  I xy y, M H ¼ rg½Axf z  I xy f þ r 0 ð¯zB0  z¯ G Þy þ I yy y. A.4. Hydrostatic actions up to third order It is noted that the above expressions are functions of geometric properties (A, Axf, Ayf, Ixx, etc.) of the instantaneous plane of flotation, which in turn depend on the instantaneous displacements of the ship (z, f, y). These properties may be approximated employing multi-variable Taylor series around the average position. Aiming at describing the hydrostatic actions up to third order, we consider the geometric functions expanded up to second order:       qA qA 1 q2 A 2 q2 A  1 q2 A 2 1 q2 A 2 A ¼ A0 þ  z þ  y þ z þ zy þ f þ y, qz 0 qy 0 2 qz2 0 qz qy0 2 qf2 0 2 qy2 0     qAxf  qAxf  1 q2 Axf  2 q2 Axf  Axf ¼ A0 xf 0 þ zþ yþ z þ zy 2 qz2 0 qz 0 qy 0 qz qy 0   1 q2 Axf  2 1 q2 Axf  2 þ f þ y , 2 qf2 0 2 qy2 0    qAyf  q2 Ayf  q2 Ayf   fþ Ayf ¼  zf þ  fy, qf 0 qz qf  qf qy  0

0

    qI xx  qI xx  1 q2 I xx  2 q2 I xx  z þ y þ z þ zy 2 qz2 0 qz 0 qy 0 qz qy 0   1 q2 I xx  2 1 q2 I xx  2 þ f þ y , 2 qf2 0 2 qy2 0

I xx ¼ I xx0 þ

I yy

    qI yy  qI yy  1 q2 I yy  2 q2 I yy  ¼ I yy0 þ zþ yþ z þ zy 2 qz2 0 qz 0 qy 0 qz qy0   1 q2 I yy  2 1 q2 I yy  2 þ f þ y, 2 qf2  2 qy2  0

0

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I xy ¼

1877

   qI xy  q2 I xy  q2 I xy  f þ zf þ fy, qf 0 qz qf0 qf qy0

where A0, A0xf0, Ixx0, Iyy0 are geometric properties of the average plane of flotation. Substitution of the given expansions in the previous general expressions of the restoring actions will result in third-order representations in terms of derivatives: "      qAyf  f2 qAxf  y2 q2 A z3 qA z2 qA   Z H ¼ rg A0 z  A0 xf 0 y þ  þ  zy þ  þ 2  qz 0 2 qy 0 qz 0 6 qf 0 2 qy 0 2  #     2 q2 A  z2 y q2 A f2 z q Ayf  f2 y q2 A y2 z q2 Axf  y3 þ 2 þ þ 2  þ ,  qz qy0 2 qf qy  2 qf 0 2 qy 0 2 qy2 0 6 0

"

  qI xx  qI xx  zf þ fy qz 0 qy 0 #     q2 I xx  z2 f q2 I xx  y2 f q2 I xx  q2 I xx  f3 þ 2  þ þ zfy þ , qz 0 2 qz@y 0 qy2 0 2 qf2 0 6

K H ¼ rg r 0 ð¯zB0  z¯ G Þf þ I xx0 f þ

  qAxf  z2 qI yy  þ zy M H ¼ rg A0 xf 0 z þ r 0 ð¯zB0  z¯ G Þy þ I yy0 y  qz 0 2 qz 0      qI xy  f2 qI yy  y2 q2 Axf  z3 q2 I yy  z2 y q2 Axf  f2 z  þ  þ  qf 0 2 qy 0 2 qz2 0 6 qz2 0 2 qf2 0 2 #    q2 I yy  f2 y q2 I yy  zy2 q2 I yy  y3 þ þ þ . qz qy0 2 qf2 0 2 qy2 0 6 

Considering that hydrostatic actions are functions with continuous partial derivatives, second and third order mixed derivatives are equal, and the following equalities are verified:     qAyf  qAxf  qA qI xx   ¼ ; ¼ , qy 0 qz 0 qf 0 qz 0     qAxf  qI xy  qI yy  qI xx  ¼ ; ¼ ,    qy 0 qf 0 qz 0 qy 0    q2 Ayf  q2 A q2 I xx  ¼ ,  ¼ qz qf  qz2 0 qf2 0 0    q2 Axf  q2 I yy  q2 A ¼ ¼ , qz qy 0 qz2 0 qy2 0   q2 Axf  q2 A  ¼  , qz qy qz2  0

0

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Table A1a Hydrostatic coefficients—Linear Heave

Roll

Pitch

Zz ¼ rgA0 Zf ¼ 0 Zy ¼ rgA0 xf 0

Kz ¼ 0 K f ¼ rg½r 0 ð¯zB0  z¯ G Þ þ I xx0  Ky ¼ 0

M z ¼ rgA0 xf 0 Mf ¼ 0 M y ¼ rg½r 0 ð¯zB0  z¯ G Þ þ I yy0 

Table A1b Hydrostatic coefficients—Second order Heave

Roll

Pitch

  Zzz ¼ rgqA qz 0

K zz ¼ 0

M zz ¼ rg

Zzf ¼ 0 Zzy

 ¼ rg qA

qy 0

Zff ¼ rg



qAyf  qf 0

Zfy ¼ 0 Zyy

 qAx  ¼ rg qy f 

 K zf ¼ rg qIqzxx 0 K zy ¼ 0

M zy ¼ rg

K ff ¼ 0

M ff ¼ rg

 K fy ¼ rg qIqyxx 0 K yy ¼ 0

0

M zf ¼ 0

M fy ¼ 0 M yy ¼ rg



qAxf  qz 0



qI yy  qz 0



qI xy  qf 0



qI yy  qy 0

    q2 Ayf  q2 I xy  q2 Axf  q2 I xx  ¼  ¼ ¼  ,  qz qy 0 qz qf0 qf qy  qf2 0 0    q2 I xy  q2 I yy  q2 I xx  ¼  ¼ , qf qy0 qy2 0 qf2 0   q2 I yy  q2 Axf  ¼ . qz qy0 qy2 0 Comparing the above results for ZH, and KH, MH with corresponding terms in Eq. (12) of the main text gives the expressions for the derivatives in terms of geometric properties of the plane of flotation, as shown in Tables A1a–c.

A.5. Expressions for the geometric derivatives It is possible and relevant to relate the derivatives given above to longitudinal distributions of hull characteristics. Paulling and Rosenberg (1959) presented a simple methodology for obtaining some of the geometric derivatives considering a second-order model, in terms of semi-breadth and flare defined at the average floating position. The same methodology was employed in Neves (2002) to obtain

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Table A1c Hydrostatic coefficients—Third order Heave Zzzz

Roll

 2  ¼ rg qqzA2 

Pitch

K zzz ¼ 0

0

Zzzf ¼ 0

K zzf ¼

 q2 A  Zzzy ¼ rg qz qy0  2  Zffz ¼ rg qqfA2 

K zzy ¼ 0

0

 2  Zyyz ¼ rg qqyA2 

0

Zyyy ¼ rg

  

qy2

0

qf2

0



q2 I yy   qf2 0

M ffy ¼ rg

K yyz ¼ 0

 q2 I yy  M yyz ¼ rg qz qy 

 2  K yyf ¼ rg qqyI xx 2 

M yyf ¼ 0

K yyy ¼ 0

M yyy ¼ rg

 2 xx  K zfy ¼ rg qqzIqy 

0

Zzfy ¼ 0

  

q2 Axf 

K ffy ¼ 0

0

0

q2 Axf 



M fff ¼ 0

0

Zyyf ¼ 0

0

q2 I yy   qz2 0

M ffz ¼ rg

 2  K fff ¼ rg qqfI xx 2 



q2 Ay  rg qf qyf 

qz2

M zzf ¼ 0 M zzy ¼ rg

K ffz ¼ 0

Zfff ¼ 0 Zffy ¼



2  rg q qzI 2xx  0

  

q2 Axf 

M zzz ¼ rg



q2 I yy   qy2 0

M zfy ¼ 0

0

the complete set of second-order derivatives. Here the procedure is extended to the present third-order model. When deriving these expressions with respect to z, f and y, it will be observed that the following rules of derivation are to be applied in the case of angular displacements in roll and pitch: q q qz ¼ ; qf qz qf

q q qz ¼ qy qz qy

and from the general transformation matrix between a rotated and a fixed frame of reference it can be deduced that in the case of roll: z ¼ y¯ sin f þ z¯ cos f;

‘

qz ¼ y¯ cos f  z¯ sin f qf

such that  qz  q q ¼ y¯ ¼ y¯ ) qf0 qf q¯z and in the case of pitch: z ¼ x¯ sin y þ z¯ cos y;

‘

qz ¼ x¯ cos y  z¯ sin y, qy

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 qz  q q ¼ x¯ ¼ x¯ ) qy0 qy q¯z  and it is also observed that qz=q¯z0 ¼ 1. Table A2 Second order geometric derivatives 



R y ¼ 2 L q¯ q¯z dx    R qAxf  y ¯ q¯ qz 0 ¼ 2 L x q¯z dx  qAyf  qz 0 ¼ 0  R 2 q¯y qI xx  ¯ q¯z dx qz 0 ¼ 2 L y  R 2 q¯y qI yy  ¯ q¯z dx qz 0 ¼ 2 L x  qI xy  qz  ¼ 0



R y ¼ 2 L x¯ q¯ q¯z dx    R qAxf  y ¯ 2 q¯ qy 0 ¼ 2 L x q¯z dx  qAyf  qy 0 ¼ 0    R y qI xx  ¯ y2 q¯ qy 0 ¼ 2 L x¯ q¯z dx2    R 3 q¯y qI yy  ¯ q¯z dx qy 0 ¼ 2 L x  qI xy  qy  ¼ 0

qA qf0

qA qz 0

qA qy 0

¼0  qAxf  qf 0 ¼ 0  R 2 q¯y qAyf  ¯ q¯z dx qf 0 ¼ 2 L y  qI xx  qf 0 ¼ 0  qI yy  qf 0 ¼ 0    R qI xy  y ¯ y2 q¯ qf  ¼ 2 L x¯ q¯z dx

0

0

0

Table A3 Third order geometric derivatives 

q2 A  qz2 0

¼0

a

 q2 A  qz qf0 ¼ 0  q2 Axf   ¼0a 2 qz  0 q2 Axf   qz qf  ¼ 0 0 q2 Ayf   ¼0 qz2  0 R q¯y2 q2 Ayf   ¯ q¯z dx þ A0 qz qf  ¼ 4 L y 0  2 R 2  y q I xx dx þ A0  ¼ 4 L y¯ q¯ q¯z qz2 0  q2 I xx  qz qf 0 ¼ 0  q2 I yy   ¼0a qz2 0  2 q I yy  qz qf 0 ¼ 0  q2 I xy   ¼0 qz2 0   2 R 2 q I xy  y dx þ A0 xf 0 ¯ y q¯ qz qf  ¼ 4 L x¯ q¯z 0

a



q2 A  qf2 0

¼4

R L

 2 y dx þ A0 y¯ q¯ q¯z

 a q2 A  qz qy0 ¼ 0   2 R q2 Axf  q¯y  ¼ 4 x¯ dx þ A0 xf 0 y ¯ 2 L  q¯z qf 0 2  q Axf a  qz qy  ¼ 0 0 q2 Ayf   ¼0 qf2  0 q2 Ayf   qz qy  ¼ 0 0  R 3 q¯y2 q2 I xx  ¼ 8 ¯ q¯z dx þ 2I xx0  2 Ly qf 0   2 R y q2 I xx  dx  A0 xf 0 ¯ y q¯ q¯z qz qy 0 ¼ 4 L x¯    2 R 2 q I yy  y  ¼ 4 L x¯ 2 y¯ q¯ q¯z dx þ I yy0 qf2 0  2 q I yy  a qz qy 0 ¼ 0  q2 I xy   ¼0 qf2 0  2 q I xy  qz qy  ¼ 0 0



q2 A  qy2 0

¼0a  q2 A  qf qy0 ¼ 0  q2 Axf   ¼0a 2 qy  0 q2 Axf   qf qy  ¼ 0 0 q2 Ayf   ¼0 qy2  0  2 R q2 Ayf  y  dx  A0 xf 0 ¯ y q¯ q¯z qf qy  ¼ 4 L x¯ 0    2 R 2  y q I xx dx þ I yy0  ¼ 4 L x¯ 2 y¯ q¯ q¯z qy2 0  q2 I xx  qf qy 0 ¼ 0  q2 I yy   ¼0a 2 qy 0  2 q I yy  qf qy 0 ¼ 0  q2 I xy   ¼0 qy2 0  R 2 q¯y2 2 q I xy  ¯ y¯ q¯z dx  I yy0 qf qy  ¼ 4 L x 0

These expressions were obtained analytically for the case of a ship with inclined wall side, corresponding to a good approximation in the case of ships of conventional forms, small displacements and smooth transversal curvatures ðq2 y¯ =q¯z2 ! 0Þ at the considered water line.

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With these results, the expressions for the non-vanishing second and third-order hydrostatic derivatives can be derived, with the integrations of sectional geometrical characteristics defined along the ship length. Tables A2 and A3 summarizes the results.

Appendix B. Characteristics of fishing vessel The hull form and main characteristics of the ship employed in the present investigation—here denominated TS—are give in Fig. B1 and Table B1, respectively. The ship was tested with two different metacentric heights, GM ¼ 0:37 m and GM ¼ 0:50 m. The respective roll-roll restoring curves in calm water (GZ curves) are given in Fig. B2.

Fig. B1. Hull form of transom stern fishing vessel.

Table B1 Ship main characteristics Denomination

Transom stern (TS)

Overall length [m] Length between perpendiculars [m] Breadth [m] Depth [m] Draft [m] Displacement [ton] Longitudinal radius of gyration [m]

25.91 22.09 6.86 3.35 2.48 170.3 5.52

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Static Stability Curve - Ship TS (GM = 0.37 m)

Static Stability Curve - Ship TS (GM = 0.50 m) 600

600

SSC-TS50

SSC-TS37 500

7th-order fit Rest. Moment [kN.m]

Rest. Moment [kN.m]

500 400 300 200

400 300 200 100

100 0

7th-order fit

0

10

20

30 phi [deg]

40

50

60

0

0

10

20

30

40

50

60

phi [deg]

Fig. B2. Restoring curves of fishing vessel TS for two different conditions (calm water).

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