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Abstract
In this letter, we propose a novel technique for dynamic ultra-high pressure calibration that measured pressure by FBG based strain sensor. Generally, the traditional method of dynamic ultra-high pressure calibration by standard sensor is costly and it is difficult to improve the accuracy. Therefore, we prefer FBG strain sensor to replace the standard sensor to calibrate the ultra-high pressure. In this proposal, the calibration process is that the central wavelength of the FBG attached to the elastic element changes rapidly with the strain of the elastic element during the drop hammer impact, synchronously. This allows the calibration accuracy to be easily increased to 0.02% and the cost to be reduced by 1/100 compared to traditional calibration techniques. The experiment results show that coefficient of linear correlation between the strain waveform and the pressure signal reaches 0.999. The strain calibration based on FBG is of great significance to the measurement and calibration of dynamic ultra-high pressure sensors.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Dynamic ultra-high pressure measurements are widely used in various fields, such as ballistics, aerospace, artillery, explosion research and petrochemical industry [1–4]. In the dynamic ultra-high pressure measurement, the core point is that the dynamic characteristics of the pressure sensor must reach a sensitivity with a certain precision, so as to ensure the accuracy and reliability of the obtained pressure signal [5]. If the required dynamic pressure data fail accurately measured, it may affect product quality and even cause a certain danger [2,6,7]. Therefore, to precisely measure the transient pressure, the pressure sensor must be dynamically calibrated not only before using but also after a period of time [8,9]. At present, the dynamic calibration method of a shock tube step force calibration device meets the requirements of dynamic pressure calibration in a small range, but it is difficult to achieve a dynamic pressure calibration above 100 MPa [10,11]. In contrast, the pressure calibration device by drop hammer can better simulate the chamber pressure curve of most weapons, and is more suitable for dynamic calibration of ultra-high pressure sensors [12]. Most of ultra-high pressure calibrations by the drop hammer are based on piezoelectric electrical acquisition devices, which are prone to sparks and explosions. Yet the ultra-high pressure dynamic pressure sensor with explosion-proof function has high measurement cost and low calibration accuracy [13,14].
Therefore, we propose a dynamic ultra-high pressure calibration system by drop hammer impact based on FBG strain sensor. FBG is usually used as a substitution for traditional sensors due to its small size, excellent dynamic characteristics, low cost, high precision, and anti-electromagnetic interference. Here, the center wavelength of the FBG assembled in the elastic element changes with the strain of the elastic element linked with the drop hammer. According to this strain calibration method, when the FBG peak wavelength changes by 5 nm and the wavelength demodulation accuracy is 1pm, the calibration accuracy can reach 2/10,000 or even higher. It enables improve calibration accuracy and reduce cost of dynamic ultra-high pressure calibration.
2. Design of dynamic ultra-high calibration device based on FBG
We now explain the design of the drop hammer strain dynamic calibration system, which is suitable for the dynamic calibration of optical fiber pressure sensor. The dynamic calibration system by drop hammer strain includes the calibration device, the fiber Bragg grating demodulator and the data acquisition computer. The FBG strain sensor that is integrated into the drop hammer, and the pressure sensor to be calibrated, are connected with the FBG demodulator. The demodulation of the FBG center wavelength is obtained by an FBG demodulator based on a high-speed scanning laser. The parameters of the demodulator are: the working wavelength range is 1525nm-1565nm, the demodulation speed of the center wavelength is 200kHz/s, and the wavelength resolution is 1pm. The demodulates the wavelength change of the sensor and then transmits the pressure data to the computer for analysis. Figure 1 shows the schematic design of the drop hammer strain dynamic calibration device.
The structure of the dynamic calibration device for the drop hammer strain is designed with reference to the mechanical relationship model for the pressure in the hydraulic cylinder and the stress and strain of the elastic element on the drop hammer. The structural design diagram of the device is shown in Fig. 2. The main body of the device is chiefly composed of the drop hammer and the hydraulic cylinder of the integrated fiber Bragg grating strain sensor. The drop hammer consists of four parts: the hammer head, the hammer body, the elastic element, and the connecting bolt. The elastic element is a hollow column structure. It is designed to connect the hammer head, the hammer body and the elastic element by use of high-strength connecting bolts. The hammer body is the main part of the drop hammer, while the center of the hammer structure has a cylindrical through hole. A clamping platform is designed for the bottom of the through hole. On assembling the hammer body, the bolts are positioned into the through holes and clamped on the clamping platform. This means that the connecting bolts can closely connect to the other components of the drop hammer. Threads are designed for the inner side of the upper part of the through hole, which enables the installation of lifting rings. Four optical fiber outlets are also designed for the hammer body. The surface of the elastic element of the FBG based strain sensor is engraved with a semicircular micro-groove with a radius of 1.5mm. The grating region of the FBG is kept at the lower position of the center of the elastic element. We use high-strength resin glue to dispense both ends of the fiber grating, and cure for 24 hours to ensure the reliability of its strength. This assembly method is easy to disassemble. The elastic element can easily be replaced according to the requirements of the experiment. The optical fiber pressure sensor, which is to be calibrated, is installed on the hydraulic cylinder. When the drop hammer falls and impacts the hydraulic cylinder, the strain generated by the elastic element is measured by the fiber Bragg grating that is solidified on the elastic element of the drop hammer. Simultaneously, the hydraulic piston compresses the pressure transmission medium and generates a dynamic pressure in the cylinder. The optical fiber pressure sensor is then calibrated.
3. Sensing mechanism analysis
3.1 Principles of fiber Bragg grating strain sensing
Fiber Bragg grating can be used as a sensitive unit for sensing because it responds to changes in the external environment through the shift of the center wavelength [15,16]. The external environment can have various factors, such as temperature, strain, displacement, refractive index [17]. This paper focuses on the relationship between the strain and the shift of the center wavelength of the fiber Bragg grating [18]provided by:
where Рe is the effective elasticity coefficient of FBG, ɛfiber is the axial strain in the fiber, and λB is the wavelength of the FBG, ΔλB is variation of FBG wavelength. As can be seen the drift of the central wavelength of the fiber Bragg grating provides a linear relationship with the axial strain [19,20]. Typically, for practical implementations of ultra-high pressure calibration applications, the amount of strain of the elastic element is calculated by the offset of the center wavelength of the fiber Bragg grating during impact [21,22].3.2 Mechanical model of the calibration device for the drop hammer
The drop hammer falls freely from a certain height, and it collides with piston on the upper part of the hydraulic cylinder, resulting in an elastic collision. After impacting on the hydraulic piston, it compresses the pressure transmitting medium in the hydraulic cylinder to generate a certain pressure. When the drop hammer and the piston travel downwards simultaneously, until the point at which the downward speed decreases to zero, all the kinetic energy of the drop hammer is transformed into the elastic potential energy of the pressure transmitting medium at this moment, the dynamic pressure value in the hydraulic cylinder reaches its maximum [23]. Due to the elastic recovery effect of the pressure transmitting medium, it produces an upward force on the drop hammer, and the drop hammer and the piston move upward simultaneously. When the drop hammer and the hydraulic piston separate, the elastic potential energy of the pressure transmitting medium is all converted into the kinetic energy of the drop hammer. After this process, the dynamic pressure waveform similar to a half-sine is formed in the hydraulic cylinder. As the dynamic calibration device for a pressure sensor, the peak value and pulse width of the device to generate a half-sine pressure pulse must be known. Therefore, when designing such a dynamic calibration device, the relationship between the structural parameters of each device and the pressure peak should be established.
During the dynamic calibration of the pressure sensors above 100 MPa and below 1000 MPa [24], the hydraulic cylinder block, the drop hammer and the hydraulic piston can be regarded as rigid. By ignoring the mass of the hydraulic piston and the frictional forces acting on various parts of the device during dynamic calibration, the compression of the pressure transmitting medium (in the hydraulic cylinder of the calibration device) is nonlinear during the compression process. Pm is the peak pressure and τrepresents the pressure pulse width, the following expressions can be used:
The above expressions, whose symbols are defined below: m is the drop hammer mass, P is the pressure in the cylinder, S becomes defined by the working area of the piston, E is given by the liquid bulk modulus of elasticity, E0 represents the constant modulus of the liquid volume, x is the piston displacement, ΔV is given by the volume change of the cylinder, V0 is the initial volume of the cylinder, α defines the liquid volume compression coefficient, and W is the drop hammer kinetic energy, relate to a mechanical model that has quasi-linear characteristics for the dynamic calibration device. The relationship between the peak of the pressure and the pressure pulse width produced by the device, and various other parameters of the device under high pressure is obtained. This provides the theoretical basis for the design of the dynamic calibration device.
A fiber-optic pressure sensor mounted on a hydraulic cylinder was dynamically calibrated using an FBG based strain sensor integrated in an elastic element. The drop hammer consists of a hammer head, an elastic element, and a hammer body connected by bolts. During the calibration process, when the drop hammer hits the piston, a representative schematic diagram of the device is shown in Fig. 3. We assume that the masses for the piston, hammer head, and hammer body inside the device express by m, M1, and M2, respectively; The elastic coefficient and damping coefficient when the hammer head is in contact with the piston define k1 and c1, respectively. In addition, the stiffness and damping coefficients of the elastic element between the hammer head and the hammer body represent k2 and c2, respectively. The relative displacements of the piston, hammer head, and hammer body define x0, x1, and x2, respectively. Where, P and P0 are defined by the pressure in the cylinder and the initial pressure in the cylinder, respectively. The friction between the piston and the cylinder is given by ƒ. The following is a mechanical analysis of each part of the model:
Fig. 3. Mechanical diagram depicting the drop hammer strain dynamic calibration device calibration device.
Taking the piston in isolation, it can be defined that:
While, when taking the hammer head as the isolation body, the following is found:
Which, for cases when the hammer body is the isolation body, is rewritten as:
In order to simplify the formula, we define a variable An. Combining and simplifying (4), (5) and (6) can obtain:
There is a certain mechanical relationship between the deformation of the elastic element and the pressure in the hydraulic cylinder. If the height of the elastic element is H, assuming $y_{1}=x_{2}-x_{1}$, the strain generated by the force of the elastic element is [25]:
A differential operator is introduced, which represents the symbol Dn. g is the gravitational acceleration constant. The relationship between the central wavelength change of the FBG and the pressure in the hydraulic cylinder can be obtained by simplifying a parallel connection, i.e:
4. Simulation and experimental analysis
4.1 Simulation analysis of elastic elements
Fig. 4. Vibration modals of the elastic elements for the (a) first-modal, (b) second- modal, (c) third- modal, (d) fourth- modal, (e) fifth- modal, and (f) sixth- modal
The static analysis of the elastic element is performed with finite element analysis software. By fixing one end of the elastic element and then applying pressure onto the other end, the equivalent strain of the elastic element can be calculated by a simulation. When the outer diameter of the elastic element and the applied pressure are fixed values, the strain increases with a decrease in the wall thickness of the elastic element. The size parameters of the elastic element are determined by the simulation data to ensure that the elastic element obtains a large strain while avoiding the phenomenon of element bending due to too thin wall thickness. To ensure that the strain hammer has excellent dynamic response, and avoid resonance effects, the natural frequency of the designed elastic element is required to be different from the modulation frequency. Then, via the simulation analysis of the natural frequency of the elastic element, the vibration modal of each order of the elastic element can be found. As seen in Fig. 4, the vibration modal of the elastic elements for the labels 1-6 are shown.
Via a modal analysis of the elastic element, its natural frequencies under different modal can be obtained. Figure 5 shows the natural frequencies of the elastic elements for the modal 1-6. Since the minimum pulse width of the measured half-sine pressure signal is not less than 2 ms, it can be estimated that the upper limit of the frequency of the half-sine pressure signal is 500 Hz. It can be seen from Table 1 that the natural frequencies (frequency range of 16.16 - 42 kHz) in different modal are much larger than the upper limit of the half-sine signal frequency. As a result, we can conclude that the natural frequency of the elastic element is suitable for dynamic calibration experiments on the drop hammer strain.
Table 1. Natural frequency of the elastic element. in different modal analysis
4.2 Dynamic ultra-high pressure calibration experiment of FBG strain sensor
Through the drop hammer strain dynamic calibration test, the strain waveform data of the fiber Bragg grating strain sensor is obtained [26]. Figure 6 provides a set of strain data waveforms for FBG based strain sensor. At this point, the corresponding standard optical fiber pressure sensor has a peak pressure of 299 MPa. It is observed that there are many aliased high-frequency signals in the output waveform of strain, and it is difficult to extract the strain waveform curve and strain peak value. Therefore, it is necessary to process the strain waveform data of the FBG sensor.
4.3 Analysis of the dynamic ultra-high pressure calibration data of the FBG strain sensor
By performing FFT [27]and low-pass filtering on the waveform data, the high-frequency components are basically filtered out, and the actual strain waveform signal is extracted [28]. The result is the blue curve in Fig. 7. It can be seen that the waveform resembles a half-sine waveform. The pulse width of the strain curve is 4.25 ms and the peak value of the strain waveform (for the sensor on the strain drop hammer) is 31.3 pm.
An optical fiber pressure sensor with a measuring range of 0-500 MPa and a measurement accuracy of 0.1% was selected as the standard sensor for the calibration experiment. In order to facilitate the comparison with the strain waveform of the fiber Bragg grating strain sensor, the value is taken as negative, and the black curve is depicted in Fig. 7. Its peak pressure is 299 MPa, and the pulse width of the pressure curve is 4.18 ms. The pulse widths of the two groups of waveforms are basically the same, and the time positions corresponding to the peak values of the waveforms are basically the same, and the deviation is small.
In the dynamic calibration experiment of drop hammer strain, in addition to the consistent pulse width of the pressure curve and the strain wavelength curve, and the same trend of curve change, it is also necessary to determine the relationship between the peak value of the strain waveform and the pressure. Figure 8 is a linear fit of the strain waveform peak value and pressure.
The correlation coefficient R2 of the linear fitting between strain and pressure waveform peak is 0.9997. This indicates that the linear correlation between the strain waveform peak and the pressure peak of the FBG pressure sensor is very high and proves the feasibility of the system. The cost of the standard ultra-high pressure calibration sensor is up to 100,000 RMB. The FBG sensor is about 100 RMB each, the elastomer is generally 100 RMB, and the total processing cost can reach 1,000 RMB. The cost of FBG-based calibration is lower than that of traditional calibration using a pressure standard sensor. According to the current measurement indicators of FBG based strain sensor: the measurement range is 5000 micro-strain (the FBG peak wavelength change is 5 nm), and the wavelength demodulation accuracy becomes 1 pm. The accuracy of the calibration sensor can be improved to 0.02% easily, and lower cost than traditional methods, which is of great significance to the measurement and calibration of dynamic ultra-high pressure sensors.
5. Conclusion
In summary, this letter presents a dynamic ultra-high pressure calibration system by drop hammer impact based on FBG strain sensor, for the first time, which calibrate pressure by fiber Bragg grating strain sensor. Dynamic calibration of ultra-high pressure is accomplished by demodulating fiber Bragg grating integrated in elastic element of drop hammer. The linear correlation coefficient between the peak value of the strain waveform and the pressure reached more than 0.999. Using the method of strain calibration dynamic impact pressure proposed in this paper, the calibration accuracy can be improved by an order of magnitude, and lower cost than traditional methods. We believe our work offers a novel method for ultra-high pressure dynamic calibration by FBG strain sensor.
Disclosures
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work re-ported in this paper.
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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