Investigating the dynamic response of a beam on elastic foundation, subjected to a moving mass is an important issue in mechanical and structural engineering analysis. Analysis of high-speed train movement on a track placed on a foundation is one of the important applications of this problem. Due to the importance of this issue, in addition to the dynamic response of the beam, the dynamic analysis of the moving mass is also of particular importance. In addition, in this case, foundation modeling has significant effect on obtaining critical velocities and also beam and mass maximum displacements. Therefore, in this study, the moving mass is modeled by mass-spring-damper oscillators as a single-degree-of-freedom and one-foot and two-foot two-degree-of-freedom cases. Furthermore, foundation modeling will be considered as linear, nonlinear and bi-linear Winkler cases. Also, for sudden changes of foundation stiffness, the non-uniform foundation will be considered and compared with the uniform foundation. In all cases, critical velocities and extreme values of beam and moving oscillator displacements will be obtained using finite element method and numerically computed using $\\alpha$-method. The effects of stiffness of different foundation cases and natural frequencies of moving oscillator on the time response of beam and moving oscillators will be investigated. In linear and non-uniform foundation model, the critical velocities were reduced compared to the linear and uniform model, when moving from the foundation with lower stiffness to the foundation with higher stiffness. In nonlinear model, compared to the linear one, the extreme displacement values decreased and the critical velocities increased. Also, in this model, by increasing the natural frequency, the number of direction changes of the oscillator displacements increased. By considering non-uniform foundation, less extreme displacements and higher critical velocities occurred when oscillator moves from the region of lower foundation stiffness to the higher one. By considering foundation with less tensile stiffness in the bi-linear model, the oscillator upward displacements increased and the critical velocities decreased. In bi-inear model, unlike the other models, using two-degree-of-freedom oscillator increased the extreme displacement when crossing from the lower-stiffness region to the more stiffness one. In all the considered models, the highest oscillator and beam displacements occurred during the presence of the oscillator at the end of the beam. Keywords: Nonlinear Foundation, Bi-linear Foundation, Moving Oscillator, Non-uniform Foundation, Critical Velocity