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Research Papers

Output-Only Identification of Nonlinear Systems Via Volterra Series

[+] Author and Article Information
Oscar Scussel

Departamento de Engenharia Mecânica,
Faculdade de Engenharia de Ilha Solteira,
UNESP—Universidade Estadual Paulista,
Avenida Brazil 56,
Ilha Solteira, São Paulo 15385-000, Brazil
e-mail: oscar.scussel@gmail.com

Samuel da Silva

Departamento de Engenharia Mecânica,
Faculdade de Engenharia de Ilha Solteira,
UNESP—Universidade Estadual Paulista,
Avenida Brazil 56,
Ilha Solteira, São Paulo 15385-000, Brazil
e-mail: samuel@dem.feis.unesp.br

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 5, 2015; final manuscript received April 12, 2016; published online May 25, 2016. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 138(4), 041012 (May 25, 2016) (13 pages) Paper No: VIB-15-1075; doi: 10.1115/1.4033458 History: Received March 05, 2015; Revised April 12, 2016

The operational modal analysis methods based on output-only measurements are well-known and applied in linear systems. However, they are not so well developed for nonlinear systems. Thus, this paper proposes an approach for nonlinear system identification using output-only data. In the conventional Volterra series, the outputs of the system are computed by multiple convolutions between the excitation force and the Volterra kernels. However, in this paper at least two time series measured in different placements are used to compute the multiple convolutions and the excitation signals are not required. The novel kernels identified can be used to characterize nonlinear behavior through a model using only output data. A numerical example based on a Duffing oscillator with two degrees-of-freedom (2DOF) and experimental vibration data from a buckled beam with hardening nonlinearities are used to illustrate the proposed method. The prediction results using output-only data are similar to the conventional Volterra kernels based on input and output data.

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Figures

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Fig. 4

Evidences of nonlinearities through the restoring force curve

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Fig. 3

Duffing oscillator with 2DOF and cubic stiffness

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Fig. 2

Computational flow chart illustrating the method

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Fig. 1

Steps of the algorithm for nonlinear system identification based on Volterra series and output-only data

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Fig. 5

Extended kernels and conventional ones of first- and third-order: (a) kernel of first-order, (b) Volterra kernel of first-order, (c) kernel of third-order (main diagonal), and (d) Volterra kernel of third-order (main diagonal)

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Fig. 6

Outputs predicted by the nonlinear models (+) in comparison with the response simulated by using Newmark method (continuous line) that corresponds to the velocity of the second mass: (a) predicted by using the output-only method (y(k;Θ′) with 98% of fit) and (b)predicted by the conventional method (y(k;Θ) with 86% of fit)

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Fig. 7

PSD of the predictions (dotted line with square) in comparison with the PSD of the response simulated (continuous line): (a) PSD of the output predicted by the output-only method and (b) PSD of the output predicted by the conventional Volterra model

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Fig. 8

Experimental setup view (a) and the schematic diagram illustrating the test rig (b)

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Fig. 9

Analysis of the nonlinear effects present in the outputs when is applied a sinusoidal input with excitation close to 37 Hz and high level of amplitude (0.10 V): (a) harmonic input with 0.10 V of amplitude, (b) PSD of harmonic input, (c) experimental output at sensor 1, (d) PSD of output y1, (e) experimental output at sensor 2, and (f) PSD of output y2

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Fig. 10

Evolution of the objective function in the identification procedure using SQP algorithm: (a) objective function using the model based on output-only data and (b) objective function using the conventional Volterra model

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Fig. 11

Kernels of the first-, second-, and third-order: (a) kernel of first-order, (b) Volterra kernel of first-order, (c) kernel of second-order (main diagonal), (d) Volterra kernel of second-order (main diagonal), (e) kernel of third-order (main diagonal), and (f) Volterra kernel of third-order (main diagonal)

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Fig. 12

Output estimated by the output-only method and by the conventional Volterra method (+) in comparison with the acceleration (sensor 2) measured (continuous line): (a) predicted by using the output-only method (y(k;Θ′) with 94% of fit) and (b) predicted by the conventional method (y(k;Θ) with 84% of fit)

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Fig. 13

Nonlinearity detection through PSD of the output predicted when is applied a sinusoidal input with high level of amplitude (0:10 V) and excitation frequency in 37 Hz: (a) PSD of the output predicted by the output-only method and (b) PSD of the output predicted by the Volterra model

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Fig. 14

Frequency contribution of the components for η=1 (continuous line), η=2 (line with triangle), and η=3 (dotted line with square): (a) output-only method and (b) conventional Volterra method

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Fig. 15

Analysis of the nonlinear contribution in the total response by increasing the level of excitation amplitude: (a) output-only method and (b) conventional Volterra method

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