The uniform motion of a mass along an axially compressed Euler-Bernoulli beam on a viscoelastic foundation is investigated. It is assumed that the mass is subjected to a constant vertical load and that the beam and mass are in continuous contact. The velocity of the mass after which the vibrations of the system are unstable is found. The instability implies that the amplitude of the mass vibrations is growing exponentially and that the problem does not have a steady state solution. It is shown that the instability starts at lower velocities as the compresional force increases. The instability occurs even for over-critical viscosities of the foundation when there is no dynamical amplification of the steady state vibrations due to resonance.