The dynamic response of a tension leg platform is studied using spectral analysis techniques. Equations of motion are derived for the rigid platform and its legs which are idealized as cables with longitudinal inertia. Attention is focused on the high-frequency resonant response of the platform in heave, pitch and roll modes with typical periods of 3 to 5 s. These modes are lightly damped, but the wave excitation at the high natural frequencies is small. Thus the response may turn out to be small. Formulas are derived for expected fatigue damage in the tethers due to high-frequency resonant vibration. The analysis includes linear damping from wave radiation and from energy dissipation in the tethers. By way of example, the results of the analysis are applied to a typical tension leg platform for petroleum production in 640 m (2100 ft) of water. The fatigue life for resonant vibration is found to be adequate.