The present work investigates dynamics of a gear-pair system involving backlash and time-dependent mesh sti!ness. In addition, the system is under the action of external excitation, caused by torsional moments and gear geometry errors. First, the equation of motion is established in a strongly non-linear form. Then, the emphasis is laid on a speci"c forcing frequency range, corresponding to conditions of simultaneous fundamental parametric resonance and principal external resonance. For these conditions, several types of periodic steady state response are identi"ed and determined by employing suitable methodologies, including techniques applicable to piecewise linear systems and to oscillators with time-periodic coe$cients. Moreover, these methodologies are complemented by appropriate procedures revealing the stability properties of the located periodic solutions. In the second part of the work, numerical results are presented. These results verify the validity and e!ectiveness of the new analytical methodology and provide information on the gear-pair dynamics. First, series of typical response diagrams are obtained, illustrating the e!ect of the mesh sti!ness variation, the damping and the forcing parameters on the gear-pair periodic response. These response diagrams are accompanied by results obtained with direct integration of the equation of motion. In this way, it is demonstrated that for some parameter combinations, the dynamical system examined can exhibit more complicated and irregular response, including crises and intermittent chaos.