# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
###### Abstract
[15] investigated the exploitation of in-plane vibrations realizing two planar MVEHs to easily integrate printed circuit boards in the design. [PERSON] et al. [16] designed an MVEH with a self-adaptive structure, capable of coil switching with active feedback control to improve conversion efficiency at different excitation frequencies and amplitudes. [PERSON] et al. [17] developed a device capable of stiffness tuning using magnetic rotors. Our research group has carried out previous works regarding MVEHs applications in the railway and biomechanical fields [18; 19]. Both numerical and experimental approaches were adopted to investigate the dynamics of the magnetic suspension and its conversion efficiency [20; 21]. The peculiarity of our devices is the use of an asymmetric magnetic suspension with only one fixed magnet at the bottom. These devices are called gravitational vibrational energy harvesters (GVEHs) since they exploit gravity actions as the restoring force due to the absence of the top fixed magnet. This configuration shows a non-linear \"softening\" behavior that reduces the system's natural frequency as the excitation amplitude increases. The realization of a two-degrees-of-freedom system has led to a performance improvement in terms of frequency broadening and power generation [22]. The purpose of this study is to provide a design and optimization procedure for GVEHs, performing sensitivity analyses based on both numerical simulations and experimental validation on multi-scaled prototypes. The purpose of the proposed procedure is to design the best-performing device suited for a specific application, having compatible size and resonance conditions tuned to the fundamental frequencies range of the random vibrations carrying the highest energy content. Some authors have investigated the design strategies for MVEHs with symmetric springs through numerical simulations only without experimental comparison on real prototypes [23]. Others performed design optimization of MVEHs both experimentally and numerically but do not consider specific application constraints for the design process, [24]. Most of the authors focus only on the maximization of the prototype output power in laboratory tests but do not follow a proper design approach since the real application constraints are not considered [25; 26]. The magnet's dimensions should not be chosen based only on power criterion, as they affect the dynamics in terms of resonance frequency, equilibrium position and overall system size. These factors are often design constraints in the real application system on which the VEH should be installed, having its own fundamental frequencies during vibration and installation limits. A numerical modeling technique for GVEHs is proposed using a combination of Ansys Maxwell FEM environment and MATLAB/Simulink. The numerical model is necessary to derive non-linear stiffness and damping characteristics and to simulate the system's dynamic behavior. The proposed modeling approach is applied to three GVEH prototypes having different sizes. Numerical results of dynamic simulations are compared to experimental data from laboratory tests on the prototypes for validation of the modeling technique. The validated model is then used to perform sensitivity analyses of the system resonance frequency and equilibrium position by varying both fixed and moving magnet dimensions. These analyses allow us to evaluate the two magnet volumes of the asymmetric magnetic suspension that comply with the design constraints of the destined application and generate the maximum possible power. Coil length and the number of turns optimization is the final step of the design process for maximizing the device performance. Section 2 shows the materials and methods used in the paper. Firstly, the configurations of the tested harvesters and their main design parameters are displayed. Secondly, the analytical model of the GVEH is described along with its equation of motion and electro-mechanical coupling of the transducing mechanism.