INTRODUCTION With the tendency towards the construction of slender buildings of relatively high flexibility and low mass the dynamic loads on the structure have become increasingly important. The sway frame building is a popular structural option where such buildings are required up to 15 or 20 storeys. In a sway frame building@ the rigidity of the connections between the columns and the beams or slabs usually suffices to provide lateral resistance. As the 20 storey height is approached@ the rigidity of sway frame buildings can be insufficient and the sway due to wind may be unacceptable to the occupants of the building. In this situation@ the introduction of shear walls interacting with the frame may increase the total rigidity@ or the designer may choose another structural system@ with frame-tube type buildings@ shear buildings and core buildings being the most common structures. For the calculation@ by the modal superposition technique@ of the response of a building to a fluctuating load induced by wind@ earthquake or mechanical vibration@ a knowledge of the free vibration characteristics is required. This Item provides a means of estimating the free vibration characteristics of rectangular sway frame buildings for use in these response calculations. In addition@ a method of analysis is presented enabling the free vibration characteristics of more complex frame structures such as lattice towers@ cranes and bridges to be computed efficiently. The way in which these data are used in the response of a building or structure under the action of a fluctuating wind load is described in ESDU 87035 and 88019. This Item is intended to complement ESDU 79005@ for the undamped natural vibration of shear buildings@ and ESDU 81036@ for the undamped natural vibration of core buildings.