The stability of vertical vibrations of an oscillator moving uniformly along a beam on an elastic half-space has been investigated. Expressions for the equivalent stiffness of the beam have been derived at the contact point with the oscillator. It has been shown that the imaginary part of the equivalent stiffness can have a sign, which is interpreted as the so-called `negative viscosity'. Frequency bands where the equivalent stiffness gives the `negative viscosity' have been analyzed. Using the expressions for equivalent stiffness the instability zones for the oscillator vibrations have been found. Instability can take place when the velocity of the oscillator exceeds the minimum phase velocity of waves in the beam. The effect of viscosity in the beam on the stability of the system has been considered. It has been shown that a small viscosity destabilizes the system.