Optimal vibration control of beams with total and partial MR-fluid treatments

This paper presents the synthesis of full state and limited state flexible mode shape (FMS) based controllers for the suppression of transient and forced vibration of a cantilever beam with full and partial magnetorheological (MR) fluid treatments. The governing equations of motion of the three layer MR sandwich beam are expressed in the state variable form comprising a function of the control magnetic field. An optimal control strategy based on the linear quadratic regulator (LQR) and a full state dynamic observer is formulated to suppress the vibration of the beam under limited magnetic field intensity. The lower flexural mode shapes of the passive beam are used to obtain estimates of the deflection states so as to formulate a limited state LQR control synthesis. The transient and forced vibration control performances of both the full state observer-based and the limited state FMS-based LQR control strategies are evaluated for the fully as well as partially treated MR-fluid sandwich beams. The results show that the full state observer-based LQR control can substantially reduce the tip deflection responses and the settling time of free vibration oscillations. The limited state LQR control based on the mode shapes effectively adapts to the deflections of the closed loop beam and thus yields vibration attenuation performance comparable to that of the full state LQR controller. The partially treated beam with MR-fluid concentration near the free end also yields vibration responses comparable to the fully treated beam, while the natural frequencies of the partially treated beams are considerably higher.