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2025 | OriginalPaper | Buchkapitel

2. Representation of Complex Structures with the Volterra Series

verfasst von : Quankun Li, Xingjian Jing

Erschienen in: Frequency Domain Nonlinear Vibration Features for Fault Diagnosis in Complex Structures

Verlag: Springer Nature Singapore

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Abstract

In this chapter, fundamental theories of the Volterra Series representing nonlinear systems with multiple inputs and multiple outputs are introduced firstly. Then, apart from the harmonic probing method, the nth-OSE algorithm and its improved version for estimating each order Volterra-type OS are introduced with detailed derivation as well.

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Literatur
Zurück zum Zitat Bedrosian E, Rice SO (1971) The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Presented at the proceedings of the IEEE. Bedrosian E, Rice SO (1971) The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Presented at the proceedings of the IEEE.
Zurück zum Zitat Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Transactions on Circuits and Systems 32(11):1150–1161MathSciNetCrossRefMATH Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Transactions on Circuits and Systems 32(11):1150–1161MathSciNetCrossRefMATH
Zurück zum Zitat Cheng CM, Peng ZK, Zhang WM, Meng G (2017) Volterra-series-based nonlinear system modelling and its engineering applications: a state-of-the-art review. Mech Syst Signal Process 87:340–364CrossRefMATH Cheng CM, Peng ZK, Zhang WM, Meng G (2017) Volterra-series-based nonlinear system modelling and its engineering applications: a state-of-the-art review. Mech Syst Signal Process 87:340–364CrossRefMATH
Zurück zum Zitat Ewins DJ (1984) Modal testing: theory and practice. Research studies press, Letch worthMATH Ewins DJ (1984) Modal testing: theory and practice. Research studies press, Letch worthMATH
Zurück zum Zitat Jing XJ (2011) Frequency domain analysis and identification of block-oriented nonlinear systems. J Sound Vib 330:5427–5442CrossRefMATH Jing XJ (2011) Frequency domain analysis and identification of block-oriented nonlinear systems. J Sound Vib 330:5427–5442CrossRefMATH
Zurück zum Zitat Jing XJ (2014a) Nonlinear characteristic output spectrum for nonlinear analysis and design. IEEE/ASME Trans Mechatron 19(1):171–183CrossRefMATH Jing XJ (2014a) Nonlinear characteristic output spectrum for nonlinear analysis and design. IEEE/ASME Trans Mechatron 19(1):171–183CrossRefMATH
Zurück zum Zitat Jing XJ, Li QK (2016) A nonlinear decomposition and regulation method for nonlinearity characterization. Nonlinear Dyn 83:1355–1377MathSciNetCrossRefMATH Jing XJ, Li QK (2016) A nonlinear decomposition and regulation method for nonlinearity characterization. Nonlinear Dyn 83:1355–1377MathSciNetCrossRefMATH
Zurück zum Zitat Jing XJ, Lang ZQ, Billing SA, Tomlinson GR (2006) The parametric characteristic of frequency response functions for nonlinear systems. Int J Control 79(12):1552–1564MathSciNetCrossRefMATH Jing XJ, Lang ZQ, Billing SA, Tomlinson GR (2006) The parametric characteristic of frequency response functions for nonlinear systems. Int J Control 79(12):1552–1564MathSciNetCrossRefMATH
Zurück zum Zitat Jing XJ, Lang ZQ, Billing SA (2008) Mapping from parametric characteristics to generalized frequency response functions of non-linear systems. Int J Control 81(7):1071–1088MathSciNetCrossRefMATH Jing XJ, Lang ZQ, Billing SA (2008) Mapping from parametric characteristics to generalized frequency response functions of non-linear systems. Int J Control 81(7):1071–1088MathSciNetCrossRefMATH
Zurück zum Zitat Jing XJ, Lang ZQ (2015) Frequency domain analysis and design of nonlinear systems based on Volterra series expansion. In: A parametric characteristic approach. Springer Jing XJ, Lang ZQ (2015) Frequency domain analysis and design of nonlinear systems based on Volterra series expansion. In: A parametric characteristic approach. Springer
Zurück zum Zitat Jing XJ (2014b) A frequency-domain approach to crack-related fault detection. In: Proceedings of 2014 IEEE Chinese guidance, navigation and control conference. Jing XJ (2014b) A frequency-domain approach to crack-related fault detection. In: Proceedings of 2014 IEEE Chinese guidance, navigation and control conference.
Zurück zum Zitat Jones JP, Billings SA (1989) Recursive algorithm for computing the frequency response of a class of non-linear difference equation models. Int J Control 50(5):1925–1940CrossRefMATH Jones JP, Billings SA (1989) Recursive algorithm for computing the frequency response of a class of non-linear difference equation models. Int J Control 50(5):1925–1940CrossRefMATH
Zurück zum Zitat Li QK, Jing XJ (2017) A second-order output spectrum approach for fault detection of bolt loosening in a satellite-like structure with a sensor chain. Nonlinear Dyn 89:587–606CrossRefMATH Li QK, Jing XJ (2017) A second-order output spectrum approach for fault detection of bolt loosening in a satellite-like structure with a sensor chain. Nonlinear Dyn 89:587–606CrossRefMATH
Zurück zum Zitat Li QK, Jing XJ (2019a) Fault diagnosis of bolt loosening in structures with a novel second-order output spectrum-based method. Struct Health Monit 19(1):123–141CrossRefMATH Li QK, Jing XJ (2019a) Fault diagnosis of bolt loosening in structures with a novel second-order output spectrum-based method. Struct Health Monit 19(1):123–141CrossRefMATH
Zurück zum Zitat Mathews VJ, Sicuranza GL (2000) Polynomial signal processing. Wiley, New YorkMATH Mathews VJ, Sicuranza GL (2000) Polynomial signal processing. Wiley, New YorkMATH
Zurück zum Zitat Peng ZK, Lang ZQ (2010) The nonlinear output frequency response functions of one-dimensional chain type structures. J Appl Mech 77(1):011007CrossRefMATH Peng ZK, Lang ZQ (2010) The nonlinear output frequency response functions of one-dimensional chain type structures. J Appl Mech 77(1):011007CrossRefMATH
Zurück zum Zitat Peng ZK, Lang ZQ, Billings SA (2007a) Crack detection using nonlinear output frequency response functions. J Sound Vib 301(3–5):777–788CrossRefMATH Peng ZK, Lang ZQ, Billings SA (2007a) Crack detection using nonlinear output frequency response functions. J Sound Vib 301(3–5):777–788CrossRefMATH
Zurück zum Zitat Rugh WJ (1981) Nonlinear system theory. Johns Hopkins University Press, BaltimoreMATH Rugh WJ (1981) Nonlinear system theory. Johns Hopkins University Press, BaltimoreMATH
Zurück zum Zitat Schetzen M (1980) The Volterra and Wiener theories of nonlinear systems. John Wiley and Sons Inc., HobokenMATH Schetzen M (1980) The Volterra and Wiener theories of nonlinear systems. John Wiley and Sons Inc., HobokenMATH
Zurück zum Zitat Swain AK, Billings SA (2001) Generalized frequency response function matrix for MIMO non-linear systems. Int J Control 74(8):829–844MathSciNetCrossRefMATH Swain AK, Billings SA (2001) Generalized frequency response function matrix for MIMO non-linear systems. Int J Control 74(8):829–844MathSciNetCrossRefMATH
Zurück zum Zitat Volterra V (1959) Theory of functionals and of integral and integra-differential equations. Dover Publications, New YorkMATH Volterra V (1959) Theory of functionals and of integral and integra-differential equations. Dover Publications, New YorkMATH
Zurück zum Zitat Wiener, (1942) Response of a non-linear device to noise. Radiation Lab MIT, CambridgeMATH Wiener, (1942) Response of a non-linear device to noise. Radiation Lab MIT, CambridgeMATH
Zurück zum Zitat Worden K (2001) Nonlinearity in structural dynamics: detection, identification and modelling. CRC Press, New YorkCrossRefMATH Worden K (2001) Nonlinearity in structural dynamics: detection, identification and modelling. CRC Press, New YorkCrossRefMATH
Zurück zum Zitat Worden K, Manson G, Tomlinson GR (1997) A harmonic probing algorithm for the multi-input Volterra series. J Sound Vib 201(1):67–84MathSciNetCrossRefMATH Worden K, Manson G, Tomlinson GR (1997) A harmonic probing algorithm for the multi-input Volterra series. J Sound Vib 201(1):67–84MathSciNetCrossRefMATH
Zurück zum Zitat Zhao XY, Lang ZQ, Park G, Farrar CR, Todd MD, Mao Z, Worden K (2014) A new transmissibility analysis method for detection and location of damage via nonlinear features in MDOF structural systems. IEEE/ASME transactions on mechatronics 20(4):1933-1947. Zhao XY, Lang ZQ, Park G, Farrar CR, Todd MD, Mao Z, Worden K (2014) A new transmissibility analysis method for detection and location of damage via nonlinear features in MDOF structural systems. IEEE/ASME transactions on mechatronics 20(4):1933-1947.
Zurück zum Zitat Zhou YL, Abdel Wahab M, Perera R (2015) Damage detection by transmissibility conception in beam-damage detection by transmissibility conception in beam-like structures. In: 4th International conference on fracture fatigue and wear, vol 1, issue 1, pp 1–14 Zhou YL, Abdel Wahab M, Perera R (2015) Damage detection by transmissibility conception in beam-damage detection by transmissibility conception in beam-like structures. In: 4th International conference on fracture fatigue and wear, vol 1, issue 1, pp 1–14
Metadaten
Titel
Representation of Complex Structures with the Volterra Series
verfasst von
Quankun Li
Xingjian Jing
Copyright-Jahr
2025
Verlag
Springer Nature Singapore
DOI
https://doi.org/10.1007/978-981-96-1702-9_2

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