Analysis of magnetic forces in two-dimensional space with applications for the tuning of vibration energy harvesting devices

Vibration-based energy harvesting is a process by which ambient vibrations are converted to electrical energy, and is of interest for supplementing or replacing the batteries of individual nodes comprising wireless sensor networks among other applications. Generally, it is desired to match the resonant frequencies of the device with the primary ambient vibration frequencies for optimal energy harvesting performance. While previous work has demonstrated the use of magnetic forces to tune the resonant frequencies of vibrating energy harvesting structures, such efforts have been limited to one-dimensional analyzes. Here frequency tuning is realized by applying magnetic forces to the device in two-dimensional space, such that the resulting magnetic force has both horizontal and vertical components. In the case of a cantilever beam, the transverse force contributes to the transverse stiffness of the system while the axial force contributes to a change in the geometric stiffness of the beam. The effective resonant frequency of the device is then a function of the contributions of the original stiffness of the beam and the two additional stiffness components introduced by the presence of the magnet in 2D space. The simulation results from a COMSOL magnetostatics 3D model agree well with an analytical model describing the magnetic forces between the magnets as a function of location. Such 2D magnetic stiffness tuning approaches may be useful in applications where space constraints impact the available design space of the energy harvester.