ELSEVIER Soil Dynamicsand EarthquakeEngineering 13 (1994) 197-212 © 1994 Elsevier Science Limited Printed in Great Britain. All rights reserved 0267-7261/94/$07.00 Analytical and numerical techniques for the dynamic analysis of non-classically damped linear systems Federico Perotti Department of Civil Engineering, University of Brescia, Via Branze 38, 25100 Brescia, Italy Communicated by G. Borm (Received 11 December 1992; revised version received 14 February 1994; accepted 15 February 1994) The paper deals with the use of complex modal analysis for computing the response of linear systemshaving non-proportional damping. The problem of the approximate evaluation of complex modes is first addressed: a second-order perturbation technique, proposed by other researchers, is adopted and modified in view of the application to the analysis of systems having a large number of degrees of freedom. Frequency domain algorithms for the computation of modal response are then tested and a technique for reducing the computational effort due to the performance of inverse Fourier Transforms is proposed. Two examples of application are finally given. INTRODUCTION Considerable research activity has been devoted, during the last years, to the dynamic analysis of linear systems having non-classical viscous damping; this situation arises, in fact, in many practical cases involving, for example, the analysis of soil-structure or structure- equipment systems. Even though the dynamic behaviour of non- classically damped systems can be studied by means of direct integration of the system equations, most of the quoted research work has been focused on modal analysis; this is not only because modal shapes and frequencies are of engineering interest themselves, but also for the efficiency demonstrated by modal decom- position in the treatment of many types of dynamical loadings. Earthquakes and gusty winds, for example, are responsible for dynamic loads which are of relatively long duration but tend to excite a limited number of normal coordinates; these properties, beside the inherent advantage of equations decoupling, make the modal approach particularly attractive for their treatment. Classical modal analysis can be rigorously applied only to systems with proportional damping; when this is not the case, approximate solutions, 1,2 based on the use of undamped modal shapes and on the forced diagonalization of the transformed damping matrix, are adopted in most practical cases. It has been 197 demonstrated, however, that these solutions can some- times lead to unacceptable errors in response compu- tations. 3'4 An alternative strategy,5 based as well on the use of classical modes, consists of direct integration of a reduced set of coupled normal coordinates; with this approach, however, some of the advantages of modal analysis (e.g. the possibility of response spectrum analysis for the seismic case) are lost. For the quoted reasons most of the recent research activity in this field has been devoted to 'exact' modal analysis, that is to the use of the normal modes of the damped system; general response calculation procedures have been derived6-11 both for deterministic and for random loading, while several research works have been focused on the more specific case of structure- equipment interacting systems. 12-15 As it concerns seismic actions, response-spectrum-based modal tech- niques have been treated by several authors. 16 18 The most important drawback of exact modal analysis of damped systems lies in the lack of numerical efficiency; damped systems eigenpairs, in fact, are complex-valued quantities, while the size of the eigenvalue problem (for an n-degree-of-freedom system) is 2n and the bandedness properties of the original structural matrices are usually lost. Moreover, the efficiency of the numerical procedures available for handling this kind of eigenvalue problem (involving matrices which are not symmetrical and positive