EXTRACTION OF UNDAMPED NORMAL MODES AND NONDIAGONAL MODAL DAMPING MATRIX FROM DAMPED SYSTEM REALIZATION PARAMETERS

1993 ◽  
Author(s):  
K. ALVIN ◽  
K. PARK ◽  
L. PETERSON
Keyword(s):  
Normal Modes ◽  
Damping Matrix ◽  
Damped System ◽  
System Realization ◽  
Modal Damping

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.

A novel energy dissipation outrigger system with rotational inertia damper

2018 ◽  
Vol 21 (12) ◽  
pp. 1865-1878 ◽  
Author(s):  
Liangkun Liu ◽  
Ping Tan ◽  
Haitao Ma ◽  
Weiming Yan ◽  
Fulin Zhou
Keyword(s):  
Damping Ratio ◽  
Mass Parameter ◽  
Inertial Force ◽  
Rotational Inertia ◽  
Analysis Model ◽  
Damping Force ◽  
Damped System ◽  
Modal Damping ◽  
Assembly Technique ◽  
Excellent Control

Rotational inertia damper, a novel damper, possessing the advantage of displacement amplification, has been employed in outrigger system for seismic mitigation. The equivalent analysis model composed by a uniform cantilever beam and an equivalent spring was proposed to simulate the rotational inertia damper outrigger system, by which the corresponding dynamic characteristic equation was derived based on numerical assembly technique. To gain the response of the damped system, finite element method and state space method have been utilized. Finally, the results show that the pseudo-undamped natural frequency ratios and system modal damping ratios are significantly influenced by stiffness parameter of the exterior column, while the mass parameter of the rotational inertia damper has little effect on them. The optimal damping ratio can be acquired for one mode, but it may be worse for the other mode in the same position equipping rotational inertia damper. Furthermore, numerical simulation results for the typical earthquake records have verified that the rotational inertia damper outrigger has excellent control performance in displacement as well as acceleration. A good agreement between damping force and equivalent force also suggests that the damping force of rotational inertia damper is predominant and the inertial force has no significant effect on the structure.

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.

Vibration Reduction in Large Flexible Systems Through Independent Modal Control

2011 ◽  
Author(s):  
Francesco Braghin ◽  
Simone Cinquemani ◽  
Ferruccio Resta
Keyword(s):  
Acoustic Noise ◽  
Dynamic Performance ◽  
Active Vibration Control ◽  
Spillover Effects ◽  
Flexible System ◽  
Damping Matrix ◽  
Modal Damping ◽  
Active Vibration ◽  
The Stability ◽  
Modal Controller

Many systems have, by their nature, a small damping and therefore they are potentially subjected to dangerous vibration phenomena. The aim of active vibration control is to contain this phenomenon, increasing the damping of the system without changing its natural frequencies and vibration modes. A control of this type can improve the dynamic performance, reduce the vibratory phenomenon (and the resulting acoustic noise) and increase the fatigue strength of the system. The paper introduces a new approach to the synthesis of a modal controller to suppress vibrations in structures: it turns from the traditional formulation of the problem showing how the performance of the designed controller can be evaluated through the analysis of the resulting modal damping matrix of the controlled system. Such analysis allows to evaluate spillover effects, due to the presence of un-modeled modes, the stability of the control and the consequent effectiveness in reducing vibration. The ability to easily manage this information allows the synthesis of an efficient modal controller. Theoretical aspects are supported by experimental applications on a large flexible system.

Stall Nonlinear Flutter of Wind Turbine Blade Modeled as Thin-Walled Composite Beam Integrated with Structural Damping

2013 ◽  
Vol 291-294 ◽  
pp. 496-500
Author(s):  
Yong Sheng Ren ◽  
Ting Rui Liu
Keyword(s):  
Structural Model ◽  
Structural Damping ◽  
Aeroelastic Stability ◽  
Damping Matrix ◽  
Modal Damping ◽  
Strip Theory ◽  
Analytical Formulas ◽  
Thin Walled ◽  
The Stability ◽  
Domain Integration

The effects of structural damping on the aeroelastic stability have been investigated for composite thin-walled blade. Structural model of the composite thin-walled blade exhibits bending-bending-twist coupling, with accounting for the presence of pretwist angle. The aerodynamic model used in the present paper is the differential dynamic stall model developed at ONERA. The structural damping of the blade is predicted based on the analytical formulas of the modal damping of thin-walled composite structure. The effect of structural damping on aeroelastic stability is taken into account by using proportional damping matrix. By means of Galerkin method, the nonlinear aeroelastic equations are reduced to ordinary equations. The general aerodynamic forces are obtained from strip theory. The resulting equations are then linearized for small perturbation about the equilibrium point and the stability characteristics are investigated through eigenvalue analysis and time domain integration.

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.

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.

On the Approximate Solution of Nonclassically Damped Linear Systems

1993 ◽  
Vol 60 (3) ◽  
pp. 695-701 ◽  
Author(s):  
J. H. Hwang ◽  
F. Ma
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Error Bound ◽  
Large Scale ◽  
Substantial Reduction ◽  
Approximation Error ◽  
Computational Effort ◽  
Common Procedure ◽  
Damping Matrix ◽  
Modal Damping

A common procedure in the solution of a nonclassically damped linear system is to neglect the off-diagonal elements of the associated modal damping matrix. For a large-scale system, substantial reduction in computational effort is achieved by this method of decoupling the system. In the present paper, the error introduced by disregarding the off-diagonal elements is evaluated, and a quadrature formula for the approximation error is derived. A tight error bound is then obtained. In addition, an effective scheme to improve the accuracy of the approximate solution is outlined.

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