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The simultaneous measurement of temperature and mean strain based on the distorted spectra of half-encapsulated fiber Bragg gratings using improved particle swarm optimization
Optics Communications 392 (2017) 153–161
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The simultaneous measurement of temperature and mean strain based on the distorted spectra of half-encapsulated fiber Bragg gratings using improved particle swarm optimization
MARK
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Zheng-fang Wang, Jing Wang, Qing-mei Sui , Lei Jia School of Control Science and Engineering, Shandong University, Jinan, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Fiber Bragg grating Spectra reconstruction Half-encapsulation Particles swarm optimization
A half-encapsulated FBG is capable of simultaneously measuring temperature and strain. However, spectrum distortion, which may be induced by overlapping or non-uniform strains, may hinder the adoption of this technique. In order to resolve this issue, an improved particle swarm optimization (IPSO) based spectra reconstructing method has been adopted in this study to estimate the temperature and mean strain according to the distorted spectrum. Also, a dynamic adaptive inertia weight adjusting strategy based on the swarm success rate has been adopted to improve the algorithm. To validate the method, a total of 48 scenarios of distorted spectra have been simulated, and the temperature and mean strain estimated by IPSO have been compared with the genetic algorithm and linearly-declined PSO. The simulation results indicated that the IPSO based reconstructing method provided a higher accuracy. Additionally, the feasibility of the proposed method has been experimentally verified using a strain tunable apparatus within a measurable temperature environment. The experimental results demonstrated that the half-encapsulated FBG with an IPSO based spectra reconstructing method was applicable for the simultaneous measurement of temperature and mean strain, even when the spectrum was distorted.
1. Introduction Fiber Bragg gratings (FBGs) have been extensively used in a wide range of applications for the measurement of temperature, and strain and force, as well as other measrands due to their advantages of small size, simplicity in encapsulation, immunity to electromagnetic inference, multiplexing capability, and wavelength-coded characteristics [1]. Nevertheless, in many applications, the temperature-strain cross sensitivity of FBG, which impacts the accuracy and reliability of the measurements, is undesirable [2,3]. Various techniques for the simultaneous measurements of strain and temperature have been reported to discriminate the two parameters. Some of these have been based on the specially designed single FBG, including an FBG with two peanut tapers [4]; special strain-function -chirped FBG [5]; dual-gratings in one fiber [6], and so on. Meanwhile, others have relied on hybrid combinations of FBG with other optical components, for example, another strain-insensitive FBG [7]; long period grating (LPG) [8]; nocore fiber (NCF) [9], and so on. However, specially designed single FBG or required external optical components in these techniques are usually have complicated fabrications, as well as fragility and high costs which
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are unsuitable for practical applications. A partially encapsulated FBG, where part or half the length of the FBG is encapsulated, is an attractive and particularly simple method by which to solve the issue of practical usage. It is easy to fabricate, and is capable of reducing the system's complexity and cost [10]. The spectrum of the partially encapsulated FBG splits into two peaks, which are referred to as the strain-sensitive peak and the straininsensitive peak in this study. The two peaks respond differently to tensile strain and temperature. In the cases where the strain-sensitive peaks shift away from the strain-insensitive peaks under uniform axial tensile strain, the partial encapsulation technique was found to be applicable, since the wavelengths of the two peaks could be easily identified. However, in practice, challenges have arisen when the spectra of the partially encapsulated FBG become easily distorted with multiple peaks when the strain-sensitive peak overlaps with the straininsensitive peak under compressive strain, or when the FBG suffers from non-uniform axial strain. In these situations, the identification of the wavelengths of the two peaks becomes impractical. This in turn hinders the adoption of the technique. Some studies have put forward results while employing this technique to measure temperature and
Corresponding author. E-mail addresses: [email protected] (Z.-f. Wang), [email protected] (Q.-m. Sui).
http://dx.doi.org/10.1016/j.optcom.2016.10.027 Received 29 June 2016; Received in revised form 9 October 2016; Accepted 13 October 2016 Available online 04 February 2017 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
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other mechanical parameters simultaneously [11–13]. However, none of these studies have considered the above-mentioned issue, or addressed the distorted spectra of the two split peaks. An evolutionary algorithm (EA) may provide a solution for the estimation of the applied temperature and strain profile from a distorted spectrum. Various evolutionary algorithms and their modifications have been developed to deal with the distorted FBG spectra, and reconstruct the strain profiles along the length of FBG. These include a genetic algorithm (GA) [14]; chaos GA [15]; genetic programing algorithm [16]; quantum-behaved PSO [17]; adaptive simulated annealing (ASA) algorithm [18]; Nelder-Mead optimization [19]; and so on. For some applications, it is significant to process the distorted spectra using evolutionary algorithms to estimate the strain distribution. However, in most cases, the accurate evaluation of the mean strain is more crucial and practical. When considering a partial encapsulation technique of FBG, it is of paramount importance to estimate the temperature and mean strain when the FBG spectra are distorted. However, to the best of our knowledge, no study has been carried out so far to resolve this issue. In this research study, the spectrum distortion phenomenon of a half-encapsulated FBG attributed to the overlap of two peaks or nonuniform strain was illustrated. A swarm success rate based dynamic adaptive PSO algorithm, namely an Improved PSO (IPSO), was introduced to reconstruct the distorted spectrum, as well as estimate the temperature and mean strain. The numerical simulations of 48 different cases of applying temperature and strain were conducted in order to validate the proposed method, and to compare its performance with the conventional evaluation algorithms (EAs). In addition to the numerical simulation validation, the feasibility of the proposed method was experimentally verified by adhering half of the FBG on the beam of a strain tunable apparatus in temperature measurable conditions. The results indicated that temperature and mean strain could be simultaneously estimated by using an IPSO-based spectrum reconstructing method, even for the distorted spectrum of the half-encapsulated FBG. The study was further structured as follows: Section 2 introduces the sensing principles of the half-encapsulated FBG, and illustrates the reasons for the spectrum distortion. Section 3 presents the improved PSO algorithms for dealing with the distorted spectra. The simulation validations are presented in Section 4, and the experiments are described in detail in Section 5. Finally, the conclusions of this study are presented in Section 6.
κ=
⎡ R0 ⎤ ⎡ RN ⎤ ⎡ f11 f12 ⎤ ⎡ R0 ⎤ ⎥⋅⎢ ⎥ = FN ⋅FN −1 ⋯Fi ⋯F1⋅F0⋅⎢ ⎥ ⎢ ⎥=⎢ ⎣ S0 ⎦ ⎣ SN ⎦ ⎣ f21 f22 ⎦ ⎣ S0 ⎦
R = S0 / R0 2 = f21 / f11
2
(5)
Under the non-uniform strain, the equivalent grating periods Λ(z) of the FBG were described by:
Λ (z ) = Λ0 [1 + (1 − Pe ) ε (z )],
(6)
0≤z≤L
where, Λ0 denotes the grating period of the FBG before the nonuniform strain was applied; L is the length of the FBG, the effective photo-elastic coefficient Pe≈0.22; and ε(z) represents the strain function along the length of the FBG. In this research study, the nonuniform strain field applied on the FBG was approximated using a polynomial. For the half-encapsulated FBG which was one of the typical partially encapsulated situations, the strain function ε(z) could be expressed as follows:
⎧ a + a2⋅z + a3⋅z 2, 0 ≤ z ≤ L /2 ε (z ) = ⎨ 1 0, L /2 < z ≤ L ⎩
(7)
Where, a1, a2, and a3 were defined as the constant coefficient, linear coefficient, and secondary coefficient, respectively. When the FBG wass simultaneously subjected to axial strain ε(z) and temperature changes ΔT, its Bragg wavelength variation Δλ, which shifts proportionally with these thermo-mechanical loadings, could be expressed using the following equation [22]: (8)
Δλ / λB = ΔλT + Δλε = KT ⋅(T − T0°C ) + Kε⋅ε (z )
Where, λB is the Bragg wavelength; ΔλT and Δλε are related to the wavelength shifts induced by the temperature and strain; and KT and Kε are the sensitivities to the temperature and strain, respectively. 2.2. Sensing principles of the half-encapsulated FBG The schematic of the half-encapsulated FBG sensor is illustrated in Fig. 1. Half of the FBG was adhered on a substrate, while the remaining half was kept free. Since the FBG was pre-stressed before being attached to the substrate, an initial pre-strain ε0 existed on the fixed half after it is half-encapsulated. Meanwhile, for the other half, since it is unbound, there was no initial strain. Under the initial pre-strain, the instinct parameters varied, such as the grating period and effective refractive index of the fixed half of the FBG, which accordingly caused the shift of the reflection wavelengths. As a result, the reflected wavelengths of the fixed half were different from those of the free half, which contributed to the reflection spectrum of the FBG becoming split
In order to reconstruct the distorted spectra of half-encapsulated FBG and evaluate the applied temperature and strain the transfer matrix method was used in this study to model the reflection spectra of the FBG under non-uniform strain. This method divided the FBG into a sufficient number of short periodic segments, and each segment was characterized by a 2×2 transfer-matrix based on the coupled-mode theory [20,21]. The transfer-matrix of the ith segment Fi was described by the following:
Optic fiber Cable Encapsulated
(1)
Non-encapsulated
Where, Δz=L/N, signifies the length of each segment; L is the length of the FBG; N is the total number of the segments; and γ = κ 2 + σ 2 . The ‘dc’ self-coupling coefficient σ, and the ‘ac’ coupling coefficient κ were expressed as:
1 1 2π − )+ δneff λ λB λ
(4)
Also, the reflection coefficient of the FBG R was given by:
2.1. Theory of fiber Bragg grating
σ = 2πneff (
(3)
The characteristics of the entire FBG were obtained by multiplying the transfer-matrices of all of the segments.
2. Principles
κ ⎡ cosh(γ⋅Δz ) + i⋅ σ ⋅ sinh(γ⋅Δz ) ⎤ − i⋅ γ ⋅ sinh(γ⋅Δz ) γB ⎥ B Fi = ⎢⎢ κ σ ⎥ i ⋅ ⋅ sinh( γ ⋅ Δz ) cosh( γ ⋅ Δz ) − i ⋅ ⋅ sinh( γ ⋅ Δz ) γ γ ⎣ ⎦
2π ν⋅δneff λ
FBG
Substrate
(2)
Fig. 1. Schematic of partially encapsulated FBG.
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Table 1 Parameters in the simulation based on transfer matrix method. Parameters
Values
Young's Modulus of FBG: E Poisson Ratio of FBG: ρ Length of FBG: L Effective refractive index: Neff Initial Grating Period: Λ0 δneff Effective photo-elastic constant: Pe Temperature sensitivity: KT Strain sensitivity: Kε Initial Strain: ε0 Wavelength resolution
72 GPa 0.17 10 mm 1.46 530.598 nm 2.6×10−4 0.22 10.1 pm/°C 1.209 pm/µε 500 µε 0.01 nm
into two peaks. The peak reflected by the fixed half was named as the strain-sensitive peak in this study, due to the fact that it was adhered on the substrate, and responsive to the applied strain. Also, the other peak reflected by the free half was referred to as the strain-insensitive peak. Both the strain-sensitive peak and the strain-insensitive peak were related to the ambient temperature. A numerical simulation was conducted for the purpose of illustrating the sensing principle of the half-encapsulated FBG. The parameters used in the simulation are shown in Table 1. The simulated responses of the spectrum of the half-encapsulated FBG to temperature and strain are shown in Fig. 2. Spectrum I shows the initial spectrum when the strain was 0 µε, and the temperature was 0 °C. Spectrum II shows the response of the FBG when a uniform axial strain of 1000 µε was applied at 0 °C. Spectrum III shows the spectrum at 0 °C without suffering from any strain. As it can be observed in Fig. 2, when a uniform axial strain was applied on the half-encapsulated FBG, the strain-sensitive peak shifted according to the applied strain, and the strain-insensitive peak remained constant. Additionally, when the ambient temperature varied, both the strain-sensitive peak and strain-insensitive peak shifted with the temperature changes. This was due to the fact that, when the half-encapsulated FBG was subjected to temperature at a constant strain, the fixed portion of the FBG responded to the both temperature and strain, whereas the unbound portion responded only to the temperature. Therefore, the temperature and uniform strain could be simultaneously obtained by observing the variations of the two peaks. 2.3. Spectra distortion of the half-encapsulated FBG
Fig. 3. Three cases of spectra distortion. (a) Overlap. (b)Non-uniform strain. (c) Combined effects.
The half-encapsulated FBG was determined to be applicable when the wavelengths of the two peaks were identifiable, and no distortions existed in its reflection spectrum. However, in practical applications, it is possible that the spectrum of the half-encapsulated FBG may become
distorted. For instance, the spectrum may be unidentifiable due to the overlap when the strain-sensitive peak shifts to the strain-insensitive peak under the compressive strain (Fig. 3(a)). Also, a distortion of the strain-sensitive peak may occur when the fixed half of the FBG is subjected to non-uniform axial strain (Fig. 3(b)), or may be due to the combined effects of the overlap and non-uniform strain on the spectrum (Fig. 3(c)). Once the spectrum becomes distorted, the measurements of the temperature and strain become impossible. The distorted spectra could be utilized to evaluate the strain gradients along the FBG for some special applications. However, in most cases, the distortion would be considered as a source of error. The main focus in most applications is the mean strain at the location where the FBG is attached. Under non-uniform strain, the mean strain ε along the length of the FBG can be expressed as follows: n
ε=
n ∑ Δli l 1 = i =1 = ⋅ ∑ εi L n⋅l n i =1
(9)
where, n is the number of subsections of the FBG; l denotes the overall elongation of the FBG; l is the length of each subsection; Δli denotes
Fig. 2. Responses of the half-encapsulated FBG to temperature and strain.
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the elongation of the ith subsection; and εi represents the strain of the ith subsection. In the following sections, the method which was capable of evaluating the mean strain and temperature based on the distorted spectrum of the half-encapsulated FBG was introduced.
(16) and (17), respectively [24].
φi = e(−i
Vni+1 = wv⋅Vni + C1⋅R1 (Pibest − Xni ) + C2⋅R2 (Gbest − Xni )
(10)
Xni+1 = Xni + Vni+1
(11)
E=
i3 Iter 3
N
∑ ( fitnessn − fitnessavg)2 n =1
∑n =1 succni (19)
N i
Where succn signifies the success of particle n at the ith iteration, and is defined as:
⎧1, if fitness Pibest i < fitness Pibest i −1 succni = ⎨ ⎩ 0, if fitness Pibest i ≥ fitness Pibest i −1
(20)
When no particle succeeded to improve the individual fitness, the ssri equaled 0; when all of the particles succeeded in improving the fitness, it equaled 1. Therefore, the swarm success rate ssr reflected the state of the swarm, and thus could be treated as a feedback to the algorithm in order to adjust the inertia weight at the ith iteration [25]. The procedure of reconstructing the FBG spectrum using IPSO is illustrated in Fig. 4. 4. Simulation validation To validate the performance of the method introduced in the previous section, numerical simulations were conducted to reconstruct the spectra of the half-encapsulated FBG under different scenarios, and also to estimate the applied mean strains and temperatures.
(13) 4.1. Numerical simulation A total of 48 different scenarios of applying strain and temperature were simulated by varying the coefficients a1, a2, a3, and T in Eqs. (7)
(14)
Where λi is the sampled wavelength of the FBG spectrum; and M is the total sampling number. Inertia weight wv is an important parameter which influences the performance of the PSO during the course of solving global optimization problems. The most commonly used strategies of adjusting inertia weight are constant, randomly-declined, or linearly-declined. However, none of these contain any information regarding the states (superiority or inferiority) of the particle in the search space, which could greatly influence the performance of the algorithm. In this study, an Improved PSO (IPSO), of which the swarm success rates (ssr) were based on a dynamic adaptive PSO, was utilized to adjust the particle's inertia weight based on the state of the particle in order to improve the convergence speed and avoid the local optimal. The strategy of updating the inertia weight wv in the IPSO was expressed as:
wvi = (wmax − wmin ) × F i × ϕi + wmin × ssr i −1
(18)
N
Where C1_ini and C1_fin are the initial and final values of the cognitive coefficient; and C2_ini and C2_fin are the initial and final values of the social coefficient. The quality of the particles was appraised using a fitness function, which was defined as the error between the reconstructed spectrum Rrec(λ) and the target spectrum Rtar(λ) (Eq. (14)) for this spectrum reconstruction problem:
(i = 1... M )
1 N
ssr i =
(12)
i3 Iter 3
∑ (Rrec (λi ) − Rtar (λi ))2
(17)
Where fitnessn is the fitness of particle n and fitnessavg is the current mean fitness of the swarm. The swarm success rate at the ith iteration ssri in Eq. (15) is defined as:
Where, Vni and Xni represent the velocity and position of particle n at the ith iteration, respectively; 1≤n≤N, N is the particle population; 1≤i≤Iter, Iter is the maximum of iteration; wv is the inertia weight representing the impact of the previous velocity; R1 and R2 are two random numbers belonging to [0,1]; Pibest is the best previous position of the particle, and Gbest is the best previous position among all of the particles until the ith iteration; The acceleration coefficients C1 and C2, which were respectively named as the “cognitive coefficient” and “social coefficient”, represented the impact of the individuals and neighbors on the particles. The time-variant acceleration coefficients were employed in this study to balance the exploration and exploitation. The acceleration coefficients C1(i) and C2(i) at the ith iteration were updated using the following equations:
fitness =
(16)
Where σ=Iter/3, and E is the group fitness expressed as:
In this research study, a Particle Swarm Optimization (PSO) based spectrum reconstruction method was adopted to process the distorted spectrum, and to estimate the temperature and mean strain. In the PSO algorithm, a swarm of particles (potential solutions) is randomly positioned in the search space and then moves through the search space with a specified velocity [23]. Each particle maintains a memory of its previous best position as well as the global best position, and adjusts its velocity and position for the optimal solution according to the following formulas:
C2 (i ) = C2 ini + (C2 fin − C2 ini ) ×
2 ⋅arctan(E ) π
Fi = 1 −
3. Reconstructing approach
C1 (i ) = C1 ini + (C1 fin − C1 ini ) ×
2 /(2⋅ σ 2 ))
(15)
Where wmax and wmin represent the maximum and minimum inertia weight values; φi and Fi are the adjustment function and diversity function at the ith iteration, respectively, which were calculated by Eqs.
Fig. 4. Procedure of IPSO based spectrum reconstruction.
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Table 2 Parameters used in the optimization algorithms. Particles population N Maximum of iteration Iter Maximum of the inertia weight wmax Minimum of the inertia weight wmin Initial value of the cognitive coefficient C1_ini Final value of the cognitive coefficient C1_fin Initial value of the cognitive coefficient C2_ini Final value of the cognitive coefficient C2_fin Percentage of cross Percentage of mutation
100 100 1.2 0.6 2.2 0.5 0.5 2.2 0.3 0.1
Fig. 6. Performance comparisons of the different algorithms. (a) Errors of temperatures. (b) Errors of mean strains.
Fig. 5. Scenarios of numerical simulations. (a) Coefficients a1 and a2. (b) Coefficients a3. (c) Applied temperature and mean strain.
4.2. Performance evaluation
and (8), as shown in Fig. 5. Fig. 5(a) and (b) depict the variations of the strain coefficients a1, a2, and a3 for each scenario. The range of coefficient a1 was from −0.0011 to 0.0011, which corresponded to −1100 µε to 1100 µε. The coefficient a2 varied from −0.21 (−210 µε/ mm) to 0.21 (210 µε/mm), and the coefficient a3 belonged to [−11, 11], which approximated [−11 µε/mm2, 11 µε/mm2]. The mean strains were obtained by Eq. (9), and the applied temperature T changed in a range of 0–65 °C. The applied temperatures and mean strains for all the scenarios are shown in Fig. 5(c). The parameters which are shown in Table 1 were utilized to calculate the FBG spectrum. For comparison purposes, both a Genetic Algorithm (GA) and Linear-declined PSO were employed to reconstruct the spectra of all the scenarios, and were compare with that reconstructed using IPSO. The parameters for IPSO, GA, and PSO in the simulation are given in Table 2.
Fig. 6 shows the reconstructed errors of the different algorithms as a function of the simulated scenarios. The a1, a2, a3, and T were treated as the parameters to be optimized, and the mean strains were calculated from the three coefficients a1, a2, and a3. Fig. 6(a) shows the comparison of the temperature errors for the different algorithms. The maximum temperature error calculated using the IPSO was 2.3 °C in all of the 48 cases, while they were 21.9 °C and −12.9 °C for the LDPSO and GA, respectively. The errors of the mean strain are illustrated in Fig. 6(b). The results showed that the maximum error of the IPSO was 55.0 µε, which was smaller than that calculated by the GA (125.6 µε) and PSO (−149.6 µε). The error range for the 48 scenarios provided by the IPSO was better than the other two algorithms. The comparisons of the error range for the three algorithms are given in Table 3. The results demonstrated that the IPSO algorithm performed better than the conventional GA and PSO in reconstructing the distorted spectra of the half-encapsulated FBG. In this study, four scenarios (No. 1, No. 12, No. 20, and No. 43) 157
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Table 3 Comparisons of error ranges for the three algorithms. Algorithms
Error range of T (°C)
Error range of ε (µε)
IPSO PSO GA
−1.5 to 2.3 −5.4 to 21.9 −12.9 to 2.3
−25.7 to 55.0 −149.6 to 125.6 −43.0 to 162.5
were randomly selected for a comparison of the original spectra with the spectra reconstructed using the IPSO. In terms of the physical meaning, in Scenario No. 1, the FBG suffered from a uniform compressive strain when the temperature was around 60 °C. Also, the No. 12, No. 20, and No. 43 Scenarios occurred when the nonuniform strains were applied on the FBG under different temperatures. In the No. 12 and No. 20 Scenarios, a small tensile strain was applied at the beginning end of the FBG, and the strains dramatically decreased along the length of the FBG. Therefore, the mean strains of the FBG for the two scenarios were negative. In the cases where the FBG was attached below the surface of a simply supported beam at the location of approximately 3/4, the load was applied to the upper surface of the beam. The temperatures of the two scenarios were different. The physical meaning for Scenario No. 43 was that, when the temperature was approximately 0 °C, a relatively large tensile strain was implemented at the beginning end of the FBG, and the strain dramatically decreased along the length of the FBG. Consequently, the value of the mean strain on the FBG was positive. In the cases where the FBG was attached to the below surface of a simply supported beam between the locations of 1/2 and 3/4, the load was applied on the upper surface of the beam. The original and reconstructed distorted spectra, using IPSO for the four scenarios, are shown in Fig. 7. The results illustrated that the reconstructed spectrum using IPSO fit well with the real spectrum. Although it was found that slight differences existed for Scenario No. 43, the reconstructed spectrum still reflected the profiles of the original spectrum. The estimated temperatures, strains, and errors for the four scenarios (No. 1, No. 12, No. 20, and No. 43) are listed in Table 4. 5. Experiments In this section, the performance of the introduced spectrum reconstruction method for the estimation of the temperature and mean strain, based on the distorted spectra of the half-encapsulated FBG, was experimentally validated. The experimental setup is elaborated in Section 5.1, and the experimental results and analysis are given in Section 5.2. 5.1. Experimental setup This study's experiments were performed in a measurable temperature environment using a strain tunable apparatus, as shown in Fig. 8. Two ends of a constant cross-section beam (material: 65 Mn) were attached on the two brackets of the apparatus. One bracket was fixed, while the other remained movable by turning a knob assembled on the end of a long screw. The dimensions of the beam (between the two brackets) were as follows: 200×8×1, and a caliper (accuracy of 0.05 mm) was utilized to measure the deflection of the beam. Then, FBG with grating lengths of 10 mm and wavelengths of 1550 nm were half adhered using epoxy resin on the beam at approximately a quarter of the length. Then, in order to precisely control the length of encapsulation, half of the FBG was packaged in a glue-resistive Teflon film when it was gelatinized to the other half. The location of the adhered FBG was then measured by the caliper, and was found to be from 52.35 to 57.35 mm. The spectrum of the half-encapsulated FBG was captured using an FBG interrogator SM125, with a wavelength resolution of 0.005 nm. To characterize the strain distribution of the cross-section beam, as
Fig. 7. Original and reconstructed distorted spectra of four scenarios. (a) Spectra of scenario No.1. (b) Spectra of scenario No.12. (c) Spectra of scenario No.20. (d) Spectra of scenario No.43.
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5.2. Experiments and analysis
Table 4 Estimated measurands and errors of the four scenarios. Scenarios
Estimated T (°C)
Error of T (° C)
Estimated ε (µε)
Error of ε (µε)
No.1 No.12 No.20 No.43
60.0 3.1 63.5 1.2
0.1 0.1 −0.5 1
−251.9 −257.9 −256.9 216.0
−1.9 3.2 3.3 −25.7
Due to the difference of the thermal expansion coefficients between the optic fiber and the substrate where the FBG was attached, the temperature responses of the encapsulated half and the free half were different. This contributed to a deviation between the temperature sensitivities of the strain-sensitive peak and that of the strain-insensitive peak. Therefore, the calibration for the half-encapsulated FBG was conducted prior to the strain tuning experiments in order to obtain the temperature sensitivities of the two peaks which were utilized in the transfer matrix method. The calibration was conducted in a temperature controlled incubator. The temperature was increased from 25 to 65 °C in increments of 10 °C. The temperature sensitivity of the straininsensitive peak (KT1) was determined to be 0.011 nm/°C, while that of the strain-sensitive peak (KT2) was 0.023 nm/°C. The increase in sensitivity was due to the thermal expansion coefficient of the substrate (65 Mn) being higher when compared with the optic fiber. After the temperature sensitivities were obtained, the next step was to ascertain the intrinsic parameters of the FBG, along with the initial strain induced by the half encapsulation. The spectrum, when the applied strain was 0 µε at 25.1 °C, was used as the target spectrum, which is illustrated by the solid line in Fig. 10. An IPSO algorithm was adopted to reconstruct the spectrum, as well as to estimate the intrinsic parameters and initial strain. The wavelength resolution of the target spectrum was 0.01 nm, which was picked up from the original spectrum in order to reduce the data amount, and accelerate the calculation process. The reconstructed spectrum is shown in Fig. 10 (dotted line). Although there was found to be a slight deviation due to the background noise of the measured spectrum, the profile of the two spectra fit very well. Table 6 shows the reconstructed parameters. In this study, three cases were experimentally validated. Case 1, in which the applied strain was 0 µε and the temperature was 40.0 °C, was conducted in a temperature controlled incubator. Case 2 and Case 3 were carried out on a strain tunable apparatus at room temperature, which was measured by a thermometer. For Case 2, the surface where the FBG was attached was downward, and a displacement of 20 mm was applied to the beam. In regards to Case 3, the surface where the FBG was attached was upward, and a displacement of 35 mm was applied on the beam. The strain data for the two cases were extracted from the FE simulation results in order to calculate the applied mean strain, as shown in Fig. 11. The dotted line in Fig. 11 represents the strain distribution along the centerline of the beam on the below surface when the displacement was 20 mm (Case 2). Also, the solid line in Fig. 11 shows the strain distribution of Case 3, when a displacement of 35 mm was implemented on the beam. The strain data where the FBG was attached, which are also highlighted in Fig. 11, indicated that the applied strain along the FBG followed a linear relationship. A linear
Fig. 8. Experimental setup. Table 5 Parameters used in the FE simulations. Length (mm) Width (mm) Thickness (mm) Yong's modulus E (GPa) Poisson ratio μ Density (kg/m3) Number of mesh
200 8 1 210 0.3 7650 6071
Fig. 9. Displacement contours when applied displacement is 35 mm.
well as to estimate the mean strain applied on the FBG, finite element (FE) simulations were performed for the beam. A 3D cuboid was built to simulate the beam, and an elastic material model with a geometric non-linearity was selected. One end of the model was fixed, and the other end could only move in a horizontal direction (x axis). Then, vertical displacements (z axis) were implemented in the middle of the beam. The strains along the centerline of the beam surface were extracted for analysis. Table 5 displays the parameters used in the FE simulations. When the displacement was 35 mm, the displacement contours were as shown in Fig. 9.
Fig. 10. Measured and reconstructed spectra at 25.1 °C when the applied strain is 0 µε.
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Table 6 Reconstructed intrinsic parameters and initial strain. Effective refractive index: Neff Initial Grating Period: Λ0 δneff Initial Strain: ε0
1.449 534.390 nm 1.85×10−4 485.6 µε
Fig. 11. Strain distribution of the beam extracted from FEM simulation.
least square fitting method was utilized to obtain the strain distribution function for the two cases. The fitted function for Case 2 was: ε2(z) =−109.5z-46.99, (0≤z≤5 mm), which meant that the FBG was subject to a compressive strain, and the strain declined along the length of the FBG. For Case 3, the strain function was ε3(z)=195.1z+232.52, (0≤z≤5 mm), which indicated that the FBG suffered from a tensile strain, and the strain linearly increased along its length. The calculated mean strains for Case 2 and Case 3 were −151.0 µε and 349.9 µε, respectively. Also, the measured temperatures were 25.1 °C for Case 2, and 25.2 °C for Case 3. The measured and reconstructed spectra for the three cases are shown in Fig. 12. For Case 1, the spectrum of the half-encapsulated FBG shifted towards the long wavelength (Fig. 12(a)). The two peaks were identifiable due to the fact that there was no strain applied on the FBG. The slight distortion of the spectrum was likely a result of the effects of adhesion under high temperatures. The reconstructed spectrum reflected the profile of the measured spectrum, and the reconstructed temperature and mean strain were determined to be 39.7 °C and 1.4 µε, respectively. Fig. 12(b) displays the spectra of Case 2. The two peaks overlapped with each other under the applied compressive strain, and the spectrum became distorted as a result of the linearly-declined strain field (non-uniform strain). The reconstructed spectrum fit well with the measured spectrum, and the estimated temperature and mean strain for this case were determined to be 24.2 °C and −136.5 µε, respectively. The comparison of the spectra for Case 3 can be seen in Fig. 12(c). One peak, the straininsensitive peak, was identifiable, while the other peak showed a serious distortion under the non-uniform tensile strain. Although a small difference existed in the two spectra, the reconstructed spectrum still indicated the variation of the measured spectrum. The calculated temperature was 26.9 °C, and the mean strain was 304.7 µε. Table 7 shows the comparison of the applied and estimated measurands. It can be seen that Case 3, whose temperature error was −1.7 °C, and strain error was 45.2 µε, provided the worst accuracy. However, it was still superior to the current technique in which the temperature and strain cannot be evaluated when the spectra are distorted. Therefore, the half-encapsulated FBG with an IPSO-based spectra reconstructing method was found to be significant to the practical applications in which the spectrum is distorted.
Fig. 12. Displacement contours when applied displacement is 35 mm. (a). Case1. (b). Case2. (c). Case3. Table 7 Comparison of applied measurands and estimated measurands of the three Cases.
Applied T (°C) Applied ε (µε) Reconstructed a1 Reconstructed a2 Reconstructed a3 Estimated T (°C) Estimated ε (µε) Error of T (°C) Error of ε (µε)
160
Case1
Case2
Case3
40.0 0.0 −0.000236 0.07 8 39.7 1.4 0.3 −1.4
25.1 −151.0 −0.000077 −0.06 −5 24.2 −136.5 0.9 −14.5
25.2 349.9 −0.000010 0.22 7 26.9 304.7 −1.7 45.2
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High Technology Research and Development Program (“863”Program) of China (2014AA110401).
6. Conclusion It has been determined that a half-encapsulated FBG is a straightforward method to simultaneously measure temperature and strain. However, an important issue has been that the spectrum of a halfencapsulated FBG tends to be distorted during practical usage, which makes the measurement impossible. This study analyzed the causes of the spectrum distortion by first examining the overlapping and nonuniform strains, as well as the combined effects. Then, a method which was capable of simultaneously estimating the temperature and mean strain based on the distorted spectrum was proposed. The method adopted an improved particle swarm optimization (IPSO) algorithm to reconstruct the spectrum, and to calculate the temperature and mean strain. The swarm success rate was based on a dynamic adaptive strategy for adjusting the inertia weight, and was utilized to balance the exploration and exploitation. Then, a total of 48 simulation scenarios were carried out to validate the proposed method. The simulation results indicated that the temperature and mean strain could be effectively estimated, even when the spectrum was highly distorted. A comparison of the GA, linearly-declined PSO, and IPSO-based reconstructing methods indicated that the IPSO provided a higher accuracy. Moreover, the feasibility of the proposed method was experimentally verified using a strain tunable apparatus in a temperature measurable environment. Three experimental cases were performed, and the strain distribution along the FBG for each case was characterized utilizing FEM simulations. A maximum temperature error of −1.7 °C and a mean strain error of 45.2 µε were obtained in the experiments for the three cases. The experimental results demonstrated that the halfencapsulated FBG with an IPSO-based spectra reconstructing method was applicable to the simultaneous measurements of temperature and mean strain, even when the spectrum was distorted.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Number: 41472260) and the National
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The simultaneous measurement of temperature and mean strain based on the distorted spectra of half-encapsulated fiber Bragg gratings using improved particle swarm optimization
MARK
⁎
Zheng-fang Wang, Jing Wang, Qing-mei Sui , Lei Jia School of Control Science and Engineering, Shandong University, Jinan, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Fiber Bragg grating Spectra reconstruction Half-encapsulation Particles swarm optimization
A half-encapsulated FBG is capable of simultaneously measuring temperature and strain. However, spectrum distortion, which may be induced by overlapping or non-uniform strains, may hinder the adoption of this technique. In order to resolve this issue, an improved particle swarm optimization (IPSO) based spectra reconstructing method has been adopted in this study to estimate the temperature and mean strain according to the distorted spectrum. Also, a dynamic adaptive inertia weight adjusting strategy based on the swarm success rate has been adopted to improve the algorithm. To validate the method, a total of 48 scenarios of distorted spectra have been simulated, and the temperature and mean strain estimated by IPSO have been compared with the genetic algorithm and linearly-declined PSO. The simulation results indicated that the IPSO based reconstructing method provided a higher accuracy. Additionally, the feasibility of the proposed method has been experimentally verified using a strain tunable apparatus within a measurable temperature environment. The experimental results demonstrated that the half-encapsulated FBG with an IPSO based spectra reconstructing method was applicable for the simultaneous measurement of temperature and mean strain, even when the spectrum was distorted.
1. Introduction Fiber Bragg gratings (FBGs) have been extensively used in a wide range of applications for the measurement of temperature, and strain and force, as well as other measrands due to their advantages of small size, simplicity in encapsulation, immunity to electromagnetic inference, multiplexing capability, and wavelength-coded characteristics [1]. Nevertheless, in many applications, the temperature-strain cross sensitivity of FBG, which impacts the accuracy and reliability of the measurements, is undesirable [2,3]. Various techniques for the simultaneous measurements of strain and temperature have been reported to discriminate the two parameters. Some of these have been based on the specially designed single FBG, including an FBG with two peanut tapers [4]; special strain-function -chirped FBG [5]; dual-gratings in one fiber [6], and so on. Meanwhile, others have relied on hybrid combinations of FBG with other optical components, for example, another strain-insensitive FBG [7]; long period grating (LPG) [8]; nocore fiber (NCF) [9], and so on. However, specially designed single FBG or required external optical components in these techniques are usually have complicated fabrications, as well as fragility and high costs which
⁎
are unsuitable for practical applications. A partially encapsulated FBG, where part or half the length of the FBG is encapsulated, is an attractive and particularly simple method by which to solve the issue of practical usage. It is easy to fabricate, and is capable of reducing the system's complexity and cost [10]. The spectrum of the partially encapsulated FBG splits into two peaks, which are referred to as the strain-sensitive peak and the straininsensitive peak in this study. The two peaks respond differently to tensile strain and temperature. In the cases where the strain-sensitive peaks shift away from the strain-insensitive peaks under uniform axial tensile strain, the partial encapsulation technique was found to be applicable, since the wavelengths of the two peaks could be easily identified. However, in practice, challenges have arisen when the spectra of the partially encapsulated FBG become easily distorted with multiple peaks when the strain-sensitive peak overlaps with the straininsensitive peak under compressive strain, or when the FBG suffers from non-uniform axial strain. In these situations, the identification of the wavelengths of the two peaks becomes impractical. This in turn hinders the adoption of the technique. Some studies have put forward results while employing this technique to measure temperature and
Corresponding author. E-mail addresses: [email protected] (Z.-f. Wang), [email protected] (Q.-m. Sui).
http://dx.doi.org/10.1016/j.optcom.2016.10.027 Received 29 June 2016; Received in revised form 9 October 2016; Accepted 13 October 2016 Available online 04 February 2017 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
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other mechanical parameters simultaneously [11–13]. However, none of these studies have considered the above-mentioned issue, or addressed the distorted spectra of the two split peaks. An evolutionary algorithm (EA) may provide a solution for the estimation of the applied temperature and strain profile from a distorted spectrum. Various evolutionary algorithms and their modifications have been developed to deal with the distorted FBG spectra, and reconstruct the strain profiles along the length of FBG. These include a genetic algorithm (GA) [14]; chaos GA [15]; genetic programing algorithm [16]; quantum-behaved PSO [17]; adaptive simulated annealing (ASA) algorithm [18]; Nelder-Mead optimization [19]; and so on. For some applications, it is significant to process the distorted spectra using evolutionary algorithms to estimate the strain distribution. However, in most cases, the accurate evaluation of the mean strain is more crucial and practical. When considering a partial encapsulation technique of FBG, it is of paramount importance to estimate the temperature and mean strain when the FBG spectra are distorted. However, to the best of our knowledge, no study has been carried out so far to resolve this issue. In this research study, the spectrum distortion phenomenon of a half-encapsulated FBG attributed to the overlap of two peaks or nonuniform strain was illustrated. A swarm success rate based dynamic adaptive PSO algorithm, namely an Improved PSO (IPSO), was introduced to reconstruct the distorted spectrum, as well as estimate the temperature and mean strain. The numerical simulations of 48 different cases of applying temperature and strain were conducted in order to validate the proposed method, and to compare its performance with the conventional evaluation algorithms (EAs). In addition to the numerical simulation validation, the feasibility of the proposed method was experimentally verified by adhering half of the FBG on the beam of a strain tunable apparatus in temperature measurable conditions. The results indicated that temperature and mean strain could be simultaneously estimated by using an IPSO-based spectrum reconstructing method, even for the distorted spectrum of the half-encapsulated FBG. The study was further structured as follows: Section 2 introduces the sensing principles of the half-encapsulated FBG, and illustrates the reasons for the spectrum distortion. Section 3 presents the improved PSO algorithms for dealing with the distorted spectra. The simulation validations are presented in Section 4, and the experiments are described in detail in Section 5. Finally, the conclusions of this study are presented in Section 6.
κ=
⎡ R0 ⎤ ⎡ RN ⎤ ⎡ f11 f12 ⎤ ⎡ R0 ⎤ ⎥⋅⎢ ⎥ = FN ⋅FN −1 ⋯Fi ⋯F1⋅F0⋅⎢ ⎥ ⎢ ⎥=⎢ ⎣ S0 ⎦ ⎣ SN ⎦ ⎣ f21 f22 ⎦ ⎣ S0 ⎦
R = S0 / R0 2 = f21 / f11
2
(5)
Under the non-uniform strain, the equivalent grating periods Λ(z) of the FBG were described by:
Λ (z ) = Λ0 [1 + (1 − Pe ) ε (z )],
(6)
0≤z≤L
where, Λ0 denotes the grating period of the FBG before the nonuniform strain was applied; L is the length of the FBG, the effective photo-elastic coefficient Pe≈0.22; and ε(z) represents the strain function along the length of the FBG. In this research study, the nonuniform strain field applied on the FBG was approximated using a polynomial. For the half-encapsulated FBG which was one of the typical partially encapsulated situations, the strain function ε(z) could be expressed as follows:
⎧ a + a2⋅z + a3⋅z 2, 0 ≤ z ≤ L /2 ε (z ) = ⎨ 1 0, L /2 < z ≤ L ⎩
(7)
Where, a1, a2, and a3 were defined as the constant coefficient, linear coefficient, and secondary coefficient, respectively. When the FBG wass simultaneously subjected to axial strain ε(z) and temperature changes ΔT, its Bragg wavelength variation Δλ, which shifts proportionally with these thermo-mechanical loadings, could be expressed using the following equation [22]: (8)
Δλ / λB = ΔλT + Δλε = KT ⋅(T − T0°C ) + Kε⋅ε (z )
Where, λB is the Bragg wavelength; ΔλT and Δλε are related to the wavelength shifts induced by the temperature and strain; and KT and Kε are the sensitivities to the temperature and strain, respectively. 2.2. Sensing principles of the half-encapsulated FBG The schematic of the half-encapsulated FBG sensor is illustrated in Fig. 1. Half of the FBG was adhered on a substrate, while the remaining half was kept free. Since the FBG was pre-stressed before being attached to the substrate, an initial pre-strain ε0 existed on the fixed half after it is half-encapsulated. Meanwhile, for the other half, since it is unbound, there was no initial strain. Under the initial pre-strain, the instinct parameters varied, such as the grating period and effective refractive index of the fixed half of the FBG, which accordingly caused the shift of the reflection wavelengths. As a result, the reflected wavelengths of the fixed half were different from those of the free half, which contributed to the reflection spectrum of the FBG becoming split
In order to reconstruct the distorted spectra of half-encapsulated FBG and evaluate the applied temperature and strain the transfer matrix method was used in this study to model the reflection spectra of the FBG under non-uniform strain. This method divided the FBG into a sufficient number of short periodic segments, and each segment was characterized by a 2×2 transfer-matrix based on the coupled-mode theory [20,21]. The transfer-matrix of the ith segment Fi was described by the following:
Optic fiber Cable Encapsulated
(1)
Non-encapsulated
Where, Δz=L/N, signifies the length of each segment; L is the length of the FBG; N is the total number of the segments; and γ = κ 2 + σ 2 . The ‘dc’ self-coupling coefficient σ, and the ‘ac’ coupling coefficient κ were expressed as:
1 1 2π − )+ δneff λ λB λ
(4)
Also, the reflection coefficient of the FBG R was given by:
2.1. Theory of fiber Bragg grating
σ = 2πneff (
(3)
The characteristics of the entire FBG were obtained by multiplying the transfer-matrices of all of the segments.
2. Principles
κ ⎡ cosh(γ⋅Δz ) + i⋅ σ ⋅ sinh(γ⋅Δz ) ⎤ − i⋅ γ ⋅ sinh(γ⋅Δz ) γB ⎥ B Fi = ⎢⎢ κ σ ⎥ i ⋅ ⋅ sinh( γ ⋅ Δz ) cosh( γ ⋅ Δz ) − i ⋅ ⋅ sinh( γ ⋅ Δz ) γ γ ⎣ ⎦
2π ν⋅δneff λ
FBG
Substrate
(2)
Fig. 1. Schematic of partially encapsulated FBG.
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Table 1 Parameters in the simulation based on transfer matrix method. Parameters
Values
Young's Modulus of FBG: E Poisson Ratio of FBG: ρ Length of FBG: L Effective refractive index: Neff Initial Grating Period: Λ0 δneff Effective photo-elastic constant: Pe Temperature sensitivity: KT Strain sensitivity: Kε Initial Strain: ε0 Wavelength resolution
72 GPa 0.17 10 mm 1.46 530.598 nm 2.6×10−4 0.22 10.1 pm/°C 1.209 pm/µε 500 µε 0.01 nm
into two peaks. The peak reflected by the fixed half was named as the strain-sensitive peak in this study, due to the fact that it was adhered on the substrate, and responsive to the applied strain. Also, the other peak reflected by the free half was referred to as the strain-insensitive peak. Both the strain-sensitive peak and the strain-insensitive peak were related to the ambient temperature. A numerical simulation was conducted for the purpose of illustrating the sensing principle of the half-encapsulated FBG. The parameters used in the simulation are shown in Table 1. The simulated responses of the spectrum of the half-encapsulated FBG to temperature and strain are shown in Fig. 2. Spectrum I shows the initial spectrum when the strain was 0 µε, and the temperature was 0 °C. Spectrum II shows the response of the FBG when a uniform axial strain of 1000 µε was applied at 0 °C. Spectrum III shows the spectrum at 0 °C without suffering from any strain. As it can be observed in Fig. 2, when a uniform axial strain was applied on the half-encapsulated FBG, the strain-sensitive peak shifted according to the applied strain, and the strain-insensitive peak remained constant. Additionally, when the ambient temperature varied, both the strain-sensitive peak and strain-insensitive peak shifted with the temperature changes. This was due to the fact that, when the half-encapsulated FBG was subjected to temperature at a constant strain, the fixed portion of the FBG responded to the both temperature and strain, whereas the unbound portion responded only to the temperature. Therefore, the temperature and uniform strain could be simultaneously obtained by observing the variations of the two peaks. 2.3. Spectra distortion of the half-encapsulated FBG
Fig. 3. Three cases of spectra distortion. (a) Overlap. (b)Non-uniform strain. (c) Combined effects.
The half-encapsulated FBG was determined to be applicable when the wavelengths of the two peaks were identifiable, and no distortions existed in its reflection spectrum. However, in practical applications, it is possible that the spectrum of the half-encapsulated FBG may become
distorted. For instance, the spectrum may be unidentifiable due to the overlap when the strain-sensitive peak shifts to the strain-insensitive peak under the compressive strain (Fig. 3(a)). Also, a distortion of the strain-sensitive peak may occur when the fixed half of the FBG is subjected to non-uniform axial strain (Fig. 3(b)), or may be due to the combined effects of the overlap and non-uniform strain on the spectrum (Fig. 3(c)). Once the spectrum becomes distorted, the measurements of the temperature and strain become impossible. The distorted spectra could be utilized to evaluate the strain gradients along the FBG for some special applications. However, in most cases, the distortion would be considered as a source of error. The main focus in most applications is the mean strain at the location where the FBG is attached. Under non-uniform strain, the mean strain ε along the length of the FBG can be expressed as follows: n
ε=
n ∑ Δli l 1 = i =1 = ⋅ ∑ εi L n⋅l n i =1
(9)
where, n is the number of subsections of the FBG; l denotes the overall elongation of the FBG; l is the length of each subsection; Δli denotes
Fig. 2. Responses of the half-encapsulated FBG to temperature and strain.
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the elongation of the ith subsection; and εi represents the strain of the ith subsection. In the following sections, the method which was capable of evaluating the mean strain and temperature based on the distorted spectrum of the half-encapsulated FBG was introduced.
(16) and (17), respectively [24].
φi = e(−i
Vni+1 = wv⋅Vni + C1⋅R1 (Pibest − Xni ) + C2⋅R2 (Gbest − Xni )
(10)
Xni+1 = Xni + Vni+1
(11)
E=
i3 Iter 3
N
∑ ( fitnessn − fitnessavg)2 n =1
∑n =1 succni (19)
N i
Where succn signifies the success of particle n at the ith iteration, and is defined as:
⎧1, if fitness Pibest i < fitness Pibest i −1 succni = ⎨ ⎩ 0, if fitness Pibest i ≥ fitness Pibest i −1
(20)
When no particle succeeded to improve the individual fitness, the ssri equaled 0; when all of the particles succeeded in improving the fitness, it equaled 1. Therefore, the swarm success rate ssr reflected the state of the swarm, and thus could be treated as a feedback to the algorithm in order to adjust the inertia weight at the ith iteration [25]. The procedure of reconstructing the FBG spectrum using IPSO is illustrated in Fig. 4. 4. Simulation validation To validate the performance of the method introduced in the previous section, numerical simulations were conducted to reconstruct the spectra of the half-encapsulated FBG under different scenarios, and also to estimate the applied mean strains and temperatures.
(13) 4.1. Numerical simulation A total of 48 different scenarios of applying strain and temperature were simulated by varying the coefficients a1, a2, a3, and T in Eqs. (7)
(14)
Where λi is the sampled wavelength of the FBG spectrum; and M is the total sampling number. Inertia weight wv is an important parameter which influences the performance of the PSO during the course of solving global optimization problems. The most commonly used strategies of adjusting inertia weight are constant, randomly-declined, or linearly-declined. However, none of these contain any information regarding the states (superiority or inferiority) of the particle in the search space, which could greatly influence the performance of the algorithm. In this study, an Improved PSO (IPSO), of which the swarm success rates (ssr) were based on a dynamic adaptive PSO, was utilized to adjust the particle's inertia weight based on the state of the particle in order to improve the convergence speed and avoid the local optimal. The strategy of updating the inertia weight wv in the IPSO was expressed as:
wvi = (wmax − wmin ) × F i × ϕi + wmin × ssr i −1
(18)
N
Where C1_ini and C1_fin are the initial and final values of the cognitive coefficient; and C2_ini and C2_fin are the initial and final values of the social coefficient. The quality of the particles was appraised using a fitness function, which was defined as the error between the reconstructed spectrum Rrec(λ) and the target spectrum Rtar(λ) (Eq. (14)) for this spectrum reconstruction problem:
(i = 1... M )
1 N
ssr i =
(12)
i3 Iter 3
∑ (Rrec (λi ) − Rtar (λi ))2
(17)
Where fitnessn is the fitness of particle n and fitnessavg is the current mean fitness of the swarm. The swarm success rate at the ith iteration ssri in Eq. (15) is defined as:
Where, Vni and Xni represent the velocity and position of particle n at the ith iteration, respectively; 1≤n≤N, N is the particle population; 1≤i≤Iter, Iter is the maximum of iteration; wv is the inertia weight representing the impact of the previous velocity; R1 and R2 are two random numbers belonging to [0,1]; Pibest is the best previous position of the particle, and Gbest is the best previous position among all of the particles until the ith iteration; The acceleration coefficients C1 and C2, which were respectively named as the “cognitive coefficient” and “social coefficient”, represented the impact of the individuals and neighbors on the particles. The time-variant acceleration coefficients were employed in this study to balance the exploration and exploitation. The acceleration coefficients C1(i) and C2(i) at the ith iteration were updated using the following equations:
fitness =
(16)
Where σ=Iter/3, and E is the group fitness expressed as:
In this research study, a Particle Swarm Optimization (PSO) based spectrum reconstruction method was adopted to process the distorted spectrum, and to estimate the temperature and mean strain. In the PSO algorithm, a swarm of particles (potential solutions) is randomly positioned in the search space and then moves through the search space with a specified velocity [23]. Each particle maintains a memory of its previous best position as well as the global best position, and adjusts its velocity and position for the optimal solution according to the following formulas:
C2 (i ) = C2 ini + (C2 fin − C2 ini ) ×
2 ⋅arctan(E ) π
Fi = 1 −
3. Reconstructing approach
C1 (i ) = C1 ini + (C1 fin − C1 ini ) ×
2 /(2⋅ σ 2 ))
(15)
Where wmax and wmin represent the maximum and minimum inertia weight values; φi and Fi are the adjustment function and diversity function at the ith iteration, respectively, which were calculated by Eqs.
Fig. 4. Procedure of IPSO based spectrum reconstruction.
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Table 2 Parameters used in the optimization algorithms. Particles population N Maximum of iteration Iter Maximum of the inertia weight wmax Minimum of the inertia weight wmin Initial value of the cognitive coefficient C1_ini Final value of the cognitive coefficient C1_fin Initial value of the cognitive coefficient C2_ini Final value of the cognitive coefficient C2_fin Percentage of cross Percentage of mutation
100 100 1.2 0.6 2.2 0.5 0.5 2.2 0.3 0.1
Fig. 6. Performance comparisons of the different algorithms. (a) Errors of temperatures. (b) Errors of mean strains.
Fig. 5. Scenarios of numerical simulations. (a) Coefficients a1 and a2. (b) Coefficients a3. (c) Applied temperature and mean strain.
4.2. Performance evaluation
and (8), as shown in Fig. 5. Fig. 5(a) and (b) depict the variations of the strain coefficients a1, a2, and a3 for each scenario. The range of coefficient a1 was from −0.0011 to 0.0011, which corresponded to −1100 µε to 1100 µε. The coefficient a2 varied from −0.21 (−210 µε/ mm) to 0.21 (210 µε/mm), and the coefficient a3 belonged to [−11, 11], which approximated [−11 µε/mm2, 11 µε/mm2]. The mean strains were obtained by Eq. (9), and the applied temperature T changed in a range of 0–65 °C. The applied temperatures and mean strains for all the scenarios are shown in Fig. 5(c). The parameters which are shown in Table 1 were utilized to calculate the FBG spectrum. For comparison purposes, both a Genetic Algorithm (GA) and Linear-declined PSO were employed to reconstruct the spectra of all the scenarios, and were compare with that reconstructed using IPSO. The parameters for IPSO, GA, and PSO in the simulation are given in Table 2.
Fig. 6 shows the reconstructed errors of the different algorithms as a function of the simulated scenarios. The a1, a2, a3, and T were treated as the parameters to be optimized, and the mean strains were calculated from the three coefficients a1, a2, and a3. Fig. 6(a) shows the comparison of the temperature errors for the different algorithms. The maximum temperature error calculated using the IPSO was 2.3 °C in all of the 48 cases, while they were 21.9 °C and −12.9 °C for the LDPSO and GA, respectively. The errors of the mean strain are illustrated in Fig. 6(b). The results showed that the maximum error of the IPSO was 55.0 µε, which was smaller than that calculated by the GA (125.6 µε) and PSO (−149.6 µε). The error range for the 48 scenarios provided by the IPSO was better than the other two algorithms. The comparisons of the error range for the three algorithms are given in Table 3. The results demonstrated that the IPSO algorithm performed better than the conventional GA and PSO in reconstructing the distorted spectra of the half-encapsulated FBG. In this study, four scenarios (No. 1, No. 12, No. 20, and No. 43) 157
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Table 3 Comparisons of error ranges for the three algorithms. Algorithms
Error range of T (°C)
Error range of ε (µε)
IPSO PSO GA
−1.5 to 2.3 −5.4 to 21.9 −12.9 to 2.3
−25.7 to 55.0 −149.6 to 125.6 −43.0 to 162.5
were randomly selected for a comparison of the original spectra with the spectra reconstructed using the IPSO. In terms of the physical meaning, in Scenario No. 1, the FBG suffered from a uniform compressive strain when the temperature was around 60 °C. Also, the No. 12, No. 20, and No. 43 Scenarios occurred when the nonuniform strains were applied on the FBG under different temperatures. In the No. 12 and No. 20 Scenarios, a small tensile strain was applied at the beginning end of the FBG, and the strains dramatically decreased along the length of the FBG. Therefore, the mean strains of the FBG for the two scenarios were negative. In the cases where the FBG was attached below the surface of a simply supported beam at the location of approximately 3/4, the load was applied to the upper surface of the beam. The temperatures of the two scenarios were different. The physical meaning for Scenario No. 43 was that, when the temperature was approximately 0 °C, a relatively large tensile strain was implemented at the beginning end of the FBG, and the strain dramatically decreased along the length of the FBG. Consequently, the value of the mean strain on the FBG was positive. In the cases where the FBG was attached to the below surface of a simply supported beam between the locations of 1/2 and 3/4, the load was applied on the upper surface of the beam. The original and reconstructed distorted spectra, using IPSO for the four scenarios, are shown in Fig. 7. The results illustrated that the reconstructed spectrum using IPSO fit well with the real spectrum. Although it was found that slight differences existed for Scenario No. 43, the reconstructed spectrum still reflected the profiles of the original spectrum. The estimated temperatures, strains, and errors for the four scenarios (No. 1, No. 12, No. 20, and No. 43) are listed in Table 4. 5. Experiments In this section, the performance of the introduced spectrum reconstruction method for the estimation of the temperature and mean strain, based on the distorted spectra of the half-encapsulated FBG, was experimentally validated. The experimental setup is elaborated in Section 5.1, and the experimental results and analysis are given in Section 5.2. 5.1. Experimental setup This study's experiments were performed in a measurable temperature environment using a strain tunable apparatus, as shown in Fig. 8. Two ends of a constant cross-section beam (material: 65 Mn) were attached on the two brackets of the apparatus. One bracket was fixed, while the other remained movable by turning a knob assembled on the end of a long screw. The dimensions of the beam (between the two brackets) were as follows: 200×8×1, and a caliper (accuracy of 0.05 mm) was utilized to measure the deflection of the beam. Then, FBG with grating lengths of 10 mm and wavelengths of 1550 nm were half adhered using epoxy resin on the beam at approximately a quarter of the length. Then, in order to precisely control the length of encapsulation, half of the FBG was packaged in a glue-resistive Teflon film when it was gelatinized to the other half. The location of the adhered FBG was then measured by the caliper, and was found to be from 52.35 to 57.35 mm. The spectrum of the half-encapsulated FBG was captured using an FBG interrogator SM125, with a wavelength resolution of 0.005 nm. To characterize the strain distribution of the cross-section beam, as
Fig. 7. Original and reconstructed distorted spectra of four scenarios. (a) Spectra of scenario No.1. (b) Spectra of scenario No.12. (c) Spectra of scenario No.20. (d) Spectra of scenario No.43.
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5.2. Experiments and analysis
Table 4 Estimated measurands and errors of the four scenarios. Scenarios
Estimated T (°C)
Error of T (° C)
Estimated ε (µε)
Error of ε (µε)
No.1 No.12 No.20 No.43
60.0 3.1 63.5 1.2
0.1 0.1 −0.5 1
−251.9 −257.9 −256.9 216.0
−1.9 3.2 3.3 −25.7
Due to the difference of the thermal expansion coefficients between the optic fiber and the substrate where the FBG was attached, the temperature responses of the encapsulated half and the free half were different. This contributed to a deviation between the temperature sensitivities of the strain-sensitive peak and that of the strain-insensitive peak. Therefore, the calibration for the half-encapsulated FBG was conducted prior to the strain tuning experiments in order to obtain the temperature sensitivities of the two peaks which were utilized in the transfer matrix method. The calibration was conducted in a temperature controlled incubator. The temperature was increased from 25 to 65 °C in increments of 10 °C. The temperature sensitivity of the straininsensitive peak (KT1) was determined to be 0.011 nm/°C, while that of the strain-sensitive peak (KT2) was 0.023 nm/°C. The increase in sensitivity was due to the thermal expansion coefficient of the substrate (65 Mn) being higher when compared with the optic fiber. After the temperature sensitivities were obtained, the next step was to ascertain the intrinsic parameters of the FBG, along with the initial strain induced by the half encapsulation. The spectrum, when the applied strain was 0 µε at 25.1 °C, was used as the target spectrum, which is illustrated by the solid line in Fig. 10. An IPSO algorithm was adopted to reconstruct the spectrum, as well as to estimate the intrinsic parameters and initial strain. The wavelength resolution of the target spectrum was 0.01 nm, which was picked up from the original spectrum in order to reduce the data amount, and accelerate the calculation process. The reconstructed spectrum is shown in Fig. 10 (dotted line). Although there was found to be a slight deviation due to the background noise of the measured spectrum, the profile of the two spectra fit very well. Table 6 shows the reconstructed parameters. In this study, three cases were experimentally validated. Case 1, in which the applied strain was 0 µε and the temperature was 40.0 °C, was conducted in a temperature controlled incubator. Case 2 and Case 3 were carried out on a strain tunable apparatus at room temperature, which was measured by a thermometer. For Case 2, the surface where the FBG was attached was downward, and a displacement of 20 mm was applied to the beam. In regards to Case 3, the surface where the FBG was attached was upward, and a displacement of 35 mm was applied on the beam. The strain data for the two cases were extracted from the FE simulation results in order to calculate the applied mean strain, as shown in Fig. 11. The dotted line in Fig. 11 represents the strain distribution along the centerline of the beam on the below surface when the displacement was 20 mm (Case 2). Also, the solid line in Fig. 11 shows the strain distribution of Case 3, when a displacement of 35 mm was implemented on the beam. The strain data where the FBG was attached, which are also highlighted in Fig. 11, indicated that the applied strain along the FBG followed a linear relationship. A linear
Fig. 8. Experimental setup. Table 5 Parameters used in the FE simulations. Length (mm) Width (mm) Thickness (mm) Yong's modulus E (GPa) Poisson ratio μ Density (kg/m3) Number of mesh
200 8 1 210 0.3 7650 6071
Fig. 9. Displacement contours when applied displacement is 35 mm.
well as to estimate the mean strain applied on the FBG, finite element (FE) simulations were performed for the beam. A 3D cuboid was built to simulate the beam, and an elastic material model with a geometric non-linearity was selected. One end of the model was fixed, and the other end could only move in a horizontal direction (x axis). Then, vertical displacements (z axis) were implemented in the middle of the beam. The strains along the centerline of the beam surface were extracted for analysis. Table 5 displays the parameters used in the FE simulations. When the displacement was 35 mm, the displacement contours were as shown in Fig. 9.
Fig. 10. Measured and reconstructed spectra at 25.1 °C when the applied strain is 0 µε.
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Table 6 Reconstructed intrinsic parameters and initial strain. Effective refractive index: Neff Initial Grating Period: Λ0 δneff Initial Strain: ε0
1.449 534.390 nm 1.85×10−4 485.6 µε
Fig. 11. Strain distribution of the beam extracted from FEM simulation.
least square fitting method was utilized to obtain the strain distribution function for the two cases. The fitted function for Case 2 was: ε2(z) =−109.5z-46.99, (0≤z≤5 mm), which meant that the FBG was subject to a compressive strain, and the strain declined along the length of the FBG. For Case 3, the strain function was ε3(z)=195.1z+232.52, (0≤z≤5 mm), which indicated that the FBG suffered from a tensile strain, and the strain linearly increased along its length. The calculated mean strains for Case 2 and Case 3 were −151.0 µε and 349.9 µε, respectively. Also, the measured temperatures were 25.1 °C for Case 2, and 25.2 °C for Case 3. The measured and reconstructed spectra for the three cases are shown in Fig. 12. For Case 1, the spectrum of the half-encapsulated FBG shifted towards the long wavelength (Fig. 12(a)). The two peaks were identifiable due to the fact that there was no strain applied on the FBG. The slight distortion of the spectrum was likely a result of the effects of adhesion under high temperatures. The reconstructed spectrum reflected the profile of the measured spectrum, and the reconstructed temperature and mean strain were determined to be 39.7 °C and 1.4 µε, respectively. Fig. 12(b) displays the spectra of Case 2. The two peaks overlapped with each other under the applied compressive strain, and the spectrum became distorted as a result of the linearly-declined strain field (non-uniform strain). The reconstructed spectrum fit well with the measured spectrum, and the estimated temperature and mean strain for this case were determined to be 24.2 °C and −136.5 µε, respectively. The comparison of the spectra for Case 3 can be seen in Fig. 12(c). One peak, the straininsensitive peak, was identifiable, while the other peak showed a serious distortion under the non-uniform tensile strain. Although a small difference existed in the two spectra, the reconstructed spectrum still indicated the variation of the measured spectrum. The calculated temperature was 26.9 °C, and the mean strain was 304.7 µε. Table 7 shows the comparison of the applied and estimated measurands. It can be seen that Case 3, whose temperature error was −1.7 °C, and strain error was 45.2 µε, provided the worst accuracy. However, it was still superior to the current technique in which the temperature and strain cannot be evaluated when the spectra are distorted. Therefore, the half-encapsulated FBG with an IPSO-based spectra reconstructing method was found to be significant to the practical applications in which the spectrum is distorted.
Fig. 12. Displacement contours when applied displacement is 35 mm. (a). Case1. (b). Case2. (c). Case3. Table 7 Comparison of applied measurands and estimated measurands of the three Cases.
Applied T (°C) Applied ε (µε) Reconstructed a1 Reconstructed a2 Reconstructed a3 Estimated T (°C) Estimated ε (µε) Error of T (°C) Error of ε (µε)
160
Case1
Case2
Case3
40.0 0.0 −0.000236 0.07 8 39.7 1.4 0.3 −1.4
25.1 −151.0 −0.000077 −0.06 −5 24.2 −136.5 0.9 −14.5
25.2 349.9 −0.000010 0.22 7 26.9 304.7 −1.7 45.2
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High Technology Research and Development Program (“863”Program) of China (2014AA110401).
6. Conclusion It has been determined that a half-encapsulated FBG is a straightforward method to simultaneously measure temperature and strain. However, an important issue has been that the spectrum of a halfencapsulated FBG tends to be distorted during practical usage, which makes the measurement impossible. This study analyzed the causes of the spectrum distortion by first examining the overlapping and nonuniform strains, as well as the combined effects. Then, a method which was capable of simultaneously estimating the temperature and mean strain based on the distorted spectrum was proposed. The method adopted an improved particle swarm optimization (IPSO) algorithm to reconstruct the spectrum, and to calculate the temperature and mean strain. The swarm success rate was based on a dynamic adaptive strategy for adjusting the inertia weight, and was utilized to balance the exploration and exploitation. Then, a total of 48 simulation scenarios were carried out to validate the proposed method. The simulation results indicated that the temperature and mean strain could be effectively estimated, even when the spectrum was highly distorted. A comparison of the GA, linearly-declined PSO, and IPSO-based reconstructing methods indicated that the IPSO provided a higher accuracy. Moreover, the feasibility of the proposed method was experimentally verified using a strain tunable apparatus in a temperature measurable environment. Three experimental cases were performed, and the strain distribution along the FBG for each case was characterized utilizing FEM simulations. A maximum temperature error of −1.7 °C and a mean strain error of 45.2 µε were obtained in the experiments for the three cases. The experimental results demonstrated that the halfencapsulated FBG with an IPSO-based spectra reconstructing method was applicable to the simultaneous measurements of temperature and mean strain, even when the spectrum was distorted.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Number: 41472260) and the National
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