Slamming clustering on fast ships: From impact dynamics to global response analysis

Ocean Engineering 62 (2013) 110–122

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Slamming clustering on fast ships: From impact dynamics to global response analysis Daniele Dessi n, Elena Ciappi CNR-INSEAN, Via di Vallerano 139, 00128 Rome, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 November 2011 Accepted 23 December 2012 Available online 28 February 2013

In this paper the statistical properties of the slamming impact process are analyzed with the help of experimental data acquired in the towing tank on a high speed ferry model. The physical model is a segmented-hull with a flexible backbone-beam equipped, among other devices, with sensors to measure the wetness of hull sections and strain gauges to estimate the induced vertical bending moment. This setup allows us to analyze the slamming process not only on the basis of the detected slamming events but also on the basis of the whipping response produced by the impacts. Moreover, due to the particular model selected for the present analysis, characterized by a V-shaped hull, bottom as well as bow flare slamming contributions are investigated. One of the major findings is the evidence that the impact statistics are largely affected by the grouping of slams into clusters thus violating the hypothesis of mutual independence between successive impacts that is at the basis of most of the statistical models. The dependence of the whipping response on the impact velocity is also investigated. Finally, the definition of a new criterion for slamming identification based on the evaluation of the whipping bending moment is discussed & 2013 Elsevier Ltd. All rights reserved.

Keywords: Slamming statistics Impact identification Whipping induced bending moment Scaled physical models

1. Introduction A slamming event has generally two distinct effects in the time-domain: an instantaneous increase of the forced loading during the water-entry and water-exit phases, and a transient vibration of the structure, also known as whipping, that superimposes inertial load cycles with decreasing intensity on the wave loading. Model-scale and full-scale measurements have shown that the impulsively induced stresses amidship due to slamming can be of the same order of magnitude as those due to the continuous wave loading (e.g., see Ramos et al. (2000)). For this reason, classification societies have considered the combination of slamming-induced loads with the continuous wave loading as one of the critical aspects in establishing the ship design criteria or rules. The consequent need to assess the ship safety with respect to given environmental and operational conditions has led to the search for the direct statistical correlation between a set of parameters, usually describing the sea state and the ship course and speed, with the relevant response variables, e.g., the maximum stresses or the stress cycle diagram (see Junker, 2009; Gao and Moan, 2008). However, less attention than in the past has been recently paid to explore the physical mechanisms underlying these

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correlation models. One of the most challenging, due to possible hydroelastic coupling and non-linearities demanding for timedomain analysis, is the connection between slamming and whipping, that may still unveil phenomena worth further reviewing of the global sea-state to stress relation. The present paper renews this insight, by investigating the slamming behavior of a fast ferry. The identification of water impacts, as well as the evaluation of their intensity, has constituted the prerequisite for probabilistic theories of the slamming occurrence. Thus Ochi (1964) paved the way to slamming statistics by stating two conditions for slamming to occur that were: (i) relative motion must exceed the sectional draft; (ii) relative velocity at the instant of re-entry must exceed a certain magnitude. These conditions successfully delimited the type of water-entry phenomena for which a probabilistic analysis of the slamming induced response was carried out in several papers, beginning with the probability distributions for the impact pressure and slamming interarrival times that were obtained by fitting experimental data by Ochi himself (1964). Following this novel approach, Ochi and Motter (1971, 1973) and Mansour and Lozow (1982) and, from the experimental side. Wheaton and et al. (1970) pointed out the relevance of the slamming induced stresses alone as well as their critical combination with the steady-state (still-water), low-frequency (wave spectrum excitation) or resonant stresses (springing). Following the assumption that the slamming phenomenon is statistically regarded as a Poisson process, Ochi and Motter (1973)

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estimated the time interval between successive slam impacts and suggested a truncated exponential probability distribution of the impact pressure. They estimated conservatively the slamming induced moment amidship by calculating the dynamic response of the ship-beam that was supposed to be loaded by sectional impact forces generated by extreme slamming pressures. Mansour and Lozow (1982) refined this probabilistic model by assuming for the slamming impacts a Poisson pulse process with independent amplitudes and interarrival times. Ferro and Mansour (1985) focused on the probabilistic combination of slamming and low-frequency stresses for which they proposed a mathematical model, and estimated the extreme value distribution of the combined wave-induced and slamming response with the Turkstra’s (1970) rule. Successively, some of the basic assumptions that had served as simplifying guidelines for all the previous studies were reviewed and even removed in order to achieve satisfactory modeling of the slamming occurrence. The seakeeping tests described later in this paper will follow this direction. Nikolaidis and Kaplan (1992) criticized the assumption that the times of occurrence of slamming impacts are independent, because of the periodic nature of relative motion. They found, as implicitly pointed out by Ochi and Motter, that the times of occurrence of the slamming or of the wave induced stress peaks are highly correlated. Jiao (1996) proceeded further in this way and derived the extreme probability distributions for the combined stresses, accounting for the process non-stationarity and the mutual dependence between the slamming and the waveinduced stresses. A probabilistic model, capable taking into account the presence of clusters of slamming events, was finally proposed by Hansen (1994). Hansen drops the hypothesis of a Poisson pulse-train process, that was considered to be asymptotically correct only as long as the draft approaches infinity. He modeled the slamming phenomenon as a Slepian process, that is a nonGaussian and non-stationary process. This provides a complete description of the original and ergodic Gaussian process after an arbitrary upcrossing into a critical interval where a clusterization of slamming events occur. Though removing the hypothesis of mutual impact independence is widely accepted in developing statistical models of slamming, there is still need for extensive experimental validation of recently introduced theories. Evidence of clustering of slams in full scale data was reported by Fu et al. (2009) which investigated the occurrence of wetdeck slamming in the case of a high-speed catamaran. However, satisfactory availability of fullscale measurements in extreme conditions is often subjected to operational constraints. Recent advancement in measurement techniques allows us to achieve a clearer picture of the whole phenomenon at model scale, beyond a mere counting of the impacts, as will be shown in the present paper. In this perspective, the use of an elastic segmented model for the seakeeping tests provides additional chances for the monitoring of the response in terms of accelerations or bending moments along the ship. In this paper, the seakeeping tests accounting for the vessel slamming behavior at high speeds clearly offered two challenging aspects demanding a deeper insight: the prevalent grouping of slamming events in clusters – not just a remote chance but almost the rule – and the presence of wedge-shaped sections in the impacting bow that makes the distinction between bottom and flare slamming events less evident. Clustering of impacts shows experimentally that the impact velocity loses partially its effectiveness in accounting for the global slamming excitation, like the vertical bending moments or the sectional stresses, though retaining its validity for the local

111

analysis. This lack of correlation suggests a need to review the impact dynamics and to interpret the energy distribution within a slamming cluster. Concerning the distinction between bottom and flare slamming, the relevant whipping recorded after bow-flare slamming events confirms the observation that the absence of a flat bottom for wedge-shaped hulls reduces the difference in the force intensity between violent water-entry phenomena with and without bow emergence. This evidence has motivated the inclusion of these events into the present investigation, due to their relevance especially for the slamming induced global response. The use of an elastically scaled physical model allows us to associate the peak value of the transient bending moment response to the slamming load responsible for this vibration. Indeed, if the relative vertical velocity of the impacting ship section can give an approximate but acceptable estimation of the pressure acting on the wetted hull panels, it is shown to be less correlated to the whipping transient loading. This observation leads to a revision of the procedure for the identification of the slamming events as well as the evaluation of their strength if global effects are taken into account. Thus, a novel procedure focused and based on the global ship response will be proposed and discussed in the paper and then compared with the wellknown Ochi conditions. The comparison will cover both the case of bow-flare slamming and the case of bottom impacts with a vertical entry speed not satisfying the Ochi threshold. The paper is divided into several sections. In Section 2 the experimental set-up is presented. The implementation of the Ochi conditions as a criterion to identify slamming events from the collected experimental data is discussed in Section 3, and a similar criterion to identify also the occurrence of bow-flare impacts is introduced. In Section 4, the application of the Ochi’s criterion for the slamming analysis of the scaled-model seakeeping tests is presented, highlighting the existence of impact clusters; moreover, clustering of slams is shown to be responsible for deviations from statistical predictions obtained with the Poisson model. In Section 5, the transient bending response peak, obtained from strain-gage data, is defined and chosen to be indicative of the global response induced by slamming; therefore, these values are related to the impact velocity to analyze the slamming-whipping correlation. The inclusion of bow-flare impacts in the present analysis is carried out in Section 6. In Section 7 a procedure to identify the slamming impacts on the basis of the intensity of the global response, thus overcoming the distinction between bottom impact and bow-flare slams, is outlined and may constitute a starting point to define new criteria for slamming identification.

2. Experimental set-up The model experiments were carried out at the INSEAN towing tank basin. The basin (220 m long) is equipped with a single-flap wave-maker capable of generating regular and irregular wave patterns. A segmented model of the fast ferry MDV3000 was constructed according to the backbone-modeling technique. The principal model characteristics are listed in Table 1, and in Fig. 1 a schematic view of the experimental set-up is shown. The vertical bending behavior of the ship was reproduced by shaping properly the elastic beam that formed the backbone of the segmented model. The elastic beam was made of an aluminium alloy with 20 elements of constant length and variable transverse section, as represented in Fig. 2. In order to shape the beam sections of the backbone, the bending stiffness and shear area distributions, obtained by collapsing the structural 3D FE model of the full-scale ship into an equivalent Timoshenko beam, were used as reference data. The model scale was 1:30, that

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implies according to Froude scaling a model frequency of the twonode bending mode in air equal to 11.1 Hz. This reference value was closely approximated in dry-vibration tests of the segmented ðmÞ model (f 1,dry C10:8 Hz). This satisfactory result was confirmed by Coppotelli et al. (2008) for the corresponding wet mode estimated with vibration ðmÞ tests using wave excitation (f 1,wet C7:4 Hz with respect a reference frequency of 7.3 Hz). The modal correspondence, upon which the hydroelastic scaling was based, involved a model design with a slightly reduced curve of the sectional inertia (in order to counterbalance the lower value attained by the shear deformability) and an optimized ballast position. The ballasts were placed by an optimizing algorithm in the neighborhood of the rear node of the first (two-nodes) bending mode, in order not to decrease too much its frequency and so to keep it close to its reference (scaled) value. The hull was divided into six segments, each one connected to the elastic beam with a vertical steel leg; therefore, an adequate spatial sampling of the true fluid loading along the hull was achieved. The longitudinal positions of the hull cuts were chosen in order to load the legs in a similar way during the seakeeping tests (see Table 2). These segments were made of fiber-glass, and the gaps between adjacent segments were made water-tight by using rubber straps. The segments are numbered from the bow (No. 1) to the stern (No. 6), as shown in Fig. 1. As a rigid-body, the physical model was free to heave, to pitch and, partially, to surge. In every test the following physical quantities were measured: (i) the absolute wave height, (ii) the draft at two specific sections on the hull, (iii) the rigid-body degrees of freedom (dofs), (iv) the vertical force on each segment and (v) the vertical bending moment on several beam sections. The incoming, absolute wave height was

measured by using two finger probes placed at fixed positions with respect to the towing tank (see Figs. 1 and 3). These positions, one in front of the model (FP0), the other on the model left side (FP2), were chosen so that the incident wave was not yet affected by the presence of the hull. The local draft was directly measured at two hull sections WP2 and WP3 (corresponding to the middle of the segments 2 and 3) by using capacitive wire probes, shown in Fig. 3. The heave, pitch and surge dofs were measured with an optical system, based on cameras placed on the carriage and on four LEDs glued on a plate carried onboard the model. The optical system indicated directly the sensed motion with respect to the model center of gravity G. The legs connecting the segments to the elastic Table 2 Mass and length at water line of the segments. Segment

1

2

3

4

5

6

Mass (kg) Length at W.L. (m)

9.250 0.682

11.050 0.615

12.050 0.779

12.600 0.820

14.400 0.820

15.550 0.567

Table 1 General characteristics of the scaled model. Scale Length between perpendiculars (m) Beam (m) Draft (mean) (m)

l Lpp B

Mass (displacement) (kg) Longitudinal position of center of gravity (m) Vertical position of center of gravity (m) Moment of inertia about center of gravity (kg m2 )

d M xG zG Jyy

1 : 30 4.280 0.6 0.137 144.74 1.621 0.259 201.93

Fig. 3. Finger probes for absolute wave elevation (on the left) and wire-probes for local draft measurement onboard (on the right).

Fig. 1. Schematic representation of the segmented-hull and the wave probes layout.

Fig. 2. View of the backbone beam, including legs (grey boxes numbered from 1 to 6), ballasts (black, thin lines), and strain-gages (small marks numbered from 1 to 12).

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 the relative velocity w_ r at the same section must be less than or equal to a given negative threshold V BS , i.e., _ r 9w ¼ d r V BS , w r

ð1Þ

where the subscript BS indicates that this threshold is relative to bottom slamming. The criterion introduced by Ochi was established considering the measured pressure on the keel plate of the MARINER, that was then correlated with the experimentally estimated impact velocity. The threshold velocity was defined as the value below which no impact pressure was observed and, for a different ship, this value should be modified according to Froude scaling law regardless of the shape of the section close to the bottom, i.e., below 1=10th of the nominal draft. Looking at the original pressure/velocity diagram that inspired Ochi’s statement, it seems that the velocity threshold simply represents the lower limit of validity of the relationship _ 2r C p , p ¼ 1=2rw Fig. 4. Strain-gages placed on the beam top-face.

beam had embedded load cells to measure the vertical force. The use of these load cells needed some care. The first requirement to fulfill was that their deformation under load induces small relative displacements of one segment with respect to the other in order to avoid contact between them. A second requirement was that the fluid moment should not affect significantly the vertical force measurement, at least with medium and long waves (with respect to the segment lengths), and during the slamming tests. Thus, the elastic deformation and measurement performance of the load-cells were initially tested reproducing the above conditions in a controlled environment. The bending moment acting upon the beam was measured in 12 points by using strain gauges glued on the top face of the beam (see Fig. 4). The calibration of the strain gauges was performed loading statically the beam and comparing the theoretical bending moments with the voltage values. The acquisition system based on a National Instruments SCXI module recorded globally 28 signals at a 500 Hz sampling rate whereas the wire probes had to be sampled at 50 Hz due to limitation of their own acquisition cards. The highest sampling rate at which most of the signals were recorded was selected to represent accurately the slam events (see Dessi and Mariani, 2008). The water-entry phase for the scaled fast-ferry model lasts generally about 0.2–0.3 s and therefore about 100 points are used for describing the rising side of a slamming impulse. It implies that in the worst case the error relative to the peak value of the curves is about 1% of the overall variation (peak to peak) of the considered variable. The error is larger in the case of the signals acquired from the wire probes but these data are considered just for comparison. A trigger was used to synchronize both the acquisition systems and the wire signals which were to be numerically re-sampled using splines.

_ r is the relative vertical where Cp is the pressure coefficient, w velocity and r is the fluid density. In fact, experimental points appear to deviate from the regression line only as they approach the area of less intense water impacts. A similar approach for the choice of the threshold value was the one introduced by Conolly (1974) a decade later. He focused on the definition of a threshold as the lower limit of slamming pressure peaks measured along the hull bottom. From this, a limit vertical velocity can be obtained as well. It is worthwhile to recall that Ochi and Motter (1973) shifted their attention also to the definition of a ship forward speed limit based on the consideration of several slamming-induced effects. In particular, they pointed out the link between slamming events and the damage that might occur in the bottom keel plates, aspects that have since been investigated more deeply successively. From the analysis of maximum stresses and/or plate deflection, they developed a rule alternative for the voluntary speed reduction of the captain to select if slamming is to be acceptable for the ship. The Ochi criterion, assumed as the basic reference in this paper, is essentially built on clear but simplified kinematic considerations established on the basis of experimental pressure observations. Despite its usefulness, it rises some questions about its practical application and its physical meaning, that are discussed in the following sections. 3.2. Measurement of water entry kinematics The experimental procedure carried out to evaluate the Ochi physical conditions necessary to discriminate the slamming occurrence is first illustrated. The relative motion wr ðx,tÞ is calculated as follows: wr ðx,tÞ ¼ wG ðtÞ þxyðtÞhðx,tÞ,

3. The kinematic criterion: application to experimental data and related issues 3.1. Applicability of the Ochi–Motter criterion The criterion for the occurrence of ship slamming has been debated for two decades since the publication of the original Ochi paper (1964). Following Ochi, the necessary and sufficient conditions for the occurrence of a slamming event are:

 the relative motion wr at a reference section must be equal to the sectional draft d at instant of slamming ts ;

ð2Þ

ð3Þ

where hðx,tÞ is the absolute wave elevation measured with the wave finger-probe, wG(t) and yðtÞ are the recorded heave and pitch, respectively, and x indicates the abscissa of the considered section (with respect to the model center of gravity G). Despite its simplicity, in some cases this procedure may be faulty, leading to possible inaccuracies in slamming identification that need to be considered. If the wave surface below the impacting bottom is affected by the ship radiated wave field, Eq. (3) is no longer valid due to these unaccounted disturbances and just provides an approximation of the real situation. Furthermore, it may also occur that the finger-probe measurement point is reached by the waves generated by the hull, so that the undisturbed wave elevation can not be measured in that position.

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Therefore, a second measurement technique, based on the direct measurement of the local draft dðx,tÞ, was exploited in this experimental program to check if the calculation of the relative motion yields a good estimation of the impact time and velocity. As anticipated in Section 2, two wire probes were installed on board in the middle of the second and third segments (numbered from the bow), where the hull was likely to experience slamming. Indeed, this experimental technique also demands suitable processing of the recorded signals. Beyond the sensor calibration that allows us to transform the voltage output into the wire wetted length, two other relevant issues need to be considered: the first concerns the orthogonal projection of the wetted length over the vertical plane of the ship to obtain the sectional draft, whereas the second relates to the lack of smoothness that the recorded signals may exhibit. The main findings of this analysis are summarized by the comparison shown in Fig. 5 relative to regular wave tests.. To enhance the comparability, the recorded local draft is shifted downward by an amount equal to the nominal draft, providing the so-called relative draft dr ðx,tÞ ¼ dðx,tÞd, that is plotted together with the (undisturbed) relative wave elevation zðx,tÞ ¼ wr ðx,tÞ (in Fig. 6, the meaning of dr is clarified). As expected, the relative wave elevation (dashed line in Fig. 5) has a rather different behavior with respect to the relative draft (solid line), since the last one accounts also for water uprise generated by slamming events and near-field effects during the exit phase. However, it is interesting to note that the time derivative of the relative wave elevation, at impact time (marked with a vertical line), provides rather the same estimation of the impact vertical velocity. This experimental evidence indicates that before the impact time the

Relative Draft (Wire Probes) Draft Derivative (Wire Probes) Rel. Wave Elevation (Abs. Wave - RB Displ.) Rel. Wave Elev. Derivative (Abs. Wave - RB Displ.) Nominal Water Exit/ Entry Threshold

Draft evaluation [m]

0.6 0.4 0.2

water surface under the impacting bow is well represented by the undisturbed wave elevation, because the high forward speed of the ship makes negligible the disturbances produced by the diffracted and radiated wave fields. The same remark applies also to the finger-probe position. Therefore, for the purpose of the present investigation, the experimentally evaluated relative motion can be assumed to implement the Ochi criterion. 3.3. Extension of the Ochi–Motter criterion to the bow-flare slamming case The hull immersion can be efficiently and accurately determined by the wire probe measurement and provides a general way to determine the wetness of the ship bottom. However, since the ship forward speed is sufficiently high in the present experimental analysis, it is reasonable to set wr ðx,tÞ ¼ zðx,tÞ Cdr ðx,tÞ and the relative motion can satisfactorily describe the waterentry of the bow. The advantage of this choice lies also in the possibility to extend this analysis in time instants after to the impact occurrence. The minimum entry velocity, usually reached around the middle of the entry phase, is relevant in the present analysis because it is assumed in the following to be symptomatic of the intensity of bow-flare slamming events.1 The kinematic condition for bow-flare slamming is then defined as _ r  rV FS , Min½w

ð4Þ

where VFS is a new (positive) threshold greater or at least equal to the Ochi threshold VBS, since for bottom slamming cases _ row _ r 9w ¼ d holds strictly, as the experimental evidence Min½w r shows. Since the minimum velocity is determined in a way consistent with the determination of the impact velocity, an experimental correlation between the impact velocity and the minimum water-entry velocity makes sense, and in particular _ r  C1:5 w _ r9 Min½w , thus suggesting that we should select wr ¼ d V FS ¼ 3=2 V BS . On the other hand, the use of the time derivative _ r ) may lead to results of the local draft in Eq. (4) (instead of w heavily affected by the hull shape due to the flow acceleration as it rises along the section sides. This feature would be not desirable when comparing bottom slamming and flare slamming events.

0

4. Cluster analysis

-0.2 18

18.5

19 Time [s]

19.5

20

Fig. 5. Evaluation of the draft obtained by direct wire-probe measurement and by reconstruction from rigid-body and absolute wave elevation measurement (x position at the middle of segment 2).

It is worth recalling that the generic slamming process Z can be represented, as already proposed by Mansour and Lozow (1982) and Ferro and Mansour (1985), by a pulse process as ðrÞ

Z ðrÞ ðtÞ ¼

N X

wiðrÞ dðttðrÞ Þ, i

ð5Þ

i¼1

where the superscript r denotes a specific towing tank run (in the present case, r ¼ 1, . . . ,11), N ðrÞ is the Poisson counting process (pertaining to run r), dðÞ is the Dirac delta function, wðrÞ and tiðrÞ are i the intensity and the impact time, respectively, associated with the ith slamming event occurred during run r (the subscript s is dropped for sake of conciseness in the impact time t whenever it does not generate ambiguity). To clarify the physical meaning of Eq. (5), in the top of Fig. 7 the process Z ð2Þ is synthetically represented as a sequence of vertical segments indicating the impact velocity, named as W ð2Þ , at the correspondent impact i times tð2Þ . The impact times tð2Þ are obtained from the timei i history of the vertical relative velocity wr(t) (slamming events are

Fig. 6. Physical meaning of variables related to wave elevation.

1 The absence of the bow-emergence prevents to identify a particular time at which the water-entry velocity has to be evaluated.

D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110–122

τ1 τ2

0.2

W5

W3 W4

W2

W6

τ3 τ4 τ5

W7

8

τ6 τ7

0

VSlam

W1

0 4 -0.2 2

-0.4

0

-0.6 10

15

20

25 Time [s]

30

35

40

-2

Impact time [s]

35 30 25 20 15 10 5 0

0

1

2

3

4

5

6 Run

7

8

9

10

11

12

Fig. 9. Impact times for each run recorded during towing-tank tests.

τ4 τ5

τ6 τ7

0.2

0.1 0.05

0.1 0

0

-0.05

-0.1 -0.2

-0.1

15

20

25 30 Time [s]

35

Wave elevation [m/s]

τ3

τ1 τ2

40

Fig. 10. Encountered wave elevation (line and points) and relative motion (solid line).

6 Relative velocity [m/s]

Relative motion [m]

0.4

10

40

Relative wave elevation [m]

marked with circles in the middle curve), whereas the impact velocities can be evaluated from the time-history of the vertical _ r ðtÞ, plotted at the bottom of Fig. 7. With relative velocity w respect to Ferro and Mansour (1985), the slamming intensities wiðrÞ , represented by the slamming impact velocities, will be shown not to be independent and equally distributed random variables. The Ochi kinematic criterion was then applied to the experimental data collected in the 11 towing-tank runs to identify the relevant statistics. The tests were carried out in irregular sea according to a Jonswap spectrum with H1=3 ¼ 5 m and T 1 ¼ 7:5 s at full scale (H1=3 C0:016 m and T 1 C 1:30 s at model scale). The measured sea spectrum is shown in Fig. 8 and compared with the theoretical spectrum of the wavemaker. Each valid acquisition lasted about 30 s, and overall time at model scale of about 360 s is equivalent to nearly 30 min at full-scale. In Fig. 9 the impact time tiðrÞ for each identified slam satisfying the Ochi conditions (see Eq. (1)) is drawn as a point with the abscissa indicating the corresponding towing-tank run. Thus, aligned dots on vertical lines indicate slamming impacts of the same test. The slamming impacts show the tendency to appear grouped into short sequences, named as slam clusters (one of them is circled). This experimental observation contradicts the hypothesis of impact independence upon which several early theories of slamming statistics were built on. The sequence of repeated slamming events seems only partially related to grouping of waves with relevant heights. In Fig. 10 the relative wave elevation (continuous line) is plotted together with

115

Fig. 7. Slamming process Z ð2Þ obtained from the seakeeping tests (towing-tank run r ¼2). Dashed lines indicate the relative motion and velocity thresholds of the Ochi criterion, respectively.

the absolute wave elevation of the encountered waves (line and points) relative to the same ship section. There is not a strict correlation between the elevation of the waves and the relative wave elevation (or relative motion) occurring shortly after the wave crest has passed. Moreover, it may also happen that the second high wave crest following and anticipating a water impact (see grouped slams at t4 t5 and at t6 t7 ) exhibits an elevation lower than the first crest. According to Ochi (1964), the probability of slamming events (satisfying the Ochi conditions) should comply the following exponential law: PrfSlamg ¼ ea eb ,

ð6Þ

2

where a ¼ d =ð2s2wr Þ and b ¼ V 2BS =ð2s2w_r Þ, with s2wr and s2w_ r the variance of the relative motion and relative motion velocity at the considered ship section for all the runs. The number of impacts per unit time, defined as l ¼ N s =T with Ns the overall number of observed slamming events in the time interval T, can be evaluated as sffiffiffiffiffiffiffiffi s2w_ r a b 1 e e , l¼ ð7Þ 2p s2wr

0.0015

psd

0.001

0.0005

0 0

0.5

1 Frequency [Hz]

Fig. 8. Measured and theoretical sea spectrum.

1.5

where l is also termed unconditional mean slamming rate. In Table 3 the predicted and experimental value of the mean slamming rate are reported, showing that Ochi theory significantly underestimates the number of expected slamming events. A measure of the tendency of the slams to appear in clusters can be evaluated with different statistical indices. The first one concerns the estimation of the time interval Dts between

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divided into L consecutive time intervals of length Dt ¼ T r =L. Thus, Dt denotes the constant length of the elementary time window to

Table 3 Predicted and experimental number of slamming impacts per unit time (unconditional mean slamming rate at model scale). Type of estimation

l

Predicted (Eq. (7)) Experimental

0.094 0.300

4

Theoretical (Poisson Law) Mixed (Poisson Law with exp. slamming rate) Experimental

pdf

3

2

1

0

0

Δ τ0

2

4 Time between slams [s]

6

8

Fig. 11. Comparison between theoretical (solid and dashed lines) and experimental (bars) probability density functions relative to the time interval between successive slams (at model scale).

successive impacts, that for a Poisson process is given by the following probability density function (pdf) f ðDts Þ ¼ lelðDts T pitch Þ ,

ð8Þ

where Tpitch is the natural pitching period. The comparison between the theoretical and experimental time intervals between successive slams is shown in Fig. 11. It is worth noting that most of the observed time intervals fall in a narrow band centered on the value denoted as Dt0 , which is closely related to the natural pitching period, i.e., Dt0  T pitch ¼ 1 s at model scale, and also to the frequency of the encountered sea spectrum peak, since Dt0 C 1=f ðpeakÞ ¼ 0:84 s. As also apparent in Fig. 9, the slamming e events inside the cluster sequences are often separated in time by near multiples of Dt0 . This experimental evidence contradicts the expectation of a smooth probability distribution over the values Dt 4 Dt0 , as should hold in the case of independently occurring impacts. Both the solid and the dashed lines represent the values given by Eq. (8), but using different unconditional slamming rates l: for the solid line the value of l is provided by Eq. (7), whereas for the dashed line the experimental value of l reported in Table 3 is used. It is evident that the poor prediction given by Eq. (8) is a consequence of a too low estimation of the unconditional mean slamming rate. A second analysis concerns the expected number of impacts ns in a given time window Dt. Its average value is formally defined as ns ¼ Dt  l. Moreover, the probability that ns ¼ k events will occur in Dt is evaluated for a Poisson process by the following probability function Prfk, Dtg ¼

ðDt lÞk Dtl e : k!

ð9Þ

Its experimental counterpart can be defined and evaluated as explained as follows. The acquisition time for each run, Tr, is first

be used for slamming counting. In each ith time interval, it may happen that one or more slamming events, say k(i) events, are observed. Then, let us denote with m(k) the number of times that k events occur in a single observation window (for instance, mð3Þ ¼ 2 means that twice three distinct slamming events occurred in distinct observation time windows). This allows us to make a histogram approximating, after normalization, the experimental pdf that accounts for the probability that k slamming events are observed in a time window of length Dt. The choice of Dt indeed affects this analysis. The shortest useful observation window for the analysis can be identified with respect to the minimum Dt, say Dtmin , considered in the experimental data of Fig. 11. Since no more than one event can occur in an observation time window such as Dt o Dtmin , it is trivial to set Dt 4 Dtmin . On the other hand, as long as Dt-T r , the overall acquisition time, all the events will be trivially included in this time window. Therefore, for sake of conciseness, we will focus on multiples of a basic interval nearly equal to the pitching period Tpitch, thus leading to Dt C 1:1 s, 2.2 s, 3.3 s, 4.4 s, 5.5 s. This ensures highlighting of the potential impact clusterization. In Fig. 12 the pdf obtained both experimentally (using the histogram method) and theoretically (provided by the Poisson model) are depicted together for the chosen different time windows. For Dt ¼ 1:1 C T pitch , it begins to be evident that the Poisson model underpredicts the possibility that more than one event may occur in the given time window. This lack of agreement is a consequence of the clustering effect not accounted for by the Poisson model. This remark is also confirmed in the comparison for Dt ¼ 2:2 C 2T pitch . As long as Dt is increased, the peak in the observed slamming distribution is moved forward to higher slamming rates and, when Dt is decreased, there is a reduced probability that any events are recorded at all in the same time window.

5. Correlation between impact dynamics and global effects 5.1. Impact severity: local analysis It is well known that there is a relationship between the number of slams per unit time and their strength, roughly obeying to the principle that more slams are observed, more intense they are supposed to be. For instance, Ferro and Mansour (1985) stated that ! V 2BS ðOchiÞ 2 2 mZ ¼ 2lc sw_ r 1 þ 2 , ð10Þ

sw_ r

is the mean value of the slamming excitation, c is the where mðOchiÞ Z constant relating the square of the impact velocity to the pressure.2 If the square of the entry velocity is assumed as the reference slamming excitation, thus setting wi ¼ W 2i in Eq. (5) and c¼1 in Eq. (10), its mean value can be experimentally evaluated as

mðexpÞ ¼ Z

Ns 1 X W2: Ns i ¼ 1 i

ð11Þ

In Table 4 the predicted and experimental mean slamming excitation are compared to each other. It is apparent that, 2

It is worth to underline that l can be evaluated with Eq. (7) as l ¼ pðN s Þ=T.

D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110–122

1 0.8 pdf

Table 4 Predicted and experimental excitation (at model scale).

Experimental Theory (Poisson Distribution)

0.6 0.4

Δt = 1.1s

117

mean

slamming

Type of estimation

mZ

Predicted Experimental

0.440 1.595

0.2 0

0

1

1

5

6 2

Experimental Theory (Poisson Distribution)

0.6 0.4

Δt = 2.2s

0

1

1

5

6

Experimental Theory (Poisson Distribution)

0.8 pdf

2 3 4 Number of slams / Δt

0.6

f ðW 2 Þ ¼ 0

1

1

pdf

2 3 4 Number of slams / Δt

5

6

Experimental Theory (Poisson Distribution)

0.8 0.6 0.4

Δt = 4.4s

0.2 0

1

1

pdf

2 3 4 Number of slams / Δt

5

6

Experimental Theory (Poisson Distribution)

0.8

0.4 0.2 1

2 3 4 Number of slams / Δt

5

ð12Þ

I¼

Ns 1X W 2, Ti¼1 i

ð13Þ

that, recalling T ¼ N s =l, leads to a stronger dependence on the mean slamming rate l

Δt = 5.5s

0

2 2 1 _2 eðV BS w r Þ=sw_r : 2s2w_r

In Fig. 13 the predicted (Eq. (12)) and experimental pdfs are shown. First, we see that the experimental curve shows a peak beyond the impact velocity threshold, whereas the theoretical curve decreases monotonically. Moreover, experiments show that impacts are likely to occur also at high velocities at which Ochi theory predicts less or even negligible slamming events. As a final remark on the discrepancies between predictions and experiments, it is worthwhile to stress that the combination of higher slamming rates with larger impact pressures is responsible for relevant local effects that, unfortunately, are underestimated by Ochi theory and successive improvements. To highlight this aspect, it is necessary to evaluate how the intensity of these slamming events, proportional to W 2i , is distributed in time. For this purpose, a severity index I , defined as the time average of the slamming excitation, is introduced, i.e.,

0.6

0

2.2 2.4 2.6 2.8

Fig. 13. Comparison between theoretical and experimental probability density functions relative to the impact velocity.

Δt = 3.3s

0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Impact velocity [m/s]

This discrepancy is further evident if the pdf of the entry velocity is taken into account. According to Ochi, it is expressed as

0.4

0

1 0.5

0.2 0

Experimental Poisson

1.5 pdf

0.8 pdf

2 3 4 Number of slams / Δt

6

I¼

Ns l X

Ns i ¼ 1

W 2i ¼ lmZ ,

ð14Þ

Fig. 12. Comparison between theoretical and experimental probability density functions relative to the number of slam impacts observed in given time windows (at model scale).

as shown also by the Ferro and Mansour formula ! V 2BS 2 2 I ¼ 2l sw_ r 1 þ 2 :

assuming the entry velocity as an indicator of the extent of the slamming excitation, the experimental value of the excitation level is about four times the predicted one.

The underestimation of l (Table 3) and of the mean slamming excitation (Table 4) strongly affects the final value of the predicted severity index I according to Ochi theory.

sw_ r

ð15Þ

118

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5.2. Impact severity: global effects It has been shown up to this point that the analysis of slamming phenomena characterized by a relevant presence of impact clusters requires a different theoretical framework to determine the impact statistics and, consequently, the evaluation of local effects. This lack of accuracy relative to the models based on Ochi theory is increased if global effects are considered. In the paper of Ferro and Mansour (1985), the presence of a transient response due to slamming, i.e., whipping, is taken into account by decomposing the total sectional load R(t) as the sum of two stationary stochastic processes. Thus, assuming RðtÞ ¼ M y ðtÞ, the vertical bending moment My(t) is expressed as ðHFÞ ðtÞ, M y ðtÞ ¼ M ðLFÞ y ðtÞ þ M y

ð16Þ

M ðLFÞ y ðtÞ

is the sum of the low-frequency contribution (LF), where directly related via the Duhamel’s integral to the local wave elevation h(t), and M ðHFÞ , the high-frequency (HF) contribution, y given as Z 1 ðMÞ M ðHFÞ ðtÞ ¼ h ðttÞZðtÞ dt, ð17Þ y

Fig. 14. Continuous wavelet transform of the vertical bending moment amidship (model scale data).

350 300

1

250

mM ¼ mZ HðMÞ ð0Þ,

ð18Þ

where HðMÞ ðoÞ is the Fourier transform of the impulse response function. According to Eq. (18), the evaluation of the average of the global response mM will significantly suffer of the approximation introduced in the prediction of mZ . However, in the present case, it is worth noting also that within a cluster the transient oscillations might have not been extinguished before the next impact, making the prediction worse. For this reason, the correlation between the impact kinematics and the induced response needs to be investigated in more detail before the averaged quantities are considered. In particular, the first peak of MðHFÞ ðtÞ, occurring immediately after the slamming y event, can be assumed to be representative of the intensity of the slamming induced response and will be correlated to the impact velocity Wi. The peak of the whipping bending moment is formally defined as ^ i ¼ max½M ðHFÞ ðtÞ M y

with tti ¼ min4 0,

ð19Þ

ðtÞ is obtained through the analysis of experimental where M ðHFÞ y data and ti represents the corresponding impact time. In order to extract, from the bending moment time history My(t), its HF part M ðHFÞ ðtÞ, the continuous wavelet transform y (CWT) has been used. The Continuous Wavelet Transform T ðwavÞ of a generic signal s(t) is defined as   Z þ1 1 tx dt, ð20Þ sðtÞc T ðwavÞ ½sðtÞða, xÞ ¼ pffiffiffi a a 1 where the function cðtÞ is termed the mother wavelet. The parameters a and x are denoted as the scaling and the shift parameter, respectively. The scaling parameter plays the role of setting the time scale or the frequency at which the signal is analyzed. A typical choice for the ‘mother wavelet’ is the so-called Morlet Wavelet, whose expression is

cðtÞ ¼ et

2

=2 io0 t

e

,

ð21Þ

with o0 ¼ 2p. The wavelet transform of the overall VBM amidship (sðtÞ ¼ M y ðtÞ in the present case) is shown as a contour plot in Fig. 14, where the different contours are the CWT levels at model scale for the test run r ¼2. From the analysis of Fig. 14, it appears

CWT

ðMÞ

where h ðtÞ is the impulse response function for the chosen HF load. Since Z(t) is not a zero-mean process, M ðHFÞ ðtÞ has a non-zero y mean, given by (cfr. Ferro and Mansour, 1985)

200 150 100 50 0

5

10

15

20 25 Time [s]

30

35

40

Fig. 15. Slice of the continuous wavelet transform at f ¼ 6:8 Hz: envelope of the HF-VBM.

that the whipping CWT peaks lie at a frequency of about 6.6 Hz, slightly smaller than the first vertical bending mode frequency, that was evaluated to be about 7.3 Hz in calm water with noforward speed (Coppotelli et al., 2008). It is worth noting that, since more than 97% of the energy is usually contained in the first bending mode for this ship (cfr. Mariani and Dessi, 2012), it is reasonable to denote the corresponding frequency as the whipping frequency. Therefore, the slice of the CWT at the response peak frequency was assumed to be representative of the highfrequency VBM response amplitude (HF-VBM for short in the following) at midship. This curve is plotted in Fig. 15 for the towing-tank run r¼ 2. ^ i appear clearly distinguishSeveral HF-VBM response peaks M able in Fig. 15 and their correlation with the associated slamming event is shown in Fig. 16, where the HF-VBM (thick solid line) is plotted together with the relative motion (thick dashed line) and the relative velocity at the position WP1 (thin solid line). From the bottom slamming analysis, several events are identified and a delta symbol is chosen to indicate the entry time instant ti . In the same figure, the vertical arrows denote, at these time instants, the entry velocity equal to W i ¼ wr ðti Þ. It is evident that the HF-VBM peaks are likely to occur after the slamming ^ i appear events and, from the analysis of this figure, their values M to be dependent on the Wi. However, there are also some waterexit events (see for instance between 23 s and 25 s) which do not cause any appreciable whipping at all or some slamming events (look around 26.5 s) with subsequent relatively small whipping. Also it may happen that a significant HF-VBM peak is not related to any water-exit event. Focusing on the first impact in the slamming sequence or on isolated impacts, it appears that the

D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110–122

200 100

350 HF-VBM peak

CWT-Rel.mot-Rel.vel.

400

CWT Relative motion Relative velocity Velocity threshold Emersion limit Vmax Identified slamming τi

300

119

0

300 250 200 150 100

-100

50

-200

0

-300

15

20

25

_ r. Fig. 16. Time history of the HF-VBM envelope, wr and w

1

2

3

4

5

W2

30

Time [s]

0

^ i ) with the square of Fig. 18. Correlation between the HF-VBM envelope peaks (M the impact velocity (W 2i ) for all the test-runs.

After-Slam VBM-1st Mode Peak

400 350 300 250 200 150 100 50 0

0

1

2

3

2

W,W

^ i ) and the square of Fig. 19. Correlation between the HF-VBM envelope peaks (M the impact velocity (W 2i ) with impact order information (different filling colors).

^ i ) and the impact Fig. 17. Correlation between the HF-VBM envelope peaks (M velocity (Wi, small circles and W 2i , large circles) for the test run r¼ 2.

analysis of the relative motion at a single section is not entirely representative of the intensity of the slamming excitation force. Hull area affected by the water impact is a relevant parameter but also the phase shift with the loads (including inertia forces) applied along the hull may play a role. To analyze this aspect in more detail, in Fig. 17 slamming impacts are represented as ^ i Þ (small circles) or with copoints with co-ordinates ðW i , M ^ i Þ (large circles). It appears, especially looking ordinates ðW 2i , M at the large circles, that the impact velocity and the HF-VBM peaks are well correlated. However, this conclusion turns out to be only partially true if all the runs are taken into account, as shown by Fig. 18. The dispersion of data shown in Fig. 18 prevents on assessment of a sharp correlation between the impact pressure at the given section (p ¼ rcW 2i ) and the resulting bending moment amidship. This is not a surprising result since for instance in Thomas et al. (2003) a similar observation about the correlation of full-scale data was made. A deeper insight into this correlation is provided by taking into account whether an impact belongs to a cluster and which position it possesses within the cluster sequence (isolated impacts are treated as impacts in the first position). In Fig. 19, the circles denoting the bottom slamming impacts are filled with different grey levels depending on the order of the impact inside the respective cluster (see the legend). Thus, focusing on the points representing impacts successive to the first leading one, it emerges that the amidship HF-VBM seems less intense if the impact is not the leading one in the sequence (represented with white points). For the interpretation of these results, it is useful to recall that the ship structure passing through a slamming cluster can be

Δα

HF-VBM

t

Fig. 20. Response to two consecutive impulsive functions for different phase shifts representing time delays between consecutive slamming events.

approximately described as a 1-d of mechanical system with an excitation given by a train of impulses, i.e., € þ2o0 zyðtÞ _ þ o20 yðtÞ ¼ dðtt 0 Þ þ dðtt 0 DaÞ þ . . . : yðtÞ

ð22Þ

The response oscillation may be enhanced or not depending on the time delay Dt between the Dirac-d functions occurring in the system, as shown in Fig. 20, where two different response curves representing the solution of Eq. (22) are plotted. It is evident (compare the continuous and the dotted curves) that the peaks after the second impulse may be quite different depending on the time intervals between the two excitations. This should explain ^ why there is a large dispersion of the points in the plane ðW 2 , MÞ, showing that the experimental evaluation of the global effects can not be covered with the single-impact analysis.

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6. Flare slamming inclusion Until now, only bottom slamming was considered in the analysis. For instance, as shown in Fig. 18, the correspondent HF-VBM overshoot has been associated with each identified bottom slamming event. In the following discussion, this process is reversed: given the HF-VBM maxima, the corresponding source of excitation is sought. Thus, all the different maxima in the HFVBM time-history are considered and plotted in Fig. 21 as circular grey points. Some of them are then marked with square boxes, indicating that a particular peak was effectively caused by a bottom slamming impact. It is evident that many of the highintensity peaks are associated with bottom slamming events but, on the other hand, some of them miss this correlation. To investigate further the nature of these HF-VBM peaks, bowflare slamming events are taken into account as an additional source of whipping excitation. Thus, in Fig. 22 circles are added to mark those peaks caused by bow-flare slamming. Though improving significantly the correlation between slamming and HF-VBM response, it is evident in Fig. 22 that there are still some peaks that do not seem correlated either to bow-flare slamming events, or to bottom slamming events. We should stress that the chances to mark more HF-VBM peaks as due to both bottom and bow-flare slamming event increase as long as one reduces the respective velocity thresholds, VBS and VFS. As a final step, bottom slamming events not satisfying the entry velocity condition were considered to improve further the correlation, drawing smaller boxes in Fig. 23 over the correspondent HF-VBM peaks. This inclusion

HF-VBM peaks

400

300

200

100

0

0

5

10

15

20 25 Time [s]

30

35

40

Fig. 21. Correlation between HF-VBM envelope peaks and bottom slamming events (squares).

HF-VBM peaks

400

300

200

100

0

0

5

10

15

20 Time [s]

25

30

35

4

Fig. 22. Correlation between HF-VBM envelope peaks and bottom (squares) and bow-flare (circles) slamming events.

Fig. 23. Correlation between HF-VBM envelope peaks and bottom (large squares), bow-flare (circles) and low-velocity slamming events (small squares).

allows to associate a corresponding slamming impact to all the relevant HF-VBM peaks. On the other hand, it appears that there are also some slamming events which seem to produce very small effects on the structure.

7. Slamming detection based on response analysis The results shown in Figs. 21–23 highlight the possibility of identifying the slamming events by observing their effects in terms of the induced response. The purpose is to provide a new criterion for the slamming detection, focused on global effects than on local effects, the latter being covered by the Ochi criterion. Of course, this approach is much more possible if one is interested in identifying every kind of slamming event responsible for impulsive excitation, not only bottom slamming events. Thomas et al. (2003) have worked in recent years about an alternative criterion to identify slamming events occurring to high-speed catamarans, equipped with strain-gages for real-time structural monitoring. In particular, they focused on wet-deck slamming analysis by setting a threshold on the slope that led to peaks in the high-pass filtered time-history of the stress s,   ds Z 0:05  s Y , ð23Þ dt max where ðds=dtÞmax is the maximum rate of change of the stress prior to the peak and s Y is the yield stress. They discussed also other alternatives to Eq. (23). In particular, concerning the possibility of using a criterion based on a maximum stress threshold, they argued that low-rate stress peaks would have been erroneously identified as slams and, on the other hand, small slamming events might not have been identified. They did not try to correlate explicitly the rate-of-stress criterion with the Ochi kinematic criterion, but they emphasized that many wet-deck impacts occurred with a normalized vertical velocity of about 0.15. Similarly, Thomas (2003) proposed a two-stage slam criteria based on combining a rate criterion with a fatigue criterion. The latter criterion will further discussed at the end of this section. In the following the approach to identify slamming events by observing its consequences is labeled as the whipping criterion, since we look for a threshold on the whipping response or its derivative, provided via a slice of the CWT at the operational 1st-mode frequency of the VBM response in Fig. 24. Referring again to the test run No. 2, in Fig. 25 the phase portrait shows the trajectory of the point representing the HF-VBM and its rate of change (more precisely, ! ðHFÞ it has been chosen OP ðtÞ  ðdM y =dt,M ðHFÞ Þ). Whilst M ðHFÞ ðtÞ is a y y ðHFÞ

positive function, the rate-of-change dMy

=dt can assume both

D. Dessi, E. Ciappi / Ocean Engineering 62 (2013) 110–122

121

Rate of change of HF-VBM [N m2 / s]

1400 1200 1000 800 600 400 200 0 0

50

100

150

200

250

300

350

40

HF-VBM [Nm2] Fig. 24. Time-history of the HF-VBM and its time derivative for the test run r ¼2.

Fig. 26. Correlation between after-slam maxima obtained from HF-VBM (whipping moment) and its time derivatives.

350

HF-VBM [N m2]

300

Table 5 Correspondence between Ochi criterion and the whipping criterion for slamming ^ i Z 50 Nm, model scale data). identification (M

250 200 150 100 50 0 -50 -1000

-500

0

500

positive and negative values, and, if one focuses on a single cycle, its positive maxima are larger in modulus than its negative minima. In general, the whipping criterion can be expressed as M ðHFÞ ðtÞ Z m, y ðtÞ

dt

Z Z,

ð24Þ

where m and Z are thresholds to be established. This definition looks like the Ochi criterion, but more complicate expressions can be also conceived. On the other hand, the criterion defined by Eq. (24) can be further simplified, as suggested by examining the phase portrait in Fig. 25, where it appears that a cycle can be approximately obtained by a uniform scaling of the others. This feature is further examined in the diagram shown in Fig. 26, where the points represent in the x-coordinate the positive maxima of the HF-VBM ðHFÞ rate of change, i.e., dM y =dt9max , and in the y-coordinate the ^ i , both positive maxima of the HF-VBM, denoted in Section 5 as M relative to a slamming impact detected with the Ochi criterion. It is interesting to note that the peak values of the HF-VBM and its rate of change are linearly correlated. For instance, this happens if the first HF-VBM oscillation is described sufficiently well with a sine function. Therefore, the two inequalities defined by Eq. (24) are not independent and the criterion can be expressed by just one of the two conditions. In the following, since the threshold m on the VBM peaks can be more physically established, only the first of Eq. (24) will be considered, i.e., ^ i Z m: M

Nmiss

g

BS BS þFS BS þFS þ LV

45 55 77

51 41 19

0.47 0.57 0.80

g¼

Fig. 25. Phase-plot of the HF-VBM for the test run r ¼2.

ðHFÞ

Nvalid

of valid correspondences, Nvalid, with respect to the number of missing correspondences, Nmiss, thus defining an index as

1000

Rate of change of HF-VBM [N m2/s]

dM y

Type of event

ð25Þ

The equivalence of the whipping criterion defined by Eq. (25) with the Ochi criterion can be established by analyzing the number

Nvalid , Nmiss þ Nvalid

ð26Þ

that will be 1 for a full correspondence and 0 if any recorded HF-VBM peak is not due to a slamming event. Of course, the index g is a function of the type of impact that have been considered, i.e., Ochi bottom slamming (BS), bow-flare slamming (FS) and lowvelocity (LV) bottom impact. Nonetheless, this index depends also on the set of the HF-VBM peaks for which a slamming correspondence is sought after, since it may not be reasonable to evaluate the correspondence for HF-VBM peaks below a certain threshold, say ^ min . In Table 5, an example of this correspondence analysis M is given. Since different cases are taken into account (BS, BS þFS, y) the sum Nvalid þ Nmiss , i.e., the total number of possible correspondences, changes from one case to another. Setting a threshold to enhance the correspondence with the Ochi criterion can be useful when the physical variables of the Ochi inequalities in Eq. (1) can not measured or experimentally evaluated, like in full-scale trials. ^ min produces poor figures for Of course, choosing as threshold M the performance of this index. It seems, in this case, much more effective to use a posteriori criterion based on dedicated scaled model tests or on numerical simulations of the elastic ship response in a seaway. Comparing the slamming events identified with the different criteria applied to experiments or simulations, ^ Ochi that meets the two criteria to be equivalent can a threshold M be established. A second way to set this threshold is based on the analysis of the global effects alone and involves the question of possible damage associated with slamming. Thus, a slamming event is counted only if it might be dangerous for the structure. For instance, the criterion can be provided by a minimum number of induced load cycles that is relevant for the fatigue life of the structure. Analyzing stress data collected on-board high-speed catamarans, Thomas et al. (2005) showed that slams do not contribute to the reduction of the fatigue life at lower stress

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levels. On the other hand, the number of load cycles is significantly increased at high stress levels, probably related to large slamming events. The particular stress level, at which the S–N curves with or without the slamming induced cycles separate, provides a criterion for defining a threshold above which the consideration of slamming events make sense. Indeed, avoiding any relationship with the kinematic criterion may be convenient for several reasons: (i) it avoids the cases where poor correlation between whipping VBM and impact velocity exists, as in the present case of relevant slamming clusterization, and (ii) it can provide a uniform criterion for every kind of slamming events (bottom and flare slamming, bow and stern slamming). An acceptable balancing between the two methods to fix the threshold can be conceived as well. For instance, fixing the threshold to 50 Nm allows us to recover most of the peaks as slamming events, with good balancing between the Ochi and whipping criteria and associated relevant load cycles.

8. Conclusions In this paper, the slamming statistics of a fast ferry have been investigated through a dedicated experimental set-up and a novel analysis of the collected data, in order to introduce a different point of view concerning the identification of slamming events. The seakeeping tests of the segmented-hull model at high Froude numbers have revealed the tendency of the bottom slamming impacts to be often grouped into clusters, where the time separation of the events inside a cluster is generally close to the pitching period. The consequent assumption of mutual dependence between the slamming events is confirmed by the lack of accuracy of statistical models based on Ochi theory that assumes event independence. Probability density functions concerning interarrival times, number of impacts for given time intervals and impact velocity show the poor agreement between theory and experiments. In addition to the interpretation of the results, the correctness of the experimental procedure to identify the slamming impacts according to Ochi kinematic conditions was considered. This disagreement indirectly confirms the existence of clusters and, shifting from local to global effects, it becomes even larger. This depends on the inaccurate prediction of the velocity/pressure statistics along the hull. Also, the short interarrival time within a cluster prevents the whipping oscillations to be damped out before the successive impact, showing again the importance of considering impact grouping. To investigate this aspect, the correlation between the square of the impact velocity and the oscillation peak of the whipping bending moment induced by the slam has been considered. The wavelet transform was successfully used to obtain the envelope of the whipping oscillations and, consequently, its maxima. Poor correlation is exhibited by experimental data; nonetheless, if the position of the impact within the cluster is taken into account, better results are obtained. In this case, the first impact appears to be more correlated to the impact velocity and, in general, more intense with respect to this. This may happen because the first impact is not affected by previously induced oscillations. The nature of the V-form of the hull sections has motivated considerations in the analysis of bow-flare slamming events,

because there is not a clear distinction between them as in the case of ships presenting a flat hull bottom. Thus, proceeding in the opposite direction, first the relevant peaks of the whipping bending moment were identified, and then the slamming events explaining these peaks were sought. It has been evident that bowflare slamming was found to be responsible for several intense HF-VBM peaks. Also, consideration of low-velocity bottom impacts, not satisfying the Ochi velocity threshold, was required to bring full correspondence. Therefore, the possibility of establishing new criteria for slamming detection, based on their global effects as suggested by the correlations shown before, is discussed. The different options for the choice of thresholds, being largely dependent on the application, indicates that this requires further studies.

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