A fiber Bragg grating vibration sensor based on crossed variable section beam

This paper presents a triaxial vibration sensor that can be used for vibration acceleration detection. The main body of the elastic structure of the sensor is a variable section beam. The fiber Bragg gratings (FBG) are applied to the surface of the crossed variable section beam. The theoretical equation for the sensitivity of this sensor is given by the theoretical analysis of material mechanics. The simulation of modal analysis was carried out by means of the finite element method. The resonant frequencies of the proposed fiber Bragg grating triaxial sensor in the Z, X, and Y directions are 363.01 Hz, 466.34 Hz, and 466.82 Hz, respectively. Finally, the sensor was calibrated and tested in each direction of vibration, and the results showed that the sensitivity of Z, X, and Y directions were 6.66 pm/g, 4.61 pm/g, and 4.61 pm/g respectively, which proved that the FBG sensor has good sensing response for vibration in three directions.


Introduction
Since the advent of fiber optic communication in 1970, fiber optic sensing has also received attention.Compared with traditional piezoelectric and mechanical vibration sensors, fiber-optic grating vibration sensors have the advantages of anti-electromagnetic interference [1], no zero drift, small size, and easiness in being multiplexed, so fiber-optic grating vibration sensors can be widely used in structural health inspection [2], earthquake detection [3][4], medical and thermal engineering [5], etc.
Several scholars have investigated different types of FBG vibration sensors in recent years.For example, FBG sensors based on L-beam [6], flexible hinge [7], diaphragm [8], cantilever beam [9], etc.They all show excellent detection performance in different vibration scenarios.However, research on three-dimensional FBG vibration sensors is still lacking.Masek et al. proposed a three-dimensional vibration sensor using six FBGs [10].Used as elastomer, the FBGs were easy to be destroyed.Li et al. designed an FBG three-axis vibration sensor by combining a copper cylinder and a cross-beam-type elastomer [11], which consists of four constant section beams.However, for the FBG pasted beam sensors, when the size of FBG could not be negligible compared to the elastomer, the internal stress of the fiber grating is not uniform, resulting in distortion of the reflection spectrum.Under the same force, the surface strain of a variable section beam is more uniform compared to that of a constant section beam.Moreover, the variable section beam has higher stiffness, so it could be used in some more intense vibration environments.

Theoretical analysis of the FBG vibration sensor
In this paper, COMSOL Multiphysics is used to model the sensor and carry out the mechanical simulation.The structure of the sensor is shown in Figure 1.The material of the cylinder in the figure is copper and the material of the crossed variable section beam structure is aluminum alloy.Five FBGs are pasted on the upper and lower surfaces of the variable section beams, and they can detect three-dimensional vibration acceleration.When the acceleration of the external environment is detected by the sensor, because of the inertia effect, it is equivalent to a force applied to the copper cylinder at the center in the same direction as the acceleration of the vibration.From the symmetric structure, it can be considered that the sensor has the same sensing characteristics for the vibration along X-axis and Y-axis directions.Taking the force in the X direction as an example, when the sensor detects the force F X along the X-axis direction, the variable section beam A bends and shears.Meanwhile, beam B twists, as shown in Figure 2 (a).Let the force at point A be FA, and at point C be FC.Considering the bending deformation of beam A, the angular displacement  of beam A can be expressed as where l is the length of the beam, h is the thickness of the beam, r 1 is the equivalent radius of the central mass block, b 1 and b 2 are the short side length and the long side length of the variable section beam, k is the shear coefficient, E is the elasticity modulus, G is the modulus of shear, and P is a constant.
Considering the torsion occurring between beam B and beam D, the torsion angle is obtained as Using the force balance equation and geometrical characteristics, According to the Timoshenko theory, the strain on the upper surface of beam A can be expressed as Similarly, when the sensor vibrates along the Z-axis direction, from the simulation results, the vibration state is shown in Figure 2 (b).According to the mechanical equilibrium equation, it can be obtained that When the sensor vibrates, the strain of the FBG is similar to that of the beam.The fiber gratings on beam A and beam C, which are symmetric about the Y-axis, are FBG1 and FBG3, respectively.The relationship between their resonant wavelength offset and surface strain can be represented by the following equations: ( ) ( ) where P e is the elasto-optical coefficient, α is the thermal expansion coefficient, and ξ is the thermo-optical coefficient.The effect of temperature on the resonant wavelength can be eliminated by subtracting the two equations.Since the resonant wavelengths of fiber gratings are all around 1550 nm, the equation could be simplified as 2 (1 ) The sensitivity of the proposed FBG sensor in the Y direction can also be derived by this calculation method.By means of the wavelength offset of FBG pasted on the upper and lower surfaces, the acceleration in the Z direction can be calculated.

Mechanical simulation of the FBG vibration sensor
To further explore the inherent vibration characteristics of the elastic structure sensor designed in this paper, modal analysis was performed using the software.and the results are shown in Figure 3. From the results of the modal analysis, the first three orders of the modal vibration pattern correspond to the states of the sensor when it vibrates along the Z, Y and X directions, and the corresponding characteristic frequencies are 363.1 Hz, 466.34 Hz and 466.82Hz, respectively.The fourth-order modal vibration pattern corresponds to a characteristic frequency of 3397 Hz, which is much larger than the characteristic frequencies of the first three orders.This means that the sensor will not show a vibration mode similar to the fourth-order mode in the measurement frequency range.

Experimental environment
An FBG string with different reflection wavelengths (1542 nm, 1546 nm, 1550 nm, 1554 nm, 1558 nm, reflectivity of 87.41%, effective grid length of 5 mm) is passed through the circular hole in the sensor structure and pasted on the beams.The wavelength drift of the five FBGs was demodulated by a Fiber Bragg Grating Interrogation Analyzer (FBGA, Bay Spec, FBGA-F-1525-1565-FA ).The sensor with FBGs attached was fixed on the shaker (SINOCERA ® , JZK-1) which was driven using a power amplifier (SINOCERA ® , YE1311). Figure 4 (a) shows the method of attaching the fiber Bragg grating.The environment of the calibration experiment is shown in Figure 4 (b).

Experimental results
In order to perform calibration experiments on the proposed acceleration sensor, the vibration frequency (100 Hz) of the shaker is kept constant.The acceleration of the shaker can be adjusted by controlling the output voltage of the power amplifier.The PC receives the resonant wavelength demodulated by the interrogation analyzer and performs signal processing such as windowing, filtering and peak tracking.Take the FBG with the resonant wavelength of 1554 nm as an example.
When the acceleration is 2 m/s 2 , 5 m/s 2 and 7 m/s 2 respectively, the variation of the demodulated wavelength shift with time is shown in Figure 5. From this figure, we can learn that the wavelength shift and acceleration of vibration correspond one to one.Similar to theoretical analysis, the larger the acceleration is, the greater the wavelength shift of demodulation is.As the measurement time is different, the three demodulated differential curves not only have differences in amplitude but also exhibit different phases.

EEICE-2023
Journal of Physics: Conference Series 2625 (2023) 012068 By measuring more wavelength shifts for different accelerations, we can draw wavelength offset-acceleration curves.Combined with the analysis in Section 2.1, the curves in Figure 6 can be obtained by the subtraction operation of the demodulated waveforms of two FBGs.The wavelength-acceleration curve shown in Figure 6 (a) is the calibration curve of the sensor vibration in the Z-axis.From this figure, the linear relationship fitted from the data is 0.00068 0.00048 y x   , and this result shows that the sensitivity of the proposed sensor vibrating along the Z direction is 6.66 pm/g.Similarly, the linear relationship fitted from Figure 6 (b) is 0.00047 0.00028 y x   , so the sensitivity in the X and Y directions is 4.61 pm/g.However, unlike the theory results, the fitted straight line does not pass through the zero point.Non-uniform pasting of fiber Bragg gratings, which leads to this result, is difficult to avoid.In addition, the measurement error of the experiment also has an impact on the calibrated sensitivity curve.
In practical engineering, 70% of the resonant frequency is usually taken as the working bandwidth.Therefore, the working bandwidths of the proposed vibration sensors in X, Y and Z directions are 20-326.44Hz, 20-326.77Hz and 20-254.11Hz, respectively.The above results indicate that in the direction of larger working bandwidth, sensitivity is lower.This reflects the mutual constraints between the dynamic range and sensitivity of beam-type sensors Several novel vibration sensors are summarized in Table 1.Compared with these sensors, the sensor proposed in this paper has more sensing dimensions and a higher dynamic range, but is still insufficient in sensitivity.

Conclusion
A fiber grating vibration sensor based on a crossed variable section beam was designed.The characteristics of this sensor can be summarized as follows: (1), as few FBGs as possible are used for sensing; and (2), the elastomer consists of variable section beams that ensure its higher stiffness, which can improve the stability of sensor operation in intense vibration environments.The resonant frequencies of this sensor in Z, X and Y directions are 363.01Hz, 466.34 Hz and 466.82Hz, respectively.The calibration experiment results indicate that the sensitivity of Z, X and Y directions were 6.86 pm/g, 4.61 pm/g and 4.61 pm/g, respectively.In summary, the sensor achieves three-dimensional vibration sensing and can be applied to low-frequency vibration signal detection.
In the follow-up study, we can balance sensitivity and the dynamic range by modifying the size of the variable section beam and mass block, so the can can suit a wide range of vibration environments.

Figure 1 .
Figure 1.(a) Simulation of fiber grating sensor structure (b) Sensor with FBGs

Figure 3 .
Figure 3. Simulation diagram of modal analysis.( a), (b), (c) and (d) are the first four orders of vibration modes of the sensor

Figure 6 .
Figure 6.Calibration curve of vibration in Z (a), X and Y direction (b).

Table 1 .
A comparison of several novel vibration FBG sensors