A Study of Errors Induced by Decoupling Approximation in Damped Linear Vibratory Systems

Abstract

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.

On The Decoupling Approximation in Damped Linear Systems

“…The following example is provided by Ajavakom (2005). Consider two fourdegree-of-freedom systems of the form (7).…”

A remark about the decoupling approximation of damped linear systems

“…Example 1. Consider two four-degree-of-freedom systems (Ajavakom, 2005) of the form (8). System 1 is governed by € q þ D 1 _ q þ X 1 q ¼ gðtÞ where the spectral matrix, the modal damping matrix, and the excitation are given by X 1 ¼ diag½ 3:95 2 3:98 2 4:00 2 4:10 2 ; ð19Þ…”

Asymptotic expansion of slightly coupled modal dynamic transfer functions

“…A thorough review of the literature related to non-diagonal modal damping matrices has been recently presented in [2,3]. We therefore restrict the portrayal of the panorama to a brief description of three major philosophies.…”

“…The modal approach is classically used in order (i) to reduce the number of unknowns (as m5n) and (ii) to uncouple the set of equations (1). The normal modes of vibration are computed in such a way that U T MU and U T KU are precisely diagonal, see (2), but nothing can in principle be said about D. Nevertheless, owing to the lack of knowledge about damping in structures, the damping matrix C is commonly assumed to take a particular formulation, as the most famous Rayleigh damping, allowing goal (ii) to be reached. However, in some circumstances as the presence of dash-pots or aerodynamic loading, a specific mathematical modeling of damping forces is available.…”