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Research Articles

Generalized Cause-and-Effect Analyses for the Prototypic Nonlinear Mass–Spring–Damper System Using Volterra Kernels

[+] Author and Article Information
Ashraf Omran

Engineer Analyst Systems Modeling,
CNH-Fiat Industrial,
Burr Ridge, IL 60527

Brett Newman

Professor
Department of Mechanical and Aerospace Engineering,
Old Dominion University,
Norfolk, VA 23529

1 The presented work in this paper was completed while the author was serving as a research scientist at the Old Dominion University Research Foundation and the work does not belong or represent his current affiliation, CNH-Fiat Industrial, at any level.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 21, 2011; final manuscript received April 1, 2012; published online October 30, 2012. Assoc. Editor: Douglas Adams.

J. Dyn. Sys., Meas., Control 135(1), 011008 (Oct 30, 2012) (13 pages) Paper No: DS-11-1218; doi: 10.1115/1.4006629 History: Received July 21, 2011; Revised April 01, 2012

This paper develops generalized analytical first and second Volterra kernels for the prototypic nonlinear mass–spring–damper system. The nonlinearity herein is mathematically considered in quadratic and bilinear terms. A variational expansion methodology, one of the most efficient analytical Volterra techniques, is used to develop an analytical two-term Volterra series. The resultant analytical first and second kernels are visualized in both the time and the frequency domains followed by a parametric study to understanding the influence of each nonlinear/linear term appearing in the kernel structure. An analytical nonlinear step and periodic responses are also conducted to characterize the overall system response from the fundamental components. The developed analytical responses provide an illumination for the source of differences between nonlinear and linear responses. Feasibility of the proposed implementation is assessed by numerical examples. The developed kernel-based model shows the ability to predict, understand, and analyze the system behavior beyond that attainable by the linear-based model.

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References

Volterra, V., 1958, Theory of Functionals and of Integral and Integro-Differential Equations, Dover, New York.
Rugh, J. W., 1981, Nonlinear System Theory: The Volterra/Wiener Approach, John Hopkins University Press, Baltimore, MD.
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Glass, J. W., and Franchek, M. A., 2002, “H Synthesis of Nonlinear Feedback Systems in a Volterra Representation,” J. Dyn. Syst., Meas., Control, 124(3), pp. 382–389. [CrossRef]
Omran, A., and Newman, B., 2010, “Nonlinear Analytical Multi-Dimensional Convolution Solution of the Second Order System,” J. Nonlinear Dyn., 62(4), pp. 799–819. [CrossRef]
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Schurer, H., Slump, H., and Herrmann, E., 1995, “Second Order Volterra Inverses for Compensation of Loudspeaker Nonlinearity,” IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, New York, Oct.15–18.
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Figures

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Fig. 1

First kernel ( 0 < ζ < 1)

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Fig. 2

Surface (ωn = 2 and ζ = 0.1) and diagonal line of the second kernel’s components (0 < ζ< 1). (a) surface of quadratic position component, (b) diagonal of quadratic position component, (c) surface of bilinear position-rate component, (d) diagonal of bilinear position-rate component, (e) surface of quadratic rate component, and (f) diagonal of quadratic rate component.

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Fig. 3

First kernel magnitude generic shape

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Fig. 16

Nonlinear step response (0 < ζ < 1). (a) Linear response to step input, (b) Quadratic position response to step input, (c) Bilinear position-rate response to step input, and (d) Quadratic rate response to step input.

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Fig. 4

Quadratic position kernel magnitude at ωn = 5 rad/s and ζ = 0.1

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Fig. 5

Quadratic position kernel magnitude at ωn = 5 rad/s and ζ = 0.4

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Fig. 6

Quadratic position kernel magnitude at ωn = 5 rad/s and ζ = 0.8

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Fig. 11

Magnitude diagonal of the component i = {qp, bpr, qr} for ζ<ζc1i

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Fig. 12

Magnitude diagonal of the component i = {qp, bpr, qr} for ζ>ζc1i

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Fig. 13

Magnitude diagonal of the quadratic position at ζ< ζc2qp

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Fig. 19

Step response for system I at F = 2 N

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Fig. 14

Quadratic position kernel phase at ωn = 5 rad/sec and ζ=0.3

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Fig. 15

Phase diagonal of the component i = {qp, bpr, qr}

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Fig. 7

Bilinear position-rate kernel magnitude at ωn = 5 rad/s and ζ = 0.1

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Fig. 8

Bilinear position-rate kernel magnitude at ωn = 5 rad/s and ζ = 0.4

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Fig. 9

Quadratic rate kernel magnitude at ωn = 5 rad/s and ζ = 0.1

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Fig. 10

Quadratic rate kernel magnitude at ωn = 5 rad/s and ζ = 0.4

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Fig. 27

Steady periodic response components for system II at F = −1.5 sin(0.5ωdt) N

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Fig. 28

Total steady time response for system II at F = −1.5 sin(0.5ωdt) N

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Fig. 17

Linear periodic input–output relationship

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Fig. 18

Nonlinear periodic input–output relationship for i = {qp, bpr, qr}

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Fig. 20

Total phase plane of system I at F = 2 N

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Fig. 21

Steady periodic response components for system I at F = 5 sin(ωdt) N

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Fig. 22

Total steady periodic response for system I at F = 5 sin(ωdt) N

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Fig. 23

Step response components for system II at F = 0.75 N

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Fig. 24

Total step response for system II at F = 0.75 N

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Fig. 25

Total steady periodic response for system II at F = 0.75 sin(0.5ωdt) N

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Fig. 26

Total steady periodic response for system II at F = −1.5 sin(0.5ωdt) N

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