Vibration analysis and optimization of fully and partially MR-fluid treated multi-layer beams

Magnetorheological (MR) fluid is known to exhibit rapid variations in their rheological properties when subjected to varying magnetic field and thus offers superior potential for applications in smart structures requiring high bandwidth. MR sandwich structures can apply distributed control force to yield variations in stiffness and damping properties of the structure, and thus provide enhanced vibration suppression over a broad range of external excitation frequencies. In this study, the governing equations of a three-layer beam structure employing MR-fluid layer as the mid-layer are presented in the finite element form and Ritz formulation. The validity of the finite element formulations is demonstrated by comparing the results with those obtained from the Ritz formulation and laboratory measurements performed on a prototype sandwich beam. Furthermore, the relationships between the magnetic field and the complex shear modulus of the MR material in the pre-yield regime is estimated through the free vibration experiment of the prototype MR sandwich beam. Simulations are performed to derive the essential properties (natural frequencies and loss factors), and the dynamic response of the multilayered structure as functions of the applied magnetic field and thickness of the MR-fluid layer under different boundary conditions. A concept of a partially-treated multi-layer MR fluid beam is proposed to achieve a compact and cost effective design. Laboratory experiments are performed on a partially treated beam and the data are used to demonstrate the validity of both the finite element and Ritz formulations. The properties of partially treated MR-fluid beams are evaluated to investigate the influences of the location and length of the treatment for different boundary conditions, and compared with those of the fully-treated beams. The damping performance of the partially treated MR multilayer beam is studied in terms of the modal damping factors corresponding to different modes that are evaluated using the principle of modal strain energy in the finite element model. The results showed enhanced stiffness and damping properties of the MR-treated beams with increasing magnetic field intensity. A design optimization problem is then formulated to achieve maximum modal damping factors by combining the finite element analysis with the genetic algorithm and sequential quadratic programming optimization algorithms. Optimal configurations of partially treated beams are identified for realizing maximum damping factors corresponding to the first five modes, individually or simultaneously. An optimal control strategy based on the linear quadratic regulator (LQR) IS subsequently formulated to achieve enhanced vibration suppression for the fully- and partially treated beams. A state estimator based on the flexural mode shapes (FMS) is introduced and implemented to synthesize a full-state feedback controller design. The validity of the proposed FMS-based LQR controller is demonstrated by comparing the results with those obtained with observer-based LQR controller. The effectiveness of both the control strategies is demonstrated by investigating the tip displacement response to an impulse and a white noise disturbance for different boundary conditions. The results show that the settling time, peak displacement and FFT amplitude corresponding to each mode of the system have significantly been reduced in the controlled structure. The closed loop power spectral density of the tip displacement response due to the white noise disturbance confirms the controllable capability of the LQR controller in semi-active control to attenuate the vibration of the MR sandwich beam under high band frequencie