Validation of a Reduced Order Magnetoelastic Beam Model

Abstract

There has been a recent interest in the use of magnetoelastic materials to develop new magnetic antenna technologies [1,2,3]. Magnetic antennas show promise for surpassing shortcomings in current technologies, such as communicating in lossy environments (e.g., underground or underwater). One commonly proposed design is a resonant composite multiferroic beam consisting of magnetoelastic, elastic, and piezoelectric layers. Appling a voltage to the piezoelectric layer causes the beam to bend, straining the magnetoelastic layer, modulating the magnetic field around the antenna, thereby transmitting information. Current analytical and finite element magnetoelastic beam models [4,5,6] appear to work well for sufficiently long and thin structures, however the analytical models often neglect or overly simplify the magnetostatic field (i.e. demag) or spatially distributed material properties such as Young’s modulus or piezomagnetic coupling coefficients. Therefore, an analysis of under what conditions reduced-order analytical models are valid is needed before they can be used as a tool for analyzing general geometries where complex magnetic field distributions are expected. This work presents a detailed comparison of a finite element model and an analytical reduced order Euler-Bernoulli beam model with the goal of determining under what conditions the reduced order model is valid, and which foundational assumptions are responsible for any discrepancies.Both the finite element model and the analytical reduced order Euler-Bernoulli beam are set up to solve the same geometry, governing equations, and boundary conditions. The antenna has a simple geometry composed of three rectangular plates; with a linear elastic layer sandwiched between piezoelectric and magnetoelastic layers. The governing equations consist of the equations of magnetostatics, electrostatics, and linear elasticity (i.e., suitable for a near-field antenna operating at low kHz frequencies). The mechanical boundary conditions are setup to model a free-free Euler Bernoulli beam.The finite element analysis is conducted in a series of steps. First, a uniform magnetic field is applied, and the bias state of the antenna is solved for using a coupled, nonlinear, magnetoelastic constitutive model (along with linear elastic, and piezoelectric behaviors). A linearized modal analysis is then performed around this bias state to determine the dynamic eigenmodes of the mechanical vibrations. Once found, the first eigenmodes is subsequently used to model the mechanical deformation, which is then fed into the nonlinear magnetic material model to obtain the dynamic magnetization changes (i.e., a one-way coupled model, with mechanics driving the magnetics). The reduced order Euler-Bernoulli beam model is constructed using a similar approach. A bias strain distribution for a composite Euler-Bernoulli beam is first computed. In contrast to the FEA model, this assumes a spatially uniform demag factor, and spatially uniform material properties. A linearized eigenmode analysis is then performed that utilizes a linear piezomagnetic approximation to obtain a fully-coupled linearized solution.Both the finite element model and the analytical model have their strengths and weaknesses. The finite element model produces more accurate results utilizing spatially non-uniform and nonlinear material properties, and an accurate description of demag. However, the speed of the analytical model greatly outperforms the finite element model making it more suitable for understanding design alternatives and optimization. This work will examine how the assumption of uniform material properties and demag fields affects the accuracy of the analytical model and determine under what conditions the approximations are valid. Fig. 1 shows an example, from the finite element model, of the spatially distributed piezomagnetic coupling in the magnetoelastic material where the maximum coupling is found to be at the end of the beam. A specific focus will be made on examining the validity of using an analytical model to optimize the geometry of a magnetic antenna. **