Abstract Characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics. The most common approach is to use “viscous damping” where the instantaneous generalized velocities are the only relevant state variables that affect damping forces. However, viscous damping is by no means the only damping model within the scope of linear analysis. Any model which makes the energy dissipation functional non-negative is a possible candidate for a valid damping model. This paper, and its companion (see pp. 63–88 of this issue), are devoted to developing methodologies for identification of such general damping models responsible for energy dissipation in a vibrating structure. This paper considers identification of viscous damping under circumstances when the actual damping model in the structure is non-viscous. A method is presented to obtain a full (non-proportional) viscous damping matrix from complex modes and complex natural frequencies. It is assumed that the damping is “small” so that a first order perturbation method is applicable. The proposed method and several related issues are discussed by considering numerical examples based on a linear array of damped spring-mass oscillators. It is shown that the method can predict the spatial location of damping with good accuracy, and also give some indication of the correct mechanism of damping.