Discovering governing equation from data for multi-stable energy harvester under white noise

It is sometimes difficult to model the stochastic differential equations for strongly nonlinear multi-stable vibration energy harvesters, especially for those under additive and multiplicative white noises, because of the existing challenges in quantifying noise intensities, nonlinear stiffness coefficients and damping coefficient. From the perspective of machine learning, a sparse identification method is devised to discover the general governing equation of energy harvester by using observed data on system state time series. With the observed data, the drift term and the diffusion term can be learned and then the stochastic differential equation can be identified. A penta-stable vibration energy harvester is taken as an example to verify the feasibility and effectiveness of the devised sparse identification method, which indicates that the method can be successfully applied to model the governing equation of a multi-stable vibration energy harvesting system under random excitation. Based on the learned data-driven stochastic differential equation for energy harvester, the stochastic dynamics can be further explored by appropriately adjusting the system parameters to improve energy harvesting performance and optimize the miniaturization design.