Diagonal Dominance and the Decoupling Approximation in Damped Discrete Linear Systems

2007 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Diagonal Element ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Damped System ◽  
Modal Damping ◽  
Principal Coordinates ◽  
Discrete Linear Systems ◽  
Modal Coupling ◽  
Decoupling Approximation ◽  
Damped Systems

The principal coordinates of a non-classically damped linear system are coupled by nonzero off-diagonal element of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely accepted that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively believed that the more diagonal the modal damping matrix, the less will be the errors in the decoupling approximation. Two quantitative measures are proposed in this paper to measure the degree of being diagonal dominant in modal damping matrices. It is demonstrated that, over a finite range, errors in the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is presented. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.

Characteristics of Modal Coupling in Nonclassically Damped Systems Under Harmonic Excitation

1994 ◽  
Vol 61 (1) ◽  
pp. 77-83 ◽  
Author(s):  
I. W. Park ◽  
J. S. Kim ◽  
F. Ma
Keyword(s):  
Harmonic Excitation ◽  
Frequency Separation ◽  
Diagonal Dominance ◽  
Normal Coordinates ◽  
Damping Matrix ◽  
Damped System ◽  
Natural Modes ◽  
Modal Damping ◽  
Modal Coupling ◽  
Damped Systems

The normal coordinates of a nonclassically damped system are coupled by nonzero off-diagonal elements of the modal damping matrix. The purpose of this paper is to study the characteristics of modal coupling, which is amenable to a complex representation. An analytical formulation is developed to facilitate the evaluation of modal coupling. Contrary to widely accepted beliefs, it is shown that enhancing the diagonal dominance of the modal damping matrix or increasing the frequency separation of the natural modes need not diminish the effect of modal coupling. The effect of modal coupling may even increase. It is demonstrated that, within the practical range of engineering applications, neither diagonal dominance of the modal damping matrix nor frequency separation of the natural modes would be sufficient for neglecting modal coupling.

Some Remarks About the Decoupling Approximation of Damped Linear Systems

2009 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Linear Systems ◽  
Approximation Error ◽  
Driving Frequency ◽  
Finite Range ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Modal Damping ◽  
Diagonally Dominant ◽  
Decoupling Approximation ◽  
Damped Systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. These unexpected drifts in errors due to the decoupling approximation can be observed at any driving frequency. Small off-diagonal elements in the modal damping matrix may not be sufficient to ensure small errors due to the decoupling approximation. Error-criteria based solely upon diagonal dominance of the modal damping matrix cannot be accurate.

A Study of Errors Induced by Decoupling Approximation in Damped Linear Vibratory Systems

2005 ◽  
Author(s):  
N. Ajavakom ◽  
F. Ma
Keyword(s):  
Coupling Effect ◽  
Vibratory System ◽  
Damping Matrix ◽  
Modal Damping ◽  
Principal Coordinates ◽  
Decoupling Approximation ◽  
Damped Systems

It is well known that an undamped linear vibratory system can be decoupled through transformation to principal coordinates. In the presence of damping, coordinate decoupling occurs only if the system is classically damped. Upon modal transformation, the system generally remains coupled by the off-diagonal elements of its modal damping matrix. A common approximation in the analysis of nonclassically damped systems is to ignore the off-diagonal elements of the modal damping matrix, which is equivalent to neglecting coupling of the principal coordinates. This procedure is termed the decoupling approximation. Intuitively, the errors of decoupling approximation should be small if the off-diagonal elements of the modal damping matrix are small. Contrary to this widely accepted belief, an example is provided to demonstrate that this criterion is not sufficient for decoupling approximation. In fact, coupling effect can even increase as the off-diagonal elements of the modal damping matrix decrease in magnitude. Discussion and explanation are provided as to why the errors increase when the modal damping matrix becomes increasingly diagonal.

Characterization of Modal Coupling in Nonclassically Damped Systems

1992 ◽  
Author(s):  
F. Ma ◽  
I. W. Park ◽  
J. S. Kim
Keyword(s):  
Large Scale ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Frequency Separation ◽  
Common Procedure ◽  
Damping Matrix ◽  
Natural Modes ◽  
Modal Damping ◽  
Modal Coupling ◽  
Damped Systems

Abstract A common procedure in the solution of a nonclassically damped linear system is to neglect the off-diagonal elements of the associated damping matrix. For a large-scale system, substantial reduction in computational effort is achieved by this method of decoupling the system. Clearly, the decoupling approximation is valid only if modal coupling can somehow be neglected. The purpose of this paper is to study the characteristics of modal coupling, which is amenable to a complex representation. An analytical formulation that facilitates the evaluation of modal coupling is developed. Contrary to widely accepted beliefs, it is shown that neither frequency separation of the natural modes nor strong diagonal dominance of the modal damping matrix would be sufficient to suppress the sometimes significant effect of modal coupling.

Approximate Solution of Nonclassically Damped Systems

1991 ◽  
Author(s):  
F. Ma ◽  
J. H. Hwang
Keyword(s):  
Approximate Solution ◽  
Large Scale ◽  
Computational Effort ◽  
Frequency Separation ◽  
Common Procedure ◽  
Damping Matrix ◽  
Complex Modes ◽  
Modal Coupling ◽  
Order Of Magnitude ◽  
Damped Systems

Abstract In analyzing a nonclassically damped linear system, one common procedure is to neglect those damping terms which are nonclassical, and retain the classical ones. This approach is termed the method of approximate decoupling. For large-scale systems, the computational effort at adopting approximate decoupling is at least an order of magnitude smaller than the method of complex modes. In this paper, the error introduced by approximate decoupling is evaluated. A tight error bound, which can be computed with relative ease, is given for this method of approximate solution. The role that modal coupling plays in the control of error is clarified. If the normalized damping matrix is strongly diagonally dominant, it is shown that adequate frequency separation is not necessary to ensure small errors.

Rayleigh Quotient and Dissipative Systems

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
A. Srikantha Phani ◽  
S. Adhikari
Keyword(s):  
Perturbation Theory ◽  
Rayleigh Quotient ◽  
Dissipative Systems ◽  
Diagonal Dominance ◽  
Dissipative Forces ◽  
Damping Matrix ◽  
Modal Damping ◽  
Nonproportional Damping ◽  
Vibrating Systems

Rayleigh quotients in the context of linear, nonconservative vibrating systems with viscous and nonviscous dissipative forces are studied in this paper. Of particular interest is the stationarity property of Rayleigh-like quotients for dissipative systems. Stationarity properties are examined based on the perturbation theory. It is shown that Rayleigh quotients with stationary properties exist for systems with proportional viscous and nonviscous damping forces. It is also shown that the stationarity property of Rayleigh quotients in the case of nonproportional damping (viscous and nonviscous) is conditional upon the diagonal dominance of the modal damping matrix.

The Proportional-Damping Matrix of Arbitrarily Damped Linear Mechanical Systems

2002 ◽  
Vol 69 (5) ◽  
pp. 649-656 ◽  
Author(s):  
J. Angeles ◽  
S. Ostrovskaya
Keyword(s):  
Stiffness Matrix ◽  
Mechanical Systems ◽  
Least Square ◽  
Simultaneous Diagonalization ◽  
Single Degree Of Freedom ◽  
Practical Applications ◽  
Single Degree ◽  
Damping Matrix ◽  
Damped System ◽  
Damped Systems

The vibration of linear mechanical systems with arbitrary damping is known to pose challenging problems to the analyst, for these systems cannot be analyzed with the techniques pertaining to their undamped counterparts. It is also known that a class of damped systems, called proportionally damped, can be analyzed with the same techniques, which mimic faithfully those of single-degree-of-freedom systems. For this reason, in many instances the system at hand is assumed to be proportionally damped. Nevertheless, this assumption is difficult to justify on physical grounds in many practical applications. What this assumption brings about is a damping matrix that admits a simultaneous diagonalization with the stiffness matrix. Proposed in this paper is a decomposition of the damping matrix of an arbitrarily damped system allowing the extraction of the proportionally damped component, which, moreover, approximates optimally the original damping matrix in the least-square sense. Finally, we show with examples that conclusions drawn from the proportionally damped approximation of an arbitrarily damped system can be dangerously misleading.

Identification of Multi-Degree of Freedom Systems With Nonproportional Damping Using the Resonant Decay Method

2004 ◽  
Vol 126 (2) ◽  
pp. 298-306 ◽  
Author(s):  
Steven Naylor ◽  
Michael F. Platten ◽  
Jan R. Wright ◽  
Jonathan E. Cooper
Keyword(s):  
Measurement Noise ◽  
Mode Shapes ◽  
Damping Matrix ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Rigid Body Modes ◽  
Nonproportional Damping ◽  
Modal Mass ◽  
Damped Systems ◽  
Modal Space

This paper describes an extension of the force appropriation approach which permits the identification of the modal mass, damping and stiffness matrices of nonproportionally damped systems using multiple exciters. Appropriated excitation bursts are applied to the system at each natural frequency, followed by a regression analysis in modal space. The approach is illustrated on a simulated model of a plate with discrete dampers positioned to introduce significant damping nonproportionality. The influence of out-of-band flexible and rigid body modes, imperfect appropriation, measurement noise and impure mode shapes is considered. The method is shown to provide adequate estimates of the modal damping matrix.

Study on Equivalent Damping Ratio of Shenzhen Baoan International Airport T3 Terminal

2012 ◽  
Vol 446-449 ◽  
pp. 871-877
Author(s):  
Yu Chen Yang ◽  
Lei Gu ◽  
Zhong Yi Zhu ◽  
Kai Qin ◽  
Lin Zhang
Keyword(s):  
Modal Analysis ◽  
Damping Ratio ◽  
Earthquake Response ◽  
Structural System ◽  
Equivalent Damping ◽  
Damping Matrix ◽  
Modal Damping ◽  
Modal Coupling ◽  
Response Equation ◽  
Energy Theory

Today, the structures constituted by different materials, increase more and more, especially the lower part is concrete, the upper is steel. For this type of structural system, modal damping matrix is non-diagonal matrix, the earthquake response equation is coupled on the modal damping matrix, the modal coupling of the non-proportional damping system leads to the traditional real modal analysis methods not be directly applied. For such structure, the changes of the damping ratio are analyzed in this article. Finally, the equivalent damping ratio of Shenzhen Airport is obtained using the energy theory.

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