Dynamic modeling and analysis of a hard-magneto-viscoelastic soft beam under large amplitude oscillatory motions: simulation and experimental studies

In recent years, hard-magnetic soft (HMS) structures have emerged with meritorious properties for potential applications in many fields, such as soft robotics, wearable devices, and stretchable electronics. Developments in reliable computationally efficient dynamic models of HMS structures, however, are vital not only to gain a better understanding of their magneto-mechanical behaviors but also to design effective control algorithms. This study proposes a time-dependent computationally efficient magneto-viscoelastic model of a cantilever HMS structure subject to an external magnetic field. The Kelvin–Voigt internal energy dissipation model is utilized to accurately capture the nonlinear time-dependent response behavior of the HMS beam subject to large magnitudes of the magnetic field at different frequencies. The Galerkin modal decomposition scheme and Bogacki–Shampine method are used to discretize and solve the proposed model. A 2D finite element (FE) model is further developed to assess the effectiveness of the proposed nonlinear model in terms of computational efficacy and accuracy. Moreover, A hardware-in-the-loop framework is developed to experimentally characterize the deflection responses of the HMS beam to steady as well as harmonically varying magnetic fields. The nonlinear deflection responses of the proposed magneto-viscoelastic model subject to magnetic flux density up to 30 mT showed very good agreements with the measured data and the FE model, while the response saturation occurred under a field exceeding \(15\mathrm{ mT}\) $15\mathrm{mT}$ . The measured and model responses were further analyzed to obtain time-histories, phase-plane, and hysteretic response characteristics of the structure considering different magnitudes ( \(5{-}30\;{\text{ mT}}\) $5-30\text{mT}$ ) and frequencies ( \(0.25{-}1 \;{\text{Hz}}\) $0.25-1\text{Hz}$ ) of harmonic as well complex harmonic variations in the magnetic field. The simulation results from the developed model aligned well with the experimental findings across all considered excitations. Moreover, the computation time varied from approximately 3.2 to 31 s, depending on the number of generalized coordinates used in the modal decomposition. The computation time of the FE model on the same computing platform was in excess of 4700s.