Investigation of a ship resonance through numerical simulation ＊

S. S. Kianejad, Hossein Enshaei, Jonathan Duffy, Nazanin Ansarifard

Australian Maritime College, University of Tasmania, Launceston, Australia

Abstract: Understanding dynamic stability of a ship on a resonance frequency is important because comparatively smaller external forces and moments generate larger motions. The roll motion is most susceptible because of smaller restoring moments. Most studies related to the failure modes such as parametric roll and dead ship condition, identified by second generation of intact stability criteria(SGISC) are performed at a resonance frequency. However, the nature of resonance, where the model experiences an incremental roll motion, has not been well understood. In this study, nonlinear unsteady computational fluid dynamics (CFD) simulations were conducted to investigate the resonance phenomenon using a containership under a sinusoidal roll exciting moment. To capture the complexity of the phenomenon, simulations were conducted over a range of frequencies to cover the resonance frequency including lower and higher amplitudes. In addition to the resonance frequency, the phase shift between roll exciting moment and roll angle, as well as the phase difference between acceleration and roll angle, were found to have significant effects on the occurrence of resonance.

Key words: Resonance, harmonic excited roll motion, natural frequency, phase shift, phase difference

Introduction

A deep understanding of a ship’s responses in calm and regular wave conditions can assist predicting ship responses in a realistic irregular wave sea condition[1]. Nowadays, improvement of computational capabilities and advancement in non-linear ship dynamics, have enabled more complex analysis of dynamic stability. Second generation intact stability criteria (SGISC) were proposed by international maritime organization(IMO) which defined several failure modes. These phenomena are parametric roll, pure loss of stability,surf-riding and broaching, dead ship condition and excessive accelerations. Resonance can be experienced in some of the failure modes such as parametric roll and dead ship condition where a ship experiences an incremental roll angle and in the worst cases a large generated roll angle endangers safety of the ship. The parametric roll occurs in the head and following sea condition, where an encounter frequency is twice the ship’s roll natural frequency. However, in the quartering and beam sea conditions, resonance occurs if the encounter frequency matches the ship’s natural frequency[2].Investigation of the parametric roll and dead ship condition was performed at resonance frequency,yet the reasons of resonance occurrence are not known. This study evaluates impact of some parameters on resonance occurrence and quantifies the variation of motion characteristics at frequencies close to resonance frequency. The literature review consists of two sections, which review two failure modes where resonance is more common to occur.The first section reviews the occurrence of resonance in the parametric roll and the second section reviews the resonance in the dead ship condition.

Investigating the parametric roll in terms of theoretical studies began in the 1950s, when linear and weakly nonlinear Mathieu’s equations were deployed by Paulling and Rosenberg[3]over a range of frequencies including roll natural frequency. They observed that at an encounter frequency of twice the natural frequency, the model generates large roll angles in a head sea. Thereafter, Soliman and Thompson[4]examined roll motion variation at head sea condition whereas Oh et al.[5]applied a coupled pitch and roll approach to investigate the parametric rolling. In their research, optimal encounter frequency of waves was twice that of roll natural frequency and results were justified experimentally. Longitudinal waves can impose parametric rolling both in head and following sea conditions, but there is greater likelihood of the parametric roll occurrence in head sea condition and it is usually coupled with heave and pitch motions[6].Considering a ship free in coupled motions in investigating parametric roll is necessary because it is dependent on the other motions especially pitch and heave motions. Neves et al.[7]suggested a 3 degrees of freedom (DOF) method, where coupled pitch and heave motions were input to a 1DOF nonlinear roll motion’s equation. Initially, the heave and pitch motions were solved as coupled and the results were used to analyse the occurrence of parametric roll both at frequencies twice and equal to roll natural frequencies. A 2-D analysis was conducted by Neves and Rodriguez[8]to couple the heave, pitch and roll motions considering second and third order non-linearities to measure the restoring moment at a frequency twice roll natural frequency. Silva and Soares[9]suggested a 3DOF (heave, roll and pitch) and a 5DOF (sway, heave, roll, pitch and yaw) numerical model in time domain condition. These numerical simulations have been deployed at a frequency of twice that of roll natural frequency using a 3-D panel code,which were combined with the time domain by tuning the impulse response functions method. Shin et al.[6]used the same method with some modifications, the hybrid singularity method based on the Rankine source and transient Green’s function were deployed in the near field and in the far field, respectively. Munif and Umeda[10]utilized several 6DOF models for numerical simulations in the time domain. In their study, for the case of following and quartering sea, the encounter frequency was taken to be equal to roll natural frequency, while in the case of head and bow sea the encounter frequency was twice that of the roll natural frequency.

A 4DOF coupled non-linear motion equation was suggested by Ahmed et al.[11], where nonlinear restoring moment, instantaneous wetted surface,hydrodynamic forces and moments were taken into account in irregular waves at a frequency twice that of roll natural frequency. Prediction of the parametric roll occurrence in the realistic irregular wave seas is more intricate, while to recognize the physics of parametric roll, numerical and experimental simulations in regular waves can be helpful. A 1DOF (roll motion equation)simulation was performed by Bulian and Francescutto[12]in long-crested head waves using Grim's effective wave amplitude concept at a resonance frequency. They proposed a framework considering the hydrostatic restoring variations in respect to the location of a wave crest. A study by Ribeiro Silva and Gudes Soares[13]has shown that linear and non-linear methods can be utilized to investigate the parametric roll in the regular head waves at a resonance frequency. They observed that the accuracy of roll angle prediction under wave-induced parametric roll was not satisfactory, because the linear method fails to take into account the effect of deck submergence and the non-linear damping term.However, the nonlinear numerical model of Ribeiro Silva and Gudes Soares[13]addresses the weaknesses of the linear method in the regular head waves. The numerical simulations of this method are in good agreement with the experimental measurements.

To address the deficiencies of non-linear method,a coupled 5DOF (sway, heave, roll, pitch and yaw)model was suggested by[14]to simulate the time domain responses of a ship at a resonance frequency and in long-crested irregular waves. The model has been developed to 6DOF using a semi-empirical equation for surge motion. In a recent study, Umeda et al.[15]conducted 5DOF numerical simulations to predict the parametric roll in oblique waves with low forward speed at a frequency twice the roll of natural frequency.The obtained results were in good agreement with experimental data.

In the dead ship condition, a ship loses her main propulsion system and becomes more susceptible to winds and waves. Themelis and Spyrou[16]studied this phenomenon based on a probability assessment including piece-wise linear method in beam sea waves at an encountered frequency equal to a roll natural frequency. Bulian and Francescutto[17]performed a 1DOF simulation, based on Monte Carlo method to investigate the dead ship condition at an encounter frequency equal to roll natural frequency of the model in irregular beam waves. Umeda et al.[18]carried out model tests in irregular beam waves condition at a constant wind to verify the uncoupled roll equation in a piece-wise linear approach. Gu et al.[19]used 3DOF simulations free in roll, drift and sway motions in dead ship condition at a roll natural frequency. The results were in good agreement with model experiments. In another approach, Kubo et al.[20]adopted a 4DOF model considering sway, heave, roll and pitch motions to study dead ship condition at a resonance frequency.They found that the 4DOF method compared to the 1DOF could exactly simulate the capsizing event and was in a better agreement with the model test data.

According to these literature reviews, the resonance occurs at a specific frequency, however,it is not well understood what the reason is for resonance. The aim of this paper is to investigate the factors that influence the resonance and to improve understanding of the resonance phenomenon using computational fluid dynamics (CFD) simulation. To achieve this aim, a container ship model was excited with a sinusoidal roll moment based on harmonic excited roll motion (HERM) technique over a range of frequencies including those higher and lower than the roll natural frequency. In addition to the encounter frequency, the effects of phase shift between the roll exciting moment and roll angle, the phase difference between the roll angle and angular acceleration at resonance, are investigated.

1. Ship geometry

A fully appended model of a post Panamax container ship (Duisburg Test Case) was employed to study the resonance occurrence in the beam sea condition. The model has a transom stern and a large bow flare area and was tested in the Hamburg Ship Model Basin (HSVA) to calculate damping coefficients and resistance. The main particulars of the ship and model are shown in Table 1 and Fig. 1. More details of the tests and hull geometry can be found in Ref. [21].

Table 1 Main particulars of ship and model, model scale factor is 59.467

Fig. 1 (Color online) Hull geometry of Post Panamax container ship

2. Numerical modelling

The following sections discuss details of the selected numerical simulation approach for this study.

2.1 Governing equations

The simulations were performed by STAR CCM+ and the associated solver resolves the integral form of RANS equations. The simulation solver uses averaged continuity and momentum equations for an incompressible flow with body-force[22]. It considers tensor form and Cartesian coordinates shown by the following set of equations:

whereu=(u,v,w) is time averaged velocity field in the cartesian coordinate,ρis the fluid density and has a constant value for water and air,p*andgrepresent the time averaged pressure including hydrostatic and gravitational acceleration,X=(x,y,z) are the cartesian coordinates,μis the dynamic viscosity,τis the Reynolds tensor and ?is the gradient operator (?/?x, ?/?y, ?/?z) and the first term in the right side of Eq. (2) reflects surface tension which has negligible effects in this study and can be removed. The selected solver is based on the finite volume approach which discretises the integral form of Navier-Stokes equations. Connection between momentum and continuity equations is made with a predictor-corrector method which is employed by Reynolds-averaged Navier-Stokes (RANS) solver.

2.2 Physics modelling

The volume of fluid (VOF) as a simple multiphase approach was utilised to simulate the free surface. While using VOF, extra modelling to capture an inter-phase interaction is not required. The VOF utilises a presumption that governing equations are the same for both single phase and multiphase, that is,they will have the same velocity and pressure magnitudes. A second-order convection scheme has also been used to capture sharp interface between the phases. The volume of fraction ()γwas used to track the fluids. The magnitude of volume of fraction for air and water is 0 and 1, while a mixture of these two fluids has an intermediate value. The distribution of volume fraction is modelled by equations below

whereuris the relative velocity vector and the last term of equation is the compression term which decreases smearing of the interface. The variation ofρa(bǔ)ndμcan be computed by:

For the turbulence model, the realizablek-εwas selected to address the uncertainty of the stress tensor. This turbulence model in calm water condition predicts the roll motion precisely and reduces simulation running time in comparison with other turbulence types such ask-ωand SST[23]. The RANS solver uses a segregated flow model, and the flow equations are solved in an uncoupled condition where convection terms were discretised by second order upwind scheme. SIMPLE algorithm was used throughout the solution. Dynamic fluid body interaction (DFBI) was employed to simulate a real ship condition in the sea, considering 6DOF motions.The DFBI model feeds the solver to compute different forces and moments which are acting on the model in calm and wave sea condition.

Courant number (Co) was applied to determine a proper time step, which is a proportion of physical time step (Δt) to the mesh convection timescale (mesh cell dimension (Δx) per mesh flow speed (U))according to the Eq. (6). The Courant number was less than one for each cell to maintain the numerical stability.

2.3 Meshing structure

Overset mesh method was employed to generate a high quality mesh configuration to simulate large roll motions. The overset mesh surrounds and holds the body and moves with the body inside a motionless background mesh in the entire domain[24]. Hence, the overset mesh method provides refined mesh structure around the body without changing the simulation’s precision as well as reducing the simulation time by reducing the number of cells. In this study, the dimension of overset region was set to capture boundary layer, flow separation during body motion,wave making and vortices around the body. To capture the complex flow specification around the body and in the free surface, the refining mesh procedure was carried out using volumetric control zones as shown in Fig. 2. To prevent solution divergence, an overlap region was considered to match the cell size between the background and overset regions.

Four types of meshers were used including trimmed, prism layer, surface and automatic surface repair meshers to generate the mesh. The trimmed mesher is an effective way of generating a better quality of mesh for simple or complex geometries,whereas the prism layer mesh was utilised to generate orthogonal prismatic cells next to the body to capture the velocity gradient and the boundary layer. The surface mesher improves the quality of a surface to provide better volume mesh. Subsequently, automatic surface repair is used to refine a range of geometric problems created after surface remeshing.

2.4 Boundary and initial condition

Selecting suitable boundary and initial conditions reduces the simulation time and increases the precision. Velocity inlet boundary condition was selected for positive x direction where incident flow run to the body. The oppositexdirection boundary was set as a pressure outlet to avoid backflow from a fixed static pressure. The sides, top and bottom boundaries can be set as slip-wall or symmetry but should be located far enough away from the model to minimise their effects on the model. While, these boundaries were considered as velocity inlet to avoid sticking of the fluid to the walls as well as generating velocity gradient because of the walls and fluid interaction. By considering the velocity inlet condition for the top and bottom overcomes the problem of deep water and unlimited air condition. Non-slip wall boundary condition was selected for the model to capture the moments and motion characteristics.

The initial flow velocity at all inlet boundary conditions was set for the simulation. For this reason,the flow in all lateral boundaries moves towards the outlet boundary. Consequently, fluid turbulence at these boundaries will be decreased. The initial hydrostatic pressure was selected for the outlet boundary.

Fig. 2 (Color online) Cross-section of meshing structure around the model

2.5 Coordinate system

Upon simulation of the fluid field around the body, exerted forces and moments on the body were computed in the earth-fixed coordinate system. The forces and moments were transferred into the local body coordinate system, which was set at the centre of gravity. In the following step, the velocity and acceleration of the body were computed by solving the motion’s equations and this data transferred back to the earth-fixed coordinate system to locate the body[25].

3. Equations of motion

The translation equation of motion (surge, sway and heave) with respect to the centre of gravity of the model in the global coordinate system is[22]

wheremis the mass of the model,fis acting force on the model and indicespandvdenote pressure and viscous terms.uis the velocity of the model at centre of gravity. The equation of rotational motion of the model (roll, pitch and yaw) in the body local system with respect to the centre of gravity is[22]

whereIandωare the tensor of mass moment of inertia and angular velocity of the model,Mis acting moment on the model andpandvrepresent the pressure and viscous parts. The equations of motion in six degrees with some simplifications and considering the correlation between different motions are as follows[26]:

whereηis the position of the model,mandIijare the elements of the generalized mass matrix,Aijare the elements of the added mass matrix,Bijare the hydrodynamic damping coefficients,Cijare restoring coefficients,FiandMiare diffraction and Froude-Krylov forces and moments andME44is an external harmonic roll exciting moment[26]. The focus of this study is on roll resonance by considering the model free in 6DOF. The Eq. (21) can be simplified to a 1DOF approach considering the effects of other motions on roll motion equation as below[23]

The equation of motion (Eq. (10)) was used to compute total mass moment of inertia from numerical simulations[24]. The magnitude of mass moment of inertia is calculated by computing the generated acceleration from the simulations. As the simulations were conducted in calm water condition, it is assumed that the dynamic restoring moment is equal to the hydrostatic restoring moment[27]. Hence, the restoring moment was calculated using Autohydro software and the exciting moment is an input to the simulations.When the magnitude of roll angle reaches its maximum at a cycle, the magnitude of angular velocity and damping moment is almost zero, and the roll added mass moment of inertia can be computed by Eq. (11). Therefore, the total roll mass moment of inertia (mass moment and added mass moment of inertia) is computed.

4. Numerical uncertainty analysis

Prior to commencing the simulation for different situations, it is necessary to specify the uncertainty in simulation results to ensure the numerical approach accurately simulates the physics. According to the verification method advised by Stern et al.[28],numerical uncertaintyUSNconsists of iterative convergence uncertaintyUI, grid-spacing uncertaintyUcand time-step uncertainty.UTare given in the following equation:

The uncertainty ofUIis negligible[24], however, the grid-spacing and time step as major sources of uncertainty were investigated for the case of 5.5 Nm roll external exciting moment at frequencies of 1.38 rad/s and 1.39 rad/s which are very close to the natural frequency of the model. The model is free in 6DOF and simulations were performed for 100 s to confirm that amplitude of the roll motion characteristics reaches steady-state condition with negligible differences at different cycles. Total resistance of the model at a speed of 1.54 m/s was calculated. The grid-spacing uncertainty of different mesh configurations was performed based on a Richardson extrapolation[24]. Three different mesh configurations with a refinement ratio ofwere considered,and the number of meshes for each case is shown in Table 2. Simulation was set on the basis of course,medium and fine mesh configurations and shown byS3,S2andS1respectively. Alteration of simulation results is calculated by the following formulas:

Table 2 The number of elements for each mesh configuration tested

The numerical convergence ratio was calculated using Eq. (10). According to the convergence ratio,four typical conditions can be predicted: (1)monotonic convergence (0＜RG＜1), (2) oscillatory convergence, (3) monotonic divergence (RG＞1), and (4) oscillatory divergenceNumerical uncertainty in cases (3)and (4) cannot be computed because of divergence.For case (2) uncertainty can be computed based on bounding error with upper limitSUand lower limitSLby using the equation below

The time step uncertainty can be computed based on a similar procedure, starting from Δt=T/29and considering a uniform refinement ratio of 2 (rT=2).The grid uncertainties were computed by the smallest time step, whilst the time step uncertainties were computed by the finest mesh configuration. The numerical simulation results for each mesh configuration at frequency of 1.39 rad/s are shown in Fig. 3, the experimental measurement (EM)[21]and magnitude of grid-spacing and time step uncertainties are presented in Tables 3-6. It is apparent that the uncertainty of grid spacing is more than the uncertainty of time step. It is seen that the uncertainty of angular velocity and drag are more than the roll angle, angular acceleration and roll moment. The flow separation, vorticity and boundary layer could be captured accurately by decreasing the time step and increasing number of mesh around the body. They have greater impact on computation of the drag and angular velocity. Hence, the simulation results for the drag and angular velocity for the coarse mesh and the larger time step have relatively larger differences against EM compared to other roll motion characteristics. The simulation results were more accurate for the fine mesh configuration. Therefore,further simulations were performed based on the fine mesh configuration and smallest time step. The+Yvalue was set less than 1 for the selected mesh configuration to ensure the pressure and shear forces were calculated precisely.

Fig. 3 Effect of different numbers of mesh on the roll responses including roll angle, velocity, acceleration and roll moment

Table 3 Grid convergence study considering 5.5 Nm exciting moment at the frequency of 1.39 rad/s

Table 4 Time step convergence study considering 5.5 Nm exciting moment at the frequency of 1.39 rad/s

Table 5 Grid convergence study considering 5.5 Nm exciting moment at the frequency of 1.38 rad/s

Table 6 Time step convergence study considering 5.5 Nm exciting moment at the frequency of 1.38 rad/s

5. Results and discussion

5.1 Verification study

The experimental tests were performed at the Hamburg Ship Model Basin to extract roll damping coefficients. The model was free in 6DOF and was excited using two masses were rotating contrarily around the vertical axis (Z) at the centre of gravity and they were encountering at both sides of the model to generate roll exciting moment which was equal to restoring moment when two masses were at one side.It means, they generate roll exciting moment at the same direction with the same magnitude, whereas, one of them generates clockwise yaw moment and another one anti-clockwise. Hence, they counteracted each other, and the yaw motion remained negligible[21]. The rotating masses method works well for small roll angle while to generate larger roll angle, heavier masses should be used which impose some unwanted motions, particularly sway motion. Hence, in this study, the model was excited using external harmonic roll exciting moment instead of rotating masses as a field function and an input to the simulations. The simulation results for the smallest time step and fine mesh configuration under 5.5 Nm roll exciting moment at two frequencies (1.38 rad/s and 1.39 rad/s)were compared against EM as can be seen in Table 7.The drag calculation was carried out to make sure the selected mesh structure has ability to capture pressure and sheer forces accurately. The maximum deviation between experimental and numerical simulations was about 3% which means the current numerical method has capability to simulate a ship motions at frequencies close to the resonance frequency.

5.2 Reasons of resonance

The roll resonance is the most dangerous phenomenon at sea and evidence suggests that it occurs when total mass moment of inertia and restoring moment have the same magnitude but act in opposite directions through an incremental roll motion. Themost influential parameter for the roll resonance is the exciting frequency, but what is not yet clear is the impact of phase shift variation between the roll exciting moment and roll angle as well as phase difference between the roll acceleration and roll angle.

Table 7 Comparison of numerical and experimental simulations’ results

To investigate causes of the resonance, the model was excited under 10 Nm harmonic roll exciting moment at different frequencies, ranging between 0.90 rad/s and 1.70 rad/s. The lower excitation frequencies were conducted at intervals of 0.20 rad/s and the intervals were decreased to 0.10 rad/s for the higher frequencies to capture more details. The time trace of roll exciting moments (M), roll angles and acceleration (Acc) are shown in Figs. 4-10. The angular acceleration for each frequency was scaled up to make comparison of the phase differences easier.

Fig. 4 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 0.90 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 0.90 rad/s

Fig. 5 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.10 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.10 rad/s

Fig. 6 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.30 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.30 rad/s

Fig. 7 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.40 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.40 rad/s

Fig. 8 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.50 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.50 rad/s

Fig. 9 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.60 rad/s. (b) the restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.60 rad/s

The significant observations from a comparison of these data are the variation of phase shift, phase difference and induced roll acceleration while changing the roll excitation frequency. At a resonance frequency, the phase difference between the roll angle and roll acceleration is 180° where the magnitude of restoring moment and total mass moment of inertia remains the same. While, at a frequency away from the resonance, the phase difference is either less or more than 180°. The magnitude of roll acceleration at the same roll angle is smaller at frequencies less than the resonance frequency and becomes larger at frequencies larger than the resonance frequency. This means that the magnitude of the total mass moment of inertia at frequencies less than the resonance frequency is smaller than restoring moment, while,this condition is reversed at higher frequencies.Section b in Figs. 4-10 shows variation of the total mass moment of inertia (mass and added mass moment of inertia) and restoring moment at different roll angles. The restoring moment was calculated using Autohydro software and assuming the dynamic restoring moment in calm water remains equal to hydrostatic condition[27]. The phase shift between roll exciting moment and roll angle at the resonance frequency is 90°, which means at zero angle the model with zero acceleration absorbs maximum energy in terms of inertia to generate larger roll angle. For a low range of excitation frequencies, the phase shift is less than 90°, whilst increasing the excitation frequency increases the phase shift.

Fig. 10 (a) The roll exciting moment (10 Nm), acceleration and roll angle trajectories at a frequency of 1.70 rad/s. (b) The restoring moment and total mass moment of inertia at different roll angles at a frequency of 1.70 rad/s

Table 8 Phase differences at last four cycles under 10 Nm exciting moment at different frequencies

It can be seen in Figs. 4 and 5 that for a low range of frequencies (0.90 rad/s and 1.10 rad/s), the induced accelerations are small with large fluctuations due to small phase shift between the roll exciting moment and roll angle. When the model has zero angle, the magnitude of exciting moment is small, and the model cannot acquire enough energy in terms of moment of inertia. Therefore, the magnitude of contributing moments to increase the roll angle is small, and the interaction between the model and generated waves (during oscillations) is the reason for the fluctuations of angular acceleration. It can also be seen in Figs. 4 and 5 that the fluctuations in the first cycles are smaller than the following cycles. The fluctuations of acceleration reduce by increasing either the magnitude of exciting moment or the excitation frequency as the magnitude of contributing moments become and dominant. The difference between roll restoring moment and total mass moment of inertia is significant and the exciting moment is not large enough to compensate this difference to increase the roll angle[30]. As shown in Table 8, the phase difference between the roll angle (restoring moment)and acceleration (total mass moment of inertia) at steady state condition is 160°-170°. As well as this,the phase shift between the roll exciting moment and roll angles is small, where in the following cycles the exciting moment overtakes the roll angle trajectories.Therefore, this incorporation cannot impose a large roll angle.

When frequency is closer to the resonance frequency of the model (1.38 rad/s) as shown in Figs.6 and 7, the generated acceleration and total mass moment of inertia are large, especially at a frequency of 1.4 rad/s, where the magnitude of restoring moment and total mass moment of inertia at different roll angles are equal. In this condition, the phase shift between exciting moment and roll angle is closer to 90° (Table 9), and phase difference between roll angle and acceleration is close to 180 degrees, resulting in,the model experiencing a larger roll motion.

At a high range of frequencies, as the frequency increases the phase shift goes beyond 90° and phase difference moves away from 180° (Tables 9 and 8). In this condition, the induced total mass moment of inertia is larger than restoring moment (Figs. 8-10)and the difference rises as the excitation frequencyincreases. The exciting moment compensates the difference and as a result, the model generates smaller roll angle.

Table 9 Phase shifts at last four cycles under 10 Nm exciting moment at different frequencies

In conclusion, for a frequency less than the resonance frequency, the external roll exciting moment and total mass moment of inertia are the main contributors for development of the roll motion, while the restoring and damping parts oppose the roll motion. However, the magnitude of total mass moment of inertia is larger than the other terms at a higher range of frequencies and due to larger phase shift, the roll exciting, damping and restoring moments oppose the development of roll motion[27].

5.3 Effects of resonance on roll motion characteristics

The changes of roll, angular roll velocity, angular roll acceleration, roll moment, yaw, sway and pitch motion of the model over a range of frequencies from 0.90 rad/s to 1.70 rad/s under 10 Nm harmonic roll exciting moment are shown in Figs. 11-17. At the beginning of each simulation, all cases have the same amplitudes, while in the following cycles the model acquires energy from the roll exciting moment and experiences larger roll angles. Increasing the frequency from 0.90 rad/s to 1.40 rad/s (natural frequency is about 1.38 rad/s) increases the maximum roll angle. At the frequencies of 0.90 rad/s and 1.10 rad/s, which are far from the natural frequency,model generates roll angles of up to 6° and 10°,respectively. The model at the frequency of 1.30 rad/s generates larger roll angle of about 17.5° at fifth cycle and thereafter remains constant. At the frequency of 1.40 rad/s, which is the closest frequency to the natural frequency, the model generates up to 20.5° roll angle. Recordings show that for higher range of frequencies including 1.50 rad/s, 1.60 rad/s and 1.70 rad/s, increasing the frequency reduces the maximum roll angle to 13°, 8° and 6°, respectively.

Increasing the roll angle variation increases the angular velocity subsequently increasing angular velocity variation, which increases the angular acceleration. The model generates smaller vorticity at 0.90 rad/s and increasing the frequency towards the natural frequency of the model increases the magnitude of vorticity in which in the case of 1.40 rad/s has the largest vorticity. It is mainly because of the larger angular velocity and further increase of the excitation frequency decreases the magnitude of vorticity[25]. The excitation frequency of 1.70 rad/s is closer to the natural frequency of the model compared to 0.90 rad/s.While, the induced roll angle for both cases is the same, because the magnitude of vorticity and damping in case of 1.70 rad /s is larger due to larger excitation frequency and angular velocity[25].

Fig. 11 Roll angle amplitudes under 10 Nm exciting moment at different frequencies

Fig. 12 Angular velocity amplitudes under 10 Nm exciting moment at different frequencies

Though, the maximum difference for the roll angle between the frequencies of 1.40 rad/s and 1.70 rad/s is large, the angular acceleration difference is small. Because, the frequency of 1.70 rad/s is from a region dominated by acceleration with relatively larger induced acceleration. It also can be seen that the which is furthermost from the resonance frequency,the maximum roll moment is about 18 Nm, which is slightly more than the roll exciting moment.

Fig. 13 Angular acceleration amplitudes under 10 Nm exciting moment at different frequencies

Fig. 14 Roll moment amplitudes under 10 Nm exciting moment at different frequencies

The distribution of total pressure on the model at different frequencies is shown in Fig.18. The model in all conditions is in upright condition which means, the hydrostatic pressure on both sides of the model is almost the same. The pressure difference between two sides of the model is related to the dynamic pressure resulted from the HERM realizations. The pressure distribution difference between two sides of the model in cases of 1.30 rad/s, 1.40 rad/s and 1.50 rad/s is larger than other cases because of larger angular velocity and dynamic pressure. Hence, the sway motion in these cases is larger than other cases (Fig.16) while they have smaller yaw angle as can be seen in Fig. 15. On the other hand, the model in cases of 1.10 rad/s and 1.60 rad/s experiences larger yaw motion because of smaller sway motion.

When the model is rolling, the pressure force of the fluid in the aft half of the model, due to larger wetted surface area, becomes larger than the forward half, resulting in a trim by stem. Increasing the roll angle increases the imposed pitch angle as shown in Fig. 17 in the cases of 1.30 rad/s and 1.50 rad/s. However, the variation of pitch angle is negligible compared to the yaw angle variation.roll angle at frequencies of 0.90 rad/s and 1.70 rad/s are similar, while the angular acceleration at frequency of 1.70 rad/s is two times higher than 0.90 rad/s.

As expected, the maximum roll moment of the model in the case of 1.40 rad/s is larger than the other cases, which accounts for 70 Nm. It is seven times the roll exciting moment, while the magnitude of roll moment decreases by either increasing or decreasing the excitation frequency. At the frequency of 0.90 rad/s

6. Conclusions

Numerical simulations were carried out using a containership model based on the HERM technique and under a 10 Nm external roll harmonic exciting moment at lower and higher frequencies than the roll natural frequency of the model. The results of this investigation show that two conditions are required for the resonance to occur. First, the phase shift between the roll exciting moment and roll angle should be close to 90° where the model can absorb maximum energy from the exciting moment. Second, the phase difference between the angular acceleration and roll angles should be close to 180°, while the total mass moment of inertia and restoring moment have the same magnitudes but in opposite directions to cancel each other.

Fig. 15 (Color online) Time histories of yaw angle under 10 Nm exciting moment at different frequencies

Fig. 16 (Color online) Time histories of sway motion under 10 Nm exciting moment at different frequencies

Fig. 17 (Color online) Time histories of pitch angle under 10 Nm exciting moment at different frequencies

Fig. 18 (Color online) Total pressure distribution of the model at different excitation frequencies

For the frequencies less than the resonance frequency, the phase shift between exciting moment and the time trace of roll angle is less than 90° and it reduces as frequency further decreases. On the other hand, at a frequency larger than the resonance frequency, the phase shift is greater than 90° and increases by increasing the excitation frequency. The phase difference between roll angle and angular acceleration at a low range of frequencies is less than 180°, whereas it is larger than 180° at a high range of frequencies.

The resonance condition is dangerous where small external forces and moments force a ship to experience larger roll motion. These findings are useful for ship designers as well as ship operators to avoid resonance and to recognize its consequences.