The problem of a beam subjected to moving masses has been an interesting subject of investigation for a long time. In most of the related studies, the researchers merely solved a simplified model of the system. Some of these simplifications include ignoring the mass of the moving objet or its inertial effects in analyses, omitting nonlinear terms, and especially considering it as a linear-invariant system; indeed the problem classifies into time-varying systems according that the moving mass position along the beam varies with time. Byaccounting the non-autonomous form of the governing equations of motion, new features of the system become clarified which may lead to a failure in practical applications if ignored. Some of these original results include the discovery of a multitude of new unstable regions in the parameter's plane of the system, occurrence of resonance and coexistence phenomena for some parameters of the system. Condition for coexistence is obtained by deriving Hill's determinants for the beam-moving mass problem which corresponds to the vanishing of instability tongues. This phenomenon occurs when instability regions narrow such that its boundaries coincide and consequently results in disappearance of the instability tongue. The significance of paying attention to this phenomenon is proved by showing the re-opening of this closure by the introduction of the slightest disturbance in the system which leads to the sudden appearance of a previously invisible region of instability. Considering small strains but large displacement, the differential equation of motion is derived in the nonlinear regime. Results of perturbation analyses show that the separating instability curves are identical to those found in the linear case. Results show that although the unstable behavior of the nonlinear system is not characterized by an unbounded response, the system behavior changes qualitatively by crossing the boundary curve and some phenomena like jump and limit cycle may happen. Moreover, it is shown that in addition to system parameters, different initial conditions will result to different response patterns. Keywords: beam- moving mass, modified Hamilton's principle, Floquet theory, dynamic stability, resonance curves, coexistence phenomenon.
