Majid Khorsand Vakilzadeh

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.

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.

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.

Calibration and Cross-Validation of a Car Component Model Using Repeated Testing

Repeated testing has been made both in the sense of testing multiple test pieces of the same type and in the sense of repeated tests on the same test piece by 13 testers at different occasions. Between the individual tests, the test subject has been dismounted from the test stand and the sensors have been re-calibrated. Statistical evaluation of these tests gives information about the spread that can be expected in modal tests. The SIMO test data and data statistics are used for traditional validation and cross-validation of a finite element model of the car component under test. Planning of the sensor layout has been made prior to testing with the use of the nominal finite element model. A model calibration is made prior to the model validation. The finite element model size is over 20,000 degrees-of-freedom and involves two calibration parameters. The work has been made as part of a structural dynamics model validation Master’s course. An open-source Matlab application has been used for calibration, validation and cross-validation.

Automated modal parameter estimation using correlation analysis and bootstrap sampling

The estimation of modal parameters from a set of noisy measured data is a highly judgmental task, with user expertise playing a significant role in distinguishing between estimated physical and noise modes of a test-piece. Various methods have been developed to automate this procedure. The common approach is to identify models with different orders and cluster similar modes together. However, most proposed methods based on this approach suffer from high-dimensional optimization problems in either the estimation or clustering step. To overcome this problem, this study presents an algorithm for autonomous modal parameter estimation in which the only required optimization is performed in a three-dimensional space. To this end, a subspace-based identification method is employed for the estimation and a non-iterative correlation-based method is used for the clustering. This clustering is at the heart of the paper. The keys to success are correlation metrics that are able to treat the problems of spatial eigenvector aliasing and nonunique eigenvectors of coalescent modes simultaneously. The algorithm commences by the identification of an excessively high-order model from frequency response function test data. The high number of modes of this model provides bases for two subspaces: one for likely physical modes of the tested system and one for its complement dubbed the subspace of noise modes. By employing the bootstrap resampling technique, several subsets are generated from the same basic dataset and for each of them a model is identified to form a set of models. Then, by correlation analysis with the two aforementioned subspaces, highly correlated modes of these models which appear repeatedly are clustered together and the noise modes are collected in a so-called Trashbox cluster. Stray noise modes attracted to the mode clusters are trimmed away in a second step by correlation analysis. The final step of the algorithm is a fuzzy c-means clustering procedure applied to a three-dimensional feature space to assign a degree of physicalness to each cluster. The proposed algorithm is applied to two case studies: one with synthetic data and one with real test data obtained from a hammer impact test. The results indicate that the algorithm successfully clusters similar modes and gives a reasonable quantification of the extent to which each cluster is physical.

On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems

Abstract In this paper, a review of Gramian-based minimal realization algorithms is presented, several comments regarding their properties are given and the ill-condition and efficiency that arise in balancing of large-scale realizations is being addressed. A new algorithm to treat non-minimal realization of very large second-order systems with dense clusters of close eigenvalues is proposed. The method benefits the effectiveness of balancing techniques in treating of non-minimal realizations in combination with the computational efficiency of modal techniques to treat large-scale problems.

A Metric for Modal Truncation in Model Reduction Problems Part 1: Performance and Error Analysis

The strength of the modal based reduction approach resides in its simplicity, applicability to treat moderate-size systems and also in the fact that it preserves the original system pole locations. However, the main restriction has been in the lack of reliable techniques for identifying the modes that dominate the input-output relationship. To address this problem, an enhanced modal dominancy approach for reduction of second-order systems is presented. Briefly stated, a modal reduction approach is combined with optimality considerations such that the difference between the frequency response function of the full and reduced modal model is minimized in \(\mathcal{H}_{2}\) sense. A modal ranking process is performed without solving Lyapunov equations. In the first part of this study, a literature survey on different model reduction approaches and a theoretical investigation of the modified modal approach is presented. The error analysis of the proposed dominancy metric is carried out. Furthermore, the performance of the method is validated for a lightly damped structure and the results are compared with other dominancy metrics. Finally the optimality of the obtained reduced model is discussed and the results are compared with the optimum solution.

A Metric for Modal Truncation in Model Reduction Problems Part 2: Extension to Systems with High-Dimensional Input Space

In the first part of this study, a theoretical investigation of an improved modal approach and a complete error analysis of the proposed modal dominancy metric were presented. In this part the problem of metric non-uniqueness for systems with multiple eigenvalues is described and a method to circumvent this problem based on spatial distribution of either the sensors or the actuators is proposed. This technique is implemented using QR factorization without solving Lyapunov equations. Moreover, the method is improved such that it is able to use the information extracted from spectral properties of the input. Also in order to make the method more effective, information extracted from the input internal structure is incorporated in the modal ranking process. It is shown that this improvement is particularly effective in problems with high-dimensional input and/or output space such as in distributed loading and moving load problems. Finally the performance of the method is validated for a high order system subjected to a high-dimensional input force. That originates from a railway track moving load problem.

Towards an Automatic Modal Parameter Estimation Framework: Mode Clustering

The estimation of modal parameters from a set of measured data is a highly judgmental task, with user expertise playing a significant role for distinguishing between physical and spurious modes. However, it can be very tedious especially in situations when the data is difficult to analyze. This study presents a new algorithm for mode clustering as a preliminary step in a multi-step algorithm for performing physical mode selection with little or no user interaction. The algorithm commences by identification of a high-order model from estimated frequency response functions to collect all the important characteristics of the structure in a so-called library of modes. This often results in the presence of spurious modes which can be detected on the basis of the hypothesis that spurious modes are estimated with a higher level of uncertainty comparing to physical modes. Therefore, we construct a series of data using a simple random sampling technique in order to obtain a set of linear systems using subspace identification. Then, their similar modes are grouped together using a new correlation criterion, which is called Modal Observability Correlation (MOC). An illustrative example shows the efficiency of the proposed clustering technique and also demonstrates its capability to dealing with inconsistent data.