Experimental and Numerical Analysis of Bow Slamming and Whipping in Different Sea States

CHEN Zhan-yang,REN Hui-long,LI Hui,ZHANG Kai-hong

(College of Shipbuilding Engineering,Harbin Engineering University,Harbin 150001,China)

1 Introduction

With increasing demands for huge dimensions and high-speed transportation,ship designers have to confront with the challenge of reducing the weight through the use of light-weight materials.These factors make hull more flexible and natural frequency lower that lead to whipping easily.The whipping phenomenon is a transient vibration induced in the ship girder by the excitation force generated from the slamming phenomenon.The necessity to study the whipping phenomena is due to the high stress which has a negative effect on the strength of the hull,with a higher reduction of the ship operation life than previously taken into account.This raises the need to develop a better understanding of the whipping responses[1].

Recently,on the international stage,the common method to study the ultra large ships which has obvious elastic effects is combination of theoretical study and model test.Actually the hull is flexible body,traditional rigid theory can not accurately reflect characteristic of the high-frequency of wave loads.Therefore,in order to predict wave loads accurately,hydroelasticity theory should be adopted.The experimental analysis can consider a first approach for studying the problem of whipping phenomena,but rigid ship model can not consider elastic effect of the hull on the load of the fluid around.Those models are not adequate for the experimental analysis of vibration phenomena,specially not for whipping phenomena,so the hydroelastic segmented ship model should be adopted to solve this problem[2].Some experimental schemes for hydroelastic responses have been developed.Model tank tests of a large cruise ship have been conducted by Cusano(2007)[3-4],they used a 4-segment model with discrete flexible joints.With respect to the midship bending moment in head seas they found that the LF and the HF contributions can be relatively well described by a Weibull-type distribution.Also Rousset(2010)[5]conducted model tests on the whipping response of a large cruise vessel.The tests are carried out in irregular waves with a 4-segment model with uniform beams.Though much effort has been paid to the development of segmented model test,uniform beams are simulated the stiffness of the hull in these papers,and the number of segments of ship models is so few that high order component in response can not be reflected clearly.

Based on the above,the traditional segmented model is improved in this paper and the method that the stiffness of the hull is simulated by variable cross-section beams is proposed.The purpose of this paper seeks to investigate several parameters in different sea states:total bending moment Mtol,high-frequency(HF)slamming moment Mslammingand low-frequency(LF)wave moment Mwave.Finally,Some computer programs of nonlinear hydroelasticity theory considering slamming loads are developed to predict wave loads in different sea states.From the comparison between experimental and theoretical results,it is showed that the linear results are in good agreement with experimental results in low sea states,but for the severe sea states,the nonlinear results are in better agreement with experimental results.

2 The hydroelasticity theory

2.1 Linear hydroelasticity theory

The method based on beam model of the hull structure dry mode and the three-dimensional linear potential flow theory are combined.The boundary conditions on the body and free surfaces are linearized and the motion equation under consideration can be established with the elastic effect of hull girder,and then the generalized coordinates of the hull and section load are obtained.

2.2 Nonlinear hydroelasticity theory

Unlike linear hydroelasticity theory,the nonlinear hydroelasticity theory takes into account the slamming forces.In this paper,the momentum slamming theory is used to predict slamming loads:

where m（x,t）is the instantaneous added mass,ZR（x,t）is vertical relative ship motion to wave,s（x,t）is instantaneous sinking area.

If the hydroelastic behaviors of ship are considered,vertical relative ship velocity to wave takes the form:

where w（x）is principal mode,is wave elevation velocity.

So slamming loads added to the equation of ship motion is obtained:

where,［a］,［b］and［c］denote generalized structural mass matrix,generalized structural damp-

Finally,the equation of motion at time t is expressed in the time domain as follows:ing coefficient matrix and generalized structural stiffness matrix,respectively;［A］,［B］and［C］denote generalized fluid mass matrix,generalized fluid damping coefficient matrix,and generalized fluid stiffness matrix,respectively;pr（t）is the r-th mode principal coordinates,FI（t）is incident wave force,FD（t）is dispersion wave force,is slamming force.

3 The hydroelastic model design

At the ship test tank an elastic multisegmented ship model for a scaled ship model of a very large fast ship is buil up.The vibration phenomena induced in the ship girder by slamming is studied.The ship characteristics are presented in Tab.1.

Tab.1 Main parameters of the model

The fiberglass-reinforced plastics model is cut into 10 elements that are connected together by the longitudinal steel beams and made watertight by rubber membranes.The beam bending moment was measured in five sections(A～E).Motion recorder,which is mounted at the position of weight center,is used to record the heave and pitch motions,see Fig.1.Nine pressure transducers are arranged on the bow flare to measure the slamming pressures.

Fig.1 Sketch map of the segmented model

Unlike the traditional segmented model that is construced with uniform beams.In present paper,the traditional segmented model is improved.In order to make the stiffness and weight distribution of model consistent with the hull,the method that the stiffness of the hull is simulated by variable cross-section beams is adopted.Since whipping is mainly two-node vibration,the first order natural frequency of the full scantling vessel is used as a basis for estimating the flexural response frequency of the scaled segmented ship model[6].

Before launching of the model,each hull segment is weighed and balanced in air in a cradle.This is to obtain the correct mass,longitudinal centre of gravity and mass moment of inertia around the transverse axes according to the specified longitudinal mass distribution.

Prior to the wave loads tests,through hammering testing in calm water is performed to obtain time traces of midship stress,see Fig.2.Spectral analysis based on FFT is performed on the data gathered to identify the natural frequency of ship model.Fig.3 shows the power spectral density of first three modes frequency results.

Fig.2 Time traces of midship stress

Fig.3 Frequency characteristics of midship stress

The wet natural frequencies of the first three modes vertical flexible modes are measured,which are close to calculation for the model of ship by Finite Element Method(FEM)and Transfer Matrix Method(TMM),see Tab.2.

Tab.2 Comparison of theoretical and experimental natural frequency of the model

4 Results and discussion

4.1 Load response variation according to the sea states

Due to the bow flare and high speed of the ship it will inevitably encounter severe slamming during its life.Slamming loads on ship may induce uncomfortable vibrations which may produce significant fatigue and local damages on the hull structure.Whipping responses play an important role in nonlinear wave loads.Slamming events are deemed to usually happen in severe sea states.However,recently some literatures say moderate seas can also generate water impacts on the ship with severe bow flare.Therefore,in order to investigate the parameters af-fecting the whipping responses of a high-speed ship subjected to slamming in different sea states,the model is tested in regular waves at three wave heights for ship speed of 9 knots(λ/L=0.9）,see Figs.4～6.

Fig.4 Time trace of VBM in v=9 kns,h=5.6 m

Fig.5 Time trace of VBM in v=9 kns,h=17.5 m

Fig.6 Time trace of VBM in v=9 kns,h=21 m

It clearly shows that time trace of HF slamming moment is free damping,especially for moderate and high sea states(h=17.5 m and 21 m),it means that the ship girder generates transient vibration induced by slamming loads.This is quite different from the steady the time trace of HF responses.Besides,the maximum always occurs in the trough at the moment.It is well-known that the slamming is a transient response,once the ship subjects to the excitation force,the ship girder oscillation amplitude is the largest,then the amplitudes of whipping moment decay because of structural and hydrodynamic damping until next slamming occurs,which is the so-called whipping[7].For each wave condition,slamming phenomenon is observed in the video.

In order to emphasize the proportion of HF slamming moment in the total bending moment in regular waves in different sea states,the mean values of the total moment,HF slamming moment and LF wave moment are calculated,which have been transformed to corresponding full scale data,see Tab.3.When wave height increases from 5.6 m to 21 m.It clearly shows that mean value of total moment increases from 24.64%to 92.02%compared with that of wave moment because of servere whipping.It is shown that there are significant increases in the proportion of slamming moment in total moment with increasing wave height.It is necessary to pay attention to bow slamming and whipping problems in design and service.

Tab.3 The mean value of VBM in different sea states

4.2 Numerical prediction of wave loads in different sea states

It has been demonstrated that through a combination of model tests and calculations it is necessary to accurately predict the effect of wave loads on the ship in different sea states.

The time-domain procedure based on linear and nonlinear hydroelasticity theory is programmed.The total midship bending moment in different sea states is calculated and compared with model test results.In the low sea states,Mtolis predicted by linear hydroelasticity theory.In the moderate and high sea states,Mtolis predicted by linear and nonlinear hydroelasticity theory.

Fig.7 Comparison between theoretical and experimental results of Mtolin low sea states(v=24 kns,h=4.143 m,λ/L=1.0)

Fig.8 Comparison between theoretical and experimental results of Mtolin moderate sea states(v=24 kns,h=5.6 m,λ/L=1.0)

Fig.9 Comparison between theoretical and experimental results of Mtolin severe sea states(v=9 kns,h=17.5 m,λ/L=0.9)

Fig.10 Comparison between theoretical and experimental results of Mtolin severe sea states(v=9 kns,h=21 m,λ/L=0.9)

By comparing the calculations based on linear and nonlinear hydroelasticity theory and the measurements,from Figs.7～10,we can see that for the low sea states,the HF response caused by the slamming is very small,so the linear hydroelasticity method can obtain accu-rate results.However,the linear theory for severe sea states can not reflect the HF components in the moment.Nonlinear hydroelasticity theory should be adopted to predict wave loads.

In addition,as the sea state increases,the HF components in the total bending moment increase,the nonlinear characteristics of response curve become more evident,especially for severe sea states(v=9 kns,h=17.5 m and 21 m).The amplitudes of theoretical calculation results are lager than those of experimental results.This is because the momentum slamming theory used to predict slamming loads in this paper can not completely describe the instantaneous effect of flow field when slamming events happen,such as air cushion effect,jet flow,and so on,but these factors that can ease the slamming force are not considered in the theoretical calculation.Therefore,the theoretical calculations in the future need to be developed in the these aspects.

5 Conclusions

Segmented model experiments based on variable cross-section beams are carried out in a towing tank.Severe slamming is observed in different sea states.Through analysis of experimental data,we can draw two conclusions concerning this ship from those results:

(1)The effects from bow slamming induced whipping vibration can become relevant to wave height:as the wave height increases,the maximum vaule of Mtolincreases compared with that of Mwavebecause of severe whipping loads;

(2)For low sea state,the HF response caused by the slamming is very small,so the linear hydroelasticity method can obtain accurate results,but for high sea states nonlinear hydroelasticity theory should be adopted to predict wave loads.

This paper only focuses on whipping responses induced by slamming in different sea states.The ultimate goal of research on whipping response is to discuss effect of whipping on the design wave load of ship,and then the long-term predicted values of nonlinear wave load are obtained,and effect of whipping on fatigue damage and strength of the hull is studied based on measured data.These will be studied in future work.

[1]Leonard Domnisoru,Daniela Domnisoru.Experimental analysis of springing and whipping phenomena[J].University of Galati,Shipbuilding Department,Romania,2000,47(450):129-140.

[2]Dudson E,Rambech H J,Wu M K.Determination of wave bending loads on a 40 knots,long slenderopen topped containership through model tests and hydrodynamic calculations with particular reference to the effects of hullfiexibility on fatigue life[C]//Proceedings of the 6th Interbnational Conference on Fast Sea Transportation,Southampos,UK,2001,2:177-190.

[3]Cusano G,Sebastiani L,Bacicci G.Hydroelastic methodologies for the study of whipping effects[C].MARINE 2007,Barcelona,Spain,2007.

[4]Cusano G,Sebastiani L,Bacicchi G.Assessment of whipping effects induced by stern/bow-flare slamming[C]//Proceedings of the 10th International Symposium on Practical Design of Ships and Other Floating Structures.Houston,USA.2007:428-436.

[5]Rousset J M,Ferrant P,Alessandrini B.Experiments on a segmented ship model in directional irregular waves[C].Proc.of the 21th IWWWFB,2010.

[6]Jason Lavroff,Michael R Davis,Damien S Holloway,Giles Thomas.The whipping vibratory response of a hydroelastic segmented catamaran model[C]//Ninth International Conference on Fast Sea Transportation FAST2007,September 2007.Shanghai,China,2007.

[7]Luo Hanbing,Qiu Qiang,Wan Zhengquan,Yang Daming.Experinmental investigation of the stern slamming and whipping in regular and irregular waves[J].Journal of Ship Mechanics,2006,10(3):150-162.