A Framework for Determining the Maximum Theoretical Power Output for a Given Vibration Energy

This paper outlines a mathematical framework to determine the upper bound on extractable power as a function of the forcing vibrations. In addition to determining the upper bound on power output, the method described provides insight into the dynamic transducer forces required to attain the upper bound. This relationship, between input vibration parameters and transducer force gives a critical first step in determining the optimal transducer architecture for a given vibration input. The method developed is applied to two specific vibration inputs; a single sinusoid, and the sum of two sinusoids. For the single sinusoidal case, the optimal transducer force is found to be that produced by a linear spring, resonant with the input frequency, and a linear viscous damper, with matched impedance to the mechanical damper. The solution to this first case was previously known, but has been used here to validate the methodology. The resulting transducer force for the input described by a sum of two sinusoids is found to be inherently time dependent. This time dependency shows that an active system can outperform a passive system. Furthermore, the upper bound on power output is shown to be twice that obtainable from a linear harvester centred at the lower of the two frequencies.