1 Introduction

As a monitoring tool, the tilt sensor plays an important role in the structural safety monitoring of large buildings such as slopes (Xu et al. 2019a; Wu et al. 2019), transmission towers (Zhang, et al. 2022; Huang 2019), bridges (Zhang et al. 2019; Saad 2021), dams (Allil et al. 2021), etc. Optical fiber sensing technology (Xu et al. 2022; Zhu et al. 2021; Abbas et al. 2021) has made great progress in the sensing field because of its advantages such as distributed measurement, strong corrosion resistance, strong electromagnetic interference resistance, simple structure (Di 2019), small size, lightweight, etc., which make up for the defects of traditional mechanical and electronic tilt sensors.

According to the working principle, the optical fiber tilt sensor can be divided into Mach–Zehnder (M–Z) interference tilt sensor (Silveira, et al. 2015), Fabry–Perot (F–P) interference tilt sensor (Yang et al. 2018; Yiyang 2018; Abbas 2020), Michelson Interferometer (M-I) tilt sensor (Wang et al. 2022), Fiber Bragg Grating (FBG) tilt sensor (Li et al. 2020; Guo et al. 2020; Yang et al. 2015), etc. In 2014, Lee et al. (2014) proposed a directional optical fiber tilt sensor based on M-Z interference. The sensor consists of two abrupt tapers with different tapered waists and iron spheres. Inclination angle sensitivities of about 335 pm/° and 125 pm/°. In 2017, Pan et al. (2017) developed a tilt sensor consisting of a pair of optical fiber collimators and a simple pendulum with an F-P filter. Within the measuring range of 6° ± 3°, the average sensitivity is 1104 pm/°. In 2022, Wang et al. (2022) proposed an in-fiber MI-based inclinometer, which is composed of misalignment spliced optical fibers with end coating. The sensitivity is 0.55 nm/° in the range from 0° to 50°. The fiber optic tilt sensors based on interferometry have relatively high sensitivity, but their structure is complex, their production incurs significant losses, and their demodulation is inconvenient.

In 2019, Xu et al. (2019b) investigated a one-dimensional (1-D) FBG inclination sensor for slope monitoring, which consisted of a pendulum structure and a circular body. The experimental results show that the sensitivity within the range of ± 3° is 406 pm/°. The 1-D FBG tilt sensor is simple in structure, but it has a limited range of measurements. In 2017, Chao et al. (2017) proposed a two-dimensional (2-D) FBG tilt sensor, which consisted of two cylindrical floats equipped with FBGs suspended in water. The tilt sensitivities of the two FBGs within the range of ± 5° were 0.132 nm/° and 0.128 nm/°, respectively. In 2020, Luo et al. (2020) proposed a 2-D temperature-insensitive tilt sensor, utilizing four fibers containing two FBGs to form a multifiber bundle. The measured sensitivity for the positive along the x-axis was found to be 0.011 nm/°, and the sensitivity for the positive along the y-axis was 0.0083 nm/°.

In this paper, we propose and demonstrate a 2-D FBG tilt sensor, which is less affected by temperature variations. The three FBGs of this sensor act as strain-sensing elements. The ends of the three FBGs are fixed to a cantilever beam structure, the angle between all three FBGs is 120°. The inclination sensitivities of the three FBGs are 19.83 pm/°, 20.26 pm/°, and 20.04 pm/°, respectively. The measurement error of the deflection angle of this fiber optic tilt sensor is ± 0.2° and the error of the direction angle is ± 0.55°.

2 Principle and analysis

2.1 FBG sensing theory

The grating period and effective refractive index change when the FBG is subjected to external strain. The central wavelength \({\lambda }_{B}\) of the FBG (Di 2019) satisfies

$$\begin{array}{c}{\lambda }_{B}=2{n}_{eff}\Lambda \end{array}$$
(1)

where \({\lambda }_{B}\) is the central wavelength of FBG \({n}_{eff}\) is the effective refraction index of the fiber core, and \(\Lambda\) is the period of the index modulation. The change in wavelength of FBG under axial strain can be expressed as

$$\begin{array}{c}\frac{d{\lambda }_{B}}{{\lambda }_{B}}=\left(1-P\right)\varepsilon \end{array}$$
(2)

where P is the optical coefficient of the strain of the fiber and \(\varepsilon\) is the applied strain. According to \({a}_{\varepsilon }={\lambda }_{B}(1-P)\), the central wavelength drift can be expressed as

$$\begin{array}{c}\Delta {\lambda }_{B}={a}_{\varepsilon }\varepsilon \end{array}$$
(3)

where \({a}_{\varepsilon }\) is the strain calibration coefficient of the FBG.

2.2 Sensor theoretical analysis

The main unit of the proposed 2-D fiber optic tilt sensor is a cantilever beam structure with an H-shaped cross-section. Three FBGs are used as strain-sensing elements. The two ends of the three FBGs are fixed on the large cylinders, with the middle portion situated in the small cylindrical region. The three FBGs are spaced at intervals of 120°. The structure of the tilt sensor is shown in Fig. 1.

Fig. 1
figure 1

Tilt sensor structure

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When the FBG is fixed, a force is applied at a specific position, causing a change in the deflection and direction angle of the sensor. The distance between the two fixation points of the FBG changes under stretch and compression as a result of the stress, which causes the central wavelength of the FBG reflection spectrum to drift. The inclination angle measurement can be achieved by monitoring the central wavelength drift of the FBGs.

The response characteristics of the tilt sensor are simulated and analyzed for different deflection angles and direction angles. The sensor structure model is established according to the parameters shown in Table 1 and then simulated by simulation software. The material is set to aluminum alloy. The upper end of the sensor is fixed.

Table 1 Parameters of the simulation model
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The tilt response characteristics of the tilt sensor at different deflection angles are analyzed. A force ranging from 0 to 2 N is applied at position A on the mass underneath FBG3 inwards perpendicular to the tilt sensor. The step length is 0.5 N. The deformation simulation is shown in Fig. 2.

Fig. 2
figure 2

The simulation diagram of the tilt sensor at 0.5 N

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The changes in the distance between the two fixed points of the three FBGs when different forces are applied are recorded separately. Figure 3 shows the linear fitting results between the changes in the distance between the two fixed points of the three FBGs and the force. When the force is increased in the range from 0 to 2 N, the changes in the distance between the two fixed points of FBG1 and FBG2 increase linearly, and their values are equal. The changes in the distance between the two fixed points of FBG3 decrease linearly. And the sum of the changes in distance between the two fixation points of FBG1 and FBG2 is equal in magnitude but opposite in direction to the change in distance between the two fixation points of FBG3. Because when a force is applied at the position of FBG3, FBG3 is compressed while FBG1 and FBG2 are stretched, and the forces of FBG1 and FBG2 are equal.

Fig. 3
figure 3

Relationship between the change in distance and the force

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The direction angle of the tilt sensor is simulated by applying a constant force of 1 N counterclockwise from the position where FBG3 is located. The step length is 20° and the rotation span is 0° to 360°. Figure 4 shows the fitting results between the changes in the distance between the two fixed points of the three FBGs and the rotation angle. The fitting results indicate that the changes in distance between the two fixation points of the three FBGs exhibit a sinusoidal relationship with the rotation angle. The amplitude of the distance changes between the two fixation points of the three FBGs is equal, and they have a phase difference of 120°.

Fig. 4
figure 4

Relationship between the changes in distance and the rotation angle

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2.3 Tilt sensor 2-D inversion algorithm

The wavelengths of the three FBGs are analyzed using a correlation algorithm. Three points \({A}_{1}\), \({B}_{1}\), and \({C}_{1}\) are taken at the same horizontal position of the three FBGs in space, forming an equilateral triangle. The plane in which this equilateral triangle is located is defined as plane\(\alpha\). The relative positions of the FBGs in the sensor are shown in Fig. 5. In plane, the center of the three points \({A}_{1}\), \({B}_{1}\), and \({C}_{1}\) is defined as O. The line passing through \({OC}_{1}\) is defined as the y-axis, the line passing through O and perpendicular to the y-axis is defined as the x-axis, and the line passing through O and perpendicular to the xy-plane is defined as the z-axis. Thus, an O-xyz Cartesian coordinate system is established. The coordinates of the three points are \({A}_{1}\left(-\frac{\sqrt{3}}{2}\mathrm{r},-\frac{\mathrm{r}}{2},0\right)\), and \({B}_{1}\left(\frac{\sqrt{3}}{2}\mathrm{r},-\frac{\mathrm{r}}{2},0\right)\), and \({C}_{1}(0,\mathrm{r},0)\), where r is the radius of the large cylinder of the cantilever beam. According to the structure design, it is known that r is 3 mm.

Fig. 5
figure 5

Relative positions of the FBGs in the sensor

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When the tilt table is tilted, the three points become \({A}_{2}\), \({B}_{2}\), and \({C}_{2}\). The plane is defined as plane β. The coordinates of the three points are \({A}_{2}(-\frac{\sqrt{3}}{2}\mathrm{r},-\frac{r}{2},\mathrm{r}tan{\theta }_{A})\), \({B}_{2}(\frac{\sqrt{3}}{2}\mathrm{r},-\frac{\mathrm{r}}{2},\mathrm{r}tan{\theta }_{B})\), and \({C}_{2}(0,\mathrm{r},\mathrm{r}tan{\theta }_{C})\), respectively. The tilt sensitivity coefficients of the three FBGs can be known through calibration experiments. The drift of the central wavelength of the FBGs when the inclination angle changes can be obtained through a demodulator. Therefore, the value of θ for each FBG can be calculated, and it is also can be known the coordinates of \({A}_{2}\), \({B}_{2}\), and \({C}_{2}\).

The angle of deflection of the tilt sensor can be obtained from the knowledge of space vectors and three-dimensional geometry as

$$\begin{array}{*{20}c} {cos\left\langle {\vec{m},\vec{n}} \right\rangle = \frac{3}{{2\sqrt {\left( {tan^{2} \theta_{A} + tan^{2} \theta_{B} + tan^{2} \theta_{C} - tan\theta_{A} tan\theta_{B} - tan\theta_{A} tan\theta_{C} - tan\theta_{B} tan\theta_{C} + \frac{9}{4}} \right)} }}} \\ \end{array}$$
(4)

direction angle can be obtained from

$$\begin{array}{*{20}c} {cos\left\langle {\overrightarrow {{OC_{1} }} ,\vec{m}^{\prime } } \right\rangle = \frac{{tan\theta_{B} + tan\theta_{C} - 2tan\theta_{A} }}{{2\sqrt {\left( {tan^{2} \theta_{A} + tan^{2} \theta_{B} + tan^{2} \theta_{C} - tan\theta_{A} tan\theta_{B} - tan\theta_{A} tan\theta_{C} - tan\theta_{B} tan\theta_{C} } \right)} }}} \\ \end{array}$$
(5)

where \(\overrightarrow{m}\) is the normal vector of plane \(\alpha\), \(\overrightarrow{n}\) is the normal vector of plane \(\beta\), \(\vec{m}^{\prime }\) is the vector of \(\overrightarrow{m}\) projected onto the plane \(\alpha\).

Their sensing coefficients for temperature are considered to be approximately the same. Since the three FBGs are in the same temperature environment, the temperature-induced θ is the same. In (4), \(tan{\theta }_{A}=tan{\theta }_{B}=tan{\theta }_{C}=0\), then

$$\begin{array}{*{20}c} {cos\left\langle {\vec{m},\vec{n}} \right\rangle = \frac{3}{{2\sqrt{\frac{9}{4}} }} = 1} \\ \end{array}$$
(6)

Because \(cos0^\circ =1\), the temperature-induced deflection angle is 0. This tilt sensor is unaffected by temperature and is temperature self-compensating.

3 Experiment

3.1 Fabrication of tilt sensor

The sensor mainly consists of a cantilever beam structure, FBG, mass block, etc. The two ends of the three FBGs are fixed on the cantilever beam structure, and the included angle between them is 120°. The structure is shown in Fig. 6. The FBGs are written in femtosecond, the length of the FBG is 5.5 mm, the reflectivity is 85.645%, and the 3 dB bandwidth is 0.26 nm. The central wavelengths of the FBGs are 1537.413 nm, 1555.807 nm, and 1550.548 nm. According to the simulation results, the structural dimensions and quantities are shown in Table 2.

Fig. 6
figure 6

Structure of the fiber optic tilt sensor

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Table 2 Materials and parameters required for fabrication
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A section of FBG with a wavelength of 1537.413 nm is placed in the middle of the cantilever beam, a preload of 3 nm wavelength is applied and the ends of the FBG are fixed with epoxy resin adhesive. Two additional FBGs with center wavelengths of 1550.548 nm and 1555.807 nm are also fixed on the cantilever beam using the same method. A black sleeve is applied to each end of the three sections of FBG and epoxy glue is dotted close to the grating region to protect the FBG from breakage. A mass is fixed at the end of the above cantilever beam. Thus, the fabrication of the fiber optic tilt sensor is completed.

3.2 Angle calibration of tilt sensor

The whole optical fiber inclination sensing system is composed of transmission fiber, optical fiber tilt sensor, inclinometer, demodulator, and computer. The completed fiber optic tilt sensor is suspended vertically in a column housing made of aluminum alloy, which is lightweight and resistant to deformation. The tilt sensor is fixed to the inclinometer. A rotary table is fixed on top of the tilt table. The measuring range of the tilt table is from − 15° to 15°, and the range of the rotary table is from 0° to 360°. The test system diagram is shown in Fig. 7. The tilt sensor is fixed to the inclinometer. Three FBGs are coupled through a 1 × 4 coupler and then connected to the demodulator. The 8-channel demodulator is equipped with a light source in the C-band wavelength range and connected to the computer to observe the central wavelength of the sensor reflection spectrum. The experiments are carried out on a precision optical platform with the tilt sensor in the same horizontal position at all times.

Fig. 7
figure 7

Experimental system diagram

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3.2.1 Angle calibration of the sensor

The calibration experiments are carried out at room temperature. The rotary table of the inclinometer is fixed and the initial scale of the tilt table is set to 0°. The tilt table is changed by 5° in turn, and the central wavelengths of 5°, 10°, 15°, − 5°, − 10°, and − 15° are recorded after the spectra are stabilized. The response characteristics of the FBGs at different inclination angles are fitted and the results are shown in Fig. 8. It can be seen that the inclination angle sensitivity of FBG1, FBG2, and FBG3 are 19.83 pm/°, 20.26 pm/°, and 20.04 pm/°, respectively.

Fig. 8
figure 8

Fitting of the response characteristics of the FBGs for different tilt angles

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3.2.2 Analysis of the sensor's directional angle response characteristics

In the direction angle response characteristics experiments, the tilt table is fixed at 15°, and the rotary table has a measurement range from 0° to 360° with a step length of 15°. The central wavelengths are recorded after the reflection spectrum is stabilized. The response characteristics of the FBGs at different rotation angles are fitted and the results are shown in Fig. 9. When the direction angle changes, the variation of wavelength follows a sine curve, which is consistent with the previous simulation results.

Fig. 9
figure 9

Fitting of the response characteristics of the FBGs for different rotation angles

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3.3 Temperature calibration of tilt sensor

The temperature has a large impact on the FBG, so temperature calibration must be performed. The fiber optic tilt sensor is put into the water bath, set the temperature to “− 20 °C, 0 °C, 20 °C, 40 °C, 60 °C” cycle. After the reflection spectrum is stabilized, the central wavelengths are recorded, and the fitting results are shown in Fig. 10. It can be found that the wavelength shifts of the three FBGs have a good linear relationship with the temperature, the temperature sensitivity of the three FBGs are 35.8 pm/°C, 36.79 pm/°C, and 36.24 pm/°C, respectively. The three temperature sensitivities are basically the same. Therefore, combined with the inversion algorithm, the effect of temperature on the tilt sensor can be eliminated. The reasons for the inconsistency of the temperature sensitivity of the three FBGs include the FBG itself, the package, etc.

Fig. 10
figure 10

Temperature responses of the tilt sensor

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3.4 Optical fiber tilt sensor testing and analysis

The tilt experiments consist of a deflection angles experiment and a direction angles experiment. In the deflection angles experiment, the rotating table of the inclinometer is fixed. Both the rotating table and the tilting table are set to 0° and the central wavelengths are recorded when the spectrum is stable. The tilt table is tilted within the range of 0° to 15° with a step length of 3° and then tilted within the range of 0° to − 15° with a step length of − 3°. The central wavelengths of the three FBGs are recorded when the spectrum is stable. Repeat the above operation three times and calculate the deflection angles and direction angles at different inclination angles using the 2-D tilt inversion algorithm.

The relative variations of the deflection angles under different tilt angles as shown in Fig. 11. When the rotary table is fixed at 0° and the tilt table is tilted from − 15° to 15°, the relative variations of the deflection angles linearly increase and within the range of ± 0.1°. In addition, the error of the direction angles is ± 0.45°.

Fig. 11
figure 11

Relative variations of deflection angles at different tilt angles

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In the direction angles experiment, the tilt table is fixed at 15°, the measurement range of the rotating table is 0°–360°, the step length is 15°, and the central wavelengths are recorded after the reflection spectrum is stable. Calculate the deflection angles and direction angles at different rotation angles using the 2-D tilt inversion algorithm. The relative variations of the direction angles under different rotation angles as shown in Fig. 12. When the tilt table is fixed at 15° and the rotary table changes from 0° to 360°, the relative change in the direction angles within ± 0.4°. In addition, the error of the deflection angles is ± 0.2°.

Fig. 12
figure 12

Relative variations of direction angles at different rotation angles

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The changes in direction angles and deflection angles are also measured at inclination angles of 10° and 5°, respectively. The results indicate a measurement error of ± 0.15° for deflection angles and ± 0.55° for direction angles, making this tilt sensor suitable for inclination measurements.

As shown in Table 3, four kinds of tilt sensors based on FBG are compared, with one 1-D sensor and three 2-D sensors. In terms of measurement range and sensitivity, sensors with higher sensitivity have a smaller measurement range, while sensors with a larger measurement range have lower sensitivity. This work has a relatively wide measuring range and relatively high sensitivity. In terms of structural design, compared to the other three sensors, this work has a simple structure and a small size. Additionally, the sensor can achieve 2-D measurement and is temperature-insensitive.

Table 3 Performance parameters of different types of FBG tilt sensors in recent years
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4 Conclusion

Through the study of FBG and optical fiber sensing technology, the 2-D optical fiber tilt sensor is simulated and fabricated, and the designed 2-D optical fiber tilt sensor is calibrated and tested for deflection angle and direction angle. The deflection angle sensitivity of the sensors FBG1, FBG2, and FBG3 are 19.83 pm/°, 20.26 pm/°, and 20.04 pm/° respectively. The optical fiber tilt sensor has a measurement error of ± 0.2° for the deflection angle and ± 0.55° for the direction angle, making it suitable for inclination angle measurements. In addition, the temperature sensitivities of the three FBGs are 35.8 pm/°C, 36.79 pm/°C, and 36.24 pm/°C, respectively. The three temperature sensitivities are basically the same. Therefore, the effect of temperature on the measured tilt angle can be eliminated. In summary, the 2-D tilt sensor based on the cylindrical pendulum and FBGs has advantages over traditional tilt sensors, including high sensitivity, the ability to measure 2-D inclination, and unaffected by temperature, which provides broad prospects for application.