A numerical study of damping

Abstract

Diagonal damping matrices were computed for three systems which have non‐proportional damping matrices. These diagonal damping matrices were computed on three bases, as follows: 1. After normalizing the equations of motion by the modal matrix, the diagonal terms are retained ignoring the non‐diagonal terms. 2. Diagonal damping matrix is established by the optimization algorithm which minimizes the mean square error of the frequency response. 3. Diagonal damping is determined from the normalized differential equation by matching the peaks of the coupled and uncoupled system. The frequency responses for the three cases of one of the three systems are presented together with a comparison of the energy dissipation.

The decoupling of defective linear dynamical systems in free motion

“…As a result, system (1) may not be decoupled by classical modal analysis in general. Consequently, much attention has been given to the development of decoupling approximation schemes [5][6][7][8][9][10][11][12][13][14][15][16] for analyzing non-classically damped systems, along with the so-called coupling or non-proportionality indices [17][18][19][20][21][22][23][24][25][26][27][28] that quantify their degree of coordinate coupling. Of course, one may also consider decoupling in the 2n-dimensional state space.…”

The Time Dimension: Semi-Discretization of Field and Dynamic Problems

“…Indeed a form of this type is necessary for the use of modal decomposition, although other generalizations are possible [39,40]. From the definition of ξ i , the ratio to its critical ratio in Eq.…”

Large errors in approximate decoupling of nonclassically damped linear second-order systems under harmonic excitations

“…in which for each i = 1, 2 ..... n, the quantity ai denotes the sum of the absolute values of the off-diagonal elements of C on its ith row, namely, j=l j~i (10) and the functions V: (0, 1) ~ IR and ~ (0, 1) --* IR are given by…”

“…When C is not diagonal, system (1) may be solved for x by either direct numerical integration or by the method proposed by Foss [9]. A simple and commonly used approximate technique of solving system (3) is to neglect the offdiagonal elements of C, and hence to decouple the system equations into a set of n decoupled equations (see, e.g., Meirovitch [1], and Thompson et al [10]). The solution of the resulting decoupled equations leads to an approximate solution for system (1).…”