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Abstract
Most of fiber-optic pressure sensors used in shock wave measurements are based on deformations of sensing elements. These approaches result in low dynamic pressure ranges for these sensors used in the air. A novel fiber-optic method based on the relationship between pressure and the acceleration of a diaphragm is proposed to obtain peak reflected pressure of shock waves in the air. The optical sensor is designed with a thin circular diaphragm as the sensing element, and the Fabry-Perot optical interferometry is used to detect the acceleration of the diaphragm. Shock tube and explosive-blast experiments prove that the proposed fiber optic method is feasible and has the advantages of no calibration, high precision and fast response time. The proposed fiber-optic pressure method has potential in practical applications for shock wave measurements.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Peak pressure and impulse are important parameters to evaluate the potential for damage due to a shock wave. In order to detect the pressure of shock waves, conventional electrical pressure sensors have been used, which typically rely on piezoresistive or piezoelectrical effects. However, most of these electrical sensors cannot be used in critical electromagnetic interferences environments like electrical explosions. Recently, optical fiber pressure sensors have been studied, which often have significant advantages including superior resistance to electromagnetic interferences, fast dynamic response, applicability in harsh environments, long-term reliability, high sensitivity and dimensional compactness [1,2]. Fiber optic extrinsic Fabry-Perot interferometer pressure sensors have several advantages over intensity-based and fiber Bragg grating pressure technology, due to their design flexibility and high sensitivity. Owing to these benefits, they have been used to measure the pressure for industrial production [3–8] and shock wave detection [9–18]. However, in these measurements, the pressure range was less than 70 MPa in the case of shock wave in water [9,10], and lower than 500 kPa for shock wave in the air [11–17]. These ranges cannot satisfy the requirement for most of the shock wave pressure measurements in the air. In addition, there is a very practical optical measurement method to obtain the shock wave pressure in the liquid, which is based on the changes in Fresnel reflection between the fiber end face and the strong discontinuous face in water [19].
In this paper, a novel fiber optic method is presented to obtain the peak reflected pressure of shock waves in the air for a wide range of explosion environments. Unlike conventional optical fiber pressure sensors that the pressure relates directly to the length of the Fabry-Perot cavity or the deformation of mechanism like diaphragm, this method is based on the Newton's second law and the pressure relates directly to the acceleration of a diaphragm. The acceleration is obtained by an interferometric measurement of the displacement using a Fabry-Perot cavity arrangement. Experimental demonstration is carried out in a shock tube and an explosive-blast experiment. Results demonstrate that the method has advantages including fast response times and high precision. But, the optic probe designed by this method also have disadvantages including one-off in pressure measurement, large size and just used to measure the peak reflected pressure of shock waves but not pressure curve. However, because the optic probe is easy to manufacture and does not require calibration, it still has value for promotion and application in measurement of shock waves.
2. Principle and design of the sensor
2.1 Principle
The principle of the measurement method is based on Newton’s second law, and the sketch of the principle is shown as Fig. 1. The pressure-sensing element is a circular diaphragm that is fixed to a shield, and the diaphragm can move under the impact of a shock wave. As we know from Newton’s second law, the pressure acting on the diaphragm is equal to the product of the mass and the acceleration, if the force between shield and diaphragm is ignored. The acceleration of diaphragm is measured by an optical interferometer with a Fabry-Perot cavity. The light beam reflected by the diaphragm’s inner surface and the light beam reflected by the optical fiber’s interface get coupled in the same optical fiber and interfere with each other. Then, the displacement of the diaphragm can be obtained by processing the interferometric signal. Finally, the pressure of shock wave can be obtained by a twice differential coefficient of the displacement.
Fig. 1 Sketch of the optical fiber pressure principle. The diaphragm moves under shock wave, and the pressure is proportional to the acceleration of the diaphragm. The F-P cavity structure formed on the surface of the fiber end and the diaphragm is used to get the acceleration of diaphragm.
As we know from the principle of the Newton’s second law, in the central area of the diaphragm, the pressure of shock wave is given by,
where h is the thickness of the diaphragm, ρ is the density of the diaphragm, and a(t) is the acceleration of the diaphragm. Only the diaphragm is considered as a rigid object for the pressure in the shock propagation direction, the Eq. (1) can be exact correct, which means the diaphragm must be thin enough in the measurement. Typical shock wave generated by shock tube has positive pressure periods of a few milliseconds. An ideal pressure sensor must have a temporal resolution approaching 1 µs in order to measure rapid pressure changes due to shock wave [12]. Therefore, in order to make the proposed optical sensor have fast response, the mechanical response of different thickness diaphragm in 1 µs is required. Using AUTODYN software, the velocity curves of 35 µm and 500 µm thickness diaphragms under 1 MPa step pressure are obtained, as shown in Fig. 2. The diaphragm material chooses stainless steel, and the velocity obtained is the back surface velocity of the diaphragm.Fig. 2 The steel diaphragm velocity obtained by numerical simulation. (a) The thickness of the diaphragm is 35 µm, and the period of the velocity oscillation is 12 ns as shown in the zoomed-in curve. (b) The thickness of the diaphragm is 500 µm, and the period of the velocity oscillation is 172 ns.
The mechanical stress waves propagate back and forth in the diaphragm after an impact. According to stress wave theory [20], the period of the stress wave propagation can be calculated by,
where C is the velocity of the stress wave. The stress wave velocity of stainless steel is about 5800 m/s. The period of the stress wave can be calculated to be 12 ns for 35 µm thickness diaphragm and 172 ns for 500 µm thickness diaphragm based on Eq. (2), which is the same as the numerical simulation result as shown in Fig. 2. If the oscillation period of the stress wave is less than 20ns, it means that there are more than 50 oscillatory cycles in 1 µs as shown in Fig. 2(a), and it can be judged that the acceleration obtained by the fitting of the velocity curve has higher precision. Therefore, only some thin diaphragms are selected to prove the feasibility of the proposed optical sensor.The time lag between the shock wave and the first suction wave propagating the surface of the diaphragm can be calculated by Eq. (2), which means the mechanical resistance of the diaphragm is the same as an infinite thickness diaphragm to the shock wave in the time of T. An increase of velocity of the diaphragm is expected as more suction waves get to the surface, resulting in the weakening of the mechanical resistance as the time increases. Therefore, the pressure obtained by Eq. (1) is equal to the reflected pressure of shock wave only in the earlier period.
2.2 Mechanical design
Under the same pressure and the same density of the diaphragm, a thinner diaphragm sustains a higher acceleration as is evident from Eq. (1). So the effective time for pressure measurement will shorten under higher acceleration. But thinner diaphragms can be more accurately treated as a rigid object. Therefore, there is a conflict about the choice of thickness h. If only the peak reflected pressure of shock wave is measured but not pressure curve, the effective measurement time after impact can even be designed to be less than 1 µs. Therefore, this conflict can be eliminated.
We chose a stainless steel film as the diaphragm material and the diaphragm thickness h ≈35 µm. The diaphragm areal density hρ is more significant than parameters h and ρ for pressure calculation considered separately. And the diaphragm areal density will be measured accurately as a whole.
Under ideal conditions, the diaphragm is affected only by the pressure, if Eq. (1) is considered. But the diaphragm can also be influenced by the gravitation, the aerodynamical resistance and the force between shield and diaphragm. The gravitation and the aerodynamical resistance can be ignored, as their impact is weak as compared to the pressure in the earlier positive pressure period. But the force between shield and diaphragm cannot be simply ignored. There are bound to be vibrations at the edge of the diaphragm because of the mechanical border effect. The vibrations tend to spread to the diaphragm center, which is disadvantageous for the acceleration measurement and even leads to the failure of the measurement.
It is generally difficult to minimize mechanical border effects, hence a diaphragm with a large radius is designed. The vibrations spread to the diaphragm center in the form of longitudinal waves and transverse waves. Transverse waves pose a disadvantage for the measurement because they are in the same vibration direction as the acceleration a(t). The time taken (Tt) for the transverse wave to spread from the edge to the center of the diaphragm can be calculated by,
where r = 8 mm is the diaphragm radius and Ct ≈3200 m/s is the transverse wave velocity of the stainless steel film. Hence, Tt ≈2.5 µs is larger than the designed effective measurement time of 1 µs, and the force between shield and diaphragm can be ignored.Based on the above parameters, a pressure sensor is designed as shown in Fig. 3. The optical probe with a single mode fiber inside is attached to the bushing by the epoxy method. The diaphragm, the gasket and the bushing are then impacted in the shield with a press bolt. A Fabry-Perot cavity structure is formed with this geometry with a cavity length identical to the gasket. The diaphragm radius is 8 mm and the diaphragm areal density is 0.26732 kg/m2.
Fig. 3 Photograph of a pressure sensor. (a) The section drawing of the sensor. These mechanical parts are used to support the diaphragm and form the F-P cavity. The hollow bushing is conducive to the stability of the optical probe. (b) The sample of the sensor.
The bushing is made of copper and it is designed to have a hollow circular structure. In this case, before the vibration propagates to the optical probe, it must spread through the solid part of the bushing. Therefore, the vibrations that spread from the shield to the optical probe are delayed, and the optical probe could be considered static in the period of measurement. All parts of the sensor can be made by conventional machining methods, and an inexpensive sensor was realized.
2.3 Optical design
A Fabry-Perot cavity structure is designed with two reflective interfaces as shown in Fig. 1. Refractive index of a typical single mode fiber’s core is 1.46, and the reflectivity of the flber-air surface is calculated to be 3.8%. The intensity of light reflected from the diaphragm back into the optical fiber is co-determined by the diaphragm reflectivity and the length of the Fabry-Perot cavity. The diaphragm is made of a stainless steel film, and its reflectivity is constant. Hence, the light intensity can be controlled by changing the cavity length. The end face of the bushing and the end face of the optical probe are flush, so the cavity length is the same as the thickness of the gasket as shown in Fig. 3(a). The gasket thickness is designed to be 0.5 mm, and the first two light beams coupling back to the fiber have much stronger interference than higher order reflections. The designed interference of Fabry-Perot cavity can be considered as a two-beam interference, and the interference signal can be expressed as a sinusoidal function [12,14], whose phase ϕ is,
where, n ≈1 is the refractive index of air, l is the Fabry-Perot cavity length, λ is the light wavelength and ϕ0 is the initial phase. Upon differentiation, the diaphragm velocity u(t) can be given by,where, ν(t) is the frequency of the interference signal. After another differentiating step, the diaphragm acceleration a(t) can be expressed as,From Eq. (1) and the Eq. (6), the pressure of shock wave is determined to be,The schematic of the pressure measurement system is shown in Fig. 4. The operating wavelength of the measurement system is 1550 nm. A diode laser, a photo detector and the pressure sensor are connected by an optical circulator. The diode laser operating precisely at 1550.23 nm can generate 0~20 mW optical power. The responsive bandwidth of the photo detector is DC~1 GHz, and the voltage signal emitted from the detector is stored using a high-power oscillograph.
Fig. 4 Schematic of the pressure measurement system. The interference signal generated by the pressure sensor passes the circulator, is converted by the detector and collected by the oscillograph.
3. Shock tube experiment and discussion
A comparison test is carried out to verify the feasibility of the optical method and obtain the precision of the designed pressure sensor. However, the static test is challenging to carry out because of the requirement of the diaphragm acceleration and effective measurement time being a few microseconds. A shock tube is used to generate a shock wave with step change, which is essential in carrying out the dynamic pressure calibration [18]. The schematic of the shock tube experiment is shown in Fig. 5. The total length of shock tube is 6 m, the inside diameter of shock tube is 82 mm, the inflatable shock tube length is 1 m, and the aluminum membrane thickness is 0.5 mm. The aluminum membrane ruptures suddenly when the nitrogen pressure is sufficiently high. This results in the generation of a shock wave in the shock tube. The shock wave impacts the designed optical sensor with 35 µm thickness diaphragm and the reference sensor at the same time. The reference sensor in this case is a piezoresistive pressure sensor (8510b, Endevco).
Fig. 5 Schematic diagram of the shock tube experiment. The shock wave forms in the shock tube. The optical sensor and reference sensor are subjected by the shock wave at same time.
The signals from the optical sensor and reference sensor are collected and stored in a same oscillograph with a sampling rate of 250 MHz. The processed signals from the optical pressure sensor are shown as Fig. 6.
Fig. 6 The optical signal of the shock tube experiment. (a) The original voltage signal of the optical pressure sensor. (b) The frequency of the voltage signal and its linear fit. The least squares method is used to fit the frequency signal between 0.4 µs and 3.0 µs. The fitting line is shown in red, and the slope k is 4.7944 MHz/µs.
The original voltage signal in Fig. 6(a) exhibits characteristics of optical interference. There are several methods to obtain the instantaneous frequency from the interference signal. A basic method based on the definition of the frequency is selected to obtain the instantaneous frequency, and the frequency expression is,
After a normalization operation on the voltage signal, the phase is calculated by an arcsine and the instantaneous frequency is calculated by Eq. (8). The normalization operation is explained as follows: The optical interference voltage signal is segmented according to the oscillating period, so that each oscillation period contains a maximum and a minimum value. The amplitude of the signal in each period is scaled up or scaled down, so that the maximum value of each oscillation period is 1, and the minimum is −1. The normalization operation is conducive to the direct use of the arcsine function. In the operation of obtaining the signal phase and instantaneous frequency, it is difficult to judge whether some data at the peak moment of the interference signal is included in this period or another. If the judgment is wrong, subsequent calculations will be affected. To solve this problem, the instantaneous frequency data near the peak moment is removed. Because the removal of frequency data is very few compared to the whole cycle, the remove operation will not affect subsequent calculations.
The frequency curve is approximately linear in the period of 0~4 µs, which means the diaphragm acceleration and the pressure is approximately constant in this period, as shown in Fig. 6(b). There are continuous disturbances in the frequency signal possibly caused by signal processing error, electromagnetic disturbances or elasticity effects of the diaphragm. However, these disturbances do not significantly impact the acceleration calculation. The frequency signal between 0.4µs and 3.0µs is fitted by the least square method, which is shown as the red line in Fig. 6(b). The adjusted coefficient of determination is 0.9875, so the frequency curve has a good linearity. The slope k is found to be 4.7944 MHz/µs, and its standard error of proportion is 0.0067/4.7944 = 0.14%. Then the peak reflected pressure calculate according to Eq. (7) is 0.9934 MPa. The pressure signals of the optical sensor and reference sensor are shown in Fig. 7.
Fig. 7 The experiment results of shock tube experiment. (a) The pressure curves of the reference sensor. The green line shows that the average of the pressure signal between 200 µs and 500 µs is 0.9994 MPa. (b) The zoomed-in pressure curves. The red line shows that the pressure value obtained by the optical sensor is 0.9934 MPa.
The shock wave generated by shock tube has the ideal step pressure characteristics [18]. The mean pressure of the reference sensor in Fig. 7 is calculated by the data in period of 200~500 µs, when the pressure curve is flatter as compared to the beginning. This mean pressure is about 0.9994 MPa that can be considered as the real reflected pressure of shock wave. The pressure obtained by the optical sensor is 0.9934 MPa, which is in good agreement with the reference sensor. The pressure curves shown by the red line and the green line almost overlap in Fig. 7(b), which could be observed repeatedly in several shock tube experiments. Therefore, it is proved that the optical sensor has high accuracy from these experiments.
The optical fiber pressure sensor with 50µm thickness diaphragm is also carried out, and the diaphragm areal density is 0.37733 kg/m2. The interference signal and frequency signal similar to Fig. 6 are obtained. The slope of the frequency curve is 3.3976 MHz/µs. Then the peak reflected pressure calculate according to Eq. (7) is 0.9937 MPa, which is also consistent with reference sensor. As a result, the pressure measurement method is proved to be feasible, not only because the optical fiber pressure sensors with different thickness diaphragm are proved to be correct, but also because all the frequency curves in experiments have good linearity as shown in Fig. 6(b). According to Eq. (5), the good linearity of frequency curve means the good linearity of velocity curve, which conforms to Newton's law of kinematics, and demonstrates that the proposed method is reasonable.
It is to be noted that the pressure value of optical sensor is calculated using only Eq. (7) and there is no calibration or correction, which means that the proposed optical sensor does not require calibration before use. In Fig. 7(b), the reference sensor has an overshoot phenomenon, which is a problem existing in conventional pressure sensors and is not conducive to the measurement of short pulse pressure. However, the proposed optical sensor avoids this problem and can obtain accurate pressure values in a few microseconds.
In Fig. 7(b), the arriving time lag of the shock wave between the reference sensor and the optical sensor is 1.3 µs, which is possibly caused by the gap between the netlike shield in the reference sensor’s surface and the pressure-sensing element of the reference sensor.
The diaphragm generally breaks in each shock tube test, because its thickness is about 35 µm. This makes the optical sensor suited for one-off pressure measurements. Fortunately, the cost of the sensor is low and the fabrication of the sensor is straightforward.
4. Explosive-blast experiment and discussion
The pressure value of shock wave generated by shock tube is finite, whereas composite explosives can be used to generate an air-blast-pressure wave. The explosive-blast experiment is similar to the shock tube experiment. The optical sensor with 35 µm thickness diaphragm and a piezoelectrical pressure sensor (113B23, PCB) are used to record the pressure of shock wave at the same distance from the explosive charge as shown in Fig. 8.
Fig. 8 Schematic diagram of the explosive-blast experiment. The shock wave is generated by explosive detonation. The optical sensor and electrical sensor are subjected by the shock wave at the same distance from the explosive charge.
An electrical detonator and a blasting fuse are used to detonate the explosive, which is beneficial for generating an ideal air-blast shock wave. The weight of the explosive is 62.0 g, and the distance between the center of the explosive and the surface of the pressure sensor is about 140 mm.
The signals of sensors are collected and stored in a same oscillograph with a sampling rate of 1 GHz. The higher sampling rate is selected to obtain the transient details of the signal. Similar to the shock tube test, the processed signals of optical pressure sensor are shown as Fig. 9. For the proposed optical sensor, the effective measurement time means that the obtained diaphragm velocity has a good linearity within that time. The effective measurement time is about 1 µs shown as Fig. 9(b), which is less than the time of the shock tube experiment. The reasons for the shorter effective measurement time are as follows: Higher pressure leads to higher acceleration for the same thickness of the diaphragm. When the velocity of the diaphragm cannot be ignored compared to the velocity of the shock wave particles, the mechanical resistance of the diaphragm will weaken, and there will be unnecessary measurement errors. In addition, when the velocity of diaphragm is high, it also causes instability and destruction of the diaphragm. Fortunately, the fiber optic pressure sensor is only used to obtain the peak pressure, so the requirement for the effective measurement time is not very strict.
Fig. 9 The optical signal of the explosive-blast experiment. (a) The original voltage signal of the optical pressure sensor. (b) The frequency of the voltage signal and its linear fit. The least squares method is used to fit the frequency signal between 0.15 µs and 0.58 µs. The fitting line is shown in red, and the slope k is 138.56 MHz/µs.
The frequency signal between 0.15 µs and 0.58 µs is fitted by the least square method, which is shown as the red line in Fig. 9(b). The adjusted coefficient of determination is 0.9919. The slope of the red line is 138.56 MHz/µs, and its standard error of proportion is 0.189/138.56 = 0.14%. Thus the peak reflected pressure of shock wave can be calculated based on Eq. (7), and the result is shown as Fig. 10.
Fig. 10 Experimental results of the explosive-blast experiment. (a) The pressure curve of the electrical sensor. There are three peaks at time 4.7 µs, 14.8 µs, and 143.0 µs. (b) The zoomed-in pressure curves. The blue curve is the exponential fitting curve of the pressure data obtained by the electrical sensor, and the peak of the blue curve is 29.5 MPa. The red line shows that the pressure value obtained by the optical sensor is 28.7 MPa.
There are three peaks in the pressure signal of the electrical sensor in Fig. 10(a), and the arrival times of the peaks are 4.7 µs, 14.8 µs and 143.0 µs. Based on the analysis of explosion mechanics and numerical simulations, the first peak is likely caused by the shock wave, the second peak could be caused by the substance of explosive, and the third peak may be a result of the reflection of shock wave from the opposite safeguard.
In this experiment, the response frequency of the PCB pressure sensor system is less than 500 kHz, and the time-averaged effect is evident in the measurements. This causes the rise of the pressure curve is slow and the peak pressure value is lower than the actual pressure of shock wave. An extrapolation method [21] is used to minimize this error, and the time-averaged effect is compensated with an exponential curve fit shown by the blue line in Fig. 10(b). The raw peak pressure value obtained by the electrical sensor is 23.1 MPa, and the extrapolated peak pressure value is 29.5 MPa. The peak pressure value obtained by the optical sensor is 28.7 MPa shown by the red line in Fig. 10(b), close to the extrapolated peak pressure value. This observation proves that the results obtained by the extrapolated method and the optical method are in good agreement. The difference of the pressure values is 0.8 MPa, which is possibly caused by the displacement error in fixing the explosive and the symmetry error of the explosive itself.
It is to be noted that the extrapolated value is not actually present in the experimental records [21], there are calculation and arbitrary errors in the pressure compensation. But the peak pressure obtained by the optical sensor is precise and free of such errors. In Fig. 9(b), total effective measurement time of the optical sensor is observed to be about 0.5 µs, which is lower than the response times of most conventional pressure sensors. Therefore, the optical sensor has fast response time and good performance in recording the peak reflected pressure of shock wave.
Although the optical fiber pressure sensor experiments in only two different thickness diaphragms, it can be predicted that the thickness of diaphragm has influences on the dynamic range and precision of the sensor. If the thickness of diaphragm is very thin, such as several nanometers, the velocity of diaphragm will rise rapidly under the shock wave. The high speed of diaphragm will result in a lower mechanical resistance to the shock wave than expected, and will also lead to the increase of air resistance to the diaphragm. In this case, measurement results and dynamic range will be lower than expected. If the thickness of diaphragm is very thick, such as several millimeters, the diaphragm will not be treated accurately as a rigid object. In this case, the precision of the sensor will decrease. The main purpose of this article is to prove the feasibility of the proposed method, and the optimal range of diaphragm thickness will be studied later by numerical simulation and experimentation.
Through the shock tube experiment and the explosive-blast experiment, the results show that it is reasonable to assume that the specific thickness diaphragm is considered as rigid object in the direction of impact. Then, the Newton's second law can be used to describe the impact process of the diaphragm, which is a necessary condition for the feasibility of the proposed method and leads to the simple relationship between pressure and acceleration. The good performance of the proposed method is due to the simple relationship between pressure and acceleration, as well as reasonable design and data processing.
Unlike the shock tube experiment, in the explosive-blast experiment, not only the diaphragm was damaged, but also the fiber probe as shown in Fig. 3 was damaged. In order to reduce the cost in future experiments, the protection will be used in front of the optical probe.
5. Conclusions and future work
This paper presents a novel fiber-optic pressure method based on the Newton’s second law and Fabry-Perot technology. Optical sensors based on this method are designed and manufactured to obtain the peak reflected pressure of shock waves. In shock tube tests, the feasibility and precision of the optical sensor is demonstrated, as compared to piezoresistive pressure sensors. The optical sensor also shows good performance in an explosive-blast experiment, compared to a piezoelectrical pressure sensor. The optical sensor measures the pressure value of 28.7 MPa, which is higher than the existing reported optical fiber pressure sensor in the air.
The good performance of the optical sensor is due to the simple relationship between pressure and acceleration, as well as reasonable design and data processing. Although the optical sensor can only be operated in one-off pressure measurements, and can only get the peak reflected pressure of shock waves, but the sensor has the advantages of no calibration, no overshoot, high accuracy and low cost. These advantages are significant to dynamic pressure sensors, and even allow the optical sensor to be used as a standard pressure sensor to calibrate other dynamic pressure sensors.
The emphasis of this paper is to prove the feasibility of the proposed method, and there is still some related work to be carried out in the future. The work includes the effect of different thickness diaphragms on accuracy of pressure measurement, specific pressure measurement range and precision, which all can be obtained through theoretical analysis and experiment. However, we still envision that the new fiber-optic pressure method has several other advantages including wide pressure range, good resistance to electromagnetic interferences and thermal-shock, and it will become an advantageous tool in shock wave measurements.
Acknowledgments
This work is supported by the School of Mechanical Engineering of Xi’an Jiaotong University and the Northwest Institute of Nuclear Technology.
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