AbstractThe paper considers the finite element analysis of the free, undamped and the forced damped vibrations of rotationally periodic structures. Associated with every natural frequency (except for those for which the deflection is the same at corresponding points on every substructure) there are a pair of orthogonal mode shapes, with eigenvectors {u} and {ü}. The complex vector {z}={u}+i{ü} is also an eigenvector of the equations of motion, and represents a rotating normal mode. The deflection of one substructure has the same amplitude as, and a constant phase difference from the deflection of the preceding substructure. It is therefore possible to analysse the complete structure by considering only one substructure, and applying appropriate complex constraints at its boundary with the following substructure, so as to and applying appropriate complex constrainsts at its boundary with the following substructure, so as to impose this phase difference. The method has been implemented in a computer program and is illustrated by analyses of an alternator end winding, a cooling tower with legs, and a wheel of turbine blades.For forced vibration, it is shown that any arbitrary oscillatory force can be decomposed into a series of rotating forces. For any one of these rotating components, there is a fixed relationship between the amplitude and phase of the force acting on one substructure, and that acting on an adjoining substructure. This relationship, which does not involve any approximation, can be used to enable a series of calculations of the response of one substructure to be performed instead of one on the whole structure. A series of calculations on an individual substructure normally requires much less computer time and storage than a single calculation on the complete structur