Estimating the number of degrees of freedom of a mechanical system or an engineering structure from the time-series of a small set of sensors is a basic problem in diagnostics, which, however, is often overlooked when monitoring health and integrity. In this work, we demonstrate the applicability of the network-theoretic concept of detection matrix as a tool to solve this problem. From this estimation, we illustrate the possibility to identify damage. The detection matrix, recently introduced by Haehne et al. [Phys. Rev. Lett. 122, 158301 (2019)] in the context of network theory, is assembled from the transient response of a few nodes as a result of non-zero initial conditions: its rank offers an estimate of the number of nodes in the network itself. The use of the detection matrix is completely model-agnostic, whereby it does not require any knowledge of the system dynamics. Here, we show that, with a few modifications, this same principle applies to discrete systems, such as spring-mass lattices and trusses. Moreover, we discuss how damage in one or more members causes the appearance of distinct jumps in the singular values of this matrix, thereby opening the door to structural health monitoring applications, without the need for a complete model reconstruction.
REFERENCES
1.
H.
Haehne
, J.
Casadiego
, J.
Peinke
, and M.
Timme
, “Detecting hidden units and network size from perceptible dynamics
,” Phys. Rev. Lett.
122
, 158301
(2019
). 2.
S. W.
Doebling
, C. R.
Farrar
, and M. B.
Prime
, “A summary review of vibration-based damage identification methods
,” Shock Vibr. Dig.
30
, 91
–105
(1998
). 3.
O. S.
Salawu
, “Detection of structural damage through changes in frequency: A review
,” Eng. Struct.
19
, 718
–723
(1997
). 4.
O.
Avci
, O.
Abdeljaber
, S.
Kiranyaz
, M.
Hussein
, M.
Gabbouj
, and D. J.
Inman
, “A review of vibration-based damage detection in civil structures: From traditional methods to machine learning and deep learning applications
,” Mech. Syst. Signal Process.
147
, 107077
(2021
). 5.
A.
Deraemaeker
, E.
Reynders
, G.
De Roeck
, and J.
Kullaa
, “Vibration-based structural health monitoring using output-only measurements under changing environment
,” Mech. Syst. Signal Process.
22
, 34
–56
(2008
). 6.
F.
Ubertini
, G.
Comanducci
, N.
Cavalagli
, A. L.
Pisello
, A. L.
Materazzi
, and F.
Cotana
, “Environmental effects on natural frequencies of the San Pietro bell tower in Perugia, Italy, and their removal for structural performance assessment
,” Mech. Syst. Signal Process.
82
, 307
–322
(2017
). 7.
Y.
Yang
and S.
Nagarajaiah
, “Output-only modal identification with limited sensors using sparse component analysis
,” J. Sound Vib.
332
, 4741
–4765
(2013
). 8.
A. Z.
Hosseinzadeh
, A.
Bagheri
, G. G.
Amiri
, and K.
Koo
, “A flexibility-based method via the iterated improved reduction system and the cuckoo optimization algorithm for damage quantification with limited sensors
,” Smart Mater. Struct.
23
, 045019
(2014
). 9.
A.
Bagheri
, A. Z.
Hosseinzadeh
, P.
Rizzo
, and G. G.
Amiri
, “Time domain damage localization and quantification in seismically excited structures using a limited number of sensors
,” J. Vib. Control
23
, 2942
–2961
(2017
). 10.
F. P.
Kopsaftopoulos
and S. D.
Fassois
, “Vibration based health monitoring for a lightweight truss structure: Experimental assessment of several statistical time series methods
,” Mech. Syst. Signal Process.
24
, 1977
–1997
(2010
). 11.
K.
Krishnan Nair
and A. S.
Kiremidjian
, “Time series based structural damage detection algorithm using Gaussian mixtures modeling
,” J. Dyn. Syst. Meas. Contr.
129
, 285
–293
(2006
). 12.
M.
Gul
and F.
Necati Catbas
, “Statistical pattern recognition for structural health monitoring using time series modeling: Theory and experimental verifications
,” Mech. Syst. Signal Process.
23
, 2192
–2204
(2009
). 13.
H.
Shokravi
, H.
Shokravi
, N.
Bakhary
, S. S.
Rahimian K.
, and M.
Petru
, “Health monitoring of civil infrastructures by subspace system identification method: An overview
,” Appl. Sci.
10
, 2786
(2020
). 14.
S. D.
Fassois
and J. S.
Sakellariou
, “Time-series methods for fault detection and identification in vibrating structures
,” Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci.
365
, 411
–448
(2007
). 15.
H.
Salehi
and R.
Burgueño
, “Emerging artificial intelligence methods in structural engineering
,” Eng. Struct.
171
, 170
–189
(2018
). 16.
E.
Figueiredo
, G.
Park
, C. R.
Farrar
, K.
Worden
, and J.
Figueiras
, “Machine learning algorithms for damage detection under operational and environmental variability
,” Struct. Health Monit.
10
, 559
–572
(2011
). 17.
J. M.
Druce
, J. D.
Haupt
, and S.
Gonella
, “Anomaly-sensitive dictionary learning for structural diagnostics from ultrasonic wavefields
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
62
, 1384
–1396
(2015
). 18.
A. C.
Neves
, I.
Gonzalez
, J.
Leander
, and R.
Karoumi
, “Structural health monitoring of bridges: A model-free ANN-based approach to damage detection
,” J. Civ. Struct. Health Monit.
7
, 689
–702
(2017
). 19.
A.
Sudu Ambegedara
, J.
Sun
, K.
Janoyan
, and E.
Bollt
, “Information-theoretical noninvasive damage detection in bridge structures
,” Chaos
26
, 116312
(2016
). 20.
O.
Shai
and K.
Preiss
, “Graph theory representations of engineering systems and their embedded knowledge
,” Artif. Intell. Eng.
13
, 273
–285
(1999
). 21.
E.
Berthier
, M. A.
Porter
, and K. E.
Daniels
, “Forecasting failure locations in 2-dimensional disordered lattices
,” Proc. Natl. Acad. Sci. U.S.A.
116
, 16742
–16749
(2019
). 22.
P.
Moretti
and M.
Zaiser
, “Network analysis predicts failure of materials and structures
,” Proc. Natl. Acad. Sci. U.S.A.
116
, 16666
–16668
(2019
). 23.
A.
Smart
and J. M.
Ottino
, “Granular matter and networks: Three related examples
,” Soft Matter
4
, 2125
–2131
(2008
). 24.
J. E.
Kollmer
and K. E.
Daniels
, “Betweenness centrality as predictor for forces in granular packings
,” Soft Matter
15
, 1793
–1798
(2019
). 25.
M.
Giger
and P.
Ermanni
, “Evolutionary truss topology optimization using a graph-based parameterization concept
,” Struct. Multidiscipl. Optim.
32
, 313
–326
(2006
). 26.
A.
Kawamoto
, M. P.
Bendsøe
, and O.
Sigmund
, “Planar articulated mechanism design by graph theoretical enumeration
,” Struct. Multidiscipl. Optim.
27
, 295
–299
(2004
). 27.
J.
Runge
, S.
Bathiany
, E.
Bollt
, G.
Camps-Valls
, D.
Coumou
, E.
Deyle
, C.
Glymour
, M.
Kretschmer
, M. D.
Mahecha
, J.
Muñoz-Marí
, E. H.
van Nes
, J.
Peters
, R.
Quax
, M.
Reichstein
, M.
Scheffer
, B.
Schölkopf
, P.
Spirtes
, G.
Sugihara
, J.
Sun
, K.
Zhang
, and J.
Zscheischler
, “Inferring causation from time series in Earth system sciences
,” Nat. Commun.
10
, 2533
(2019
). 28.
K. R.
Pilkiewicz
, B. H.
Lemasson
, M. A.
Rowland
, A.
Hein
, J.
Sun
, A.
Berdahl
, M. L.
Mayo
, J.
Moehlis
, M.
Porfiri
, E.
Fernández-Juricic
, S.
Garnier
, E. M.
Bollt
, J. M.
Carlson
, M. R.
Tarampi
, K. L.
Macuga
, L.
Rossi
, and C.-C.
Shen
, “Decoding collective communications using information theory tools
,” J. R. Soc. Interface
17
, 20190563
(2020
). 29.
N.
Boers
, J.
Kurths
, and N.
Marwan
, “Complex systems approaches for Earth system data analysis
,” J. Phys.: Complex.
2
, 011001
(2021
). 30.
E.
Bullmore
and O.
Sporns
, “Complex brain networks: Graph theoretical analysis of structural and functional systems
,” Nat. Rev. Neurosci.
10
, 186
–198
(2009
). 31.
P.
van Overschee
and B. L.
De Moor
, Subspace Identification for Linear Systems: Theory–Implementation–Applications
(Springer Science & Business Media
, 1996
).32.
M.
Porfiri
, “Validity and limitations of the detection matrix to determine hidden units and network size from perceptible dynamics
,” Phys. Rev. Lett.
124
, 168301
(2020
). 33.
A.
Cancelli
, S.
Laflamme
, A.
Alipour
, S.
Sritharan
, and F.
Ubertini
, “Vibration-based damage localization and quantification in a pretensioned concrete girder using stochastic subspace identification and particle swarm model updating
,” Struct. Health Monit.
19
, 587
–605
(2020
). 34.
M.
Döhler
and L.
Mevel
, “Subspace-based fault detection robust to changes in the noise covariances
,” Automatica
49
, 2734
–2743
(2013
). 35.
S.
Greś
, M.
Döhler
, P.
Andersen
, and L.
Mevel
, “Subspace-based Mahalanobis damage detection robust to changes in excitation covariance
,” Struct. Control Health Monit.
28
, e2760
(2021
).36.
E.
Viefhues
, M.
Döhler
, F.
Hille
, and L.
Mevel
, “Statistical subspace-based damage detection with estimated reference
,” Mech. Syst. Signal Process.
164
, 108241
(2022
). 37.
T. K.
Caughey
, “Classical normal modes in damped linear dynamic systems
,” J. Appl. Mech.
27
, 269
–271
(1960
). 38.
39.
D. S.
Bernstein
, Matrix Mathematics: Theory, Facts, and Formulas
(Princeton University Press
, 2009
).40.
41.
J. E.
Gentle
, Matrix Algebra, Springer Texts in Statistics (Springer, New York, 2007), Vol. 10, 978–980.42.
W.
Gawronski
and T.
Williams
, “Model reduction for flexible space structures
,” J. Guid. Control Dyn.
14
, 68
–76
(1991
). 43.
Z.-S.
Liu
, D.-J.
Wang
, H.-C.
Hu
, and M.
Yu
, “Measures of modal controllability and observability in vibration control of flexible structures
,” J. Guid. Control Dyn.
17
, 1377
–1380
(1994
). 44.
E. D.
Sontag
, Mathematical Control Theory: Deterministic Finite Dimensional Systems
(Springer Science & Business Media
, 2013
), Vol. 6.45.
M.
Tyloo
and R.
Delabays
, “System size identification from sinusoidal probing in diffusive complex networks
,” J. Phys.: Complex.
2
, 025016
(2021
). 46.
J. H.
Tu
, C. W.
Rowley
, D. M.
Luchtenburg
, S. L.
Brunton
, and J. N.
Kutz
, “On dynamic mode decomposition: Theory and applications
,” J. Comput. Dyn.
1
, 391
(2014
). 47.
X.
Tang
, W.
Huo
, Y.
Yuan
, X.
Li
, L.
Shi
, H.
Ding
, and J.
Kurths
, “Dynamical network size estimation from local observations
,” New J. Phys.
22
, 093031
(2020
). 48.
B.
Peeters
and G.
De Roeck
, “Reference-based stochastic subspace identification for output-only modal analysis
,” Mech. Syst. Signal Process.
13
, 855
–878
(1999
). 49.
M.
Basseville
, F.
Bourquin
, L.
Mevel
, H.
Nasser
, and F.
Treyssède
, “Handling the temperature effect in vibration monitoring: Two subspace-based analytical approaches
,” J. Eng. Mech.
136
, 367
–378
(2010
). 50.
I.
Kovacic
, R.
Rand
, and S.
Mohamed Sah
, “Mathieu’s equation and its generalizations: Overview of stability charts and their features
,” Appl. Mech. Rev.
70
, 020802
(2018
). 51.
A. H.
Nayfeh
and B.
Balachandran
, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods
(John Wiley & Sons
, 2008
).52.
R. L.
Harne
and K.-W.
Wang
, Harnessing Bistable Structural Dynamics: For Vibration Control, Energy Harvesting and Sensing
(John Wiley & Sons
, 2017
).53.
K.
Leet
, C.
Uang
, and A. M.
Gilbert
, Fundamentals of Structural Analysis
(McGraw-Hill
, 2008
).54.
G.
Sobczyk
, “Generalized Vandermonde determinants and applications
,” Aportaciones Mat. Ser. Comun.
30
, 203
–213
(2002
).55.
Note that the conditions of Theorem 2 require persistence of excitation that is obviously not satisfied with the chosen input that leads to a rank-deficient Hankel matrix; however, such a condition is not needed for all the claims of Theorem 2, in particular, for the one we are interested in on the determination of .
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