Numerical and experimental investigations into the nonlinear dynamics of a magneto-elastic system

Abstract

We present an analytical, numerical and experimental treatment of the magneto-elastic system given by Moon and Holmes [1] for a cantilevered ferromagnetic beam between two magnets. A single-mode approximation is assumed, and the nonlinear quantities in the governing ODE of the modal amplitude that arise due to magnetic forces and moments are computed numerically using an algorithm presented by Derby [6] by modelling the magnets as ideal solenoids. This numerical model enables us to compute the static force distribution as a function of beam displacements for fixed experimental parameters. The magneto-elastic system is also built in order to compare the model with experimental data. A strain gauge is used to measure the deflection of the beam, and an electronic circuit is also constructed to collect and process data to obtain time series for the position and velocity of the beam. We find that the model displays good agreement with experiment for the static case where no external forcing is present. The predictions of buckled equilibria positions, and also of the natural frequency of vibrations about those positions from the model are close to those that of experimental results. The topological structure of the static force distribution as computed from the model is found to be similar to that of the Duffing oscillator in the case of a double-well potential, as seen by both having similarly-structured basins of attraction in the static case. Furthermore, we observe cases of single and triple well potentials from the model as well for certain parameters as well, and bifurcation diagrams and bifurcation sets are presented for varying physical parameters. Catastrophes involving stable and unstable surfaces of equilibria are also investigated, and we observe surfaces similar to those seen in cusp catastrophes[14]. In the presence of external forcing of the system, the experimental data as well as the numerical model display qualitative agreement with the theory of the Duffng oscillator [3], and we present examples of "strange attractor" structures over the Poincare Map for moderate forcing amplitudes from both experimental and numerical results. Bifurcations of the fixed points of the Poincare Map are also investigated using the numerical model. Discrepancies between the model and experimental data are highlighted, and limitations of the model are discussed in relation to experimental results as well as with consideration to the Duffng oscillator as an analytical model of the system.