Modeling for IFOG Vibration Error Based on the Strain ... - MDPI

sensors Article

Modeling for IFOG Vibration Error Based on the Strain Distribution of Quadrupolar Fiber Coil Zhongxing Gao, Yonggang Zhang * and Yunhao Zhang College of Automation, Harbin Engineering University, Harbin 150001, China; [email protected] (Z.G.); [email protected] (Y.Z.) * Correspondence: [email protected]; Tel.: +86-150-4510-9122 Academic Editor: Vittorio M. N. Passaro Received: 24 May 2016; Accepted: 1 July 2016; Published: 21 July 2016

Abstract: Improving the performance of interferometric fiber optic gyroscope (IFOG) in harsh environment, especially in vibrational environment, is necessary for its practical applications. This paper presents a mathematical model for IFOG to theoretically compute the short-term rate errors caused by mechanical vibration. The computational procedures are mainly based on the strain distribution of quadrupolar fiber coil measured by stress analyzer. The definition of asymmetry of strain distribution (ASD) is given in the paper to evaluate the winding quality of the coil. The established model reveals that the high ASD and the variable fiber elastic modulus in large strain situation are two dominant reasons that give rise to nonreciprocity phase shift in IFOG under vibration. Furthermore, theoretical analysis and computational results indicate that vibration errors of both open-loop and closed-loop IFOG increase with the raise of vibrational amplitude, vibrational frequency and ASD. Finally, an estimation of vibration-induced IFOG errors in aircraft is done according to the proposed model. Our work is meaningful in designing IFOG coils to achieve a better anti-vibration performance. Keywords: fiber elastic modulus; quadrupolar fiber coil; vibration model; asymmetry of strain distribution

1. Introduction Interferometric fiber optic gyroscope (IFOG) is an optical sensor based on Sagnac effect, which is used to determine the angular velocity of the carrier [1]. Over the past few decades, it has been developed as a leading instrument in sensing of rotational motion and has been manufactured around the world for a wide range of both military and commercial applications [2,3]. Some commercial IFOG with high performance has been reported in recent years [4,5], the bias stability specification over environment is 0.0005˝ /h with the scale factor less than 10 ppm and an angle random walk (ARW) < 8 ˆ 10´5.0 /h1/2 . A key component in IFOG is fiber coil whose winding quality will affect the performance of gyroscope directly. Thus far, numerous researches have been performed to improve the adaptability of IFOG in the harsh environment. The theoretical model for thermal induced rate error of IFOG with temperature ranging from ´40 ˝ C to 60 ˝ C is established in [6], which can be potentially used in the temperature compensation for gyroscope. To improve the thermal performance of the fiber-optic gyroscope, the approach of using air-core photonic-bandgap fiber is present in [7] and it is demonstrated by experiments in [8] that the thermal sensitivity is 6.5 times less than the conventional fiber coil. Furthermore, modeling for rate errors of IFOG induced by moisture diffusion is present in [9]. In practical applications for high precision IFOG, mechanical vibration is a common and inevitable environmental factor that typically encountered in the inertial navigation platforms, and it will degrade the performance of the gyroscope seriously. Generally, the anti-vibration performance of IFOG can be effectively improved by optimizing the mechanical structure design, avoiding the resonant Sensors 2016, 16, 1131; doi:10.3390/s16071131

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structure design, avoiding the resonant frequency, and tightening the optical components specificallyand [10–12]. Nevertheless, these preliminary methods cannot reveal the principles of frequency, tightening the optical components specifically [10–12]. Nevertheless, these preliminary vibrationcannot error inreveal essence. other words, a systematic to analyze vibration error of study IFOG methods the In principles of vibration error instudy essence. In otherthe words, a systematic hasanalyze not been addressed the thoroughly main purpose of our work. to thethoroughly vibration error of IFOGand hasthat not is been addressed and that is the main purpose The vibrational problem of IFOG is rather complicated and tough in practical engineering. In of our work. otherThe words, it is veryproblem difficult of to IFOG establish a mathematical model describe engineering. and analyze vibrational is rather complicated andprecisely tough intopractical theother vibration of gyroscope to various disturbances as imperfect fiber and In words,error it is very difficult to due establish a mathematical modelsuch precisely to describe andcoil analyze inaccurate parameters for computation [13]. Furthermore, work in [14] focus on the the vibration error of gyroscope due to various disturbancesprevious such as imperfect fibermainly coil and inaccurate vibration error from the perspective of controlprevious equationwork of IFOG asmainly well asfocus signal in parameters for computation [13]. Furthermore, in [14] onprocessing the vibration circuit, butthe notperspective consideredof the error equation induced by imperfect winding of processing fiber coil. To bestbut of our error from control of IFOG as well as signal in the circuit, not knowledge,the thiserror is the first report to investigate IFOG vibration error terms of strain distribution considered induced by imperfect winding of fiber coil. To thein best of our knowledge, this is in the (QAD) fiber coil. The paper organized as follows: The theory closed-loop the firstquadrupolar report to investigate IFOG vibration errorisin terms of strain distribution in theofquadrupolar IFOG isfiber introduced Section 2. The mathematical of IFOG rate error induced vibration in is (QAD) coil. Theinpaper is organized as follows:model The theory of closed-loop IFOG isby introduced established in mathematical Section 3. Themodel computational results on by thevibration model are provided ininSection Section 2. The of IFOG rate errorbased induced is established Section 4. 3. Finally, the conclusion andbased discussion givenare in provided the last section. The computational results on theare model in Section 4. Finally, the conclusion and discussion are given in the last section. 2. Description of Closed-Loop IFOG 2. Description of Closed-Loop IFOG The details of the optical components and the signal processing circuit constructed for The details of are the introduced optical components theachieve signal processing circuit constructed forwavelength closed-loop closed-loop IFOG in Figureand 1. To higher output power and better IFOG are introduced in Figure 1. To achieve higher output power and better wavelength stability, a broadband Er-doped super fluorescent fiber source (SFS) is utilized as the lightstability, source. aThe broadband Er-doped fluorescent fiber source (SFS) is utilized as The 3-dB 3-dB coupler splitssuper the input light into two waves to propagate in the the light fiber source. coil. The Sagnac coupler splits the input of light two waves to propagate thea polarization fiber coil. The Sagnac interferometer interferometer consists an into integrated-optic circuit (IOC) in and maintaining fiber (PMF) consists of an integrated-optic circuitto(IOC) andthe a polarization maintaining (PMF) coil withcircuit QAD  . The fiber coil with QAD winding pattern detect angular rate signal processing winding pattern to detect the ratehigh-speed Ω. The signal processing circuit gate incorporates pre-amplifier, incorporates pre-amplifier, ADangular converter, field programmable array (FPGA) and DA AD converter, high-speed field programmable gate array (FPGA) and DA converter. Moreover, the converter. Moreover, the gyroscope sensitivity is maximized by applying a square wave modulation gyroscope is maximized applying wave modulation the IOC atfiber the frequency in the IOCsensitivity at the frequency of 1/ 2by, where  aissquare the propagation time ofinlight in the coil. The of 1{2τ, where τ is the propagation time lightAcquisition in the fiber coil. Thecard gyroscope output acquired gyroscope output data are acquired byofData (DAQ) and then sentdata intoare a computer by Data Acquisition (DAQ) card and then sent into a computer to be stored and post-processed. to be stored and post-processed.

Figure 1. Schematic diagram of digital closed-loop interferometric fiber optic gyroscope (IFOG).

Based on onthe thecontrol controltheory theory and Figure 1, response the response of closed-loop IFOG to the Based and Figure 1, the of closed-loop IFOG is linearistolinear the angular  angular rate and the dynamic model can be simplified, as shown in Figure 2. G is the gain of the rate Ω and the dynamic model can be simplified, as shown in Figure 2. G is the gain of the gyroscope gyroscope control system, which is mainly determined by Sagnac coefficient, photo-detector, control system, which is mainly determined by Sagnac coefficient, photo-detector, pre-amplifier, and

1 controller in FPGA. M is the feedback pre-amplifier, AD converter. 1 is the AD converter. and inintegral FPGA. M is the feedback coefficient, which is coefficient, influenced S is the integral controller S by DA converter and half-wave voltage of IOC, and 1/M is the closed-loop scale factor of IFOG. which is influenced by DA converter and half-wave voltage of IOC, and 1/M is the closed-loop scale In addition, e´τs denotes the delay unit of the control system. factor of IFOG. In addition, e  s denotes the delay unit of the control system. The transfer function of Figure 2 is given as follows.

Fpsq “

Ypsq K “ e´τs Ωpsq Ts ` 1

(1)

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(t )  Sensors 2016, 16, 1131

1 S



Y(t )

e s

3 of 12

Here, K “ 1{M and T “ 1{GM. The frequency-amplitude characteristic as well as frequency-phase characteristic of Equation (1) is illustrated as $ &

b 1 pωT q `1

|Fpjωq| “ofK digital closed-loop IFOG. Figure 2. Simplified model 2

(2)

% θrFpjωqs “ ´arctanpωTq The transfer function of Figure 2 is given as follows.

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Y( s ) K  s F (s)   e ( s ) 1Ts  1

(t ) 

S

3 of 13

(1)

Y(t )

e s

Here, K  1/ M and T  1/ GM . The frequency-amplitude characteristic as well as frequency-phase characteristic of Equation (1) is illustrated as 1   F ( j )  K (T ) 2  1   [ F ( j )]   arctan(T )  Figure 2. Simplified model of digital closed-loop IFOG.

(2)

Figure 2. Simplified model of digital closed-loop IFOG.

-5

o

 (F(jw))( )

|F(jw)|(dB)

For a high precision IFOG fabricated in our laboratory, the closed-loop scale factor K is The transfer function of Figure 2 is given as follows. T is in 816 LSB/(°/h) the time constant measured as 1.6 ms. this case,scale the normalized For a highand precision IFOG fabricated our laboratory, the In closed-loop factor K is ˝ Y( s ) K curve and f   [ F ( j  )] curve (i.e., Bode diagram) are shown in Figure 3. It is f  20 lg F ( j  ) s 816 LSB/( /h) and the time constant FT( s )is measured ase 1.6 ms. In this case, the normalized  (1)  s ) that Ts  1thediagram) ((i.e., f ´ 20lgseen |Fpjωq| and f ´ θrFpjωqs curve Bode are shown in Figure It isa clearly fromcurve the frequency characteristic curve output signal of gyroscope will3.have clearly seen from the frequency that frequency the output signal of gyroscope will have a half-attenuation −3 dBT characteristic in1/ GM Figure 3)curve atfrequency-amplitude the of 99.47 Hz. Therefore, M and . The characteristic as well the as Here, K  1/(i.e., half-attenuation (i.e., ´3 error dB in of Figure 3) with at the high frequency of 99.47 Hz.beTherefore, the vibration-induced vibration-induced rate IFOG frequency will dramatically attenuated after frequency-phase characteristic of Equation (1) is illustrated as rate errorEquation of IFOG(1). with high frequency will be dramatically attenuated after passing Equation (1). passing 1  0  F ( j )  K 0 2 (T )  1 (2)   [ F ( j )]   arctan(T ) -20  For a high precision IFOG fabricated in our laboratory, the closed-loop scale factor K is -40 T 816 LSB/(°/h) and the time constant is measured as 1.6 ms. In this case, the normalized -10 Bode diagram) are shown in Figure 3. It is f  20 lg F ( j ) curve and f   [ F ( j )] curve (i.e., -60 clearly seen from the frequency characteristic curve that the output signal of gyroscope will have a -15 -80 half-attenuation (i.e., −3 dB 10in Figure 3)10 at the 10frequency of 99.47 Hz.10 Therefore, 10 the 10 10 10 (a) Frequency(Hz) (b) Frequency(Hz) vibration-induced rate error of IFOG with high frequency will be dramatically attenuated after passing Equation (1). Figure 3. Normalized Bode diagram (a) Frequency-amplitude characteristic curve; and (b) 0

1

2

3

0

1

2

3

Figure 3. Normalized Bode diagram (a) Frequency-amplitude characteristic curve; and (b) Frequency-phase Frequency-phase characteristic curve. characteristic curve. 0 0

|F(jw)|(dB)

3. Establishment 3. Establishmentof ofVibration VibrationModel Model

-20

-5

o

 (F(jw))( )

In general, general, the the fiber fiber coil coil has has aa tensile tensile strain strain induced induced by by tensile tensile stress, stress, and and the the strain strain distribution distribution In -40 is asymmetrical asymmetrical with is with respect respect to to the the midpoint midpoint of of the the fiber fiber coil coil due due to to imperfect imperfect winding winding techniques. techniques. -10 According to the theory of material mechanics, the elastic modulus of the material a constant if According to the theory of material mechanics, the elastic modulus of the materialisisnot not a constant -60 the fiber strain exceeds a certain range. In this case, different fiber strain has different fiber elastic if the fiber strain exceeds a certain range. In this case, different fiber strain has different fiber elastic modulus after winding and use in 4. -15 -80 results modulus after winding and we we will will use the the measurement measurement results from from10[15], [15], as as illustrated illustrated in Figure Figure 4. 10 10 10 10 10 10 10 Note that the unit of the strain μ denotes the variation of 0.0001% and the curves are fitted by Frequency(Hz) Note that the unit of (a) theFrequency(Hz) strain µ denotes the variation of 0.0001% and the(b)curves are fitted by six-order six-order polynomial to guarantee the computational Figure 4that reveals that the relationship polynomial to guarantee the computational precision.precision. Figure 4 reveals the relationship between Figure 3. stress Normalized Bode diagram (a) Frequency-amplitude characteristic curve; and (b) and between fiber and fiber strain is nonlinear if the stain is more than 3000 μ approximately, fiber stress and fiber strain is nonlinear if the stain is more than 3000 µ approximately, and the Frequency-phase characteristic curve. the corresponding elastic modulus becomes variable respect to fiber the fiber strain. corresponding elastic modulus becomes variable withwith respect to the strain. 0

1

2

3

0

1

2

3

Specifically, the fiber coil Model without spool is solidified by UV-curing adhesive and fixed in the 3. Establishment of Vibration metal housing by metallic cover, as present in Figure 5. The movement of the QAD coil under In general, fiberincoil has a tensile strain induced tensilethat stress, the strain distribution vibration is alongthe z-axis a global coordinate system. Weby assume the and acceleration applied to the is asymmetrical with respect to the midpoint of the fiber coil due to imperfect winding techniques. coil is described as Asinpωtq, where A is vibration amplitude and ω is angular vibration frequency. According to the theory of material mechanics, the elastic modulus of the material is not a constant if the fiber strain exceeds a certain range. In this case, different fiber strain has different fiber elastic modulus after winding and we will use the measurement results from [15], as illustrated in Figure 4. Note that the unit of the strain μ denotes the variation of 0.0001% and the curves are fitted by six-order polynomial to guarantee the computational precision. Figure 4 reveals that the relationship

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200

34 32

Modulus(GPa)

150

Stress(MPa)

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30

4 of 12

28

Then the velocity and displacement of the coil motion26are given as ´Acospωtq{ω and ´Asinpωtq{ω 2 50 by integrating Asinpωtq one time and two times, respectively; that is 24 0 1000

22

7000 d2 z dt2

1000 2000 3000 4000 5000 6000 7000 “8000 Asinpωtq (b) Strain( ) dz dt “ ´Acospωtq{ω 2 Figure 4. Measurement results of the fiber mechanical properties (a) The tensile stress versus the z “ ´Asinpωtq{ω

2000

3000

4000

5000 (a) Strain( )

6000

tensile strain; and (b) the elastic modulus versus the tensile strain.

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8000

(3)

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Specifically, the fiber coil without spool is solidified by UV-curing adhesive and fixed in the 34 metal housing by metallic cover, as present in Figure 32 5. The movement of the QAD coil under 150 vibration is along z-axis in a global coordinate system. We assume that the acceleration applied to 30 the coil is described as A sin( t ) , where A is vibration amplitude and  is angular vibration Modulus(GPa)

Stress(MPa)

200

100

28

frequency. Then the velocity and displacement of the coil motion are given as  A cos(t ) /  and 26

2  A sin( by integrating A sin( t ) one time and two times, respectively; that is 50 t ) /  24

d2z 0 A sin(t221000 ) 2000 3000 4000 5000 6000 7000 8000 1000 2000 3000 4000 5000 6000 7000 2  8000 dt (b) Strain( ) (a) Strain( ) dz (3) mechanical  A cos(t )properties / Figure 4. Measurement Measurementresults resultsofof fiber (a) tensile The tensile stress versus the Figure 4. thethe fiber mechanical properties (a) The stress versus the tensile dt tensile strain; the modulus elastic modulus versus the tensile2 strain. strain; and (b)and the (b) elastic versusz the A sin(strain. t) /   tensile

Specifically, the fiber coil without spool is z solidified by UV-curing adhesive and fixed in the y 5. The movement of the QAD coil under metal housing by metallic cover, as present in Figure Metallic cover vibration is along z-axis in a global coordinate system. We assume that the acceleration applied to x amplitude and  is angular vibration the coil is described as A sin( t ) , where A is vibration frequency. Then the velocity and displacement of the coil motion are given as  A cos(t ) /  and

 A sin(t ) /  2 by integrating A sin( t ) oneFiber time coil and two times, respectively; that is d2z  A sin(t ) dt 2 Metal housing dz A sin(t ) (3)   A cos(t ) /  dt Figure 5. Diagram Diagramof ofthe themechanical mechanical vibration applied to the The vibration direction is vibration applied to the fiberfiber coil. coil. The vibration direction is along z   A sin(t ) /  2 along the z-axis and perpendicular thecoil. fiber coil. the z-axis and perpendicular to the to fiber

z

y number N, the index of each layer and each For a QAD coil with layers number M and loops For a QAD coil with layers number M and loops number N, the index of each layer and each loop loop could be denoted by m (mMetallic = 1, 2 …cover M) and n (n = 1, 2 … N). The diameter of the fiber is denoted could be denoted by m (m = 1, 2 . . . M) and n (n = 1, 2 . . . N). The diameter of the fiber is denoted by d f . The dynamic model to illustrate the movement of each loop caused by vibration is shown in by d f . The dynamic model to illustrate the movementxof each loop caused by vibration is shown Figure 6. To facilitate thethe analysis, a alocal in Figure 6. To facilitate analysis, localcoordinate coordinatesystem systemisisestablished establishedinineach each loop loop and and the the displacement along vibration direction is denoted by z . In this model, each fiber loop is regarded Fiber coil displacement along vibration direction is denoted by znn . In this model, each fiber loop is regarded as a mass-spring-damper mass-spring-damper system system with with stiffness stiffness coefficient coefficient kk and and damping damping coefficient coefficient η. η. Here, Here, ηη is is a constant for the fiber, while k is different in terms of the distance to the coil bottom. The The relationship relationship between fiber stress σand and fiber strain fiber strain ε is is Metal housing A sin( t) σ “Eε (4)   E (4) Figure That is5. Diagram of the mechanical vibration applied to the fiber coil. The vibration direction is That is ∆L along the z-axis and perpendicular to the fiberF coil. “E (5) ∆S F LL  E number N, the index of each layer and each (5) For QAD coil with layers number M and Thena we have S loops L E∆S = 1, 2 … N). The diameter of the fiber is denoted loop could be denoted by m (m = 1, 2 … M) F “and n (n ∆L “ k∆L (6) L by d f . The dynamic model to illustrate the movement of each loop caused by vibration is shown in Figure 6. To facilitate the analysis, a local coordinate system is established in each loop and the displacement along vibration direction is denoted by zn . In this model, each fiber loop is regarded as a mass-spring-damper system with stiffness coefficient k and damping coefficient η. Here, η is a constant for the fiber, while k is different in terms of the distance to the coil bottom. The relationship between fiber stress  and fiber strain  is

  E

(4)

coefficient k in loop n is given as k

E ( R22  R12 ) 2n  1 df 2

(7)

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where R1 and R2 are inner radius and outer radius of the fiber coil, respectively, and E is elastic modulus of the fiber.

zn

yn

xn

A sin(t ) z n

yn

xn

A sin(t ) Figure 6. 6. Dynamic Dynamic model model for for each each fiber fiber loop loop under under vibration. vibration. Figure

Considering that the relative velocity and relative displacement between fiber loop n and the coil We assume that fibers in Figure 6 are modeled as close-packed with adhesive filling the interstitial dz 1 1 volume. In this case, the πpR2 ´ R21 q.Hence, stiffness coefficient k in  [ is calculated A cos(t )] by t )] , thethe bottom are described as coiln area and∆Sz“ motion equations of fiber n  [  22 A sin( dt   loop n is given as 2 ´ R2 q mcoil EπpR 2 mass 1 and loop n is given in Equation (8). Here, mcoilk “ is the coil is the mass of each loop. (7) 2n´1 N d 2

f

2

zn dzand ( R22the  R12fiber ) Aouter t ) E of A sin(t ) cos(radius where R1 and R2 are m inner and E is elastic coil d radius n  [ ] [ zn coil, respectively, ]0    2 2 (8) n 2 1  dt   modulus of the fiber. N dt df 2 Considering that the relative velocity and relative displacement between fiber loop n and the coil dzn 1 1 bottom are described as ´ r´ Acospωtqs and z ´ r´ the motion equations of fiber Asinpωtqs, n 2 Equation (8) could dt be rewritten as Equation (9) after rearrangement and being divided by ω ω mcoil loop n is given in Equation (8). Here, m is the coil mass and is the mass of each loop. coil mcoil N . N EπpR22 ´ R21 q dzn Asinpωtq mcoil d2 zn Acospωtq ` ηr ` s ` rzn ` s“0 (8) 2n´ 12 z N dt2 N dt ωn2 d dz ω 22 d nf n 2  A cos(t )  2 A sin(t )  2  2n  n z n (9)  mcoil  dt dt mcoil Equation (8) could be rewritten as Equation (9) after rearrangement and being divided by N .

ω2 Nη d2 z n dzn Acospωtq ´ n2 Asinpωtq “ ` 2ξωn ` ωn2 zn (9) mcoil ω dt ω dt2 c Eπ p R22 ´ R21 q In the above equation, ωn “ mN is the natural resonant frequency, which is usually 2n´1 ´

coil

2

df

more than one magnitude larger than ω; and ξ “ d 2

η 2 mcoil EπpR2 2 ´R1 q N 2n´1 d f 2

is the damping ratio. It is clearly

seen that ωn and ξ are both determined by loop number n. The solution to Equation (9) is given in Equation (10). zn ptq “ ´

A

a

mcoil 2 ωn4 ` N 2 η 2 ω 2 sinpωt ` α ` βq b ` C1 e´ξωn t sinpωn 2 mcoil ω 2 2 2 2 2 2 pωn ´ ω q ` 4ξ ωn ω

b 1 ´ ξ 2 t ` γq

(10)

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Nηω

nω where α “ tan´1 p m ω2 q and β “ ´tan´1 p ω2ξω 2 2 q. Since the second term of Equation (10) will n ´ω coil n attenuate rapidly after a short period time, the steady state solution of Equation (9) will be the first term of Equation (10). Therefore, the force acting upon coil loop n under vibration is described as Equation (11), which is different in different n and ω.

f nvib

a m 2 ω 4 ` N 2 η 2 ω 2 {N mcoil d2 zn ptq “ “ b coil n Asinpωt ` α ` βq “ Fn sinpωt ` α ` βq 2 N dt 2 pωn2 ´ ω 2 q ` 4ξ 2 ωn2 ω 2

(11)

In order to calculate the nonreciprocity phase shift induced by vibration, the whole fiber coil is divided into M ˆ N segments, and the length of each segment si based on the cross section of the coil present in Figure 6 is given in Equation (12). In this equation, m is the index of each QAD layer, i is the index of each segment. ? si “ 2πrR1 `

3d f pm ´ 1qspi “ 1, 2, ¨ ¨ ¨ , M ˆ Nq 2

(12)

According to the material mechanics, the stress employed in each segment should be the force divided by coil area, as shown in Equation (13). The relationship between the index of each loop n and the index of each segment i are also given in Equation (13). Pivib ptq “

Fn sinpωt`α` βq π p R2 2 ´ R1 2 q

pi “ 1, 2, ¨ ¨ ¨ , M ˆ Nq

Here, $ ’ n“1 ’ ’ ’ & n“2 ’ ’ ’ ’ %

i “ 1, N ` 1, ¨ ¨ ¨ , pM ´ 1qN ` 1 i “ 2, N ` 2, ¨ ¨ ¨ , pM ´ 1qN ` 2 .. . n “ N i “ N, 2N, ¨ ¨ ¨ , MN

(13)

Depending on Figure 4b and Equation (13), the vibrational strain induced by mechanical vibration can be written as $ & εvib t“0 i ptq “ 0 (14) Pivib ptqµ vib % ε i ptq “ ten vib t ě ∆t Erε i `ε i pt´∆tqs

vib ten In Equation (14), Erεten i ` ε i pt ´ ∆tqs is elastic modulus of the fiber versus tensile strain ε i plus vibrational strain εvib i pt ´ ∆tq in segment i, ∆t is discrete step in time domain, and µ is Poisson’s ratio of the fiber. Therefore, the overall strain in vibrational condition will be the summation of tensile strain and vibrational strain, that is vib ε i ptq “ εten (15) i ` ε i ptq

The light with wavelength λ will have a phase shift ϕi after passing through a fiber with length si and refractive index ne f f _i , i.e., ϕi “ 2πne f f _i si {λ. In vibrational condition, the length and the refractive index of the fiber coil in segment i will change, as described by Equations (16) and (17) [16], and the corresponding phase shift can be described by Equations (18). si ptq “ si ` si ε i ptq ne f f _i ptq “ ne f f _i ` ϕi ptq

“

2π λ ne f f _i ptqsi ptq

“

ne f f _i 3 rP11 µ ´ P12 p1 ´ µqsε i ptq 2

ne f f _i 3 si 2π rP11 µ ´ P12 p1 ´ µqsε i 2 ptq 2 λ ne f f _i si ` k i ε i ptq ` « 2π λ ne f f _i si ` k i ε i ptq

(16) (17) (18)

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πn

3

e f f _i where P11 and P12 are photoelastic coefficients of the fiber and k i “ 2π si rP11 µ ´ P12 λ ne f f _i si ` λ p1 ´ µqs. Note that the last term of Equation (18) is ignored since it is very small as compared with the former three terms. Substituting Equation (13) into Equations (14) and (15), Equation (18) could be rewritten as Equation (19). Note that we only consider the situation t ě ∆t, which means εvib i ptq exists

F sinpωt`α` βq

i µ 2π π p R2 2 ´ R1 2 q ϕi ptq “ ne f f _i si ` k i tεten ` u i vib λ Erεten i ` ε i ptqs

(19)

Depending on Equation (19), the light will have a phase shift ϕcw and ϕccw after passing through the whole optical path in clockwise and counterclockwise directions respectively, as described by Equations (20) and (21). In this case, the equivalent rate error of IFOG induced by mechanical vibration can be calculated by Equation (22) according to Sagnac effect. ϕcw ptq “

Mÿ ˆN Fi sinrωpt ` s1 `s2c`¨¨¨si q ` α ` βsµ 2π u k i tεten ne f f _i L ` i ` vib λ πpR2 2 ´ R1 2 qErεten i ` ε i ptqs

(20)

i “1

ϕccw ptq “

2π λ ne f f _i L ` s

Mř ˆN

`s

i“1

k Mˆ N `1´i tεten M ˆ N `1 ´ i `

(21)

`¨¨¨`s

MˆN`1´i FMˆN`1´i sinrω pt` MˆN MˆN´1c q`α` βsµ u 2 2 ten vib π p R2 ´ R1 qErε MˆN`1´i `ε MˆN`1´i ptqs

Ωe ptq “

rϕcw ptq ´ ϕccw ptqsλc 180 ˆ 3600 o p q 2πLD π h

where c is velocity of the light in vacuum, D is mean diameter of the fiber coil, and L “

(22) Mř ˆN

si is the

i “1

length of the coil. For further analysis and discussion, the definition of asymmetry of strain distribution (ASD) is given in Equation (23). According to the definition, ASD has no unit and can be used as a specification to evaluate the winding quality of the fiber coil. The lower ASD, the better the coil will be. We can see from Equation (11) to Equation (22) that the vibration error Ωe ptq is determined by the factors that16, illustrated in Figure 7. Sensors 2016, 1131 8 of 13 MˆN 2 M N 2

ÿ |ε i ´ ε Mˆ N `1´i | ASD “ ASD  i“1  i   M  N 1 i

(23) (23)

i 1

ASD

E

A Fi

e (t)

 ki Figure 7. Factors that influence IFOG vibration error. Figure 7. Factors that influence IFOG vibration error.

Based on Figure 7, we can see that the vibration error e (t ) is determined by internal factors Based on Figure 7, we can see that the vibration error Ωe ptq is determined by internal factors include ASD , E and k , as well as external factors include A and  . For a given vibration include ASD, E and k i , as i well as external factors include A and ω. For a given vibration condition, condition, e (t ) can be significantly suppressed if the fiber strain satisfies Equation (24) as follows Ω e ptq can be significantly suppressed if the fiber strain satisfies Equation (24) as follows (i.e., ASD “ 0). E (i.e., ASD Besides, be suppressed effectively the fiber Eelastic 0 ).can e (t ) can also Besides, Ωe ptq also besuppressed effectively if the fiber elasticifmodulus keepsmodulus constant or keeps slightly constantinorthe varies slightly in the the fiber whole range of less the than fiber3000 strain (i.e., less 4), than μ in in varies whole range of strain (i.e., µ in Figure as 3000 shown Figure 4),(25). as shown in Equation However, should be pointedin out thatloop the kparameter in each Equation However, it should(25). be pointed outitthat the parameter each i is asymmetrical loop ki is asymmetrical with respect to the coil center since the characterization of QAD pattern. Hence, the residual errors will exit even when the coil fully meet Equation (24), as will be seen from the simulation result in Section 4.

 iten   Mten N 1 i (i  1, 2, ,

M N

)

(24)

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with respect to the coil center since the characterization of QAD pattern. Hence, the residual errors will exit even when the coil fully meet Equation (24), as will be seen from the simulation result in Section 4. MˆN q 2

(24)

Erεten i s “ constant pi “ 1, 2, ¨ ¨ ¨ , M ˆ Nq

(25)

ten εten i “ ε Mˆ N `1´i pi “ 1, 2, ¨ ¨ ¨ ,

4. Computation Geometric parameters and material properties of the fiber coil for theoretical computation are listed in Tables 1 and 2 [6,16], respectively. Figure 8 shows the standard representation of the QAD fiber coil, where CW and CCW refer to the fiber layer wound in clockwise and counterclockwise directions. Sensors 2016, 16, 1131 9 of 13 60

50

QAD layer

40

CW

CCW

30

20

10

0

0

200

400

600

800

1000

1200 1400

1600 1800

2000

Fiber length(m)

Figure 8. Standard representationof ofthe the quadrupolar quadrupolar (QAD) winding pattern. Figure 8. Standard representation (QAD) winding pattern.

Considering that the computation of the model is based on the measurement data of the fiber Table 1. Geometric parameters. strain, a stress analyzer NBX6055PM produced by Neubrex Corporation is utilized to obtain the strain distribution of the fiber coil, as shown in Figure 9. The sampling interval of the stress analyzer Valuesdistribution is asymmetrical with along the fiber coil is 5 cm. FigureParameters 9 clearly reveals that the strain Fiber coil length L 2058 In m this case, the specification ASD respect to the midpoint of the fiber coil due to imperfect winding. Layers number M 52 is essential to evaluate the asymmetry of the coil, and the calculated result in terms of Equation (23) Loops number N 87 is 0.22. Inner radius R 68 1 As aforementioned, the vibration error e (t ) is mainlymm dependent on A ,  and ASD . Outer radius R2 76.8 mm Therefore, the undesired error of IFOG induced by vibration is actually described by the function as Fiber diameter df 169 µm follows. Based on Equation (26), the simulations are performed according to different parameters, A ,  and ASD . Table 2. Material properties.  e (t )   e ( A,  , ASD, t ) (26) Parameters

4000

Values

Refractive index neff_i Damping coefficient η 3800 Poisson‘s ratio µ Photoelastic coefficients P11 and P12 3700 Fiber coil mass mcoil 3600 Light wavelength λ Strain( )

3900

ASD=0.22

1.45 0.01 0.17 0.121 and 0.270 69.94 g 1550 nm

3500

3400 Considering that the computation of the model is based on the measurement data of the fiber 3300 strain, a stress analyzer NBX6055PM produced by Neubrex Corporation is utilized to obtain the strain distribution of the fiber coil, as 3200 shown in Figure 9. The sampling interval of the stress analyzer along the fiber coil is 5 cm. Figure 9 clearly reveals that the strain distribution is asymmetrical with respect to 3100 0

200

400

600

800

1000 1200 1400 1600

1800 2000

Fiber length(m)

Figure 9. Strain distribution of the QAD fiber coil.

Firstly, the open-loop vibration error is calculated through Equations (3)–(22), that is Equation (28) as follows, and the computational results are present in Figure 10. Here, the vibration amplitudes in Figure 10a range from 0 g to 10 g, the vibration frequency is fixed at 50 Hz, and the

along the fiber coil is 5 cm. Figure 9 clearly reveals that the strain distribution is asymmetrical with respect to the midpoint of the fiber coil due to imperfect winding. In this case, the specification ASD is essential to evaluate the asymmetry of the coil, and the calculated result in terms of Equation (23) is 0.22. As aforementioned, the vibration error e (t ) is mainly dependent on A ,  and ASD . Sensors 2016, 16, 1131

of 12 Therefore, the undesired error of IFOG induced by vibration is actually described by the function9as follows. Based on Equation (26), the simulations are performed according to different parameters, A ,  and ASD . the midpoint of the fiber coil due to imperfect winding. In this case, the specification ASD is essential  e the (t ) calculated  e ( A,  , ASD, t ) in terms of Equation (23) is 0.22. to evaluate the asymmetry of the coil, and result (26) 4000 ASD=0.22 3900 3800

Strain( )

3700 3600 3500 3400 3300 3200 3100

0

200

400

600

800

1000 1200 1400 1600

1800 2000

Fiber length(m)

Figure9.9.Strain Straindistribution distribution of of the Figure the QAD QADfiber fibercoil. coil.

Firstly, the open-loop vibration error is calculated through Equations (3)–(22), that is As aforementioned, the vibration error Ωe ptq is mainly dependent on A, ω and ASD. Therefore, Equation (28) as follows, and the computational results are present in Figure 10. Here, the vibration the undesired error of IFOG induced by vibration is actually described by the function as follows. Based amplitudes in Figure 10a range from 0 g to 10 g, the vibration frequency is fixed at 50 Hz, and the on strain Equation (26), theissimulations are performed according different parameters, and ASD. in the coil shown in Figure 9 with ASD The vibration frequenciesA,inω Figure 10b  0.22 . to range from 0 Hz to 500 Hz, the vibration amplitude is fixed at 1 g, and the strain in the coil is also Ω ptq “ Ωe pA, ω, ASD, tq (26) shown in Figure 9 with ASD  0.22 . e

Firstly, the open-loop vibration error is calculated through Equations (3)–(22), that is Equation (28) as follows, and the computational results are present in Figure 10. Here, the vibration amplitudes in Figure 10a range from 0 g to 10 g, the vibration frequency is fixed at 50 Hz, and the strain in the coil is shown in Figure 9 with ASD “ 0.22. The vibration frequencies in Figure 10b range from 0 Hz to 500 Hz, the vibration amplitude Sensors 2016, 16, 1131 is fixed at 1 g, and the strain in the coil is also shown in Figure 9 with 10ASD of 13 “ 0.22.

e

open-loop o

( /h)

0.3 0.2 0.1 0 f=50Hz ASD=0.22 -0.1

0

1

2

3

4

5

6

7

8

9

10

(a) Amplitude(g)

e

open-loop o

( /h)

0.4 0.3 0.2 0.1 A=1g ASD=0.22 0

0

100

200

300

400

500

600

(b) Frequency(Hz)

e

open-loop o

( /h)

0.08 0.06 0.04 0.02 A=1g f=50Hz 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(c) ASD

Figure 10. Standard deviation of vibration errors for open-loop IFOG (a) Open-loop errors versus

Figure 10. Standard deviationerrors of vibration errors (c) forOpen-loop open-loop IFOG (a)ASD. Open-loop errors versus amplitude; (b) Open-loop versus frequency; errors versus amplitude; (b) Open-loop errors versus frequency; (c) Open-loop errors versus ASD. To simulate the relationship between e (t ) and different ASD in Figure 10c, we assume the strain in the coil from starting point to midpoint is identical with Figure 9, while the strain from the ending point to midpoint is given in Equation (27).

 M  N 1 i   i   (i=1,2,Λ,

M N ) 2

(27)

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To simulate the relationship between Ωe ptq and different ASD in Figure 10c, we assume the strain in the coil from starting point to midpoint is identical with Figure 9, while the strain from the ending point to midpoint is given in Equation (27). ε Mˆ N `1´i “ ε i ` ∆εpi “ 1, 2, ¨ ¨ ¨ ,

MˆN q 2

(27)

where ∆ε “ 0µ, 20µ, 40µ, ¨ ¨ ¨ , 200µ, and the corresponding ASD can be obtained in terms of the definition in Equation (23). The vibration amplitude and frequency are fixed at 1 g and 50 Hz respectively. Secondly, the closed-loop vibration error is achieved through Equations (1) and (28) as described by Equation (29) as follows, where L´1 denotes Laplace inverse transformation, and the results are present in Figure 11. It should be noted that the computational results of Equations (28) and (29) are present in the form of sinusoidal function, which has the same frequency with vibration. Figures 10 and 11 only show the standard deviation of the errors. Ωe open´loop ptq “ Ωe ptq Ωe closed´loop ptq “ Ωe ptqL´1 p Sensors 2016, 16, 1131

(28)

1 e´τs q Ts ` 1

(29) 11 of 13

e

closed-loop o

( /h)

0.3 0.2 0.1 0 f=50Hz ASD=0.22 -0.1

0

1

2

3

4

5

6

7

8

9

10

(a) Amplitude(g)

0.04 0.02

e

closed-loop o

( /h)

0.06

A=1g ASD=0.22 0

0

100

200

300

400

500

600

(b) Frequency(Hz)

0.04 0.02

e

closed-loop o

( /h)

0.06

A=1g f=50Hz 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(c) ASD

Figure 11. Standard deviation of vibration errors for closed-loop IFOG (a) Closed-loop errors versus

Figure 11. Standard deviation of vibration errors for closed-loop IFOG (a) Closed-loop errors versus amplitude; (b) Closed -loop errors versus frequency; (c) Closed -loop errors versus ASD. amplitude; (b) Closed -loop errors versus frequency; (c) Closed -loop errors versus ASD.

To provide a quantitative estimation of the performance of IFOG in an environment with high level vibration, the errors of figures IFOG are computed oninthe model under the spectralIFOG We can see from the above that vibrationbased errors both open-loop andpower closed-loop density (PSD) that obtained by accelerometers mounted in NASA F-15B aircraft [17]. As has been 10c increase with the raise of A, ω and ASD. However, there are still residual errors in Figures studied [17], “the lateral was found be dominant for most of theeach flight and 11c wheninASD 0. As has axis beenacceleration discussed before, this to is because the coefficient k i in loop is conditions. Hence, we will simulate the vibration in lateral direction when the aircraft takeoff. The asymmetrical with respect to the coil center due to the characterization of QAD pattern. Moreover, PSD between 10 Hz and 1 kHz is plotted in Figure 12 with blue line and the corresponding vibration someerrors model’s error may exit which can affect theoretical values in Figures 10 and 11. The reasons are also plotted with black line. To facilitate the calculation based on the mathematical model, are parameters in Tables 1 and are not(RMS) fully of accurate and the strain shown in Figure 9 we firstly calculate root mean2 square vibration amplitude in distribution each frequency according to couldPSD, be influenced by temperature. Nevertheless, the results are, to some extent, consistent with the and then the random vibration could be transformed into the deterministic signal.

experimental measurements in [11]. 0

0.07

10

2

-2

10

o

PSD(g /Hz)

10

IFOG errors( /h)

0.06 -1

0.05 0.04 0.03

clos

e

0.02 A=1g f=50Hz 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(c) ASD

11.1131 Standard deviation of vibration errors for closed-loop IFOG (a) Closed-loop errors versus SensorsFigure 2016, 16, 11 of 12 amplitude; (b) Closed -loop errors versus frequency; (c) Closed -loop errors versus ASD.

To To provide provide aa quantitative quantitative estimation estimation of of the the performance performance of of IFOG IFOG in in an an environment environment with with high high level vibration, the errors of IFOG are computed based on the model under the power spectral density level vibration, the errors of IFOG are computed based on the model under the power spectral (PSD) obtained by accelerometers mountedmounted in NASAinF-15B aircraft As [17]. has been studied densitythat (PSD) that obtained by accelerometers NASA F-15B [17]. aircraft As has been in [17], the lateral axis acceleration was found to be dominant for most of the flight conditions. Hence, studied in [17], the lateral axis acceleration was found to be dominant for most of the flight we will simulate thewe vibration in lateral the aircraft takeoff. between 10The Hz conditions. Hence, will simulate thedirection vibrationwhen in lateral direction whenThe the PSD aircraft takeoff. and kHz is plotted in Figure blueinline and 12 thewith corresponding vibration errors are also plotted PSD1between 10 Hz and 1 kHz12iswith plotted Figure blue line and the corresponding vibration with black line. To facilitate the calculation based on the mathematical model, we firstly calculate root errors are also plotted with black line. To facilitate the calculation based on the mathematical model, mean square (RMS) root of vibration amplitude in of each frequency according PSD, and thenaccording the random we firstly calculate mean square (RMS) vibration amplitude in to each frequency to vibration be random transformed into the deterministic signal.into the deterministic signal. PSD, and could then the vibration could be transformed 0

0.07

10

IFOG errors( /h)

0.06 o

10

2

PSD(g /Hz)

-1

-2

10

0.05 0.04 0.03 0.02

-3

10

1

10

2

10

3

10

0.01 1 10

(a) Frequency(Hz)

2

10

3

10

(b) Frequency(Hz)

Figure12. corresponding standard errors in in Figure 12. PSD PSD of of vibration vibration in in F-15B F-15B aircraft aircraft and and the the corresponding standard deviation deviation of of errors IFOG (a) (a) PSD PSD of of vibration; vibration; and and (b) (b) IFOG IFOG errors errors versus versus PSD. PSD. IFOG

The result indicates that vibration errors are mostly generated by low-frequency vibration The result indicates that vibration errors are mostly generated by low-frequency vibration between between 10 Hz and 60 Hz, and the largest error in IFOG is approximately 0.065°/h. Moreover, the 10 Hz and 60 Hz, and the largest error in IFOG is approximately 0.065˝ /h. Moreover, the total value total value of vibration error is about 0.12°/h by integrating the PSD. In this case, it is essential to of vibration error is about 0.12˝ /h by integrating the PSD. In this case, it is essential to improve the improve the mechanical design of gyro to isolate the vibration between 10 Hz and 60 Hz. This will be mechanical design of gyro to isolate the vibration between 10 Hz and 60 Hz. This will be meaningful meaningful to achieve a better navigation precision when the aircraft takeoff. to achieve a better navigation precision when the aircraft takeoff. 5. Conclusions In this paper, the mathematical model of IFOG under vibration is established based on the strain distribution of the QAD fiber coil. Theoretical analysis indicates that the asymmetrical strain distribution in the fiber coil and the variable fiber elastic modulus in large strain situation are two dominant reasons that give rise to rate errors of IFOG in vibrational environment. Moreover, the errors will increase with the raise of vibrational amplitude, frequency and ASD. Finally, an estimation of vibration-induced IFOG errors in NASA aircraft is done based on the proposed model. In summary, our work is meaningful to achieve a more robust (against mechanical vibration) sensor performance for high precision IFOG working in harsh environments. Acknowledgments: This work was supported by the National Natural Science Foundation of China under Grant No. 61001154, 61201409 and 61371173; China Postdoctoral Science Foundation No. 2013M530147, 2013T60347; and China Scholarship Council (CSC). Author Contributions: Zhongxing Gao conceived the main idea and wrote this paper. Yonggang Zhang performed the experiments. Zhongxing Gao and Yunhao Zhang analyzed the data. All of the work was done under the supervision of Yonggang Zhang. Conflicts of Interest: The authors declare no conflict of interest.

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