Performance enhancement of a vibration energy harvester via harmonic time-varying damping: A pseudospectral-based approach

Energy harvesting technologies are attracting increasing attention in the last decades, mainly thanks to the rapid advancement of sensors miniaturization, wireless sensors network, the internet of things, and the growing awareness on energy efficiency. The main objective of current research on energy harvesters is to enlarge the natural response bandwidth in order to increase the otherwise insufficient performance away from resonance. While popular approaches consider the inclusions of specific nonlinearities, this paper investigates the benefits of time variations of the damping coefficient of a linear power take-off system. A rigorous mathematical framework is firstly introduced, based on optimal control and pseudospectral decomposition, providing proofs and conditions for the existence of an unique optimal solution. Furthermore, a convenient algebraic formulation for the calculation of the steady-state response is provided, applicable to a wider family of time-varying systems. Such a tool is used to extensively study harmonic variations of the control damping parameters, discussing various combinations of orders and orthogonal terms. The optimization is first performed unconstrained, then forcing passivity, hence considering both active and semi-active controls. It is found that, in both cases, the best performance is achieved with damping variations at twice the exciting frequency, including both cosine and sine terms in the control law, while lower and higher harmonics are of less relevance.