A Framework to Find the Upper Bound on Power Output as a Function of Input Vibration Parameters

This paper outlines a mathematical framework necessary to determine the optimal transducer force for a given vibration input. 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. This relationship also yields a theoretical maximum energy output for a system with a given proof mass and parasitic mechanical losses, modeled as linear viscous damping. This relationship is then applied to three specific vibration inputs; a single sinusoid, the sum of two sinusoids, and a single sinusoid with a time dependent frequency (chirp). For the single sinusoidal case, the optimal transducer is found to be a linear spring, resonant with the input frequency, and a linear viscous damper, with matched impedance to the mechanical damping. The resulting transducer force for the input as a sum of two sinusoids is found to be inherently time dependent. This time dependency shows that an active system (not only dependent on the states of the system) can outperform a passive system (dependent only on the states). The final application, for a swept sinusoidal input, results in a transducer of a linear viscous damper, with matched impedance to the mechanical damping, as well as a linear spring with a time dependent coefficient. INTRODUCTION Recent work in vibration energy harvesting has focused on ways to improve power output from vibration sources that are not modeled as a single sinusoidal input. Much of this work has investigated the use of nonlinearities as a way to increase energy output [1] [2] [3]. These nonlinearities are usually of the form of a nonlinear spring, such as a Duffing oscillator. Daqaq et al. showed that for Gaussian white noise the energy generation was not a function of the transducer’s potential function. That is to say, that the restoring force of the system does not affect the power generation for a Gaussian white noise vibration input. When Daqaq examined the case for filtered white noise, where some frequencies are more represented than others, he was forced to assume a form for the potential function in order to estimate a solution [4]. Hoffmann et al. showed that for certain vibration inputs a nonlinear mono-stable or bi-stable oscillator could greatly outperform a linear system. However this work had to assume a form for the restoring force before the parameters could be optimized for power generation. Select results from this study are shown in Table 1 [5]. Stepped Input Swept Input Three Inputs Bounded White Noise Linear 100% 100% 100% 100% Nonlinear Monostable 0% +479% 0% +52% Nonlinear Bistable +11% +364% +11% +33% Table 1. Select results from Hoffman et al [5]. These example works, and others, give useful insight to the potential uses of nonlinearities for harvesting from complex vibration inputs. However these works do not give a clear relationship between the parameters that define the input vibration and the transducer. By using methods from the Calculus of Variations this work will find the unconstrained and globally optimal relationship between the input vibration, and force that must be produced by the transducer. This relationship will also define an upper limit for power generated for a given vibration 1 Copyright © 2014 by ASME Proceedings of the ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2014 September 8-10, 2014, Newport, Rhode Island, USA