Examination of the validity of proportional damping approximations with two further numerical indices

Nair, S. Sasidharan; Singh, Rajendra

doi:10.1016/0022-460x(86)90274-9

Cited by 24 publications

(9 citation statements)

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“…Thus, after solving the decoupled system (2), defined by the coefficient matrices (12), with initial conditions (18), the response xðtÞ of system (1) may be recovered directly via transformation (19). All parameters required for decoupling are obtained through solution of the quadratic eigenvalue problem (7).…”

Section: Decoupling Of Non-defective Systemsmentioning

confidence: 99%

“…As a result, system (1) may not be decoupled by classical modal analysis in general. Consequently, much attention has been given to the development of decoupling approximation schemes [5][6][7][8][9][10][11][12][13][14][15][16] for analyzing non-classically damped systems, along with the so-called coupling or non-proportionality indices [17][18][19][20][21][22][23][24][25][26][27][28] that quantify their degree of coordinate coupling. Of course, one may also consider decoupling in the 2n-dimensional state space.…”

Section: Introductionmentioning

confidence: 99%

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The decoupling of defective linear dynamical systems in free motion

Kawano¹,

Morzfeld²,

Ma³

2011

Journal of Sound and Vibration

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a b s t r a c tIt was demonstrated in two earlier papers that there exists a real, linear, time-varying transformation that decouples any non-defective linear dynamical system in free vibration in the configuration space. As an extension of this work, the present paper represents the first systematic effort to decouple defective systems. It is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues. While any defective system can be decoupled with the eigenvalues kept invariant, the geometric multiplicities of these eigenvalues may not be preserved. Several numerical examples are provided to illustrate the theoretical developments.

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Section: Decoupling Of Non-defective Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The decoupling of defective linear dynamical systems in free motion

Kawano¹,

Morzfeld²,

Ma³

2011

Journal of Sound and Vibration

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“…Over the years, various types of decoupling approximation were employed in the analysis of damped systems [3,8,11,13,16,21,26,29,51,57]. Different indices of coupling were also introduced to quantify coordinate coupling [2,33,34,43,45,47,54,62]. However, a solution to the "classical decoupling problem" has not been reported in the open literature.…”

Section: The Classical Decoupling Problemmentioning

confidence: 99%

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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“…Over the years, various types of decoupling approximation were employed in the analysis of damped systems [10][11][12][13][14][15][16][17][18][19]. Different indices of coupling were also introduced to quantify coordinate coupling [20][21][22][23][24][25][26][27][28]. However, a solution to the ''classical decoupling problem'' has not been reported in the open literature.…”

Section: Problem Statementmentioning

confidence: 99%

The decoupling of damped linear systems in free or forced vibration

Morzfeld

Imam

2010

Journal of Sound and Vibration

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a b s t r a c tThe purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the ''classical decoupling problem.'' Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.

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Examination of the validity of proportional damping approximations with two further numerical indices

Cited by 24 publications

References 1 publication

The decoupling of defective linear dynamical systems in free motion

The decoupling of defective linear dynamical systems in free motion

The Transformation of Second-Order Linear Systems into Independent Equations

The decoupling of damped linear systems in free or forced vibration

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