Research Papers

Informative Data for Model Calibration of Locally Nonlinear Structures Based on Multiharmonic Frequency Responses

[+] Author and Article Information
Yousheng Chen

Department of Mechanical Engineering,
Linnaeus University,
Växjö SE-35195, Sweden
e-mail: yousheng.chen@lnu.se

Vahid Yaghoubi

Department of Applied Mechanics,
Chalmers University of Technology,
Gothenburg SE-41296, Sweden
e-mail: yaghoubi@chalmers.se

Andreas Linderholt

Department of Mechanical Engineering,
Linnaeus University,
Växjö SE-35195, Sweden
e-mail: andreas.linderholt@lnu.se

Thomas J. S. Abrahamsson

Department of Applied Mechanics,
Chalmers University of Technology,
Gothenburg SE-41296, Sweden
e-mail: thomas.abrahamsson@chalmers.se

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 21, 2015; final manuscript received May 1, 2016; published online June 2, 2016. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 11(5), 051023 (Jun 02, 2016) (10 pages) Paper No: CND-15-1389; doi: 10.1115/1.4033608 History: Received November 21, 2015; Revised May 01, 2016

In industry, linear finite element (FE) models commonly serve as baseline models to represent the global structural dynamics behavior. However, available test data may show evidence of significant nonlinear characteristics. In such a case, the baseline linear model may be insufficient to represent the dynamics of the structure. The causes of the nonlinear characteristics may be local in nature and the remaining parts of the structure may be satisfactorily represented by linear descriptions. Although the baseline model can then serve as a good foundation, the physical phenomena needed to substantially increase the model's capability of representing the real structure are most likely not modeled in it. Therefore, a set of candidate parameters to control the nonlinear effects have to be added and subjected to calibration to form a credible model. An overparameterized model for calibration may results in parameter value estimates that do not survive a validation test. The parameterization is coupled to the test data and should be chosen so that the expected covariance matrix of the parameter estimates is made small. Accurate test data, suitable for calibration, is often obtained from sinusoidal testing. Because a pure monosinusoidal excitation is difficult to achieve during a physical test of a nonlinear structure, a multisinusoidal excitation is here designed. In this paper, synthetic test data from a model of a nonlinear benchmark structure are used for illustration. The steady-state solutions of the nonlinear system are found using the multiharmonic balance (MHB) method. The steady-state responses at the side frequencies are shown to contain valuable information for the calibration process that can improve the accuracy of the parameters' estimates. The model calibration made and the associated κ-fold cross-validation used is based on the Levenberg–Marquardt and the undamped Gauss–Newton algorithm, respectively. Starting seed candidates for calibration are found by the Latin hypercube sampling method. The candidate that gives the smallest deviation to test data is selected as a starting point for the iterative search for a calibration solution. The calibration result shows good agreement with the true parameter setting and the κ-fold cross validation result shows that the variances of the estimated parameters shrink when multiharmonics nonlinear frequency response functions (FRFs) are included in the data used for calibration.

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Fig. 1

The parts constituting the proposed model calibration method are schematically shown

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Fig. 2

Thirty-three degrees-of-freedom numerical model representing the ECL Benchmark setup. The photo of the test bed is taken from Ref. [15]. The beam is modeled using twelve CBAR elements, to the left of the discrete spring. It has bending stiffness (EI = 672 N·m2), mass per unit length (m = 3.36 kg/m),and total length (Lb = 0.593 m), and 2DOFs at each node. The taut membrane, to the right of the discrete spring, has density (ρ=7850  kg/m3) thickness (t = 0.005 m), width (b = 0.03 m), and free length Lm. Young's modulus of the membrane is parameterized as parameter number 2 ( p2). The membrane is modeled by four CBAR elements with 3DOFs at each node. The right-most node of the membrane is fully constrained after the elongation caused by the pretension. The system pretension is p1 and the periodic loading is f(t). Responses at the tip of the cantilever beam are denoted q10. The loading of the displaced membrane is shown in the insert. The restoring force from the membrane fa sin(ϕ) acting at the cantilever beam end is a nonlinear function of the displacements because of the large membrane rotation ϕ. S0 is the elongation of the membrane due to the pretension force. The rotational dofs of the numerical model are removed by Guyan reduction, which result in a eighteen degrees-of-freedom numerical model.

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Fig. 3

The δ -level multiharmonic nonlinear FRFs, here accelerances, for the load amplitudes 0.2 N (left figure) and 1 N (right figure). Solid lines show FRFs of the linear model, whereas the dotted lines show the nonlinear FRFs.

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Fig. 4

The correlation index Cij for gradient data associated with the full data set and the five parameters. The correlation between parameters number 1 and number 3 is clear.

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Fig. 5

The calibration results from the tenfold cross-validation for three different noise levels. The ten estimated parameter realizations and their mean values and variances are normalized to the true parameter values; one indicates a perfect match between the estimated parameter value and the corresponding true parameter value. The bars represent the normalized estimated parameter value for each calibration. The lines indicate the standard deviation for each parameter. The estimated normalized p1, p2, p4, and p5 from the first-order nonlinear FRFs objective function and the multiharmonic nonlinear FRF objective function are shown in (a)–(d), respectively. (a) The estimated normalized p1, (b) The estimated normalized p2, (c) The estimated normalized p4, and (d) The estimated normalized p5.




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