scholarly journals On the residual motion in damped vibrating systems

2013 ◽  
Vol 40 (1) ◽  
pp. 5-15
Author(s):  
Ranislav Bulatovic
Keyword(s):  
Positive Definite ◽  
Symmetric Positive Definite ◽  
Damping Matrix ◽  
Stiffness Matrices ◽  
Residual Motion ◽  
Vibrating Systems ◽  
Symmetric Systems

In this paper, linear vibrating systems, in which the inertia and stiffness matrices are symmetric positive definite and the damping matrix is symmetric positive semi-definite, are studied. Such a system may possess undamped modes, in which case the system is said to have residual motion. Several formulae for the number of independent undamped modes, associated with purely imaginary eigenvalues of the system, are derived. The main results formulated for symmetric systems are then generalized to asymmetric and symmetrizable systems. Several examples are used to illustrate the validity and application of the present results.

A Symmetric Positive Definite Inverse Vibration Problem With Underdamped Modes

1995 ◽  
Author(s):  
Ladislav Starek ◽  
Daniel J. Inman ◽  
Deborah F. Pilkev
Keyword(s):  
Positive Definite ◽  
Inverse Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Symmetric Positive Definite ◽  
Vibration Problem ◽  
Damped System ◽  
Stiffness Matrices ◽  
Inverse Eigenvalue ◽  
Inverse Vibration ◽  
Vector Differential Equation

Abstract This manuscript considers a symmetric positive definite inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices. The inverse problem of interest here is that of determining real symmetric, positive definite coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include the definiteness of the coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a non-proportional damped system which will have symmetric positive definite coefficient matrices.

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.

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