Continuum Mech. Thermodyn. (1997) 9: 165–173 c  Springer-Verlag 1997 Original Article Torsional vibration of a shape memory wire Stefan Seelecke Technical University Berlin, Institut f¨ ur Verfahrenstechnik, Sekr. HF2, Straße des 17. Juni 135, 10623 Berlin Received May 7, 1997 The paper studies the rotational motion of a rigid mass suspended by a shape memory wire. The wire represents a torsional spring, which causes the mass to oscillate. At the same time, however, the wire induces damping through the hysteretic phase transitions that it undergoes during the oscillation. Due to the strong temperature dependence of the wire’s thermome- chanical properties, a complex behavior can be observed. The oscillation is shown to differ substantially depending on whether the wire is quasiplastic (low temperature) or pseudoelastic (high temperature). This provides a mechanism for an active control of the system. By an appropriate heating strategy, it is possible to compensate for the damping, and the oscillation can thus be stabilized. 1 Introduction The thermomechanical properties of shape memory alloys are to a large extent determined by phase transi- tions from austenite to several martensitic variants and vice versa. Depending on the temperature, the material exhibits a strongly differing behavior which is termed quasiplastic at low and pseudoelastic at higher tem- perature, see for example [1], [2] and [3] for an overview. Both cases, however, have in common that they are characterized by the occurence of hysteretic phenomena – a subject of great interest to many thermo- dynamicists, experimental as well as theoretical [4], [5], [6], [7] and [8]. Most work done in the field has been devoted to quasi-static behavior, whereas the present work focusses on some dynamic aspects of shape memory hysteresis. It investigates the frequency and damping properties of an oscillatory motion. A simple one-dimensional case, yet allowing for all three possible phases, is the torsional vibration of a thin wire. As shown in Fig. 1, the wire holds a rigid mass suspended. It does not only act as a complex torsional spring, but – due to its hysteretic behavior – it also damps the angular motion Φ (t ) of the mass. In the following section, we give the equations of motion for the mass and the model equations for the shape memory wire. As we want to describe a transient process, we need a model that is capable of reproducing the kinetics of the phase transitions. For this reason, we use the tensile-compressive model developed by M¨ uller and Achenbach [9], [10] and [11] and modify it slightly in order to make it applicable to the case of torsion. The resulting system of equations is then solved numerically in section 3.1 for two fixed temperatures corresponding to quasiplasticity and pseudoelasticity, respectively. Section 3.2 deals with the control of the oscillation. A heating strategy is presented that counteracts the damping of the wire and stabilizes the oscillatory motion.