Degree Candidacy: Master of Science in Mechanical Engineering
Date: Tuesday, November 19, 2013
Time: 2:00 PM
Location: 215 EIB
Advisor: Dr. Mohammed Daqaq
Committee Members: Dr. Ardalan Vahidi and Dr. Georges Fadel
Title: Nonlinear Energy Harvesting Under White Noise
While purposeful introduction of stiffness nonlinearities into the dynamics of energy harvesters is aimed at enhancing performance under non-stationary and random excitations, most of the conclusions reported in the current literature are based on the steady-state response which assumes a harmonic fixed-frequency excitation. As a result, we still do not have a clear understanding of how the nature of the excitation influences the output power, or what role stiffness nonlinearities play in the transduction of energy harvesters under random excitations.
To fill this gap in the current knowledge, this thesis investigates the response of nonlinear mono- and bi-stable energy harvesters to environmental excitations that can be approximated via a white noise process. For the mono-stable case, statistical linearization is utilized to analytically approximate the statistical averages of the response. The influence of the nonlinearity and the symmetry of the restoring force on the mean power is investigated under optimal electric loading conditions. It is shown that the nonlinearity has no influence on the output power unless the ratio between the time constant of the harvesting circuit and the period of the mechanical oscillator is small. In such case, a mono-stable harvester with a symmetric nonlinear restoring force can never produce higher mean power levels than an equivalent linear harvester regardless of the magnitude or nature of the nonlinearity. On the other hand, asymmetries in the restoring force are shown to provide performance improvements over an equivalent linear harvester.
For energy harvesters with a bi-stable potential function, statistical linearization, direct numerical integration of the stochastic differential equations, and finite element solution of the Fokker-Plank-Kolmogorov equation governing the response probability density function are utilized to understand how the shape and symmetry of the potential energy function influence the mean output power of the harvester. It is observed that, both of the finite element solution and the direct numerical integration provide close predictions for the mean power regardless of the shape of the potential energy function. Statistical linearization, on the other hand, yields non-unique and erroneous predictions unless the potential energy function has shallow potential wells. It is shown that the mean power exhibits a maximum value at an optimal potential shape. This optimal shape is not directly related to the shape that maximizes the mean square displacement even when the time constant ratio, i.e., ratio between the time constants of the mechanical and electrical systems is small. Maximizing the mean square displacement yields a potential shape with a global maximum (unstable potential) for any value of the time constant ratio and any noise intensity, whereas maximizing the average power yields a bi-stable potential which possesses deeper potential wells for larger noise intensities and vise versa. Away from the optimal shape, the mean power drops significantly highlighting the importance of characterizing the noise intensity of the vibration source prior to designing a bi-stable harvester for the purpose of harnessing energy from white noise excitations. Furthermore, it is demonstrated that, the optimal time constant ratio is not necessarily small which challenges previous conceptions that a bi-stable harvester provides better output power when the time constant ratio is small. While maximum variation of the mean power with the nonlinearity occurs for smaller values of the time constant ratio, these values do not necessarily correspond to the optimal performance of the harvester. Finally, it is shown that asymmetries in the potential shape of bi-stable harvesters do not improve the mean power unless the symmetric potential function is designed away from its optimal parameters.