The Quadratic Eigenvalue Problem

Tisseur, Françoise; Meerbergen, Karl

doi:10.1137/s0036144500381988

Cited by 1,173 publications

(959 citation statements)

References 124 publications

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“…For these systems, a quadratic eigenvalue problem [4] must be solved to obtain a coherent modal description of the problem, using left and right complex modes. In this paper, we propose to extend the symmetric properness condition to non symmetric systems.…”

Section: Introductionmentioning

confidence: 99%

On the properness condition for modal analysis of non-symmetric second-order systems

Ouisse

Foltête

2011

Mechanical Systems and Signal Processing

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To cite this version:Morvan Ouisse, Emmanuel Foltete. On the properness condition for modal analysis of non symmetric second order systems. Mechanical Systems and Signal Processing, Elsevier, 2011, 25 (2) Emmanuel FOLTÊTE AbstractNon symmetric second order systems can be found in several engineering contexts, including vibroacoustics, rotordynamics, or active control. In this paper, the notion of properness for complex modes is extended to the case of non self-adjoint problems. The properness condition is related to the ability of a set of complex modes to represent in an exact way the behavior of a physical second order system, meaning that the modes are the solutions of a quadratic eigenvalue problem whose matrices are those of a physical system. This property can be used to identify the damping matrices which may be difficult to obtain with mathematical modeling techniques. The first part of the paper demonstrates the properness condition for non symmetric systems in general. In the second part, the authors propose a methodology to enforce that condition in order to perform an optimal reconstruction of the "closest" physical system starting from a given basis complex modes. The last part is dedicated to numerical and experimental illustrations of the proposed methodology. A simulated academic test case is first used to investigate the numerical aspects of the method. A physical application is then considered in the context of rotordynamics. Finally, an experimental test case is presented using a structure with an active control feedback. An extension of the LSCF identification technique is also introduced to identify both left and right complex mode shapes from measured frequency response functions.

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Section: Introductionmentioning

confidence: 99%

On the properness condition for modal analysis of non-symmetric second-order systems

Ouisse

Foltête

2011

Mechanical Systems and Signal Processing

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“…In this case R max = 0.052 m, f max approximate 3 GHz. According to the recursion step introduced by (17), two solutions are presented in Table 1. Solution 1 carries segmentation from f L , whereas Solution 2 performs reversely.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…Compared to the mature level of the techniques for the linear system [17], the development of second-order techniques is a relatively new topic, but has raised growing interest, including Second Order Arnoldi [18], Second order dominant pole identification [19] and Quadratic Arnoldi algorithm [11]. In this paper, Quadratic Arnoldi algorithm (Q-arnoldi) are adopted as the numerical method, in terms of its remarkable feature of favoring the convergence to the particular eigenvalues using the strategies of implicit restarting or purging [17].…”

Section: Dominant Pole Identification Using Q-arnolimentioning

confidence: 99%

“…In this paper, Quadratic Arnoldi algorithm (Q-arnoldi) are adopted as the numerical method, in terms of its remarkable feature of favoring the convergence to the particular eigenvalues using the strategies of implicit restarting or purging [17]. Known as a memory-efficient and structure-preserving, the core step of Q-arnoldi is to project a large quadratic system into a subspace of much smaller dimension via Krylov recurrence relation [11].…”

Section: Dominant Pole Identification Using Q-arnolimentioning

confidence: 99%

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An Automatic Model Order Reduction of a Uwb Antenna System

Zhang

Lee²

2010

PIER

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Abstract-This paper proposes an efficient and automatic means of achieving a reduced model of a transfer function for UWB antenna design. According to the formulation of a transfer function, we have derived two factors, which are critical in determining the radiation pattern and input impedance respectively. Their special formula allow us to establish a reduced model automatically using the Model Order Reduction (MOR) techniques of a second order system. The process is free of any human factors and suitable to any antenna systems, thus enabling a direct and efficient interface with the optimization process in the design of a UWB antenna system. In addition, the proposed way of establishing a transfer function of the whole antenna system has successfully cascaded the entire system into separate subsystems, thus offering deeper insights in analyzing a UWB antenna system.

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“…In many applications, the study of spectral properties of matrix polynomials is a central topic [8,21,22,24,34,38]. A common technique to find the eigenvalues of a matrix polynomial is converting to a linear problem using the following method.…”

Section: Linearizations Of Matrix Polynomialsmentioning

confidence: 99%

Duality of matrix pencils, Wong chains and linearizations

Noferini

Poloni

2015

Linear Algebra and its Applications

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We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L 1 , L 2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials.

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The Quadratic Eigenvalue Problem

Cited by 1,173 publications

References 124 publications

On the properness condition for modal analysis of non-symmetric second-order systems

On the properness condition for modal analysis of non-symmetric second-order systems

An Automatic Model Order Reduction of a Uwb Antenna System

Duality of matrix pencils, Wong chains and linearizations

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