An Index of Damping Non-Proportionality for Discrete Vibrating Systems

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.

Section: Defective Oscillatory Motion (Complex Eigenvalues)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The decoupling of defective linear dynamical systems in free motion

Kawano¹,

Morzfeld²,

Ma³

2011

“…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

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.

“…Over the years, various types of decoupling approximation were employed in the analysis of damped systems [12][13][14][15][16][17][18][19][20][21]. Different indices of coupling were also introduced to quantify coordinate coupling [22][23][24][25][26][27][28][29]. However, a solution to the ''classical decoupling problem'' has not been reported in the open literature.…”

Section: Problem Statementmentioning

confidence: 99%

“…However, elements of the modal vector s j (t) in Eq. (27) generally possess different phase angles f jk . As a consequence, the system components do not pass through their equilibrium positions at the same time.…”

Section: Non-classically Damped Modesmentioning

confidence: 99%

The decoupling of damped linear systems in oscillatory free vibration

Imam

Morzfeld

2009

TitleThe decoupling of damped linear systems in oscillatory free vibration AbstractThe purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in oscillatory free vibration. Based upon an exposition of how viscous damping causes phase drifts in the components of a linear system, the concept of non-classically damped modes of vibration is introduced. These damped modes are real and physically excitable. By synchronizing the phase angles in each damped mode, a time-varying transformation is constructed to decouple damped oscillatory free vibration. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the first part of a solution to the ''classical decoupling problem'' of linear systems. r