Abstract

A new iterative matrix algorithm has been applied to improve the precision of temperature and force decoupling in multi-parameter FBG sensing. For the first time, this evaluation technique allows the integration of nonlinearities in the sensor’s temperature characteristic and the temperature dependence of the sensor’s force sensitivity. Applied to a sensor cable consisting of two FBGs in fibers with 80 µm and 125 µm cladding diameter installed in a 7 m-long coiled PEEK capillary, this technique significantly reduced the uncertainties in friction-compensated temperature measurements. In the presence of high friction-induced forces of up to 1.6 N the uncertainties in temperature evaluation were reduced from several degrees Celsius if using a standard linear matrix approach to less than 0.5°C if using the iterative matrix approach in an extended temperature range between −35°C and 125°C.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  6. O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
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2017 (1)

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg grating sensors for the oil industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

2016 (1)

B. Hopf, A. W. Koch, and J. Roths, “Iterative matrix inversion technique for simultaneous strain and temperature sensing in an extended temperature range,” Proc. SPIE 9916, 99160O (2016).

2015 (1)

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

2012 (1)

M. Willsch, T. Bosselmann, M. Villnow, and W. Ecke, “Fiber optical sensor trends in the energy field,” Proc. SPIE 8421, 84210R (2012).

2005 (2)

O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
[Crossref]

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

2004 (2)

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

1999 (1)

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

1997 (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

1996 (1)

S. W. James, M. L. Dockney, and R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fibre Bragg grating sensors,” Electron. Lett. 32(12), 1133 (1996).
[Crossref]

1994 (1)

L. Reekie, J. P. Dakin, J.-L. Archambault, and M. G. Xu, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

1956 (1)

S. Spinner, “Elastic Moduli of Glasses at Elevated Temperatures by a Dynamic Method,” J. Am. Ceram. Soc. 39(3), 113–118 (1956).
[Crossref]

Araujo, F. M.

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

Araújo, F. M.

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
[Crossref]

Archambault, J. L.

M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Simultaneous measurement of strain and temperature using fibre grating sensors,” in 10th Int. Conf. on Optical Fiber Sensors (1994), pp. 191–194.
[Crossref]

Archambault, J.-L.

L. Reekie, J. P. Dakin, J.-L. Archambault, and M. G. Xu, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Bao, W.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg grating sensors for the oil industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Barton, J. S.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

Bosselmann, T.

M. Willsch, T. Bosselmann, M. Villnow, and W. Ecke, “Fiber optical sensor trends in the energy field,” Proc. SPIE 8421, 84210R (2012).

Burnell, G.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

Carvalho, J. P.

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

Cavaleiro, P. M.

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

Chen, G.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Dakin, J. P.

L. Reekie, J. P. Dakin, J.-L. Archambault, and M. G. Xu, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Simultaneous measurement of strain and temperature using fibre grating sensors,” in 10th Int. Conf. on Optical Fiber Sensors (1994), pp. 191–194.
[Crossref]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Dockney, M. L.

S. W. James, M. L. Dockney, and R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fibre Bragg grating sensors,” Electron. Lett. 32(12), 1133 (1996).
[Crossref]

Ecke, W.

M. Willsch, T. Bosselmann, M. Villnow, and W. Ecke, “Fiber optical sensor trends in the energy field,” Proc. SPIE 8421, 84210R (2012).

Farahi, F.

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

Ferreira, L. A.

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
[Crossref]

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

Frazão, O.

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
[Crossref]

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Hopf, B.

B. Hopf, A. W. Koch, and J. Roths, “Iterative matrix inversion technique for simultaneous strain and temperature sensing in an extended temperature range,” Proc. SPIE 9916, 99160O (2016).

Ibrahim, S. K.

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

James, S. W.

S. W. James, M. L. Dockney, and R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fibre Bragg grating sensors,” Electron. Lett. 32(12), 1133 (1996).
[Crossref]

Jia, H.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Jiang, X.

G.-H. Lv, S.-H. Shang, X. Jiang, O. Jin-Ping, C. Yang, C. D. Li, and W. Xu, “FBG temperature and pressure sensing system for hot water pipeline of petrochemical factory,” in Proceedings of the 1st Asia-Pacific Optical Fiber Sensors Conference (IEEE, 2008), pp. 1–3.

Jin-Ping, O.

G.-H. Lv, S.-H. Shang, X. Jiang, O. Jin-Ping, C. Yang, C. D. Li, and W. Xu, “FBG temperature and pressure sensing system for hot water pipeline of petrochemical factory,” in Proceedings of the 1st Asia-Pacific Optical Fiber Sensors Conference (IEEE, 2008), pp. 1–3.

Jones, J. D. C.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Koch, A. W.

B. Hopf, A. W. Koch, and J. Roths, “Iterative matrix inversion technique for simultaneous strain and temperature sensing in an extended temperature range,” Proc. SPIE 9916, 99160O (2016).

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Li, C. D.

G.-H. Lv, S.-H. Shang, X. Jiang, O. Jin-Ping, C. Yang, C. D. Li, and W. Xu, “FBG temperature and pressure sensing system for hot water pipeline of petrochemical factory,” in Proceedings of the 1st Asia-Pacific Optical Fiber Sensors Conference (IEEE, 2008), pp. 1–3.

Lindner, E.

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

Liu, L.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Lv, G.-H.

G.-H. Lv, S.-H. Shang, X. Jiang, O. Jin-Ping, C. Yang, C. D. Li, and W. Xu, “FBG temperature and pressure sensing system for hot water pipeline of petrochemical factory,” in Proceedings of the 1st Asia-Pacific Optical Fiber Sensors Conference (IEEE, 2008), pp. 1–3.

MacPherson, W. N.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

Maier, R. R. J.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

McCulloch, S.

R. R. J. Maier, W. N. MacPherson, J. S. Barton, J. D. C. Jones, S. McCulloch, and G. Burnell, “Temperature dependence of the stress response of fibre Bragg gratings,” Meas. Sci. Technol. 15(8), 1601–1606 (2004).
[Crossref]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[Crossref]

Qiao, X.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg grating sensors for the oil industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Reekie, L.

L. Reekie, J. P. Dakin, J.-L. Archambault, and M. G. Xu, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Simultaneous measurement of strain and temperature using fibre grating sensors,” in 10th Int. Conf. on Optical Fiber Sensors (1994), pp. 191–194.
[Crossref]

Rong, Q.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg grating sensors for the oil industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Roths, J.

B. Hopf, A. W. Koch, and J. Roths, “Iterative matrix inversion technique for simultaneous strain and temperature sensing in an extended temperature range,” Proc. SPIE 9916, 99160O (2016).

Santos, J. L.

O. Frazão, J. P. Carvalho, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Discrimination of strain and temperature using Bragg gratings in microstructured and standard optical fibres,” Meas. Sci. Technol. 16(10), 2109–2113 (2005).
[Crossref]

O. Frazão, L. A. Ferreira, F. M. Araújo, and J. L. Santos, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).
[Crossref]

P. M. Cavaleiro, F. M. Araujo, L. A. Ferreira, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using Bragg gratings written in germanosilicate and boron-codoped germanosilicate fibers,” IEEE Photon. Technol. Lett. 11(12), 1635–1637 (1999).
[Crossref]

Schuster, K.

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

Shang, S.-H.

G.-H. Lv, S.-H. Shang, X. Jiang, O. Jin-Ping, C. Yang, C. D. Li, and W. Xu, “FBG temperature and pressure sensing system for hot water pipeline of petrochemical factory,” in Proceedings of the 1st Asia-Pacific Optical Fiber Sensors Conference (IEEE, 2008), pp. 1–3.

Shao, Z.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg grating sensors for the oil industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Spinner, S.

S. Spinner, “Elastic Moduli of Glasses at Elevated Temperatures by a Dynamic Method,” J. Am. Ceram. Soc. 39(3), 113–118 (1956).
[Crossref]

Tatam, R. P.

S. W. James, M. L. Dockney, and R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fibre Bragg grating sensors,” Electron. Lett. 32(12), 1133 (1996).
[Crossref]

van Roosbroeck, J.

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

Villnow, M.

M. Willsch, T. Bosselmann, M. Villnow, and W. Ecke, “Fiber optical sensor trends in the energy field,” Proc. SPIE 8421, 84210R (2012).

Vlekken, J.

J. van Roosbroeck, S. K. Ibrahim, E. Lindner, K. Schuster, and J. Vlekken, “Stretching the limits for the decoupling of strain and temperature with FBG based sensors,” Proc. SPIE 9634, 96343S (2015).

Wang, W.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Willsch, M.

M. Willsch, T. Bosselmann, M. Villnow, and W. Ecke, “Fiber optical sensor trends in the energy field,” Proc. SPIE 8421, 84210R (2012).

Xu, L.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Xu, M. G.

L. Reekie, J. P. Dakin, J.-L. Archambault, and M. G. Xu, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Simultaneous measurement of strain and temperature using fibre grating sensors,” in 10th Int. Conf. on Optical Fiber Sensors (1994), pp. 191–194.
[Crossref]

Xu, W.

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Figures (12)

Fig. 1
Fig. 1 Schematic of the sensor element: A small piece of an 80 µm diameter fiber (ca. 15 mm) is positioned between a 125 µm cladding fiber. FBGs are located at each side of the splice joint. Temperature and force are supposed to be the same in both FBGs. Due to the different cladding diameters, the force sensitivity is about 2.4 times higher for the 80 µm fiber than for the 125 µm diameter fiber, while temperature sensitivities are nearly the same for both fiber types. The signals of both FBGs can be used to determine temperature and force acting on the fiber.
Fig. 2
Fig. 2 Spectra of the FBGs in the 80 µm and 125 µm fiber at 24°C: The Bragg wavelength of 1544 nm corresponds to the grating in the fiber with 80 µm cladding diameter, the Bragg wavelength of 1549 nm to the grating in the fiber with 125 µm.
Fig. 3
Fig. 3 Measurement setup using a tunable laser interrogator (SM125, Micron Optics) for Bragg wavelengths determination: a) Temperature calibration was carried out on vertically suspended sensor elements in a climatic chamber. b) Weights were applied to the fiber for force calibration. A calibrated Pt100 was used for temperature reference.
Fig. 4
Fig. 4 Temperature sensitivity: Change of Bragg wavelength ΔλB,k = λB,k - λ0,k as a function of temperature and the corresponding linear fits and the 3rd order polynomial fits The values of the FBG in the 80 µm fiber are depicted in green, the values of the 125 µm fiber in blue.
Fig. 5
Fig. 5 Force sensitivities aF,k at room temperature: Change of Bragg wavelength ΔλB,k = λB,k - λ0,k as a function of the applied force and the corresponding fits of linear functions. The values of the FBG in the 80 µm fiber are depicted with green dots, the values of the 125 µm fiber with blue triangles.
Fig. 6
Fig. 6 Measurement setup for temperature dependency of the force sensitivity. Temperature stabilization was obtained a) with a Peltier element (TED 200C: Thorlabs GmbH) in the temperature range between 4°C and 50°C, b) with a tube furnace (ROS 20/250/12: ThermConcept GmbH) at 97°C.
Fig. 7
Fig. 7 Relative change in force sensitivity bF,k as a function of temperature: The values of the FBG in the 80 µm fiber are depicted with green dots, the values of the 125 µm fiber with blue triangles.
Fig. 8
Fig. 8 Schematic of the dimensions and locations of the sensor element in the PEEK capillary and measurement setup.
Fig. 9
Fig. 9 Bragg wavelength shifts ΔλB,k = λB,k - λ0,k of the tandem sensor element as a function of time. The values of the FBG in the 80 µm fiber are depicted in green, the values of the 125 µm fiber in blue and the temperature measured with a calibrated Pt100 in red. The scale markings for temperature correspond to the wavelength difference if assuming a temperature sensitivity of 10.4 pm/°C.
Fig. 10
Fig. 10 Nonlinear iterative matrix approach: The determined temperatures of the linear matrix method with constant matrix elements (blue circles) are compared to the results of the iterative matrix method with temperature dependent force and temperature sensitivities after three iteration cycles (blue dots), plotted against the reference temperatures of the Pt100 (red line: identity). The temperatures evaluated with the iterative matrix method match the values of the Pt100 reference in the whole temperature range.
Fig. 11
Fig. 11 Nonlinear iterative matrix approach: The determined force values of the linear matrix method with constant matrix elements (blue circles) and the values obtained with the iterative matrix method using temperature dependent force and temperature sensitivities after three iteration cycles (blue dots) are plotted against the reference temperatures of the Pt 100. There is no evidence for friction-induced forces for temperatures below 55°C. Higher temperatures led to increasing forces with 1.6 N at 125°C.
Fig. 12
Fig. 12 Deviations of the FBG-based temperature values from reference temperature evaluated with different approaches: Circles: Linear matrix approach with constant matrix elements. Triangles: Iterative matrix solution with nonlinear temperature characteristics and constant force sensitivity. Dots: iterative matrix solution with nonlinear temperature characteristics and temperature-dependent force sensitivities.

Tables (1)

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Table 1 Force and temperature sensitivities

Equations (7)

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Δ λ B (ΔT)= λ B,0 (α+ξ)ΔT= K T ΔT,
Δ λ B (F)= λ B,0 1 p eff AE F= K F F.
( Δ λ B,80µm Δ λ B,125µm )=( K T,80µm K F,80µm K T,125µm K F,125µm )( ΔT F ),
( ΔT F )= 1 D ( K F,125µm K F,80µm K T,125µm K T,80µm )( Δ λ B,80µm Δ λ B,125µm ),
( Δ T j F j )= 1 D(Δ T j1 ) ( K F,125µm (Δ T j1 ) K F,80µm (Δ T j1 ) K T,125µm (Δ T j1 ) K T,80µm (Δ T j1 ) )( Δ λ B,80µm Δ λ B,125µm ),
Δ λ B,k (ΔT)=( a T,k + b T,k ΔT+ c T,k Δ T 2 )ΔT K T,k (ΔT)ΔT
Δ λ B,k (F)= a F,k ( 1+ b F,k ΔT )F K F,k (ΔT)F

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