Design, Simulation and Optimisation of a Fibre-optic 3D Accelerometer

Optics & Laser Technology 49 (2013) 137–142

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Design, Simulation and Optimisation of a Fibre-optic 3D Accelerometer Zhen Yang a, Xiao-Yong Fang a,n, Yan Zhou a, Ya-lin Li a, Jie Yuan b, Mao-Sheng Cao c,n a b c

School of Science, Yanshan University, Qinhuangdao 066004, China School of Information Engineering, Central University for Nationalities, Beijing 100081, China School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e i n f o

abstract

Article history: Received 30 July 2012 Received in revised form 22 October 2012 Accepted 6 November 2012 Available online 28 January 2013

Using an inertia pendulum comprised of two prisms, flexible beams and an elastic flake, we present a novel fibre-optic 3D accelerometer design. The total reverse reflection of the cube-corner prism and the spectroscopic property of an orthogonal holographic grating enable the measurement of the two transverse components of the 3D acceleration simultaneously, while the longitudinal component can be determined from the elastic deformation of the flake. Due to optical interferometry, this sensor may provide a wider range, higher sensitivity and better resolving power than other accelerometers. Moreover, we use finite element analysis to study the performance and to optimise the structural design of the sensor. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Optical fibre sensor Acceleration sensor design 3D acceleration measurement

1. Introduction An acceleration sensor can be used in various applications and devices, such as in an inertial gyroscope [1], large mechanical structure testing [2,3], mechanical impact testing [4], an underwater acoustic intensity sensor [5], wireless MEMS-IDT[6], mine detection [7], well logging [8] and earthquake monitoring [9]. Currently, the acceleration sensors used in these applications are mainly capacity type [10], piezoelectricity type [11–13], strain type [14] and laser interferometric [9,15]. Compared with traditional piezoelectric, or piezoresistive, and with capacitance sensors, optical acceleration sensors have higher sensitivity and accuracy; however, there are disadvantages to these sensors, such as a larger size and harder integration. Optical fibre sensors have attracted considerable attention in recent decades due to their small size, light weight, wide band, strong anti-interference ability, etc. Due to these properties, studies have been extensively performed on temperature sensors [16], strain and pressure sensors [17,18], electric field and magnetic field sensors [19,20], current and voltage sensors [21,22], displacement sensors [23], and acoustic sensors [24]. In recent years, sensors that use the theory of optics to measure non-optical quantities have become an active research topic. Fibre acceleration sensors may help to resolve the disadvantages of optical type sensors. However, fibre-optic acceleration sensors are

n

Corresponding author. Tel.: þ86 335 8387569; fax: þ 86 335 8057027. E-mail addresses: [email protected], [email protected] (X.-Y. Fang), [email protected] (M.-S. Cao). 0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.11.011

rarely reported, and most of them obtain acceleration information using a piezoelectric method [3,8,14,25]. In 2002, we used a cube-corner prism-covered orthogonal holographic grating to improve the technique of Pond and Texeira [26] to measure the 2D angle [27]. A new design for a 2D acceleration laser sensor was later proposed [15]. Here, we present a fibre-optic 3D accelerometer based on this 2D acceleration laser sensor. Because this sensor can simultaneously measure the three components of acceleration, it can reduce the volume used in aerospace applications, such as in spacecraft or satellite. Consequently, this sensor could simplify the structure of navigation and guidance systems. The objective of this paper is to report the structural design and measuring principle of the fibre-optic accelerometer. Furthermore, we use finite element analysis to obtain a numerical simulation of the measuring range, sensitivity and resolving power, which can be useful in optimising the sensor design.

2. Structure and Principle 2.1. Structure of 3D Acceleration As shown in Fig. 1, the fibre-optic 3D accelerometer is composed of a sensor, an optical system and an electronic device. The acceleration information is accurately collected and transferred by the sensor and the optical system, both of which are shown in Fig. 1(b). The sensor is fixed to the moving object on the detection side, while the light source and the electron devices are on the user side. Between the sensor and the user side

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UserSide Optical System 2

1 3

4

All-fibre optical path 5 6 7 8

Detecting Side Sensor

10 9

Calculating Circuit

13 11 12 Counting Circuit

Output

Electronic Device

1. LD, 2. Y-type fibre coupler, 3. All-fibre Michelson interferometer, 4. 3dB fibre coupler, 5. 6. 7. & 8. Fibres, 9. & 10.Prism, 11. 12.& 13. Phototubes

grating. When incident light passes through the grating, it is divided into three diffracting light beams (  1, 0, þ1). After the diffracting light beams are reflected inside the prism and pass through the grating again, each of them is diffracted along orthogonal directions, as shown in Fig. 3. Here, we focus on the emergent light beams ( þ1, 0) and (0, þ1), which are coherent and able to generate an orthogonal interference pattern. When a prism swings, the variation in the incident angle will lead to a change in the optical path difference between ( þ1, 0) and (0, þ1). The interference fringes will then move, and the number of moving fringes contains the transverse acceleration information. Similarly, the prism without the grating will produce interference fringes containing longitudinal acceleration information. For a sensor acceleration a ¼axi þ ayj þ azk, the prisms oscillate under an inertia force –Max and –May. The corresponding oscillation angles are yx and yy, which can be defined by [21] ( yx ¼ Ax N x ð1Þ yy ¼ Ay Ny where M is the mass of the prism, Ax and Nx are the angle-width and moving number of the interference fringes received by phototube 11, and Ay and Ny are the angle-width and moving number of the interference fringes received by phototube 12. The inertia force –2Maz generates a deformation of the flexible metal flake along the z-direction. Let z(0) be the longitudinal displacement of the flake centre, which can be defined by

Fig. 1. (a) System chart of the 3D acceleration sensor and (b) its photo.

is an all-fibre path, which could effectively avoid EMI in certain technical situations. On the user side, a laser device (LD) that can provide good coherence is chosen as the light source. The LD light is divided into two optical paths by a Y-type fibre coupler. One of the paths is directed to prism 9 by fibre 5. According to references [26,27], 2D angle information is contained in the interference pattern that is generated by the emergent light from prism 9. Fibres 7 and 8, which are orthogonal to each other and below prism 9, will transfer the angle information to phototubes 11 and 12. This process may eventually lead to the measurement of the transverse accelerations. The other optical path enters an all-fibre Michelson interferometer, where the path is again divided by a 3 dB coupler and directed to prism 10 through fibre 6. Due to the total reverse reflection of prism 10, the emergent light returns to the all-fibre Michelson interferometer. Finally, phototube 13 obtains a set of interference fringes, which contain information regarding the 3-dimensional displacement. This fringe information can be used to measure the longitudinal acceleration. In the electron device, light signals are turned into electronic signals by the phototubes. The signals are then subdivided in a division circuit and conducted to a counting circuit, where the moving number of interference fringes caused by the optical path difference is recorded. The treated signals are transformed into acceleration data by a calculating circuit in real time. Finally, the data are outputted and displayed. 2.2. Measuring Principle of 3D Acceleration The structural design of the sensor is shown in Fig. 2. The sensing unit mainly consists of an elastic metal flake. Both sides of the flake are fixed to cube-corner prisms by micro-flexible beams. At the joints between the beams and the prisms, two mass compensations are included to increase the measuring range. One of the prisms is covered with an orthogonal holographic

zð0Þ ¼

l 2

Nz

ð2Þ

where Nz is the moving number of the interference fringes received by phototube 13. The length, radius and mass of the micro-flexible beams are l, r, and m. For a beam with one end fixed, the shear force FS can be written as dF S ¼ PðzÞ dz

ð3Þ

where P(z) is the transverse load distributed along the beam. According to the theory of Timoshenko, the oscillation angles can be obtained from Eq. (3) and are   8 pffiffi > 3Max lH  4 2  þ 2ð12 þ2v2Þ > y ¼ x > 2E < pr k l m 3r2 þ l   ð4Þ pffiffi > 3 Ma lH y >  4 2  þ 2ð12 þ2v2Þ > : yy ¼ 2E 2 m 3r þ l

pr k l

where H is the height of the prism and E and v are the elastic modulus and Poisson ratio of the metal flake. k2 is the shear ratio and should be 0.89 for a beam with a circular cross-section. From Eqs. (1) and (4), the transverse accelerations can be written as 8 0:89Epr 2   > > < ax ¼ pffiffi3MH ð1 þ v=lÞ þ ð1:78=ð3r2 þ l2 ÞrÞ Ax Nx ð5Þ 0:89Epr 2   > > : ay ¼ pffiffi3MH ð1 þ v=lÞ þ ð1:78=ð3r2 þ l2 ÞrÞ Ay N y where r is the density of the sensing unit. For a flexible flake, it is well known that the longitudinal strain Q(z) can be expressed as 3

Q ðzÞ ¼

Eh   r4 zðzÞ 12 1v2

ð6Þ

where h is the flake thickness and z is the distance between the strain point and the flake centre. Because Q is mainly due to the inertial force of the prism –2Maz, Eq. (6) can be solved using

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139

Fiber 6 Cube-corner prism Sensing unit Y-Coupler

Mass compensation

3dB

Prism with grating

LD

Electronic Part

Fiber 5, 7 & 8 Moving Object

Fig. 2. Sensor structure, the illustration is the photo sensing unit.

these functions are defined by the material parameters and the applied load. In conclusion, the integrated expression of the 3D acceleration can be written as

Grating

8 > > > ax ¼ > > > < ay ¼ > > > > > > : az ¼

(0,+1)

MH



MH

1:6mT Er 2

ð1 þ v=lÞ þ ð1:78=ð3r2 þ l2 ÞrÞ



1:6mL Er

 Ax N x

2

ð1 þ v=lÞ þ ð1:78=ð3r2 þ l2 ÞrÞ 3

4:2mL lEh M ð1v2 ÞR2

 Ay Ny

ð9Þ

Nz

3. Performance Analysis and Optimisation

(+1,0)

3.1. Simulation Experiment

Laser

Cube-Corner

Fig. 3. Simulation of the prism with the grating.

the compatible equation: zðzÞ ¼

3Maz R2 3

8pEh

2

ð1v Þ 1

z2 R2

!2 ð7Þ

where R is the flake radius. From Eqs. (2) and (7), the longitudinal acceleration can be written as

3.2. Measuring Range

3

az ¼

4plEh

3Mð1v2 ÞR2

The theory of finite element analysis (FEA) is used to discretise a constant entity into finite elements, which are connected at their nodes. At the microcosmic level, the external force transfer among the elements is determined by the interaction of their nodes. The meshed model of the sensing unit is shown in Fig. 4(a). The displacement distribution of the sensing unit due to a random applied 3D static load is shown in Fig. 4(b). From its nodal solution, we can determine the transverse maximum deformation of the beam ox (or oy) and the longitudinal maximum deformation of the flake oz. Converting these deformations into yx, yy, and z(0), we can obtain the coupling function values using Eqs. (1), (2) and (9). Fig. 5 shows the dependence of the coupling function coefficients on the load and on the size of the sensing unit, where the beam length l ¼1.0, 1.25, 1.5, 1.75, and 2.0 mm; the flake thickness h¼0.10, 0.13, and 0.15 mm; and the prism heights H¼10 mm. As shown in Fig. 5(a) and (b), mT increases while mL decreases with increasing l, but neither function varies with a change in the applied load. This result indicates that, for a specific sensor structure, the coupling functions will degenerate into three coupling coefficients. Thus, we can obtain the coupling coefficients using a load test and can add a static correction to the calculating circuit to exactly measure the 3D acceleration.

Nz

ð8Þ

In Eqs. (3) and (6), the stress analyses for the prism and for the flake were performed separately; however, the deformations of the prism and the flake should affect each other. To accurately express the acceleration using Eqs. (5) and (8), we include a transverse coupling function mT and a longitudinal coupling function mL to describe the mutual influences. The variables of

For elastic materials, their elastic and inelastic behaviours are distinguished by the yield strength scalar. Once the load stress exceeds the yield point, this stress may cause the plastic deformation of the material. The acceleration corresponding to this maximum load can be regarded as the measuring range of the sensor. In FEA, the yield point can be determined using eigenvalue buckling analysis. After a series of iteration operations, we obtain

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Fig. 4. FEA model and simulation of the sensing unit, where (a) is the meshed sensing unit and (b) is the displacement distribution for random static.

Extremum of a / km s-2

800

h = 0.10 mm

600

h = 0.13 mm h = 0.15 mm

400

200

0 51

01

52

0

Radiusof the diaphragm / mm Fig. 6. The longitudinal measuring range dependence on the beam size.

Fig. 5. Coupling coefficients from the FEA simulation as a function of (a) load and (b) beam length.

Table 1 Maximal incident angles for different prism and grating parameters. Grating Constant (mm)

Prism Material

the extremum of the applied load that the sensing unit can endure. The curves of the longitudinal acceleration extremum az-max, which vary with R and h, are shown in Fig. 6. The curves converge with increasing R, and a small change in h produces a larger measuring range difference when R is small. Because of the inaccuracy in h due to the machining, the flake should have a large radius to avoid the considerable influence of the thickness on the measuring range. The flake radius is set to approximately 2 cm, and the longitudinal measuring range is defined as 25 km/s2. For the transverse accelerations, their extremums ax-max and ay-max are influenced not only by the stress that the beam can endure but also by the allowable incident angle. It should be noted that the incident angle should not exceed the critical angle of the cube-corner prism. Based on the Zemax model shown in Fig. 3, we obtain the maximal incident angles ymax for different grating constants and prism materials, which are listed in Table 1. We can obtain the transverse acceleration extremums ax-max and ay-max using Eq. (4). Additionally, the extremums corresponding to the maximal allowable stress of the beam can be obtained from FEA. Comparing these extremums as a function of flake size in Fig. 7, we observe that the two curves intersect at a point with an acceleration of approximately 49 km/s2. The acceleration extremum is clearly determined by the allowable incident angle to the left of this point. To the right of this point, the extremum is determined by the allowable stress. Because the longitudinal measuring range is defined as 25 km/s2, we use the result that

Type/number

Refractive index

0.1

0.2

0.3

0.4

0.5

FK54 BK7 F2

1.44 1.52 1.62

37.31 39.11 38.91

32.61 39.01 38.71

29.51 39.01 38.51

26.71 33.51 34.01

20.41 26.61 26.71

was calculated from the allowable incident angle as the transverse measuring range. 3.3. Sensitivity Because the sensitivity is an important parameter of an accelerometer, it is necessary to research possible influences, such as structural or material, on the sensitivity. This research will help improve the measuring accuracy of the sensor. An experimental measurement with a vibration exciter is plotted in Fig. 8, which indicates that longitudinal measuring provides less resolving power than transverse measuring. This disparity can be decreased by modulating the spatial frequency of the grating. Because the sensor relies on the interference fringes for acceleration sensing, we compared the luminance of the fringes in Fig. 9, where (a) indicates the effect of l on the transverse precision and (b) indicates the effect of R on the longitudinal precision. Fig. 9(c) and (d) show the effect of H on both the transverse and longitudinal precision. From these data, we observe that a longer beam and a larger flake produce up to

Z. Yang et al. / Optics & Laser Technology 49 (2013) 137–142

141

Fig. 7. The transverse measuring range dependence on the flake size.

Fig. 8. Acceleration curves of the transverse measurements and the longitudinal measurement.

2 times and 4 times higher precision for the sensing unit, while a bigger prism can increase the transverse measuring precision up to 3.2 times and the longitudinal precision up to 8 times. The sizes of the beam and the flake can be optimised to improve the sensitivity. A high sensitivity can also be obtained by increasing the number of frequency multiplications.

3.4. Sensor Error

Fig. 9. Effects of the structural parameters on the interference fringes.

From Eq. (9), once the size of the sensing unit is determined, the sensor error can be defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

Da DA DN 2 ¼ 2 þ3 ð10Þ a A N where DN/N¼1/Z. DA/A is determined by the non-linear relation between the interference fringe angle-width and the incident angle and can be written as

DA A

¼

2cos2 ymax ðcos2ymax sin2ymax Þ Dy cosymax ðcosymax cos2ymax Þðsin2ymax þcos2ymax Þ

þ

sinymax cos2ymax ðsin2ymax þ cos2ymax Þ2cosymax Dy cosymax ðcosymax cos2ymax Þðsin2ymax þ cos2ymax Þ

ð11Þ

Because the error of the 2D angle measurement is at most 0.011 and the maximal angle is 351, DA/A E 0.0027. Obviously, for

Z o100, the error is mainly due to the resolving power of the optoelectronic device. 4. Conclusion In conclusion, this paper presents a new optical method for measuring acceleration. Using two cube-corner prisms as an inertia pendulum, the 2D transverse accelerations can be measured. One of the prisms can be used to simultaneously obtain the longitudinal acceleration. This fibre-optic acceleration sensor can accurately measure the 3D acceleration of a moving object, an alldirectional application of optical interferometry. Using FEA software, we studied the resolving power, sensitivity, measuring range and accuracy of the sensor. The analysis demonstrates that

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the sensor has high sensitivity, high precision and a wide measuring range. Furthermore, the sensor can be easily used for different application needs through the choice of the design parameters.

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