#### Date of Award

8-2009

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Legacy Department

Mechanical Engineering

#### Advisor

Daqaq, Mohammed F.

#### Committee Member

Haque , Imtiaz-ul

#### Committee Member

Jalili , Nader

#### Abstract

Energy harvesting is a process by which otherwise wasted ambient energy can be captured and transformed into a useful form. Historical examples of this concept include windmills, sailing ships, and waterwheels. Modern technologies and current energy needs have brought this same concept to a smaller scale wherein small wireless devices with minute energy consumption can be operated autonomously. Today, many critical electronic devices, such as health-monitoring sensors, pace makers, spinal stimulators, electric pain relievers, etc. require minimal amounts of power to function, which permits utilizing energy from their environment to power them. By exploiting the ability of active materials and some mechanisms to generate an electric potential in response to mechanical stimuli, it is now possible to harvest energy from the environment to power these devices and run them autonomously.

This thesis focuses on issues related to the modeling and response of energy harvesters to time-varying frequency excitations. Specifically, in the first part of this effort we address the accuracy and convergence of reduced-order models (ROMs) of energy harvesters. Two types of energy harvesters are considered, a magnetostrictive rod in axial vibrations and a piezoelectric cantilever beam in traverse oscillations. Using Generalized Hamilton's Principle, the partial differential equations (PDEs) and associated boundary conditions governing the motion of these harvesters are obtained. The eigenvalue problem is then solved for the exact eigenvalues and mode shapes. An exact expression for the steady-state output power is attained by direct solution of the PDEs. Subsequently, the results are compared to a ROM attained following the common Rayleigh-Ritz procedure. It is observed that the eigenvalues and output power near the first resonance frequency are more accurate and has a much faster convergence to the exact solution for the piezoelectric cantilever beam. In addition, it is shown that the convergence is governed by two dimensionless constants, one that is related to the electromechanical coupling and the other to the ratio between the time constant of the mechanical oscillator and the harvesting circuit. It is shown that the number of modes necessary for convergence should be obtained at the maximum electric loading for the piezoelectric harvester and at the minimum electric loading in the magnetostrictive case. Using these results, critical conclusions are drawn in regards to the design values for which the common single-mode ROM is accurate.

The second part of this work addresses the response of energy harvesters to harmonic excitations of time-varying frequency. Specifically, we consider a piezoelectric stack-type harvester subjected to a harmonic excitation of constant amplitude and a sinusoidally-varying frequency. Such excitations are very common and can result from rotating machinery operating at variable speed. We analyze the response of the harvester in the fixed-frequency scenario, then use the Jacobi-Anger's expansion to analyze the response in the time-varying frequency case. We obtain analytical expressions for the harvester's response, output voltage, and power. In-depth analysis of the attained results reveals that the solution to the more complex time-varying frequency can be understood through a process which ``samples'' the fixed-frequency response curve at a discrete and fixed frequency interval then multiplies the response by proper weights. Extensive discussions addressing the effect of the excitation parameters on the output power is presented leading to some initial suggestions pertinent to the harvester's design in the time-varying frequency case.

#### Recommended Citation

Seuaciuc-osorio, Thiago, "On the Modeling of Energy Harvesters and Analysis of their Response under Time-Varying Frequency Excitations" (2009). *All Theses*. 613.

https://tigerprints.clemson.edu/all_theses/613