FBG displacement sensor with hyperbolic flexible hinge structure

This paper proposes a design method for a fiber Bragg grating (FBG) displacement sensor with a hyperbolic flexible hinge structure. A compact FBG displacement sensor with strong micro displacement measurement capability was fabricated. The final assembly testing showed that the sensor achieved good linearity with a linearity of 0.53%F·S at a range of 50 mm. The sensitivity was measured as 24.45 pm mm−1, hysteresis error as 1.1%FS, and repeatability error as 0.659%. The testing results demonstrated that the fabricated FBG displacement sensor exhibited strong practicality and good stability, making it suitable for long-term displacement monitoring of slope safety.


Introduction
Displacement measurement is of great importance in slope safety monitoring, as it reflects the minor deformations of slopes.Accurate displacement measurement, including linear and angular displacement, is crucial for slope safety monitoring [1].
In recent years, there have been increasing research reports on the application of fiber optic sensing technology and fiber Bragg grating (FBG) sensors in slope safety monitoring [2][3][4][5][6][7][8][9][10][11].Existing FBG displacement sensors can be categorized into those based on cantilever beams, fixed-end beams, Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.rope-type sensors combining cantilever beams and rotational bearings [12], composite structures combining wedge sliders and cantilever beams [13], as well as displacement sensors with hydraulic [14] or I-beam structures [15].These sensors have been widely used in geotechnical engineering, breaking the monopoly of traditional displacement sensors.However, sensors based on cantilever beam structures or composite structures involving cantilever beams and other transmission devices increase the complexity of the sensors.They also introduce mechanical friction, reducing sensitivity and durability.Moreover, the installation of FBG on cantilever beams may cause uneven adhesion, resulting in chirping effects and affecting the accuracy of measurement results.
FBGs convert the relationship between the wavelength drift and strain into displacement sensing, making them suitable for displacement monitoring due to their simplicity, high sensitivity, flexible installation, and adaptability to different environments [16][17][18].They have achieved good monitoring results in geotechnical engineering and hydraulic infrastructure construction.Based on the theory of FBG sensing and the principle of triangle amplification, this paper proposes a FBG displacement sensor with a hyperbolic flexible hinge structure.The sensor is designed to be simple in structure, without mechanical friction, highly sensitive, and suitable for slope displacement monitoring.The FBG is installed using a two-point installation method in the maximum displacement amplification region of the hinge structure.This sensor can be applied in the field environment of slope safety monitoring, providing a new monitoring solution for slope instability.

Sensor working principle
The developed sensor consists of a hyperbolic flexible hinge structure, a spring, and a displacement transmission rod.Under the movement of the transmission rod, the measured displacement is converted into the deformation of the FBG element.The hyperbolic flexible hinge structure is shown in figure 1, and the principle of triangle amplification is shown in figure 2. The length of the hypotenuse L of the triangle remains constant.When it undergoes a downward displacement B in the vertical direction, it will generate a rightward displacement A in the horizontal direction.The horizontal displacement A also causes a slight change in the angle α.The relationship between the displacement amplification factor X and the angle α is given by equation ( 1): It can be expressed by the following equation: l/2t = Tensile stiffness/Rotational stiffness where l represents the length of the flexible hinge cut, t represents the cutting thickness.This equation is used to evaluate the influence of cutting thickness on the stiffness of the flexible hinge structure.When the cutting thickness is small, the tensile stiffness of the flexible hinge is relatively high, while the rotational stiffness is low, making the overall structure more flexible.As the cutting thickness increases, the tensile stiffness of the flexible hinge decreases, while the rotational stiffness increases, making the overall structure more rigid.The above equation is only used to describe the effect of stiffness on the flexible hinge structure.The application of force causes the flexible hinge to generate tensile stiffness (Cx) and rotational stiffness (Caz), The relationship between them is provided by equations ( 2)-( 4): Here, the tensile stiffness is determined by the applied force and the resulting displacement, while the rotational stiffness is determined by the applied torque and the resulting angular displacement.E is the elastic modulus parameter of the material.
Both the tensile stiffness and rotational stiffness affect the angular displacement and linear displacement of the structure, which are metrics for measuring the precision and performance of hinge structures.By keeping the cut length (l) constant for four hinge structures and varying the cutting thickness (t), the influence of the s parameter on the four hinge structures is analyzed.Given parameters L 1 = 20 mm, L 2 = 10, R 2 = 5, R 1 = 5, a = 10, w = 10 mm, and F = 20 N. The elastic modulus E is set to 110 GPa, and the Poisson's ratio µ is 0.33.In the ANSYS model, one end of the flexible hinge is fixed, while the other end is free and subjected to axial force and applied load.
The calculation results, as shown in figure 3, indicate that when the notch length remains constant and the cutting thickness increases, the rotational stiffness and tensile stiffness of the hyperbolic and inverse rounded corner beam flexible hinges are significantly greater than those of the direct circular and straight beam flexible hinges.The rotational stiffness and tensile stiffness of all four types of flexible hinges increase with the increase of structural parameters.This suggests that, with a fixed notch length of the flexible hinge, the rotational stiffness and tensile stiffness of the flexible hinge increase gradually as the cutting thickness decreases.Therefore, under the same loading conditions, the rotational angle displacement and axial linear displacement of the hinge structure also increase.Thus, when the notch length of the flexible hinge is determined, increasing the cutting thickness will reduce the travel range of the flexible hinge.
As shown in figure 4, under the same dimensional parameters, the analysis of the four types of hinge structures reveals that the rotational stiffness and tensile stiffness of the hyperbolic flexible hinge are the largest, while the rotational stiffness and tensile stiffness of the straight beam flexible hinge are the smallest.This indicates that, under the same bending moment or loading conditions, the axial linear displacement of the straight beam flexible hinge is the smallest, making it suitable for applications with smaller measurement ranges.On the other hand, the rotational angle displacement or axial linear displacement of the hyperbolic flexible hinge is the largest, making it suitable for applications with larger measurement ranges.Additionally, for the same axial linear displacement or deflection angle, the straight beam flexible hinge experiences the smallest force conditions and bending moment, while the hyperbolic flexible hinge experiences the largest force conditions and bending moment.Therefore, the straight beam flexible hinge is suitable for scenarios with small torque driving, while the hyperbolic flexible hinge is suitable for scenarios with large torque driving.After determining the relationship between the stiffness and cutting thickness of the four hinge structures, the influence of bending moment and axial force on the amplification and transmission of small displacements by the flexible hinge needs to be considered.This will affect the overall displacement amplification performance of the hinge structure.Therefore, it is necessary to explore the sensitivity of the four hinge structures to axial force and bending moment.The stiffness ratio is the rotational stiffness divided by the tensile stiffness, and the relationship between the stiffness ratio and the parameter s is shown in figure 5.A smaller stiffness ratio indicates that the flexible hinge is more likely to produce axial linear displacement and less likely to produce rotational angle displacement.At the same time, the stiffness ratio λ reflects the sensitivity of the output displacement of the flexible hinge under the combined action of bending moment and axial force.The higher the stiffness ratio, the higher the sensitivity of the flexible hinge to rotational angle displacement, indicating that it is more likely to produce rotational angle displacement.Conversely, a higher sensitivity to linear displacement indicates a higher sensitivity to produce linear displacement.
As the structural parameter s varies, the stiffness ratios of the four types of flexible hinges decrease.This indicates that when the notch length of the flexible hinge remains constant, the sensitivity of axial displacement in the main output displacement of the flexible hinge gradually decreases, while the sensitivity of rotational angular displacement increases.In other words, as the structural parameter increases, the main form of output displacement for the four types of flexible hinges transitions from axial displacement to rotational angular displacement.
From figure 5, it can also be observed that the stiffness ratios of the straight beam and straight circular flexible hinges are significantly larger than those of the hyperbolic and reverse rounded corner straight beam flexible hinges.This indicates that when subjected to the same bending moment and axial force, the main form of output displacement for the straight beam and straight circular flexible hinges is rotational angular displacement, while the hyperbolic and reverse rounded corner straight beam flexible hinges are more prone to axial displacement.
In order to achieve a larger displacement amplification gain and provide a greater displacement range for the displacement sensor in the hinge sensing mechanism, it is advisable to choose a flexible hinge structure that primarily generates axial displacement.This will maximize the output displacement sensitivity of the flexible hinge structure.Based on the above calculation results, in order to obtain a higher sensitivity to axial displacement, the hyperbolic flexible hinge is selected as the sensing structure for the displacement sensor.

Optimization of hyperbolic-type flexible hinge structure
The output sensitivity of the displacement sensor primarily depends on the dimensions of the hinge structure.Using 65Mn steel as the machining material, with a Young's modulus of 2.1 × 10 5 MPa and a Poisson's ratio of 0.33, assuming the initial values of the given parameters are as follows: lateral tension of 50 N, a = 20 mm, b = 5 mm, w = 5 mm, and a specified stiffness ratio of 1.By varying the value of s, the displacement gain and maximum stress of the flexible hinge structure are calculated.
As shown in table 1, as the cutting thickness of the hinge decreases, the output displacement of the hinge increases, and the displacement gain gradually increases, consistent with the analysis results mentioned above.When considering the 10th set of data, the maximum stress within the mechanism approaches the allowable stress of the material.To ensure the maximum displacement gain, the 8th set of data is selected, determining a cutting thickness (t) of 5 mm, and the parameters a and b as 10 mm and 5 mm, respectively, for the notch.The total length of the edge of the hyperbolic-type flexible hinge structure is 40 mm, and the distance from the vertex of the hyperbolic-type flexible hinge structure to the long side is 10 mm.This structure effectively increases the sensitivity of displacement detection, with a displacement gain of 3.4144.The four corners are designed with rounded edges to enhance the overall stability of the structure while meeting the requirements of the elastic body.
The schematic representation of the simulation analysis is depicted in figure 6.From the relationship between the longitudinal displacement and the lateral displacement and tension in figure 7, it can be observed that the longitudinal displacement of the sensor has a linear relationship with the tension, with a high linear correlation coefficient (R 2 ) of 0.9995.The longitudinal displacement of the sensor is consistently about 3.2 times the lateral displacement, indicating a microdisplacement amplification factor of 3.2.This differs from the theoretically calculated value of 3.4144, which can be attributed to the consideration of the thickness of the spring leaf in the hyperbolic-type flexible hinge structure and the dimensions at both ends.These parameters were not taken into  account in the theoretical calculation.The error rate between the calculated amplification gain and the results of finite element simulation analysis is 6.7%.Based on the structural parameters and stiffness ratio, the design and optimization of the hyperbolic-type flexible hinge structure is feasible and accurate.

Assembly and testing of displacement sensor
In order to achieve a rigid connection between the fiber optic grating and the hinge structure, the photosensitive adhesive Ergo8500 was chosen as the adhesive.The photosensitive adhesive can only cure under ultraviolet light, which provides time to apply pre-stress to the fiber optic grating and avoid rapid bonding with the hinge structure.The amplified mechanism of the hyperbolic-type flexible hinge, with the fiber optic grating securely attached, is fixed onto the sensor base.One end of the mechanism is fastened to the bottom of the sensor using a GB/T5782-2000 hexagon head bolt, while the other end is connected to an M4/12 fixed lead-sealing screw.The lead-sealing screw passes through a through-hole and connects to one end of an M2/10/50 stainless steel elastic spring.The other end of the elastic spring is connected to the displacement transmission rod.The assembled physical appearance of the displacement sensor is shown in figure 8.
The testing platform consists of a fiber optic grating displacement sensor, a broadband light source, an optical circulator, an AQ-6370D spectrometer, and a computer.The light emitted by the broadband light source is transmitted through optical fibers and the circulator to reach the displacement sensor.When the sensor is subjected to external forces, the effects are transmitted back through the reflected light signal, which passes through the circulator and into the spectrometer.The collected data is then imported into a computer for further processing and analysis.
To eliminate the influence of temperature, an additional fiber optic grating is installed inside the displacement sensor, which is solely affected by temperature.By processing the wavelength drift of the sensing grating's central wavelength in comparison with the reference grating's central wavelength, the temperature effect can be eliminated.The temperature compensation test is conducted using a temperature control box from Zhongke Meiqi.The central wavelength of the sensing fiber optic grating is 1547.540nm, and the central wavelength of the reference grating is 1546.190nm.Both FBGs are manufactured by Shenghai Optoelectronics Co., Ltd, with a reflectivity of 99.7%, a side-mode suppression ratio of 18 dB, and a 3 dB bandwidth of 13.1 nm.
Install the sensors inside the temperature control chamber.Maintain the room temperature without heating until the wavelength stabilizes.Then, vary only the temperature inside the control chamber while keeping all other parameters constant.Establish six temperature points at 0, 10, 20, 30, 40, and 50 • C. Once each temperature point is reached, maintain it for 2 min to enable the central wavelength of the fiber optic grating to stabilize before increasing the temperature to the next point.Conduct temperature tests at 10 • C intervals in both positive and negative directions while recording the wavelength drift of the displacement sensor.
Establish the relationship between temperature and the drift of the central wavelength.Normalize the temperature data concerning the baseline output value of the central wavelength at 0 • C during the processing.The following section presents the temperature compensation test results for the displacement sensor.Table 2 shows the experimental data.
Figure 9 portrays the fitted curves representing the wavelength values corresponding to each temperature point for the two optical fiber gratings.
Equation ( 5) expresses the fitting formula for FBG1, with a correlation coefficient R 2 of 0.9997, y = 0.14 + 18.92x. ( Equation ( 6) presents the fitting equation for FBG2, which has a correlation coefficient R 2 of 0.9995, Linear regression applied to the data yields equation (7), The output formula for the displacement sensor, derived by excluding external temperature variations, is depicted as equation (8), To validate the static performance of the hyperbolic flexible hinge structure within the FBG displacement sensor, we devised a static calibration system employing the TD8411 rotary platform magnetic field distribution test instrument, fabricated by Tianheng Measurement & Control Company.A comparative analysis was undertaken, utilizing the DS-01 displacement sensor, a product of Chang Wei Optoelectronics Company.The schematic depiction of this experimental arrangement is presented in figure 10 for visual reference.
The hyperbolic flexible hinge structure FBG displacement sensor and commercial displacement sensor were attached to the clamping end of the displacement loading platform.Following that, the displacement adjustment knob was turned along the X-axis with a 10 mm step size.The knob was sequentially pulled from the 0 mm mark to the 50 mm mark in the positive direction.A pause of 3-5 s was observed at each displacement point until the center wavelength of the fiber grating stabilized.The data for the center wavelength of the fiber grating was recorded.The displacement was then reduced from 50 mm to 0 mm during the return journey after recording.The entire process was repeatedly performed three times.Figure 11 shows the center wavelength values of the fiber grating corresponding to the displacement points recorded during the three experimental trials.
After the arithmetic average and linear fit were performed on the data from the cyclic testing in the graph, as shown in figure 12.The fitted curve is given by y = 0.08 + 24.45x, with an R 2 value of 0.9998.This indicates that the displacement sensitivity of the sensor is 24.45 pm mm −1 .The sensitivity of the commercial displacement sensor is 9.82 pm mm −1 , which is lower than that of the hyperbolic flexible hinge structure displacement sensor.As discernible from the above diagram, the full-scale output of the sensor is 2.427 nm.Therefore, the linearity of the developed sensor can be calculated as follows: Linearity = (Maximum difference/Full-scale output) × 100% = (0.013 nm/2.427nm) × 100% = 0.534%.

Test data for three displacement cycle
Similarly, the linearity of the commercial displacement sensor is 2.38%FS.
Hysteresis refers to the phenomenon where the inputoutput characteristic curve of the sensor during three cycles of displacement testing is not coincident.The difference between the two curves and the hysteresis error are used to measure the hysteresis phenomenon, also known as the return error.The hysteresis error was calculated by taking the absolute difference between the average calibration points of two cycles for each displacement point in the positive and negative directions.The maximum hysteresis is 0.021 nm, occurring at a displacement of 20 mm, and the hysteresis error is 1.1% F•S.In the same vein, the hysteresis error of the commercial displacement sensor is 0.79%FS.
Repeatability error is an indicator that assesses the consistency of the center wavelength difference in the optical fiber grating within the displacement sensor across multiple measurements.The maximum non-repeatability error is 0.016 nm.Therefore, repeatability error of the sensor can be determined to be 0.659%.Based on a comparable computation, the commercial displacement displays a scale repeatability error of 1.76%.
All in all, the two types of displacement sensors exhibit distinct pros and cons in terms of linearity, repeatability error, and hysteresis error.However, the sensitivity of the hyperbolic flexible hinge structure FBG displacement sensor is 2.48 times that of the commercial displacement sensor.As a result, the hyperbolic flexible hinge structure FBG displacement sensor demonstrates superior displacement measurement performance, making it well-equipped for the task of monitoring slope displacements.
The stability of the sensor was tested by stretching the rod to a certain displacement and holding it fixed for over 10 min before returning it to the zero position.Figure 13 shows the wavelength drift over time during the entire experimental process.It can be observed that the wavelength drift exhibits good stability.Upon zooming in on the data from the 4th to 6th minute, it is found that the fluctuation in wavelength is within 4 pm, which is within the fluctuation range of the demodulator's own wavelength measurement accuracy.This indicates that the sensor possesses good stability.
The ambient temperature of the slope remains within the range of −30 • C to 40 • C throughout the year.In order to further corroborate the long-term temperature stability of the sensors within this range, the temperature of the temperature control chamber was incrementally raised from −30 • C to 40 • C, with intervals of 10 • C. Each increment was maintained for a duration of 6 h, and once the temperature reading within the chamber had achieved stability, the central wavelength drift was recorded at hourly intervals.The experiment spanned approximately 48 h, and the curve depicting the central wavelength drift of the dual optical fiber gratings within the sensor is illustrated in figure 14.
As depicted in figure 14, it is evident that the sensor's output voltage exhibits a gradual fluctuation within the temperature range of −30 to 40 • C.

Conclusion
In conclusion, this study has developed a fiber optic grating displacement sensor for slope safety monitoring, taking into account the requirements of slope monitoring.By combining the advantages of fiber optic grating sensing, a hyperbolic flexible hinge structure with high motion sensitivity was selected.Numerical analysis and ANSYS simulation were conducted to analyze the structure, leading to the development of a fiber optic grating displacement sensor for slope safety monitoring.
The test results demonstrate that the displacement sensor exhibits good measurement capabilities for microdisplacements.With a measurement range of 50 mm, the sensor shows a linear range of 0.53% F•S, a of 24.45 pm mm −1 , a hysteresis error of 1.1% F•S, and a repeatability error of 0.659%.The experimental data analysis confirms that the sensor possesses excellent measurement accuracy for micro-displacements, meeting the requirements of precision and long-term stability in slope safety monitoring.
In summary, this research has successfully developed a fiber optic grating displacement sensor for slope safety monitoring.The sensor combines the advantages of a hyperbolic flexible hinge structure and fiber optic grating sensing technology, providing high sensitivity and good linearity.Numerical analysis, simulation, and a series of tests have demonstrated the feasibility and performance of the sensor.The experimental results indicate that the sensor has excellent capabilities for measuring micro-displacements and meets the requirements of precision and long-term stability in slope safety monitoring.
The development of this fiber optic grating displacement sensor provides an effective tool for slope safety monitoring.It can be widely applied in monitoring and analyzing slope displacement, deformation, and stress parameters, enabling the timely detection of abnormal slope conditions.This sensor contributes to the safety assessment and management of slopes by providing important data support.Furthermore, the design and development process of this sensor can serve as valuable experience and reference for the application of fiber optic grating sensors in other fields.Future work can focus on further optimizing the design and performance of the sensor to meet broader application requirements and exploring its potential applications in other engineering domains.

Figure 2 .
Figure 2. The principle of triangular amplification.

Figure 5 .
Figure 5. Relationship between stiffness ratio λ and s.

Figure 7 .
Figure 7. Relationship between tension and displacement.

Figure 8 .
Figure 8. Physical structure of the displacement sensor with hyperbolic flexible hinge.

Figure 12 .
Figure 12.Fitting of mean values for three experiments.

Figure 14 .
Figure 14.The output center wavelength drift curves of the sensor in different temperature ranges.

Table 1 .
Optimization calculation data for dimensions of hyperbolic flexible hinge.

Table 2 .
Temperature compensation test data sheet.