1 Introduction

The need to ensure sufficient energy for an expanding global population and the transportion required to deliver large amounts of cargo globally has fueled an increase in the design of larger ships so that companies can benefit from economy of scale(Temarel and Hirdaris,2009; ISSC,2012). In addition,the trend in faster and larger sea vessels has been accelerated by the rapid increase in global trade(Kim et al.,2009). For example,the continued growth of China’s economy requires an increasing supply of ores,and thus necessitates the design and optimization of super-large ore carriers(Hu et al.,2012). A greater number of high-strength steel plates have been used in ship hulls to reduce ship weight,and this has consequently resulted in a reduction in the structural rigidity of ships(Hirdaris and Temarel,2009; Hu et al.,2012). However,large ships are relatively more flexible,and their hull structural natural frequencies belong to a range of the encounter wave frequencies in an ordinary sea spectrum(Senjanovic et al.,2008),which results in resonance. Therefore,this causes a springing response in large ships(Hirdaris et al.,2009; Kim et al.,2009; Choi et al.,2010),and it is critical to make a precise estimation of springing-induced hull girder wave loads,so that such loads are incorporated at the design stage to prevent hull girder failure due to fatigue damage caused by springing vibrations(Jang et al.,2007,Hirdaris et al.,2014). In addition,there is a critical need to determine the effect of discontinuities,such as deck openings,on ship strength(Hirdaris et al.,2006),and it is also important to determine the wave-induced hydroelastic response of large ships to improve classification rules and ensure ship safety(Senjanovic et al.,2008).

Several prior studies have been conducted to investigate the springing and whipping characteristics of containerships(Miyake et al.,2009; Choi et al.,2010; Miyakeet al.,2010; Lee et al.,2011; Hong et al.,2012; Barhoumi and Storhaug,2014; Hong et al.,2015; Shin et al.,2015). Springing loads result in enormous resonant loads because the bending and torsional stiffness of containerships are both generally small due to the open deck. However,although higher-order springing and whipping responses of containerships were also previously studied during the joint industry project known as Wave-Induced Loads on Ships(Lee et al.,2011; Hong and Kim,2014),to the authors’ knowledge there is limited available literature and experimental data on the springing response of very large ore carriers.

In general,ore carriers have a higher stiffness than containerships,which necessitates the need to also investigate higher order springing. In this study therefore,a 550 000 DWT ore carrier prototype with two different stiffness schemes,i.e.,scheme A and scheme B,is investigated to determine the effect of stiffness on the springing response. The full load condition is considered,and the ship prototype is designed with a softer stiffness than a real ship to focus on the springing phenomenon. Tank model experiments are conducted for the two stiffness schemes,and numerical analyses are conducted using both rigid body theory and hydroelasticity theory,based on an in-house code known as Compass Wave Loads Calculation System(WALCS): the WALCS-Basic module for rigid theory and a linear elastic WALCS-LE module for hydroelasticity theory(Bishop et al.,1986; Hirdaris and Temarel,2009; Li,2009). A comparison of the springing response in regular waves is then made between the numerical simulation and experimental tests results.

2 Numerical simulation 2.1 Hydroelasticity theory

The classical approach used to determine ship motion and wave loads assumes that the ship hull acts as a rigid body. However,this approach is not reliable enough for use with ultra-large ships,due to the mutual influence of wave loads and the structure response(Senjanovic et al.,2008; Temarel and Hirdaris,2009). A large ship’s hull girder is elastic,and the action of waves causes it to distort(Bishop and Price,1979); therefore,the analysis of wave load and ship deformation is a coupled hydroelastic problem(Senjanovic et al.,2008). Elastic deformations of the body depend on hydrodynamic forces and vice versa(Hirdaris and Temarel,2009; Korobkin et al.,2011),and thus an analysis of ship motion and wave loads needs to consider the hull deformation effect in waves(Bishop and Price,1983; Cui et al.,2007).

Linear hydroelasticity is based on both beam theory and three dimensional linear potential flow theory(Bishop et al.,1986,Hirdaris and Temarel,2009; Wu and Cui,2009),where the boundary conditions are assumed to be linear on both the body and free surface. According to a theory by Rayleigh,any distortion of a beam may be expressed as an aggregate of distortions in its principal modes(Bishop and Price,1979). In hydroelasticity,the ship’s structure is considered as a beam,and the structural deflections of a floating slender ship expressed by the free undamped “wet” bending modes of the hull in water(Newman,1994). The hydrodynamic force,F^h,is divided into two parts: F^R,which depends on the structural deformations,and F^DI,which represents the pure excitation(Senjanovic et al.,2008)as

$ {F^h} = {F^R} + {F^{DI}} $

(1)

Using modal superposition,the vector of the wetted surface of deformations H(x,y,z)can be presented as a series of dry natural modes h_r(x,y,z),

$ \begin{array}{l} \boldsymbol{H}\left({x,y,z} \right)= \sum\limits_{r = 1}^N {{\boldsymbol{h}_r}\left({x,y,z} \right)} \\ {\rm{ }} = \sum\limits_{r = 1}^N {{\xi _r}\left[ {h_x^r\left({x,y,z} \right){\boldsymbol{i}} + h_y^r\left({x,y,z} \right){\boldsymbol{j}} + h_z^r\left({x,y,z} \right){\boldsymbol{k}}} \right]} \end{array} $

(2)

where h_r(x,y,z)denotes the general motion/deformation mode,and ξ_r denotes unknown coefficients for the r^th mode.

The total velocity potential,φ,is defined using the Laplace differential equation,where the velocity potential is decomposed into incident wave potential,φ_I,diffraction potential,φ_D,and radiation potential,φ_r,components as follows,

$ \varphi = {\varphi _I} + {\varphi _D}{\rm{ + i}}\omega \sum\limits_{r = 1}^N {{\xi _r}{\varphi _r}} $

(3)

where ω is wave frequency,and i is the imaginary unit.

The linearized wave pressure is given by the Bernoulli’s equation,

$ p = {\rm{i}}\omega \rho \varphi - \rho gz $

(4)

where ρ is density,g is acceleration due to gravity,and z is depth.

The hydrodynamic forces are then obtained by integrating the pressure over the wetted surface. The excitation and radiation components are determined by considering the pressure term associated with the velocity potential,iωρφ,while the hydrostatic component is determined by considering the hydrostatic term,-ρgz.

The boundary conditions used for the velocity potential are as follows(Bishop and Price,1979,Cui et al.,2007),

$ \left\{ \begin{array}{l} {\nabla ^2}{\varphi _j} = 0{\rm{ in the fluid}}\\ - {\omega _e}^2{\varphi _j} + g\frac{\partial }{{\partial z}}{\varphi _j} = 0{\rm{ on }}z = 0\\ \\ \frac{{\partial {\varphi _j}}}{{\partial n}} = \left\{ \begin{array}{l} \left[ {i{\omega _e}{u_j} + \left({ - U\frac{\partial }{{\partial x}}} \right){u_j}} \right] \cdot n{\rm{ }}j = 1,2,....,m\\ - \frac{{\partial {\varphi _I}}}{{\partial n}}{\rm{ }}j = m + 1 \end{array} \right.{\rm{ on }}S\\ \left\{ \begin{array}{l} \frac{{\partial {\varphi _j}}}{{\partial Z}} = 0{\rm{ }}Z = - h\\ \nabla {\varphi _j} = 0{\rm{ }}Z \to - \infty \end{array} \right.\\ \mathop {\lim }\limits_{R \to \infty } \sqrt R \left({\frac{{\partial {\varphi _j}}}{{\partial R}} - {\rm{i}}{k_0}} \right)= 0{\rm{,}}R = \sqrt {{x^2} + {y^2}} \end{array} \right. $

(5)

where ω_e is the wave encounter frequency,U is the ship’s mean forward speed,u is the displacement vector,S is the mean wetted body surface,n is the unit normal vector,Z is the sea floor condition,and x,y,z are coordinates in equilibrium coordinate system.

The boundary value problem for hull-girder hydroelasticity is given by the condition(Cui et al.,2007):

$ \frac{{\partial \varphi }}{{\partial n}} = \left[ {\boldsymbol{\dot u} + \left({\boldsymbol{W} \cdot \nabla } \right)\boldsymbol{u} - \left({\boldsymbol{u} \cdot \nabla } \right)\boldsymbol{W}} \right] \cdot \boldsymbol{n}{\rm{ on }}S $

(6)

where W is the steady flow velocity vector relative to the equilibrium coordinate system,

$ \boldsymbol{W} = U\nabla \left({\bar \varphi - x} \right)$

(7)

and φ is the steady potential produced by a body moving with unit speed along the x-axis.

Furthermore,the condition for the incident wave potential and the diffraction potential on the body surface is

$ \frac{{\partial {\varphi _D}}}{{\partial n}} = - \frac{{\partial {\varphi _I}}}{{\partial n}}{\rm{ on }}S $

(8)

The body surface condition for each potential is then given by

$ \frac{{\partial {\varphi _r}}}{{\partial n}} = \left[ {i{\omega _e}{\boldsymbol{u}_r} + \left({\boldsymbol{W} \cdot \nabla } \right){\boldsymbol{u}_r} - \left({{\boldsymbol{u}_r} \cdot \nabla } \right)\boldsymbol{W}} \right] \cdot \boldsymbol{n}{\rm{ }}r = 1,2,...,m $

(9)

where the vector,u_r,is the r^th principal mode of the hull.

The boundary integral equation in a source form is given by(Dai,1998)as

$ \begin{array}{*{20}{l}} {2{\text{Π }}{\sigma ^{(j)}}\left( p \right) + \iint\limits_S {{\sigma ^{(j)}}\left( q \right)\frac{\partial }{{\partial {n_p}}}G\left( {p,q} \right){\text{d}}{S_q}}{\text{ = }}} \\ {\left\{ \begin{gathered} - \frac{\partial }{{\partial {n^{\left( p \right)}}}}{\varphi _I} \hfill \\ \left[ {i{\omega _e}{u_j} + \left( { - U\frac{\partial }{{\partial x}}} \right){u_j}} \right] \cdot n \hfill \\ \end{gathered} \right.\left( {p \in S} \right)} \end{array}$

(10)

where G(p,q)is the zero speed free surface Green function,p represents the field,q represents the source point,σ is the source strength distribution on the body,and n_p is the normal vector at field point.

The coupled dynamic motion equation in matrix form is given by

$ \left\{ {\begin{array}{*{20}{l}} { - \omega _e^2\left({\left[ \boldsymbol{m} \right] + \left[ {\boldsymbol{A}\left({{\omega _e}} \right)} \right]} \right)+ {\text{i}}{\omega _e}\left({\left[ \boldsymbol{d} \right] + \left[ {\boldsymbol{B}\left({{\omega _e}} \right)} \right]} \right)+ } \\ {\left({\left[ \boldsymbol{k} \right] + \left[ \boldsymbol{C} \right]} \right)} \end{array}} \right\}\left\{ \boldsymbol{\xi} \right\} = \left\{ \boldsymbol{F} \right\} $

(11)

where ω_e is encounter frequency,[m] is structural mass,[A(ω_e)] is added mass,[d] is structural damping,[B(ω_e)] is hydrodynamic damping,[k] is structural stiffness,[C] is restoring stiffness,{ξ} is the modal amplitudes,and {F} is the wave excitation force.

The solution to Eq.(11)determines the modal amplitudes. The displacements,bending moments,and stresses are then obtained by modal superposition(Hirdaris et al.,2003). The expressions that are used to determine the vertical displacement,vertical bending moment,and longitudinal direct stress by the modal superposition method are given by

$ \begin{gathered} \boldsymbol{u}(s;\omega)= \sum\limits_{r = 1}^N {{\xi _r}(\omega){\boldsymbol{u}_r}(s)} \hfill \\ {\boldsymbol{M}_y}(x;\omega)= \sum\limits_{r = 1}^N {{\xi _r}(\omega){\boldsymbol{M}_{yr}}(x)} \hfill \\ \boldsymbol{\sigma}(s;\omega)= \sum\limits_{r = 1}^N {{\xi _r}(\omega){\boldsymbol{\sigma} _{xr}}(s)} \hfill \\ \end{gathered} $

(12)

where u_r,M_yr,and σ_xr are respectively the modal vertical displacement,vertical bending moment,and longitudinal direct stress for the r^th mode shape,with the corresponding principal coordinate ξ_r evaluated in regular waves. The position on the vessel is defined as s=(x,y,z)for three dimensional structural models,whereas s denotes the position along the vessel for the two dimensional beam finite element models.

2.2 Hydroelastic model set-up

A numerical analysis was performed in the frequency domain to determine the wave loads and ship motion in regular waves in infinite water depths. The numerical code was based on three-dimensional frequency-domain velocity potential theory and takes into consideration the elastic deformation of the ship hull girder. Modal calculation of the hull girder in a wet condition was conducted using the transfer matrix method and the numerical solution was based on the zero speed free surface Green function approach with the assumption of high encounter wave frequency and a low forward speed.

The hydrodynamic mesh of the ore carrier is shown in Fig. 1. The mesh has 1 632 panels; 120 of these are located in the stern section and 72 in the bow section. The target minimum number of panels should be six for a given computational wavelength; this value was determined after conducting a large number of numerical experiments. Therefore,when the ratio of wavelength to ship length is low(a high wave frequency),the number of panels should be very large. The above number of panels used in this study was determined in relation to making a compromise between accuracy and computational resources(time and memory)for the numerical simulation. In the mid-region,the grid number per cross-section was set as 24,the bow centerline had 12 grid points,the stern centerline 20 grid points,and the grid aspect ratio was 2.

Using an in-house developed code,both linear hydroelasticity theory and rigid body theory were then adopted for the numerical simulations to investigate the motion and load responses for the two stiffness schemes. We also aimed to determine the effect of stiffness on the springing response of the ship’s hull. The hydroelastic model was divided into 20 blocks with 21 cross-sections(Hirdaris and Temarel,2009). Simulations were conducted for head waves at infinite depths with a forward ship speed of 24.5 kn. Although this speed is higher than the typical maximum of 15 knots for ore carriers,it was selected to achieve resonance because of limitations in the maximum wave frequency that could be produced in the tank. The critical damping coefficient was taken as 0.05(Bishop and Price,1979; Hirdaris et al.,2003); this was chosen based on experience gained during previous analyses. Wave frequencies ranging from 0.075 rad/s(0.012 Hz)to 0.976 rad/s(0.155 Hz)in steps of 0.023 rad/s(0.0037 Hz)were then investigated for both schemes of the ship’s prototype.

3 Experimental design 3.1 Towing tank facility

The tests were conducted in the towing tank at Harbin Engineering University(the tank is 108 m long,7 m wide,and 3.5 m deep). A hydraulically driven wave maker of the single flap type was used to generate regular waves,where the maximum regular wave height that can be generated is 38 cm. A ramp wave absorber was situated at the opposite end of the tank from the wave maker to minimize wave reflection and calm the wave down quickly before the following test run. The model was towed by a carriage with a maximum speed of 6.5 m/s at full capacity. The model’s motion was measured by a seaworthy instrument using five degrees of freedom.

3.2 Model design

The principal dimensions of the prototype ore carrier and the experimental model are shown in Table 1(where CG denotes center of gravity,LCG denotes longitudinal position of center of gravity,and AP denotes after-perpendicular). The full load condition was investigated for motion and load responses in regular waves.

Using a scaling ratio of 1:70,a fiberglass reinforced plastic segmented hull model connected by a steel backbone system at its neutral axis was fabricated for the experiment. Fig. 2 shows a photo of the model ship prior to experiments.

The model hull was composed of nine segments(Chen et al.,2012)mounted on a steel backbone system to link them,as shown in Fig. 3. The segments were used to transfer the hydrodynamic loads resulting from the water waves onto the steel backbone. The joints between the segments were covered and made water-tight using silicone rubber membrane strips,which were elastic and thus do not interfere with the motion of segments. Two backbone systems were designed for the ore carrier tests,scheme A and scheme B,respectively. Each backbone system was composed of four beams with tubular sections,which were arranged as shown in Fig.. The central part was designed with two parallel beams to create space for mounting the sea worthiness instrument at the centerline of the model.

For the backbone of scheme A,the cross-sectional dimensions of beams 1 and 4 are 40 mm×40 mm×3 mm(width,height,and thickness,respectively),while for beams 2 and 3 they are 20 mm×40 mm×3 mm,respectively. For the backbone of scheme B the cross-sectional dimensions of beams 1 and 4 were 60 mm×60 mm×3 mm,respectively,while for beams 2 and 3 they were 30 mm×60 mm×3 mm,respectively. Strain gauges were then used to measure the vertical bending moment loads on the backbone at cut sections S1 to S8. The mass and stiffness distribution of the model from stern to bow is given in Table 2

3.3 Hammer test

To check the natural frequencies of the backbone system,impact hammer tests were performed both in dry and wet conditions prior to conducting towing tank tests. Fig. 4 shows the time histories of the vertical bending stress amidships during the hammer test in a wet condition with calm waters. Using spectral analysis based on Fast Fourier Transform,the measured data were then processed to determine the natural frequencies of the model.

The vertical natural frequencies of the model were also simulated using the finite element method in a wet condition,and Table 3 shows the measured and simulated wet natural frequencies for the two-node vertical bending vibration mode.

The test and numerical results showed good agreement,and any slight variations were attributed to errors inherent in the model design.

3.4 Testing schemes

The most important springing frequency for hull girder vibration is its 2-node vertical natural vibration frequency(Skjørdal and Faltinsen,1980; ABS,2010). In this respect,for a hull with a two-node natural frequency of ω_n,moving with a constant forward speed of U,and a heading of β in regular waves of frequency ω,the wave encounter frequency,ω_e,is given by(Kim et al.,2010; Lee et al.,2011)as

$ {\omega _e} = \omega \left({1 - \frac{{U\omega }}{g}\cos \beta } \right)$

(13)

where ω_e is the wave encounter frequency(rad/s),ω is wave frequency(rad/s),U is the ship’s forward speed(m/s),β is the heading angle in waves(°),and g is acceleration due to gravity(m/s²).

The first,second,and third harmonics of wave frequencies of encounter are ω_e=ω_n,2ω_e=ω_n,and 3ω_e=ω_n,respectively. Resonance occurs when the wave encounter exciting frequency or its harmonics approach or is equal to the natural frequency of the ship hull girder(Lee et al.,2011). Harmonics are therefore important in determining the testing scheme that is used to investigate springing-induced wave loads. The towing tank testing scheme was determined prior to the experiment,and is shown in Table 4. Tests were run at a towing speed of 1.5 m/s in head seas for wave heights and various wave frequencies(as shown in the table). For the backbone A scheme,the tested wave height used was 5 cm,which corresponds to a wave height of 3.5 m for the prototype ship. The outcome of the WALC-LE program is linear,and thus for comparison purposes the experimental wave height needs to be as small as possible. However,when the wave height is very small,the wave-induced loads are also very small,and the response of model is not sufficiently significant. Therefore,as a compromise between the linear response required and the recognizable/measureable experimental load responses,a wave height of 5 cm was selected. For the backbone B scheme,a wave height of 10 cm was selected,because the stiffness involved is slightly larger than that of scheme A.

4 Comparison of experimental and numerical results 4.1 Motion response in regular waves

The motion Response Amplitude Operators(RAOs)derived using numerical and experimental methods for heave and pitch modes are shown in Fig. 5 and Fig. 6 for schemes A and B,respectively. The experimental results were transformed into ship prototype values using the similitude rule and then compared with numerical results(this was conducted for all results discussed in this paper). As can be seen from Figs. 5 and 6,the results of motion response obtained using hydroelasticity theory are in good agreement with those obtained by the model test. In the low frequency range,both pitch and heave motions obtained using hydroelasticity theory are closer to the test results than those obtained using rigid theory(motion response using rigid body theory is larger). However,both pitch and heave numerical motion responses are in good agreement with the test responses in the high frequency range,and a comparison between pitch and heave RAOs shows that the pitch values are generally more in agreement with the test. Rigid body theory gives the maximum pitch response of 1.122 at an encounter frequency of 0.338 rad/s. A comparison of pitch responses shows that hydroelasticity theory gives a maximum response of about 1.00 at an encounter frequency of 0.267 rad/s for scheme A as shown in Fig. 5(a),and of about 1.06 at an encounter frequency of 0.314 rad/s for scheme B,as shown in Fig. 6(a). For heave,rigid body theory gives a maximum response of 1.533 at an encounter frequency of 0.469 rad/s; hydroelasticity theory gives a maximum of 1.214 at an encounter frequency of 0.456 rad/s for scheme A as shown in Fig. 5(b) and 1.344 at encounter frequency of 0.464 rad/s for scheme B as shown in Fig. 6(b).

4.2 Wave loads in regular waves 4.2.1 Load response in frequency-domain

The results of wave-induced vertical bending moment RAOs using numerical methods(both rigid body theory and hydroelasticity theory)were then compared with results of the experimental method. Figs. 7(a) and (b) show a comparison of the VBM RAOs amidships for schemes A and B,respectively,during motion in head seas at a speed of 24.5 kn. It can be seen that the trends of the results from the two numerical methods generally agree with the test results,but that some errors exist. The first peak represents the low frequency wave load and subsequent peaks represent high frequency springing loads. The first and the second peaks for scheme A occur at wave frequencies of 0.62 rad/s(0.099 Hz)and 1.15 rad/s(0.183 Hz),respectively,whereas for scheme B the first peak occurs at 0.65 rad/s(0.103 Hz),the second peak at 1.2 rad/s(0.191 Hz),and a third peak occurs at 1.75 rad/s(0.279 Hz). Results of hydroelasticity theory are closer to those of the test result than rigid theory. In the low frequency range,results using rigid theory overestimate the load response,while those using hydroelasticity theory show a better agreement with test results. In addition,there is no high frequency springing response for the rigid theory result,whereas the springing response is evident in results using the hydroelasticity method and the test,as shown by the peaks in the high frequency range. Furthermore,the results obtained using scheme B are more consistent when using the various methods. To check whether the flexible body solution converges to a rigid body solution,model rigidity was set very high. A comparison between the flexible-body solution and the rigid body solution shows that the bending moment load response curves are in good agreement in the low frequency range(see the similar curves in Fig. 7),but that the springing response continues to exist in the high frequency range for the flexible-body solution,even when model rigidity is very high.

4.2.2 Load response in time-domain

To investigate the springing response occurrence in the time domain,the Power Spectrum Density(PSD)was obtained by conducting a spectral analysis on the vertical bending stress time histories obtained from model tests using the Fast Fourier Transform(FFT)technique. Figs. 8 and 9 show the vertical bending stress time histories and PSD for scheme A and scheme B,respectively,at a model speed of 1.5 m/s,where the incident head wave frequencies are 0.53 Hz and 0.55 Hz for scheme A and scheme B,respectively(corresponding to wave encounter frequencies for the model of 0.8 Hz and 0.841 Hz,respectively); these values were chosen because they are approximately half of the two node natural frequencies.

The rugged curves of the stress time histories are indicative of the occurrence of springing,where the high frequency springing load is superimposed onto the low frequency wave loading. The first peak in the PSD diagrams indicates the encounter frequency,and the second and third peaks are the 2nd and 3rd harmonics of the encounter frequency; these represent the occurrence of a wave-induced springing frequency response during the model experiment. The frequencies obtained from Fig. 8(b)and Fig. 9(b)are shown in Table 5. For scheme A,the frequencies correspond to 0.097 Hz,0.192 Hz,and 0.289 Hz for encounter frequency,2nd harmonic,and 3rd harmonic,respectively,for the ore carrier ship prototype; while for scheme B the values correspond to 0.100 Hz,0.201 Hz,and 0.301 Hz for encounter frequency,2nd harmonic,and 3rd harmonic,respectively. The value of the encounter frequency was chosen as it was half that of the 2-node bending natural frequency. The second harmonic is coincident with the 2-node springing frequency.

5 Conclusions

This paper presents experimental and numerical investigations of the springing response for a 550,000 DWT ore carrier using models with softer stiffness. The segmented model towing tank test results are presented,together with results of the numerical analysis using both rigid and linear hydroelasticity theories. Conclusions made are as follows:

1)A comparison between the experimental results for both backbone schemes shows that the stiffness of the hull girder has little influence on the motion response of the ship. Therefore,softer stiffness models can be used to investigate the response of the VLOC in a sea way.

2)The results of both hydroelasticity theory and experimental results for the two schemes are generally in good agreement,whereas rigid theory overestimates responses in the low frequency range. The use of hydroelasticity theory is thus preferable when investigating motion and wave load responses of large ships.

3)Time histories that were recorded during the towing tank tests and the high frequency load were superimposed onto the linear wave load,which represents the occurrence of the springing phenomenon. In addition,the second peaks of power spectrum density functions are indicative of the springing response and the respective springing frequencies for this ore carrier.

It is considered that further experimental research is required to investigate higher order springing characteristics of the VLOC. In addition,knowledge of the ore carrier’s higher order springing characteristics would be useful to enable an improvement in classification rules.