Structural Dynamic Modeling and Optimization for Vibration Control of a Composite Cantilever with Magnetostrictive Shunt Damp-er

Structural vibrations can cause excessive noise and damage. Piezoelectric, electromagnetic and magnetostrictive (e.g., Terfenol-D, Galfenol) transducers can convert unwanted vibration into electrical energy, thus they can be used as energy harvesters for powering other devices and as shunt dampers for vibration control. The vibration control method using the dampers with shunt circuits becomes more attractive because such lightweight devices do not cause high load effects and require little or no power supply. To increase broadband damping, it is necessary to establish an accurate structural dynamic model of the dampers and optimize parameters of the shunt circuits, which have been extensively studied in vibration control of piezoelectric and electromagnetic shunt dampers [1]–[4]. Recently, magnetostrictive shunt dampers [5]–[9] begin to receive attention. The experimental results show that the dampers can effectively suppress vibration [5]–[7] and the attenuation of transmissibility of 5–15 dB occurs in the wide frequency range from 400 to 800 Hz [5]. Different expressions and models of the dampers are also presented. The electrical energy loss expressions [8] of a damper with shunt resistance Rs and parallel shunt resistance-capacitance (Rs-Cs) circuits are derived by a purely electrical model. Based on the linear piezomagnetic equations, the storage modulus and loss factor expressions [5] of a damper with negative inductance Ls are derived, a damper’s electromechanical linear model [9] is established, and general analytical expressions for storage modulus and loss factor [9] of the damper with Rs, Ls and series Rs-Cs circuits are derived. However, the above expressions and models are limited because they neglect structural dynamic behaviors of the devices. Although a dynamic compliance model [9] of a vibrating structure is presented, the validity of the model is not proven. A structural dynamic linear model [7] of a composite cantilever with magnetostrictive shunt damper is established. However, three key model parameters [7] are not related to conventional magnetostrictive properties, thus have to be experimentally determined. A generalized structural dynamic model and optimization for vibration control using the magnetostrictive shunt damper have not been reported. In this paper, the structural dynamic modeling framework of a composite cantilever with the magnetostrictive shunt damper is studied. Based on the structural dynamic equilibrium theory, the linear piezomagnetic equations, the law of electromagnetic induction and the circuit theory, the actuation and sensing equations are established to describe the dynamic behaviors of the mechanical-magneto-electro coupled system. The transmissibility, the effective mechanical-mechanical-electro coupled coefficient, the effective storage stiffness and the loss factor of the system are derived from the proposed model. The stability for the system with Rs, Cs and series Rs-Cs circuits are analyzed. The optimal tuning ratio fopt, the optimal electrical damping ratio ζeopt (the corresponding optimal resistance Rsopt and optimal capacitance Csopt) of the series Rs-Cs circuit are obtained by using H2 optimization criteria to minimize the RMS value of transmissibility. The results calculated by the proposed model for the damper are shown in Fig.1 and Fig.2. In Fig.1, comparisons between the calculated and measured results show that the proposed model can accurately describe the transmissibility frequency response curves of the system with positive resistance Rs and positive capacitance Cs, and can predict the changing laws of the resonant frequency and the peak transmissibility with Rs and Cs. Fig.2 (a) shows that to ensure a better damping performance, Rs should have a limitation of [– 60 Ω, 100 Ω] under the stability condition Rs ≥ – Rc (Rc = 460 Ω is the inherent resistance of the damper’s coil [7]). In Fig.2(b), we can discuss the effects of different electrical damping rations of the series Rs-Cs circuit on the transmissibility. Fig.2(c) shows the damping ration of the peak transmissibility under the series Rs-Cs circuit with optimal resistance Rsopt = 55 Ω and optimal capacitance Csopt = 10 μF reaches 90.89%, which is higher than 70.44% under Cs circuit with optimal capacitance Csopt = 5 μF, and 53.29% under Rs circuit with optimal resistances Rsopt = [– 60 Ω, 100 Ω]. The proposed model has clear physical meaning, and can describe the vibration control behaviors of the magnetostrictive shunt damper, thus has very strong practicability.