head seas described by the one-dimensional ISSC spectrum. The linear analysis revealed .... m11 = m22 = m33 = m, m44 = Ixx, m55 = Iyy, ...... Additional information about the code can be picked up from the program manual [29] and from the ...
Estimation of Ship Dynamic Wave Loads in Irregular Sea ¨ ding S. Sutulo, H. So 2003 Abstract A description of mathematical models used for estimation of loads on a ship hull subject to the ship motions, slamming and whipping in irrregular seas is presented. The primary seakeeping model is linear and the nonlinear loads (including those caused by slamming) are computed with the help of the momentum theory with empirical corrections introduced. The vibrations model is based on the beam theory with modal expansions and numerical integration of the modal equations. The algorithm was implemented as a Fortran 90 code SLAVIB which computes also time histories of longitudinal distributions of equivalent static loads, vertical shear forces and bending moments.
1
Introduction
We focus on the vertical wave-induced loads on a ship hull imagined as a longitudinal girder. The loads can be imagined as superposition of several components: 1. “Linear” loading as computed by linear seakeeping codes (as, for instance, STRIP [2]), although accounting for nonlinearities would improve the accuracy [3]. 2. Springing vibrations induced by linear or low-order nonlinear loading. It is caused by short wave components. It can be studied by linear or low-order nonlinear hydroelastic analysis. 3. Shock loads observed when the forward part of the bottom was emersed and then enters the water again, and/or when a strongly flaring hull part immerses. A kind of impact is observed in both cases; this is called (bottom or bow-flare) slamming. 4. The slamming generates whipping vibrations which differ from the continously excited springing vibrations. 5. The loading is influenced by green water on deck.
1
A brief survey of some existing approaches to predict wave loads, with emphasis on slamming loads and vibrations, is given below. The classic von K´arm´an’s and Wagner’s solutions to the impact problem (the references can be found, say, in the book [4]) laid foundation for most of the applications-oriented studies on the ship slamming, the present contribution being no exception. Faltinsen [5] stressed that Wagner’s solution worked well even with moderate deadrise angles α although the discreapancies can be larger for the sections with large curvature which is assumed zero in Wagner’s theory. Also, he pointed out that at small deadrise angles one shouldn’t put too much emphasis on the peak pressure as, for instance, an air cushion can be created at approximately α < 3◦ which will change the pressure distribution and reduce the pressure peak dramatically. In general, the sectional slamming force seems to be fully defined by the section entrance velocity although this is questionable at very high Froude numbers of advance. As to the influence of the hull’s local elasticity, Faltinsen [6] mentioned it as another reason for making often meaningless the designers’ wish to know values of the rigid-body peak pressure. At the same time, these pressure values can become quite meaningful in bowflare slamming as local elastic effects are negligible at high deadrise angles. 3D effects can be relatively important in bow slamming (not considered here), but in bow-flare slamming they can only cause 10–20 percent reduction of the pressure. Another comprehensive survey of water impact and related problems was given by Korobkin [7], who put more attention on theoretical formulations and on some subtle effects like cavitation and compressibility of the water. The latter is significant in the case of a rigid body with a flat-bottom entering an undisturbed fluid. In this case, neglecting air and compressibility effects would result in an infinite slam force. This fact, probably, inspired Korobkin to develop an acoustic theory of water impact. Some theoretical aspects of the air-cushion and elasticity effects are also covered in the review [7]. However, the usual incompressible-fluid approach is adequate if the “stiffness” of the body is substantially smaller than that of the fluid, which is the usual case in ships. Under papers using this assumption, especially well-known is the contribution by Zhao and Faltinsen [8] where a boundary-element algorithm was used to solve the Wagner problem. The presented results have been used for comparisons in a number of later studies. In his 1993 Georg Weinblum Memorial Lecture Faltinsen [13], focusing on the slamming pressure distribution, pointed out that Wagner’s solution can underpredict maximum slamming pressures at deadrise angles exceeding 45◦ . A comparison of different theories with emphasis on that by Zhao and Faltinsen was made. It was also mentioned that the maximum slamming pressure occurring at the first contact of the body with the fluid will be always limited by the acoustic pressure ρwce , where ρ is the water density, w is the entrance velocity of the section, and ce the velocity of sound in water. Another numerical solution is given in [9] where the case of a flat girder entering quiet water was treated using the FE method. The fluid compressibility was taken into account in acoustic approximation. The results have confirmed the importance of accounting for the structural elasticity in this particular case of a flat parallel impact. However, sophisticated numerical methods still havn’t found direct applications in seakeeping codes where preference is still given to simpler momentum theories (the Wagner-like
2
approach). The structural elasticity (or plasticity) and water compressibility are normally neglected, as they are not pronounced in the great majority of real entry situations (non-flat structures, non-flat water surface, more complicated kinematics). Moreover, the actual entry conditions can only be predicted with considerable uncertainty; thus using sophisticated algorithms seems to make no sense. This situation encouraged Vorus to develop a compromise Flat Cylinder Theory [10] which is simpler than the method by Zhao and Faltinsen, but here the final results are still obtained by a numerical method instead of an explicit formula. Other studies aim to compute all kinds of wave loads simultaneously. Here we mention only those in which slammimg and vibration effects are taken into account. In a relatively early study by Guedes Soares [11] emphasis was given to whipping responses. The ship motions were determined with a linear strip theory, and only the case of regular oncoming waves was considered. The output of the seakeeping code was used to obtain the relative motions of the sections, and the total loading (referred to, however, as the slamming loading) was estimated with a simple momentum theory using sectional infinite-frequency added masses but without accounting for the water pile-up. These loads were further used as input for a vibration analysis. The latter was performed with the FE method applied to a Timoshenko beam modelling the ship hull. An implicit time integration scheme was used in combination with the modal transformation. The vibrations were analysed only after the start of the slam which was supposed to occur when any part of the hull was entering the water after having been completely out of it. (The popular concept of Ochi, i.e. requiring a critical slamming entry velocity, was ignored because it seems to neglect minor slams.) Thus only bottom slamming was considered. The numerical calculations carried out for the Mariner ship showed that accounting for the whipping effect can substantially increase the estimated maximum values of the midship bending moment: from 500 to 2500 MNm in sagging, and from −500 to −1700 in hogging for 12 m wave height. This method was modified by Ramos and Guedes Soares [12] who applied simplified estimation methods for the slamming force. Empirical approximations for the slamming pressure were tested, and some semi-empirical procedure was finally chosen. A much more consistent and complicated hydroelastic mathematical model for ship motions and vibrations was developed by Wu and Moan [14]. A linear seakeeping and vibration frequency-domain formulation constituted the foundation of the theory. Transition to the time domain, which is better suited for nonlinear effects, was done using the Fourier transform. The Vlasov beam model (which includes rotary inertia and shear deformations) was supposed to be applicable to monohulls and catamarans (with additional assumptions in the latter case). Nonlinear effects were introduced as corrections to the basic linear model. They are caused by large amplitude heave and pitch while the structural deformations remain small. Linear wave theory was assumed applicable. The sectional nonlinear hydrodynamic force is computed using the usual momentum theory without accounting for the pile-up effect. The authors argue that this effect is small for high deadrise angles typical for the V-shaped bow, and it can be compensated by trapped air and 3D effects. In this method the hydrodynamic and hydrostatic forces are splitted into a linear part and a nonlinear correction. The influence of the nonlinear component on the ship motions is approximately taken into account. Numerical calculations were carried out in irregular head seas described by the one-dimensional ISSC spectrum. The linear analysis revealed a high-frequency resonance of the bending moment corresponding to the first vibration mode. 3
It was found also that four elastic modes were sufficient in all cases, and that hydrodynamic damping due to forward speed dominated over structural damping at high forward speed (35 kn). At the same time, the linear hydroeleastic effect (springing) was found negligible because of very low wave energy at high frequencies. Simulations with the nonlinear force included showed a certain reduction of the relative bow displacements, an increase of the sagging bending moment, and a reduction of the hogging moment. Later numerical experiments with this algorithm were performed by Wang and Moan [15] for the S175 container ship. In this case, the slamming did not affect substantially the statistical distribution of the bending moment, although instantaneous values could be changed considerably; and the main nonlinear factors were the hydrostatic and Froude– Krylov components. In this particular case, the hydrostatic component of the extreme sagging moment compensated approximately the slamming component. A similar model was presented by Xia and Wang [16]. Here, the Timoshenko beam was used to represent the ship hull girder. The nonlinear hydrodynamic force was handled in a simplified way. The authors concluded that two elastic shape modes were sufficient. Results obtained for the S175 ship agreed fairly well with experimental data although some underestimation was observed at higher Froude numbers. Another computer code LAMP (Large Amplitude Motion Program) was developed by Weems et al. [17]. The hydrodynamic part is based on a time-domain 3D formulation with partial account for the nonlinearities; i.e. the body boundary condition and the free-surface condition are satisfied on the instantaneous wetted body surface, but the latter condition was linearized. Various non-potential forces (including viscous roll damping, rudder and propeller forces) are included into the mathematical model. However, the model is not truly hydroelastic as the nonlinear hydrodynamic loads (slamming) and vibrations are not supposed to affect the rigid-body motions. Part of the code is based on the strip method and beam theory. Three methods were used to compute slamming loads: an empirical model; the orthodox momentum approach; and a simplified version of the nonlinear boundary-element model of Zhao and Faltinsen [8]. In the last case, also horizontal slamming forces were determined. For the vibratory response a FE beam model was used. The resulting whipping bending moments were added to the rigid-body moments obtained from ship motion computations. The results agreed well with measured data. Recently, a comparison of several nonlinear seakeeping codes providing information on dynamic loads has been undertaken by Watanabe and Guedes Soares [18]. Unfortunately the explored seven computer programs did not include the three codes described above. All the benchmark computations were carried out in regular incident waves of various steepness for the container ship S175. The comparative study leads to the following conclusions: 1. Relative displacements of the forward perpendicular are predicted with large scatter, and nonlinear formulations showed no advantages over linear ones. The influence of the hull elasticity seems to be insignificant. 2. The nonlinearities were important for the vertical bending moment. 3. The differences between different codes were unacceptable: The computed midship bending moment varied from 140 to 267 MNm (linear value 131) for the wave length λ equal to ship length L, and from 110 to 342 MNm (linear: 110) for λ/L = 2. The 4
wave steepness was 1/20 in the both cases. Concluding from the above discussion, it seems premature to include too many sophistications into a new code. We prefer a relatively simple but robust program which accounts for all major effects but handles them in the simplest possible way.
2
Main Assumptions
The objective of the assumptions listed below is to simplify the task as much as possible while accounting for all the essential effects. • A monohull moderate-speed displacement ship is supposed to travel with constant speed at an arbitrary heading angle across a stationary three-dimensional irregular seaway. • The ship is slender so that the strip theory can be used. • All motion amplitudes are moderate so that a linear theory is sufficient. Nonlinear effects (including the slamming) have influence only on the loads. • Although the ship has six degrees of freedom, it is assumed that only heave and pitch affect significantly the slamming, nonlinear loads and vibrations. • The eigenfrequencies and the mode shapes of the ship vibrations are supposed to be known (pre-calculated). These assumptions allow to compute the ship motions in the frequency domain using the program STRIP [2]. A simulation is necessary only for nonlinear loads and vibrations.
3
Description of the Mathematical Model
3.1
Overall Structure
Our mathematical model deals with • the 3D irregular seaway, • the rigid-body ship motions in waves (linear, 6DOF, slender body, frequency domain) • the kinematics related to water entry • the nonlinear loads • the springing and whipping vibrations and loads equivalent to these vibrations
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3.2
Input Process Generation
The irregular sea is supposed to be stationary and three-dimensional with an angle µ between the main direction of propagation and the x-axis. The sea is described by means of a twodimensional energy spectrum Sζ (ω, µ). The random process in the body frame of a moving ship can then be represented as ζ(x, t) =
N OM
Re[ζn∗ei(ωen t−kn x cos µn ) ],
(1)
n=1
where
2Sζ (ωn , µn )∆ωn ∆µn · cos εn , Im(ζn∗ ) = 2Sζ (ωn , µn )∆ωn ∆µn · sin εn , Re(ζn∗ ) =
(2)
NOM is the number of harmonic wave components for which Kr¨ oger recommends 25 [21], S¨ oding [22] 40–80 harmonics, and Faltinsen [23] 1000. The last two authors insist also that every ωn be chosen randomly, uniformly spread within the ω intervals ∆ω which, together, cover the frequency region of interest. The same should be done for the wave component encounter angles µn . ωen is the encounter frequency ωen = ωn − kn V cos µn ,
(3)
and εn is the random phase uniformly distributed within the interval [0, 2π].
3.3
Linear Mathematical Model for Ship Motions
The linear frequency-domain mathematical model for the ship motions is largely that of [19] and [20], but a concize description is given here to facilitate references and bring a certain completeness into the description. The body-fixed frame of reference Cxyz is the same as in [20]; i.e. the origin C is at the intersection of the midship section plane, the centerplane, and the base plane; the horizontal x axis points forward, the z axis downward, and the y axis to starboard. Any time-dependent real quantity P varies with time like P = Re(P ∗ eiωe t ), where the asterisk is the complex amplitude symbol. If we introduce the column matrices (vectors) U = (ξ0 , η0 , ζ0 , φ, θ, ψ)T , the vector of ship displacements, and E = (XE , YE , ZE , KE , ME , NE )T , the vector of excitation forces, their complex amplitudes will be connected to each other by the following linear matrix equation: ∗ CU∗ = E∗ ζM ,
(4)
where C = [cij ], i, j = 1, . . . , 6 is the complex “dynamic stiffness” matrix which is represented as follows: C = S − H − ωe2 M, (5) where S = [sij ] is the matrix of static (restoring) forces, H = [hij ] is the matrix of hydrodynamic reactions related to ship motions, and M = [mij ] is the ship’s inertial matrix; all these matrices are of size 6 × 6. 6
Supposing a symmetric mass distribution, the inertial matrix contains the following nonzero elements: m11 m44 m15 m24 m26 m35 m36
= m22 = m33 = m, = Ixx , m55 = Iyy , m66 = Izz , = m51 = mzG , = m42 = −mzG , = m62 = mxG , = m53 = −mxG , = m63 = −Ixz .
(6)
(7)
The standard notations are used here for the ship mass m, its centre of gravity coordinates, and for the gyration moments. If the latter are given for axes through the centre of gravity (indicated by index G), they must be transformed to the coordinate origin: Ixx Iyy Izz Ixz
= = = =
2 IGxx + mzG , 2 2 IGyy + m(xG + zG ), 2 IGzz + mxG , IGxz + mxG zG .
(8)
The non-zero elements of the static matrix are: s13 s23 s35 s53 s64
= ρgAT , = ρg∇ − gm, = −ρgAxW , = −ρgzT AT − ρgAxW , = ρg∇x − gmxG ,
s15 s33 s44 s55
= −ρgxT AT − ρg∇ + gm, = ρgAW , = −ρg∇z + gmzG , = −ρg∇z + ρgAxx W + gmzG ,
(9)
where ρ is the density of water; g the acceleration of gravity; AT and xT the transom area1 and its x coordinate; ∇ is the displacement, and ∇x , ∇xx and ∇z are its moments; AW is the waterplane area, and AxW , Axx W are its moments. Most of the listed geometric parameters are computed by means of integration along the ship length L: ∇ = L AS dx, AW = L BW dx, AxW = L xBW dx, (10) 2 x ∇z = L AS zS dx, Axx W = L x BW dx, ∇ = L xAS dx, where AS (x) is the section area, zS (x) the z coordinate of its centroid, and BW (x) the breadth of the waterline, all at position x. The hydrodynamic matrix H is decomposed: H = H1 + H2 , where H1 = [h1 ij ] is calculated by means of the strip theory: ∗ ∂ D(−iωe + V H1 = )(AW)dx, ∂x L 1
(11)
(12)
the area of the transom below the time-averaged waterline at the ship side, including the stern wave; the aft face of the transom is supposed to be clear from water.
7
∗ where the symbol L means an integration which, for terms involving ∂∂x , excludes positions where the integrand increases discontinually. D is the (6 × 3) dimension transformation matrix (from 3D “sectional” to 6D “ship” generalized forces), A is the (3 × 3) matrix of complex added masses of a cross section, and W is a (3 × 6) matrix for transformation from 6D “ship” motions to 3D motions of ship cross-sections. The mentioned matrices are (the zeros follow from the presupposed symmetry): ⎤ ⎡ 0 a24 (x) a22 (x) a33 (x) 0 ⎦, A(x) = ⎣ 0 (13) a42 (x) 0 a44 (x) where any element is a complex sectional added mass which can be represented as aij = µij +
νij . iωe
(14)
µij is the usual frequency-dependent real added mass, and νij is the damping coefficient. The a values are computed using subroutine YEUNGA. Further,
⎡ ⎢ ⎢ ⎢ D=⎢ ⎢ ⎢ ⎣
and
0 0 0 1 0 0 0 1 0 0 0 1 0 −x 0 x 0 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(15)
⎡
⎤ 0 iωe 0 0 0 iωe x − V ⎦. 0 W = ⎣ 0 0 iωe 0 −(iωe x − V ) 0 0 0 0 0 iωe
(16)
The component H2 accounts for longitudinal forces on the hull: H2 = ωe2 m11 LLT ,
(17)
where L = (1, 0, 0, 0, z0, 0)T with z0 corresponding approximately to a point half-way from the waterplane to the bottom. The longitudinal added mass is supposed to be real; it can be estimated empirically as m . (18) m11 = π L3 /∇ − 14 The column matrix of excitation forces is also supposed to consist of E1 = (e11 , . . . , e16 )T and E2 = (e21 , . . . , e26 )T which give transverse and longitudinal loads respectively: E = E1 + E2 , where
iV ∂Ex7 = D Ex0 + Ex7 + e−ikx cos µ dx, ω ∂x L dAx −ikx cos µ e = Lα dx + LT αAxT e−ikxT cos µ , dx L
E1 E2
(19)
∗
8
(20) (21)
with
α = −ρge−k(zx +T ) .
(22)
Ex0 = (e0x2 , e0x3 , e0x4 )T and Ex7 = (e7x2 , e7x3 , e7x4 )T are (3 × 1) column matrices of the FroudeKrylov and diffraction forces respectively which are provided by the program YEUNGA; zx is the vertical coordinate of the section centroid, and subscript T relates to the transom. The expressions (12) and (20) are computed directly with derivatives approximated by finite differences. Where, in the afterbody, the hull sections change strongly within a short length (especially in front of the propeller in single-screw ships and at the end of a skeg in twin-screw ships) the integration is interrupted for short or vanishing x intervals, to take account of the flow separation at these locations. For the same reason, the integration starts also immediately before, not immediately behind an immersed transom. This has substantial influence only ∂ on the terms involving derivatives ∂x which otherwise would be extremely large within these intervals. Relative motions of a point of the ship. Let Xs be the 3 × 1 column matrix of absolute linear displacements of a point with body-fixed coordinates x, y, z. The complex amplitudes of the displacements are then given by
with
X∗s = W(x, y, z)U∗ ,
(23)
⎤ 1 0 0 0 z −y 0 x ⎦. W(x, y, z) = ⎣ 0 1 0 −z 0 0 1 y −x 0
(24)
⎡
The complex amplitudes of water particle displacements at the sea surface due to waves are X∗w = Fw ζw∗ e−ik(x cos µ−y sin µ) ,
(25)
where Fw = (i cos µ, −i sin µ, 1)T . Finally, the complex amplitude of the vector of relative displacements Xr = (xr1 , xr2 , xr3 )T will be X∗r = X∗s − X∗w . (26) Wave induced velocities and accelerations The vector Yw∗ = (vwx , vwy , vwz )T of the complex amplitudes of the wave velocity components at a point (x, y, z) is Yw∗ = iωe−k(z+T ) X∗w
(27)
where T is the time-averaged value of the draft of the x axis below the water surface. The corresponding vector of complex amplitudes of accelerations is Z∗w = −ω 2 e−k(z+T ) X∗w .
9
(28)
3.4 3.4.1
Frequency Transfer Functions and Output Processes Basic Ship Motions
The vector Φ = (φ1 , . . . , φ6 )T of frequency transfer functions (FTF) for the ship motions is the vector of complex amplitudes of the ship reactions to a regular wave of unit amplitude. Thus, Φ is determined by solving CΦ = E∗ . (29) 3.4.2
Relative Displacements
The FTF vector of local displacements is Φs (ω, x, y, z) = W(x, y, z)Φ(ω),
(30)
and the FTF vectors of water surface displacements and relative motions are Φw (ω, x, y, z) = Fw e−ik(x cos µ−y sin µ) , Φr (ω, x, y, z) = Φs (ω, x, y, z) − Φw (ω, x, y, z).
(31) (32)
The process (time function) of the relative vertical displacement of a point (x, 0, 0) on the x axis is N OM Re[φr3 (ωn , x)ζn∗eiωen t ]. (33) xr3 (x, t) = n=1
3.4.3
Vertical Wave Acceleration
The FTF vector of the water particle’s acceleration is Φwa (ω, x, y, z) = −ω 2 e−k(z+T ) Φw (ω, x, y, z).
(34)
Then, the vertical wave acceleration process at a point (x, 0, z) is zw3 (x, z, t) =
N OM
Re[φwa3 (ωn , x, z)ζn∗ eiωen t ].
(35)
n=1
3.4.4
Linear Vertical Loads
In regular waves and assuming linearity, the vertical time-variable loading (force per length) γ3 (ω, x, t) can be represented as γ3 (ω, x, t) = Re φγ3 ζ0 eiωe t , (36) where the vertical loading FTF φγ3 (ω, x) can be decomposed as γH γE + φγS φγ3 = φγM 3 3 + φ3 + φ3 .
10
(37)
Here M stands for inertia (mass), S for hydrostatics, H for motion-induced hydrodynamics, and E for excitation2 . The first component is 2 φγM 3 (ω, x) = m(x)ωe [φ3 (ω) − xφ5 (ω)],
(38)
where m(x) is the ship mass per length. The hydrostatic loading FTF is φγS 3 (ω, x) = −ρgBW (x)[φ3 (ω) − xφ5 (ω)].
(39)
To describe the hydrodynamic component, let us introduce the vector d
G ≡ (g1 , . . . , g6 )T = D(−iωe + V
∂ )(AW ) · Φ ∂x
(40)
(see eq. (12)). Then φγH 3 (ω, x) = g3 . Finally, the excitation FTF component is
iV ∂e7x3 −ikx cos µ γE 0 7 e . φ3 (ω, x) = ex3 + ex3 + ω ∂x
3.5
(41)
(42)
Mathematical Model for Nonlinear Vertical Loading
The vertical hydrodynamic forces observed on a ship hull in longitudinal motions have only partly been considered in the previous subsection. In the most interesting and critical case of heavy seas and large amplitude motions the loads can only be adequately described with a nonlinear model able to account for substantially nonlinear effects of slamming. The approach followed here is based on the concept of relative motion and, hence, the effective station immersion velocity w(x, t) is one of the most important defining parameters. Also, the water surface deformations disregarded in the linear theory cannot be neglected anymore as the water elevation (piling up) can sensibly affect the hydrodynamic forces. 3.5.1
Estimation of Immersion Velocity and Water Pile-Up
Let us introduce the following submergence parameters of an arbitrary section with abscissa x (see also Fig. 1): 1. T00 (x) — the distance between the x-axis in the ship’s equilibrium position and the still water surface; 2. T0 (x) — the distance between the x-axis in any instantaneous position of the ship and the water surface with account for undistorted oncoming waves; 2
Of course, the inertia forces are not real active forces and according to the d’Alembert principle the sum of all the listed components is identically zero for the whole body but this doesn’t hold for every particular strip as the shear forces are then to be added to the balance equation.
11
3. T (x) — the same as T0 but with additional account for the water pile-up caused by the hull’s immersion. All these quantities are positive when the x-axis is below the free surface. It is evident, that T0 (x, t) = T00 (x) + xr3 (x, t).
(43)
and the submergence parameter T will be commented later. Entrance velocity. The section’s effective immersion velocity w depends on the ship’s motions (heave and pitch), on the wave motion of the incident waves, on the ship’s speed of advance (a z-projection of this speed at a non-zero instantaneous pitch angle), and on the changes in the position and the shape of the section currently located on the observation plane being pierced by the moving hull. A consistent account for the last component is difficult because the section under consideration behaves like a deformable contour for which no translation velocity can be defined unambiguously. Fortunately, in the present case of a relatively slow ship in rough seas this component is not of major importance and it is sufficient to estimate it approximately. One can write
Dr(x, t) , (44) Dt where D/Dt is the time derivative for a point moving with velocity V backwards along the ship: ∂ ∂ D/Dt = −V , (45) ∂t ∂x and r(x, t) is the vertical distance between the wave profile and some fixed characteristic (reference) line inside or on the hull: w=
r(x, t) = T0 (x, t) + zref (x),
(46)
where z = zref (x) is the explicit equation of this line. The most obvious and popular choice is the keel line but it must not include any vertically oriented elements or appendages with small thickness (deadwood, stern post, stabilising fin etc.) which do not produce any vertical hydrodynamic forces within the assumed model (see the line ABOCD in Fig. 1). A more complicated situation arises in the vicinity of a protruded bulbous bow as in this case two consecutive entrance events can happen: entrance of the bulb and — later on — entrance of the above water-bow structure. While the keel line will be good to describe the entrance of the bulb, this is not the case with the bow itself as in this case the stem will apparently become the better choice. But the stem line is much steeper than the keel line under the bulb and the assumption of a 2D sectional flow is definitely violated in this region. The 3D effects will reduce the magnitude of the hydrodynamic loads in this region and this will more than compensate for a possible underestimation of the loads stemming from the underestimation of the reference line’s slope. At the same time, to eliminate doubts cocerning dangerous understimation, it seems to be necessary to give the user a possibility to set up the line manually which will allow the user to estimate the upper boundary of the load (see the line CF in Fig. 1. However, comparative calculations carried 12
out even at a very high (33 knots) speed of advance did not show any substantial influence of this issue. We will need also the pile-up immersion velocity wp defined by the similar equation wp =
Drp (x, t) , Dt
(47)
where rp is defined as r but with T instead of T0 . Account for the pile-up. S¨oding [25] has proposed the following empirical relationship linking the pile-up submergence observed at a sufficiently high entrance velocity T inf with T0 (x, t) : A T inf = T0 + 0.6 , (48) b where the sectional area A and waterline breadth b refer to the submergence T inf . The formula (48) is in fact a nonlinear algebraic equation with respect to the pile-up submergence T inf . S¨oding [25] also suggested the following iterative formula for estimating T inf : inf T inf = Tpre +
inf bpre (T0 − Tpre ) + 0.6Apre , inf bpre − (T0 − Tpre )/ tan α − 0.6bpre
(49)
where b is the instantaneous waterline breadth, and A(x, t) — the instantaneous section submerged area, without the pile-up effect, and the subscript pre refers to the previous iteration. For α, which has no influence on the result of the iteration, the average deadrise angle α ¯ of eq. (57) can be used. At the start of the iteration process it is assumed that b = b0 and A = A0 , where the subscript 0 refers to the absence of the pile-up. The equation (48) presumes that the pile-up is independent from the entry velocity (in fact it is valid for a fast entry). It is evident that in general T = T inf as, for instance, a body emerging from the water may cause a negative pile-up and any deformations of the free surface are negligible at low entrance velocity. A simple method to account for the entrance velocity’s influence was developed and it leads to the following calculation sequence transforming T inf into the final submergence parameter T (x; w) (T (1) is the intermediate variable):
1. T
(1)
(x) =
T inf (x) T0 (x)
where wth = FnT th
at w > wth otherwise,
(50)
gT
(51)
is the threshold entrance velocity, and FnT th is the threshold entrance Froude number based on the ship’s draught T ; 2. The actual submergence increment due to the pile-up is FnT (x) 1 − u− (1) , log ∆T (x) = F [T (x) − T0 (x)] tanh − FnT th 1 + u− 13
(52)
where FnT = √wgT , and u− is a real number close to unity (say, 0.95). The operator F : y(t) = F{x(t)} is a smoothing filter for which we used a Butterworth low-pass filter [26] described in the third-order case with the help of the following set of finitedifference equations: y(tk ) = y(tk−1) + y1 (tk−1 )∆t, y1 (tk ) = y1 (tk−1 ) + y2 (tk−1 )∆t, (53) y2 (tk ) = y2 (tk−1 ) + [ωc3 x(tk−1 ) − ωc3 y(tk−1) − 2ωc2 y1 (tk−1 ) − 2ωc y2 (tk−1 )]∆t, where y1 and y2 are the auxiliary state variables; ∆t = tk − tk−1 is the time interval; ωc = 2π/TCU F is the cutoff frequency. The cutoff period TCU F must be expressed as a fraction of some characteristic period for which the natural heave period Tζ was chosen i.e. TCU F = τCU F Tζ where τCU F is the dimensionless cutoff period. Plots of the both unfiltered and filtered ∆T are shown on Fig. 5 and Fig. 6. The filtered pile-up shows some overshoots which can be removed by increasing the cutoff period but we preferred not to do it as these overshoots don’t impair the smoothness of the final time history and smaller values of the cutoff period help to keep the peak values practically unreduced. 3. Finally, the estimated full submergence accounting for the pile-up effect is T (x) = T0 (x) + ∆T (x).
(54)
Comparative time histories for T0 (x, t) and T (x, t) are shown in Fig. 7 and Fig. 8. The described method of accounting for the entrance velocity in the pile-up effects assessment is not based on any rigorous theory but on the observations by Ochi (see [1, P. 232–233]) that there is some threshold entrance velocity above which the entrance is percepted as a slam. It was further assumed that just then a pronounced pile-up is observed and the th threshold entrance Froude number was roughly estimated as FnT th = √wgT = 0.45. Then, this initially stepwise entrance velocity correction was spread over some velocity interval and then further filtered to produce a smooth function. Of course, this method is rather simplistic and further studies of the pile-up phenomenon are desirable. 3.5.2
Estimation of Vertical Force on a Strip
The basic formula for the total hydrodynamic force (per length) FHD caused by immersion of a hull section is, according to von K´ arm´ an and Wagner:
2 π b(x, t) D FHD (x) = −k ρ w(x, t) , (55) Dt 2 2 where k is a coefficient which accounts for the flatness of the section shape: π k = 0.75( − α ¯ ), 2
(56)
and the “non-local deadrise angle” α ¯ is defined as α ¯ = arctan 14
2A . b2
(57)
In equation (55) the expression ρ π2 [b(x, t)/2]2 stands for the instantaneous sectional added mass. The formula holds exactly if, first, the section shape is a half-immersed ellipse, and, second, that gravity-induced forces can be neglected. The hydrostatic forces, which are to be added to FHD , must take into account the Smith effect: the acceleration of gravity g must be replaced with g − zw3 (x, z0 , t), where the wave particle acceleration zw3 is taken at the average position (x, 0, z0 ) of the section centre of gravity. With all corrections for the motion asymmetry (no shock is expected at the section’s withdrawal, the pressure cannot drop below the water vapour pressure pv , and the stationary part of the pressure cannot exceed the total head) introduced by S¨ oding [25] the formula for the total loading F becomes
π 2 Dw Db2 F = min −kρ b − ρ(g − zw3 (x, z0 , t))A0 , (pa − pv )b + w+ 8 Dt Dt
∂b2 π 1 2 ∂w 2 + f+ −kρ V b + w+ (58) − ρbV , 8 ∂x ∂x 2 where f+ (x) = max (x, 0), w if wp > 0 and w > 0, w+ = 0 otherwise
(59)
and where pa is the atmospheric pressure and wp , as before, is the immersion velocity with account for the pile-up. The physical sense of using w+ instead of w is that when being withdrawn from the water, the section doesn’t receive from the fluid the additional momentum corresponding to alterations of the submerged shape. Time histories for all the variants of the immersion velocity are plotted in Fig. 9. Time histories of the total external force F (x, t) and of its corrected hydrodynamic component
Db2 π 2 Dw F101 = −kρ b + w+ (60) 8 Dt Dt are shown in Fig. 10. Of particular interest is the impact component Db2 π (61) FIM P = −kρ w+ 8 Dt which is responsible for the slamming and whipping. This component appeared to produce discontinuities of numerical nature in spite of the described preliminary filtering. Therefore, it is desirable to filter it as well if this component and/or the total force without account for elasticity are to be plotted. At the same time, it is preferrable to leave it unfiltered if the total loading is used as vibration excitation. The time histories for the impact component in both filtered and unfiltered variants are shown in Fig. 11 and Fig. 12. The total hydrodynamic loading F approximately accounts for all relevant components including the time-averaged hydrostatic support force. Then, the total vertical load within the rigid-body approach FRB will be represented as FRB (x, t) = F (x, t) + γ3M (x, t), 15
(62)
where γ3M is the inertial and gravitational loading assumed according to d’Alembert’s principle. (The cosine of the pitch angle is taken as 1). It should not be determined from the linear motions because these do not match accurately the nonlinear force F . Thus, we determine the nonlinear vertical acceleration of the ship’s centre of gravity (at xG ), including the gravitational component (−g), 1 az (t) = F (x, t)dx, (63) m L and the pitch acceleration aθ (t) = −
1 IGyy
(x − xG )F (x, t)dx.
(64)
L
¿From these follows the vertical acceleration (including gravity acceleration) at position x, a(x, t) = az (t) − (x − xG )aθ (t),
(65)
and the inertial and gravitational load γ3M (x, t) = −m(x)a(x, t).
(66)
Due to the hull’s elasticity and distributed inertia, the rigid-body loading will induce vibrations, and the resulting stresses can become substantially different from those determined as if the load FRB (x, t) were applied quasi-statically. But it is possible to determine some additional equivalent statical loading Fs (x, t) which is due to vibrations considering FRB (x, t) as a driving force distribution. The corresponding theory will be given later.
3.6
On Slamming Detection
The method for estimating the hydrodynamic forces and vibrations used in this study doesn’t imply any slamming scenarii and doesn’t require detection of any particular slamming events. However, such information (number of slams observed during some specified time period or similar) can be of interest, and it can be obtained in course of the simulation. It is assumed that a slam occurs (and the slams counter is incremented) when the ship slips from the smooth-ride state to the slamming regime. The latter means that the slam is in progress i.e. at least at some station x the following condition is satisfied: FIM P (x, t) > Π0 = π0 ρgL, BW (x)
(67)
where Π0 is a threshold value of the mean impact pressure and π0 is the corresponding nondimensional parameter which must be in general tuned to experimental data but a value π0 = 0.001 can be assumed as a rough but reasonable estimate. Although a correct absolute value of the slamming frequency is then not guaranteed, it makes possible comparing different hull shapes from this viewpoint.
16
3.7
Mathematical Model for Vibration
Ship vertical vibrations excited by the seaway are determined using a beam model for the ship girder. Free undamped elastical deformations z0 (x, t) are governed by the equation µz¨0 + K(z0 ) = 0,
(68)
where K is the linear stiffness operator and µ(x) = m(x) + µ33∞ (x).
(69)
m(x) is the ship mass per length, µ33∞ the vertical added mass per length for high oscillation frequency. The forced vibration is treated as a superposition of modes: a0j (t)ζj (x), z0 (x, t) =
(70)
j
where a0j (t) are mode factors, and ζj (x) mode shapes. Substituting eq. (70) into eq. (68) shows that each a0j satisfies an un-coupled second-order linear homogeneous ordinary differential equation with constant coefficients and absent first derivative and, hence, can only be of the form aj (t) = ajm eiΩj t , where the amplitudes ajm can all be assigned unity values as they can be embedded into the modal shapes, and Ωj are modal (natural) frequencies. It can be shown also that the modal shapes ζj (x) must all be orthogonal on the ship length with the weight µ(x) and that they possess the property K(ζj ) = Ω2j µ(x)ζj .
(71)
The same mode shapes can be used for expanding the solution to the more general problem of forced and damped vibrations z(x, t) which are described by the equation µ¨ z + Kz + γ
∂ (Kz) + β z˙ = FRB (x, t), ∂t
(72)
where γ and β are respectively the structural and hydrodynamical damping coefficients. Substituting into this equation the solution taken in the form similar to (70) aj (t)ζj (x), (73) z(x, t) = j
and, applying the orthogonality property, we obtain the following equation for the amplitude functions aj (t): (74) a ¨j + ν¯Ωj a˙ j + Ω2j aj = fj (t), where the integrated damping factor ν¯ describing the decay of the modes’ free oscillations is introduced instead of γ and β. Exact values of ν¯ are not easy to determine and they can depend on the eigenfrequencies but for our purposes it is sufficient to use very crude estimates taken from literature (see e.g. [27]). The value 0.05 was used in our calculations.
17
The forcing functions are
fj (t) =
L
FRB (x, t)ζj (x)dx . µ(x)ζj2 (x)dx
(75)
L
As the right-hand side of (74) can be discontinuous, high-order numerical integration methods are not appropriate. Thus the explicit Euler method is used: bj (tk ) = bj (tk−1 ) + ∆t · [fj (tk−1 ) − νj bj (tk−1 ) − Ω2j aj (tk−1 )], aj (tk ) = aj (tk−1 ) + ∆t · bj (tk−1 ),
(76)
where bj (t) ≡ a˙j (t) and νj = ν¯Ωj . The vibrations follow directly from the aj (t) while the corresponding stresses may be obtained from the equivalent static load Fs (x, t) = K(z). But it may be easier to transform this by using eqs. (73) and (71) to Fs (x, t) = µ(x) aj (t)Ω2j ζj (x). (77) j
The eigenfrequencies Ωj and the corresponding mode shapes ζj (x) follow from structural computations. Only for testing purposes or for some qualitative estimates one could use the modes of a prismatic Bernoulli (Euler) beam with buoyancy effects neglected. The ship modal shapes can then be represented as [27], [28]: ζj (x) = αj + βj x + ζ0j (x), where the coefficients αj and βj must be chosen to satisfy the equilibrium conditions µ(x)ζj (x)dx = 0, xµ(x)ζj (x)dx = 0, L
(78)
(79)
L
and the prismatic beam shapes are: ⎧ cosh ωj ξ cos ωj ξ 1 ⎪ ⎨ 2 cosh 1 ωj + cos 1 ωj , j = 1, 3, . . . 2 2 ζ0j (ξ) = ⎪ sinh ω ξ sin ω j jξ ⎩ −1 , j = 2, 4, . . . + 1 1 2 sinh ω sin ω 2
j
2
(80)
j
Here ξ = x/L and the first four values of the dimensionless frequency parameter ω are [27]: ω1 = 4.73, ω2 = 7.85, ω3 = 11.0, ω4 = 14.4. The corresponding natural frequencies are then EI 1 2 Ωj = ω , (81) µ L2 j where E is Young’s modulus of elasticity, I is the mean girder moment of inertia of the hull, and µ is the mean value of µ(x). An alternative is to use empirical formulae suggested by Kumai (for the first harmonic) and by Johannessen and Skaar (higher harmonics up to number j ≤ 4 − 5). These formulae can be found in [28]. However, it is important to warn against using this analytic modal shapes option for any real world estimates as the errors can be substantial and unpredictable. 18
3.8
Total Vertical Loading, Shear Forces, and Bending Moments
The total vertical force per length v3 on the ship hull girder in irregular seas can be set equal to either the rigid body loading FRB (x, t), or to the equivalent static loading Fs (x, t): FRB (x, t) for the rigid hull, v3 (x, t) = (82) Fs (x, t) for the elastic hull. Although no springing and whipping effects are accounted for in the former case, both options may be of interest. The time-dependent vertical shear force distribution is given by bow v3 (ξ, t)dξ, V3 (x, t) =
(83)
x
and the relevant bending moments by bow M3 (x, t) = V3 (ξ, t)dξ
(84)
x
Using these relations, the bending moment amidships is positive in hogging condition. A considered but not realized alternative to this approach is to isolate the linear part of the wave-induced loading (or shear force and bending moment) which is computed with better accuracy by the strip method used before the nonlinear load simulation. However, it is not clear whether the resulting estimates would be more accurate.
4 4.1
Remarks on Implementation Numerical Methods
Although the results will be obtained in the time domain, the method does not require a numerical solution of an initial value problem for estimation of the hydrodynamic forces but involves mainly computation of the finite-difference approximations for various derivatives. Sets of ordinary differential equations aris only in connection with the filtering and with the vibrations computations. The explicit Euler method is used in both cases. While the numerical filtering could be performed with the basic time step, a smaller step was required in the vibrations analysis. The basic time step and the filters’ cutoff periods are adjusted in accordance with the approximate natural period of heave estimated by means of the formula 2CV T T00 Tζ = 2π , (85) g where CV T is the vertical prismatic coefficient for which a value 0.8 can be used as a universal constant. Then, there is no need to adjust the integration step and filtering periods when a smaller ship or a model is investigated. 19
Two kinds of the derivatives of some function f () are dealt with in this aplication: the time derivative ∂f and the longitudinal spatial derivative ∂f . In the former case the function ∂t ∂x is always known for the current time ti and for previous time instants ti−1 , ti−2 , . . .. The first-order estimate of the time derivative related the best to the function value f (t¯i ) = 1 [f (ti ) + f (ti−1 )] will then be given by the finite difference fraction 2 ∂f f (ti ) − f (ti−1 ) ≈ , ∂t ∆t
(86)
where ∆t = ti − ti−1 is the time increment, which is chosen constant for each simulation. Spatially, the functions are initially known at discrete values of the argument xi related to the sections of the hull (i = 0, . . . , nx − 1, where nx is the number of sections). Function values in the middle between xi and xi−1 , where all the sectional loads are applied, are approximated by averaging f (xi ) and f (xi−1 ), while the derivatives at the same point are determined as ∂f f (xi ) − f (xi−1 ) . (87) ≈ ∂x ∆xi As the iterated spatial derivatives approximated with the central differences are again estimated in the sections 1...nx − 1, they are averaged again to get estimates in the middle of the slices. As to the two extreme slices, it is assumed that the derivatives there remain equal to those at the corresponding pre-extreme sections. The middle-value rectangular rule is applied in all the cases when numerical integration is needed.
4.2
Structure and Peculiarities of the Simulation Program
The algorithm was coded in Fortran 90. It contains modules for 1. the Donelan wave energy spectrum; 2. modelling of waves; 3. hull shape input and preparations; 4. nonlinear load simulations; 5. the vibratory response; 6. the Butterworth filters; 7. auxiliary procedures; and 8. definitions of constants and kinds of variables. Additional information about the code can be picked up from the program manual [29] and from the commented source files.
20
5 5.1
Simulation Results Description of the Sample Ship
A number of test calculations have been carried out to verify the program’s results. All the computations were done for the ship I which had already served as a test ship in the study by Zhou and S¨oding [24]. The ship is 190 meters long, has the beam equal to 28 meters and has a draught 6.81 m in full load. Her displacement at this draught is 19031 m3 . The computations were carried out at the zero speed and the speed 32.9 knots (the same as used by Zhou and S¨oding), firstly in head regular sea with the wave height 8 m and, then, in the head irregular sea described by a Donelan spectrum with the significant wave height equal to 8 m as well. A trapezoidal schematised longitudinal mass distribution was assumed. The body lines, as they follow from the EUMEDES description, are shown on Fig. 2. The program SLAMVORM [29] was used to obtain discretisations for the preliminary linear transfer functions calculations with the STRIPmod program (Fig. 3) and for the slamming simulation in time domain with the SLAVIB program (Fig. 4).
5.2
Preliminary Calculations: Qualitative Study and Comparison
The hydrodynamic calculations were carried out for a 39-sections discretisation (numbered from 0 to 38) while only 21 sections were used for vibrations analysis and for the final presentation of all the results. This relatively scarce spacing was chosen as identical to that used by Zhou and S¨oding [24] and this facilitated validation of the present code. The code is able to produce output for any of the 20 strips but only strips from 15 to 20 were chosen to avoid excessive congestion of the plots. All the results involving the vibratory analysis presented here are based on the canonic mode shapes for a prismatic Bernoulli beam, as the real mode shapes for the given ship were not available to us. Hence, these results should not be considered as providing any valuable quantitative estimates. The rigid ship load distribution time histories FRB (x, t) are presented in Fig. 13 while there equivalent static counterparts Fs (x, t) accounting for the ship’s elasticity in Fig. 14. Time histories of the elastic vertical displacements z(x, t) at different stations of the ship’s hull including those due to whipping are given in Fig. 15. At any time instant, the code is automatically catching the maximum and minimum values of certain dynamic parameters over the hull’s length and fixing the strips where these extrema are reached. The Fig. 16 shows such values for the loads (both FRB and Fs ) while the stations where these values are achieved are marked in Fig. 17. The maximum and minimum elastic displacements and their positions are shown in Fig. 18 and Fig. 19. The same kind of information related to the shear forces is given in Fig. 20 to Fig. 23, and for the bending moments in Fig. 24 to Fig. 27. Finally the bending moments time histories in the midship section are presented in Fig. 28. The simulated wave is shown in Fig. 29, whereas the time histories for the impact force, hydrodynamic force and total force on a strip are given on Fig. 30, Fig. 31, and Fig. 32 respectively. Finally, time histories for the bending moment are presented in Fig. 33. 21
Comparisons with other data. The results obtained by Zhou and S¨oding [24] correspond to the rigid-body formulation and they can be compared to ours in that part. However, the value for the hydrodynamic load F101 equal to 650 kN/m for the strip no. 36 (with no. 39 being the extreme bow) was never approached in our calculations. However, those values for the strip no. 36 seem to be strange as the highest value at the next strip no. 35 was only 160 kN/m. Our maximum observed values were 42 kN/m when unfiltered although we believe that in reality those sharp peaks will not happen in practice and the real amplitude will be even smaller. The total external loading F for the same strip was found to have the double amplitude 350 kN/m in our calculations and practically the same in those done by Zhou and S¨oding if the excessive peak is removed. Good agreement for the total force was found for other strips as well. As to the data on the shear force and bending moment, they are absent in [24] and it was only possible to check them qualitatively against corrresponding published data on the S175 ship. Our data were in reasonable agreement with the data given in [3] and [15], especially if the dimensions corrections are introduced, with the main difference that ship I is in hogging condition in still water with the assumed mass distribution while the S175 is in sagging condition.
5.3
Preliminary Calculations: Discretization Influence
The simulation algorithm has main three discretisation parameters: longitudinal defined by the number of transverse strips nx , vertical defined by the number of vertical slices nz , and temporal defined by the main time step ∆t. The finer is the grid and the time discretisation, the more accurate is the result within the adopted mathematical model and algorithm but the program is then becoming more time consuming. To choose optimal spatial and temporal spacings, several simulations were carried out in regular waves with different values of the mentioned parameters. Maximum and minimum values of the bending moment V5 (without any account for vibrations) were chosen as reference. Results of these simulations are assembled in Table 1. Table 1: Influence of Discretization nx 50 100 200 200 200 200 400 400
nz 100 200 400 400 400 400 400 600
∆t, s 0.01 0.01 0.01 0.005 0.002 0.001 0.005 0.005
V5max , kNm 0.1498e7 0.1520e7 0.1515e7 0.1516e7 0.1516e7 0.1516e7 0.1516e7 0.1517e7
V5min , kNm Relative CPU time -8 1 -4570 2.44 -1757 7.41 -1570 14.07 -1464 34.67 -1428 70.83 -1807 28.1 -1809 40.34
Keeping in mind that the method itself is approximate, one can come to conclusion that the influence of the discretization will be negligible at nx = 200, nz = 400, and ∆t = 0.01s. 22
The absolute CPU time was then equal to 1240 s with the PC PIII/550MHz simulating 100 s of the process in regular waves.
5.4
Main Block of Calculations
The results presented in this subsection were obtained at the same speed and wave conditions as the previous ones but with the fine discretisation (200–400–0.01) and using far more realistic data on the eigenfrequencies and modal shapes which are orthogonal with the given mass+added mass distribution. 51 sections instead of 21 were used for the STRIP transfer functions calculations and vibrations calculation (in fact, the number of sections was slightly lower for STRIP as extreme stern and bow sections not immersed into the water in equilibrium condition don’t participate in linear analysis). The regular waves calculations were carried out for the same wave height 8 m, 11.3 s period head sea, and Froude number 0.39. The time histories for midship bending moment in the rigid body and elastic formulations are given on Figures 34–35 and also some numerical results are given in a special file:
SLAVIB Nonlinear Loads and Vibrations Simulation: Ishiguro Schiff voll beladen Main Particulars: Length......... 191.230 m Beam........... 28.009 m Draught........ 6.810 m Trim........... 0.000 m Displacement... 19154.4 m^3 Speed.......... 16.90 m/s = 32.9 kn Froude number.. 0.390 Regular waves: Height: 8.000 m Period: 11.300 s Propagation angle: 180.0 deg
Simulation time:
400.0 s
Results are presented for
50 strips numbered from 1 (stern) to
50 (bow)
Extreme bending moments (still water + waves; elastic hull): --maximum equivalent statical value: 0.1211E+07 kN*m at station --minimum equivalent statical value: -5843. kN*m at station Extreme bending moments (still water + waves; rigid hull): 23
29 6
--maximum value: --minimum value:
0.1183E+07 kN*m -7779. kN*m
at station at station
28 5
Midship extreme equivalent bending moments (still water + waves; elastic hull): --in hogging: 0.1157E+07 kN*m --in sagging: 0.3589E+06 kN*m Midship extreme bending moments (still water + waves; rigid hull): --in hogging: 0.1141E+07 kN*m --in sagging: 0.4706E+06 kN*m
Simulation started on Simulation finished on Elapsed CPU time:
2003.01.23 2003.01.23
at at
17:52:37.8 19:05:50.7
4392.957 seconds
The irregular sea simulation was carried out at similar conditions: significant wave height 8 m and mean period 11.3 s. There were used 100 harmonics (10 frequencies and 10 angles). The midship bending moment time histories are presented on Figures 36–37. And on Figures 38–39 shown are time histories of maximum and minimum vibrational displacements and their corresponding positions along the ship’s hull. Here are the numerical values obtained as the result of the simulation:
SLAVIB Nonlinear Loads and Vibrations Simulation: Ishiguro Schiff voll beladen Main Particulars: Length......... 191.230 m Beam........... 28.009 m Draught........ 6.810 m Trim........... 0.000 m Displacement... 19154.4 m^3 Speed.......... 16.90 m/s = 32.9 kn Froude number.. 0.390 Irregular waves (the Donelan spectrum): Significant height: 8.000 m Peak period: 11.300 s General propagation angle: 180.0 deg
24
Simulation time:
400.0 s
Results are presented for
50 strips numbered from 1 (stern) to
50 (bow)
Extreme bending moments (still water + waves; elastic hull): --maximum equivalent statical value: 0.1723E+07 kN*m at station --minimum equivalent statical value: -0.1561E+06 kN*m at station
26 38
Extreme bending moments (still water + waves; rigid hull): --maximum value: 0.1652E+07 kN*m at station 26 --minimum value: -0.1576E+05 kN*m at station 39 Midship extreme equivalent bending moments (still water + waves; elastic hull): --in hogging: 0.1714E+07 kN*m --in sagging: 0.1220E+06 kN*m Midship extreme bending moments (still water + waves; rigid hull): --in hogging: 0.1646E+07 kN*m --in sagging: 0.3735E+06 kN*m
Simulation started on Simulation finished on Elapsed CPU time:
2003.01.23 2003.01.23
at at
15:49:59.3 17:09:30.9
4771.562 seconds
In fact, irregular sea simulations must be substantially longer, at least 1200–1500 real-time seconds and carried out with increased number of harmonics (say, 25 frequencies instead of 10) but the both factors will increase considerably the required CPU time.
6
Conclusions
A Fortran 90 code SLAVIB for simulating nonlinear ship loads in irregular sea has been developed and tested. The program is supposed to be used together with the modified STRIP program computing transfer functions on the basis of a linear ship mathematical model. Two runs of the program SLAVIB are required with an intermediate run of the STRIPmod program. An improvement could be made by re-arranging the code such that it would require a single run of the code only. A more substantial modification of the code STRIP would be required to provide a better modelling of the irregular seaway by using a set of random couples wave angle, wave length instead of a set of random angles together with a set of random lengths, as is done at present.
25
References [1] Paulling J.R. Strength of Ships. In: Principles of Naval Architecture V.1/Ed. E.V. Lewis. Jersey City, NJ: SNAME, 1988. P. 205–300. [2] S¨oding H. Beschreibung des Programms STRIP. Institut f¨ ur Schiffbau der Universit¨at Hamburg: 1994. 7 p. [3] Fonseca N., Guedes Soares C. Time-Domain Analysis of Large-Amplitude Vertical Ship Motions and Wave Loads. Journ. Ship Research, 1998, V. 42, No. 2, pp. 139–153. [4] Payne P.R. Design of High-Speed Boats: Planing. Fishergate Publ., Annapolis, 1996. 233p. [5] Faltinsen O. Hydrodynamics of High Speed Vehicles. In: Advances in Marine Hydrodynamics/ Ed. M. Ohkusu. Comp. Mech. Publ., Southampton, 1996. P. 133–175. [6] Faltinsen O. Slamming on Ships. 9th Congress of International Maritime Association of Mediterranian, 2–6 April 2000, Ischia (Italy): Proceedings V. 1. P. 25–36. [7] Korobkin A. Water Impact Problems in Ship Hydrodynamics. In: Advances in Marine Hydrodynamics/ Ed. M. Ohkusu. Comp. Mech. Publ., Southampton, 1996. P. 323–371. [8] Zhao R., Faltinsen O. Water Entry of Two-Dimensional Bodies. J. Fluid Mech., 1993, vol. 246, pp. 593–612. [9] Bereznitski A., Boon B., Postnov V. Numerical Study of the Hydroelastic Effect on the Impact Loads Due to Bottom Slamming on Ship Structure. Proceedings of ETCE/OMAE2000 Joint Conference Energy for the New Millenium. February 14–17, 2000, New Orleans, LA, USA, 9 pp. [10] Vorus W.S. A Flat Cylinder Theory for Vessel Impact and Steady Planing Resistance. Journ. Ship Research, 1996, V. 40, No. 2, pp. 89–106. [11] Guedes Soares C. Transient Response of Ship Hulls to Wave Impact. Intern. Shipbuild. Progr., 1989, V. 36, No. 406, pp. 137–156. [12] Ramos J., Guedes Soares C. Vibratory Response of Ship Hulls to Wave Impact Loads. Intern. Shipbuild. Progr., 1998, V. 45, No. 441, pp. 71–87. [13] Faltinsen O. On Seakeeping of Conventional and High Speed Vessels. Journ. Ship Research, 1993, V. 37, No. 2, pp. 87–101. [14] Wu MK., Moan T. Linear and Nonlinear Hydroelastic Analysis of High Speed Vessels. Journ. Ship Research, 1996, V. 40, No. 2, pp. 149–163. [15] Wang L., Moan T. Stochastic Analysis of Nonlinear Wave Load Effects in Ships. In: Applications of Statistics and Probability/ Ed. Melchers & Stewart, Balkema, Rotterdam, 2000, pp. 861–868. [16] Xia J., Wang Z. Time-Domain Hydroelasticity Theory of Ships Responding to Waves. Journ. Ship Research, 1997, V. 41, No. 4, pp. 286–300. 26
[17] Weems K., Zhang S., Lin W.-M., Bennett J., Shin Y.-S. Structural Dynamic Loadings Due to Impact and Whipping. Proceedings PRADS’98 (Practical Design of Ships and Mobile Units)/ Ed. M.W.C. Oosterveld and S.G. Tan. Elsevier Science B.V. 1998. P. 79– 85. [18] Watanabe I., Guedes Soares C. Comparative Study on the Time-Domain Analysis of Non-Linear Ship Motions and Loads. Marine Structures, 1999, V. 12, pp. 153–170. [19] S¨oding H. Bewegungen und Belastungen der Schiffe im Seegang. Institut f¨ ur Schiffbau der Universit¨at Hamburg: Vorlesungmanuscript Nr. 18. 1982. 50 p. [20] S¨oding H. Berechnung der Bewegungen und Belastungen von SWATH-Schiffen und Katamaranen im Seegang. Institut f¨ ur Schiffbau der Universit¨at Hamburg: Bericht Nr. 483. 1988. 26 p. [21] Kr¨oger, H.-P. Rollsimulation von Schiffen im Seegang. Schiffstechnik Bd. 33 1986. P. 187–205. [22] S¨oding H. Ermittlung der Kentergefahr aus Bewegungssimualtionen. Schiffstechnik Bd. 34 1987. P. 28–39. [23] Faltinsen O. Sea Loads on Ships and Offshore Structures. Cambridge University Press, 1990. [24] Zhou Y., S¨oding H. Berechnung von Stoßkr¨aften auf schnelle Schiffe. Manuscript 1999. 19p. [25] S¨oding H. Computation of Forces due to Slamming and in Planing. Manuscript 1999. 14p. [26] Carlson G.E. Signal and Linear System Analysis. John Wiley & Sons, Inc., 1998. [27] Kurdyumov A.A. Ship Vibration. Sudpromgiz Publ., Leningrad, 1961. [28] Vorus W.S. Vibration. In: Principles of Naval Architecture V.2/Ed. E.V. Lewis. Jersey City, NJ: SNAME, 1988. P. 255–316. [29] Sutulo S. Dynamic Ship Loading Estimation Code: Description of the Program SLAVIB. Manuscript, 2003, 16p.
27
Figures
Figure 1: Some immersion parameters and definition of the keel line: W0 L0 : still-water waterline; W L: actual waterline in waves
28
Figure 2: Body lines of the sample ship: EUMEDES representation
Figure 3: Body lines of the sample ship: sections generated for STRIPmod (51 nominal sections)
29
Figure 4: Body lines of the sample ship: sections generated for SLAVIB (200 sections, 400 waterplanes
15 14
∆Tunfiltered ∆T
13 12 11
∆Tunfiltered, ∆T
10 9 8 7 6 5 4 3 2 1 0 -1
10
20
30
40
Time
Figure 5: Instantaneous submergence increment due to the pile-up: unfiltered and filtered (section 36)
30
6 ∆Tunfiltered ∆T
5
∆Tunfiltered, ∆T
4
3
2
1
0
30
32
34
36
38
Time
Figure 6: Instantaneous submergence increment due to the pile-up: a zoomed view (section 36)
30
T0 T
25
20
T0, T
15
10
5
0
-5
10
20
30
40
Time
Figure 7: Instantaneous submergence time histories: with and without pile-up, unsmoothed and smoothed (section 36 with section 38 being the last one)
31
18
17.5
T0 T
T0, T
17
16.5
16
15.5
15 11
12
13
Time
Figure 8: Instantaneous submergence time histories: a zoomed view (section 36)
5
0
w, wp, w+
-5
-10
-15
-20
-25
w wp w+ 10
20
30
40
Time
Figure 9: Entry velocity time histories (section 36)
32
0 -50 -100 -150
F, Ffilt, F101
-200 -250 -300 -350 -400
F Ffilt F101
-450 -500 5
10
15
20
25
30
35
40
Time
Figure 10: Time histories for the total force on the strip 36 (filtered and unfiltered) and for the hydrodynamic component F101 (unfiltered)
50 FIMP FIMPfilt
40
FIMP, FIMPfilt
30
20
10
0
-10
0
10
20
30
40
Time
Figure 11: Time histories for the impact component of the force on the strip 36: filtered and unfiltered
33
50 FIMP FIMPfilt
FIMP, FIMPfilt
40
30
20
10
0 28
29
30
31
32
33
34
Time
Figure 12: Time histories for the impact component on strip 36: a zoomed view
Rigid-body load, kN/m
600
400
200
0
-200
-400 0
F15 F16 F17 F18 F19 F20 10
20
30
40
time
Figure 13: Time histories for the total loading on the strips 15 to 20 (with 20th strip being the last): the ship is assumed to be rigid; head regular waves hw = 8 m, Fn = 0
34
800
Equivalent static load, kN/m
600
400
200
0
-200
-400
0
F15 F16 F17 F18 F19 F 10 20
20
30
40
time
Figure 14: Time histories for the equivalent static load on the elastic ship
0.3
z15, z16, z17, z18, z19, z20
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 0
z15 z16 z17 z18 z19 z 1020
20
30
time
Figure 15: Time histories for the vibration displacements
35
1600 1400 1200
mxdyn, mndyn, mxsta, mnsta
1000 800 600 400 200
mxdyn mndyn mxsta mnsta
0 -200 -400 -600 -800
-1000 -1200 -1400 -1600 -1800
0
10
20
30
40
time
Figure 16: Time histories for the maximum and minimum (along the ship hull) values of the rigid-body and equivalent static load, kN/m
imxdyn, imndyn, imxsta, imnsta
20 18 16 14 12 10 8 imxdyn imndyn imxsta imnsta
6 4 2 0
0
10
20
30
40
time
Figure 17: Time histories for the positions where maximum and minimum values of the rigid-body and equivalent static loads are obtained
36
0.35 zmax zmin
0.3 0.25
mxvib, mnvib
0.2 0.15 0.1 0.05 0 -0.05 -0.1 0
10
20
30
40
time
Figure 18: Time histories for the maximum and minimum vibrational displacements
20
izmax izmin
18
imxvib, imnvib
16 14 12 10 8 6 4 2 0
0
10
20
30
40
time
Figure 19: Time histories for the positions of extreme displacements
37
40000 30000
mxshear, mnshear
20000 10000 V 3max V 3min
0
-10000 -20000 -30000 -40000
0
10
20
30
40
time
Figure 20: Time histories for the maximum and minimum values of the shear force (elastic hull)
18 16
imxshear, imnshear
14 12 10 8 6 4 iV3max iV3min
2 0
0
10
20
30
40
time
Figure 21: Time histories for the positions of extreme shear force (elastic hull)
38
3.0E+04
mxshearrb, mnshearrb
2.0E+04
1.0E+04
V 3RBmax V 3RBmin
0.0E+00
-1.0E+04
-2.0E+04
-3.0E+04
0
10
20
30
40
time
Figure 22: Time histories for the maximum and minimum values of the shear force (rigid hull)
20
iV3RBmax iV3RBmin
18
imxshearrb, imnshearrb
16 14 12 10 8 6 4 2 0
0
10
20
30
40
time
Figure 23: Time histories for the positions of extreme shear force (rigid hull)
39
2.5E+06
V 5max V 5min
2.0E+06
mxbend, mnbend
1.5E+06
1.0E+06
5.0E+05
0.0E+00
-5.0E+05
-1.0E+06
0
10
20
30
40
time
Figure 24: Time histories for the maximum and minimum values of the bending moment (elastic hull)
20 18
imxbend, imnbend
16 14 12 10 8 6 4 iV5max iV5min
2 0
0
10
20
30
40
time
Figure 25: Time histories for the positions of extreme bending moment (elastic hull)
40
2.5E+06
V 5RBmax V 5RBmin
mxbendrb, mnbendrb
2.0E+06
1.5E+06
1.0E+06
5.0E+05
0.0E+00
-5.0E+05
-1.0E+06
0
10
20
30
40
time
Figure 26: Time histories for the maximum and minimum values of the bending moment (rigid hull)
iV5RBmax iV5RBmin
20
imxbendrb, imnbendrb
18 16 14 12 10 8 6 4 2 0
0
10
20
30
40
time
Figure 27: Time histories for the positions of extreme bending moment (rigid hull)
41
MidVBMelastic, MidVBMrigid
2.0E+06
1.5E+06
1.0E+06
5.0E+05
MidVBMelastic MidVBMrigid
0.0E+00
-5.0E+05
-1.0E+06
0
10
20
30
40
Time
Figure 28: Time histories for the bending moment at the midship section: for rigid and elastic hull
4 3 2 1
Wave18
0 -1 -2 -3 -4 -5 -6
0
50
100
time
¯ 1/3 = 8.0m; Donelan spectrum; process at the station 18 Figure 29: Irregular head sea: h (out of 21); Fn = 0
42
50
40
FIMPfilt
30
20
10
0
-10
0
25
50
75
100
Time
Figure 30: Impact force time history in irregular sea: strip 36 (out of 38), Fn = 0.39
0
-10
F101
-20
-30
-40
-50
-60
0
25
50
75
100
Time
Figure 31: Time history for the hydrodynamic force F101 on the strip 36 in head irregular sea
43
0 F Ffilt
-50 -100
F, Ffilt
-150 -200 -250 -300 -350 -400
0
50
100
Time
Figure 32: Time histories for the full force acting on the strip 36 in head irregular sea: unfiltered and filtered
44
MidVBMelastic MidVBMrigid 1.6E+06
MidVBMelastic, MidVBMrigid
1.4E+06 1.2E+06 1E+06 800000 600000 400000 200000 0 0
50
100
Time
Figure 33: Time histories for the midship bending moment in head irregular sea: rigid-body and elastic formulations (preliminary variant: 39 sections discretisation and prismatic Euler beam eigenfrequencies and modal shapes)
45
1.2E+06
MidVBMelastic, MidVBMrigid
1.1E+06 1.0E+06 9.0E+05 8.0E+05 7.0E+05 6.0E+05 5.0E+05 4.0E+05 3.0E+05 2.0E+05
MidVBMelastic MidVBMrigid
1.0E+05 0.0E+00
0
50
100
150
200
Time
Figure 34: Time histories for the midship bending moment in head regular sea: rigidbody and elastic formulations (final variant: Fn = 0.39, 201 sections × 401 waterplanes discretisation and variable section Timoshenko beam eigenfrequencies and modal shapes)
46
1.2E+06
MidVBMelastic, MidVBMrigid
1.1E+06 1.0E+06 9.0E+05 8.0E+05 7.0E+05 6.0E+05 5.0E+05 4.0E+05 3.0E+05 2.0E+05
MidVBMelastic MidVBMrigid
1.0E+05 0.0E+00
180
185
190
195
200
Time
Figure 35: Time histories for the midship bending moment in head regular sea: rigid-body and elastic formulations (final variant, zoomed)
2.0E+06
MidVBMelastic, MidVBMrigid
MidVBMelastic MidVBMrigid 1.5E+06
1.0E+06
5.0E+05
0.0E+00
0
50
100
150
200
250
300
350
400
Time
Figure 36: Time histories for the midship bending moment in head irregular sea: rigid-body and elastic formulations (final variant)
47
MidVBMelastic MidVBMrigid
1.4E+06
MidVBMelastic, MidVBMrigid
1.3E+06 1.2E+06 1.1E+06 1.0E+06 9.0E+05 8.0E+05 7.0E+05 6.0E+05 5.0E+05
270
280
290
300
310
320
330
Time
Figure 37: Time histories for the midship bending moment in head irregular sea: rigid-body and elastic formulations (final variant, zoomed)
0.25 zmax zmin
0.2
mxvib, mnvib
0.15
0.1
0.05
0
-0.05
0
100
200
300
400
time
Figure 38: Time histories for the maximum and minimum vibrational displacements in head irregular sea
48
izmax izmin
50
imxvib, imnvib
40
30
20
10
0
0
100
200
300
400
time
Figure 39: Time histories for the positions of extreme displacements in head irregular sea
49
