Superwide-Range Fiber Bragg Grating Displacement Sensor Based on an Eccentric Gear: Principles and Experiments

Displacement is the most basic physical quantity reflecting structural deformation and warning of structural failure in the field of structural health monitoring. Excessive displacement not only affects the normal use of a structure, but also causes additional stress, by which a structure’s capacity can be significantly reduced. Therefore, accurate measurement of structural displacement exhibits an inherently great significance for the safety of engineering structures and for reducing maintenance costs.

A fiber Bragg grating (FBG) is a kind of photosensitive element with good and reliable performance in strain measurement. FBGs have been widely used in a variety of fields because of advantages such as small size, high sensitivity, anti-electromagnetic interference, convenient layout, easy to use for quasi-distributed measurements (Sun et al. 2016; Ho et al. 2016). Some scholars have proposed a method for monitoring steel corrosion in concrete with FBG sensing technology (Li et al. 2016, 2017b). Sun et al. (2015) proposed a new way of real-time, long-term monitoring of underground pipelines based on FBG strain sensors, and the feasibility of the method was proved by numerical simulation. Huynh and Kim (2017) installed FBG sensors in the prestressing tendons of prestressed concrete (PSC) girders, and prestress loss and temperature variation events were monitored. Kong et al. (2017) used FBG sensors to design a reliable sensor protection measure and a bridge scour monitoring system was also developed. The functionality of the proposed system was verified by tests. Ho et al. (2017) proposed a smart anchor plate based on FBG sensors to monitor the load level of rock bolts, and the FBG-instrumented anchor plate was able to observe the load of the plate with good repeatability in the experimental observations.

In recent years, many scholars have developed various types of FBG displacement sensors. A FBG displacement sensor that can be directly utilized to measure sub-micrometer displacement and also to support multipoint distributed detection was developed, and tests showed that the sensor had the same displacement measurement value as a commercial laser displacement sensor (Li et al. 2017a; Li and Ren 2017). However, in the field of road and bridge construction, the horizontal and vertical displacement of subgrade surfaces and the displacement of bridge bearings can reach dozens of millimeters, and ground deformation and underground displacement can reach a few meters. Therefore, there is an urgent to develop a wide-range FBG displacement sensor. Wang et al. (2007) developed a FBG displacement sensor with a wide range capacity based on the structure of cantilever beam and spring. Two fiber Bragg gratings with similar temperature sensitivity coefficients were packaged in a pair at the same position, with one on the upper surface and the other on the lower surface of a cantilever beam. The string was connected to the free end of cantilever beam. After pulling the string, the change in displacement was obtained according to the change in the wavelength of the fiber grating on the cantilever beam. This design compensated for the influence of temperature. He et al. (2010) designed a FBG displacement sensor with the advantages of a simple and reliable structure, stable performance, and accurate monitoring. The sensor was mainly composed of a measuring probe, transfer member, spring, and cantilever beam. The movement of the measuring probe drove the transfer member acting on the free end of the cantilever beam, resulting in variations in the wavelength of the FBG that was packaged on its surface. The change in displacement was obtained, and the spring provided restoring force for the measuring probe. Cui et al. (2011) proposed that fiber Bragg gratings be packaged in the same position but on the upper and lower surfaces of two cantilever beams. This FBG displacement sensor also had temperature compensation capacity. The structure of the two cantilever beams not only eliminated the influence of temperature but also avoided the influence on sensor accuracy resulting from the lateral deviation of the slider. All previous research has focused on the development of FBG displacement sensors based on cantilever beams. However, the elastic range of cantilever beams that has been used as an elastic element and the effective strain of FBGs are limited; thus, the range of measurement is limited. Designing a FBG displacement sensor with the advantages of a wide range and a simple and exquisite structure is a prerequisite for meeting the need for large displacement measurement in engineering practice.

This paper proposes a design for a superwide-range FBG displacement sensor based on eccentric gear; the sensor was verified in a hysteresis test on a steel frame–reinforced concrete infill wall. By the periodic rotation of the gear, this device achieved the purpose of enlarging the range of the sensor without enlarging the deformation of the cantilever beam. The fabricated sensor was applied in a crack monitoring test on a shape memory alloy (SMA) strengthened concrete beam, providing a new method of displacement monitoring for the structural health monitoring field.

The developed superwide-range FBG displacement sensor was based on an eccentric gear, which was composed of a gear, rotating shaft, rack, spring, metal case, and a cantilever beam with a FBG packaged on its surface. A cross-sectional view of the superwide-range FBG displacement sensor based on eccentric gear is shown in Fig. 1, and a schematic diagram is shown in Fig. 2. When the rack moves by a displacement the length of the arc $s$, the gear rotates the length of the arc, and the rotating shaft passing through the eccentric point of the gear drives the cantilever beam to produce a deflection of $\delta$. This causes a change in the strain so that the center wavelength of the FBG attached to the cantilever beam is changed. We can determine the displacement by the central wavelength variation that demodulated by the demodulator. The spring provides restoring force for the rack; that is, when the sensor stops working, the rack is restored to its original state. The circular motion of the gear is periodic, and the circular movement takes the gear circumference as its period; therefore, the change in wavelength is also cyclical. The number of turns of the gear can be obtained from the scale of the rack, thereby allowing the displacement to be obtained. No matter how long the displacement measured, the maximum deflection of the cantilever beam is two times as long as the eccentricity of the gear. Therefore, the sensor achieves an enlarged range of displacement measurement without enlarging the deformation of the elastic element.

The theoretical derivation is as follows.

When the rack moves by a displacement the length of the arc $s$, the gear also rotates by the same arc length. According to Fig. 2, this is obtained as

Using Eqs. (1) and (2), we can be obtain

For the cantilever beam fixed at one end, $\sigma$ is the maximum stress at the joint of the cantilever beam and the eccentric gear; $F$ is the resultant force perpendicular to the cantilever beam at the contact of the eccentric gear and the end of the cantilever beam; and $ϵ$ is the strain of the FBG packed on the surface of cantilever beam. Through basic principles of material mechanics (Sun 2009):where $b$ = section width of the cantilever beam; $h$ = thickness of the cantilever beam; $E$ = elastic modulus of the cantilever beam; and $I$ = cross-sectional moment of inertia of the cantilever beam.

Using Eqs. (4) and (6), we can obtain:

Using Eqs. (3) and (7), we can obtain:

From the working principle of a FBG (Sun et al. 2017), if temperature is neglected, the working principle expression of the FBG is established aswhere $K_{ϵ}=1-{\mathrm{P}}_e$; ${\mathrm{P}}_e$ = valid elastic-optic constant of the FBG; ${\lambda }_{\mathrm{B}}$ = Bragg central wavelength; and $\mathrm{\Delta }{\lambda }_{\mathrm{B}}$ = central wavelength variation of the FBG.

Using Eqs. (8) and (9), we can obtain:

Eq. (10) is the optical displacement-wavelength equation for the wide-range FBG displacement sensor based on eccentric gear. When the eccentricity of the eccentric gear is changed, the relationship between the change in wavelength and the rack is changed. Therefore, the sensitivity of the sensor can be adjusted. When the temperature is constant, the change in displacement is calculated by the change in the central wavelength. The superwide-range FBG displacement sensor based on an eccentric gear is shown in Fig. 3.

The metal case for the superwide-range FBG displacement sensor with an eccentric gear was split into two parts and the rack was fixed on the right-hand side of the metal case. The refitted sensor is called the FBG relative displacement sensor based on an eccentric gear and it is shown in Fig. 4. This device can be used for monitoring structural cracks; this is described in detail in the following section.

Furthermore, the cantilever beam is subjected to horizontal tension or pressure in the course of the rotation of the gear. In order to eliminate the horizontal force of the cantilever beam and make the central wavelength variation of the FBG packaged on the cantilever beam only affected by the deflection of the cantilever beam, an arc track was installed on one side of the metal case of the sensor. The motion of the rotating shaft is restricted to the width of the arc track, and the track of the arc track corresponds to the motion trajectory of the rotating shaft under the action of vertical force. The influence of the horizontal force of the cantilever beam on the central wavelength variation of the FBG was avoided by setting the arc track.

The test devices adopted were a vernier caliper and a FBG sensing interrogator. The demodulator was developed by Micron Optics, Atlanta, Georgia (model sm130). The maximum acquisition frequency of the instrument was 1,000 Hz, and the resolution was less than 1 p.m. The central wavelength of the FBG in the test was 1,532 nm, and the peak reflectivity was more than 90%. The influence of temperature on the change in the central wavelength was neglected due to the short duration of the calibration test. The addendum circle diameter of the gear in the sensor was 46 mm, the eccentricity was 2 mm, and the displacement measured by the rotation of the gear was about 144.5 mm. After turning a circle, the gear rotates repeatedly, and the change in wavelength is also repeatable; the measured displacement can be obtained by recording the number of rotational circles. In order to conveniently record the number of gear rotations, a label was pasted on the surface of the gear when the sensor was installed. The displacement of the rack was measured by the vernier caliper, and the displacement was increased from 0 to 5 mm each time. At the same time, the change in the central wavelength of the FBG was recorded, and the backward stroke was tested when the displacement reached 145 mm. Three groups of tests were completed, and the results are shown in Fig. 5. In the elastic range, the strains of the cantilever beam and the FBG were regarded as the same.

Fig. 5 shows that when the number of turns of the eccentric gear was not greater than one circle, the fitting equation of the central wavelength and displacement was $y=\mathrm{1,532.6046}+0.1832\sin (2.4906\times -0.5235)$; the adjusted $R$-square was up to 0.99. The results show that the design of the wide-range FBG displacement sensor based on an eccentric gear is reasonable. In Fig. 5, the wavelength values correspond to different displacement values. Therefore, we can obtain the displacement by recording the angle of gear rotation. For example, when the wavelength is 1,532.70 nm, the displacement calculated by the formula is 12.8 mm. If the angle of gear rotation is greater than 90°, one quarter of the circumference of the gear should be added to the displacement value. When the rack is infinitely long, the displacement $x$ of the sensor is obtained by recording the number $n$ ($n$ is natural number) of turns of the eccentric gear and the central wavelength $y$. The displacement $x$ is ($144.5n+x_c$) mm, where $x_c$ is obtained by the angle of gear rotation and the formula $x_c=(\arcsin ((\mathrm{y}-\mathrm{1,532.6046})/0.1832)+0.5235)/2.4906$. Therefore, in theory, the FBG displacement sensor can achieve an unlimited range of displacement measurement.

In order to ensure the feasibility for engineering and the practicality of the superwide-range FBG displacement sensor based on an eccentric gear, static performance tests are essential in order to analyze the hysteresis and repeatability errors of the FBG displacement sensor.

Table 2 shows the data measured by the FBG relative displacement sensor based on an eccentric gear and the crack integrated test instrument.

Table 1 shows that the shrinkage of the crack measured by the FBG relative displacement sensor based on an eccentric gear was larger than the shrinkage measured by the crack integrated test instrument, and the maximum deviation was 7.8%. The reason for this is that when the SMA concrete beam was cracked after compression, it had a certain deflection; the relative displacement between cracks measured by the FBG relative displacement sensor based on an eccentric gear was the sum of the shrinkage of the crack and the displacement between the left- and right-hand parts of the sensor under the recovery of the deflection of the concrete beam. Therefore, the displacement measured by the FBG relative displacement sensor based on an eccentric gear was larger. However, the test showed that the crack measurement results of the FBG displacement sensor and the crack integrated test instrument were in good agreement and had a consistent trend of change. According to the test results, it is proved that the FBG displacement sensor can be used to accurately measure changes in displacement.

This paper developed a superwide-range FBG displacement sensor based on an eccentric gear. This sensor can not only meet the needs of long-term displacement monitoring of engineering structures but also can be used, in theory, to measure infinite large displacements. The wavelength-displacement fitting curve of the superwide-range FBG displacement sensor based on an eccentric gear was obtained by a calibration test. The hysteresis error was 3.76% and the repeatability error was 4.93%, which indicates that the design is reasonable. The feasibility of the sensor was verified in a hysteresis test on a steel frame–reinforced concrete infill wall. The refitted sensor was applied in the self-repairing performance test of an SMA strengthened concrete beam, successfully monitoring the change in the crack width. The test showed that the sensor is of high reliability, has a flexible layout, and is suitable for measuring large displacements in engineering structures.

This research was supported by the National Natural Science Foundation of China (Grant No. 51578347), Natural Science Foundation of Liaoning Province (Grant No. 2015020578), and the Taishan Scholar Priority Discipline Talent Group program funded by Shandong Province.