General isospectral flows for linear dynamic systems

Garvey, Seamus D.; Prells, Uwe; Friswell, Michael I.; Chen, Zheng

doi:10.1016/j.laa.2003.12.027

Cited by 20 publications

(10 citation statements)

References 16 publications

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“…This has been done in [15, sect. 4.2] and [8]. Garvey et al [8] go even further with these block symmetric pencils, using them as a foundation for defining a new class of isospectral transformations on matrix polynomials.…”

Section: Other Constructions Of Block Symmetric Linearizationsmentioning

confidence: 99%

“…4.2] and [8]. Garvey et al [8] go even further with these block symmetric pencils, using them as a foundation for defining a new class of isospectral transformations on matrix polynomials. Since Lancaster's construction of pencils is so different from ours there is no a priori reason to expect any connection between his pencils and the pencils in DL(P ).…”

Section: Other Constructions Of Block Symmetric Linearizationsmentioning

confidence: 99%

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Symmetric Linearizations for Matrix Polynomials

Higham

Mackey

et al. 2007

SIAM J. Matrix Anal. & Appl.

115

169

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Abstract.A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil-a process known as linearization. Two vector spaces of pencils L 1 (P ) and L 2 (P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P . For arbitrary polynomials we show that every pencil in DL(P ) is block symmetric and we obtain a convenient basis for DL(P ) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P ) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P ). When P is symmetric, we show that the symmetric pencils in L 1 (P ) comprise DL(P ), while for Hermitian P the Hermitian pencils in L 1 (P ) form a proper subset of DL(P ) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of DL(P ) together with some new, more concise proofs.

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Section: Other Constructions Of Block Symmetric Linearizationsmentioning

confidence: 99%

Section: Other Constructions Of Block Symmetric Linearizationsmentioning

confidence: 99%

Symmetric Linearizations for Matrix Polynomials

Higham

Mackey

et al. 2007

SIAM J. Matrix Anal. & Appl.

115

169

View full text Add to dashboard Cite

show abstract

“…The above τ k yields the intermediate results (3.19) and (3.20) into (3.17) now provides the sought-after expression for the components 20) of the synchronized mode. By time-shifting every component, we again obtain equation (3.11) with…”

Section: Phase Synchronization Of Two Real and Distinct Eigensolutionsmentioning

confidence: 88%

“…Recall that two systems with identical eigenvalues and multiplicities are termed strictly isospectral. Since the property of being strictly isospectral is reflexive, transitive and symmetric, strictly isospectral systems generate an equivalence class [20]. It is easy to verify that system (1.1) and the decoupled systems (4.3) are strictly isospectral regardless of the pairing of real eigensolutions during phase synchronization.…”

Section: Nonlinearity and Non-uniqueness In Decouplingmentioning

confidence: 98%

“…Based upon the notion of structure-preserving transformations (SPT), Garvey and others [9,10,[18][19][20] introduced an alternative method for decoupling homogeneous systems. There, algorithms employing linear coordinate transformations in a higher dimensional space (the state space) are utilized to compute a real and second-order diagonal system.…”

Section: Introductionmentioning

confidence: 99%

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The Transformation of Second-Order Linear Systems into Independent Equations

Morzfeld¹,

Ma²,

Parlett³

2011

SIAM J. Appl. Math.

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Linear second-order ordinary differential equations arise from Newton's second law combined with Hooke's law and are ubiquitous in mechanical and civil engineering. Perhaps the most prominent example is a mathematical model for small oscillations of particles around their equilibrium positions. However, second-order systems also find applications in such diverse areas as chemical engineering, structural dynamics, linear systems theory or even economics. Very large second-order systems appear, for example, in mathematical modeling of complex structures by finite-element methods.In general, any system of second-order equations is coupled. Each equation is linked to at least one of its neighbors and the solution of one of the equations requires the solution of all equations. The "classical decoupling problem" is concerned with the elimination of coordinate coupling in linear dynamical systems. The decoupling transforms the system of equations into a collection of mutually independent equations so that each equation can be solved without solving any other equation. In "The Theory of Sound" in 1894, Lord Rayleigh already expounded on the significance of system decoupling. Since then, the problem has attracted the attention of many researchers.Mathematically, the system of differential equations is defined by three coefficient matrices. The equations are coupled unless all three matrices are diagonal. The "classical decoupling problem" is thus equivalent to the problem of simultaneous conversion of the coefficient matrices into diagonal forms. Current theory emphasizes simultaneous diagonalization of the coefficient matrices by equivalence or similarity transformations. However, it has been shown that no time-invariant linear transformations will decouple every second-order system. Even partial decoupling, i.e. simultaneous conversion of the coefficient matrices into upper triangular forms, is not ensured with time-invariant linear transformations.The purpose of this work is to present a general method and algorithm to decouple any second-order linear system (possessing symmetric and non-symmetric coefficients). The theory exploits the parameter "time," characteristic of a dynamical 2 system. The decoupling is achieved by a real, invertible, but generally nonlinear mapping. This mapping simplifies to a real, linear time-invariant transformation when the coefficient matrices can be simultaneously diagonalized by a similarity transformation. A state-space reformulation of the mapping is also derived. In homogeneous systems the configuration-space decoupling transformation is real, linear and time-invariant when cast in state space. In non-homogeneous systems, both the configuration and associated state transformations are nonlinear and depend continuously on the excitation. The theory is illustrated by several numerical examples. Two applications in earthquake engineering demonstrate the utility of the decoupling approach. i I would like to thank my family for all of their love and support.

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Bibliography

2013

Structural Dynamic Analysis With Generalized Damping Models

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General isospectral flows for linear dynamic systems

Cited by 20 publications

References 16 publications

Symmetric Linearizations for Matrix Polynomials

Symmetric Linearizations for Matrix Polynomials

The Transformation of Second-Order Linear Systems into Independent Equations

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