This paper concerns the formulation of lumped-parameter models for rigid footings on homogenous or stratified soil. Such models only contain a few degrees of freedom, which makes them ideal for inclusion in aero-elastic codes for wind turbines and other models applied to fast evaluation of structural response in the time domain. However, the order of a lumped-parameter model must be sufficiently high to provide the correct dynamic stiffness of the ground. The aim of the present analysis is to examine the quality of low-order lumped-parameter models with respect to the prediction of the maximum response during excitation and the geometrical damping related to free vibrations of a hexagonal footing. The optimal order of a lumped-parameter model is determined for each degree of freedom, i.e. horizontal and vertical translation as well as torsion and rocking. In particular, the necessity of coupling between horizontal sliding and rocking is discussed.