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Identification of cubic stiffness nonlinearity by linearity-conserved NARMAX modeling

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Abstract

A modeling technique to estimate a NARMAX model is developed to identify nonlinearities which are contained in linear-based nonlinear systems. Considering great contributions by linear parts of the NARMAX model on describing nonlinearities, a linear model, which is estimated from small amplitude input and the corresponding output is taken as the linear part of the NARMAX model. Hence, the capabilities of the model to predict nonlinear behaviors for any input within stable region are fairly improved, and multiplicity problem in selecting a nonlinear regression model is also resolved. As an illustration, one degree of freedom system with cubic stiffness is identified in terms of NARMAX modeling technique using the procedure proposed in this work and conventional one, respectively. By extraction higher order FRFs from the NARMAX models, dominant nonlinearities of the system are predicted, and the results by the two methods are compared with analytic one, which shows the priority of the modeling technique proposed.

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Abbreviations

⌢:

Estimated value

AIC :

Akaike's information criterion

ERR i :

Error reduction ratio of thei-th model term

FRF :

Frequency response function

H n(f1,…, fn):

nth order frequency response function

N :

Number of data in the processing

NARMAX :

Nonlinear Auto-Regressive Moving Averge with eXogenous input

e(t) :

Residual Sequence

f i :

ith frequency component in generalized frequency response function

n :

degree of nonlinearity

p, q, r :

order of output, input and residual sequence

u t(=u(t)):

input sequence

y t(=y(t)):

output sequence

α i (t):

ith term in NARMAX model

θi :

ith NARMAX model parameter

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Authors and Affiliations

  1. Department of Mechanical Engineering, Center for Noise and Vibration Control, KAIST

    Han-Kee Jang & Kwang-Joon Kim

Authors
  1. Han-Kee Jang
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  2. Kwang-Joon Kim
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Jang, HK., Kim, KJ. Identification of cubic stiffness nonlinearity by linearity-conserved NARMAX modeling. KSME Journal 8, 332–342 (1994). https://doi.org/10.1007/BF02953362

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  • DOI: https://doi.org/10.1007/BF02953362

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