Thomas Abrahamsson

Brief paper: Constrained state-space system identification with application to structural dynamics

Constrained identification of state-space models representing structural dynamic systems is addressed. Based on physical insight, transfer function constraints are formulated in terms of the state-space parametrization. A simple example shows that a method tailored for this application, which utilizes the non-uniqueness of a state-space model, outperforms the classic sequential quadratic programming method in terms of robustness and convergence properties. The method is also successfully applied to real experimental data of a plane frame structure.

Spread in Modal Data Obtained from Wind Turbine Blade Testing

This paper presents a pre-study for an on-going research project in experimental dynamic substructuring, initiated by the SEM Substructuring Focus Group. The focus group has selected a small wind turbine, the Ampair 600W, to serve as test bed for the studies. The turbine blades are considered in this study. A total of 12 blades have been tested for modal properties in a free-free configuration. The data has been acquired and analysed by students participating in the undergraduate course “Structural Dynamics – Model Validation” at Chalmers University of Technology. Each blade was tested by different students as part of their required course work to account for spread in modal properties between the blades. A subset of the blades were tested independently multiple times to account for variability in the test setup. Furthermore, correlation analysis of test data was made with Finite Element model eigensolution data of the blade.

The Modal Observability Correlation as a Modal Correlation Metric

The historical development of the Modal Assurance Criterion (MAC) originated from the need of a correlation metric for comparingexperimental modal vectors, estimated from measured data, to eigenvectors that have been determined from finite element calculation. For systems with well separated eigenvalues with many system degrees-of-freedom (DOF) represented in the eigenvectors it is normally easy to distinguish eigenvectors associated to different eigenvalues by low MAC correlation numbers. However, for eigenvectors with a sparse DOF sampling it may be hard to distinguish between vectors by MAC correlation numbers. To reduce the problem of distinguishing between eigensolutions, this paper advocates the use of a new correlation metric based on the observability matrix of the diagonal state-space realization. This is instead of using a metric based on the eigenvectors only.

Modal Reduction Based on Accurate Input-Output Relation Preservation

An eigenmode based model reduction technique is proposed to obtain low-order models which contain the dominant eigenvalue subspace of the full system. A frequency-limited interval dominancy is introduced to this technique to measure the output deviation caused by deflation of eigenvalues from the original system in the frequency range of interest. Thus, the dominant eigensolutions with effective contribution can be identified and retained in the reduced-order model. This metric is an explicit formula in terms of the corresponding eigensolution. Hence, the reduction can be made at a low computational cost. In addition, the retained low-order model does not contain any uncontrollable and unobservable eigensolutions. The performance of the created reduced-order models, in regard to the approximation error, is examined by applying three different input signals; unit-impulse, unit-step and linear chirp.

Automated Modal Analysis Based on Frequency Response Function Estimates

Given measured data as estimated frequency responses of a quasi-linear system, there is a variety of system identification methods that identify a state-space model that gives good correlation to the data. Such methods are the N4SID and the PolyMAX methods. Using these methods, a key problem is to select the proper model order. In this work we investigate a method for the automatic detection of proper model order. The method is based on the statistical evaluation of an ensemble of state-space models all identified from the same basic set of frequency response functions, but with different realizations based on a bootstrapping scheme. We apply the method to real test data.

Calibration and Validation of a Car Subframe Finite Element Model Using Frequency Responses

A finite element model of a car front subframe has been calibrated against test data. Stepped-sine testing has been used to give frequency response function estimates on an ensemble of seemingly identical subframes. Therefore, the deviation between test data and simulation results can be compared in a meaningful way by the outcome of model calibration and cross-validation. Emphasis has been put on the preparation of the test pieces for high fidelity testing and on bettering the chances of getting a calibration outcome that provides insight into the physical processes that govern the subframe dynamics. The front subframe model has more than 200,000 degrees-of-freedom and 17 model calibration parameters. The efficiency of the calibration procedure under these conditions is reported. To achieve efficiency, a calibration with a smooth deviation metric is used together with a damping equalization method that eliminates the need for matching of experimental and analytical eigenmodes. The method is combined with surrogate model frequency response evaluation based on model reduction for increased speed. The Matlab based open-domain software tool FEMcali that employs the Levenberg-Marquardt minimizer with randomized starts has been used for calibration and an unregularized Gauss-Newton minimizer has been used in the cross-validation.

A Parallel Solution Method for Structural Dynamic Response Analysis

With the continuous improvements of technology in and around multi-core CPU:s and GPU:s there is a strong desire to exploit this technology in its full potential. For structural dynamics problems, the domain decomposition is a very mature technique that is well adapted to parallel computations in multi-core machines as it is almost trivially parallelizable. However, competing alternatives with model reduction without parallel computation has also reached an extremely high level of maturity and are thus highly competitive. In this paper, a domain decomposition method, in a procedure named the split-stitch-spread (S3) procedure, is proposed to do transient analysis of large finite element models in parallel. In the method, the structure splits into model substructures with elastic interfacial substructures coupling them together. Each of them can be sent to different computer cores to do time discretization. The model substructures stitch to each other by using interfacial forces and as a result, the systems’ state sequence will be obtained. The solution can then be spread into the substructures and response quantities can be evaluated in parallel processing. The method is applied to a multi-story building subjected to earthquake loading and the results are compared with mode displacement method as a model reduction method with focus on computational efficiency.

Modeling and Calibration of Small-Scale Wind Turbine Blade

The SEM Substructuring Focus Group has chosen an Ampair 600W wind turbine to be used as a test bed in the continued efforts to further experimental and experimental-analytical substructure coupling techniques. To assess such coupling techniques, validated models of the parts, the substructures considered, play a crucial role. This paper describes the modeling, calibration and validation of a Finite Element (FE) model of a blade for the test bed turbine. Orthotropic composite material modeling is used to set up the model, which is calibrated and validated based on results from an ambitious measurement campaign including both non-destructive testing for dynamic properties and dedicated destructive tests for deduction of material properties. The measurement campaign is carefully described in the paper.

An efficient simulation method for structures with local nonlinearity

In general, simulating the nonlinear behavior of systems needs a lot of computational effort. Since researchers in different fields are increasingly targeting nonlinear systems, attempts toward fast nonlinear simulation have attracted much interest in recent years. Examples of such fields are system identification and system reliability. In addition to efficiency, the algorithmic stability and accuracy need to be addressed in the development of new simulation procedures. In this paper, we propose a method to treat localized nonlinearity in a structure in an efficient way. The system will be separated by a linearized part and a nonlinear part that is considered as external pseudo forces that act on the linearized system. The response of the system is obtained by iterations in which the pseudo forces are updated. Since the method is presented in linear state space model form, all manipulations that are made on these, like similarity transformations and model reduction, can easily be exploited. To do numerical integration, time-stepping schemes like the triangular hold interpolation can be used to the advantage. We demonstrate the efficiency, stability and accuracy of the method on numerical examples.

FEM Calibration with FRF Damping Equalization

A finite element model calibration procedure that uses frequency response function data and relies on damping equalization is presented. In this, the dampings of the finite element model and the corresponding experimental model are set equal before calibration. The damping equalization is made to avoid the mode pairing problem that normally needs to be solved in other model updating procedures. It is demonstrated that one particular use of frequency response data gives a calibration deviation metric that is smooth in the variation of model parameters and give a large radius of convergence to the calibration minimum. The method is combined with model reduction for increased speed and employs a minimizing procedure that employs randomized multiple starting points in the parameter space to get to the calibration solution. The performance of the calibration procedure is demonstrated by two numerical examples.