THREE DIMENSIONAL STRAIN MEASUREMENT BY A BRAGG GRATING SENSOR SUBJECTED TO AXIAL AND TRANSVERSE LOAD SIMULTANEOUSLY A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Tadamichi Mawatari November 2006 UMI Number: 3242591 Copyright 2007 by Mawatari, Tadamichi All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.
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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 © Copyright by Tadamichi Mawatari 2007 All Rights Reserved il I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Nelson) Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. a (Kosuke Ishii) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Sheppard) Approved for the University Committee on Graduate Studies. iii Abstract The objective of this thesis is to establish mathematical models for predicting strains in the axial and other two orthogonal (transverse) directions of a multi-parameter Bragg grating sensor which is subjected to an axial and transverse loadings simultaneously. Previous work by other researchers has established models for predicting axial strains, but not a combination of those three strain components. Test specimens consisted of polarization maintaining Bragg grating sensors created in optical fibers with bow-tie stress applying parts.
Each of the sensors had two Bragg wavelength peaks at around 1300 nm, and two others around 1550 nm, and those four peaks moved due to external stimuli such as applied loads and/or temperature changes. The change in wavelength from a certain reference state, which is called a wavelength shift denoted by AA, was considered as the key to predict the strains due to the extemal stimuli experienced by the sensor. The specimens underwent thermal, axial, transverse, and combined loading tests, and the relations between the wavelength shifts and external stimuli were investigated. In order to predict the strains experienced by the sensor from the data of wavelength shifts obtained in the experiments, several mathematical models were investi gated.
Those mathematical models were categorized into linear and non-linear models. The linear model assumed a linear relation between the wavelength shifts and the loads applied, while the non-linear model tried to deal with more general conditions since a non-linear IV relation was observed in some of the data. The computed strains were compared with ones obtained from finite element analyses, and the accuracies of the mathematical models were studied. Some researchers hypothesize that the non-linearity between wavelength shifts and applied loads is caused by the rotation of optical axes of the sensor due to large transverse loads.
The related studies were reviewed, and some computations were conducted to understand a possible cause for the non-linear behavior. If a multi-parameter sensor is properly embedded in a material, it is expected that one might be able to measure the multi-axial strains in the material from data on wavelength shifts. An example of an experiment that could check this possibility was formulated, and the related computational procedures were discussed. Also, to improve the ability of the multi-parameter sensor to determine transverse strains, some different cross-sectional geometries of an optical fiber sensor were suggested.
Acknowledgements This thesis was only possible with the support of many individuals. First of all, 1 would like to thank my advisor, Professor Drew Nelson. His academic advice, encouragement, patience, and continuous financial support are greatly appreciated. Second, I would like to thank other reading committees, Professor Kosuke Ishii and Professor Sheri Sheppard, for helpful suggestions on my thesis.
Third, I would like to thank Mr. Stephen Kreger and Mr. John Seim of Blue Road Research (BRR), a fiber optic R&D firm in Fairview, Oregon for conducting experiments that provided data for this research. Furthermore, I would like to extend my thanks to my friends with whom | enjoyed my student life for many years.
Finally, I would like to thank my parents for supporting me during the entire of my life in Stanford. Vi Table of Contents Abstract iv Acknowledgements vi Table of Contents vii List of Tables XV List of Figures xvii 1.1 Objective Of theSi. cà HH HH HH HH HH HT TH Hay 1 1.2 Optical fiber — How 1t FunCfiOTS. sàn HH TH HH TH HH HH ng dư 1 1.3 Optical fIber — HOW I{ ÍS HAđ€,.
HH HH HH HH TH ng nàn 4 1.1 Fabrication Of Dr€ÍOTim. - -- ús- Ăn HH ng ng rệt 5 1. HH HT HH HH TT TH cv 6 1.4 Optical ÍIb€T S€TISOTS.- HH TT HH HH gà TT Hàn iệc 8 1.1 Fabry-Perot Interferometer (FPÏ. HH HH kh ve, 8 1.1 Intrinsic Fabry-Perot Interferometer (TFPI).2 Extrmsic Fabry-Perot Interferometer (EFPI]).2 Bragg Grating S€TSOT.5 Polarization-Maintaining Optical FIb€rS.
-- hnHHHg nh HH HH nen ngư. Multi-parameter sensor 18 2. HT HT HH HH TH TH TT TH ng Hy 18 2.2 Multi-paraImet€T S€TSOF. SH HH HH TH TT ng ca tấp 20 Vii 2.1 Basic concept of single-parameter SCNSOT.2 Basic concept of mulfI-paramefer S€TSOT.
Experiments and finite element analysis results 29 3. Ác HH“ HH HH HH HH HH nàn Hệ 29 3.1 SK specimen f©SfS.-- «TH HH Họ HH nh HH ng 31 3. cu HH «HH HH HH ng gu 33 3.2 SK Axial loading f€SE.- - HH ng yên 36 3.3 SK Transverse loading f€s.4 SK Combined loading fest. -- LH HH ng HH kp 69 3.1 C2 Axial loading test dala.2 C2 Transverse loading test data.3 C2 Combined loading test dafa.
cà kg HH 11111111316 76 3.1 C3 Axial loading test dafa.2 C3 Transverse loading test đafa.3 C23 Combined loading test data.2 Finite element anaÏyS1S. HH HH TH HH HH Hiện 34 3.1 Basics coordinate system for finite element analysis results. Hàng TH HT Hà Hàng tiện 86 3. -- HH nh HT nh HH Hà vi, 86 3.4 Comparison with closed form solufion.3 Determination of fast and SIOW aX€S.
SH HH ki, 88 4. 3-by-3 linear model for strain analysis 93 4.1 Determination of the second and third columns of the K matrix.2 Determination of the first column of the K matrix.3 Reducing the K matrIX tO 3-DV-3 SIZ€. LH HH TH HT nh HH TH Tà gu ch 98 4.1 Data shiÍt in transverse loading t€SfS.2 Data shift in combined loading test.3 Strain analysis and r©SuÏfS. ng HH HT kg.1 SK Axial loading f†€S.2 SK Transverse loading test.3 SK Combined loading tesfs.1 SK Combined loading test — Analysis I.2 SK Combined loading test — Analysis ÏI.3 SK Combined loading test — Analysis III.
HH HH HH HH TT cán nàng HH Hư 131 4.2 C2 Transverse loading f€sf.3 C2 Combined loading f€St.1 C2 Combined loading test — Analysis I.2 C2 Combined loading test — Analysis II.3 C2 Combined loading test — Analysis III. ốẽ ốố ố ốố .1 C3 A xial loading f€SI.- HH HH Hee, 160 4.2 C23 Transverse loading ©Sf.3 C3 Combined loading test 0.1 C3 Combined loading test — Analysis !.2 C3 Combined loading test — Analysis IT.3 C3 Combined loading test — Analysis III. - HH HH TT TH To TH TH HT HH nu 185 4.5 Other maihematical mod€Ì§. ch HH HH HH TH HH ngà key 186 4.1 Model 1: Over-deterministiC sysfem.2 Model 2: DeterministiC sYSf€m.3 Model 3: Singular value decomposition (SVP).
Non-linear model for strain analysis: Foliated Quasi Inverse (FQI) method. HH HH TH TH TT TT HH HT Hàn HH 203 5.1 Problem de€SCTIptIOH. càng HT TH HH nh HH Hư Hy 203 5.1 Mathematical model for wavelength shift: AA=[AA,,AA,, ix 5.2 Mathematical model for strain: e=[£¡,¿,£;, T].5 Outline of this chapf€T.2 5ystem COndi[IOTNS.- HH Hà 1H11 11 Hà Hàn ng Hệ 209 5.1 Relationships observed in experimenIs.2 Reasonable assumptions from mechanics of materials .4 Available experimental đata.5 Formulae to be determined.3 Construction of matrix representation .1 Basic formulae for A and Bio.2 Basic formulae for Aw.3 Basic formulae for E_.4 Matrix forms of A and E.4 Best approximation of matrix represenfation.1 Theoretical properties of the best approximation .2 Experimental properties of the best approximation.3 Selection of the function for the best approximation.4 Selection of the basis.5 Determination of knof sequence.6 Derivation of composition mmatfiX.7 Derivation of companion MAatriX.8 Formulation into Mini-max problem .5 Foliation for an underdetermined system .1 Definition of ÍOliatiOH.--- cà xe cv, 230 5.2 Definition of wavelength and strain foliation.3 Foliation and simultaneous algebraic equations.6 Algorithm for the best apprOXimafOn.1 Determination of the maximum pOInt.2 Determination of the minmimum pOIt.3 Algorithm for the best approximation - Summary -.7 ĐBragg grating senSOr th€OFY.- ng HH ng nh 243 5.1 Mathematical representation for wavelength foliation.2 Mathematical representation for strain foliation.3 Derivation of the generalized K matrix and its inverse.4 Mathematical representations for mechanical loading.1 Mathematical representation for axial loading.2 Mathematical representation for transverse ÏOAdITBE. Ăn HH HH H4 ky 250 5.5 Global coordinate system and 1{s SÍTuCfUTe.1 Introducing orthogonal global coordinate system —.2 Definition of the coordinate transformation @ .3 BGS space and properties of data type.6 Local coordinate system and Its structfure.1 Local coordinate system in BGS A(Ð).{s ho TH K1 TH HC TT TH Ti TH 018 8 1411011550 264 5.1 Erasing Noise TYype Í.2 Erasing Noise Type Í|.9 Basic structure of the theory for Bragg grating sensor.2 ResultS Of aïtaÏYS1S.
HH TH TH HT nọ TH gà Hà Ty 271 5. HT HH HH Hà HH TK TH Hàn Hàn Hà nàn 271 5.1 SK Axial loading test.2 SK Transverse loading teSt.3 SK Combined loading tesi.1 SK Combined loading test — Analysis I.2 SK Combined loading test — Analysis H.3 SK Combined loading test — Analysis II. Án LH HH HH TH TT Tà nọ TH TH ng 1e 294 5.1 C2 Axial loading f€SỂ. - -- QnnH HH re.2 C2 Transverse loading †€sf.1 C2 Combined loading test - Analysis Ï.2 C2 Combined loading test — Analysis IT.3 C2 Combined loading test - Analysis HỊ.1 C3 Axial loading f†€SỂ.2 C3 Transverse loading test .3 C3 Combined loading test.1 C3 Combined loading test — Analysis 1.2 C3 Combined loading test - Analysis I].3 C3 Combined loading test - Analysis ITl.3 SUMMALY n nố ố ốố ốốốốốốGAÕ.
Comparison of results in strain analysis 341 6.1 Computation Of ET OFS. cv HH HH TH HH TH HT KT Hàn ĐH HH 341 6.- HQ HH HH ng HH nh nụ 342 6.1 AXial loading f€SE.à HH HH HH Ha TH HH HH vê 342 6.2 Transverse loading †€SiL.- ong TH HH Hà Hy 343 6. nh HH TH H1 1 Hy HH nh nvệy 345 6. HH HH HH TH nàng thay 345 3/6, 1n nh 4.
con TH HT TH HH Hà ru 348 7. Rotation of optical axes 351 7.1 Non-linear behavior of AÀ, vs. HQ LH HH ngư 351 XH W0 00 vác ch.3 Stress analysis for Rotation Of OptICal aX€§. Án“ HH HT HH HT nh 366 8.1 Strain analysis by muÌti-param€f€T S€TSOTS.1 Multi-parameter sensor embedded 1n a host material.2 Transforming sensor strains to material strains.3 Strains measured by strain rOS€ff©S.
