Active control of beam with magnetostrictive layer
Introduction
Vibration suppression of structures using smart materials has received considerable attention by many researchers. Shape memory alloys, piezo-electric, electro-rheological fluids are commonly used smart materials for sensors and/or actuators. The use of piezo-electric materials for smart structural applications has been extensively studied. Of late, magnetostrictive materials are being considered as promising candidates for their dual applicability as sensors and actuators. With the availability of magnetostrictive material Terfenol-D, in various forms including powders, magnetostrictive materials are emerging as a highly attractive material for smart structure applications.
As early as 1972, studies have been carried out on electromagnetic-mechanical coupling problems. For instance, Wallerstein and Peach [1] have studied magnetoelastic buckling of beams and plates of magnetically soft materials. Then, Miya et al. [2] carried out experiments and theoretical studies on magnetoelastic buckling of ferromagnetic structures. They have also developed a finite element formulation for analysis of buckling [3]. Review of magneto elasticity of thin plates and shells has been carried out by Ambartsumian [4]. While Hudson et al. [5] carried out elastic and magnetomechanical coupling modeling for polymer-bonded Terfenol composites. Recently, one of the significant contributions to the modeling of the structural-magnetic strain developed in a magnetostrictive transducer has been made by Dapino et al. [6]. They have calculated and validated the strains developed by considering both the rotation of the moments within the material in response to the applied field and the elastic property of the material. Earlier, Krishnamurthy et al. [7], Reddy and Barbosa [8] have also studied the use of magnetostrictive material for vibration control of flexible beams. They have considered the magnetostrictive material to be one full layer in the laminate, with partly covered actuating coils. The actuating coils were modeled as solenoids. The current input to the solenoid is considered as a function of space and time. Though it is mathematically possible to model the current in the solenoid in this manner, it is not physically reasonable to provide a spatially varying current throughout the length of the coil. The magnetic field induced by the solenoid is generally less spatially varying within the solenoid length, but reduces to an insignificant amount, at a short distance from the end of the solenoid. This residual magnetic field will stimulate the magnetostrictive layer. This effect is not considered in their papers. Hence in the present study, the beam is discretized into finite elements and the magnetostrictive layer of the size of a single element along with its actuating coil is located at various positions to demonstrate the effective damping characteristics of this material.
Magnetostrictive layer, Terfenol-D, is an alloy of terbium, iron and dysprosium. The magnetostrictive layer responds to the magnetic and mechanical stimuli. The magnetostrictive layer expands when excited with a magnetic field allowing it to be used as embedded actuators. On the other hand when the layer is mechanically strained, it generates an induced voltage across a sensing coil placed near the vincity. The magnetomechanical coefficient depends upon the pre-stress and magnetic field. For the purpose of illustration, in this work, the authors have assumed perfect orientation.
Section snippets
Formulation
A typical conventional aluminum beam having a small Terfenol-D layer on the top, with a current carrying coil, enclosing the beam, as shown in Fig. 1, is considered. The magnetostrictive layer is of nonlinear nature at moderate or high magnetic drive levels. To model the magnetostrictive layer with a linear constitutive relation, the necessary low magnetic field regimes are obtained by applying a biasing current to the surrounding coils. The low magnetic field intensity induces an actuation
Numerical results and discussion
An aluminum beam of square cross-section of size 10 mm×10 mm, with a length of 1 m is considered. On the top of the beam a layer of Terfenol-D material is placed. The properties of the magnetostrictive material, are Em=26.5 GPa, ρm=9250 kg/m3 and d=1.67×10−8 m/A. The effective radius of the coils, rc, enclosing the beam is taken to be 10 mm, with a coil density, no turns/m. The coils are of 38 AWG copper wires with a density of 8844 kg/m3. The mass of the coil per unit length assuming no=104 is
Conclusions
In the present study, a finite element formulation is derived for beam with a magnetostrictive layer on top. The mathematical finite element model developed is consistent with the physics of the problem. A parametric study has been carried out for different boundary conditions, different number of discretized elements as well as for different locations of the actuating coils. This study indicates that the best location of the actuating coil to enhance damping depends on both the boundary
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