The objective of this paper is an analytical and numerical study of the dynamics of a beam with attached masses. Specifically, a finite inextensible beam that rests on a uniform elastic foundation and carries an accelerating mass is considered. Of interest is the dynamics of the beammass system due to the motion of the moving mass. The influence of various parameters such as forward force, retard force and friction upon the performance of the beam are investigated. The mechanics of the problem is Newtonian. Based on the assumption that when the moving mass is set in motion the mass is assumed to be rolling on the beam, the mechanics, including effects due to friction and convective accelerations of the interface between the moving mass and the beam, are determined. The problem of the system is nonlinear, due to the presence of friction and the convective acceleration. In the modeling, the mass can be accelerated by a force. Meanwhile, the mass is capable of reducing speed and being brought to a stop at any position on the beam by applying a retard force to the mass and/or increasing the friction between the mass and the beam. The force is assumed to be tangential to the deformed configuration of the beam. By employing the Galerkin procedure, the partial differential equations which describe the transient vibrations of the beam-mass system are reduced to an initial value problem with finite dimensions. The method of numerical integration is used to get convergent solutions.