Time optimal semiactive vibration damping of a mechanical oscillator with variable damping coefficient

Abstract

In this paper the semiactive vibration control of a single degree of freedom oscillator is considered. The oscillator comprises a magnetic viscous damper, providing a damping force which can be easily changed in real time. A semiactive control law acting on this parameter is developed, with the aim of damping as fast as possible the free oscillations of the structure. The application of the Pontryagin minimum principle formally proves that, if the goal is to reach in minimum time a small invariant set around the origin (that is, the rest position), the optimal control law for the considered bilinear system consists of a switching (bang-bang) feedback, operating at either the lowest or the highest available level of the damping force. Simulation results are presented, showing that the proposed vibration control scheme always outperforms a benchmark case of a mechanical oscillator with constant damping coefficient equal to the critical value. I. INTRODUCTION Semi-active vibration control refers to the possibility of varying, in real-time, some of the properties of passive damping devices, for improving their effectiveness. This task is achieved without the need of a significant external power. Moreover, since the involved damping devices are passive, they can only extract energy from the system, thus avoiding possible instability problems. As an outcome of the massive research effort devoted in the last decades to the development of smart materials and technologies, several passive devices are now available for the realization of semi-active control schemes. In most of the practical applications, semi-active control schemes are obtained by either varying the stiffness or the damping coefficient of a vibrating structure. The former option can be achieved by using combinations of mechanical springs (1), (2), pneumatic springs (3), (4) or piezoelectric devices (5), (6), whose mechanical stiffness can be modified by simply changing the shunt impedance applied to their electrodes. The latter option concerns the use of adjustable dampers such as magnetorheological devices (7), (8), whose damping coefficient can be modified by changing the external voltage applied to the magnetorheological fluid contained in the device. Alternatively, friction dampers can be adopted (9), (10), able to operate between a stick state, in which the velocity is zero, and a slip state, in which motion is permitted and the damper exerts a constant friction force. Dampers with controllable damping coefficients can also be obtained by exploiting the magnetic induction mechanism (11), (12). In this case, viscous damping forces are generated by the relative motion between a magnet and a conductive plate or a coil (magnetic dampers), and can be changed by changing the distance between magnet and plate or by changing the electric impedance connected to the coil. These strategies have been extensively applied by researchers in