Bounds on the vibrational energy that can be harvested from random base motion

This paper is concerned with the development of upper bounds on the energy harvesting performance of a general multi-degree-of-freedom nonlinear electromechanical system that is subjected to random base motion and secondary applied periodic forces. The secondary forces are applied with the aim of enhancing the energy harvested from the base motion, and they may constitute direct excitation, or they may produce parametric terms in the equations of motion. It is shown that when the base motion has white noise acceleration then the power input by the base is always πS0M/2 where S0 is the single sided spectral density of the acceleration, and M is the mass of the system. This implies that although the secondary forces may enhance the energy harvested by causing a larger fraction of the power input from the base to be harvested rather than dissipated, there is an upper limit on the power that can be harvested. Attention is then turned to narrow band excitation, and it is found that in the absence of secondary forces a bound can be derived for a single degree of freedom system with linear damping and arbitrary nonlinear stiffness. The upper bound on the power input by the base is πMmax[S(ω)]/2, where S(ω) is the single sided base acceleration spectrum. The validity of this result for more general systems is found to be related to the properties of the first Wiener kernel, and this issue is explored analytically and by numerical simulation.