A complete physical 3D model from first principles of vibrational-powered electromagnetic generators

The dynamic behavior of a vibrational electromagnetic generator using a magnetic levitation architecture was theoretically and experimentally studied in great detail, when operating under a wide range of three-dimensional excitations. We developed a complete rigorous physical model from first principles based on the theory of electrodynamics of continua, centered on the laws of electrodynamics and balance of mass, linear momentum, angular momentum, energy and entropy. Local electromagnetic and gravitational body forces, couples and powers were considered, and the surface tractions were divided into constraint and friction components, as well as those due to external mechanical energy sources. The balance of linear momentum, angular momentum and circuit equations resulted in up to 13 non-linear differential equations describing the dynamics of the levitating-magnet and container, relating input forces and torques with output displacement, constraint forces and voltage. The balance of energy yielded a consistent equivalence between the time rate of change of the internal kinetic and potential energies of the generator and the output power, associated with the external circuit, Ohmic losses and friction losses, as well as the input mechanical power being supplied to the system by the environment. Both the input and output powers were proven to tend to increase equally when operating the generator under resonant conditions. The levitating generator was shown to be sensitive to axial translational and centrifugal inertial forces, each one effectively resulting in a uni-stable or bi-stable system. The dynamical response yielded multiple initial conditions dependent steady-states, hysteretic frequency output and chaotic characteristics. Relevant guidelines to optimize the energy conversion efficiency of energy harvesters are provided. This model was validated by experimental tests, including general 3D motions combining translations and rotations: cross-correlations exceeding 90% were achieved. Such Newtonian and Langrangian modelling approaches hold great potential to be easily adapted to a wide range of other electromagnetic generators, with multiple degrees-of-freedom and operating under various environments, such that significant advances in energy technologies can be supported.