The analysis of free and forced vibration phenomena pertinent to the behaviour of structural frameworks requires the investigation of mathematical models which adequately represent input loads, structural stiffness, and output response. Research conducted over the last five years has centred upon the second of these three aspects. An understanding of the stiffness relationships which dictate dynamic response requires both philosophical consideration, and rigorous mathematical investigation. It is this philosophical and mathematical investigation which is reported in this Thesis.The Thesis is presented in three parts.Part A deals with preliminary investigations and miscellaneous developments which are preparatory to the main findings reported in later parts. The material presented in Part A is partly an historical review, but also contains original developments, and examples which serve to illustrate aspects of the more general themes presented in Part B.Part B reports the main body of original investigations. A generalized approach to matrix formulations of the free vibration phenomena is developed. A solution of the forced vibration problem, using an original development from the modal superposition method, is developed in detail. All necessary mathematical results, sufficient to provide a viable solution technique, are presented in explicit form.These techniques have led to the development of a User-. oriented computer package, details of which are presented in Part C. The general development is supplemented throughout by investigations of alternative, less rigorous techniques and illustrations of various aspects of the relevant phenomena.Part C contains a description of the FORTRAN-IV computer package, FORVIB, and fully descriptive Users' manual. Example analyses of both free and forced vibration are included using framed structures which include multi-storey frames, a light-weight airport terminal building frame and a cross-frame from a major off-shore structure.The Thesis concludes with a brief summary of the major findings.