Frequency Response of Nonproportionally Damped, Lumped Parameter, Linear Dynamic Systems

Abstract

The frequency response of nonproportionally (or nonclassically or non-Raleigh) damped linear lumped mass systems is discussed here. The exact method, via Laplace transforms, is presented and the practical difficulties revealing the necessity of approximate methods are emphasized. Modal analysis is used to transform the governing equations of motion to the respective modal coupled equations of motion. The Laplace transform is performed to transfer to the frequency domain. The modal coupling is then analyzed through coupling terms. A criterion in the form of nonproportionality indices is developed, in order to measure the extent of the modal coupling and to predict the error introduced by neglecting this coupling either partially or completely. An attempt to interpret the frequency spectrum of the modal coupling is also made. A specific application to a six degree of freedom system, with two sets of closely spaced undamped natural frequencies and moderate damping is given as an illustrative example. Useful information about the accuracy, the applicability, and the advantages of the proposed method over the exact method, as well as over the common procedure of ignoring the modal coupling, are derived.

The decoupling of defective linear dynamical systems in free motion

“…Consequently, it is generally not possible to extract an order n transformation from the state equation (28) in a concise form, such as in Eqs. (19) and (45), that recovers the response xðtÞ directly. However, in principle, one may isolate such a transformation from Eq.…”

“…Thus, after solving the decoupled system (2), defined by the coefficient matrices (12), with initial conditions (18), the response xðtÞ of system (1) may be recovered directly via transformation (19). All parameters required for decoupling are obtained through solution of the quadratic eigenvalue problem (7).…”

“…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 decoupling of damped linear systems in oscillatory free vibration

“…Over the years, various types of decoupling approximation were employed in the analysis of damped systems [12][13][14][15][16][17][18][19][20][21]. Different indices of coupling were also introduced to quantify coordinate coupling [22][23][24][25][26][27][28][29]. However, a solution to the ''classical decoupling problem'' has not been reported in the open literature.…”

The decoupling of damped linear systems in free or forced vibration

“…Over the years, various types of decoupling approximation were employed in the analysis of damped systems [10][11][12][13][14][15][16][17][18][19]. Different indices of coupling were also introduced to quantify coordinate coupling [20][21][22][23][24][25][26][27][28]. However, a solution to the ''classical decoupling problem'' has not been reported in the open literature.…”