An analysis has been made of a possible mechanism of reactor instability due to interaction of the effects of longitudinal vibration of the fuel elements, heat generation, thermal expansion, and neutron kinetics. The analysis shows that under certain conditions, if there were no frictional damping forces, an instability would exist in a reactor of the type that has solid fuel elements that run continuously the length of the core. Self-sustained oscillations would build up until the fuel elements were stressed beyond their yield point. It is further shown that the frictional damping must be greater than a certain critical value to prevent an instability of this type. The Enrico Fermi Fast Reactor, taken as an example, is found to have sufficient damping to assure that the instability will not exist.