Characteristics of bow-flare slamming and hydroelastic vibrations of a vessel in severe irregular waves investigated by segmented model experiments

2.1. Test model

A 1:50 scaled geosim model was manufactured by using Fiberglass-Reinforced Plastics (FRP) to allow the measurement of slamming pressure and hull global responses experienced by a large vessel. Main particulars of prototype and the model are summarized in Table 1. More detailed parameters regarding the model can be found in the authors’ previous work [23].

As shown schematically in Fig. 1, the model was configured into seven separate segments, and cuts at stations #2, #4, #6, #8, #10 and #12 were provided. Gaps between each pair of segments were sealed using latex rubber which is elastic and waterproof. The segments were connected by a steel elastic backbone and the hull sectional bending load at each cut station can be measured by the strain gauges placed on the top and bottom surfaces of the backbone. The backbone was designed with varying cross-section so as to match the natural vibrational frequency and longitudinal stiffness distribution of vertical bending responses of the prototype. The backbone was rigidly fixed on the bottom bases of the segments at stations #1, #3, #5, #7, #9, #11 and #13 by using the dedicated fixing plates.

A monoblock segment from station #12 to #20 was adopted to house the propulsion system. The propulsion mechanism includes motors, shafts, cross-connect gearboxes, and propellers. During the tests, the thrust of the model was achieved by the propellers. The model is equipped with a total of four propellers to achieve the maximum testing speed. Twin rudders were also installed on the model to simulate the wake flow distribution near the propellers. Ballast iron blocks were installed inside the model to achieve the target center of gravity (COG) and radii of gyration so as to be similar with ship prototype.

Fig. 1Global view of model arrangement

a) Photograph of the overall model

b) Backbone beam system

2.2. Test equipments

The experiments were conducted in the laboratory of high-speed hydrodynamic basin of Aviation Industry Institute No. 605, which is located in Jingmen, China. The dimensions of the tank are 510 m long, 6.5 m wide and 4 m water deep. The maximum speed allowable of the electric powered towing carriage is 16 m/s, and it has a speed accuracy of 0.001 m/s. During the tests, the speed of the carriage served as a reference, and the speed of the self-propelled model was controlled by an experienced staff via a dedicated motor control module. Both regular and irregular waves can be generated by a computerized single-flap type wave maker located at one end of the tank. A wave absorbing beach is positioned at the opposite side of the wave maker to prevent the reflecting of waves. The sketch of the ship hydrodynamic tank laboratory is shown in Fig. 2.

Fig. 2Sketch of the high-speed hydrodynamic tank laboratory

Motions of the model were measured by an in-house-developed four-degree-of-freedom (4-DOF) seaworthiness instrument, which also attached the model to the carriage. Pitch and heave motions were obtained by the mV/V signals of potentiometers mounted on heave stick and pitch pivot, respectively. The two sticks were installed respectively at stations #8.5 and #13.5 of the model, and they move freely in vertical and longitudinal directions. The heave motion at COG of the model can be derived by interpolation method using the data of the two known positions. Lateral movements of the model were restrained by the heave sticks. Therefore, the model was free to heave, pitch, roll and surge while being constrained in yaw and sway during the tests. View of the towing tank facility and the model setup are respectively shown in Figs. 3, 4.

The model was fully equipped with technique instruments to fulfill the required experimental measurement. A total of 10 pressure sensors were utilized and arranged on the bow area of the model. Nine were installed at the alternately wet and dry area to measure the bow-flare slamming pressure. They were arranged in the centerline plane (P1-P3 at bow front edge) as well as in three transverse planes (P4-P5 at station #0.6, P7-P8 at station #1.5 and P9-P10 at station #2.4). Another sensor was installed at the bow bottom (P6 at station #0.6) to measure the bottom fluctuating/slamming pressure. The locations of the pressure sensors are shown in Fig. 5. The pressure sensor comprises a pressure cell and a transducer amplifier, and is shown in Fig. 6. Technical data regarding the pressure sensors are listed in Table 2.

Fig. 3Towing carriage facility

Fig. 4Photograph of test setup

Fig. 5Arrangement of pressure sensors

Fig. 6Pressure sensor

Sectional vertical bending moment (VBM) at each of the six cut sections was measured by strain gauges placed onto the backbone surface. A full-bridge circuit was adopted to achieve high measurement accuracy and high stability. The measured strain signal was transformed to VBM by considering the Young modulus and geometric cross-section parameters of the backbone beam. Calibration tests were also conducted on the beam prior to the tank tests by applying known static loads and recording the corresponding strain values to validate the calculated conversion coefficients.

Vertical accelerations at two points, the station #1 at the bow and the station #19 at the stern, were measured by axis accelerometers mounted on centerline of the primary deck. The accelerometer has an output coefficient of 2000 mV/g. Technical data regarding the accelerometers utilized are listed in Table 3.

During the tests, sampling frequencies of motions, loads and accelerations were set at 50 Hz, while the sampling frequency of pressures was set at 1000 Hz in order to capture the peak value of slamming. All the acquired signals were transmitted to a data collector in the operational room onboard the carriage via a local signal cable.

2.3. Test scheme

Tests were conducted corresponding to two control strategies, i.e. wave condition and vessel sailing speed. Control parameters of the scaled model were determined according to the similitude law in the knowledge of ship hydrodynamic experiment. In order to study the slamming and global responses of the vessel under different sea states, tests were conducted corresponding to real ship significant wave height from 6 m to 20 m. For each of the full-scale significant wave height, the corresponding wave mean period was determined according to the most probable one from the scatter diagram of the North Atlantic sea states. ISSC double-parameter spectra were adopted to simulate the time series of irregular waves. The testing speed for the majority of cases is 18 knots corresponding to real ship, which is the designed cruising speed. In addition, three speeds, i.e. 18 knots, 24 knots and 30 knots, were investigated for 12 m significant wave height. The test conditions studied in this paper are listed in Table 4.

It is noted that all the cases were tested for head wave conditions. For each of the test condition, one run was conducted in the ultra-long distance tank. With the exception of the acceleration and deceleration regions, the steady-run testing region is over 300 m for each of the test scheme. At least 150 waves were encountered by the model during the steady-run region to ensure the statistical representation of the measured data.

4.1. Identification of the resonance frequencies

Impact hammer tests were conducted when the model was in still water without forward speed to determine at which frequencies the vertical resonances occur. As a matter of fact, it has already been verified that forward speed has very little effects on the frequency of the first-order bending mode [16]. During the mode test, a soft hammer was used to hit the bow center area of the model and recorded the decay oscillation time series using strain gauges placed on the backbone beam. In order to visualize the frequency-domain information in detail, FFT method was used to process the measured time series data. Since some measuring locations may be at or near the vibrational nodes, it is better to refer the frequency-domain outcomes of stress at all the recorded positions along the model so as to determine the resonance frequencies comprehensively [24].

The measured decay curves at different stations and the corresponding frequency-domain results by FFT are shown in Fig. 11, and the results are given in model scale. As seen from the results, there are obvious peaks at the two node vertical vibration frequency 24.16 rad/s (3.87 Hz) and the three node vertical vibration frequency 58.92 rad/s (9.38 Hz).

Fig. 11Hammer test results of vertical vibrations in calm water

a) Decay curves of sectional stress

b) Frequency-domain results by FFT

4.2. Analysis of motion, acceleration and load responses

In order to obtain the frequency-domain information as well as the statistical significant values of motions, accelerations and loads, an in-house code based on auto-correlation function algorithm was adopted. The basic equations regarding the auto-correlation function method are summarized in the following statements [25].

As is well known, the variance spectrum can be obtained by carrying out the Fourier transformation of the auto-correlation function, which is expressed as follows:

where $S\left(\omega \right)$ denotes the frequency spectrum, $\omega$ denotes the frequency, $R\left(\tau \right)$ denotes the auto-correlation function, and $\tau$ denotes the time interval between two given points.

According to the theory of probability, the auto-correlation function of variables is expressed as:

where $E$ denotes the acquiring of mathematical expectation, and $\eta \left(t\right)$ denotes the value in the time series at the time $t$.

The auto-correlation function can be written as follows by assuming that the signal processed is ergodic and stationary random process:

where $T$ denotes the time duration of the measured time series.

The time interval is $\mathrm{\Delta }t=T/n$ after dividing the time duration $T$ into $n$ partitions. The mean representative time of the classified intervals in the order from the beginning to the end over the whole process range are $t_1$, $t_2$,…, $t_n$, respectively. Then, using $\mathrm{\Delta }t$, $2\mathrm{\Delta }t$,…, $k\mathrm{\Delta }t$ to replace $\tau$ and substituting them into Eq. (3), the following equation can be obtained:

where $m$ denotes the largest interval number.

The auto-correlation values of $R_k$ for $k=$0, 1,…, $m$ can be obtained by Eq. (4), and then substituting these values into Eq. (1). Besides, trapezoidal quadrature method is used to obtain the spectrum:

Since the initial spectrum by Eq. (5) is not smooth, the Hamming smoothing is adopted to process the initial spectrum. The equations for smoothing can be derived as:

where $S^{\text{'}}$ denotes the final spectrum after smoothing.

In addition, the significant amplitude value can be obtained based on the acquired spectral result, written as:

where $A_{1/3}$ denotes the significant amplitude value, and $m_0$ denotes the 0th moment of the spectrum, i.e. the area below the spectral curve.

Fig. 12Motion and acceleration spectra for different conditions

a) Pitch

b) Heave at COG

c) Bow acceleration

d) Stern acceleration

4.2.1. Motion and acceleration responses

Fig. 12 shows the obtained frequency spectra of motion and acceleration by the above-mentioned spectral method. The results are given corresponding to model scale. In these figures, spectral curves of all the test schemes for each signal are plotted together for the sake of comparison, and the $x$ axis denotes the encounter frequency (${\omega }_e$). The results indicate that the peak frequency of curves decreases with increasing significant wave height, however, it rises with increasing sailing speed. This can be explained by the fact that the dominant encounter frequency varies with different incident wave states or different sailing speeds. Another feature of the curves is that the peak value increases for both increasing significant wave height and sailing speed. Moreover, it is noted that – unlike the pitch and heave spectra – there exist HF components at around the two node vibrational frequency of the acceleration spectra, which is in accordance with the phenomenon observed from the results in Fig. 9.

The statistical significant amplitude values of motions and accelerations obtained by using of Eq. (7) are plotted in Fig. 13-14, respectively for varying significant wave height cases and sailing speed cases. It can be clearly seen from Fig. 13 that the statistical significant amplitude value is almost proportion to the significant wave height for all of the four signals. On the other hand, as seen from Fig. 14, heave and acceleration values increase linearly with increasing sailing speed, however, the pitch shows an opposite trend. This is attributable to the fact that the encounter frequency departs from the resonant frequency of pitch when speed is increased from 1.309 m/s (18 knots at full-scale) to 2.182 m/s (30 knots at full-scale), therefore, the largest pitch appears at the speed of 1.309 m/s.

Fig. 13Significant amplitude values of motion and acceleration responses against the wave height

a) Pitch and heave at COG

b) Accelerations at bow and stern

Fig. 14Significant amplitude values of motion and acceleration response against the speed

a) Pitch and heave at COG

b) Accelerations at bow and stern

4.2.2. Load responses

Fig. 15 shows the obtained frequency spectra of VBM ranged at stations #2, #4,…, #12. The results are given corresponding to model scale. Some common characteristics regarding the HF components can be found between the load spectra and the acceleration spectra: HF components are mainly congregated at around the two node vibrational frequency. The HF energy in the load spectra is more pronounced than that in the acceleration spectra. In addition, there are more HF components at or near the bow area due to bow-flare slamming, and the curves around the bow area are more irregular than those at stations near the amidships area.

Figs. 16, 17 respectively show the changes of statistical significant amplitude values of VBM obtained by Eq. (7) varying with the significant wave height and the sailing speed. The results in Fig. 16(a) and 17(a) indicate that the significant amplitude values increase with increasing significant wave height and sailing speed. There is a noticeable trend that the values increase faster under higher seas than moderate seas, which can be attributed to the nonlinear wave loads acting on the vessel. On the other hand, as seen from Fig. 16(b), the largest significant amplitude value took place at station #12 for cases of significant wave height less than 320 mm (16 m at full-scale), while it moves towards the bow direction for cases of higher significant wave height, i.e. it occurred at station #10 (amidships section) for cases under significant wave height 360 mm (18 m at full-scale) and 400 mm (20 m at full-scale).

Fig. 15Load spectra at different stations

a) Station #2

b) Station #4

c) Station #6

d) Station #8

e) Station #10

f) Station #12

Fig. 16Significant amplitude values of VBM for different wave states

a) Against the significant wave height

b) Against the station

4.3. Analysis of slamming pressure characteristics

During the tests, the experimental process was also recorded by a dedicated video camera – at left front side of the model – fixed on the carriage. Playback of the video recordings reveals the following phenomenon: when the ship sailed with large amplitude coupled pitch and heave motions, violent bow flare impact would occur due to the relative motion of the model with respect to the incoming waves. During a slamming cycle, once the bow was at the trough of a wave it emerged from the water, and then hit the incoming wave crest at a relative high velocity.

Fig. 17Significant amplitude values of VBM at different speeds

a) Against the speed

b) Against the station

Slamming phenomena under different conditions captured from the video recordings are shown in Fig. 18. To summarize, only slight slamming event happened when the significant wave height was less than 160 mm (8 m at full-scale). Slamming became more pronounced with increasing significant wave height. When the significant wave height was greater than 320 mm (16 m at full-scale), the bow had already pierced the water and thereby resulted in severe slamming and shipping water on deck. In addition, from carefully observation of the video recordings for different sailing speeds, the slamming phenomenon became more severe with increasing speed from 1.309 m/s (18 knots at full-scale) to 2.182 m/s (30 knots at full-scale) for significant wave height of 240 mm (12 m at full-scale), which can be observed from Fig. 18 (d)-(f).

The frequency of slamming event occurrence at P1-P3 was counted from the steady-run region of each time series recorded. Fig. 19 shows an overview of the statistical description of slamming occurrence frequency varying with significant wave height and sailing speed. Fig. 19(a) indicates that the water impact frequency at P3 fluctuated between 40 and 45 over all the cases with the sailing speed of 1.309 m/s (18 knots at full-scale). This can be attributed to the fact that almost every pitch motion would result in water entry of P3 since it is very near to the waterline. The impact frequency shows a linear growth against the increasing significant wave height at P1. The impact frequency at P2 experiences a sharp rise from the significant wave height of 120 mm (6 m at full-scale) to 160 mm (8 m at full-scale), and then it rises gradually, finally with the significant wave height 400 mm (20 m at full-scale), it achieved the same frequency as that of P3. Fig. 19(b) shows that the impact frequencies at P2 and P3 witness a slight and steady growth from the speed of 1.309 m/s (18 knots at full-scale) to 2.182 m/s (30 knots at full-scale). However, the frequency at P1 shows a small fluctuation with the increasing of speed.

4.3.1. Distribution of largest pressure peaks

In order to study the distribution of the pressure peak values, the maximum peak values at sensors P1-P5 and P7-P10 and the largest peak-to-peak value at sensor P6 during the steady-run region of the measured time series were extracted. Fig. 20 shows the largest pressure peaks against the significant wave height and the sailing speed for all the sensors.

As seen from Fig. 20(a), for sensors P3-P10 the largest pressure peak value rises with increasing significant wave height. However, the curves at P1 and P2 revealed a fluctuation of trend with increasing significant wave height. For instance, the slamming pressure peak value at P1 for significant wave heights of 360 mm (18 m at full-scale) and 400 mm (20 m at full-scale) is even smaller than that of 320 mm (16 m at full-scale). This can be attributed to the randomicity and nonlinear effects of severe slamming. In spite of the sharp fluctuations of curves at P1 and P2, the trend is obviously upwards. Fig. 20(b) shows the largest pressure peak value against the increasing sailing speed from 1.309 m/s (18 knots at full-scale) to 2.182 m/s (30 knots at full-scale) for significant wave height 240 mm (12m at full-scale). It is of interest to note that increasing sailing speed has little influence on the peak pressures at P4-P10. Moreover, the peak pressures at P5, P6 and P10 even decrease slightly with increasing sailing speed. The peak pressures at P2 and P3 show a pronounced rise with increasing sailing speed. However, the largest peak pressure at P1 reaches the maximum value at the speed of 1.746 m/s (24 knots at full-scale).

Fig. 18Snapshots captured from the video recordings

a) 120 mm_1.309 m/s

b) 160 mm_1.309 m/s

c) 200 mm_1.309 m/s

d) 240 mm_1.309 m/s

e) 240 mm_1.746 m/s

f) 240 mm_2.182 m/s

g) 280 mm_1.309 m/s

h) 320 mm_1.309 m/s

i) 360 mm_1.309 m/s

j) 400 mm_1.309 m/s

Fig. 19Statistical results of slamming frequency

a) Against the significant wave height

b) Against the sailing speed

Fig. 20Distributions of largest pressure peak values

a) Against the wave height

b) Against the speed

4.3.3. Probability density distribution of peak values

To investigate the characteristics of probability density distribution of pressure peaks, the peak values of the four typical pressure sensors of P1-P3 and P5, which have relatively strongly nonlinear and large magnitude features, were extracted from the steady-run region of time series. For the sake of simplicity, only the most severe condition – significant wave height of 400 mm (20 m at full-scale) and sailing speed of 1.309 m/s (18 knots at full-scale) – is selected for this study.

The peak values at P1-P3 and P5 at different slams are shown in Fig. 22. A total of 143 slamming events have been taken place during the run. It is clear that the association between the peak values of the four sensors is weak. For example, the ratios of the four pressure values vary for different slamming events, i.e. for some of the slams the largest pressure happened at P1, while for others it happened at P2 or P3.

Fig. 21Spatial distribution of notable enormous pressure peaks

a) 120 mm_1.309 m/s

b) 160 mm_1.309 m/s

c) 200 mm_1.309 m/s

d) 240 mm_1.309 m/s

e) 240 mm_1.746 m/s

f) 240 mm_2.182 m/s

g) 280 mm_1.309 m/s

h) 320 mm_1.309 m/s

i) 360 mm_1.309 m/s

j) 400 mm_1.309 m/s

Graphical representations of the statistical peak values with histograms are shown in Fig. 23. Each figure presents the frequency of occurrence of peak values in different intervals counted from the results in Fig. 22. In addition, the corresponding fitted results by the Weibull function are also plotted in the figures. The statistical histograms show acceptable agreement with the fitted Weibull distribution. However, since the available samples are limited, the results show imperfect distribution in the figures.

Fig. 22Pressure peak values at different slams

Fig. 23Histograms of pressure peak value distribution

a) Sensor P1

b) Sensor P2

c) Sensor P3

d) Sensor P5