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Abstract
To meet the needs of vibration monitoring with special requirements for the size and quality of accelerometers, a miniaturized fiber Bragg grating accelerometer based on flexible hinges is proposed in this paper. The sensor uses a flexible hinge as an elastic body, and the suspended arc package realizes the miniaturization of the accelerometer. At the same time, the grating prepared by bending-resistant optical fiber successfully solves the problem of light loss in arc-shaped packaging. The structural model and principle of the accelerometer are introduced, and its sensing characteristics are analyzed theoretically and by simulation. The physical size of the prepared accelerometer is 17 mm × 12 mm × 10 mm, and its mass is only 4.44 g. The experimental results show that the resonant frequency of the accelerometer is about 900 Hz, the sensitivity is 26.962 pm/g in the flat range of 20-400 Hz, and the lateral interference is less than 5%. The accelerometer is suitable for medium and low frequency vibration monitoring in narrow spaces in aerospace and other fields.
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1. Introduction
Compared with traditional electronic sensors, fiber Bragg grating (FBG) sensors have attracted more and more attention for their advantages of anti-electromagnetic interference, corrosion resistance, long-distance transmission, and easy realization of distributed measurement [1–3]. In recent years, FBG acceleration sensors have been increasingly applied in important fields such as mechanical equipment [4], railways [5], [6], bridges [7] and oil industry [8]. According to the encapsulation method of FBG, FBG acceleration sensor can be divided into two types: full paste type and suspension type. The fully pasted FBG acceleration sensor pastes the FBG on the surface of the elastomer or embeds the elastic material for measurement. The fully pasted FBG is prone to chirp and uneven strain. The suspended FBG acceleration sensor realizes the measurement of acceleration by fixing the two ends of the FBG on the cantilever beams [9–12], diaphragms [13–15], or flexible hinges [16–21].
Suspended FBG acceleration sensors based on cantilever beams are often large in volume and mass, and are mostly used for low-frequency measurement. The diaphragm-based FBG acceleration sensor has strong anti-lateral interference ability, but the diaphragm structure limits the size of the sensor. Flexible hinges have many advantages such as no friction, easy processing, small size, etc., and are widely used in FBG acceleration sensors. Zhang et al. [18] proposed a 2-D medium–high frequency FBG accelerometer based on a universal flexure hinge. The resonance frequency of the sensor in the Y-axis direction is 1060 Hz, and the sensitivity is only 12 pm/g. The distance between the two fixed points of FBG is 12 mm, and the horizontal dimension is larger. Yan et al. [19] proposed an FBG accelerometer based on parallel double flexible hinges, with a sensor measuring range of 30–200 Hz and a sensitivity of up to 54 pm/g. Han et al. [20] proposed a FBG vibration sensor based on orthogonal flexure hinge structure. The sensor has a sensitivity of 41.2 pm/g in the flat frequency range of 20-800 Hz. Wang et al. [21] realized a three-dimensional acceleration measurement based on composite flexible hinges. The sensor's Z-axis sensitivity is 20.3 pm/g, and the measurement frequency range is only 0-250 Hz. In the flexure hinge based acceleration sensors, the FBG is usually placed above the flexible hinge, and the two ends of the FBG are respectively connected to the mass and the base. In order to ensure long-term and reliable fixation, each end of the FBG is usually glued to fix the length of at least 8 mm. The suspended linear encapsulation limits the miniaturization of the sensor size.
Some achievements have been made in the research of the miniaturized FBG accelerometers. Wang et al. [22] proposed a miniaturized FBG accelerometer based on a thin polyurethane shell, which reduces the volume of the sensor by inserting the mass into the inner of the shell. The size of the sensor is Φ23 mm×17 mm, and the sensitivity on the flat frequency response range is 54 pm/g, but the resonance frequency is only 480 Hz. The measuring direction of the sensor is the same as that of the optical fiber suspension direction. The packaged sensor has a larger size and poor practicability. Wang et al. [23] proposed a high frequency FBG accelerometer based on the esteel tube-mass block elastic structure. The sensor has a high resonance frequency of 3806 Hz and the size of the sensor is Φ20 mm×52 mm. Li et al. [24] proposed an ultra-small FBG accelerometer with sensor sensitivity and resonance frequency of 244 pm/g and 90 Hz, respectively. Sensors that directly use optical fibers as elastic deformation elements have poor impact resistance. Wei et al. [25] proposed a miniaturized FBG vibration sensor, the size of the sensor is 27 mm × 11 mm × 22 mm, the mass of the mass block is 5 g, but the sensitivity is only about 12 pm/g. Moreover, the structure of the sensor is relatively complicated, and the fixed length of the optical fiber is only 3 mm, which leads to the low reliability of the sensor.
In existing studies, the weight and size of the FBG accelerometer are often large, which limits the application of the sensor in the field of vibration monitoring. In view of this situation, we propose a miniaturized FBG accelerometer based on flexible hinge. The sensor uses a flexible hinge as the elastic body, and the suspended arc encapsulation realizes the miniaturization of the sensor. Using the characteristics of bending-resistant optical fibers, bending-resistant fiber gratings are prepared to solve the problem of optical loss caused by the miniaturized arc packaging method. In this article, the sensor’s sensing characteristics are theoretically analyzed, and the resonance frequency and sensitivity of the sensor are verified by simulation analysis. Finally, the sensor is experimentally studied. The accelerometer is small, high in resonance frequency, greater sensitivity, and strong in lateral anti-interference ability, and shows excellent application prospects in the field of vibration monitoring.
2. Design of the accelerometer
2.1 Structure model and principle of the accelerometer
The detailed design of the proposed miniaturized FBG accelerometer is illustrated in Fig. 1. This sensor is mainly composed of a semicircular mass block, a straight circular flexure hinge and an arc base. The radius of the semicircular mass and the arc base are the same. There are arc grooves on the surface of the mass and the base. Both ends of the FBG are fixed in the optical fiber groove by glue. The FBG is suspended above the straight circular flexible hinge, and the base and the package shell are connected by threads. To adjust the sensitivity and resonance frequency of the sensor, the width of the semicircular mass is greater than the width of the flexible hinge.
When an acceleration excitation along the measurement direction is generated externally, the masses will rotate slightly relative to the base around the center of flexible hinge, driving the FBG to stretch or compress and converting the vibration acceleration into FBG strain. This leads to a shift in the FBG central wavelength. Vibration signal information can be obtained from the shift of the FBG central wavelength. In this design, the suspended arc encapsulation further reduces the lateral size of the sensor, and the sensitivity of the sensor can be adjusted by changing the width of the semicircular mass.
2.2 Derivation of sensitivity formula
When the acceleration of the external excitation signal acts on the sensitive direction of the sensor, the mass will slightly rotate relative to the center of flexible hinge under the action of inertial force. The mechanical model of the sensor is shown in Fig. 2.
The torque balance equation of the system can be expressed as:
where m is the mass of the mass block, a is the vibration acceleration in the Z-axis direction of the sensor, e is the distance from the mass center of the mass to the center of the straight circular flexure hinge, kf is the elastic coefficient of the optical fiber, Δl is the displacement of the FBG fixed point in the X-axis direction, h is the vertical distance between the optical fiber and the center of the flexure hinge, K is the stiffness of the straight round flexure hinge, and θ is the rotation angle of the mass relative to flexible hinge.The distance from the mass center of the mass to the center of the straight circular flexure hinge e can be obtained by the following:
where r is the radius of the straight circular flexible hinge, the distance between the center of mass of the semicircular mass and the center of the circle b can be obtained by the following:The elastic coefficient of the optical fiber kf can be expressed as follows:
where l is the bonding span of the optical fiber, and Af and Ef are the cross-sectional area and elastic modulus of the optical fiber, respectively.The stiffness of the straight round flexure hinge is given by [26]:
Since θ is very small, we can take sinθ ≈ θ. From the geometric relationship, we have the following:
When deformation occurs between spans, the strain ɛ corresponding to the grating can be expressed as follows:
The FBG accelerometer is a dynamic measurement device. The FBG accelerometer analyzes the acceleration signal based on the transient relative wavelength shift of the FBG center wavelength. As the quasi-static change, temperature has almost no effect on the change of the FBG center wavelength in a short time. Therefore, the temperature sensing characteristics of FBG are ignored in the theoretical analysis. The effects of strain on the Bragg wavelength shift can be expressed as follows:
Combining (1), (6), (7) and (8), the sensitivity of the sensor can be expressed as follows:
2.3 Derivation of resonance frequency formula
The resonance frequency is an important indicator to account for the performance of an accelerometer. Suppose the moment of inertia of the mass around the center of flexible hinge is J and θ is the generalized coordinate for obtaining the Lagrangian function:
The strain potential energy of the optical fiber can be obtained by the following:
The elastic potential energy of the hinge can be expressed as follows:
The kinetic energy of the mass block can be described as follows:
The Lagrangian equation for the conservative force can be written as follows:
Substituting (10–13) into (14), the dynamic equation for the system can be calculated as follows:
The resonance frequency of the system can be obtained by the following,
According to the moment of inertia formula and the parallel axis theorem, the moment of inertia can be described as follows:
The structure and material characteristics of the FBG accelerometer are shown in Table 1. Based on the data in the Table 1, the theoretical resonance frequency and sensitivity of the sensor are 1249.46 Hz and 26.77 pm/g, respectively.
Table 1. Parameters for the FBG accelerometer structure and material properties
For the bending-resistant fiber grating in the acceleration sensor, the special bending-resistant fiber G.657.B produced by YOFC is selected. It is prepared by optical fiber hydrogen carrier, UV laser mask method and other processes, and the main performance parameters are shown in the Table 2.
Table 2. Main optical and mechanical parameters for the bending-resistant FBG
3. Simulation analysis of the accelerometer
According to the size parameters given in Table 1, a three-dimensional model of the sensor was established through SOILDWORKS software and imported into ANSYS software. The material properties of the model were set according to the material characteristic parameters in Table 1. The flexure hinge part was divided into hexahedral grid, and the other parts of the sensor were tetrahedral grid. Cylindrical support constraint was imposed on the cylindrical hole on the base. The optical fiber was modeled by spring constraints with the same elastic coefficient. The sensor was subjected to modal analysis and sensitivity analysis.
3.1 Modal analysis
Figure 3 shows the first-order and second-order modes of the accelerometer based on modal analysis method, respectively. As shown in Fig. 3(a), the first-order mode shape of the sensor involves the semicircular mass block rotates slightly relative to the base around the center of the flexible hinge. The first-order resonance frequency is 1269.4 Hz, which is quite consistent with the theoretical value (1249.46 Hz). As shown in Fig. 3(b), the second-order mode shape of the sensor involves the semicircular mass block is twisted laterally around the center of the straight circular flexure hinge. The second-order resonance frequency is 4200.1 Hz, which is 3.31 times the first-order natural frequency, indicating that the miniaturized accelerometer has a good ability to resist lateral vibration interference.
3.2 Sensitivity analysis
The simulation calculates the displacement change (Δl) of the fixed point of the FBG corresponds to different acceleration amplitudes. The wavelength drift of FBG corresponding to different acceleration amplitudes is calculated by using (7) and (8) formula. Figure 4 is the curve of FBG wavelength drift corresponding to different acceleration amplitudes. According to the slope of the curve in Fig. 4, the sensitivity of the accelerometer along the main axis is 26 pm/g, which is consistent with the theoretical calculation value.
4. Experimental characterization of the sensing properties
4.1 Experimental system compositions
Figure 5 shows the physical product of the bending fiber Bragg grating accelerometer, in which the metal elastomer of the accelerometer is fixed to the base through threaded connection, and the circular arc mass block is kept in a suspended state during the fixation. The size of the sensor is 17 mm × 12 mm × 10 mm, and the mass of the mass block is only 4.44 g. The experimental system for the FBG accelerometer is shown in Fig. 6. The experimental system consists of two parts: a vibration sensing system and a demodulation system. The vibration sensing system (LAN-XI, made by the Danish B&K company, including a vibration exciter 4808, power amplifier 2718, and standard reference acceleration sensor 8305) is responsible for controlling the excitation generated by the vibration exciter, including the frequency and amplitude of the excitation. The demodulation system is responsible for collecting the central wavelength signal of the reflected light from the sensor grating under excitation. The demodulation system consists of the FBG demodulator and signal acquisition software. The FBG demodulator was self-developed based on an FPGA-IRS demodulation module produced by BaySpec Inc. The maximum sampling frequency was 8 kHz, and the resolution was 0.1 pm. The sensor was fixed onto the vibration base of the vibration exciter, the sensor grating was connected to the demodulator through a transmission fiber, and the central wavelength shift of the sensor was obtained by the FBG demodulator. The experiment was carried out at room temperature (25 °C).
4.2 Amplitude-frequency characteristic
The experimental device shown in Fig. 6 was used to perform the excitation experiment. To study the amplitude-frequency characteristics of the sensor, the amplitude of the excitation acceleration was kept at 1 g, and the excitation frequency was varied in the range of 20 Hz to 1300 Hz with a step size of 50 Hz. The relationship between the central wavelength shift of the FBG and frequency under different excitation frequencies was measured, and the amplitude-frequency response curve for the sensor is shown in Fig. 7. The experimental results show that the resonance frequency of the accelerometer is approximately 900 Hz. The amplitude-frequency characteristic curve has better flatness below 400 Hz, and the operating frequency range of the sensor is 20-400 Hz.
4.3 Sensitivity characteristic
Through the vibration exciter, sinusoidal excitation signals of 50 Hz, 100 Hz, 200 Hz, 300 Hz and 400 Hz were given to the sensor, and the acceleration amplitude increased from 0.5 g to 3 g with a step length of 0.5 g. Each group of experiments was repeated three times. The variation curve for the wavelength shift with acceleration at different frequencies is shown in Fig. 8. The Bessel formula was used to calculate the standard deviation for the sensor. First, all subsamples’ standard deviations σ were calculated, where i = 1, 2 …N, and N = 6 is the number of calibration points. Then, the standard deviation of the sensor was calculated using $\sigma = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {\sigma _i^2} }$ . Finally, the repeatability error was calculated as eR= ησ / λFS, where η = 3, which is the coverage factor, and λFS is the maximum wavelength shift. As shown in Fig. 9, when the frequency of the excitation signal is 50 Hz, 100 Hz, 200 Hz, 300 Hz, and 400 Hz, the repeatability errors for the sensor are 3.16%, 3.46%, 2.03%, 2.30% and 1.98%, respectively.
Fig. 9. The liner fitting straight lines of the wavelength drift with acceleration at different frequencies.
The three sets of experimental data obtained at different frequencies were then averaged, and the least-squares fitting method was used to acquire the linear fitting lines, as shown in Fig. 8. The experimental results show that the wavelength shift for the sensor has a good linear relationship with the input acceleration amplitude. At excitation signal frequencies of 50 Hz, 100 Hz, 200 Hz, 300 Hz, and 400 Hz, the sensitivity of the sensor is 24.19 pm/g, 24.27 pm/g, 26.16 pm/g, 28.16 pm/g, and 32.03 pm/g, respectively. Within the working frequency of 20 - 400 Hz, the sensor sensitivity showed an average value of 26.962 pm/g.
4.4 Cross-interference characteristic
In addition to resonance frequency and sensitivity, anti-interference ability is also an important indicator for accelerometers. During the experiment, the sensor was fixed onto the vibration exciter according to the measurement direction and the transverse direction of the measurement direction, and a sinusoidal excitation signal with a frequency of 200 Hz and an amplitude of 1 g was applied to the sensor. The acceleration response curves for the sensor were obtained in the main and lateral directions, as shown in Fig. 10. The main direction wavelength shift for the sensor is 25.8 pm, the lateral wavelength shift does not exceed 1.1 pm, and the lateral interference degree for the sensor is less than 5%, which shows that the designed sensor has a good lateral anti-interference ability.
4.5 Discussion and analysis
The experimental values of the resonance frequency and sensitivity of the sensor are different from the theoretically calculated values. The main reasons are as follows:
- 1) There is a deviation between the actual processing size of the sensor and the theoretical design size.
- 2) Errors introduced in the sensor manufacturing process, such as FBG pre-stretching, may cause deformation of the elastomer structure.
- 3) The sensor is connected to the housing by screws, and there may be a gap between the two, resulting in the actual rigidity of the sensor being less than the theoretical rigidity.
- 4) Deviations caused by sensor packaging lead to poor sensing characteristics.
- 5) In addition to the above reasons, the calculation deviation caused by the theoretical calculation formula, the inconsistency of the experimental material properties and the theoretical value, the strain generated by the elastomer cannot be completely transmitted to the FBG, and other reasons can also cause this inconsistency.
The inconsistency between the theoretical resonance frequency and sensitivity of the miniaturized FBG accelerometer and the experimental results can be observed in many studies [25], [26]. Therefore, special attention should be paid to the control of errors in the research of miniaturized FBG accelerometers. Table 3 summarizes the characteristics of FBG acceleration sensors based on different structures, including resonance frequency, sensitivity, volume and weight.
Table 3. Summary of the characteristics of reported FBG accelerometers
5. Conclusion
A miniaturized FBG accelerometer based on flexible hinge is proposed in this paper. The accelerometer uses a flexible hinge as an elastic body, and a suspended arc encapsulation realizes the miniaturization of the sensor. Using the characteristics of bending-resistant optical fibers, bending-resistant fiber gratings are prepared to solve the problem of optical loss caused by the miniaturized arc packaging method. In this article, the sensor's sensing characteristics are theoretically analyzed, and the sensor characteristics are simulated and analyzed. The size of the accelerometer is 17 mm × 12 mm × 10 mm, and the mass of the mass block is only 4.44 g. The experimental results show that the resonance frequency of the accelerometer is about 900 Hz, the sensitivity is 26.962 pm/g in the flat range of 20-400 Hz, and the lateral anti-interference degree is less than 5%. The accelerometer shows an excellent application prospect in the field of small-space vibration monitoring.
Funding
Scientific research project of Hainan Sanya Yazhouwan Science and Technology City Administration Bureau (SKJC-2020-01-016).
Acknowledgments
This research is supported by Scientific research project of Hainan Sanya Yazhouwan Science and Technology City Administration Bureau (SKJC-2020-01-016).
Disclosures
The authors declare no conflicts of interest.
Data availability
Datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.
References
1. B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9(2), 57–79 (2003). [CrossRef]
2. R. D. Sante, “Fibre Optic Sensors for Structural Health Monitoring of Aircraft Composite Structures: Recent Advances and Applications,” Sensors 15(8), 18666–18713 (2015). [CrossRef]
3. T. Li, J. Guo, Y. Tan, and Z. Zhou, “Recent Advances and Tendency in Fiber Bragg Grating-Based Vibration Sensor: A Review,” IEEE Sens. J. 20(20), 12074–12087 (2020). [CrossRef]
4. R. P. Linessio, K. D. M. Sousa, T. D. Silva, C. A. Bavastri, P. F. D. C. Antunes, and J. C. C. D. Silva, “Induction Motors Vibration Monitoring Using a Biaxial Optical Fiber Accelerometer,” IEEE Sens. J. 16(22), 8075–8082 (2016). [CrossRef]
5. A. Catalano, F. A. Bruno, M. Pisco, A. Cutoloa, and A. Cusano, “An Intrusion Detection System for the Protection of Railway Assets Using Fiber Bragg Grating Sensors,” Sensors 14(10), 18268–18285 (2014). [CrossRef]
6. K. Yüksel, D. Kinet, V. Moeyaert, G. Kouroussis, and C. Caucheteur, “Railway monitoring system using optical fiber grating accelerometers,” Smart Mater. Struct. 27(10), 105033 (2018). [CrossRef]
7. P. F. C. Antunes, HFT Lima, NJ Alberto, H Rodrigues, PMF Pinto, JL Pinto, RN Nogueira, H Varum, AG Costa, and P. S. B Andre, “Optical Fiber Accelerometer System for Structural Dynamic Monitoring,” IEEE Sens. J. 9(11), 1347–1354 (2009). [CrossRef]
8. X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors 17(3), 429 (2017). [CrossRef]
9. D. Feng, X. Qiao, H. Yang, Q. Rong, R. Wang, Y. Du, M. Hu, and Z. Feng, “A Fiber Bragg Grating Accelerometer Based on a Hybridization of Cantilever Beam,” IEEE Sens. J. 15(3), 1532–1537 (2015). [CrossRef]
10. F. Liu, Y. Dai, K. Joseph, and M. Yang, “A Low Frequency FBG Accelerometer with Symmetrical Bended Spring Plates,” Sensors 17(12), 206 (2017). [CrossRef]
11. O. P. Parida, J. Thomas, J. Nayak, and S. Asokan, “Double-L Cantilever Based Fiber Bragg Grating Accelerometer,” IEEE Sens. J. 19(23), 11247–11254 (2019). [CrossRef]
12. S. Li, Z. Feng, Q. Qi, R. Wang, R. Zhou, and X. Qiao, “Fiber Bragg grating accelerometer based on symmetrical tilting cantilever beams and solder glass packaging for harsh environment,” Opt. Fiber Technol. 65, 102579 (2021). [CrossRef]
13. T. Li, C. Shi, Y. Tan, R. Li, Z. Zhou, and H. Ren, “A Diaphragm Type Fiber Bragg Grating Vibration Sensor Based on Transverse Property of Optical Fiber with Temperature Compensation,” IEEE Sens. J. 17(4), 1 (2016). [CrossRef]
14. T. Li, Y. Tan, X. Han, K. Zheng, and Z. Zhou, “Diaphragm Based Fiber Bragg Grating Acceleration Sensor with Temperature Compensation,” Sensors 17(12), 218 (2017). [CrossRef]
15. Q. Liu, X. Qiao, Z. Jia, H. Fu, H. Gao, and D. Yu, “Large Frequency Range and High Sensitivity Fiber Bragg Grating Accelerometer Based on Double Diaphragms,” IEEE Sens. J. 14(5), 1499–1504 (2014). [CrossRef]
16. Y. Dai, G. Yin, B. Liu, G. Xu, and J. M. Karanja, “Medium-high frequency FBG accelerometer with integrative matrix structure,” Appl. Opt. 54(11), 3115–3121 (2015). [CrossRef]
17. L. Wei, L. Yu, J. Wang, D. Jiang, Q. Liu, and Z. Liu, “An FBG-Sensing Two-Dimensional Vibration Sensor Based on Multi-Axis Flexure Hinge,” IEEE Sens. J. 19(10), 3698–3710 (2019). [CrossRef]
18. Y. Zhang, W. Zhang, Y. Zhang, L. Chen, T. Yan, S. Wang, L. Yu, and Y. Li, “2-D Medium–High Frequency Fiber Bragg Gratings Accelerometer,” IEEE Sens. J. 17(3), 614–618 (2017). [CrossRef]
19. B. Yan and L. Liang, “A Novel Fiber Bragg Grating Accelerometer Based on Parallel Double Flexible Hinges,” IEEE Sens. J. 20(9), 4713–4718 (2020). [CrossRef]
20. H. Song, Q. Wang, M. Liu, and Q. Cai, “A Novel Fiber Bragg Grating Vibration sensor Based on Orthogonal Flexible Hinge Structure,” IEEE Sens. J. 20(10), 5277–5285 (2020). [CrossRef]
21. H. Wang, L. Liang, X. Zhou, and B. Tu, “New Fiber Bragg Grating Three-Dimensional Accelerometer Based on Composite Flexure Hinges,” Sensors 21(14), 4715 (2021). [CrossRef]
22. J. Wang, Y. Zeng, C. Lin, Z. Hu, G. Peng, and Y. Hu, “A Miniaturized FBG Accelerometer Based on a Thin Polyurethane Shell,” IEEE Sens. J. 16(5), 1210–1216 (2016). [CrossRef]
23. X. Wang, Y. Guo, L. Xiong, and H. Wu, “High-Frequency Optical Fiber Bragg Grating Accelerometer,” IEEE Sens. J. 18(12), 4954–4960 (2018). [CrossRef]
24. K. Li, G. Liu, Y. Li, J. Yang, and W. Ma, “Ultra-Small Fiber Bragg Grating Accelerometer,” Appl. Sci. 9(13), 2707 (2019). [CrossRef]
25. L. Wei, D. Jiang, L. Yu, H. Li, and Z. Liu, “A Novel Miniaturized Fiber Bragg Grating Vibration Sensor,” IEEE Sens. J. 19(24), 11932–11940 (2019). [CrossRef]
26. Y. F. Wu and Z. Y. Zhou, “Design of flexure hinges,” Eng. Mech. 19(6), 136–140 (2002).
