Investigation of ship hydroelasticity

ARTICLE IN PRESS

Ocean Engineering 35 (2008) 523–535 www.elsevier.com/locate/oceaneng

Investigation of ship hydroelasticity Ivo Senjanovic´a,, Sˇime Malenicab, Stipe Tomas˘ evic´a a

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lucˇic´a 5, Zagreb, Croatia b Bureau Veritas, Paris, France Received 5 June 2007; accepted 23 November 2007 Available online 3 December 2007

Abstract The importance of hydroelastic analysis of large and flexible container ships of today is pointed out. A methodology for investigation of this challenging phenomenon is drawn up and a mathematical model is worked out. It includes definition of ship geometry, mass distribution, structure stiffness, and combines ship hydrostatics, hydrodynamics, wave load, ship motion and vibrations. Based on the presented theory, a computer program is developed and applied for hydroelastic analysis of a flexible segmented barge for which model test results of motion and distortion in waves have been available. A correlation analysis of numerical simulation and measured response shows quite good agreement of the transfer functions for heave, pitch, roll, vertical and horizontal bending and torsion. The tool checked in such a way can be further used for reliable hydroelastic analysis of ship-like structures. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ship hydroelasticity; Mathematical model; Segmented barge; Numerical simulation; Model tests

1. Introduction Sea transport is rapidly increasing and larger fast merchant ships are built. Large ships are relatively more flexible and their structural natural frequencies can fall into the range of the encounter frequencies in an ordinary sea spectrum. Therefore, the wave induced hydroelastic response of large ships becomes an important issue especially for improving the classification rules and ensuring ship safety. For ships with closed cross-section and ordinary hatch openings such as tankers, bulk carriers, general cargo vessels etc., the lowest natural frequencies are usually associated with the vertical bending. On the other hand, for ships with open cross-section, such as container ships, the lowest elastic natural modes are those of coupled horizontal and torsional vibrations. This coupling is highly pronounced due to the fact that the torsional (shear) centre is below the keel. The classical approach to determine ship motions and wave loads is based on the assumption that the ship hull Corresponding author.

E-mail addresses: [email protected] (I. Senjanovic´), [email protected] (S. Malenica). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.11.008

acts as a rigid body (Principles of Naval Architecture, 1988). The obtained wave load is then imposed to the elastic 3D FEM model of ship structure in order to analyse global longitudinal and transverse strength, as well as local strength with stress concentrations related to fatigue analysis (Hughes, 1988). The above approach is not reliable enough for ultra large ships due to mutual influence of the wave load and structure response. Therefore, a reliable solution requires analysis of wave load and ship vibration as a coupled hydroelastic problem (Bishop and Price, 1979). This is especially important for impulsive loads such as ship slamming that causes whipping. The methodology of hydroelastic analysis is shown in Fig. 1, according to Remy et al. (2006). It includes definition of the structural model, ship and cargo mass distributions, and geometrical model of ship surface. First, dry natural vibrations are calculated. Then modal hydrostatic stiffness, modal added mass, damping and modal wave load are determined. Finally, wet natural vibrations are obtained as well as the transfer functions (RAO—response amplitude operator) for determining ship structural response to wave excitation.

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Fig. 1. Methodology of hydroelastic analysis.

2. Structural model

and the modal stiffness and mass can be determined (Senjanovic´, 1998a)

The hydroelastic problem can be solved at different levels of complexity and accuracy. The best, but highly time-consuming way is to consider 3D FEM structural model and 3D hydrodynamic model based on the radiation–diffraction theory (Salvesen et al., 1970). Such an approach is recommended only for the final strength analysis since it is time consuming. However, in the preliminary strength analysis it is more rational and convenient to couple 1D FEM model of ship hull (Senjanovic´ and Grubisˇ ic´, 1991) with 3D hydrodynamic model. In both cases of the FEM approach, the governing matrix equation of dry natural vibrations yields (Bathe, 1996):

½k ¼ ½dT ½K½d;

½m ¼ ½dT ½M½d.

(3)

Since the dry natural vectors are mutually orthogonal, matrices [k] and [m] are diagonal. Terms ki and O2i mi represent deformation and kinetic energy of the ith mode, respectively. Note that generally the first six natural frequencies Oi are zero with corresponding eigenvectors representing the rigid body modes. As a result, the first six diagonal elements of [k] are also zero, while the first three elements in [m] are equal to structure mass, the same in all directions x, y, z, and the next three elements represent the mass moment of inertia around the coordinate axes. 3. Geometrical model of wetted surface

ð½K  O2 ½MÞfdg ¼ f0g,

(1)

where [K] is stiffness matrix, [M] is mass matrix, O is dry natural frequency, and {d} is dry natural mode. As solution of the eigenvalue problem (1), Oi and {d}i are obtained for each the ith dry mode, where i ¼ 1,2, y, N, N is total number of degrees of freedom. Now natural modes matrix can be constituted ½d ¼ ½fdg1 ; fdg2 . . . fdgi . . . fdgN 

(2)

For determining pressure forces acting on the wetted surface, it is necessary to specify panels and their position in space. Wet surface is given by offsets of waterline ordinates at body plan stations. If the strip method is used for pressure calculation, then the panels bounded with two close stations can be used. If 3D radiation–diffraction theory is used for hydrodynamic pressure determination, the wetted surface mesh can be created in such a way that panels of more regular

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Fh has to be split into two parts: the first part FR depending on the structural deformations, and the second one FDI representing the pure excitation: F h ¼ F R þ F DI .

(8)

Furthermore, the modal superposition method can be used. Vector of the wetted surface deformations H(x, y, z) can be presented as a series of dry natural modes hi(x, y, z): Hðx; y; zÞ ¼ Fig. 2. Rational mesh of wetted surface.

N X

xi hi ðx; y; zÞ ¼

i¼1

þ shape and refined subdivision in the area of the free surface are achieved (Malenica et al., 2004). Such a rational mesh is shown in Fig. 2, where the panel rows follow the ship body diagonals similarly to the structural elements of ship outer shell. In this way, efficiency and accuracy of the hydrodynamic calculation is increased.

hiy ðx; y; zÞj

þ

N X

xi ½hix ðx; y; zÞi

i¼1 i hz ðx; y; zÞk,

ð9Þ

4. Dry modes of wetted surface

where xi are unknown coefficients. Vectors hi(x, y, z) related to wetted surface are obtained from the structural dry modes as explained in the previous section. The potential theory assumptions are adopted for the hydrodynamic part of the problem. Within this theory the total velocity potential j, in the case of no forward speed, is defined with the Laplace differential equation and the given boundary values:

As mentioned in Section 2, structural dry modes can be determined by 1D or 3D FEM analysis. If 1D analysis is used, the beam modes are spread to the ship wetted surface as follows: vertical vibration

Dj ¼ 0 qj ¼0 nj þ qz qj ¼ ioHn qn

dwvi ðz  zN Þi þ wvi k, dx horizontal vibration

hi ¼ 

dwhi yi þ whi j, dx torsional vibration

(4)

hi ¼ 

(5)

hi ¼ ci ðz  zS Þj  ci yk,

(6)

where w is hull deflection, c is twist angle, y and z are coordinates of the point on ship surface, and zN and zS are coordinates of neutral line and shear centre, respectively. If strong coupling between horizontal and torsional vibration occurs, as in the case of container ships, the coupled mode yields:   dwhi dci hi ¼  yþ u¯ i þ ½whi þ ci ðz  zS Þj  ci yk, (7) dx dx where u¯ ¼ u¯ ðx; y; zÞ is the cross-section warping function reduced to the wetted surface (Senjanovic´ and Fan, 1992, 1993). 5. Hydrodynamic model Harmonic hydroelastic problem is considered in frequency domain and therefore we operate with amplitudes of forces and displacement. In order to perform coupling of the structural and hydrodynamic models, it is necessary to express the external pressure forces in a convenient manner (Malenica et al., 2003). First, the total hydrodynamic force

within the fluid at the free surface; z ¼ 0

(10)

on the wetted body surface; S;

where n is the wave number, n ¼ o2/g, o is wave frequency, n is the wetted surface normal vector, and i is the imaginary unit (Nahin, 1998). Furthermore, the linear wave theory enables the following decomposition of the total potential (Salvesen et al., 1970): j ¼ jI þ jD  io

N X

xj jRj ,

(11)

j¼1

where jI ¼ i

gA nðzþixÞ e , o

(12)

where jI is incident wave potential, jD is diffraction potential, jRj is radiation potential, and A wave amplitude. Now, the body boundary conditions (Eq. (10)) can be deduced for each potential: qjD qj qjRj ¼  I; ¼ hj n. qn qn qn

(13)

It is necessary to point out that the diffraction and radiation potentials should also satisfy the radiation condition at infinity. Once the potentials are determined, the modal hydrodynamic forces are calculated by pressure work integration over the wetted surface. The total linearised pressure can be found from Bernoulli’s equation: p ¼ iorj  rgz.

(14)

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First, the term associated with the velocity potential j is considered and subdivided into excitation and radiation parts (Eq. (11)): ZZ DI F i ¼ ior ðjI þ jD Þhi n dS, (15) S

2 FR i ¼ ro

ZZ

N X

xj

j¼1

jRj hi n dS.

(16)

S

Thus, Eq. (15) represents the modal pressure excitation. Now one can decompose Eq. (16) into the modal inertia force and damping force associated with acceleration and velocity, respectively 2 F ai ¼ ReðF R i Þ¼o

N X

ZZ Aij ¼ r Re

xj Aij ;

j¼1

jRj hi n dS,

(Malenica, 2003) ZZ ZZ ~ h~ i n~ dS~ þ rg Z ¼ rg Zhi n dS. FH i S

S

Each of the above quantities can be presented in the form ð.g . .Þ ¼ ð. . .Þ þ dð. . .Þ, where d denotes the variation. By neglecting small terms of higher order, one can write for the modal hydrostatic force (Eq. (19)): ZZ H F i ¼ rg ðdZhi n þ Zdhi n þ Zhi dnÞ dS. (20) S

Variation of the particular quantity can be determined by applying the notion of directional derivative Hr, where H is given with Eq. (9) and r is Hamilton differential operator:

ð17Þ Hr ¼

S

N X

xj hj r ¼

j¼1

F vi ¼ i ImðF R i Þ ¼ io

jRj hi n dS,

  N X q q q xj hjx þ hjy þ hjz . qx qy qz j¼1

(21)

As a result,

xj Bij ;

j¼1

ZZ Bij ¼ ro Im

N X

(19)

dZ ¼ ðHrÞZ ¼ Hk; ð18Þ

S

where Aij and Bij are elements of added mass and damping matrices, respectively. Determination of added mass and damping for rigid body modes is a well-known procedure in ship hydrodynamics (Principles of Naval Architecture, 1988). Now the same procedure is extended to the calculation of these quantities for elastic modes. The hydrostatic part of the total pressure, rgz in Eq. (14), is considered within the hydrostatic model.

dhi ¼ ðHrÞhi ;

dn ¼ ðHrÞn.

(22)

Determining the variation of the body surface normal vector, dn, according to Eq. (22) is a rather difficult task. Therefore, a relatively simpler procedure, taken from (Malenica, 2003), is shown in Appendix A. By using Eqs. (21), (22) and (A.12), the modal hydrostatic force (Eq. (20)) can be presented in the following form: FH i ¼

N X

xj C H ij ,

(23)

j¼1

where 6. Hydrostatic model Hp Hh Hn CH ij ¼ C ij þ C ij þ C ij

(24)

6.1. General Hydroelastic analysis is performed by the modal superposition method. Modal forces represent work of actual static and dynamic forces on rigid body and elastic mode displacements. Thus, modal restoring forces consist of time-dependent modal pressure forces and modal gravity forces. They include effect of large displacements. 6.2. Pressure forces In order to specify contribution of the hydrostatic part of the total pressure, rgz in Eq. (14), to hydrostatic stiffness it is necessary to determine the change of the modal hydrostatic force as the difference between its instantaneous value and the initial value for the vibration mode hi of the body wetted surface Z ¼ Z(x, y)

is the i, jth element of the hydrostatic stiffness matrix, composed of static pressure, surface mode and normal vector contributions, respectively ZZ C Hp ¼ rg hjz ðhix nx þ hiy ny þ hiz nz Þ dS (25) ij S

C Hh ij



i i qhix j qhx j qhx þ hy þ hz ¼ rg Z nx qx qy qz S ! qhiy qhiy j qhiy j j þ hy þ hz þ hx ny qx qy qz  i i i  j qhz j qhz j qhz þ hy þhz þ hx nz dS qx qy qz

ZZ

hjx

ð26Þ

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C Hn ij

# ! qhjy qhjz qhjy qhjz þ ny  nz hix ¼ rg Z nx  qy qz qx qx S    j  qhj qhx qhjz qhj þ þ  x nx þ ny  z nz hiy qy qx qz qy " ! # ) j j j j qhy qhx qhx qhy nx  ny þ þ þ  nz hiz dS. qz qz qx qy ZZ

("

ð27Þ Note that the wetted surface coordinate Z is measured from the waterplane. Based on the constitution of the above coefficients, it is evident that the hydrostatic stiffness matrix is not diagonal. In the literature various definitions of hydrostatic stiffness matrix can be found. Somewhat simple formulae are derived by Molin (2003) using a slightly different approach. Those formulae lead to the same result as the more classical method in which some integral transformations are applied in order to simplify the final expressions (Newman, 1994), as elaborated by Molin (2003). Quite different expressions are presented by Huang and Riggs (2000). The advantage of the present method is that the derived formulae are general and applicable for a complex body as well as for its parts. This is important for determining the local hydrostatic action as internal loads, transfer of load to an FEM structural model, etc. 6.3. Gravity forces The above expressions represent only the action of the hydrostatic pressure, and the gravity part has to be added in order to complete the total restoring coefficients. Similarly to the pressure part, Eqs. (19) and (20), a change of the generalised modal gravity force associated with a particular mode yields (Malenica, 2003): ZZZ m F i ¼ g dhi k dm, (28) V

where V is body volume and dm is differential mass. By employing Eq. (22) and further Eq. (21), one can write: ZZZ N X m F i ¼ g ðHrÞhiz dm ¼  xj C m (29) ij , V

j¼1

where Cm ij

ZZZ  i i i j qhz j qhz j qhz þ hy þ hz ¼g hx dm. qx qy qz

(30)

V

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It is important to point out that the above expressions for the hydrostatic and gravity coefficients are general and therefore valid not only for the elastic modes but also for the rigid body modes as well as for their coupling. Since the rigid body modes are a special case of elastic modes, the corresponding modal restoring stiffness is reduced to the hydrostatic coefficients (Principles of Naval Architecture, 1988). 7. Hydroelastic model After the structural, hydrostatic and hydrodynamic models have been determined, the hydroelastic model can be constituted. For that purpose, let us impose modal hydrodynamic forces (Eqs. (16), (17) and (18)) and hydrostatic and gravity forces (Eqs. (23) and (28)) to the modal structural model (Section 2): ð½k  o2 ½mÞfxg ¼ fFgDI þ fFga þ fFgv þ fFgH þ fFgm .

ð32Þ

Furthermore, all terms dependent on unknown modal amplitudes, xi, can be separated on the left hand side. Thus, the governing matrix differential equation for ship motions and vibrations is deduced: f½k þ ½C  ioð½d þ ½BðoÞÞ  o2 ð½m þ ½AðoÞÞgfxg ¼ fFg,

ð33Þ

where all quantities are related to the dry modes, [k] is structural stiffness, [d] is structural damping, [m] is structural mass, [C] is restoring stiffness, [B(o)] is hydrodynamic damping, [A(o)] is added mass, {x} is modal amplitudes, {F} is wave excitation, and o is encounter frequency. Structural damping can be given as a percentage of the critical value based on experience. As it is well-known, added mass and hydrodynamic damping depends on the frequency. The solution of Eq. (33) gives the modal amplitudes xi and displacement of any point of the structure obtained by retracking to Eq. (9). The wet natural modes can also be determined by solving the eigenvalue problem extracted from Eq. (33): f½k þ ½C  o2 ð½m þ ½AðoÞÞgfxg ¼ f0g.

(34)

Now damping is neglected since its influence on the eigenpair is very small. The solution of Eq. (34) gives natural frequencies of ship motion and vibration in water and the corresponding so-called wet natural modes. Since added mass is a frequency-dependent function, it is evident that an iteration procedure has to be employed to solve Eq. (34). Therefore, the wet modes are not orthogonal. Also, there are no more zero natural frequencies and pure rigid body modes due to their coupling with the elastic modes (Malenica et al., 2003). 8. Hydroelasticity of a flexible barge

6.4. Restoring stiffness

8.1. Barge characteristics Finally, the complete restoring coefficients read: m C ij ¼ C H ij þ C ij .

(31)

The experimental model of a flexible barge (Fig. 3) consisting of 12 pontoons is considered in references

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Fig. 3. Barge cross-section.

Fig. 4. Geometry of bow pontoon.

(Remy et al., 2006; Malenica et al., 2003). The front pontoon no. 12 differs somewhat from the others (Fig. 4). The pontoons are connected by a steel rod above the deck level, as shown in Fig. 3. So, the deformation centre is above the gravity centre. This is the opposite situation to the one in the case of container ships, but anyway strong coupling between horizontal and torsional vibrations is achieved. The main characteristics of the prismatic barge are the following (Remy et al., 2006; Malenica et al., 2003): Young’s modulus of rod Shear modulus of rod Moment of inertia of rod cross-section

E ¼ 2.1  1011 N/m2 G ¼ 0.808  1011 N/m2 4

Polar moment of inertia of rod cross-section

It ¼

Bending stiffness of rod Torsional stiffness of rod Pontoon length Barge length (pontoons+clearances)

EI ¼ 175 Nm2 GIt ¼ 135 Nm2 l ¼ 0.19 m L ¼ 2.445 m

a Iy ¼ Iz ¼ ¼ 8:33  1010 m4 12

a4 ¼ 16:67  1010 m4 6

Total mass (pontoons+equipment) Distributed mass Radius of gyration in roll Polar moment of inertia of distributed mass Distance of gravity centre from torsional centre Polar mass moment of inertia about torsional centre Radius of inertia

M ¼ 171.77 kg m ¼ M/L ¼ 70.253 kg/m ix ¼ 0.225 m J 0t ¼ mi2x ¼ 3:556 kg m c ¼ 0.144 m

J t ¼ J 0t þ mc2 ¼ 5:013 kg m

r¼

qffiffiffi Jt m

¼ 0:267 m

8.2. Dry vibrations Dry vertical natural vibration and coupled horizontal and torsional vibrations are performed by the general computer program developed for this purpose (Tomasˇ evic´, 2007). The program is based on the theory presented by Senjanovic´ and Grubisˇ ic´ (1991). The barge is modelled by 50 beam finite elements. As a result of dry vibration

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Table 1 Natural frequencies of dry vibrations, oi (rad/s) Mode no.

Vertical

Coupled horizontal and torsional

1 2 3 4 5

5.21 12.37 21.08 30.67 40.87

5.32 7.92 12.70 15.56 21.56

Fig. 7. The first dry natural mode of coupled horizontal and torsional vibrations, o1 ¼ 5.32 rad/s.

Fig. 5. The first dry natural mode of vertical vibrations, o1 ¼ 5.21 rad/s.

Fig. 8. The second dry natural mode of coupled horizontal and torsional vibrations, o2 ¼ 7.92 rad/s.

Fig. 6. The second o2 ¼ 12.37 rad/s.

dry

natural

mode

of

vertical

vibrations,

calculation, the modal stiffness and modal mass in Eq. (33) are obtained. The values of the first five natural frequencies are listed in Table 1. The first two natural modes of wetted surface for vertical and coupled horizontal and torsional vibrations are shown in Figs. 5–8, respectively. Natural vertical vibrations for a prismatic pontoon can be also determined analytically in a simple way (Timoshenko and Young, 1955). Analytical solution of coupled horizontal and torsional vibrations can be obtained by direct integration of governing differential equations or by

the energy approach. However, in both cases, the solution is rather complicated (Senjanovic´ et al., 2007). For illustration, natural frequencies of coupled horizontal and torsional vibration of the prismatic barge, determined analytically by direct integration and Ritz method, and numerically by 1D FEM analysis, are compared in Table 2. Quite good agreement of the results is achieved. 8.3. Hydrostatic and hydrodynamic parameters Restoring stiffness matrix in Eq. (33) is determined by the developed program (Tomasˇ evic´, 2007) based on the theory considered in Section 6. Added mass, hydrodynamic damping and wave excitation in Eq. (33), depending on dry modes and wave frequency, are determined by program Hydrostar (Hydrostar, User’s manual, 2006). Structural

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Table 2 Natural frequencies of coupled horizontal and torsional vibrations of prismatic barge, oi (rad/s) Mode no.

Analytical

FEM

1 2 3 4 5 6 7 8 9 10

5.727 7.884 16.123 14.392 23.591 26.225 37.532 31.958 44.013 45.268

5.728 7.887 16.135 14.399 23.633 26.257 37.702 32.062 44.102 45.604

1, 3, y: symmetrical modes; 2, 4, y: anti-symmetrical modes.

Fig. 9. Modal amplitude of vertical vibrations, w ¼ 1201.

is sufficient to describe accurately enough the barge response in waves. That is confirmed by convergence of the modal amplitudes xi of normalised modes (maximum displacement 71). For illustration, Figs. 9 and 10 show xi values for vertical and coupled horizontal and torsional vibrations, respectively. The first three modes in Figs. 9 and 10 are related to surge, heave and pitch, and sway, roll and yaw, respectively, while the remaining modes are elastic. All diagrams are determined for heading angle w=1201, and each of them is a result of single wave length in the range l/L=0.5–2. 8.4. Barge response The model tests of the considered barge were conducted in the BGO–First Basin, Toulon, France. Detailed description of the barge, equipment, measuring procedure and the obtained results is given in references (Remy et al., 2006; Malenica et al., 2003). The barge is constructed to be quite flexible in order to expose high level of hydroelastic phenomena. The success of this intention can be seen in Fig. 11, where the barge distortion follows more or less the wave surface. The model tests were performed in irregular waves generated by JONSWAP spectra. Numerical calculation of the barge response to waves is performed for a set of heading angles. For verification of the described methodology and numerical simulation only the transfer functions for w ¼ 601 are shown and compared to the measured ones. This particular heading angle was chosen since it includes complete coupling of the rigid and flexible modes of the vertical, and horizontal and torsional vibrations, which are highly exited in the case of quartering seas. Figs. 12–14 show RAOs (Response Amplitude Operatortransfer function) in the wave period domain, T, for the average value of heave, pitch and roll respectively, measured at the every second pontoon. In Figs. 15–17,

Fig. 10. Modal amplitude of coupled horizontal and torsional vibrations, w ¼ 1201.

damping of the pontoon joints to the rod, and of the rod itself is quite low in the considered case. The mesh of the wetted surface used in the calculation can be seen in Figs. 5–8. All necessary vibration parameters are determined for a set of dry modes consisting of six rigid body modes, the first five vertical elastic modes and 10 coupled horizontal and torsional modes. The chosen number of elastic modes

Fig. 11. Barge in waves.

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Fig. 12. Average heave transfer function, w ¼ 601.

Fig. 13. Average pitch transfer function, w ¼ 601.

Fig. 14. Average roll transfer function, w ¼ 601.

vertical and horizontal bending and barge torsion are shown. These quantities are defined as difference of the rotation angles at the last and first pontoon.

RAOs are related to the wave height spectrum. Thus, the heave RAO, as translation motion, converges to unity for higher T values. Contrary, RAOs for all others motions,

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Fig. 15. Vertical bending transfer function, w ¼ 601.

Fig. 16. Horizontal bending transfer function, w ¼ 601.

Fig. 17. Torsional transfer function, w ¼ 601.

which are rotational, converge to zero (Principles of Naval Architecture, 1988). The relatively large scattering of the measured transfer functions is mostly caused by irregular waves. The

obtained numerical results agree quite well with the average of the measured ones. Higher discrepancies between the calculated and measured values occur in area of the response peaks, i.e. at resonances where damping

ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 523–535 Table 3 pffiffiffiffiffiffiffi Additional modal damping, percentage of critical value ncr ¼ 2 km Motion

Adjustment to resonant peak (%)

Decay test (%)

Heave Roll Pitch Vertical vibration Horizontal vibration Torsional vibration

5 5 5 5 9 7

6 4 6 7.5 2.5 2.5

plays the main role. Figs. 12–17 show the final calculation results determined by adjusting damping to achieve minimal discrepancies. The modal damping is adjusted in such a way that the value of diagonal elements in damping matrix is increased for a percentage of critical damping, depending on the type of motion, i.e. rigid body, vertical, horizontal and torsional vibrations, in order to achieve resonant response. These values differ somewhat from those determined by decay test, as may be seen in Table 3 (Tomasˇ evic´, 2007). This adaptation can be physically explained. In the hydroelastic analysis, namely, the segmented barge is considered as a monohull. The influence of this assumption on restoring forces and water inertia forces is negligible. However, additional resistance to the pontoon motion is induced between their heads. It depends on the relative angular velocity of the pontoon adjacent heads. 9. Conclusion The investigation methodology of hydroelasticity of ship structures is presented. A mathematical model which combines ship geometry, mass distribution, 1D hull model, dry natural vibrations, hydrostatic model and hydrodynamic model is worked out. The numerical procedure and developed computer program are validated for the case of a very flexible barge for which coupling between rigid and elastic modes is highly pronounced as well as coupling between horizontal and torsional vibrations. Quite good agreement between simulated and measured barge response for this very sensitive non-linear experimental model with large amplitudes of rigid and elastic modes of the same order is achieved. Thus, one may conclude that the presented methodology and mathematical model are reliable enough to be applied for hydroelastic analysis of ship structures for which the amplitude ratio of the elastic and rigid modes is somewhat lower. The application of the developed tool is especially important for hydroelastic analysis of large container ships which are very flexible from the torsional point of view, and therefore exposed to high level of coupling between horizontal and torsional vibrations. This is a subject of further investigation in which special attention has to be paid to investigation of damping based on full-scale

533

measurement since transfer of model test results to ship structure is unreliable. Further development is directed to the investigation of the hydroelastic response to impulsive load, i.e. slamming and whipping, sloshing, underwater explosions, etc. in time domain. The final target is to analyse the influence of ship elasticity on wave load in order to check validity and applicability of the present classification rules for large container ships. In the meantime, direct strength calculation as a hydroelastic task should be applied.

Appendix A. Variation of body surface normal vector A body surface can be represented in the x-, y-, z-space in implicit, explicit, parametric or vectorial form (Bronstein et al., 2004; Kreyszig, 1993). In the considered problem, the vectorial representation by position vector is preferable (widely used in the shell theory (Novozˇilov, 1964; Senjanovic´, 1998b): r ¼ xðu; vÞi þ yðu; vÞj þ zðu; vÞk,

(A.1)

where u and v are parameters that form coordinate mesh on the surface. Therefore, these parameters are called curved or Gauss coordinates. It is well-known in the differential geometry that the arcs of differential surface are (Fig. A.1)   qr qx qy qz dsu ¼ du ¼ i þ j þ k du, qu qu qu qu   qr qx qy qz dv ¼ i þ j þ k dv. ðA:2Þ dsv ¼ qv qv qv qv For differential surface one finds: dS ¼ dsu  dsv ¼ N du dv,

(A.3)

where components of the normal vector are as follows: qy qz qz qy  , qu qv qu qv qz qx qx qz  , Ny ¼ qu qv qu qv qx qy qy qx  . Nz ¼ qu qv qu qv Nx ¼

ðA:4Þ

If the body surface is deformed by the mode h, the deformed surface is represented by the resulting position vector: r~ ¼ r þ h. The corresponding arcs read:   q~r qr qh d~su ¼ du ¼ þ du qu qu qu   q~r qr qh dv ¼ þ dv d~sv ¼ qv qv qv

(A.5)

ðA:6Þ

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Fig. A.1. Definition of normal vector variation dn due to modal deformation h.

and the deformed differential surface:

one can for instance write:

dS~ ¼ d~su  d~sv   qr qr qh qr qr qh qh qh  þ  þ  þ  ¼ qu qv qu qv qu qv qu qv

qhx qhx qx qhx qy qhx qz þ þ , ¼ qu qx qu qy qu qz qu

du dv.

ðA:7Þ

The first product in Eq. (A.7) is dS, while the last product is a small quantity of higher order and can be therefore neglected. Thus, one finds for variation of differential surface:   qh qr qr qh dðdSÞ ¼ dS~  dS ¼  þ  qu qv qu qv du dv ¼ dN du dv. The above vector products take form    qhy qz qhz qy qh qr qhz qx qhx  ¼   iþ qu qv qu qv qu qv qu qv qu   qhx qy qhy qx  k þ qu qv qu qv    qr qh qhz qy qhy qz qhx qz qhz  ¼   iþ qu qv qv qu qv qu qv qu qv   qhy qx qhx qy  k. þ qv qu qv qu

ðA:8Þ

etc. for the other derivatives of hx, hy, hz per u and v. Furthermore, by substituting those derivatives of type from Eq. (A.11) into Eq. (A.9) and summing up all terms, one finds according to Eq. (A.8) variation of the normal vector    qhy qhz qhy qhz dN ¼ þ Ny  Nx i Nx  qy qz qx qx     qhx qhx qhz qhz Nx þ þ Nx j þ  Ny  qy qx qz qy     qhy qhx qhx qhy Nx  Ny þ þ þ  N x k ðA:12Þ qz qz qx qy as it is elaborated by Malenica (2003). Since

 qz j qv

dN du dv ¼ dn dS

(A.13)

the same form of expression (A.12) is also valid for the variation of the unit vector, dn, with components nx, ny and nz.

 qx j qu

References ðA:9Þ

Since h ¼ hx ðx; y; zÞi þ hy ðx; y; zÞj þ hz ðx; y; zÞk

(A.11)

(A.10)

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ARTICLE IN PRESS I. Senjanovic´ et al. / Ocean Engineering 35 (2008) 523–535 Huang, L.L., Riggs, R.R., 2000. The hydrostatic stiffness of flexible floating structure for linear hydroelasticity. Marine Structures 13, 91–106. Hughes, O.F., 1988. Ship Structural Design. SNAME. Hydrostar, User’s manual, 2006. Bureau Veritas, Paris. Kreyszig, E., 1993. Advanced Engineering Mathematics. Wiley. Malenica, Sˇ., 2003. Some aspects of hydrostatic calculations in linear seakeeping. In: Proceedings of the 14th NAV Conference, Palermo, Italy. Malenica, Sˇ., Molin, B., Remy, F., Senjanovic´, I., 2003. Hydroelastic response of a barge to impulsive and non-impulsive wave load. Hydroelasticity in Marine Technology, Oxford, UK, pp. 107–115. Malenica, Sˇ., Senjanovic´, I., Chen, X.B., 2004. Automatic mesh generation for naval and offshore hydrodynamic simulations, SORTA’04, Plitvice Lakes, Croatia. Molin, B., 2003. Hydrostatique d’un corps deformable, Technical note, Ecole Superieure d’Ingenieurs de Marseille, Marseille,pFrance. ffiffiffiffiffiffiffi Nahin, P.J., 1998. An Imaginary Tale—The Story of 1. Princeton University Press. Newman, J.N., 1994. Wave effects on deformable bodies. Applied Ocean Research 16, 47–59. Novozˇilov, V.V., 1964. Thin Shell Theory. P. Noordhoff Ltd., Groningen, The Netherlands. Principles of Naval Architecture, 1988. SNAME. Remy, F., Molin, B., Ledoux, A., 2006. Experimental and numerical study of the wave response of a flexible barge. Hydroelasticity in Marine Technology, Wuxi, China, pp. 255–264. Salvesen, N., Tuck, E.O., Faltinsen, O., 1970. Ship Motion and Sea Loads, Transactions, Vol. 70. SNAME, pp. 250–287. Senjanovic´, I., 1998a. Finite Element Method in Ship Structures. University of Zagreb, Zagreb (in Croatian). Senjanovic´, I., 1998b. Plate and Shell Theory. University of Zagreb, Zagreb (in Croatian). Senjanovic´, I., Fan, Y., 1992. A higher-order theory of thin-walled girders with application to ship structures. Computers & Structures 43 (1), 31–52. Senjanovic´, I., Fan, Y., 1993. A finite element formulation of ship crosssectional stiffness parameters. Brodogradnja 41 (1), 27–36.

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Senjanovic´, I., Grubisˇ ic´, R., 1991. Coupled horizontal and torsional vibration of a ship hull with large hatch openings. Computers & Structures 41 (2), 213–226. Senjanovic´, I., C`atipovic´, I., Tomasˇ evic´, S., 2007. Coupled horizontal and torsional vibrations of a flexible barge. Engineering Structures 30 (2008), 93–109. Tomasˇ evic´, S., 2007. Hydroelastic model of dynamic response of Container ships in waves. Ph.D. Thesis, FSB, Zagreb (in Croatian). Timoshenko, S., Young, D.H., 1955. Vibration Problems in Engineering. D. Van Nostrand. Ivo Senjanovic´, D.Sc. Professor of Naval Architecture at the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. Teaching several courses on strength and vibration of ship and offshore structures, submarines, etc. Investigation fields are ship stability, launching, numerical methods, numerical simulations, non-linear dynamics, ship strength and vibration, shell theory and design of pressure vessels, http:// mahazu.hazu.hr/Akademici/ISenjanovic.html.

Sˇime Malenica graduated in 1990 in Naval Architecture at the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. He obtained his Ph.D. degree in 1994, at the University Paris VI, on the problem of nonlinear water wave diffraction. He has published more than 70 papers both in scientific journals and at the specialised conferences. He is the actual head of hydro-structure section in the Bureau Veritas Research Department leading several national and international research projects. His main fields of research concern the hydro-structural interactions due to the sea waves (linear, nonlinear, frequency domain, time domain, ships and offshore, etc.). Actual topics concern the hydroelasticity issues for ships (springing, whipping, slamming, sloshing, etc.).

Stipe Tomas˘ evic´, D.Sc. Lecturer at the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. Support in education on the finite element method and software application. Investigation of ship strength, vibrations and fatigue.