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Abstract
Jerk is directly related to a physical mutation process of structural damage and human comfort. A fiber optic jerk sensor (FOJS) based on a fiber optic differentiating Mach–Zehnder interferometer is proposed. It can directly measure jerk by demodulating the phase of interference light, which avoids the high-frequency noise interference caused by differentiating the acceleration. The sensing theory and sensor design are given in detail. The experimental and theoretical results agree, demonstrating that the FOJS has a high sensitivity, an ultralow phase noise floor, a wide measuring range, and good linearity. The impact test shows that the FOJS can directly measure jerk and has good consistency with a standard piezoelectric accelerometer. The FOJS has potential applications in earthquake engineering, comfort evaluations, and railway design. This is the first time that directly measuring jerk with an optical sensor is reported.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Jerk, which is the derivative of acceleration, is closely related to motional mutations or rapid changes in forces. The physical manifestation of jerk is mostly structural damage, material fatigue, or human comfort. For nearly two decades, jerk is important for comfort evaluations [1,2], motion control [3–6], earthquake engineering [7–9], and other applications. Especially in earthquake engineering, the jerk of strong ground motion and its response spectra reflect the maximum impact response of structures with different periods under strong ground motion. Measuring jerk is critical for designing ground motion response spectra [10–12], analyzing nonstationary characteristics of ground motion frequency [11], and designing earthquake resistant structures [8,9].
The most common method of jerk measurement utilizes differentiation circuits or numerical differentiation to differentiate the acceleration measured by accelerometers [13]. This method easily amplifies high-frequency noise interference because the differential is sensitive to noise. It also requires complex filter processing [14]. To measure jerk directly, jerk sensors have been designed and reported. Tamura et al. combined a vibratory gyroscope and a cantilever to develop a jerk sensor that measures jerk directly by measuring the output voltage [15]. Tsuchiya et al. developed a jerk sensor based on a servo-type system and feedback control that outputs jerk directly as a voltage value [16]. Fujiyoshi et al. proposed and developed a micro jerk sensor with viscous coupling on a micro electromechanical system (MEMS) [17]. Henmi et al. designed a jerk sensor based on piezoelectric devices that measured jerk directly by measuring the change rate of the charge [18]. Yang et al. designed a jerk sensor with a coupled oscillator sensing device based on absolute motion [19]. All the jerk sensors discussed above are based on the piezoelectric effect, the principle of electromagnetic induction or electromechanical coupling, which are electric jerk sensors and are highly sensitive to electromagnetic interference, high temperatures, electric leakage, and other factors. They are not suitable for long-term, long-distance, and real-time work in large buildings or extreme environments.
In this paper, we propose a fiber optic jerk sensor (FOJS) based on a differentiating Mach–Zehnder interferometer. An optical method to measure jerk directly is proposed. The sensing theory introduces the principle for detecting jerk. The sensor design analyzes the performance of the FOJS, and the experimental results agree with the theory. Furthermore, by conducting an impact test, we demonstrate that the FOJS can measure jerk directly.
2. Sensing theory
The proposed FOJS is based on a differentiating Mach–Zehnder interferometer, as shown in Fig. 1. We use an amplified spontaneous emission (ASE) light source with low coherence and the Mach–Zehnder interferometer to construct a differentiating optical path. The direct jerk measurement is achieved by measuring the absolute phase of the interference light. The laser from the ASE light source passes through the attenuator and isolator and then exits both arms of the 3×3 coupler. Because the fibers between the 3×3 coupler and 1×2 coupler are unequal in length, only two beams of light can interfere: clockwise through the 3×3 coupler → jerk sensing probe → 1×2 coupler → delay fiber → Faraday rotator mirror (FRM) → delay fiber → 1×2 coupler → 3×3 coupler and counterclockwise through the 3×3 coupler → 1×2 coupler → delay fiber → FRM → delay fiber → 1×2 coupler → jerk sensing probe → 3×3 coupler. When the jerk sensing probe detects a jerk signal, the fiber winding on the probe stretches and modulates the phase of the interferometer.
Ideally, the laser from the ASE source is split into two equal beams of light by the 3×3 coupler. Both light beams follow the same optical path but in opposite directions. One light beam passes through the jerk sensing probe first, and its phase modulated by the jerk sensing probe is φ(t). The other light beam passes through the delay fiber first, and its phase modulated by the jerk sensing probe is φ(t + T). T is the delay time of the delay fiber and is expressed as
where LD is the length of the delay fiber; n is the effective refractive index of the optical fiber core; and c is the velocity of light in vacuum.Given that both types of light are modulated by the jerk sensing probe, the phase of interference light can be expressed as [20]
where ψ is the phase of interference light; φ(t) and φ(t + T) are the phases modulated by the jerk sensing probe.According to Eq. (2), we could measure the derivative of the phase modulated by the jerk sensing probe by measuring the phase of interference light. The phase modulated by the jerk sensing probe can be expressed as
where ξ is the fiber’s strain-optic correction factor; λ is the central wavelength of the ASE source; and ΔL(t) is the change in fiber length.We find that the phase of the interference light is proportional to the velocity of the fiber length change, and their relationship can be expressed as
The jerk sensing probe not only modulates phase, but also is a vibration system for measuring the jerk. We can describe the vibration system with the expression of
where γ is the damping coefficient of the jerk sensing probe; ω0 = 2πf0 is the angular frequency of the jerk sensing probe; f0 is the resonant frequency of the jerk sensing probe; x is the displacement of the inertial mass relative to the probe; and y is the displacement relative to the ground reference frame.Given that the displacement of the inertial mass relative to the probe is the change in fiber length, we can combine Eqs. (4) and (5). Therefore, the transfer function between the phase of interference light and the jerk can be expressed as
The amplitude characteristic of the FOJS can be expressed as
The static jerk sensitivity of the FOJS can be expressed as
When the frequency of the jerk signal detected by the FOJS is lower than the resonant frequency, the static sensitivity shows a linear relationship between the jerk and the phase of interference light. This means that the FOJS can measure the jerk directly by demodulating the phase of interference light.
Three photodetectors (PDs) are used, along with a demodulation algorithm based on a 3×3 coupler. The phase of the interference light can be demodulated by the algorithm based on the 3×3 coupler. The signals detected by the three photodiodes can be expressed as [21]
For the FOJS, the maximum signal frequency of the power spectra density (PSD) of the modulated phase φ(t) is given by
The maximum signal frequency of the modulated phase is limited to 1/2πT. There is generated by the fiber optic differentiating Mach–Zehnder interferometer, which ensures the linear working of the FOJS.
3. Sensor design
3.1 Sensing probe
The jerk sensing probe is an improved version of the accelerometer we previously designed [22], as shown in Fig. 2(a). The jerk sensing probe has a double-diaphragm structure to improve lateral interference sensitivity. Fiber winds on the jerk sensing probe and is used as an arm of the differentiating Mach–Zehnder interferometer. When the jerk sensing probe senses the jerk, the movable cap and mass will vibrate and change the tensile velocity of the fiber length. The jerk will modulate the phase induced by the change in fiber length and be demodulated from the phase of the interference light.
Fig. 2. Jerk sensing probe of the FOJS and its encapsulation. (a) Schematic diagram of the jerk sensing probe; (b) Schematic diagram of the overall encapsulation. The third optical fiber at one end of the 3×3 coupler is cut, which is not shown.
To protect the jerk sensing probe and enhance its flexibility, the jerk sensing probe, 3×3 coupler, and 1×2 coupler are enclosed inside a stainless steel shell, as shown in Fig. 2(b). The steel shell is specially designed to isolate the acoustic and air disturbance. The fixed base of the jerk sensing probe is rigidly fixed to the steel shell by the support. The other arm of the Mach-Zehnder interferometer is a shorter-length optical fiber, which is fixed on the support of the steel shell with glue and is not affected by the sensing probe. The 3×3 coupler and 1×2 coupler are fixed on both sides of the support. The optical fibers are routed through the through-holes and protected by armored cables. The 3×3 coupler and 1×2 coupler are connected to the photodetectors and delay fiber, respectively. For long-term, long-distance, and real-time jerk measurements, we protect the delay fiber with the armored cable and shield it in an isolation box to reduce the acoustic, vibration and temperature disturbance. It has more advantages and potential than electric jerk sensors for strong ground jerk measurements in high-rise buildings, large bridges, and extreme environments.
3.2 Sensitivity
The equivalent stiffness coefficient of the jerk sensing probe is expressed as [22]
where N is the number of turns of optical fibers; l is the length of the optical fiber wrapped in a half circle; E, L, b, and h are the Young’s modulus, length, width, and thickness of the rectangular diaphragm, respectively; Ef and A are the Young’s modulus and cross-sectional area of the optical fibers, respectively; and r is the contact radius between the rectangular diaphragm and mass.We noted that the static jerk sensitivity of the FOJS described in Eq. (8) is a product of the delay time and the static acceleration sensitivity of the fiber optic Mach–Zehnder interferometer-based accelerometer. The parameters of the FOJS are shown in Table 1. The jerk sensing probe has a static acceleration sensitivity of 292.51 rad/g (1 g = 9.8 m/s2). The delay fiber was detected by an optical time-domain reflectometer, and its length was 2.021 km. Therefore, the static jerk sensitivity of the FOJS is calculated to be 590.68 µrad/(m/s3).
Table 1. Designed parametersa
3.3 Minimum detectable jerk
For the FOJS, the minimum detectable jerk is the resolution of the FOJS, and it is dependent on the sensitivity and self-noise of the FOJS. Because of the differentiating effect in the fiber optical path, thermal noise, optical system faults, light source interference, and other interferences are suppressed or eliminated. The phase noise floor of the FOJS can be set to self-noise, and the minimum detectable jerk can be expressed by
where ψnoise is the phase noise floor of the FOJS.Given that the phase noise floor of the 3×3 demodulation algorithm is 10−4 rad/√Hz at >1 Hz, the minimum detected jerk of the FOJS can be calculated to be 0.169 m/s3.
3.4 Dynamic range
When the maximum detectable jerk is demodulated, the dynamic range of the FOJS can be expressed as
The FOJS is demodulated by the 3×3 demodulation algorithm that has a large demodulation phase range, such as hundreds of phase radians. Therefore, combined with the static jerk sensitivity, the maximum detectable jerk of FOJS is up to 106 m/s3 theoretically. Further, it indicates that the proposed FOJS has a potential of a large dynamic range greater than 120 dB.
3.5 Experiment setup
To calibrate the FOJS, we need to test it with a standard jerk sensor. However, standard jerk sensors are lacking. We noticed that when given an acceleration with a sinusoidal period, the corresponding jerk can be calculated. When the acceleration signal is a = Dsin(2πfmt), the corresponding jerk signal can be expressed as
where D is the amplitude of the acceleration signal and fm is the frequency of the sine vibration. We found that the peak-to-peak value of the jerk is 2πfm as large as the acceleration value.The experimental setup is shown in Fig. 3. A standard piezoelectric accelerometer (Lance LC0406) and the stainless-steel shell that encloses the jerk sensing probe are installed on a vibration exciter (BK 4808). The standard piezoelectric accelerometer is used to measure the acceleration provided by the vibration exciter and convert it to jerk. We used a signal generator to generate sine vibrations of different amplitudes and frequencies to make the exciter vibrate. A demodulator is used to acquire and process the sensing signals, which integrates photodetectors, analog input modules (NI-9215) and embedded controllers (NI cRIO-9034). The sensing signal of the standard piezoelectric accelerometer is transmitted to the demodulator through a charge amplifier (BK 2692-A-OI1). The interference lights of the FOJS are detected by three photodetectors, are acquired by the NI-9215 module and are processed by the cRIO-9034. A terminal is used to demodulate and display the demodulation results. The sampling rate of demodulation is set as 20 kHz. We tested the sensitivity, phase noise floor, dynamic range, linearity, and impact test of the FOJS.
Fig. 3. The experiment setup. The red lines, black lines, and blue lines represent optical fibers, electric wires, and communication cables, respectively. PZT, piezoelectric accelerometer.
4. Results and discussions
4.1 Sensitivity
The axial amplitude characteristic of the proposed FOJS was tested from 5 Hz to 200 Hz. The experimental results and a comparison with the calculated theoretical value are shown in Fig. 4. The average axial static sensitivity is 590.23 µrad/(m/s3) (−64.60 dB re. 0 dB = 1 rad/(m/s3)) in a frequency range of 6.3 Hz to 100 Hz. The axial sensitivity fluctuates by ±1 dB. The frequency range of 6.3–100 Hz can be used as the working bandwidth. The maximum cross insensitivity is approximately 29 dB. This shows that the FOJS has good anti-transverse interference performance. Furthermore, Fig. 4 shows that the resonant frequency of the proposed FOJS is about 160 Hz, which is determined by the mechanical structure of the sensing probe. Although the FOJS should avoid working at the resonant frequency of 160 Hz for a long time, it can still measure and analyze the jerk signal at the frequency point of 160 Hz in combination with the axial sensitivity.
4.2 Phase noise floor and minimum detectable jerk
The proposed FOJS was placed in the vibration and temperature isolation box to test its phase noise floor. The phase noise was recorded from 12 a.m. to 8 a. m. A one-hour record was analyzed, and its time-domain waveform and PSD are shown in Fig. 5.
Fig. 5. Phase noise floor of the FOJS. (a) The time-domain waveform; (b) The frequency spectrum of the phase noise floor.
Figure 5(a) shows that the phase noise floor changed little in the time-domain waveform, indicating that the FOJS suppresses long-period interferences, such as temperature and frequency fluctuations. This is because the two light beams have an equal optical path, and the phase demodulated from the interference light is proportional to the derivative of the phase modulated by the jerk sensing probe. From the PSD, as shown in Fig. 5(b), the phase noise floor of the FOJS is low in the frequency bandwidth of 0.01–200 Hz, which is less than 10−4 rad/√Hz. Furthermore, the phase noise floor gradually decreases as the frequency decreases from 200 Hz to 0.01 Hz. The phase noise floor increases after 0.005 Hz. A possible cause is the lack of sample points, which makes the PSD of the phase noise floor unreliable after 0.005 Hz.
The minimum detectable jerk can be measured based on the phase noise floor and axial sensitivity. Therefore, when the phase noise floor is 6×10−5 rad/√Hz at 80 Hz, the minimum detectable jerk is 0.10 m/s3. If the FOJS is placed in a seismic observation chamber with a more stable temperature and atmosphere, the phase noise floor may be lower than 10−6 rad/√Hz, and the minimum detectable jerk will be close to 0.01 m/s3.
4.3 Dynamic range
We gradually increased the amplitude of signal at 80 Hz and tested the output phase. When the total harmonic distortion (THD) is about 10%, the signal has a maximum output phase. Figure 6 shows the results processed by a fast Fourier transform (FFT). The maximum output phase is about 4.87 rad/√Hz. From the axial sensitivity, the maximum detectable jerk can be calculated as 8251.02 m/s3. The dynamic range of the FOJS is then calculated to be 98.33 dB.
Fig. 6. Frequency spectrum recorded by the FOJS when the THD is about 10%. Inset view shows the details of the higher harmonics.
The dynamic range is an important index of the FOJS and is limited by the maximum demodulated phase of interference light. From the inset view in Fig. 6, the higher harmonics distort the signal. At the beginning of our analysis, we mistakenly thought that it was due to the limitation of the differentiating interferometer. However, we found that the higher harmonic was mainly caused by the 3×3 demodulation algorithm [23] and the sensing probe. We will optimize the demodulation algorithm and sensing probe structure to enlarge the dynamic range in future work.
4.4 Linearity
We generated a vibration signal at 80 Hz and varied its amplitude to test the linearity of the FOJS. The result is shown in Fig. 7. The output phase and the input jerk have a good linear relationship. The goodness of fit is 0.9999, indicating that the FOJS is working linearly.
For the FOJS, its linearity would be influenced by the maximum signal frequency. Equation (10) shows that the frequency of detectable jerk signals should be less than 1/2πT. If the maximum frequency is close to 1/2πT, the error is controlled to 5% [20]. Within the working frequency range, the FOJS has a good linearity.
From the above measurement results, the FOJS has a measuring range of 0.10–8251.02 m/s3. As shown in Table 2, the FOJS can be used in a variety of fields, including comfort evaluation, railway design and seismic analysis.
Table 2. Typical jerk values in different fieldsa
4.5 Impact test
To verify whether our FOJS could directly measure jerk, an impact test was carried out. We hit the vibration exciter with a small hammer to generate random impacts. Impact signals were recorded by the standard piezoelectric accelerometer and the FOJS, respectively. To show clear impact signals, we use the same band pass filter to process the signals. Figure 8 shows the recorded impact signals after filtering. The waveform shown in Fig. 8(a) is the voltage signal measured by the standard piezoelectric accelerometer and represents the acceleration. And Fig. 8(b) is the numerical differentiation of the voltage signal, which represents the differentiation of the acceleration. It has been filtered to filter out the noise error amplified by the differential. From Fig. 8(c), we can find that the phase signal recorded by the FOJS and the differentiation of the voltage signal have the same waveform trend. In fact, unfiltered differentiation of the voltage has lots of noise errors, whereas the FOJS can avoid the noise errors caused by the differentiation.
Fig. 8. Recorded impact signals. (a) Acceleration measured by the piezoelectric accelerometer; (b) Differentiation of the acceleration; (c) Jerk measured by the FOJS.
Combining the voltage sensitivity, the differentiation of the voltage signal recorded by the standard piezoelectric accelerometer was converted into the jerk. The working frequency range of the standard piezoelectric accelerometer is 0.1 Hz to 2 kHz. In this frequency range, we use the charge amplifier to set the sensitivity of the accelerometer to 10 V/g in this impact test. The phase signal recorded by the FOJS was converted into jerk using axial sensitivity. We compare the two impact jerks and show the results in Fig. 9. As shown in Fig. 9(a), the correlation coefficient between the FOJS and the differentiation of the piezoelectric accelerometer is 0.9926, and this high correlation is significant. The two signals have nearly the same variation trend and are in good agreement at each sampling point. In addition, we compared the two signals in the frequency-domain. The PSD is shown in Fig. 9(b). The two impact jerk signals have the same dominant frequency and similar spectrum waveforms. The results indicate that the FOJS can directly measure jerk.
Fig. 9. Comparison of the impact signals recorded by the FOJS and the piezoelectric accelerometer, respectively. (a) High correlation between the two impact jerk signals; (b) Power spectrum of the two impact jerk signals.
5. Conclusion
A FOJS based on a fiber optic differentiating Mach–Zehnder interferometer is proposed. The sensing theory, sensor design and experiments all prove that the FOJS is feasible. The experimental results show that the FOJS has a high sensitivity in the working frequency bandwidth, an ultralow phase noise floor, a wide measuring range to accommodate different fields, and good linearity. An impact test was carried out, which proved that the FOJS can measure jerk directly. The FOJS can be used to research earthquake engineering, evaluate comfort, and design railways, among other things.
Funding
National Key Research and Development Program of China (2019YFC1509500); National Natural Science Foundation of China (61875185, U1939207).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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