Exact Receptances of Nonproportionally Damped Dynamic Systems

1993 ◽  
Vol 115 (1) ◽  
pp. 47-52 ◽  
Author(s):  
B. Yang
Keyword(s):  
Exact Solution ◽  
Asymptotic Stability ◽  
Dynamic Systems ◽  
Matrix Inversion ◽  
Iteration Procedure ◽  
Exact Method ◽  
Modal Data ◽  
Damping Matrix ◽  
Solution Methods ◽  
Damped Systems

This paper presents an exact method for evaluating the receptances of nonproportionally damped dynamic systems. Based on a decomposition of the damping matrix, an iteration procedure is developed. This method does not require matrix inversion, and completely eliminates the error caused by undamped modal data. Compared to the existing exact solution methods, the proposed method can save more time in the computation. Also, the study yields a convenient way to determine the asymptotic stability of nonproportionally damped systems.

Further Results on Iterative Approaches to the Determination of the Response of Nonclassically Damped Systems

1993 ◽  
Vol 60 (1) ◽  
pp. 235-239 ◽  
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Rate Of Convergence ◽  
Iterative Methods ◽  
Dynamic Systems ◽  
Positive Definite ◽  
Asymptotic Convergence ◽  
Iterative Schemes ◽  
Symmetric Positive Definite ◽  
Damping Matrix ◽  
Damped Systems

Iterative schemes for obtaining the response of nonclassically damped dynamic systems have been shown to converge when the damping matrix possesses certain specific characteristics. This paper extends those previous results which pertain to symmetric, positive-definite damping matrices. The paper considers a more general decomposition of the damping matrix, and extends the applicability of these iterative methods to all positive definite, symmetric damping matrices. This decomposition is governed by a parameter α. Conditions on α under which convergence is guaranteed are provided. Estimates of a which yield the fastest asymptotic convergence as well as estimates of the value of this asymptotic rate of convergence are also provided.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

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.

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.

Exact solution methods for the multi-period vehicle routing problem with due dates

2019 ◽  
Vol 110 ◽  
pp. 148-158 ◽  
Author(s):  
Homero Larrain ◽  
Leandro C. Coelho ◽  
Claudia Archetti ◽  
M. Grazia Speranza
Keyword(s):  
Exact Solution ◽  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Due Dates ◽  
Routing Problem ◽  
Solution Methods

Observability and Distinguishability Properties and Stability of Arbitrarily Fast Switched Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Najah F. Jasim
Keyword(s):  
Asymptotic Stability ◽  
Dynamic Systems ◽  
Switched Systems ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
External Disturbances ◽  
Fast Switching ◽  
Nonlinear Switched Systems ◽  
Arbitrary Switching

This paper addresses sufficient conditions for asymptotic stability of classes of nonlinear switched systems with external disturbances and arbitrarily fast switching signals. It is shown that asymptotic stability of such systems can be guaranteed if each subsystem satisfies certain variants of observability or 0-distinguishability properties. In view of this result, further extensions of LaSalle stability theorem to nonlinear switched systems with arbitrary switching can be obtained based on these properties. Moreover, the main theorems of this paper provide useful tools for achieving asymptotic stability of dynamic systems undergoing Zeno switching.

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.

Nonclassically Damped Dynamic Systems: An Iterative Approach

1990 ◽  
Vol 57 (2) ◽  
pp. 423-433 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Ramin S. Esfandiari
Keyword(s):  
Dynamic Systems ◽  
Iterative Scheme ◽  
Sufficient Conditions ◽  
Iterative Technique ◽  
Computationally Efficient ◽  
Rigorous Results ◽  
Second Order Differential Equations ◽  
Time Histories ◽  
Broad Variety ◽  
Damped Systems

This paper presents a new, computationally efficient, iterative technique for determining the dynamic response of nonclassically damped, linear systems. Such systems often arise in structural and mechanical engineering applications. The technique proposed in this paper is heuristically motivated and iteratively obtains the solution of a coupled set of second-order differential equations in terms of the solution to an uncoupled set. Rigorous results regarding sufficient conditions for the convergence of the iterative technique have been provided. These conditions encompass a broad variety of situations which are commonly met in structural dynamics, thereby making the proposed iterative scheme widely applicable. The method also provides new physical insights concerning the decoupling procedure and shows why previous approximate approaches for uncoupling nonclassically damped systems have led to large inaccuracies. Numerical examples are presented to indicate that, even under perhaps the least ideal conditions, the technique converges rapidly to provide the exact time histories of response.

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