Research on a fiber Bragg grating hydrostatic level based on elliptical ring for settlement deformation monitoring

Jianyu Yang; Jianyu Yang; Lina Yue; Qing Zhang; Qing Zhang; Nianwu Deng

doi:10.1364/OE.467734

1. Introduction

In order to monitor the settlement and uplift of each point in the engineering structure, the connecting pipe principle can be used. The hydrostatic levels are arranged at the reference point and each measured point. When the measured point variations in height relative to the reference point, the hydrostatic level can sensitively identify its corresponding liquid pressure. The measured system based on hydrostatic level has been widely used in bridge deflection deformation [1], tunnel convergence deformation [2], building uneven settlement [3] and underground synthesis pipe gallery uplift [4] and other engineering fields. Compared to radar-based sensors using radio frequency or ultrasound waves, the application of the connecting pipe principle for measuring structural settlement avoids the influence of airborne dust impurities on the monitoring system [5].

To achieve the measurement of water level changes, according to the principle that the resonant frequency of the tuning fork part of the sensor varies with the depth of immersion, the tuning fork is excited using a piezoelectric actuator with its frequency recorded [6]. Other common sensors to identify the water level height are thermal dispersion-based sensors [7] and capacitance-based sensors [8]. Although the sensitivity and linearity of the above sensors are extremely high, the accumulation of substances on the probe of the sensor during long-term monitoring can affect their measurement accuracy, reducing the reliability of this system. In addition, the traditional technology has the disadvantage of low accuracy in the vibration environment and the inability of multiplexing.

Optical fiber sensing technology has the excellent advantages of small size, wavelength coding, distributed measurement, convenient network, anti-electromagnetic interference and reliability in structural health monitoring [9]. Numerous academics have considered the application of optical fiber sensing principles from Mach-Zehnder Interferometer (MZI), Fabry-Perot Interferometer (FPI), Brillouin Optical Time Domain Reflectometry (BOTDR), weak Fiber Bragg Grating (wFBG), Fiber Bragg Grating (FBG) as well as special Fiber Bragg Grating (sFBG) to measure the water level height for settlement monitoring [6]. To begin with, since the fiber MZI-based water level sensors are immersed in liquid, the refractive index around the core mode and cladding mode of the fiber are perturbed by the medium, which lead to variations in the interference pattern [10]. In addition, the fiber FPI-based structure has an ultra-high sensitivity response to variation in micro-pressure. A femtosecond laser was used by Roldán to inscribe FPI cavities in optical fibers that were mounted on diaphragms measuring water pressure with a resolution of 4 µm [11]. However, the technique of demodulation based on the interference principle was expensive which was hard to be applied. The BOTDR sensing optical cables are woven in the monitoring poles by Wu through special supports, which could convert the deformation information of the structure such as settlement into variations of local strain [12]. The engineering application of BOTDR technology was still facing the issues of how to improve the spatial resolution and realize its temperature self-compensation. Further, wFBG belongs to distributed fiber optical sensing technology, which has the advantages of high capacity, multi-parameter as well as ultra-long-distance transmission [13]. Compared with the BOTDR technology, wFBG can achieve dynamic measurement of strain that currently has excellent application prospects in structural health monitoring. Li proved the effectiveness of wFBG by burying wFBG optical cable during tunnel construction with the deformation measurement technique of laser level as an auxiliary research method [14].

Generally, only strain and temperature can cause the FBG central wavelength shift. By encapsulating the FBG in a special mechanical core, changes in physical parameters such as inclination, vibration, displacement, pressure, and settlement can cause a linear shift in the FBG central wavelength within the range of material linear elasticity and small structural deformation. At present, the common structural style of FBG hydrostatic level is mainly open buoy structure. The basic structural components of the buoy-type FBG hydrostatic level mainly include double FBGs, sensitive elements (such as equal-strength beam [2]), buoy and dowel rod. The buoyancy applied to the buoy immersed in the liquid is transferred from the dowel rod to the FBG attached to the equal-strength beam. When settlement occurs at the monitoring point, the draft depth of the buoy at this point relative to the reference point becomes higher. In turn, the central wavelength of the FBG is allowed to produce a linear shift [1,15]. In order to realize the function of temperature self-compensation, FBGs are usually pasted on the upper and lower sides of the equal-strength beam. The effect to temperature can be canceled by subtracting the wavelength of the double FBGs, as well as the sensitivity can be achieved [16]. Vorathin designed an FBG water level sensor with a sensitivity of 3.3 pm/mm [17]. Its structure consisted of rubber diaphragm with aluminum cantilever beam and so on. And then Vorathin considered the large difference in stiffness between rubber and aluminum. The high elastic modulus of aluminum limited the deformation of the rubber diaphragm. Therefore, PVC polymer is selected as the material of the cantilever beam [18]. As well as the equal-strength cantilever beam replaced the previous rectangular structure. Plastics with low elastic modulus improved the sensitivity of the sensor. In addition, the equal-strength beam structure could better avoid the chirp phenomenon. Its sensitivity could reach 6.39 pm/mm. However, the problem of plastic susceptibility to aging had limited its application in long-term health monitoring. Rodrigues replaced the equal-strength beam with a structure of channel steel with better strain uniformity [19], with a sensitivity of 5.4 pm/mm. It had been successfully applied to the deflection deformation monitoring of the Lezira Bridge. Ameen encapsulated the FBG between a rubber diaphragm and a plastic diaphragm, where the dual diaphragm structure transmitted the variation of uniform water pressure to the FBG [20]. Within the range of 0∼70 cm, its sensitivity was 0.98 pm/mm. Bhowmik investigated the sensitivity of polymer FBGs with different diameters to variations in water level pressure by etching technology [21]. Among them, the sensitivity of etched polymer FBG with a core diameter of 55 µm was 0.75 pm/kPa, which was much greater than the radial pressure sensitivity of an ordinary FBG (-4.33 pm/MPa) [22]. Marques considered the polymer FBG was encapsulated in a rubber diaphragm on this basis [23], whose sensitivity can reach 5.4 pm/mm compared to the literature [20,21]. Notably, sFBG such as tilted FBG or chirped FBG was inscribed on the optical fibers that can distinguish micrometer level variations [24]. Differing from the principle of etched polymer FBG, tilted FBG and chirped FBG measure the variation of the surrounding refractive index to monitor the liquid level height. However, this technique was limited by the grating physical length, resulting in a limited range. In addition, to the above structure of FBG hydrostatic levels, mechanical elastomers such as the Borden tubes [25], the diamond rings [26], and the hinged levers [27] had been proposed as sensitive elements for FBG hydrostatic levels.

In this study, in order to realize structural settlement monitoring, an FBG hydrostatic level based on elliptical ring is proposed. The designed hydrostatic level is essentially an FBG pressure sensor with high sensitivity and low range. The relationship between the size of the elliptical ring and its structural deformation is accurately described using the force method and Mohr integral. Furthermore, by constructing a complete temperature self-compensation model of the elliptical ring and diaphragm, the effect of temperature self-compensation can be achieved by using single FBG. After determining the parameters of the sensor, the performance of sensitivity, resolution and temperature self-compensation for the FBG hydrostatic level are tested to verify the reliability of the structure based on the elliptical ring and diaphragm.

2. Principle and structural analysis of FBG hydrostatic level

2.1 Structural design of FBG hydrostatic level

The core element of the designed hydrostatic level is FBG. When a broadband light source passes through the FBG, the narrowband spectrum consistent with the Bragg central wavelength λ_B can be reflected, as well as the spectrum of the remaining bands transmits. Linear offset of λ_B caused by strain Δɛ and temperature ΔT acting on FBG, and the relationship between the changes can be expressed as:

(1)$$\Delta {\lambda _\textrm{B}}\textrm{/}{\lambda _\textrm{B}}\textrm{ = }(1 - {P_\textrm{e}})\Delta \varepsilon + \textrm{(}{\alpha _\Lambda } + {\alpha _\textrm{n}})\Delta T,$$

where P_e, α_Λ and α_n are the equivalent elastic-optic coefficient, thermal expansion coefficient and thermo-optic coefficient of the optical fiber, respectively. FBG has high sensitivity to strain and temperature. The strain sensitivity coefficient of the FBG with the central wavelength λ_B of 1550 nm is about 1.21 pm/µɛ, as well as the temperature sensitivity coefficient is about 10.6 pm/°C. However, FBG cannot be directly used to measure the settlement deformation of the structure. Therefore, it is necessary to design the corresponding mechanical elastomer. It can be understood that the settlement change of the structure causes the deformation of the mechanical elastomer. Further, this deformation is transmitted to the FBG, causing it to produce a strain change and wavelength change.

The novel FBG hydrostatic level is composed of a section of FBG, deformed diaphragm, dowel rod, elliptical ring, and external shell. Among them, the dowel rod and elliptical ring are made of TA2 titanium alloy material, and the other structural metal materials are 316 stainless steel. The structure of the sensor is shown in Fig. 1. FBG is pasted on the long axis direction of the elliptical ring. Laser welding is used to connect the deformed diaphragm to the external shell and dowel rod. The lower part of the deformed diaphragm must be provided with a vent hole, so that the lower part the deformed diaphragm of the sensor is consistent with the external atmospheric pressure. The function of the drainage valve is to exhaust the air in the shell, allowing the liquid in the connecting pipe system to fill the housing. The force transmission path of the hydrostatic level is that when the shell is filled with liquid, the liquid pressure on the deformed diaphragm is transmitted from the dowel rod to the elliptical ring. The compression of the short axis leads to the stretching of the long axis, resulting in the positive strain, which leads to the shift of the central wavelength λ_B. The settlement variation of each measuring point relative to the reference point can be judged by monitoring the offset of λ_B.

Fig. 1. Structural diagram of FBG hydrostatic level.

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The design of FBG hydrostatic level has the following advantages: the traditional float type FBG hydrostatic level must be connected to the atmosphere, resulting in a large amount of water evaporation in practice, which requires timely replenishment of water into the connecting pipe system. The designed elliptical ring is well isolated from the atmosphere. Extremely low moisture evaporation after closing the drain valve of each sensor during application. Long-term safety monitoring of structures can be achieved. The FBG is constantly in tension when subjected to liquid pressure in the diaphragm. Therefore, the pre-stretched amount is relatively low when making the sensor paste FBG, which reduces the production difficulty. The traditional FBG sensor eliminates the temperature effect by using a two section FBG arrangement, with one section measuring settlement and the other providing temperature compensation. Whereas, through the design of elliptical ring structure, combined with the low coefficient of thermal expansion of TA2 titanium alloy, the function of single FBG temperature self-compensation can be realized, which greatly reduces the manufacturing cost of the FBG hydrostatic level.

2.2 Deformation analysis of elliptical ring

In this paper, the elliptical ring structure is used as a sensitive elastomer for the FBG hydrostatic level. The force method and Mohr integral are applied to calculate the structural deformation of elliptical ring under the action of concentrated force. By analyzing the relationship between the lateral deformation and vertical deformation at each point of the elliptical ring, it provides a basis for constructing an accurate mathematical model of temperature self-compensation, that also provides a basis for the sensitivity calculation of the hydrostatic level.

Taking the elliptical ring as the research object, its force deformation is shown in Fig. 2(a). In the FBG hydrostatic level, the point A of elliptical ring is connected to the bottom of the external shell. Therefore, point A is used as the fixed terminal constraint. When a concentrated force P acts on point C, each point of ABCD moves to AB′C′D′. Suppose the radius of the long axis and the radius of the short axis of the elliptical ring are a and b respectively, the elliptical equation, the length of the microelement ds and the angle β can be expressed as:

(2)$$\left\{ \begin{array}{l} x = a\sin \theta ,\;y = b\cos \theta \\ ds = \sqrt {d{x^2} + d{y^2}} = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } d\theta \\ \sin \beta = \frac{{dy}}{{ds}} = \frac{{ - b\sin \theta }}{{\sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } }}\textrm{, }\cos \beta = \frac{{dx}}{{ds}} = \frac{{a\cos \theta }}{{\sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } }}\textrm{ }, \end{array} \right.$$

where θ denotes the angle between the line connecting the origin and any point of the elliptical ring and the positive y-axis. Since the elliptical ring satisfies the symmetry requirement, the symmetric structure is analyzed by taking its 1/2 model, namely arc AC, under the action of the concentrated force P. The isolated body model is shown in Fig. 2(b). Among them, the shear force Q_C at point C is 0, thus the elliptical ring displacement calculation is simplified to a quadratic super-stationary problem. Its equivalent system is shown in Fig. 2(c). The excess constraint unknown forces are the axial force N_C and bending moment M_C, which can be found by the force method. Since horizontal and rotational displacements at point C of the elliptical ring are 0, the typical equation for the force method can be obtained as:

(3)$$\left\{ \begin{array}{l} {\delta_{11}}{N_C} + {\delta_{12}}{M_C} + {\Delta _{1P}} = 0\\ {\delta_{21}}{N_C} + {\delta_{22}}{M_C} + {\Delta _{2P}} = 0. \end{array} \right.$$

Fig. 2. Diagram of elliptical ring analysis. (a) Deformation of elliptical ring by concentrated force; (b) 1/2 isolated body schematic; (c) quite system of elliptical ring.

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In the isolated body model of Fig. 2(b), the internal force at each point G on the elliptical ring satisfies the following equation.

(4)$$\left\{ \begin{array}{l} {N_G}\cos \beta - {Q_G}\sin \beta = {N_C}\\ {N_G}\sin \beta + {Q_G}\cos \beta ={-} \frac{P}{2}\\ {M_G} = {M_C} - \frac{P}{2} \cdot a\sin \theta - {N_C} \cdot b(1 - \cos \theta ). \end{array} \right.$$

According to Cramer's rule, the expression for the axial force and bending moment at point G can be written as:

(5)$${N_G} = \frac{{\left|{\begin{array}{*{20}{c}} {{N_C}}&{ - \sin \beta }\\ { - P\textrm{/2}}&{\cos \beta } \end{array}} \right|}}{{\left|{\begin{array}{*{20}{c}} {\cos \beta }&{ - \sin \beta }\\ {\sin \beta }&{\cos \beta } \end{array}} \right|}} = {N_C}\cos \beta - \frac{P}{2}\sin \beta ,\textrm{ }{Q_G} = \frac{{\left|{\begin{array}{*{20}{c}} {\cos \beta }&{{N_C}}\\ {\sin \beta }&{ - P\textrm{/2}} \end{array}} \right|}}{{\left|{\begin{array}{*{20}{c}} {\cos \beta }&{ - \sin \beta }\\ {\sin \beta }&{\cos \beta } \end{array}} \right|}} ={-} \frac{P}{2}\cos \beta - {N_C}\sin \beta.$$

As shown in Fig. 3, when the horizontal left unit force 1 acts on point C, the internal force at each point G on the elliptical ring can be expressed as:

(6)$${\bar{N}_{\textrm{1}G}} = \cos \beta ,\textrm{ }{\bar{Q}_{\textrm{1}G}} ={-} \sin \beta ,\textrm{ }{\bar{M}_{\textrm{1}G}} ={-} b(1 - \cos \theta ),$$

when the clockwise unit torque 1 acts on point C, the internal force at each point G on the elliptical ring can be expressed as:

(7)$${\bar{N}_{\textrm{2}G}} = \textrm{0},\textrm{ }{\bar{Q}_{\textrm{2}G}} = \textrm{0},\textrm{ }{\bar{M}_{\textrm{2}G}} = 1.$$

Fig. 3. Force map of isolated body under generalized unit force. (a) Horizontal left unit force 1; (b) clockwise unit torque 1.

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The main coefficients and the secondary coefficients of the force method equation are obtained with the Mohr integral as:

(8)$${\delta _{ij}} = \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{N}}_{iG}}{{\bar{N}}_{jG}}}}{{{E_1}A}}} ds + k\int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{Q}}_{iG}}{{\bar{Q}}_{jG}}}}{{{G_1}A}}} ds + \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{M}}_{iG}}{{\bar{M}}_{jG}}}}{{{E_1}I}}} ds\textrm{ (}i = 1,2;\textrm{ }j = 1,2).$$

The free terms Δ_1P and Δ_2P of the force method equation are:

(9)$${\Delta _{kP}} = \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{N}}_{kG}}{N_G}}}{{{E_1}A}}} ds + k\int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{Q}}_{kG}}{Q_G}}}{{{G_1}A}}} ds + \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{M}}_{kG}}{M_G}}}{{{E_1}I}}} ds\textrm{ (}k = 1,2).$$

In formulas (8) to (9), E₁A, G₁A and E₁I are the tensile stiffness, shear stiffness and bending stiffness of the elliptical ring section, respectively. E₁ and G₁ are the elastic modulus and shear modulus of titanium alloy TA2 respectively. k is the section coefficient related to the shape of the curved beam. For the rectangular cross section, k = 6/5.

Substituting the main coefficients, secondary coefficients and free terms into the force method equation, the internal force at each point G of the elliptic ring is obtained. The solution of internal force on the elliptical ring can be obtained by substituting it into formulas (4). The deformation of the elliptical ring under the action of concentrated force P is actually consistent with the quite systematic deformation in Fig. 2(c). Solving the displacement of statically determinate structures can greatly simplify the calculation work. The vertical displacements of point C and the lateral displacements of point B on the ring can be obtained by Mohr integral. In the quite system, when unit force 1 acting vertically downward at point C, the expression for the internal force at each point G on the elliptical ring are:

(10)$${\bar{N}_{3G}} = \sin \beta ,\textrm{ }{\bar{Q}_{3G}} = \cos \beta ,\textrm{ }{\bar{M}_{3G}} = a\cos \theta ,$$

when unit force 1 acting laterally at point B, the expression for the internal force at each point G on the elliptical ring can be obtained:

(11)$${\bar{N}_{4G}} = \cos \beta ,\textrm{ }{\bar{Q}_{4G}} = \sin \beta ,\textrm{ }{\bar{M}_{4G}} = b\sin \theta.$$

Since point A is a fixed constraint, the compression Δb between points A to C is consistent with the vertical deformation Δl_AC of point C.

(12)$$\Delta {l_{\textrm{AC}}} = \Delta b = \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{N}}_{3G}}{N_G}}}{{{E_1}A}}} ds + k\int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{Q}}_{3G}}{Q_G}}}{{{G_1}A}}} ds + \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{M}}_{3G}}{M_G}}}{{{E_1}I}}} ds = P \cdot J.$$

The transverse tension Δl_BD between points B to D of the elliptic ring satisfies:

(13)$$\Delta {l_{\textrm{BD}}} = 2 \times \Delta a = 2 \times (\int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{N}}_{4G}}{N_G}}}{{{E_1}A}}} ds + k\int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{Q}}_{4G}}{Q_G}}}{{{G_1}A}}} ds + \int_{\mathop {AC}\limits^ \cap } {\frac{{{{\bar{M}}_{4G}}{M_G}}}{{{E_1}I}}} ds) = P \cdot W.$$

The above calculations are based on the assumptions of linear elasticity and small deformation for the structural mechanics analysis. The deformation of each point on the elliptical ring is linearly related to the concentrated force P. Since the result of the integral expression is complex, the symbols J and W are defined in formulas (12) and formulas (13) as the structural flexibility of the vertical and lateral deformations of the elliptical ring.

In order to analyze the relationship between structural deformation and dimensional parameters of elliptical ring, the section shape of elliptical ring must be determined. The section of the elliptical ring of FBG hydrostatic level is chosen to be rectangular in shape, as well as its thickness δ and width h are set to 0.6 mm and 1 mm respectively. Under the action of the concentrated force P, define the ratio of the elongation Δl_BD between points B to D and the contraction Δl_AC between points A to C as η. Using Maple for analysis, it can be seen that the ratio of the short axis radius b and the long axis radius a of the elliptical ring is highly linearly correlated with η in the range of small deformations in line elasticity. Its linear relation can be expressed as:

(14)$$\eta = \textrm{1}\textrm{.0196} \times \frac{b}{a} - 0.0\textrm{723}\textrm{.}$$

2.3 Free thermal deformation of elliptical ring

For the titanium alloy TA2, the expansion or contraction deformation under temperature change can also affect the temperature self-compensation model of the elliptical ring. Therefore, its free deformation during temperature change is analyzed. The free thermal deformation diagram of the elliptical ring is shown in Fig. 4. After free thermal deformation, each point of ABCD moves to A′′B′′C′′D′′. The length change of the free expansion of the elliptical ring points B to D and points A to C can be expressed as:

(15)$$\left\{ \begin{array}{l} \textrm{2}\Delta a^{\prime} = {\alpha_{\textrm{Ti}}}{l_{\textrm{BD}}}\Delta T = 2{\alpha_{\textrm{Ti}}}a\Delta T\\ \textrm{2}\Delta b^{\prime} = {\alpha_{\textrm{Ti}}}{l_{\textrm{AC}}}\Delta T = 2{\alpha_{\textrm{Ti}}}b\Delta T \end{array}. \right.$$

Since the thermal expansion deformation also satisfies the assumption of small deformation, it can be assumed that the eccentricity of the elliptical ring remains the same before and after the free thermal deformation. The following formula can be obtained:

(16)$$e = \frac{{\sqrt {{a^2} - {b^2}} }}{a} = \frac{{\sqrt {{{(a + \Delta a^{\prime})}^2} - {{(b + \Delta b^{\prime})}^2}} }}{{a + \Delta a^{\prime}}}.$$

Fig. 4. Free thermal deformation of elliptical ring.

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Both sides of the equation are squared with Δb² and Δa² omitted for higher order infinitesimal values. The relationship between the elongation Δa′ and Δb′ as well as the major axis radius a and minor axis radius b of the elliptical ring can be obtained:

(17)$$\frac{b}{a} = \frac{{\Delta b^{\prime}}}{{\Delta a^{\prime}}}.$$

2.4 Temperature self-compensation model of FBG hydrostatic level

As shown in Fig. 1, the elliptical ring is encapsulated in the shell of the FBG hydrostatic level, whereby the heat deformation of the elliptical ring needs to be coordinated with the mutual contact of the dowel rod, diaphragm and external shell of the sensor. The material of the elliptical ring and the dowel rod in the hydrostatic level is TA2 titanium alloy, as well as the material of the diaphragm and external shell is 316 stainless steel. The coefficient of thermal expansion of TA2 titanium alloy α_Ti (8.5×10⁻⁶/°C) is smaller than that of 316 stainless steel α_Fe (18.1×10⁻⁶/°C).

The principle of temperature self-compensation of FBG hydrostatic level is as follows:

1. Take the case of elevated temperature as an example. The extension of the elliptical long axis and the temperature variation increase the central wavelength λ_B of FBG. At the same time, the increase in temperature also causes the external shell of the hydrostatic level to expand, resulting in stretching in the direction of the elliptical short axis. According to the analysis of structural deformation of elliptical ring, according to the analysis of structural deformation of elliptical ring, the tension in the short axis direction will cause compression in the long axis direction, which reduces the central wavelength λ_B of FBG. The deformation-coordination equation can cancel each other with the increase of FBG center wavelength caused by the temperature increase with the establishment of temperature self-compensation model.
2. It can be seen from formula (14) that when the value of b/a is increased, the more round the elliptical ring structure is, the larger the ratio $\eta $ is. The further the elongation A between the points B and D of the elliptical ring under the action of the unit concentration force, the greater the elongation A. Since the FBG is pasted between points B and D of the elliptical ring, it can be understood that the higher the sensitivity of the FBG hydrostatic level. Accordingly, its application range will be reduced.
3. The length variation Δb′ of the elliptical short axis in formula (17) is analyzed. Compared with the elliptical ring with larger values of b/a, the former becomes larger in terms of the change for length Δb in the direction of the elliptical short axis per unit temperature change. The designed hydrostatic level can achieve temperature self-compensation with a single FBG. If there is a large length Δb′ in the direction of the elliptical short axis during the free thermal phase, it is not convenient to use the difference in the thermal expansion coefficients of the materials in order to achieve temperature self-compensation. Consequently, this can cause inconvenience to the temperature self-compensating design of the sensor.

In summary, the temperature self-compensation function of the FBG hydrostatic level can be realized by reasonably adjusting the three parameters such as long axis radius a, short axis radius b and dowel rod length c of the elliptical ring.

The schematic diagram of the elliptical ring in the hydrostatic level with the diaphragm and external shell is shown in Fig. 5. Its structure is symmetric about y-axis. The FBG is defined to be pasted at points B and D. The connection points of the dowel rod to the diaphragm and external shell are E and F.

Fig. 5. Structural diagram of elliptical ring in the FBG hydrostatic level.

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Assuming the amount of temperature change as ΔT, the central wavelength change of FBG is mainly composed of the following three parts.

1. Thermal expansion deformation of elliptical ring:
The length change of the elliptical ring subjected to thermal expansion satisfies formula (15).
2. The central wavelength shift of FBG with thermal effects: $(18)$$\Delta {\lambda _\textrm{B}} = \textrm{(}{\alpha _\Lambda } + {\alpha _n})\Delta T \cdot {\lambda _\textrm{B}}.$$$
3. The deformation-coordination of the elliptical ring due to the different thermal expansion coefficients of the materials:
The upper and lower side of the dowel rod at E and F are connected to the diaphragm and the external shell, which is made of 316 stainless steel. When the temperature change is ΔT, the length change between the bottom of the external shell and diaphragm is $\Delta {l^{\prime}_{\textrm{EF}}} = 2{\alpha _{\textrm{Fe}}}(b + c)\Delta T$.Since the thermal expansion coefficient of 316 stainless steel α_Fe is greater than that of TA2 titanium alloy α_Ti, the difference between it and the expanded deformation $\Delta {l_{\textrm{EF}}} = 2{\alpha _{\textrm{Ti}}}(b + c)\Delta T$ is:
$(19)$$\Delta {l^{\prime}_{\textrm{EF}}} - \Delta {l_{\textrm{EF}}} = 2({\alpha _{\textrm{Fe}}} - {\alpha _{\textrm{Ti}}})(b + c)\Delta T.$$$
Formula (19) can be interpreted as applying forcible displacements at the points E and F, resulting in shortening between the points B and D of the elliptical ring. Because of the structural flexibility of the elliptical ring is much greater than that of the dowel rod, the forcible displacement is considered to act entirely on the elliptical ring. Therefore, the displacement change between points B and D is:
$(20)$$\Delta {l^{\prime}_{\textrm{BD}}} = \eta \Delta {l^{\prime}_{\textrm{AC}}} = \eta (\Delta {l^{\prime}_{\textrm{EF}}} - \Delta {l_{\textrm{EF}}}) = 2\eta ({\alpha _{\textrm{Fe}}} - {\alpha _{\textrm{Ti}}})(b + c)\Delta T.$$$

However, the displacement between points B and D calculated by formula (15) and formula (20) are exactly opposite directions. Combining with formula (18), the FBG wavelength shift on the elliptical ring is:

(21)$$\begin{aligned} \Delta {\lambda _\textrm{B}}\textrm{/}{\lambda _\textrm{B}}&\textrm{ = }(1 - {P_e})\frac{{\Delta {{l^{\prime}}_{\textrm{BD}}} - 2\Delta a}}{{2a}} - \textrm{(}{\alpha _\Lambda } + {\alpha _n})\Delta T\\ &\textrm{ = }(1 - {P_e})\frac{{2\eta ({\alpha _{\textrm{Fe}}} - {\alpha _{\textrm{Ti}}})(b + c)\Delta T - 2{\alpha _{\textrm{Ti}}}a\Delta T}}{{2a}} - \textrm{(}{\alpha _\Lambda } + {\alpha _n})\Delta T. \end{aligned}$$

Suppose $\Delta {\lambda _\textrm{B}} = 0$ of the above formula, the temperature self-compensation model of elliptical ring in FBG hydrostatic level is obtained. When the structural dimensions of the elliptical ring satisfy formula (22), a single FBG can realize the function of temperature self-compensation when the external temperature changes.

(22)$$(1 - {P_e})\frac{{2\eta ({\alpha _{\textrm{Fe}}} - {\alpha _{\textrm{Ti}}})(b + c)\Delta T - 2{\alpha _{\textrm{Ti}}}a\Delta T}}{{2a}} - \textrm{(}{\alpha _\Lambda } + {\alpha _n})\Delta T\textrm{ = 0}\textrm{.}$$

2.5 Sensitivity analysis of FBG hydrostatic level

The sensitive elements in the FBG hydrostatic level are mainly diaphragm, dowel rod and elliptical ring. The sensitivity of the FBG hydrostatic level is obtained by studying the deformation-coordination equation between the diaphragm and the elliptical ring in the sensor. The diaphragm transmits the water pressure to the elliptical ring through the dowel rod resulting in tensile strain on the FBG attached to the elliptical long axis.

A circular diaphragm with radius R as shown in Fig. 5. The load and boundary conditions are symmetrical about the y-axis, which belongs to the axisymmetric deformed problem. Therefore, the deflection of the circular diaphragm is only related to the variable R, as well as has nothing to do with θ. The function of uniform water pressure above the diaphragm bearing the FBG hydrostatic level is defined as q(y). q(y) satisfies q(y)=ρ_wgy. The thin circular diaphragm satisfies the Kirchhoff assumption of small deflection bending theory. Acting by the uniform water pressure q(y), the central part of the circular diaphragm (R = 0) is the location where it produces the maximum deflection. The deflection value is [28]:

(23)$${w_{q,\max }}(y) = \frac{{3{\rho _w}gy\textrm{(1} - {\nu ^2}\textrm{)}{R^4}}}{{16{E_2}{t^3}}}.$$

As shown in Fig. 5, the lower part of the diaphragm in the FBG static level has the dowel rod and elliptical ring. When the uniform water pressure acts on the diaphragm, the dowel rod and elliptical ring prevent the diaphragm from deforming. The mechanical model can be regarded as the lower part of the diaphragm is subjected to concentrated force P. The deflection value at the center of the diaphragm (R = 0) is [28]:

(24)$${w_{P,\max }} = \frac{{3P{R^2} \cdot \textrm{(1} - {\nu ^2}\textrm{)}}}{{4\pi {E_2}{t^3}}}.$$

Ignoring the axial deformation of the dowel rod, the deflection value at the center of the circular diaphragm (R = 0) under the action of both the uniform water pressure q(y) and the concentrated force P can be obtained:

(25)$${w_{\max }}(y) = {w_{q,\max }}(y) - {w_{P,\max }} = \frac{{3{\rho _w}gy \cdot (1 - {\nu ^2}){R^4}}}{{16{E_2}{t^3}}} - \frac{{3P{R^2} \cdot (1 - {\nu ^2})}}{{4\pi {E_2}{t^3}}}.$$

According to the analysis of the deformation of the elliptical ring, the expression of the vertical deformation Δl_AC of the elliptical ring under the action of the concentrated force P is shown in formula (12). The deformation-coordination equation of the FBG hydrostatic level is Δl_AC= w_max(y). Furthermore, formulas (12), (13) and (25) are combined. The relationship between q(y) and Δl_BD can be obtained:

(26)$$\Delta {l_{BD}}(y) = \frac{{3{\rho _w}gy \cdot (1 - {\nu ^2}){R^4} \cdot W}}{{16{E_2}{t^3} \cdot [J + \frac{{3{R^2} \cdot (1 - {\nu ^2})}}{{4\pi {E_2}{t^3}}}]}}.$$

Thus, the FBG central wavelength shift Δλ_B can be expressed as:

(27)$$\Delta {\lambda _\textrm{B}} = (1 - {P_e}){\lambda _\textrm{B}}\varepsilon = (1 - {P_e}){\lambda _\textrm{B}}\frac{{\Delta {l_{BD}}(y)}}{{{l_{BD}}}} = (1 - {P_e}){\lambda _\textrm{B}}\frac{{3{\rho _w}gy \cdot (1 - {\nu ^2}){R^4} \cdot W}}{{16{E_2}{t^3} \cdot [J + \frac{{3{R^2} \cdot (1 - {\nu ^2})}}{{4\pi {E_2}{t^3}}}] \cdot 2a}}.$$

The sensitivity S of the FBG hydrostatic level is the ratio of the FBG central wavelength shift Δλ_B to the water level height y.

(28)$$S = \frac{{\Delta {\lambda _\textrm{B}}}}{y} = (1 - {P_e}){\lambda _\textrm{B}}\frac{{3{\rho _\nu }g \cdot (1 - {\nu ^2}){R^4} \cdot W}}{{16{E_2}{t^3} \cdot [J + \frac{{3{R^2} \cdot (1 - {\nu ^2})}}{{4\pi {E_2}{t^3}}}] \cdot 2a}}.$$

The higher the sensitivity of the FBG hydrostatic level, the greater the resolution range of settlement. However, the range of its application is reduced. The expected sensitivity of the designed FBG hydrostatic level is 12 pm/mm. Analysis of formula (28) shows that the main factors affecting the sensitivity of the FBG hydrostatic level include the thickness δ, width h, long axis radius a and short axis radius b of the elliptical ring as well as the radius R and thickness t of the diaphragm. Conditions such as ease of fabrication, mechanical processing, and sensitivity of the sensor are considered comprehensively. In this paper, the thickness δ and width h of the elliptical ring are set to be 0.6 mm and 1 mm, respectively. According to the analysis of structural deformation of elliptical ring, increasing the long axis radius a and short axis radius b of elliptical ring, with the increase of its structural flexibility J and W in response. However, when the radius of the long axis is larger, the length of the overhanging FBG on the elliptical ring is larger. Not only the sensitivity of the sensor decreases, but also its volume size increases. The research of Gu [29] and Rajan [30] showed that when the stiffness difference of the combined structure is excessively large, the sensitivity calculation of the sensor is not fully deformation coordinated according to formula (25). Similarly, the size of the elliptical ring is excessively small and its nearby guide slot and dowel rod cause a significant impact on the temperature self-compensation performance of the sensor. In summary, the long axis radius a and short axis radius b of the elliptical ring are set to be 10 mm and 8 mm, respectively. Substituting into the elliptical ring temperature self-compensation model of formula (22), it can be determined that the length c of the dowel rods is 17 mm.

The diaphragm plays an important role in transmitting the uniform water pressure in the FBG hydrostatic level. Figure 6 discusses the effect that the radius R and thickness t of the diaphragm have on the sensitivity S of the sensor. Each variable satisfies $R \in [15\textrm{ mm},\textrm{ }40\textrm{ mm}]$ and $t \in [0.01\textrm{ mm},\textrm{ }1\textrm{ mm}]$. The dimensional parameters that satisfy the sensitivity from 0 to 51 pm/mm are presented in the form of cloud maps. The grid lines in Fig. 6 mean the influence of a variable on sensitivity when another variable is controlled. It can be seen that the sensitivity S of the sensor increases with the increase of the diaphragm radius R and the decrease of the thickness t. And the smaller the diaphragm thickness, the greater the slope value of sensitivity and radius. However, if the diaphragm radius R is excessively increased and thickness t is reduced for increasing the sensitivity S of the sensor, the stiffness ratio of the diaphragm in the entire sensor is low. The stiffness of diaphragm is much lower than the stiffness of elliptical ring, which may lead to the design of the FBG hydrostatic level not achieving the effect of temperature self-compensation.

Fig. 6. Sensor sensitivity and the variation of dimensions in diaphragm.

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According to Fig. 6, an S-shaped function relationship between diaphragm thickness t and sensor sensitivity S can be found. When the thickness t is larger than 0.6 mm, the change in thickness does not have a clear effect on the sensitivity. However, when the thickness t is in the range of 0.2∼0.6 mm, its sensitivity increases rapidly. After the thickness t is less than 0.2 mm, the growth of sensitivity S enters a stagnant phase. the thickness t is set to be 0.2 mm at the extreme point of the growth rate. Substituting 0.2 mm and the expected sensitivity of the sensor (12 pm/mm) into formula (28) determines the radius R to be 30 mm. After the above analysis, the relevant parameters of the FBG hydrostatic level are shown in Table 1.

Table 1. Relevant parameters of FBG hydrostatic level

View Table

3. Testing and discussion of FBG hydrostatic level

3.1 Sensitivity test

The brand of optical fiber being used in the FBG hydrostatic level was the SMF-28-ULTRA model which was manufactured by Corning Incorporated. The acrylate coating of the optical fiber was stripped off. Furthermore, the optical fiber was placed in a hydrogen environment to increase its photosensitivity. The incident light of the ultraviolet spectrum was divided into two lights by the phase mask with the period of 1065 nm. The interference pattern from the two lights inscribed the periodic distribution of the refractive index of the germanium-doped fiber core to satisfy the first-order Bragg condition. The initial central wavelength of the obtained FBGs were about 1558.0 nm, where the grating length were 10 mm and their reflectivity were increased to 90%. In addition, to prevent the FBGs from chirping, they were pasted on the long axis direction of the elliptical rings in the hydrostatic levels. The position of the specific pasting is shown in Fig. 5. Noteworthy, the FBG interrogator being used in this thesis was the model GC-97001C, which was developed by Wuhan University of Technology. The wavelength resolution of the FBG interrogator was 0.1 pm, with the accuracy of 0.3 pm. Its sampling frequency could reach up to 2000 Hz.

The experimental testing system was shown in Fig. 7. The four packaged FBG hydrostatic levels and the bucket were installed on the rack. The displacement variation of the rack caused by the loading process was ignored. The FC/APC plugs of the hydrostatic levels were connected to the GC-97001C interrogator to read the FBG central wavelength. The numbers of the four FBG hydrostatic levels were defined by the distance from the rack as HL_A, HL_B, HL_C and HL_D. As shown in Fig. 1 and Fig. 7(a), the hydrostatic level had two outlet valves and a drainage valve. In this test, the water outlet hole on one side was sealed with a water plug valve. Further, ϕ 8 mm silicone hoses were connected to the other side of outlet valves, bucket and tee valve, thus forming a complete connector system. Before the loading test, the central wavelength of the four FBG hydrostatic levels were 1559.3341 nm, 1559.1493 nm, 1559.6228 nm and 1559.3961 nm in sequence.

Fig. 7. Experimental testing system. (a) FBG hydrostatic level; (b) connecting pipe system.

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The sensitivity test of the FBG hydrostatic level was carried out at ambient temperature. The sampling frequency of the FBG interrogator was set to 5 Hz. The loading method used a measuring cylinder or syringe to fill the bucket, thus loading the water level pressure. The cross-sectional area of the bucket was 8107.3 mm². If all the hydrostatic levels were loaded with a water level difference of 10 mm, 81.1 mL of water needed to be injected into the bucket. At this time, the FBG center wavelength shift of each FBG hydrostatic level should be increased by 120 pm according to the theoretical analysis of formula (28). The loading range of the sensor was set from 0 to 100 mm in steps of 10 mm water level height. The water level pressure loading and unloading test was performed three times cyclically, recording the wavelength shift of the four FBGs in real time.

The curves of FBG central wavelength for loading and unloading of the four FBG hydrostatic levels were shown in Fig. 8. The FBG wavelength shift showed a step-shaped rise and fall, while the step-shapes were similar for the three cycles. It could also be seen that the response time was faster. After the water was injected, the FBG wavelength can be stabilized within three seconds. Compared to the loading curve, the jump in the unloading curve was larger since it was more difficult to pump water from the bucket using a syringe. The fiber grating can sensitively capture the vibration signal during manual operation. Therefore, during signal post-processing, smooth bands should be selected for analysis.

Fig. 8. Real-time curve of loading and unloading FBG central wavelengths. (a) HL_A; (b) HL_B; (c) HL_C; (d) HL_D.

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The FBG wavelength shifts corresponding to each water level height in the three repeated experiments (6 groups) were averaged as well as fitted by ordinary least squares. The variation of the central wavelength shifts of the FBG hydrostatic levels with the water level height (or volume of loaded water) was represented in Fig. 9. The results showed that the sensitivities of the four FBG hydrostatic levels from HL_A to HL_D were 12.01 pm/mm, 10.16 pm/mm, 11.73 pm/mm and 12.58 pm/mm, respectively. The linearity of all the sensors in the range of 0 to 100 mm was above 0.9996.

Fig. 9. Sensitivity test of FBG hydrostatic levels.

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Repeatability error δ₁ characterized the deviation of the output characteristic curve of the same sensor under several repeated measurements. In formula (29), ${\bar{\lambda }_{FS}}$ was the average value of FBG wavelength shift at its full scale in 3 cyclical experiments. Δ_max is the maximum deviation between the forward stroke value and its average value or the reverse stroke value and its average value. According to the data, the repeatability errors of four FBG hydrostatic levels from HL_A to HL_D can be calculated to be 1.44%, 1.30%, 1.26% and 3.41%, respectively.

(29)$${\delta _1} = \frac{{{\Delta _{\max }}}}{{{{\bar{\lambda }}_{FS}}}} \times 100\%.$$

Hysteresis error δ₂ refers to the phenomenon that the forward stroke curve did not coincide with the reverse stroke curve of the sensors. It was mainly related to the secure assembly of the components as well as the material properties of the mechanical elastomer. The area of the hysteresis curve loop in the forward and reverse stroke of a well performing sensor should be as small as possible. In formula (30), λ_pi and λ_ti are the test values of the forward stroke and reverse stroke, respectively. According to the data, the hysteresis errors of four FBG hydrostatic levels from HL_A to HL_D can be calculated to be 1.67%, 1.29%, 1.61% and 1.30%, respectively.

(30)$${\delta _2} = \frac{{{{|{{\lambda_{pi}} - {\lambda_{ti}}} |}_{\max }}}}{{{{\bar{\lambda }}_{FS}}}} \times 100\%.$$

3.2 Resolution test

The resolution of a sensor referred to its ability to perceive the smallest variation in physical quantity. It determined the effective number of measured data, which was also used as the main performance parameter of the sensor. In general, the sensitivity of FBG sensors were inversely proportional to their resolution. However, the hydrostatic level was based on the connecting pipe principle, which used the variation of water level pressure to monitor settlement. There were many factors that affect its resolution. The measured data of FBG hydrostatic level was analyzed by statistical methods such as dichotomy, analysis of variance (ANOVA) and analysis of correlation to obtain its effective resolution range.

The resolution test of the FBG hydrostatic level was carried out at ambient temperature. A volume of water was fed into the bucket several times in succession using a measuring cylinder or syringe (8 mL was measured, corresponding to a water level of approximately 1 mm). The FBG central wavelengths of the four hydrostatic levels were recorded. Then this data was averaged to obtain the wavelengths shift compared to the initial state. Furthermore, double factor ANOVA without repetition was used to test whether the variation of water volume as well as the different of sensors had a significant effect on its wavelength shift. The significance level α was taken as 0.01. F_α=3.36. In addition, the linear correlation threshold was set as 0.9. If the variation of water volume had a significant effect, and its linear correlation was greater than the threshold, it was considered that the FBG hydrostatic level can distinguish a small variation of 1 mm in settlement. Then the volume of water was adjusted to half, 4 mL. The above process was repeated to determine if the sensor was still able to discriminate variation. If the variation of water volume had no significant effect, or its linear correlation was less than the threshold, the volume of water needed to be adjusted to multiple, 16 mL. The above process was repeated, and so on. The flow chart was shown in Fig. 10.

(a) The four FBG hydrostatic levels in Fig. 7(b) were loaded with 2 mL water for 8 times in succession, corresponding to the water level of approximately 0.25 mm, with a total of 16 mL water. As shown in Fig. 11(a). The linear correlations of the sensors were all greater than 0.9. F_c= 257.37 > F_α. It was considered that the sensors can distinguish a small variation of 0.25 mm in settlement.
(b) The four FBG hydrostatic levels in Fig. 7(b) were loaded with 0.8 mL water for 8 times in succession, corresponding to the water level of approximately 0.1 mm, with a total of 6.4 mL water. As shown in Fig. 11(b). The linear correlations of the sensors were all greater than 0.9. F_c= 139.66 > F_α. It was considered that the sensors can distinguish a small variation of 0.1 mm in settlement.
(c) The four FBG hydrostatic levels in Fig. 7(b) were loaded with 0.4 mL water for 8 times in succession, corresponding to the water level of approximately 0.05 mm, with a total of 3.2 mL water. As shown in Fig. 11(c). The linear correlations of the sensors were all greater than 0.9. F_c= 73.67 > F_α. It was considered that the sensors can distinguish a small variation of 0.05 mm in settlement.
(d) The four FBG hydrostatic levels in Fig. 7(b) were loaded with 0.2 mL water for 8 times in succession, corresponding to the water level of approximately 0.025 mm, with a total of 1.6 mL water. As shown in Fig. 11(d). The linear correlation of the sensors was particularly low. It was considered that the sensors cannot distinguish a small variation of 0.025 mm in settlement.

Fig. 10. Flow chart of effective resolution of FBG hydrostatic level.

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Fig. 11. Experimental results of effective resolution of the four FBG hydrostatic levels. (a) 2 mL/time of water; (b) 0.8 mL/time of water; (c) 0.4 mL/time of water; (d) 0.2 mL/time of water.

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According to the analysis results, the effective resolution range of the FBG hydrostatic levels based on the connecting pipe system was 0.025∼0.05 mm after processing after the data was processed by the average algorithm.

3.3 Temperature self-compensation test

To verify the temperature self-compensation performance of single fiber grating of the elliptical ring, the bare fiber Bragg grating (referred to as FBG1), the elliptical ring with fiber grating attached (referred to as FBG2), and the encapsulated FBG hydrostatic level (referred to as FBG3) were placed in a temperature test chamber of KB-TH-S-80Z. The temperature variable range was set to 10∼70 °C and the temperature interval step was set to 10 °C in the test. To ensure a consistent temperature within the chamber of the hydrostatic level, each temperature point was kept for at least 1 hour before recording the central wavelength of the FBG.

The relationship between the central shifts of FBG1, FBG2 and FBG3 with the variation of temperature was shown in Fig. 12. The fitted sensitivities to temperature of FBG1, FBG2 and FBG3 were 10.24 pm/°C, 12.53 pm/°C and 1.46 pm/°C, respectively. At the meantime, the linear correlations of the data were 0.9989, 0.9958 and 0.9928, respectively. The experimental results showed that the elliptical ring had an extremely low response to temperature after being encapsulated in the external shell. It was proved that the designed FBG hydrostatic level can realize the temperature self-compensation of single fiber Bragg grating.

Fig. 12. Temperature test of FBG1, FBG2, and FBG3.

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Although the designed FBG hydrostatic level did not reach 0 pm/°C from the theoretical analysis, we knew that when monitoring the settlement at various points on the structure, a reference point as far away as possible from the settlement must be located and installed. The response in temperature of the measured points and the reference point were consistent when they were in the same gradient of temperature variation. The data of the FBG hydrostatic levels as the measured points should be subtracted from the data of the reference point, thus successfully avoiding the variation of settlement caused by temperature sensitivity. The full sense of temperature self-compensation was achieved in commercial applications.

3.4 Error discussion

After the experiments, it could be found that the FBG hydrostatic level had high sensitivity and resolution. Meanwhile its repeatability was excellent and had a low sensitivity to temperature. There was a slight difference between the theoretically calculated sensitivity result (12 pm/mm) and experimental results of the 4 sensors. In addition, in the temperature self-compensation test of FBG hydrostatic level, the sensitivity to temperature was 1.46 pm/°C after the package was completed, which did not reach 0 pm/°C in the theoretical analysis. There were five main reasons for the error.

1. The elliptical ring was affected by the nearby guide slot and dowel rod when deformed with force. However, it was difficult to consider the above factors when theoretically deriving the lateral and vertical deformation of the elliptical ring. From the qualitative point of view, the deformation of the elliptical ring would be obstructed by the nearby guide slot and dowel rod, resulting in the actual sensitivity of the sensor being lower than the theoretical value, which also led to a certain deviation in the temperature self-compensation model of the elliptical ring.
2. The thickness dimension of the elliptical ring in FBG hydrostatic level was thin (0.6mm), which was difficult to be machined. There might be initial deviations in the actual dimensions during the wire cutting process.
3. The FBG hydrostatic level was encapsulated with 353ND glue in order to connect the single fiber Bragg grating to the elliptical ring. The actual strain transfer efficiency of the sensor was less than 100%, while the strain loss of the glue layer had not been considered in the theoretical analysis.
4. In the settling experimental testing system in Fig. 7(b), the installation of the bucket and the FBG hydrostatic level had an initial tilt angle. At the same volume of water, the level change caused by a vertically oriented bucket was greater than that caused by a tilted placement. The reason was that tilted placement would cause an increase in its cross-sectional area, resulting in a decrease in the sensitivity of the FBG hydrostatic level during the test.
5. The steps of making FBG hydrostatic level were made manually, such as the deviation of the angle between the encapsulated FBG and the direction of the elliptical long axis, as well as the syringe injection into the volume of water in the bucket with deviations. In addition, the systematic errors of the experimental equipment and the interference of the external environment could lead to different degrees of error in the results.

4. Conclusions

An FBG hydrostatic level that can achieve temperature self-compensation by applying single fiber grating was designed in this paper. The temperature self-compensation model of elliptical ring was established by the different coefficient of thermal expansion between TA2 titanium alloy and stainless steel 316. The temperature tests showed that the sensitivity to temperature of the encapsulated sensor was only 1.46 pm/°C, which was much lower than that of the bare fiber Bragg grating as well as conventional FBG sensors. The core of the sensor were two parts: elliptical ring and diaphragm. Compared with the conventional cross-compensation method with double FBG, the structure of this FBG hydrostatic level could significantly reduce its cost. Extensive experimental data showed that their sensitivity of the four sensors were about 12 pm/mm, which were close to the theoretical analytical results. The proposed theory and method could be effectively used in the design of the FBG hydrostatic level. Their linearity were as high as 0.9996. Meanwhile, the repeatability error and hysteresis error were within a reasonable range. In addition, the resolution of the experimental data was analyzed using the dichotomy with statistical methods. The effective resolution range of the FBG hydrostatic level based on the connecting pipe system could reach 0.025∼0.05 mm, with excellent reliability. Finally, a comprehensive analysis of the reasons for the errors in the experiment was presented from multiple perspectives, including the derivation of the theory, fabrication and loading tests. Ideas were provided for the innovation and improvement of the sensor. The designed FBG hydrostatic level could be applied to the settlement monitoring of engineering structures, after composing the connecting pipe system.

Funding

National Natural Science Foundation of China (51278387).

Acknowledgments

Lina Yue and Nianwu Deng thank the School of Science, Wuhan University of Technology for the theoretical work.

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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Symbol	Meaning	Value	Unit
t	Thickness of diaphragm	0.2	mm
R	Radius of diaphragm	30	mm
a	Radius of the long axis of the elliptical ring	10	mm
b	Radius of the short axis of the elliptical ring	8	mm
c	Length of dowel rod	17	mm
E₂	Elastic modulus of stainless steel 316	206×10³	MPa
v	Poisson's ratio of stainless steel 316	0.3	/
E₁	Elastic modulus of Titanium alloy TA2	106.4×10³	MPa
G₁	Shear modulus of Titanium alloy TA2	40×10³	MPa
δ	Section thickness of the elliptical ring	0.6	mm²
h	Section width of the elliptical ring	1	mm⁴
k	Shear Factor for rectangular Sections	1.2	/
η	Deformation scale factor for elliptical ring	0.81568	/
S	Theoretical sensitivity of FBG hydrostatic level	12	pm/mm

Research on a fiber Bragg grating hydrostatic level based on elliptical ring for settlement deformation monitoring

Abstract

1. Introduction

2. Principle and structural analysis of FBG hydrostatic level

2.1 Structural design of FBG hydrostatic level

2.2 Deformation analysis of elliptical ring

2.3 Free thermal deformation of elliptical ring

2.4 Temperature self-compensation model of FBG hydrostatic level

2.5 Sensitivity analysis of FBG hydrostatic level

3. Testing and discussion of FBG hydrostatic level

3.1 Sensitivity test

3.2 Resolution test

3.3 Temperature self-compensation test

3.4 Error discussion

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (1)

Equations (30)

Optics Express