TOWARDS IDENTIFICATION OF A GENERAL MODELOF DAMPING S. Adhikari and J. Woodhouse Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ (U.K.) ABSTRACT Characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics. In spite of a large amount of research, understanding of damp- ing mechanisms is not well developed. A major reason for this is that unlike inertia and stiffness forces it is not in general clear what are the state variables that govern the damping forces. The most common approach is to use ‘viscous damp- ing’ where the instantaneous generalized velocities are the only relevant state variables. However, viscous damping by no means the only damping model within the scope of linear analysis. Any model which makes the energy dissipation func- tional non-negative is a possible candidate for a valid damping model. This paper is devoted to develop methodologies for identification of such general damping models responsible for energy dissipation in a vibrating structure. The method uses experimentally identified complex modes and complex natural frequencies and does not a-priori assume any fixed damping model (eg. , viscous damping) but seeks to determine param- eters of a general damping model described by the so called ‘relaxation function’. The proposed method and several re- lated issues are discussed by considering a numerical exam- ple of a linear array of damped spring-mass oscillators. NOMENCLATURE M mass matrix K stiffness matrix G(τ ) matrix of kernel functions y(t) generalized coordinates ωj , xj j -th undamped frequency and mode λj , zj j -th complex frequency and mode ˆ uj , ˆ vj real and imaginary parts of zj μ relaxation parameter of damping C damping coefficient matrix ℜ(•), ℑ(•) real and imaginary parts of (•) (•) ∗ complex conjugation ˆ (•) measured quantity of (•) 1 INTRODUCTION In the context of experimental modal analysis, by far the most common damping model is so-called ‘viscous damping’, a lin- ear model in which the instantaneous generalized velocities are the only relevant state variables that determine damping. This model was first introduced by Rayleigh [1] via his famous ‘dissipation function’, a quadratic expression for the energy dissipation rate with a symmetric matrix of coefficients, the ‘damping matrix’. Complex modes arise with viscous damping when it is non-proportional [2] . Practical experience in modal testing also shows that most real-life structures possess com- plex modes − as Sestieri and Ibrahim [3] have put it ‘ ... it is ironic that the real modes are in fact not real at all, in that in practice they do not exist, while complex modes are those practically identifiable from experimental tests. This implies that real modes are pure abstraction, in contrast with complex modes that are, therefore, the only reality!’ Although with viscous damping models, linear systems show complex modal behaviour, it is by no means the only damping model within the scope of linear analysis. Any causal model which makes the energy dissipation functional non-negative is a possible candidate for a damping model. Unfortunately, most of the studies on damping identification reported in the literature consider viscous damping only. Such a priori selec- tion of viscous damping in identification procedure rules out any possibility of detecting the need for a different damping model. In this paper we consider identification of non-viscous damping models in the context of general multiple degrees-of- freedom linear systems. A key issue in identifying non-viscous damping is− what non- viscous damping model to consider whose parameters have to be identified? There have been detailed studies on material damping or specific structural components. Lazan [4] , Bert [5] and Ungar [6] have given excellent accounts of different mathe- matical methods for modeling damping in (solid) material and their engineering applications. The book by Nashif et al [7] presents more recent studies in this area. Currently a large body of literature can be found on damping in composite ma- terials where many researchers have evaluated a materials specific damping capacity (SDC). Baburaj and Matsuzaki [8] and the references therein give an account of research in this area. Other than material damping a major source of energy dissipation in a vibrating structure is the structural joints [9, 10] . In many cases these damping mechanisms turn out be locally non-linear, requiring an equivalent linearization technique for a global analysis [11] . One way to address this problem is to use non-viscous damp- ing models which depend on the past history of motion via convolution integrals over kernel functions. The equations of motion of free vibration of a linear system with this type of