Abstract
Finite element model (FEM) is a broadly used numerical tool in structural damage detection. In such applications, damage parameters in FEM are estimated by minimizing the differences between experimental modal analysis data and the corresponding FEM model prediction. Very limited works exist on analyzing the identifiability of the FEM used in such applications. In this paper, the identifiability of FEM-based structural damage detection is investigated for undamped elastic beams. We theoretically proved that damage severity at a given location in a uniform beam is identifiable by reformulating the FEM into a linear time invariant (LTI) system. A numerical algorithm is also proposed for checking the identifiability issue of multiple damage locations. Numerical case studies are provided to validate the effectiveness and usefulness of the proposed framework.
Issue Section:
Research Papers
References
1.
Hamzeloo
, S. R.
, Shamshirsaz
, M.
, and Rezaei
, S. M.
, 2012
, “Damage Detection on Hollow Cylinders by Electro-Mechanical Impedance Method: Experiments and Finite Element Modeling
,” C. R. Mec.
, 340
(9
), pp. 668
–677
.2.
Stubbs
, N.
, Broome
, T. H.
, and Osegueda
, R.
, 1990
, “Nondestructive Construction Error Detection in Large Space Structures
,” AIAA J.
, 28
(1
), pp. 146
–152
.3.
Stubbs
, N.
, and Osegueda
, R.
, 1990
, “Global Non-Destructive Damage Evaluation in Solids
,” Int. J. Anal. Exp. Modal Anal.
, 5
, pp. 67
–79
.4.
Stubbs
, N.
, and Osegueda
, R.
, 1990
, “Global Damage Detection in Solids—Experimental Verification
,” Int. J. Anal. Exp. Modal Anal.
, 5
, pp. 81
–97
.5.
Shiradhonkar
, S. R.
, and Shrikhande
, M.
, 2011
, “Seismic Damage Detection in a Building Frame Via Finite Element Model Updating
,” Comput. Struct.
, 89
(23–24
), pp. 2425
–2438
.6.
Mayes
, R.
, 1991
, “Error Localization Using Mode Shapes: An Application to a Two Link Robot Arm
,” Sandia National Labs., Albuquerque, NM, Report No. SAND-91-2297C; CONF-920234–1.7.
Jafarkhani
, R.
, and Masri
, S. F.
, 2011
, “Finite Element Model Updating Using Evolutionary Strategy for Damage Detection
,” Comput. Aided Civil Infrastruct. Eng.
, 26
(3
), pp. 207
–224
.8.
Lee
, J. J.
, Lee
, J. W.
, Yi
, J. H.
, Yun
, C. B.
, and Jung
, H. Y.
, 2005
, “Neural Networks-Based Damage Detection for Bridges Considering Errors in Baseline Finite Element Models
,” J. Sound Vib.
, 280
(3
), pp. 555
–578
.9.
Wu
, J. R.
, and Li
, Q. S.
, 2006
, “Structural Parameter Identification and Damage Detection for a Steel Structure Using a Two-Stage Finite Element Model Updating Method
,” J. Construct. Steel Res.
, 62
(3
), pp. 231
–239
.10.
Pandey
, A. K.
, Biswas
, M.
, and Samman
, M. M.
, 1991
, “Damage Detection From Changes in Curvature Mode Shapes
,” J. Sound Vib.
, 145
(2
), pp. 321
–332
.11.
Chen
, J.-C.
, and Garba
, J. A.
, “On-Orbit Damage Assessment for Large Space Structures
,” AIAA J.
, 26
(9
), pp. 1119
–1126
.12.
Chen
, J.-C.
, and Garba
, J. A.
, 1988
, Structural Damage Assessment Using a System Identification Technique. Structural Safety Evaluation Based on System Identification Approaches
, Vieweg+ Teubner Verlag
, Berlin
, pp. 474
–492
.13.
Sanayei
, M.
, and Onipede
, O.
, 1991
, “Damage Assessment of Structures Using Static Test Data
,” AIAA J.
, 29
(7
), pp. 1174
–1179
.14.
Sanayei
, M.
, Onipede
, O.
, and Babu
, S. R.
, 1992
, “Selection of Noisy Measurement Locations for Error Reduction in Static Parameter Identification
,” AIAA J.
, 30
(9
), pp. 2299
–2309
.15.
Liu
, P.-L.
, 1995
, “Identification and Damage Detection of Trusses Using Modal Data
,” J. Struct. Eng.
, 121
(4
), pp. 599
–608
.16.
Lim
, T. W.
, 1990
, “A Submatrix Approach to Stiffness Using Modal Test Data
,” AIAA J.
, 28
(6
), pp. 1123
–1130
.17.
Lim
, T. W.
, 1991
, “Structural Damage Detection Using Modal Test Data
,” AIAA J.
, 29
(12
), pp. 2271
–2274
.18.
Lim
, T. W.
, 1994
, “Structural Damage Detection of a Planar Truss Structure Using a Constrained Eigenstructure Assignment
,” 35th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference
, Paper No. AIAA-94-1715-CP, pp. 336
–346
.19.
Zimmerman
, D. C.
, and Kaouk
, M.
, 1994
, “Structural Damage Detection Using a Minimum Rank Update Theory
,” ASME J. Vib. Acoust.
, 116
(2
), pp. 222
–231
.20.
Zimmerman
, D. C.
, Simmermacher
, T.
, and Kaouk
, M.
, 1995
, “Structural Damage Detection Using Frequency Response Functions
,” 13th International Modal Analysis Conference
, Vol. 2460
, pp. 375
–381
.21.
Zimmerman
, D. C.
, Kaouk
, M.
, and Simmermacher
, T.
, 1995
, “On the Role of Engineering Insight and Judgement Structural Damage Detection
,” 13th International Modal Analysis Conference
, Vol. 2460
, pp. 414
–420
.22.
Chang
, J.-D.
, and Guo
, B.-Z.
, 2007
, “Identification of Variable Spacial Coefficients for a Beam Equation From Boundary Measurements
,” Automatica
, 43
(4
), pp. 732
–737
.23.
Ju
, F. D.
, Akgun
, M.
, Paez
, T. L.
, and Wong
, E. T.
, 1982
, “Diagnosis of Fracture Damage in Simple Structures: A Modal Method
,” New Mexico University, Albuquerque Bureau of Engineering Research, Paper No. CE-62 (82) AFOSR-993-1.24.
Nguyen
, V. V.
, and Wood
, E. F.
, 1982
, “Review and Unification of Linear Identifiability Concepts
,” SIAM Rev.
, 24
(1
), pp. 34
–51
.25.
Franco
, G.
, Betti
, R.
, and Longman
, R. W.
, 2006
, “On the Uniqueness of Solutions for the Identification of Linear Structural Systems
,” ASME J. Appl. Mech.
, 73
(1
), pp. 153
–162
.26.
De Angelis
, M.
, Lus
, H.
, Betti
, R.
, and Longman
, R. W.
, 2002
, “Extracting Physical Parameters of Mechanical Models From Identified State-Space Representations
,” ASME J. Appl. Mech.
, 69
(5
), pp. 617
–625
.27.
Sun
, H.
, Luş
, H.
, and Betti
, R.
, 2013
, “Identification of Structural Models Using a Modified Artificial Bee Colony Algorithm
,” Comput. Struct.
, 116
, pp. 59
–74
.28.
Reynders
, E.
, 2012
, “System Identification Methods for (Operational) Modal Analysis: Review and Comparison
,” Arch. Comput. Methods Eng.
, 19
(1
), pp. 51
–124
.29.
Simani
, S.
, Fantuzzi
, C.
, and Patton
, R. J.
, 2013
, Model-Based Fault Diagnosis in Dynamic Systems Using Identification Techniques
, Springer Science & Business Media
, Berlin
.30.
Rugh
, W. J.
, 1996
, Linear System Theory
, Vol. 2
, Prentice Hall
, Upper Saddle River, NJ
.31.
Segerlind
, L. J.
, and Saunders
, H.
, 1987
, Applied Finite Element Analysis
, Wiley
, New York
.32.
Meurant
, G.
, 1992
, “A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices
,” SIAM J. Matrix Anal. Appl.
, 13
(3
), pp. 707
–728
.33.
Nabben
, R.
, 1999
, “Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices
,” SIAM J. Matrix Anal. Appl.
, 20
(3
), pp. 820
–837
.34.
Demko
, S.
, Moss
, W. F.
, and Smith
, P. W.
, 1984
, “Decay Rates for Inverses of Band Matrices
,” Math. Comput.
, 43
(168
), pp. 491
–499
.35.
Eijkhout
, V.
, and Polman
, B.
, 1988
, “Decay Rates of Inverses of Banded M-Matrices That are Near to Toeplitz Matrices
,” Linear Algebra Appl.
, 109
, pp. 247
–277
.36.
Yamamoto
, T.
, and Ikebe
, Y.
, 1979
, “Inversion of Band Matrices
,” Linear Algebra Appl.
, 24
, pp. 105
–111
.37.
Balageas
, D.
, Fritzen
, C.-P.
, and Güemes
, A.
, eds., 2006
, Structural Health Monitoring
, Vol. 493
, ISTE
, London, UK
.38.
Wang
, S. Q.
, and Li
, H. J.
, 2012
, “Assessment of Structural Damage Using Natural Frequency Changes
,” Acta Mech. Sinica
, 28
(1
), pp. 118
–127
.39.
Salawu
, O. S.
, 1997
, “Detection of Structural Damage Through Changes in Frequency: A Review
,” Eng. Struct.
, 19
(9
), pp. 718
–723
.40.
Lo
, S. S.
, Philippe
, B.
, and Sameh
, A.
, 1987
, “A Multiprocessor Algorithm for the Symmetric Tridiagonal Eigenvalue Problem
,” SIAM J. Sci. Stat. Comput.
, 8
(2
), pp. s155
–s165
.41.
Trefftz
, C.
, Huang
, C. C.
, McKinley
, P. K.
, Li
, T. Y.
, and Zeng
, Z.
, 1995
, “A Scalable Eigenvalue Solver for Symmetric Tridiagonal Matrices
,” Parallel Comput.
, 21
(8
), pp. 1213
–1240
.42.
Fernandez
, F. A.
, Davies
, J. B.
, Zhu
, S.
, and Lu
, Y.
, 1991
, “Sparse Matrix Eigenvalue Solver for Finite Element Solution of Dielectric Waveguides
,” Electron. Lett.
, 27
(20
), pp. 1824
–1826
.43.
Gruber
, R.
, 1980
, “HYMNISBLOCK—Eigenvalue Solver for Blocked Matrices
,” Comput. Phys. Commun.
, 20
(3
), pp. 421
–428
.44.
Demmel
, J. W.
, Marques
, O. A.
, Parlett
, B. N.
, and Vömel
, C.
, 2008
, “Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers
,” SIAM J. Sci. Comput.
, 30
(3
), pp. 1508
–1526
.45.
Peng
, D.
, Middendorf
, N.
, Weigend
, F.
, and Reiher
, M.
, 2013
, “An Efficient Implementation of Two-Component Relativistic Exact-Decoupling Methods for Large Molecules
,” J. Chem. Phys.
, 138
(18
), p. 184105
.46.
Beyn
, W. J.
, 2012
, “An Integral Method for Solving Nonlinear Eigenvalue Problems
,” Linear Algebra Appl.
, 436
(10
), pp. 3839
–3863
.47.
Davidson
, G. G.
, Evans
, T. M.
, Jarrell
, J. J.
, Hamilton
, S. P.
, Pandy
, T. M.
, and Slaybaugh
, R. N.
, 2014
, “Massively Parallel, Three-Dimensional Transport Solutions for the k-Eigenvalue Problem
,” Nucl. Sci. Eng.
, 177
(2
), pp. 111
–125
.48.
Barrett
, W. W.
, 1979
, “A Theorem on Inverse of Tridiagonal Matrices
,” Linear Algebra Appl.
, 27
, pp. 211
–217
.49.
Engwerda
, J. C.
, Ran
, A. M.
, and Rijkeboer
, A. L.
, 1993
, “Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A * X − 1A = Q
,” Linear Algebra Appl.
, 186
, pp. 255
–275
.50.
Adams
, R. D.
, Cawley
, P.
, Pye
, C. J.
, and Stone
, B. J.
, 1978
, “A Vibration Technique for Non-Destructively Assessing the Integrity of Structures
,” J. Mech. Eng. Sci.
, 20
(2
), pp. 93
–100
.51.
Qian
, G.-L.
, Gu
, S.-N.
, and Jiang
, J.-S.
, 1990
, “The Dynamic Behaviour and Crack Detection of a Beam With a Crack
,” J. Sound Vib.
, 138
(2
), pp. 233
–243
.52.
Gillich
, G.-R.
, Praisach
, Z.-I.
, and Negru
, I.
, 2012
, “The Relationship Between Changes of Deflection and Natural Frequencies of Damaged Beams
,” Advances in Remote Sensing, Finite Differences and Information Security: (F-And-B ′12),(REMOTE ′12), (ISP ′12)
, E.
Scutelnicu
, L.
Lazic
, and P. F.
de Arroyabe
, eds., Wseas LLC
, Stevens Point, WI
, pp. 38
–42
.53.
Hearn
, G.
, and Testa
, R. B.
, 1991
, “Modal Analysis for Damage Detection in Structures
,” J. Struct. Eng.
, 117
(10
), pp. 3042
–3063
.54.
Hassiotis
, S.
, 2000
, “Identification of Damage Using Natural Frequencies and Markov Parameters
,” Comput. Struct.
, 74
(3
), pp. 365
–373
.55.
Li
, R. C.
, 2013
, “Rayleigh Quotient Based Optimization Methods for Eigenvalue Problems
,” Summary of Lectures Delivered at Gene Golub SIAM Summer School
, World Scientific Publishing Company
, Singapore
.56.
Kato
, T.
, 2012
, Perturbation Theory for Linear Operators
, Springer Science & Business Media
, Berlin
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