This research project deals with developing a deeper understanding of the behavior of systems with multiple tuned vibration absorbers and to use this knowledge to offer improved guidelines for the design of absorber systems. From a fundamental dynamics viewpoint the work concerns the dynamic response of structures to which are attached multiple tuned subsystems for the purpose of vibration attenuation although the focus of research is on applications to rotating machinery, the techniques developed are more widely applicable, for example, to translational vibration absorber systems. In the area of multiple absorbers, recent results for a special absorber system have shown that the desired unison motion can become dynamically unstable, resulting in a drastic reduction in operating range. The thrust of the analytical portion of the proposed work will focus on the dynamic performance of more general multiple absorber systems, including: the dynamic performance of multiple absorber:systems with general paths (including: the mistuned circular paths most commonly found in industrial applications), the effects of imperfections in the absorber paths, and the effects of rotor flexibility. this research will significantly expand the knowledge base for tuned absorbers as applied to rotational systems and will offer an improved understanding of the dynamics of a general class of systems which have attached tuned substructures. Such understanding can be distilled into improved design guidelines that will include dynamics of a general class of systems which have attached tuned substructures. Such understanding can be distilled into improved design guidelines that will include dynamic bifurcation's and their effects on system performance. Exchange of information with industrial liaisons will be carried out in order to focus the research in the proper direction and to transfer the understanding gained by the proposed research. Specifically, manufactures of automotive engines (Ford Motor Company) and light aircraft/helicopter engines (Textron Lycoming) will be kept appraised of the results of this research and consulted for direction.