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Abstract
A fiber optic accelerometer with a high sensitivity, low noise, and compact size is proposed for low-frequency acceleration sensing. The sensor is composed of a 20 mm diameter spherical outer frame and a three-dimensional spring-mass structure as the inertial sensing element. Three Fabry-Pérot interferometers (FPI) are formed between flat fiber facets and cubic mass surfaces to measure the FPI cavity length change caused by acceleration. The dynamic signal sensing of the designed accelerometer is performed, which shows a high acceleration sensitivity of 42.6 dB re rad/g with a working band of 1-80 Hz. An average minimum detectable acceleration of 4.5 µg/Hz1/2 can be obtained. The sensor features simple assembling, small size, light weight, and good consistency. Its transverse sensitivity is measured to be less than 3% (-30 dB) of the sensitive axis. The experimental result indicates that the proposed accelerometer has application potential in areas such as seismic wave detection and structural health monitoring.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Accelerometers and vibration sensors are of great significance in modern human activities such as natural disaster forecast [1–3], structural health monitor [4,5], medication [6,7], and so on. In these applications, usually focused on low frequency vibrations of several tens of hertz [2,5], the properties of the vector vibration signal not only indicate severity and direction of the source movement but also show modal features which are closely connected to the working status or integrity of the source. As an effective way of sensing acceleration or vibration, spring-mass structure has been widely explored in the past decades. By precisely converting acceleration to structure’s inertial displacement and stress, the spring-mass structure has its advantages of high transducing efficiency, flexible design, and compact size, therefore it is adopted by both conventional electric and emerging optical accelerometers. For commercially used electric accelerometers, piezoelectric [8], conductive [6], and capacitive [9,10] sensors are applied as a convenient method to measure the displacement and stress caused by acceleration. However, these kinds of sensors suffer from temperature depending performance, bulky cables, high transmission loss, and vulnerability to electromagnetic (EM) jamming, placing restrictions on harsh environment operations. In comparison, the optical accelerometer based on optical fiber stands out for its EM immunity, light weight, and chemical stable nature [11], attracting more and more attention from many researchers over the world.
As the optical acceleration sensing techniques develop, many kinds of fiber accelerometers based on different mechanical structures and optical principles have been reported. The early-used intensity method is mainly based on light coupling efficiency of input and output fibers attached to cantilevers [12–15], which is easy to accomplish, but also inaccurate because it is vulnerable to light source jitter or environmental disturbance. The wavelength method using fiber Bragg grating (FBG) [16–19] or fiber laser [20,21] to detect acceleration-induced structural stress, features simple demodulation system and multiplexing abilities, but the installation of fiber grating is of poor repeatability since glue is applied to FBG-spring-mass structure and the linear stress distribution of FBG are hard to precisely control. In vector sensing purposes, FBGs integration efficiency is relatively low, which lays burden on limited small platform conditions [1,16,17,20]. FBGs are also sensitive to temperature, which possibly introduces low frequency noise that needs compensation [19,22,23]. The phase method of acceleration detection is another widely used technique that recovers acceleration in high accuracy with the assistance of interferometers such as Michelson interferometer (MI) [24–27], Fabry-Pérot (FP) interferometer [28–31], and so on. Since MI accelerometers use long fiber winding with bulky compliant cylinders or flexural disks, they are usually space consuming and have complicated fabrication requirements to keep a good consistency, especially in multi-axis constructions [25,28]. Fabry-Pérot interferometer (FPI) is one of the most ideal designs that are suitable for narrow space applications. With the common light path of FPI, the interference arm and the interference optical path difference (OPD) can be easily controlled and be made short, enhancing the resistance to phase noise or polarization fading of the sensing system. Fiber FP accelerometers reported are commonly based on mesh diaphragms with embossment [32,33] fabricated by machining or micromachining techniques. The planar nature of diaphragm makes it hard to achieve spatial vector sensing by a single structure, which means that fabrication complexity and sensor volume will increase in multi-axis integration [34]. Therefore, by combining 3-dimensional spring-mass suspension and FPI, a fiber accelerometer of high sensitivity, low noise, and compact multi-axis sensing can be realized.
In this article, a miniature tri-axis accelerometer based on fiber FP interferometers is demonstrated. The main components of the sensor are an outer frame and a three-dimensional spring-mass structure as inertial sensing element. The cube mass is at the center of the sensor restricted by 6 replaceable metal elastic springs to form a spatially symmetric structure that contains 3 acceleration-sensitive axes. The rigid outer frame of the accelerometer is built by two mutually perpendicular rings and 6 frame support parts fixed by square-shaped plugs and slots. The design is to hold the inertial sensing parts of the sensor with maximum space utilization, in which fibers are inserted to form FP interferometric cavities between fiber facets and cubic mass surfaces. The proposed spherical sensor has a diameter of 20 mm and weighs less than 7 g. The simplified structure avoids bulky compliant cylinders, long fiber winding, or FBG gluing, granting a robust, light-weight, and small-sized tri-axis sensor design with high sensitivity and good consistency. In order to analyze the performance of the proposed sensor scheme, numerical calculations are conducted with the help of a spring mechanical model of the transverse stiffness coefficient, and parameters of the sensing structure are chosen to acquire an acceleration response that mainly covers the low frequency region around 100 Hz. Exposed to dynamic acceleration, sensors with different parameters are tested and their sensing performances are compared. As a result, the frequency response characteristics of the demonstrated accelerometers are in accord with numerical simulation. Using a proper configuration of the sensor, an acceleration sensitivity level higher than 42.6 dB re rad/g is observed with a 1-80 Hz flat response operating band during the experiment. The average minimum detectable acceleration (MDA) reaches 4.5 µg/Hz1/2 within the working band. The experimental results indicate that high axes consistency can be achieved by the proposed scheme. Moreover, the sensor exhibits a vector signal sensing directivity and the transverse sensitivity is less than 3% (-30 dB) compared to the sensitive axis. In general, the demonstrated compact tri-axis accelerometer is proved to be promising in low frequency vibration sensing areas.
2. Sensing principle and sensor configuration
The schematic structure of the proposed tri-axis fiber FP accelerometer is shown in Fig. 1(a). A cubic mass is located at the center of the sensor as inertial sensing element. The center mass and the outer frame of the sensor are connected by three pairs of elastic springs lying along sensor’s x, y, and z axes with six frame supports. Two ends of each elastic spring are fixed on cylindrical platforms mounted on the corresponding cube mass surface and the front side of frame support. The elastic springs are placed with a slight pre-compression to make sure the location of the inertial mass is restricted and its movement is limited within the area around the geometric center of the whole structure of the sensor. The sensor outer frame is comprised of two rigid rings that are perpendicular to each other. One of the rings is separated into two halves to connect to the other one using plugs on the back side of the frame support. The square-shaped plugs on the frame support and slots on the rings can prevent the outer ring frame from deforming or loosening, thus the misalignment of the outer frame is greatly reduced. This combination aims to reduce sensor weight and machining fabrication difficulty while ensuring an easy and firm assembly of the inside parts of the sensor such as block mass and elastic springs. Fibers can be inserted from three orthogonal directions into the holes in the outer frame, and then be faced with the center mass surfaces directly. Shown in Fig. 1(b), the inserted fibers are cleaved to have bare flat facets with no reflective coatings, which are parallel to the corresponding center mass surfaces. In this way, three FP interferometers are formed between the three facet-surface pairs to build the main optical sensing paths of the proposed tri-axis accelerometer. Exhibited in Fig. 1(c), the assembled sensor is as small as a 20 mm diameter sphere and weighs less than 7 g. The center mass surfaces are polished by abrasive paper for sufficient light reflectivity to form an FP interferometer with the fiber end facet. The reflection of the fiber facet and the center mass surface is expected to be low so that the output spectrum of the FP interferometer can be simplified as two beam interference with the following expression
where I1 and I2 are light power reflected by fiber end face and mass surface. λ, n and L0 represent wavelength of light, refractive index of FP cavity medium, and the initial FP cavity length, respectively. The assumption of weak reflection on fiber facet and mass surface can be verified by the following experiment, and the model expressed by Eq. (1) is proved to be well matched with the test result. Due to the tri-axis spatial symmetric suspension of center mass, the two reflective surfaces of FP cavity can stay parallel during the center mass movement caused by vibration of different directions or amplitudes, thus maintaining a strong FP interference for signal demodulation and reducing crosstalk from transverse vibration.
Fig. 1. (a) The schematic of fiber optic accelerometer design. (b) The schematic of the optical sensing paths of the tri-axis accelerometer, including three FPIs on the x, y, and z axes. (c) The picture of the proposed tri-axis fiber optic accelerometer. Inset: The connection of the two rings of sensor’s outer frame.
When dynamic acceleration is applied to the sensor, the outer frame vibrates synchronously with the given signal, while the center mass experiences lagging because of inertia. The phase difference between the outer frame and the center mass leads to a change of their relative displacement, which is equal to the cavity length change of the FP interferometer inside the sensing structure. To analyze the mechanical property of the designed sensor, the model of one-dimensional spring-mass oscillator is introduced. The mass of the cubic mass is M and the equivalent spring stiffness coefficient of the oscillator system is K. According to Newton’s Second Law, the frequency dependent relationship of the acceleration excitation a and the relative displacement between frame and mass Δz can be expressed by
According to Eq. (3), acceleration response of the oscillator system can be predicted if the key parameters ξ, M, and K are given. In our scheme, the open space between the sensor parts makes the squeeze film damping effect small enough to be neglected. Without measures to increase the total damping, it is expected to be too small to have obvious effect on the sensor’s responsivity away from resonance, which is proved by subsequent experiment. In the frequency range that is distant from the resonant frequency, the sensor response depends mostly on M and K instead of ξ. M is easily calculated for the regular-shaped mass and its known material density, but the system equivalent spring stiffness coefficient K cannot be directly obtained in our situation since the influence of elastic springs on the transverse direction of vibration cannot be ignored, shown in Fig. 2(a). The transverse stiffness coefficient of the elastic spring Kt is deduced by Castigliano's second theorem and the equivalent spring stiffness coefficient of the sensor can be expressed by
where K0, E, and l are the elastic spring stiffness coefficient, the Young’s Modulus of spring wire, and the spring wire length, respectively. The cross-section inertial product of the spring wire I=π(2r)4/64 is determined by the radius of spring wire r. θ represents the winding angle of spring wire, shown in Fig. 2(b). After the elastic springs are installed into the sensor, θ can be considered as constant to estimate Kt and K.Fig. 2. (a) The equivalent spring-mass system of tri-axis sensing unit. (b) The winding angle of spring wire.
With the above discussion, the model of tri-axis spring-mass oscillator is concluded and the prediction of the sensor’s frequency response curve can be done by numerical simulation using Eq. (3). As shown in Fig. 3(a), by individually changing elastic spring stiffness K0 and mass M, the sensitivity of the proposed sensing structure is calculated and the flat response region within ±3 dB sensitivity fluctuation is estimated. The total damping of the sensor structure is ignored in the simulation.
Fig. 3. (a) Theoretical estimation of the frequency response under different sensor parameters by changing K0 while M = 1.76 g and changing M while K0 = 174 N/m. (b) The assessment of frequency response using the Sa-f integration factor Ж (red), and ratio of the interference contrast after and before gravity induced initial deviation Δz0 (blue).
As K0 decreases or M increases, the sensitivity of the sensor becomes larger while the resonant frequency declines. To obtain high sensitivity within a working band as wide as possible, a factor Ж defined as the integration of sensitivity Sa(ω) in the working frequency band is introduced and its curve can be drawn with a changing resonant frequency of the accelerometer in Fig. 3(b). The red curve of Ж goes up as the resonant frequency decreases, showing that more sensitivity enhancement is gained although the working band narrows. However, the initial deviation Δz0 = g/ω02 of the center mass from the sensor’s geometric center becomes larger because of the existence of gravity (g = 9.8 m/s2). Too much variation of Δz0 will cause the deterioration of the symmetry of sensor structure as the sensor rotates, which consequently increases transverse crosstalk.
As for the influence on the light path, a large Δz0 will damage the FP interference contrast as the spatial loss is related to the cavity length. The spatial loss of light transmission in the FP cavity can be expressed by Eq. (6) [35].
For the sake of a relatively wider working band, transverse crosstalk resistance, and demodulation practicability, compromise must be made during the parameter optimization, implying that it is impractical to simply improve the acceleration sensitivity of the sensor without limit. The model applied in our study provides a theoretical instruction to the design of the accelerometer, and it can be verified by experiments.
During the experiment, the idea of the ring frame provides open space for spring and mass replacement at any time to change key parameters of the sensing element conveniently, without damaging the light path or disassembling the outer structure. The elastic springs of varied stiffness coefficient K0 are changed instead of the block mass M for a more efficient acceleration responsivity control of the sensor. K0 is set in a certain range to ensure that the needed elastic springs are available and the low frequency region can be mostly covered by the working band of the sensor. In comparison, if accurate frequency response control is necessary, a precise redesign of size, shape, and density of the center mass will be a more applicable way.
The material selection of the sensor parts will have impact on the sensor performance as well. For instance, materials of high density can be used for sensitivity improvement, while the difference of the thermal expansion coefficient of the sensor parts should not be too large, considering a better temperature performance. In practice, brass is used to fabricate the center mass and outer frame for its high density and low machining complexity, ensuring considerable M with a small size.
The proposed sensing scheme is flexible for different applications. In ultra-low frequency sensing areas like gravity and earthquake monitoring, the sensor can be modified into a single-axis accelerometer with the same working principle. As the sensor is fixed to a certain direction during the whole working process, the sensitivity can be further enhanced since the influence of gravity induced cavity length change Δz0 is no longer a problem. In higher frequency signal sensing applications like acoustic detection and engine surveillance, optical sensitivity improvement method such as high-finesse FP interferometer should be adopted in the sensor to make wide frequency range signal sensing with high acceleration sensitivity possible. In our specific design, the main parameters of sensor setup are listed in Table. 1.
Table 1. Parameters of the tri-axis fiber FP accelerometer design
As a result, the sensor design is totally free of bulky compliant cylinders, complicated fiber winding or FBG gluing, granting superiorities of simple structure, miniaturization, and consistency. On the whole, the demonstrated accelerometer has advantages of compact size, light weight, easy fabrication with replaceable parts, making it suitable for practical technical uses.
3. Experimental analysis and discussion
To test the proposed accelerometer’s sensing performance, the inertial sensing elements and the outer frame are fabricated and the single mode fibers (SMF) with protection ferrule are installed and fixed on the outer frame in three orthogonal directions corresponding to x, y, and z axes of the sensor. The depth of each SMF insertion is carefully adjusted to control the length of FP cavity formed by fiber end facet and the center mass surface. A broadband source of light is injected into the FP interferometer, whose output spectrum is observed by an optical spectrum analyzer (OSA, YOKOGAWA 70C). The reflectivity of the fiber facet and the reflectivity of center mass surface are measured to be about 3.46% and 30.31%, respectively. Considering the spatial loss of light in the FP cavity, the condition of low-finesse FP interference can be met. Figure 4(a) shows the output spectrum of the sensing unit on x axis. The free spectral range is 4.1 nm and the contrast is high according to the collected data, indicating that the short FP cavity length is about 293 µm and the polished brass surface is sufficient for FP interference formation. Equation (1) is adopted to fit the reflective spectrum of the FP interferometer in the sensor. The fitting curve drawn in the inset of Fig. 4(a) proves the feasibility of the model of two-beam interference approximation. The OSA used here is to assist the FP cavity adjustment by showing the static reflective spectrum of the sensor. Since the wavelength scanning speed of the OSA is not high enough for dynamic signal readout, the OSA is not applied to the acceleration demodulation system in the following sensing tests.
Fig. 4. (a) The output signal of the sensor on one axis (inset) and its corresponding spatial frequency spectrum (blue). (b) The schematic of the acceleration demodulation system.
When a sensor is prepared, it is connected to the demodulation system and its sensing characteristics are examined by experiment. A demodulation method based on white light interferometry is applied for the sensor’s dynamic acceleration signal readout. The schematic of the demodulation system is shown in Fig. 4(b).
The broadband amplified spontaneous emission (ASE) source sends light into the FP interferometer of the proposed fiber sensor through an optical circulator. The output signal of the sensor then heads back to the circulator and is collected by a spectrum interrogator (I-MON 512HS, Ibsen Photonics). The interrogator serves as an optical spectrometer with a high time sampling rate up to 17 kHz, whose volume phase grating separates light of different wavelengths spatially so that the dispersed light can be converted into electric signal by a CCD array. The fiber sensor is bonded with a standard piezoelectric accelerometer (YMC 271A01) and placed on a vibration table (YMC VT-500). A data acquisition card (DAQ, B&K 3160-A-022) is used for providing a sinusoidal signal to power the vibration table with a power amplifier. When the vibration table shakes, the induced acceleration can be received by the proposed sensor and the standard accelerometer spontaneously. The standard sensor’s readout is sent back to DAQ as reference acceleration for calibration, while the modulated fiber sensor spectrum goes to the interrogator for demodulation. Exposed to a set of different acceleration signals, the performance of the proposed fiber accelerometer can be characterized.
Elastic springs with varied stiffness K0 are installed in the fiber accelerometers to study their sensing performance. The abbreviations K-174, K-275, K-410, and K-686 stand for accelerometers with K0 = 174 N/m, K0 = 275 N/m, K0 = 410 N/m, and K0 = 686 N/m, respectively. To obtain the demonstrated fiber accelerometers’ frequency response, acceleration signals at frequencies starting from 1 Hz are applied during the experiment. Signals from the fiber accelerometer and the standard accelerometer are processed on a personal computer (PC) so that by Fourier transformation, the amplitude of fiber sensor phase change and the amplitude of dynamic acceleration are acquired, respectively. Then the sensitivities of fiber accelerometers within the testing frequency band can be calibrated. As shown in Fig. 5(a), experimental results are marked with scattered points while the corresponding theoretical fittings are drawn in dotted curves. The high pass filtering of the standard accelerometer signal in data acquisition is the main reason for slightly higher sensitivities acquired at 1 Hz which causes some calibration error. The light green area at the back of data in Fig. 5(a) outlines the upper frequency bounds of ±3 dB sensitivity fluctuation of fiber accelerometers tested, indicating a negative correlation between the sensor’s sensitivity and the flat response bandwidth. The theoretical prediction accords well with reality especially when the signal frequency is far lower than the sensor’s resonant frequency. In the most acceleration-sensitive scenario, the fiber accelerometer K-174, whose sensitivity is over 42.6 dB re rad/g (∼135 rad/g), has the narrowest working frequency band from 1 to 80 Hz. Using the “ringdown” method [31,36], the total damping of the sensor is tested to be as low as 0.015 kg/s. The total damping has negligible influence to the sensor response in the working band, confirmed by the damping-ignored fittings curves in Fig. 5(a).
Fig. 5. (a) The frequency response of the fabricated sensors with different K0. (b) Experimental result of the minimum detectable accelerations of the sensors.
For noise characterization, demodulation results of sensors with and without acceleration signal are recorded. Under laboratory environment, the background noise level is regarded to be unchanged in a short period of time. Therefore, signal to noise ratio (SNR) and MDA calculation can be done by the collected data. In spite of the EM immunity of the fiber optic sensor, the data acquisition and processing devices can be disrupted by power line interference, thus the power frequency 50 Hz as well as its harmonic frequency 100 Hz is skipped in the analysis. The MDA curves of the tested sensors are illustrated and the average MDA levels are indicated by transparent horizontal lines in Fig. 5(b), showing the overall noise levels on the sensor working bands. As the sensitivity rises, better SNRs can be obtained, granting lower MDAs accordingly. For sensor K-174, the average MDA on the working band is about 4.50 µg/Hz1/2. The low noise level attributes to turbulence resistance of short FP interference arms that suffer no polarization fading or temperature changing. In addition, short optical path difference of the FP interferometer also reduces phase noise from light source. As a result, low frequency noise caused by the sensor itself is effectively suppressed by the firm mechanical structure and its built-in FP interferometers. Due to the limitation of testing environment, the experimental results are expected to be improved and to be closer to the actual noise levels of the sensors in more vibration-isolated surroundings.
The consistency of the proposed accelerometers is investigated by experiments. Comparison tests of the sensitivity of sensor axes are conducted and the sensors’ maximum responsivity differences between axes are shown in Fig. 6(a). A sensitivity difference no more than 0.5 dB can be seen in the operation frequency range on axes of sensor K-174. The symmetrical design of sensing element is the main reason for its good consistency between axes. It is noteworthy that the consistency of the sensor decreases when elastic springs of greater stiffness coefficient are applied, meaning that there may be greater internal force that distorts the mechanical properties of the elastic spring owing to fabrication tolerance and pre-compression, causing more impact on the uniformity of the axes response.
Fig. 6. (a) The consistency test result of the sensors. (b) Sensor K-174 directivity test result at 5 Hz.
Transverse sensitivity is another essential property in accelerometer assessment. The sensitivity of every proposed sensor is measured while the sensor itself is rotated so that the angle between sensor’s sensitive axis and vibration direction can be changed. During this process the sensitivity varies as vibration direction changes, thus the directivity curves are drawn and the transverse sensitivities are obtained when the vibration angle reaches 90 or 270 degrees. As a result, the transverse sensitivities of sensor K-174, K-275, K-410, and K-686 are about 2.92%, 2.41%, 2.30%, and 1.62% of their main axis sensitivities, respectively. Figure 6(b) shows the directivity of the sensor K-174, revealing typical 8-shaped directivities of the vector sensor. The differences observed between the transverse sensitivities in the two directions perpendicular to the sensitive axis are mainly caused by insufficient accuracy in the sensor pose angle control. Although the sensitivity of the demonstrated sensor structure can be further enhanced by reducing elastic spring stiffness, greater transverse crosstalk is expected to be introduced due to the symmetry degradation that comes from the initial mass deviation Δz0 we discussed in Section 2. Generally, the proposed sensors present low transverse crosstalk below 3% (-30 dB) of their main axis sensitivities in the spatial acceleration measurement.
Compared with the sensing properties of our scheme, performances of some representative accelerometers reported by other researchers are listed in Table. 2. The most prominent feature of our design is its compact size with efficient tri-axis sensing integration, which is beneficial to the use in harsh environments or small platforms where space is strictly limited. During the experiment, the setup of sensor K-174 has a better overall performance among the tested sensors. The proposed sensor exhibits simple structure, high acceleration sensitivity, low transverse crosstalk, and relatively low noise in low frequency region.
Table 2. Performance of fiber accelerometers reported by researchers
4. Conclusions
In summary, we demonstrate a compact tri-axis accelerometer based on fiber FP interferometers for low frequency vibration sensing. With proper configuration, the sensor has a diameter of 20 mm and its acceleration sensitivity reaches 42.6 dB re rad/g on the working band of 1-80 Hz. The proposed sensing structure features firm assembly, high sensitivity, and small size. An average MDA of 4.5 µg/Hz1/2 is achieved and the transverse crosstalk of the sensor is less than 3% (-30 dB). The experimental result fits well with theoretical analysis, showing that the designed sensor is promising in the areas of seismic sensing, structural health monitoring, etc.
Funding
National Natural Science Foundation of China (61775070); NSFC-RS Exchange Programme (62111530153); The Royal Society International Exchanges 2020 Cost Share of United Kingdom (IEC\NSFC\201015); Science, Technology and Innovation Commission of Shenzhen Municipality (2021Szvup089); Science Fund for Creative Research Groups of the Nature Science Foundation of Hubei (2021CFA033).
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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