A brief review is presented of methods of analysis of dynamic systems, including direct integration methods and mode superposition techniques and concepts associated with these methods. The component element method is then employed to model the problem of masses moving over continuous structures. The time-varying relationships between the moving masses and the structures are expressed via component elements, e.g. springs, dampers, stops, etc., connecting the two. Central to the analysis is a change of basis whereby the generalized masses of the structure are uncoupled, and an explicit numerical scheme for direct integration of the equations of motion. The validity and accuracy of this approach are demonstrated by analyses of comparatively simple systems. Its generality and versatility are demonstrated by combining it with finite element techniques. By incorporating finite element methods, the analysis of very complex structures subjected to complex systems of live loads is possible.