A Thermomechanical Example of Auto-Oscillation

Abstract

A system which progresses repeatedly through a cyclic sequence of physical configurations (continuous or discrete), in the absence of any periodic excitation, is said to be in the state of auto-oscillation. In this paper the author examines a thin thermoelastic rod projecting from the warmer of two parallel rigid walls, their separation being slightly wider than the length of the rod. Assuming heat transfer by conduction only, it is found that for certain combinations of physical parameters the rod does auto-oscillate (repeatedly makes and breaks contact with the cooler wall at finite frequency) provided that part of the rod near the cooler wall has a nonpositive thermal coefficient of expansion. This last condition is absolutely necessary since otherwise one encounters certain thermomechanical pathologies. The only other way that these pathologies could be eliminated would be to employ the concept of imperfect contact as suggested by Barber, which may also be interpreted as allowing heat transfer by radiation.