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.

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.

scholarly journals The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix EquationX+AHX-1A+BHX-1B=I

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Iterative Methods ◽  
Matrix Equation ◽  
Positive Definite ◽  
Numerical Results ◽  
Iterative Schemes ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

We present two inversion-free iterative methods for computing the maximal positive definite solution of the equationX+AHX-1A+BHX-1B=I. We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. We also present some numerical results to compare our proposed methods with some previously developed inversion-free techniques for solving the same matrix equation.

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.

Sign in / Sign up

Export Citation Format

Share Document