Research on Whipping and Springing Responses of Hull based on Different Experimental Method and Nonlinear Hydroelastic Time-domain Theory

CHEN Zhan-yang,GUI Hong-bin

(Department of Naval Architecture and Ocean Engineering,Harbin Institute of Technology at Weihai,Weihai 264209,China)

Research on Whipping and Springing Responses of Hull based on Different Experimental Method and Nonlinear Hydroelastic Time-domain Theory

CHEN Zhan-yang,GUI Hong-bin

(Department of Naval Architecture and Ocean Engineering,Harbin Institute of Technology at Weihai,Weihai 264209,China)

Abstract:The growth in size of hull may result in serious whipping and springing phenomenon that can increase both the ultimate load and fatigue damage on the hull structure.In order to study the vibration responses of hull better,the self-propulsion and towing segmented ship model experiments of a 10 000-TEU containership were carried out in a towing tank.The effect of self-propulsion and towing test methods on whipping and springing responses in different sea states is analyzed.In order to consider the effect of various vibration frequency components on load responses,a nonlinear hydroelastic method considering the wave memory function is proposed.A method is proposed for time-domain retardation function to solve limitation in the calculation of damping coefficient at high frequency.Finally,the measured results on the central hull are compared with those calculated by the linear hydroelastic theory and nonlinear hydroelastic theory,and it indicates that the nonlinear hydroelastic method proposed in this paper can predict the vibration responses better.

Key words:whipping;springing;slamming;hydroelastic;retardation function

0 Introduction

When the ship is sailing in waves,the vibration caused by waves can be divided into two types:One type is called whipping that belongs to a transient vibration caused by slamming and often happens in moderate and high sea states;The other type is named springing,which is a kind of steady forced vibration when the encounter frequency is equal to the frequency of the ship itself.

Recently 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[1].These factors make hull more flexible and natural frequency lower that lead to whipping and springing easily.In order to study vibration phenomena of hull bet-ter,the hydroelastic theory has been widely used.Considering the coupling effect of elasticity effect and loads around flow field,hydroelastic theory can fully utilize the flow field information around the floating body to predict the motion of rigid body,besides,deformation,even shear and bending moments of floating body structure.It has a broader development prospect for dealing with strength check,fatigue life calculation and structural dynamic response analysis for floating structure.There are a large number of researches on hydroelastic methods at home and abroad in recent years,such as Zou[2],Hasheminejad[3],Lu[4],Liu[5],Faltinsen[6],Jiang[7],and Panciroli[8].

However,for the hull vibration,especially the whipping phenomenon caused by slamming occurs mostly in the high sea condition,therefore,if the linear method is still used,which will bring the error.Although the nonlinear hydroelastic method has been used for a long time,these methods only consider 1～2 kinds of nonlinear factors[9-11],which are unable to meet the prediction demands of hull load responses in the high sea condition.Besides,to solve the nonlinear problem,the time-domain method should be used.Nevertheless,in most of the existing timedomain methods,the hydrodynamic coefficients of single frequency can be considered[12-14],which cannot consider the effect of double frequency component on the vibration responses simultaneously.Therefore,it is necessary to propose a nonlinear hydroelastic method to predict vibration responses of hull accurately.

In addition,the experimental analysis can be considered as a first approach for studying the problem of vibration phenomena,but the traditional rigid ship model is not adequate for the experimental analysis of vibration phenomena and elastic effect of the hull on the load of the fluid around,specially not for whipping and springing phenomena,so the hydroelastic segmented ship model should be adopted to solve this problem.Some experimental schemes for vibration responses have been developed[15].However,some disadvantage could be found,such as towing test method[16];too few segments[17];uniform backbone beams[18].The objective of the experiment is to reproduce the flexible characteristics of the hull in the vertical,horizontal and torsion deflection modes.So the perfect experimental model and test method are of importance.

Therefore,for the investigation of whipping and springing responses of hull,the self-propulsion and towing segmented ship models system are adopted first.Then,experimental results in different sea states were described and analyzed.Finally,a three-dimensional(3-D)nonlinear time-domain hydroelastic method considering the nonlinearity of instantaneous body and slamming for load responses of large size ships has been proposed.

1 The segmented ship model experiment

1.1 Model design of 10 000-TEU containership

Experiments have been carried out by using a segmented scaled model of a 10 000-TEU containership in the seakeeping basin in Harbin Engineering University.The vibration load responses of the hull in head waves are studied.A scale of 1:75 is chosen as a compromise be-tween the necessity of having enough volume and weight margin to adjust the weight distribution and the capabilities of the towing tank in terms of waves and speed.The ship characteristics are presented in Tab.1.

Tab.1 Main parameters of ship and model

The fiberglass-reinforced plastics self-propulsion model is cut into 7 rigid segments,and the towing ship model is cut into 10 rigid segments(see Fig.1).All the segments are connected together by the longitudinal flexible beams.The gap between adjacent segments is designed as 10 mm and sealed by thin latex.The reason to make the gap relatively large is to prevent contact of the segments.The beams are instrumented with strain gauges to measure the vertical bending moments at the position of each of the cuts.

Fig.1 Towing ship model and self-propulsion ship model

The propulsion system used in self-propulsion test is arranged in the stern of self-propulsion model,which is used to maintain the forward speed when the motion is disturbed by waves.The propulsion system of the self-propulsion model is achieved by using four five-blade screw propellers,rudders,gear boxes and electric motors.The electric motors provide excellent controllability during the experiment.The cross-connection gear boxes are important components of this plant,because they drive two shafts using one electric motor.The connection method of the propulsion system can be seen in Fig.2(a).The speed of the motor is regulated by the control system onboard the model.In towing test,the motion signals of ship model are usually measured by the 6 degree of freedom(6-DOF)seaworthiness instrument.The 6-dof seaworthiness instruments used in towing test are shown in Fig.2(b).

Fig.2 The test system of ship model

Since whipping and springing are mainly two-node vibration,in order to reproduce the first few deformation modes as accurately as possible,the first order natural frequency of the hull is used as a basis for constructing the flexible beams.The natural frequency of hull is determined by the mass and stiffness distribution,since the stiffness of different hull frames is not the same,in the analysis of structure vibration responses,the hull girder should be regarded as a variable cross-section beam.Unlike the traditional segmented model that is constructed with uniform beams,in order to make the stiffness and weight distribution of model consistent with the hull,the method that variable cross-section beams are used to simulate the stiffness of the hull is adopted.The rectangular backbone beams are shown in Fig.3.The responses of the ship model in waves are shown in Fig.4.

Fig.3 The variable cross-section rectangular backbones

Fig.4 The responses of the ship model in waves

Prior to the wave loads tests,the model is tested whether it meets the design requirements.By applying the excitation force to the ship model in calm water,the time traces of stress amidships are obtained.Spectral analysis based on Fast Fourier Transformation method(FFT)is performed on the data gathered to identify the natural frequency of ship model.The measured results and calculation results based on Finite Element Method(FEM)are compared,see Tab.2.

As shown in Tab.2,the measured data in self-propulsion test are smaller than those in towing test.The error of first order result is 4.94%,and only the former two order nature frequencies can be measured.This is because the population system is arranged in the stern of model,whose weight and stiffness distributions cannot keep consistent with those of prototypeexactly.The ship model is cut into 7 segments.The high order deformation cannot be obtained.However,if the model is cut into 10 segments,the elastic characteristic of model can be reflected obviously,experimental results are closer to the calculation results.The error of first order result is 1.64%.

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

1.2 Analysis of springing responses

For the springing phenomenon,there have been many researches showing that when the encounter frequency of regular wave is approximate to the first order natural frequency of ship model in waves,the stress peak will come out.In the test,if the gap of the two frequencies is larger than 5%,the stress peak will disappear quickly.Therefore,we can conclude that springing indeed belongs to a kind of resonance.

In order to reproduce this phenomenon in the experiment,the encounter frequency ωeis calculated:

where θ is wave direction angles;λ is wave length;v is ship speed;c is wave speed.

Taking advantage of the generation mechanism of springing,the wave length or frequency that produces the springing can be obtained.As the springing is a kind of sympathetic vibration,the bandwidth of response frequency is so harrow that the small change of wave length or ship speed can enable the sympathetic vibration disappear quickly.Although the requirements that produce the springing can be obtained theoretically,in the test process,only 28 times of springing appears out of 102 times of towing.It is seen that the test of springing is quite difficult.

Fig.5 shows the time traces of midship total bending moments,which is divided into low frequency(LF)wave moments and high frequency(HF)springing responses by filter theory.It shows that there is obvious springing phenomenon in the time traces of total bending moments.In addition to the LF wave moments components related with the wave encounter frequency,there are also stable HF load components according to the change of first natural frequency.But the HF springing responses are not caused by slamming,because in this load case the wave height is not high,and the slamming phenomenon is not observed in the video either.

Although the springing belongs to high frequency vibration,the peak and valley of the springing responses are roughly equal,which is quite different to other nonlinear high frequency moments.Besides,the bending moments amplitudes of other frequencies in this wave height are very small,only when wave frequency reaches the first order natural frequency of hull,doesthe responses amplitudes raise obviously,which is due to the resonance.

1.3 Analysis of whipping responses

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.

Fig.6 shows that time trace of whipping responses is free damping,especially for high sea states,it means that the ship girder generates transient vibration induced by slamming loads.This is quite different from the time traces of springing responses of Fig.5.Besides,the maximum always occurs in the trough at the moment.The slamming is a transient response,when the hull subjects to the excitation load,the structure vibration amplitude is the largest,then the amplitudes of whipping responses decay due to structural and hydrodynamic damping until next slamming occurs.

Since the slamming occurred in the serious sea state,the negative effect of whipping on ultimate strength of hull is usually studied.On the contrary,for the springing,ship designers have to confront with the challenge of structural fatigue damage of very large hull,Therefore,It is necessary to pay attention to distinguish the springing and whipping.

Both springing and whipping belong to hull high frequency vibration,but they are quite different.To be specific,the whipping derives from transient impact loads caused by slammingthat is a kind of transient vibration.However,the springing originates from the effect of continuous fluid loads,whose time-domain curves are stable(Fig.5).From the aspect of the generation mechanism of the two vibrations,a method of distinguishing the springing and whipping is presented in this paper:separating HF bending moments from the original total bending moments,then the category of vibration can be judged according to the shape of time-domain curves.If the high frequency curve is stable,it is the springing(Fig.5(b)).On the contrary,the HF curve shows delay appearance,it refers to the whipping(Fig.6(b)).

1.4 Effects of different experimental ways on vibration responses

Fig.7 shows the vertical bending moments measured from self-propulsion test and towing test in high sea states.As the curves in the figures indicate,the results from these two kinds of test are in good consistency.For LF wave moments,both of them are almost the same;but for the HF whipping responses,the results of towing test is much larger than those of selfpropulsion test,and these differences are more remarkable with the rise of the sea states.

This is primarily because that the speed of ship model in towing test is provided by the seaworthiness instrument.However,in practice the ships will subject to obvious speed loss when the ships sail in waves.The higher the sea condition is,the more serious the consequence caused by speed loss will be.However,this law cannot be presented in towing test,even though when suffered from waves,the ship model is still caught by seaworthiness instrument and sails according to the predetermined speed,which is the direct reason that the high frequency vibration responses of towing test is larger than those of self-propulsion test.

Fig.7 Comparisons of time trace results between methods(v=9 kns,h=17.5 m,λ/L=1.0)

When the ship is sailing in waves,the ship will subject to various severe sea states.Both self-factors(such as speed)and external factors(such as wave height)may lead to various degrees of damage to the hull.Hence,how to analyze the change trend of various load responses components in different parameters is important premise of assessing the fatigue life and structural strength.

The self-propulsion and towing test results of various load response components(λ/L=1.0)against wave heights and sailing speeds,which have been transformed to corresponding full scale data,are shown in Figs.8-9.

From the results in Figs.8-9,it can be seen that each load component increases with the increasing wave height and speed.For the low sea state,the HF vibration responses(MHF)caused by slamming are lower than LF wave moments(MLF).But with the wave height becoming higher,the slamming phenomenon tends to be more obvious,and the HF bending moments begin to increase due to nonlinear effects.But the proportion change is linear with the speed of ship model.From what has been discussed above,it is known that the wave height and speed are two main factors to influence HF load responses,and the effect of wave height is more obvious.

In terms of the total bending moments as well as HF bending moments,the results from the towing test are higher than those from the self-propulsion test.By contrast,the LF bending moments from the self-propulsion test are higher,which is mainly because there is no restriction of the seaworthiness instrument in the self-propulsion test.

Fig.8 VBM components amidships at various wave heights(v=9 kns)

Fig.9 VBM components amidships at various sailing speeds(h=5.6 m)

2 Numerical model and computational method

The traditional rigid theory considers the hull as rigid body,which has only 6 freedom degree motions.Actually the hull is flexible body,traditional rigid theory cannot accurately reflect high frequency characteristic of the whipping and springing.Therefore,in order to predict wave loads accurately,hydroelastic theory should be adopted.

2.1 The equation of nonlinear motion considering the memory effect

The nonlinearity of instantaneous body and slamming have been taken into consideration in this paper.The equation of nonlinear motion considering the memory effect is expressed in the time domain as follows:

2.2 Calculation of nonlinear fluid forces

For the calculation of the wave exciting force,the original method is the integration on the instantaneous average wet surface.In this paper,through the interception of instantaneous grid,the incident wave force and dispersion wave force can be written as:

where φ0and φdare instantaneous incident potential and diffraction potential per wave amplitude respectively,which are obtained by Green’s function;is wave amplitude;is the normal vector,which is defined positive when pointing into body from the boundary surface;is the r-th principal modes of the structure.

The hydrostatic restoring forces have been directly computed through the composition method including the integration of hydrostatic pressure from instantaneous hull wet surfaceand hull gravity in this paper.

where wkis the k-th mode vertical displacement;is weight collection degree of each station.

Unlike linear hydroelasticity theory,the nonlinear hydroelasticity theory takes into accountthe slamming forces in this paper.The momentum slamming theory is used to predict slamming force:

To analyze the vibration responses of hull,the high frequency component of load responses cannot be ignored.Therefore,in order to consider the effect of various frequency components on load responses,the retardation function is used.The radiation force of the hull can be expressed as:

where r=1～6 denotes motion modes of rigid hull;r≥7 denotes motion modes of flexible hull.Then the process of solution of μ andis introduced in detail.

We have the relations between time domain and frequency domain,

Eqs.(8)and(10)are the so-called Kramer-Kronig relations.As seen from Eq.(8),it is not difficult to see thatis referred as infinite frequency added mass.

Through the above derivation,the retardation function has the following characteristics:

However,due to limitation in the numerical difficulties,such as the numbers of theandto be calculated,the higher-order quadratures,the wave frequency range,the damping coefficient at ω→∞ is difficult to be obtained.If the damping coefficient at high frequency needs to be solved,the computational cost will be too high.But the numerical quadrature is still very likely to give inaccurate results due to the nature of the methods.It is hard to obtain both computational accuracy and efficiency.Therefore,a method for the flexible hull is proposed in this paper to solve time-domain retardation functionFor the integral in Eq.(10),the entire frequency rangeis first divided intois the truncated frequency.Then Eq.(10)can be expressed as,

The contribution to the retardation function from the domaincan then be written as:

Secondly,the contribution from rangewill be analyzed.Sincewill vanish as ω→∞,it is approximated with an exponentially decay function in this range[19]:

where α and β are undetermined coefficient,in order forto vanish as ω→∞,β must be larger than 0,the contribution from rangecan be expressed as:

Finally,the integral of time-domain retardation functionwill be solved.

2.3 The solution of nonlinear equation

For the solution of the second-order differential motion Eq.(2),the fourth-order Runge-Kutta method has been adopted in this paper,which is explicitly single step with fourth-order accuracy.in each time can be obtained.

Then based on mode superposition principle,the time-domain displacementbending momentand shearing forceof each cross section of ship can be obtained.

2.4 Comparison between the calculations and experimental results

In order to verify the method and program presented in this paper,the time-domain procedure based on linear and nonlinear hydroelastic theory is programmed.Four typical conditions are selected for the comparison of the results.The time traces of the numerical as well as the experimental results of total bending moments(Exp.data)are shown in Fig.10.

By comparing the calculations based on linear and nonlinear hydroelastic theory and the measurements,from Fig.10,we can see that the nonlinear numerical results show better agreement with the experimental ones.The linear calculation results cannot reflect HF vibration effect caused by slamming.So the higher the sea states are,the more obvious the advantages of nonlinear results are.

Fig.10 Comparisons between experimental results and calculated results of MTotal

In addition,as the sea state increases,the HF load components in the total bending moments increase,the nonlinear characteristics of response curve become more evident,especially for severe sea states.Therefore,the necessity to predict the whipping responses is due to the high stress which has a negative effect on the strength of the hull.This raises the need to develop a better understanding of the whipping responses.Since the linear method cannot simulate the vibration responses,the whipping responses are predicted by nonlinear hydroelastic method and compared with model test results in Fig.11.

Fig.11 Comparisons between theoretical and experimental results of whipping responses

3 Conclusions

In this paper,the whipping and springing responses of a large containership in regular waves are studied by experimental and numerical methods.Through analysis of experimental and numerical data,the following conclusions can be drawn:

(1)Based on the elastic beam model and nonlinear hydroelastic method,the generation mechanisms of whipping and springing responses are analyzed,a method of distinguishing the springing and whipping is presented;

(2)Wave height is the main factor that can aggravate HF vibration responses.As the wave height increases,the results from these tow test methods increase.But the proportion of LF wave moments decreases.The proportion of HF load components increases dramatically because of severe slamming.In addition,the LF bending moments from the self-propulsion test is higher.By contrast,HF bending moments from the towing test is higher than those from the selfpropulsion test,which is mainly because there is no speed loss in the towing test;

(3)Compared with the linear hydroelastic method,the nonlinear hydroelastic method presented in this paper can reflect the HF vibration characteristics of hull load responses better.The method for the retardation function that obtains the accurate results does not suffer the numerical difficulties of the numerical quadrature.Besides,since the proposed method does not require uniform spacing of the frequency discretization,the number of the frequencies needed to compute the hydrodynamic coefficients is reduced significantly,thus the computational cost is also reduced.

This paper only focuses on whipping and springing responses based on the segmented model experiment and nonlinear hydroelastic analysis.The ultimate goal of research on whipping response is to discuss effect of whipping and springing on fatigue damage and strength of the hull.These will be studied in future work.

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基于不同试验方法和非线性水弹性时域理论的鞭振弹振响应研究

陈占阳，桂洪斌
（哈尔滨工业大学（威海）船舶与海洋工程学院，山东 威海 264209）

船体主尺度增大会导致严重的鞭振和弹振现象，这会增大船体结构的极限载荷和疲劳损伤。为了深入探究船体的振动响应，文中在拖曳水池对某万箱集装箱船分别进行了分段模型的自航和拖航试验。分析了不同海况下自航和拖航这两种试验方式对鞭振和弹振响应的影响。为计及不同振动频率成分对载荷响应的影响，提出一种考虑波浪记忆效应的非线性水弹性方法。文中提出了一种求解延时函数的方法，能够解决高频区域的阻尼系数的计算限制。最后，船舯弯矩试验结果分别和线性与非线性理论结果进行了比较，发现文中提出的非线性方法能够更好地预报弹性船体的振动响应。

鞭振；弹振；砰击；水弹性；延迟函数

U661.73

A

陈占阳（1984－），男，博士，哈尔滨工业大学（威海）船舶与海洋工程学院讲师，通讯作者；桂洪斌（1967-），男，博士，哈尔滨工业大学（威海）船舶与海洋工程学院教授，博士生导师。

10.3969/j.issn.1007-7294.2017.09.009

Article ID： 1007-7294（2017）09-1145-15

Received date：2017-04-07

Foundation item:Supported by National Natural Science Foundation(Grant No.51509062),the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.201727)and Science and technology development projects of Weihai for supporting this work(2015DXGJMS009)

Biography：CHEN Zhan-yang(1984-),male,Ph.D.,lecturer,corresponding author,E-mail:chen_1228@163.com;GUI Hong-bin(1967-),male,professor/tutor.