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Abstract
In order to realize the multi-point load measurement of ship hull during wave experiments, an FBG pressure sensor was developed to monitor ship bottom liquid level. The principle is to measure the responses of optical fiber sensing units caused by hydraulic pressure. By utilizing a designed steel diamond structure and reasonable selected material, the sensor realized the temperature self-compensation function of a single FBG. The theoretical models and experimental methods are analyzed in detail. Lots of sensing tests showed that, the sensor had a sensitivity of 58.94 pm/kPa and a precision of 1.7 Pa. The temperature sensitivity was only 2.7 pm/°C, which basically achieved the effect of temperature self-compensated. Finally, 15 sensors were installed at the bottom of the hull and a series of wave load experiments were carried out, which further showed that the pressure sensor had good measuring ability and wide application prospect.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid progress of economy, the marine shipping industry is becoming more and more prosperous; however the severe sea conditions cause marine accidents frequently, causing heavy damage to life, property and even the ecological environment. It also makes the hull inspection and specification making departments of various countries put forward higher requirements for the strength research of the hull. When sailing in waves, the hull will be subjected to complex loads, which will lead to local deformation of the hull, and the deformation of the hull also affect the flow field. Therefore, it is of positive significance to explore the load distribution and structural response of the hull in the waves for evaluating the strength of the hull [1,2].
At present, the hull load distribution was mainly predicted by numerical methods such as the frequency domain Rankine panel method (RPM) [3] and the computational fluid dynamics (CFD) method [4], but there were few experimental studies that can better reflect the real situation. Kong Shuai [5] inverted the load of the ship hull by measuring the local structural strain, and Suyuthi [6] measured the total load of the ship structure by using the inertial sensor. The above methods were indirect method and cannot reflect the load distribution of the hull, and cannot retain more thorough understanding of hydrodynamic features. The method based on pressure measurement was the most intuitive method to test the wave load distribution of the hull, such as, Masashi Kashiwagi [7] had carried out relevant research on the load distribution of the rigid hull at different speeds, he had improved many times and finally used up to 333 sensors.
Fiber Bragg grating (FBG) pressure sensor has the advantage of distributed measurement, and is a good carrier for hull load distribution measurement [8–11]. According to the ship test requirements, the model ship is generally the main one, with small ship height and shallow draft, which requires the pressure sensor to have higher sensitivity. However, at present, there is no special FBG pressure sensor for this demand. Liu Xiaobin designed a sensor based on diaphragm and hinge structure, and its pressure sensitivity is 5.227pm/kPa [12]. Arnaldo Leal-Junior embedded the grating in a polyurethane diaphragm, and the pressure resolution reached 1.75 Pa [13], but it was strongly influenced by temperature. Allwood obtained a pressure sensitivity of 116 pm/kPa by sticking an FBG onto a flat rubber diaphragm, but the pressure range was only 15 kPa [14]. Lyu achieved a high sensitivity sensor of 77.3 pm/cm with resolution of 0.129 mm to measure liquid density [15]. These sensors all used the second FBG for temperature compensation. In large-scale measurement, higher requirements are put forward for the multiplexing capacity of FBG demodulator. For example, the literature [7] used 333 FBG pressure sensors, which enabled the demodulator to collect dates of 666 FBGs.
In this paper, combined with the actual needs of ship testing, a high-sensitivity FBG pressure sensor was designed, and the sensor realized the temperature self-compensation function of a single FBG through ingenious structure design and reasonable material selection, which can release twice the demodulation capacity. Finally, the prototype of the pressure sensor was manufactured and the performance test was carried out. Then the sensor was applied to the experimental model ship and a series of wave load experiments were carried out, which further showed that the pressure sensor had good measuring ability and wide application prospect.
2. Experiment system and sensing principle
2.1 Structure of FBG pressure sensor
Figure 1 shows the overall structure of the FBG pressure sensor, which can be used for liquid level monitoring of ship. The main components included An FBG, diamond shaped elastomer, diaphragm, shell, mounting base and cover slab. The wavelength of FBG is pre stretched by about 1 nm, and the two ends of FBG are pasted in the groove of elastomer with 353ND adhesive produced by EPO-TEK, the 353ND adhesive needs to be cured for 5 minutes at 120°C. The diaphragm is fixed on the mounting base by laser welding. It is necessary to ensure that all contact positions are welded to avoid water intrusion into the sensor. One end of the elastomer is connected with the cover slab through two screws, and the other end is welded at the center of the diaphragm. The optical fiber is led out from the outlet hole of the cover slab. Finally, the assembly composed of cover slab, elastomer, diaphragm, mounting base and optical fiber is installed in the shell, and the shell is welded with the cover slab and mounting base respectively.
Fig. 1. FBG pressure sensor. (a) Structure diagram (b) Schematic diagram of pressure and structural deformation.
The sensor was connected to the hull in the form of threads. With the deepening of the water intake of the ship, the water pressure causes the diaphragm to deform, which in turn forces the elastomer to deform and drives the FBG to shift in wavelength. Conversely, when the water intake decreases, the wavelength of FBG tends to return to the original wavelength under the reaction force of the elastomer and the diaphragm. Considering that FBG is sensitive to strain and temperature at the same time, temperature compensation of FBG pressure sensors is necessary. By ingenious structural design and material selection, the single FBG temperature self-compensation of the sensor was realized; the detailed theory and method were introduced in Section 2.2.
According to the design in this paper, the point D of the diamond structure was fixed, and the point C was rigidly connected with the diaphragm. When the diaphragm was subjected to the external pressure p, it will deform and drive the diamond structure to change. After the force was applied, it will move from ABCD to A′B′C′D′. Suppose the change between points AB was $\Delta l_{AB}^{}$ and the change between points CD was $\Delta l_{\textrm{CD}}^{}$, then
The elastic deformation of metal belongs to small deformation. $\Delta l_{AB}^2$ and $\Delta l_{\textrm{CD}}^2$ are high-order infinitesimal quantities, so they can be ignored, then
As the diamond structure is symmetrical, the force analysis of the BD section is as follows:
where, F is the force of the diaphragm on the diamond structure, and θ is the angle of the diamond structure.The BD section can be regarded as a cantilever beam structure,
The normal deflection ν and axial expansion u of the point B can be expressed as
where, $A = bh$ is the cross sectional area, $I = \frac{1}{{12}}b{h^3}$ is the area moment of inertia, b is the width, h is the height.It can be obtained from the above formulas
Neglecting the effect of axial expansion, $\varDelta {l_{AB}}$ and $\varDelta {l_{CD}}$ can be expressed as
In this paper, the thickness-to-diameter ratio $\frac{t}{{2R}}$ of the elastic diaphragm satisfies the thin plate theory ($\frac{\textrm{1}}{{\textrm{100}}}\mathrm{\sim }\frac{\textrm{1}}{{\textrm{80}}}\mathrm{\ < }\frac{t}{{\textrm{2R}}}\mathrm{\ < }\frac{\textrm{1}}{\textrm{8}}\mathrm{\sim }\frac{\textrm{1}}{\textrm{5}}$), and the deformation satisfies the theory of small deflection ($\mathrm{\omega } \le \frac{\textrm{1}}{\textrm{5}}\textrm{t}$), the diaphragm can be regarded as flexible structure with hard core, when the water pressure p and the diamond structure force F act on the diaphragm alone, the deflection at the center of the diaphragm is respectively:
Therefore, the final deformation $\Delta l_{\textrm{CD}}^{}$ of the diaphragm can be regarded as the result of the combined action of pressure p and the reaction force F of the diamond structure:
2.2 Sensing principle
In this paper, the fiber is a standard single-mode fiber with a diameter of 125µm and acrylic coating, Fiber Bragg grating (FBG) is a periodic refractive index structure written in a germanium-doped silica fiber core by an ultraviolet beam, and the length of FBG is about 10 mm, the reflectivity is greater than 90%, the bandwidth is about 0.3 nm. When light from a wide-band source passes through an FBG, the Bragg wavelength satisfying the resonance conditions can be expressed as [16]:
where ${\lambda _\textrm{B}}$ is the Bragg wavelength of the FBG, and ${n_{neff}}$ and $\Lambda $ are the effective refractive index and grating period of the FBG, respectively. The effective refractive index and grating period are the main factors affecting the Bragg wavelength. Due to the influences of external environmental factors such as strain and temperature, these two physical parameters will change, which will lead to changes in the FBG center wavelength. The effects of strain and temperature on the FBG center wavelength shift can be expressed as [17]:The FBG is fixed at the AB of the diamond structure, so the stretched amount of FBG is the same as $\Delta l_{AB}^{}$, and the wavelength shift $\Delta \lambda $ is
According to the formulas (10), (11), (12), (13) and (19), the sensitivity of the pressure sensor is
It can be seen from the Eq. (20) that the pressure p has a linear relationship with the wavelength shift $\Delta \lambda $.
The changes in the CD direction and AB direction of the diamond structure caused by temperature changes are respectively,
where L1 is the effective length of the shell, L2 is the effective length of the diamond structure, α1 is the thermal expansion coefficient of the shell, and α2 is the thermal expansion coefficient of the diamond structure.In order to offset the influence of temperature on the wavelength shift of FBG, there should be
That is, when the size and material of the pressure sensor satisfy the above Eq. (23), the temperature self-compensation can be realized. In this paper, 316 stainless steel is selected as the shell material and 4Cr13 as the elastomer material. The thermal expansion coefficients are 16×10−6/°C and 10.5×10−6/°C respectively. L1 is 30 mm, L2 is 27 mm, lAB is 12.5 mm, lCD is 8.3 mm.
2.3 Sensing system of ship bottom pressure
The research object of this paper is a three-segment hull structure, with a total length of 6.75 m, each cabin length of 2.25 m, a molded width of 1.0 m, and a molded depth of 0.3 m. The middle section of the hull is a large opening section, and the fore and stern section is the loading section. In order to ensure that the area where the collapse occurs is the middle section in the experiment, the fore and stern section should be properly strengthened during the construction of the hull, the plate thickness is 3 mm, the thickness of the middle section is 1.5 mm. The middle section is connected with the fore and stern section by bolts, and rubber seals and waterproof glue are used to ensure good water tightness. A number of openings are left on the deck of the fore and stern section for placing fixed weights or installing sensors. The deck and bottom of the fore and stern section are evenly arranged with three stiffeners, the longitudinal size is 20×3 mm, the transverse frame size is 50×3 mm, and the plate thickness is 3 mm; the middle section has a large opening on the deck, and 3 stiffeners are evenly arranged on the bottom, the longitudinal size is 20×1.5 mm, the transverse frame size is 30×1.5 mm, and the plate thickness is 1.5 mm. The hull material is made of 3105-O aluminum alloy, its yield strength is 40 MPa, elastic modulus is 6.9×104 MPa, and Poisson's ratio is 0.3.
The motion and structural response of the hull in the waves are dynamic problems that change with time. According to the D'Alembert principle, the dynamic problems of all particle systems can be transformed into static problems by applying inertial force at every moment. That is, at any moment when the mass point is under force to move, the main force, binding force and inertial force acting on the mass point are balanced with each other. For a ship sailing in waves, every moment of its entire motion process can be regarded as a balanced state under the action of wave pressure and inertial force. Therefore, if the wave pressure and inertial force at each moment of the hull's motion in the waves can be accurately obtained, the structural response of the hull in the waves can be transformed into a quasi-static problem for solution. In order to realize the real-time collection of the pressure state of the ship hull under the wave, this article uses the FBG pressure sensor introduced above to monitor. The layout of pressure sensors is shown in Fig. 2, 15 sensors are arranged on the axial centerline of the bottom of the ship, 3 sensors are arranged in the bow and stern section, and 9 sensors are arranged in the middle section. It should be noted that all sensors are not arranged at equal intervals, but the closer to the midship, the denser they are.
Fig. 2. Sensor layout on the ship. (a) Experimental test site; (b) Schematic diagram of sensor position
3. Results and discussion
3.1 Properties of FBG pressure sensor
In order to ensure that the FBG pressure sensor can work reliably for a long time in the harsh field environment of the ship, a comprehensive performance test is required. First, all FBG pressure sensors were put into a thermostat for aging experiment in the temperature range of 0°C∼80°C, and keeping every temperature node for one hour with a step of 10°C. The heating and cooling are repeated ten times, to make the adhesive on FBG steadier. Then a series of performance tests were conducted, including pressure calibration test, temperature compensate test.
3.1.1 Pressure calibration
In order to ensure the accuracy, sensitivity and other performance of the sensor to meet the actual engineering requirements, it is necessary to calibrate all FBG pressure sensors with the same structure design before they were put into usage. In this subsection, the pressure calibration experiments of the sensors were finished. As shown in Fig. 3, The FBG pressure sensor and the standard pressure sensor (range 0∼100 kPa, accuracy class 0.05) were installed on the pressure calibration table, the calibration experiments were carried out in a laboratory with constant room temperature. The FBG interrogator was connected with the sensor through optical fiber, and was used to receive and record the reflection wavelength shift of the FBGs undergoing external pressure. In this research, the FBG interrogator was self-developed based on FPGA-IRS demodulation module produced by BaySpec Inc., the maximum sampling frequency was 5 kHz, the repeatability was ±2 pm, and the solution was 0.1 pm. The operation steps of the calibration experiments were as follows: First, the calibration of pressure was performed from 0 to full range (30 kPa) with a step of 5 kPa, maintaining each pressure point for 3∼5 s and then unloading the pressure back to 0 for the same interval. The pressure loading and unloading test process was repeated three times. Experimental data were collected at a frequency of 10 Hz by the FBG interrogator. Figure 4 gives the time-history diagrams of the wavelength shift of the sensor during three pressure cycle test, by which, the dependence relationship of the wavelength shift with the pressure in loading and unloading process could be obtained and shown in Fig. 5.
Fig. 5. Wavelength shift versus pressure during three periods of pressure cycle test. The blue solid lines are from linear fittings.
According to the fitting curves in Fig. 5, it can be known that the FBG pressure sensor has good linearity, its sensitivity Sl was 58.94 pm/kPa. Combined with the long-term test of FBG interrogator by the research group, the interrogator accuracy Ia is about 3 pm, the resolution Ir is 0.1 pm, so that the pressure measurement accuracy of the sensor Da can be obtained ${D_a} = {{{I_a}} / {{S_l}}} = 0.05KPa$, and the resolution Dr can be obtained ${D_r} = {{{I_r}} / {{S_l}}} = 1.7Pa$. Bessel formula was used to calculate the standard deviation of the sensor. First, all subsample standard deviations σUi and σDi in the loading-unloading calibration points were calculated respectively, where i = 1, 2 … N, and N = 7 is the number of calibration points. Then the standard deviation of the sensor was calculated as $\sigma = \sqrt {\frac{1}{{2N}}\left( {\sum\limits_{i = 1}^N {\sigma_{{\textrm{U}_i}}^2\textrm{ + }\sum\limits_{i = 1}^N {\sigma_{{\textrm{D}_i}}^2} } } \right)} = \textrm{12}\textrm{.021}$ pm. Finally, the repeatability error were calculated as eR = ησ / λFS= 2.02% (η = 3, which is coverage factor). In addition, according to the data of the three-cycle tests, the maximum wavelength shift λFS was 1785 pm. The maximum hysteresis ΔHmax was 18.8 pm, which occurred at a pressure of 10 kPa in the third cycle, so the hysteresis error eH in the three-cycle process was calculated as eH = ΔHmax / λFS= 1.05%.
3.1.2 Temperature compensate testing
The wavelength changes of FBGs are affected by both axial strain and temperature. When a FBG sensor was used to measure a pressure, it is necessary to remove the effect of temperature, so it should be discussed about the temperature characteristics of the designed sensor. In order to prove that the pressure sensor designed in this paper was insensitive to temperature, the packaged pressure sensor, diamond structure with FBG and bare FBG were simultaneously placed in the thermostat (resolution: 0.5°C; accuracy: 1°C), and their wavelength shifts were collected in real time at a frequency of 20 Hz by the FBG interrogator. The thermostat used in this test was produced by Chongqing Sida Test Equipment Co., Ltd., the temperature range was 0°C-150 °C, the resolution was 0.5°C, the accuracy was 1°C, and the working room size was 400mm×500mm×600 mm. The time-history curves of the wavelength shifts of FBGs with temperature were shown in Fig. 6. It can be seen that from 10 °C to 80 °C, the temperature sensitivity of the diamond structure with FBG was 27.1 pm/°C and that of the bare FBG was 10.1 pm/°C. However, the temperature sensitivity of the pressure sensor was only 2.7 pm/°C, which basically achieved the effect of temperature self-compensated.
Fig. 6. Wavelength shift of FBG under different carriers. (a) Wavelength shift versus time curves; (b) Wavelength shift versus temperature.
3.2 Pressure testing on ship bottom
In order to lay a good experimental foundation for studying the successive collapse behavior and accurately predicting the ultimate bearing capacity of hull in waves, the experimental process in this paper includes four static load monitoring stages and three dynamic load monitoring stages. The experimental conditions and sequence are shown in Table 1. The static load experiment was to put the counterweight in the bow and stern of the hull respectively, and monitor the pressure state when the counterweight reached the set value. Dynamic load experiment was to set different wave heights and monitor the pressure state at different wave heights.
Table 1. Experimental conditions and sequence
During the experiment, the pressure value was monitored in a continuous data acquisition mode, that is, the demodulation instrument continuously collected the reflected wavelengths of the sensors during the experiment. The conversion relation between water pressure p and wavelength change was as follows:
According to the formula, $\lambda$ is the reflection wavelength collected by the demodulator, ${\lambda _0}$ is the initial wavelength, S is pressure sensitivity of the sensor.
Figure 7 shows the pressure value of No. 2 sensor in the experimental process. According to its distribution characteristics, it can be seen that the pressure value was relatively stable in the static load stages, and its fluctuation was basically the error of the demodulator itself. In the dynamic loading stages, it presents the characteristics of sinusoidal curve with the same period as waves. In 30 min ∼ 40 min and 60 min ∼ 70 min, the irregular fluctuation curve was caused by jitter when placing counterweights. The counterweight is placed manually, which makes the hull shake irregularly in the water.
3.2.1 Static loading testing
The pressure measured values were intercepted and averaged in each static load stage, and the pressure value of 15 sensors for four different static load states were shown in Fig. 8.
Let x and y be the 2D coordinates of any point Q on ship bottom shell along the horizontal and gravitational directions respectively, then the pressure at the point Q(x, y) is:
where p0 is the pressure at water surface, h is the water depth from the free and static water surface, then h-y is the immersion depth of bottom shell point Q(x, y) in water.Since the free and static water surface can be considered as a horizontal plane, the contour curve of immersion depth of ship in water is the deformation curve of ship bottom shell.
As can be seen from Fig. 8, with the increase of ballast mass, the water pressure of the hull was also increased. The pressure at the prow and stern was larger than the amidships, which means the amidships are arched. The pressure at each measuring point was not a smooth transition, especially in the intermediate section. This shows that the increase of ballast mass and the action of wave load can make the hull locally deformed.
3.2.2 Dynamic loading testing
In the dynamic loading experiments, the period of wave loading is 2s, the wavelength equals to ship length 6.75 m. Figure 9 shows the time dependence of pressures measured by Sensor 13# (at prow), 16# (at amidships), and 2# (at stern) under dynamic loading D2, where the insets are the enlargement figures in time interval 60-80 second. It t can be seen that these pressure sensors give better sine curves different amplitudes.
Fig. 9. the time dependence of pressures under dynamic loading D2. (a) Sensor 13# (at prow); (b) Sensor 16# (at amidships); (c) Sensor 2# (at stern)
After FFT, Fig. 10 gives the amplitude frequency with a base frequency of f1 = 0.50 Hz, which is consistent with the frequency of wave loading can be obtained. The amplitude at base frequency f1 is much greater than the amplitudes at the second and third frequencies.
Under the driving of water wave loading, the pressure at the ship bottom can be expressed as:
where η is the deviation of water depth with the action of water wave source.So when the wave trough is amidships, the wave peak is prow and stern, and vice versa. Under the dynamic load condition, Fig. 11 (a) shows the pressure values of each pressure measuring point when the wave peak is amidships, and Fig. 11 (b) shows the situation when the wave trough is amidships. It can be seen from the figure that when the wave peak is amidships, there will be more hogging deformation than under static load; When the wave trough is amidships, the hull deforms is to sagging instead of hogging.
Fig. 11. the pressure values of each pressure measuring point. (a) the wave peak is amidships; (b) the wave trough is amidships
4. Conclusions
In this article, a FBG pressure sensor was developed to monitor ship bottom liquid level. A designed steel diamond structure and reasonable selected material was to realized the temperature self-compensation function using a single FBG. The test data and error analysis show that, the sensor has a sensitivity of 58.94 pm/kPa and a precision of 1.7 Pa. The repeatability error and hysteresis error are 2.02% and 1.05% respectively. The temperature sensitivity was only 2.7 pm/°C. Four static load monitoring tests and three dynamic tests were measured, the test results were in good agreement with the theory. It provides an effective detection and monitoring means for studying the successive collapse behavior and accurately predicting the ultimate bearing capacity of hull in waves.
Funding
National Natural Science Foundation of China (61735013, 61905184); Natural Science Project of Xiaogan (XGKJ2020010062); Project of Department of Education in Hubei Province (B2020152).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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