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.

Symmetric Model Correction of Mechanical Systems

1993 ◽  
Author(s):  
Marca Lam ◽  
Daniel J. Inman ◽  
Andreas Kress
Keyword(s):  
Analytical Model ◽  
Model Updating ◽  
Symmetric Model ◽  
Simple Method ◽  
Modal Data ◽  
Model Correction ◽  
Systems Model ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Damped Systems

Abstract This work examines the model updating problem for simple nonconservative proportionally damped systems. Model correction, also called model updating, refers to the practice of adjusting an analytical model until the model agrees with measured modal data. The specific case examined here assumes that natural frequencies and modal damping ratios are available from vibration tests and that the measured data disagrees in part with the modal data predicted by an analytical model. Most model correction schemes tend to produce updated damping and stiffness matrices which are asymmetric. The simple method presented here focuses on retaining the desired symmetry in the updated model.

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.

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.

Updating the Damping Matrix Using Frequency Response Data

1995 ◽  
Author(s):  
Cécile Reix ◽  
Alain Gerard ◽  
Christian Tombini
Keyword(s):  
Frequency Response ◽  
Weighted Least Squares ◽  
Element Model ◽  
Mode Shapes ◽  
Minor Effect ◽  
Sensitivity Matrix ◽  
Damping Matrix ◽  
Stiffness Matrices ◽  
Weighted Least Squares Estimation ◽  
A Minor

Abstract This paper presents a method for the updating of the damping matrix of a linear dynamic system. For this dynamic study, it is presumed that the characteristic mass and stiffness matrices are perfectly known thanks to the updating of the experimental and calculed frequencies and mode shapes as from a finit element model. Furthermore, it is accepted that damping has only a minor effect on the frequencies and mode shapes of a structure (a hypothesis that has been verified for structures with low damping). It is proposed to adjuste the coefficients of the [D] hysteretic damping matrix as from the superposition of the experimental and analytical Frequency Response Functions (FRF). The frequencies and mode shapes are extracted from the solutions of the caracteristic equation (3) resulting from the classic dynamic equation. An analytical FRF is calculed and then used to establish the sensitivity matrix, translating the influence of the updating parameters on the FRF. To update the [D] matrix, we use a non-linear weighted least squares estimation.

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.

Structural Modification Software Development for Design Optimization

1995 ◽  
Author(s):  
Xiaoye Gu ◽  
Alison B. Flatau
Keyword(s):  
Structural Response ◽  
Acoustic Power ◽  
Modal Parameters ◽  
Constrained Layer Damping ◽  
Experimental Frequency ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Structural Responses ◽  
Damping Materials ◽  
Modal Mass

Abstract This paper presents a method for obtaining structural matrices from experimental frequency response function (FRF) data and using these structural matrices to predict the response of the structure to modifications at various locations. The approach taken is designed for subsequent use in optimizing structural modifications to efficiently reduce radiated acoustic power. A series of programs were written for identifying the structural matrices (mass matrices, stiffness matrices and damping matrices) from the measured FRF data. These matrices are used to obtain the modified response of the structure resulting from adding linear springs at different locations on the structure. Experimental results from a beam are presented to verify these programs. Work is in progress on extending this method to incorporate modifications to the structure produced by constrained-layer damping materials. The programs for obtaining the structural matrices and the structural response are composed of approaches used by several prior authors. Potter and Richardson’s [1,2] method is used for obtaining the relative modal parameters (modal mass, modal stiffness and modal damping). Luk and Mitchell’s [3,4] pseudo-inverse method is employed to obtain the structural matrices for cases when the number of modes measured is much less than the number of test points. A method for deriving the absolute value of modal parameters from the measured FRF data is also developed using modal analysis theory. Linear springs are added at various positions to modify the structure. The structural matrices are used to predict the modified structural responses scaled to displacement per unit force. A series of linear spring modifications are modeled and implemented experimentally to verify these programs.

On Frequency-Domain System Identification

1997 ◽  
Author(s):  
Eric L. Blades ◽  
Roy R. Craig
Keyword(s):  
System Identification ◽  
Frequency Domain ◽  
Mode Shapes ◽  
Finite Element Models ◽  
Identification Algorithm ◽  
Parameter System ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Modal Model ◽  
Vibration Tests

Abstract Over the past three decades vibration tests have frequently been conducted on structures and machines for the purpose of validating (updating) finite element models. Such testing is usually referred to as experimental modal analysis. The results of the vibration test are generally presented as a modal model consisting of natural frequencies, mode shapes, and modal damping factors, and the algorithms most frequently used to reduce the test data to modal-model form are time-domain algorithms. Alternatively, direct-parameter system identification may be performed, in which case the direct-parameter model consists of system mass, damping, and stiffness matrices. This paper discusses several features of a new frequency-domain direct-parameter system identification algorithm. Simulations based on a 52-DOF “payload simulator” are used to illustrate the performance of this algorithm.

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.

Dynamics of Multibody Tracked Vehicles Using Experimentally Identified Modal Parameters

1996 ◽  
Vol 118 (3) ◽  
pp. 499-507 ◽  
Author(s):  
Toshikazu Nakanishi ◽  
Xuegang Yin ◽  
A. A. Shabana
Keyword(s):  
Equations Of Motion ◽  
Kinematic Chain ◽  
General Purpose ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Tracked Vehicles ◽  
Revolute Joints ◽  
Modal Damping ◽  
Modal Mass ◽  
The Individual

The mode shapes, frequencies, and modal mass and stiffness coefficients of multibody systems such as tracked vehicles can be determined using experimental identification techniques. In multibody simulations, however, knowledge of the modal parameters of the individual components is required, and consequently, a procedure for extracting the component modes from the mode shapes of the assembled system must be used if experimental modal analysis techniques are to be used with general purpose multibody computer codes. In this investigation, modal parameters (modal mass, modal stiffness, modal damping, and mode shapes), which are determined experimentally, are employed to simulate the nonlinear dynamic behavior of a multibody tracked vehicle which consists of interconnected rigid and flexible components. The equations of motion of the vehicle are formulated in terms of a set of modal and reference generalized coordinates, and the theoretical basis for extracting the component modal parameters of the chassis from the modal parameters of the assembled vehicle is described. In this investigation, the track of the vehicle is modeled as a closed kinematic chain that consists of rigid links connected by revolute joints, and the effect of the chassis flexibility on the motion singularities of the track is examined numerically. These singularities which are encountered as the result of the change in the track configuration are avoided by using a deformable secondary joint instead of using the loop-closure equations.

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.

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