Statistical energy analysis (SEA), a vibration analysis approach to estimate the dynamic response values for coupled systems, based on vibrational energy flow concepts, is studied from the basic principles and a systematic theoretical development of SEA, using modal analysis approach, is provided in this thesis. SEA relies on a combination of theoretically and empirically determined parameters (e.g. internal loss factor, coupling loss factor, modal densities and coupling damping when the coupling between subsystems is nonconservative), to facilitate the predictions. In this thesis, it is one of these SEA parameters, that is, coupling damping which is investigated. SEA is applied to nonconservatively coupled oscillators, rods and beams with arbitrary coupling strength, using a modal analysis approach. The power flow between coupled subsystems, power input to these subsystems, energy levels, power dissipated in the subsystems and in the coupling are all formulated. The energy balance equation between nonconservatively coupled subsystems and the power dissipated in the coupling in terms of modal energies of the subsystem is developed. It is shown that in the case of nonconservatively coupled dynamical systems, the power flow between the two subsystems is related to the difference of energies of these subsystems and also to the average energies of the individual subsystems. It is also shown that coupling damping has a significant quantitative effect on the energy flow between coupled subsystems but qualitatively the influence is of minor significance. Periodic structures are studied to provide a broader understanding for the vibration analysis of complex mechanical structures. Floquet's theory is applied for the analysis of simple and compound periodic structures. In the case of simple periodic beams, the analysis becomes more complex when each segment is composed of different material, shape and size. The complex mathematical formulation is confirmed assuming uniform beams and the results obtained from Floquet's theorem are validated with those obtained by 'ANSYS'.