On the application of discrete-time Volterra series for the damage detection problem in initially nonlinear systems
Open accessAbstract
Nonlinearities in the dynamical behavior of mechanical systems can degrade the performance of damage detection features based on a linearity assumption. In this article, a discrete Volterra model is used to monitor the prediction error of a reference model representing the healthy structure. This kind of model can separate the linear and nonlinear components of the response of a system. This property of the model is used to compare the consequences of assuming a nonlinear model during the nonlinear regime of a magneto-elastic system. Hypothesis tests are then employed to detect variations in the statistical properties of the damage features. After these analyses, conclusions are made about the application of Volterra series in damage detection.
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Shiki et al. used a discrete Volterra model to represent the behaviour of a magneto-elastic beam with nonlinearity even in a reference state.
“Shiki et al. [256] used a discrete Volterra model to represent the behaviour of a magneto-elastic beam with nonlinearity even in a reference state.”
From Review on the new development of vibration-based damage identification for civil engineering structures: 2010–2019 (2021) · cites this paper as [256]
Experimental studies showed that the nonlinear index could detect structural changes in both regimes of motion, while the linear one failed during the nonlinear regime.
“Experimental studies showed that the nonlinear index could detect structural changes in both regimes of motion, while the linear one failed during the nonlinear regime.”
From Review on the new development of vibration-based damage identification for civil engineering structures: 2010–2019 (2021) · cites this paper as [256]
Cites
The Volterra and Wiener Theories of Nonlinear Systems
Schetzen, M. (1980) · Medical Entomology and Zoology · Cited by 2,404 · see citing works
Bendat, J. S., & Piersol, A. G. (2010) · Wiley series in probability and statistics · doi:10.1002/9781118032428 · Cited by 1,281 · see citing works
Past, present and future of nonlinear system identification in structural dynamics
Kerschen, G., Keith Worden, B., Vakakis, A. F., & Golinval, J. (2005) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2005.04.008 · Cited by 1,148 · see citing works
Nonlinear System Theory: The Volterra / Wiener Approach
Rugh, W. J. (1981) · Medical Entomology and Zoology · Cited by 943 · see citing works
Damage diagnosis using time series analysis of vibration signals
Sohn, H., & Farrar, C. R. (2001) · Smart Materials and Structures · doi:10.1088/0964-1726/10/3/304 · Cited by 572 · see citing works
Nonlinearity in Structural Dynamics
Keith Worden, B., & Tomlinson, G. R. (2001) · Institute of Physics eBooks · doi:10.1887/0750303565 · Cited by 418 · see citing works
Identification and Control Using Volterra Models
Doyle, F. J., Pearson, R. K., & Ogunnaike, B. A. (2002) · Communications and control engineering/Communications and control engineering series · doi:10.1007/978-1-4471-0107-9 · Cited by 334 · see citing works
Modelling and Identification with Rational Orthogonal Basis Functions
Heuberger, P. S. C., Wahlberg, B., & den Hof, P. M. J. V. (2005) · Springer eBooks · doi:10.1007/1-84628-178-4 · Cited by 322 · see citing works
Transient synthesis in the time domain
Kautz, W. H. (1954) · Transactions of the IRE Professional Group on Circuit Theory · doi:10.1109/tct.1954.1083588 · Cited by 270 · see citing works
Reynders, E., Wursten, G., & De Roeck, G. (2014) · Structural Health Monitoring · doi:10.1177/1475921713502836 · Cited by 237 · see citing works
A review of nonlinear dynamics applications to structural health monitoring
Worden, K., Farrar, C. R., Haywood, J., & Todd, M. (2008) · Structural Control and Health Monitoring · doi:10.1002/stc.215 · Cited by 212 · see citing works
Self-citationSilva, W. (2005) · Nonlinear Dynamics · doi:10.1007/s11071-005-1907-z · Cited by 210 · see citing works
Harmonic analysis of the vibrations of a cantilevered beam with a closing crack
Ruotolo, R., Surace, C., Crespo, P., & Storer, D. (1996) · Computers & Structures · doi:10.1016/0045-7949(96)00184-8 · Cited by 168 · see citing works
Introduction to Experimental Nonlinear Dynamics
Virgin, L. N. (2000) · Cambridge University Press eBooks · doi:10.1017/9781139175227 · Cited by 165 · see citing works
Structural damage detection method using frequency response functions
Bandara, R. P., Chan, T. H. T., & THAMBIRATNAM, D. P. (2014) · Structural Health Monitoring · doi:10.1177/1475921714522847 · Cited by 150 · see citing works
Nonlinear System Identification for Damage Detection
Farrar, C. R., Keith Worden, B., Todd, M. D., Park, G., Nichols, J. M., Adams, D. E., Bement, M., & Farinholt, K. M. (2007) · doi:10.2172/922532 · Cited by 117 · see citing works
Self-citationChatterjee, A. (2010) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2010.02.026 · Cited by 104 · see citing works
Post-beamforming second-order Volterra filter for pulse-echo ultrasonic imaging
Phukpattaranont, P., & Ebbini, E. S. (2003) · IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control · doi:10.1109/tuffc.2003.1226543 · Cited by 65 · see citing works
Fault diagnosis approach based on Volterra models
Tang, H., Liao, Y. H., Cao, J., & Xie, H. (2009) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2009.09.001 · Cited by 63 · see citing works
Damage detection in initially nonlinear systems
Bornn, L., Farrar, C. R., & Park, G. (2010) · International Journal of Engineering Science · doi:10.1016/j.ijengsci.2010.05.011 · Cited by 56 · see citing works
Nonlinear structural damage detection based on cascade of Hammerstein models
Rébillat, M., Hajrya, R., & Mechbal, N. (2014) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2014.03.009 · Cited by 52 · see citing works
Nonlinear Modeling of Causal Interrelationships in Neuronal Ensembles
Zanos, T. P., Courellis, S. H., Berger, T. W., Hampson, R. E., Deadwyler, S. A., & Marmarelis, V. Z. (2008) · IEEE Transactions on Neural Systems and Rehabilitation Engineering · doi:10.1109/tnsre.2008.926716 · Cited by 51 · see citing works
Yelve, N. P., Mitra, M., & Mujumdar, P. M. (2013) · Structural Control and Health Monitoring · doi:10.1002/stc.1604 · Cited by 46 · see citing works
Nonlinear identification in structural dynamics based on Wiener series and Kautz filters
da Silva, S., Cogan, S., & Foltête, E. (2009) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2009.05.017 · Cited by 41 · see citing works
Non-linear model updating of a three-dimensional portal frame based on Wiener series
da Silva, S. (2010) · International Journal of Non-Linear Mechanics · doi:10.1016/j.ijnonlinmec.2010.09.014 · Cited by 36 · see citing works
Detecting nonlinear behaviour using the Volterra series to assess damage in beam-like structures
Surace, C., Ruotolo, R., & Storer, D. (2011) · PORTO@Iris · Cited by 33 · see citing works
da Rosa, A., Campello, R. J. G. B., & Amaral, W. C. (2009) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2009.2031721 · Cited by 31 · see citing works
Identification of nonlinear structures using discrete-time Volterra series
Shiki, S. B., Lopes, V., & da Silva, S. (2013) · Journal of the Brazilian Society of Mechanical Sciences and Engineering · doi:10.1007/s40430-013-0088-9 · Cited by 26 · see citing works
Self-citationKim, Y. (2014) · Applied Ocean Research · doi:10.1016/j.apor.2014.11.002 · Cited by 25 · see citing works
Analysis of Local State Space Models for Feature Extraction in Structural Health Monitoring
Overbey, L. A., & Todd, M. D. (2007) · Structural Health Monitoring · doi:10.1177/1475921706072079 · Cited by 24 · see citing works
Self-citationDamage detection with small data set using energy-based nonlinear features
Mohammadi-Ghazi, R., & Büyüköztürk, O. (2015) · Structural Control and Health Monitoring · doi:10.1002/stc.1774 · Cited by 24 · see citing works
Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations
Cheng, C., Peng, Z., Zhang, W. M., & Meng, G. (2013) · Nonlinear Dynamics · doi:10.1007/s11071-013-1182-3 · Cited by 23 · see citing works
Nonlinear features for damage detection on large civil structures due to earthquakes
Chanpheng, T., Yamada, H., Katsuchi, H., & Sasaki, E. (2012) · Structural Health Monitoring · doi:10.1177/1475921712437182 · Cited by 18 · see citing works
Tawfiq, I., & VINH, T. (2003) · Mechanical Systems and Signal Processing · doi:10.1006/mssp.2002.1499 · Cited by 17 · see citing works
Locating Nonlinear Components in Periodic Structures using Nonlinear Effects
Peng, Z., Lang, Z., Chu, F., & Meng, G. (2010) · Structural Health Monitoring · doi:10.1177/1475921710361327 · Cited by 11 · see citing works
Damage Detection in Nonlinear Structures Using Discrete-Time Volterra Series
Shiki, S. B., Lopes, V., & da Silva, S. (2013) · Key engineering materials · doi:10.4028/www.scientific.net/kem.569-570.876 · Cited by 10 · see citing works
Self-citationNonlinear Features for SHM Applications
Nichols, J. M., & Todd, M. D. (2008) · Encyclopedia of Structural Health Monitoring · doi:10.1002/9780470061626.shm049 · Cited by 8 · see citing works
Self-citationSynthesis of multi-variate Volterra systems by a topological assemblage scheme
Carassale, L., & Kareem, A. (2014) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2014.06.004 · Cited by 3 · see citing works
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- Authors
- Shiki, S. B., da Silva, S., & Todd, M. D.
- Affiliations
- Universidade Estadual Paulista (Unesp) (BR)
- Departamento de Engenharia Mecânica, Faculdade de Engenharia, UNESP—Universidade Estadual Paulista, Ilha Solteira, Brasil
- University of California San Diego (US)
- Venue
- Structural Health Monitoring
- Publisher
- SAGE Publications
- Published
- 2016-09-24
- Volume
- 16
- Issue
- 1
- Pages
- 62-78
- ISSN
- 1475-9217 (ISSN-L)
- 1741-3168
- OpenAlex
- W2514620008
- Cited by
- 35
- References
- 38
- Citations by year
- 2026: 2 · 2025: 3 · 2024: 4 · 2023: 2 · 2022: 4 · 2021: 5 · 2020: 2 · 2019: 7 · 2018: 4 · 2017: 2
- FWCI
- 5.89
- Type
- journal article
- Language
- en
- Open access
- green
- doi.org
- Locations
- Structural Health Monitoring (publishedVersion)
- LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) (publishedVersion, other-oa)
- Americanae (AECID Library) (submittedVersion, other-oa)
- eScholarship (California Digital Library) (submittedVersion)
- Topics
- Physical Sciences › Engineering › Civil and Structural Engineering › Structural Health Monitoring Techniques
- Physical Sciences › Engineering › Mechanics of Materials › Ultrasonics and Acoustic Wave Propagation
- Physical Sciences › Engineering › Control and Systems Engineering › Vibration and Dynamic Analysis
- Keywords
- damage detection, nonlinear structural dynamics, prediction error, hypothesis testing
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