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  • 1.
    Chen, Yousheng
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Model calibration methods for mechanical systems with local nonlinearities2016Doctoral 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.

  • 2.
    Chen, Yousheng
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Model Calibration of Nonlinear Mechanical Systems Using Multi-Harmonic Frequency Response Functions2014Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In industry, linear finite element models are commonly employed to represent global structural behavior. It is crucial that the computational models are able to accurately represent the structures’ behavior. This cannot always be achieved by the use of linear models.

    When vibrational test data show significant nonlinear characteristics, an initial linear finite element model may be judged insufficient in representing the structural behavior. Although an initial model can give a good foundation for the understanding of the dynamic behavior of the structure, the parameters that capture the nonlinear effects are most likely not included. Therefore, a set of candidate parameters controlling the nonlinear effect have to be added. The selection of such candidates is a delicate task which solution is preferably supported by engineering insight into the characteristics of the structure.

    One part of this work is on the selection of parameters, among all possibly uncertain properties, together with the forming of the objective function to be used for calibration. To obtain precise estimates of the parameters, the objective function data have to be informative with respect to the selected parameters. Further the parameters have to be identifiable. To improve these qualities, a multi-harmonic sinusoidal excitation was designed since the corresponding steady-state responses at the sub- and super- harmonics were shown to contain valuable information for the calibration process. Model calibration of nonlinear systems made by minimizing the differences between predicted and measured multi-harmonic frequency response functions.

    Further, in the calibration, multi-harmonic frequency response functions need to be calculated recurrently in order to reach convergence; therefore a fast simulation scheme was required. The performance of computations of multi-harmonic frequency response functions calculated using time domain as well as frequency domain simulation techniques were studied and compared.

    Finally, the proposed calibration method was validated by use of experimental testing on a replica of the Ecole de Lyon nonlinear benchmark structure. It was shown in the validation results that the predictions stemming from the calibrated model matched the experimental data well.

  • 3.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Ahlin, Kjell
    Xielalin Consulting, Sweden.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Bias errors of different simulation methods for linear and nonlinear systems2015In: 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.

  • 4.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    An Efficient Simulation Method for Large-Scale Systems with Local Nonlinearities2016In: 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.

  • 5.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Experimental Validation of a Nonlinear Model Calibration Method Based on Multiharmonic Frequency Responses2017In: Journal of Computational and Nonlinear Dynamics, ISSN 1555-1415, E-ISSN 1555-1423, Vol. 12, no 4, 041014Article in journal (Refereed)
    Abstract [en]

    Correlation and calibration using test data are natural ingredients in the process of validating computational models. Model calibration for the important subclass of nonlinear systems which consists of structures dominated by linear behavior with the presence of local nonlinear effects is studied in this work. The experimental validation of a nonlinear model calibration method is conducted using a replica of the École Centrale de Lyon (ECL) nonlinear benchmark test setup. The calibration method is based on the selection of uncertain model parameters and the data that form the calibration metric together with an efficient optimization routine. The parameterization is chosen so that the expected covariances of the parameter estimates are made small. To obtain informative data, the excitation force is designed to be multisinusoidal and the resulting steady-state multiharmonic frequency response data are measured. To shorten the optimization time, plausible starting seed candidates are selected using the Latin hypercube sampling method. The candidate parameter set giving the smallest deviation to the test data is used as a starting point for an iterative search for a calibration solution. The model calibration is conducted by minimizing the deviations between the measured steady-state multiharmonic frequency response data and the analytical counterparts that are calculated using the multiharmonic balance method. The resulting calibrated model's output corresponds well with the measured responses.

  • 6.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers tekniska högskola.
    Frequency Response Calculations of a Nonlinear Structure a Comparison of Numerical Methods2014In: Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 / [ed] Gaetan Kerschen, Springer, 2014, 35-44 p.Conference paper (Refereed)
    Abstract [en]

    Mechanical systems having presence of nonlinearities are often represented by nonlinear ordinary differential 5 equations. For most of such equations, exact analytic solutions are not found; thus numerical techniques have to be used. 6 In many applications, among which model calibration can be one, steady-state frequency response functions are the desired 7 quantities to calculate. 8 The objective of this paper is to compare the performance of computations of nonlinear frequency response functions 9 (FRFs) calculated directly within the frequency domain, using the Multi-Harmonic Balance method, with the time-domain 10 methods Runge–Kutta, Newmark and Pseudo Force in State Space (PFSS). The PFSS method is a recently developed state- 11 space based force feedback method that is shown to give efficient solutions. 12 The accuracy and efficiency of the methods are studied and compared using a model of a cantilever beam connected to a 13 non-linear spring at its free end.

  • 7.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Xia, Yuying
    University of the West of England, UK.
    Friswell, Michael I.
    Swansea University, UK.
    A Pretest Planning Method for Model Calibration for Nonlinear Systems2016In: Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016, Springer, 2016, Vol. 3, 371-379 p.Conference paper (Other academic)
    Abstract [en]

    With increasing demands on more flexible and lighter engineering structures, it has been more common to take nonlinearity into account. Model calibration is an important procedure for nonlinear analysis in structural dynamics with many industrial applications. Pretest planning plays a key role in the previously proposed calibration method for nonlinear systems, which is based on multi-harmonic excitation and an effective optimization routine. This paper aims to improve the pretest planning strategy for the proposed calibration method. In this study, the Fisher information matrix (FIM), which is calculated from the gradients with respect to the chosen parameters with unknown values, is used for determining the locations, frequency range, and the amplitudes of the excitations as well as the sensor placements. This pretest planning based model calibration method is validated by a structure with clearance nonlinearity. Synthetic test data is used to simulate the test procedure. Model calibration and K-fold cross validation are conducted for the optimum configurations selected from the pretest planning as well as three other configurations. The calibration and cross validation results show that a more accurate estimation of parameters can be obtained by using test data from the optimum configuration.

  • 8.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Xia, Yuying
    University of the West of England, UK.
    Friswell, Michael I
    Swansea University, UK.
    Validation of a model calibration method through vibrational testing of a mechanical system with local clearance2016In: Proceedings of ISMA2016 International conference on noise and vibration engineering and USD2016 International conference on uncertainty in structural dynamics / [ed] Sas, P; Moens, D; VanDeWalle, A, Leuven, Belgium: Katholieke University Leuven , 2016, 2581-2595 p.Conference paper (Refereed)
    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.

  • 9.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Nasrabadi, Vahid
    Chalmers University of Technology.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Informative Data for Model Calibration of Locally Nonlinear Structures Based on Multi-Harmonic Frequency Responses2016In: Journal of Computational and Nonlinear Dynamics, ISSN 1555-1415, E-ISSN 1555-1423, Vol. 11, no 5, 051023Article in journal (Refereed)
    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.

  • 10.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Engineering.
    Nasrabadi, Vahid
    Chalmers.
    Linderholt, Andreas
    Linnaeus University, Faculty of Science and Engineering, School of Engineering.
    Abrahamsson, Thomas
    Chalmers.
    Model calibration of locally nonlinear structures using information from sub and super harmonic responses2012In: International Conference on Noise and Vibration Engineering 2012 (ISMA 2012): Proceedings of a meeting held 17-19 September 2012, Leuven, Belgium. Including USD 2012 / [ed] P. Sas, S. Jonckheere & D. Moens, Leuven: Katholieke Universiteit Leuven, Department of Mechanical Engineering , 2012, 2451-2464 p.Conference paper (Other academic)
    Abstract [en]

    Large linear finite element models are commonly used in industry to represent global structural behavior and such models are often validated by use of data from vibrational tests. The validated models serve to predict the structural responses due to dynamic loads. Hence, it is important to have models that are able to represent the structural dynamics within the given operating envelope. When test data show proof of non linear behavior, a linear model may not be able to represent the dynamics well enough and thus a modification of the model is required. The main part of the structure may have a linear characteristic whereas localized physical processes can be sources of the observed nonlinearities. Model calibration of such locally nonlinear structures is studied in this paper. Specifically, the calibration process including the selection of appropriate data to be used for calibration of the model parameters chosen is treated. Here, synthetic test data stemming from a model of the Ecole Centrale de Lyon (ECL) nonlinear benchmark are used.

  • 11.
    Chen, Yousheng
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Yaghoubi, Vahid
    Chalmers University of Technology.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Model Calibration of a Locally Non-linear Structure Utilizing Multi Harmonic Response Data2014In: Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 / [ed] Gaetan Kerschen, Springer, 2014, 97-109 p.Conference paper (Refereed)
    Abstract [en]

    Model correlation and model calibration using test data are natural ingredients in the process of validating computational models. Here, model calibration for the important sub-class of non-linear systems consisting of structures dominated by linear behavior having presence of local non-linear effects is studied. The focus is on the selection of uncertain model parameters together with the forming of the objective function to be used for calibration. To give precise estimation of parameters in the presence of measurement noise, the objective function data have to be informative with respect to the parameters chosen. Also, to get useful data the excitation force is here designed to be multi-harmonic since steady-state responses at the side frequencies are shown to contain valuable information for the calibration process. In this paper, test data from a replica of the Ecole Centrale de Lyon (ECL) nonlinear benchmark together with steady-state solutions stemming from calculations using the Multi-Harmonic Balancing method are used for illustration of the proposed model calibration procedure.

  • 12.
    Linderholt, Andreas
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Chen, Yousheng
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology.
    Time Domain Dynamic Simulations of Locally Nonlinear Large-Scale Systems2016In: Presented at Aerospace technology congress: The Swedish Aeronautics Congress in a Globaised world, October 11-12, 2016, 2016Conference 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.

  • 13.
    Linderholt, Andreas
    et al.
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Chen, Yousheng
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Orlowitz, Esben
    University of Southern Denmark, Denmark.
    Brandt, Anders
    University of Southern Denmark, Denmark.
    A study of the coupling between test data accuracy and life prediction2014In: Proceedings of ISMA2014 including USD2014: 26th International Conference on Noise and Vibration Engineering and 5th International Conference on Uncertainty in Structural Dynamics, 15-17 September 2014, Leuven, Belgium / [ed] P. Sas, H. Denayer & D. Moens, Leuven: Katholieke Universiteit Leuven, Department of Mechanical Engineering , 2014, 415-428 p.Conference paper (Other academic)
    Abstract [en]

    Computational models are regularly used to predict responses and the life of structures. Often, the computational models are validated by use of data stemming from vibration testing. However, the damping values are almost always taken from test alone and hence cannot be validated. Experimentally obtained damping values are well known to have relatively high uncertainties. In the present paper we investigate how life time predictions are affected by modal damping uncertainties by simulating responses due to random loads. The simulations are based on experimental modal analysis results of a structure consisting of two steel plates which are joined by bolts, tested under several different experimental support conditions. The results illustrate the importance of careful consideration of experimental test setups, as these can strongly affect the obtained damping estimates. This, in turn, can lead to large differences in predicted life time.

  • 14.
    Yaghoubi, Vahid
    et al.
    Chalmers University of Technology, sweden.
    Chen, Yousheng
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Linderholt, Andreas
    Linnaeus University, Faculty of Technology, Department of Mechanical Engineering.
    Abrahamsson, Thomas
    Chalmers University of Technology, Sweden.
    Locally Non-Linear Model Calibration Using Multi Harmonic Responses: Applied on Ecole de Lyon Non-Linear Benchmark Structure2013In: Topics in Nonlinear Dynamics, Volume 1: Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013 / [ed] Gaetan Kerschen, Alex Carrella, Douglas Adams, Springer, 2013, 113-123 p.Conference paper (Other academic)
    Abstract [en]

    In industry, linear FE-models commonly serve to represent global structural behavior. However, when test data are availa-ble they may show evidence of nonlinear dynamic characteristics. In such a case, an initial linear model may be judged being insufficient in representing the dynamics of the structure. The causes of the non-linear characteristics may be local in nature whereas the major part of the structure is satisfactorily represented by linear descriptions. Although the initial model then can serve as a good foundation, the parameters needed to substantially increase the model’s capability of representing the real structure are most likely not included in the initial model. Therefore, a set of candidate parameters controlling nonlinear effects, opposite to what is used within the vast majority of model calibration exercises, have to be added. The selection of the candidates is a delicate task which must be based on engineering insight into the structure at hand.The focus here is on the selection of the model parameters and the data forming the objective function for calibration. An over parameterized model for calibration render in indefinite parameter value estimates. This is coupled to the test data that should be chosen such that the expected estimate variances of the chosen parameters are made small. Since the amount of information depends on the raw data available and the usage of them, one possibility to increase the estimate precision is to process the test data differently before calibration. A tempting solution may be to simply add more test data but, as shown in this paper, the opposite could be an alternative; disregarding low excessive data may make the remaining data better to dis-criminate between different parameter settings.Since pure mono-harmonic excitation during test is an abnormality, the excitation force is here designed to contain sub and super harmonics besides the fundamental one. Further, the steady-state responses at the side frequencies are here shown to contain most valuable information for the calibration process of models of locally nonlinear structures.Here, synthetic test data stemming from a model representing the Ecole Centrale de Lyon (ECL) nonlinear benchmark are used for illustration. The nonlinear steady state solutions are found using iterative linear reverse path state space calculations. The model calibration is here based on nonlinear programming utilizing several parametric starting points. Candidates for starting points are found by the Latin Hypercube sampling method. The best candidates are selected as starting points for optimization.

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