Abstract In a companion paper (see pp. 43–61 of this issue), it was shown that when a system is non-viscously damped, an identified equivalent viscous damping model does not accurately represent the damping behaviour. This demands new methodologies to identify non-viscous damping models. This paper takes a first step, by outlining a procedure for identifying a damping model involving an exponentially decaying relaxation function. The method uses experimentally identified complex modes and complex natural frequencies, together with the knowledge of the mass matrix for the system. The proposed method and several related issues are discussed by considering numerical examples of a linear array of damped spring-mass oscillators. It is shown that good estimates can be obtained for the exponential time constant and the spatial distribution of the damping.