Validation of the Global Hydroelastic Model for Springing & Whipping of Ships

11th International Symposium on Practical Design of Ships and Other Floating Structures Rio de Janeiro, RJ, Brazil 2010 COPPE/UFRJ Validation of the Global Hydroelastic Model for Springing & Whipping of Ships Q. Derbanne 1), Š. Malenica 1), J.T. Tuitman 2), F. Bigot 1) & X.B. Chen 1) 1) BUREAU VERITAS - Research Department, Paris, France 2) TNO, Delft, Netherlands Abstract Hydroelastic response of the ship can represent an important part of the overall structural response. This is true both for the extreme ship structural response as well as for the fatigue loads of some structural details. Several full scale monitoring campaigns on the container ships clearly show an important high frequency contribution coming from the global wave induced ship vibrations. Correct numerical modeling of these phenomena is extremely difficult and only approximate models exist up to now. During last several years an efficient hydroelastic model based on the state of the art numerical techniques was developed within the cooperation in between TNO and Bureau Veritas. The model couples the 3D hydrodynamics with either non-uniform beam or 3DFEM structural dynamics. This model showed to be very efficient in terms of the CPU time and in terms of qualitative picture of the time histories of the structural responses. At the same time, several experimental campaigns were undertaken and are in progress, in order to provide the data for validation of the numerical models. The present paper will give an introduction in the numerical method and present some results of these comparisons. Introduction Increased size of the recent Ultra Large Container Ships (ULCS) and LNG ships, reactualizes the hydroelastic wave induced type of ship structural responses. Usually the hydroelastic ship response is separate into springing and whipping, springing being defined as the steady ship vibrations induced by non-impulsive wave loading and whipping as a transient vibratory ship response induced by impulsive loading such as slamming. Whipping can be induced by different types of impulse forces, however in this paper only slamming induced whipping is considered. Fig. 1 shows typical measurements at sea of these two phenomena. Keywords Whipping; springing; slamming; hydro elasticity; seakeeping; structural loading 331

(2) After solving the different BVP-s the resulting pressure is calculated using Bernoulli s equation and integrated over the wetted surface in order to obtain the corresponding forces, so that the following coupled dynamic equation can be written: Fig. 1:Typical springing (up) and whipping (down) response (3) Hydroelastic Model in Frequency Domain The general methodology for hydroelastic seakeeping model is rather well known and the first developments can be attributed to Bishop & Price (1979). In their work they used a Timoshenko beam model as simplified model of the structure and strip theory for seakeeping part. Since then several more or less sophisticated models were proposed: Wu & Price (1986), Wu & Moan (1996), Xia & Wang (1997), Korobkin (2005). Below we give a brief introduction in the basic principles of the model used in this study. The 3DBEM model for the seakeeping is coupled to a 1D- or 3DFEM model of the ship structure. A more detailed description of the applied 3DBEM model can be found in Newman (1994) and Malenica et al (2003). Linear hydroelastic Model In contrast to the well known rigid body seakeeping model, the hydroelastic model basically extends the motion representation with the additional modes of motion/deformation chosen as a series of the dry structural natural modes. We write: where h i }( x, y, z) (1) { denotes the general motion/ deformation mode which can be either rigid or elastic. The above decomposition leads to the additional radiation boundary value problems (BVP) for elastic modes, with the following change in the body boundary condition: where: The solution of the above equation gives the motion amplitudes and phase angles by which the problem is formally solved. Note that the motion equation includes both 6 rigid body modes and a certain number of elastic modes. Several technical difficulties need to be solved before arriving to the above motion (Eq. (3)). Certainly the most difficult one is the solution of the corresponding hydrodynamic BVP. In this paper we do not enter into the detailed description of the methods used to solve the seakeeping problem at forward speed and we just mention that these difficulties remain the same, both for the rigid and elastic body. It is fair to say that the numerical methods which are used to solve the seakeeping problem nowadays are not fully ready yet for a general combination of speed, heading and frequency. However, most of the methods have approximate solutions to account for the forward velocity. The method used in this paper is the so called encounter frequency approximation which was reasonably well validated for rigid body case. 332

The second technical difficulty concerns the application of the body boundary condition (Eq. (2)) for the general mode of motion. The goal is to apply the modal deformation, computed on the structural mesh of the ship, to the hydrodynamic mesh used to solve the hydrodynamic problem. This interpolation problem can be solved through many ways. The procedure used here is explained in Tuitman & Malenica (2009). Fig. 2 shows an example of the transfer of modal displacements for a torsion mode. was successfully explained using the weakly nonlinear second order theory. In principle similar kind of methods should also be applied in the case of the ships, but unfortunately the state of the art in ship hydrodynamics do not allow for that so that only approximate and empirical methods can be used. However, due to their huge dimensions and particular structural properties, the case of ULCS is quite particular because their first structural natural frequencies become quite low so that it can be excited in a linear sense too! This means that the linearly excited springing might become dominant which justify the linear model for its assessment. The hydroelastic model in time domain which will be presented in the next section includes the linear and some non-linear springing response. The Froude- Krylov loading is the only non-linear wave excitation in this model. Some non-linear springing due to this Froude-Krylov is observed in the results of the time domain calculations, but to the authors opinion a more sophisticated non-linear model might be necessary to correctly calculate the non-linear springing response. Stress Once the motion Eq. (3) solved, all necessary quantities (motions, accelerations, stresses,...) can now be derived from the calculated modal response. Indeed, the modal decomposition remains valid for any quantity and, in particular, we can write for the stress distribution: (4) Fig. 2: First torsional mode of ULCS and the transfer of modal displacements from structural (up) to hydrodynamic mode (down) Springing The solution of the hydroelastic motion Eq. 3 includes the linear springing response automatically. However, there is no clear justification for considering springing as a linear phenomenon, at least as far as the wave excitation part is considered. Indeed, experience shows that many type of ship suffer from more or less important structural response around their first structural natural frequencies even if there is no important wave energy around these frequencies. The explication for existence of the structural response can be explained only by introducing the nonlinearity into the hydrodynamic model. The well known example is the springing of the tendons of the TLP platforms where the hydrodynamic excitation σ represents the spatial distribution of the stresses corresponding to each mode of motion/ deformation. Equation similar to Eq. (4) remains valid in the time domain model too. The only change concerns the replacement of ω by t. i where ( x, y, z) It is important to note that the rigid body modes do not contribute to any stress, and that the above expression (Eq. (4)) is slowly convergent in general. This is especially true for the structural details which are affected by the local structural effects. An approach presented in Malenica & al. (2008) separates the dynamic response and the quasi-static response. The global structural dynamic response of the ship is well described by the first few lowest modes. The other modes will have a purely quasi-static response 333

which will be calculated, after separation, by the socalled direct method which calculates the structural response after the rigid body hydrodynamic pressure is transferred on the wetted finite elements. The advantage of this decomposition is to remove the convergence problem of the pure modal approach, and to clearly identify the dynamic part as a correction of the quasi static part. A typical stress RAO is shown in Fig. 3. It is interesting to observe that the dynamic part of the response contributes to the stresses even for the frequencies well away from the natural frequency. This might seems a bit strange, but one should remind that these are very flexible ships. The dynamic amplification of the flexible modes will have an influence even at low frequencies. All this suggests the importance to utilize a hydroelastic analysis for ULCS s even when springing and whipping are not explicitly considered. where overdots denote the time derivatives and: (5) It was shown by Ogilvie, that the impulse response functions can be calculated from the frequency dependent damping coefficients: (6) Fig. 3: typical stress RAO Non Linear Time Domain Simulations In contrast to the linear springing calculations which can be performed in frequency domain, the nonlinear springing and whipping simulations require the time domain simulations. The procedure used in this paper is based on the method proposed in Cummins (1962) and elaborated in Ogilvie (1964). This method uses the frequency domain hydrodynamic solution and transfers it to the time domain using the inverse Fourier transform. In this way the following time domain motion equation is obtained: After the impulse response functions K ij have been calculated, the motion equation (10) is integrated in time using the Runge Kutta 4th order scheme. The main advantage of the time domain method lies in the fact that we can introduce non-linear components in the excitation forces Q(t). There are many nonlinear effects which are missing in the linear model and the state of the art in the numerical seakeeping, does not allows for the consistent inclusion of all of them. In this paper we include the so-called Froude- Krylov correction for the wave excitation and the non-linear slamming forces. Froude-Krylov Approximation It is well known that the linear hydrodynamic model evaluate the pressure up to z = 0 only so that we can end up with the negative pressure close to the waterline, as well as with zero pressure above the waterline and below the wave crest which is unphysical. The so-called Froude-Krylov approximation is usually employed to correct this pressure distribution close to the waterline. The procedure used here is described in detail in Tuitman & Malenica (2009). We should keep in mind that several approximations are possible and, in spite of the quite complicated considerations, it is impossible to remain fully consistent. It is also important to note that several variants of the Froude- Krylov method are in use and the simplest one consists in taking only the incident wave part of elevation in 334

the definition of the relative wave elevation η R (and thus neglecting the diffracted and radiated part). This choice highly simplifies the numerical implementation and that is the reason why it is most often employed. In that case we need to be extremely careful when interpreting the results. This, lets call it incident wave Froude-Krylov approximation, is likely to be valid only in the case of rather long waves where the diffraction and radiation parts play less important role in the overall contribution to relative wave elevation η R. However, in the case of ULCS which have quite a low block coefficient i.e. rather slender form the incident wave Froude-Krylov approximation can be reasonably justified for head or almost head wave conditions for which the springing and especially whipping are most likely to occur. Course Keeping The non-linear part of the force in Eq. 5 will cause a mean loading at the ship. The ship will not hold course and speed due to this mean force. This mean force is not an error in the calculation. The mean force by the Froude-Krylov calculation is a part of the added resistance in waves (RAW) and slamming forces cause also a loading in the horizontal direction. The drift of the ship in the time domain calculation should be compensated in order to obtain a proper calculation. Springs and dampers are often added to keep the ship at the correct positions. Another solution is to model the complete resistance, propulsion and maneuvering system of the ship. Both methods require quite some tuning to be able to make a realistic simulation. To the authors opinion the correct nonlinear surge, sway and yaw response will not affect the structural loading much. Also, one could question if using a spring, damper system would give the correct non-linear horizontal motions. In the present approach Lagrange multipliers are added to Eq. (5). These Lagrange multipliers impose the surge, sway and yaw motions as is calculated using the frequency domain solution. This gives reasonable horizontal motions without having to do any tuning and this approach is unconditional stable. Whipping The time domain calculation will automatically include the linear and non-linear springing. In order to calculate the slamming induced whipping response, the slamming forces should be calculated during the seakeeping calculation. This forces are added to the force vector {Q(t)} of Eq. (5). The hydrodynamic modeling of slamming is extremely complex and still no fully satisfactory slamming model exists. However, the 2D modeling of slamming is well mastered today and 2D models are usually employed to assess the slamming loads on ships. Within the potential flow approach, which is of concern here, several more or less complicated 2D slamming models exist, starting from simple von- Karman model and ending by the fully non-linear model. In between these two models there are several intermediate ones such as Generalized Wagner Model (GWM) 11) and Modified Logvinovich Model (MLM) 5) which are implemented in the presented method. We will not go into too much details of these models and we refer to Malenica & Korobkin (2007). The advantage of the MLM method when compared to GWM method lies in the lower requirements of CPU time. Most often, at least when rigid body impact is concerned, two methods give comparable results as shown in Fig. 4. The GWM method is used for the presented examples because this method has a wider range of validity and the number of slamming calculations remains limited. If many calculations, for example a complete scatter diagram, have to be preformed the MLM method might become more attractive. The validity of the MLM calculation can be checked using the GWM results for the some cases. Fig. 4: Slamming forces on typical 2D section obtained by GWM and MLM method 335

The use of 2D methods implicitly requires the employment of the so-called strip approach for 3D simulations. The usual procedure consist in cutting the 3D mesh into a certain number of 2D sections as shown in Fig. 5. For each particular section, independent calculation of slamming loading are performed and the overall slamming is obtained by summing up different contributions. The implementation of this procedure is different for, the most often used, beam structural model and for 3DFEM model. More details can be found in Tuitman & Malenica (2009). Pressures obtained on the slamming sections are integrated on the elastic modal deformations (including rigid modes). For each mode this gives the slamming force {Q(t)} to be added to Eq. (5). Fig. 6: Flexible model Fig. 5: 3D hydrodynamic mesh and corresponding 2D sections The whipping response is automatically computed when solving Eq. 5 in time domain with the slamming forces taken into account in the right hand side. Validation with Model Tests Springing Tests on a very elastic ship The aim of these tests was to build a very flexible unrealistic ship model, in order to validate the hydroelastic numerical model presented here. With such an elastic model, the amplitudes of the elastic deformations of the model are in the same order of magnitude as the 6 rigid body motions. The tests have been performed by Ecole Centrale de Marseille within the European project TULCS. The model is made of 12 segments attached to a squared section steel rod. The shape of the segments, although simplified, is the one of a containership. Because of their U shape sections, these ships are very flexible in torsion, and there is a strong coupling between the horizontal bending mode and the torsion mode. In order to have a similar behavior of the elastic model, the steel rod is located just under the baseline of the ship in order to have a center of torsion as low as possible. Fig. 6 shows the model on the wave basin and the detail of the two aft segments and the steel rod. The length of the model is 4.42m. The tests were done without forward speed, in regular and irregular sea states, with head seas and oblique seas. The frequency domain numerical model described in this paper has been used to compute the motions of the segmented model, including the hydroelasticity. The modal analysis of the structural model shows 6 modes in vertical bending, horizontal bending and torsion in the range of wave frequency (up to 7.5 rad/s). The hydrodynamic coupling terms introduced a modification of the ranking of the natural modes, because the hydrostatic stiffness and the added mass mays change significantly the natural frequencies. For this very elastic ship the resonant rigid modes (roll, pitch and heave) are merged with the elastic modes, and there are 8 natural frequencies in the range of the wave frequencies. 336

Table 1: modes Mode Natural frequency in water (rad/s) Natural frequency in air (rad/s) 2 nodes horizontal bending 1.45 1.91 Roll 2.18-3 nodes horizontal bending 2 nodes vertical bending 3.78 4.44 5.14 1.79 [rad/m] 35 30 25 20 15 10 5 Vertical bending, 180 heading Measured Hydro elastic computation Quasi static computation 0 1 2 3 4 5 6 7 8 Frequency [rad/s] 3 nodes vertical bending 5.45 4.45 Fig. 7: Vertical bending RAO, heading 180 Torsion 5.80 5.74 Heave 6.01 - Pitch 6.54-35 30 25 Vertical bending, 150 heading Measured Hydro elastic computation Quasi static computation Convergence tests have been done on the number of modes to be used in the simulation. The conclusion is that as far we want to compute global responses of the ship (global deformation, vertical bending ), 19 modes (6 rigid and 13 elastic) are enough. Of course this is not enough to model very local deformations due to local pressure. The motions of six segments have been measured with an optic system. The relative motion between the aft segment and the fore segment is used as a measure of the deformations of the model. For instance, the vertical bending is defined as the difference of the fore segment pitch angle and the aft segment pitch angle. As a comparison, the motions predicted by a rigid model are presented in the same graphs. In this so-called quasi-static model, we first compute the seakeeping rigid body motions, and then the structural deformations under the loads coming from the seakeeping results. [rad/m] 20 15 10 5 0 1 2 3 4 5 6 7 8 Frequency [rad/s] Fig. 8: Vertical bending RAO, heading 150 The vertical bending deformations computed by the hydroelastic model and the measured deformations are very close (Fig. 7 and Fig. 8). However, because the model is very elastic, the deformations computed by the quasi-static model are largely over-predicted. We can not observe any resonance peak in vertical bending. 337

[rad/m] [rad/m] 12 10 8 6 4 2 Horizontal bending, 150 heading Measured Hydro elastic computation Quasi static computation 0 1 2 3 4 5 6 7 8 Frequency [rad/s] 14 12 10 Fig. 9: Horizontal bending RAO, heading 150 8 6 4 2 Torsion, 150 heading Measured Hydro elastic computation Quasi static computation 0 1 2 3 4 5 6 7 8 Frequency [rad/s] Fig. 10: Torsion RAO, heading 150 The resonance of the horizontal bending mode (Fig. 9) and the torsion mode (Fig. 10), which are strongly coupled, is well predicted by the numerical model around 5.8 rad/s. We can see however a secondary peak in torsion around 3.8 rad/s that is not present in the measurements. This frequency is the natural frequency of the 3 nodes horizontal bending mode which includes a little bit of torsion. A more detailed analysis of the modes contributing to this secondary peaks shows that the two contributing modes are the 3 nodes horizontal bending mode and the torsion mode (due to coupling effects). The over prediction of the amplitude of this secondary peak might be explained by the viscous damping which is not well known. For vertical and horizontal bending, viscous damping doesn t play an important role, because the potential damping is big enough and no strong resonance around the natural frequency is observed. However the torsion mode, as rolling, is a very resonant mode, and viscous damping is an important parameters for the height of the resonance peaks. The comparisons presented here seem to show that the numerical hydroelastic model can predict with a good accuracy the elastic deformations of the model both at the wave frequencies away from the resonant frequencies and around the resonant frequencies. Whipping Tests on a Cruise Ship Some model tests on an elastic segmented model of a cruise ship have been performed to measure the influence of aftbody slamming induced whipping to the vertical bending moment amidship. These tests have been done with two hull forms, which are identical in their main part except around the skeg and the duck tail. Tests have been done without speed in an irregular sea state modeled by a Bretschneider spectrum of 7m significant height. The model tests lasted 83 minutes (real scale) for the hull 1 and 28 minutes for the hull 2. Identical conditions have been simulated with the numerical tools described above. For a better convergence of the statistical values, the numerical simulations have been done for 24h. When comparing the results with the model tests, one could consider that the numerical results are more converged than the test results. Test results, as numerical results, are time history of the vertical bending moment including whipping. Test results are filtered to obtain the bending moment without any whipping contribution, while numerical calculations are redone without any slamming force. The numerical simulations have been done with 13 modes (6 rigid and 7 elastic). A convergence study showed that higher modes do not affect the vertical bending moment amidship. Of course the modal representation is not suitable for the local response to the slamming pressure (local deflection of plates and stiffeners at the bow) because it would need too much modes. In that case we should use the method proposed by Malenica & al. (2008) where the global structural dynamic response of the ship is described by the first few lowest modes, and the response of the other modes is computed in a quasi static way. 338

Bending moment with slamming time (s) without slamming Fig. 11: Time history of bending moment (numerical simulations) Results are analysed through an up-crossing analysis based on the cycles of the filtered signal (without whipping). Results are presented in Fig. 12. Exceedence per hour 1000 100 10 1 0.1 Hull 1 wave Hull 1 total Hull 2 wave Hull 2 total Sagging bending moment model tests model tests model tests model tests Fig. 12: Influence of whipping on the sagging vertical bending moment: comparison between test results and simulations The following remarks can be derived: Results without whipping are in very good accordance and don t show any difference between the hull forms. The little difference between hull 1 and 2 on the model test results is probably explained by the use of two different wave historic during the tests. The numerical simulations, in turn, have used exactly the same waves, in order to minimize the statistical difference due to the finite length of the simulations. Test results are a bit short to derive conclusion on the extreme events. However the concordance with the numerical results is still good. Results with whipping are also in very good accordance. The numerical model, despite its strong assumptions, predicts the correct ranking between hull 1 and hull 2. Results in irregular sea states are usually synthesized in statistical values A 1 / n (mean of the amplitudes exceeded with a risk of 1 / n ). Those values have been computed with and without whipping. Their ratio gives the increase of bending moment due to whipping. The whipping part increases with N, and can reach important values, that are confirmed by the model tests (more than 40% for hull 1). Some other comparisons have been done for a 9m sea state and for hogging bending moment. They show less good agreement than those presented here. More comparisons are still needed in order to validate properly the numerical model presented here. 60% 50% 40% 30% 20% 10% 0% Increase insagging due to whipping Hull 1 Model tests Hull 2 Model tests A 1/3 A 1/10 A 1/100 A 1/1000 Fig. 13: Statistical values for sagging bending: increase due to whipping Conclusions The method for hydroelastic analysis using the 3DFEM 3DBEM coupling procedure is presented. Both frequency and time domain approaches are discussed. The overall problem is extremely difficult and only approximate solution is possible up to now. In spite of numerous technical difficulties the present model seems to produce quite realistic results. The first comparisons with model test are quite good. Some other validations are needed before employing this numerical tool in practice to improve the design procedures. Several research project are ongoing, including some model tests on segmented elastic models (as shown in Fig. 14), measurements at sea, and numerical simulations. These projects will allow more detailed validation of the tools presented in this paper in the two traditional fields of the naval engineer: extreme loading and fatigue damage loading. The very first results from these projects seem to show that the hydroelastic phenomena, which have been neglected 339

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