Numerous mechanical systems contain surfaces in partial or full sliding contact, and therefore, prone to friction-induced vibration instability. These include systems containing mechanical switches, brakes, clutches, gears, rolling contact bearings, journal bearings, robot end-effector grasp and motion, oil drills, etc. The prominent dynamic features of a mechanical system, subject to friction-induced vibration, can be captured by an appropriate equivalent mass-on-belt model. It is the goal of this research to provide a comprehensive study of friction-induced vibrations in mechanical systems by using their equivalent mass-on-belt models. Friction-induced vibration is manifested through three mechanisms termed Stribeck effect, mode-coupling and sprag-slip. Mechanical systems prone to vibrations by one or more of the three mechanisms of instability are considered and studied in detail. The mechanical systems fall into one of two groups. A system in the first group is the pseudo-rigid-body mass-on-belt representation of a compliant bistable linkage mechanism characterized by substantial geometric nonlinearity and nonlinear elasticity. A system in the second group is a mass-on-belt model that accounts for mass-belt contact stiffness. Such a system is excited primarily through mode coupling. In the first group of mechanical systems super and subcritical pitchfork bifurcation as well as Hopf bifurcation are observed. The normal force and spring pre-compression are bifurcation parameters leading to the subcritical pitchfork bifurcation and the belt velocity corresponds to the Hopf bifurcation. It is found that for a low damping and negligible spring nonlinearity, one equilibrium point dominates the steady-state response. Otherwise, the phase plane is split into two separate planes associated with the corresponding fixed point. The boundary is dictated by structural damping and spring nonlinearity. It is shown that the destabilizing mechanism in the bistable mechanisms is the Stribeck effect of friction. The dominant mode of instability for the second group of mechanical system is mode coupling instability. In this group intermittent loss of contact between the mass and the moving belt within a periodic cycle is allowed. Addition of a vibration absorber consisting of a second mass suspended from the first mass by a spring provides effective passive control of friction-induced instability due to mode-coupling. The research concludes with the study of a two mass system in which both masses are in contact with a belt and the friction force is characterized by the three regimes of lubricated contact that include boundary lubrication, mixed boundary and hydrodynamic lubrication and full hydrodynamic lubrication as sliding speed is increased. It is shown that such systems can experience periodic, quasi-periodic and chaotic vibration response.