Sadegh Rahrovani

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.

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.

An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 2: Symplecticity and Global Error Analysis

In the first part of this study an exponential integration scheme for computing solutions of large stiff systems was presented. It was claimed that the integrator is particularly efficient in large-scale problems with localized nonlinearity when compared to general-purpose methods. Theoretical aspects of the proposed method were investigated. The method computational efficiency was increased by using an approximation of the Jacobian matrix. This was achieved by combining the proposed integration scheme with the developed methods for model reduction, in order to treat the large nonlinear problems. In this second part geometric and structural properties of the presented integration algorithm are examined and preservation of these properties such as area in the phase plane and also energy consistency are investigated. The error analysis is given through small scale examples and the efficiency and accuracy of the proposed exponential integrator is investigated through a large-scale size problem that originates from a moving load problem in railway mechanics. The superiority of the proposed method in sense of computational efficiency, for large-scale problems particularly system with localized nonlinearity, has been demonstrated, comparing the results with classical Runge–Kutta approach.

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.

An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 1: Theoretical Investigation

In the first part of this study an exponential integration scheme for computing solutions of large stiff systems is introduced. It is claimed that the integrator is particularly effective in large-scale problems with localized nonlinearity when compared with the general purpose methods. A brief literature review of different integration schemes is presented and theoretical aspect of the proposed method is discussed in detail. Computational efficiency concerns that arise in simulation of large-scale systems are treated by using an approximation of the Jacobian matrix. This is achieved by combining the proposed integration scheme with the developed methods for model reduction, in order to treat the large nonlinear problems. In the second part, geometric and structural properties of the presented integrator are examined and the preservation of these properties such as area in the phase plane and also energy consistency are investigated. The error analysis is given through small scale examples and the efficiency and accuracy of the proposed exponential integrator is investigated through a large-scale size problem that originates from a moving load problem in railway mechanics. The superiority of the proposed method in sense of computational efficiency, for large-scale problems particularly system with localized nonlinearity, has been demonstrated, comparing the results with classical Runge–Kutta approach.

Bayesian Treatment of Spatially-Varying Parameter Estimation Problems via Canonical BUS

The inverse problem of identifying spatially-varying parameters, based on indirect/incomplete experimental data, is a computationally and conceptually challenging problem. One issue of concern is that the variation of the parameter random field is not known a priori, and therefore, it is typical that inappropriate discretization of the parameter field leads to either poor modelling (due to modelling error) or ill-condition problem (due to the use of over-parameterized models). As a result, classical least square or maximum likelihood estimation typically performs poorly. Even with a proper discretization, these problems are computationally cumbersome since they are usually associated with a large vector of unknown parameters. This paper addresses these issues, through a recently proposed Bayesian method, called Canonical BUS. This algorithm is considered as a revisited formulation of the original BUS (Bayesian Updating using Structural reliability methods), that is, an enhancement of rejection approach that is used in conjunction with Subset Simulation rare-event sampler. Desirable features of CBUS to treat spatially-varying parameter inference problems have been studied and performance of the method to treat real-world applications has been investigated. The studied industrial problem originates from a railway mechanics application, where the spatial variation of ballast bed is of our particular interest.

Stability Limitations in Simulation of Dynamical Systems with Multiple Time-Scales

This paper focuses on the stability properties of a recently proposed exponential integrator particularly in simulation of highly oscillatory systems with multiple time-scales. The linear and nonlinear stability properties of the presented exponential integrator have been studied. We illustrate this with the Fermi–Pasta–Ulam (FPU) problem, a highly oscillatory nonlinear system known as a test benchmark for multi-scale time integrators. This example is also illustrative when studying the numerical resonance and algorithmic instability in the multi-time-stepping (MTS) methods, such as in exponential and/or trigonometric integration schemes, since it has no external input force and therefore no real physical resonance.