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Abstract
We propose a directional bending sensor by mechanically bonded FBGs in a radially placed triangular shape to measure the bending moment and its applied direction. Bending measurement and the determination of its direction is essential to get the maximum bending stress and to figure out the deflection shape by the curvature information in a wide range of engineering applications. The proposed sensor configuration is original in terms of a pointwise sensing scheme based on the mechanically bonded scheme of three optical fibers and its simple fabrication process only using optical fibers. This set-up made of three FBG sensors provides a simple relationship of the bending moment to the independent strain measurement of each FBG. We demonstrate two sensor configurations such as the self-bending measurement and the structural bending measurement in a surface-mount (or embedded) measurement. Through the verification experiments, the proposed bending sensor is successfully shown to measure the applied bending moment and its direction.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Measurement of bending moment is an important part of structural integrity assessment. Many civil and mechanical structures including airplanes, ships, pipeline, etc. can be modeled by a slender beam member. Bending moment characterizes the mechanical behavior of the modeled beam structure and determines the maximum stress in which the structural failure happens. For instance, the bending moment diagram roles a useful design tool in the preliminary design of slender structures. The Euler-Bernoulli theory is applicable to analyze the slender beam behavior with the assumption that the plane sections remain plane. In a pure bending case, the maximum stress is only proportional to the applied moment and the distance from the neutral axis.
In the measurement of bending, there are two unknowns, the magnitude of the moment and its direction, which implies one pointwise sensor is not sufficient by nature. Several kinds of fiber bending sensors have been developed to date, including asymmetric FBGs [1], a long period grating [2], tilted fiber gratings [3,4], fiber interferometers [5,6]. A special asymmetric structure or mode of the optical fiber results in the direction-dependent property. Recently Bao et al. proposed an eccentric FBG inscribed over part of the depressed cladding fiber core for the vector measurement [7] and Chen et al. demonstrated the bending sensor consists of both the fiber core and the surrounding cladding of a section of a side-hole fiber [8]. In particular, Wang et al. proposed a vector sensor based on 3-core multimode fiber structure [9], which is similar to our proposed scheme.
There have been several applications of bending or curvature measurement using FBGs; structural monitoring and medical devices. The deflection of the underground cavern was estimated using a radially placed FBG sensing bar [10]. In particular, the biomedical application for measuring bending deflections of the needle being inserted into tissues is an active research topic of FBG sensors to track and control the trajectory of the needle [11,12]. Park et al. demonstrated the set of FBG sensors, located along the needle, estimated the bent profile as well as the temperature compensation [13]. A similar sensor configuration was applied to control flexible surgical instruments [14], in which three FBGs are radially placed at intervals of 120 degrees. Henken et al. showed an error analysis of FBG-based shape sensors for medical needle tracking [15]. 4 multicore optical fiber with FBGs was used to assess the shape of flexible medical instruments [16]. In addition to the medical application, the motion capture using the optical fiber 3D sensor [17], the shape sensing of snake-like robot [18] and the curvature monitoring of pipe line [19] were reported.
In this paper, we propose a mechanically bonded FBGs in a triangular shape to measure the bending moment and its applied direction using a similar configuration to the Refs. [13–15]. However, the proposed sensor configuration is original in terms of a pointwise sensing scheme based on the mechanically bonded scheme of three optical fibers and its simple fabrication process only using optical fibers. This set-up made of three FBG sensors provides a simple relationship of the bending moment to the independent strain measurement of each FBG. In particular, we examine two sensor configurations, the self-bending measurement, and the structural bending measurement in a surface-mount (or embedded) configuration. Through the fabrication procedure of the proposed sensor and the verification experiment by applying rotational displacement load, we demonstrate the derived relationship to obtain the magnitude of the moment and its applying direction as a directional bending sensor in a wide range of applications.
2. Principle and sensor fabrication
2.1 Measurement principle
To measure the structural bending moment, i.e. the magnitude and the direction, the strain information at three vertexes of the equilateral triangle is utilized. Three single-mode optical fibers in which Bragg grating is written are mechanically bonded in the triangular shape as shown in Fig. 1. This configuration of mechanically bonded fibers is called a triangular bending FBG sensor. Using the fabricated bending sensor, two configurations are proposed; the self-bending measurement and the detached (or embedded) measurement.
First, consider the configuration of the sensor itself measures the bending moment applied to it. The centroid of the triangular bending FBG sensor through which the neutral axis in pure bending passes lies in the centroid of the equilateral triangle made of lines connecting the center of each optical fiber. The second moment of the cross-section area of triangular bending FBG sensor with regards to the neutral axis is given by
In pure bending, the strain at the beam section is described by
where M is the applied moment at the section, y is the distance from the neutral axis and E, I are the modulus of the elasticity and the second moment of the cross-section area, respectively. With regards to the inclined neutral axis, the strain of each FBG sensor is expressed asNext, the structural bending sensor configuration that is mounted on the surface of the structure or embedded inside it is considered. As shown in Fig. 2, the triangular bending FBG sensor is placed by the finite distance from the neutral axis. When the moment is present by the external force, the neutral axis is determined by its applied angle as well as the sectional geometry. With unknown variables, the distance, d, and the inclination angle $\theta $, the measured strain at the center of each FBG is given by
2.2 Fabrication and error estimates
Three single-mode optical fibers in which FBG is inscribed are prepared to fabricate the proposed triangular-shaped sensor for bending measurement. Using the microscope, two optical fibers are bonded by the adhesive in Fig. 3(a) and the remaining fiber is finally bonded to build a triangular shape as shown in Fig. 3(b). In the section view of the fabricated sensor in Fig. 3(c), they are well-arranged as anticipated. The center wavelength of each FBG for fabrication is 1538.10 nm, 1547.44 nm, and 1568.04 nm, respectively.
After fabrication of the bending sensor, the alignment angle between the optical fiber lines was measured using a microscope. The measurement results were 120 degrees, 114 degrees and 126 degrees, respectively.
To assess the bending measurement error by the fabrication in self-bending measurement, an arbitrary misalignment is applied in angular and radial directions in Eq. (3). The bending angle is calculated by Eq. (5) and is compared to the values without misalignment. Similarly, Eq. (7) and Eq. (9) are utilized for the error estimation in surface-mount bending measurement. The measured angle error in self-bending case and surface-mount configuration is summarized in Table 1. In the angular misalignment, the measured angle error in surface-mount condition is worse than the case of self-bending measurement and in the radial misalignment case, the surface-mount case shows a better results. However, it should be noted that both misalignments occur at the same time in the actual fabrication. Interestingly, the measured angle error in surface-mount condition is independent of the mount distance, d.
Table 1. Measured angle error by angular and radial misalignments
3. Experiments
3.1 Self-bending measurement
To verify the proposed sensor experimentally, the triangular bending FBG sensor is placed on the rolling stage. For the self-bending measurement, 30 mm section is vertically hung as a cantilever in which three FBGs are inscribed in 10 mm at the root part of the cantilever in Fig. 4. In addition, this vertical test structure helps to minimize errors due to its own weight. While rotating the proposed sensor, the 5 mm displacement is applied at the end tip of the sensor. It acts as a displacement load in the beam structure to apply the bending moment. This rotation experiment is conducted 5 times at every 10 degree intervals. It takes long time to get all the data for a 360-degree total. In this case, there is a high possibility of an experimental error caused by the temperature change around the FBG sensors [20–22]. Therefore, the temperature compensation system of the FBG sensor is required. In this experiment, additional FBG sensors of the same Bragg wavelengths used in bending sensors are connected together in an optical drive system. The FBG sensors are placed in a strain-free state and installed so that they are affected by temperature only. Finally, when measuring strain in the bending sensor, the signal from the separately attached FBG sensor is used for temperature compensation.
In Fig. 5, the measured strain by each FBG is plotted. As expected in Eq. (3), there is a phase difference of 2pi/3 between each sensor’s strain measurements, showing a sinusoidal curve. Using Eqs. (5) and (6), the direction of the applied moment and its magnitude are calculated. In the calculation of the moment, the normalized moment, only ${\varepsilon _A}/\textrm{cos}\theta $ is considered in Eq. (6), since other terms are constant.
In Fig. 6 and Fig. 7, the measurement results of the moment direction and its amplitude are shown. The mean offset value of the angle difference between the applied angle and measured direction of the neutral axis is 16.03° (slightly different from the y-intercept value of −14.821°) with a standard deviation of 1.44°. The representing normalized moment value by the strain divided by the angle cosine shows an average of 167.23 with a standard deviation of 2.66. The error in the measurement may be added from the fabrication arrangement of each FBG that can be assured from the measured angles in an asymmetric shape. Practically, it is hard to arrange in perfect 2pi/3 phase and the same central distance, therefore, the measurement includes an angular geometrical error.
3.2 Structural bending measurement (surface-mount configuration)
The structural bending measurement in the surface-mount configuration is considered. As hown in Fig. 8, the triangular bending FBG sensor is attached to the aluminum round bar of 2.38 mm diameter. The 65 mm length of the round bar is fixed at the top and the displacement load is applied at the 35 m position from the fixture to verify the proposed bending sensor rotating the round bar with the bending sensor. Since the second moment of inertia of the round bar is small and constant with regards to the rotation angle, the theoretically expected strain of each FBG should be a sine curve with different amplitudes and phases. Experimentally measured strain of each FBG by the 0.5 mm tip displacement is shown in Fig. 9. Experimental results show some discrepancies from the perfect sine curve due to installation inaccuracies and fabrication errors.
Fig. 8. Directional bending sensor for structural measurement and application of displacement loads.
Similar to self-bending measurement, the measurement of the moment direction in the surface-mount configuration is calculated from each FBG’s strain. In Fig. 10, the experimentally calculated bending direction versus the applied rotation angle is plotted using Eq. (9). For the applied 0.5 mm tip displacement, the mean value of the angular offset is 14.14° (slightly different from the y-intercept value of −14.544°), and the standard deviation is 9.99°. As explained previously, the magnitude of the applied moment can be calculated by the relationship in Eq. (8), however, the fabrication error, i.e. the small departure from the exact triangle shape seems to be sensitive to moment magnitude. Error compensation strategy for the fabrication error is under development. The beauty of the proposed method is that Eq. (9) is independent of the second moment of area. So the value of inclination angle obtained from the experiment is correct regardless of the second moment of area. However, the calculation of the moment requires an accurate value of the second moment of area. In Fig. 8, the difference between maximum and minimum values of the second moment of area is about 5%, which may cause an error in the moment calculation. The error by the second moment of FBG sensors becomes negligible if the main structure is sufficiently larger than the FBG sensors.
4. Conclusion and discussions
We propose a directional bending sensor by a mechanically bonded FBGs in a radially placed triangular shape to measure the bending moment and its applied direction. This set-up made of three FBG sensors provides a simple relationship of the bending moment to the independent strain measurement of each FBG, in which two sensor configurations such as the self-bending measurement and the structural bending measurement in a surface-mount (or embedded) measurement are experimentally demonstrated. In self-bending measurement, the bending direction is accurately measured with the error standard deviation of 1.44° and in the surface-mount set-up, the bending direction is obtained with the error standard deviation of 9.99°. Once the bending direction is obtained, the bending magnitude applied to the structure is easily calculated using the proposed equations. Moreover, The proposed directional bending sensor is applicable in various engineering fields from large structures to small medical devices.
Funding
National Research Foundation of Korea (2016R1D1A1A09917611); Korea Institute of Energy Technology Evaluation and Planning (20194010201800).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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