# A Low Frequency FBG Accelerometer with Symmetrical Bended Spring Plates

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle and Structure of the Sensor

## 3. Mathematical Model and Calculations

_{BH}is the horizontal displacement of point B; E is the elastic modulus of beryllium bronze, which is the material of the bended spring plate; I is the cross-section inertia moment of the bended spring plate; I = bt

^{3}/12, b is the width of the plate; t is thickness of the plate; m = ρbcd; ρ is the density of the mass, and the mass is made of the ordinary carbon steel; c is the length of the mass; d is the height of the mass; a is the exciting acceleration; l = k + c/2; k is the length of the horizontal segment; r is the radius of the bended segment; h is the height of the vertical segment; F is the horizontal force of the fiber at point B; E

_{f}is the elastic modulus of the fiber materials; A

_{f}is the cross-section area of the fiber; l

_{f}is the distance from point B to the center of the FBG. Based on the simultaneous Equation (1), we can get:

_{e}is elasto-optical coefficient. In general, the elasto-optical coefficient of the germanium-doped silica fiber is 0.22. So, we get the sensitivity of the sensor:

_{11}is the horizontal displacement of the barycenter of the mass under the action of the unit horizontal force, δ

_{12}is the horizontal displacement of the barycenter under the action of the unit vertical force, δ

_{21}is the vertical displacement of the barycenter under the action of the unit horizontal force, δ

_{22}is the vertical displacement of the barycenter under the action of the unit vertical force, ω is the angular velocity of the barycenter, ∆x is the horizontal displacement of the mass center under the inertia force, ∆y is the vertical displacement of the barycenter under the inertia force.

_{11}, δ

_{21}, δ

_{12}and δ

_{22}can be obtained by the unit load method [18] and the superposition principle. Because the derived procedure of these parameters is very complex, omitting the intermediate procedure, we can get:

_{1}is the pulling force of the FBG under the action of the unit horizontal load, F

_{2}is the pulling force of the FBG under the action of the unit vertical load. They can be calculated by:

^{2}are obtained:

## 4. Optimization Design of the Parameters

_{m}. For ensuring the sensor works in a lower frequency range, we can restrict the value of f

_{m}to the range of 50 to 60 Hz. Additionally, structure size of the sensor should be controlled in a relatively reasonable range. For example, the total length of the sensor is restricted to less than 50 mm, the total height is restricted to within 30 mm, the space between the bottom surface of the mass and the top surface of the base cannot be less than 2 mm, and so on. Finally, the constraint conditions can be expressed as:

## 5. Experimental Results and Discussion

^{2}, and an excitation frequency from 5 Hz to 60 Hz, we were able to acquire the data of the FBG wavelength shifts from the interrogator. Figure 5 shows the frequency response curve of the FBG accelerometer.

^{2}and excitation frequency was 5 Hz or 10 Hz, as shown in Figure 6. It can be seen from the figure that the time domain waveforms had a good response to the external excitation in the work band. In the practical applications, the frequency spectrum can be obtained by Fourier transform from the time domain waveforms. Then the frequency information of the measured vibration body can be analyzed.

^{2}, as shown in Figure 7. It can be seen that the FBG accelerometer demonstrated fairly good linearity related to the applied acceleration. Through the slope of the straight line, we estimated the sensitivity as 1067 pm/g at 5 Hz, 1084 pm/g at 10 Hz, 1126 pm/g at 15 Hz and 1166 pm/g at 20 Hz, respectively. From Figure 6 and Figure 8, it can be seen that the response sensitivity of the accelerometer had little correlation with the excitation frequency. For practical use, we must implement the calibration of the sensor based on the shake table test results.

^{2}and the excitation frequency set at 20 Hz, the wavelength response in the work and transverse directions was demonstrated as shown in Figure 8. As a result, we realized that the transverse response amplitude is about 6.9% of the work direction amplitude. The prototype FBG accelerometer has a preferable anti-interference capacity.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Simplified structure diagram of the sensor and the meaning of each symbol. x, y: the horizontal and vertical directions; F: force at point B; h: the height of CD segment; r: the radius of the bended segment; k: the length of horizontal segment; m: mass; c, d: the length and height of the mass; a: the exciting acceleration; dθ: the infinitesimal angle.

**Figure 3.**The influence of major parameters on the sensitivity and eigenfrequency. The influence of thickness t (

**a**); the influence of width b (

**b**); the influence of length c (

**c**); the influence of height h (

**d**).

**Figure 8.**Cross-sensitivity of the sensor: response under excitation acceleration of 1 m/s

^{2}and frequency at 20 Hz.

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**MDPI and ACS Style**

Liu, F.; Dai, Y.; Karanja, J.M.; Yang, M.
A Low Frequency FBG Accelerometer with Symmetrical Bended Spring Plates. *Sensors* **2017**, *17*, 206.
https://doi.org/10.3390/s17010206

**AMA Style**

Liu F, Dai Y, Karanja JM, Yang M.
A Low Frequency FBG Accelerometer with Symmetrical Bended Spring Plates. *Sensors*. 2017; 17(1):206.
https://doi.org/10.3390/s17010206

**Chicago/Turabian Style**

Liu, Fufei, Yutang Dai, Joseph Muna Karanja, and Minghong Yang.
2017. "A Low Frequency FBG Accelerometer with Symmetrical Bended Spring Plates" *Sensors* 17, no. 1: 206.
https://doi.org/10.3390/s17010206