An excellent title delineating a comprehensive theory and providing the foundation required to build a good math basis to perform the analysis of a model of a physical systems under vibration. The volume is illustrating how exactly the physics of the problem in question is utilized for the development of the more specific analytical framework. The author of this volume has elucidated a general basic theory which would apply to continuous/discrete systems and also includes proofs of the important results, particularly ones instructive for the proper understanding of the result. The volume starts with the discussion of the physics of systems consisting of the particles and rigid/deformable bodies together with the physics and math for analysis of the single freedom degree systems. He has also illustrated the analysis and development of the linear operations commonly used in solving numerous problems and formulating the equations that do govern the responses of the non-linear systems in terms of different operators. The author has also paid particular attention to the free response of the above mentioned systems and non-self-adjoin systems and explored the proven and effective methods of the correct determination of the forced response plus the approximate solutions methods for continuous systems.