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Model calibration methods for mechanical systems with local nonlinearities
Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Most modern product development utilizes computational models. With increasing demands on reducing the product development lead-time, it becomes more important to improve the accuracy and efficiency of simulations. In addition, to improve product performance, a lot of products are designed to be lighter and more flexible, thus more prone to nonlinear behaviour. Linear finite element (FE) models, which still form the basis of numerical models used to represent mechanical structures, may not be able to predict structural behaviour with necessary accuracy when nonlinear effects are significant. Nonlinearities are often localized to joints or boundary conditions. Including nonlinear behaviour in FE-models introduces more sources of uncertainty and it is often necessary to calibrate the models with the use of experimental data. This research work presents a model calibration method that is suitable for mechanical systems with structural nonlinearities. The methodology concerns pre-test planning, parameterization, simulation methods, vibrational testing and optimization.

The selection of parameters for the calibration requires physical insights together with analyses of the structure; the latter can be achieved by use of simulations. Traditional simulation methods may be computationally expensive when dealing with nonlinear systems; therefore an efficient fixed-step state-space based simulation method was developed. To gain knowledge of the accuracy of different simulation methods, the bias errors for the proposed method as well as other widespread simulation methods were studied and compared. The proposed method performs well in comparison to other simulation methods.

To obtain precise estimates of the parameters, the test data should be informative of the parameters chosen and the parameters should be identifiable. Test data informativeness and parameter identifiability are coupled and they can be assessed by the Fisher information matrix (FIM). To optimize the informativeness of test data, a FIM based pre-test planning method was developed and a multi-sinusoidal excitation was designed. The steady-state responses at the side harmonics were shown to contain valuable information for model calibration of FE-models representing mechanical systems with structural nonlinearities.

In this work, model calibration was made by minimizing the difference between predicted and measured multi-harmonic frequency response functions using an efficient optimization routine. The steady-state responses were calculated using the extended multi-harmonic balance method. When the parameters were calibrated, a k-fold cross validation was used to obtain parameter uncertainty.

The proposed model calibration method was validated using two test-rigs, one with a geometrical nonlinearity and one with a clearance type of nonlinearity. To attain high quality data efficiently, the amplitude of the forcing harmonics was controlled at each frequency step by an off-line force feedback algorithm. The applied force was then measured and used in the numerical simulations of the responses. It was shown in the validation results that the predictions from the calibrated models agree well with the experimental results.

In summary, the presented methodology concerns both theoretical and experimental aspects as it includes methods for pre-test planning, simulations, testing, calibration and validation. As such, this research work offers a complete framework and contributes to more effective and efficient analyses on mechanical systems with structural nonlinearities.

Place, publisher, year, edition, pages
Linnaeus University Press, 2016. , 145 p.
Series
Linnaeus University Dissertations, 262
Keyword [en]
model calibration, finite element modelling, nonlinear structural dynamics, pre-test planning, multi-sinusoidal excitation, vibrational testing, cross validation
National Category
Mechanical Engineering
Research subject
Technology (byts ev till Engineering), Mechanical Engineering
Identifiers
URN: urn:nbn:se:lnu:diva-57638ISBN: 978-91-88357-37-3OAI: oai:DiVA.org:lnu-57638DiVA: diva2:1040158
Public defence
2016-10-25, N1017, Växjö, 09:30 (English)
Opponent
Supervisors
Available from: 2016-11-10 Created: 2016-10-26 Last updated: 2016-11-22Bibliographically approved
List of papers
1. Informative Data for Model Calibration of Locally Nonlinear Structures Based on Multi-Harmonic Frequency Responses
Open this publication in new window or tab >>Informative Data for Model Calibration of Locally Nonlinear Structures Based on Multi-Harmonic Frequency Responses
2016 (English)In: Journal of Computational and Nonlinear Dynamics, ISSN 1555-1415, E-ISSN 1555-1423, Vol. 11, no 5, 051023Article in journal, Editorial material (Refereed) Published
Abstract [en]

In industry, linear FE-models commonly serve as baseline models to represent the global structural dynamics behavior. However, available test data may show evidence of significant nonlinear dynamic 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 modelled in it. Therefore, a set of candidate nonlinear property parameters to control the nonlinear effects have to be added and subjected to calibration to form a credible model. The selection of the calibration parameters and the choice of data for a calibration metric form a coupled problem. An over-parameterized model for calibration may result 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 co-variances of the chosen parameter's estimates are made small. Accurate test data, suitable for calibration, is often obtained from sinusoidal testing. Because a pure mono-sinusoidal excitation is difficult to achieve during a test of a nonlinear structure, the excitation is here designed to contain sub and super harmonics besides the fundamental harmonic. 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 parameter estimates. The nonlinear steady-state solutions can be found efficiently using the multi-harmonic balance method. In this paper, synthetic test data from a model of a nonlinear benchmark structure are used for illustration. The model calibration and an associated K-fold cross-validation are 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 realization 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 K-fold cross validation result shows that the variance of the estimated parameters shrinks when adding sub and super harmonics to the nonlinear frequency response functions.

Place, publisher, year, edition, pages
ASME Press, 2016
Keyword
model calibration, Fisher information matrix, identifiability, multi-harmonic response, cross-validation
National Category
Applied Mechanics
Research subject
Technology (byts ev till Engineering), Mechanical Engineering
Identifiers
urn:nbn:se:lnu:diva-34671 (URN)10.1115/1.4033608 (DOI)000383104400023 ()
Available from: 2014-06-04 Created: 2014-06-04 Last updated: 2016-11-01Bibliographically approved
2. An Efficient Simulation Method for Large-Scale Systems with Local Nonlinearities
Open this publication in new window or tab >>An Efficient Simulation Method for Large-Scale Systems with Local Nonlinearities
2016 (English)In: Special topics in structural dynamics, 34th IMAC / [ed] DiMiao, D; Tarazaga, P; Castellini, P, Springer, 2016, Vol. 6Conference paper (Other academic)
Abstract [en]

In practice, most mechanical systems show nonlinear characteristics within the operational envelope. However, the nonlinearities are often caused by local phenomena and many mechanical systems can be well represented by a linear model enriched with local nonlinearities. Conventional nonlinear response simulations are often computationally intensive; the problem which becomes more severe when large-scale nonlinear systems are concerned. Thus, there is a need to further develop efficient simulation techniques. In this work, an efficient simulation method for large-scale systems with local nonlinearities is proposed. The method is formulated in a state-space form and the simulations are done in the Matlab environment. The nonlinear system is divided into a linearized system and a nonlinear part represented as external nonlinear forces acting on the linear system; thus taking advantage in the computationally superiority in the locally nonlinear system description compared to a generally nonlinear counterpart. The triangular-order hold exponential integrator is used to obtain a discrete state-space form. To shorten the simulation time additionally, auxiliary matrices, similarity transformation and compiled C-codes (mex) to be used for the time integration are studied. Comparisons of the efficiency and accuracy of the proposed method in relation to simulations using the ODE45 solver in Matlab and MSC Nastran are demonstrated on numerical examples of different model sizes.

Place, publisher, year, edition, pages
Springer, 2016
Series
, Conference Proceedings of the Society for Experimental Mechanics Series, ISSN 2191-5644
Keyword
Efficient time integration, Triangular-order hold, State-space, Locally nonlinear systems, C-code/mex
National Category
Mechanical Engineering
Research subject
Technology (byts ev till Engineering), Mechanical Engineering
Identifiers
urn:nbn:se:lnu:diva-51448 (URN)10.1007/978-3-319-29910-5_27 (DOI)000381977000027 ()978-3-319-29910-5 (ISBN)978-3-319-29909-9 (ISBN)
Conference
34th International Modal Analysis Conference (IMAC XXXIV), Orlando, FL, 25-28 January 2016
Available from: 2016-03-28 Created: 2016-03-28 Last updated: 2016-11-01Bibliographically approved
3. Bias errors of different simulation methods for linear and nonlinear systems
Open this publication in new window or tab >>Bias errors of different simulation methods for linear and nonlinear systems
2015 (English)In: Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015 / [ed] Gaetan Kerschen, Springer, 2015, 505-520 p.Conference paper (Other academic)
Abstract [en]

Responses of mechanical systems are often studied using numerical time-domain methods. Discrete excitation forces require a transformation of the dynamic system from continuous time into discrete time. Such a transformation introduces an aliasing error. To reduce the aliasing error, different discretization techniques are used. The bias errors introduced by some discretization techniques are studied in this paper.

Algebraic expressions of the bias error obtained for some discretization methods are presented. The bias error depends on the assumption of the characteristics of the load between two subsequent time steps; here the zero-order, first-order and Lagrange second-order assumptions are studied. Different simulation methods are also studied for numerical evaluation of the derived theoretical bias errors. The discretization techniques are implemented for Runge-Kutta, the Digital Filter method and for the Pseudo Force in State Space method.

The study is carried out for both a linear and a nonlinear system; two numerical examples assist in evaluating the theory. Perfect matches between the numerically estimated bias errors and the theoretical ones are shown. The results also show that, for the nonlinear example, the fourth order Runge-Kutta method is less accurate than the Digital Filter and the used single step Pseudo Force in State Space method.

Place, publisher, year, edition, pages
Springer, 2015
Series
, Conference Proceedings of the Society for Experimental Mechanics Series, ISSN 2191-5644
Keyword
Bias error, numerical methods, digital filter, state space, frequency response function
National Category
Mechanical Engineering Other Mechanical Engineering
Research subject
Technology (byts ev till Engineering), Mechanical Engineering
Identifiers
urn:nbn:se:lnu:diva-40230 (URN)10.1007/978-3-319-15221-9_44 (DOI)000381753700044 ()978-3-319-15220-2 (ISBN)
Conference
The 33rd IMAC Conference and Exposition on Structural Dynamics, February 2–5, 2015, Orlando, Florida
Available from: 2015-04-02 Created: 2015-02-18 Last updated: 2017-01-10Bibliographically approved
4. Validation of a model calibration method through vibrational testing of a mechanical system with local clearance
Open this publication in new window or tab >>Validation of a model calibration method through vibrational testing of a mechanical system with local clearance
Show others...
2016 (English)In: Presented at the International Conference on Noise and Vibration Engineering, September 19-21, 2016, Leuven, Belgium, Leuven, Belgium: ISMA , 2016Conference paper (Other academic)
Abstract [en]

Nonlinear finite element models are often validated using experimental data. A previously proposed calibration method, which concerns pre-test planning, multi-sinusoidal excitation and an effective optimization routine, is improved with an extended version of the pre-test planning. The improved method is validated using a test structure with a clearance type nonlinearity. From the pretest planning, an optimal configuration for the data acquisition is determined. The multi-harmonic nonlinear frequency response functions (FRFs) of the structure under test are then generated by a multi-sinusoidal excitation. Model calibration is conducted by minimizing the difference between the experimental multi-harmonic nonlinear FRFs and their analytical counterparts. The uncertainties of the estimated parameters are assessed by a k-fold cross validation, which confirm that the uncertainties of the estimated parameters are small when the optimal configuration is applied.

Place, publisher, year, edition, pages
Leuven, Belgium: ISMA, 2016
National Category
Other Mechanical Engineering
Identifiers
urn:nbn:se:lnu:diva-57032 (URN)
Conference
International Conference on Noise and Vibration Engineering, September 19-21, 2016, Leuven, Belgium
Note

Ej belagd 20161031

Available from: 2016-10-04 Created: 2016-10-04 Last updated: 2016-11-04Bibliographically approved

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