This paper presents a new formulation for critical damping of structures with elastically supported visco-elastic dampers. Owing to the great dependence of damper performance on the support stiffness, this model is inevitable for reliable modelling of structures with visco-elastic dampers. It is shown that the governing equation of free vibration of this model is reduced to a third-order differential equation and the conventional method for defining the critical damping for second-order differential equations cannot be applied to the present model. It is demonstrated that the region of overdamped vibration is finite in contrast to that (semi-infinite) for second-order differential equations and multiple critical damping coefficients exist. However, it turns out that the smaller one is practically meaningful. Copyright (C) 2001 John Wiley Sons, Ltd.